EPA-600/5-73-007
September 1973
Socioeconomic Environmental Studies Series
Critique of Role of Time
Allocation In River Basin Model
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
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EPA-600/5-73-007
September 1973
CRITIQUE OF ROLE
OF TIME ALLOCATION
IN RIVER BASIN MODEL
by
Philip G. Hammer, Jr.
Grant No. R-801521
Program Element 1HA096
Project Officer
Philip D. Patterson
Environmental Studies Division
Washington Environmental Research Center
Washington, D. C.
Prepared for
Washington Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D. C. 20460
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D. C. 20*03 - Price 13.40
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ABSTRACT
This critique presents a review of the theory of
time use in consumer behavior and applies this review
to an evaluation of time and location assignment pro-
cedures for population units in the Social Sector of
the River Basin Model. Time allocations in this model
serve to describe population unit work, travel and
leisure behavior and to link them to the operations of
other sectors through spatial movements, flows of goods
and services, and participation in the institutional
activities. The objective of the simulations contem-
plated by the model is the identification of the impacts
of man's activities on his environment.
ii
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CONTENTS
Abstract i
Acknowledgments Iv
Sections
I Introduction 1
Part One: Review of the Theory of Time Use 6
II Classical Theory and the River Basin Model 7
Classical and Modern Utility Theory of Consumer
Behavior 8
Individual Versus Groups Behavior and Preference 17
III The Demand for Leisure Model and Its Aspatial
Extensions 21
The Classical Demand for Leisure and Supply of
Labor Model 22
Consumption Activities in the Labor-Leisure Model 30
Consumption Technology as Universal 38
Two Applications: Household Work and Overtime 42
Activity Frequency in Time & Money Allocations 51
IV Time and Frequency in Travel Behavior 59
Journey-to-Work in the Labor-Leisure Model 60
Trip Frequency in Route and Mode Selection 70
Trip Frequency for Leisure Activities 74
Multi-Purpose Trip-Making 86
Some Applications in the Theory of Trip Frequency 92
V Residential Location and the Use of Space 112
The Industrial Location Analogy 113
Residential Space Use and Bid-Price Curves 119
Location by Occupation (Wage Rate) 125
Activity Pattern, Residential Location and Space
Use 141
VI Summary 149
Part Two; The River Basin Model 152
VII The Structure of the Model 153
An Overview of Model Operation 154
Social Sector Inputs and Allocations 161
VIII Dissatisfaction and Migration 177
The Quality of Life Index 178
Allocating to Minimize Dissatisfaction 191
Migration Procedures in MIGRAT 198
iii
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CONTENTS (cont'd)
IX From Time Allocations to Time Expenditures 210
Employment 211
Adult Education 238
Charges Against the Time Budget 244
Educational and Political Status 254
Park Use Assignment 266
Summary 276
X Monetary Consumption Expenditures 278
Consumption Expenditures 278
Shopping Assignments 285
Financial Accounting for Population Units 305
XI Some Concluding Thoughts on Simulation 312
Static Optimization for Population Units 315
The Objective Function in the River Basin Model 335
iv
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ACKNOWLEDGMENTS
The sole author of this work is Philip G. Hammer,Jr.,
Center for Urban and Regional Studies, The University
of North Carolina, Chapel Hill, North Carolina. This
project has been financed in part with Federal Funds
from the Environmental Protection Agency under grant
number R-801521. The contents do not necessarily
reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or
commercial products constitute endorsement or recom-
mendation for use.
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SECTION I
INTRODUCTION
The River Basin Model is a computer simulation model
which focuses on man - environment interactions and mutual
interdependencies. As such it must model the internal be-
havior of both natural and man-made systems as well as
identify and replicate the linkages between them. At the
core of the human sphere is the household, as represented
in the River Basin Model through the population units of
the Social Sector. Households form the core not so much
because they initiate action - indeed, they often tend to
follow the lead of other sectors or institutions - but
rather because their well-being, now and in the future, is,
or should be, the ultimate objective of the operations of
man-made systems. Dependent as they are on the state of the
natural environment for so much of this well-being, house-
holds and the institutions designed to serve them must
recognize the delicate relationships between their activities
and the state of that environment. The River Basin Model
has been developed to simulate those relationships in
order to promote that recognition.
The River Basin Model takes the differential use of
time in activities as the essence of household behavior,
and quite rightly so, for it is in connection with such
allocation of time that households make demands of the
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other sectors as citizens and consumers and participate
in their operations as directors and employees. Associ-
ated with the devotion of household time to activities
are flows of energy from participants through the physical
instruments they manipulate and the resultant movement of
people, goods and money through the system to assure that
the proper actors and instruments are in the right place
at the right time for activity performances. Not only are
these exchanges and movements related to the ultimate
time-use of goods in activities, but also they are them-
selves time-consuming processes necessary for overcoming
the separation of spatially as well as temporally extensive
activities. Various intensities of human activities over
durations make simultaneous demands of the natural environ-
ment, altering its structure through the use and pollution
of its resources.
The concern of this critique is the appropriateness
of this view of time use as the crux of household behavior
and as the essential link between that behavior and the
operations of other institutions in the human system. From
the remarks above it should be obvious that this review is
begun with a bias favorable to the usefulness of the house-
hold time allocation perspective. It is hoped that in the
course of the critique the arguments for the perspective
will become apparent, along with a recognition of the
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problems which do arise in its application. However, no
attempt will be made here to present a formal argument as
to the general validity of this perspective, although much
attention will be given to micro-economic consumer theory
with time use.
This critique, then, will evaluate the use of the
household time allocation mechanism in the River Basin
Model for describing the behaviors of households and relating
them to the behaviors of other sectors. Most of the atten-
tion will go to detailed examinations of the time and lo-
cation assignment procedures used in the model, and there-
fore the critique tends to be short-sighted. That is, it
does not attempt any new model-building e'fforts. Instead,
it characterizes and criticizes the present structure of
the model and offers suggestions for its improvement in
terms of comprehensiveness and behavioral realism. Finally,
although it is not responsible for development of a new
simulation procedure to replace the gaming format now used,
the critique will show how the consistancy of tne present
assignment structure with micro-economic theory (given
certain modifications) suggests using the static consumer
optimizing model in that dynamic simulation.
The critique is divided into two parts. The first is
a review of the literature on utility theory for consumer
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behavior in which time is a dimension (stock). This review
is condensed from the author's doctoral dissertation (forth-
coming) and will serve both as a frame of reference for
evaluating Social Sector manipulations in the River Basin
Model and as a source of suggestions for the development of
the simulation procedure for population unit decision-
making. The second part consists of that evaluatioh of the
model and ends with some thoughts on that simulation. It
had been hoped that some simple numerical sensitivity
tests of certain parts of the model might be undertaken,
but this did not prove feasible.
The use of the time allocation, or time budget, tech-
nique was found to be innovative and appropriate for the
tasks demanded of it in the ciodel. There are many ways in
which it could be tightened up to close gaps this reviewer
thinks exist in its structure at present, but most of these
would require only minimal effort. Much research does need
to be done to check out the reasonableness of the parameter
values assumed in the model. At this time there is no pub-
lished information adequate for this prupose, but the model
might be fit to data such as that for the Washington study
referred to in this critique. The Quality of Life Index
seems to lack the cotiprehensiveness over choice variables
required of a utility function, and its form should be re-
considered in the development of the procedure for simulating
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time allocations. These are technical matters, however, and
do not qualify the general applicability of the time use
mechanise for population units.
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PART ONE
REVIEW OF THE THEORY OF TIME USE
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SECTION II
CLASSICAL THEORY AND THE RIVER
BASIN MODEL
The view of household behavior presented in the
River Basin Model is similar to that of classical micro-
economic theory of consumer behavior. In both the
consumer has scarce resources to be allocated to various
consumption activities, and it is presumed in both that
it is possible to assign a numerical value to any par-
ticular allocation of resources to consumption to re-
present the (relative) "utility" or satisfaction arising
from that allocation. In the classical theory the
perspective is a static one. The consumer makes his
allocational choice so as to maximize his utility once
and for all under conditions of perfect knowledge of the
state of the environment and of the outcomes of his
choices. As a simulator of systems* behavior over
time, the River Basin Model takes a dynamic perspective,
with households making repeated allocations in a chang-
ing environment about which they have only limited
knowledge. However, in its present form the River
Basin Model incorporates constructs from the static
classical theory. Therefore, it is felt that a proper
-------
critique of the model should begin with a thorough re-
view of those areas of and extensions to classical
theory which are relevant to the model. Moreover* in
moving from a gaming approach to internal machine
simulation of household behavior* the classical theory
and its extensions should supply hints for the con-
struction of appropriate mechanisms for determining
revised consumer resource allocations in a changing
environment. For these reasons* and to provide a
general frame of reference in discussing consumer be-
havior, this chapter and several which follow will be
devoted to the presentation of static optimizing models
of consumer behavior in the classical vein.
Classical and Modern Utility Theory of Consumer
Behavior
Central to the utility theory of consumer behavior
is the "utility function" or "preference function"
which assigns relative weights or "preferences" to
individual components of consumption and produces a
single value as the utility of that set of components,
that set representing as it does an allocation of scarce
resources to different consumption activities. In the
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classical theory a consumption activity was simply the
purchase of a certain amount of a particular good.
Later extensions to the theory consider use of goods
explicitly as well as their purchase. The amounts of
the goods purchased were the arguments to the utility
function. In the classical theory there was generally
one scarce resource, money (or income), to be allocated
to the purchases of goods in various amounts. The
problem for the consumer was to spend his income in
such a way that his utility was maximized. In mathe-
matical form the problem was to
Maximize U = 7X,(x. ) (objective
function)
n
Subject to Y~ V Pixi = ° (income
L
constraint)
where u is the utility level
i) is the utility function defined over
the x.^
x^ is the amount of the itn good pur-
chased* i = 1, . . . n
^ is the price of the in good
is the amount of income to be spent
-------
As stated here* the problem assumes that all in-
come will be spent. Normally it is assumed that the
marginal utility of a good is positive
>0
i '
so that more of the cjood is preferred to less, and
that utility increases at a decreasing rate (74 is
a single-valued, monotonic function),
ax!
With all income spent, maximizing U subject to the
budget constraint is equivalent to maximizing the
following Lagrangian form of the objective function*
L = U + X(Y- "
The maximal level of U is achieved by that allocation
of income Y to goods x. which renders the partial
derivatives of L with respect to the x^ all equal to
zero. These "first order conditions" are
Higher order conditions for a maximum will not be
presented in these simple expositions of the theory
and its extensions.
10
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BL - S7X. - ^.p. = 0 1=1, . . . , n
where X is the Lagrangian multiplier or the "shadow
price" on the income resource. These conditions* plus
the budget constraint, produce a system of n + 1
equations which can be solved (if appropriate condi-
tions are met) for the n + 1 unknowns x. and X. X is
called the shadow price on income (money) because it
is the value of the change in utility at the optimum
allocation of the x. with a small change in income*
dx.
X() p, -TT-) • X
dY
1
Modern versions of the classical theory consider
U a relative index, with arbitrary origin and scale
and specific to the individual. This view of 21 as an
ordinal utility function precludes comparison of
utility levels between individuals, in contrast to the
assumption of cardinal utility of much classical
theory which would allow such interpersonal compari-
sons. Modern theory also views the utility maximize-
11
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tion problem as a general non-linear program in which
there may be several resource constraints, a constraint
may be slack (some resource not completely exhausted
at the optimum), and a choice variable (such as the
x.) need not be undertaken at a positive level. For
example, the consumption problem above might be
generalized as
Maximize U =K.(x.) (objective function)
Subject to x. ^ 0 i = 1, . . • , n
(non-negativity con-
straints)
Yk -Gyxj) ^ 0 k = 1, . . . , m
(budget constraints
for resources k)
where the ^fc(Xi) are functions of the choice variables
x. and may be thought of as uses of the resources Y^»
The Lagrangian form of the objective function is now
L = U + \ (Y. -
-------
= 0 if xi > 0
< 0 if x, = o
i = 1, . . . , n
- Y - (P U.) > 0
~ zk k i
= 0 if X > 0
fc k = 1, . • * , m
> 0 if X = 0
KL
The lastm conditions are merely the budget constraints,
which we note will have multipliers or shadow prices
which vanish when there are unused resources.
There are two concepts in this model of consumer
behavior which should be identified here. The first
is the "marginal rate of substitution" of an argument,
say x., for another, x., at the optimum. This is
J i
given by the slope of the indifference curve or utility
contour in their direction at that point, and is equal
to the negative ratio of their marginal utilities:
dx.
tf+ = MRS (j for i) * -
For "normal" arguments with positive marginal utility,
this slope is negative in keeping with the concept of
13
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"substitution"! a positive slope indicates comple-
mentarity. From the first order conditions, this
k
In the classical case, with only one constraint, the
negative of the marginal rate of substitution be-
tween two goods i and j is simply the ratio of their
2
prices. In the general multi-constraint problem,
marginal rates of substitution involve shadow prices
on resources as explicit weights.
This brings up the second concept, which is the
relative shadow price of one resource in terms of the
shadow price of another resource at the optimum.
Such a relative shadow price of resource k in terms
of resource k» is given by the ratio
•There, (frx.,) = Jp^
and
= p., etc.
14
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which we know to be equal to the ratio of the mar-
ginal utilities of external marginal changes in the
levels of the stocks
dU/dYk,
This ratio gives in effect the negative of the mar-
ginal rate of substitution between the stocks, or the
amount of one stock which is a utility-equivalent of
a unit of the other at the margin. One point should
be stressed, and that is that under the ordinal utility
interpretation shadow prices for resources are in
utility units per unit of resource. Therefore, their
absolute magnitudes are arbitrary. Thus it is pre-
cisely the ratio of shadow prices which is of signi-
ficance and forms a metric for making interpersonal
comparisons .
Theoretical Constructs in the River Sasin Model
Households^ in the social sector of the River
Basin Model are faced with the allocation of two scarce
resources (although, as we will see, these are highly
Actually, groups of households as "population units
See the discussion to follow.
15
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interdependent allocations), these resources being
time and money. The model introduces as a choice
parameter a variable called the "dollar value of time
in travel" which proports to measure the marginal rate
of substitution of money (in travel costs) for travel
time in route and modal selection for the journey to
work. This selection can influence in turn job assign-
ment and residual time for use in other activities,
and, of course, both gross and net income for use in
the purchase of goods for activities. Thus this sub-
stitution should also reflect more general tradeoffs
between leisure time and money for expenditures.
Classical and modern utility theory give spe-
cific relationships for such parameters, which are,
after all, ratios of shadow prices and marginal
utilities. Although the gaming version of the River
Basin Model does not force any particular choice of
values for such parameters, a simulation version of
the social sector in the model could incorporate
choice of parameter values which are at least explicit-
ly suboptimal and deviate from optimal values in a
systematic fashion, if indeed they are not themselves
optimal values for a static perfect knowledge model.
16
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However, even in the static perfect knowledge case
there is considerable variation in optimal parameter
values according to the way in which consumer beha-
vior is visualized and functional relationships
identified. For this reason a good deal of effort will
be given over in this report to demonstrate the types
of shadow price relationships between time and money
which would develop under different specifications of
consumer behavior and satisfaction. In concluding
sections, recommendations will be made for the applic-
cation of the theory and its constructs to a dynamic
simulation of the sector.
Individual Versus Group Behavior and Preferences
Behaving units in the social sector of the River
Basin Model are "population units" or groups of house-
holds with specific demographic structures, whereas
micro-economic utility theory of consumer behavior
refers to the individual, or at most the household,
as the behaving unit. However, treatment of popula-
tion units in the River Basin Model seems for all
practical purposes independent of actual population
unit definition and thus susceptible to arbitrary
17
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scaling of population unit size without appreciable
effect on the functioning of the model, subject to
Jj-
scale adjustments in other sectors.
In extrapolating from micro-economic theory of
the individual consumer to the behavior of groups
of consumers, then, we face the "aggregation problem"
in developing preference functions and behavior di-
mensions for groups as collections of individuals.
On the other hand, it is frequently remarked that the
attempt to model individual consumer behavior will
be as futile as the attempt to model the behavior of
individual atomic particles has been and that under-
standing is best achieved by viewing the behavior of
groups of systems of individual units as a whole.
However, aggregation of postulates for individual be-
havior to those for behavior of groups as collections
of individuals, however this aggregation is accomplish-
ed, cannot be expected alone to provide a view of the
"whole" which is fundamentally different from - and
free of the defects of - the view of the parts sepa-
rately. mherefore, a "solution" to the aggregation
This should become apparent in the expositions of
Part Two.
18
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problem for group preferences would not in itself re-
solve disputes about the reasonableness of postulating
the existence of preference functions for individuals.
The value of analyzing collective rather than in-
dividual behavior lies in the possibility of finding
simplifications which might present themselves as
"central tendencies" for the aggregate. In the River
Basin Model, population unit behavior is no more easily
analyzed than would be the behavior of a single house-
hold, were it not for the limited capacity of the
computer in handling a large number of behaving units.
Aggregation, then, appears to have been a practical
rather than behavioral matter, and efficiency or model
feasibility are certainly justifiable reasons for
aggregation. Preference analysis has much to offer
in its own right, in terms of its ability to provide
constructs and indices the values of which can be of
considerable interest to planners despite their limi-
tations .
It is our view here, then, that the population
units of the social sector of the River Basin Model
are in fact behaving units indistinguishable from
19
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"individual consumers" and that the "aggregative"
preference function approach of the model can be re-
viewed constructively by reference to and comparison
with classical theory for individuals and its recent
extensions. If this type of structure - treating
behaving units deterministically - is thought to be
inappropriate for machine simulation of the sector,
then some fundamentally different approach from pre-
ference analysis should be pursued, such as a truly
probabilistic structure. Development of such a model
is not the charge here, only the critique of the present
structure. As stated above, suggestions for determi-
nistic machine simulation based on the present pre-
ference function approach will be offered in conclusion
to this critique o'f the River Basin Model.
^Since consumer demands are not automatically met in
the model, outcomes to consumers are not strictly
deterministic. However, machine simulation with pre-
ference functions would most probably involve deter-
ministic derivation of those demands themselves each
time period. A "truly probabilistic structure" would
not only determine outcomes stochastically, as is the
case to some extent in the model as it stands, but
would also produce probabilistic demands on the part
of the consumers (population units) and thus yield
probability distributions for outsomes and the state
of the system as a whole.
20
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SECTION III
THE "DEMAND FOR LEISURE" MODEL
AND ITS ASPATIAL EXTENSIONS
Among models for simulating the urban system the
River Basin Model is unique in its placing of time on
an equal footing with money in resource allocational
problems facing the consumer, or population units in
the Social Sector. Indeed, time allocation even takes
precedent over monetary allocation in the model's view
of consumer work, social, and recreational behaviort
The special contribution that time budget analysis can
make to the study of pollution and environmental sys-
tems in the urban area is the translation of institu-
tional ties between the Social and Economic sectors
(as measured by money flows) into extensive (time)
measures of consumer demand for environmental qualities
in various use intensities (activities) leading to sub-
sequent determination of the polluting and quality-
reducing implications of those demands.
It is the objective of this and the chapters
immediately following to examine the place that time
allocation and the detailing of consumer (household)
activities has recently found in the micro-economic
theory of consumer behavior, where once the only "ac-
21
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tivity" implied was the instantaneous acquisition and
consumption of physical quantities. This chapter will
introduce the time "budget and the analysis of different
household activities in the utility theory of consump-
tion in an aspatial context. The next chapter will
consider trip-making behavior, and the following one
will bring up residential location and space use with
a time budget perspective. In each case we will be
especially interested in the marginal rates of sub-
stitution between variables and in the shadow prices
on time and money, since these constructs have rele-
vance for the operation of the River Basin Model.
The Classical Demand for Leisure and Supply of Labor
Model
One of the simplest models in the theory of con-
sumer behavior, and historically the first to use time
as a static dimension (a stock) in that behaviort is
the theory of the demand for leisure, or conversely,
the theory of the supply of labor offered by the worker.
This theory ties together consumer demand for goods for
consumption and worker participation in the labor mar-
ket, selling his labor, as measured in time units.
22
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The individual attempts to find the allocation of time
to work and leisure which maximizes his utility as a
function of income earned (goods consumed) and time
spent for leisure or work at a given wage rate per
hour. This is a two constraint, or two budget (time
and money), model. When the wage rate is varied, the
trace of the amount of labor the consumer-worker will
offer to work, as a function of that wage rate, is
called the individual's "labor offer curve." This
curve is used in the theory of wages for aggregation
into market functions for the supply of labor avail-
able to firms at the wages they offer.
Leisure itself is treated somewhat ambiguously in
this model. Some writers consider leisure time a
proxy for personal inputs of effort or energy in con-
sumption as the use of goods, such inputs requiring
time for their implementation. Other theorists view
leisure as an alternative source of utility distinct
from the utility of income as a proxy for consumption.
•*-Most texts of the theory of wages describe the work-
leisure model and market relationships in some detail.
See, for example, Richard Perlman, Labor Theory,
(New York: John Wiley), 1969.
23
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In the first case we have in effect time and goods as
inputs to a production function for consumption, with
utility defined over the resulting level of consump-
tion. In the second case leisure time and income are
direct sources of utility with no reference to con-
sumption as an explicit activity. These divergent
interpretations lead to virtually the same specifica-
tion of the model in general termsf however. In either
case, work time could be an argument to the utility
function in addition to income or consumption (goods
and leisure), but it is usually omitted from that
function. That simplification has important implica-
tions, and the fact that widely differing results can
be obtained under treatments of work time is often
overlooked in applications of this model.
Under the production function approach to consump-
tion, the work-leisure problem for the consumer may be
written as followsi
Maximize U ='U(tw, tlf x)
Subject to tw, t-^ x > 0
W'tvi - x > 0 money budget
T - tw - t-^ > 0 time budget
24
-------
where n
x = £ PIXI is gross consumption expendi-
^=^ tures, the sum of goods pur-
chased times their prices.
t is work time
w
w is the wage rate
wtw is the income earned
T is the total stock of time
(length of the time period)
In this statement of the problem, there can "be less
consumption expenditure than income earned. In the
alternative leisure time interpretation, total income
Y would replace expenditures x in the utility function
and in the money budget, and the money budget constraint
would become a definitional equation. Whether or not
this change in interpretation would imply a different
type of utility function, in terms of its properties,
is a matter for further speculation.2 The non-linear
2For example, in trading off the utility of income
against the utility of leisure time with no reference
to consumption activities per se, there may seem to be
no reason to expect satiation effects to ever set in.
But with consumption as a function of leisure time and
goods inputs, it is easy to see how satiation points
could be reached at which there is so little time for
consumption relative to the amounts of goods to be con-
sumed that no further increase in utility with an in-
crease in goods can occur without a concurrent increase
in leisure time. The phenomenon of "conspicuous con-
sumption" of additional goods without "using" them
could postpone or eliminate satiation, of course. The
point is that the outcome of the model can vary con-
siderably under these different specifications.
25
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programming problem then could be simplified by sub-
stitution to eliminate the money budget equation and
either the income or work time variable.
In the problem as stated above, the first-order
conditions for a maximum are
+ XW - 0 < 0
w
< o - o if t > o
i
< 0 if t » 0
<
}
< 0 if t =0
« 0 if x > 0
< 0 if x =0
- 0 if X > 0
X " Wtw - ~ ( < 0 if X =0
= 0 if Q > 0
t + ti - T < 0 <
V * ^1 4-^u / < 0 if 0 =0
The parameters X and 9 are the shadow prices associa-
ted with the money and time budget constraints, re-
spectively. Each will be positive if its constraint
is binding (an equality at optimal allocation of re-
sources). As such, they represent the marginal utili-
ties of money and time at the optimum, being the
26
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increases in utility for slight "relaxations" of the
constraints (through additions to stocks)i
dU a~U dx 3*bL dt ai(. dt
dT " ax dT + Bt dT + at, dT
dS Bx dS dtw dS 3tx dS
Where S is the amount by which consumption expendi-
tures exceed earned income.^ These shadow prices are
also often referred to as the marginal values of time
and moneyt the unit of value (utility) being implicit*
The ratio of 0 to X is the ratio of marginal
utility per unit time to marginal utility per dollar,
in terms of stocks. As such it is often referred to
as the "dollar value of time." In general, this
valuation need not refer to specific uses of time or
dollars, although its value depends on those uses.
In this model, if at the optimum there is not satia-
tion in leisure or consumption expenditures, so that
these arguments, and the partials of the utility
3Alternatively, S is a stock of other non-wage income.
In the statement here S is assumed to be zero, but
its inclusion in the money budget at a positive level
would not alter the first order conditions.
27
-------
function with respect to them are positive, then this
dollar value of time equals the marginal rate of sub-
stitution of leisure time for consumption expendi-
IL
tures
6 _ dU/dT SU/atj
* = dU/dS = SU/Bx
and represents the full opportunity cost of spending
a unit of time in leisure rather than in non- leisure
(income production for the purchase of goods). This,
then, is here the (marginal) value of that leisure
time, and from the first order condition for work it
may also be written
If work time is desirable (positive marginal
utility), then this opportunity cost of leisure and
thus its value is greater than the wage rate, while it
is less if work time is undesirable at the optimum
(negative marginal utility). The traditional theory
does not define utility over work, so that that par-
Strictly speaking, the negative of that marginal rate
of substitution.
28
-------
tial derivative vanishes and the marginal unit of
leisure is valued at the market price for labor time,
the wage rate. This familiar - but very special -
result is widely applied as a rule of thumb in valuing
time but does not appear to have much behavioral
justification. Introduction of other activities, such
as travel, qualifies that result even further, as we
shall see.
The labor offer curve itself may be obtained for
t as a function of w by substituting for the shadow
Tl
prices and w in the first order condition for t«
This can be denoted as some function g of w, since at
the optimum the other arguments and marginal utilities
may be solved for as functions of w and other constant
parameters. Since
... aUdx all dt aU dt,
dU _ _ _ . -- w + -- i = xt... > 0 ,
d^ - 3x dw + Stw dw * ati dw w - '
a higher wage is always preferred by the consumer-
worker to a lower one. Whether or not he will in-
crease his labor offer with an increase in the wage
29
-------
at a particular level of that wage depends on the
nature of g, which in turn depends on the nature of the
utility function. Traditional theory hypothesized a
"backward bending" curve, where t increased with w
V?
up to a point, then decreased with further wage in-
creases, although not enough to ever cause a decrease
in consumption expenditures.
Consumption Activities in the Labor-Leisure Model
A straightforward modification of the demand for
leisure model is the disaggregation of gross expendi-
tures x into individual physical amounts for each good,
x^, i = 1, . . . , n. When x is replaced by the x^
separately in the utility function and by its alge-
braic equivalent, Y Pixj» ^n ^ne money budget con-
straint, where p. is the price of good i, then the
first order conditions for the amounts purchased are
< 0 if x = 0
o
ax.
These conditions are no more than the first-order
conditions of the classical problem without time,
which now may be seen to fit into the demand for
leisure model with no obvious inconsistencies.
30
-------
The next simple extension would be to identify
particular activities within the leisure time com-
ponent and to make specific the amount of time, t, ,
and of each good i, XAT,» used in each activity k = 1,
. . . , m. Suppose that the utility of a good can vary
by its use, so that the x are used separately in the
utility function in place of the gross amounts x. -
X x., .-* With the individual times t, also in the
K
utility function in place of t, , the first-order con-
ditions for these arguments become (k = 1, • • • , m)
- - Xp, < 0 = 0 if x.k > 0
Sxik 7
< 0 if xik = 0
- 9 < 0 ( = 0 if tk > 0
k "~ I
K Q < 0 if tk = 0
This means that marginal utilities of time in all
activities (non-work) undertaken must be equal, that
the ratios of the marginal utilities of goods however
Again, this treatment may be interpreted as subsuming
in the utility function production functions for each
activity in time and goods, with utility defined over
the "levels" of the activity as given by those pro-
duction functions. That interpretation, however,
implies separability properties which are not pre-
requisite to the theory.
31
-------
used are still equal to the ratios of their prices,
and the ratio of the marginal utilities of an activity
time to a good input to that activity is equal to the
ratio of the dollar value of time (as the price of
time) to the price of the good.
A simplification to this type of extension to the
leisure-labor model that was made first by Becker? and,
p
later by DeVany, is the assumption of fixed propor-
tions in mixes of goods in activities. What is done is
to define each of a set of consumption activities in
terms of an identifiable "composite commodity" unique
to that activity.-' An activity becomes by definition
°For examination of higher-order conditions in simple
cases similar to thesef see the Mathematical Appendix
to Staff an B. Linder, The Harried Leisure Class* (New
York: Columbia University Press), 1970.
?Gary S. Becker, "A Theory of the Allocation of Time,"
Economic Journal, Vol. 75> September 1965» PP» ^95-51?.
8 Arthur DeVany, "Time in the Theory of the Consumer,"
Professional Paper No. 36, Center for Naval Analyses,
Arlington, Virginia, June 1970.
usage of the term "composite commodity" differs
from that in comparative statics, where a set of goods
can be treated as a composite commodity if their prices
change proportionately.
32
-------
the "consumption" of the appropriate composite good.
This one-to-one correspondence between activities and
mixes of goods avoids confusion which might arise in
the variable proportions model above concerning the
identifiability of activities having, at the optimum,
similar input structures. On the other hand, to re-
move input substitution within activities seems to
overly restrict the range of choices open to the con-
sumer.
If the amount of the composite good k consumed
in its activity is denoted A, , then the requirement
of proportional inputs for good i can be written as
Xiks cik\
The scale of the composite goods is arbitrary, thus
so is the scale of the c coefficients.
A related simplification made by both Becker and
DeVany is the assumption that the consumption of each
"unit" of the composite good k takes a fixed amount of
time, b^. Thus
Again, the scale of the b^ is arbitrary. Indeed, any
one good or time could have been picked as "numeraire"
33
-------
and used in place of the composite goodsf which are»
in effect, indexes for the levels of the consumption
activities. These proportionality relations may be
used to restate the budget constraints in terms of the
Ak by substitution. With utility defined over these
activity (or composite good) levels, first order con-
ditions for a maximum become
-o«*kio
-------
sumption time per unit of k is valued at the dollar
value of time, (9/X), and the resulting value is added
to the composite market price P. for the good to give
its full or shadow price in terms of both of the con-
sumer's scarce resources, time and money.
Work may be included among these activities, or
the stock of income may be taken as given. In the
latter case, there is a good chance that one constraint
is slack. Indeed, with fixed proportions this is more
than likely, meaning that one shadow price would be
zero and that all shadow prices and orve set of real
prices, either the Pfe or the b^, would be eliminated
from the marginal utility ratios above. With income
generation within the model, however, both constraints
are likely to be binding. Under the traditional assump-
tion of no (dis-) utility to work, the dollar value of
time is then
W -
There the value of time in dollars is the wage rate
corrected for the costs of inputs to work supplied by
the consumer-worker himself. With no such inputs of
35
-------
physical goods (ciw - 0, all i), bw can be set to
unity and all other t>k*s scaled according, since that
scale is arbitrary, yielding the familiar 6/X = W.
In any case, the full prices -77^ are then constants in
the system, and the structure of the Becker-DeVany
activity analysis in A, and 77-^ is strictly analogous to
traditional non-time analysis in physical quantities
x. and their prices p..
The chain rule for partial derivatives may be used
to restate the first order conditions in terms of the
original arguments xik and tk for comparison with the
variable proportions case presented previously.
In the variable proportions case, the production
function interpretation implies relations such as
Ak =#k(V x.k)
with the#k as the production function for activity
k in terms of time and goods inputs, in variable
proportions. When utility is defined over the A ,
simple substitution of the production functions ^
into the utility function yields a (separable)
utility function in t, and x.,, as used previously.
36
-------
3tk ~" BAk 3tk "" bk 3Ak
Now the ratio of the marginal utilities for two goods
used in the same activity is equal to the inverse ratio
of their input coefficients, rather than equal to the
ratio of the prices, as in the variable proportions
case. Ratios of marginal utilities for arguments in
different activities will involve ratios of the ac-
tivities* full prices divided by the appropriate input
coefficients as weights.
The assumption of fixed proportionality seems un-
necessary in these theoretical models, although it does
show under what circumstances the time and money model
is analogous to traditional non-time analysis, and it
may offer computational simplicities for empirical
application. Judging from the substantial variation in
times spent on activities by consumers surveyed in
time budget studies, it seems premature to posit strict
12.
See, for example, Philip G. Hammer, Jr., and F. Stuart
Chapin, Jr., Human Time Allocationi A Case Study of
Washington, P.O., (Chapel Hill: Center for Urban and
Regional Studies, University of North Carlina), March
1972.
37
-------
proportionality of inputs, or at least to interpret
such proportionality as any more than an average
tendency. It may be that input substitution in ac-
tivity performance needs to be constrained. However,
the problem can be handled by postulating production
functions or utility functions exhibiting diseconomies
or satiation for arguments which are too high relative
to the levels of other inputs to their activities.1^
Theorists do not like to consider satiation, as its
existence can complicate the conditions for an optimum.
Prom a behavioral viewpoint, however, it is a likely
phenomenon, certainly at the level of the individual
consumer in the short run. This point will come up
again in considering activity frequency as well as
duration.
Consumption Technology as Universal
Activity definition per se is an extremely diffi-
cult problem, conceptually and empirically, for there
may be cognitive distinctions not reflected by differences
li
•'Indeed, DeVany specifically challenges Becker's
assertation that proportionality relationships con-
stitute proper production functions for activities.
His solution is to posit one-to-one correspondence
between goods and activities, which seems hardly more
than a technical sleight-of-hand.
38
-------
in input allocations to activities. Becker's approach
has been to posit technological relationships among in-
puts to activities as if these were independent of
14
consumer preferences for activities. The obvious
advantage of this approach is that activity definition
becomes universal even if preference is peculiar to the
individual. Kelvin Lancaster has gone one step fur-
ther and suggested that the experiencing of activity
performance is multi-dimensional and that the intensity
of the activity on each dimension contributes to the
utility of the activity. Preferences, then, are di-
rectly for types of stimulii as represented by those
dimensions, and only indirectly for activities in
terms of their ability to impart those stimulii.
1 Another input to activities which has been suggested
is "energy" or "effort", although identification of
the appropriate stock level and use coefficients, and
even "production" of energy in other activities,
might be difficult and quite particularistic. See
Mark Menchikfs unpublished paper, "Towards a Generali-
zation of the Theory of Consumer Behavior," (Phila-
delphia: Regional Science Research Institute), April
27/1967.
Kelvin Lancaster, "A New Approach to Consumer Theory,"
Journal of Political Economy. Vol.7*h W6) ??
39
-------
In Lancaster^ scheme, then, an activity is
characterized net only "by its input structure, as with
Becker, but also by its output structure in terms of
the stimulii it imparts. Lancaster assumes that the
intensity level Zm of stimulus (or "characteristic")
m is a linear function of the activity levels At
2mH emkAk ».!,...
k
Utility, in turn, is defined over the stimulus in-
tensities Z rather than over the activity arguments.
He follows Becker in assuming proportional goods in-
puts to activities —
xik 3
— and assumes that the coefficients em. as well as the
are Univers3-l parameters independent of individual
preferences for characteristics.
The optimization problem for the consumer becomes
the problem of choosing levels for activities so that
utility over characteristics is maximized, subject to
the money budget constraint and the technological con-
sumption relations above. With more activities to
choose from than there are characteristics and with the
40
-------
possibility of experiencing the same characteristics
in many different activities at varying intensities,
something less than the full set of activities need be
undertaken in maximizing utility even if utility is
(as he presumes) a positive monotonic function of all
characteristics. Because of the technical complexities
thus created in the general case, optimization there
is intractable analytically, although special cases
can be handled analytically. It is therefore difficult
to say r-.uch about the nature of optimal solutions.
Lancaster does show how an "efficiency frontier" can be
constructed to weed out activities that are strictly
inferior to others in characteristics production under
the money budget constraint and thus to identify among
all possible activity combinations a much smaller sub-
set which will contain the optimal solution. Choice
among this reduced set along the efficiency frontier
is called the "efficiency choice."
Time as an input to activities and a resource
stock constraint is not included in Lancaster's scheme.
Incorporation of time is possible, but the addition of
the second budget constraint complicates the analysis
because it further breaks up an already jagged effi-
41
-------
ciency frontier into additional segments for which
only one constraint or the other is binding, with
"both binding only at corner points,if income is held
constant. With variable income (i.e., work as an ex-
plicit activity), there can be segments on the frontier
where both constraints are binding. These properties
are demonstrated by an example worked out in the Appen-
dix -to this report, although they have not been demon-
strated for the general case. One interesting property
of the example chosen is that when both constraints
are binding, the dollar value of time which results
is independent of preferences and thus of the actual
location of the optimal solution on that segment of the
efficiency frontier where both constraints are binding.
This is not expected to be a property of the general
model, however.
Two Applications: Household Y/ork and Overtime
There are numerous theoretical applications that
can be made of the aspatial time and money model as it
has been developed here. An example is the "home pro-
duction" of consumer goods'or services. This can be
42
-------
viewed as a problem in labor force participation
or as a problem in the maintenance of consumption
goods for activities as an alternative to purchase of
17
new goods. Suppose that such a good or service xs
is available on the market at price p_ per unit, and
s
that it may also be produced by the individual himself
according to the production function ^(x^g* ts), with
inputs of goods xis and time ts. The Xj_s are amounts
of goods i bought on the market at price p^, of course.
Let the good or service, whether purchased or produced,
be homogeneous in quality and used as an input xsk to
other activities k. Then the consumer optimization
problem in variable proportions will have the additional
In this problem, the a.uestion is whether a person's
time is worth more on the labor market or in house-
hold tasks, as, for example, in the decision for
potential secondary wage earners in a household to
enter the labor market. One statement of this problem
is given in O.K. Erownlee and Hohn A. Buttrick, Pro-
ducer, Consumer, and Social Choice, (New York» McGraw-
Hill), 19"5B» Chapter 13 •
^Linder, op. cit«, offers a different version of the
discussion here, in which he treats "personal work"
as a contributor to real income along with work for
wages in the labor market.
43
-------
constraint
+ xs - jj) xsk > 0
specifying thst the uses of the good or service s in
activities k not exceed supply via purchase (xs) or
production (Qs(xis, ts)).
When these new arguments - xs, x^s, ts, and xsk •
are introduced into the utility function and into the
appropriate constraints, additional first order condi-
tions for a maximum in the variable proportions model-
are found as
dx
o
s
s
= o if xs > 0
< 0 if x = 0
- < o C = 0 if xsk > 0
xsk /
( < 0 if xsk = 0
0 if xis = 0
at! -
- 0 if t > 0
< o if t- = o
-5C - a (TT. -h ^ 4. VY <- n C ~ ° if °c! > °
"xs -"slXis, ^s^+y.XskS ° ^ s -
k
< 0 if Os = 0
44
-------
where $ > 0 is the shadow price on the use-supply
constraint for activity s. First order conditions for
other arguments are the same as in the variable pro-
portions model above. It is presumed that the partials
of Qs with respect to tg and the xis are all non-negative,
Obviously, if the good or service s is not used
in an activity k then the marginal utility of xsk must
be non-positive. If this were true for all activities,
then § would be zero and the good or service would be
purchased or produced only if the appropriate arguments
had sufficiently positive marginal utility, as for any
other good or time input. Marginal utilities of x^s
and ts, like those for work inputs, need not be positive
for the activity to be undertaken.
There are many different situations which can
develop here. Let's take here only the simple case
where the only input to home production of the service
s is time, in terms of which the service is measured
(as is x_, but that is someone else's time). Then the
S
first order conditions for the xis drop out, and the
condition for ts becomes
45
-------
If some xsk > 0 at the optimum, then for both the home
production and purchase on the market to occur the
following relation must hold between the appropriate
marginal utilities and the dollar value of time (ob-
tained by substitution for
I S VU dtl7
\ ( dto " Bxs S
X = PS
We note, too, that in this case 0
o
r * r ^ 'w ; < o if ts = o
which can hold simultaneously if
46
-------
PS = X = W + dl
where the right hand side of this relation is the
first order condition for work time. If the consumer
is also neutral to work so that that marginal utility
vanishes, then both conditions can hold only when
p = w, assuming 0. If W > p_, there will be only
S S
purchase and no self production. The opposite is true
if w < ps. The deciding factor is simply whether or
not payment in kind (terms of pg as the opportunity
cost of home production) is greater than the wage rate
so that home production returns more than work.
Again, there are other situations which can de-
velop under other assumptions on the marginal utilities.
Further, if there is a source of income other than work
(e.g., a spouse's wages), there is the possibility that
there will be no work undertaken if ps is relatively
high. This consideration is used in applying this
model to the study of the labor force participation
rate of married women faced with the choice between
work and housework. In particular, labor offer curves
can be derived to show at what wage level relative to
p_ such a person would enter the labor market and what
O
work time would be offered.
47
-------
The consumer's ability to vary his work time at
will has seemed to many theorists an unrealistic postu-
late of the labor-leisure model, and some attention
has been given to models in which there was a fixed
work shift, with the possibility of overtime work.
1 R
The classic work is that of Moses , but a slightly
different version will be developed here. Suppose that
the worker must work a shift of length to at wage W per
unit time but may also work overtime at the wage rate
W =
-------
proportions model)
|3i + XKW - 0 + 0 = 0
dtw
where y>> 0 is the shadow price on the new work time
constraint. This condition is an equation, since the
problem requires t > 0.
w
If at the optimum t > to, then overtime is under
taken, and tp= 0. Moses terms such a worker a "work
preferer." Note that in such a case
so that the dollar value of time is based on oCw
rather than w alone. Here tfw is the marginal wage
rate even though all income is not earned at that rate.
This valuation of time may differ from a specification
of the model in which wi^ was considered fixed income
so that that portion of work time, t0, would not figure
into the utility function. Optimal tw would also differ
from that work time in the free-choice model at rate o(w
for all work time, since the bundle of goods obtainable
for a given work time would differ, and that would
affect time and money allocations.
49
-------
If tw = t0 then lp> 0, and we have a "leisure
preferer" (Koses1 term) whose shift forces him to work
as much or more than he would choose otherwise. The
role of this shadow price t0, an increase in tQ will
cause a change in the proportions of total work time
tw which are remunerated at rates w and oCw* dt^/d^Q
need not "be unity, as it must be when (/>> 0. With
- 0,
- = W(l -Ot)X
dto
This will be negative when o(> 1, since it would then
mean a reduction in income for a given amount of work
Readjustment of t could be in either direction, as
VF
with wage changes in the simple labor-leisure model.
A change in the wage rate w here could also cause a
50
-------
change in tw in either direction. In general, then,
seems to represent a measure of the cost (in utility)
of a forced suboptimal allocation of work time.
would give a dollar equivalent of this cost.
Activity Frequency in Time and Money Allocations
A dimension which is rarely given comprehensive
treatment in theories of consumer "behavior is activity
frequency, giving the number of activity occurrences
or episodes to which time and money (goods) are allo-
cated.20 It seems intuitively obvious that individuals
are not completely indifferent to the way a given amount
of time in an activity is organized into discrete epi-
sodes. For example , the four day work week may not
be preferred to the five, even for a given amount of
work time and a fixed wage rate, if there is more dis-
satisfaction created with longer hours worked per day
with less leisure time on those days than is gained
through transportation savings (fewer work trips) and
20
Activity frequency has been studied in travel beha-
vior, as described in the next chapter, but usually
without reference to non-travel behavior. For further
discussion and some empirical findings, see Hammer
and Chapin, op. cit.
51
-------
more continuous leisure time on the weekend.
As is implied in this example, actual sequence of
activities is a factor in the utility derived along
with frequency-time-money relations. Sequencing, how-
ever, poses problems in optimization which are much
too complex for the purposes here. It is felt, on the
other hand, that activity frequency is one aspect of
activity organization which can and should be repre-
sented in an optimizing model of consumer behavior.
For transportation and public faciiity planning purposes
there would seem to be a need to be able to explain
traffic and facility use patterns in terms of under-
lying activity time, frequency and money tradeoffs.
Such explanation would be crucial in relating changes
in travel patterns to changes in facility use and to
the organization of household activity in general.
Again, to look at durations of individual episodes, as
one aspect of sequencing, is to delve into more detail
than can be expected to provide some simple insights
into behavior. In addition, the optimizing models here
must treat frequency as a continuous variable, allowing
"fractional episodes." However, if frequency can be
viewed as an average tendency from a sampling of time
periods, this continuous feature of the variable should
52
-------
cause not more conceptual difficulty than it does in
the case of physical goods, themselves more discrete
than continuous. Finally, there is also the implica-
tion of homogeneity of activity episode or occurrence.
It is proposed, then, that the frequency f^ of an
activity k be introduced to the utility function along
with the time (tk) and goods (x^k) inputs. As with
time and goods, it is expected that satiation will be
a relevant property of utility relationships involving
activity frequency. It should be easy to see, for
example, how there could be points at which activity
frequency is so great relative to the total time spent
on the activity that the frequency could not be further
increased without a loss in utility unless the total
time spent on the activity were also increased (see
footnote 2, this chapter). That is, for that frequency
there is a finite "best" total time (or average time
per occurrence), all else equal. Such satiation would
be reflected in utility contours, or "indifference sur-
faces," which are concave to the origin in the vicinity
of the origin, indicating substitution possibilities
between arguments, but which turn away from the axes
and the origin beyond the "satiation points," showing a
53
-------
shift to complementarity relations between arguments
to the utility function.
Models with activity frequency explicit can vary
greatly according to the types of activities considered.
Occasionally a new constraint must be added, such as
constraints which assure that for out-of-home activities
there are enough trips to "the activity location to per-
mit the number of activity occurrences taking place
there. These will be considered further in the next
chapter. For out-of-home activities involving facility
use there may be admission prices charged, so that fre-
quency variables appear in the money budget. However,
for activities at home and many "free" ones out-of-home,
there will be no direct cost to frequency per se. This
means that the first order conditions in such cases
would simply be non-positivity constraints on the mar-
ginal utilities*
= 0 if fk > 0
< 0 if f. = 0
Obviously, the marginal utility would be zero only at
a satiation point for the frequency relative to the
levels of other activities.
54
-------
With an admission price charged for out-of-home
activities, optimum frequency levels should be some-
thing less than the corresponding satiation points,
since with such prices q^, the first-order conditions
< o f - o if fk , o
< 0 if fk = 0
demand positive marginal utility if the frequency is
greater than zero. In connection with admission prices
for out-of-home activities, it should be mentioned that
there could be charges by the hour for facility use in
place of or in addition to admission prices. This would
place the time argument for the activity in both the
time and money constraints, so that the first order
condition for that argument would contain both shadow
prices, as is the case with work.
In passing it should be mentioned that when more
than one shadow price appears in first order conditions,
it can be difficult to derive a system of equations in
the unknown levels of the arguments from which those
shadow prices have been eliminated by substitution.
This can make analytical solution of the system for a
specified utility function very complicated. Although
this is a practical matter and does not qualify the
55
-------
theorems derived about behavior, it does explain why
practical applications of the theory often go to ex-
treme lengths to eliminate extra constraints and shadow
prices from the consumer's optimization problem.
Since by definition the total time spent on an
activity is the product of the frequency of occurrence
and the average time per occurrence, substitution of
that product for total activity time can put the problem
in terms of those average times (denoted here tk) in-
stead of the totals. This does not change the optimi-
zation problem, of course, but it may make manipulation
of the first order conditions easier for theorem de-
rivation. For example, the first order conditions for
the classical labor-leisure model with activity frequency
explicit become
- X < 0 $ . 0 if x > 0
< 0 if x = 0
< o
< o
< 0 if t-, = 0
56
-------
- xw) < o ( = o if fw >
0 if fw = 0
0 if tw > 0
J± . f (e - xw) < o 7 . - = 0
jt L W
In this case, the leisure and work activities have
marginal rates of substitution of frequency for average
time inversely proportional to those arguments
MRS(fk for tk) = k = k = 1, w
fk
There are also explicit statements for the dollar value
of time here in terms of the marginal utilities of
frequencies arid marginal rates of substitution for con-
sumption expenditures:
£ = 1 , w
X ti
Again, if there is indifference to work time (average,)
and frequency, the dollar value of time equals the wage
rate.
57
-------
This has been only a brief introduction to the
role of activity frequency in optimizing models of in-
dividual consumer behavior. More examples will be
presented in the following chapters. No attempt will
be made to cover systematically the wide range of model
specifications that can arise with activity frequency,
however. To delve into the implications of alternative
frequency/time pricing arrangements for the use of fa-
cilities is of considerable interest in its own right,
but it would contribute little to our critique of the
River Basin Model. The present discussion, on the other
hand, should provide a broad frame of reference for dis-
cussing that model.
58
-------
SECTION IV
TIME AND FREQUENCY IN
TRAVEL BEHAVIOR
Time is of special relevance in considering the
spatial behavior of consumers because in many instances
time' may become a more restrictive factor in movement
over space than is money. "Space" is used in a very
general and often artificial sense in these applica—
tions. It is standard procedure to view space as a
"featureless plain" permitting movement in any direction
at rates which are independent of direction and place
per se. The alternative to this procedure is to allow
movement only over given paths with travel times and
costs specific to the links taken between nodes in the
path. Although appropriate for simulation and opera-
tional modeling, this network approach is usually not
helpful in deriving theorems from a classical type
model of consumer behavior.
Straight-line movement between points is the first
case of time use with spatial implications that we will
consider. Such movement is presumed to occur because
of the spatial separation of opportunities for different
types of activities. The second case to be taken up
(in the next chapter) is the actual use of space itself
59
-------
"by the consumer as, for example, an input in its own
right to activities requiring spatial extent, plus the
choice of location(s) for assembling those inputs (i.e.,
the residence). Indeed, it is the extensive quality of
space use that forces the differentiation of space by lo-
cation, as space is scarce at locations and can accomo-
date only so much activity. Such differentiation is
reflected in variable prices for space, or "land,"
viewed as a primary input to activity. As such an in-
put, land is presumed to be a homogeneous good, like
other goods in the traditional scheme. In this chapter,
then, we will take up the movement over space in con-
sumer behavior, leaving the extensive use of space by
the consumer to the next section.
Journey-to-Work in the labor-Leisure Model
In considering movement over space the logical
place to begin is the journey-to-work, given the cen-
tral! ty of the work activity in time-inclusive models
of consumer behavior. Perhaps the simplest case in-
volves a given residence and place of work, with travel
time and work time standing in some strict relationship.
The arch typical case would have travel time and work
60
-------
time in a fixed proportion. This has realism only
under the presumption of a fixed work shift time and
a fixed trip duration, so that the proportionality of
work and travel time is "based on the necessity of a
round trip for each shift worked. Obviously, the time
variable for work and travel in such a case is really
a proxy for frequency, but we will postpone explicit
introduction of frequency for the moment. Activity
times here will be gross rather than average.
Let us add travel cost kd and time td of tra-
versing the distance d in the journey- to-work to the
money and time budgets, respectively, in the labor-
leisure model*
Wtw - x - kd > 0
Obviously, if these were constants independent of
other times and expenditures, the loss of utility for
external changes in their levels would be measured by
the resource shadow prices i
dU dU dx .
dkd &x dkd atw dkd dtx dk
61
-------
Likewise,
dU _ aU _ 0
dtd ~
Note that provision is made here for inclusion of these
arguments into the utility function, even when they are
not choice variables for the consumer. The point is
that treatment of consumer behavior should be ex-
haustive, covering all uses of time and money even if
ar«.
some^beyond the control of the consumer. Here the sha-
dow prices have a negative sign, since increases in
these arguments are depletions to stocks. Here the
relative value of a marginal reduction in travel time
versus a marginal reduction in travel cost is
- dU/dtd _ e -
- du/dtd x -
This dollar value of time in travel equals the general
dollar of time 6/X only with neutrality toward travel
time and cost*
The cases that we are working up to will treat
travel as an activity at least partially under the con
trol of the consumer, with travel time and cost in the
utility function as a general case. As stated above,
62
-------
•the simplest situation would be that with proportion
ality between total travel time and total work time
w
- with ct > 0 the constant of proportionality. For
simplicity t assume, too, that travel cost is propor
tional to travel time:
The budget equations for the labor-leisure become, on
substitution,
w - x > 0
T - (1 + «)t - t > 0
w
td and kd can be eliminated from the utility function
by substitution, so that the optimization problem is
in form strictly analogous to the original labor-
leisure problem. The differences are a revised utility
function, so that the marginal utilities could have
different functional forms, and different weights on
t in the budget constraints.
Tl
Without writing out all of the first-order equa-
tions here, the dollar value of time can be given as
63
-------
9 aiV/dt W - 6cr
X dU/ax l + a \(1 + «)
The marginal utility of work includes the effects of
travel time and cost, effects that are usually thought
to yield disutility.1 Even with neutrality to work
and travel, so that the partial vanishes, the dollar
value of time have should be significantly less than is
value without travel.
Changes in utility with marginal changes in the
travel parameters
-------
marginal reduction in the travel time compared to that
of a marginal reduction in the unit money cost for
travel is
- dU/dty _ 1/0 , ft^
- dU/d0 ~ a\ r*
Presuming that tw > t^, so that
-------
to-work. Thus, if there are n such alternatives,
j = 1, . . . , n, there would be n pairs of constraints
in the labor-leisure model
- x - kj > 0
T - t, - tx - tj > 0
where the k^ and t^ are constants giving the travel
j j — — — — — —
cost and time, respectively, for route/mode combina-
tion j. The shadow prices for the pairs of constraints
are the set of pairs, Xj and 63.
The first order conditions for leisure and work
time and consumption expenditures are
= 0
<0iftw=0
, ,0
j / < 0 if x = 0
order
The first.conditions for a route/mode combination j
are the budget constraints t
66
-------
Wtw < 0 } = 0 if X.. > 0
< 0 if X . = 0
0
- T < 0 \ := 0 if 9j > 0
< 0 if 9. =0
If a route is chosen, its shadow prices are positive?
otherwise, they are zero. If the time and price pairs
are all different, then only one route/mode combination
should be chosen. The results thus obtained are basic-
ally those of the simple labor-leisure model with con-
stant travel cost and time, there being but one pair of
positive shadow prices X and 9. This discrete choice
formulation is not unlike the structure developed for
travel selection in the River Basin Model. Moses and
Williamson use this technique as a basis for discussing
"diversion prices" for influencing changes in travel
choices. However, unless solutions were worked out for
each route/mode combination as if it were the only
choice available, so that shadow prices and utility
levels could be compared, there is very little we can
say about the nature of the optimal solution here.
67
-------
3y way of comparison, suppose that route/mode
choice were continuous --that is, suppose it were possible
to trade travel time against travel cost in a continu-
ous fashion according to some functional relationship
between the two. For example, let
be added to the system as a constraint. Here the
function 0
< o if td = 0
68
-------
= o if kd > o
Mote that this problem, like the Moses-Williamson prob-
lem, treats the travel choice independently of the
work time allocation. Here $ > 0 is the shadow price
on the travel constraint. It will be zero if at the
optimum the consumer enjoys travel so much that he
spends more time than is necessary on it (the marginal
utility of travel .must then be positive and equal to
0). If there is disutility to travel time, so that the
marginal utility of travel time is negative, then |)
must be positive. Finally, when there is complete
neutrality to travel (so that partials with respect to
time and cost vanish), optimal mode/route selection
will be related to the dollar value of time through
(in terms of kd and td) as
a _ . &. > o
X - *kd
the partial of 9" being negative. In such a fictitious
situation as this, we would have precise valuation of
marginal preferences and valuations concerning travel
69
-------
alternatives, something we do not have in the discrete
choice case.
Trip Frequency in Route and Mode Selection
There is considerable loss of clarity in the models
above with the failure to make frequencies explicit.
For example, the Moses-Williamson journey-to-work route/
mode choice problem can be restated to involve the num-
ber of trips, f., by route/mode type j as an explicit
*J
choice variable, with fixed parameters or* and 3 i as the
J J
time and money costs per trip, respectively. Then
give the total time and money allocations to mode/route
alternative j, used f. times. Total travel time and
u
cost over all alternatives are the sums of these com-
ponents :
t- =7 * f
0 L J j
J
70
-------
These can be entered directly into the time and money
"budget constraints without having to specify separate
budget constraints for each alternative. However,
if there is some minimal number of trips f which must
be taken, then there should be an additional constraint
- f > 0
Then, in the simple labor-leisure model, along with
the usual first order conditional for t-^, tw, and x
will be the set of conditions for the frequencies f .•
o
of using mode/route selection j»J
\ = 0 if f . > 0
i < 3
. X8 . - e*. + ) < o / < o if f = o
where $ is the shadow price on the trip frequency con-
straint.
Obviously, some trips must be made in this model.
With positive marginal utility to some trip type, there
could be more trips than necessary, so that jo = 0.
3 Variables t^ and k^ are presumed replaced in the utili-
ty function with f ^ by substitution. The f* may be in
the utility function by their own right as well, but
their marginal utilities must now also reflect time
and cost effects.
71
-------
More than likely, however, $ will be positive. It is
apparent that more than one trip type may be utilized
at the optimum, but it is perfectly possible for one
to be so far preferable to others that it is the only
one taken. In the case of neutrality to trips by
whatever mode, then there will be only one positive
trip frequency. With neutrality to travel, $ will be
positive, so that that trip frequency would equal f .
Then the ratio /\ gives the dollar value of trip fre-
quency,
where j is the optimal route/mode. However, if > 0,
then a reduction in the frequency f (viewed as a "stock")
is desired, since
_ dx SU_ dtx 5U dtw r dU, df.
df ax df ati df dt™ df * . 3fT df
J "
< 0
This means that more trips are required than would be
chosen otherwise.
For a mode/route selection j that is utilized,
however many are utilized, the marginal utilities of
increases in the time and cost parameters are given as
12
-------
dU _ Q f.
, — W -L -,
d °
= . xf . +
de. 0 * as.
J o
The partials with respect to the arguments must be
included, since they entered the utility function by
substitution (see footnote 3 above). If these partials
are negative, so are the marginal utilities of the in-
creases. With neutrality toward the ». and Q*, then
J
-------
frequency fw represents the number of shifts worked.
Therefore, in this model we can replace the constant
f in the frequency constraint with the choice variable
fw. If we optimize with respect to fw as well as tw
and the other variables in the labor-leisure model,
plus the trip frequency by mode and route, we get the
additional first order condition (presuming f... is an
W
argument to the utility function)
= 0 if fw > 0
< 0 if f = 0
W
Obviously, to get positive f... here requires positive
W
marginal utility if § > 0, which means that the con-
sumer prefers to break work time t up into separate
Inf
episodes even though it requires more trips than he
would otherwise make. If trip-making is so pleasurable
that 0 a* 0, the solution for fw is at a satiation point
relative to the other arguments.
An alternative specification is to assume that
work shift length is fixed at s time units. Then
1^ = sfw can be substituted into the utility function
and the money and time equations. The revised first
order condition for f is
ri
74
-------
- S(9 - Xw) - 0
W
< 0 if fw = 0
W
remembering that the condition for tw is now meaning-
less and that the marginal utility of fw incorporates
effects of the utility of work time. In this speci-
fication it is no longer necessary for the marginal
utility of f (with time effects) to be non-negative
to get a positive solution. With such pleasure to
travel that $ = 0, we have the traditional result ex-
pressed in frequency terms, such that neutrality to
work produces the familiar &/\ = W.
In this version, the dollar value of time can be
related to work frequency and trip frequency in mode/
route choice j under the presumption that j6 > 0 accord
ing to
+T
X w
where, if j is the only alternative chosen, then fw =
since ^ > 0. If the partials vanish the dollar value
of time equals the ratio of the net income after trans
portation earned per trip to the total time (work plus
travel per trip. The marginal utility of an external
75
-------
increase in the required shift length S is
f = (XW - 9) fw + f
where S entered the utility function through substi
tution for
Trip Frequency for Leisure Activities
In the models above with travel frequency explicit,
it is apparent that when $ = 0 and the number of trips
exceeds that required for the journey- to-work, then
there is the occurrence of trip-making for pleasure.
The present treatment makes no distinction between such
purposes, but it is simple enough to separate the journey-
to-work from pleasure trips and enter both in the utility
function. In such a case the frequency constraint on
the journey- to -work should be a definitional equation,
so that p could take either sign. This is equivalent
to stating simply that there cannot be a journey- to -work
without an occurrence of work, which seems logical
enough.
Frequencies for pleasure trips need not be con-
strained in such a manner. Johnson, for example, treats
pleasure trip frequencies exactly as DeVany treats goods,
76
-------
with proportional time and cost coefficients. That
obviously makes them specific to origin and destina-
tion. Chiswick disaggregates them by type to allow
specification by route and mode as well as purpose and
origin and destination, although he does not do this
explicitly. Neither, however, faces up to the problem
of distinguising from the properties of trips and the
properties of the activities which take place at their
destinations. If we thus allow consumers to spend time
at destinations of trips before returning home, then
the study of trip frequency behavior can be integrated
with the disaggregation of out-of-home leisure activi-
ties by type. Finally, there could be differentiation
among these out of home leisure activities according to
pricing policies. Use of commercial facilities may be
possible at a flat admission rate per occurrence (e.g.,
a swimming pool or a zoo), at an admission rate with a
time limit (e.g., a movie), at a fixed rate per hour
K. Bruce Johnson, "Travel Time and The Price of leisure,"
Western Sconomic Journal, Vol. k, Spring 1966, pp. 135-
--
^Barry R. Chiswick, "The Economic Value of Time and The
Wage Rate: A Comment," Western Economic Journal,
77
-------
(e.g., a tennis court)* or at some combination of
these. Free activities would include visiting in
houses and use of public facilities for which no
charge is made.
By way of illustration, let leisure time t^ be
disaggregated into activity times spent in leisure
activities at home, tm, m = 1, . . . , and times t^
spent in out-of-home activities, k = 1, . . . Let
their corresponding frequencies of occurrence be fm and
Tt will be presumed that use of consumption goods
x^, i = 1, . . . , occurs entirely in home in the ac-
tivities m. This is not necessary but makes things
simpler. Let pleasure trips be undertaken for the pur-
pose of their out~of-home leisure activities, on the
presumption that travel is homogeneous with respect to
trip purpose. This will be reconsidered below. fd will
denote the number of pleasure trips taken. The assump-
tion, will also be made that all activities occur at a
single place, for example, the "city center," so that
trip time & and cost 8 per trip are constant. Likewise,
a single route/mode alternative is assumed. Generali-
zation to route/mode choice could be introduced in the
manner above. Generalization to other activity locations
78
-------
will be brought up later. Finally, to make these re-
sults comparable to those in the discussion of ac-
tivity frequency in the previous chapter, leisure out-
of-home activity times here will be average times t^.
The analysis to follow will not explicitly consider
work time, the journey-to-work, or in-home activities.
In general, their treatment would be much the same as
in the specifications considered above and in the last
section of the previous chapter.
In summary, then, utility will be defined here
over activity times t^ and frequencies fk (and fd for
travel; since td =
-------
travel time for that
leisure
where a is travel time per trip
B is travel cost per trip
q, is admission price for k
rk is unit time price for k
As "before, trip frequency must have a constraint in
addition to the time and money budgetsi
The first order conditions which result for
out-of-home leisure activities are
0 if fk > 0
- X(q, + r t) - Gt - d < 0
Hk k fc ^ r "" / < 0 i
iffk = 0
< o if tk = o
and for travel frequency
X8 -Gff+ b < 0 0 if fd > 0
'
< 0 if fd = 0
80
-------
where is the shadow price on the frequency constraint.
Its interpretation is analogous to that given above
for the frequency constraint in mode/route choice for
the journey-to -work.
Assuming that an activity k is undertaken, the
(negative of) the marginal rate of substitution for
frequency for average time in that activity is
The term in parentheses on the right gives a ratio of
full prices for frequency and time spent in the ac-
tivity, being the sums of their "market" prices (if
there are any) and the shadow prices on the resource
used (with trips viev/ed as a stock allocated to ac-
tivities). The dollar value of a trip as a resource
is given by the first order condition for trip fre-
quency as a function of trip cost (direct plus time
valuation) less the marginal utility of trip frequency;
4 = o - «I,,-,-,
Md
For > 0 (no extra pleasure trips), that marginal
utility could well be negative.
81
-------
Mote that for free activities to occur their mar-
ginal utilities must be set equal to 0. With satiation
in such activities, they could occur with $ = 0. How-
ever, if any one has strictly positive marginal utility,
then $ must be positive, so that no extra trips are
made. That would mean that even if trips were pleasur-
able, the free activity would be performed enough to
assure that no extra trips need be taken to satisfy the
(relative) desire for travel.
An additional complication which might arise for
activities charging admission (frequency) prices qk but
not time prices (rk = 0) is that there may be for such
activities a maximum limit on the duration of an
occurrence. A movie would be an example. In terms of
gross activity time the constraint would be
or, on the average,
There would be a shadow price t^k > 0 for every ac-
tivity with such a constraint, and the marginal rate
of substitution relation would become
82
-------
qk
(f)fk + (
with ^k/X as the dollar cost of the average time con-
straint. The multiplicity of these constraints makes
analytic solution of the problem considerably more
complicated.
A difficulty with the specification above is the
assumption of homogeneity of trips regardless of purpose.
This means, for example, that if there are extra pleasure
trips made, so that $ = 0, then all types of trips are
equally pleasurable. The simple solution is to dis-
aggregate trips by purpose (destination activity) and
enter them separately in the utility function. This
means replacing the single trip frequency constraint
with a constraint for each activity
f dk - f k 2- 0
where fdk is the number of trips made to perform fk
episodes of activity k. If this relation is forced
to be a definitional equation, presuming that a trip
cannot be made for an activity without an occurrence
83
-------
of the activity, then the f,. arguments can be elimi-
nated from the problem by substitution, remembering
that activity frequency now incorporates trip pre-
ference in its utility relations. This is acceptable
under fixed activity locations and mode/route selections.
Here, then, travel time and cost parameters per trip,
&•£ and RlK, must be specific to activity. Then the first
order conditions for out-of-home leisure activities
(without average time constraints) are
- 6(tk
= 0 if f > 0
k ~
< 0 if f = 0
K
if
_
< 0 if t = 0
1C
There is no longer a general travel argument f.. The
new marginal rate of substitution relations are
84
-------
where the term in parentheses gives the ratio of the
revised "full" prices.
This interpretation of the problem, incorporating
the effects of travel into the utility of the activity,
is a familiar one in the literature on travel behavior*
although such trip-activity frequency and time rela-
tions are virtually never made explicit. As mentioned
before, it would be possible to introduce route and
mode choice for each activity in the manner presented
for the journey-to-work, although the resulting opti-
mization problem would become formidable. Likewise,
the work activity and its journey can be explicit com-
ponents. With average time constraints the shadow prices
FU will appear again.
Pleasure trips for their own sake have dropped out
of the model but may be reintroduced quite simply and
made specific to location. Thus, let fdk now represent
the number of trips to location k as round trip excur-
sions purely for the ride. The utility of this trip
need have nothing to do with the activity associated
with location k, although its time and cost parameters
are the same as those for the trip for the activity,
being &^ and 8^, respectively. Then
85
-------
= 0 if fkd > 0
< 0 if fkd = 0
Not only must the trip have, positive marginal utility
to occur, but that marginal utility must be greater
than the marginal 'utility of the frequency ffe of the
activity at k, since, by substitution, if fdk, fk > 0,
then
Cidk °ak "°k "k
(assuming that f, and tk have positive marginal utility).
"• . K. . •"•
Otherwise, it would be better either to do the activity
at k on the trips or else not to make those trips.
Multi-Purpose Trip-Making
The presumption in these developments so far con-
cerning trip-making is that all trips are -single-purpose
round trips and that only.one occurrence of an activity
takes place per trip. The complexity of the problem
increases manyfoid when these presumptions are relaxed..
No attempt will be made h-ere to do this for a "general
case." It should be mentioned, however, that "composite
activity trips" could be defined in a manner analogous
to the definition of composite goods, on the assumption
86
-------
of fixed .proportions of times in the component non-
travel activities. Then the activity-trip combination
could be treated as above.
Another approach is to define the multi-purpose
trip as a distinct type of trip without forcing any
particular relation among activities on the trip other
than the requirement that one occurrence of each take
place whenever the multi-purpose trip is made. As a
simple example, let's look at a particular version of
the simple labor-leisure model. Let there be the work
activity and a leisure activity with frequencies f
and f-p respectively. Following an earlier specifica-
tion, let the work shift length per occurrence be fixed
at s and remunerated at rate w per unit time. Further,
let the leisure activity be the generalized "costless"
•i
leisure of the classical model, with consumption ex-
penditures x being a separate item. Finally, let there
be three types of trips, k = 1, 2, 3, where
f-j_ = frequency of trips for leisure only
f2 = frequency of trips for work only
fo = frequency of trips for work and leisure
87.
-------
The locations for work and leisure are given but may
be different. Therefore, let the time and cost para-
meters for the trip of type k be denoted o?^ and a^t
respectively.
The labor-leisure optimization problem is now to
Maximize U = ~U(fk fj, fw, t^ x)
Subject to fk, fp fw, tlf x > 0
wsfw -.
- sfw -
(f2 -t- f3) - fw > o
The last two constraints are necessary to assure that
the activity occurrences are matched by the appropriate
number of trips. Denote their shadow prices 0u and
0W, respectively. The first order constraints for a
maximum are
f-X <_ 0
< 0 if x = 0
88
-------
+ \ws - 6s - $ < 0 C = 0 if fw > 0
w
< 0 if fw = 0
> 0
= 0
= 0 if t1 > 0
<°ift . o
0 if f > 0
o if f = o
0 if f2 > 0
o if f = 0
= 0 if fj > 0
< Q if = 0
On the assumption of no pleasure trips per se
other than what is implicit in the generalized leisure
activity, the two frequency constraints will be de-
finitional equations. These could be used to eliminate
f, and fw from the problem, but since we will be in-
terested in the interpretation of the utilities of the
fk, that will not be done here. With these constraints
89
-------
as equations* the shadow prices , and $ could take
either sign, positive if the frequency of trips for
an activity yields relatively more utility than does
the frequency of the activity itself, negative other-
wise.
Mo attempt will be made here to explore all
possible cases. It should be apparent that different
trip-making patterns are possible according to the
nature of the utility function. In general, if all
three types of trips are undertaken, then there is the
relation
e (P! + s2) - 83
Since it is unlikely that the joint purpose type of
trip would have less cost but greater time (or greater
cost but less time) than the sums of the respective
costs and times of the two single purpose trips, the
right-most term would have to be sufficiently positive
to make the whole expression positive (the first term
on the right being negative)l Thus the relationship
between the marginal utilities of the frequencies here
must be in the same direction (yield the same sign) as
90
-------
their time parameters and in the opposite direction
of their cost parameters, if all three types of trips
are to be undertaken. This would be reversed if the
supposed relationship between times and costs for multi-
versus single purpose trips does not hold.
With neutrality to all trips, so that the marginal
utilities vanish everywhere, it is not likely that the
equation above can hold, so that at least one type of
trip is not made. It also means that the shadow
prices on trip frequencies will bepositive and can
generally be solved for as functions of 6 and X. If
there is also neutrality toward work, then ^w = s(xw-G)
so that the dollar value of time 0/X is less than the
wage rate.
If there is not neutrality to trip-making, it is
germaine to ask what the interpretation of the utility
of the dual purpose might be vis-a-vis the utilities of
single purpose trips. One view is that there may be
reflected in the utility of the dual (multi-) purpose
trip some effect due to the sequencing of those ac-
tivities on that trip over and above the utility of
the travel per se. In the example above, there could
be deviation of the (marginal) utility of the dual-
91
-------
purpose trip from what might accrue simply to a
weighted sum of the single purpose trips, such devia-
tion being due to an altered salience of the activities
when performed in such tight sequence, i.e., without
the usual interruption associated with the single purpose
trip set. Such a "mix effect" is only a proxy for se-
quencing effects, of course, and in this specification
the effect is attributed to travel and not to dimen-
sions of those activities themselves, as perhaps it
should be. In any case, it is not thought unreasonable
to attempt a better understanding of travel behavior
by directly addressing in a model such as this the
question of activity mix in (multi-purpose) trip pattern.
Some Applications in the Theory of Trip Frequency
Trip times, frequencies and costs are the stuff
of transportation theory and modeling, of course, and
there have been numerous empirical and theoretical
investigations into the factors affecting trip behavior.
Because of the complexity of that behavior, most of
those investigations have turned from micro-study of
the household to study of travel patterns at a higher
level of aggregation. This has substantial advantage
92
-------
from a practical point of view, but often it does
not reveal much about the social welfare implications
of those travel patterns. Recently, however, there
have been several renewed attempts to tie trip-making
behavior back to micro-economic theory of consumer be-
havior, for small groups of households if not for
individuals per se. There has always been interest in
the dollar value of time as a normative index for use
in assessing changes in the transportation system, and
planners need to be able to trace the impact of planning
decisions back through the whole activity structure of
households in terms of time as well as money allocations
and readjustments in facility use in the face of change.
There are three developments to be discussed here
in this context. One concerns the application of the
Lancaster activity-characteristics approach to the de-
mand for "abstract" transportation modes. This is an
attempt to construct a method for anticipating consumer
reaction to the availability of a novel type of trans-
portation mode, in terms of the mix of characteristics
of that mode relative to the mixes of characteristics
for existing modes. The second development is an
attempt to find in a model of consumer trip-making
93
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behavior the origin of the "gravity law" for the dis-
tribution of trip ends in terms of the force of attrac-
tion of opportunities for activities over the friction
of distance. The third development is an example of
the derivation of similar entropy-type formulations
for trip-making behavior among attracting opportunities
in which marginal utilities of trips by purpose are
made explicit and estimatable. These three developments
all make or imply extensions of the micro-economic
theory of behavior of individuals in frequency, time
and money to the behavior of groups of such individuals.
These illustrations, then, are of particular relevance
to the River Basin fciodel, where just such an extension
is made.
The study of the demand for travel by abstract
modes is an attempt to present a theoretical frame-
work in which demand for (public) travel modes can be
analyzed in terms of the specific service characteris-
tics of each. With travel demand by mode interpreted
as demand for travel service components first, and
thus for modes per se only indirectly, the demand for
a non-existent or "abstract" mode can be estimated from
statistical demand functions for existing modes in
94
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terms of their service characteristics by extrapola-
tion. Moreover, by testing the sensitivity of demand
to changes in the service mix of the abstract mode,
we have an instance of the design of an "optimal" mode
as a selection of service parameters in a continuous
rather than a discrete choice of modes, much as dis-
cussed in a previous section of this chapter.
Although the original formulators of this approach,
Baumol and Quandt, did not present the individual
consumer equilibrium position on which their method
rests but focused instead on aggregate demand, it
appears that the underlying model of individual consumer
behavior is basically an application of a Lancaster-type
utility model to the Moses-Williamson problem of route/
mode selection in traveling between two given points.
For example, a simple model of this type could be a
labor-leisure model stated in terms of frequencies in
fe R. E. °v?ndt and W. J. Baumol, "The Demand for Ab-
stract Transport Modes: Theory and Measurement,"
Journal of Regional Science, Vol. 6, Winter 1966,
pp. 13-26
'Baumol and Quandt address themselves to inter-city
trips, but the method is equally applicable to intra-
city travel.
95
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the manner of the development of this and the previous
chapters, where uses of different transportation modes
are viewed as separate activities generating charac-
teristics in the Lancaster fashion as do other v/ork
and consumption activities. One version might be the
choice of journey-to-work frequencies ffc by mode k,
work frequency fw for shifts of length s (so that gross
work time is sf ), leisure time t » and consumption
expenditures x such as to
Ffexiinize U =U(z )
m
Subject to:
"production" of characteristics levels z in
activities* m
emkfk + emwfws
for each characteristic m = 1, . .
budgets
T ' «kfk - sfw -
k
V/Sfw - - x
96
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frequency constraint for work trips
. - f >
k w —
k
v/here e are coefficients for the yielding
m* of characteristics in activities
«k, 3^ are the time and cost of a work
trip
w is the wage rate per unit time
s is the work shift length per
occurrence, so that income
Y= swfw
Por simplicity leisure here is a composite, at-home
activity and the intermediate use of goods in activities
after Becker and Lancaster has been omitted. Expansion
of this model along other lines could follow the di-
rections of previous sections, and no further analysis
will be carried out here. Problems in obtaining analy-
tical solutions to such problems were mentioned pre-
viously.
Baumol and Quandt presume that on the basis of
such individual optimization behavior individual demand
functions for the f, can be derived and aggregated into
K ""
class demand functions for those mode/route alternatives
in terms of the various trip characteristics (service
levels) coefficients emk, prices pk, and times
97
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Strictly speaking, other parameters of the optimization.
problem should also be arguments to the demand functions,
but they are not specified by Baumol and Quandt, who,
after all, do not specify the individual optimization
problem. Here, such parameters would be w, s, and the
other characteristics coefficients. In any case, the
result would be the aggregate demand functions
where the arguments cover all subscripts m >=. 1, . . •
A
and k = 1, . . • , k, • . .
To make the abstract mode application, Baumol and
Quandt postulate that the functions 8k are the same
for all modes. The reasoning is that if utility is
defined directly OVar characteristics, then the only
source of differences in demands for modes lies in
differences in characteristics coefficients and time
and cost parameters, the functional relations between
these being the same for all travel demand functions.
Thus, the general demand function $ which results yields
a demand estimate for any mode, real or abstract, on
the insertion of the appropriate parameter values re-
presenting that mode. The function d" , then, treats
the travel choice problem as a problem in selection
98
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among "continuous modes." The authors present some
statistical results based on presumed forms of the
demand function, adding other parameters to reflect
characteristics of the origin and destination locations.
Certain further modifications have been suggested in
0
subsequent articles.
The central weakness in this and many travel de-
mand models is the independence of travel behavior
from other activity behavior. This results precisely
because the model-builders do not start with a com-
prehensive model of individual behavior. By using an
expanded version of the model above such problems could
be avoided. It seems intuitively obvious that changes
causing shifts in travel demands will also cause changes
in the activities, for which those trips are made. Yet
models which do not allow identification of the latter
types of changes are not likely to represent adequately
the former types. There.are practical reasons for these
See Reuben Gronau and Roger E. Alcaly, "The Demand
for Abstract Transport Modes: Some Misgivings," pp.
153-157* and R. E.' Quandt and V,'. J. Baumol, "The De-
mand for Abstract Transport Kodes: Some Hopes," pp.
159-162, in Journal of Regional Science, Vol. 9,
April 1969,
99
-------
omissions, of course. Unlike much of economic theory,
trip theory is readily applied in problems of urban
and regional transportation planning. Complete time
budget information necessary for analyzing travel be-
havior in the context of non-travel behavior is ex-
tremely expensive to collect and analyze. Moreover,
there are many problems yet to be resolved in activity
definition and measurement. On the other hand, much
current data on trip patterns by destination land use,
purpose, and time of day could be recast in time bud-
get form for out-of-home activities.
As has been mentioned at several points, mode
choice decisions of the type above could be extended
to cover trips for:different purposes within the me-
tropolitan area, where the non-travel activities at
destinations which define trip purpose have their own
time and money components. The series of articles
initiated by Kiedercorn and Beckdolt on a utility
model derivation of the "gravity law" of spatial in-
teraction makes a start in this direction but still
stops short of an integrated vie\v of travel and non-
travel behavior.
100
-------
Q
In their original article Niedercorn and Bechdolt
see utility as a function of both the numbers of trips
from an origin to destinations (for unspecified pur-
poses) and the number of "opportunities" available
there, the latter being used to weight the former in
the utility function* This has the consequence that
the optimum number of trips to a destination tends to
vary directly with the number of opportunities available
there, as in the gravity law formulation. Unfortunate-
ly, Niedercorn and Bechdolt assume independent de-
termination of the total time and total money to be
budgeted to travel. This has the consequence that
usually only one constraint - either the time budget or
the money budget but not both—is binding at the opti-
mum, whereas models examined in this presentation
presume that any slack resource would tend to be put
to use in some other (non-travel) activity. In any case,
the authors show that with certain utility functions,
the inverse power relation between trip frequency and
J. H. Niedercorn and B. V. Bechdolt, Jr., "An Economic
Derivation of the 'Gravity Law* of Spatial Inter-
action," Journal of Regional Science, Vol. 9, August
1969, pp. 2?3-282.
101
-------
distance (in terms of time or cost, depending on
which constraint is binding) typical of the gravity
lav/ can result in utility maximization.
In subsequent articles Mathur10 and then Allen 1
suggested that the characteristics of the trips be
introduced to the utility function in the manner of
Lancaster's work. Indeed, this results in a formula-
tion quite similar to that presented in connection
with the Baumol and Quandt approach but without the
work and leisure components. Unfortunately, the slack
budget situation is perpetuated, because they presume,
like Kiedercorn and Bechdolt, a prior independent de-
termination of the maximal amounts of time and money
to go into travel. That a solution is possible with
Vijay Iv'.athur» "An Economic Derivation of the 'Gravity
Law* of Spatial Interactioni A Comment," Journal of
Regional Science, Vol. 10, December, 19?0, pp. 4-03-
^05. See also Kiedercorn and Bechdolt 's "Seply"
which follows, pp.
W. Bruce Allen, "An Economic Derivation of the 'Gra-
vity Lav/' of Spatial Interaction: A Comment on the
Reply," Journal of Regional Science, Vol. 12, April
1972, pp. 119-126. Again, Niedercorn and Bechdolt
follow with "A Further Heply and A Reformulation,"
pp. 127-136.
102
-------
binding time and money constraints is demonstrated
by the example presented in the appendix to this paper.
The characteristics approach of Kathur and Allen allows
greater detailing of travel phenomena, but it does not
rectify the limitations of the specification of travel
behavior as separable from non-travel behavior.
The third develorjment in this set of applications
comes in some recent work of Beckmann, Golob and Gus-
tafson. It, too, focusses on trip frequencies and
trip purpose in the context of time and money constraints
and uses a specific (separable and additive) form of the
utility function to lead to empirical statements about
trip behavior. It does not incorporate the Lancaster
approach to activity characteristics and is mentioned
here primarily because of its method of handling binding
time and money constraints. In particular, it represents
a less than satisfactory (in this writer's view) way of
determining shadow prices in the two constraint case.
Martin J. Beckmann, Thomas P. Golab, and Richard L.
Gustafson, "An Economic Utility Theory Approach to
Spatial Interaction," paper presented at the 1972
North American Meeting of the Regional Science Asso-
ciation, Philadelphia, November 11, 1972.
103
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Beckmann et. al. set up the trip-making problem
similar to the travel components of models presented
in this paper and to the original Niedercorn-Bechdolt
model in general formi
Llaximize U = *U(f jk)
Subject to f ji, > 0
v/here la and T. are the money and time stocks available
d
for travel, f .v give the number of trips chosen to
JK
location j for purpose k (presumably a non- travel ac-
tivity), and cf± and jjj are the travel time and cost to
j j
location j, respectively t This is a straightforward
application of the basic consumer model to trip fre-
quency but ignoring time and money aspects of non- travel
behavior. The first order conditions for a maximum are
. X8 . Oar. < 0 = ° if f jk * °
< 0 if f .. = 0
104
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The marginal rate of substitution between trips to
the same location for different purposes is thus
(minus) unity, while all those between trips to two
given locations for v/hatever purposes are equal.
At this point, the authors make further simpli-
fications and postulates. First, they imply that both
constraints will be binding. Although this can of
course happen, it is not likely with fixed stocks r.I
and T, unless those stocks stand in (near-) optimal
d
proportions. Thus, the Kiedercorn-Bechdolt assumption
of a slack constraint may be more correct technically,
given the specification of these models which omit
non-travel behavior. Second, they "merge" (my term)
the cost parameters and shadow prices by defining the
"total" interaction cost" rk of a trip to a location
as the weighted sum of
-------
Then the first order conditions
in the single shadow price ^s X + 6 can be fitted
statistically by traditional methods once a utility
function is specified, preruming the existence of
data for r^.
The authors claim that r^ is specific only to
origin and destination of the trip, which is true only
if all people at the origin have identical preferences
and stocks set aside for travel, for 6/X will be spe-
cific to relative preferences and stocks. Yet they do
not impose that restriction on preferences in their
investigations. In their empirical work, where re-
gressions are carried out by income class, origin,
destination, and purpose (working and shopping), they
actually use travel time for r^, which implies that
X = 0. Moreover, they then go on to interpret ty as
the "marginal utility of income," which could happen
only if 9 = 0 and r. = 3^. Indeed, even under those
106
-------
conditions (^= \)» it would be the marginal utility
of M, not income. M itself is never identified in
the regressions, either as a parameter or control
variable, nor is T^. Thus it would seem that there
is no way to check whether their empirical determina-
tion of parameter values would be consistent with the
budget constraints.
Given this beginning, the method used by the
authors to derive regression equations and obtain es-
timates of the values of marginal utilities is quite
ingenious. However, because the simplifications of
the theory as described above are considered inadequate
that method will not be investigated further here.
All three developments presented here suffer badly
in theory (and presumably also in application) from
the isolation of travel from non-travel behavior and
consequently also from the inadequate treatment of the
budget constraints and their shadow prices. In passing,
it is interesting to note that while the literature
generally ascribes disutility to travel, these examples
impute positive utility to trip frequency. This is not
unreasonable if it is "purpose," not travel as such,
which is desired. The characteristics approach helps
to separate these effects, but it would seem that the
107
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inclusion of choice variables for the destination
activities per se is by far the best way to clarify
the situation.
To give an example of the way in which activity
and travel behavior can be treated, one author (who
asks not to be quoted) considers the problem of making
trips to shop. Disaggregating consumption expenditures
x into components x, specific to (purchased at) loca-
"" K.
tions k, he defines utility over the x , the trip
ic
frequencies to shop, f, , and work and leisure time,
~ K
t and t-i, respectively. The utility of x, can in-
w •*• K
elude the effects of the goods per se and the time
spent in shopping for them, since shopping time is
assumed proportional to the goods bought. Likewise,
the utility of the ffc can include travel effects and
shopping frequency effects. Then the consumer's op-
timization problem is to
Maximize U = U (xfc, ffc, tw, t-^
Subject to
108
-------
wtw - / Pkfk " /- P>x* - /. rk f Jc\ > 0 (money)
vf I- * ^^ ix
where 8k is the cost of a trip to k
, is the price of a unit of good k
*K atoHn
-------
Such considerations will be brought up again in the
next chapter, where this mor'el will be reviewed in
more detail. Although limited to treatment of only
one type of destination activity -note that there are
no trips for work or leisure —this example does show
how time/money tradeoffs may be made between trip fre-
quency and destination activity choice variables. For
example, the (negative of) the marginal rate of sub-
stitution of frequency for a good is equal to the ratio
of the marginal costs
(Bk - rkxk/f{) +
k (pk -i- rk/fk) +
with time valued at 6/X. Note that an increase in
frequency means a savings in storage costs, thus the
negative sign of the change in storage costs with an
increase in frequency.
To incorporate all of the dimensions and activity
choice specificities introduced in the various models
of this and the previous chapter into a single model
would result in an optimization problem of considerable
proportions, and no attempt to do so will be made here.
In many cases the first order conditions for arguments
110
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would be little changed, so that the separate models
become but components of the larger one. In some cases,
however, there may be substantial modifications re-
quired. The separate models serve our didactic purposes
adequately, however, and we will not attempt to include
them all in the discussion of the final topic to be
addressed in the theory of individual consumer behavior,
residential location and the use of space, to which we
turn in the next chapter*
111
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SECTION V
RESIDENTIAL LOCATION AND
THE USE 0? SPACE
While time and money allocations to activities
are explicit choice variables to the social sector
(households) in the River Basin Model, residential lo-
cation and the purchase of residential space (housing)
are not. Therefore, the discussion of these topics
will be more cursory than treatment of other topics
in the previous chapters, since optimizing behavior
in residential location and use of space will be of
less relevance to the critique to be undertaken. More-
over, it will be suggested that most of the models
already discussed can be turned into location models
by treating travel costs and time and housing (as one
of the generalized goods inputs to leisure and activi-
ties) price as functions of distance or location, then
maximizing utility over location. Choice of residential
location and the space (housing) to be purchased there
thus becomes a choice of the optimum optimorum —•
picking the best activity/consumption pattern from
the optimal patterns for each location as derived pre-
viously for the general case (location).
112
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For these reasons, the discussion to follow will
not be as comprehensive as to activity pattern as
previous discussions, although it should be emphasized
that the optimal activity pattern is very much a func-
tion of residential location in these models. Indeed,
the purpose of including this chapter is to suggest
the importance of residential location in time and
money allocations and to show how some theorists have
approached residential location problems. Their mech-
anisms might find application in a revised simulation
procedure for the Social Sector of the River Basin
Kodel.
The Industrial Location Analogy
The general statement by Isard in 1956 of the prob-
lem of the choice of residential location as a function
of shopping pattern marks the beginning of modern theory
of residential location. As presented by Isard, the
consumer ;Location problem is strictly analogous to the
V/eberian location problem for industrial plant location,
where the objective involves finding a location central
Walter Isard, Location and Space Economy, (Cambridge:
tf.I.T. Press), 1956.
113
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to sources of inputs for production such that the ob-
taining of inputs and the production and distribution
r>f the product maximizes profit. The household,
accordingly, is faced with finding a location central
to sources of goods inputs (i.e., retail establishments)
to its "production" of utility, which is to be maximized
subject to budget constraints. In addition to direct
outlays for goods, the consumer must bear the costs
of procuring those goods as functions of distances
traveled, frequencies of trips, and amounts of goods
purchased.
So far this model is similar to the shopping model
of the last chapter, except that it lacks a time budget
and does not introduce storage costs. What makes it
different, of course, is the simultaneous choice of
the residential location along with the pattern of goods
purchased and trips made for the purchases. Rather than
presenting Isard's original formulation, detailing the
additional assumptions he makes for trip cost and fre-
quency relations,2 we will instead take up the shopping
i
'Isard makes trip frequency dependent on purchase level
and travel costs incorporate Implicitly some fixed
value of time component.
114
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model of the last chapter which includes a time budget
and storage costs and cast it in the form of a location
model like Isard's. The results will not be too dif-
ferent and will contain the desired time elements. The
first order conditions for a maximum for this model are
xw - e < o ^ = o if tw > o
' < o if tw = o
= o if tl > o
< 0 if t- = 0
< 0 if fk = 0
_w ^ 0 5=0ifxk-°
k k ( < 0 if XR = 0
The first two are the traditional relations of the
labor-leisure model. It was pointed out previously
that increased frequency can reduce storage costs but
add to travel costs. Goods now have procurement and
storage costs (the latter in terms of the average per
trip) in addition to direct purchase price. The mar-
ginal utility of an external change (increase) in the
storage requirement cost r. is
K
115
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dU = £U dt^ a"tl dtj r £U_ dfk
^ drk k'Wk» drk *xk, drk
/x \
= -*u
To make this a location model after Isard, the
travel time and cost parameters a-,, and 6k must be con-
sidered as functions of the distances from the locations
k to the residential location. If those locations can
be identified in terms of a set of location variables
or coordinates, then the distances can be written as
functions of the location variables identifying the re-
sidential location. If 1. is one of those location
j
variables, j = 1, . . » , then the optimal location can
be derived by maximizing utility with respect to the 1 -
J
as well as other choice variables. Since the 1. are not
j
included here in the utility function, the first order
condition for 1. would be
J
116
-------
Finding an analytic solution for the optimal location
in terms of the 1. is no simple matter, and various
J
approximation methods have been developed for Weberian-
type problems,-' though not in a time budget context.
Most models of residential location start with
one dimensional space, or location along a line, and
then generalize (usually informally) to two-dimensional
space, where there may not be an analytic solution of
a simple nature. For example, if in the shopping model
the source locations k for goods are strung out along
a line, with d^ being the distance from an end of the
line to location k and d being the (variable) distance
to the residential location, and if the travel times
and costs are proportional to distances traveled —
^ = «K - dl
Rk = 9|dk - d
— then there is a single first order condition for the
ODtimal value of d:
See, for example, Leon Cooper, "An Extension of the
Generalized Weber Problem," Journal_of Regional
Science, Vol. 8, Winter 1963, pp. 181-197.
117
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which is oust
>d k)dk
-------
•the detailing of consumer activities and frequency
relations.
Residential Space Use and Bid Price Curves
In the early work on residential location and
space use following Isard's formulation, trip frequen-
cies were not made explicit and goods were considered
in an aggregated form, save for housing. Out-of-home
activity was not specified by type; instead, it was
simply presumed that expenditures for travel to the
"center" along a line for whatever purpose must be
made. The only variable in determining that travel
cost was the distance to the residential location to
the center. On the other hand, while trip pattern was
aggregated, actual purchase of space, or "housing" or
"land", for residential use was introduced. Variation
in its price per unit over distance from the center
thus replaced travel frequency as the central factor
in selection of optimal residential location. In the
Isard-type model, the collapsing of all destination
opportunity locations to a single "center" would lead
to optimal residential location at that center, there
being no interplay of land prices and travel costs
over distance.
119
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It was Alonso who first formulated the tradeoff
of residential space and distance through land prices
and travel costs in the choice of the optimal residen-
tial location. In his model utility was defined over
housing (land) xTfl and all other goods, the composite x «
Transportation costs, k(d), were an increasing function
of the distance d from the center, without specifica-
tion of trip frequency or purpose. Housing (land) price
per unit was likewise a function of distance, p(d),
here a decreasing function.* Income Y is fixed.
Alonso fs consumer optimization problem can be
stated as
Maximize U = H (x , x. )
m ii
Subject to
Y - xm - p(d)xn - k(d) > 0
First order conditions for a maximum are
William Alonso, Location and Land Use. (Cambridge*
Harvard University Press), 1964.
Alonso subsequently demonstrates how a bidding procedure
for land would lead to such a price schedule. See
below.
120
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= xp(d)
where X is the shadow price for money. At the optimal
distance, the first order condition for distance —
called the "location equation" — requires that the
marginal savings for reduced land price with distance—
— be set equal to the marginal costs of transporta-
tion at that location —
(assuming no (dis-) utility to distance), where XK is
here the optimal amount of housing.
A Drovocative construct set up by Alonso is his
"bid rent curve" for land. This curve is the function
(?(d) which gives the price the consumer would bid at
any location d to be indifferent between location at d
Alonso notes that the marginal utility of d need not
be definitionally zero if d is considered a proxy for
travel in U.
121
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and location at the center at some price p0« That
is, (?(d) traces an indifference -^rve over land price
with distance, relative to the price p at the center,
which may "be considered a parameter to u(d). £P(d)
takes into account variations in the optimal levels of
xh and xm with distance. In fact,(?(d) is precisely
that function for which the first order condition
holds at all distances, x,n being the optimal for that
distance, relative to a price p at the center and
specific to a certain level of utility. (lA)(sV/Sd) may
be added to the right-hand side if it is not defi-
nitionally zero.
Since a bid price curve is specific to a utility
level and a price p at the center, the family of
curves for various PO defines a set of indifference
curves for various utility levels. Utility will vary
inversely with p , since a. greater p means that at
the center (when d = 0)
122
-------
dp0
where x and x. _ are optimal levels when d is forced
rno no
to be zero. hus if there is some market price structure
(?_(£) for land, the consumer would maximize his utility
by locating at the distance for which the price struc-
ture intersects the lowest possible of his bid price
curves. In general, this would be a point of tangency
from below of that bid price curve with the price struc-
ture, given that the necessary higher order conditions
for such a tangency hold. 3y demonstrating that bid
price curves may be steeper (more negative) for lower
income groups than higher ones at most distances, Alonso
explains spatial stratification of residence by income
in rings around the center, with income usually incres-
ing with distance. Using the bid price construct, Alonso
develops a hypothetical bargaining procedure by which
the market price structure itself might be generated.
It should be apparent that a time budget could
easily be added to Alonsofs framework in much the same
way as it was introduced to non-residential models in
the previous section. The simplest modification would
123
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be the introductior of leisure time t^ to t^e utility
function, the specification of a travel time function
t(d), with dt(d)/*d > o, and the addition of the time
"budget constraint
T - tx - t(d) > 0
The first order condition for leisure time is simply
ikt- e
»*i "
and the revised location equation can be written (in
bid price form) as'
^•+1-
.h ^ Bd
The inclusion of the marginal cost of travel time as
the valuation of the marginal travel time at the dollar
value of time (6/x) should make bid price curves steeper
(more negative), all else equal, fiote that (0/X) may
vary with distance along a bid price curve. This is
a particularly interesting phenomenon, for one wonders
whether the dollar value of time should increase or
decrease with distance. The answer obviously depends
o
'As before, (dls)utility to distance through travel may
be added to this expression if appropriate.
124
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on the relative scarcity of time and money at a dis-
tance, and that is a matter of stock levels, prices
and preferences there.
To place residential location squarely in the
context of the labor-leisure model, income could "be
allowed to vary through the choice of work time, t •
Here work time would be explicit in the budgets for
time and money and possibly also in the utility function.
Then a change in location would entail not only changes
in leisure and travel time and in money outlays for
housing, other goods and travel, but it would also
involve a change in the amount of time actually worked,
thus the income stock. There would be a complementarity
between optimal residential location and associated bid
price curves and the labor offer schedule, for the
optimal location as well as the labor offer will vary
as the wage rate varies. In the traditional case of no
(dis-) utility to work, the dollar value of time equals
that wage rate and does not vary along a bid price.
Location by Occupation (Wage Rate)
It is apparent that the entire development of the
previous chapters, in which the labor-leisure model was
125
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expanded to incorporate more detail in activity choice
through time and money allocations, could be retraced
Q
and location equations derived for each model. This
would "be tedious, and the general method for carrying
out such derivations should become apparent here. Vie
will focus instead on some developments in the litera-
ture which are relevant to the discussion of the River
Basin I-iodel, with some digression later on activity
patterns. Despite the plea of previous chapters for
the inclusion of frequency variables in time and money
allocation models, no attempt will be made here to in-
troduce frequency choice into each model considered.
Again, the method of previous chapters should suggest
how that might be done.
was perhaps the first to consider in theory
the systematic effect of transportation time (through
its implication for leisure time) as well as transporta-
o
For example, the labor-leisure version of the Alonso
model above could next be extended to cover the Moses
problem for overtime work. Such a treatment has, in
fact, just recently appeared in the literature. See
Hiryuki Yamada, "On The Theory of Residential Location:
Accessibility, Space, Leisure, and Environmental Quality,"
Papers of the Regional Science Association, Vol. 29,
1972, pp. 125-135.
126
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o
tion cost on residential location. What is of par-
ticular relevance for the River Basin .Model is that his
approach is a group location model, rather than one for
the individual consumer, and it focuses* on the role of
occupation and v/age structure in residential location.
As such it gives a range of locations for the group,
rather than the optimum location for any individual,
and this turns out to create some inconsistencies in
the model.
In effect, Wingo presumes an Alonso-type location
problem for the individual, with a time budget but
holding actual work time and the time budget fixed.
Instead of optimizing utility for the individual, he
falls back on the labor offer curve as an independent
function for computing a rent schedule over distance as
a function of gross work time (work plus travel for the
journey-to-work) and net wage (computed over gross work
time), along with travel costs. This rent schedule re-
presents the rents that could be charged workers (con-
"lowdon T.7ingo, Jr., Transportation and Urban Land,
(v'/ashington: Re so urces for the Future, Inc.), 196!,
and "An Economic Model of the Utilization of Urban
Land for Residential Purposes," Papers of the Regional
Science Association, Vol. ?, 1961, pp. 191-205.
127
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sinners) distributed around a v/ork place. As such
it deals directly with determination of the market
price (rent) structure without locating individuals
explicitly. However, to do this it must presume a
type of partial optimization for the individual in
terms of his work/leisure/travel relations, and the
nature of this "optimization" is the weak point of his
development.
V/ingo poses the location problem for a group of
workers who are homogeneous, having the same .preferen-
ces for consumption and leisure since they share a
common labor offer curve. Their common employer wishes
to hire them at a given shift length of tw = s hours
per worker. Frequency of work and journey-to-work are
implicit, but the implication is that they are equal
and fixed. The questions are, what wage must the
plant pay to get shifts of s from each worker - it
must pay all workers the same wage — and what rent
differentials will develop as these workers distribute
7
themselves in space around the plant.
The classical wage theory has shown how a labor
offer curve can be developed from a utility analysis in
128
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which income or consumption and leisure time are
evaluated in terms of the contribution to the con-
sumers utility. Indeed, the labor offer curve, giving
the equilibrium work time as a function of the wage
rate, is a solution of the first-order conditions (in-
cluding the budget constraints) for maximal utility.
Wingo is interested in reversing the roles of these two
components, wage rate and work time, by holding work
time fixed and finding the equilibrium wage rate.
Therefore, he takes the inverse function of the labor
offer curve to yield
w = a(s)
Since preferences are assumed identical for all workers
in the group, they all share this same view of the work
time-wage relationship.
The first question Wingo raises concerns the wage
the worker must be paid in order to give up leisure time
for income when he must make a journey-to-work as well
as work the shift of length s. At this point, Wingo
makes an important assumption — that the function
gives the remuneration necessary to induce the worker
to yield a marginal unit of leisure. As we know, w equals
the value of that remuneration only when there is no
129
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(dis-) utility to the argument of Q . Moreover, w is
the remuneration per unit time which must be applied
to the argument of q as representing all non-leisure
time; otherwise it would be a function of more than one
independent variable. This, in turn, implies indifference
among uses of non-leisure time (which, in this case, all
yield zero utility). Under these conditions a becomes,
as vVingo calls it, the "marginal value of leisure func-
tion," with non-leisure time as the argument.
Consider, then, a worker living d units from his
place of employment. With a journey- to-work taking
t(d) time, he must yield s + t(d) units of leisure to
complete s units of work. According to the marginal
value of leisure function, he would be willing to do
that although <& gives the Marginal value of lei-
sure only with.no (dis-) utility to non-leisure, w as
some function <\ can still be determined with such
(dis-) utility to non-leisure; it just^.cannot be given
that interpretation. Likewise, $ (or 0, ) need not be
a function" solely of non-leisure time as a whole but
may have as arguments various components (such as work
and travel) separately. In these cases, of course, &
cannot be used"as the valuation of leisure time, butd
some other function (9/X) =•§(...) can still be de-
termined. To simplify his model, V/ingo is overly-
restrictive.
130
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this if he received a "manifest wage" w(d) per unit
time over the total non-leisure period, where
w(d) = a (s + t(d))
Since the worker at d would actually be paid a market
wage w (d) for each of s hours, that market wage may
be determined by equating the incomes
w(d)(s + t(d)) = wni(d)s
or
t(d)
wm(d) = (1 + s )• Q(s + t(d))
In the absence of the journey-to-work the market wage
needed to elicit s hours of work need only be the "pure
wage"
w = (s)
which is independent of location.
The increment in income necessary to get the workers
to overcome the time cost of the journey (in utility
dollar equivalents for leisure time cost) then, is
V(d) = s(w (d) - w)
m
r t(d) -i
= s (1 + s ) Q(s + t(d))-0(s)
131
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where, since a is presumed to be an increasing function,
a (s + t(d)) >Q(s) if t(d) > 0
If the dollar travel costs for the journey-to-work are
k(d), then the total travel costs c(d) (including time
valuation) are, for a worker at d
c(d) = v(d) + k(d)
The market wage w (d) above is that market wage
m
per hour of work required to induce the worker at d
to yield s hours of work and t(d) hours of travel for
the journey to work. Obviously, ^(d) varies with the
distance (location) d. The plant, however, must pay
all workers the same market wage, w , and that wage must
be adequate to induce the most distant worker to make
the journey and work s hours. That is, if that "farthest
distance" is d = d, then the actual market wage wm paid
by the plant must be at least as great as wm(d), the
necessary market wage for the worker at d. Since it
need not be greater* then
\ A A
w (d) > w (d) for all d < d, since t(d) < t(d) and
Q(s + t(d)) < d(s + -t(d)).
132
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A ,A
wm = Wm(d)
—the actual market wage equals the acceptable market
wage for the "marginal" (most distant) worker.
This wage relationship means that a worker at
A
d < d receives an increment in income
s(w - w (d)) > 0
m m
in excess of what is required to induce him to make
his journey-to-work and to work s hours. Moreover*
he pays an amount
k(d) - k(d)
less in transportation costs than does the most distant
worker. V/ingo reasons that this total—the income in-
crement plus the travel cost savings—may "be realized
by the landowner as an increment in rent that can be
charged this worker relative to the rent paid by the
marginal worker (at d). That is, the landowner can
increase the rent at d by this amount over that at d
without forcing the worker here into an untenable po-
sition with respect to his income/leisure balance —
he is still willing to travel t(d) distance to work s
133
-------
hours for v; v/age per hour worked. This increment to
m °
rent is called "position rent" and is equal to
c- iv i r N ~"l
R(d) = \s(vL - w_(d))l + ]k(d) - k(d)
L, ni in j 1^1 j
- v;) - (wm(d) " w
- v(d)]
A
= c(d) - c(d)
i.e., the difference in total travel costs, including
dollar valuation for the leisure time lost in travel.
Total rent at a location d, then, is the sura of
position rent R(d) at d and "location rent" R which is
the common component to rent for workers at all loca-
tions d < d. At d, of course, position rent is zero,
and total rent equals location rent. The location rent
component for these workers is based on other factors
such as supply of land and rents being paid at succes-
sive locations by workers at other plants. The resulting
rent schedule over distance, when extended over various
employment groups, is not unlike .the locational strati-
fication resulting from the Alonso bid-rent process
modified to include a time budget. It has the additional
perspective coupling rent determination to labor market
134
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mechanisms. However, by breaking the consumer resource
allocation problem into parts via specification of an
independent marginal value of leisure function, prob-
lems are created which do not come up in the modified
Alonso simultaneous determination model. On the other
hand, that Alonso model subsumes the V/ingo model as a
special case.
One obvious problem with the V/ingo model is that
the schedule of total rents is produced without regard
to land consumption for residential purposes. Wingo
tries to rectify this by positing an independent de-
mand function for housing in terms of density and unit
price, but this begs the question, since that demand
function should not be independent of locational choice.
V/ingo apparently is led into believing that it can be
independent because workers in a group are thought to
be indifferent among locations under the rent schedule,^2
since that rent schedule assures a proper income/leisure
balance for all workers (at v/hatever locations) accord-
ing to their (common) marginal value of leisure function.
12
See Transportation and Urban Land, p. 65.
135
-------
It is true that there is some type of equilibrium
between income (wage rate and work time plus travel)
and leisure for ail workers, but there is no assurance
that the rent schedule produces a constant level of
utility over locations. First, there is ambiguity
as to which components of income yield utility, and con-
sequently there is ambiguity as to the nature of time-
money equilibrium represented by the marginal value of
leisure function. That is, it is not known whether that
equilibrium is based on the utility of gross income in-
dependent of allocation to uses, income net of rent and
travel, or allocations to uses of income separately.
Second, there is the fact that the marginal value of
leisure function itself, like the labor offer curve
from which it is derived, is not an indifference curve.
Points on the curve for that function give the wage
rate for whio h a certain amount of work (plus travel
time, for Wingo) is forthcoming but give no indication
of utility level, which may vary over the curve.
For there to be indifference among locations, then,
the rent schedule not only must not jeopardize the de-
sired work level s per worker at all locations but must
also produce the same level of utility everywhere. This
136
-------
has strong implications concerning the utility of space
consumption and/or rent and travel expenditures, either
directly or indirectly through their implications for
other expenditures as residuals, since both require-
ments cannot be met simultaneously without such effects.
This brings up the first point, that the marginal value
of leisure function is .ambiguous on such effects. In-
deed, it implies indifference to those expenditure
components; otherwise Y/ingo could not use it as an in-
dependent (of location) function. Yet indifference
among expenditure components, including rent, would
preclude indifference among locations under this rent
schedule, since income is everywhere the same, while
leisure time varies inversely with distance. Thus,
utility differences among locations would make this
rent schedule unstable, so that Alonso-type bargaining
would lead to new rent levels.
In aggregating behavior through simplification of
individual behavior, V/ingo creates inconsistencies
which would not arise in an individual-oriented model
such as Alon^o's modified for a time budget. Moreover,
the leisure time component in the latter can be detailed
to include any of the activity time and frequency per-
137
-------
spectives discussed in previous chapters, something
not possible in 7/ingo's model. One approach which
mo4«\
uses the time budget version of the Alonso^to try to
derive the type of rent schedule that Wingo desires
based on travel costs is that of Harris, Tolley and
Harrell. ^ They include in the utility function an
"amenities" good x& which is a function of distance
but bears no price, and they exclude work time and
travel time and cost from the utility function. The
dollar value of time thus is equal to the wage rate at
equilibrium. With the further assumption that travel
time and costs are proportional to distance —
t(d) = ad
k(d) = ed
— marginal travel costs with time valuation become
constant as fe + <*w). The authors then reason that for
reasons of travel a consumer would be willing to give
13
R. K. S. Harris, G. . Tolley, and C. Harrell, "The
Residence Site Choice," Review of Economics and Sta-
tistics, Vol. 50, Ivlay 1968.
138
-------
up in rent an extra amount
A
(d - d}{0 -f wff)
to move from location d to a closer location d, where
A
d < d» As for Wingo, this rent increment is the sum of
travel cost savings and the dollar value of travel time
savings, although here work time may vary. To express
this as a "bid price curve per unit housing, relative
to some location d and the rent there, the authors di-
vide the increment "by rx, , where r is the capitalization
rate and. x^ the housing purchased. This curve is, after
all, a particular form of the general case considered
previously.
Real world deviations from the travel-based curve
derived above were thought by the authors to provide
an estimate of the amenity effects omitted from that
curve. These they try to identify from the location
equation for their model — which is
— rearranged and modified as above to become
139
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_ + -/3
A
where (P is a statistical estimate of the market rent
function. Thus their measure of amenity effects is an
aggregate measure.
It is evident that if amenities yield positive
marginal, they must increase with distance if the right-
hand side is positive and decrease with distance if it
is negative. There does seem to be some circularity
here, for amenity may be more exactly a function of the
resulting settlement pattern than of distance per se,
so that the observed pattern need not represent an
equilibrium
Activity Pattern, Residential Location and Space Use
Residential space, or housing or land, may be
identified as one of the goods - or even as a set of
those goods, if differentiated by type or component
to a residential "bundle" - used as inputs to activities
in the variable proportions or in the fixed proportions
(3ecker-DeVany-Lancaster) models of the previous chap-
ters. As residential location changes, the price(s) of
140
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the residential good(s) can be expected to change, re-
sulting in changes in the use of goods of all kinds,
including residential, in activities. Other dimensions
of activities, such as time and frequency, will change
with changes in goods allocations, and also with changes
in unit time and money costs of travel as residential
location changes. Residential location change, then,
means change in relative accessibility to opportunity
for activity and leads to readjustment of the entire
activity pattern. Finally, there may be location-
specific amenity or "environmental" variables which
impute a utility to a location over and above its
accessibility effects for allocations from limited bud-
gets of time and money.
As became apparent in the discussion of Isard's
location model in terms of shopping pattern, when ac-
tivities (or opportunities for them) are distributed
over space rather than along a line, determination of
an optimal residential location—and consequently the
optimal activity pattern and space use—becomes ex-
tremely difficult even in theory, so that it may not
be possible to characterize the nature of such optimal
solutions (in terms of first order conditions) in
141
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sufficient detail even for didactic purposes. This
is not to mention the fact that in the real world re-
sidential "bundles" tend to "be discrete, location-
specific entities distributed over a plane that is far
14
from uniform. Wilson has subjected consumer opti-
mization to additional "space budget" and other inter-
activity and space-time constraints in an activity
choice model in time and money in an attempt to recog-
nize (in continuous fashion) some of the limits to
freedom in residential activity allocations. ^ Hoy/ever,
See J. Herbert and B. H. Stevens, "A I/Iodel for the
Distribution of Residential Activity in Urban Areas,"
Journal of Regional Science, Vol. 2, I960, pp. 21-
36, for consideration of discrete residential choice.
Their model was a provocative input to model develop-
ment in the Penn-Jersey Transportation Study. Britton
Harris has Dursued many of the issues raised in that
study in his own research. See, for example, Britton
Harris, et. al., "Research on An Equilibrium Model of
Metropolitan Housing and Locational Choice« Interim
Report," (PhiladelDhia: Institute for Environmental
Studies, University of Pennsylvania), Inarch 1966.
^A. G. \7ilson, "Some Recent Developments in loLcro-
Sconomic Approaches to Modelling Household Behavior,
With S-oecial Reference to Spatio-Temporal Organiza-
tion, ""in his Papers in Urban and Regional Analysis,
(London: Pion Limited), 1972, pp.^216-236.His review
covers some of the developments cited in these chapters,
but he does not treat activity frequency and its travel
implications. He does stress aggregation problems in
optimizing models of consumer behavior and has de-
veloped a probabilistic approach.
142
-------
these do not serve to clarify the nature of optimal
solutions to such problems.
By v .y of illustration, we can derive the location
equations for the set of location variables 1. in the
variable proportions model with time (t,)t frequency
(f, ) and goods (x^» i = 1, » . . * n) inputs to ac-
tivities k = 1, . • • « Assuming a single residential
good i = h, the first order condition for the j**1
location variable (e.g»t a coordinate), j = 1, . . . , is
0
KU1.I
X
where the activity frequency ffe is also the number of
trips made for that activity at cost 8. per trip and
taking «k time per trip, the latter both being functions
of location. Such conditions tell us that at the op-
timal location the marginal utility with a location
variable just equals the full marginal costs—through
trip pattern for activities and housing inputs -with
that location variable. (For work to be included among
143
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the k, the appropriate cost function 8 could be
corrected for the shift wage.) Information on the
nature of the optimal location is minimal without
specification of the environment of choice, of course.
These conditions are analogous to those in the Isard
model, with the addition of the housing costs and a
utility to location.
Becker has worked out an example for the fixed
proportions activity model. First, let us consider a
more general statement than his example and relate it
to the variable proportions case by defining xtv. as the
amount of travel good i = t used in activity k, where
this can be assumed to be proportional to the number of
trips made:
xtk = ^kfk
As for all goods in the Becker model, this good is
proportional to activity level —
Xtk = ctkAk
(meaning that the utility of Ak must include travel
effects, if any) — so that
144
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Further, if travel time and cost per trip are o-^ and
8k» respectively, then the time and cost per unit good
t are (of^/Yfc) and 8k/Yk)* This means that the time
input coefficient b^ for activity k covers activity
time only, to which travel time must be added. With
these modifications the first order conditions (the
location equations) "become
ail . __ii _^ _^ _
_ m 1 * l*^»- _^ ~* K _— _ *^ ™ I ^ * II **1» "™" »
ai4
where inputs of housing to activities are given through
chk'
Becker simplifies the problem by taking housing
and travel out of the proportionality relationships
and posing the Alonso problem of location along a line,
with no frequency considerations and trips aggregated
into the general time and cost functions t(d) and k(d)
with distance d from the center as the argument. As a
labor-leisure problem, the budgets are
145
-------
- t(d) * °
- k(d) ^
k i/h
where b, gives activity time per unit and c., gives
input of good i to activity k per unit, good prices
being the p. . By postulating neutrality toward work,
he eliminates work and one budget by substitution for
ty;. The resulting budget is
wT - \(A) - p(d)x - k(d) - wt(d) > 0
where Becker denotes "total consumption" as T\(A). It
is easy enough to show that
picik> Ak =
k i
where t, is non-travel leisure time and xm is "other
expenditures" as in Alonso's model* Quite naturally,
then, his location equation —
1
" "
— is that of the modified Alonso model (and the Harris-
Tolley-Harrell model) with w «(e/x)due to the neutrality
of work.
146
-------
rlo further attempt will be made to detail loca-
tion equations for various specifications of activity
models, for the reasons given previously. It is hoped
that this discussion will provide some background to
location problems which can serve as a frame of re-
ference in reviewing location-activity relationships
in the River Basin Model. The rather extensive review
of Wingo's model was undertaken precisely because its
reliance on dollar valuation of time in location/journey-
to-work/rent relationships is not unlike the method of
the River Basin Model.
147
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SECTION VI
SUMMARY
That the consumer optimization problem can be
made as comprehensive — and complex — as desired
should be evident from the diversity of behavioral
situations reflected in the models presented in these
chapters* It should also be evident that such com-
plexity can be at the expense of clarity and operational
feasibility. Indeed, the statements and derivations
as presented here give only minimal information about
the solutions to the problems they describe. It is
true that the generality of th6 first-order optimi-
zation conditions is a positive factor, in that it leads
to rules-of-thumb concerning relations "at the margin"
which should hold for all utility functions of the types
postulated. But to say anything oi significance about
actual patterns of trip and activity frequency and time
and money allocations requires going beyond the general
cases here to specification of the utility function
itself. Choice of that function often seems far too
arbitrary in the literature. Selection of a specific
utility function should only follow the careful detailing
of postulates about behavior, whereas behavioral impli-
149
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cations of different functional forms tend to be
treated in rather cavalier fashion in most applications.
The models described heretofore, then, sketch the
range of behaviors relevant to River Basin Model con-
sider r-tions. They illustrate how the general principle
of consumer optimization — that a course of action
should be pursued until its marginal utility equals its
marginal cost in market and resource (shadow) prices —
would be manifest in various choice situations. Simu-
lation in the River Basin Ilodel is to involve incremen-
tal choice-making under imperfect knowledge, rather than
the perfect knowledge simultaneous determination of
these consumer optimization models. "It is also aggre-
gated, with "population units" in place of individuals.
It has been argued that in the River Basin Model dis-
tinction as to scale or size of the behaving unit in
the social sector is trivial. Therefore, even without
global optimization the "marginal utility equals mar-
ginal cost" principles of the consumer models can be
used, as rules-of-thumb to suggest the direction and
relative extent or proportions of incremental changes
to budget allocations that the consumer (or population
unit) might make in response to outcomes of previous
150
-------
allocations. These rules can be framed in the language
of the consumer models, that is, in terms of marginal
rates of substitution and relative values of resources
(i.e., the dollar value of time).
As it now stands, the River Basin Model does indeed
view Social Sector (population unit) behavior in much
this way, although in a gaining context. In complete
simulation of the Social Sector, decision-maker input
procedures must be replaced with mechanisms for internal
determination of incremental population unit allocations.
These mechanisms might well be based on just these.
rules-of-thumb from the utility-maximizing models of
consumer behavior. It is not the charge of this research
to develop these simulation procedures, although there
will be occasion to comment on their development later.
It is hoped that this review will suggest behavior
components to such mechanisms. At this point, we will
turn to a direct critique of the River Basin Ivjbdel,
where the current structure of the model will be reviewed
in terms of the consumer theory and other research.
151
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PART TWO
THE RIVER BASIK MODEL
152
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SECTION VII
THE STRUCTURE OP THE MODEL
In the chapters to follow the operation of the
River Basin Model as it affects the population units
of the Social Sector will be thoroughly reviewedt
Source material for this review are the program list-
ings of torch 11, 1972, called SOURCE, and the NEY/LIB
update of ISarch 6, 1972, plus the "River Basin ivlodel"
reports in the U.S. Environmental Protection Agency's
Water Pollution Control Research Series prepared by
Snvironmetrics, Inc., and listed as Project # 16110
FRU, Volumes 1 through 12.
In this critique, every attempt will be made to
adhere to the original subprogram names, variable
identifications, and programming formats from the source
listings to facilitate cross reference. As there are
between three and four hundred subprograms, however,
some liberties will be taken in the description and
analysis, so that functions may be attributed to manag-
ing subprograms instead of to the computational sub-
programs they call for actual system operations.
Similarly, since the objective here is the review of
the model's operation as it affects the Social Sector,
153
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not program debugging, there may be technical accuracies
which slip through in this presentation. These should
not be of real consequence for our purposes.
An Overview of Model Operation
In this chapter the structure of the .River Basin
Model will be outlined. The purpose here is to provide
a sketch of the operating modules of the model and the
manner in which they tie together. In subsequent chap-
ters, those modules (or groups of subprograms) most
relevant to population unit behavior will be reviewed
in detail. The behaving "sectors" of the model are
presented here in their gaming format, as originally
specified in the model. Koweve~. the move to complete
simulation of a rector can be accomplished by replacing
the decision-maker input procedures for that sector
each round (section "A" in Table 1, below) with sub-
routines which adjust sector demand and operating
parameters in response to previous round outputs
according to behavioral rules for sector choice-making
(as, for example, from rules-of-thumb extrapolated from
the consumer theory to population unit behavior in the
Social Sector). From that point on, the model can
154
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operate much as it does at present.
Model structure, in terms of major subprograms,
is outlined in Table 1 below. An attempt has been to
separate the managing subprograms (in the left-hand
column) from the major operating subprograms they
call (in the center and right-hand columns). These
operating subprograms are numbered opposite the calling
subprogram in the order they are called. The actual
subprogram name is given in parentheses following a
simple sectoral or functional identification for that
subprogram. The managing subprograms are themselves
listed in the order in which they are called (A through
M) by the main program, CTYMAIN. Operating subprograms
are separated into columns according to whether or not
they directly determine Social Sector behavior. Those
subprograms jointly determining social and other sector
behavior are listed between the center and right-hand
columns, with dashes to either side to indicate this
common relevance. Those subprograms of special import
to the Social Sector and thus falling in the center
column will be considered in detail below and in the
following chapters, usually grouped to have a common
calling subprogram. Although the water system is a
155
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focal point in the River Basin Model, since population
units have so little direct say over water-related
matters there will be little reference in this cri-
tique to the water system outside of its role in the
determination of the level of the Environmental Index
for population units.
The general procedure for a round's operation of
the model is fairly obvious from Table 1. First, each
sector specifies the desired levels of operation of
activities over which it has control (A). For the
Social Sector this means the desired work and consump-
tion patterns for population units, including time and
money allocations and relationships with institutions,
such as political and organization ties. These desired
levels for the round can be viewed as extensions or
adjustments to previous desires (or "demands") in
light of the degree of "success" realized, however
success is defined by the behaving units of the sec-
tors. Next, some population units are redistributed
by location within the system, partly in response to
dissatisfaction with environment and activity mani-
festations (B). After certain adjustments to the
system (C and D), employment relationships are re-
156
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TABLE li Sequence of Operations in a Round of the
River Basin Model (via CTYMAIN)
en
Program Management
At System up-date
from input de-
cisions (SETEDT,
EDTMAIN, EDIT)
B. Migration
(MIGMAIN)
Ct Water System
(GAILMAI)
Social Sector Activities
5
Activities of Other Sectors
time allocations
(TALOC)
6. dollar value of time
(TMVAL)
?. vote (CASTER)
8,
10. summary(TIMET)
14. boycotts (BUYCT)
1?,
19-
dissatisfaction,
migration (MIGRAT)
1. land purchase (PU)
2. land use alterations
(BUILD)
3* financial operations
(NCHPVT, SCHPVT)
4. utilities (CHGUTS)
cash transfers - - - -
(CASHT)
summary (LOSTT, SPPTST)
rail system (RAIL)
tax rates (TAXEZ)
assessment (AS)
redistricting (REDIST)
16. water, facilities
(WRPRC, WRBLD)
external inputs - - - -
(ODDS, ENDS)
18. bus routes (ADD BUS)
summary - - - -
9
11
12
13
1. water system
(SETSTF, GAILMW)
-------
H
(J\
00
Program Management
D. Post-Demolition
Operations (KLEAR)
E. Employment and
Transportation
Assignments
(EMPMAIN. TRCMAIN)
System Loads and
Further Alloca-
tions (ALCMAIN)
G. Commercial
Transactions
TABLE 1 - Continued
Social Sector Activities
r~
1
Activities of Other Sectors
5<
6,
educational strati-
fication (EDORD)
- - - - 5(
2. depreciation* value
ratios (DEPREC)
3. assessed value (ASVSET)
4. job opportunities
(SETEMP)
full time employ- - - - -
inent and trans-
portation (EMP,
TRTRC)
part-time employ- - - - -
ment and trans-
portation (EMPRT)
7. business capacities
(SETCAP)
adult education
(NSPACK)
recreation and park
usej voter regis-
tration and education
level (TMALC)
employment - - - -
summary
(EKPSUM)
2. municipal service loads
(LOADMS)
3. municipal service quality
(MS QUAD
**. school loads (LOADSC)
1. construction contracts
(CONAG)
2. prices, costs, taxes
(PRCSET)
-------
H
(J1
Program Management
H. Terminals
(TRMMAIN)
I. Output for
Private Sectors
(FRYMAN)
J. Water System Out-
put (WATRMAI)
K. Public Depart-
mental Output
(PUBMAIN)
L. Other Govern-
mental Output
(BSHMAIN)
M. Summary Statistics
(IDMAIN)
TABLE 1 - Continued
Social Sector Activities
3
Activities of Other Sectors
demand for - - - -
goods
(SECTOR!)
- - - - 4-. shopping assign- - - - -
ments (OPCM)
- - - - 5» summary - - - -
(COMDIG)
1. terminals (TERMS)
1. social (WRYOU) 2. other (miscellaneous)
- - - - 3« summary - - - -
(PRINTY)
1. water system (WATOUT)
1. utilities (UTS)
2. municipal services (PWS)
3. planning and zoning (PZ)
b. school TSCHOUT)
1. chairman (CHFO)
2. highway (HYWAY)
3. bus and rail (BSRROT)
1. government (GOVMNT)
2. demographic and - - - -
economic (IDEMEC)
-------
considered (S). Workers in population units may seek
new or better jots or be forced from previous ones
according to their ability to complete (education),
their willingness to travel (time versus money), and
changes in the level and distribution of jobs available.
The new employment relationships result in' new trans-
portation loads and institutional operating capacities.
Under these new capacity constraints, population units
attempt to realize their demands for facility use in
terms of time (?) and money (G), while institutions
make similar transactions among themselves in keeping
with their demands and system capacities. Finally,
the impacts of these actions on the system are assessed
and summarized (H through Jl) •
At the end of this sequence for a round, then,
the system is in a new "state," and the behaving units'
experience has been determined accordingly. The model
is thus ready for a new set of (incremental) demands
to initiate another round. The sequence of outcomes
to behaving units over time traces the behavior of the
system as a whole and in all its parts. Below, the
nature of Social Sector inputs (allocations) for a
round will be explained and commented upon. Successive
160
-------
chapters will take up migration, employment, and
consumption pattern.
Social Sector Inputs and Allocations
The "inputs" for population units of the Social
Sector each round are the choice variables open to
those units. Therefore, the specification of these
inputs defines the behavioral alternatives open to the
(partial) control of the behaving unit, and this is
true whether decisions concerning input levels are
made externally (gaining) or internally (machine simu-
lation). There are five sets of choice variables for
population units here and five subprograms for en-
tering them: allocation of leisure time (TALOC),
setting of a dollar value to time in transportation
(TKVAL), casting votes for political offices (CASTER),
transferring cash from savings to other sector accounts
(CASHT), and instigating boycotts of facilities or
institutions (BUYCT).
Before discussing these, let us make explicit
some choices which are not open to population units
in this model, using as a frame of reference the
optimizing models considered earlier. First, and
161
-------
most important, population units are not allowd to
allocate money to uses independent of time alloca-
tions t Indeed, money expenditures are largely pro-
portional to time expenditures, the only tradeoff
provided between the two coming in the role of dollar
valuation of time in travel to be explained later.
From the viewpoint of puretheory, this would lead to
situations where there is a great likelihood of a
slack constraint and a zero shadow price to time or
money, meaning either a zero or infinite valuation of
time in dollars. As it turns out, there is no money
budget constraint at all, nor any loss of utility to
dissaving. Under such circumstances in pure theory
there can be no shadow price to money, and the shadow
price of time is in terms of utility equivalents.
These are issues which will be treated later in greater
detail. The dictates of the pure theory are not the
final standards for judgment, of course, especially
when there are so many deviations from the pure choice
situation. Still, these issues force careful consi-
deration of the type of behavior that is being implied
in the specification of the problem and in the defini-
tion of the choice or instrumental variables.
162
-------
Other choices not open to population units here
but receiving attention in the consumer theory as it
has been developed are activity frequencies, including
numbers of trips, and residential location and space
use. As in many non-frequency optimizing models, one
might suppose that fixed proportionalities for activity
time and frequency (in terms of a given time per epi-
sode) and for activity cost and frequency (in terms of
goods inputs or admission prices or fares per episodes)
lie at the heart of postulated proportionalities be-
tween time and money expenditures, but the model is
not explicit in this regard. Space use is considered
a function of the type of population unit, based on
income, and is held fixed for a population unit where-
ever it resides. Interestingly, the classification of
the population unit never changes — thus neither does
its use of housing — even though its income and
education can vary (only within strict limits, it is
true, except in the cases of under-employment and
unemployment). Migration from a residential location
is partly probabilistic and partly behavioral, as is
the choice of the new residential location. However,
the behavioral rules involved are divorced from other
163
-------
consumer choices, unlike the optimizing model, where
they interrelated. In the River Basin Model, re-
sidential location does have implications for other
choices and outcomes, of course. Again, these are
all issues which will come up in the review of the
appropriate operating subprograms.
In TALOC, then, each population unit specifies
the times it desires to allocate to each of five
leisure time activities. This information is stored
in the vector
TIME (K, 1)
where the index K gives the uses
1 = part time work
2 = public education time (adult)
3 = private education time (adult)
^ = political activity time
5 = recreation time
The index "1" indicates that these values are times
requested; thus these time allocations are demands.
At the end of a round there will be output another
vector, TIME (K, 2), giving the actual times in these
activities that the population unit was able to realize.
These TIME (K, 2) values are thus time expenditures as
differentiated from the allocations, where the model
will assign times such that
164
-------
TIME (K, 2) < TIME (K, 1)
— that is, expenditures will not exceed allocations.
They will be less than allocations if there is some
reason, due to the behavior of the system, that the
population unit cannot achieve its objectives. Thus
they will be as close to the allocations as the opera-
tion of the system will allow. TIME (K, J), then, is
a matrix of times for allocations (demands) and ex-
penditures (outcomes). There is also a K = 6 level,
which gives travel time to work for J = 1, not a choice
variable, and involuntary time (defined below) for
J = 2. Again, TALOC simply inputs the choice values
for K <_ 5, J = 1. There are subscripts for the class
and location of the population unit, but these will be
suppressed in this review for simplicity unless needed.
Both time allocations and time expenditures are
subject to a time budget constraint whereby each adult
in the population unit has 100 time units to devote
to leisure activity. All adults in the population
unit (composed of 500 persons) are presumed to allocate
165
-------
their leisure time identically. Regular job work time
is BO time units per worker and is not included in
the (leisure) time budget. It might be noted that
although the labor force participation rate for a popu-
lation unit — and thus its total time at work —
varies by class of population unit, the stock of lei-
sure time does not change. This seems a little strange.
for it means that some adults have less total time than
others. If allocation times do not total in excess of
100 units per adult, neither will time expenditures for
those activities (K < 5). However, expenditures must
include two non-allocational items, illness and travel
time, both of which are determined in activity assign-
ment procedures based on system capacities. These are
deducted from the leisure time stock of 100 units before
other time expenditures are determined (see TMALC).
This is the prime justification in viewing population
unit behavior as the behavior of an individual and
using the optimizing model of utility theory as a
frame of reference. Strictly speaking, not only all
adults in a population unit but also all population
units of the same class on a parcel (residential
location) have the same leisure time allocation.
166
-------
Therefore, the residual time stock for leisure ac-
tivity time expenditures may be considerably less
than 100 units, and there may not be enough time to
meet all allocations even if the system has the capa-
city to permit them. On the other hand, if expendi-
tures are so limited by system capacities, or if a
significant amount of time were not allocated, the
sum of time expenditures, plus illness and travel,
may leave some time stock unused. Such slack time is
"involuntary time," TIME (6, 2), although failure to
allocate enough time would certainly appear to be
voluntary. Such a failure is not likely to happen
without a money budget constraint, however.
The relative sizes of regular work and leisure
time stocks — bO and 100 units, respectively — means
that of the total 180 time units, the working adult
2
This is the way involuntary time is computed in TMALC
(see below). Manual #4, p. 90 states that involun-
tary time equals the difference between allocated and
expended times in extra job and education acticities.
The TMALC method can yield zero involuntary time
when the manual method does not, and vice versa.
167
-------
spends bb% in his full time job. This does not appear
to be too different from the real world situation.
In Y/ashington, D.C., for example, an adult working a
4-0-hour work week can be expected to have about 3^
hours in the leisure categories K = 1, . . . , 5 above,
plus about 5-5 hours in the journey-to-work, per week.-'
Of the total, then, almost exactly y>% is work. Sleep-
ing, eating, childcare and other routine activities
are not included. 3y giving working and non-working
adults alike the same leisure time stock, the River
Basin I-;odel implies that the non-workers match the job
time of the workers with additional household and child-
care activities, but non-workers in the Washington
study had well over ten more hours more in leisure
each week than workers .had. Moreover, "lower" class
population units not only have higher labor force par-
ticipation rates but also higher birth rates (number of
children) per population unit, meaning that "higher"
class population units must either have more leisure
3
The metropolitan Washington, B.C. time budget study
will be used as a frequent reference. See Hammer and
Chapin, op. cit.
168
-------
or else be spending an inordinate amount of time in
housework and childcare, whereas they are the most
able to hire help. Such implications and possible in-
consistencies as these should be analyzed in detail
and adjusted to fit real-world patterns before planning
simulations are made based on time allocation proce-
dures. Elsewhere in this critique many other sugges-
tions will be made for bringing the River Basin Model
time budget more in line with reality.
Subprogram TMVAL simply inputs the dollar value
the population unit attaches to a unit of time in tra-
vel, and TIMST prints out a record of these inputs.
This parameter serves as a shadow price on travel time;
increasing the parameter increases the chances that
the system will assign the population unit to a faster
but more expensive transportation route/mode alternative
for the journey-to-work, and decreasing it biases assign-
ment toward a slower but cheaper alternative. The
parameter, then, represents a preference in time-money
substitutions, and in theory it should be consistent
with other preferences for time uses and with relative
prices or costs of other activities. At present there
are no such consistency requirements, and we will be
169
-------
especially interested in seeing what kinds of anomalies
can arise. Since the objective function is the same
for all population units, if each tries to maximize
satisfaction differences in the dollar value of time
parameter should directly reflect differences in prices
and costs — travel and non-travel — by residential
4
location. Actual transportation costs paid do not
include this shadow price, of course. By manipulating
the parameter the population unit attempts to minimize
divergence of the activity time expenditures it realizes
from the activity demand allocations it chose in pre-
sumably attempting to maximize satisfaction. In com-
plete simulation the parameter should be a simulatane-
ously determined index rather than an independent choice
variable.
The only direct money manipulation open to the
population unit is the transfer of cash from savings
to the accounts of other sectors, as personal loans or
contributions. There is no direct allocation of money
Similarly, in theory these price and cost differences
are the only reasons for differences in time alloca-
tions (demands). See discussion of the objective
function in MIGRAT (HSDSSW).
170
-------
by the population unit for household consumption ex-
penditures; instead, current personal expenditures to
cover transportation, health, education, taxes, and
consumer goods and services are determined in popula-
tion unit activity time and location assignments, so
that time expenditures represent money expenditures
as well. The only constraint on cash transfers by
the population unit is that it not exceed savings.
Population unit cash transfer decisions are input in
CASHT, and these may have to be limited in complete
simulation of the Social Sector because they are so
closely tied to over-the-table bargaining. Otherwise,
we may end up with mechanistic vote- or political
favor-buy ing.
Voting and boycotting are the final two choice-
areas open to the population unit. CASTER enters
population votes for political officers, presumably
cast in reaction to their previous behavior and current
policies. Before voting can be completely automated,
some index of the salience of a candidate to the popu-
lation unit in terms of that behavior and those policies
must be constructed. The total number of votes which
can be cast by a population unit is limited to its voter
171
-------
registration level as a function of time spent in
political activity in the previous round (see TMALC).
While time allocations are in effect labor supply
and consumer demand statements without specific re-
ference to source of employment or supply >r to spatial
wage or price differences, boycotts permit population
units to exert pressure for change in the operation
or behavior of particular units in other sectors by
intentionally constraining the opportunity choice-set
to be considered by employment or other activity
spatial-temporal system assignment procedures. Units
boycotting identify the units to be boycotted in the
subprogram BUYCT, and those boycotted units will not
be included among opportunities for activity assign-
ments for the boycotting units.
There are already certain built-in preferences
or restrictions on the behavior of population units
by class which amount to class-wide permanent boycotts
of other types of behaving units. For example, low
class population units cannot use private schools, and
high class ones cannot (will not) live with low class
ones. The second case here appears to be a behavioral
"social boycott," but the first is ambiguous, as it
172
-------
could mean either discrimination by private schools
or financial inability by population units. The latter
would best be handled by introducing a money budget
constraint for population units and then allowing low
income population units to lobby for price changes in
education as normal political (voting) or boycotting
behavior. These examples point out the need to sepa-
rate behavioral propensities of behaving units from
functional constraints on their behavior. Functional
constraints which could be relaxed through changes in
the system should not be confused with behavioral pro-
pensities (preferences) and thus permanently built into
system simulations. This issue will come up several
times in this critique. It suggests that as many of
these propensities as possible be made characteristics
of preferences (objective functions) differentiated by
class rather than presented as "permanent boycotts."
As with voting, boycotting may be hard to simulate
internally because of its bargaining aspects, it being
a reasonable course of action for a population unit
only if others can be convinced to do likewise. Other-
wise, the opportunity costs in time and money to the
boycotting unit may be too high with little chance of
173
-------
successt In simulation threshhold levels for insti-
gating boycotts would have to be established as functions
of relative price levels and/or service quality levels
if boycotting is to be retained and carried out auto-
matically. Population units which are unsuccessful
in boycotting would be prone to leave the system, so
that boycotting could be viewed as a last-ditch effort
before outmigration occurs for dissatisfied population
units. Widespread dissatisfaction and outmigration
would be dampened because increased boycotting would
improve chances for system change (see MIGEAT). Voting
behavior could have the same effect. Again, there
would have to be mechanisms for identifying the re-
sponses appropriate for desired changes in the system,
degree of success being solely a matter of the number
of population units with similar enough experience to
evoke the same response.
This discussion of current input procedures has
set up the population unit choice problem and raised
some issues relevant to the machine simulation of this
choice-making. We will turn in the next several chap-
ters to system assignment procedures. These procedures
in most cases can be operative under a wide range of
174
-------
allocational simulation procedures, since they require
from population units only the choice inputs described
here, however derived. Similarly, there are some prob-
lems with these procedures which would "be evident re-
gardless of the nature of input derivation. Finally,
there are some instances in which the nature of the
choice-making simulation interacts with the assignment
operation, so that modification of the latter is de-
pendent on the selection of the former. These aspects
will "be drawn together in the concluding chapter to
the critique.
175
-------
SECTION VIII
DISSATISFACTION AND &IC-RATIOK
The River Basin :'7odel contains an explicit "ob-
jective function" for the population units of the Social
Sector, and all population units are thought to share
the preferences represented by this function. In this
case, not only do all population units have the same
preferences but also view utility or satisfaction with
a scale and origin to the function which is common to
all, so that interpersonal (inter-population unit) com-
parisons of utility level are possible. Indeed, such
comparisons are actually used in the migration procedures
to screen out the "most dissatisfied" population units
for migration purposes.
This chapter will discuss the objective function,
which it is presumed that population units set levels
to their choice variables (including time allocation)
to optimize, and the migration procedures, which uti-
lize the level of the objective function as one among
several factors in migration behavior. The level of
the objective function, of course, gives the "value"
of the outcomes of a round's operation of the system
to the population unit in terms of the satisfaction of
177
-------
experience that round. Migration is the only direct
consequence the population unit might have to face
purely because of its satisfaction level, and as such
it is an extension of its choice behavior. That is,
the decision to migrate is a part of normative popu-
lation unit behavior that is simulated internally in
the model. Following chapters will examine just hoW
the outcomes to the population unit are determined
through activity assignment procedures.
The Quality of Life Index
The objective function is called the Quality of
Life Index because it has as arguments measures of
environmental quality, of which manifest life style
in terms of time expenditures is a component. It is
also called the "dissatisfaction index" because it
actually measures increasing levels of dissatisfaction.
^•Perhaps the behavior is more truly descriptive than
normative, in that a certain percentage of the most
dissatisfied simply tend to migrate. It would be
normative, like allocational decisions, if a popula-
tion unit would be better off to migrate when its
satisfaction reached a certain level.
178
-------
Thus, optimizing behavior would be to select choice
variables hopefully leading to outcomes which would
minimize the level of the indext In reviewing the
composition of the index it will become apparent that
most arguments to the function for an individual are
beyond the direct control of the population unit.
That is, while time allocations have a direct role in
determining time expenditure outcomes, other compo-
nents of the index can only be influenced indirectly
by the population unit in its political (voting) or
boycotting behavior.
That time expenditures are only a few of the many
components in the Quality of Life Index, as computed
?
for population units in subprogram HSDSSW," can be seen
in its structure as the sum of four separate indices«
the Pollution Index and the Neighborhood Index (to-
gether, the Environmental Index) and the Health Index
and the Time Index (together, the Personal Index).
These indices are given equal weight, but only the
Time Index has as arguments actual time units. The
others are functions of quality and service indices
20r HSDSST, in the event that the water system is not
included.
179
-------
which do reflect population unit time and money ex-
penditures, but for whole classes of users in addition
to the population unit for which the index level is
being computed.
Because these other indices for a population unit
depend on the activities of other units, and because
control can be exercised only politically or through
boycott in concert with other population units, then
if the Time Index is not sufficiently weighted rela-
tive to those others in the Quality of Life Index, time
allocations and resulting expenditures for a population
unit may fluctuate wildly from round to round in re-
sponse to small changes in system prices and parameters
but having very little direct impact on the dissatis-
faction level for that population unit. Such time
fluctuations lead to fluctuations in consumer demand
and then to oscillations throughout the economy and
the system. Population unit control over its destiny
must be adequate to stabilize its behavior. Otherwise,
the time budget approach is best abandoned. Because
voting and boycotting behavior may be difficult to
simulate internally, they cannot be counted on as sources
of that control.
180
-------
In order to assess the sensitivity of the Quality
of Life Index to time expenditures, and thus tenta-
tively to time allocations on which they are based,
the component indices will be defined below without
much comment on non-time arguments, the Quality of
Life Index being the sum of the values of the compo-
nent indices for a population unit.
1. PLIN» Pollution Index for a parcel "I"
PLIN =
(WTQL(I) - 3«5p if the parcel has sur-
face water
N
if the parcel does
not have surface
water, where J= 1,
. . . , K denotes
adjacent parcels
where WTQL(J) is the water quality rating
of parcel J (or I),
WTQL(J) = maximum ( POLN(J, K) )
oer K ^ '
v/here POLN(J, K) is the quality
level for pollutant of type K on
parcel J, K = 1, . • • » ? (in-
cluding coliform bacteria)
0 < FOLN (J, K) < 9
thus
-(3-5)3 <
< (5*5)
181
-------
2. HD* Neighborhood Index for a population unit of
class "X" on a parcel "I" in political juris-
diction "J"
HD = CTYDST(l) + CTYDST(2) + RNF + QIF +
+ JURFAC (J, K)
where
2.a. CTYDST(1)» Municipal Services Quality Index
CTYDST(l) = ?UBFAC(1, K)* AMAX1(0.,
PU3VAL(1, I) - PUBMINU, K))
where PU3VAL(1, I) < 200 is the municipal
services level (100 times the ratio of
demand to supply capacity), PU3MIN(1, K)
is the acceptable utilization level, and
the difference is overutilization; AI£AX1
takes the maximum of zero and overutili-
zation; PUBFAC(l) is a scale factor
2.b. CTYDST(2): School Quality Index
CTYDST(2) = PU3FAC(2, K)* AKAX1(0.,
PUBVAL(2, I) - PUBMIN(2, K))
where the arguments are analogous to those
above for municipal services
current datas
for all L = 1, 2 and K = 1, 2, 3
PUEFAC(L, K) = 1
PUB:.IIK(L, K) = 100
therefore, for L = 1, 2
0 < GTYDST(L) < 100
(See subprogram DSPREC for computation
of supply capacities.)
182
-------
2.c. RKFs Rent Charged Index for a population
unit of class "K" on parcel "I"
RNF - RNTPAC(K)* AMAX1 (0., RENT(I)
- RKTMIN(K))
where R2NT(I) is the rent charged per
space unit, RNTMIN(K) is a "typical"
rent per space unit for this class, and
AI'/IAXl takes the maximum of zero and the
"excess rent" charged; RNTFAC(K) is a
scale factor for class "sensitivity" to
excessive rent
*}
current data:-^
R2NT(I) < 210
K _
RNTFAC(K)
RNTMIN(K)
1
3
140
2
2
150
3
1
165
therefore
0 < RNF < (210, 120,
2.d. QIF: Residential Quality Index for a
population unit of class "K" on
parcel "I"
QIF = QIFAC(K)* AIv!AXl(0., QIMAX(K) - QI)
where QIMAX(K) is the lowest quality index
(lowest quality) the unit would accept
on moving into new housing, QIFAC(K) is
a scale factor, and QI is the residen-
tial quality,
QI = r.'IAXO(QUAL(l), MNQI(I))
where KNQI is the maintenance level set
o
^These data are from manual #3* p. 130. HSDSSW gives
183
-------
by the owner (if MNQI > QUAL, then
property is being updated), and QUAL
is the condition of the property, being
a function of age, municipal services,
flood and fire factors (see subprogram
DEPRSC; if iMNQI < QUAL, the property
is being allowed to deteriorate? I4AXO
picks the maximum of maintenance and
quality levels?
AKAXl picks the maximum of zero and the
deficiency in quality of the residence
current data*
K * 1
QIFAC(K)
QIMAX(K)
0.5
140
0-5
180
0.5
200
0 < MNQI(I) < 200
0 < QUAL(I) < 200
therefore
0 < QIF < (70, 90, 100)
2.e. JURFAC(J, K)» Tax/Welfare Index for a
population unit of class
"K" on parcel "I" in juris-
diction "J"
If K = 1,
JURFACCJ, 1) a VfLFAC* AMAX1(0.,
WLFMAX - PCSH(30, J))
where WLFMAX is the minimal acceptable
unemployment compensation rate and
PCSH(30, J) is that rate for the
jurisdiction, AMX1 takes the maxi-
mum of 0 and the rate discrepancy,
and WLFAC is a scale factor
If K > 1,
2
JURFAC(J, K) F (TAXFAC(L, D*
184
-------
AMAX1(0., TAXS(L, J) - TAXMIN(L, K)))
where TAXS give the tax rates in mils
for land and development (L = 1)
and income and consumption (L = 2),
TAXMIN (L, K) are the respective
maximal "acceptable" tax rates by
class, AIvlAXl takes the maximum of
zero and the excess tax rates, and
TAXFAC are scale factors
current data:
WLFAC = ^-.OiWLFMAX =20
TAXFAC(L, K) = (.125, -250)
by L = 1, 2, for all K
TAXMIK(L, K) = 0.0, all L, K
for K = 1, JURFAC(J, 1) < 80
for K > 2, for example, if the rates
were 20#, then JURFAC(J, K) = 75;
obviously, TAXS(L, J) < 1000,
L = 1, 2, so that JURFAC(J, K) < 375
2.f. Summary:
0 < HD < (655, 7»5» 720) for K = 1, 2, 3
HLIN: Health Index for a population unit of class
"K" on parcel "I"
3
HLIN = F HLVAL(L)
LSI
where
3»a. HLVAL(l)* Municipal Services Component
HLVAL(l) is simply the Municipal Services
Quality Index CTYDST(l) above with the
scale factor PUBFAC(1, K) = 0.25, so
HLVAL(l) < 25
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3-b. HLVAL(2)t Residential Crowding Index
HLVAL(2) = AMAX1 (0., AMIN1 (25, (CROWD
- 100)* 1.25))
where
3
CROWD * PPROPd, K)* CRDFAC(K))/
CAP (L,, LEV)
gives the per cent of capacity reached
in the residential structure, PROP(I, K)
being the number of population units
there of classK, CRDFAC(K) being the
amount of space used by a population
unit of class K, and CAP(L, LEV) being
the space capacity for the building of
type L on parcel I operated at level
L2V
current da tat
CRDFAC(K) = (1.0, 1.33. 2.0)
HLVAL(2) is such that
0 < HLVAL(2) < 25 if CROWD < 120
HLVAL(2) = 25 if CROWD > 120,
since AKIN1 takes the minimum ol' 25
and the per cent overutilization, and
AMAX2 takes the maximum of that and
zero.
3.c. KLVAL(3)s Coliform Bacteria Index
HLVAL(3) = AMAX1 (0., AMIN1(50, (COLF(I)
- 1000)* (.0025)))
where COLF(I) is the coliform bacteria
count (in parts per million gallons of
surface water. In the manner of
version is from HSDSSW. Manual #4, p. 90 states
HLVAL(3) * COLP(I)A, with a maximum of 50 if
COIP(I) > 200.
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HLVAL(2), HLVALO) is such that it
ranges from zero to fifty if COLF(I)
< 21,000 and equals fifty if COLF(I)
exceeds that limit.
3«d. Summary
0 < KLIN < 100
Summary of Environmental and Health Indices
PUN + HD + HLIN < 1121.375
It would be very unlikely that these indices
would reach anywhere near such a great value. On
the other hand, if arguments consistently exceeded
"desirable" levels by about 10#, with tax rates
near 100 mils, the value for this sum should be in
the area of 100 to 200. Almost all arguments can
range in size from zero to two-hundred, and they
tend to yield little or no dissatisfaction in their
middle range and below (or above, depending on the
direction of the effect). The weights in the in-
dices attached to the deviations above the middle
range vary from one-half to two or so for those
arguments; these give the rough changes in the in-
dex sums with a change in those arguments when they
create dissatisfaction. There are two exceptions.
One concerns the tax and welfare rates, where weights
187
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can drop to one-eighth but absolute values can
range much higher. The other exception concerns
water quality components, which have non-linear
effects.
TIM: The Time Index for a population unit on
parcel "I"
TIM = TIME(6, 2) + 5* TIMS(6, 1)
- (.01)* PRKFAC* TIME(5> 2)
where TIME(5» 2) is recreation time expenditure*
TIME(6, 1) is travel time to workt
TIK2(6, 2) is involuntary (slack) time,
and PRKFAC is a function of the park use index
PKAL(I) for parcel I (see subprogram TMLC) —
PRKFAC = AMAX1(0., AMIN1(100, 200
- PKAL(I)))
— so that PRKFAC is 100 if PKAL(I) < 100 and there
is no overcrowding, or it is equal to the degree
of overcrowding, 200-PKAL(I), when PKAL(I) > 100
and there is overeroding. The scale factor .01
converts this percentage factor to a value < 1.0.
TIM takes its least value when recreation
time is maximal, travel and involuntary time are
minimal, and there is no park overcrowding. This
would be achieved by allocating all 100 time budget
units to recreation. Subprogram TMALC (see later
chapters) would take out time only for travel to
188
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work and illness and assign the residual to re-
creation; involuntary time would "be zero. Under
these circumstances, recreation time would "be 100
less travel and illness, the latter "being one-
tenth of the Health Index HLIN. Then,
TIM * 6* TBE(6, 1) + .1HLIK - 100
The lowest limit to TIM, then, comes when there
is unemployment (no travel) and minimal health
index, at which time TIM = -100. With employment
(full time only), and assuming that about 20$ of
leisure time goes to travel to work as the Wash-
ington Study suggests, then TIM would be .1HLIN
+ 20 here, or 20 if HLIN = 0.
Obviously, the upper limit to TIM comes when
there is no recreation time and maximal travel and
involuntary time. This can be achieved (under the
TKALC procedure, but not according to the manuals)
by making no time allocations at all. Then in-
voluntary time would be 100 less travel and illness,
so that
TIM = 100 - .1HLIN + 4* TIME(6, 1)
Interestingly, this is maximal when HLIN = 0.
Under the 2Q% travel time assumption the value of
189
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TIM would then be 180. The absolute maximum
would come when involuntary time was zero, so that
travel is 100 less illness. Then
TIM = 500 - .5HLIN
which has as upper limit the value of 5°°'
It is important to note that there are no
monetary expenditure items in these indices.
Moreover, work (full or part time), education and
political time do not figure directly into the in-
dices. Their only utility lies in amassing cash
(for transfers) and political/coercive power for
influencing the levels of arguments in Environ-
mental and Health Indices, influences which are
tentative at best and, as it has been pointed out,
will be difficult to simulate effectively because
of their bargaining aspect. Since there is no
money budget constraint and no penalty in the in-
dices for dissaving, there is no mandatory reason
for working and undertaking travel other than to
provide funds for transfers. Finally, as will be-
come evident in later chapters, there are other
activities in the model for which no activity or
travel times are computed.
190
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Allocating to Minimize Dissatisfaction
In the gaining version of the River Basin Model's
Social Sector there is no requirement that a popula-
tion unit (decision-maker) make allocations to minimize
dissatisfaction in the Quality of Life Index. The
very specification of such a "welfare function" im-
plies that attempts at minimization would "be rational
and desirable behavior, of course, and it is assumed
that if the function is retained in simulated decision-
making, then optimum-seeking behavior should be postu-
lated and the appropriate rules of thumb derived. An
attempt at a statement of such rules is made in a
concluding chapter to this critique.
If a population unit were to try to minimize
total dissatisfaction (denoted PDST) represented in
the Quality of Life Index, it would attempt to under-
take "marginal analysis" of the type applied to
utility-maximizing models of consumer behavior.
Since the Quality of Life Index is basically a linear
function (the pollution component is non-linear, and
recreation time interacts vjith park crowding* as the
exceptions), the existence of a single budget (re
time) suggests that only a few, and possibly only one,
191
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argument under the control of the population unit need
be allocated at a non-zero level to minimize dissatis-
faction. That education (skill) level and voter re-
gistration cannot fall below certain limits for a class
further undercuts the importance of those activities.
This may not be unrealistic, of course, as the Washing-
ton study shows very little interest in these activities
is exhibited among the adult population.
Normative allocations in the River Basin Model
would all be subject to lack of knowledge of outcomes,
of course. Marginal changes in dissatisfaction with
changes in arguments are simple enough to identify
here, but it is not so simple to see how environmental
and health factors will change with population unit
time (or money) allocations, especially when effects
may be lagged several rounds in the Model's operation.
It is true, however, that there are explicit functions
in the Model which relate the levels of these arguments
to levels of parameters and choice variables in other
sectors. If these parameters can be affected by pop-
ulation unit allocations, then these technical functions
can be used to replace non-time arguments in the Quality
of Life Index of the population unit with functions of
192
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its own allocations. For example, water quality could
be expressed as a function of treatment level, treat-
ment level as a function of tax revenue, and tax re-
venue as a function of tax rate and tax base (income,
for example). Then if each population unit can specify
a desired direction of change in the tax rate and is
defined weight in influencing change in proportion to
its political activity, the change in the tax rate being
the aggregate of these weighted directional changes for
all population units in the jurisdiction, water quality
as a variable is replaced in the dissatisfaction index
by these population unit choice variables, so that mar-
ginal returns can be evaluated, conditional on the
similar activities and incomes (work) of other popula-
tion units.
In the absence of such cost/time relations in the
dissatisfaction (Quality of Life) index, the population
unit, without the option of bargaining, would be prone
to try to minimize the Time Index, ignoring the others,
even though the discussion above shows that the range
of values accorded the Time Index is roughly that of
each of the others under normal circumstances. It
has already been brought out that when the population
193
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unit is unsure of the impact political pressure -will
have, and when meeting a balanced money budget or
saving money for transfers is of dubious merit, then
there is little value to a second job and to education
and political activity. Similarly, boycott behavior
could lead to increased travel time and cost (although,
with no money constraint the increased money cost it-
self yields no dissatisfaction) without the promise of
significant change in health or environmental variables.
Under these conditions a population unit might be
expected to:
1) not request secondary work! this helps keep
travel time down, and there is no need to
avoid dissaving by earning extra income;
2) not request education or political activity,
letting education (skill) level and voter
registration drop to their minimal levels for
the class? this weakens competitive position
for jobs, but again the loss of income does
not lead to dissatisfaction; it might lead to
greater travel time in job assignment (see
EMP), but 3) suggests how to offset thisj
194
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3) set the dollar value of time parameter very
high to reduce travel time at the expense of
cost, which, again, causes no dissatisfaction;
this could make it hard to find an acceptable
job, under the acceptable limits of travel
costs to income earned (see Bi€P), but un-
employment is no problem when there is no
money budget constraint—indeed, it would
eliminate travel time to work?
*0 minimize involuntary timej according to
manual # 9, p. 90, this would mean minimizing
the disparity between allocations and expendi-
tures realized for second job and education,
and this would be done by steps 1) and 2)
above, making involuntary time equal to zero;
under the TMALC computation of involuntary
time as residual time not expended, involun-
tary time would be minimized by allocating
all 100 units to activities without making
allowance for the non-choice items of travel
and illness; then, even if all demand allo-
cations are not met in the operation of the
system, preemption of time for travel and
195
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illness reduces the chances that there will
be slack time; if steps 1) and 2) are followed
and only recreation is allocated time (of 100
units), then involuntary time is assured to
be zero.
As was suggested in the previous section, then,
optimizing the Quality of Life Index by minimizing TIK
seems to lead to the allocation of all leisure time to
recreation. The fact that this would seem to be so
universally the best strategy undercuts the usefulness
of choice behavior and the time budget. The most
straightforward improvement would be either to impose
a strict money budget or attribute dissatisfaction to
the level of dissavings, or both. Since recreation
time entails consumption expenditures (see OPCK), allo-
cation to recreation cannot be made with abandon under
a money budget but requires attention to the job situa-
tion, and the same would be true with dissatisfaction
to dissavings. Education to improve competitiveness
and income becomes a consideration again. In addition,
it night be wise to set a limit on the choice of the
dollar value of time as some function of the relative
tines allocated to recreation and to the second job.
196
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Theory suggests that these should be consistent, sub-
ject to wage and cost rate parameters, since the dollar
value of time is precisely a measure of the marginal
rate of substitution between time and money. Indeed,
in simulation the dollar value of time to be used in
transportation assignments might be a derived rather
than a choice variable, being just such a function of
relative time allocations.
It seems that the Time Index itself should be
expanded to allow education and political activity to
be seen as having consumption value in their own right.
It has already been suggested that an attempt be made
to relate environmental and health arguments to popu-
lation unit arguments such as income, political time,
and so forth, although it is admitted that this may
be difficult to do. Finally, there are other activity
times and expenses that need to be introduced to the
time and money budgets and, possibly, to the Quality
of Life. These will be discussed later, and include
travel time and cost for shopping and recreation and
shopping time itself. As to the' nature of the Quality
of Life Index itself, its linearity increases the possi-
bility of getting all-or-nothing type solutions such as
197
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the recreation allocation case above. Other forms
should be considered. These issues will be raised again,
Migration Procedures in MIGRAT
After sector decisions are made and allocations
are input to the system and the infrastructure/land-
use pattern updated, but before actual assignment pro-
cedures are undertaken to translate allocations into
actual flows, exchanges or expenditures, the demogra-
phic structure of the Social Sector is modified through
population unit migration via the KIGRAT subprograms.
This means that population unit composition and situa-
tion can change between the making of allocations and
the determination of expenditures of time, giving
dynamism and uncertainty to choice-making. The Social
Sector is in a position of continual lag. Migration
can be initiated by the level of dissatisfaction of a
population unit for the previous round, and it is
MIGRAT that calls HSDSSW for computing that level.
This means that although the population unit knows the
outcomes (expenditures) of the previous round before
making current allocations, it does not know the
associated level of dissatisfaction, and thus it does
198
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not know whether or not the last round's allocations-
expenditures led to an improvement or deterioration
of its dissatisfaction position before it must make
new allocations. This seems to be unnecessary; HSDSSW
can be called as part of WRYOU (Social Sector Output)
at the end of the previous round. Indeed, in simula-
ting Social Sector decision-making this should be
done, since incremental changes in the dissatisfaction
components for specific arguments in the previous round
may be needed for computation of current adjustments
to allocations.
There are six sources for population change in
the model, and MIGRAT calls up subprograms to compute
the extent of such change and to redistribute the popu-
lation over the area. Subprogram MOVOUT determines
the numbers which vacate housing for reasons of un-
or under-employment (these leave the system), dis-
satisfaction (with information provided by KSDSSW and
GETCUT), or random movement. Next, subprogram UKCRWD
determines the number who move for reasons of residen-
tial overcrowding, usually associated with demolitions.
Finally, INI.iIG computes the number of in-migrants to
the system and the extent of natural increase. Sub-
199
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programs SETUP (using PICKRS) and KOVIN do the actual
matching of movers with residential opportunities and
the consequent redistribution of the population, and
JANGUW and IvIIGSILM provide printed details and summaries
of the migration operations. These procedures will
be reviewed briefly with an eye toward their implica-
tions for assignments and dissatisfaction levels for
movers•
Table 2 summarizes the selection procedure for
movers from among the existing population. The estimate
of a mover incidence of 20 to 2$% is only a rough guess.
Most of these remain in the system—about 3f° of the
original population leave the system because of em-
ployment problems. Note that since unemployment or
underemployment does not necessarily mean more dis-
satisfaction because of the way the Quality of Life
Index is defined, there may be little overlap of pop-
ulation units falling into those categories of movers.
(Underemployed workers are those who must take jobs
open to lower status groups; see 2I£P.) INKIG identifies
natural increase as 1.5# of the population and in-
migration as 1% of the population plus one population
unit for every job which went unfilled the previous
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TABLE 2: Determination of Population
Units Changing Residences
(MIGRAT)
1. Total population at beginning of round (100$)
l.a. 20$ most dissatisfied in each class:
random one-half move
l,b. 80$ least dissatisfied in each class:
random % (1, 5» 7) move in each class
2. Remaining approximately 85-90$ of population
2.a. under and unemployed: random % (15» 25»
33) move in each class (leave system)
2.b. fully employed: no movers
3. Remaining approximately 80-85$ of population
3.a. living on excessively crowded parcels
(over 120$ crowded): enough move to bring
crowding down to 120$
3.b. living on other parcels: no movers
J*. Resulting 75-80$ of original population retains
same residential location
round. This should roughly offset outmigration from
the system for job reasons and perhaps provide for
a net increase. These population units will also be
seeking housing, along with movers within the system.
Note that since there are maximal allowable limits to
the ratio of journey-to-work costs to income earned,
201
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there can be unemployment and unfulfilled jobs simul-
taneously (see EMP).
fevers staying within the system are selected
for consideration in random order by SETUP, which calls
PICKRS to choose the new residential location for the
population unit. In order to locate on a parcel, the
housing quality there must lie within a specified
range for the class of the population uniti
Class Housing Quality Range
low (1) 20 to 70
middle (2) 40 to 100
high (3) 71 to 100
Note that these ranges preclude low and high population
units -frowi WOVXtVQ into the same parcel simultaneously.
Movers will be located into parcels as long as the
percent overcrowding does not exceed 120. The actual
selection of a location as "best" for a population
unit from among those which are acceptable under these
conditions is that location having the lowest Environ-
mental Index. If the reason for the move had been
random factors or dissatisfaction, there is the addi-
tional condition that this index value must be less
than the index value at the previous residence, even
202
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if that value was not a crucial factor in the total
dissatisfaction? if that condition cannot be met* the
population unit must outmigrate from the system.
We see, then, that the specific factors leading
to dissatisfaction or employment problems are not con-
sidered in choosing a nev; location. Moreover, a local
mover keeps his previous place of employment, even
though transportation time and cost — which could have
led to the move in the first place — are likely to be
much greater in the nev/ location. Reassignment to a
new job is possible in EMP» but the procedure biases
assignment toward current job, and no distinction is
made there between recent movers and non-movers. More-
over, on moving into a parcel the population unit adopts
the time allocations of population units of its class
3
already on the parcel, even though these may be quite
inconsistent with the work/travel pattern that popula-
tion unit retains. Under these circumstances, it seems
In addition, the educational level and voter registra-
tion for the in-migration are averaged with those for
current residents in I.IOVIN. Financial information is
apparently not transferred unless there are no previous
residents. That is an unnecessary omission.
203
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a matter of pure chance that the population unit will
have improved its situation. Indeed, it would seem
that a population unit could easily enter a continuous
cycle of movement, ending only in its exit from the
system. The fact that 20$ or so of the population
units may change residence each round may reflect
American life—a move every five years, on the average
—but it may also make the model quite unstable.
The basic problem with the migration procedures,
then, is that they do not systematically match the
reason for moving with an appropriate rule of thumb
for locating. This precludes any intential tradeoffs
by the moving population unit. The utility maximizing
models of residential location have been used to demon-
strate how activity pattern (time use) can vary as ac-
tivity (and goods, including land) costs vary by
location. Systematic consideration of such tradeoff
possibilities allows the population unit to improve its
situation in the move. Granted, all moves need not be
rational, and the dynamism created by unpredictable
dollar value of time paramter is important in job
assignment, and changes in its value could accelerate
or dampen such a cycle.
204
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change has its realistic features. Yet it does not
seem reasonable to divorce completely migration be-
havior from other population unit choice-making.
Here are some suggestions for matching location
rules with migration reasons. First, it must be de-
cided whether population units are to move for reasons
of "push" alone, or "push" and "pull." At present
there is mainly the "push" factor: the decision to
migrate depends only on the immediate circumstances
of the population unit at its residence with no regard
to circumstances elsewhere. An optimizing approach
would also consider the "pull" of better circumstances
at other locations. The present method considers "pull"
only to the extent that location in the system will
not occur for certain movers unless some conditions are
met; however, if they are not met (i.e., not enough
"pull"), the movers still move, but out of the system
(i.e., "push" is still the sufficient condition for the
move). It is suggested, then, that potential movers be
identified first by strength of "push" factors and
subsequently by strength of "pull" to other locations.
Only those with the greatest "push" should leave the
system if no within-system location with sufficient
205
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"pull" is found; others should simply stay put in that
case. For practical reasons low "push" population units
should not be allowed to consider moving even though
"pull" may be sufficient at some other location. In
theory, a population unit would move whenever it could
improve its situation regardless of its current level
of well-being, but this is not likely to be feasible
in an operational model.
The second issue is the definition of the "push"
factors. Theory suggests that the only reason for
moving is dissatisfaction. For example, an unemployed
population unit that is fairly satisfied should be no
more likely to move than an equally satisfied employed
population unit. The need for identifying employment
reasons in the River Basin Model stems from the omission
of a money budget constraint and/or additional arguments
to the dissatisfaction index reflecting money expendi-
tures and (dis-) saving and income levels. V,?ith those
charges, dissatisfaction need be the only criterion to
define sufficient "push" levels (by class), with the
possibility of retaining some random movement. After
all, a utility function should encompass all motivation-
al dimensions for behavior, by definition*
206
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Given sufficient "push" in terms of dissatis-
faction, there would be two ways of proceeding. One
would be to identify the component index — Pollution,
Neighborhood, Health or Time -- contributing the most
to dissatisfaction (i.e., the one that is the largest)
as the predominant "push," or reason for moving. It
was suggested that just as movers were differentiated
by type of "push," so should the "pull" factor for
loeating the mover be of the appropriate type. There-
fore, the "pull" factor to be considered for loeating
would be just that same set of variables defining the
"push." ICEGRAT's use of the Environmental Index is an
example of the way the mover dissatisfied with the
environment would be located at that parcel at which
the strength of the environmental "pull" is the greatest.
An example where transportation time in TIM is the push
factor would lead to locating where the "pull" of re-
that rent level is an argument to the dissatis-
faction (neighborhood) index. Therefore, a move to
seek a reduction in rent is part of the migration
push-pull in terms of dissatisfaction. Space con-
sumption is fixed, however. This is a case where an
expenditure item is in the "utility function."
207
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duced transportation time is greatest. In this in-
stance migration to reduce travel time to work is the
complement to job assignment in EMP which attempts to
maximize the excess of income over travel costs. The
side conditions on crowding and space availability
could be retained (the River Basin Kodel does not allow
substitutions involving residential space). The pre-
sent structure of PICKRS could be easily modified to
handle these changes.
It should be remembered that in theory a mover
would evaluate all "pull" factors relative to all "push"
factors — that is, it would locate to minimize dis-
satisfaction. The second approach, then* would be
an explicit "marginal analysis" by actually picking
that location with space available which minimizes
dissatisfaction. Involuntary and recreation time are
the only two items in the Quality of Life Index that
could not be varied by location to find the optimal
one. The data for all other components has already
Other population unit outcome variables which might be
added, such as consumption expenditures, might have to
be held constant, too. Some, like savings and travel
costs, could be allowed to vary, however.
208
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been computed, and it would be hardly more difficult
to evaluate total dissatisfaction by location than
Environmental Index level, which is currently done.
Indeed, all that is involved is the simultaneous con-
sideration of travel time, the Environmental Index, and
the Health Index. This seems quite straightforward in
a modified FIGKHS.
It is felt that these changes are relatively easy
to implement without completely altering the structure
of MIG-RAT. Indeed, in some ways it is simplified,
since there would only be dissatisfaction to consider.
These changes would also lessen the problem of in-
consistencies between work/travel relationships carried
over from the old location and the time allocations
adopted at the new one, since choice of the new loca-
tion must take into account the current place of employ-
ment. It should be remembered, of course, that times
are presumed allocated as optimally as possible by
population units, subject to the uncertainties and
dynamism of system operation; otherwise, there would
be little justification for moving in the first place,
other than random factors. Although activity and
residential location would still be sequential and not
209
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simultaneous as in the neo-classical models, still
they would be more closely related and explicitly
normative.
Finally, only uncertainty that is felt to be
arbitrary has been removed in these modifications.
Population units wouid still make decisions on the
basis of previous round costs, prices and opportuni-
ties. Just as for non-movers, so also would movers
be ignorant of changes in those parameters and the
behaviors of other units in the current round* re-
taining the disequilibrium lag qualities of the model.
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SECTION IX
FROM TIME ALLOCATIONS
TO TIME EXPENDITURES
There are three types of locational assignments
"beside migration to be made for Social Sector popula-
tion units in the River Basin Model. These concern
employment, shopping, and park use. Travel times and
costs are involved in some of these assignments. In
addition to locational determinations there are time
determinations to be made for most of these activities,
plus time to be spent on education and politics. This
chapter will consider all of these except shopping,
which will be given detailed treatment in the next
chapter on monetary expenditures. In that chapter*
modifications will be suggested which will affect the
procedures discussed in this chapter.
Full-time employmemt is the first of these assign-
ments to be handled in the model, following migration.
EDORD stratifies the population by education level,
and BMP and TRTRC make employment assignments based on
education (skill) and dollar value of time in travel
and update the transportation system. Subprogram EMPRT
handles part-time employment assignments and time ex-
211
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penditures, and NSPACK does the adult education time
apportionment. Finally, TMALC sets time expenditures
for illness, political activity, and recreation. It
also determines park use and updates education level
and voter registration. These procedures will be
discussed in turn.
gitrploynient
The importance of educational level, a proxy for
skills, is seen by the fact that SLIP takes population
units in order of education, highest first as determined
by EDORD, in making job assignments. This means that
more skilled population units will have first shot at
the best jobs. Of course, "best" is relative to the
location of the population unit, which must bear travel
time and money costs to get to work, so that the best
job for a population unit need not be the one paying
the highest gross wage. Education level is a function
for past expenditures of time in education, so that
there is a return to the investment in education in
terms of added income due to improved competitive posi-
tion (place in the education list).
212
-------
The River Basin Model is socially static, however.
First, no population unit can move out of the educa-
tional level range of its class, either upwards through
additional education, or downwards by foregoing educa-
tion. These ranges are mutually exclusive by class.
Second, jobs are specific to population unit class
such that no population unit can take a job open to
a higher class? however, "underemployment" can occur
when a population unit cannot find an acceptable
job in its class and preempts one of a lower type.
Salaries are commensurate with job class. These effects
considerably dampen the return on the investment in
education. One wonders whether building such restrictions
on population unit class into the model is desirable. Be-
ginning with a particular distribution of income over
the population, the model should be capable of producing
a radically different distribution over time due to
social mobility. If there is to be class bias in pre-
ference for education, this should be accomplished by
entering education time in the dissatisfaction index
with a weight which is a function of class. Beyond
this, educational attainment should only be restricted
by financial capabilities to invest. The wealthy would
213
-------
still tend to get more education, "but social mobility
v;ould not be precluded, and investment in education
v/ould be more attractive.
The actual procedures employed by El IP in tracing
transportation routes and making assignments and by
TRTRC and its entries from BMP in providing trip in-
formation and updating the system are too complex to
be treated *here in detail. In brief, several passes
are made through the education-ordered list. First,
v/orkers who cannot find a better job within their
transportation range are reassigned to their old job,
a bias tov/ard keeping that job being built into the
procedure. Only then does competition for jobs begin
among those workers which might like to change, but a
change is made only if a better job still remains un-
taken by the time the worker is considered on the
education list. V/hat will be of major interest here is
not this process of competition but rather the way in
which jobs are evaluated. It is this evaluation pro-
cedure that is tied to time-use preferences and alloca-
tions through the dollar value of time in travel
parameter, and the travel time derived here is a major
component in dissatisfaction and a preemptor of time
for leisure purposes.
214
-------
Full-time employment subprogram EI.IF is an "op-
timizer" in that it assigns the population unit's
workers to the best job it can* "best" being defined
as that job which yields maximal net income after
deducting travel costs and a valuation for travel time,
subject to a constraint on the travel cost/income ra-
tio. Full-time job shift length is fixed at 80 units
for all full-time jobs, wherever the location and
whatever the employer, so that work time per se cannot
be traded off against other variables. However, any
job location might be reached by alternative trans-
portation route and mode combinations with different
travel times and costs. Therefore, time and cost can
be traded against each other not only by varying the
destination of the trip (and thus the gross income
earned) but also by alternating transportation route
and mode combinations. All these factors must be
taken into account in the job selection, and it is the
dollar value of time parameter which influences the
direction of time-money tradeoffs in this selection.
Again, in theory this parameter should also reflect the
relative scarcities of time and money in general; that
is, it would be inconsistent to save travel time by
215
-------
bearing higher costs when all leisure time is not
allocated to activities, nor would it be consistent
to reduce costs by taking slower routes or modes
if there was no binding constraint on spending. With
no money budget constraint, rational choice would
always require setting the parameter at an infinite
level to obtain as fast a mode as possible. Therefore,
the money budget should be added to make the employ-
ment assignment procedure consistent with other popula-
tion unit behavior.
For the population unit of class K residing on
parcel I, then, the full-time job selection problem
can be viewed as the choice of a job type K° from among
A n
those types K <_ K at location J° among all job loca-
tions J, and a trip type L° among those types L be-
tween I and J? such that (K°, J°, L°) maximizes
\
SALFACU, K, J)* SAL(K, J) - CK(I, J, L)
subject to
AC(I, J°, L°) - :/DCCOST(K)* SAL(K°, J°) < 0
A
where SAL(X, J) is the salary offered by the em-
ploye^ at J to workers of population
class K or above, where K >. X; K° is
the class o£ job ultimately chosen,
so that if K° < K the population unit
is underemployed
216
-------
SALFAC(I, K, J) is 1.1 if theApopulation
unit worked at J in job K last round,
and 1.0 otherwise; this scale factor
"biases choice toward, the previous job
held
CK(I, J, L) = AC(I, J, L)
+ VALTIM (I)*AT(I, J, L)
is the shadov; transportation cost be-
tween I and J via trip type L, AC being
the dollar cost of the trip and AT
its tine, and VALTIM being the dollar
value the population unit attaches to
time in travel
MXCOST(K) is the greatest proportion of in-
cone which any population unit of class
K is deemed willing to pay for trans-
portation
The expression maximized, then, is the net shadow
income or salary, which is the actual salary paid less
actual transportation costs and the shadow costs of
travel time.1 Since salaries increase with job type,
K° will be the highest paying job type with unfilled
positions at employer location J°, but it need not equal
K nor be the highest type for which jobs are available
somewhere. Trip type L° will minimize total shadow
trip costs CK(I, J, L) between I and J, but these need
1At some places in the manuals, it is_implied that the
shadow costs of travel time are not included,
appears to include them, however.
217
-------
not be the lowest in the system. Likewise, of course,
SAL(I, K°, J°) need not be the greatest available;
it is the net shadow income which is greatest at J°
because of the salary/travel balance there, relative
to I. Finally, note that there is a constraint to
the problem, whereby at the optimum job location the
real transportation costs cannot exceed a certain pro-
portion of the salary earned. This is a somewhat
arbitrary constraint, and it is necessary only in the
absence of an explicit money budget constraint on
consumer behavior. Its purpose is to indicate that
workers balk at excessive travel costs and would even
refuse to work if no job could offer high enough a
salary relative to travel costs. Again, it would be
preferable for such behavioral tendencies to result
directly from dissatisfaction minimizing behavior
without the insertion of rigid conditions like this
one. YJith a money budget, expenditure items in the
dissatisfaction index, and a dollar value of time
parameter related to leisure time allocations this
could be achieved.
There are simple ways to handle unemployment.
For example, a dummy job location could be introduced
218
-------
with little or no travel time or costs and a salary
equal to the unemployment compensation in the juris-
diction (or the travel could be that required to visit
a governmental office). Then the choice to become
unemployed is an explicit job choice related to other
time allocations through the dollar value of time,
rather than being a rigid determination. Note, how-
ever, that as presently defined the switch from em-
ployment to unemployment would create a direct reduction
in dissatisfaction of an amount equal to a weighted
sum .of the welfare compensation and the local tax rates.
See discussion of the Keighrobhood Index, above. It is
2
expected that this effect would have to be modified.
Shadow costs of travel time are not actually paid,
of course, only real costs are. For any location J,
selection of trip type L° for that location is such
that (ignoring the KXCOST constraint)
CK(I, J, L°) - CK(I, J, L) < 0
for any other trip type L / L°. This means that
2However, the effect on involuntary time could create
an increment to dissatisfaction. See the discussion
below.
219
-------
< VALTIM if L° is faster than
L(AAT > 0)
AAC <
AAT ) > VALTIIvl if L° is slower than
L&AT < 0)
where AAC and AAT are the changes in travel costs
and time in switching from L° to L ^ L°, i.e., those
for L less those for L°. Since VALTIM > 0, L° can
be slov/er than L only if its is much cheaper (AAC > 0),
just how much depending on VALTIM, in relative terms.
If L° is faster than L and also cheaper, then the value
of VALTIM had no effect in the selection of L° over L,
but if L° is faster but more expensive (AAC < 0), then
VALTIM indicates just how much faster (in relative
terms) it must be.
The relationships above apply only if AC < MXCOST*
SAL for both L° and L jf L°. The trip selection prob-
lem can be illustrated graphically by letting the
axes represent travel cost and time and then locating
the trip types as points in the interior. An iso-
shadow cost contour would be a line with slope
•VALTIM; to minimize that cost is to select the trip
type for which the intersecting contour is the closest
to the origin. However, the constrained optimal trip
220
-------
type must also be located within the cost constraint,
represented by a horizontal line at AC = KXCOST*SAL.
The optimal trip will be a boundary point (those points
connected by dotted lines below). Thus in the figure
below, L would be
Figure 1
wcosr
2
the unconstrained optimum, L is the constrained opti-
2
rnum, and relations between L and some other feasible
point I? would be as described above • A lower (absolute)
value to VALTII-! would mean a flatter cost curve and
possible switch to L over L . From round to round in
the model times and costs change, and a change in VALTIi-i
221
-------
might not produce the expected change in trip selection.
These effects refer to a given location Jj changes in
these parameters can cause assignment to a new job
location as well as a change in trip type and net in-
come.
Assuming that the optimal type of trip L would "be
chosen between I and J, for any location J* and that
the best available job type K would be taken at any
given J, then at the optimal location J°
ACK - SAL(K°. J°)ASALFAC
* - SALFAC(t J)
in -comparison with any other (job) location J, J ^ J »
where A SALFAC - 0.1 if J° is the same job location as
the previous round, + 0.1 if J is that same job loca-
tion, and 0.0 otherwise. The A.'s are changes in
switching from J° to J, that is, the argument for J
less that for J°. This means, of course, that the
change in salary in moving from the optimal location
is less than the change in total shadow travel costs
adjusted for changes in the salary factor. If J is the
old job location a greater salary difference is required
before a switch in locations is made (i.e., this in-
equality ceases to hold) than if neither is the old job
222
-------
location. Conversely, if J is the old. job location,
then an even smaller salary difference is required.
The weight SALFAC was introduced to the assign-
ment procedure to dampen the tendency to change jobs
every time salaries changed or travel times and costs
changed due to changes in congestion levels at various
places in the transportation network. Actual salary
received ignores this scale factor, of course. How-
ever, a real-world interpretation that could be made
would be to consider this as representing the costs of
changing jobs, such as lost time at work, expenses in
making new arrangements, etc.. Under this interpreta-
tion it would be appropriate to set SALFAC at unity for
the old job and less than unity for a new job. Fur-
ther, a real charge might be made against the salary
received to extract those costs. (Similar costs might
also be charged against rent differentials in migra-
tion.) The operation of EMP would remain the same,
of course.
In its structure the subprogram EMP appears to be
entirely consistent with dissatisfaction-motivated be-
havior for population units, provided that the valuation
of time in travel is a true representation of the re-
223
-------
lative values (in terns of dissatisfaction) of time
and money in other activities as well for the popula-
tion unit, and subject to revision of the MXCOST con-
straint. The sensitivity of job assignment to VALTBi
depends on the nature of the transportation and salary
alternatives, of course. Consider a set of jobs
offering about the same salary. If the time-cost
ratios for the "best" trip type to each are fairly
constant, then the jobs will cluster along a line v/ith
positive slope in travel tine and cost space (Figure
2.a.). Changes in the slope of the shadow cost contour
(by changing YALTIM) would have to be drastic to cause
a switch in jobs. On the other hand, if the travel
time-cost ratios vary considerably (Figure 2.b.), then
a slight change in VALTIM could bring about a change
in jobs. Such factors as congestion can quickly alter
these ratios. With different salaries offered, the
cases become nore complicated, of course.
In passing it should be pointed out that the im-
pacts of job assignment on the transportation system
are lagged. That is, traffic flows are not altered
during the job assignment process, so that travel times
and costs are fixed for its duration. After assignment,
224
-------
Fi R
s 2
^v
AT
a. similar time-
cost ratios
b. disparate time-
cost ratios
however, TRTRC computes new flow levels and congestion
factors through CONGES, which it calls. An iterative
procedure is used to re-evaluate journey-to-work trip
patterns, reassigning population units to revised least
shadow cost trip types if necessary in the fact of con-
gestion, but making no changes in the job assignments.
Actual travel times and costs assessed the population
units are those which result from these iterations, and
225
-------
they could differ substantially from those on which
the job assignments were based. Since work time it-
self is fixed at 80 units, these travel times and
costs are the first expenditures to be determined for
the population unit in the round. It should be men-
tioned that consistent treatment would add 80 units
to the stock of leisure for unemployed population units.
However, since the population unit can only allocate
100 units, this 80 would necessarily add to involuntary
time as computed in TMALC. The resulting dissatis-
faction would mitigate the welfare/tax effect mentioned
above. These matters need to be rationalized.
As it has been pointed out, the population unit
factors most directly affecting its full-time job
assignment are the dollar value of time in travel and
competitiveness in terms of skill (education) level,
and only the former can be set independently the same
round. Estimating returns to education will be con-
sidered later in connection with subprogram TMALC.
Education's effect, of course, is to expand the number
of job opportunities open to the population unit by
giving the population unit a higher priority in the
assignment list. Before further discussion of the
226
-------
dollar value of time in travel parameter is undertaken,
we will examine the assignment procedure for part-
time employment (moonlighting). As a completely dis-
cretionary activity, its time allocation should be
directly related to relative time/money values and
thus to the dollar value of time. Therefore, it will
be important that these arguments be consistent here
and also in relation to travel behavior for full-time
employment. Before part-time job assignment is begun,
auto and income taxes and welfare payments are computed
through EMPTAX. These will be updated after part-time
work determinations are made.
Part-time employment is an activity for which
population units make an explicit time allocational
request, and assignment subprogram SKPRT will not
assign more time to this activity than the population
unit requests. SLIPRT is similar to EMP in many ways,
as it assigns population units to jobs on the basis
3
of maximal net (shadow) income after transportation,
Again, the manuals and the source listing seem to dis-
agree on whether or not the income to be maximized
deducts the shadov; cost of travel time.
227
-------
where transportation routes (but not modes, as auto
trips are mandatory here) are selected on the basis
of least cost when time valuation is included. Again,
transportation costs are not allowed to exceed a cer-
tain percentage of the salary obtained. Here there
is no reassignment of travel routes due to congestion.
There are several significant differences be-
tween EI'.IPRT and EWPf however. First, there is the
limit that time expenditures cannot exceed the alloca-
tion requested. Second, and more fundamental, there
is the method of job and time assignment itself. Here,
requests for part -time work are considered in incre-
ments of ten time units (or the remainder of the allo-
cation, if less than ten). As in EMP, best educated
population units are handled first, and the increments
of time for each population unit are assigned separately
to jobs, each increment being treated as if it were a
separate job request. A population unit, then, may
have more than one part-time employer and make several
part-time job work trips. If assigned in succession,
increments for a population unit would go to the same
job as long as the demand for labor there were unmet.
However, the order of assignments here is to attempt
228
-------
to assign single increments for all population units
in the list requesting them "before assigning additional
increments for any.one population unit, running through
the education-ordered list as many times as is necessary
until no more assignments can be made. This procedure
greatly increases the chances that different increments
for a population unit will go to different jobs. This
procedure has the effect of reducing the differences
in competitiveness for jobs which would be in effect
if request were considered in their entirety instead
of incrementally. The result will be less disparity
in incomes earned and travel costs borne. This may
be admirable in a normative sense, but it does not
seem very realistic.
In each job assignment for the population unit
the role of the dollar value of time is the same as
in BMP, permitting the population unit to "bias assign-
ments one way or the other in terms of relative travel
time and cost. Separate trip times and costs are summed
to give totals for part-time work. As with full-time
work, no allocations are made for travel time. This
time may be allowed for by not allocating all 100 time
budget units, but this is not necessary since no dis-
utility arises from allocation-expenditure disparities
229
-------
-
as long as all time is expended. There is the equiva-
lence of 80 units of part-time work and one full-time
job (which takes 80 time units). Consequently, network
travel times and costs for full-time jobs are scaled
by one-tenth for each ten unit time increment assigned
to a part-time job. Comparable adjustment is made for
salaries, which are then summed to give part-time income.
The implication, then, is that there is a given number
of round trips to be made per unit of time worked, in
full- or part-time jobs, each trip having the same time
jobs, each trip having the same time and cost (specific
to route and mode, of course). Note, however, that
part-time assignments are made after the TRTRC (CONGES)
system updating for congestion, so that actual times and
Again, the manuals and source listings disagree on this
point. In any case, since pre-emptions for travel tend
to come largely from recreation where there is no dis-
parity -oenalty, the net effect may be the same (see
~
^Although salaries bear this proportionality relationship,
employers can make independent requests for full-time
and part-time labor.
6These implicit frequency relationships are of importance
in considering relative times and costs of travel in
other activities.
230
-------
costs by link may be different from those in full-
time assignment.
If there are not enough acceptable jobs (in terms
of the travel cost/salary rule, which perhaps should
be dropped with the addition of a money budget constraint,
etc.), some of the population unit's allocation of time
to part-time work will go unexpended. As with all ac-
tivity time allocations, such a residual increases the
chance that there will be slack time remaining and
dissatisfaction due to involuntary time. Part-time jobs
are strictly limited to the appropriate population unit
class; no underemployment is allowed. Competitiveness
(education level) is thus especially important here, for
dissatisfaction can be direct as well as indirect
(through the loss of income? there is no involuntary
time component associated with full-time work at pre-
sent) . EMPTAX is? called to compute appropriate incre-
ments to the income and automobile taxes, as in EivlF.
EIvIFRT returns TII-2(1, 2} as the amount of part-t:me v:orh
obtained. The total journey-to-work travel time from
EIvIP and SKFRT is TIKE(6, 1). These times will be used
in further computations in TI-'iALC. Phoney items (incomes,
travel costs, and taxes) will be reviewed in the next
231
-------
chapter on the money budget in connection with subpro-
gram WRYOU.
Consistency between job assignment and leisure time
allocation hinges on the determination of the dollar
value of the time spent in travel. In this discussion
of the model, we have not been careful to distinguish
between a general dollar value of time in leisure and
the dollar value of time in travel, but in theory they
can be different. The best way to illustrate this and
to show the relation between optimal work/travel con-
ditions and leisure time allocations is to set up a
simple Simulataneous solution, utility maximizing prob-
lem similar to the ones presented in the first part of
this report but oriented toward the type assignments
or choices considered above.? This will be a "fictitious"
problem, in that location and route/mode variables are
treated as continuous variables evaluated with perfect
knowledge. The appropriate problem is to
maximize U = 1A(x, t,, tr)
A more complete example for the River Basin Model is
set out in a later chapter, with explicit leisure ac-
tivities and activity costs.
232
-------
subject to
80w, + t<,w^ - 803, - t~99 - x > 0
^ L * *
(money budget)
100 - 80*., - \ 0
1 2 2 * -1 (time budget)
t - 80*, - t-« > 0
r -L z ^ (travel time)
where x is consumption expenditure (income after
transportation when the money budget
constraint is binding) ,
t is time in (non- travel) leisure,
t is total travel time ( journey-to-work,
r full and part-time)
w. , a. and 8. are wages, travel times and
ill
travel costs per unit of time spent in
work of time i, i = 1 for full time and
i = 2 for part-time,
t? is the amount of part-time work time
chosen.
As in the River Basin Model there is no (dis-)
utility to work per se (which is 60 units for full time
jobs) but there is for travel to work. First order
conditions for x, t_ and t are
1 r
- x
•>
< 0 if x = 0
233
-------
at
- 9 = if *
.
1 < 0 if « 0
- 82) -
< 0 if t2 = 0
where X is the shadow price on money (income), 6 is
the shadow price on non-work leisure time (stock of
100 units), and ^ is the shadow price on travel time-
These three conditions would hold for optimal alloca-
tion of x, 1;^ and t for any given choice of locations
for full and part-time work. The third condition can
be written as
T ^
which means that at the optimum only if the dollar
value of time in leisure
£ . W2 " B2
\ 1 + <*2
will there be a positive amount of time allocated to
part-time work when there is disutility to travel (see
below) . The term on the right is a net wage rate
(after travel) per unit of time (in work and travel)
234
-------
for part-time work. Remember that (9/x) is (minus)
the marginal rate of substitution of leisure for con-
sumption expenditures*
Finally, if there is part-time work, then the dollar
value of time in travel ( VA) is proportional to the
difference between that net wage for the part-time job
and the dollar value of time in leisure*
^ itself turns out to be the negative of the marginal
utility or travel, since the first order condition for
tr is
= 0 if t > 0
r ~
< 0 if t =0
r
where, if ^ ^s "^° ^e positive, there must be dis-
utility to travel. In the River Basin Model there is
just such dis-utility (dissatisfaction), although the
linear function and the absence of utility for consump-
tion expenditures and of the money budget constraint
235
-------
in that model undermines the consistency desired
here* as it has been pointed out several times. The
dollar value of time in travel, then, is just (minus)
the marginal rate of substitution for travel for
money. If travel yields disutility at the optimum,
then this is in effect complementarity rather than
substitution.
These derivations give the origin of the dollar
value of time in travel and its relation to leisure
versus income tradeoffs. It remains to be shown how
that parameter figures into job selection and choice
of trip type. This can be done here only in a sugges-
tive fashion. Let 1. be a continuous "location"
variable for work of type i (i = 1 for full-time work
and i = 2 for part-time), and let m^ be a similar
variable for choice of trip type (mode/route combi-
nation) . V/age w. is a function of location variable
1., and travel times and costs
-------
The first says (after dividing through by X) that
at the optimal job (location) the change in the net
wage rate after travel costs per unit work time are
deducted should equal the change in travel time
valued at the sum of the dollar values of time in
leisure and in travel. Similarly, the second con-
dition says that at the optimum a change in the travel
cost with a marginal change in the trip type (as a
fictional continuous variable) should equal the nega-
tive of the>marginal change in travel time valued
associated at the sum of the two dollar evaluations
o
of time. When t2 > 0 one of these evaluations may be
expressed in terms of the other, as above. Finally,
the two conditions for work type i can be combined to
Q
Presumably the partials of travel cost and time with
trip type are of opposite sign at the optimum; one
cannot be reduced there without increasing the other.
237
-------
show that at the optimum the ratios of dollar changes
and time changes for location (job) and trip type
should be equal:
aw. a*i
Obviously, the assignment procedures in EMP and
EMPRT produce results which can at best only approxi-
mate a perfect knowledge, continuous variable solution
such as this. The point of this discussion, however,
is that the best strategy for the population unit is
to use its best estimates of travel times and costs
and wage rates — as perhaps those of the previous
round modified for trends or anticipated changes —
in jointly determining the times to be allocated to
part-time work and other leisure activities and the
value to be given to VALTIM to produce consistent
trading off of travel time and costs and job salaries.
Prom first-order conditions such as those above (or
the more detailed ones in a chapter to follow) rules of
thumb for simulation of this decision-making by popu-
238
-------
lation units can be developed. In passing, we have
discovered that a proper interpretation of VALTIM
attributes to it both a valuation of time in leisure
as an opportunity cost and a valuation of time in
travel per se as a dollar evaluation of the dissatis-
Q
faction arising there.
Adult Education
The subprogram which processes population unit
requests for time in public adult education is NSPACK
(night school). It does not make final time assign-
ments (those are done in TIvIALC) as EMPRT does; instead,
9
Coimoare
MC ^
with - MT — VALTIM
^
for trip choice, and for job choice, compare
with
ASAL - AAC ^ VALTIM* AAT (assuming SALFAC = 1
and ASALFAC =0).
239
-------
it modifies time request on the basis of night school
capacity in the population unit's political juris-
diction. The ultimate time expenditure comes from a
further modification in TMLC on the basis of time
availability in the population unit's time budget.
There is no spatial assignment per sef as there is
assumed to be only one facility per jurisdiction and
that is the sole opportunity for the population of the
jurisdiction. Locational factors come up only in lo-
cation of the facility (political pressure by users)
and migration (population unit location relative to
opportunities as "pull" factors). Private education
is assumed to be available in sufficient capacity so
that no modification in requests is warranted prior to
TMALC's treatment.
It is assumed that the highest class population
units will not allocate time to public adult education
and the lowest will not allocate time to private adult
education (the former type is free and the latter
available only at a cost; all population units support
public education through taxes). These presumptions
are built into HSPACK as rules, and they do have some
240
-------
descriptive validity. Once again, however, it is
argued that such effects should be demonstrable out-
comes of the choice model — that is, differences in
investment in education by class should come from
differing preferences (which could reflect different
subjective discount rates) and/or differing financial
capability relative to cost-return differences by
10
type of education — without the necessity of ruling
out any particular type of choice by fiat. One modi-
fication which may help in this regard is to assign
a greater return to private education; at present both
have identical effects on skill level (see TMALC).
As with most types of facilities in the River
Basin Model, educational facility capacity is a function
of the level of employment, and employment is constrained
by the number of jobs offered, in connection with the
size of the physical plant and its condition, public
finances and policy, etc. Specifically, the capacity
1 Time budget analysis is directly applicable to analysis
of the returns to education. Gary Becker, among others,
has made a start in this direction. See his, Human
Capital (New York: Columbia University Press),,
241
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for public adult education in a jurisdiction is
3
NCAP = 5*(£ K* OBTN(K, 2))
where OBTN(K, 2) is the number of part-time employees
of population unit class K in the facility for the
jurisdiction. The scaling factor for adjusting time
allocations of population units in the jurisdiction is
RATIO = MINI (1.0, NCAP/ILD)
where ILD is the sum of allocations (requests)
TIMB(If 2, 1) in the jurisdiction. Then for a popu-
lation unit on parcel I in the jurisdiction the ten-
tative time expenditure in public adult education is
the scaled allocation
TIME (I, 2, 2) = RATIO*TIME(I, 2, 1)
This expenditure is subject to further modification
for time availability in TMALC. The sum of these
assignments
USE = V TIME(I, 2, 2) = RATIO* IID
I
= MINI (ILD, NCAP)
242
-------
is the minimum of the capacity and the summed re-
quests and is stored as the total demand on the fa-
cility. (Apparently this figure is not corrected
later when final population unit time expenditures
are determined in TMALC.) These are very simple
supply-demand relationships where excess demand brings
about proportional reduction of all demands until
gross demand equals supply. Supply (capacity) can
only be increased through increased tax revenues and
political pressure for altered public policies affect-
ing the employment level.
In the current version of the model there is no
charge against time or money stocks for travel for
purposes of public education, if undertaken. Since
there is but one public education facility location
in each jurisdiction, the computation of such charges
would be quite simple, and they should be included.
As with work, these charges should be determined on
a per trip basis and scaled to be proportional to
education time received (in terms of the final time
expenditure to be derived in TMALC). And as in that
case, the scaling factor will include implicitly a
parameter giving the length of an educational episode
243
-------
each occurrence (trip) so that frequency is really
the underlying choice variable. Mode and/or route
selection could be of the type considered for work,
reflecting the parameter VALTIM. Unless dissatis-
faction of travel is thought to differ by trip purpose,
TIM£(I, 6, 1) travel time should be incremented to
include the education activity trips. Similarly,
EMPTAX should be called to compute the auto tax, if
any, to be added into the population unit financial
statement.
There is no direct satisfaction accruing to time
spent in education at the present. Its only benefit
is indirect through the increase in the skill level it
provides and thus the possible improvement in competi-
tiveness for jobs it yields. This improved competitive-
ness becomes manifest as increased net income over
successive rounds, as the returns to the investment
in education. There is a limit to this improvement
(see TKALC); moreover, it may take educational expendi-
tures just to hold even, as skills are assumed to
decline with time. Again, there is a lower limit to
this decline. Since estimating the future income
and its present value is not easy to do in this model,
244
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further discussion of the feasibility of adult educa-
tion will "be postponed until a full-blown optimization
model is presented in a subsequent chapter. It can
be expected that for education to be undertaken the
marginal returns in (the present value of a future
stream of) income to education must be positive and
equal the marginal costs incurred, including the
opportunity cost of leisure foregone, direct costs
in taxes (if variable), and the dissatisfaction of
the associated travel. These, of course, involve the
shadow prices (and dollar values) of time in leisure
and travel, and consistent rules for simulation in
time allocation to education can be developed as part
of the general population unit choice strategy.
Charges Against the Time Budget
Subprogram TMALC checks part-time work, travel
(to work) and public education time assignments against
the stock of time available. It also determines ex-
penditures for illness, private education, political
activity, and recreation, again with an eye to time
availability. The education time expenditures are used
245
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to update the skill level of the population unit and
the political time expenditures to update voter regis-
tration, both done in TMALC. TMALC also determines
park use in recreation activity, and recreation time
becomes a factor in consumer expenditures (see OPCM).
Some of these time arguments, and the involuntary time
also computed in TMALC, are those used in HSDSSW in
determining the Time Index measuring the dissatisfaction
resulting from the outcome of time allocation/expendi-
ture assignments. These manipulations will be discussed
in turn below. It should be pointed out here that there
are omissions in time expenditures which should be
corrected. Such omissions include time for shopping
and for travel for shopping, political activity, public
and private adult education and recreation. Including
these would involve some changes in TMALC, and these
will be introduced here and in the next chapter.
The first computation made for the population unit
on a parcel by TMALC is that for time spent in illness.
This is simply one-tenth of the Health Index, HLIN,
for the parcel of the population unit. Under extremely
adverse circumstances this could take up a sizeable
portion of the leisure time budget (see HSDSSW). It
is indicative that illness time is to be charged against
246
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the stock of leisure (100 units) only, not against
full-time work. Just as work time under unemployment
should be credited to leisure time (see BMP), so should
illness as a "leisure" activity have an impact on
full-time work behavior; otherwise, the individual
bears most of the costs of illness and institutions
employing them bear little. A simple formula would
be to reduce full-time work (and thus the salary earned
and the travel time and costs incurred) by four-ninths
of the illness time, and to charge the remaining five-
ninths against the leisure stock, these being in the
proportion of full-time work time to total leisure time.
direct
Finally, as will be seen in WRYOU, there is no^money cost
to illness for the population unit. Charges proportional
to time could be included easily enough, as the costs of
medical attention (c$. HLTCST).
The first time budget accounting made by TMALC is
the deduction of illness—
KLTVAL = HLIN/10
Illness does cut into time available for part-time work,
hence labor supply, and for private education and re-
creation, hence demand for services. Lost income in
part-time work likewise reduces consumption expenditures
(consumer demand) when a money budget is included. If
work times are modified for illness, establishment em-
ployment and capacity will need revision.
247
-------
—then travel to full- and part-time work (TIME(6, 1)),
and time obtained in part-time work (TIME(1, 2)) from
the stock of 100 time units.12 The residual is
REMTIM = 100 - HLTVAL
- TIKE{6, 1)
- TIKE(1, 2)
Only if this remaining time is positive will there
be time available to assign positive time expenditures
to any other leisure activities. It does not appear
that travel or part-time work are readjusted in the
(unlikely) event that this remainder is negative.
Theoretically, this should be done by reducing the time
in work and travel appropriately for the population
unit«s part-time job increments, starting with the one
assigned to it last, and such reductions should be
charged against the esipoyer's labor input as well as
against the worker's time and net income. Such read-
justments may become tedious and be deemed not worth
the effort required, however. Again, part-time work
obtained in EI2PRT for the population unit will not ex-
ceed the amount requested:
12
Questions have been raised elsewhere about the appro-
priateness of a uniform stock of 100 units regardless
of employment or labor force participation status.
248
-------
TIMEd. 2) < TIMEdt 1)
The strict inequality holds if not enough acceptable
part-time jobs were available for the population unit.
The next accounting handled by TIvIALC involves time
in public education. NSPACK may have modified the
population unit allocation for this activity for capa-
city limitations, and TMALC checks on leisure time
availability, setting
PUBED = MINO(REMTIM, TIKE(2, 2))
TI:S(2, 2) = PUBED
so that the new public education time expenditure is
the minimum of the remaining stock of leisure time and
the capacity-modified request. This is the general
method to be used in checking time expenditures against
13
the remaining stock of leisure time. J The next time
allocation to be considered is that for private adult
education, remembering that the model will not allow
the highest class groups to choose public adult educa-
tion. First, the residual leisure stock is decremented
for Tiublic education as
TO
JIf REMTIivl should ever become non-positive, the model
will treat all subsequent time expenditures as being
of zero level.
249
-------
REMTIM = REMTIM - PUBED
and private educational time expenditure is determined
as the minimum of this residual and the original allo-
cation!
PRVED * MINO(REMTIM, TIME(3» D)
TIMEO, 2) = PRVSD
There is no capacity constraint on private education,
which is provided at a cost to the population unit
(see WRYOU). As with public education, there have
been no travel costs or times computed for private
adult education. Unlike public education, however,
private education seems to be a fictitious industry:
there is no facility for it in the model and therefore
no travel can be determined. This omission is not
crucial, but it does leave a gap in the system. In
any case, travel should be computed for public educa-
tion, as suggested previously. If the time budget
limits educational expenditures, then travel for edu-
cation would have to be scaled down together with public
education time. These times in education are used to
update the educational (skill) level of the population
unit (below).
250
-------
The last two time activity expenditures to be
computed are those for political activity and recrea-
tion. Neither expenditure subjects the original
allocation to any restriction other than time avail-
ability. Thus the remaining stock is calculated and
the expenditures set in each casei
REMTIM = REMTIM - PRVED
POL = IvIINO( REMTIM, TIME(4, 1))
TIME(^, 2) = POL
REMTIM = REMTIM - POL
REGR = MI NO (REMTIM, TIME(5> D)
TIME(5, 2) = RECR
The subprogram then computes the final stock residual
not expended and stores it as "involuntary timei"
REMTIM » REMTIM - RECR
TIME(6, 2) = REMTIM
It is this involuntary time which creates direct dis-
satisfaction in the Time Index. Note that if in-
voluntary time is positive, then time allocational
requests were met for at least private adult education,
political activity, and recreation; this is,
TIME(K, 2) m TIME(K, 1)
for K » 3, fc, 5. Because they involve external con-
straints, time expenditures for part-time work and
251
-------
public education may be less than the times allocated
to them even if there is slack leisure time.
The discussion of HSDSSW in the previous chapter
explained how involuntary time could be minimized by
leaving no time for illness or travel and allocating
all time not desired in education, part-time work or
politics to recreation. (Some confusions on the de-
finition of involuntary time were also brought up there.)
This type of allocation procedure is almost certain to
produce a restriction on recreation time, but since
that would yield no positive involuntary time and since
there is no penalty to allocation-expenditure dis-
crepancy for recreation, the results are desirable.
One qualification is that with a money budget constraint
introduced to the problem the population unit may not
be able to afford so much recreation (see OPCM, next
chapter) if it must constrain its allocations in terms
of money as well as time. The simplest solution lies
in the inclusion of a dissavings component in both the
money budget and the dissatisfaction index. Then the
money budget will always be binding by definition, and
allocations can proceed as at present relative only to
a fixed time stock, with the knowledge that there will
252
-------
be a loss of satisfaction if dissaving is positive.14
With a money budget the time subprogram TMALC and the
money bookkeeping subprogram V/RYOU would have to be
integrated in the event that there were an absolute
limit to dissavings. Then time and money assignments
would proceed jointly, ceasing if either time were
exhausted or the dissavings limit reached. In case
of the latter, involuntary time would measure the slack
(unexpended) time left. Without a limit to dissavings
the present separation of time and money accounting
could be maintained.
These issues raise a further question about the
time assignment, and that concerns the order in which
leisure activities are processed for assignment in
TMLC. To force a strict ordering such that excessive
transportation times and illness are all incurred at
the expense of recreation alone as long as it is non-
zero may be quite unrealistic. Indeed, it is likely
In simulation, the rules derived for making those time
allocations would of course reflect prices, costs,
wages, and dissatisfaction for dissaving. The point
is that money allocations would follow automatically
just as they do at present.
253
-------
that demand for recreation "becomes exceedingly in-
elastic while it is still a major component of leisure
time, so that reductions in other activities "become
increasingly probable. Perhaps the issue is one of
interpretation. If education, political and part-time
work activities are seen to involve committments
which are irrevokable for the course of a round ("the
short run")* then the current priorities are appropriate,
forcing readjustments to be made in allocations between
rounds. From a behavioral point of view, on the other
hand, it seems safe to say that few people, individually
or in the aggregate, would adhere strictly to commit -
ments which entailed great losses of satisfaction in
other areas, even in the short run. The tenuous role
of education and political activity (and even of work
when there is no money budget or dissatisfaction for
dissavings) in yielding satisfaction further suggests
that they, not recreation, would be reduced by the popu-
lation unit. A final issue which is related is that
there is something unreal about forcing the population
unit to stick to a recreational allocation in the event
that it has slack time and money. Involuntary time
really makes sense only when a money constraint on ex-
254
-------
penditures is reached. With extra money or credit
that time should go to recreation.
It is recognized that these procedures in TMALC
and other subprograms which have received criticism
here were developed with an eye to operationalism as
much as to behavioral realism. It is not suggested
that changes be made without weighing the costs of
such changes against the/presumed improved realism to
be obtained. Time allocation in a simulation model
is novel and provocative, and every effort should be
made to isolate behavioral issues in the simulation
and to resolve them as realistically as possible for
modeling purposes.
Educational and Political Status
In addition to determining time expenditures,
TKALC updates the educational and political status of
the population unit on the basis of those time expendi-
tures, and it assigns the population unit to parks for
recreational activities. The increment to the educa-
tional level of a population unit of class K as a
function of time in education is simply the product
of the total time in adult education this round times
255
-------
a productivity factor for that time. The education
level ILEV of the population unit is thus incremented
to be
ILEV = ILEV + (PUBED + PRVED)/TUE(K)
where TUS(K) is the inverse of the productivity factor
for class K. Current data are (6, 6, 8), so that the
highest class (K = 3) has the lowest productivity fac-
tor (1/8). Each class is restricted to a range of
educational levels, however, and these ranges do not
overlap; hence the lack of social mobility in the
model (see the discussions of NSPACK and EMP). Thus
ILEV must be limited to the appropriate rangei
ILEV = MINO(NELEV(K +1),
(ILEV - NSLEV(K))*9/10 + NELEV(K))
Here NELEV(K) is the minimum education level of
class X (and NELEV(K + 1) is its maximum level. Current
data for NELEV(K) are (0, 40, 20, 100), where K = 4,
is a "dummy" class. There is a built-in decrement to
ILEV for the passage of time. This is represented by
the use of the factor 9/10 rather than unity to scale
the difference between the lower limit and the level
adjusted for time in education. The result is that
256
-------
the new educational level is the minimum of the upper
limit for the class and the lower limit plus 9/10
of the adjusted amount above the lower limit. If no
time is allocated to education (PUBED + PRVED = 0) the
decrement in ILEV due to the passage of time is
AlLZV = - (ILEV - NSLEV(K)) < 0
Since this decrement gets (absolutely) smaller as
NELEV(K) is approached, strictly speaking it is possible
to approach NSLEV(K), but it cannot be reached if at
any time ILEV were greater than MSLEV(K). In addition,
to maintain a constant educational level (£ILEV = o),
it takes
PUBED + PRVED = (ILEV - NELEV(K) )*TUE(K)/9
units of education each round. ^ Again, this approaches
zero as ILEV approaches NELSV(K).
The cost of public education to the population
unit is not easy to compute, since there is not a one-
to-one correspondence between government revenue sources
equation produces values roughly one-half those
given in the examples on p. b8 in manual # 4. The
reason for the discrepancy is not clear.
257
-------
(taxes) and expenditures. However, a unit of private
education time costs 3000, so that to increase the
education (skill) level by one unit through private
education costs
3000*TUE(K)*(10 + ILEV - MELEV(K))/9
Since class type constricts the educational level of
the population unit, the list produced by EDORD is
automatically ordered by class as well as by education
level within classes. This means that a population
unit cannot alter its competitive positive with re-
spect to population units of a higher class. Note
that the relative values of TUB(K) imply decreasing
returns to scale on the investment in education;
however, this is of no practical consequence to pop-
ulation units, since TUB(K) is constant within the
educational range for any given population unit. The
decreasing returns phenomenon also works in reverse,
so that it takes less education to maintain a constant
level at higher levels.
Earlier in this chapter an argument was given for
removing these class limits on education level. To
retain decreasing returns to scale in that event -
indeed, to make it a real factor in the time allocation
258
-------
problem - TUE should be made an increasing function
of IL2V regardless of class. Again* it should be
pointed out that PRVED and PUBED yield the same re-
turns but at widely different costs. This effect may
need modification. One approach would be to make the
rate of return to public education a decreasing func-
tion of the demand/supply ratio but allow demand to
exceed supply (capacity), i.e., not scale down the
time allocations. Then private education could become
economically feasible in comparison with public edu-
cation when the quality of the latter declines through
over-utilization. Another approach would t>e simply
to assign excess demand for public education to private
education, so that there are direct as well as indirect
costs to the population unit for overutilized facilities.
These changes are relatively minor but could be helpful
in making education a more dynamic force in the model.
Time expended in political activity increases the
percent of the population unit's members who are re-
gistered to vote. This effect only lasts one round,
however. Moreover, the base of eligible persons is a
constant and varies by the class of the population unit.
The increment to the registration rate of the population
unit is
259
-------
IREG = P3 + PS*(POL - PT)/100
where PB is a base rate, PS is a variable rate and PT
is a base value relative to the political time ex-
penditure POL. P3 and PS are likewise functions of
POL. The -oarameter values are
POL
0 < <
10 <<
50 <<
60 <<
100
10
50
60
100
PB
0
7
10
15
15
PS
70
7
50
7
10
PT
0
10
50
60
100
There are some marked pecularities about IREG.
First, since PS/100 gives the slope of the graph of
IREG for a range of the values of POL, it is apparent
that the ranges of POL between 0 and 10 and between
50 and 60 are of special importance. There the slopes
are .7 and .5, respectively, whereas in the other
ranges the slope is but a fraction of those values,
being .0?. Second, since POL < 100, IREG for POL = 100
is only 15 (the same as for POL = 60), but between
260
-------
POL = 60 and FOL = 99 IREG rises linearly to IREG =
17«73» Presumably, the relation of the PB values to
those for POL (or PT) is meant to indicate decreasing
returns to political activity. By breaking IREG
into a series of linear segments anomalies have been
created. It would have been simpler just to use a
single non-linear function over the v/hole domain of
POL.16
The updated voter registration is computed as
VTRS = ((IREG + 100)* NREG(K))/100
where KREG(K). gives the base number of registered
voters (a minimum) for class K population units, the
data being (100, 140, 200) for K = 1, 2, 3. These
figures indicate that the minimal registration in
the highest class population urit is twice that of the
lowest and over bOfa more than that of the middle group
Even with maximal political time the lowest group can
increase its registration by only 18 voters. There
seems to be an inconsistency when the same maximal
This is true in many instances in the River Basin
Model and only adds to the complexity of the pro-
gramming.
261
-------
effort can yield such disparate results "by class;
it can hardly be argued that lower class population
units should behave against their own desires more
than upper class ones. Again, the suggestion being
offered here is that differences in voter registration
be allowed to arise from differences in time alloca-
tions rather than being built into the model. Then,
if observable differences in registration rates are
seen to stem from differences in the choice-making
environment by class, the model will still be able to
reproduce them without ruling out other possibilities.
Under rational time allocation it can be expected that
for political activity to be undertaken the marginal
returns however measured must be positive and not be
less than the marginal costs involved, if only the
dollar value of the leisure (recreation) foregone.
Measurement of such returns is difficult, of course,
and they could incorporate "preferences" plus sub-
jective probabilities that the activity will lead to
desired results. These problems will be considered
again in this report.
262
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Park Use Assignment
The last task of subprogram TivlALC is the assign-
ment of population units to parks for their recreational
activity. This assignment is of special relevance
in tine allocation, since the park use indices that
result are components in the park factor by which re-
creation time expenditures for a population unit are
weighted in the Time Index. A population unit is thoughl
to make use of only those parks located within a five-
by-five parcel grid centered on the residence of the
population unit. The effective space available at a
park is the "normalized" supply there; e.g., for park
J in the grid centering on parcel I, the supply
PRKN(J, D is
PRXN(J, 1) = PARK(J, 1) + 2*PARK(J, 2)
where PARK(J, L) gives the amount of undeveloped
(L = 1) and developed (L = 2) land in the park. Hence
developed land is twice as effective, or has twice
the capacity, as undeveloped land. The sum of the
normalized supply over all parks J in the grid gives
the total park supply available for population units
at the central location I:
263
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FIND(I) = ) PRKN(J, 1)
t~j
J
in grid
on I
Persons at I make use of park J in the grid on I in
proportion to the share of the park supply or capacity
available to I that J presents. Therefore, the number
of people at I using park J is
PEOPLE(I, J) = 500*(FRKN(J, 1)/FLND(I))
*(£ NMPU(I, K))
K
where NMPU(I, K) is the number of population units
(of size 500) of class K living at I. The total de-
mand for (use of) park J is found by summing these
assignments to J over all parcels I in the grids of
which J lies, i.e., all I in the five-by-five parcel
grid centering on Ji
PRKN(J, 2) = V PEOPLSd, J)
I
in grid
on J
Finally, a park use index is computed for each park
J as the ratio of demand for that park through these
264
-------
assignments to the capacity of the park. For park J
the park use index PKUSE(J) is
PKUSE(J) = PRKN(J, 2)/(PRKN(J, 1)* .01)
The supply figure in the denomenator is multiplied by
.01 to convert it from a percentage, in which it was
originally computed, to a proportion.^
The park use index used in HSDSSW to scale re-
creation time for a population unit on parcel I is
PKAL(I), the minimum of the park uses indices
PKUSE(J) for the parks J in the grid centered on I.
There are several shortcomings to the derivation of
these indices and their use in this fashion in the
dissatisfaction index. Perhaps the most serious is that
park use is completely independent of recreational time
expenditures, being a function only of park space and
population. Conceivably, then, a population unit in
listing of TMALC has these factors multiplied to-
gether instead of divided, as in manual # 11, p. 63.
In addition, manual # 11, pp. 52-53, specifies that
the denomenator should be multiplied by 250, whereas
this is not done in TMALC. It is definitely warranted
to properly limit PKUSE(J).
265
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an area of high population density per unit of park
space could have its recreation satisfaction scaled
down even when little use was made of the area's
parks. It seems apparent, then, that the demand
(assignment) variables FEOPLE(I, J) should be weighted
sums of recreation time expenditures, not population,
i.e.,
(PRKNU, 1)/FLND(I))*(V TIIvTE(I, K, 5» 2)*NKPU(I,K)
K
where TIMS(I, K, 5, 2) is the recreation time obtained
by population units of class K on parcel I and NMPU(I,K)
is the number of those population units. Other con-
stants of proportionality may have to be introduced to
give comparable scales for supply and demand (e.g.,
population unit size, 500).
At the heart of this procedure are the concept
of the "park11 as the locus of all recreational ac-
tivity and the restriction of assignment to an arbitrary
grid around the residence. The Washington Study shows
clearly that most leisure time is spent at home, 6J%
on weekdays and 56# on weekend days (see Table 2 below).
Less tnan half of the out-of-home leisure time is spent
on activities which utilize public or private facilities
266
-------
for recreation-type activities (the "other recreation"
category in Table 2). That the average travel time
on these recreation activities is only about ten
minutes is deceiving, for that is the average over
doers as well as non-doers and represents about 12%
1 tj
of the total time spent on the activities. To tell
whether or not an activity occurs close to home ws need
the travel time per occurrence, which unfortunately
has not yet been computed in that study. Therefore,
while we can say that "park" (or facility)-related
activities involve less than one-fifth as much time as
is spent in in-home leisure and in socializing in other
residences, we do not know whether or not those park/
facility activities take place in the neighborhood, as
is assumed in the River Basin Model*
There is some question, then, as to the appro-
priateness of taking "park" use as a proxy for all
social/recreational leisure behavior. This point will
not be belabored further here, except to make two
18In the Washington Study, the average travel time to
movies was about forty minutes for participants,
indicating a fair distance traveled.
267
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TABLE 2: Relative Leisure Time Expenditures in
the Washington, D.C. Metropolitan Area
Proportion of Leisure
Time* Spent in ... Weekdays Weekends
the residence .63 »5^
out of the residence:** .37 «44-
... in socializing at
other residences .08 .09
... in education and
training .04 .01
... in part-time work .02 .02
... in travel to full-time
work .10 .01
... in political, civic,
organizational activities .01 .01
... in religious activities .01 .10
... in other recreation .11 .20
Total: 1.00 1.00
* Categories have been grouped to be consistent with
River Basin. Mo del definitions.
** Travel times are included and the proportions ob-
viously are calculated over non-participants as well
as participants in activities, as for a population
unit.
SOURCE* Hammer and Chapin, op. cit.« pp. 50
268
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points. The first is that it is always possible to
treat residential land as a kind of "park" in which
visiting occurs. The capacity of such land for re-
creation could be a function of the number of resi-
dential space units provided there, which in turn
depends on the mix of housing types and their levels
of development. Capacity could be further specified
by population unit class if assignments were made
class-specific to represent within-class socializing.
unit
Given the number of people a spaceAcould serve in re-
creation (visiting), the capacities per % of parcel in
various residential uses could be computed for compari-
son with such figures for parks (see manual # 11,
p. 52). The second point to be made here is that if
use of developed park space is thought of as including
the patronizing of private facilities and establish-
ments, a suitable method of pricing such use must be
developed. It is true that there is a component to
consumption expenditures in OPCH which is proportional
to recreation time (see the next chapter). However,
the prices charged depend on assignments to facilities
that are part of the economic sector and have nothing
to do with parks or their locations.
269
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This brings us to the park use assignment pro-
cedure itself. In comparison to those procedures used
in the model for employment and shopping (see OPCM,
next chapter), park assignment is extremely crude.
Its greatest inadequacy is that assignment of popula-
tion units at I to a park J in its grid reflects
the supply of park land at J but not the demand for
that land. Since demand is unknown until assignments
are made, it would seem that an iterative procedure
is called for (as with shopping) unless some priority
for ordering population units is established (as with
employment).
A second problem is that assignments are made with-
out regard to location; that is, there is no attenua-
tion in assignment for the distances which would have
to be traveled. A unit of park capacity is assumed
to be just as attractive on the edge of the grid as
it is on the adjacent parcel. The procedure may be
realistic in its apportionment of demand to many oppor-
tunities rather than assignment to a sincle one (cf.
OPCM), and there are gravity models and iterative
balancing procedures which would serve this purpose
well, allowing attenuation with distance and feedback
270
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effects of crowding (demand/supply relationships
at "destinations," or parks) on individual demands
(at "origins " or residences). Because such changes
mean substantial modification of the assignment pro-
cedure, no further attempt will be made to spell out
how they should be accomplished. Instead, we will
turn to some more straightforward issues and suggested
modifications.
It was suggested that demands and assignments for
park use be carried out in units of recreation time
rather than population, which means a revision in the
composition (though not form) of the park use indices
computed for parks and population units. There would
be two ways to apply the revised park use indices.
The first is simply to apply them in PKAL for dissatis-
faction as is done at present. The second would be
to use them to scale down recreational time expenditures
until an acceptable use index level is reached whenever
there is excessive overcrowding. Two examples are the
treatment of public education time, which is reduced
until demand no longer exceeds supply, and the treat-
ment of residential densities in migration, where out-
migration is forced until demand exceeds supply by
271
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no more than some "acceptable" percent. In the first
case the use index would be brought to unity for the
population unit, so that the use of the index in
scaling dissatisfaction in recreation would no longer
be necessary, the dissatisfaction coming directly from
the reduction in recreation time and the possible in-
crease in involuntary time. " In the second case,
there could be all three effects with park overutili-
zation: less recreation time, more involuntary time,
and an altered scale factor for recreation time in
computing dissatisfaction.
The recreation time reduction suggested here is
not reassignment, in that there would be no redistri-
bution of the excess demands at parks or alterations
of assignments at parks having adequate capacity. To
19
If the recreation time decrements were reassigned to
T>arks having PKUSE < 1 (or .MAXUSE), and the process
were repeated until either all parks in I's grid had
use indices not greater than one (MAXUSE) or I had no
further decrements in recreation time, then we would
have a simple reassignment procedure. Any time still
unassigned at that point would become involuntary.
272
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balance supply and demand at park J where there was
overutilisation, for example, the assignment PEOPLE(I,J)
from each parcel I in the grid on J could "be reduced
by the amount
(1.0 - 1.6/PKUSE(J))*PEOPLE(I, J)
Total reduction in recreation time at I would be the
sum of any such reductions at parks J in its grid,
and involuntary time would be increased by this amount.
If a permissible level of overcrowding were specified,
say IJAXUS2, then the deflation factor would be
(1.0 - MXUSE/PKUSE(J))
Because all of these effects operate in the same
direction to increase dissatisfaction, the manner of
applying the use index may seem of minor importance.
It is true that undeveloped park space is not likely
to be regulated as to the extent of overutilization
permitted, whereas educational facilities (and even
developed park space, for that matter) might be.
However, from a behavioral viewpoint the determination
of recreation time should not be independent of park
capacity. There is another reason for adjusting re-
273
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creation time, and that has to do with the fact that
recreation time is a factor in determining consumer
expenditures for goods and services. By not reducing
recreation time for overcrowding we may force the popu-
lation unit to maintain an artificially high level of
(recreation-based) consumption expenditure.
There is one last issue to be brought up with
respect to the park use index PKAL(I) for a population
unit on parcel I, however the park use indices PKUSE(J)
for parks J are determined. With PKAL(I) being the
minimum of the PKUSE(J) in the grid on I, it need not
reflect the nature of the actual assignment of I's
recreational demand. It would be entirely possible to
have predominant assignment to one park in the grid
where there was overcrowding, while the park use index
for those population units is based on supply and
demand at some less utilized park.20 The most straight-
forward modification of PKAL(I) would be to define it
as the weighted sum
20
The disparity represented by this effect is greatly
reduced if there is reassignment, of course.
274
-------
PKAL(I) =(£ PKUSE(J)*PEOPLS(I, J))/TOTAL
J
in grid
on I
where TOTAL is the sum over J or the PEOPLE(I, J)
assignments. Following earlier suggestions this would
be equal to the total recreation time expenditures of
population units at I.
In connection with the discussion of iterative
assignment procedures the matter of transportation for
the use of parks was brought up. It seems only reason-
able that such transportation be identified and times
and costs assessed the population unit. In iterative
assignment based on travel it would be expected that
the dollar values of time in leisure and travel would
play roles similar to those in job and shopping21
assignments. Short of the development of such assign-
ment procedures for recreation, it should still be
possible to make appropriate charges for travel. Travel
time and cost to each park could be determined any of
several ways, including the prior specification of route and/
21
xAt present, they have no role in shopping, but one
is suggested in the next chapter.
275
-------
or mode (such as in part-time work) and, at the other
extreme, the use of VALTIM to pick the minimum shadow
cost trip type. To produce consistent time budget ex-
penditures the follov;ing alteration might be made in
the assignment procedure: assign gross times to parks
rather than net (non-travel) times. Everywhere in the
River Basin .Model travel times and costs to destinations
are assumed proportional to the time spent there, on
the implicit assumption of a given amount of time at
the destination for each trip. Thus, if AT(I, J) is
the travel time (however determined) to go from I to
the park at J, the total travel time is AT(I,J)*
PEOPLS(I, J)'. where PEOPLE(I, J) is the recreation time
to be assigned to park J and is now
PEOPEEd, J) =* (PRKN(J, I)/FLND(I))
*<7 TIME(I, K, 5, 2)*NWPUd* K))/
K
(.1 + AT(I, J))
Obviously, the sum of I*s travel and recreation
components for park I equal the total recreation
assignment under the non-travel procedure presently
276
-------
used (modified for time assignment rather than people).
Now, however, the park use index for a park J, when
computed by the same formula as before, will exclude
the travel component. This will reduce actual utili-
zation of parks at greater distances from population
units. Total travel costs to J from I would be
AC(I, J)*PEOPLE(I, J). The travel times and costs
should be summed and charged against the appropriate
budget for the population unit. The recreation ex-
penditure finally determined would be TIMB(I, K, 5, 2)
decremented for the travel times. If population units
attempt to optimize in allocating time, the optimal
allocation to recreation would include an estimate of
the travel component, analogous to the example pre-
sented earlier for part-time work. Consequently, the
dollar valuations for time in leisure and travel v/ould
now reflect recreation travel parameters (and ultimately
those for shopping, plus prices) as well as those for
part-time work and wages. These complications to the
optimization problem will be considered later.
277
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Summary
This rather lengthy chapter has made numerous
suggestions for modifying time and location assignment
procedures for the Social Sector of the River Basin
Kbdel. Improved correspondence with reality has been
the major objective. Admittedly, the achievement of
that objective need not produce a "better" model in
terms of its ability to simulate the whole system,
of which the Social Sector is but one part, efficiently
and at a reasonable cost. These suggestions are not
meant to imply that the present Social Sector is thought
inadequate. Quite to the contrary, the treatment of
the Social Sector is exciting and capable of producing
new insights into the interaction of households with
"the system." Indeed, it is precisely because the
implications are so provocative that so many issues
arise in this view of behavior — time use — about
which we know so little. These suggestions are offered,
then, with full realization that the need for simplicity
in such a pioneering model as this may preclude under-
taking many of the modifications involved, some of which
are rather picky attempts at consistency and comprehensive-
ness.
278
-------
On the other hand, most modifications suggested in
this report require the alteration of no more than
several statements in the computer program, and it is
felt that such changes would be well worth the effort
involved.
279
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SECTION X
MONETARY CONSUMPTION
EXPENDITURES
The operations carried out thus far for a popu-
lation unit in a round —migration, employment, and
leisure activity — provide most of the information
needed to construct a financial statement for the
population unit. The remaining activity assignment
to be made is that for shopping. On its completion
sources of incomes and expenses for the round can
be identified and the accounting procedures undertaken.
Shopping assignment and financial bookkeeping methods
will be reviewed in this chapter, and this review will
tie up our examination of the detailed workings of
the Social Sector as they pertain to time and money
allocations and expenditures.
Consumer Expenditures
Assignment of population units to shopping es-
tablishments is made for the purchase of consumption
goods and services. Before this assignment is attempted,
the consumer demand for these goods and services must
be determined. This is done in subprogram SETCOM, which
computes consumer demand for a population unit as the
sum of a base component which is fixed and a variable
280
-------
component which is a function of the recreation time
obtained by the population unit. This demand, then,
does serve to measure population unit inputs of goods
and services to "recreation" (broadly defined); again,
hov/ever, this is not made specific to the recreation
facility itself, so that payments will not accrue to
such facilities directly from population units. The
fixed component varies by population unit class and
reflects differences in their demographic composition
and, possibly, differences in input mix to recreation
by class. If recreation were dissagregated by type,
it would be desirable to relax assumptions about
differing input mixes and to let the model replicate
such differences through preference/income/cost re-
lationships.
For population units of type K on parcel I, demand
for consumption goods (T = 1) or services (T = 2) is
found in SSTCOK as
IUN = i.'iULT*(3AS + DIF*REN)
where BAS = EASCON(K, T) gives the fixed base
consumption rate per population unit
of class K for consumption of type T
281
-------
REK = RENCOK(K» T) gives the variable con-
sumption rate per unit time in recrea-
tion per population unit for consumption
of type T
DIP = TIiVIS(If K, 5, 2) is the recreation time
obtained by a population unit of class
K on parcel I
MULT « NUMPU(I, X) is the number of population
units of class K on parcel I
In the current version of the model, the data are
BASCON REHCOK
Class (K)
low (1)
middle (2)
high (3)
T « 1
21
28
34
T = 2
7
11
16
T = 1
.025
.050
.100
T = 2
.000
.050
.075
These data can be determined as
3ASCOK(K, T) = (22 - 9*T) + (14.5 - 6*T)*K
+ (-1.5 + T)*K*K
REKCON(K. T) = (.025)*((5 -
+ (1.5 - T}*K*K)
for T = 1, 2, and K = 1, 2, J. From the table it
is apparent that services are less sensitive to re-
creation time than are goods in an absolute sense,
These data are from manual # 4, p. 86. The source
listing for SETCOK gives different values for RENCOtf.
282
-------
since their values on RENCON are lower i
ARENCON(X, T) * 4* (K - i) - K*K < o
T "~
Similarly, the fixed consumption of services is lower*
A,BASCON(K, T) = - 9 - 6K + K*K < 0
However* a greater proportion of the consumption of
services is due to recreation than is true for goods
for the upper two classes (but not the lowest, which
uses no service in recreation)*
A(DIF*REKCON(K, T)/BASCON(K, T))
=(DIF/BASCOH(K, I)*BASCON(K, 2)))
(-^3 + 19-5*K + 23
- ?.0*K*K*K)
> 0 for K = 2, 3 but < 0 for K = 1
Interestingly, consumption rates for goods plus
services are simple linear functions of class. That is,
V 3ASCON(K, T) = 1? + 11*K
T
RENCON (K, T) = (.025)* (-2 + 3*K)
T
283
-------
The proportion of total consumption which is due to
recreation time can be computed from this ^s approxi-
mately 1/10, 1/5, and lA for K = 1, 2, 3, respectively,
when DIP = 100, its upper limit under the time budget.
These proportions all decrease as DIP drops below 100,
but the order by class remains. Finally, we note that
all consumption rates increase with class*
ABASCON(K, T) =(13.0 - 5*T)
K + (-3.0 + 2*T)*K > 0
T\RE!'ICON(K, T) = (.025)* (3* (T - 1)
+ (3 - 2*T)*K) > 0
However, the rates of increase vary by type. Fixed
consumption increases at an increasing rate for ser-
vices but at' a decreasing rate for goods —
< 0 for T = 1
A2BASCON(K, T) = 2*T-3
K > 0 for T = 2
— while the variable consumption rate increases at
an increasing rate for goods but at a decreasing rate
for services —
284
-------
? > 0 for T = 1
$ REKCON(K, T) = (.025)* (3 - 2*T)
K < 0 for T = 2
Since there have "been no studies for which com-
parable data on time and money expenditures for house-
holds have "been collected and analyzed, it is not
possible to make a direct evaluation of the validity
of the trends assumed in SETC033, although for all
practical purposes they seem reasonable. The only
qualification that might be made is one that has been
made several times before, that it would be preferable
to develop the model so that such trends would be out-
put rather than input* thus providing an explanation
for their origin in reality. For illustrative purposes.
Table 3 below has been set up to give total consumption
expenditures as a proportion of income under the default
tines for part-time work and recreation (manual # ^,
t>. 8*0, minimum and maximum limits on prices of consumer
goods and services (50 and 150, respectively), and
minimum and maximum limits on salaries by class, which
can be summarized as
(11 - 5*K + 6*K*K)» 100
and (50 - 25*K*(K - 1))*100
285
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to
CO
TABLE 31 Hypothetical Consumption Expenditures As
Proportions of Income
Consumption Units Expenditures as Proportion of Income
Proportion Low Price High Price
Class K Total Variable Low Income High Income Low Income High Income
1 28.50 .02 .85 «23 2.55 -61
2 40.00 .03 .62 .15 1»85 .46
3 51-75 -03 '4-3 «11 1*29 O2
-------
respectively. That only 355 of consumption stems from
recreation time in this example is disturbing, for it
means that if these default time allocations are
typical, recreation time has virtually no direct cost.
Inclusion of transportation costs for recreation will
help in this regard.
Shopping Assignments
The consumption levels determined by SSTCOK are
measured in "consumption units" for which a price per
unit is charged at retail establishments. Establish-
ments are differentiated as to the provision of goods
or services, and population units are assigned to the
different types of establishments separately. Ob-
viously, consumption unit prices may differ by type of
consumption as well as by establishment. The subpro-
gram making the shopping assignments is OPCivl. Since
the procedure is the same for goods as for services,
it will not be necessary to specify the type of con-
sumption purchase involved.
The commercial optimizer, OPCM, attempts to assign
population units to retail establishments to minimize
the "shadow" cost per unit of consumption for the popu-
287
-------
lation unit. This shadow cost is a function of trans-
portation cost, crowding at the establishment, price
charged, and preference for shopping at the same es-
tablishment as in the previous round. Since crowding
reflects interdependence in population unit assign-
ments, the procedure is iterative and assignments are
adjusted until certain conditions are satisfied. There
are no time dimensions to this assignment procedure,
even though the manuals (# 2, p. 121) claim that the
crowding component is a proxy for excessive shopping
time. Because the introduction of time elements will
be advocated below, this procedure will be examined
in more detail than was afforded the discussion of
employment assignments.
In subprogram SETCAP the effective capacity of
an establishment had been determined as
CAP = EM* BASE
where EM is the ratio of the number of workers hired
to the number required for the given level of develop-
ment, i.e., the "employment effect," and BASE is the
base capacity as a function of the type (LUS) and
level (LSV) of development and of the value ratio (VR)
288
-------
indicating maintenance and depreciation effects.
Effective capacity will be the "supply" factor in
demand/ supply relations for crowding. The first
operation performed by OPCM is calculation of the
least cost transportation route from each buyer (popu-
lation unit) to each seller (establishment). As with
part-time work, only automobile travel is allowed, but
route choice is made with transportation subprogram
TSCAN. It is unclear whether the dollar value of time
parameter VALTIM is used in this choice. There is no
reason why it should not be; indeed, it should be used,
and its value should reflect shopping travel require-
ments if time allocations are to be consistent.
The next step is the computation of the "base
perceived usage" of each establishment. These para-
meters are used by population units in their assess-
ment of shopping opportunities, rather than actual
usage levels, in order to minimize oscillations in
assignments over iterations and to force a convergence
such that few population units would choose to change
establishments on further iteration. IDMDW, the base
perceived usage of an establishment J on an iteration,
is defined as a weighted sura of S3RVE(J), its level on
-------
the previous iteration, and SERVMG(J), the actual
level of usage at the end of that iteration. The
functional relationship is
IDMDW - (SERVNG(J)*1000/IBANG
+ (1 - 1000/IBANG)*SERVE(J»
IBANG is an iteration parameter starting at a value
of 1000 on the first iteration and incremented by
whenever an iteration seems to be diverging rather than
converging on a solution. Over successive iterations
actual usage has less weight and perceived usage more.
On an iteration a population unit I is assigned
to that establishment which offers it the lowest "sha-
dow cost" of shopping per consumption unit. The shadow
cost of an establishment J is basically the money cost
per consumption unit, being the price plus transporta-
tion cost, times a function of the ratio of the per-
ceived usage to the establishment's capacity*
LSD = SHADOW*(PUG + LS)
where LSD is the shadow cost,
PCU = 100*CMWD(J, 2) with CMWD(J, 2) as
the price of a consumption unit (in
hundreds) at establishment J,
290
-------
LS = AC(I, J) is the transportation cost
of the9least-cost route via auto from
i to jr
SHADOW is a function of the ratio of perceived
usage to capacity.
The demand/supply factor which is the argument to
SHADOW is the variable RATIO:
RATIO = (IDMOW + COKSUM)/CAP
Here the population unit's own demand CONSUM is added
to perceived usage if it did not shop here the last
iteration; otherwise COKSUM is excluded from the cal-
culation.-^ The function SHADOW of RATIO is
SHADOW = Al -(- (RATIO - A2)*A3
2
It Is unclear whether IS is just AC or AC + VALTIM*AT
to include time valuation. The latter is more appro-
priate. In addition, it is implied that AC is of
appropriate scale to represent the cost of however many
trips are associated with one consumption unit's pur-
chase. However, no such scale factor is identified.
CONSUL is the variable IUN computed in SSTCOK. It is
stored in COr.iR(I, K, 3» T) where K is the class of this
population unit on I and T = 1, 2 denotes consumption
good (1) or service (2).
291
-------
where the A's are themselves step functions of RATIO.
SHADOW is something akin to a polynomial in RATIO,
increasing with RATIO at an increasing rate in much
of its range above .75* When RATIO < .75, SHADOW is
unity. Thus there is no crowding effect when RATIO < «75»
Since perceived usage of each establishment tends
to stabilize over successive iterations in the direction
of the predominant trend of actual usage, RATIO and
thus SHADOW also stabilize. Thus the chances increase
that an establishment offering minimal shadow costs LSD
to a population unit one iteration will also offer it
minimal ISO the next, tthen most population units no
longer switch establishments, iterations cease, and
the population units are charged for the costs of their
consumption purchases and transportation according to
costs and prices for the establishments to which they
v/ere assigned. It should be noted that no charge is
made for travel time, even though that variable may
have been used in least-cost route selection if evalua-
tion was made for the value of time via VALTIM. Again,
such valuation and charges against time should be made.
There is a problem of circularity, however, since total
travel costs and times are functions of consumption —
292
-------
AC*CONSUM and AT»CONSUMf respectively, — and CONSU&I
is a function of recreation. The determination of
recreation time may have exhausted the time stock,
leaving none for this travel. On the other hand, travel
time cannot be deducted from the budget prior to de-
termining recreation time. This problem will be addressed
below.
Although the iteration procedure used here appears
to be an appropriate one, it could have been tied more
meaningfully and directly to time utilization. In
particular, even though SHADOW is supposed to be a
proxy for shopping time effects (through crowding) no
determination of shopping time nor charge for it against
the time budget is made. Crowding implicitly forces a
trade-off of money (for higher prices and/or travel
costs) and thus an implicit dollar valuation of crowd-
ing (i.e., shopping time), despite the fact that VALTIM
is a variable designed to do just that while being con-
sistent with similar tradeoffs for other activities,
and despite the fact that the population unit yields
no time to shopping.
To show how crowding forces a valuation of shopping
time, consider the following utility-maximizing problem
293
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based on the River Basin I^del shopping assignment
modified for time use and ignoring part-time work;^
maximize U = 2<(x, t^ t , t , te)
subject to 803 - px - 9tr > 0 (money)
100 - tr - ts - tx - t - te > 0 (time)
t^ - ort^ > 0 (travel time)
r s ~~
t_ - ex > 0 (shopping time)
S
x - (at, + b) > 0 (consumption
goods and
services)
where S is the full-time work wage rate and t^ p.nd t
are politican and education times.
Here consumption goods and services x (purchased
at price p) must not be less than a fixed amount b plus
a variable amount at, based on leisure ("recreation")
time tj_, as in the River Basin Model. Two further re-
lations have been added (it is proposed below that these
be used in shopping assignment): travel time tr cannot
Inclusion of all activities in a utility-maximizing
model will be presented in the next chapter. See also
the shopping model of Part One of this report and the
part-time model re EMPRT.
-'it is necessary to make t and t explicit here to avoid
a trivial solution having no more variables than con-
straints.
294
-------
"be less than some proportion v of shopping time tg,
and shopping time cannot be less than some proportion
c of consumption level x. These relationships are
analogous to relationships used previously in the River
3asin IJodel. It will be assumed that these three con-
straints are definitional equations, so that their re-
spective shodov/ prices,
-------
(the negative of) the marginal rate of substitution
of consumption for money. Parameter c represents
crowding, for it determines how much shopping time is
required for the purchase of a unit of goods and ser-
vices. Shopping time valuation in dollars is the
general dollar value of time plus the dollar value of
time in travel weighted by the travel/shopping time
factor rr.
In choosing the optimal retail establishment,
there will be first-order conditions for location
variables 1 , where the travel parameters a and 9 and
price p and crowding factor c are functions of such
variables. Such a first order condition can be written
to give an additional equation for the evaluation of
shopping time in dollars as
Finally, under the fiction of a continuous trip type
variable m, optimal choice of route mode yields the
condition
296
-------
relating the dollar valuation in travel time to the
marginal changes in travel parameters. These two
conditions will be brought up again below.
Next, let us revise the shopping assignment vari-
ables to reflect the time relationships developed
above. Basic to this approach is the redefinition of
variables in the time dimension. First, there is the
modification of the function CAPCTY used in SETCOM to
determine effective capacities CAP so that these ca-
pacities will be in customer shopping time units. As
before, these effective capacities are "normal" design
limits based on employment and condition of the es-
tablishment. Only a simple scaling should be necessary
for this modification. Let actual usage of the es-
tablishment last iteration, SERVNG(J), and perceived
usage last iteration, SERVE(J) also be in tine units
(see below), and let perceived usage this iteration be
defined as before —
IDMOW = (SERVNG(J)*1000/I3ANG
+ (1 - 1000/IBANG)*SERVE(J))
— in terms of the iteration parameter IBANG. Then we
have perceived utilization rate PRATIO as
297
-------
PRATIO « (IDlvDW -l- SHOP)/CAP
where the increment SHOP to perceived usage (perceived
gross shopping time at the establishment) is the popu-
lation units own shopping time requirement and is
added only if the population unit did not shop here
the previous iteration.
PRATIO is now used to determine PTCU, the "per-
ceiVed" shopping time per consumption unit, in the
same manner as SHADOY* was derived in the River Basin
i.iodel, as an increasing (step.) function of PRATIO:
PTCU = 31 + (PRATIO - 32)*B3
As before, the 3's may be functions of PRATIO. Note,
however, that the population unit shopping requirement
SHOP is also "perceived", since it is the population
to
unit's estimate of the time it will take^shop for its
consumption demand, CONSUI-", and therefore it is based
on PTCU. Specifically,
SHOP = PTCU*CONSUL
3y substitution of this relation, we get PTCU as
As pointed out previously, programming and analysis
would be simplified with continuous rather than step
functions.
298
-------
PTCU = (Bl + (IDMDW/CAP - B2)*B3)/(1 - CONSUK*33/CAP)
Of course a new function for PTCU in terms of IDMDW,
CAP and COMSUM could be specified; the old form was
kept mainly for illustrative purposes.
It is proposed that shopping assignments now be
made to minimize a unit cost function which gives ex-
plicit dollar valuation to time in shopping and travel
and includes travel cost as well as the price of goods,
and also retains the perceived crowding effect (now
based on time) to stabilize the iteration procedure.
The new "shadow cost," then, is
LSD = (PCU + AC)+ VALTIM*(PTCU + AT)
Assignment for a population unit at I would thus be to
establishment (location) J for which LSD is minimal.
As PTCU stabilized over iterations, so would choice of
the optimal establishment. At the end of an iteration
the actual amount of goods or services purchased at J,
GCU, is the sum of the CONSUM for those I assigned there.
Since the time budgets of population units must
be charged for actual shopping time as well as travel
299
-------
time (AT* SHOP-), the actual shopping time per con-
sumption unit must be determined. The actual utili-
zation rate for the establishment this iteration is
ARATIO = GTII/v/CAP
v/here GTIM is the new gross utilization (and will be
SERVNG(J) for the next iteration). It seems only
reasonable that actual unit shopping time ATCU be the
same function of ARATIO that PTCU is of PRATIO . That
is, population units do not estimate shopping time in
correctly but only base sucb estimates on distorted
perceptions. Moreover, since
GTII* =
(ATCU*COr:SUM(lJs£ ATCU*GCU
I shopping
at J
ATCU can be computed as
ATCU = (Bl - 32*33)/(l - B3*GCU/CAP)
once assignments are made. The shopping time for a
population unit is thus ATCU*COKSUK, and this should be
7
As explained before, it is assumed that AC and AT are
scaled to represent levels per trip times the (constant)
number of trips to be made per consumption unit bought.
300
-------
charged against the time budget.8
Under such a procedure as this the change in sha-
dow cost LSD in switching from the optimal establish-
ment to another would be
= (&PCU + AAC) + VALTIM»(&PTCU + AAT) > 0
This means that at the optimum
< VALTIM if
(APTCU + MT) > 0
j
/APCU + AAC
«. „ / / >
otherwise
This means that if the change brings a net increase in
travel and shopping time, then any dollar savings
which might arise will be less than the dollar value
of the extra time. If the change brings a net re-
duction in travel and shopping time, then there will
be an increase in costs which exceeds the dollar value
of those time savings. Manipulations of the shadow
price relations in the utility-maximizing example con-
sidered above suggest that VALTIM reflect both the
8In addition, 2I3PTAX should be called to compute auto-
mobile and sales taxes.
301
-------
shadow price of shopping and the shadow price for
leisure time in general.'
One problem which has been overlooked here is
that shopping assignments are made after all time bud-
get determinations have been completed. Specifically,
there may not be enough time left in the budget (in-
voluntary time) to cover the shopping and travel times
required for the consumption that recreation time con-
After substitution for VA we have
which is comparable to
+ AAC
for changes in location (establishment). For trip
choice v/e have
versus VALTIM
302
-------
tributes to.10 There are several ways to balance the
time budget. One is to reverse the time assignment
priorities and take whatever time is necessary away
from, first, political activity and then education.
otVxer
No modifications than the proper decrementing of those
A
amounts is required. If this does not create enough
time, part-time work increments of 10 units could be
freed, the last assigned taken first, until the time
budget balanced. This would change the employment
situation, of course, and may not be desired for that
reason. On the other hand, the "quitting" of a job is
not necessarily unrealistic. Modifying the employment
lists is a simple matter.
A second method would be to make assignments first
only on the basis of fixed consumption components. This
is likely to take care of as much as three-quarters of
the consumption unit assignment and could bs done before
other time assignments are made, with proper charges
for travel and shopping time. Here there are several
additional options. After other time assignments are
made, the assignment of variable consumption based on
100bviously, if there is enough involuntary time reserved
for these, it can be decremented by their amounts.
303
-------
recreation time could be made independent of the first
assignment, by another run of the procedure with time
budget balancing carried out as in the first alterna-
tive above. Or there could be a second run of the
"scaled-dov;n" or "simultaneous" varieties suggested
below. These second-runs could utilize usage levels
carried over from the first run for fixed composition
but would allow choice of a second establishment as well.
Finally, the second run could be foregone, and variable
consumption assigned (after recreation time determina-
tion) to the same establishment as the first run (fixed
consumption), using total utilization rates to de-
termine the crov;ding effect on shopping time. Once
again, time budget balancing could be carried out by
the first method above or by the "scale-down" procedure
below. Mote that here the time changes forced upon
other activities would be very much smaller than in
the first method for total consumption, since most
travel for shopping and shopping itself would already
have been absorbed.
The third method is a "scale-down" method. All
population units having time budget violations after
shopping assignment would have their recreation times
304
-------
reduced until their budgets balanced. Such reduction
would reduce travel and shopping along with recreation,
so that the total reduction required of recreation
would be less than the amount by which the total stock
of time (100) was exceeded. Since variable consumption
is such a small portion of the total, this may not
balance the time budget even when no recreation time
remains. Another way to scale down would be to reduce
other leisure activities (politics or education or both)
at the same rate as recreation until the budget balanced.
Finally, recreation alone could be scaled down
and, if exhausted, the first method above applied to
the remaining activities. In such scaling, utilization
._
at .establishment would drop, thus making shopping more
N
efficient. If desired the recreation times of other
population units could be scaled upwards accordingly,
subject to their time budget limitations.
The last method is to develop a "simultaneous"
solution, whereby recreation time could be modified as
shotvDing assignment iterations proceed. Any of the
methods above could be used to balance the time budget
for a population unit vdthin an iteration upon completion
of its shopping assignment for that iteration. In this
305
-------
manner the consumption variables themselves could
fluctuate from iteration to iteration but would con-
verge on stable levels as the iteration process itself
converged. A last alternative would be a deterministic
simultaneous solution, as follows. There are the
relations
SHOP a CONSUM»PTCU
CONSUM = BASCON+ RECON*TIME(5, 2)
TRAV = AT*SWOP
SHOP + TRAV + TIME(5, 2) < REMTIM
before the final level of recreation time TIM£(5, 2)
is set in TMALC, plus the equation for PTCU as a func-
tion of PRATIO, or more exactly, of IDIvlDW and CAP
(which are both constant for the iteration) and CONSUM.
This is a set of five equations and five unknownsi
PTCU, TIME(5, 2), SHOP, CONSUM, AND TRAV. If REMTIM
is large enough, then, there is a deterministic solution
for PTCU for an establishment J for which the time bud-
get will be balanced. A different recreation, shopping
and travel time would be computed for each establish-
ment. Assignment to that one which minimizes the
At present that value of PTCU is one of the roots
of a quadratic equation. A simplified formula could
be developed if desired.
306
-------
shadow cost would also produce the optimal time set.
The iteration procedure itself could be carried out
as before. If REMTIM cannot handle SHOP plus TRAY
even when recreation time is zero, a decrement vari-
able could be created to keep track of the amount of
the disparity. When iteration ceases, any population
unit with such a decrement at its final shopping
assignment would have that decrement charged against
its political time, a simple bookkeeping matter.
Note that choice of a time-budget balancing method
means possible integration of parts of T&1ALC with
parts of SETCOM and OPCM, and a reconsideration of the
time priorities assumed in TMALC. It should be em-
phasized that the description of these changes is much
more complicated than the making of the changes them-
selves. For example, the modified assignment procedure
with simultaneous deterministic solution of recreation
time could be inserted with a change of about fifty
programming statements.
Financial Accounting for Population Units
For completeness, the accounting equations for
deriving monetary expenditures are listed here, with
307
-------
comments only if some change should be made to make
them consistent with the modifications in time assign-
ment procedures suggested previously. It should be
remembered that all population units of a class on a
parcel are assumed to make the same time allocations
and decisions, and thus while time assignments are
usually made for population units singly, the final
time and money accounts present expenditures for the
population units of a class on a parcel as a group as
well as "per capita." Many of these computations are
carried out in the summary subprograms WRYOU and PRINTY,
but others are performed elsewhere. The relevant sub-
program name is enclosed in parentheses. The resi-
dential parcel in question, denoted I, will not be an
explicit subscript unless necessary. K denotes class
of the population units and J a location.
1. Cash transfers (CASHT)
ITEST = CSBL(I, K) + MISI(I, K)
New cash balance ITEST is the old one, CSBL, plus
net transfers ("miscellaneous income") MISI
308
-------
2. rent (DEPREC)
RNPD = NF*RN*BASE(K)/BASE(1)
KP = RPOP(I, K) number of population units
RM = RENT(I) rent per space unit
BASS(K) space consumption parameter
-
health (WRYOU)12
HLTCST = BASE(K) + HINO(C, 100)*ONE(K)
+ IvIAXC(C - 100, 0)*TWO(K)
C = COLF(I) coliform count
BASE(K) = 1000*2**! (data)
ONS(K) = BASE(K)/50 (data)
TWO(K) = 3ASE(K)/20 (data)
HLTCST reflects only part of illness time,
since the latter, as proportional to Health
Index HLIN, contains arguments other than GOLF.
income for full-time work (EMP)
SAISUM = SAL(J, K)*KWK(I, J, K)
J
SAL(J, K) salary paid class K at job J
NWK(I, J, K) number persons working at J
= WIKCLS(K)*NP(I, J, K)
V/INCIS(K) labor force for class K (data)
MP(I, J, K) number of population units of
class K and I v/orking at J
l ?
There is some question as to whether or not health
cost is actually charged against the money budget.
309
-------
5» income taxes for full-time work (EKPTAX)
TAXSUM = SAISUK*(£ RAT(J))
J
RAT(J) tax rate of appropriate type or
jurisdiction J (including automobile tax)
6. transportation cost for tull-time work (EMP)
AUCSUM = £AUC(J)*NWK(!, J, K)
J
AUC(J) = AC(I, J, L) the cost of going from
I to J by least shadow cost mode L
7» income from part-time work (EMPRT)
BASET(l) s Y T PERCNT*3STSAL(L, J, K)*
£^ NPK(I, J, K, L)
BSTSAL(L, J, K) salary of part-time job J
assigned on the Lth iteration
PERCNT = .125-»V/IKCLS(K) scales salary to
equivalent of one-eighth ^job for 10 time
unit increment per iteration per worker
NPI\7(I, J, K, L) number of population units
assigned to part-time job J on iteration L
Note that no correction is made if last job assign-
ment for a population unit involves time less than
ten units. Allocation could be constrained to
multiples of ten.
8. income tax for part-time work (EKPTAX)
TAXSUK(l) = T 7 PERCNT*BSTSAL(L, J, K)*
^J1 KPN(I, J, K, L)*RAT(J, L)
Excludes automobile tax.
310
-------
9« transportation cost for full-time work (SJVIPRT)
BASET(2) = T PERCNT*BSTCST(L, J)*NPK(I, J, K,L)
L J
BSTCST(L, J) = AC(I, J, 1) the cost of going
from I to J (for job of iteration L)
via automobile by least shadow cost route
10. automobile tax for part-time work
TAXSUM(2) = Y V PERCNT*BSTCST(L, J)*
L j NPN(I, J, K, L)*RAT(J)
Unlike auto tax for full-time work, which is
based on income, this tax is based on trans-
portation COStt
11. welfare (SMP)
WELPAR = UN*WINCIS(I)*BUFF(30)
UN = UNEK(I, K) number of unemployed popula-
tion units of class K
BUFF(30) unemployment compensation in I's
jurisdiction
12. adult education (private; public is free) (YJRYOU)
BDCOST(l) = NP*TIME(I, K, 3* 2)*3000
Note absence of transportation cost and (auto)
tax. These could be
TRVED(L) = NP*TIKS(I, K, L, 2)*AC(I, L)
EDTAX(L) = KP*TI&as(I, K, L, 2)*RAT
AC(L) is the travel cost by auto from I to edu-
cation facility L for the jurisdiction (L = 2
Dublic, L = 3 private) by the least shadow cost
route, scaled to be per time unit as for work. 13
13Travel times and costs in the River Basin Model are
used as given for full-time work. Therefore, per
unit time in other activities they -can be scaled by
the factor 1/80.
311
-------
13 • children's education (privates public is free)
(LOADSC)
EDCOST(2) = SCOUT*CPED(K)
CPED(K) cost of private education per pupil
for class KJ (125, 175. 300) = 150 +
25* I* (3*1-5^2
SCOUT number of children in private school
= NP*P1TOP(K)*(1 - AMT(J, K))
PITOP(K) number of school children for a
population unit of class K
AMT(J, K) proportion of children of class K
which can be assigned to the public school
of jurisdiction J as a function of facility
value-ratio, teacher skills (class in em-
ployment) , and pupil teacher ratio
This expenditure can be altered by the population
unit only through collective influence on" the
Public Sector (political activity) to change AMT.
All lower class children are assigned to public
education.
1^. Political activity: no cost (a travel component
could be determined).
15. Consumption Expenditures (see discussion of SETCOM
and OPCK)* these are stored as COMM(I, K, 3, L)
for L = 1 (goods) and L = 2 (services) gross for
the K class at I.
16. Sales tax for consumption (EMPTAX)
SLTX(L) = COMM(I, K, 3, L)*RAT(L) L = 1,2
1?. Transportation costs for consumption (OPCM)
TRCOK(L) - COI'.1K(I, K, 3, L)*AC(I, J) L = 1,2
J is the establishment location for good service L.
Travel and shopping times should be added to the
present model. Travel time and cost should be
based on shopping time instead of consumption ex-
penditures.
312
-------
18. Automobile Tax for consumption (shopping) (EMPTAX)»
at present this is computed like sales tax, but
it should be based on shopping time (transporta-
tion time)!
Subprogram Y7RYOU takes these expenditures for
population units of a class K on a parcel I and de-
rives the cash balance (net savings) on a per popula-
tion unit basis. These are averages for the class on
the parcel and will cover population units in various
circumstances (e.g., employed and unemployed) even
though all made the same original time allocations.
Total income is averaged as TEM. Rent, travel, con-
sumption, education and taxes are averaged to give
expenses TEf.TPP. The new cash balance, then, is
SAV = TEK TEMPP + CSBL(I, K)
and is stored as the new value of CSBL(I» K), where
the old value in this equation has been adjusted for
cast transfers this round.
313
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SECTION XI
SOME CONCLUDING THOUGHTS ON SIMULATION
This report has reviewed the literature oy» utility
theory for major activity time-oriented developments,
and it has analyzed the time-related assignment pro-
cedures of the River Basin Model with reference to those
developments. It is the conclusion of this report that
the role of time use in Social Sector behavior is of
fundamental importance and that this aspect of the River
Basin Model makes it unique in its ability to represent
in a modelling context the crucial dimensions of de-
cisions of real-woriA households. This report was not
charged v/ith the actual development of simulation pro-
cedures for time allocation decisions, only with the
analysis of the manner in which those decisions are
translated into time and money expenditures in the
model. It is felt that this analysis has been thorough
and that the basic structure of Social Sector operations
is appropriate for its primary task of inter-relating
the behavior of households with the behaviors of in-
stitutions and agencies in other sectors, all in the
delicate balance between human and non-human environ-
mental systems.
314
-------
The numerous modifications that have been suggested
to make activity assignments consistent and compre-
hensive all lie well within the present structure of
the model, and it is possible to carry out these modi-
fications with a minimum of reprogramming of the model.
Indeed, it has been demonstrated that with these modi-
fications the model's structure for activity assignment
embodies processes in which the static optimizing
behavior of households (population units) is modified
by the uncertainties and feedbacks of a dynamic en-
vironment. It seems reasonable, therefore, that the
rules by which allocational decisions of population units
will be made in simulation should be derived from the
static optimization theory. The dynamism of the assign-
ment procedures (subprograms) means that such allocations
will always be before the fact, since they are based on
knowledge of the past and only estimates of the present
and future, so that the expenditures assigned are always
likely to be at least slightly out of equilibrium for
the population unit. From round to round the population
unit seeks to modify its allocations in response to a
changing environment, but it is never able to completely
anticipate all the changes that are forthcoming.
315
-------
In the interest of facilitating the development
of allocational rules for a simulation subprogram to
replace the time allocation input subprogram TALOC, a
Social Sector-oriented optimization problem will be
presented below. Given some method by which the popu-
lation unit estimates the prices and costs and other
operating parameters of the system for the round, the
solution of the first-order conditions for the optimi-
zation problem will give relations to be used as rules
for population unit allocation of the leisure time stock
and determination of dollar values of time for input
to the assignment subprograms. The treatment here is
somewhat superficial, for problems remain in the simula-
tion of inter-unit decisions, such as boycotts and
political pressure and voting, not to mention recon-
sideration of the nature of the objective function
(dissatisfaction index, or Quality of Life Index) and
other behavioral assumptions which have been accepted
below without comment. These include proportionality
assumptions between certain activity times and between
activity times and costs, reflecting as they do implicit
1A simple method would be to modify the actual parameters
of the last round for trend effects or for the effects
of outcomes to the appropriate institutions.
316
-------
assumptions concerning activity frequencies and the
average time and cost of each episode. It would be
desirable to make such relations explicit and to com-
pare their implied values against empirical findings,
as might be derived from the data on the Washington
Study. At present, those data are not in the proper
form for comparisons other than those presented else-
where in this report* however.
Static Optimization for^Population Units
The analysis to be presented here gives an example
of the decisions a population.might make to maximise
satisfaction as it perceives the state of the system.
It is necessary to distinguish, then, between the values
of parameters used here, based as they are on perceptions,
and the actual values to be used in assignment procedures.
The point is that the best strategy for the population
unit is. to allocate to maximize satisfaction according
to the state of the system as it sees it, with possible
modification for probabilistic distributions for the
likelihoods of different states (not considered below).
The time and money budget constraints reflect the account-
ing equations of TMALC and WRYOU, respectively, and will
317
-------
be used as presented there without elaboration. To
simplify various income tax rates have been aggregated
to single parameters for full and part-time work? they
are specific to job location and residence since some
components could vary by those locations. Automobile
tax rates have also been presented as specific to lo-
cation, although this may not be necessary. Rent, health
and child education expenditures have been aggregated to
the variable M, which does not vary with time allocations.
All time uses are explicit in the time budget. Two of
those are fixed, full time work at 80 and illness as
proportional to HLIN. The total time stock, then, will
be 180 units to allow the possibility of using work time
for leisure in the event of unemployment.
In a first look at the optimization problem the
objective function will be a general utility function
but one which is separable as to the effects of savings,
travel, recreation, and involuntary time as a group
versus the effects of education, political activity, and
cash transfers as a group. That is, utility U will be
the sum of the function ZC of the first group of argu-
ments and the function 91 °? i^16 second group. In this
form U. is a generalized time index and ^ is a generalized
composite population, neighborhood and health index. The
318
-------
separation of arguments in this manner is meant to
indicate that utility is affected directly through 7X
"but that Q is only a representation of the present
value of lagged effects of its arguments on the future
state of the system and the population units position
relative to it. Savings level has been included in cX
because without it or without a limit to dissaving (also
included as a constraint) the money budget will have
little effect. Additional constraints on the composition
of the travel time aggregate and the derivation of
shopping time are left explicit rather than removed by
substitution because explicit shadow prices for them are
desired. Travel is not differentiated by purpose in Ik .
The optimization problem heref then, is a composite of
the ones presented earlier in connection with SMPRT and
OPCM, although consumption expenditures x have been
omitted from the utility function here and from the
money budget by substitution with a function of recrea-
tion time, in keeping with the River Basin Model. For
further explanation of the relationships included in the
problem, see the previous discussions of the River Basin
i.'iodel assignment procedures.
The optimization problem for the population unit
becomes
319
-------
maximize U - "U,(t^t t^ t?, V) +
subject to
the time "budget (shadow price 9)i
9
180 - 7 t_. =0
.w -1-
1=0
where tQ = 80 full-time work;
t, = part-time work;
t_ = adult public education;
t = adult private education;
t, = political time;
Q
t = recreation;
t. = transportation;
o
t = shopping (goods);
t0 = shopping (services);
o
t = illness;
t, Q = involuntary.
The money budget (shadow price X)»
(income) SQ80 + S^ - £ k
(income - roSQ80 -
taxes)
320
-------
( auto - } r . a . t .
*• ai Pl x
taxes)
/ 8 8
(expen- „ ~
ditures) - > p.t, - ) (a + b.tJP. - M.
£0 i=7 1 X 5 *
1X6
(sales
taxes
_
) - ) (a. + b. tK)r.
" "
(net
savings) - V
a 0
travel time (shadow price f )s
8
«,---"
shopping times (shadow prices 9 ., i = 7, 8}
S 1
limit to dissavings (shadow price tp )i
V + Z > 0 where Z > 0
To simplify, the dissavings limitation constraint can be
removed by substituting V=V+Z, orV=V-Z, in the
money budget and in "lA- • V is then the amount by which
321
-------
the dissavings limit Z exceeds dissavings.
In addition to the constraints there are the
following first-order conditions for time, savings
and cash transfer allocations:
- 0 + X
< 0 if t = 0
= 0 if t > 0
1 ~
U
- 0 - X5(l + r) - Wt < 0 < 0 if = 0
= 0 if t. > 0
i = 2, 3
- 6 - X« 3,,(1 + rojl) + 3000* - cu^ < 0
0^ - «4 0
8
y i
=7
i=7 / > 0 if t > 0
vp 0
o
322
-------
- 9 - X»«(1 + rai) - ^ + ^s. < 0 ^ < 0 if ^ = 0
= 0 if t. > 0
i —
i = 7, 8
- Q < 0
< 0 if t = 0
- x < o < ° if ki = °
i i J 9
J 0 if k. ^ 0*
J
- X < 0
It is quite obvious that unless involuntary time
t10 yields positive marginal utility (which it does not
in the River Basin Model) no time will be allocated to it.
The same applies to (dis-) savings V and to cash trans-
fers when the change in their level is negative, meaning
increased outflow or reduced net inflow. These relations
are based on the assumption that 9 and X are both posi-
tive. Since the constraints are equations, they could
2
Only cash transfers from the population unit (k.< 0)
are choice variables. *-^/^k. represents an incre?se in
the outward transfer.
323
-------
take on negative values, "but that is not likely.
can also take either sign, but disutility to travel is
likely to assure that it will be positive. If it is
positive then the marginal utility of education OP
politics would have to be positive for the allocation
of time to that activity, and the shadow price on shop-
ping would have to be positive as well if shopping occurs.
That is reasonable, since shopping* like travel, uses time
without adding utility. Finally, with all shadow prices
positive, the marginal utility of recreation would have
to be positive for time to be allocated to it. These
signs are necessary but not sufficient conditions, of
course.
From these first-order conditions dollar valuations
for time in various activities can be derived. .That for
travel is related directly to its marginal utility and
the general dollar value of the time in leisure it pre-
empts:
/£\ 1
^ \ I X
3
This version follows the River Basin Ifodel in omitting
shopping from the utility function. However, it might
be desirable to treat it like travel^ creating dissatis-
faction.
324
-------
Travel is practically mandatory, so that this will be
an equation. The dollar value of time in shopping of
type i = 7, 8 is
This is the sum of the value of the time it takes from
leisure, the value of the time it requires of travel,
and the direct travel cost it creates, including taxes.
Since there is a consumption component independent of
time, this too will be an equation. For the dollar
value of time in leisure in general,
f 9\ S
where the right hand side is the net part-time salary
(wage) after travel and taxes and after deduction of the
dollar value of the time spent in travel. If that net
wage is great enough to equal the dollar value of time
in leisure, there will be part-time work.
In choice of travel mode or route for an activity
i, there is the first-order condition equating
325
-------
where m. is the fictitious continuous mode variable
for activity i. In the choice of optimal locations
for activity i, through the proxy location variable
k
there are the following conditions*
ao
work (i =s 0, 1) where by convention r = = O i
j ao ^ j_
o
as. dr. as
.
i
shopping (i = 7, B) •
.
(a, ,bit)(_i +
If allocations are spatially static—- that is, the popu-
lation unit does not consider locational changes or
anticipate any that might occur—then these conditions
are extraneous to a simulation procedure for population
unit decision-making. On the other hand, rates of change
with location could be approximated by use of wage, cost,
and price differences utilized in the actual assignment
procedures of the previous round.
326
-------
where t = c.x = c.(a. + b-t ) as a function of eon-
i i 11 -"• 5
sumption x and thus of recreation t , so that this is
There could also be a "location equation" for the re-
sidence in connection with voluntary migration (see
Chapter IV and KIGRAT)i
8
i=0
Here ^/^l represents the change in the value (in
dissatisfaction) of the residential environment in terms
of price and cost changes as well as amenities (see the
definition of the indices in KSDSSW), and 3iV^l also
reflects changes in expenditure parameters, especially
rent.
Recreation activity is distributed to locations in
a pro-rated fashion rather than in an optimizing fashion.
327
-------
Its travel time and cost here are average values based
on relative capacities and demands at parks, since they
form the "basis for pro-rating the recreation time to
locations (see TMAIC). If the pro-rating is to be done
as part of this optimising process, then t^ would have
to be disaggregated to t-, for recreation times at
5J
parks 3. In place of the first order condition for
t,, there would be a set of conditions, one for each
5
t ., with travel cost jj . and time «<-• coefficients
-?J Ja -?J
specific to parks. If the partials SU/at-. were func-
j J
tions of the park demands and capacities, then the
formula for pro-rating t^ would be a property of the
optimal solution. Education and political activity
locations are fixed.
Allocations representing a static optimum are not
likely to lead to an actual optimum, of course. For
example, there is obvious interdependence between allo-
cations, shadow prices, and activity locations; yet
in making allocations the population unit does not know
for sure where its activities will be assigned, since
it does not know the exact levels of system parameters
or competing allocations of other population units for
the round. Therefore, shadow prices based on allocations
328
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for an assumed set of parameter values may actual^ lead
to assignments to other locations with different para-
meter values \vhich cause the associated time allocations
to be non-optimal. Over rounds the system may stabilize,
so that parameters change little and the population unit
shadow prices and time allocations converge on values
which minimize the chances for location changes. Of
course, the opposite could happen, the system exploding
because wildly fluctuating system parameters throw
shadow price estimates for population units into con-
fusion, precipitating radical location reassignment
and extreme disequilibrium of time allocations. These
dynamic properties are important in studying the be-
havior of the system, and it is felt that the static,
short-term optimization of population unit decisions
is appropriate here, for it creates a situation in
which the population unit is continually making lagged
readjustments in its behavior in the face of a constantly
changing environment. This seems very much to be the
real-world circumstance for households.
Given the population units guesses of the "best"
or "most likely" locations of its activities and thus
of the parameter values which are relevant to them, the
329
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population unit would use these to set the dollar
values of time and make the time allocations. As
described above, there will be different values of
VALTIM to be used in different procedures. In fact,
the objective function of each assignment procedure
indicates exactly which shadow prices should be used
in the time valuations. In work assignments the ob-
jective is to maximize the net wage after travel with
a valuation to travel time included* That is, in EMP
and EMPRT
Kaximize SAL - (AC +' VALTIM(I)*AT)
for I as 0,1 for full- and part-time work. In the de-
rivations above it has been shown that this should
"be corrected for income and automobile taxes r* and
J-
Maximize (1 - r. )SAL - ((1 + r . )*AC
3. ax
+ YALTIM(I)*AT)
In this context the proper time valuation is
VALTBI(I) = r I = o, 1
X
the dollar value of time in travel as determined by
the first order conditions to be consistent with (Q/X)
and the entire set of time and money allocations.
330
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For shopping activities the objective would be
written as
Minimize PCU(I) - (AC + VALTIM(I, 1)*AT
+ VALTIIKI, 2))*TCU(I))
where travel parameters are per unit of shopping time
and TCU(I) gives the shopping time coefficient for
purchases of type 1=7 (goods), 8 (services). With
modification for sales and automobile taxes this is
Minimize (1 + r.)*PO!!(I) - ((1 + r .)*AC + VALTlM(i,l)*AT
i ai
•f VALTIM(I, 2))*TCU(I)}
The valuation of travel time for shopping is the same
as for work —
(L)
VALTIMd, 1) = -2-E
A.
— while the valuation of shopping time itself is its
shadow price in dollars
VALTIIvI(I, 2) a
This is a change in the objective function of OPGM to
include valuation of shopping time. It should also be
noted that these assignment problems are in effect
"duals" to the "primal" problem of maximizing satis-
faction.
331
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These different values for VALTIK, then, are the values
which should .be applied in the objective functions
above in their respective assignment procedures in the
Pdver Basin Model.
With an explicit utility function, the time allo-
cations may be solved for by eliminating the shadow
prices from the first-order conditions. Then the values
of the shadow prices may be found by substituting the
time allocations back into the first-order conditions.
If the utility (satisfaction or dissatisfaction) index
is given the ordinal interpretation, then only the re-
lative values of the shadow prices can be found uniquely.
This is adequate for computing the time valuations needed
for the VALTIM.
iProin the first order conditions it can be seen that
the marginal utilities of travel and recreation will
be especially important in determining the values of the
shadow prices. These determinations bring us to the
nature of the objective function itself. A proper
discussion of utility functions and their properties
cannot be attempted here. V/ith-two budget constraints
solutions can be tedious, even with such familiar func-
tions as the Cobb-Douglas (which leads to quadratic first-
332
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order conditions in the arguments here, on elimination
of the shadow prices). A quadratic function would not
be too difficult to work with and can be specified to
exhibit properties of complementarity as well as sub-
stitution along indifference curves.
Attention to a proper role for component £J- is
important, for without it education and political action
become useless appendages to the model. Since the
arguments to ^ --education, politics, cash transfers—
cannot affect satisfaction directly for the current
round, Q-must represent a present value of future re-
turns to those allocations. There are several types of
returns. One type is the improvement of Pollution,
Neighborhood, and Health Index levels in succeeding
rounds due to such things as political pressure or cash
contributions. These returns are more or less indepen-
dent of future time allocations. There are other types
of returns xvhich will interact with future time allo-
cations, however. Such things as increases in the
supply or capacities of public facilities and improve-
ments in the transportation system, again from political
pressure or cash contributions, will directly affect the
future allocation of time and the levels of the Time
333
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Index. Investments in education can yield returns in
terms of better paying jobs, and this may alter time
allocations. There may be future cost changes as a
result, of course, such as increased tax rates, and
these will modify changes in time allocations and thus
the present value of the future returns to current
allocations.
Suppose, then, that utility (satisfaction) com-
ponents U. and 8" are subscripted by round k, k = 0
being the present round, and that {k is the no n- time
k
component ?tyk of the same round plus the discounted
level of the total utility (satisfaction) Uk+1 of the
next round. Then
where d is a subjective discount factor. $ty /vfc. for
education or politics, for example, would be the weighted
sum of the partials UT^/^ + Vftj/^) where the 1A. k
are stated in terms of the parameterized forms ofthe
optimal time allocations of round k and the parameters
of both *Uk andtyk are written as explicit functions of
the t. of the present round.
334
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To attempt to simulate such an "investment analy-
sis" for population units would not seen feasible, or
perhaps even desirable, for this model, so that some
proxy or construct will be necessary for $• . Indeed,
even with perfect knowledge, finding solutions to such
dynamic optimization problems can be quite complicated •
However, it might be possible to work out a simple two-
period "time preference" analysis which could serve the
purpose. Although this will not be done here for the
River Basin Model, consider the simple two-period work-
leisure-education problem:
Maximize U = 7X.(x , t.., , d)
simultaneously over both periods k = 0, 1
subject to
Vwk ' X* " ptek *• ° (money)
- tlk - tek> 0 (time)
for each period k = 0, 1
where x is consumption, t,v leisure, t education
K -i-K ek
time* and t is work time, for period k. d is the
discount or "time preference" factor, and w-, - w + rt
x o eo
gives the second period wage rate as a function of the
first period rate and education (assume t , = 0). Then if
335
-------
is the Cobb-Douglas utility function such that
*
the optimal "investment" in education in the first
period is
V ,
W
Analytical solutions such as this could be worked out
for the more general problem above with assumptions
about the values of parameters for th$ next round
(e.g., "no change"). Of course, the populaton unit
allocates for the current round only and is not held
to these tentative allocations for the next round.
336
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The .Objective Function in the River Basin Model
In the River Basin Kodel the Time Index is linear
in travel, recreation and involuntary time, and no
other choice variable enters into the other indices
making up the dissatisfaction index. In this case the
optimization problem becomes a linear programming prob-
lem. The Time Index is the only index directly affected,
and it is
TIM . t? + 5t6 - gt5
where g is the park utilization factor PRKFAC* .01.
If the dissavings limit is not binding then V > and
X = Sty/dV = 0, so that the dollar valuations of time
would be infinite. The population unit would want to
take the fastest mode or shortest trip (in time) possible
regardless of cost. The same applies to cash transfers:
if any are made, \ = 0. These situations are unlikely,
so it will be assumed that \ > 0.
Since shopping and travel both contain some fixed
elements, both will be undertaken. Since -~W.= TIM we
have
^ = e + 5
ra.H. i , ?. 8
337
-------
This leaves five linear conditions in \» Q and con-
stants, and one in 9 and a constant. No more than one
need hold as an equation, and it is unlikely that more
than two will* This suggests that at most only two
of the t. , i » 1 to 5 and i « ?, will be positive.
Recreation will occur only if it creates less dissatis
faction in shopping and travel than it adds directly
in satisfaction, so that shadow prices 9 and x v/ill
fulfill
8
8
8
L + ra5)85 +,L biO?i + ri + Ci8i(1 + ra;
Bart-time work may be undertaken to support extra re-
creation or just to help cover fixed costs, and con-
sequently
0 « (1 + &)e + *» - X S(l - r) - P(I + r
6
Then 6 > 0 and X > 0 and with &V&t. = 0 for i »= 2, 3, 4
there will be no education or politfcal activity.
338
-------
V'ith both activities pursued there will be a solution
for 8 and A from these two equations, and the values
for VALTH".(i) can be determined. With no education or
politics (see footnote 2), t and t can be solved
directly from the budget constraints after substitution
for shopping and travel.
Ther^ are other possibilities representing the
other corner solutions of the linear program, of course.
For example, if travel and shopping due to recreation
create net dissatisfaction, there may be no recreation
i£ this net is greater than what is created by other
activities or involuntary time. There could be part-
timirork just to eat up involuntary time in such a
case, leaving V > 0 and \ = 0. On the other hand, if
the optimal solution means no recreation or part-time
work, then there could be involuntary time (v/ith & < 0)
or politics and/or education (with 6 < 0 and/or (J/r < 0).
Because this is a linear program these possibilities
must be systematically evaluated and narrowed down
without the convenience of a simple algebraic deter-
mination.
There are some properties of the Time Index which
could introduce non-linearities into the problem. Con-
339
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sider the definition of PRKFAC (g above) as
PRKPAC = AMIiaCLOO, 200 - PRKVAL)
where PRKVAL is the park-use index for the population
unit. It happens that PRKVAL is
(other demand t
at park J) + ->
(capacity of ^ (capacity of
park J) *J park J')
J
when recreation time rather than consumption is used
in assignment, J is the park with the lowest PRKVAL,7
and the other demand specifies use by population units
other than this one. Obviously, multiplication of t by
PRKFAC in TIE can create a quadratic term. It should
be noted that the coefficient of t -squared would be
positive but very small.
Use of shopping time as proportional to consumption
(purchase) but as a function of crowding introduces
non-linearities in the cost of shopping, and thus of
consumption and recreation, through the proportionality of
If PRKVAL is computed as a mean of indices for parks
weighted by the population unit's use, the first numera-
tor and denominator will each also be a sum of J*.
' 34o
-------
travel to shopping time. At a facility the shopping
time coefficient v/as defined as
C = B + 3 RATIO
where demand (other demand) + C,x
RATIO = = i-1
supply supply
Therefore,
B (supply) + 3 (other demand)
1 2
GI =
sut)ply - 3 x
2 1
and C X., v/here x. = a.. + b t » creates a non-linearity
in t- in the time budget.
A final example is the scaling down of public
education time allocation for a population unit by the
capacity-demand ratio when demand exceeds capacity.
This means that in such a situation if t is the time
to be requested for public education, the budget should
be charged (and returns based on) an amount
\
capacity
/(other 7" +
demand) 2 j
t
2
341
-------
This does mean that the sum of reauests could exceed
100 units, and therefore to enforce the budget alloca-
tion constraint may force expenditure on involuntary
time. However, since the population unit need not
actually request time for travel and shopping but should
only know their liKely values in setting the proper
dollar time evaluations, the education request could be
scaled upward appropriately, likewise* recreation time
can be over-requested to be safe. This should not be
done with other activities under the current priorites
of TKALC, since this may preempt time from recreation.
In closing this section it should be reiterated
that the "optimality" of this solution is artificial,
not only because the dynamism of the model creates un-
certainties in the environment and discrete rather than
continuous assignments and adjustments> but also because
it may not be feasible to allow the population unit to
evaluate all opportunities even under limited accuracy
in perception. The simplest method is to consider only
the locations utilized the past round, so that times and
dollar valuations are essentially readjustments for
changes in the system the previous round. Reassignment
to new locations can still occur, of course. The com-
342
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putational effort could be reduced with a simple
non-linear dissatisfaction function, which may also
o
be justified for behavioral reasons, and it is ad-
vised that additional variables be introduced to motivate
a wider range of behavior. The static model, however,
does seem to be an appropriate core of a simulated
decision-process.
g
Very little attention has been paid in this report to
the nature of activity time and money substitutions,
chiefly because there exists no real information on
them.
343
«0A GOVERNMENT PRINTING OFFICE: 1974 582-413/78 1-3
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SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
'.t. Restart flo.
3. Accession No.
w
4, Title
Critique of Role of Time Allocation
in River Basin Model
$.
7. Authcr(s)
Philip G. Hammer, Jr.
9. Organization
Center for Urban & Regional Studies
The University of North Carolina
Chapel Hill, North Carolina
}• -form i;
ffo.
W, Project No.
II. Contract/ Grant No.
801521
1 T?f» .' Kept., anrf
Period Co'-e
is, jSr
I S. Supple menf ary No tes
Environmental grotectign^ Agency _____________
Environmental Protection Agency Report
Number EPA-600/ 5-73-007, September 1973
Final Report
IS. Abstract
This critique presents a review of the theory of time use in consumer behavior
and applies this review to an evaluation of time and location assignment pro-
cedures for population units in the Social Sector of the River Basin Model.
Time allocations in this model serve to describe population unit work, travel
and leisure behavior and to link them to the operations of other sectors through
spatial movements, flow of goods and services, and participation in Institutional
activities. The objective of the simulations contemplated by the model is the
identification of the impacts of man's activities on his environment.
llz. Descriptors
Time Use Allocation; Consumer Behavior; Labor-Leisure Model
t7b. Identifiers
17c. COWRR Field & Group
A variability
Security
'Sapor >
11.
tto. of
Pages
Pr», *
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C. 2O24O
Abstractor Philip G, Hammer, Jr. j institution University of North Carolina
WRSIC )Q2 (REV, JUNE 1S71)
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