tlEAl
WATER POLLUTION CONTROL RESEARCH SERIES • 16110 FIO 01/72
Use of General Equilibrium
in Regional
Water Resource Planning
U.S. ENVIRONMENTAL PROTECTION AGENCY
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes
the results and progress in the control and abatement
of pollution in our Nation's waters. They provide a
central source of information on the research, develop-
ment, and demonstration activities in the Environmental
Protection Agency, through inhouse research and grants
and contracts with Federal, State, and local agencies,
research institutions, and industrial organiEations.
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Chief, Publications
Branch, Research Information Division, Research and
Monitoring, Environmental Protection Agency, Washington,
D. C. 20460.
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USE OF GENERAL EQUILIBRIUM IN REGIONAL
WATER RESOURCE PLANNING
by
Georgetown University
Washington, B.C. 20007
for the
Office of Research and Monitoring
ENVIRONMENTAL PROTECTION AGENCY
Project #16110 FIO
January 1972
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $1.50
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EPA Review Notice
This report has been reviewed by the Environmental
Protection Agency and approved for publication.
Approval does not signify that the contents necessarily
reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or
commercial products constitute endorsement or recommenda-
tion for use.
ii
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ABSTRACT
This report presents a methodology for applying general equilibrium
analysis to a regional economy. An interaction or trade mechanism is
presented which will force a regional economy into equilibrium with
the economy in which it is embedded in the sense that the relative
prices will be identical in these economies for their common commodi-
ties . Specifically one or more consumers in the regional economy is
assumed to hold money resources which are allocated to the purchase
(sale) in the surrounding economy according to the plan determined
by the algorithm presented in this report.
The assumptions required for applying general equilibrium analysis to
a region are stated. A technique is presented by which public goods
can be treated in a general equilibrium framework. Since one of the
conclusions from a general equilibrium application is the optimum
supply of any commodity, this technique can determine the optimum
supply for a region for such public goods as clean water (air, etc.).
There is an equilibrium price for a public good for each consumer and
these different prices determine what each consumer should be taxed
for the public good.
The results of some numeric computations are presented indicating how
the use of the algorithm works with a regional economy which has a
public good.
iii
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This report was submitted in fulfillment of Project Number 16110 FIO
under the sponsorship of the Federal Water Pollution Control
Administration.
iv
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CONTENTS
Section
I Conclusions 1
II Recommendations 3
III Introduction 5
IV General Equilibrium Analysis in a Regional Economy 9
The General Equilibrium Model
The Representation of a Public Good
in General Equilibrium Analysis
Allocation of Money Resources to a
Regional Economy
A Numerical Example
V The Algorithm of the Fundamental Theorem 49
The Fundamental Theorem
Application of the Fundamental Theorem
to the General Equilibrium Model
Application of the Fundamental Theorem
to the Allocation Model
An Alternative Method of Solution of the
Allocation Model
VI Acknowledgments 113
VII References 115
VIII Appendices 117
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FIGURES
PAGE
1 The set P formulated algebraically for the case of
n=3, G=5, Q^-0, Q2=l, and Q3=-l 51
2 A geometric view of the mapping from a set P, of
allocations to the unit simplex of prices relative
to some hypothetical allocation model 104
vi
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TABLES
No.
Specification of a Hypothetical Regional Economy
with a Public Good 38
Results of Allocation Model Applied to the
Economy of Table 1 under the Assumption
that Only the Factors of Production Can
Be Traded 41
Results of Allocation Model Applied to the
Economy of Table 1 under the Assumption that
Unskilled Labor is Localized to the Regional
Economy and the Locally Produced Good Can
Be Traded 43
vii
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SECTION I
CONCLUSIONS
1. General Equilibrium analysis appears to be a feasible tool for
investigating the optimal level of public goods in a regional economy
and the optimal allocation of public funds to obtain the desired level.
2. In as much as pollution can be interpreted as detracting from a
public good (water, air, etc.), this analysis provides a methodology
for investigating the externalities associated with various forms of
production.
3. This analysis provides a method which can determine the optimum
method of controlling water (air, etc.) pollution in a region.
Specifically it could determine whether pollution should be controlled
by requiring the producer to adopt a new (and presumably more costly)
technology or whether the existing technology should be retained and
methods of cleaning up after the pollution be adopted.
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SECTION II
RECOMMENDATIONS
1. Appropriate data should be obtained so that the techniques
described in this paper could be applied to a realistic situation.
It is only in this way that the real applicability of this analysis
can be tested.
2. Estimates of utility and/or demand functions are crucial to this
analysis. Further investigations of methods of estimation and their
applicability should be pursued.
3. Alternative computational techniques should be investigated to
increase the efficiency of the algorithm. One such technique is to
search only the polar cone of prices in the general equilibrium
solution.
4. If pollutants are treated as "bads" with costly disposal, their
prices will be negative. Negative prices should be investigated to
see if they provide a more realistic approach to the treating of
pollution of public goods and show more clearly the externalities of
production with pollution.
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SECTION III
INTRODUCTION
The possibility of computing a set of general equilibrium prices for
an economy with fairly restrictive yet standard specifications has
only recently been achieved by Herbert Scarf. Such a model does not
at first appear relevant to applied economics since it requires the
specification of all the interlocking parts of an economy. If some
segment of an economy were isolated for analysis, there would be no
guarantee that this segment would be in equilibrium with the rest of
the economy. In fact, for a developed economy, general equilibrium
would have to apply to a whole country if foreign trade were ignored,
or to the whole world if trade between countries were taken into
consideration. Under these conditions the task of specifying meaning-
fully even the major parts of an economy is simply prohibitive.
While examining the possibility of applying general equilibrium
analysis to regional water resource planning, it became increasingly
clear that if some appropriate form of interaction could be specified
between a regional economy and a larger economy in which it is
embedded, it would then be possible to simplify drastically the
specification requirements for general equilibrium analysis. Any
regional or small economy for which it would be feasible to specify
the interlocking parts required for general equilibrium analysis would
then be artificially bounded by the rest of the economy at least for
purposes of analysis. Assuming that the larger economy is already in
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some form of general equilibrium, then all its prices can be considered
as general equilibrium prices. To be in equilibrium with the larger
economy, the regional or smaller economy must then reach the same set
of equilibrium prices for those commodities common to both economies.
What is proposed in the present study is an adjustment mechanism
which brings the regional economy as close to equilibrium with the
larger economy as is consistent with the given specification of the
regional economy.
The adjustment mechanism consists in allowing a net inflow of commodi-
ties from the larger economy into the regional economy whose value is
fixed in terms of the prices of the larger economy. The source of
such money resources can be government subsidies, foreign aid, or any
economic source of liquid funds private or public. The term net
inflow is used to indicate that the adjustment mechanism also permits
the sale of commodities from the regional to the larger economy. The
adjustment mechanism determines the optimum allocation of the money
resources to the commodities which can be traded between the two
economies, so that their relative prices in the smaller economy
become as close as possible to their relative prices in the larger
economy.
The remarkable fact which the present study proceeds to demonstrate
is that precisely the same algorithm that computes general equilibrium
prices can be used to compute the optimum allocation of the money
resources. This algorithm is embedded in the proof of what will be
called the Fundamental Theorem.
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Section IV of this report will introduce the model and the assumptions
behind the model. General Equilibrium analysis will be presented
first, followed by the treatment of public goods in a general equili-
brium framework. The methodology for isolating a regional economy
for analysis with the tools of general equilibrium will then be
presented as the allocation model. Section IV will conclude with
the model applied to a hypothetical regional economy. The purpose of
this model is to demonstrate the workings and possible results of the
model and not to present a realistic application.
A regional economy which contains a public good lends itself ideally
to the allocation model. Indeed, since there can be no market price
associated with a public good, the optimum supply of such a commodity
must be approached from a purely theoretical basis. When a region is
forced into equilibrium with the economy in which it is embedded, the
general equilibrium solution produces a set of shadow prices of the
public good, one price for each consumer, at which the supply of this
public good is optimized subject to the given specification of the
regional economy.
Section V will present in detail the theory justifying the solutions
to the models presented in Section IV. Specifically the description
and proof of the algorithm will be presented as the Fundamental
Theorem. This theorem will then be applied to the general equilibrium
model to solve for a set of equilibrium prices and also to the allo-
cation model to determine the optimum allocation of money resources to
a region so that the region will be in equilibrium with the economy in
which it is embedded.
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With this methodology, it is hoped that General Equilibrium analysis
can become a convenient tool for applied economics. As the numerical
example shows, actual computations of equilibrium solutions can be
obtained, and comparative studies by parametric changes in the speci-
fication of the regional economy are made possible with this method-
ology. All of the models of this study have been programmed and the
details of the program can be found in Appendix B.
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SECTION IV
GENERAL EQUILIBRIUM ANALYSIS IN A
REGIONAL ECONOMY
The use of general equilibrium analysis in applied economics is
limited by the fact that most economies are so large that the speci-
fication requirements are prohibitive. This section will present a
methodology which makes the analysis of a regional economy with the
tools of general equilibrium analysis feasible. The assumptions of
the general equilibrium model will be presented first, followed by
a description of a technique for treating public goods in a general
equilibrium framework. The methodology for applying general equi-
librium in a regional economy will follow and a hypothetical numerical
example will be presented which demonstrates the feasibility of this
analysis.
The General Equilibrium Model
General equilibrium theory postulates an economy with a finite number
of consumers and producers who own, exchange, and transform a finite
number of physical input and output commodities. Each consumer is
assumed to have a well defined system of preferences for any set of
physical commodities which could be offered to him in the sense that
he can rank all such commodity bundles in a consistent way. It can
be assumed that such a preference ordering could be represented
mathematically in the form of a utility function from which demand
functions can be derived specifying the quantity of every good which
will be demanded at any set of prices once the consumer's income is
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given. The derivations of the demand functions are made under the
assumption of perfect competition so that each consumer is a price
taker and his choices do not affect individually the price structure.
As a practical matter all that is needed are demand functions which
may in many cases be easier to estimate than the utility functions from
which they are derived.
For this paper we will assume that each consumer has a specified
preference ordering represented by a utility function of either the
activity, Cobb-Douglas , or C.E.S. type. Each consumer is furthermore
assumed to have a fixed bundle of commodities w at the beginning
of the analysis from which the consumer derives his income. Specif-
ically, if p is the set of prices prevailing in the economy and
n
there are n commodities, £ p, w, represents the income of the
k-1
consumer.
In the case where the consumer's preference ordering is represented
by a utility function of the activity type, the demands will always be
in fixed proportions. Specifically if b. is the element in the
activity vector associated with commodity j , the demand for commodity
j will be
x. = qb . where q
This demand function is homogeneous of degree zero in the prices and
satisfies the consumers budget constraint. This type of ordering
10
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reflects the attitude of a consumer who is confronted with goods which
are all perfect complements to each other, an ordering which could very
well apply to a government having a fixed set of priorities which must
be carried out at all cost.
A second type of representation of a consumer's preference ordering
is a Cobb-Douglas type utility function. In this case a consumer's
utility for a commodity bundle x, U(x) can be represented as follows:
b
U(x) = A H (x. ) where £b. = 1
k k k k
Such a function is linear homogeneous and gives rise to a single valued
demand function for any commodity j for any price system p. This
demand function is derived in Appendix A and is of the following form:
provided p £ 0, in which case x. is a free good which does not
appear on the market. This function is homogeneous of degree zero in
the prices as is required of all demand function. The Cobb-Douglas
utility function is the limiting case of the C.E.S. utility function
when the elasticity coefficient is equal to unity. An interesting
property of this type of utility function is that a consumer who orders
his preferences accordingly, spends a fixed proportion of his income
on each commodity no matter which price system prevails in the economy.
Specifically he spends b . of his total income on commodity j .
J
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Perhaps a more realistic mathematical representation of a consumer's
preference ordering is given by a function of the C.E.S. (constant
elasticity of substitution) type. This may be written as follows:
i ^ I/*
'I (V
Ik k
where e = l/(l-a) is the constant elasticity of substitution between
the input commodities. This function is linear homogeneous and gives
rise to a single valued demand function for any commodity j for any
price system p. This demand function is derived in Appendix A and
is of the following form:
b . Jp, w,
Xj =
provided p ^ 0 in which case x. is a free good which does not
appear on the market. This function is again homogeneous of degree
zero in the prices.
The fact that all the consumer demand functions are homogeneous of
degree zero in the prices means that only relative prices need be
considered. Indeed, any scale of the price system would affect income
as well as the purchase prices, and the consumers would have identical
demands. Specifically we will assume that all the prices have been
scaled to add up to unity. It should be noted that by construction all
of the elements in the consumer demand function will be nonnegative.
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In applying general equilibrium analysis to the real world one need
not have available the preference ordering for every consumer. If
one assumes a homogeneity of preferences among consumer groups, such
as unskilled labor, professional labor, etc., one may use these
consumer groups as consumers. This simplification is probably
necessary in any real application.
The most difficult data to obtain in applying general equilibrium
analysis is clearly the parameters of these utility (or demand)
functions. Particularly with regard to public goods, actual demand
can never be estimated directly as there is no market price for such
a commodity (this will be discussed in the following part of this
section). More analysis is needed to determine how such parameters
can be obtained by means of surveys or questioning public officials.
Up to this point only the consumption side of the economy has been
discussed. The regional economy is further assumed to possess a
finite number of production processes. In the real world a firm or
company may possess one or more of these processes. One may interpret
a production process as an available technology. Essentially, a pro-
duction process is a method of transforming one or more commodities
into one or more different commodities.
For this paper it is assumed that each production process can be
specified either as an activity vector, a single output linear
homogeneous Cobb-Douglas production function, or as a single output
linear homogeneous C.E.S. production function. Each of these
13
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specifications displays constant returns to scale, for example, if
all inputs were doubled, output would also be doubled. The assumption
of constant returns to scale can be shown to imply that at equilibrium
each production process operated at a positive level will be associated
with zero profits. Thus ownership of the means of production need not
be specified since the problem of distribution of profits among con-
sumers has been assumed away.
While the assumptions of constant returns to scale may appear to be
a very strong assumption, in practical applications a process which
does not display constant returns to scale could be approximated by
a set of constant returns to scale production processes. Again the
assumptions of perfect competition prevail on the input and product
markets so that the operators of a production process are assumed to
be price takers. If a price system p prevails on the market, the
operator of each production process will calculate its profits at
these prices. For a production process of the activity type profits
are very easy to calculate. First the following convention is adopted:
let a. be the 1^ element in an activity vector. If a. < 0,
commodity i is an input to the production process; if a. > 0,
commodity i is an output to the production process; if a. = 0,
commodity i is not involved in this production process. With this
convention the profit associated with activity a at the price system
n
p is p-a = I p^.
Ki*™" -L
Now when production is specified by a single output linear homogeneous
Cobb-Douglas production function written:
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bk r
P(x) = A n (x ) K with ^bk = 1
k k
or by a single output linear homogeneous C.E.S. production function
written:
i \ I/a
POO =
Vk
where e = I/ (1-a) is the constant elasticity of substitution, there
is a two step procedure involved for determining profits. First at
the price system p, the optimal activity vector must be determined.
Mathematically there are two methods by which one may determine the
optimal input-output (activity) vector at unit output. The first
method is to minimize costs C = Zp, XL subject to being on the unit
cC
isoquant P(x) = 1. The second method is to assume some arbitrary
cost constraint C; maximize output subject to this cost constraint;
determine the level of output so obtained; and then to adjust by a
change of scale the input demands to produce unit output. Both of
these methods are applied to these two production functions in
Appendix A and they both give the following identical results; for
the Cobb-Douglas production function the jth element in the derived
optimum activity vector can be determined by:
bk bk
x - b. n (p, ) k/ P.A n o> ) K
J J k J k
provided p. + 0 in which case x. is a free input good which does
J J
not appear on the market, while for the C.E.S. production function
15
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the 2 element will be given by:
I/a
provided p. ^ 0 in which case x. is a free input good which does
not appear on the market. Since these are input coefficients, they
will be entered into the derived activity vector with a negative sign.
Since these input demand functions have been determined for a unit
level of output, the positive 1 is entered as the coefficient of the
commodity being produced. Again if we represent the j^ element of
such a derived vector by a , the profit associated with the derived
-* n
optimal activity will be p«a = £ Pi,ai« Clearly at any price system
k-1 k k
p, profits can be calculated for every production process and one of
these processes will maximize profits at such a price system (if more
than one process maximizes profits, one process can be chosen arbi-
trarily) . We will see that this will be necessary in the computation
of equilibrium prices.
Note that these input demand functions are homogeneous of degree zero
in the prices. As in the case of the consumer demand functions, only
relative prices need be considered in determining these input demands.
Hence, the only price systems that need be considered in the analysis
are those whose components add up to unity.
The production sector will also contain a method of free disposal of
any commodity. Specifically, if there are n commodities, there
will be n disposal activities, one for each commodity. The disposal
16
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activity for commodity j will be of the form (0, ..., -1, ..., 0)
with the -1 in the j™ position and zero in all other components.
We will also assume that production is irreversible, i.e., there is
no production process or combination of production processes from
which one can retrieve the inputs to production from the outputs of
production. In a mathematical statement, the assumption of irreversible
production is equivalent to saying that the set of production activi-
ties and derived production activities is positively linearly inde-
pendent. This assumption guarantees that for any fixed amount of
resources in the economy, the production set will be bounded.
The general equilibrium problem can now be interpreted in the following
way. A price is quoted in the market place for every commodity involved
in trade. Every economic agent is fully aware of each and every price.
Each consumer sells all of his initial holdings at this price system
to some central agency and obtains money in exchange. The producers
buy some of these commodities as inputs to production and transform
them into ouput commodities which consumers then purchase on the product
market. The producers choose the level of operation at which their
profits are maximized. Now it may happen that at this set of prices
the consumers want to buy more of a particular good than is available,
or some good will be in excess supply after all the purchases are
made. If this happens the prices quoted in the market cannot be
equilibrium prices. A new set of prices is called and the same process
is repeated. A general equilibrium price system will then be that
17
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vector of prices at which: (1) each consumer maximizes his preferences
subject to his budget constraint, (2) each producer maximizes his
profits, and (3) the market is cleared and consumers spend all their
income, i.e., Walras1 law holds. The assumption of free disposal
guarantees that all the prices will be nonnegative. Indeed, if a
free disposal activity is operated at a positive level at equilibrium,
the commodity being disposed of is relatively abundant and will have
a zero price since there is no demand for it and it can be disposed of
at no cost. :
The general equilibrium solution determines the optimum level of
i
operation of each of the possible production processes. Those pro-
duction processes which are inefficient will have an optimum level of
operation of zero and should be shut down. As a specific example, there
may be a number of different technologies available for cleaning up the
pollution in a river. The general equilibirum solution would determine
which technology or combination of technologies should be employed.
In some applications one may wish to have one of the consumers be the
government. This has the advantage that one can incorporate the
preference ordering of the central planners into the analysis. One may
wish to weight more heavily the preferences of the government than its
income alone permits. This may be accomplished by taxing the income
of the other consumers and letting that income be spent by the govern-
ment. A simple percentage income tax for each consumer has been
programmed into the model. The details on how to use this will be
found in Appendix B.
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In concluding this section on the assumptions of general equilibrium
analysis, the flexibility of this analysis should be noted. First, a
time dimension (dynamics) can be introduced by dating the commodities;
for example, bread at time i could be considered different from
bread at time j. Such a technique permits the introduction of
production technologies which require more than one time period to
complete. It also permits the study of implicit interest rates and
interest costs on.commodities available for future consumption.
Secondly, a spacial dimension can be added for selected variables; for
example, water at location i can be considered a distinct commodity
from water at location j. Finally, quality distinctions can be made
in commodities by treating different quality ranges as separate
variables. For example, water of a very good quality could be defined
as a different commodity from water of a poor quality.
l
The Representation of a Public Good in
General Equilibrium Analysis
Today there is much concern for the betterment of society. Many of
the needs of society fall in a class of commodities which have come
to be called public goods. A public good can be defined as any
commodity which is consumed equally by all members of the society
and of which each member consumes the total amount in existence
[3, p. 49]. Examples of public goods are such items as national
defense, clean air, sanitation services, etc. The basic problem is
that different individuals generally prefer different amounts of this
type of good. How much public goods should the government provide is
always a difficult question.
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From the point of view of general equilibrium analysis, a public good
associated with a given consumer must be considered as distinct from
the same public good associated with any other consumer. In production,
the same number is then repeated as many times as there are consumers
in the economy. In consumption, the consumer has a positive demand
only for the public good associated with him, and a zero demand for
the same public good associated with any other consumer. For example,
if there are ten consumers and one public good in the economy, the
commodity vectors must contain ten distinct components associated with
the public good, one for each consumer. If a production activity
produces one unit of a public good, this production activity must
represent ten outputs, one for each consumer. The production processes
used in the algorithm of the Fundamental Theorem can thus be specified
directly as activity vectors or be derived from a one commodity con-
tinuous linear homogeneous production function of its inputs.
Any general equilibrium solution associates a relative price with each
commodity in the economy. Clearly then, there is a price for each
public good for each consumer. Since in general, different consumers
have different demands for the same public good, the relative price of
each public good will differ for each consumer. The price system is
such that each consumer demands the entire amount of the public good
associated with him. Indeed, general equilibrium price systems clear
the market, i.e., supply equals demand.
The resultant prices associated with a public good are usually shadow
prices, since in general consumers do not pay directly for them;
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public goods are usually paid for by taxes. The prices of public goods
reflect the desire or preference of each consumer for the public good
and hence, can serve as a basis for a tax system based on consumer
preferences.
In general, the mathematical representation of an economy with public
goods is as follows. A similar representation has been developed
independently by Duncan K. Foley [2]. There are n private goods,
q public goods, m non-disposal production activities and r
consumers in the economy.
Each consumer is endowed with an initial bundle of resources, that is,
a finite amount of the private goods which he can buy and sell at the
current prices on the market and from which he derives part of his
income, and a finite amount of the public goods which can be bought
and sold on the market and from which each consumer derives part of
his income based on his own price of a public good. A typical initial
bundle of resources for consumer t is an rq + n dimensional column
vector with zeros in (r-l)q of the first components and nonnegative
numbers in the itth component for i = 1, ..., q and in the last n
components, for example ^ = (0, ..., w , ..., 0; ...; 0, ..., w ,
..., 0; w ., , ..., w . ..). Furthermore, each consumer has well-
q+l,t q+n,t
defined preferences for private goods and for public goods alike,
represented by a utility function homogeneous of degree one in the
commodities. Each consumer's preferences are assumed independent of
the preferences of all other consumers and no consumer can consume
the public good with another consumer's label. Thus each consumer's
21
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utility and demand functions contain no other public good but that
with his own label.
The productive sector of the economy is specified by a set of m
activity vectors whose negative elements represent inputs and whose
positive elements represent outputs. Each production activity which
produces public goods repeats the positive component corresponding to
a public good r times, one for each consumer. A typical production
activity with public goods is an rq + n column vector of the form
a = (a, ..., a; ...; a> .... a.; at •••» a> where
n, ..., lr ... ql> .... qj. q+1t
= a*[ > 0 if the kth public good is produced in the jth
production activity, a~1 = . . . = a~ < 0 if the k"1 public good
is used as an input in the jtl1 production activity, a-* = ... =
a^ = 0 if the k^ public good is not involved in the j production
activity, a • j •* 0 represents an output of private good i,
a . < 0 represents an input of private good i, and a . = 0 if
private good i is not involved in the j*1 production activity.
At equilibrium, there exists an rq + n dimensional price vector
p = (PU, ..., plr; ...; Pql, .... Pqr; Pq+r ..., Pq+n> such that
for each production activity a operated at a positive level we
have that:
r n
k=l t=l i=l
q r n
Pkt + Pq+iaq+i
k=l t« 1 1-1
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q . n
k-1 1=1
r
since a^ = a^ = ... = a^ and where E p, = p. is the unit value
of the k^ public good.
Now for each consumer t we have by Walras ' law that:
q n q n
k-1 1-1 k=l i=l
so that:
n t n q r t 1
Pq+iDq+i(p) = l Pq+iVi,t ~ I Pkt Dk(p) ' Wkt|
1-1 i-1 k-1
and the tax burden each consumer t is willing to assume is given by
£
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Now by Walras' law the society as a whole must use up all of its
income so that adding the above equations over all the consumers
we obtain:
n r n r r q
V V T^t / \ V V V V
) p ) D (p) = ) p )w — ) )
" q+i q+i ** q+i " q+i,t " *•
1=1 t-1 1=1 t-1 t-1 k-1
Let 1C D i j (p) = D .j (p) be the market demand for private good i,
r
Z w . = w . be the initial amount of commodity i in the
Dq+i(p) = E Vi I Vi.t' E * pkt Pk(p) - wkt
. .
t=1 » q+i
regional economy, and since at equilibrium the supply of public goods
must equal its demand and the total amount of the public good is
consumed by each consumer, we obtain:
q
I Pq+iVi(P) = ^ ViVi " *
1-1 1=1 k-1
where E p, = p, and w, = E w, . This st?.tes that
CC K. ic . . iCt
r
p, D, (p) - w,
l * L *
q
Z p, D, (p) - w, is the total tax burden the society is willing to
k
assume for the optimum quantity of the kth public good. This implies
that p. is the specific tax for the k1-*1 public good for society
1C
as a whole and then (PI;L ..... plr; ...; pkl, ..., pfcr; ...;
p ..... p ) is the specific tax structure in which each consumer's
benefits exactly equal his tax burden.
24
-------
Allocation of Money Resources^
To a Regional Economy
The use of general equilibrium analysis in applied economics is
limited by the fact that most economies are so large that the
specification requirements are prohibitive. Isolating a small region
for analysis is restricted by the fact that there is no guarantee
that the region will ever be in equilibrium with the economy in which
it is embedded.
This subsection proposes a method of utilizing the powerful tools of
general equilibrium analysis in studying a small regional economy
whose boundaries are determined solely on the basis of the objective
of the analysis and on the feasibility of the required specifications.
It opens up the possibility of applying general equilibrium analysis
to economic areas such as cities, small geographic areas, and so on.
Specifically the Fundamental Theorem is applied to an interaction
mechanism which brings the small, artificially bounded economy into
equilibrium with the economy in which it is embedded. This is an
entirely new application of the Fundamental Theorem.
In order to simplify the nomenclature, the small economy identified
for analysis will hereafter be called the "regional" economy and the
economy in which this small economy is embedded will be called the
"larger" economy. Since by assumption the regional economy is embedded
in the larger economy, the regional economy is in a non-optimum state
if the relative prices of the commodities which are in both the
25
-------
regional and the larger economies differ in these two economies.
Indeed, if the relative prices do not match then it is feasible for
some entrepreneur to improve his own position without hurting anyone
else simply by buying those commodities which are relatively inex-
pensive in one economy and selling them in the other economy. Clearly
if the relative prices in the regional economy are exactly matched
with the corresponding relative prices in the larger economy, this
type of arbitrage is not feasible and the regional economy is in a
Pareto optimal state relative to the larger economy. The exact
matching of these relative prices will be the goal of the interaction
mechanism.
In most cases, it will be possible to exactly match the relative
prices in the regional economy with the relative prices of the
corresponding commodities in the larger economy. Whenever the inter-
action mechanism does not bring the regional economy into full equili-
brium with the larger economy, it still produces the highest level of
utility attainable for each consumer in the regional economy con-
sistent with general equilibrium in the regional economy itself, i.e.,
a Pareto optimal state consistent with the structure of the economy.
This will be discussed in detail below.
To summarize, we say that the regional economy is optimized with respect
to the larger economy if the relative prices in the regional economy
are as close as possible to the relative prices of the corresponding
commodities in the larger economy.
26
-------
The novelty of this optimization procedure arises from the fact that
the regional economy is treated as an integral part of the larger
economy. For this reason, the concept of optimization relates to
equating relative prices in the two economies. Were the two economies
considered as distinct, such as the economies of two countries, then
political and trade barriers would in general preclude the matching of
relative prices. In this case, there is a true cost of trading between
the two countries and the adjustment mechanism proposed in this sub-
section no longer applies.
We will assume that the regional economy can be completely described
by a set of conditions which are sufficient for the application of the
algorithm to compute approximate equilibrium prices and quantities.
Specifically assume that the regional economy consists of a finite
number of commodities and of a finite set of producers and consumers.
Each producer is completely described by a finite set of productive
activities which display constant returns to scale. These can be
represented either by activity vectors or by single output production
functions linear homogeneous in all inputs. Each consumer is initially
endowed either with a limited amount of physical resources or money
or both. Moreover, each consumer possesses a well-defined utility
function linear homogeneous in all commodities, from which he derives
his demand for each commodity as a function of the prices in the
regional economy.
Trade with the larger economy is introduced in the following way. At
least one consumer initially holds a positive amount of money. This
27
-------
assumption could be relaxed by requiring one or more of the consumers
to obtain an initial quantity of money by selling some of their initial
resources to the larger economy. The money can be used only to increase
the initial quantity of physical resources by purchases from the larger
economy at existing prices there. Physical resources initially held
by consumers can also be sold at existing prices in the larger economy.
This transaction will result in an increase in the amount of money
initially held by the consumers; however, the consumer must immediately
spend this additional money on physical resources in the larger
economy. In other words, all monetary transactions are completed
before equilibrium prices and quantities are computed in the regional
economy. Thus, each consumer's wealth constraint depends only on the
quantity of physical resources he finally holds after all monetary
transactions have been completed.
The rule for allocating the money resource to purchases and sales of
commodities in the larger economy is defined as follows. There exists
a central decision-maker who dictates how each and every consumer in
the regional economy is to allocate his money holdings in trades with
the larger economy. Hereafter this central decision-maker, who may
or may not be one of the consumers in the regional economy, will be
referred to as the government. The dictates of the government will
be represented by a set of real numbers, one for each commodity, which
add up to one. This set of numbers is then called an allocation of
initial money resources.
28
-------
The numbers which define an allocation of initial money resources can
be negative, positive, or zero so long as they add up to one. A
negative number a. represents an instruction from the government to
the consumer to sell in the large economy the number of units of
commodity i which corresponds to a multiplied by the amount of
money he holds initially, at the price of commodity i in the larger
economy. This quantity of commodity i is now subtracted from the
initial amount of commodity i the consumer initially holds. If,
however, the consumer is instructed to sell more units than he holds
initially, the excess value creates an exogenous demand for commodity
i in the regional economy at the price of commodity i in the regional
economy. This exogenous demand can be interpreted as a demand coming
from the larger economy. This transaction is performed first and the
proceeds from the sale increase the amount of money with which the
consumer can buy other commodities in the larger economy. A positive
number a represents an instruction from the government to the
consumer to buy in the larger economy the number of units of commodity
j which correspond to a. multiplied by the amount of money he holds
initially at the price of commodity j in the larger economy. This
additional quantity of commodity j is now added on to the initial
amount of commodity j the consumer initially holds. A 2ero number
a, is an instruction from the government to the consumer to keep the
initial quantity of commodity k unchanged.
For example suppose there are three commodities in the regional economy
with corresponding per unit prices p. = $1.00, p_ = $2.00, p3 » $3.00,
29
-------
respectively, in the larger economy, and the consumer owns $100 in
money resources. Let the allocation of money resources dictated by
the government be given by the numbers a1 = 0.4, a_ = - 0.3, and
a = 0.9. Then the consumer is instructed to sell (a2>($100) =
(- 0.3)($100) = $30 worth, i.e., 15 units of commodity 2, in the
larger economy. He now has $130 in money resources with which he will
buy (a )($100) = (0.4)($100) = $40 worth, i.e., 40 units of commodity
1 and (a3)($100) = (0.9)($100) = $90 worth, i.e., 30 units, of
commodity 3 from the larger economy. The consumer now has 40 more
units of commodity 1, 15 less units of commodity 2, and 30 more units
of commodity 3 in his initial vector of physical resources. If,
however, he owned initially less than 15 units of commodity 2, say
only 10 units, he now has 0 units of commodity 2 in his initial vector
of physical resources, and there has been created in the larger economy
a demand for 5 units of commodity 2 from the regional economy at the
price of commodity 2 in the regional economy.
The underlying assumption is that the regional economy is so small
with respect to the larger economy that the larger economy can supply
any amount of any commodity which consumers in the regional economy
may wish to buy from it. Thus the minimum size of the larger economy
is indirectly specified by the magnitude of the numbers in the allo-
cation of money resources and by the amount of money initially held
by all consumers in the regional economy.
A negative number in the allocation of money resources and the amount
of money initially held by all the consumers in the regional economy
30
-------
determine the quantity of the corresponding commodity that must be
sold in the larger economy. Thus, given the amount of money initially
held by all the consumers in the regional economy, the admissible
lower bound of any negative number in the set of all allocations of
money resources specifies the maximum quantity of the corresponding
commodity which the larger economy can buy from the regional economy.
The extent to which the larger economy can buy from the regional
economy will be called the absorptive capacity of the larger economy.
Clearly, the choice of admissible allocations of money resources in
the regional economy will depend on the absorptive capacity of the
larger economy.
To each allocation of money resources there corresponds a uniquely
well specified regional economy. Indeed, after the initial trans-
actions have taken place with the larger economy, the consumers in the
regional economy own a new set of physical resources and have demand
functions homogeneous of degree zero in all prices; the exogenous
demand which may have been created by trade with the larger economy is
simply added onto the total demand for each commodity; the set of
production processes display constant returns to scale as required.
Thus, the conditions for applying the algorithm to compute a set of
equilibrium prices and quantities are all verified. Consequently, to
each allocation of money resources, there will correspond a unique
set of equilibrium prices. .
We will now say that the regional economy is optimized with respect to
the larger economy if the relative prices of some or all the commodities
31
-------
are as close as possible to the relative prices of the same commodities
in the larger economy. The rationale behind this concept of optimal!ty
is that the corresponding general equilibrium will then minimize the
dislocative effects upon the larger economy of any development taking
place in the regional economy. It is a constrained optimum subject
to the absorptive capacity in the larger economy.
Not every commodity in the regional economy need be a commodity in the
larger economy. Not every commodity in the regional economy need be
marketable in the larger economy. For example, water in a river basin
is localized in that particular region through which the river flows
and is usually marketable there only. Such commodities will have no
corresponding number in the allocation vector since there is no price
in the larger economy to compare its price with. Thus, the set of
relative prices in the regional economy which will be compared with
the corresponding relative prices in the larger economy will only be
a subset of the set of equilibrium prices computed by applying the
general equilibrium algorithm to the regional economy. This presents
no difficulty, however, since relative prices can always be scaled
differently without changing their relationship to one another. The
algorithm which will be devised in the next section to find the appro-
priate allocation of money resources will simply require that the
sets of prices to be compared be scaled to add up to unity in both the
regional economy and the larger economy.
Heuristically, there are two basically distinct situations arising
from the analysis of the possible results of this model of money
32
-------
resource allocation by government fiat. In the first instance, the
only commodities that are common to both the regional and the larger
economy are inputs to production in the regional economy. If there
is sufficient absorptive capacity for these commodities in the larger
economy, then an appropriate allocation of money resources to the
factors of production will result in matching exactly their relative
prices in both economies.
Indeed, the price of any factor of production can be made as high as
desired in the regional economy by making that particular commodity
sufficiently scarce. Any negative allocation of money resources to
a particular commodity is a mandate of the government to sell a
corresponding quantity of the commodity. If this quantity exceeds
the initial stock held by any consumer with money wealth, an exogenous
demand is created to the amount by which the quantity to be sold
exceeds the initial stock. Clearly, larger negative allocations
produce a greater scarcity and a higher exogenous demand which,
together with the endogenous demand, result in higher prices. Thus
if an appropriate quantity of any factor of production is sold in the
larger economy, its price can be made arbitrarily high. The only
restriction is the absorptive capacity of the larger economy.
The price of any factor of production can be made as low as desired
in the regional economy by making that particular commodity suffi-
ciently abundant. Any positive allocation of money resources to a
particular commodity is a mandate of the government to buy a corre-
sponding quantity of the commodity from the larger economy at the
33
-------
fixed price prevailing in that economy. Clearly, larger positive
allocations produce a greater abundance which results in lower prices.
If there are sufficient money resources in the regional economy,
enough of any particular commodity can be purchased in the larger
economy to make its relative price in the regional economy as low as
desired.
A completely different situation arises when there are one or more
commodities common to both the regional and the larger economy that
are outputs to production in the regional economy. In this case, it
is not always possible to match exactly the relative prices in both
economies. Consider, for instance, a regional economy with a pro-
ductive activity all of whose components (inputs and outputs) are
common to both the regional economy and the larger economy, and which
is profitable at the prices prevailing in the larger economy. It
will then always be better for the regional economy to operate the
profitable activity at a higher level of operation, buying more
inputs and selling more outputs in the larger economy. This is a
consequence of the assumption of constant returns to scale in
production which can then be bounded only by a fixed supply of
physical input resources. By allowing negative allocations of the
money resources thereby increasing the positive allocations, there
is no bound on the quantity of physical input resources that the
regional economy can acquire, except the absorptive capacity of the
larger economy. Thus, each profitable productive activity will be
scaled up to the bounds defining the absorptive capacity of the larger
34
-------
economy. The relative prices in the regional economy will be as close
as possible to the relative prices in the larger economy subject to
the absorptive capacity of the larger economy, but they will not be
matched exactly.
The opposite situation arises when all the productive activities in
the regional economy are composed of commodities common to both
economies but no productive activity is profitable at the relative
prices prevailing in the larger economy. This says that production
in the regional economy is completely inefficient. The consumers in
the regional economy would be better off to sell their factor resources
and buy the outputs at the given fixed prices in the larger economy.
If there is enough absorptive capacity in the larger economy production
in the regional economy will grind to a halt. With sufficient
absorptive capacity in the larger economy, relative prices in both
economies will be exactly matched.
A more complex but perhaps more realistic situation develops when
some but not all of the outputs in the production acitivites are
common to both the regional economy and the larger economy. Since
at least one price is left to vary independently, an activity which
is profitable at one allocation of the money resources is not
necessarily profitable at some other allocation. Specifically,
consider an activity which produces joint outputs such that one
output is localized within the regional economy. If excluding the
localized commodity, the activity is profitable at the relative
prices prevailing in the larger economy, the situation reduces to
35
-------
the case previously considered and the results depend entirely on
the absorptive capacity of the larger economy. If not, because of
constant returns to scale, as the level of operation of any such
activity is expanded, the supply of the localized output increases
and its prices falls within the regional economy. Equilibrium is
attained when the price of the localized commodity is reduced to
the point that the productive activity yields exactly zero profits in
the regional economy and the relative prices in the regional economy
of the commodities common to the larger economy are identical to the
relative prices of the same commodities in the larger economy.
A similar situation arises when one of the factors of production is
a physical resource localized to the regional economy. In this case,
as production expands by purchasing other factor inputs and selling
outputs at a profit in the larger economy if need be, the localized
factor input becomes relatively more and more scarce so that its
price increases until all corresponding productive activities are
no longer profitable at the relative prices in the regional economy.
Again, provided there is sufficient absorptive capacity in the larger
economy, equilibrium will be attained so that the relative prices of
the commodities common to both economies are matched exactly and the
localized factor price is increased to the point that the most
profitable activities all yield zero profits.
The problem for the government is now clear. It must determine the
optimum allocation of the money resource so that wherever possible
the prices of those commodities common to both the regional and
36
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larger economies will be identical in both places. The algorithm
presented in the next section will determine for the government this
optimum allocation.
A Numerical Example
To illustrate the applicability of the allocation model to an economy
with a public good, consider a small hypothetical economy which displays
alternatives similar to those present in the real world. The specifi-
cations of the economy are presented in Table 1. The economy is
assumed to have four factors of production: unskilled labor, technical
labor, steel, and cement; an intermediate good labeled capital; a
consumer good which is locally produced; and a public good, water,
which can be consumed by each consumer without affecting the quantity
or quality available to any other consumer. This public good is
therefore indexed separately as three distinct commodities, one index
for each consumer. Units of water are interpreted either as quality
units or as quantity units. If, for instance, the water is assumed
to be used in a river with a relatively fixed flow, or in a lake, then
the units of measurement can be quality standards (such as units of
dissolved oxygen or some other method of classifying water purity);
if the water is simply drawn from a lake or from a river, then the
units can be measures of quantity. No attempt is made to formulate
a real economic problem so that the reader may supply his own economic
interpretation.
37
-------
TABLE 1
SPECIFICATION OF A HYPOTHETICAL REGIONAL ECONOMY WITH A PUBLIC GOOD
Un- Tech- Elasti-
Economic Function skilled nical Local Water Water Water city or
Agent Type Labor Labor Steel Cement Capital Good Cl C2 C3 Scale
Production Functions for Each Firm in the Economy
Capital Good
Production C.E.S. 0.3 0.5 0.7 0.2 -1.0 0.0 0.0 0.0 0.0 0.3
Local - High
Pollution
Local - Moder-
u> ate Pollution
00 Local - No
Pollution
Dam
Pond
Chemical Cobb-Douglas 0.1 0.5 0.2 0.2 0.0 0.0 -1.0 -1.0 -1.0 1.0
Resources Available to Each Consumer in the Economy
Consumer 1
Consumer 2
Consumer 3
Utility Functions for Each Consumer in the Economy
Consumer 1 Cobb-Douglas 0.0 0.0 0.0 0.0 0.0 0.6 0.4 0.0 0.0 1.0
Consumer 2 C.E.S. 0.0 0.0 0.0 0.0 0.0 4.0 0.0 6.0 0.0 3.0
Consumer 3 C.E.S. 0.0 0.0 0.0 0.0 1.0 4.0 0.0 0.0 6.0 0.8
Money Resources Available to Each Consumer
Consumer 1 Consumer 2 Consumer 3
$100 $200 $300
C.E.S.
Activity
Activity
Activity
Activity
Activity
>b -Douglas
0.3
-1.2
-1.4
-0.5
-0.8
-2.3
0.1
0.5
-0.5
-1.3
-0.5
-0.8
0.0
0.5
0.7
0.0
0.0
0.0
-0.9
-0.9
0.2
0
0
0
0
-1
-1
0
.2
.0
.0
.0
.0
.0
.2
-1
__•!
-2
-4
0
0
0
.0
.0
.0
.0
.0
.0
.0
0.0
4.0
4.0
4.0
0.0
0.0
0.0
0.0
-2.0
-0.5
0.0
1.0
1.0
-1.0
0.0
-2.0
-0.5
0.0
1.0
1.0
-1.0
0.0
-2.0
-0.5
0.0
1.0
1.0
-1.0
80.0
0.0
0.0
0.0
40.0
0.0
0.0
0.0
50.0
0.0
0.0
60.0
0.0
0.0
1.0
3.0
4.0
2.0
4.0
0.0
0.0
0.0
4.0
0.0
0.0
0.0
4.0
-------
The production sector contains alternative methods of production.
There is only one production process which manufactures the capital
necessary for producing the local consumer good. This process allows
some input substitution and is represented by a single output linear
homogeneous C.E.S. production function with an elasticity of substi-
tution of 0.3. Three production processes exist for the production
of the local consumer good, each represented by a fixed activity
vector. The first process can be considered a high pollution process
since it uses up 2 units of water for each unit level of operation.
It is, however, relatively inexpensive in terms of its other inputs
since it requires only one unit of capital. The second process produces
a moderate level of pollution using only 0.5 units of water at unit
level of operation. This process is more expensive, for besides
requiring an additional unit of capital, it also has a higher technical
labor input requirement. The third process is the most expensive in
terms of capital investment. However, it produces no pollution at all.
Now the public good, water, is demanded by all three consumers and
there exist three different methods of producing this public good.
The first method, identified as a dam, requires approximately an even
amount of the four factors of production to produce one unit of water.
The pond method is a highly non-technical method requiring relatively
high levels of unskilled labor. The third alternative is a chemical
process which is technically oriented and exhibits a unitary elasticity
of input substitution. This process is represented mathematically
as a Cobb-Douglas production function.
39
-------
Each consumer is assumed to have a fixed supply of physical resources
as presented in Table 1. Note that each consumer is initially endowed
with the same number of units of the public good. These are represented
as the 4.0 of Water Cl available to Consumer 1, 4.0 of Water C2
available to Consumer 2, and the 4.0 of Water C3 available to Consumer 3
The demand sector is represented by utility functions, one for each
consumer, of either the Cobb-Douglas or C.E.S. type. Note again that
by definition of a public good each consumer demands only his own
public good. Each consumer has a demand for the public good and the
local good; the third consumer also has a demand for capital.
This small economy is assumed to be embedded in some larger economy
with fixed dollar prices for those commodities which can be traded
with the regional economy. Each consumer in the regional economy
possesses some dollars which he can use in trade with the larger
economy. It is further assumed that a government exists which issues
a mandate to each consumer as to how he should allocate his money
resource. The objective of this mandate is to equate the relative
prices in the regional and in the larger economies of the commodities
which can be traded between the two economies.
The allocation model is applied to this problem under two separate
assumptions as to which commodities can be traded between the two
economies. Table 2 presents the case where the four factors of
production can be traded while in Table 3 it is assumed that unskilled
labor cannot be imported or exported while the local consumer good can.
40
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TABLE 2a
RESULTS OF ALLOCATION MODEL APPLIED TO THE ECONOMY
OF TABLE 1 UNDER THE ASSUMPTION THAT ONLY
THE FACTORS OF PRODUCTION CAN BE TRADED
Allocation Grid = 50, General Equilibrium Grid « 100
Absolute Price
Relative Price
True Boundaries
Computational Boundaries
Equilibrium Allocation
Equilibrium Prices
Unskilled
Labor
$3.00
.250
-20
-5
-.049
.252
Technical
Labor
3.00
.250
-20
37
.785
.252
Steel
3.00
.250
-20
10
.223
.239
Cement
3.00
.250
-20
0
.040
.258
Final General Equilibrium Solution
Firms in Final Equilibrium Solution Levels of Operation
Capital Good Production
Local - Moderate Pollution
Chemical
46.5
18.9
82.5
Commodities
Unskilled Labor
Technical Labor
Steel
Cement
Capital
Local Good
Water-Consumer 1
Water-Consumer 2
Water-Consumer 3
Gen. Equil.
Computational
Boundaries
0
0
0
0
0
0
0
0
0
Equilibrium
Relative
Prices
.087
.087
.082
.089
.183
.181
.053
.109
.129
Estimated
Final
Demand
8.1
0.0
0.0
0.0
10.1
86.9
79.1
78.6
80.0
Estimated
Net
Supply
6.7
4.4
-5.4
2.3
9.8
84.6
77.1
77.1
77.1
Note 1: The barycenter averaging technique was used for averaging the
equilibrium price simplices.
Note 2: The Levels of Operation are the weights associated with the
respective activity vectors in the final nonnegative solution
to Bx=w. The Estimated Net Supply was determined by multi-
plying the levels of operation by the activity vectors cal-
culated at the Equilibrium Relative Prices.
41
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TABLE 2b
RESULTS OF ALLOCATION MODEL APPLIED TO THE ECONOMY
OF TABLE 1 UNDER THE ASSUMPTION THAT ONLY
THE FACTORS OF PRODUCTION CAN BE TRADED
Allocation Grid = 100, General Equilibrium Grid = 100
Absolute Price
Relative Price
True Boundaries
Computational Boundaries
Equilibrium Allocation
Equilibrium Prices
Unskilled
Labor
$5.00
.333
-40
-26
-.168
.335
Technical
Labor
3.00
.200
-40
85
.875
.201
Steel
1.00
.067
-40
40
.411
.066
Cement
6.00
.400
-40
-20
-.118
.398
Final General Equilibrium Solution
Firms in Final Equilibrium Solution
Capital Good Production
Local - No Pollution
Chemical
Commodities
Unskilled Labor
Technical Labor
Steel
Cement
Capital
Local Good
Water-Consumer 1
Water-Consumer 2
Water-Consumer 3
Gen. Equil.
Computational
Boundaries
11
4
0
13
13
15
3
6
8
Equilibrium
Relative
Prices
.136
.081
.027
.161
.153
.171
.062
.091
.118
Levels of
103.9
23.2
85.9
Estimated
Final
Demand
16.8
0.0
0.0
5.9
12.9
109.1
93.9
103.5
Operation
Estimated
Net
Supply
15.7
-.4
2.0
5.2
12.1
101.8
89.9
89.9
95.9
89.9
Note 1: The barycenter averaging technique was used for averaging the
equilibrium price simplices.
Note 2: The Levels of Operation are the weights associated with the
respective activity vectors in the final nonnegative solution
to Bx=w. The Estimated Net Supply was determined by multi-
plying the levels of operation by the activity vectors calcu-
lated at the Equilibrium Relative Prices.
42
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TABLE 3
RESULTS OF ALLOCATION MODEL APPLIED TO THE ECONOMY OF TABLE 1
UNDER THE ASSUMPTION THAT UNSKILLED LABOR IS LOCALIZED
TO THE REGIONAL ECONOMY AND THE LOCALLY PRODUCED
GOOD CAN BE TRADED
Allocation Grid = 100, General Equilibrium Grid - 100
Technical
Labor
Absolute Price $2.00
Relative Price .190
True Boundaries -40
Computational Boundaries 66
Equilibrium Allocation .689
Equilibrium Prices .195
Steel
2.00
.190
-40
23
.246
.184
Cement
2.00
.190
-40
7
.085
.197
Local
Good
4.50
.430
-40
-15
-.020
.423
Final General Equilibrium Solution
Firms in Final Equilibrium Solution Levels of Operation
Capital Good Production 58.2
Local - Moderate Pollution 23.9
Chemical 102.0
Commodities
Unskilled Labor
Technical Labor
Steel
Cement
Capital
Local Good
Water-Consumer 1
Water-Consumer 2
Water-Consumer 3
Gen. Equil.
Computational
Boundaries
9
5
6
6
16
16
3
8
10
Equilibrium
Relative
Prices
.106
.083
.079
.084
.181
.181
.057
.106
.123
Estimated
Final
Demand
0.0
0.0
0.0
0.0
11.2
104.0
94.
98.
,1
,2
Estimated
Net
Supply
-1.7
3.4
-2.8
1.6
11.5
101.8
94.
94.
,1
,1
95.6
94.1
Note 1: The barycenter averaging technique was used for averaging the
equilibrium price simplices.
Note 2: The Levels of Operation are the weights associated with the
respective activity vectors in the final nonnegative solution
to Bx = w. The Estimated Net Supply was determined by multi-
plying the levels of operation by the activity vectors calcu-
lated at the Equilibrium Relative Prices.
43
-------
In all cases it is assumed that capital cannot be purchased or sold
in the larger economy. The public good, water, is also treated as
localized to the regional economy.
As pointed out in the last subsection, with sufficient absorptive
capacity in the larger economy it will always be possible to match the
relative prices of the factors of production if they are the only
commodities traded between the two economies. Two different experiments
are conducted under these specifications assuming different factor
prices in the larger economy. In the first experiment each factor
price is assumed to be $3.00. Table 2a presents the results of this
calculation. The equilibrium allocation indicates that unskilled labor
should be exported, while technical labor, steel, and cement are to
be imported into the region. With equal factor prices of $3.00 the
regional economy should produce 18.9 units of the locally produced good
using the technique of moderate pollution. The chemical process should
be used at the 82.5 level to produce the public good. At the final
equilibrium prices the derived unit output activity for this chemical
process is (-0.337, -1.685, -0.711, -0.657, 0.0, 0.0, 1.0, 1.0, 1.0).
Similarly the unit output activity vector is calculated for the pro-
duction of the capital good. At equilibrium this vector is (-0.375,
-0.624, -0.888, -0.248, 1.0, 0.0, 0.0, 0.0, 0.0) and it is operated
at level 46.5. Notice that the relative prices of the public good are
substantially different for the different consumers. For the first
consumer the relative price is only 0.053 while for the second consumer
it is more than double that amount at 0.109, and for the third consumer
44
-------
it is even higher at 0.129. At these different relative prices however,
each consumer demands approximately the same number of units, i.e.,
approximately the net supply of 77.1.
Table 2b presents the same economic problem with different factor prices
in the larger economy. A somewhat better matching of relative prices
is achieved in this case using an allocation grid size twice as large
as that used in the calculations of Table 2a. At this set of factor
prices the regional economy should produce the local consumer good
with the no pollution technology. The reason for this is that capital
has become relatively less expensive, falling from 0.183 to 0.153.
At this lower price of capital, the more capital intensive no pollution
process becomes the most efficient. The larger net supply of water
and of the local consumer good is due to the fact that the local
economy is now wealthier with the value of its initial physical
resources measured at the new prices prevailing in the larger economy.
In this experiment, the local economy is optimized by exporting
unskilled labor and cement and importing technical labor and steel.
The allocation model is also applied to this economy under a different
assumption as to which commodities can be traded between the two
economies. Specifically it is assumed that all the factors of pro-
duction except unskilled labor, and the locally produced consumer good
can be traded. Under this specification and with the prices prevailing
in the larger economy as presented in Table 3, the allocation model
indicates that the three factors of production should be imported and
the locally produced consumer good should be exported. Production of
45
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the local consumer good with the moderate pollution technology is
carried out to the point where unskilled labor becomes relatively
scarce and its price rises until the moderate pollution activity has
exactly zero profits. A small part of this output is then sold in the
larger economy.
Considerable computational efficiency is achieved by choosing compu-
tational boundaries for the general equilibrium solutions. Once it is
known approximately where the allocation solution is to be found on the
unit allocation hyperplane, small adjustments in the vector of initial
physical resources do not greatly affect the equilibrium primitive set
of prices provided there is enough continuity of the mapping. There-
fore, at each iteration of the allocation model the entire price
simplex need not be searched for the fixed point, but only that region
in which the solution is expected to be found. This technique has the
disadvantage that at some iteration of the allocation model an equilib-
rium primitive set of prices may contain a computational boundary
element. If such an equilibrium price system is associated with an
allocation in the final primitive set of allocations, the entire
solution must be recomputed lowering sufficiently the general equili-
brium computational boundaries so affected.
This section has applied a technique for determining the optimum pro-
duction methods in a regional economy confronted with the problem that
at least some of its methods of production "pollute" in the sense that
they use up a public good. The quality of water or air, or the amount
of other types of public goods produced depends on the interaction
46
-------
between the demand for the public good and the demand for the product
which uses up the public good in production. Which production tech-
nology to employ depends to a large extent on the techniques available
to clean up the polluted public good.
One source of money for a region is the government. The government can
be treated as an additional consumer with money wealth and a demand
function reflecting its desire for cleaner water. Given the amount of
money allocated to this region, the allocation model will then determine
the optimum distribution of this money resource to the various comroodi-
. i !
ties in the region. Hopefully, this methodology will make it possible
to use the powerful tools of general equilibrium analysis in the search
for the optimal allocation of money resources to regional developmental
projects.
47
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SECTION V
THE ALGORITHM OF THE FUNDAMENTAL THEOREM
This section will acquaint the reader with the algorithm which has been
used both to compute equilibrium prices and to solve the allocation
model. This section is not completely original as it is based on the
work of Hansen and Scarf [5]. The originality consists in demonstrating
that the vectors in P, as defined for the Fundamental Theorem in
K.
computationally convenient form need not be taken only from the unit
simplex as Hansen and Scarf have done; rather P. can be vectors on
an appropriately bounded subset of any hyperplane which is parallel
to the unit simplex, and the conditions of the Fundamental Theorem
will still hold. This extension is needed in the application of the
algorithm to the allocation model.
There will be four parts to this presentation. The first part will
contain a statement and proof of what will be called the Fundamental
Theorem assuming degeneracy, an assumption which leads to a compu-
tationally convenient form of the algorithm. The second part will
apply the Fundamental Theorem to the computation of general equilibrium
prices. The third part will apply the Fundamental Theorem to the
computation of an optimal allocation of money resources in a regional
economy with an almost equivalent solution presented in part four.
The Fundamental Theorem
The first concept which must be understood is that of a primitive set.
Let the vectors of a set P all be chosen from the G-hyperplane in
49
-------
n-dimensional Euclidean space and be all vectors of the form (k , k?,
n
..., k ) where Z k. = G, where G is any fixed integer and each
11 • -• -L
i=l
k. is an integer bounded from below by some integer Q. unrestricted
in sign. Clearly the G-hyperplane is the translate of a maximal
proper linear subspace parallel to the unit hyperplane. Figure 1 gives
the geometrical representation of a set P for the case n = 3, G = 5,
Q1 = 0, Q- = 1, and Q_ = -1. Only the points at the vertices of the
smaller triangles belong to the set P. Note carefully that there are
many vectors in P with identical i^ coordinates. Geometrically
all the points on a line parallel to any side of the larger triangle
have one identical coordinate. Thus P so defined does not satisfy
the assumption of nondegeneracy made in [9], and the replacement
operation described there cannot be applied directly. There must be
a method of comparing identical elements in a particular coordinate
to determine their order. If an ordering can be devised which is a
complete ordering, then primitive sets can be defined from this alge-
braic formulation of. P even if the nondegeneracy assumption is
violated. Specifically, Scarf and Hansen have chosen the following
tie-breaking rule:
Definition: Let x = (x_, ..., x ) and y = (y,, ..., y ). We define
x. to be larger than y. (written x > y ), if and only if, the
vector (x. . . . . , x , x. ..... x . n ) is lexicographically larger than
i n i i-i
(y ..... , y , y , ..., y ). We also define y.^ to be smaller than
x (written y < x.), if and only if, the vector ,(y , ..., yn>
y1 , ..., y. -) is lexicographically smaller than (x ...... x , x-, ...,
50
-------
(0,1,4)
(1,1,3)
(0,2,3)
(2
(0,3,2)
(3,1,1)
(4,1,0)
(0,5,0)
(5,1,-1) (4,2,-1) (3,3,-l) (2,4,-l) (1,5,-1) (0.6.-1)
Fig. 1. — The set P formulated alge-
braically for the case of n=3, G=5,
Q2=l, and Q3=-l.
-,
51
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We may now define a primitive set; a subset Y of n distinct
column vectors from P is said to be primitive if every column vector
contains exactly one row maximizer according to the ordering >, and
there is no vector belonging to the set P which is smaller (with
respect to the ordering >) in all components than the row maximizer
(with respect to the ordering >) of the set Y. For example, the
vertices of any little triangle of Figure 1 provide a geometric
example of a set of points from a set P taken from the G-hyperplane
which determine a primitive set. To be consistent with the geometric
objects called primitive sets in the case of nondegeneracy, only those
little triangles whose sides are parallel to the sides of the large
triangle are primitive sets. For those little triangles whose sides
are not parallel to the sides of the larger triangle, it is the
smallest triangle surrounding it whose sides are parallel to the
sides of the large triangle, which forms the geometric primitive
set. For example, the little triangle with vertices (2,2,1), (2,3,0),
and (1,3,1) will not in itself be a primitive set. However, these
vertices determine the primitive set whose vertices are (1,2,2),
(3,2,0), and (1,4,0). Since we will be dealing only with the vertices
which determine primitive sets, i.e., the vertices of the little
triangles, we will hereafter refer to the little triangles as primitive
sets.
In defining a primitive set Y above, it was specifically required
that each column of Y contain exactly one row maximizer (according
to ordering >). This defining property is needed to guarantee that
52
-------
each primitive set in P will be determined by a unique subset of
n elements from P. For example, without this property, the three
points in Figure 1, (3,2,0), (2,2,1), and (1,3,1) would determine
the same primitive set as the three points (2,3,0), (2,2,1), and
(1,3,1), i.e., the primitive set with vertices (1,2,2), (3,2,0), and
(1,4,0). This can be readily seen by drawing lines parallel to the
sides of the large triangle through the three points in each set.
The reader can easily verify that there are other three-point sets
determining the same primitive set if the definition does not require
specifically that each column of a primitive set Y contains exactly
one row maximizer (according to the ordering >).
Anticipating the results of the following lemmata, it will be shown
that the replacement operation, replacing a vector in a primitive
set with another vector from the set P to form a new primitive set,
is equivalent in geometric terms to lifting up one of the vertices of
a little triangle in Figure 1 (the vector to be removed from the
primitive set) and pivoting the entire triangle over the two remaining
points, until the vertex which was lifted lies on top of the opposite
vertex of the adjacent triangle. This vertex will then be the unique
replacement vector. Specifically consider in Figure 1 the primitive
set (3,2,0), (2,2,1), and (2,3,0) and the operation of removing the
point (3,2,0) from this set. The unique replacement from P will be
the vector (1,3,1).
While the theory of primitive sets with the unique replacement operation
will be developed with the elements of this set P on the G-hyperplane,
53
-------
it should be noted at the outset that in most applications known so
far, and in particular for the applications of this paper, the set P,
K-
actually used in the Fundamental Theorem will consist of vectors
taken from the unit hyperplane. In fact, the theory developed for
the set P will be shown to apply even more generally to a set P,
1C
whose elements are taken from some H-hyperplane, where H is any
n
rational number, i.e., if pGP, £ p. = H. This will be accom-
k i=l 1
plished by letting X = H/G and showing that the isomorphism which
maps (k1 , .. ., k ) G P into (Xk1 , . .., Xk ) €E P, will preserve
x n .L n &
primitive sets. With this mapping the elements of P, will become
closer to each other with larger values of G so that G will be
called the grid size for the H-hyperplane.
We will now state and prove a series of lemmata which will characterize
primitive sets in this algebraic formulation of P and which will
demonstrate the technique of finding the unique replacement vector
from P when a vector is removed from a primitive set. After
demonstrating that this theory applies more generally to a set P,
it
derived from the set P by the isomorphism introduced above, the
Fundamental Theorem will be proved using this algebraic formulation
of P, . The proofs are adapted from Hansen and Scarf [5].
To simplify the presentation, define the matrix K to be an n x n
matrix with integral entries such that k.. ^ Q. for i = 1, 2, ..., n
and j « 1, 2, ..., n and whose column sums are identical, i.e.,
n
Z k = G for some fixed integer G. Let the column vectors of K
1=1 XJ
represent a primitive set. We will assume for the following three
54
-------
lemmata that the columns of K have been arranged in lexicographically
decreasing order. Let us define I*(j) as indicating the position of
the unique row maximizer for the j"1 column according to the complete
ordering >. By the assumption that the columns of K form a primitive
set, (l*(j)} can be considered a permutation of the integers (1, 2,
..., n). We will use the following convention: if j = 1, j - 1 = n:
if j = n, j + 1 = 1; if I*(j) = 1, I*(j) - 1 = n, and if
I*(j) = n, I*(j) +1 = 1.
As an example of a matrix K whose column vectors form a primitive
set, consider the following:
0
1
17
2
0
1
16
3
-1
2
16
3
-1
1
17
3
with G = 20. The permutation giving the row maximizers for this
matrix is I*(l) = 1, I*(2) = 4, I*(3) - 2, and I*(4) - 3. The
next four lemmata will prove that this is indeed a primitive set.
Lemma 1; No row of K consists of identical elements.
Proof: Let us assume that all the elements in the itn row are
identical and let I*(j') = i and I*(j*) = i + 1, i.e., column j1
has the row maximizer for row i and column j* has the row maximizer
for row i + 1. Now by assumption the (i + l)tn element in column
j* is greater, with respect to .>- than the (i + l)th element in
55
-------
column j ' . Thus since the elements immediately above them in row i
are identical, the ith element in column j* must be greater with
respect to ^, than the i^ element in column j'. But this
contradicts the assumption that I*(j') = i, i.e., that column j1
had the row maximizer for row i. Therefore no row of K consists
of identical elements. Q.E.D.
Lemma 2; In any specific row of K, no two elements can differ by
more than one unit.
Proof; Let us look at row i and assume that k^. , £ k ^ - 2 for
some columns j1 and j* where I*(j*) = i. Now construct a vector
(kljf, ..., \.lty ~ 1. fc±ji + 1» ki+l,j« ..... knjf)' in P< This
vector will have its mth component smaller (with respect to >)
than the greatest (with respect to >) element in row m for all
m, which contradicts the assumption that the set K was primitive.
To see this, write in juxtaposition the column vectors j1, j*, the
new vector, and the vector j where I*(j) » m, starting at row m.
56
-------
j' J* J New
kmj' kmj* kmj kmj'
ki-l,J*
V V
knj' knj*
klj' klj*
km-l,j*
There are three situations to consider: m * i-1, m = i and m / i-1
or i. In the latter case, where m ? i-1 or i, the first element
k in column j will, by definition, be greater with respect to >
than the element k ., in column j'. Since the constructed vector
mj'
is identical to the j'^ vector down to row i-1 and at that point
is one unit smaller, then clearly the rat-h element in the constructed
vector is smaller with respect to > than the row maximizer k ..
m mj
The second case to be considered is where m m i-1. In the above
juxtaposition of vectors, this situation occurs when row i-1 is
placed at the top and all the rows above it are displaced to the
57
-------
bottom. Again k. . . is assumed to be the row maximizer, so that
1~1>J
k. is larger with respect to > than k. ., in the jth
i~-L»j i—l i—l,j
vector. But the element k. 1 . , - 1 in the (i-l)*-"- position of
i~l > j
the constructed vector is strictly smaller numerically than the element
k. ., in the j1*-" vector and is consequently smaller with respect
to . >.. than it. Because the ordering . >- is transitive, the
J-""".L I™".!.
element k . ., + 1 in the constructed vector is smaller with respect
i~i > J
,> than the row maximizer k. ,. Finally, the last case to be
i—l i—l,j
considered is where m = i, i.e., where the juxtaposition of vectors
begins with the ith row at the top and all the rows above it are
displaced to the bottom. In this case column j* and column j are
identical since they both contain the row maximizer for row i. Thus
k. .A equals k.. and since k.., is numerically smaller than k..
by at least two units, the i*-" element in the constructed vector
is numerically smaller by at least one. Again, because it is numeri-
i
cally smaller k. . , must be smaller with respect to :>. Thus in all
cases the itn coordinate in the constructed vector is smaller (with
respect to >) than the row maximizers for all i = 1, 2, ..., n.
This contradicts the assumption that the set K of vectors is a
primitive set. Therefore, in any specific row of K, no two elements
can differ by more than one unit. Q.E.D.
Lemma 3; All the rows of K are decreasing with respect to :> in
the sense that k. .^ > ... £ k ^ k £ ... £ k.^ ,)fc_1 for all i,
where j* depends on i.
58
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Proof; The proof will be by induction on i, the row index. First
when i = 1, row 1 is decreasing with respect to > by the assump-
tion that the columns of K were arranged in lexicographically
decreasing order. Now suppose that the lemma is true for row m-1,
i.e., row m-1 of K is decreasing with respect to the ordering
>, . By the assumption that the columns of K form a primitive set,
this row must have a row maximizer at some column, say column j.
Thus by Lemma 1 and Lemma 2 we have k = k in=...=k ,.*,
J m-1, 3 m-l,j+l m-l,3*-l
and k -,..1-l = k -,.-=. . . = k n =...=k , . - for some
m-l,3*-l m-l,j* m-l,n m-l,j-l
j*. The elements in row m below each set of equal elements in row
m-1 must be decreasing with respect to > since row m-1 is by
assumption decreasing with respect to >, . Thus k . > k .,_>...
m-± mj m m,j+l m
> k and k ..>...> k >...>k . _. Note that we need
m m,3*-l mj* m m mn m m m,j-l
not consider the situation where k . . .. > k , . for then column j
m,jx-l m mj*
would contain the row maximizer for the mtn row. But since by
assumption column j already contains the row maximizer for row m-1
this contradicts the assumption that the columns of K formed a
primitive set. To prove that row m is strictly decreasing with
respect to > we need only show that k . .. > k . . By transitivity
m J m,j-l m mj
of the ordering :?, if this is true, column j* will contain the
row maximizer for row m. Assume then that k . > k . n . Construct
mj m m,j-l
a vector in P from column j-1 whose ifc component is strictly
smaller than the maximizer (with respect to |) of the corresponding
i.th row for an i =s i} 2 ..... n. Specifically, this vector is
*• i •; i ' • • • > k _ . — 1, k - . _ + 1 , k , , , . • . , k , . ) . To
1,3-1 m-2,j-l m-1, j-1 m,j-l n,j-l
59
-------
see this note that all the components of this new vector except the
(m-l)*-" are smaller with respect to the ordering :> than the
components of column j-1, which are in turn less with respect to
:> than (in one component it is equal to) the corresponding row
maximizers. Hence, except possibly for the (m-l)^ position, the
constructed vector is less with respect to 3- than the corresponding
row maximizers . Now by Lemma 1 and Lemma 2 k 1 . 1 is numerically
m—J. , J"~-L
one unit less than k , . . Thus k n . n + 1 is numerically equal
m-l,j m-l,j-l J n
to k
Consequently in order to compare these two components with respect
to the ordering >1 one must compare the components immediately
below them in row n. But by assumption k . > k . .. ; thus
mj m m,j-l'
k n . n + 1 <, k . . . Now k , . is the row maximizer for row
m-1. Thus all the components of the new vector are smaller with
respect to ? than the corresponding row maximizers, which contradicts
the assumption that the set of column vectors of K were primitive.
Thus the assumption that k . > k . 1 is false, which demonstrates
r mj m m,j-l
that the components of row m are strictly decreasing with respect
to > starting at column j*. Therefore all the rows of K are
decreasing with respect to > in the sense that when I*(j*) - i,
k. .* J ... <• k. :> k. , > ... > k. .. , for all i = 1, 2, ..., n.
ijx i i in i ll l i i>J J-
Q.E.D.
The following lemma characterizes primitive sets taken from the set P
as defined above.
60
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Lemma 4: Let K be an n x n matrix and I* be the permutation on
the n first positive integers defined above. Then the columns of
K represent the vectors of a primitive set, if and only if, there
exists a permutation I*(j) of the integers (1, 2, ..., n) and a
rearrangement of the columns of K such that the jth column of
K is identical with column j-1 except for the entry in row I*(j),
which is one unit larger, and the entry in row I*(j)-l which is one
unit smaller.
Proof; The necessity of the conditions follows immediately from the
previous three lemmata. Indeed, by rearranging the columns of K in
lexicographically decreasing order, Lemma 3 has shown that the mth
row of K is decreasing with respect to > starting at some column
j* where I*(j*) = m. In the proof of Lemma 3 the components of
row m-1 were shown to decrease by one unit at column j*, i.e.,
k™ i 4* i ~ ! = k n ... Since I*(j*) = m, the mth component
ui— j.,33*—i m—1,]*
in column j* must be one unit greater than the mth component in
column j* - 1, i.e., k ,. , + 1 - k ..... But this is precisely the
m,3*—1 mj"
conditions of the lemma.
The sufficiency of the conditions will be demonstrated by induction
on m, the size of the matrix K. If K has only one row, the lemma
is trivially true. Instead of demonstrating directly that the truth
of the lemma for K of size m-1 implies its truth for K of size
m, we will prove the contrapositive. Specifically, we will assume
that the conditions of the lemma hold for a matrix K of size m
61
-------
which does not represent a primitive set of column vectors; we will
then prove that the derived matrix K" of size m-1 which satisfies
the conditions of the lemma cannot represent a primitive set of
column vectors .
Let K be a matrix of size m which satisfies the conditions of the
lemma. Let k = (kn , k_, .... k ) be a vector in P which prevents
1 L m
the column vectors of K from representing a primitive set, i.e.,
k < max(k ) . Since k G P, k must have integral entries such that
J ^ m
k. ^ Q. and I k = G, the unique column sum of K.
11 i=l 1
Now k. must be equal to the row maximum with respect to the ordering
>, i.e., k = max(k. .) for some i. Otherwise, comparing k with
1 i j 1J
an arbitrary column vector of K, say the qtn, we have: k. £
max(b. .) - 1 £ k. . But if i = I*(q), max(k. .) - 1 < k. so that
j ij iq j ij iq
m m
G = Z k, < Z k, =G: a contradiction.
Suppose then that k = max(k .)• Construct from the vector k a new
vector k1 = (k.^ k2 ..... kp-l» kp + kp+l' k p+2' " ' ' km^ °f °ne
less dimension than k. Derive the (m-1) x m matrix K1 from the
matrix K by a similar construction for each column vector of K.
Every component of k1 is less with respect to the ordering ? than
the corresponding components of each column vector in the reduced
matrix K1 of dimension (m-1) x m. To see this, write in juxta-
position the column vector k' and an arbitrary column vector of K'
with the ±th row on top, where I*(j) ™ i.
62
-------
Column vector j
ku
K. _ K. - ,
p-i P-I.J
k 4-k , n k .+k .- .
p p+1 pj p+l.j
k k .
m mj
kl klj
Since k was originally less with respect to the ordering <• than
k , it is because k
-------
the p1-*1 component of the new vectors made up of the sum of the p*-*1
and (p+l)tn components. But, since k = max(k .), k ^ k . so
P j PJ P PJ
that if the tie had not been broken up the component (p-1), k = k ,
and the tie cannot be broken at the p^ component either. If then
the tie were originally broken at the (p+l)th component, it would
now be broken by the sum of the pth and the (p+1)^ component.
Consequently, every component of k' is less with respect to the
ordering > than the corresponding components of each column vector
in the reduced matrix K' of dimension (m-1) * m. This matrix will
appear as follows:
k 1 1
p-1,1
kpl*kp+l,l •
kp+2,l
* * k i j
P-1.J
pj p+1,
• • k
• • • k ,
p-l,m
. • • • k +k . .
J pm p+l,m
• • • if
p+2,m
k. •••k. '"'k
mJ. mj mn
such that every component of the vector k' in P is smaller with
respect to the ordering > than the corresponding row maximizer.
Now consider the column j* such that I*(j*) » p+1. Both the
(j*_l)th and the j*th coiumns of K1 are identical. To see this
64
-------
recall that K satisfies the conditions of the lemma which state
that two adjacent columns will differ only in two rows. In this case
column j* and j*-l of K will differ only in rows p and p+1.
Thus k , - ,. will be one unit larger than k . - .. n and k ...
* B *-l pj*
will be one unit smaller than k ^ - . Hence, the sum of the p*-*1
and the (p+l)tla components will be identical in column j*-l and
in column j*. Consequently the matrix K" derived from K' by
omitting column j* satisfies the conditions of the lemma; it does
not represent a primitive set of column vectors. Since the size of
K" is m-1, the induction hypothesis is proved. Therefore, the
conditions of the lemma are sufficient to characterize primitive sets
given the complete ordering £. Q.E.D.
Lemma 5; Let K represent a primitive set as defined in Lemma 4.
Then the unique vector (k' , ..., k1.)1 defined by:
and
k' - k + 1 for i - I*(j) - 1 and i - I*(j+l)
k' = k otherwise;
is the unique replacement for column j. If I*(j) - I*(j+l) - 1,
then k' » k.. - 2 for this common value of i, and if
I*(j) - 1 " I*(j+l), k' » k. + 2 for the common row. If any
k' < Q. then there is no unique replacement from P which will
form a primitive set with the remaining vectors after the column
vector j is removed.
65
-------
Proof; All that is needed is to show that the new matrix satisfies
the conditions of Lemma 4, since the replacement technique will always
generate a unique vector. By Lemma 4 any two adjacent columns of a
primitive set will differ in only two rows. Hence if one compares
three columns j-l5 j and j+1, they will differ by at most four
rows, namely I*(j)-l, I*(j), I*(j+l)-l, and I*(j+l). Assuming that
these are distinct the submatrix will appear as follows:
J-l
j
J+1
I* (j+1)
a+1
b
c+1
d
a+1
b
c
d+1
a
b+1
c
d+1
where a, b, c, and d are integers bounded from below by their
respective Q..
The replacement technique stated in the lemma will modify column j
and produce a new submatrix as follows:
J-l J
" a+1
b
c+1
d
a
b+1
c+1
d
a ""
b+1
c
d+1
This new matrix satisfies the conditions of Lemma 4 with a new per-
mutation I*' defined by I*'(J) = I*(J+D and I*1 (j+1) = I*(J)
and equal to I* otherwise.
66
-------
When I*(j)-l = I*(j+l) the three columns differ in only three
rows. In this case the submatrix becomes:
j-l
a+1
b+1
c
j
a+1
b
c+1
J+l
a
b+1
c+1
In this case the replacement technique will produce a new submatrix
as follows:
j
J+l
a+1
b+1
c
a
b+2
c
a
b+1
c+1
which also satisfies the conditions of Lemma 4 with the same permutation
given above.
The last possibility is when I*(j) = I*(j+l) - 1. In this case the
submatrix becomes:
J-l
J
J+l
I*(J)-1
I*(J)
I* (J+l)
a+1
b
c
a
b+1
c
a
b
c+1
which the replacement technique changes to:
J+l
a+1
b
c
a+1
b-1
c+1
— i
a
b
c+1
67
-------
As before, this satisfies the conditions of Lemma 4 with the same
permutation, unless b = QT*/.\j then there is no unique replacement
from P which will form a primitive set with the remaining vectors
after the column vector j is removed. Q.E.D.
The replacement operation defined in Lemma 5 is reversible in the sense
that if the vector which was the unique replacement is removed from the
resultant primitive set, its unique replacement is precisely the vector
which was removed originally.
We noted above that for the applications in this paper the elements
of P, would be taken from the unit hyperplane. The lemmata one
K.
through five have characterized the set K and the unique replacement
of primitive sets from P for a set P from the G-hyperplane, i.e.,
the components of the vectors summed to G.
Note that the sign of the integer G has never been specified in any
of the preceding lemmata, and that the proofs do not depend on it.
This implies that primitive sets can also be defined on hyperplanes
whose components sum up to a negative value of G. We wish to show,
moreover, that for any positive or negative rational number H, one
can define primitive sets from the elements of P on the H-hyperplane
by an appropriate mapping of primitive sets from the G-hyperplane,
where G is any integer positive or negative. In order to prove
this, we will first demonstrate an alternative definition of primitive
sets taken from the set P defined on the G-hyperplane.
68
-------
Lemma 6: Let P be the set of all vectors in n-dimensional Euclidean
space of the form (k.. , ..., k ) where each k. is an integer
n
bounded from below by some integer Q. and such that E k. = G
1 i-1 1
where G is an integer. A set K of n column vectors from P
represents the vectors of a primitive set, if and only if, every column
vector of K contains exactly one row minimizer according to the
ordering > and there is no vector belonging to the set P which is
greater (with respect to the ordering >) in all components than the
row minimizer (with respect to the ordering >) of the set K.
Proof; To prove Lemma 6 we will show that the conditions of the Lemma
will provide a characterization of the columns of the matrix K that
is equivalent to the characterization of primitive sets obtained in
Lemma 4, and that these conditions are sufficient for such a character-
ization. The proof will proceed in six steps.
For the first three steps we will assume that the columns of K satisfy
the conditions of the Lemma and that they have been arranged in lexi-
cographically increasing order. I(j) will be defined to be the row
of K which contains the single row minimizer (with respect to the
ordering >) for column j.
1. No row of K consists of identical elements. This follows
immediately from the proof of Lemma 1 by making the following changes:
replace the symbol "I*" with "I", the words "maximizer" with
"minimizer" and "greater" with "less."
69
-------
2. In any specific row of K, no two elements can differ by more than
one unit. This follows immediately from the proof of Lemma 2 by making
the following changes: replace the symbol "I*" with "I," the
words "maximizer" with "minimizer," "greater" with "smaller," "smaller"
with "greater," and "greatest" with "smallest"; change the assumption
k. . $ k. ., - 2 to k. . 5 k. . . + 2; and replace the constructed
ij ij* 13 ij*
vector with (k , , . . . , k , + 1, k , - 1, k , ..... k ,).
-"-J •«- -"-jj -LJ -L~J-jJ "J
3. All the rows of K are increasing with respect to > in the sense
that kljjt < ... < kin <: k±1 | ... <: ^.j*.! f°r all i where j*
depends on i. This follows immediately from the proof of Lemma 3
by making the following changes: replace the symbol "I*" with
"I," the words "decreasing" with "increasing," "maximizer" with
"minimizer," "smaller" with "larger," and "less" with "greater";
change the direction of all inequalities; everywhere that a 1 is added
to a component, subtract a 1 and everywhere that a 1 is subtracted
from a component, add a 1.
4. Let K be an n x n matrix and let I be the permutation on the
n first positive integers defined above. Then the columns of K
satisfy the conditions of Lemma 6, if and only if, there exists a
permutation I(j) of the integers (1, 2, . . . , n) and a rearrangement
of the columns of K such that the j*-*1 column of K is identical
with column j-1 except for the entry in row I(j), which is one
unit smaller, and the entry in row I(j) - 1 which is one unit
larger.
70
-------
As was shown in the proof of Lemma 4, the necessity of these conditions
follows immediately from the first three parts of this proof. The
proof of the sufficiency of these conditions also follows immediately
from the sufficiency proof in Lemma 4 by making the following changes:
replace the symbol "I*" with "I," the words "maximum" with
"minimum," "less" with "greater," "larger" with "smaller," and
"maximizer" with "minimizer," change the direction of all inequalities;
and change the expression "max(k..) - 1" to read "1 + min(k ),"
and the expression "max(k..)" to read "min(k..)."
J J
5. The characterization of every set K of n column vectors from
P which satisfies the conditions of Lemma 6 is equivalent to the
characterization of primitive sets in Lemma 4. Indeed, let K be
a primitive set as in the proof of Lemma 4. Then the columns of K
are arranged in lexicographically decreasing order. Moreover, row
I*(j) and I*(j) - 1 for any two columns j and j-l of K appear
as follows:
j-l J
1 f a+1 a "1
Now if the columns of K are rearranged in lexicographically increasing
order, any such adjacent two columns will simply be interchanged and
appear as follows:
j-l J
I(j)-l f a a+1
Kj) lb+1 b
71
]
-------
where the permutation I* indexing row maximizers has been exchanged
for the permutation I indexing row minimizers. Since any two
columns of the set K in Lemma 4 and any two columns of the set K
in Lemma 6 differ only in two components, the rearrangement above
shows clearly that the set K in Lemma 4 is identical to the set K
in Lemma 6 except for the lexicographical ordering in the opposite
direction. Consequently, the characterization of any set K of n
column vectors from P which satisfies the conditions of Lemma 6 is
equivalent to the characterization of primitive sets in Lemma 4.
6. Both characterizations by indexing permutations as described in
Lemma 4 and Lemma 6 are necessary and sufficient for the set K to
satisfy their corresponding set of conditions respectively. Hence
the two sets of conditions are equivalent. Consequently, since the
set K in Lemma 4 is a primitive set, the set K in Lemma 6 is
also a primitive set. Therefore, a set K of n column vectors
from P represents the vectors of a primitive set, if and only if,
every column vector of K contains exactly one row minimizer
according to the ordering £ and there is no vector belonging to
the set P which is greater (according to the ordering >) in all
components than the row maximizer (with respect to the ordering >)
of the set K. Q.E.D.
It is interesting to note that with primitive sets defined in terms
of row minimizers, the replacement operation is completely analogous
to that of Lemma 5. Specifically, replace I* with I and reverse
operation of addition and subtraction in the statement of Lemma 5.
72
-------
We will now demonstrate that the elements of a set P, can be taken
k
from any hyperplane which is parallel to the G-hyperplane, where the
components of a vector of the hyperplane sum to any rational number H.
Lemma 7;
Let 1. H be a rational number which will identify the hyperplane from
which the elements of a set P, should be taken in the sense
i n 1
that if pj G P, , then E pV = H.
k i=l 1
2. G be an integer whose absolute value will determine the number
of elements k of P, .
3. P be the set of all vectors in n-dimensional Euclidean space
of the form (k-, ..., k ) where each k. is an integer
1 n ± n
bounded from below by some integer Q. and such that £ k. = G.
1 i=l 1
We will say that if p G P, p is on the G-hyperplane.
4. X = H/G be a rational number.
Then, the mapping of (k , ..., k ) into (Xk , ..., Xk ) of the
elements of P into the elements of P, will preserve the primitive
1C
sets as characterized by Lemma 4 and will preserve the unique replace-
ment operation on primitive sets as characterized by Lemma 5.
Proof; The mapping of (k,, ..., k ) into (Xkn, ..., Xk ) of an
" ' J. n J. n
element on one hyperplane to an element in a parallel hyperplane is an
isomorphism for any rational number as can easily be verified.
In order to demonstrate that the results of Lemma 4 and Lemma 5 are
carried over to elements of P. taken from the H-hyperplane, all that
need be demonstrated is that this mapping preserves primitive sets.
73
-------
Specifically, let P be a set of vectors as defined in the Lemma and
let K be a matrix whose columns are vectors from P and such that
they form a primitive set with respect to the ordering >. It must
be shown that the columns of the matrix XK form a primitive set
from P, with respect to the same ordering.
Let K be such a matrix whose columns form a primitive set from P
with respect to the ordering >. Then, by definition, every column
of K contains exactly one row maximizer with respect to the ordering
and there is no vector from P which is smaller (with respect to the
ordering <>) than the row maximizer with respect to the ordering >
in all components. Furthermore, by Lemma 6, every column of K con-
tains exactly one row minimizer and there is no vector from P which
is greater (with respect to the ordering :>) than the row minimizer
with respect to the ordering >. Now XK must be a primitive set
from P , i.e., each column of XK must contain exactly one row
K.
maximizer and there must not exist any vector from P, which is
K.
smaller (with respect to the ordering >) than the row maximizer
with respect to the ordering > in all components. In order to see
this assume that the column vectors of XK do not form a primitive set»
i.e., either there is a column vector from XK which does not have a
row maximizer with respect to the ordering ? or else there exists
a vector pj e P^. such that p^ < max^Xk.. I j=l, ..., n} for all
i. First assume that X > 0. If there is a column of XK which does
not have a row maximizer, then the corresponding column of K will
not have a row maximizer, which contradicts the assumption that the
74
-------
columns of K formed a primitive set. Thus there must exist a
E P, such that p^
,
K.
then pj = — £ max^k.^. I j=l , . .., n} for all i so that the
column vectors of K do not form a primitive set which again contra-
dicts the assumption that the columns of K did form a primitive
set. Now assume that X < 0. If there is a column of XK which
does not have a row maximizer, then the corresponding column of K
will not have a row minimizer which contradicts the assumption that
the columns of K formed a primitive set. Again if the column
vectors of XK do not form a primitive set it must be that
p^ £ max^Xk. I j=l, ..., n} for all i. But then p! =
'' /
— > min ^k I j=l, ..., nj for all i which again contradicts the
assumption that the column vectors of K formed a primitive set. Thus
the column vectors of XK must form a primitive set and the mapping
(k , ..., k ) into (Xk_ , . . . , Xk ) of the elements on the G-hyper-
plane to elements on the H-hyperplane preserves primitive sets. Clearly,
by a similar reasoning, the inverse mapping from P to P also
K.
preserves primitive sets. Q.E.D.
We have thus shown an alternative formulation of the elements in P,
k
which are readily generated algebraically. We now have the machinery
which is necessary to state and prove the central theorem of the paper.
The Fundamental Theorem;
Let 1. P, be the set of all vectors in n-dimensional Euclidean space
of the form (Xk- , . . . , Xk ) where each k. is an integer
In i
75
-------
bounded from below by some integer Q , G is an integer
such that Ek. = G, and X = H/G so that if p-' e P ,
n i x k
Z PI = H.
2. B be a matrix whose jtn column corresponds to the j^
column of P, . This correspondence will be arbitrary except
when an element of P, contains a Xk. such that k, = Q
1C X XX
(a boundary element of the set P, ). In this case the vector
in B will be the unit vector with a one at the position of
the first such k..
3. w be a nonnegative vector in n-dimensional space.
Assume that the set of nonnegative vectors x satisfying the system
of equations Bx=w is bounded. Then there exists a primitive set
P » P » •••> P from P, as characterized by Lemma 4 and Lemma 7
such that the columns i-, •]„, ..., j of B form a feasible basis
i L n
for the system Bx=w.
Proof; Any primitive set as characterized by Lemma A will serve as
a starting primitive set which has n - 1 of its vectors each having
a component on a boundary so that these n - 1 vectors are associated
with n - 1 different unit vectors, and the remaining vector, call
k*
it p , of the primitive set has no component on the boundary. Georoe'
trically such a situation can be seen in Figure 1. Any of the little
triangles marked with an S will have their vertices satisfying the
above criteria. Consequently finding a starting primitive set from
P, will always be possible and also very easy. Also the columns 1 W
n of the matrix B will form a feasible basis for Bx=w, with column
k*
k not associated with p
76
-------
There are n - 1 columns of B both in the feasible basis and
uniquely associated with columns of a primitive set from P, at
1C
the beginning of the algorithm and the technique of the algorithm
will be to alternate between simplex pivots and the replacement
operation on the primitive set described in Lemma 5 so that at any
intermediate stage there will remain n - 1 columns of B both in
the basis and associated with the primitive set from P. . The
algorithm will terminate at the time that all the columns of B
which are in the basis form a nonnegative solution to Bx=w and are
also associated with a primitive set in P, . This is, of course, the
statement of the Fundamental Theorem. So what remains to be shown is
that such a position will be reached.
At the outset it should be noted that the assumption that the set of
nonnegative vectors x satisfying the system of equation Bx=w is
bounded, guarantees that at any stage of the algorithm any vector
not in the feasible basis can be brought into the feasible basis by
a simplex type pivot. Indeed, the only circumstance in which it is
not possible to carry out a simplex pivot is when there is no positive
element in the column vector to be introduced into the basis, i.e.,
no pivot element can be found. Suppose at some stage of the algorithm
the j column of B contained only nonpositive elements. This
n _
implies b. = Z b. .b. where $>.)-. is the current basis and
b.. £ 0 for all i = 1, 2, ..., n. Now w « Z w.b. in terms of
n n
that basis so that w = Z w b. + x. (b - Z b..b.) for arbitrary
77
-------
n _
x. > 0. Thus w = E (w. - x.b )b -x.b is a nonnegative solution
J i=l ^" ^ J 3
to Bx=w for any x. > 0 since (w - x.b. ) > 0 for i = 1, 2,
•J sJ J
..., n. Since x. > 0 is arbitrary the set of nonnegative solutions
x to Bx=w is unbounded. Thus if it is not possible to carry out
a simplex pivot, the set of solutions x to Bx=w is unbounded.
Consequently, the boundedness of nonnegative solutions x to Bx=w
guarantees that a simplex pivot can be carried out at any stage of
the algorithm.
At the starting position there is only one possibility for the algorithm
namely to introduce column k* of B into the basis. There is no
k*
replacement in P, for the vector p since this is the special case
referred to in Lemma 5, i.e., when k' < Q.. Now when column k*
of B is introduced in the basis a unique vector b will drop out
of the basis. If d = k the algorithm would terminate since each
of the vectors of the primitive set would be associated with columns
of B which form a feasible basis for the nonnegative solution to
Bx=w. If this is not the case the algorithm will drop p out of
d*
the primitive set and some unique vector p will be introduced
d*
into the primitive set. At this stage only p of the primitive
set is not associated with a column of B in the feasible basis. The
algorithm thus alternates between simplex pivots and the unique replace'
ment operation on primitive sets and always guarantees that it is the
unit vector with a one at the kth position which is in the feasible
basis but is not associated with a vector in the primitive set, until
the algorithm terminates.
78
-------
At every intermediate stage of the algorithm there are only two
possibilities. Either a column of B is introduced)replacing a column
currently in the basis or else a vector in the primitive set is
withdrawn and a unique replacement from P, is added. The operations
are alternated so that if a particular position is reached by a simplex
pivot on the B matrix then the next step of the algorithm is a re-
placement operation on the primitive set. On the other hand, if the
particular position is reached by a replacement operation on a primi-
tive set, then the next step of the algorithm is a simplex pivot on
the B matrix.
The unique replacement vector generated by the procedure of Lemma 5
will always be from the set P, itself, that is, we will never reach
a position which is the exceptional case in Lemma .5 above, i.e., when
k' < Q . Only in the last case presented in the proof of Lemma 5
could a replacement vector be such that it contains a , k' < Q .
This would occur if b = QT*/.N. For convenience we will rewrite the
•L'Hj.)
matrix for this case when I*(j) » I*(j+l) - 1 and we are removing
column j.
j-1 j j+1
a+1
b
c
a
b+1
c
a
• b
c+1
What we wish to show is that this situation with b a boundary element
could never occur in the algorithm of the Fundamental Theorem. While
this argument will be carried out for primitive sets defined from the
79
-------
set P on the G-hyperplane, by Lemma 7 this argument applies to
primitive sets defined from the set P, of this corollary. Specif-
ically we will assume that b is a boundary element and that this
situation was arrived at in the algorithm of the Fundamental Theorem,
and show that this leads to a contradiction.
At the outset we should recall that columns j-ls j and j+1 are
identical except in the rows listed and that any column with a component
on the boundary will be associated with the unit vector with a one at
the position of the first or highest boundary element coordinatewise.
Since b is a boundary element it must be the first boundary element
in column j-1, for if it were not, column j-1 would have another
boundary element in some row above row I*(j)-l, since (a+1) cannot
be a boundary element. Suppose that the highest such boundary element
appeared in row p where p < I*(j)-l. Since column j+1 is
identical to column j-1 in row p, column j+1 must also have the
same first boundary element. Thus both column j+1 and j-1 will
be associated with the same unit vector in the basis after column j
is replaced, which is impossible in the algorithm of the Fundamental
Theorem. Thus b is the first boundary element in column j-1.
Now if a is not a boundary element, we have a similar problem. The
b element in column j+1 will be the first boundary element and again
column j+1 and j-1 will be associated with the same unit vector.
Thus, since by assumption this situation has been arrived at by the
algorithm of the Fundamental Theorem, the element a must be a boundary
80
-------
element. This means that both column j and column j+1 are
associated with the same unit vector. However, column j is being
replaced so that column j+1 must have just been introduced. Had
column j+1 not just been introduced then column j+1 in its present
form and column j would have both been associated with the same unit
vector and this unit vector would have been in the basis twice, and
this is clearly impossible.
Thus we must examine the situation before the previous numbers in column
j+1 were modified by the technique of Lemma 5 to produce the current
column j+1. In this case we have:
j+2
e
d+1
a
b
I*1 (j+1)
j-1
e+1
d
a+1
b
c
j
e+1
d
a
b+1
c
j+1
e
d+1
a
b+1
c
I*1(j+2)
where we have written I*1(j+1) 1 I*'(j) - 1.
We have used the permutation I*1 to note that at this position of
the algorithm some of the columns (specifically j and j+1) will
contain different row maximizers than in the original positions listed.
Now I*1(j+1) need not be different from I*'(j)-l and we will
consider this case without rewriting the entire submatrix. If
I*1(j+1) » I*'(j)~l then the row containing the d's and the row
containing the a's are identical. Rewriting in terms of the a's
the four elements of the row I*'(j+1) - I*'(j)-l are (a+1), (a),
81
-------
(a+1), and (a+1). If e is not a boundary element in row I*'(j+l)-l,
then the b in column j+2 will be the first boundary element and
column j+2 will be associated with the same unit vector as is column
j-1. But this is impossible so that in the case where I*1(j+1) =
I*'(j)-l the e in row I*'(j+l)~l must be a boundary element.
In order to see that the matrix above represents the situation before
column j+1 was modified, apply Lemma 5 to column j+1 and obtain the
following submatrix:
j-1
e+1
d
a+1
b
c
j
e+1
d
a
b+1
c
j+1
e+1
d
a
b
c+1
j+2
e
d+1
a
b
c+1
which is the original situation with the appropriately added rows and
columns. Note that if I*'(j+1) = I*'(j)-l the row containing the
d's could be dropped and the four elements in the row containing the
a's would be changed to (a+1), (a), (a), (a+1). Again this reflects
the starting situation for this row.
There are three cases to be considered in the situation before column
j+1 has been modified. The first situation is when the column j,
j+1, and j=2 differ in four rows. This is the case illustrated in
the matrix above. The other two situations occur when these three
columns differ in only three rows, i.e., when I*'(j+2) = I*'(j+l)-l
or when I*'(j+2)-l = I*1(j+1). We will treat these last two cases
82
-------
first. First if I*'(j+2) = I*'(j+l)-l the row depicted by e's
and the row depicted by c's are identical. If I*'(j+l) ± I*'(j)-l
as depicted in the matrix the component a in column j+2 and j
will be the first boundary element and both columns will be associated
with the same unit vector in the feasible basis, which is impossible.
Furthermore if I*'(j+1) = I*'(j)-l then the b component in column
j+2 and j-1 will be the first boundary element, and again, this
is impossible in the algorithm of the Fundamental Theorem. Hence
I*'(j+2) cannot equal I*'(J+D-1. Also I*1 (j+2)-l = I*' (j+1) is
not possible since we know that I*'(j+2)-l = I*'(j), which would
then imply that I*'(j) = I*1(j+1) which is impossible by the definition
of I*1 as a permutation indexing the row maximizers. Indeed, no two
columns of a primitive set can have the same row maximizer. Thus it
must be that the three columns j, j+1, and j+2 will differ in
four rows as depicted in the matrix above.
Given the matrix above depicting the situation before row j+1 was
modified with the four distinct rows in columns j, j+1, and j+2,
we will show that I*'(J+1)-1 < I*'(J)-1, i.e., that row I*'(j+l)-l
is above row I*'(j)~l arid that the element e is a boundary element.
Note first that d cannot be a boundary element since b is the
first boundary element in column j-1. Now if I*1(j+1) £ I*'(j)-l
as depicted in the matrix and if I*'(j+l)-l were greater than
I*'(j)-l, i.e., if row I*'(j+l)-l were below row I*'(j)-l» or
if the e component in row I*(j+l)-l were not a boundary element,
then the a component in column j+2 and j would be the first
83
-------
boundary element in both columns. By the same argument as given above
this is impossible. Also if I*' (j+1) = I*'(j)-l and row I*'(j+l)-l
were below row I*'(j)~l and the e component were not a boundary
element, then the b component in columns j+2 and j-1 would be
the first boundary elements, which is impossible.
Consequently e must be a boundary element and I*'(j+l)-l < I*'(j)-l,
and since all the columns differ only in the elements shown and b is
the first boundary element in column j-1, the e must be the first
boundary element in column j+1 and j+2, and both of these columns
must be associated with the same unit vector. Arguing as before,
since column j+1 is being removed, it must be that in the previous
iteration of the algorithm of the Fundamental Theorem the current
column j+2 was entered by means of Lemma 5.
Moving back one more step to the situation before column j+2 was
modified we have:
j-1 j J+1 j+2 j+3
I*" (j+2)
I*" (j+D
g+1
f
e+1
d
a+1
b
c
g+1
f
e+1
d
a
b+1
c
g+1
f
e
d+1
a
b+1
c
g
f+1
e
d+1
a
b+1
c
g
f+1
e
d+1
a
b
c+1
I*" (j+3)
where the matrix has been written with I*" (j+2) f I*"(j+l)-l. Now
the argument is exactly analogous to the previous iteration when column
j+1 was being modified to show that g must be a boundary element in
84
-------
row I*"(j+2)-l, which must be above row I*"(j+l)-l. Column j+2
and j+3 are then both associated with the same unit vector and
column j+3 must have been introduced in the previous iteration of
the algorithm of the Fundamental Theorem.
At some stage of retracing the steps of the algorithm of the Funda-
mental Theorem, since there are only a finite number of columns in a
primitive set, we must conclude that j+n = j-1 and that column j+n
must have a boundary element in some row above row I*(j+l). But
we have shown that if b is a boundary element in row I*(j+l) of
column j+1, coordinatewise, it must be the first boundary element
of column j-1. Hence, we have a contradiction and our assumption that
b could be an element on the boundary is fallacious.
Under no circumstances then, can the replacement operation in the
algorithm of the Fundamental Theorem lead to a new vector outside the
boundaries which define the set ?k.
Many of the column vectors of the matrix B can be identical, i.e.,
the functional mapping from the elements of P, onto the set of column
vectors of B need not be one to one. This will not cause any problem
in the algorithm for if a particular column vector is in the basis and
the algorithm attempts to introduce the identical vector into the
basis, then the column vector which was in the basis originally will
be removed. This is easy to see since a vector in the basis at some
stage of the algorithm will be represented by a unit vector with a one
at some particular row. All identical vectors will also be represented
85
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by the same unit vector so that the algorithm will simply introduce
a unit vector into the basis. The pivot row will clearly be the row
which contains the one and since the vector is already in the form
of a unit vector, the remaining columns of B will not be affected.
The pivot row will be divided by one and the nonpivot rows will have
a zero element added to each column. Since this simplex pivot
operation does not affect the matrix B, it need not be carried out
in practice if there is an efficient way of keeping track of which
vectors are in the basis. It will suffice to remove from the primitive
set the vector which is associated with the identical column of B
under consideration. There is no harm in actually performing the
simplex pivot step, except for the possibility of introducing unneeded
rounding errors into the computations when performing the simplex pivot
on the matrix B.
The algorithm cannot cycle in the sense that it cannot return to the
same primitive set and consequently to the same n vectors of B,
n-1 of which are part of a feasible basis. This is because at the
start of the algorithm there is only one operation that the algorithm
can perform, namely to introduce column k* of B into the basis
with the result that the column vector b will be dropped from the
basis, i.e., that the column d from the primitive set will not be
associated with a vector in the feasible basis. Since this operation
produces a unique result and since both the simplex pivot and the
replacement operation are reversible, the only way to arrive at the
starting position is by means of the situation which occurs when b
86
-------
k* d
is not in the basis but b is, and b is being introduced. But
this will clearly not happen at the second step of the algorithm since
the next operation will be a replacement operation on P, , in parti-
K.
cular removing p from the primitive set. Thus, unless some inter-
mediate stage of the algorithm were repeated, the program could never
return to the starting position.
Any intermediate stage of the algorithm is reached either by a replace-
ment operation on primitive sets or a simplex pivot. If it is reached
by a simplex pivot then the next step will be a replacement operation
on primitive sets and vice versa. Since both of these operations are
reversible there are exactly two ways of reaching a particular position
in the algorithm, the way the position was reached, and the reverse
of the operation leading out of that position. Now if any intermediate
stage were to recur, it would imply that there was some third situation
which would lead to the intermediate stage. But this is clearly
impossible due to the uniqueness of both the simplex pivot and the
replacement operation on primitive sets. Consequently, the algorithm
cannot cycle.
Furthermore, there are a finite number of primitive sets which can be
formed from P, , a finite set of k vectors, and a finite set of
feasible bases to the problem Bx=w. Hence, the algorithm must
terminate in a finite number of steps. Q.E.D.
In order to guarantee that there is a solution provided by the
Fundamental Theorem, the assumption was made that the set of nonnegative
87
-------
solutions to Bx=w is bounded. No condition has been specified
directly on the matrix B which will guarantee that this assumption
will be met. For applications of the Fundamental Theorem it is impor-
tant that the matrix B be appropriately formulated. The following
lemma provides the required necessary and sufficient condition on the
matrix B.
Lemma 8; The set of nonnegative solutions x to Bx=w is bounded if
and only if the column vectors of B are positively linearly independent-
Proof^: First if the column vectors of B are positively linearly
dependent, then there exists a vector y ^ 0 such that By=0. But
then for any scalar A > 0, BAy=0. Now if Bx=w, it is also true
that Bx + BAy = w which says that (x+Ay) is a solution for any
A > 0. Thus since at least one coordinate of y is strictly greater
than zero, the set of nonnegative solutions to Bx=w is unbounded.
Conversely, let S be the set of nonnegative solutions to Bx=w, let
A(S) be the asymptotic cone of S, and assume that S is unbounded.
Now S is a closed convex set and, since x G S is nonnegative,
S C ft. Thus A(S) C A(ft) . By the assumption that S is unbounded
we have that A(S) f {0}. It can be proved that for any x £ R ,
A(S) = A(S - {x}) and also that if x £ S and S - {x} is closed,
convex and contains the origin, then, A(S) = A(S - {x}) C S - {x}.
Now since A(S) f {0}, there exists y e A(S) , y 5 0. Thus
y G A(S) C (S - {x» for any x £ S. Furthermore, Ay e A(S) C (S - {x})
for all A > 0 so that (x+Ay) £ S for all A > 0. Now fix A and
88
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choose a X1 > X. We have that both B(x+Xy) = w and that B(x+X'y) = w.
Thus B(x+X'y) - B(x+Xy) = B(X'-X)y » 0. Let z = (X'-X)y. Now z 5 0
since 'X'-X) > 0 so that Bz = 0 is a nonnegative linear combination
of the columns of B which equal zero. Hence the column vectors of
B are positively linearly dependent. Q.E.D.
In many applications, the Fundamental Theorem with Degeneracy is
readily programmed for digital computers. In particular, if the column
vectors of the matrix B associated with the elements of the set P,
k
can be calculated from the elements of P when required, then there
is no need to store in the memory of the computer the entire matrix B.
By the use of the revised simplex method, only those columns of B
associated with the unit vectors need be stored in the computer. When
a column of B is to be entered into the basis at some stage of the
algorithm, then that vector is transformed into the representation of
the current basis by multiplying the vector by the matrix in the columns
of original unit vectors. It is this transformed vector which is
entered into the basis by means of the conventional linear programming
pivot. Correspondingly, the elements of P, need not all be stored
in the computer. In fact all that need be stored at any one time is
the current primitive set. The unique replacement vector can be
generated algebraically and can be stored in the same position as the
vector it replaced.
The set P. in this algebraic formulation is defined on a specific
hyperplane H (in this paper H = 1 in all the applications) simply
by defining G, the grid size and all the Q., the boundaries. In
89
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a particular application one might have some a, priori knowledge where
on the H-hyperplane the solution will be found. There are two compu-
tational techniques which will greatly enhance the efficiency of this
algorithm. First, leaving the Q. at their true values (determined
by the particular application of the algorithm), one may start the
algorithm at the starting primitive set closest to the assumed answer.
In Figure 1 such starting primitive sets are the little triangles
marked with an S. This technique has the advantage that it will
always find a true solution to the problem in the sense that the
algorithm will never terminate on an artificial boundary. The second
technique is to pick some computational boundaries Q. which are
greater than or equal to the true boundaries. By choosing larger Q.'s,
one essentially chooses a smaller set P, in the geometric sense and
K.
one which has fewer elements (given the same value of G). If the
algorithm terminates at a primitive set whose vectors do not contain
an element on a computational boundary, then this result will usually
be the same as if the true boundaries had been used. This must be
qualified since it is conceptually possible for some problem to have
more than one solution. By restricting the area of search, one might
arrive at a different but true solution. However, if the algorithm
terminates at a primitive set whose elements contain any component on
a computational boundary, then the solution will not be a true solution.
The solution must be recalculated lowering sufficiently the computa-
tional boundary on which the previous solution fell so that the final
primitive set will not contain any component on a computational
boundary.
90
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Application of the Fundamental Theorem
To the General Equilibrium Model
It was shown in Section IV that only relative prices need be investi-
gated for determining the equilibrium price vector. We will restrict
our attention to those relative price systems which sum to unity.
Consequently we will define P, for the Fundamental Theorem by
K.
setting H = 1, G to some appropriate level depending on the degree
of precision desired in the result, and Q. = 0 for all i. With
X = 1/G, the boundary elements of the set P, are then XQ. =0.
l£ i
Since XQ. forms a lower bound for the ±th price, the assumption
that Q. = 0 is equivalent to the assumption that price i will be
nonnegative. The associated matrix B in the Fundamental Theorem
will be constructed as follows. As is required in all applications
of the Fundamental Theorem the column of B associated with any
p€P, which has an element XQ., in this case zero, will be the
unit vector with a one at the position of the first XQ.. We will
interpret the negative of this unit vector to be the disposal activity
which disposes of the ith commodity. The column j of B associated
with a pj G P which does not have an element on the boundary Q,
K i
is constructed as follows.
Take p'' G P, and compute the profit of each production process at
this set of prices. Recall from Section IV that for those production
processes which are represented as either a single output Cobb-Douglas
or C.E.S. production function this is a two step procedure. First the
optimum activity vector must be determined at the price system p^
91
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then the profits calculated with this optimum activity vector. In
the case that the production process is already specified as an
activity, the optimum input-output vector is simply this activity
vector. Now if the profits associated with the production process
which maximizes profits is strictly positive, enter into column j
of B the negative of the optimum activity vector. On the other hand,
if the maximum profit is negative or zero, calculate the demands for
each consumer at the prices p , and enter the market demand vector
in column j of B, i.e., the sum of the individual demands.
By construction, there is a one-to-one correspondence between the
vectors in P, and the vectors in B. Now, let w be the nonnegative
vector of resources initially owned by all the consumers in the economy.
Since the B matrix is composed of optimal activity vectors and non-
negative demand vectors, the assumption that the vectors in the pro-
duction sector are positively linearly independent presented in
Section IV guarantees that the set of nonnegative solutions to Bx=w
is bounded. Indeed, if the set of nonnegative solutions is unbounded,
by Lemma 8, the column vectors of B would be positively linearly
dependent, i.e., there would exist a nonnegative, nonzero vector x'
1 2
such that Bx' = 0. Now this solution can be rewritten Ix ' + B'x ' +
3
B"x ' = 0 where I is the identity matrix, the negative of the
disposal activities, B1 is a matrix composed of only the demand
vectors in B, and B" is a matrix composed of only the optimal
activity vectors in the production sector. Factoring out I from the
left two terms we obtain I[x1? + B'x '] + B"x ' = 0 and let
92
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1 2
z = [x ' + B'x '] which is clearly a nonnegative vector. Let
A' = [I, B"] be the matrix of the negative of the optimal activity
vectors in B and let z' = [z,x '] be a nonnegative vector. Then
from above A'z' = 0. Consequently -A'z1 =0 so that the columns
of A are positively linearly dependent and production is reversible:
a contradiction. Hence, all the conditions of the Fundamental Theorem
are verified. Therefore, there exists a primitive set P^l, pj^, ...,
pJn in p such that the columns j,, J9, ..., j form a feasible
K «L £ Ii
basis for the system Bx=w.
If we let a , be the ith element in the optimal activity vector of
the kth productive process and D.(p) be the market demand for
commodity i at price system p, the basic feasible solution to the
system Bx=w is given explicitly by:
(1) - £a±kx + SD^p^y = w± for all i = 1, 2, ..., n
where the index j is a more convenient labeling of the index j , x.
is the positive level at which the k"1 productive activity having
the largest strictly positive profit at prices p'' is to be operated,
and y is the positive weight given to the demand functions whenever
at prices p'' the largest profit derived from the productive activities
is either negative or zero.
Since each of the p s corresponding to positive a.'s or y.'s
belong to a primitive set, the p^'s are relatively close to each other.
In fact, the larger the number of price vectors selected from the unit
simplex as potential equilibirum prices, i.e., the larger the value of
93
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G, the closer the price vectors in the primitive set. Thus by re-
fining ever more the grid of potential prices on the unit simplex, a
common price vector p is obtained in the limit. Note that by construc-
tion of the matrices P, and B if one of the first n columns of
k
B, which corresponds to a disposal activity, is included in the solu-
tion, say the ifch column, the corresponding price p will be zero.
Some average of the price vectors in the primitive set can then serve
as a good approximation to the common price vector p. If the demand
functions are continuous in a small neighborhood of the primitive set,
equation (1) then becomes:
(2) - Ea x. + (Ey.)D.(p) = w. for all i = 1, 2, ..., n
. IK. j j i i
and the following relations hold:
(a) Ep.a.. £ 0 for all k whenever y. > 0 for some j;
. i IK. j
(b) Ep.a., ^ 0 whenever x. > 0 for j corresponding to
_£ i ik. 3
the k1-*1 productive activity, including disposal activities.
It is clear from (2) and (b) above that if the y.'s are all equal to
zero - Za x. = w for all i = 1, 2, ..., n and Ep.a ^ 0 for all
. ilc j i * i IK.
k for which x. > 0 so that 0 £ EZp.a x. = - Ep.w.. This last
J -H 3 i
relation leads to a contradiction if by assumption w > 0 for all
i = 1, 2, ..., n. Indeed a set of elements from P, each with a
component on a boundary, i.e., each with a zero component, cannot be
primitive. Thus p. > 0 for at least one i so that Ep.w. > 0:
1 ± i i
a contradiction. The assumption w. > 0 for all i - 1, 2, ..., n
94
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can be relaxed to w > 0 for all i = 1, 2, ..., n. Indeed, since
a primitive set from P1 cannot have all of its elements associated
JC
with slack vectors, at least one of the positive x.'s corresponds to
a productive activity which is not a disposal activity whenever all the
y.'s are equal to zero. But then for this productive activity, say
the kth activity, 0 < Ep.a., by continuity of the supply functions
so that, while £p.w. ^ 0, 0 < ZEp a x. = -Zp.w.: a contradiction.
.11 „ i IK. 3 ^11
Therefore y. > 0 for at least one j. It follows immediately from
(a) and (b) above that Ep.a., = 0 whenever x. > 0.
£ i IK J
Multiply each term in (2) by p and add over the index i to obtain:
(3) (£y.,)2p.,D (p) = Zp w
, j . i i , i i
J •*• -^
If then Walras' law holds, that is, if the consumers spend all of their
income, which can only be derived in this economy by selling their re-
sources on the market, Zp D.(p) = Zp w so that (Zy ) = 1. Substi-
tuting (£y.) = 1 in (2) it becomes clear that supply equals demand in
all the commodity markets.
Thus at some average of the price vectors in the primitive set each
consumer maximizes his utility from which process the demand functions
are derived, all productive processes which are operated at a positive
level yield a maximum profit equal to zero, and all commodity markets
are cleared. Therefore general equilibrium is attained and approximate
equilibrium prices are computed.
95
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In any actual computation of equilibrium prices the algorithm terminates
with a primitive set of prices. An average of these prices is needed
in order to calculate the equilibrium supply and demand. For an
averaging of these prices Scarf has chosen a technique of using linear
programming to find that average price which minimizes the maximum
deviation between demand and supply. Specifically he formulates the
linear program:
Maximize XL + xg + ••• + XR
Subject to:
^1 -v- .!•> • • t -^ fl -y —PL V ~ * * * ~ cl X ^ W-,
dllxl lnxn alln+l 1m n+m 1
d21Xl + •" + d2nxn ' a21Xn+l a2mXn+m < W2
dnlXl + — + dnnXn '
P11X1 + '*' + Plnxn
P21X1 + ''' + P2nXn
p,x, + • • • + p x
ml 1 mn n
{n
S D,, / (n+l)|, where D^ is the demand for
itS*. XIV
commi
todity i at the price system pj on the final primitive set and
where p±k - p±k -( \ Pik / (n+l)J where P±j is the profit of
activity i at the same price system p
96
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This linear program produces a weight to be applied to each price in
n
the final simplex. Let V = Ex be the value of the linear program.
x i=l i
Then . , . > - 1 is the weight to apply to the i price. The sum
of these weights is clearly unity.
An alternative averaging scheme has been programmed which is compu-
tationally faster and appears to give better results when computing
equilibrium prices in the allocation model. This method finds the
bary center of the points in the final primitive set of prices, i.e.,
it calculates the unweighted average of the price vectors .
Application of the Fundamental Theorem
To the Allocation Model
Section IV has described how to each allocation of money resources
imposed by the government there corresponds a set of equilibrium prices
in the regional economy. The problem for the government is to decide
which allocation of the money resources will result in a set of equilib-
rium prices in the regional economy which are as close as possible to
the set of relative prices in the larger economy for those commodities
common to both economies .
The solution to this problem is a straight forward application of the
Fundamental Theorem. Recall that the Fundamental Theorem considers
two matrices, P, and B, in the n-dimensional Euclidean space such
that the jtn column of B is associated with the j1 column of
P, , and a nonnegative vector w such that the set of nonnegative
vectors x satisfying Bx^ is bounded. Whenever these conditions
97
-------
are satisfied, there exists a primitive set p 1, ..., p n in P,
K.
such that the columns j , ..., j of B form a feasible basis for
Bx=w. Recall also that the concept of a primitive set in P can be
K-
interpreted intuitively as a set of vectors relatively close to each
other.
In the present application of the Fundamental Theorem, the matrix P
will consist of a bounded subset of column vectors on the unit hyper-
plane derived from a set P from the G-hyperplane, where G is a
positive integer, by normalizing the vectors of P to sum to unity.
These vectors will represent allocations of the money resources to
each commodity common to both economies, i.e., the components of each
vector in P, define the proportion of the money resources to be spent
K.
on each commodity common to both economies. The matrix B will consist
of column vectors representing price systems and obtained in the follow-
ing way. Each allocation on a boundary of P, will be arbitrarily
K-
associated with the unit vector with the one in the coordinate which
has the first boundary element. Each non-boundary allocation in P,
will be associated with the prices of the corresponding commodities
normalized to sum up to unity and computed from the general equilibrium
model of the regional economy with the initial vector of physical
resources derived from this particular allocation. The vector w
will consist of the set of prices in the larger economy corresponding
to the commodities common to both economies and normalized to sum up
to unity.
98
-------
Since the column vectors in the matrix B represent either unit vectors
or equilibrium price systems, their components are all nonnegative and
consequently the columns of B are positively linearly independent.
Hence, the set of vectors x such that Bx=w is bounded. All the
conditions of the Fundamental Theorem are satisfied. Therefore, there
exists a primitive set of allocations in P, such that a nonnegative
linear combination of the equilibrium prices associated with each
allocation in the primitive set is equal to the normalized prices in
n i i 1 i
the larger economy, i.e., Z x.bj(aj) = w where bj(aj) is the price
j=l J
J
system in B corresponding to the allocation a in P , and
x. > 0 for all j .
This linear combination guarantees that the vector of prices in the
larger economy will always be in the convex hull of the price systems
associated with the allocations in the final primitive set. Indeed,
the nonnegative weights x. must sum up to unity. To see this, recall
that each b (a~*) G B and w are normalized to sum up to unity, i.e.,
n n n .
Z b-?(aj) = 1 and Z w. =1. Now I x.bj(aj) = w means that:
X
n 1 i
Z x.b^(aj) = w. for all i = 1, 2 ..... n.
J-l J X
Thus:
Z x = Z w. = 1
n
and Z x. = 1 as asserted above.
J-l J
99
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Now, increase the grid size indefinitely to obtain a finer partition
of the allocation hypersurface. To each grid size G, there corre-
(1 c* *? f nc* "\^
a , a , . . ., a J which
converges to a unique allocation a. To each grid size G, there
also corresponds a nonnegative linear combination of the price systems
-------
continuous on P, . The pricing mechanism may indeed by very sensitive
to the initial vector of physical resources. However, while the
conditions which would guarantee the continuity of b are not known,
computational experience seems to indicate that the mappings b~* are
in fact continuous for allocations interior to the allocation hyper-
surface .
Assuming again the continuity of the mappings b for interior allo-
cations, the prices which correspond to nonboundary allocations will
be matched exactly in the sense that they will be proportional to the
corresponding prices in the larger economy and hence in the same
proportion to each other. To see this let K be the set of indices
such that if j E K, a will contain a boundary element where a
is an element in the final simplex of allocations for a grid size G
for all G > N where N is fixed and arbitrarily large. Let M be
the set of indices indicating the first boundary element for each
JO
such aj where j G K. Then:
G-*»
lim xGbj VG) + I Hm x°bJ VG)
G-*» J j£
I xlm+ 7 x
JGK J J£K J
w
since b-'(a) » b(a) for all j£K and m G M. Now for i £ M,
101
-------
x.I =0 so that b.(a)I £ x. I = w. . Thus since M contains
J X \^K Jj l
the indicies of those prices associated with boundary elements, the
prices b.(a) not associated with boundary allocations are propor-
tional to the prices in the larger economy w. where Z x. is the
1 j£K J
factor of proportionality.
It was indicated above that even in this case the equilibrium allocation
will be a Pareto optimal allocation of the money resources. No en-
trepreneur can take advantage of the price differential for those goods
whose prices are associated with boundary allocations. By the assump-
tion that the absorptive capacity has been met, the entrepreneur
cannot export more of that commodity to the larger economy. Restricting
the exporting of such a commodity will make the regional economy poorer
and hence, at least one consumer in the regional economy will be worse
off. Thus there will be no way to trade with the commodities associated
with boundary allocations. It was shown above that all the remaining
commodities have the same relative prices as prevail in the larger
economy. Hence, there can be no arbitrage available for the entre-
preneur in these commodities. Consequently, even in the case where
prices are not exactly matched between the regional economy and the
larger economy, the allocation of money resources will be Pareto optimal-
Figure 2 illustrates graphically a hypothetical solution in which at
least one price in the regional economy will not be matched exactly
with the corresponding price in the larger economy. The allocation
hypersurface P, is the unit simplex of allocations where Q. = 0
102
-------
for all i = 1, 2, 3. The non-boundary allocations map into a subset
B of the unit price simplex. The vector w, representing the relative
prices in the larger economy, does not belong to the set B. Consider
12 3
the primitive set determined by the vectors a , a , and a , where
a is a boundary element in P. . The boundary allocation a maps
K-
into the unit vector p with the one in the second coordinate on
the unit price simplex, i.e., the vertex labeled 2. The algorithm
guarantees that for each grid size the relative price vector w in the
larger economy will always be in the convex hull of the prices p ,
23 123
p , and p corresponding to the allocations a , a , and a in the
final primitive set. If the set of prices B on the unit simplex, in
which all non-boundary allocations map, contains the relative price
vector w, every price in the regional economy will be matched exactly
to the corresponding price in the larger economy, i.e., the limiting
allocation a will map on the price vector w.
As was already discussed in the third part of Section IV, the
situation in which it is impossible to match exactly all the prices in
both economies arises for either of two reasons. Either the numerical
value of Q is too high for one or more i, i.e., there is not
sufficient absorptive capacity in the larger economy; or else some
productive activity involving all commodities in both economies is
profitable at the prices prevailing in the larger economy. Theoreti-
cally, the first problem can always be remedied by appropriately
increasing the absorptive capacity of the larger economy. It is only
the latter situation that can never be remedied since there is no bound
103
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Allocation Simplex
Price Simplex /
Fig. 2.—A geometric view of the mapping
from a set P, of allocations to the unit simplex
of prices relative to some hypothetical allocation
model.
104
-------
to the productive capacity of the regional economy. In practice,
however, even the first situation can remain whenever the absorptive
capacity of the larger economy is rigidly determined.
In general, it would be computationally prohibitive to define the
allocation hypersurface P, by its "true" boundaries specifying the
"true" absorptive capacity of the larger economy. The set P, would
then be very large. For computational convenience, "computational"
boundaries are then chosen, each no smaller numerically than the
"true" Q., and a solution is found. So long as the final simplex
of allocations does not contain a vector on a "computational" boundary,
the result will be the same as with the lower "true" boundary. Other-
wise, the "computational" boundary should be lowered. The procedure
consists of choosing the "computational" Q 's such that the resulting
set of allocations P, will be as small as possible and still map
K.
into a subset of the price simplex which contains the vector w of
relative prices in the larger economy. This can always be done unless
some productive activity involving all commodities in both economies
is profitable at the prices in the larger economy. A useful technique
is to choose a crude grid size G and solve the model with a moderate
size P, such as setting Q = 0 for all i. In general, the final
primitive set of allocations will have a number of allocations with
components on the boundary Q.. One can then decrease the numerical
value of Q. and compute the model over and over again, increasing
the grid size whenever the final primitive set of allocations contains
no boundary element. Such a procedure can often increase appreciably
the computational efficiency of the allocation model.
105
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An Alternative Method of Solution
Of the Allocation Model
The allocation model is a two-stage application of an algorithm which
approximates fixed points of a continuous mapping [10]. Its inter-
action mechanism approximates fixed points of fixed points, i.e.,
fixed points of the composition of two mappings: a mapping of the
money resources into a vector of initial physical resources and a
mapping of the initial physical resources into a vector of equilibrium
prices. From a computational point of view, the allocation model would
be greatly simplified if the composite mapping could be identified.
While this composite mapping has not yet been discovered, it will be
shown that a direct application of the general equilibrium algorithm
to the regional economy specified in a very particular way provides
results identical to those obtained with the allocation model.
In his dissertation, Terje Hansen describes an economy with a foreign
trade sector [4]. Specifically, he introduces a new commodity which
he calls foreign exchange and two new sets of activities. The first
set consists of the import activities, each of which converts some
amount of the foreign exchange commodity and of some other commodities
into a unit amount of the commodity to be imported. The other set
consists of the export activities, each of which produces some amount
of foreign exchange by using up one unit of the commodity to be ex-
ported and some amount of some other commodities. The general equilib-
rium algorithm then computes the relative price of foreign exchange as
106
-------
well as the optimum levels of operation of these import and export
activities. If there is no final demand for foreign exchange, all
the foreign exchange gained from exports is spent on imports. Since
foreign exchnage is used only in trade, the optimum levels of operation
of the trade activities determine the optimum allocation of foreign
exchange between the commodities imported and exported.
The foreign exchange commodity in the Hansen trade model is exactly
analogous to the money resource in the allocation model. The quantity
of foreign exchange given up in importing, or gained in exporting, one
unit of any traded commodity is precisely the price of the commodity
in the world market in terms of foreign exchange. Similarly, the
quantity of the money resource used in buying, or gained in selling,
one unit of the commodity in the larger economy is precisely the price
of the commodity in the larger economy in terms of the money resource.
In essence then the trade activities define the prices prevailing in
the larger economy and their optimum levels of operation determine the
optimum allocation of the money resource to the commodities involved
in trade between the regional economy and the larger economy.
The objective of the allocation model is to match as closely as
possible the relative prices of the commodities in the regional economy
with the corresponding relative prices in the larger economy. A
foreign trade sector added on to the general equilibrium model of the
regional economy would accomplish an exact matching of relative prices
in both economies provided each trade activity involves no other commod-
ity but the traded commodity itself and the money resource. To see
107
-------
this, consider the following "frictionless" trade activities. Let
M. be the price of commodity i in the larger economy, that is, the
amount of the money resource involved in the trade of one unit of
commodity i. Then, the frictionless import activity for commodity
i is represented by the vector (0, ..., 1, ..., -M.) with the 1
in the ifck component and its frictionless export activity is repre-
sented by the vector (0, ..., -1, ..., M ) with the -1 in the
itn component. Suppose, for example, the general equilibrium solution
says to import commodity r and to export commodity t at the rela-
tive prices p and p , respectively, with p the equilibrium
price of money. These two trade activities must then just break even,
i.e., they must exhibit zero profits. Hence,
p - p M =0 and
*r *m r
P Mr
so that — = — . Thus the relative prices in both economies are
Pt Mt
exactly matched.
The problem with this approach is that the algorithm which approximates
fixed points does not apply to the regional economy with frictionless
trade. Indeed, the set of solutions to the corresponding system Bx=w
is then unbounded. This follows immediately from the fact that the
frictionless import and export activities for any given commodity i
are positively linearly dependent, that is, there exist positive
scalars x and y such that x»(0, ..., 1, ..., -M.)• +
y(0, ..., -1 M.) = (0, .... 0 0); indeed, take x = y
108
-------
for any x > 0. Since the frictionless import and export activities
of a given commodity are positively linearly dependent, they cannot
be together in the same basic feasible solution. However, since they
are positively linearly dependent, the frictionless import and export
activities of a given commodity can be in a non basic feasible solu-
tion. Thus, in a regional economy with frictionless trade, the set
of solutions to the corresponding system Bx=w is unbounded and the
approximation algorithm fails to apply. The algorithm cannot decide
whether to import or to export the given commodity.
In order to find the general equilibrium solution of a regional economy
with trade whenever bounded solutions do exist, either frictions must
be built in the trade activities or some ja priori knowledge must be
employed to exclude from the specification of the model either the
import or the export activity for each commodity which can be traded in
the larger economy.
If frictions are built in the trade activities, the relative prices of
commodities in the regional economy cannot match the relative prices of
the corresponding commodities in the larger economy, except by accident.
The discrepancies reflect the "external" costs of trade. Each import
and export activity with friction has a negative number -a, in the
k*-" component representing the amount of commodity k needed to import
or to export one unit of the traded commodity. Suppose, for example,
the general equilibrium solution says tc import commodity r and to
export commodity t at the relative prices p and p , respectively,
109
-------
with p the equilibrium price of money and p, the equilibrium
in &
price of commodity k needed either to import commodity r or to
export commodity t or both. Then,
Pr - Vk - PmMr = ° and
-Pt -
pr ~ JW M p M
KL T* TT T*
so that - = — . Clearly, — = — if and only if Ep, a = 0
p + Zp a Mt Pt Mt k k k
6 J J J
and Ep.a. = 0, that is, the relative prices in the regional economy
J J J
exactly match the relative prices in the larger economy if and only if
the "external" costs of trade are zero in each trade activity; other-
wise, the discrepancy reflects the "external" costs of trade.
If a. priori information is given concerning which f rictionless trade
activities optimize the allocation of money resources in the regional
eonomy, the other trade activities could be excluded from the specifi-
cation of the regional economy with trade. The relative prices in each
economy would then be exactly matched. This a^ priori information can
be obtained by trial and error computations with the trade model,
excluding at each try the trade vector corresponding to the one which
is included in the solution at positive level. This procedure can be
very tedious since there are 2 different combinations to try, where
n is the number of commodities which can be traded. The same informa-
tion can be obtained directly by applying the allocation model.
110
-------
The trade model with frictionless trade activities is very important
from a computational point of view. It is faster to compute than the
allocation model and yields a closer approximation for similar grid
sizes. A suggestion then is to compute the allocation model with
sufficient, even though crude, grid sizes to determine which commodities
are to be imported and which commodities are to be exported. Once this
information is known, apply the trade model to obtain relatively
quickly a more refined estimate of the optimum allocation of the money
resources. Whenever it is possible to specify the regional economy with
trade activities all having enough friction so that the algorithm can
effectively distinguish between import and export activities, the trade
model is far superior to the allocation model.
Ill
-------
SECTION VI
ACKNOWLEDGMENTS
Rev. J. Eugene Poirier, S.J. and Dr. John M. Harrington, Jr. were
Principal and Co-Principal Investigators of this project and co-
authors of this report.
Dr. Herbert E. Scarf kindly gave permission to use his computer
program for the computation of equilibrium prices.
The support of the project by the Water Quality Office, Environmental
Protection Agency and the help provided by Mr. Mark A. Pisano, the
Grant Project Officer, who provided a great deal of insight and
optimism in the seminal discussions from which this research grew,
is acknowledged with sincere thanks.
113
-------
SECTION VII
REFERENCES
1. Debreu, Gerard. Theory of Value. New York: John Wiley & Sons,
Inc., 1959.
2. Foley, Duncan K. "Lindahl's Solution and the Core of an Economy
with Public Goods." Econometrica, XXXVIII (January, 1970), 66-72.
3. Foley, Duncan K. "Resource Allocation and the Public Sector."
Yale Economic Essays, VI (Spring, 1967), 45-98.
4. Hansen, Terje. "On the Approximation of a Competitive Equilibrium."
PhD dissertation, Yale University, 1968.
5. Hansen, Terje and Scarf, Herbert. "On the Applications of a Recent
Combinatorial Algorithm." Cowles Foundation Discussion Paper No.
272 (April, 1969).
6. Harrington, John M., Jr., "General Equilibrium and Regional
Allocation of Money Resources: A Methodology for Implementing
General Equilibrium Analysis." PhD dissertation, Georgetown
University, 1971.
7. Kuhn, Harold W. "Simplicial Approximation of Fixed Points."
Proceedings of the National Academy of Sciences, LXI (1968).
8. Scarf, Herbert. "An Example of an Algorithm for Calculating
General Equilibrium Prices." The American Economic Review, LIX
(September, 1969), 669-677.
9. Scarf, Herbert. "On the Computation of Equilibrium Prices."
Ten Economic Studies in the Tradition of Irving Fisher. New York:
John Wiley & Sons, Inc., 1967.
10. Scarf, Herbert. "The Approximation of Fixed Points of a Continuous
Mapping." SIAM Journal of Applied Mathematics, XIV (September,
1967), 1328-1343.
11. Scarf, Herbert. "The Core of an N Person Game." Econometrica,
XXXV (January, 1967), 50-69.
115
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SECTION VIII
APPENDICES
Page No,
A. Mathematical Derivation of Demand Functions 119
Derivation of Demand Function from Generalized
C.E.S. Utility Function 119
Derivation of Demand Function from Generalized
Cobb-Douglas Utility Function 122
Derivation of Input Demands for Unit Output
from a Generalized C.E.S. Production Function 124
Derivation of the Input Demands for Unit Output
from a Generalized Cobb-Douglas Production Function .... 130
B. The Computer Program 135
Technical Notes on the Computer Program 135
A User's Guide to the Computer Program ... 136
A Listing of the Computer Program 147
117
-------
APPENDIX A
MATHEMATICAL DERIVATION OF DEMAND FUNCTIONS
Derivation of Demand Function from
Generalized C.E.S. Utility Function
Given a linear homogeneous utility function for a consumer of the form
I/a
U(x) =
The consumer has initial endowment of commodity k, w, which at a
price system p, forms the "income" Zp,w, . In order to maximize
k
utility subject to the budget constraint
Let
I/a
t / \ I \' /i \ J-~ei / \ a. i .
L(x) = I £(b,_) (x,_) 1 +
k
v.
and take all the first partial derivatives
and
5L(x) P v
IT" = fkwk - fpkxk '
k k
Setting the first partial derivatives equal to zero and simplifying
where possible
119
-------
(1-a)/a
a) (ioo <0al (b.)1-3^.)*-1 - xPj
and
a-1
Dividing both sides of (1) by (x.) we obtain
Q
Now dividing both sides by (p.)
,(!-*)/* i_a _a i_a i_a
(bj) Pj = A(p.) (x.)
, , ,1-a
= X(p x )
Now taking the 1-a root of entire equation
\l/a
-a/(1-a) _ ,1/d-a)
-A p.x.
Now summing over all the j's
by (2).
Now take both sides to the 1-a power
120
-------
l-a)/y _-//i-.'A1-a f- A1'3
v l(xkr| Ubkpk
* \K
k
Thus
/ *\ \ -\ I \' /t \ J-~3. f \ ca. i i \v / \ ~*»-/ V J-—ci i I I r^
(3) * - Kbv> (^) K(PT,> SPiW,
\k
Now substituting (3) into (1) we obtain
,
k(pk)
k k k
1~a a
Dividing both sides by £(bfc) ~a(xk)a] we obtain
j .
1-a
Dividing both sides by (b.) we obtain
a-l / v,_ . ,_ v-a/(l-a)\ /v \ „ ^a-1
Taking the a-l root
Xj
121
-------
Now let e = , then
b
x. =
i kk
J k k k
V
Derivation of Demand Function from Generalized
Cobb-Douglas Utility Function
Given a linear homogeneous utility function for a consumer of the form
b
U(x) = A n(x^ ) with £b = l
k k
The consumer has an initial endowment of commodity k, w, which at
K.
a price system p, forms the "income" Zp, w, .
k k k
In , order to maximize utility subject to the budget constraint
let
b r
L(x) = An(x,) + A IPkwk -
k [k k
and take all the first partial derivatives
and
b.-l b
.. / v"l •,-, / vK. •»
= Ab (x ) J n (x ) - Ap
[Pkwk - ?PkXk
1C rC
122
-------
Setting all partial derivatives equal to zero we obtain
V1 bk
(i) Ab.(x ) J n (*) K = XP
j j k7tj TC
and
k k
Multiplying both sides of (1) by x. we obtain
(3) Abj iKx^ = APjxj .
Adding all these equations over j gives
Now from (2) and the linear homogeneity property Zb . = 1 we obtain
j J
b
A k
which on solving for X gives
A k
(4) X
Now substituting (A) into (3) gives
bi
K A
bk k
k
123
-------
which on simplifying and solving for x. gives
x,
Derivation of the Input Demands for Unit Output with
a Generalized C.E.S. Production Function
Given a generalized linear homogeneous C.E.S. production function of
the form
.1-a
There are two methods of determining the input demands for unit output.
The first is to minimize the costs C = Zp, x, subject to being on the
iC
unit isoquant, i.e., P(x) «= 1. The second method is to assume some
cost constraint C; maximize output subject to this cost constraint;
determine the level of output so obtained; and then to adjust the input
demands to produce unit output. The second method is only possible
since the function is linear homogeneous.
Method 1
Let
L(x)
3L
1 -
a
-------
3L(X) -i-| Z
-------
Taking the 1-a power
kvtV
Solving for X
(3) A-li'Vkl I^V1"*0*^'
, rv rv i I i_ *v ik-
Substituting (3) into (1)
1-af
., v
(xk)
(x.) = p.
a-lj
and
(4)
x. =
L-a/U-a)
Now substituting this result into (2)
ICb )
j J
1-a
I/a
126
-------
Now
k k
brought outside of E. We obtain
j
-a/ (1-a)
i, (Pi,) do not depend on j and can thus be
l-a
b.
,l/(l-a)
I/a
= 1 .
Thus
,-a/d-a)
)> )
j J
-
(
J
I/a
-L/cl
Substituting this into (4) we obtain
L-(l/a)
Xj "
(P)
I/a
,1-e
a/a
127
-------
where e »
1-a *
Method 2
Since the production function is identical to the generalized C.E.S,
Utility Function, we can replace the budget constraint Zwp with
k k k
the assumed cost constraint C so that we obtain immediately from
the results above that production is maximized when
Now if this is substituted into the production function, the output
will be
P(x)
1-a
i I/a
-a/(1-a)
Since C and Zb, (p.) v do not depend on j they may be
k fc
brought outside the Z sign:
j
P(x)
•-k
,1-a
J
L(Pj)
1/1-a
I/a
1-a
1
L(Pj)
I/ (1-a)
I I/a
128
-------
Thus if each x. is divided by this output, the resulting output will
be unity since the production function is linear homogeneous. Thus
for unit output
x
X
P(x)
,1-a
a/(1-a)
-------
which is the identical result as obtained by the first method.
Derivation of the Input Demands for Unit Output for a
Generalized Cobb-Douglas Production Function
Given a generalized linear homogeneous Cobb-Douglas production function
b
P(x) = A IKx. ) where £b = 1 .
k k k k
There are two methods of determining the input demands for unit output.
The first is to minimize the costs C = £p,x, subject to being on the
k
unit isoquant, i.e., P(x) = 1. The second method is to assume some
arbitrary cost constraint C; maximize output subject to this cost
constraint; determine the level of output so obtained; and then to
adjust the input demands to produce unit output. The second method is
only possible since the function is linear homogeneous..
Method 1
Let
L(x) - £pkxk + x[l - A II(xk)bk] ,
ai/ x b--1 bi
0L(X) 1 AI / \ J TT/\k
a' = P, - AAb (x ) J H (JO
j J J 3 *
and
) . A
= x - A
Setting partial derivatives equal to zero gives
V1 b
(1) P. = AAb.Oc.) J H (x.)
J J 3 k
130
-------
and
(2)
A n(x.) - i .
k k
Multiplying both sides of (1) by x. gives
p x = XAb iiy^,/ ,
J J 3 k
which by (2) reduces to
Pjx.
Ab
or
(3)
Xj =
J - X J J-
Multiplying equations together over j
n(x.) 3 = n x 3 n| —
j 3 j j
U4
Multiplying both sides by A and noting that II X J
j
n
which by (2) and Eb
j
j
1 gives
b
'^1 ^
j
j
131
-------
and
1
X =
J 1Kb )bj
- J
b.
= J J .
b.
A n(b ) 3
and replacing j with k since it is an arbitrary index
(A)
A n(b
k
Now placing (4) into (3) gives
\\ f bi
n
-------
Method 2
Since the production function is identical to the generalized Cobb
Douglas utility function, we can replace the budget constraint 2»
k
with the assumed cost constraint C so that we obtain immediately
from the results above that production is maximized when
b.C
x. = — •*—
Now if this is substituted into the production function the output
will be
Now C does not depend on k and can be brought outside the II
k
P(x) = AC" H —
which since Zb = 1 give
k R
P(x) = AC n —
Thus if each x, is divided by this output, the resulting output will
be unity since the production function is linear homogeneous.
133
-------
Cb
x.
Cb.
x! =
P(x)
AC n hr
k pk
P.AC n,
J k\pk
n(p
AP n(b
3 k
n (P )
3 k
(P.) J A n(b )
bk
K
which is the identical result as obtained by the first method.
134
-------
APPENDIX B
THE COMPUTER PROGRAM
Technical Notes on the Computer Program
The computer program is written in the FORTRAN IV language with the
technique of dynamically dimensioned arrays. The required memory size
will depend on the size of the array X in the main routine (the
value of NSPACE). If this size is increased then the program can
handle larger problems and will consequently use more computer memory.
For example, with NSPACE = 10,000, the program uses about 100k bytes
on an IBM 360 Model 40.
The program contains the Fundamental Theorem programmed twice, once
for the general equilibrium solution and once for the allocation
solution. The subroutines ALLOC and PIVOTA perform the linear program-
ming and the replacement operation on primitive sets respectively for
the allocation solution while the subroutines SCARF and PIVOT do the
same operations for the general equilibrium solution. The subroutines
TERM, DEMAND, and SUPPLY apply only to the general equilibrium solution
and perform the final linear programming averaging step, deriving
demand functions and deriving supply functions, respectively.
In theory the matrix B will contain k columns, one for each vector
of P, . However, the use of the revised simplex method allows one to
store in the computer only n + 1 vectors of B, the n columns
associated with the slack vectors and the column associated with w,
the right hand side of Bx=w. By multiplying the actual vector of B
135
-------
to be entered into the basis by the matrix of the columns associated
with the slack vectors, one obtains that column of B in the current
basis. The pivot row can then be determined directly in the usual
fashion using the column associated with w which is already in the
current basis. In the computer program the n + 1 columns of B are
stored as BINV and PBINV for the general equilibrium and the allocation
solution respectively.
The index associated with each vector in the current primitive set
from P, is the row number which the associated column of the matrix
K-
B pivoted on when being entered into the feasible basis for the
nonnegative solution of Bx=w. The program keeps track of this index
in the vector NAME and LNAME for the general equilibrium and allocation
solutions respectively. When each column is associated with a column
of B in a basis, then the column associated with w contains the
nonnegative weights x which provide the nonnegative solution to
Bx=w.
A User's Guide to the Computer Program
The computer program is written so that it will either compute the
general equilibrium prices for an economy or else compute the optimal
allocation of a money resource. All aspects of the model as discussed
in this paper have been programmed. All the information relating to
the economy must be supplied on a set of cards. For reference purposes,
the following is an outline of the cards needed to operate this program,
in the order they must be supplied to the program. An asterisk next
136
-------
to an item means that the card(s) must be supplied only for an
allocation run.
A. Header card
B. Commodity Labels
C. Production Labels
D. Consumer Labels
E. Description of the Production Sector
F. Physical Resources
G. Description of the Demand Side of the Economy
H. Boundaries for the General Equilibrium Solution (Optional)
*I. Commodity Selection for the Allocation Solution (Optional)
*J. Boundaries for the Allocation Solution
*K. Absolute Prices in the Larger Economy
*L. Money Resources
After one set of data is read and the model processed, the program will
attempt to read another set of data, beginning with the Header Card.
If there is none there, the program will terminate.
A. Header Card
The header card is always the first card read by the program in any
general equilibrium run or any allocation run. All the numbers punched
on this card must be punched without decimal points and moved as far to
the right in the field as possible. For example if a model contained
14 commodities, the number 14 must be punched in columns 1 to 5 as far
to the right as possible, i.e., in columns 4 and 5. The columns marked
137
-------
with an asterisk apply to the allocation model only and must be left
blank for a general equilibrium run.
The following restriction forms the limit on the size of the problem
which can be handled by this program. As in the program let N be
the number of commodities, NACT be the number of production processes,
including disposal activities, NPLAY be the number of consumers, and
LN be the number of commodities involved in the allocation run. Let
Kl = (NACT + 1)(2NACT + 1) and let K2 = (2N)(N + 1). If column 45
has a one punched then let Kl = 0. Let K = max{Kl, K2}. Let NSPACE
be the size of the array X which contains all the other arrays. For
the program as listed in this Appendix, NSPACE = 10,000. For a
general equilibrium run, the size of the numbers punched on the header
card is bounded by the following algebraic formula:
(11 + 3N + 2NPLAY)N + 4NPLAY + 6NACT + K + 18 s? NSPACE .
For an allocation run the size of the numbers punched on the header
card is bounded by the following algebraic formula:
(11 + 3N + 2NPLAY)N + 5NPLAY + 6NACT + K + (14 + 2LN)LN + 25 £ NSPACE .
The following is a description of each field on the header card:
Col 1- 5 Number of Commodities in economy for general
equilibrium run or in regional economy for an
allocation run.
Col 6-10 Number of Production Processes including
disposal activites.
138
-------
Col 11-15 Number of Consumers in the economy.
Col 16-20 Grid Size for the general equilibrium model.
The program assumes a value of 100 if this
field is left blank.
Col 21-25 Starting corner for the general equilibrium
model. For a general equilibrium run if this
field is left blank the program assumes a value
of 1. For an allocation run, if this field is
left blank the program will start at that corner
closest to the equilibrium price at the previous
iteration.
Col 26-30 Maximum number of iterations permitted in the
general equilibrium solution. If this field is
left blank the program assumes a value of 10,000.
Col 31-35 General equilibrium boundary option. If this
field is left blank the program will assume
Q = 0 for all i. If any number is punched
in this field the program will read in the Q..
See H below.
Col 36-40 Tax option. Punch any number in this field if
the consumers are to be taxed at a fixed rate
for each consumer. The tax rate will appear on
a resource card as if it were an additional commodity.
The government will always be the last consumer.
139
-------
Col 41-45 Averaging option for final primitive set of
prices. Leave field blank if Scarf's linear
programming technique should be used. Punch
a 1 (one) if the barycenter average is
desired.
*Col 46-50 Number of commodities in the allocation solu-
tion. If this field is left blank, the program
will assume that this number is the same as the
number of commodities in the regional economy
as specified in Col 1-5. This number may, how-
ever, be specified and less than the number punched
in Col 1-5, i.e., some commodities can be localized
to the regional economy only. In this case, either
the commodities in the allocation solution must
be the first commodities in the regional economy
and then column 66-70 of the Header card may be
left blank or else a number must be punched in
Col 66-70 of the Header card and then a card must
be supplied indicating which commodities are to
be included in the allocation solution. See I
below.
*Col 51-55 Grid size for allocation solution. This field
must be filled in for any allocation run.
*Col 56-60 Starting corner for allocation solution. If
this field is left blank the program will assume
a value of 1.
140
-------
*Col 61-65 Maximum number of iterations permitted in the
allocation solution. If this field is left
blank the program assumes a value of 10,000.
*Col 66-70 Commodity Selection option for Allocation Model.
Punch any number in this field if a card should
be read in specifying which commodities should
be included in allocation model (see I below).
This field may be left blank and no card supplied
to the program if the commodities in the alloca-
tion model are the first commodities of the
regional economy.
*Col 71-72 Print Option. Punching a number from 1 to 10
will control the information printed. If this
field is left blank only the final general
equilibrium solution will be printed. Punch a
1 for the entire printout of the general
equilibrium solution at each iteration. The
higher the number the less that will be printed.
B. through D. Label Cards
Each commodity, production process, and consumer must be given an
eight character descriptive title (or the appropriate number of
blank cards supplied). Each type of label must begin on a new card.
For example, the name of the first commodity will appear in Col 1-8
of the first commodity label card, the name of the first productive
process will appear in Col 1-8 of the first production process label
141
-------
card and so on. As many cards of each type as are needed must be
supplied. For example, if there were fifteen consumers, one would
need two consumer label cards. The first card would contain the
labels for the first ten consumers. Card two would contain the
labels for the additional five consumers, i.e., only Col 1-40 of
card two would be filled in.
The label cards must be inserted in the deck in the order specified
immediately following the Header Card.
E. Description of the Production Sector
Each non-slack activity must have one or more cards, the number
depending on the number of commodities in the general equilibrium
solution, specifying the parameters of the production function.
Col 1 of the first card for each production process will have a single
digit specifying the type of specification of the production process.
The following codes have been programmed.
0 or blank Activity Analysis
1 Cobb-Douglas Production Function
2 C.E.S. Production Function
Except for this number, all other numbers punched on these cards must
have a decimal point punched. Each field on these cards is ten columns
wide with the specification digit assumed to take up the entire first
ten columns of the first card. Col 71-80 of each card is reserved for
identifying the card and any alphabetic or numeric data may be punched.
Thus the first card will appear as follows:
142
-------
Col 1 Specification Digit
Col 2-10 Blank
Col 11-20 First parameter
Col 21-30 Second parameter
Col 61-70 Sixth parameter
Col 71-80 Comment
while the second and subsequent card will appaar:
Col 1-10 Parameter
Col 11-20 Parameter
Col 61-70 ' Parameter
Col 71-80 Comment
It is suggested that the parameter information be punched left
adjusted (with a decimal poin't punched) i.e., starting in Col 11, 21,
31, etc.
Activity Analysis
The parameters in an activity analysis formulation of production will
be specified with the following convention: inputs will be specified
as negative numbers and outputs will be specified as positive numbers,
143
-------
Cobb-Douglas and C.E.S. Formulations
The demand parameters for both the Cobb-Douglas and C.E.S. single
output production functions will be specified as positive numbers.
In order to identify which variable is the output, -1 will be
punched in the parameter field for that commodity. It is possible
to represent the production of a public good as formulated in this
paper by specifying -1 for all the commodities which will be the
public goods. In the derived unit output activity vector the unit
output number 1 will be repeated for each such commodity.
The scale factor A for the Cobb-Douglas specification and the
constant elasticity of substitution will be punched as a parameter
in the position for commodity n + 1. For example, if an economy
had four commodities then the elasticity of substitution number would
be punched in Col 51-60.
F. Physical Resources
One or more cards (depending on the number of commodities in the
general equilibrium solution) must be supplied for each consumer in
the model specifying the quantity of each commodity which is available
to that consumer at the beginning of the general equilibrium solution.
Each commodity is assigned a field of ten columns with the last ten
columns in each card reserved for identifying information. For example,
the quantity of commodity one is punched in columns 1-10 of the first
card while the quantity of commodity nine is punched in columns 11-20
of the second card. Again, all numbers must be punched with a decimal
point.
144
-------
G. Description of the Demand Sector
Each consumer must have one or more cards, depending on the number
of commodities in the general equilibrium solution, specifying the
parameters of the demand function. Except that all the demand para-
meters must be nonnegative, these cards are punched with the same
format as the production sector. The same three types of functions
are available with the same codes, namely demand functions derived
from utility functions of the activity type, Cobb-Douglas type, and
C.E.S. type.
H. Boundaries (the Q ) for the General Equilibrium Solution
This card (or cards depending on the number of commodities in the
general equilibrium solution) must be included only if some number
is punched in Col 31-35 on the Header Card. This card specifies the
computational boundaries which define the set P, from which the
tc
price systems are taken. If this card is not supplied, the program
will automatically set each Q. = 0. The boundary numbers are punched
without decimal points and moved as far to the right in each field as
possible. Each field is five columns wide so that, for example, the
boundary number for commodity one is punched in Col 1-5, for commodity
two in Col 6-10 and so on for as many cards as are necessary.
I. Commodity Selection for the Allocation Solution
This card (or cards depending on the number of commodities in the
allocation solution) must be included only if some number is punched
in Col 66-70 on the Header Card. Punch on this card the sequence
145
-------
number of each commodity to be included in the allocation model.
There must be exactly the same number of numbers punched as is
specified in Col 46-50 on the Header Card. These numbers must be
punched in ascending order. The format for this card is exactly the
same as in H above.
J. Boundaries (the Q.) for the Allocation Solution
This card (or cards depending on the number of commodities in the
allocation solution) must be included with every allocation run.
Punch on this card the computational boundaries which define the set
P from which the allocations are taken. This card is punched with
1C
the same format as in H above.
K. Absolute Prices in the Larger Economy
This card (or cards depending on the number of commodities in the
!
allocation solution) must be included with every allocation run.
Punch on this card the absolute prices of the commodities in the
larger economy which are included in the allocation run. This card
should be punched with the same format as applies to punching physical
resources in F above.
L. Money Resources
This card (or cards depending on the number of consumers in the economy)
must be included with every allocation run. Punch on this card the
quantity of money resources which are available to each consumer.
This card should be punched with the same format as applies to
punching physical resources in F above.
146
-------
A Listing of the Computer Program
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
(,
c
c
c
PROGRAM ro ALLOCATE HONEY RESOURCES AMD/OS COMPUTE EQUILIBRIUM
PRICES
THIS PROGRAM F PRMS PAWT PF TNT C I SSFH F A T I ON CF JOHN M.
MAHRIflGTU'li JR. WHICH WAS SUBMIU'll TO FHE GRADUATE SCHCOL CF
(..FORGE FUWI UNIVERSITY IN PARTIAL F ULF I LLNF NT CF THE REQUIREMENTS
FOR A PH.D. IH tCPNCMFCS.
THE GEfifHAL CQl) t L U!.< I L.M P'iKTICN OF THE PROGRAM WITH ACTIVITY TYPE
PRODUCTIU.-4 AND C.F.S. TYPf- I. EM A NO FUNCTIONS WAS CHTAINFC FROM
OR. HEUIJtKI A SCARF , COWLES FOUNDATION, YALE UNIVERSITY ON
JUIIF 3, 1-J6«V
ALL OTiiCR ASPTCTS OF THF PROGRAM WFRE PROGRAMMED AT GEORGETOWN
UNIVERSITY flY JOHN M. HAKKINGTON, JR.
THE OcVrLOf-MfNT OF THIS PRTGRAM WAS PAtC FOR IN PART BY A GRANT
FRD.M Tt.E UDE-RAL WATER POLLL'IION CONTROL AGtNCY OF THE CEPARTMFNT
OF THfi INTER I Ok.
irrj x ( 100001
COMMrj'J/SPACF/ X
CUMMIJN K'j,K.6,M,NN,f AX I , L.Mi L NNi I.MAX I ilrtO.ISFL.KPS.I
*i INLirx , JfjuT, IDXS, IDXC) I , I[)X'j?,NG,L I VOt X , L J OUT , IAVC
1 HEAD(Kr. . I0,f "JO'IOnOI H , NAC T ,NPL AY , M i NN , MAX I , I HO, I TO, I A VG, LN, LM,
10 FfiKMAf ( I'.I'i.'j 12 )
IF ILN.f:0.u) L'J-N
HP*H*l
L'.'P=L'J« 1
C
C
11*1
WT
D
I Is- I2*N*NAC:i
I J=(I3/?)*?*l
if (K2.GT.K | )
IF I I AVG. F.v.l)
A AND
--- NAME
ri
I5
P
I<>
X
MAXU=N
T
MA
H
HO=-( I 10/21*2*1
0
111 = 1 lO<)
ALOC0070
147
-------
--- E
C
C ---
C ---
C ---
C ---
C ---
C ---
C ---
C ---
C ---
112=11 ItNPlAY
! 12-( I l?/2> *2*l
MCE
I 13=1 1 2 + NHLAY
I 13=1 I 13/2) *2»1
10
Il*2»1
AOHE
ALOC0071
ALOC0072
ALOC007 1
I 17 = 1 I 17/21 *2« 1
CQMLB
llfl=I 17*2*N
I 19=11 <
FlkHLB
120=1 19*2*MACTMN
MAME
12 1=I20 + N
C --- ALOC
!22=( 122/21*2*1
C --- EXOO
I23=I2?+N
123=( [23/21*2+1
C --- IH
C --- TT
I 2 i = I ? «. » H
C
C
C
C
C ---
C ---
C ---
l -- 1 2 •>
IJ6-I25
IF(LM.EO.O) CO TO
PP
I2fa=I25*2»LN
LI1A
AP
I28=I27«LN
12C=( 128/2)*2«1
P,P
--- LIO
llO-( I 30/2 )*2U
Ltl
1 31 = 1 30 + LN
I3l»( I 31/?J*2»l
LMAME
132=1 31+LN
I32=(I 32/21*2*1
DOL
I33=I32*NPLAY
ALOC0075
ALOC0076
AIOC0077
ALOC007B
ALOC0079
ALOC0080
ALOC0081
At OCOOR2
ALOCOOB3
ALOC008'.
ALOC0085
ALOC0086
ALDC0087
ALOC0088
ALncoosq
ALOC0090
ALOCOCm
ALOC0092
ALOC0093
ALOC0095
ALOC0096
ALOC0097
ALOC009R
ALOC009Q
ALOC0100
ALOC0101
ALOC0102
ALOC0103
ALOCOlO't
ALOC0105
AI.OC0106
ALDC0107
ALOC0108
ALOC0109
ALOC0110
ALDCOII I
ALOC0112
ALOC0113
ALOC011S
ALOC0115
ALOC0116
ALOC0117
ALOC0118
ALOC0119
ALOC0120
AI.OC0121
ALOC0122
ALOC0123
ALOCOl2«i
AL000125
ALOC0126
ALOC0127
ALOC0128
ALOC0129
ALOC0130
ALOC0131
ALOC0132
ALOC0133
ALOC013<.
ALOC0135
ALOC0136
ALOC0137
ALOCOlBfl
AIOC01AO
ALOCOl^l
ALOC01A2
ALOCOl'i3
ALOCOUA
ALOC0145
148
-------
C PIHNV
I3<»«n3*2HN*LNP
C LIO
I35M I 35/ii 1*2*1
C LNAME
I36=(I3ft/2)*2+l
C IA
137=136*LN
I37=(I 37/21*2*1
C ALOCLR.
I38=137*2*LN
50 IF! I38.GT.NSPACE) GO TO 100
CALL INPUT (X< 11) ,X( I2),X(I3),X(I3>,XU,XUll),X(I12),XtP )« A ( f.AC F f , ',TP I , NAN"- I f>AC T 1 , P ( N)-
*X(NI , T (NAG I ) , MAIN, N),V. (UCLA Y,f,P >, IJ ( MM. A Y,'. I , ''. I NPL AY 1, V f f ('Pf I.HI .LMAILN.LM, APILM ,R"tLIJ I ,L rC.(LM , LI 1 (LN) ,
* LMAMF (LN) , (jfJl (MPL AYI ,PilI'iV(LN, LNP ) ,L IIUL'II , I A(LN) , ALrCLU(LN)
REAL* 3 XTYI'I (', ) , :< M !',F , X I-, ', NG
DIJUHLE
Dfiufar
PfU C I SI i!'l
PRJCISIIT;
SL.M,.^
*, IM/rX, JOUT , ir,XSi IUXS L , I()Xc\?,('i';,LIfJr,EX,LJ(:UT, IAVG
COMMD'I/LHL/XI YPI , X Ar^f- ,XF, r*,r,
III! if.. Nf.'.)). A-,:;. ( KHS. F u. u) i KPS= I o
IF (MA< I . F!.-. 0'
- IF(H.EC.O) K-^100
3 FORMAT ( • 11MIS IS A GFNfRAL f-QLI I LI HR IUM RUN A DESCRIPTION
*TMl ECONO'tY I riLLOX',' )
IF (LM .NE.U) «RIIr IK6,',)
4 FORMAT ('UIMS IS AN ALLOCATION RUN •/
*'OA OF. SCKIPIION OF IMF: R t r, I!;») ft L ECONOMY FOLLOWS :•)
WRITEIKO.til H.NACT, SPLAY
5 FORMAT ( •OMJHHFK (JF CQMMCDIIIF.S
* ' , 1 3/
* , • NUMIKR OF PROOUCTILN PROCESSES (INCLUDING niSPHSAL ACT
*TIES) .. ',I3/
* • fiUMBGR OF
* '.Ill
Atocoias
AinCOl86
ALOC0187
ALOC018R
ALOC0190
ALOC0192
OF ALnCOt95
ALOC0196
ALOC0197
ALCC019P
ALOC0199
ALOC0200
ALDC0201
ALOC0202
I VI ALOC02Q3
ALOC020A
IOXS1=0
IDXS2=0
IF I (LM.FO.O).ANO. (NN.EQ.OH NN=1
IF(NN.KO.O) NN=-NP
IFILN^.EC.ri) IMN=I
RfAlMK'j ,•)!)'. ) I CTiMLIU I ) , I» 1 ,N )
905 FORMAT ( 10.U' :)
RF AD(KS, «J')'i) (FIRMLiill) , I » 1 , NACTCN )
RKAIHK1), 9 '.)'.> ) ( !'LAYL(«( I ) , 1 = 1 »NPL AY)
13 WRI TE(K6,')03)
903 FORMAT ( • OI'ROOUCT IfIN FUNCTIONS FCR EACH FIRM IN THE FCDNflMY1/)
WRITEIKft.'lOf.KCfXLni II , l=-l tN) .XAORf
906FOKK'AH« ',' I-IRM -t^X,' TYPE < ,9 ( 4X, An I / ( 22X, 9 (4X, An I I I
ALOC0206
ALOCO?07
ALOC020R
ALOC0209
ALOC0210
ALOC02U
ALOC0212
ALOCO?13
ALOCO?15
A LOCO? 16
ALC!CO?17
ALPC071')
ALCIC0220
149
-------
907
c —
901
902
!<•
B03
606
807
C ---
BOH
10
100
800
C —-
101
810
16
C
160
170
175
180
18i
186
166
C ---
21
C ---
VP.J
FORMAT I • 'I
DO 14 J^'IP.'.ACr
JJ = J-'l
«FAD IN T-ic piinucMON r,Fcrrq PF THF rc;--jrifY
RF. AD(K5,'J Jl 1 ! I YPU JJ) , ( P, ( I , .)) , I = I , N ) , ACKF (JJ I
FOR«AT ( 1 1, n,' no. o/i rr\o.r> > >
ALOC0221
ALQCOP22
ENG=XTYPE ILL)
hRI TEIK6.9JJ I F I P,KL 0 ( J J ) , (: fiC.. (Ft ( I , J ) . I = 1 . N I , »C«E( JJ>
FtiKMAT ( ' • , Ati, :iX,4K,9Fl2.2/l^2X,9F 12.2) >
CONTINUE
W31 TE(K6,10JI
FORMAT* "O^CSrr.)!>CF<; AVALIAUI.F TO EACH CONSUMER IN THF ECCNOMY'/I
WS 1 I F { K6 , h .If,) ! C CM.L I! ( I ) , I - 1 , M
FORMAT ( • CflNSLW as ',l?X,rP<<,X,A
NT=N
READ IN THF PHYSICAL RFSOUP.IF S AVAILABLf TO THF ECONOMY
00 10 Isl.Nr-LAY
READ (Kb, IOC) ( h( I ,.)) , J = 1,M)
WkITC(K6,aCu!) PlAYLri(l|,r.'(1,J),J = lfNT)
FORMAT ( • • , Ad, 1 3x,9F12.2/( ??/. ,->l 12.2) )
CONTINUE
FHKHAT< 7f 10.0)
W«I7E(K6,505)
i-crfMAT (• OUTIL ! rr FLNCTKINS rnp FACH CCNSU^FR IN THE ECONC
WKI1H (K6,>iO')J I •; >'l •) ( I I , I- 1 ,M , XAORT:
FORMAT!' CONS.. 'M KS' , ^ X , ' TYCI • , 9 ( <,X , AR ) / I ?,'X , 9 ( <• X, A" ) ) )
WR I TE t K6, 10?)
KFAt) IN Tn[- [!' MA'.'D FUNCTION fftr, AMI: TFRS FDK r-ACH CCN^UMFR
nr. 16 i^i.fiPt ;.f
Rf A(1K',, 101 I "M\ \ I, IM I ,,) I, J= 1,N) ,FJ( I I
FORMAT! I 1,'JX, f,r I 0.0/1 7FIO.LM)
IL^ HPEI11*1
ENG^-XTYPEILL)
bRI IE ( K6, 310) PI AYI 01 1 ),CMG, (ll( I, J ) ,J- 1 ,N) ,C I I )
F fix MAI I ' •,Aft.iy.i«lf9Fl2.2/(22X,9F12.2))
COM INUF
RFAI) IN HflU'JL!.'. R IPS
IF ( IliU.EC.O) -u r'l l->0
KF A I) I K S , 1', 0 ) ( ] !! ( I ) , I ~ 1 , N J
0(j TO 160
uo !•>•; 1 = 1,'.
1 H ( I ) = 0
Cr\T I'lUF
WK II F ( K6 , I MJ ) ( H' ( I I , I ' I . N I
FflKMAT ( 'O.U'U':1. AKIPS F(M GFN. f Ct 11.1 fl« , I 6, R I 1 2 / ( 22X , 911 2 ) I
IF (LW.FO.U) Or rn 7-)0
It ( IS' L.£ J.O) .,0 K; 1 ?0
CO TO IflO
i;n 1 75 1 = 1, LN
IAI I 1 = 1
CONTINUE
1)0 195 I = It IN
L=IA( I I
AIOCLO< I )=COML3IL)
CHNTINUE
WRITF. «K6t l«M I -M-miM I »tl=l ,t.N)
F-dKf-'AT ( 'OCtirnr ,(Y IN ALOC K(K;EL • ,9 I 4Xf Al! )/ ( 22X • 9CVX t A8 I ) )
RFAtXKbi 1-tU) IL U - t t ) , I '• 1 fLN)
Ki< 1 I f! I K6 , 1 (16 I I I I (3 II ) , I I . I N I
fCKMAT MQlsriAt AMI'S FIW ALLOCATION • , I 6,01 12/ ( 2?X , 91 12 1 )
R.FAO IN Af.snn Tf PH. IC!:S
hfAnK^i, 1 JOH AiM I ) , 1*1 .I.N)
V,rtl II IK<, ,71 ) (-'. ' ( I), I- 1,1 N)
FOKKAF I 'OAIISCI ;j|(- PHItFS : ' , 4X , -5F 1 2.2/ I 22X , 9H 2.,° ) )
COMPUTE RtLATlvr PRICI:S
ALOC022A
ALOC022S
ALDCO?27
ALOC022H
ALOCO??'J
ALDCOP30
ALOCO?:U
ALOC0232
ALOC0233
ALOC023A
AIOC023')
ALOC0236
ALIK0237
ALOC0238
ALOC0239
ALOC02<.0
AlOC02<.l
ALOC02<.2
ACOC0243
ALOC0244
ALOC02't7
ALOC02A8
ALOCOP49
ALOC02SO
ALOC0251
Al.OCO?r).l
ALOC0254
ALOC0206
ALOC02^7
ALOC020H
UO 200 I = l,l.N
SUM^SUH-tAPl I )
ALOC0260
ALPC0261
ALPC026?
ALOC0263
ALDC026',
AI.HC0265
ALI)CO?6fi
ALOC0267
ALOC020n
ALOC 0.76-5
ALOC0270
aLOCO?71
ALOCO?7?
ALOC0273
AIOC027'.
ALDC0270
ALOC0276
ALOC0277
ALDC0278
ALOCO?79
ALOC0280
ALOC0281
ALOC02B2
ALOC02n3
ALDC0284
ALOC020D
ALDC0786
ALOC0287
ALOC0788
ALOC07R1)
ALOC0290
ALOC0291
ALOC0292
ALOC02T3
ALOC0294
ALOC0290
150
-------
2CO C'JNTIN'-E
DO 210 I* ItlN ALOC02W
RP( ll=iP( I )/<;ui ALOC02<5b
210 CONTINUE ALOC02'}')
KR1 TE( ».6,22) (RPC I I . 1=1 .LfJ) A LOCO TOO
22 FOKMAT('OKFLAr!VF PHICFS : • , 4X , 9F 1 2 . 3/ ( Z2X .9F12.3M ALOC0301
C — REAU i•; MONtY Kf.cnL'KCns ALOCOSO?
RFAOJK5, 100) (HQL< M,r = l,NPLAY) A LOCO 303
ViR J Tf < K6.X3) ALOC030,?<.) I PLAYLfilM, t = l,M'LAY) ALOC0306
2'* FORMAT ( '0', 10('.X,Af>)/( • • , IC(',X,AB) ) ) ALOC0307
WiUTM *fa,ir.) IDOL ( I ) , IM.NPL AY| ALOC0303
25 FORMAT('0',10F12.P/M «,IOF12.?.M ALQC03G9
250 WRITE< r-6,^1) ALDC0310
251 FORMAT ('OTHIS MODEL IS RUN KITH THE FOLLOWING OPTIONS :') ALOC03H
lE(LM.'.fc.O) WXI TF (Kh,25;>) ALOC0312
2'j2 FORMAT ! ' GFUfi THf AL LOC.AT I Cjr< (CUTE*) SOLbflON :') ALOC0313
IFUM.'lE.ul WRIIFtKA.Z'jl) LN, I f , LNN.Lf AXI ALOCOll'.
2i3 FORMAT (' O'MU^BIK OF CHI-JTin I T I fS 't!5/ ALOC0315
* ' OHIU SIi[r ',T5/ ALOC0316
» • STARTING CCf?'*TR 'itS/ ALOC0317
* • MAXIMUM MJCiUR OF irtHATIONS ALLCuCD .. ','51 ALOC031R
IF1LH.NE.O) i.KI TF|K6,2I>7) ALOC031')
257 FORMAT ('Or-CK THf- GFf.FRAL EGUlLlftRTUM (INNPR) SOLUTITN :') ALOC0320
WRITEC K6,?l>3) M , M , VI, M AX [ ALDC0321
Wf'.I TCI -!6, i^3l IAVG ALOC03??
353 FORMAT ( ' AVFKAMN''. Kf?THOO 'tT5) ALOC0123
HS1TCI 6 FORMAT ('OUf-N. f: C i L . BOUNDARY CPTION ... ',!!>/• ALOC0127
* • CONillMFR TAX HPTI'-N "rl1)/ ALOC032S
* • cuf"-vii). n, ALTC s>'LrciiOM CPTN 'ii^>/ ALOco^1?
« • PRINT TPrir-t ',[«.) ALOC0330
CALL ALL (jr. I (WT.H., CINV,/. ,SAMF,)>,x,T , « A , W. C, E , >"f , I A, ALOCLf. ) AlOC033n
OMENS I (IN WTCil , M '. .N'VCT l.i't.'iV (,\, \» 1, M f.AC F P , M P ) , NftM.F (f.AC T ) , P t N ) , ALOC0339
*X(N), T (VAC I ) , M/1. (N,M , r,l>i.')LAvV.>' I . ii I',"L A Y, M , r-t'^L A Y ), «n r I «!PL AY ) , ALOCO V,0
* IOIN) , I 1 (Ml, I I Yl" I'vACU'M i Aill'.F IN.VCTH', ) ,Ci "-'LH t.'.) , PL AYL'> t T'L AY I', ALnCOl^il
* F IRMt" (\.\C I'";) ff-'.HL-(M,AL.'.'C(\PLAY,N) , h XC '. ( M , I H. (;; ) , T T I M , ALOC03'.2
* LNAWC ( LfJ) .PP(l'J) ,L"A( LN.L'iJ.APILNI .KiM LN) .LIOII.M ,1 I I (LN) , AUOC03', 5
« LHAMi I LfJ) ,001 I 'ii'i AY) , Pni»iV(LN, LNP) ,L 1C- (LN 1 , IAILN) , *l'"Cl", ( L N I ALOCO^'t',
RF.AL*n XTYPr ( t, ) .XACMF.XP ,I-\G ALOCO^^'i
DUUUL'r fRI CIS I Tl « I \V,P,X, T ,Cr;.«LB r PLA YLB, f I RMLH , PP , RP , PP. I NV , ALOCLRALOCO *t7
DU U'> I=l,L.'J ALOCO VjH
•PBINV! I ,LfJP|--!
-------
LJOUT=LNN
LINDEX-1
LLJP.Xl-0
106 CAl L PlVOrA(Li"A,LIO,LI 1,PP,AlCCLB,LIRi
c „_ CM[:CK FOR AN ALLwCAI'KN C'J IMF! riOUf.CRY - IF
c ASSOCIATED nMM A SLACK CtU I L I MIUM PKICE.
330 DO 140 I=L,LN
is FOUND IT
IFILIP.LE.L tl3< III 00 TC 341
340 CONTINUE
IF(KPS.GE.71 GO TO 349
WRITFIKfc, (5?) LINOfcX, (PtM M, l=l,LN>
352 FORMAT CO' , 15,' ALLOCATION: • , 1 Of-10. 4 /( 2 \ X , 10F 10 . 4 ) I
GO TO 349
341 LIPIV=I
LTAMEt LJOUT)=I
DO 345 J=l,LN
IF(J.EU.LJOurI GO TC 345
IF(LM.AMF. (J1.EC.Ll'MV) GO TO 347
345 CONTINUE
00 346 I=I,LN
P(I 1=0.0
346 CONTINUE
PILIPIV 1 = 1.
KK.Ol
TFIKPS.GE.21 GO TO 400
WRITE|K6,i44) LIPI V
3<,4 FORMAT COSLACK PKlCf '.15,' BEING INTRODUCED AS AN ACTIVITY'!
GO Tl) 4 CO
C COMPUTE PHYSICAL RESOURCES EIU;M ALLCCATFD MONEY RESOURCES
349 X H = L M
DO 348 I=1,NPLAY
Uf.l 34 H J= l,N
ALUCII,J)=0.0
34b umnut.
ULJ 350 1-1,'il'lAY
00 350 J=1,LN
L=IA(J1
AL1JCII ,L>-IOC!L( M«PPU))/tAP(JI*XM)
350 CONTlNOf
00 J55 J =!,'.'
ALOCII,J)-ALUC
-------
370
371
C ---
380
390
395
39u
400
403
40fc
C ---
409
420
761
C ---
C ---
425
430
435
440
IF(LM.EC.O) RFTDRN
VECTOR P WHICH C-XITS FKO!H SCARF IS THE. EQUILIBRIUM PRICE WHICH
is ASSici/.rrD KITH IMF CURRENT ALLOCATE.
T WILL CCWIAIN THE VFCTOR P EXPRESSED IN THE E-TH BASIS WHICH
SHOULD PE ENTERED INTO THE DASIS.
IV'NN
(110=0.0
IF (NVAaV.EO.O) GO TC 371
00 370 1=1, M
IFIPUI.LT. BIG) GO TQ 370
BIG-PI I)
IV*I
CONTINUE
NN-IV
SCALE- THE APPRORIATE LN ELFt'ENTS OF P TC SUM TO ONE.
SUM-0.0
DO 390 I = ltLN
L=IA(I)
IFIPIL I.GE.O.OI SUM=SUM»P(L )
CONTINUE
00 395 l=liLN
L*IAtl1
IFIPIL). LT. 0.0) P(l) = 0.0
IF (P(L l.GT.0.0) PI M'PID/SljM
CONTINUE
WRI TEtKft, 198) (P( I ) , 1 = 1,LN)
FOKfATI 'Of INAL t CU I L I f! . PR I CCS ' . 1 1 F 1 0. 3/ 1 2 1 X • 1 1 F 10. 3 ) )
IFI IE'JO.EU.0) GU TC 400
RETURN
1)0 406 I = ltLN
T(I)=0.0
00 403 J»1,LN
TII)»TII)«PBINVII,J)*PIJI
CONTINUE
CCNTINUF
SFAPCH Ff/R THI: PIVOT RHW.
f,= lOOODCO.
UC! 4?0 l = liLN
IFITI I I.Lf .0.0) r.O TO 420 -
IFI IPfHNVI I.LNPI/TII I I .GE.R) GO TO 420
R=PGMV< IiLNP)/T( I)
LIPIV WILL RE THE ROW
MATRIX OF ORIGINAL SLACK VFCTORS
CQNTI'JUF-:
IF- (K.LT.'JOUOOn. I GO TO 425
WHITE !'.)
590 [FILIPIV.FO. LNN) GO TCI 600
ALOC0446
ALOC0447
ALOC0448
ALOC0449
ALOC0450
ALOC0451
ALOC0452
ALOC0453
ALOC0454
ALOC0455
ALOC0456
ALOC0457
ALOC045H
ALOC0459
ALOC0460
ALOC0461
ALOC046Z
ALOCOA63
ALOC046'.
ALOCOA65
ALOCOA66
ALnCO«.b7
ALDC046'5
ALOCO'.70
ALOCO«71
ALOC0472
ALOC0^73
ALOC0474
ALOC0475
ALOCO'i76
ALOC0477
ALOCOA78
ALOCOAHH
ALOC0481
ALOCO'.fl?
ALHC04H3
ALOC0484
ALOCO'.B6
ALOC0487
ALOC048P
ALOC04ft<»
ALOC0490
ALOCO'.1)!
ALOC040?
ALOC04'*?
ALOCO/,94
ALOCO'.95
ALQC049h
ALOC0497
ALOC0491
ALOC0499
ALOCO'iOO
ALOC0502
ALPC0503
ALOC0504
ALOC0505
ALOC050h
ALOC0507
ALOCOSOfl
ALOC0509
ALOCOSIO
ALOC0011
ALOC0512
ALOC0513
ALOC0514
ALOC051-J
ALOC0516
ALOC0517
ALOC051H
ALOC0519
ALOC05?0
153
-------
IFILINOEX.!C.LMAXII GO TO fcCO
GO TO IQt.
fcOO VWITFIK6, 70S) L I NTiFX , lj;r X 1
70'j FflkPAT ! «tjr«» THE M,wf(EK OF ITERATIONS I", IHt ALLOCATION SOLUTION
*s ', I ti,' :;/ WHICH •, i •>, • f'. i • t. / x T Y P f , x .\(;31;, x F , i K-G
imr^ji x.of. 11 on ro !'j
IFILIVIJfX.ft-.-l ! OP, TO ',9->
DO 10 1'lfL'J
1F( I.Frj.l/.N) GO 10 b
I SU«= ( SUM + L Iii( I I
•3 DP 10 J«1.I.M
10
DO
J=liLN
LMAIJ, Jj^L^AI J, Jl-1
LIO(J)=J
L!l ( J)=J-1
Llll 11---LN
iriKPS.GE.
Gf! TU (.qO
CO TO
35
+1
lF(LJCfUr.*-C.I N) JP'l
LOO = LI OILJ'JUI J
LIO--L r t (L oouri
L01--LIO( JP)
Lii=Li i (.J;M
LM A ( L 00 , L jribT 1 - 1. y- M LOO , L .100 1 ) + I
LMA(L I I tLjii'ul I L I'M 1.1 i .1 J:"!l, I ! « 1
LMA(LOltLJr:llTI=l.l'iMl.Ol , L .JCi: T ) - I
LMA(UOfLJ(iUr|-lMA(LlOiLJf;ul)-l
ALOC0030
ALOC0531
ALOCO'i3?
ALOCOSBi
ALOCOS3H
ALDCO'jJb
ALOC0539
ALOCO"i<,0
ALOCOSAl
ALOCO'i'.?
ALOCOSA3
ALDCOS'.^
ALOC05A6
ALC1C05A7
ALf)C0577
4LOC057«J
ALOC0579
ALOCO«;aO
ALOCOSB2
ALOCO"i83
ALOCO^ifl*
ALOCO*:
ALOC0091
ALOCOS92
ALOCOr.9J
ALOCO'i9
-------
LIOtL.U)UTl = LOl
LI1 (LJflUTMLl 1
LIOIJPMLOO
LIKJP) = L10
GO TO 550
490 WRITEIK.6.492)
492 FORMAT! '01NITIAL 1LLOCATION SIMPLEX*/!
CO TO ',00
495 IF Pl ftYLfSFIRHL^.^AMf , ALflC.EXCGi IB, Tl )
DIMFMSICN WFCn.Bi (,,hACT) ,V. : INV I N, IIP 1 1 A (SACTP.N rP) , NAMf ( NACT ) , P I N )
*XJN»f T(NftCn1MAr4,^).V-(NPLA¥,NP>,n(MPLAV,?J),E«l!PI.AV),MPE(HPLA¥),
* IO(N) , II CM, IT¥('( INACFMNt t AOCF (NACTM'Jl ,COULB«»II , t'L «YLR (NPL A Y ) ,
* FlRHUttlNAf. TMdl ,JU.HCIN) , AL C!C ( NPL AY , M , EXCICINJ , IP(N) ,TT(N)
RCAL*H Xf ¥PEI<>J,XAf;«r, XEtFNG
OOURLR PRFCISION HI NV.P, X, T .Cfl^LB, PLAYLB.F IRKLP.
DOURLC PKfClSltjM H
COMMON K5,K6,M,NN,KAXl,LM,Lf^N,LfAXIiIHC,lSFL.K.PS,ITr
«, I MCE X i JOUT, IUXS, lUXSli ir)X.S?,NO,LCNrjFX,LJf!UT, IftVG
COKKn'J/DIM/Nt NSCTt M'LAYtL'JtfiPtLNP.NAC fMNt N TPi NAC TP
CCIK.MO'J/LBL/Xl YPF,XAC*fc,XE,rN3
IMUFX*0
INDEX1=0
INUEX2=0
DO ?Q J=1,N
UO !•> 1 = 1 t'4
B(I,JI=0.
fltNVI I , Jl = 0.
CflNTlNIJF
B(J,J)=-l.
flf\V(J,J)=l.
ALOC059t
ALOC0597
ALOC059B
ALOC0599
ALOC0600
ALOC0601
ALOC0602
ALOC0603
ALOC060'.
ALOC0605
ALOC0606
ALOC0607
ALOC0608
ALOC0609
ALOC0610
ALOC0611
ALOC0612
ALOC0613
ALOC0614
ALOC0615
A LOG 06 16
A LOG 06 1 7
4LOC061 B
ALOC0619
A LOG 0620
ALOC0621
ALOC0622
ALOC0621
ALOC0624
ALOC0625
ALOC0626
ALOC0627
ALOC0628
ALOC0629
ALOC0630
ALOC0631
ALOC0632
ALOC0633
ALOC0634
ALOG0635
ALOC0636
ALOC0637
ALOC063R
15
19
ALOC0640
20 CONTINUE ALOC9641
CALL Or^AKCIwrtRrl'. t 'IV , A , NAMT , P, X , T, H A , W , D, E , MPF. , I Oi 1 1 , I T YPE , ALOC06A2
* AOKC.CnMLR.PLAYI ''I , f- 1 1'.Nl. (I , MAMF , / LCC , E XdG, I R , T T ) A I OC 064 3
C --- JOU1 IS THE CCLli"1! CF THE PHIPIUVf. Sri OF PRICFS WHOSE ASSCC I ATEDALCCOfc*'.
C --- SUPPLY OH DEMAND VKTCR tS CUI CF THE BASIS. ALOC06A5
jnuT=-*J*J ALOC06A6
106 CALl I'l VOT (MA, 10, I 1 , P ,f.n«Lil t IP,) ALOC06A7
IP = PC;M*.5 ALOC064n
IFI IP.LF..IIl(rjM) ) 1R TO 591 ALOC0649
C --- A\Y PK1CE VfCTlIK WITH A CfWONTNT ON A POUNOAKY WILL HF. ASSCCI ATEOALOC0650
C --- WI1H THE SLACK. ACIIVIIY WITH A 1 AT THF. POS11IGN OF TMF. FIRST SUCHALDC0651
UO 1HO J-liN ALOC0652
IP=P(J|f.5 ALOC0653
IFI1P.LE.IBI Jl I GO TO 171 ALOC065A
160 CONTINUE ALOC0655
C --- FIND THF PRODUCT iri'l PKCCESS HITP THE LARGEST PROFIT. JCOL WILL DFALOC0656
C --- THF: PROCESS. ALOC0657
PROFI I =-1000000. ALOC065P
00 170 J = M>,';ACT ALOC0659
CALl. SUCPI YH-. r,P ,t'INV,A,N'AMf ,P,X,T,KA,K,D,F,MPF,IC,I1, I T YPE . ALOC0660
* AOC.Et COMLb|PLAYL»,FIkMLn,HAMh , ALOC,F:XOGt IBt IT, J) A 1. OC 066 I
SUN-0. ALOC0662
ALOC0663
ALOC0664
Ufl lf>9 J =
169 SUM-SUH»XI I I
I I
IMSUM.LT.PROUT) C.C: TO 170
JC()L=J
170 CUNUNiUF
Gil TO 1 12
ALOC066'.
ALOC0666
ALOC0667
ALOC066R
ALOC0669
C --- THl VFCICr< HAKE CUNIAINS Tt!F PKCOUCTinN PROCESS NUrPEHS WHICH ARE ALOC0670
155
-------
c —
171
172
320
C
329
335
ASsnciAiFp WITH FACH f^tcc VECTCR r. INF. CUKRPNT p'LB, f AMI , ALfJC , F XOC, IB, tT)
GO TO 400
POSITIVE PROFIT SITUATION.
DO 355 J=UN
IF( J.F.O. JOUT) GO TC 335
THIS 13 THf: TEST TT p,v-vr-u THE SAME ACTIVITY FROM BEING IN THE
BASIS Hi;HI: THAN G'lCF.
IFlUCOL.LE.NI.AKO.(MAMUJt.EO.JCOL)) GC TO 502
JJ=MA*F(J)-N
IFIJJ.Lr-.O) GO TO 3J5
IF( (ITYPE1 JJ) .EO.O) .Ar=-*( IJ
340 CONTINUE
400 DO 40t 1=l,N
C T WILL CH'.'TAI'J THE VECTOR X EXPRESSED IN THE CURRENT BASIS WHICH
C SHOULD Kfc ENTERED INTO (HE BASIS.
TI I 1 = 0.
DO 403 J=l,N
T! I ) = T ( I MHINVt I , J)*Xl J)
403 tCNTI.NUE
406 CONTINUE
409 K=10&OOCO.
C SEARCH rr,J
TtlF PIVOT RCW. IPIV WILL BE THE PIVCT RCW.
iin 4 ? n i = i, N
IFITITI.LC. .L'OOOOOl) Gtl TO 420
IF I tf !•-"( I ,hP)/T( I I l.r.r.RI GO TC 420
ft«HINVII,NPI/I(I)
I PIV=I
CONTINUE
C ECONO.'.Y HILL S'AVF. AN U\nf\;N()FD SOLUTION IE ALL THE ""LEVf-JTS T,F THfALnC0715
f. VECKm TO PF tUTEKLC IM 0 TI'F. fiASIS A«J- NEGATIVE, 0» IF THE. RATIO ALOC07V6
1,ITYPE,ALOC0693
ALOC06'34
ALUC0695
ALOC0696
ALOC0697
ALOC0698
ALQCOIS99
ALOC0700
ALUC0701
ALOC070?
ALOC0703
ALOC0704
ALOC0705
ALOC0706
ALOC0707
ALDC0708
ALOC0709
ALDC0710
ALOC0711
ALOC071?
ALOC0713
ALOC0714
: Ci M I V E .
TERMS
Ml'LlES THftl THEY ART ALMCiSI ALL
IF IR.L I .'MUOJO. I r.C TO 42^
WHITE (6, U,\ )
GO TO 6CO
C RECALCULATE EUNV, THE PATHIX OF ORIGINAL SLACK VECTPRS,
C 01- THE CURRENT BASIS.
425 00 43*> I- UN
IFt I.EC.IPIV) GO TO 435
oo 4 10 J=i,np
RlNV(I,J)-»INVCUJ)-fMNV(IPlV,J)*T(I)/T(IPIV)
430 CONTINUE
435 CONTINUE
C ftl:CALCI/LATE, THf PIVOT ROW IN TERMS CF THC CURRENT BASIS,
DO 440 J=1,NP
BINV(|P1V,J)-B1NV(IPIV,J)/T(1PIV)
440 CflfJIlNUE
DO i C 0 J = U N
IFI J.CO, JIHIT ) GO TO 500
IF('lAr GO TO AGO ALOC0742
IMOFX= lNnr:X*l ALOC0743
U-"( INOFX.i Q.MAXI ) GC TO (,00 ALOC0744
GO TO 106 ALOC0745
ALOC0717
ALOC071S
ALOC0719
ALOC0720
ALOC0721
ALOC0722
ALOC0723
ALOC0724
ALOC0725
ALOC0726
ALOC0727
ALOC0728
ALOC0729
ALOC0730
ALOC0731
ALOC0732
ALOC0733
ALOC0734
ALOC0735
ALOC0736
156
-------
c — THIS is THE SITUAIICV V.HCN THP TI-HMINATIQN is CUE TC A ZERO PRICE ALOC0746
C --- AT PCS I THIN UN AND Hf^CF THT PRtf.F COKNER JOUI IS ASSOCIATED WITH ALOC0747
C --- Til! SLACK VECTOR hi IN ft 1 AT RCK NN ALOC0748
599 KftKrj jr.UTI=NN ALOC0749
NAMK(JGUn=NN ALOC0750
600 IOXS=!OXS+INDfX ALOC0751
IUXS?=IOXS?*INDE-X;> ALOC0752
IOXSl = IOXSl*INDF.Xl ALOC0753
If (KPS.GE.4) CO m 6S1 ALQC0754
KRITE(K6t 7G5) I NUF X , I 'ir.FX2 . 1NDFX1 ALOC0755
70S FfJKI-'AT « 'o**»»* "IMF; -NUMBER CF ITERATIONS IS ',15, ' THF NUMOER ALOC0756
*Cr LINEAR PROGRAMING 1'IVOTS IS',15,' CF WHICH', 15,' INTRCDUCEO ALOC0757
*ACI IVI TIES' J ALOC0758
C --- PRINT Vf.CTORS AND LEVELS ASSOCIATED WITH FINAL SIMPLEX. ALOC0759
WRITf {K6.62U ALOC0760
623 FORMAT ( 'OVFCTPKS ftSSCCIATCO WITH EQUILIBRIUM SIMPLEX AND THEIR LEVALOCOT6I
*ELS OP OPI-KATION'/I ALOC0762
WKITFI K6,f>?l ) ALOC0763
621 FORMAT ( '0' ,2X, 'PRICF' , IX , • ACT I V I TV /UNHAND ' , 3X , ' LEVF.t ' , 7X, • VECTOR OALOC076'i
*F SUPPLY OR DFMANI) CALCULATED ON THE EOULIBRIUM SI^PLEX'/I
HRITE(K6,(,?4I ICOMI.T5T I ) , 1 = 1 ,M
624 FniU'.AT(41X,Ar),AX,A^,AX,AOt1',Aal
WRI IF( K6,62bl
625 FORMAT! • ')
Of) 6?7 l = l,NACT
T( I 1 = 0. 0
627 CONTINUE
DO 650 K«lfN
630
ALOC0765
ALOC0766
ALOCQ767
ALOC0768
ALDC07«>q
ALOC0770
ALOC0771
ALOC0772
ALOC0773
622
650
651
805
810
DO 630 1=1, N ALOC0775
P ALOC0776
CONTI'JUE ALOC0777
IF ALOCOR05
820 CCJMTINUF ALOCOR06
0(1 S50 J-l.N ALOC0807
Dt) M?b 1=1, N ALOC0808
P|||''U(|,j| ALDC0809
825 CONTINUE ALOCOfUO
CALL DEMANn(WT,n,niNV,A,NAMC,P,X,T,MA,W,0,E,MPe, I 0, II , I T YPC, ALnCOBl 1
* AORE,COMLn,PLAYLn,rinMLR,MAft,ALnC,EXOG, IB,TT J ALOCOfll?
L»J*NACT
DO MO 1 = 1 iN
AH,L)=X(II
CONTINUE
PRICF PN THP FINAL PRICITIVE SET, PUT INTO COLUMN
A) THt NPIH KOW , THE PROFITS ASSUCIATED WITH EACH
830
C ---
C ---
C —
FOK FACM
SIAHTINO
ACTIVITY.
on H',O
ALOCOB13
ALOCOB14
ALDCOHl'j
ALOCOR16
L ALOC0817
PRHOUCTALnCOHlfl
ALOCORI-)
Atocofl2o
157
-------
A( 1 ,L) -0.
CALL SUCf'LYUT.n.PMV, A.fi IMP,?, X , T.fA, W,n, E.^PE
* A(iru:,cr:'i.!<,iH. AYL;*,K TH^LH, ^AMT , ALUC.EXCG, IB, T i, i )
on «ib K=I,M
A( I,L 1 =M I ,LHP(K(*X(K)
835 CONTINUE
840 CONTINUE
AIN'ACTP.L I--1.
850 CONTINUE.
DC (160 I=l,NACr
T( I )=0.
00 B55 J^l.N
L=J
655 CONTINUF
860 CONTINUE
DO 8/0 I = 1,N ACT
DO fl65 J = l ,N
L*=J+NACT
AN = N
A( I ,L)=At I ,L )-T t I )/(l.«AN)
865 CONTINUE
870 CONTINUE
900 IN[JFX = -1
CALL C I VOJ( MA, 10, I UPtCOMLP., I IM
CALL TERM I h T , H , :1I'JV , A , 'l.'.MF , P, X , T, MA, W , D, F, MPE
* AuREiCOMLR,PLAYL'1,Fl.:NLH,r.-.Mr , ALOC , FXOC, I lit T 1 )
CALL OCVA'aK hf ,". t ", INV, A.'JAHf- , P, X, I,KA,w, 0,fr,HPF.
* A(iRF,COMLR,Pl.AYL 1,F1 ';ML I1,, " A"l: , «LHC , t XOG , IU.TT I
7fcl FORMAT (20X, 'THI' fCCNOMY HAS AN LiNflflUNUCL SOLUIICN'I
701 FORMAT ( 1X.20F6.? )
KETUK-H
END
SUHROUT INT PIVOI (f.A.IO, I l.i'.Cf "LP, I fit
01 MISSION "AtNiNI . ICIN) , I 1 ( M t IMN) iCO^L U { N I , I 'U N »
RFAL'rt XTYPF('t) , X ACHF , Xf , CNG
DOUF1LF PKrCISKiN P.CrVL!*
CtVnTiN K5, Kft, V,f/;, VAX! ,1 ",l ».N. l.HAX I ,tn.O,ISFL,KPc, ,ITT
*, ni'Ex, jcui , u xs, irx', i , inx!>^,f.f,,i. I:;I:EX,LJI:UT , i AVG
CO.'-'uON/n |M/N,'J.',(, T .'.PL AY, I N,M', Lf,P
COHf ON/ L i' L / x r Y c r , < ;.r K r. , x F , r NT,
IF( IfinPX.GT.l ) GO 1C V>
IF( If,OtX.I:Q.-l ) GC 10 '.'V>
ISUK=0
un 10 i = it'j
IF ( I.EC.N::) GO 10 5
ISUK-I SU"« IIM I )
•j un lo J-liN
MAI I, J )=!« IH( I )
10 CimiNUF
CO I'i J-l.N
MAINNt J) ='"*?-N-ISUM
MAI J, J )-MA( J, J)-l
1 0 ( J ) = J
II ( J)=J-l
15 CDNTIN'JE
11 ( l) = M
IFIKPS.GE.?) GO m 550
GO TO A 90
35 CONTINUE
JP=JOUT»l
IF ( JOUT.FO.N) JP^l
LOO=IOI joun
1.10=1 1 ( JOUT )
L01=IO( JP)
LI 1 = 1 I (JP)
MAtLOOi JOUT ) -t-A(LOO, JUUT )* I
HA (LI I ,.|OUf l-^-'Ml. 1 I i-lCLT >» 1
l , JOUT 1-1
ALHCOP.21
. 10, II, ITYPE,ALOCOP72
ALOCOR23
ALOCO?J2
-------
A -30 V.f! FOKKAI i 'ormiAL PRJCT SIMPLEX'/)
GO TO '>ro
A<>5 IFIK.PS.^E . J) GO 10 550
WKlTf I K6,'. :lft)
AV6 FORMAT ( -Or. wUtLlBRIUP PRICF SIMPLEX'/)
500 DC 'Jl0 I --If 'I
WRITE (r,6, TOT) COMLHII) , (MAII, J) ,J=1,K)
703 FORMIC ' ,AO,AX,20I6/ t 13X.2016))
510 CONTINUE
550 00 560 l = l,N
' P( I )*MA( 1 , JOUU
560 CONTINUf
RETURN
END
SUBROUTIMH
* AORF.U'|"I
TERM
(V,T,f',H I 'JV , A , N AMF , P, X , T , P A , W, D , F . fPT , I 0, 1 1 1 I TYPE
, r t. *•• i. p-,f •"•••! , AUK: , r xcc . in, TT)
HI (N| , illN.'iACI ) t I- KiVt.S.NP), A(!,ACn»,MP),NAfr ItiACT I, PIN)
*X{N), T IN ACT ) , MM N, m, «( M'L A Y,NP ),;>('. "LAY, «J) , F ClPl A V 1 , M> F ( NPL A Y ) t
* 10(N) , I 1 I'-!) , I TYPr; ('JACIW'M , .1C!fU: ClACrH'il ,C()HLB(M ,PL 'YLR
* Flkf Lf»(NAf. 1 MM) , HAVP t '• ) .AICCI. \PLAY.f.) , f XClGCi) , If, IN) . fT IN)
ftfAL*?. xiyn'c. > .x'vrd-.xr .r^.c
uoufiLt; PI'.I cismN »• I'./.P.X, r .crftf., PL-\VL!',f IRMLR
CQ('ll''0>J K'j.ffj, S'.N'J, CSX I ,1 I-'.LNN.LKAXI , I lit, , I SEL.KPS, I TC
*, IMOf X, Jf.JI , II.XS, IL'Xll , ID'3^,'.n ,1. K.'CCX.LJCIU, IAVG
CUI-KON/Uir'/N.f.'ACI ,t.!'L^Y,L•J,^P,LNP,^^ACl ^'^,•II^>,NACTP
COM''OM/Ll.iL/XTYPF, XALXT ,Xh , hNG
NACT1 - 2*NACI
S=1./AN
IFIIAVC.KO.O) GO TO I
[)U 't [ = 1 , N
5UM=0.0
on 3 J=I,N
R r = »
I ) -I
Of) ^0 J- 1 ,'i.\CIT
STARCH Fti-l A MFCATIVF ft fulfil
[F (A riA.CH>, J) .Lh.-.COl ) GC TO
CUNUNUr-
GC) Id 11')
IMP Pivot r.niuHN is JPIV
I HE I'.OITRH ROW.
R=IOOUOOOO.
>Dn AO 1-1 , NACT
IK(A( i , jciv) .LF-..OOCI ) co rr AO
IF! (A( I .M'Pl/Al ! , JPIVI I .G':.R) CO TO AO
k^A(l,NTP)/A( I, JPIV)
IF(R.C.T.<3 ~>f><)<)'i'-). ) hRITHKIi , A I I
roi'.^AT I 'iJi.:Mnll/lL:i:0 llfJ'-Art
IF (R.c. r .'J j'>9Tr(v. i i*A(!,JPIV)/AUI>IV,JPIV)
cii'iir-iui:
A I 1 , J I' I V I = . 0
CMNTI'lUr
OH l')0 J--1.NTP
IF ( j.ro.j'Mvi on TO iso
A( IPIV, J) = A( IPIV, JI/AI IP IV, JPIV)
ALOCOR96
ALOC08<)7
ALOC0098
ALOCOR99
ALOC0900
ALOC0901
ALOC090?
ALOC0903
AinCQtJOA
ALOC0905
ALOC0906
ALOC0907
ALOC0908
ALOC0909
ALCC0910
,ALnC09ll
ALOC0912
, ALOC0913
A LOG 09 1 A
A UK. 09 15
ALOC0916
AI.OC0917
ALOC0918
ALOC0919
AUJC0920
ALOC0921
ALOC092?
ALOC0923
ALI1C092A
A L DC 092 5
A L 00 09? 6
ALOC0927
ALCIC092B
AI.OC0930
ALflf.0931
ALI1C093?
ALO(.0933
ALOC091A
ALCC0935
ALOC0937
ALOC09AO
A tor, 09 A?
A i or. en A 3
ALOC09AA
ALOC09A5
ALOC09A6
ALOCO-IA t
ALOC09A8
AUJC09A9
ALOC0950
ALOC0951
ALOf-0952
ALOC095 J
ALOC095A
ALOC0955
ALOC0956
AinC0957
AU-1CO'')58
Atnr.0959
A(I PI V,JPIV) = l.
IMF VECTOR HAMK CONTAINS THE INCF.X OF WHICH COLUMNS ARF
Ainr,096i
ALfir.0962
ALnC.0963
ALHCO'J&A
ALOC 0'I65
ALUC0966
Al.f)COrJ67
AL(,'C09t,fl
ALOC0969
ALOC0970
159
-------
c BASIS is s?eutNri4L rvn^k HY C.VJT VECFCHS. Ainc'nri
NAPH tPIVI = JP|V AtnCOir?
CO TO 10 ALOCO')7J
199 DO ?QO 1=1,N ALDC09A4
PI I ) = 3. ALOC0975
200 CONTINUE Al.OC.0976
0;i 220 J=l,N ALOC0977
L*J( I )*(R*S)/AM ALOC0985
210 CONTINUE ALOC0986
220 CCHTMUF ALOC09B7
225 IFIKPS.CE.6) RETURN &LOC0988
HRITE(6,901) ALOC0989
901 FORMAT ( (0':OU1L I^KIU" PR1CFS'/) ALOC0990
WRI TEt K&.'JIO) (CGPLr1. II ) , 1 = 1 ,NI ALGC0991
910VOKKArC • , IX, t«l 1 <,A.-,I I ALOC.0992
WRIir(6,900l(P(rI,t=l.M ALDC0993
900 FORMAT ( '0* , l<.n.)/ I ' ',1<,F9.JI) ALOC099<.
905 IF (KP',.r,C.t>) KtTl/i;> ALOC09TJ
WRl TE'(f»,902 J ALOC0996
902 FOKMATI'O ACTIVITY1 , 1 IX , • LF Vt-L ' , 9X., • PRCF I T • , 1C X , 'ACTIVITY VECTOR ALGC0997
*AT ECU 11 I PR Rl" PMlCrr.'/l ALOC0998
WiUTF(KC,,91',| , 1 = 1 ,N) A LOG 1031
930 FHKHATI21X, 1H ?x,A '.)) ALOC1032
HKIIE(Kfr,93U(wT(1 I ,I-l,MI ALOC1033
931 FOKVAI I 'OTDTAl SUPPLY INtT) ' , 11 F 10. 3/12 1 X, I IF 10. 3 11 ALOC103'.
WRITE!Kfr,0611 ALOC103S
961 FORKATI' •I ALOC1036
»EfUKM ALOC1037
EN1) ALOC1031
SUHROUI |f;r riEIJ'AKjlH,i«f'MNV,.\.' .HI N.NACT) , p. [ HV I K, W ) , A ( SAC IP , \ TP ) , '.' -*f (»;AC. 1) , P I f! I , -M.nC I 04 I
*XtM ,T I NM,T I , :'(V(N. N I ,'„ ( ;PI. AV ,;,P ), M (r,^>L AY,N I , f- I'lPl f. Y 1, f ff (^PLAY I , A LOG 1042
* 10 (Ml . II ("I, 1 fY!" I'.'.C i'"J) i 4">Rr (MAC IM'J) .CfJCl. !*[')) , P t, AVfU'JPl. A Y I , A LOCI 04 3
» F IRMLBIN'.r. TWO ,f'Vvl. I •; I , ALCC 1 NPLAY , M , t Xl)G(H) , IillfU ,TT(M A LOG 1044
REAL*n XlYi'C(4l,X4URr.,XC,rNG ALCC104S
160
-------
COUKLr PR'-CI'IT'l IU':'/,P,X, r ,CrfL1, PLAYLn,F IJM'.P i
DPUPLE PiU CI3I'"'I C, V jf,<>.:n',
CUM^G'J ¥•->,*<>, ••' ,ttt-i, f!> X I , I.M.L'.'i.LP AX I . !:!!";, I ill , KPS, ITT
*, ir.cr x, JGJI , re;,( .D.ro.oi. A\
IF (f'P( ( I ) .1 (J./'l
190 CO
>. = RCrN»DI Ii J)*(*( Jl
;I. ((!.-!"( 1 1) .1 F.O i ) r,n TP 190
KUI .N«n( I , J)*(P(J »**( 1 ,-EU
l'Pf- I 1 ) .FO.l I HUt-K» 1.
(;=i:Mi!'/^i)i N
DC' 2UO J = l,rj
T(Jl--o,
ir-INPt I I) .r'J.ol ft J J = C*nt t, J)
IF (PI j i .i.t .0. ci r,n in ?co
iFifpf ( i i.f c. 1 1 r ( j )= «.U)( i, Ji I/IPI j) i
IF UP? 1 1 ).; ,j.,'i 1 1 j i- i<.»ij( i ,j i i/tPU) Kli I ) )
X(JI=X(J)/OI i'l.AYLC.I II f ( F( J) i J'liM
QOO FOR^ATC ufMAf.u or ' , AH,?X, uno.i/(?lx,HFio.T) )
250 COM I'4UF.
c — ADD PI f xfr, I'jnus DCVANO
Un 124 I=1,N
X( I i»x ( I )irxn>->( 1 )
325 CBNIINUE
IFIKPS.C.H.1/! KETinN
IFI INDfX.KJ.-l ) •*•'. I rE(K6,91f,| (X(J) , J'l,N»
915 FORMAT ( 'OffUAL OK"ANO» , flXt 1 If 10. i/ ( 2 1 X , 11F 10. j J I
RRTURN
I -NO
suiu-nui I'n: siu'Pi.Yur.n.ruMV.A.'i.M'p.p. •>., T.IM.V, ,t., r.Mnr, is, 1 1, ITYPF
* AOkS.CCMLI'. tPLAYL'l.) I .'."I. B, \> M'l , A LUC t fc XflG, I " , f i . I J )
DIKFNS I t'N WT I?;} t'H \,.VJA(. T ) ,f. I^Vl NiNP I i MNftCTP.NTM ) ,t,!Wt IN. '.CD , P I M
*X (Ni).l I'JftC I ) , •'. (Mi ••. ) ,v. Cif'L .'.Y,f >' I .HC.PLAY.M .F l':»'L'iY>tl'--' (""LAV I t
* IOCNI i I UN I , I! 'r( i'.Af. I."N» »A(IHE CJAC iMM) iCi:.ul.n (M « t'LIYL" i'if'LAYl .
» F I RHLrt I MAC TUN), MARTINI , AL OC (J.l'l A Y ,.N I f EXOG (N ) , IB (M til (M
I'.r AL*>) XIYPf (/,» ,X.,' ' f , Xf-.F'jC
Uf)Ul,L£ PULClSIdN iil'IV.P.X, I ,CO»'Ln,PLAYLfi,l
ALOClOAq
ALOC1050
ALOClO'.l
A LOG 10^'
ALOClOi J
ALOClOS'i
ALOC1056
ALOC1057
ALOC105P
ALOC1059
ALOC1060
A LOG 1 061
ALOC106?
ALOC1061
ALOC1065
ALOC1067
ALOC10f,8
ALDC1069
ALOC1070
ALOC 1071
A LOG 107?
ALOC1073
A LOG 10 7'.
ALOC1075
ALOC1076
ALOC1077
ALOC107S
A L OC 1 0 7 9
ALOCIOBCJ
ALCCIO'H
ALncio<;
ALCClOf.'j
ALOClOiUi
ALOG10<)7
ALOClOftl
ALOC 1019
ALOCIO'JO
ALOC10T1
ALnC109?
A LOG 1 09 '?
ALOC 10')'.
ALOCIOTO
ALOC1C97
ALOClC'-ifi
ALOCltOy
ALOC11CI
ALOC1102
ALnCU03
AlOGHC'.
ALOCllO1)
ALOC 1106
ALOC1107
ALOC1 10"
ALOC 11 09
ALOCU10
ALOC1 111
ALOCI 11?
.AI.OC1 11 1
ALOClll'i
,ALC:CI ii'i
ALDCl I |'j
ALOC 1 1 1 7
ALOCI 1 If
Ai.tici m
ALOC11?0
161
-------
OllUULE PRECISION PP,PA,C.UM
COMMON K=0.0
IFIPII ).LE.O.O) GO TO 350
350
3iO
I)**AORE( JM))«
-------
.Accession Number
Subject
Field & Group
6CG
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
c Organization
Georgetown University
Title
Use of General Equilibrium in Regional Water Resource Planning
|Q /lutflO/fs.)
!
Poirier, J. Eugene, S.J.
Harrington, John M. , Jr.
- — , —
11
16
Date
12
Pages
162
Protect Number
16110 FIO
21
, c Contract Number
Note
22 I Citation
23 I Descriptors (Starred First)
~^-J
*Regional Analysis
*Resource Allocation
*Water Pollution Control
Water Resources
Computer Programs
25 I Identifiers (Starred First)
*Algorithm,*Externalities, *Shadow Prices, Regional, Public Goods
27 \Abstract
This report presents a methodology for applying general equilibrium
analysis to a regional economy. An interaction or trade mechanism is presented
which will force a regional economy into equilibrium with the economy in which
it is embedded in the sense that the relative prices will be identical in these
economies for their common commodities. Specifically one or more consumers in
the regional economy is assumed to hold money resources which are allocated to
the purchase (sale) in the surrounding economy according to the plan determined
by the algorithm presented in this report.
The assumptions required for applying general equilibrium analysis to a
region are stated. A technique is presented by which public goods can be treated
in a general equilibrium framework. Since one of the conclusions from a general
equilibrium application is the optimum supply of any commodity, this technique
can determine the optimum supply for a region for such public goods as clean
water (air, etc.). There is an equilibrium price for a public good for each
consumer and these different prices determine what each consumer should be
taxed for the public good.
The results of some numeric computations are presented indicating how the
use of the algorithm works with a regional economy which has a public good.
This report was submitted in fulfillment of Project Number 16110 FIO under
the sponsorship of the Federal Water Pollution Control Administration.
Abstractor
J. Eugene Poirier, S.J.
Institution
Georgetown University
(REV. OCT. !»«»)
SEND rot WATER RESOURC ES SC I ENT I FIC INFORMATION CENTER
U S. DEPARTMENT OF THE INTERIOR
WASHINSTON, D.C. 20240
»U.S. GOVERNMENT PRINTING OFFICE: 1972 484-483/88 1.3
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