United States
Environmental Protection
Agency
Office of Air Quality
Planning and Standards
Research Triangle Park NC 27711
EPA-450/5-84-003
June 1984
Air
Agricultural Sector
Benefits Analysis
For Ozone:
Methods
Evaluation and
Demonstration
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AGRICULTURAL SECTOR BENEFITS ANALYSIS FOR OZONE:
METHODS EVALUATION AND DEMONSTRATION
Submitted to:
Office of Air Quality Planning and Standards
U.S. Environmental Protection Agency
Research Triangle Park,
North Carolina 27711
Submitted by:
Raymond J. Kopp
William J. Vaughan
Michael Hazilla
Resources for the Future
1755 Massachusetts Avenue, NW
Washington, DC 20036
June 15, 1984
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DISCLAIMER
This report has been reviewed by the Office of Air Quality Planning
and Standards, U.S. Environmental Protection Agency, and approved for
publication as received from Resources for the Future. The analysis and
conclusions presented in this report are those of the authors and should
not be interpreted as necessarily reflecting the official policies of the
U.S. Environmental Protection Agency.
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ACKNOWLEDGMENTS
In the preparation of this report RFF received comments and assistance
from several individuals. Thomas Walton provided comments and suggestions at
all stages of the research. Other EPA staff members providing comments
include: Alan Basala, Pam Johnson, David McKee, John O'Connor, Harvey Richmond
and Larry Zaragoza. EPA consultants Duncan Holthausen, Jan Laarman and V.
Kerry Smith also commented on the work plan and draft final report.
We are especially appreciative for the comments of an early stage of our
work provided by Richard Carson of the University of California at Berkely and
for the assistance rendered by David Fawcett of the United States Department of
Agriculture in manipulating the Firm Enterprise Data System. Without David's
assistance this project could never have been completed.
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CONTENTS
Chapter Page
Executive Summary i
1 Introduction 1
2 An Overview of Agricultural Production 6
3 Empirical Methods of Assessing the Impacts of Changes
on Agricultural Production Due to Photochemical Oxidants. . . 10
3.1. Alternative Approaches 10
3.2. Theoretical Review of Production Duality Models 20
3«3. Modeling the Impact of Environmental Variables
on Agricultural Production 36
4 Welfare Gains (Losses) from Decreased (Increased)
Ozone Concentrations: A Review of Consumer and
Producer Surplus 46
5 The Regional Model Farm 58
5.1. Introduction 58
5.2. Simple Heuristics of the Regional Model Farm (RMF). ... 61
5.3. Analytics of the Regional Model Farm
and Welfare Calculations 64
5.4. Welfare Calculations 79
5.5. Operationalizing the Welfare Calculation 83
5.6. Conclusion 87
6 The Estimation of Dose-Response 99
6.1. Introduction 99
6.2. Statistical Considerations in Fitting
Dose-Response Functions 102
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Chapter Page
6.3. Crop Yield-Ozone Dose Model Specification:
The Single Variable Case 103
6.4. NCLAN Reported Dose-Response Functions 124
6.5. Re-estimating the NCLAN Dose-Response Functions 131
6.6. RFF Box-Tidwell Dose-Response Function Estimates 149
6.7. Averting Behavior as Embodied in Variety Switching. ... 160
6.8. Concluding Remarks. 161
7 Yield Changes Using EPA Ozone Scenarios ..... 168
8 Some Welfare Exercises Using the Regional Model Farm 185
8.1. Introduction 185
8.2. Maintained Assumptions Used in the Illustrative
Welfare Exercises 186
8.3. Benefit Calculations with Elastic Demand 188
8.4. Welfare Estimates Under EPA/OAQPS Supplied
Ozone Scenarios 190
8.5. Concluding Remarks 193
9 Sensitivity Studies 202
9.1. Introduction 202
9.2. Harvest-Nonharvest Cost Differential 202
9.3. The Problem of Varietal Switching 204
9.4. Alternative Estimates of Crop Demand Elasticity 208
9.5. Alternative Dose-Response Equations 210
9.6. Concluding Remarks 229
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Chapter Page
10 Agenda for Future Research 230
10.1. Introduction 230
10.2. Further Analysis of Ozone Using Biological
Dose-Response Functions 231
10.3. Non Dose-Response Function Approaches
to the Agricultural Impacts of Ozone 235
References 237
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EXECUTIVE SUMMARY
The U.S. Environmental Protection Agency (EPA) is currently beginning
work on the Regulatory Impact Analysis (RIA) for the reconsideration of the
ozone National Ambient Air Quality Standard (NAAQS). The RIA provides
background information that includes benefits, costs and other information
for alternative standard specifications.
In preparation for the RIA, EPA required an applied model that could use
agricultural sector biological dose response information, agricultural cost
of production data and air quality information to estimate changes in
producer and consumer welfare due to changes in ozone exposures for
agriculture. The air quality information and exposure response information
which will be used in the RIA are not yet available; therefore, preliminary
air quality information is used. Also, exposure yield functions were
estimated from information contained in summary National Crop Loss Assessment
Network (NCLAN) reports. The exposure yield data in these NCLAN summary
reports is aggregated while the data which will be used by NCLAN to develop
the dose response information for the RIA is more detailed.
The research described in this report is an attempt to incorporate the
dose-response information obtained from NCLAN into an economic model of
agricultural production. The result of this work is an assessment model
capable of describing the change in societal welfare emanating from the
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agricultural production of soybeans, wheat, corn, cotton, peanuts, sorghum
and barley in response to changes in rural ambient ozone concentrations.
The economic assessment model discussed in this report exploits a very
important hypothesized biological relationship between ozone and crop pro-
duction, namely, ozone neutrality. This term implies that the optimal ratio
of factors of production is invariant with respect to ozone concentrations.
This means that an agricultural production function shifts in a way that does
not influence the optimal mix of productive factors. The assumption of a
neutral production function shift is implicit in the design of NCLAN ozone
experiments where the experimental focus is on crop yield.
The assessment model has the ability to calculate a measure of the
change in societal welfare which is equal to the change in the sum of
consumer and producer surplus evaluated at current 1978 ambient and
alternative ozone concentrations. Throughout the text of this report, this
measure of the change in societal welfare (either positive or negative) due
to alternative ozone exposures is termed net consumer and producer surplus.
The term net does not imply that the costs of the regulatory action have been
considered — indeed they explicitly have not.
The simple diagram below illustrates the calculation of net consumer and
producer surplus as executed by the assessment model. The curve D represents
the demand for a particular crop and the curve SQ the crop's supply curve
conditioned on a given ozone concentration. Equilibrium price and quantity
are P_ and QQ respectively. Consumer surplus is the area A and producer
surplus is the area B •«• C. If ozone concentrations fall the supply curve
shifts to S1 and the new equilibrium price and quantity become P1 and Q1
respectively. The new consumer surplus is equal to the area A + B + E + F
ii
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Price
P, ~
Quantity
Figure 1. Net Consumer and Producer Surplus Calculation
A
B+C
A+B-f-E+F
C+D+G
Consumer Surplus for Demand Curve D.. and Supply Curve S~
Producer Surplus for Demand Curve D and Supply Curve S-
Equilibrium Price for S- and DO; P.. = Equilibrium Price for D- and S1
Equilibrium Quantity for SQ and DQ; Q = Equilibrium Quantity for Dn and S
Consumer Surplus for Supply"Curve S and Demand Curve Dn
Producer Surplus for Supply Curve S.. and Demand Curve D-.
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and producer surplus is equal to C + D + G. The change in societal welfare
is equal to the net gain in consumer plus producer surplus which is equal to
the area D + E + F + G.
To calculate the change in welfare the assessment model must determine
the shape and placement of the demand curve D, the shape and placement of the
original supply curve SQ and the manner in which SQ shifts in response to
ozone changes. The demand information is borrowed from the United States
Department of Agriculture (USDA) estimates and is discussed in Chapter 8.
The shape of the supply curve SQ is obtained from a model developed solely
for this purpose, while information describing the shift in the supply curves
comes .from aggregate experimental data collected by NCLAN, and from NCLAN
dose-response equations published in Heck et al. (1984a, 1984b).
The economic model generating the supply functions for particular crops
is named the Regional Model Farm (HMF) which reflects the regional nature of
the data base providing the prime informational input to the model. The RMF
is designed around the biological hypothesis of ozone neutrality. Ozone
neutrality has the desirable property that all factor demand intensities
(ratios of factor inputs) are invariant with respect to changing ozone
concentrations. Since ozone neutrality does not induce factor substitution,
and holding factor prices constant during the analysis, we are able to treat
the production function underlying our supply function as Leontief.
The informational component of the RMF is derived from • the Firm
Enterprise Data System (FEDS). Operated by USDA, FEDS provides agricultural
analysts with sample operating budgets which describe the entire cost
structure for producing an acre of a particular crop in a specific region of
the continental U.S.. The budget is representative of the average
iv
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agricultural practice in that specific region and is verified with a battery
of farm level surveys. A single budget for the production of soybeans in
southeastern North Carolina, for example, may include cost information on as
many as 200 inputs to agricultural production, the average yield per acre to
be expected and the total number of acres planted in the region.
For each of the FEDS producing areas we assume that the FEDS budget for
a particular crop type represents both the cost and yield existing for that
budget year, for given prices of inputs, outputs, and ambient ozone concen-
trations. Since the FEDS budgets are on a per acre basis we assume constant
returns to scale in order to aggregate across all of the planted acres
covered by a single budget. Further, we assume that during the analysis
input prices do not change.
With these assumptions in place the construction of aggregate supply
functions for particular crops is straightforward. First, given constant
returns to scale, marginal cost is equal to average cost. For a particular
crop/region budget we divide the total cost of producing an acre of the crop
by the yield per acre and thus generate an estimate of the marginal cost per
crop unit. Repeating this calculation for all regions growing the same crop
produces an array of marginal costs of production across the entire
continental U.S.. When the marginal cost of production in each region is
mapped against the output of that region we have a region specific supply
curve for each crop. Ranking these regional supply curves by marginal cost
from lowest to highest and then aggregating across regions yields the
aggregate supply function for the specific crop. This aggregation produces a
stepped supply curve such as that depicted in Figure 2.
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$ A
Aggregate marginal cost
MCD
MCC
MCB
MCA
0
*
i
T '
l
i
" t
i
i
• !
:
1 i «
! i
.• .
...... |
i
i
i
i
i
•
i
L
^/^ function or supply curve
r "*
1
1
I
i , decline in ozone
; i
i
• i
i
i
i
i
i
i
i
l
i
1
i
i
Qi Qo Qo Q/
1 L J '* Crop Output
Figure 2. Aggregate supply curve for regions A, B, C, D for crop Q.
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The stepped supply curve generated by the RMF is analogous to S^ on
Figure 1. To obtain S^ we employ the NCLAN experimental evidence. Essen-
tially NCLAN is- a network of research sites that among other research tasks
performs controlled experiments designed to identify the dose-response
relationship between ambient ozone concentrations and the yield of particular
crops. Using the data generated by these experiments we have estimated
dose-response functions•that explain a measure of crop yield as a,function of
ozone. The functional specification we utilize permits the estimated
dose-response relationship to be linear or take on a wide variety of
nonlinear, forms. Using these dose-response functions it is possible to
explain the shift, in the crop supply functions when ozone concentrations
change and thus determine the new supply function analogous to S- in
Figure 1. To examine the sensitivity of our welfare calculations to the
functional specification of our estimated dose-response functions we have
performed a parallel analysis using functions recently made available by
NCLAN in Heck et al. (198Ma and 1984b).
The accuracy of any welfare estimates generated by the assessment model
is linked to: (1) the accuracy of the RMF in defining the baseline crop
specific supply curves; (2) assumed characteristics of-the demand side of the
market, specifically elasticity of demand; (3) the biological dose-response
functions defining the shift .in the supply curves, and (4) the reliability of
county level ozone concentration estimates. By the time the RIA for ozone .is
undertaken, NCLAN will have extensively studied the dose-response functions
and EPA will have prepared final estimates of the county level ozone
concentrations. In this report we analyze, through the use of sensitivity
analysis, assumptions which underlie the cost structure of agricultural
vii
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production as perceived by the RMF, the implicit assumptions regarding the
demand elasticities utilized in the welfare calculations, and alternative
specifications of the dose-response equations.
On the basis of model results presented in Chapters 7 and 8 and the
sensitivity analysis presented in Chapter 9, the RMF approach to the
calculation of agricultural benefits for ozone seems far superior to the vast
majority of competing approaches discussed in Chapter 3. As the sensitivity
analysis suggests, the RMF welfare calculations are robust with respect to
demand elasticity assumptions and will benefit from the continuous refinement
of the NCLAN dose-response information and EPA ambient air quality data.
The majority of the results presented in this report were completed and
transmitted to OAQPS in a final report dated September 30, 1983. Those
results were based on a set of five dose-response equations estimated by RFF
staff using published, aggregate NCLAN experimental data for five crops:
soybeans, corn, wheat, cotton and peanuts. In May of 1984 NCLAN released
dose-response functions estimated from the original, unpublished,
disaggregate experimental data using a flexible functional specification
(WYBUL) for several crops. As part of the Regulatory Impact Analysis for the
ozone NAAQS RFF staff examined the sensitivity of the results presented in
the original September 1983 report to the use of the RFF estimated
dose-response functions by recalculating several welfare estimates employing
the new NCLAN functions. The use of the NCLAN functions provided for broader
crop coverage and permitted the inclusion of sorghum and barley in addition
to the original five crops.
viii
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FOOTNOTES
1. Consumer surplus is the difference between what a consumer would be
willing to pay for each unit of a good rather than do without it and what the
consumer actually pays for each unit of the good. Producer surplus is the
difference between what each producer is paid for each unit of the good and
what he would accept rather than foregoing sale of the good.
2. Demand elasticity is a measure of how responsive quantity demanded
is to a change in the price of a good. It is defined as the percentage in
quantity demanded divided by the percentage change in price.
ix
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CHAPTER 1
INTRODUCTION
In preparation for an eventual ozone Regulatory Impact Assessment (RIA),
EPA required an applied model that could use agricultural sector, biological
dose-yield information and air quality information to estimate changes in
producer and consumer well-being. In other words, the changes in economic
surplus due to changes in ozone exposure for agriculture. This report
presents a preliminary version of such a model. The air quality information
and exposure response information which will be used in the RIA are not yet
available.
The research described in this report is an attempt to incorporate the
natural science information obtained from NCLAN research into an economic
model of agricultural production. The result of our work is an economic
assessment model of agricultural cost and production designed to examine the
impact of ground level ozone concentrations on the production of seven field
crops: soybeans, wheat, corn, cotton, peanuts, sorghum and barley. The
model draws its economic information from the Firm Enterprise Data System
(FEDS), developed by the United States Department of Agriculture (USDA), and
thus contains the information necessary to assess ozone impacts at a fine
level of regional disaggregation.
The econonlic assessment model discussed in this report exploits a very
important hypothesized property of the biological relationship between ozone
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and crop production, namely, "neutral factor productivity enhancement"
(NFPE). This term implies that the optimal mix of factors of production is
invariant with respect to ozone concentrations. This can be understood by
visualizing an agricultural production function which shifts neutrally with
changes in ozone concentrations. The assumption of a neutral production
function shift is implicit in the design of NCLAN ozone experiments since the
major focus of the experiments is on yield. If one were to believe that
ozone differentially impacts productive factors implying a nonneutral
production function shift then one would design experiments which
systematically varied input quantities in addition to ozone and would lead to
dose-input functions as well as dose-yield functions.
For the purposes of our present study we maintain a partial NFPE
hypothesis. That is, we assume that all preharvest factors of production
have their productivities affected equally by changes in ozone concentration.
However, we find little evidence to support a similar view regarding factors
of production involved in•harvesting activities. Therefore, in our model we
admit the possibility that the productivity of harvest production factors may
not be affected by changes in ozone concentrations. For example, an increase
in yield associated with a decrease in ozone will result in productivity
enhancement for preharvest factors of production but may not enhance harvest
factors. Therefore marginal harvest cost might be unchanged and total
harvest cost increased.
It is not reasonable to assume that all environmental pollutants will
shift agricultural production functions neutrally. For example, some pre-
liminary greenhouse evidence suggests that acid precipitation has the effect
of reducing fungicide retention on plant surfaces, thus requiring more
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frequent applications. Reductions in the acidity of precipitation would thus
result in "biased factor productivity enhancement" (BFPE) and would imply a
nonneutrally shifting production function.
The above example highlights the importance of recognizing that the
biological relationship between environmental factors and crop production has
a great deal to do with economic model construction, if that model is
designed to incorporate natural science information in an economically
meaningful fashion. A biological relationship which results in BFPE requires
an exceedingly more complex economic model than a relationship characterized
by NFPE. For a more detailed discussion of these concepts see Kopp and
Vaughan (1983).
One serious limitation of any economic model which requires biological
dose-response functions is the nature of the available functions themselves.
Since they are particular to the conditions at the individual site where the
experiments were conducted, and the ceteris paribus controls of the experi-
ments, their results are not easily generalizable to any particular crop
grown over broad geographic regions of the country. This is indeed a serious
problem because complete dose-response surfaces for plant species and
cultivars which reflect variations in soil type, weather, and farming
practices are not available. Even ignoring variations in operating practice,
plant response to ozone is potentially a function of soil and climatic
conditions (light quality and intensity, temperature, relative humidity,
wind, and the concentrations of pollutants other than ozone at the field
level) and other complicating factors such as pests and plant disease (Leung
e_t al., 1978). Although field experiments continue, an expert panel
concluded in 1977 that:
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A complete understanding of the many factors that affect the
response of vegetation to oxidant pollutants is probably
impossible. An understanding of the individual factors is
possible, however, and much is already known; but the inter-
actions between some of these many factors are unclear (NAS,
1977, p. 513).
As we shall discover below, the lack of completely and exhaustively
specified dose-response functions by species and cultivar adds uncertainty to
the economic evaluation of the gains or losses to agriculture of alternative
ozone standards because the economic models have to be driven by imperfect
versions of such functions. A possible alternative outside the scope of this
study is to rely instead on microtheoretic models of farm production,
estimated from real world data. Here, ozone concentrations would be included
with other pollutants and weather conditions as explanatory variables, along
with the usual economic variables. This approach is briefly discussed in
Chapter 3 but we caution this approach would require very detailed ozone
information.
However, no matter which approach to quantitative economic modeling is
undertaken, they all include estimation of the firms' (and the aggregate)
supply function by crop in order to generate welfare impacts. Before the
modeling methods for doing so are discussed, we present in Chapter 2, in a
purely descriptive way, an overview of the agricultural production system.
Next, we briefly outline some of the approaches available to quantitatively
model this system, and tie each of them to the particular benefit measures of
ozone control each is able to produce. Having established the frame of
reference we describe in greater detail several of the modeling alternatives
which, in our opinion, are reasonable and might be pursued in this project.
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Chapter 3 discusses the various empirical methods for assessing the
impacts of photochemical oxidants on agricultural production activities,
while Chapter U provides a brief review of the economic concepts of consumer
and producer surplus as used in the analysis of public policy. Chapter 5
presents a detailed discussion of the economic assessment model constructed
for the analysis of ozone impacts. Chapter 6 provides a lengthy but neces-
sary discussion of the biological dose-response functions imbedded in the
economic assessment model. Using a set of EPA specified ozone concentration
scenarios, changes in yield for the five crops considered in this study are
reported in Chapter 7- The results of sensitivity studies on crucial model
parameters are reported in Chapter 8. Again using the EPA ozone scenarios,
the impacts of alternative concentrations on agricultural cost and production
by crop are discussed in Chapter 9. Chapter 10 contains suggestions for
future research.
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CHAPTER 2
AN OVERVIEW OF AGRICULTURAL PRODUCTION
Agricultural production processes can be distinguished from conventional
theoretical constructs of single product manufacturing processes on two
general grounds. First, agricultural processes typically result in multiple
outputs being produced by a single firm (Mittelhammer ejt al., 1981) and,
second, agricultural production is affected by inputs from the natural system
(weather) outside of the control of the producer (Weaver, 1980). Ozone,
which may adversely affect plant yield, is only one element in the set of
potentially important variables affecting agricultural production processes.
There are four broad types of agricultural production activities:
(1) crop production
(2) animal raising
(3) dairy farming
(4) combined operations (for 1, 2 and 3 above)
A generalized schematic is given in Figure 2-1 which shows that purchased and
natural (precipitation, sunlight, etc.) factor inputs are transformed by pro-
duction processes into desired product outputs, along with nonmarketed out-
puts discharged to the environment as residuals. A specific representation
of crop production activities appears in Figure 2-2 which illustrates the
types of choices regarding technology and output mix the profit maximizing
farmer must make.
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Nature's Inputs
(essentially unpriced)
Agricultural Production Activities
Factor Inputs
and Input Prices
(including land)
1. Crop Production Operations
2. Animal Raising Operations
3. Dairy Farming Operations
4. Combined Operations
Nonproduct Outputs Effects on Land
Product Output
"Food & Fiber"
Residuals
Boundary of
Activity Model
of Agriculture
l>
Product Outputs either:
1. go directly to consumer
markets, or to
2. off-site intermediate
processing operations
and then to consumer
markets
Various combinations of 1. 2, and 3
Figure 2-1. Representation of a "fientiral ized" model of agricultural activities
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00
Factor Inputa and Prlcea
Suedu, bulba, planta
Haclilnery and equipment, i.e.,
automobiles, trucka, trac-
tors, cornplckera, balera,
coablnea
Fertilizers, I.e., couaerclal
fertilizers, line
Peutlcldea, I.e., defoliants.
Insecticides, herbicides
Knergy, I.e., kuh, gasoline,
dleael fuel
Labor, I.e., farm labor,
cuntract labor, nachlnc
labor
Water
Building materials
Land - uoll and topographic
characterlatica
Nature'* Inputa
Sunlight, length of growing season, precipitation,
ground water, and other climatic factors (o.g., wind)
Crop Product Ion Operations
Unit Opuratlonu/Other Production Variables:
1. Crop Mix, I.e., field corn, barley
vegetables
2. Crop Rotation Pattern and Schedule, l.u.,
continuous corn, corn-corn-alpha-com
3. Tillage Method, I.e., minimum tillage, no
tillage, uhallow plow, harrow, dluk
It, Plowing Practices, I.e., contouring,
grading roua, ridge planting
5. FurtlllzutIon Practlceu - mix uf
furtlllzeru; application raccu, uuthuda,
and achedulea
6. Peatlcide Pfactlcua - uilx of puut Iclduu;
application ralua, uelhuda, and bcheduleu
1. Irrigation Muthoda/Syuteoiti - If any, l.u.,
aprlnklura, line canala
H. llarveallng Technology
9. On-alte Crop Proceaulng, I.e., uauhlng
and packing
10. Other
Product Output
• Cropu liarveuted
• Fluid corn for all
purposes. I.e., for
grab), for ullage
• Soybeans for all pur-
poses. I.e.. for seed
grain, ullage
• Uheat for grain
• Other small grulnu
• Suybuana
• lluy
• Peanuts
• Tobacco
• Putatoeu
• Vegetables
• OrcluirU cropu
• Cruenhouae products
• Other cropu
I Product Outputa either
I
1.
!_k'-
D
go directly to con-
sumer markets, or
to
off-site Intermedi-
ate processing
operations. I.e.,
canning, vegetable
oil processing &
refining, etc.
(these operations
are considered
aeparale activities
and analyzed
accordingly)
Figure 2-2. Kupruaunldtlon of agrlcultuiu) crop producing uctlvlllua.
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Crop producing operations involve the planting, growing and harvesting
of crops. There are three general types of cropping operations: nonirri-
gated — where all water inputs are from natural precipitation; irrigated —
where the majority of water inputs are transported to and applied on cropland
by man; and orchard growing operations (tree fruits and grapes) which can be
either irrigated or nonirrigated. The production function for each type of
cropping activity is also different. Each activity uses different combina-
tions of factor inputs (types and amounts) and unit operations to produce
different product outputs.
The basic forces which determine crop production possibilities on any
farm are soil and climate characteristics since they specify the range of
crops for which production is technically feasible. Given these feasibility
constraints the farmer selects the input menus, crop mixes and rotation
patterns that, based on factor prices and output market values, maximize his
profits (Heady and Jensen, 1954). Multiproduct outputs from a single farm
may be observed in any given year either because the farmer has elected to
reduce his risk by diversification or because he has chosen to grow a combi-
nation of crops in a rotation sequence rather than a single crop continuously
over time, or because his farm includes different soil types — or even
different microclimates.
Modeling such a complex system is a major undertaking which cannot
generally be performed either within a short time frame or at modest expense,
as we shall discover from the discussion to follow.
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CHAPTER 3
EMPIRICAL METHODS OF ASSESSING THE IMPACTS OF CHANGES
ON AGRICULTURAL PRODUCTION DUE TO PHOTOCHEMICAL OXIDANTS
3.1. ALTERNATIVE APPROACHES
Ozone concentrations potentially affect the firm's production function
— the technically feasible quantity of output producible from any specified
input set — and, by implication, its cost function. Hence the problem of
estimating the dollar impacts of a policy which lowers (raises) ambient ozone
concentrations ultimately becomes a problem of agricultural supply analysis,
given that one has some knowledge of the demand side, or can make plausible
assumptions about demand response (elasticity).
In agricultural economics there exists a long, and intellectually rich
tradition of efforts to quantitatively represent various aspects of the agri-
cultural production system described briefly in the preceding pages. (For a
review, see Judge, _e_t _al., 1977).
In fact, several of the alternative approaches to empirical agricultural
supply analysis were well understood more than twenty years ago (Nerlove and
Bachman, I960). However, in the decade of the sixties significant advances
were made in operationalizing optimization models of farm behavior (Hall,
Heady and Plessner, 1968). In the seventies duality theorems have been
successfully utilized in facilitating the applied econometric analysis of
farm profits within the context of the neoclassical model of the competitive
10
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firm (Yotopoulos and Lau, 1979). Duality theory has expanded the frontiers
of applied econometric analysis of firm behavior beyond the production func-
tion approach to show that cost or profit functions are equally adequate
representations of the firm's technology. Further, hypotheses concerning
homogeneity and separability of the multiproduct firm's cost function (see
below for definitions) can be statistically tested under the cost function
approach using flexible functional forms (Brown, Caves and Christensen,
1979), as is also the case for the profit function (Lau, 1972).
Yet curiously a quite recent catalogue of methods available to place
economic values on crop yield changes attributable to atmospheric pollution,
Leung _e_t al. (1978), ignored these advances entirely. Their use has only
recently been suggested by Crocker _et _al. (1981) and no empirical applica-
tions of the cost or profit function approaches have yet been undertaken to
analyze the welfare impacts of ozone on agricultural production. The small
set of econometric production function studies which have been done to
analyze the ozone problem all simplistically impose nonjointness on the pro-
duction function, omit or improperly measure inputs, and ignore the simul-
taneous equation bias problem — caveats mentioned in the literature over
twenty years ago (Plaxico, 1955; Griliches, 1957; Hildebrand, 1960; Walters,
1963; Hoch, 1958; Hoch, 1976).
Our own review of the literature on this subject suggests at least six
feasible routes of applied analysis. All of the methods outlined below
differ in terms of data requirements, complexity, and the extent to which
they are firmly grounded in economic theory:
I. Rule-of-Thumb Models
1. Biologists "Valuation"
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II. Economic Optimization Models
1. Linear Programming Models of Crop/
Livestock Production
2. Quadratic Programming Models of Crop/
Livestock Production and Output Demand
III. Econometric Models
1. Models Utilizing Experimentally Derived Dose-Response
Functions
a. Aggregate Econometric Agricultural Supply and
Demand Models
b. Microtheoretic Econometric Agricultural Supply
and Demand Models
2. Models Utilizing Statistically Derived Associations
Between Pollutant Concentrations and Production Activity
Variables
a. Microtheoretic Econometric Agricultural Supply
and Demand Models with Pollutant Arguments
In brief, the Biologists Valuation model simply makes output a function
of ozone concentrations via a dose-response function, and values changes in
output due to changes in ozone concentrations at the reigning output price
crop-by-crop.
The linear programming (LP) model of crop production selects the cost
minimizing set of production activities subject to a. specified bill of goods
to be produced and constraints on the availability of certain critical inputs
like land. Biological dose-response functions are used to alter the quantity
of output producible from the set of inputs required for each production
activity to mimic the effect of varying ozone concentrations. Quadratic
programming (QP) models of agriculture use a production activity.matrix just
like that of the linear programming problem. The principal difference is
that (linear)'demand functions for product outputs are an integral part of
12
-------
the model, so output quantities and prices are endogenous. The criterion
function is a quadratic function which represents either (a) the maximization
of producers' plus consumers' surplus or (b) the maximization of producer
profits. (Although the linear programming problem can be set up as one of
profit maximization, the criterion function is linear because output prices
are exogenously fixed, not endogenously solved for as in the Q-P model.)
The Aggregate Econometric supply and demand model involves the
econometric estimation of price response functions for producers and demand
functions for buyers from aggregate historical data. The link between
economic theory and the specification of the models is generally somewhat
loose. On the supply side, assumptions about optimizing behavior (cost
minimization or profit maximization) need not .be made in order to estimate
the equations of the system. Experimentally derived dose-response functions
supply exogenous information in order to shift the intercepts of the crop
supply curves to reflect alternative ozone standards. In contrast the
Microtheoretic Econometric models specify an objective function for the firm
and derive the models to be estimated from this specification under perfectly
competitive conditions. The parameters of the estimated microtheoretic
models are made functions of pollutant concentrations and thus changes in the
concentrations alter the model parameters and serve to shift relevant supply
functions. Experimentally derived dose-response functions may be employed as
estimates of the true but unknown functions embedded in the model.
The last modeling approach builds directly upon the microtheoretic
econometric model discussed above but estimates the parameters of the
pollutant induced supply shift jointly with the supply parameters themselves.
Such models are securely grounded in economic theory and can be structured
13
-------
such that ozone concentrations are embodied as arguments in the functional
specification. Hence, there is no need for independent information on
biological dose response, given such influences are already contained in the
real world data from which the function is estimated.
General properties of each of these six approaches are catalogued in
Table 3-1. The first category, normative versus positive, is meant to dis-
tinguish normative models which indicate what "ought to be" from positive
models describing "what is." Although this distinction is a simplification
(Friedman, 1935) for our purpose we can say that normative models produce
solutions which describe the way the world should behave given our assump-
tions. Particularly, the optimizing models (LP, QP) often produce prescrip-
tive, solutions for competitive equilibrium prices, input quantities demanded,
output quantities produced, and the spatial allocation of production. Some-
times such solutions are at odds with observed reality. One can never be
sure if discrepencies between the model solutions and reality are a result of
the incorrect or inaccurate modeling of production activities, improper
constraints, or just the fact that the real world operates suboptimally due
to market interference or distortions (Oury, 1971).
In contrast, the econometric models reflect by the very nature of the
data employed to develop them, historical reality over space and time. Thus
they cannot perfectly capture the effects of new technologies developed
outside of the time (or space) span of the data, nor can they tell us much
about the effect on the production technology of changes in institutional
rearrangements which are not translated into changes in market prices. They
take the institutional setting as a given (Yotopoulos and Lau, 1979).
-------
3-1. AI.TKHHATIVK
ESTIMATION HUUKUi 1-tJR
Nornutlvu
or
positive
node!
I. RuJe-of-thuab module
a. Blologlata' valuation ?
II. Optimisation aodela
a. Linear programing Normative
b. Quadratic programing Noraatlve
III. Econoaetrlc nod flu
a. Aggregate supply/demand Foaltlve
b. Hlcrotheoretlc supply/ Foaltlve
deaand
c. Ueoclaaalcal econometric Foaltlve
projection, coat, or
profit function
tcuiiuuilc Avert lily
theory of lieliuvlur tu
thu Clru oicune
None None
Cost Centrally not
minimization dlluwud
or profit
maximization
subject tu
constraints
Net aoclal Centrally not
benefit «. 11 owed
piaxlnlzatlon
(producers '
plua
consumers'
subject to
constraints
Some recognl- Generally not
tlon or syio- allowed
•etry restric-
tions on croaa-
prlce terms
Fully counts- Generally not
tent with nl lowed
optimisation
via duality
theorems
Fully conola- Reflected
tent with in the data
optimization
via duality
theorems
Biological
f tine t tuns
Initial
condition
Required aa
Initial
co ud i t ion
Kequlrcd au
initial
condition
Muquired an
initial
condll Ion
Hequlrud au
initial
condition
Nut ruijulrud-
reflectcd in
producer
cliulccu
Output
L; denuind
cunditlonti
Exogenouuly
fixed prlcea
Exogenous ly
fixed
quant It lee
(coat aln) or
exogenoualy fixed prlcea
(profit MX)
Eitdogenoua
equll Ibriun
price/
quantity
deturmlnu-
t Ion - deouind
functionu
Incorporated
in the model
Endogenous
ccjulllbrlua
price/
quantity
dclermlttatlon
fcindo^cnuuu
uqull ibr lum
i|uaiir lly
del urmlnaL iun
i:«|Ui 1 ibriua
price/
ijiidnt lly
dutcriulit.il Ion
Ueneflt
measure
Producers '
aurplua
Nut produceru1
and
conauiaera '
uurpluu
Net producers'
und
conauuera '
surplus
Net producers'
and
consumers'
surplus
Net producers'
and
cunbunera'
NUL producers'
and
cunnumeru '
burplus
-------
The second property in Table 3-1, economic theory of the firm, depicts
the extent to which the approaches are consistent with and grounded in that
theory. The Biologists Valuation method is devoid of theoretical content.
Both Programming methods are theoretically grounded but assume on the
production side that there are constant returns to scale, infinitely elastic
supplies of variable input; divisibility of production processes; additivity
of two or more processes; and a finite set of process alternatives. Further,
the QP model assumes linear demand functions for product outputs (Naylor and
Vernon, 1969).
The Aggregate Econometric method requires little in the way of theory,
except for some general specification of the variables affecting supply and
demand price and a conceptualization of the aggregate system as either simul-
taneous or recursive in the estimation step. All Microtheoretic approaches
are, as previously mentioned, fully consistent with the theory of the firm
(Varian, 1978, Chapters 1 and 4).
The third and fourth properties in Table 3-1, averting behavior and
biological dose response, are intimately connected. Any economic model which
requires as input an experimentally generated biological dose-response
function based on a few varieties of a single species as input necessarily
precludes the possibility of producer substitution among varietal seed or
plant inputs in response to changes in ozone concentrations. Suppose, for
example, that the dose-response function for a single crop is based on a
single variety, labelled YI in Figure 3-1. Also, assume there are other
varieties which are more resistant to ozone (V2, v3) for which experimental
dose-response functions are unavailable but are familiar to the farmer- If
all varieties 'require exactly the same amounts of cooperant inputs per unit
16
-------
Commonly observed
range of cone.
OZONE CONCENTRATION
Figure 3-1. Dose-Response Functions for Varietals of a Given Crop
V = Variety 1
V = Variety 2
V. = Variety 3
17
-------
output we would observe almost no "damage" due to ozone over the policy range
0-0* in the real world, since costs would be relatively unaffected by ozone
concentrations except for extreme changes. (The heavily shaded envelope
yield function in the figure). However, if V drives our economic model, the
benefits of ozone reductions will be falsely, and perhaps vastly, overstated.
This is a potential pitfall of all economic models requiring experimentally
generated single-variety biological dose-response functions. Only the last,
fully statistical, microtheoretic approach is free of this problem. But, it
does require accurate farm level ozone measures which, unfortunately, do not
exist.
The final two properties included in Table 3-1, output demand conditions
and benefit measures are also linked. To fully understand the implications
of each modeling route in these areas, a lengthier treatment is required. We
devote Chapter 4 to such considerations.
The remainder of this chapter contains an in-depth discussion of what we
have termed the microtheoretic approach. On several grounds the
microtheoretic approach is to be preferred to all others. It has the ability
to incorporate biological information (see Kopp and Vaughan (1983)) or to
estimate the parameters of biological functions directly from observed
producer behavior. Moreover, since the economic assessment model we will
present in Chapter 5 is a member of the general microtheoretic family
structure we feel the lengthy discussion is valuable.
The microtheoretic approach provides the analyst with a set of extremely
powerful research tools since the approach captures both the physical-
engineering aspects of production and the behavior of economic agents who
manage the production activity. The neoclassical theory of the firm provides
18
-------
the theoretical foundations and a set of organizing principles which insure
the internal consistency of any analysis of production activity conducted
using the microtheoretic (M-T) methodology.
Before we begin our discussion one point must be well understood.
Utilization of the M-T approach dictates strict adherence to economic theory.
Any deviation from the theory can cast the entire analysis in doubt. This
implies that model construction and estimation be devoid of ^d hoc appendages
or generalizations and that each step in the empirical analysis comply first
with theoretical strictures before any subsequent steps are undertaken or
policy conclusions drawn. We raise this caveat to emphasize the observation
that much applied work masquerading under the guise of neoclassical-
econometrics is inconsistent with underlying economic theory and thus the
results cannot claim to possess the explanatory power which the theory pro-
vides. Since theoretical consistency is vitally important to the confidence
one can place in empirical results our presentation of the M-T approach shall
be fairly formal. This formality is necessary so that the subtleties of the
theoretical dictates can be identified and their importance in the construc-
tion of economic models revealed.
During our preliminary discussion of the M-T approach we will draw no
distinction between agricultural production and any other type of production
activity. We do this to simplify the presentation and to focus on the more
general elements of the approach. In subsequent discussion we shall focus on
the specific modeling of agricultural production in an environment containing
airborne pollutants.
19
-------
3.2. THEORETICAL REVIEW OF PRODUCTION DUALITY MODELS
The bulk of the theoretical results presented in this section are drawn
in whole or part from three survey papers: Diewert (197*0, (1978) and Rosse
(1970). All proofs are omitted and only the major empirical properties of
various functions are presented. For more theoretical detail the interested
reader is directed to the extensive bibliography found in Diewert (1978).
We begin our theoretical discussion by identifying the production unit
as a firm which combines n factors of production to produce m kinds of out-
put, utilizing a given technology, which specifies the physical transforma-
tion of inputs to outputs. The multiple output nature of technology compli-
cates the analysis; however, since most agricultural production units produce
more' than one output it would be pointless to present theoretical models
based on a single output assumption. We take as given the primal technology
set T which identifies all feasible input-output combinations. The set T is
formally defined as:
T = {(x,y)|(x,y) is a feasible production choice} (1)
where x is an n x 1 vector of inputs and y an m x 1 vector of outputs.
T has the following properties:
T.1 T is a closed set
T.2 T is convex
T.3 T exhibits free disposability of inputs
Clearly, the technology set T is of fundamental importance since any physical
effect on production, attributable to an environmental variable (air pollut-
20
-------
ants for example) must impact production through an alteration in the
technology set T.
Given the technology set T we may express the firm's production possi-
bilities as the maximum of output yA the firm can produce given that it
produces fixed quantities of the remaining m - 1 outputs and fixed inputs.
We define the maximal output rate for output i as:
gi(x , y ) = maxtyj (x,y) e T, x = x , Xy = y } (2)
where *y = (y. v v
j v j 1 § jit •••» y < _ 11
(x°,y°) specifies a point in T
The transformation function may now be defined as
G(x,y) = -y. .+ g.Cx/y) if (x,y) e T (3)
- 0 Otherwise
The transformation function G indicates the distance in output space between
a specified input-output set (x,y) and the closest efficient output vector
producible by the same input vector. If G(x,y) ='0 then the transformation
function defines all those technically efficient input-output combinations.
If for any (x°,y°), g(x°,y°) > 0 the (x°,y°) is a technically inefficient
21
-------
input-output combination and thus G can serve as a measure of its technical
inefficiency — smaller positive values of G indicating greater efficiency.
For any G(x°,y°) < 0, (x°,y°) is an infeasible production choice, i.e.,
outside the technology set T and beyond the frontier of the transformation
function G. The ability of G to serve as a measure of efficiency will be
discussed in later portions of this section when we discuss the actual
modeling of ozone's impact on agricultural production. In the case of a
single output the transformation function reduces to the familiar single
output production function notion.
The transformation function has the following properties:
G.1 G is continuous
G.2 G is monotonic, i.e., G is nondecreasing in x and nonincreasing
in y
G.3 G is quasi concave in every convex subset of X cross Y
Given properties G.1-G.3, Rosse (1970) and Diewert (1974) have demon-
strated that technology set T may be retrieved (defined) in terms of the
transformation function G as shown below.
T = {(x,y) e G(x,y) = 0} (4)
Thus, a duality exists between the primal notion of a technology set and the
notion of a transformation function. This duality insures that the produc-
tion possibilities of a firm facing a multiple input, multiple output tech-
nology can be fully described by a transformation function; and further, that
22
-------
any impact realized upon the technology set T due to the effect of an envi-
ronmental set of variables will be mirrored in the transformation function.
We now introduce the minimum cost function which can be defined
equivalently by (5) or (6).
C(p,y) = min{p'x|(x,y) e T} (5)
or
C(p,y) = oiin{p'x|G(x,y) = 0} (6)
where p is an n x 1 vector of input prices
"'" indicates vector transposition
The cost minimization problem models the firm's decision making process as
the firm chooses optimal quantities of the variable factors of production
while facing given rates of output and fixed factor prices. At a cost
minimum the optimal factor demands are consistent with a firm which is both
technically and allocatively efficient, i.e., a situation in which the firm
is operating on the transformation function frontier (G(x,y) = 0) and is
employing factors of production in the correct factor intensities (allocative
efficiency).
23
-------
The minimum cost function has the following properties:
C.1 C is continuous
C.2 C is monotonic, nondecreasing in y
C.3 positive linear homogeneous (PLH) in p
C.4 strictly quasi concave
Given properties C.1-C.4 of the cost function, the frontier transformation
function (i.e., G(x,y) = 0) and the efficient input-output combinations of
the technology set T may be retrieved from knowledge of the cost function
alone. Once again this duality implies that impacts on the technology set
may be perceived and examined through the cost function.
If C satisfies C.1-C.4 and is differentiable then the following result
due to Shephard (1953) holds.
(7)
* th
where xi(p>y) is the cost minimizing quantity of i input needed to produce
the vector y with given input prices p. Thus one may find the optimal factor
demand equations by simple differentiation (Shephard's lemma) or as the solu-
tion to the following optimization problem.
* >
xj,(p,y) - min{p'x|G(x,y) = 0}
In the latter case one would posit a functional expression for the transform-
ation function- G and solve the minimization problem in terms of x. Unfortun-
-------
ately, unless simple (i.e., restrictive) functional forms for G are chosen
the solution vector is often not analytically derivable. On the other hand,
if one chooses the cost function approach one need only postulate an expres-
sion for the cost function and simply apply Shephard's lemma.
If the cost function satisfies C.1-C.4 then impacts on the technology
set T are transmitted to the optimal factor demands (7). Since the optimal
demands are a direct reflection of resource usage the demand equations pro-
vide a convenient vehicle for assessing resource gains or losses associated
with impacts on the technology set T.
Differentiation of the cost function with respect to each y. produces a
set of interdependent marginal cost functions. Given perfect competition
assumptions, these marginal cost functions can be used to characterize the
supply responses of individual production units and thus provide another
vehicle for benefit calculation purposes.
We now wish to extend the generality of our discussion to permit a
subset of our input vector x to be composed of quasi-fixed stocks of inputs
(capital is the usual example) and to examine models which are capable of
explaining both the firm's input and output choices. That is, we are
interested in deriving models capable of producing short-run factor demand
and output supply equations.
To begin our analysis we require some additional notation. Partition
the input vector x into two exhaustive and mutually exclusive subsets
xv(x1, ..., xs) and xf(xa+1, .... xn) where xv are the freely variable inputs
and v? the quasi-fixed stocks. Let pv stand for the s x 1 vector of variable
input prices and p the m x 1 vector of output prices. Now define variable
25
-------
profit as IT = p y - pv xv. Finally,- we amend the properties of our tech-
nology set T by adding T.4 (constant returns to scale).
We now introduce the variable profit function defined as:
ir(pv,p ,x ) = max{p y - pv xv|(xv,x ,y) e T} (8)
The variable profit function models the firm's decision making process as it
seeks to maximize total variable profits by choosing cost minimizing quan-
tities of variable inputs and profit maximizing levels of output all condi-
tional on levels of quasi-fixed stocks and subject to the constraints of the
technology set.
The variable profit function has the following properties:
P.1 IT is PLH in pv and p
P. 2 IT is convex in pv and p for every x*"
P.3 IT is PLH in xf
P.4 IT is nondecreasing in x for every pv, p
P.5 IT is concave in x for every pv, p
P.6 TT is increasing in p and decreasing in pv
Given properties P.1-P.6 of the variable profit function and properties
T.1-T.M of the technology set there exists a duality between the profit func-
tion and the technology set (see Diewert (1974), pp. 137) which permits char-
acteristics of the technology set to be perceived via the profit function.
Further, if IT is differentiable an analog to Shephard's lemma, known as
Hotelling's lemma, applies to variable profit functions. Specifically, dif-
26
-------
ferentiation of the profit function with respect to input and output prices
generates optimal factor demand and output supply equations respectively.
v * f
*, * = x. (p ,p ,x ) : optimal factor demands (9)
"I
v * f
9ir(p ,,p ,x ) = y*(pV)p*>xf) . optimal output supplies (10)
The profit function is an extremely powerful tool for the analysis of
firm' behavior since it provides both factor demand and output supply equa-
tions. Moreover, given its duality with -the technology set, impacts on the
technology set are immediately transmitted to the supply equations permitting
straightforward consumer surplus calculations.
Summarizing briefly, we have demonstrated how the results of duality
theory are capable of linking models of producer behavior to characteristics
of the underlying physical relations between inputs and outputs; and
similarly, how alterations in those physical relations are transmitted to
observable economic relations in the form of demand and supply equations. We
hope that this theoretical development emphasizes a remark we made in the
introduction to this section regarding theoretical consistency. For example,
the power of the variable profit function to define optimal demands and
supplies rests on the stated properties P.1-P.6 of the function. If an
empirically estimated profit function violates even one of the properties,
Hotelling's lemma produces nonsense rather than economically defensible and
27
-------
useful functions. When examining the results of empirical studies employing
dual relationships one must always begin with the uninteresting examination
of the theoretical consistency of the estimated functions in terms of their
required properties. Only if these properties are met should one give any
attention to subsequent empirical results.
The preceding discussion has demonstrated that there exist several pos-
sible models of production which could conceivably be employed to examine the
impacts of exogenous factors (e.g., airborne pollutants) on the engineering
features of production technologies. Utilizing the results from duality
theory one may construct transformation, cost or profit function models
through which one can perceive the manifestation of these exogenous factors
on input demand and output supply functions. Having quantified these
perceptions it is a straightforward, albeit time consuming, task to calculate
social benefits.
If one reflects for a moment on the assumed properties of the technology
set T one realizes how extraordinarily general these assumptions are. We
make no assumptions regarding the associations between groups of inputs,
groups of outputs or groups of inputs and outputs. We assume nothing about
substitution possibilities or the aggregation of inputs and outputs.
Unfortunately, this high degree of generality is compromised as soon as we
attempt to empirically implement the theoretical models since we must choose
functional structure (i.e., specific function specifications) for the trans-
formation, cost or profit functions. As soon as one imposes structure on
these functions one begins to make a priori statements regarding the
engineering features of the underlying technology. Since these a priori
statements can impact the qualitative and quantitative manifestations of
28
-------
exogenous factors impacting the underlying technology, we want to set out
clearly the relationships between functional structure and resulting a priori
statements.
We shall limit our discussion of functional structure primarily to the
concept of separability. Essentially, separability concerns the decomposi-
tion of a function into groups of subfunctions. If a function can be so
decomposed the function is said to be separable. The impact of separability
is to impose additional structure on the function which one can perceive by
an examination of the function's derivatives (we shall assume that the
functions we are concerned with are twice differentiable).
To formally define separability we introduce a simple function F of N
arguments x.,, .... xn.
F(x) = F(x,f .... x_) (11)
The variable indices of x form the set I = [1, ..., n]. Partition I into m
exhaustive and mutually exclusive subsets. I = [I1 , ..., Im]. The partition
A
I forms m subsets of the arguments of F(x). If F(x) can be written
-f ^ ±,, 1 , 1 . 2, 2. m, m.. /i«\
F(x) = F(g (x ), g (x ) g (x )) (12)
then F(x) is said to be weakly separable in the partition I. If F(x) can be
written
F(x) = F*(gV) + g*U2) gm(xm)) (13)
29
-------
then F(x) is said to be strongly separable in the partition I. If F(x) is
strongly or weakly separable then the g functions maybe interpreted as aggre-
gator functions which permit consistent aggregation of the arguments in each
A
subset of the partition I. Thus we have the first important result —
consistent aggregation of subsets of the arguments of a function requires
separability.
The impact of separability on the associations among arguments of the
function can be clearly seen by an examination of functional derivatives. If
the function F(x) is weakly separable with respect to the partition I then
the ratio of the partial derivatives of F with respect to two arguments
within a single subset.is independent of the magnitude of arguments outside
that subset; i.e.,
for all i,j e ir, k t ir (14)
where F^Fj = 3F/3XJ, 3F/3Xj respectively.
If the function F(x) is strongly separable with respect to the partition I
then the ratio of partial derivatives of F with respect to two arguments each
within different subsets is independent of the magnitude of arguments outside
of either subset; i.e.,
«i- ) = 0 for all i e Ir, j e Is, k t Ir U Is (15)
j /
30
-------
To realize the economic importance of separability one need only think
of F(x) as a production function. First, only if F is separable is it pos-
sible to aggregate inputs in such a fashion that the value of each aggregate
is invariant with respect to the levels of inputs outside the aggregate.
Thus, without separability no aggregation is possible. Second, if F is
weakly separable in the partition I then the marginal rate of substitution
between any two inputs within a single subset is independent of the level of
any input outside that subset. Third, if F is strongly separable in the
A •
partition I then the marginal product of any input in a subset is independent
of any input outside that subset. The second and third results imply two
more* Fourth, if F is weakly separable in the partition I then the Allen
partial elasticities of substitution between two elements of one subset and
an element outside that subset are equal, i.e.,
o = o for i,j e Ir and k £ Ir (16)
IK J K
This implies for example that if all energy inputs to a production process
formed one subset and all capital inputs another, then for example, the
elasticity of substitution between electricity and factory equipment is
exactly equal to the substitution between coal, natural gas or fuel oil and
factory equipment. Finally, if F(x) is strongly separable in I. then Allen
partial elasticities of substitution between any two inputs in different
subsets and a third input not in either subset are equal, i.e.
31
-------
o.. = o., for i E Ir, j e I3, k t lr U IS (17)
IK 3
If each input formed its own subset then all Allen elasticities of substitu-
tion would be equal. This result is characteristic for example of the
multifactor CES and the Cobb-Douglas production functions.
It is readily apparent that separability can constrain the associations
among economic variables; thus in choosing a functional structure for our
economic models one will want to constantly be aware of the ramifications of
the chosen structure. We outline briefly below the consequences of differing
types of separability on the transformation G.
Input to Output Separability
If the minimum cost function can be written as a multiplicatively sepa-
rable function in a two subset partition, one subset containing all input
prices and the other all output variables then the transformation function is
said to be separable, i.e., if the cost function can be written
C(pV,y) = (pV)iKy) (18)
The transformation function can be written
G(x,y) = -F(x) + H(y) = 0 (19)
Input-output separability implies that consistent aggregates of inputs and
output can be formed. This in turn implies and is implied by the result that
32
-------
the marginal rate of technical substitution between any two inputs is inde-
pendent of the level of any particular output and that the marginal rate of
output transformation is independent of the level of any particular input.
To determine whether the underlying technology set was indeed input-output
separable one would test econometrically the parametric restrictions which
would be implied by multiplicative separability.
Again if the transformation function is separable then the profit
function can be written in a multiplicatively separable form, i.e.
ir(pv,p*) = 8(pVH(p*) (20)
and a test for transformation function separability could be carried out by
testing parametric restrictions on the profit function which would yield
multiplicative separability.
Nonjointness of Production
The transformation function is said to be nonjoint in inputs if there
exists individual production functions for each output, i.e.
G(x,y) =0 is nonjoint in inputs if y, - f.(x. , ..., x. ) (21)
i < 1, ..., m
and G(x,y) - 0 is nonjoint in outputs if there exists individual factor
requirements functions, i.e.
33
-------
G(x,y) = 0 is nonjoint in outputs if x = B^Y^ • •••» 1
i = 1 , ..., n
If G is nonjoint in inputs and outputs then C(y,pv) may be written
c(pv,y) = y^1(pv) + y24>2(pv), •••,
implying that the cost of producing all outputs is the cost of producing each
output separately. The corresponding profit function is,
* m n *
ir(pV,p ) = Z Z a, -pVp. (24)
which implies that all production functions are identical up to a multiplica-
tive constant.
Clearly, if the transformation is nonjoint the modeling of the technol-
ogy is greatly simplified since the multiple output nature of the production
activity can be decomposed into a set of individual output production pro-
cesses. It would be difficult to imagine that such nonjointness exists in
agriculture and even input-output separability may not exist. ' Assuming such
separable structures when indeed they do not exist causes econometric speci-
fication error which can seriously distort the empirical results. As a rule
of thumb one would assume as little separability as possible.
-------
Separability of- the Input and Output Subsets
If G is input-output separable, and the input subset is weakly separable
in the partition 1^ containing r subsets while the output subset is weakly
separable in the partition I containing s subsets then G may be written
G(x,y) = -F*(f1(x1), f2(x2), .... fr(xr)) (25)
, h2(y2), ..., hs(y3))
and the corresponding cost function, partitioned in a similar fashion, may be
written
C(pVy) = [*U1(pv1), 4>2(PV2), ..., /(p^))] (26)
And finally the corresponding profit function
Ce*(e1(pv1), 92(pv2) er(pvr))3 (27)
r *t 1 *1 , 2. *2. 3, *S,,1
[T (T.(p ), T (p ),..., T (p ))]
35
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The properties of weakly separable functions discussed in the opening
paragraphs of this section may now be applied to interpret the ramifications
of separability on the input and output subsets. Strong separability for all
three functions merely imposes additivity on all the subfunction components
f(O, h(»), $(•), i|K')i 9(-), T(«). Again implications of strong separabil-
ity may be deduced from our previous discussion.
The importance of separability cannot be overemphasized since it imposes
a priori restrictions on the associations among inputs, outputs and between
inputs and outputs. To keep these restrictions as minimal as possible we
will want to choose functional structure for our econometric models with an
eye toward separability considerations as well as empirical tractability.
3-3. MODELING THE IMPACT OF ENVIRONMENTAL VARIABLES ON AGRICULTURAL
PRODUCTION
Conceptually, there are only two substantive differences between the
microtheoretic modeling of an agricultural production activity and a
manufacturing activity. First, since land is immobile and varies in degree
of productivity, the land input into an agricultural production activity is
necessarily nonhoinogeneous. Thus, in modeling agricultural production the
heterogeneity of the land input must be taken into consideration. Second,
unlike the majority of modern manufacturing processes, agriculture is highly
susceptible to the influence of nonmarket factors, e.g., climate variations,
biological infestation and natural and man-made atmospheric pollutants, to
suggest a few. Since these are nonmarket factors, they cannot be treated
symmetrically with the marketable inputs to agricultural production; however,
they are not totally beyond the control of the economic agents organizing
36
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agricultural production. Variation in annual rainfalls can be combated with
market inputs such as irrigation capital, biological infestations with vari-
eties of fungicides, herbicides and pesticides, and atmospheric pollutants
with resistant varietals.
Dealing with the heterogeneity of the land input requires an index of
land productivity or regionally specific production models where land
homogeneity may be claimed. Land quality indexes have been constructed for
many years and thus this requirement does not pose an insurmountable problem.
Given such an index 6^, where i counts the types of land employed, e.g., land
used for grazing versus land used for cash crop production, one merely
employs a factor augmentation perception of the technology set T where each
land-component of the input vector is scaled by the appropriate 6.. Clearly,
the number of land types must be less than the number of observations on
agricultural production. If land were a variable factor in the long run,
assuming reasonably competitive markets, then a cost or profit function
approach would not require the 6^^ index since the varying productivities
would be capitalized into the service price of the land.
Modeling the nonmarket forces affecting agriculture poses a more diffi-
•
cult problem. For simplicity, let us consider only environmental influences
and designate the vector of such influences E. For any given vector E, the
technology set is determined by the physical and biological relations of
agriculture. Thus, as E varies so does the technology set T, and therefore
the set T is a function of the vector of environmental influences. This
causality between E and T implies a transformation function of the form where
E impacts the manner in which x is transformed into y.
37
-------
G(x,y,E) = 0 (28)
Our empirical problem is essentially one of quantifying the impact which
E has on the transformation of x into y. As we shall demonstrate, there
exist at least two possible approaches to this problem. The first would be
an econometric procedure where the impact of E on the x,y transformation is
estimated from observed nonexperimental data. The second approach, and the
one we employ in this study is to use experimental natural science
information to form the link between E and the x,y transformation.
The manner in which E affects the x,y transformation determines how it
should be modeled within the G function. The simplest impact E could have
would impose function separability such that G is written
G(x,y,E) = G*(H(x,y) + (E)) = 0 (29)
This form of direct separability of G implies that the frontier transforma-
tion function is neutrally displaced inward and outward as the components of
E change. This direct separability of G implies cost and profit functions of
the form
C(pV,y,E) = (C*(pv,y) + (E)) (30)
and
38
-------
ir(pV,p*,E) = (ir*(pV,p*) + (E)) (3D
The economic assessment model we shall present in Chapter 5 is based on
a slightly simplified form of the cost function in Equation 30.
Specifically, the cost function is limited to a single output such that
vector y has only a single element. We are justified in utilizing 30 due to
the hypothesized neutral impact which ambient ozone has on the productivity
of production factors. In the economic jargon of Chapter 1 we term this
neutral factor productivity enhancement.
•It is, of course, quite possible that E has' a nonneutral effect on
inputs but neutral on outputs or a neutral effect on inputs and a nonneutral
effect on outputs. In these two cases the transformation would have to be
input-to-output separable and appear as
G(x,y,E) = -F(x,E) + G(y) = 0 input nonneutral (32)
or
G(x,y,E) = -F(x) + G(y,E) = 0 output nonneutral (33)
The corresponding cost and profit functions are written
C(pV,y,E) * <(>(pV,E)i|Ky) input nonneutral (34)
39
-------
and
u ^
ir(pV,p ,E) = 9(pV,E)t(p ) input nonneutral (35)
or
V V f \
C(p ,y,E) = (p H(p ,E) output nonneutral (3°)
and
ir(pV,p ,E) = (p )t(p ,E) output nonneutral (37)
Finally, if E affects both inputs and output nonneutrally then G must be
input to output nonseparable and we are back to the fully nonseparable forms
of the transformation, cost and profit functions. As we choose functional
structure for G, C and we restrict the paths along which the impact of E on
the technology set can be perceived; thus, we restrict E's impact on input
demand and output supply functions and in turn restrict E's impact on social
benefit calculations.
Let us now consider modeling the impact of E on agricultural production
via the microtheoretic econometric approach using a minimum cost function in
which we embed a vector of environmental variables. For ease of exposition
-------
let all inputs be variable. Then X is an n x 1 vector of variable inputs, Y
an m x 1 vector of outputs, and E an s x 1 vector of environmental variables,
which the economic agents operating the agricultural technology take as
constant. The agents also know how the vector E affects their technology set
T. Finally, to make the analysis nontrivial we assume E varies across
agricultural production units.
If we do not impose separability of any type on the transformation func-
tion then the joint output minimum cost function may be written
TC = C(P,Y,E) (38)
where TC = minimum total cost
and the factor demand equations are
3C(P,Y,E) m *(p(Y>E) i=1,...,n (39)
9Pi i
To estimate the factor demand equations and thereby estimate the effect of E
on the resource cost of agricultural production, we must specify a functional
form for C(P,Y,E). To maintain as much generality as possible we choose the
multiple output transcendental logarithmic cost function (translog). The
translog has the exceedingly desirable property that it can be interpreted as
a second order local approximation to any underlying cost function. In
addition, the translog can be expressed as a fully nonseparable function.
41
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Using the notation above we write the nonseparable translog as
m n s
InTC = a + Za.lnY. + IB ,lnP. + Z6.1nE (40)
0 i 1 J J K K
mm nn ss
+ 1/2ZZp.,lnY.lnY. + 1 /2ZZY. .InP .InP . + 1 /2ZZ
-------
other variables which can affect cost. In the marginal cost equation (42)
these variables are factor prices, output and our other environmental
influence variables such as weather. In this context statistical control is
different from experimental control. We simply cannot hold the values of
these variables constant as we can in a laboratory experiment; therefore, we
must provide the statistical model with quantitative measurements of all
relevant variables. This greatly complicates the model, adds large numbers
of parameters and increases problems of collinearity among the independent
variables.
Second, in the physical world some variables move with one another due
to social or physical relations existing between them. If E and some set of
other variables which explain cost move with another then the statistical
model will be incapable of distinguishing their individual impacts. In
laboratory controlled experiments, we can force orthogonality between these
variables but when we must rely on natural experiments we are subject to the
whim of man and nature.
Finally, there must be some variation in the variables of interest. If
we are concerned with the impact of a change in E on the agricultural supply
function we must observe variation in E. In the case of air pollutants this
is a serious problem. Given, the regulated nature of pollutants, their
concentrations can be very uniform over large areas. In the case of ozone,
for example, a pollutant with an experimentally proven impact on crops, its
concentration across much of the rural corn belt is probably so uniform that
it is doubtful any meaningful statistical association could be identified.
For the above reasons and several other more subtle and technical
issues, the identification of the physical dose-response mechanism with a
-------
raicrotheoretic econometric model can be difficult. Moreover, even if
accomplished model verification is largely impossible — one simply cannot
observe the predicted welfare changes. Since the welfare estimates are
directly linked to the dose-response relation more confidence can be obtained
if this relationship is identified and empirically quantified under
controlled experimental conditions.
The NCLAN experimental studies have focused on the effect of various air
pollutants on crop yields. The potential differential impact which these
pollutants might have on inputs to the agricultural production activity has
not yet been studied by NCLAN. This is consistent with the belief that ozone
(the prime pollutant of interest) has a neutral effect on all nonharvest
production inputs (NFPE). To examine this issue briefly consider a simple
single output production function for preharvest activities as given below.
Y = f(X, ..., x. E) (43)
If E does indeed affect all inputs neutrally then one may write (43) as
F(XI, .... xn) , (44)
where given a fixed vector of x,(E) can be interpreted as a do.se-response
function and dose-response functions developed by NCLAN $(E) used as proxies
to the true function <)>(E).
For concreteriess let us assume (43) is a Cobb-Douglas and replace (E)
with its NCLAN proxy $(E).
44
-------
n a
SH * * WW •
Y = (f(E)nxi1 (45)
The production function (45) has a. dual representation as the minimum cost
function
~ -1/r n ai -1/r n ai 1/r
C = r(E) /r (Ha.1) 1/r (HP.VrjY1^ (46)
and a corresponding marginal cost function
(47)
Thus using (E) changes in E can be theoretically transformed into
appropriate shifts in the agricultural supply functions thus permitting
welfare calculations. This method of employing NCLAN biological relations
embedded in microtheoretic cost functions underlies our economic assessment
model.
45
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CHAPTER 4
WELFARE GAINS (LOSSES) FROM DECREASED (INCREASED) OZONE CONCENTRATIONS:
A REVIEW OF CONSUMER AND PRODUCER SURPLUS
Suppose we have a single agricultural crop produced by a number of farms
under perfectly competitive market conditions. For this crop, decreases in
ozone concentrations shift each farm's isoquants in input space toward the
origin and increases have the opposite effect. The result in price-quantity
space of a decrease in ozone concentration is then a shift in individual
marginal cost curves downward; and vice versa for an increase in ozone con-
centration. A similar effect will be observed in the aggregate supply curve,
which is the horizontal sum of the individual marginal cost curves under a
given ozone regime.
Now, there are four alternative sets of assumptions about demand and
supply elasticities under which the benefits of changes in ozone
concentrations are customarily measured. These are developed in Table 4-1
below.
Case I is the most restrictive and embodies a peculiar set of
assumptions — namely that marginal cost is zero up to some point, Q.,
proportional to ozone concentration, and infinite thereafter; while aggregate
demand is perfectly elastic at the reigning market price. This is what is
implied when one applies dose-response relationships to existing quantities
produced and values the quantity changes at the reigning market price. This
procedure has been employed by Heck £t_ al_., n.d., and several earlier studies
46
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TABLE 4-1. ALTERNATIVE ASSUMPTIONS ABOUT SUPPLY (E ) AND DEMAND (E )
ELASTICITIES USED TO OBTAIN WELFARE EFFECTS
OF ALTERNATIVE OZONE STANDARDS
Case I Case II Case III Case IV
(Biologists' Valuation)
Aggregate Q < Q.:E = » at zero price E>0 E>0 E>0
— - X 3 S3 3
Supply
Q > Q.:Ea - 0
1 3
Q. = f (ozone)
Aggregate E. = « E. = » E. = 0 0 < E. < »
Demand
-------
so si
B,
Figure 4-1. Case I.
48
-------
Figure 4-2. Case II.
49
-------
Figure 4-3. Case III.
50
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•£}
'0
Figure 4-4. Case IV.
51
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criticized for it by Adams _et al., 1982. Case I is what we have previously
labeled Biologists Valuation. This calculation may be justified as a first
order approximation to the change in consumers' surplus arising from a policy
change, and hence is not totally devoid of economic content. (See Deaton and
Muellbauer, 1980, p. 185, and Varian, 1978, p. 221).
Graphically, Case I is displayed in Figure 4-1 where we assume a
decrease in ozone concentrations and linear supply and demand curves. In
this case the discontinuous supply curve is perfectly elastic up to Q ana;
inelastic thereafter. Lowering ozone concentrations shifts the point of
discontinuity out to Q^. Producers' surplus (rents accruing to owners of
factors of production) before the change is represented by area A. After
the change is area AI + B-,, so the welfare gain, area BI, accrues entirely to
producers.
While Case I relies entirely on the biological dose-response function
and existing market prices and quantities for outputs, Cases II and III
attempt to quantitatively model the behavior of producers to achieve an
aggregate representation of the supply function. The methods of so doing
encompass the Mathematical Programming and Microtheoretic routes discussed
previously. Whatever the route, the aggregate demand side is ignored. Two
extreme simplifying assumptions can be made about demand, once the supply
function has been estimated.
The first is that demand is perfectly elastic at some reigning output
price, P = Pg. in this case, shown in Figure 4-2, there is no consumers'
surplus, either before or after the change. Prior to the change, Q is
demanded and produpers' surplus is area C2. After the change, Q1 is demanded
and producers' surplus is area C2 + Ag + B2. The net gain, A2 + B2, accrues
52
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wholly to producers. It also represents the decrement in resource costs
required to produce the prepolicy level of output (area A2) t plus the pro-
ducers' surplus on the output increment (Q. - QQ)f or area B2.
Case III, shown in Figure 4-3 is perhaps a more realistic one for agri-
cultural commodities, since it posits perfectly inelastic demand. Before the
policy consumers' surplus is the (infinite) area E and producers' surplus is
area D + c After the policy, consumers' surplus is area E^ + D^ + 83 and
producers' surplus is area C + A,. The net gain, therefore, is area A 3 +
B3, which also represents the resource cost savings in production of QQ
occasioned by the policy. This savings is distributed between consumers and
producers, where consumers gain D3 + BS and producers gain -Dg + A3 ~ i.e.,
area D is a transfer from producers to consumers. Note also that the entire
area A_ + Q- in case III is equivalent to area ^ in Case II, the discrepancy
between the total benefits in the two instances being the area B« in panel 2.
Figure 4-3 depicts the most general case, one which would be repre-
sented, say, by an aggregate quadratic programming model of market equilib-
rium for agriculture. In this type of model, a Linear Programming represen-
tation of production activities is linked to a linear aggregate demand
function, with the objective of maximizing the sura of producer and consumer
surpluses (Takayama and Judge, 1964).
With our assumption of a single product, the total surplus before the
ozone reduction is area E^ + DJ, + C4, of which E^ is consumers' surplus and
D^ + C4 is producers' surplus. Afterwards, the total surplus expands to En +
Djj + BH + FH + GH + Cjj + A4, of which E^ + DJJ + B^ + F^ is consumers' surplus
and C^ + A4 + GI, is producers' surplus. The net gain of ozone reduction is
therefore A^ + B^ + F^ + G^. This net gain is allocated between consumers as
53
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a gain of D^ + BJ, + F^ and producers as a gain of -D4 + AH + G4 where, again,
^1} is a transfer from producers to consumers.
Note that the area A^ + BJ, is identical to A3 + 83 which itself equals
A2. Therefore, the only difference between the estimate from Case IV and
that from Case III is the welfare triangle F^ + GM in Figure 4-4. Further,
the area F^ + c4 is encompassed in (i.e., less than ) the area B2 in Figure
4-2. Thus, for the single product case we can unambiguously rank the
estimates of welfare gain across Cases II through IV, assuming equal welfare
weights apply to the affected producer and consumer groups (Just _e_t al.,
1982, Ch. 8): Case III S Case IV £ Case II. Thus, for a single product, the
linear programming solutions may be adequate representations of benefits
vis-a-vis the more complex quadratic programming solution. But we can say
nothing very useful about Case I relative to the other three cases.
If we move to the multiple output case the conclusions drawn above still
hold if: (1) supply functions for each crop are independent of the level of
output of the other crops in the multiproduct system (production exhibits
nonjointness) and (2) demand functions for each crop are independent of the
prices of other market crops. The same thing is true if (1) above holds and,
instead of (2) we have a multiple price change situation which produces an
estimate of consumers' surplus gain which is independent of_ the path of
integration, money income held constant. (Silberberg, 1978; Just _et al.,
1982).
More specifically, assume we have a set of Marshallian money income
demand curves of an n good system of the sort x _ x^ (P ..., P , M). The
sum of consumer surpluses due to changes in prices with money income, M, held
constant is the line integral:
54
-------
CSM = - M M
. . - ;Exdp.
This is the area under all of the (linear) demand curves over the relevant
limits of integration. It will not be independent of the path of price
changes unless the partial derivatives of the uncompensated demand functions
across commodity pairs, x^, Xj Witn respect to prices Pj and Pj_ are equal,
that is 3x^/3pj = axj/aP^ By definition this is a property of compensated
Hicksian demand functions. But, for the integral of a set of Marshallian
demand functions to be path independent the special case of a homothetic
utility function is required. This implies that the income elasticities of
demand for all goods in the system are equal to unity (for a proof, see
Silberberg, 1978).
In general, then, equality of cross-price terms is not a general
property of Marshallian demand functions, so the Marshallian surplus measure
associated with multiple price changes will not be unique in the sense that
it is independent of the assumed sequence of those changes. There are two
more exact surplus measures, equivalent variation (EV) and compensating
variation (CV) which can be obtained as the integrals under a set of Hicksian
(not Marshallian) demand curves.
Compensating variation is the minimum amount a consumer would have to be
compensated after a price change (i.e., from initial price vector P° to
terminal price vector P1 ) and be as well off as he was before the change
(i.e., remain at initial utility level W(J. Equivalent variation is the
amount of income, given the original price vector P°, that would leave the
55
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consumer as well off as he would be with the price vector P and its
attendant utility level y1. In terms of money expenditure, e, itself a
function of prices and reference utility levels:
CV = e(P1, u°) - e(P°,
EV = e(P1, u1) - e(P°,
Note that the difference between the two concepts is the reference utility
level (y0 or y1 respectively), and that either can be positive or negative,
depending on the way prices change (see Varian, 1978, Ch. 7).
If Hicksian demand curves could be parameterized (which they generally
cannot be) either CV or EV could be obtained from the areas under such curves.
In general, CV is independent of the price path, but EV is not (Silberberg,
1972; Mohring, 1971). In any case, it can be shown (Willeg, 1976) that under
reasonable assumptions, Marshallian surpluses provide a good approximation to
CV and EV.
When we move toward a less restrictive set of assumptions the practical
estimation of welfare changes becomes much less tractable. If we allow
marginal costs of production for any crop to be a function of its output level,
the output levels of other crops, and input prices, and at the same time are
faced with a set of path dependent Marshallian demand functions, there is no
uniquely defined net social product maximum (i.e., sum of producer and consumer
surplus across all final commodity outputs). Therefore, in this situation
there is no way to estimate the "benefits" of an air quality scenario since, if
56
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total pre and post -scenario benefits are undefined so is the change in them
occasioned by the scenario. (For a proof in the Q-P context, see Yaron jst _al.,
1965.) In the model we present in Chapter 5 we assume path independence and
nonjointness of production.
57
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CHAPTER 5
THE REGIONAL MODEL FARM
5.1. INTRODUCTION
Recalling from Chapter 3 the practical problems posed by explicit incor-
poration of ozone variables in the econometric functions and the availability
of off-the-shelf biological information from NCLAN (National Crop Loss
Assessment Network), it was decided that the use of a biologically driven
microtheoretic assessment model was the most appropriate vehicle for analyz-
ing ozone impacts on agriculture. This model was named the Regional Model
Farm (a name that reflects the model's data base more than anything else).
Reconsider for the moment the simple agricultural production function
for a single crop developed in Chapter 3- Denote the output of this crop Y
and let the 1 x n. vector x represent inputs
Y = f(x) (1)
Employing the notation of Chapter 3» where E is a vector of environmental
variables, which we shall reduce to a scalar measuring ozone concentrations,
and (E) a function of E, we rewrite (1) to permit E to affect the production
of Y
Y = f(x,
-------
If E neutrally affects the production function then (2) can be written as
Y = f (x)(E) (3)
and the corresponding cost function is written as
C = (C(PX,Y) EI , as shown in Figure 5-1. The lines PP and P'P1 are isocost
lines at constant input prices and the points A and A1 depict the cost mini-
mizing equilibrium quantities of x1 and x2 under the two ozone regimes. From
the figure one can see that neutral shifts in the production function, due to
changes in ozone concentrations, imply in the case of ozone reductions, pro-
portional decreases in all inputs while leaving the mix of inputs unchanged.
This hypothesized ozone neutrality (NFPE) has the desirable property
that with constant factor prices all factor demand equilibriums lie on a ray
from the origin and that ray may be determined from a single observed factor
demand equilibrium. Since the neutrality of ozone will not induce any factor
substitution, and if we hold factor prices constant, we may treat the
production and cost functions (3) and (U) as if they were generated from a
Leontief production process. This is precisely what we do in the construc-
tion of the RMF.
59
-------
P'
P'
Figure 5-1. A Neutral Shift in the Production Function Due to Change
in Ozone Exposure
60
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5.2. SIMPLE HEURISTICS OF THE REGIONAL MODEL FARM (RMF)
The estimation of social welfare gains from agricultural activity occa-
sioned by a reduction in ambient ozone concentrations using the RMF requires
three distinct pieces of information. First, the physical (biological)
relationship between ambient ozone concentrations and the growing character-
istics of crop types must be known and expressed as a functional
relationship. Such relationships are generally known as dose-response
functions and in their 'simplist form relate a measure of crop yield to a
given ozone concentration. In their most sophisticated form they are
implicit functions of a set of growing characteristics which include not only
yield but such things as insecticide and fungicide retention and a host of
causal variables which include all relevant pollutants, indexes of insect
infestation, moisture availability, pathogen concentrations, etc..
The second piece of required information is a characterization of the
cost structure of agricultural production. Since resources are limited the
welfare of society increases when the same level of a particular output can
be produced with a decreased level of resources. If these resources exchange
in regular markets then a measure of the resource costs of production neces-
sary to supply a given level at output will permit us to measure resource
2
savings. If resource savings are to be appropriately measured at the firm
level one must capture the value of all resources purchased in the market by
the firm and then these resources must be aggregated to scalar value.
Finally, the value of the resources required to produce an additional unit of
output must be derived. This per unit resource cost is termed the marginal
cost of production and when expressed as function of output becomes the out-
61
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put supply function of a perfectly competitive firm. As we shall demon-
strate, the supply function provides the necessary information on resource
savings to estimate social welfare gains.
The final piece of information required by the welfare analysis concerns
the demand for agricultural products. Under the restrictive assumptions
regarding demand, such as perfectly elastic or inelastic demand relations,
explicit knowledge of the demand function is not required. However, if
demand has any elasticity greater than zero in absolute value and less than
infinity some knowledge of the demand relationship is required for accurate
benefit estimation. In this study we will use USDA estimates of demand
elasticity. Since these are national estimates they abstract from
transportation cost. Indeed our study assumes that transportation cost has a
minimal effect on the welfare calculations and we therefore ignore it.
The structure of the RMF is derived from its underlying data base
identified as the Firm Enterprise Data System (FEDS). Operated by the U.S.
Department of Agriculture, FEDS provides agricultural analysts with sample
operating budgets which describe the entire cost structure for producing an
acre of a particular crop in a specific region of the continental U.S.. The
budget is representative of the average agricultural practice in that
specific region and is verified with a battery of farm level surveys every
two years. A single budget for the production of soybeans in southeastern
North Carolina, for example, may include cost information on as many as 200
inputs to agricultural production, the average yield per acre to be expected
and the total number of acres planted in the region. The FEDS divides the
U.S. into over 200 producing areas; thus when we examine the cost of
producing wheat, for example, we will be considering the variation in
62
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production cost for over 160 wheat producing areas of the U.S.. This
extremely fine disaggregation of the cost structure of production by region
and crop is one of the major strengths of the RMF since it will permit
calculation of benefits for each region. These regional benefit calculations
will not be subject to regional aggregation biases and can permit a detailed
analysis of how the social welfare gains will be regionally distributed.
For each of the FEDS producing areas we assume that the FEDS budget for
a particular crop type, represents both the cost and yield existing for that
budget year, for given prices of inputs, outputs, and ambient ozone concen-
trations. Since the FEDS budgets are on a per acre basis we assume constant
returns to scale in order to aggregate across all of the planted acres
covered by a single budget. Further, we assume in the analysis that input
prices do not change in reaction to a change in ozone concentrations.
With these assumptions in place the construction of aggregate supply
functions for particular crops is straightforward. First, given constant
returns to scale marginal cost is equal to average cost and equal to a
constant. For a particular crop/region budget we divide the total cost of
producing an acre of the crop by the yield per acre and thus generate an
estimate of the marginal cost per crop unit. Repeating this calculation for
all regions producing the same crop produces an array of marginal costs of
production across the entire continental U.S.. When the marginal cost of
production in each region is mapped against the output of that region we have
a region specific supply curve for each crop. Ranking these regional supply
curves by marginal cost from lowest to highest and then aggregating across
63
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regions yields the aggregate supply function for the specific crop. This
aggregation produces a stepped supply curve such as that depicted in Figure
5-2.
Consider for a moment Figure 5-2. Output level Q^ represents the total
quantity of crop Q produced by regions A through D. Region A is the lowest
cost producer with a marginal cost of MCA and a production rate of CL.
Region B is the next least cost producer with a marginal cost of MCB and an
output rate of Q2 - QI. xhe integral of the marginal cost function from 0 to
QH is the total cost of producing Q^. If the yield per acre in each region
increases, due to say a decline in ozone concentrations, then the step
function shifts downward as illustrated by the dashed function in Figure 5-1 .
Once again the integral of the dashed function from 0 - Q^ j_s tne total cost
of producing Q^. The difference between these two•integrals is the saving in
resources occasioned by the reduction in ozone concentrations. The actual
resource saving calculations made by the RMF are somewhat more complex than
this simple description conveys but the technique is essentially the same.
5.3. ANALYTICS OF THE REGIONAL MODEL FARM AND WELFARE CALCULATIONS
Analytics
The most straightforward way to think of the RMF is in terms of a
Leontief production function for each region/crop combination. The Leontief
production function is given below.
Q = min(x1/ai, X2/a2, .... x^) (5)
-------
Cost A
in
MC
MC
MC
MC
D
r
i
-t
i
i
i
i
Aggregate marginal cost
function or supply curve
_i
0
Figure 5-2. Aggregate supply curve for regions A, H, C, I) for crop Q.
-------
where: xi are the physical quantities of the n factors of production
ai are technological constants conditioned on a set of variables
(e.g., climatic conditions, soil characteristics, ozone
concentrations, etc.)
Q is the output rate of a single crop in a single area
The objective of our analysis is to derive the marginal cost function associ-
ated with this production function. Assuming the economic agents controlling
production seek to minimize cost, they face the following optimization
problem which specifies the minimum cost of producing Q subject to the
Leontief technology.
min: EP.x. (6)
ST: Q = minCx^, x2/a2, ..., xn/an)
The solution to the above problem implies
x1/a1 = x2/a2 = ... = xn/an = Q (7)
The optimal factor demands then are
(8)
66
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Inserting the optimal demands in the objective function leads to the cost
function below.
C = Q(ZP a.)
Differentiating (9) with respect to output leads to the marginal cost
function
MC = 3/3Q = ZPiai (10)
i
The graphs of Equations 9 and 10 are displayed as Figures 5~3 and 5-4.
For any particular producing region in FEDS there is an upper bound on
acreage planted, thus one of the factor inputs is constrained by an upper
limit. The competitive profit maximizing farm operator confronted by a land
constraint will first attempt to obtain the maximum output possible from the
limited amount of land available and choose combinations of factor inputs in
such a fashion that the cost of producing the maximum output is minimized.
We can examine these sequential decisions in a two stage optimization frame-
work. Let us first assume that all inputs have upper bounds, then we first
seek to maximize output subject to these constraints.
max: Q= minCx^, x2/a.2 ..... xn/an) (11)
X. < x.
67
-------
Slope = MC = IP a± = AC
0'
Figure 5-3. Total cost function.
MC
Figure 5-4. Marginal cost function.
68
-------
where: the upper bounds on the n inputs are denoted by xt
The solution to this problem yields a max Q* equal to the smallest
next minimize the cost of producing Q*.
. We
min: EP.x
(12)
ST: Q* = min(x1/a1, x2/a2, .... xn/an)
The optimal factor demands are
Then the dual cost function is
iff Q ^ Q*
otherwise
and the associated marginal cost function is
(13)
MC
EP,a, iff Q * Q*
otherwise
(15)
69
-------
The graphs of Equations 14 and 15 are displayed as Figures 5-5 and 5-6.
In the realistic case of agricultural production land is the only input
subject to binding constraints. If we denote land as x and its maximum
s
upper bound by
-------
.x?
i. i
TC
0|
Figure 5-5. Total cost function.
ZP.a.
Figure 5-6. Marginal cost function.
71
-------
maximum Q = Q = Ya/ja«, (19)
s s
optimal factor demands xi = a^Q = ^(xg/Sag) (20)
Since
Q = (1/5)Q* => x. = x* (21)
total cost of producing the larger output (Q > Q*) is identical to the cost
of producing Q*. The graphs of the total cost and marginal cost functions
before and after a change in conditioning variables are given by Figures 5~7
and 5-8 respectively. Recalling from the previous section, the resource
saving resulting from the hypothesized decrease in ozone concentrations is
equal
Q* Q*
AW = / MC*(Q) - / MC(Q) (22)
0 0
In the analysis above we made the simplifying assumption that the pro-
ductivity of all preharvest inputs will be affected equally by a change in
ozone concentrations. Certainly this is not the case for harvest inputs. If
declines in ozone concentrations increase yields per acre it is difficult to
see how harvest costs per acre would not rise. Thus preharvest cost per
bushel can fall while harvest cost per bushel remains unaffected.
72
-------
TC
TC
Figure 5-7. Total cost function.
MC
MC
Figure 5-8. Marginal cost function.
73
-------
Let us assume that the change in conditioning variables (ozone concen
trations) affects only a single input xr> where input xs is still the con
strained input.
Let maximum output from (16) be
Q* = min(x/af X/a> _., X/a) = 7/a (23)
Then, given the stated conditions above, max output Q will vary depending on
the relation between x^ and XQ and the value of 6. These variations are
displayed below.
*
If r = s then Q = "xr/6ar = Q (24)
In this case the constrained input is the same input experiencing the produc-
/N
S\
tivity increase, thus the new output level Q will be the same output level as
that attained if all inputs experienced an equal productivity increase.
However, we will see later that the structure of cost will be different.
If r if s then Q = x /a = Q* (25)
w O
In this instance the productivity of the constrained input is not affected by
the ozone change thus no increase in agricultural output will be forthcoming;
however, costs of production will be lessened due to the enhanced
productivity of x
-------
If 6 = 1 then Q = Y /a = Q* (26)
s s
Naturally if there is no productivity enhancement for any of the productive
factors output remains unchanged.
To examine the cost of production we must now consider the optimal
factor demands under each output scenario (2U)-(26). Under (24) we have
optimal demands x. = a,Q for i 5^ r,s (27)
x = a_Q for r = s (28)
which implies that total cost is equal to
TC
+ Pr(6arQ) . (29)
r, s
In this scenario the maximum output obtainable is the same as that which
would result if the productivity of all inputs was enhanced as in the case in
Equation 20. However the cost given by (29) exceeds that calculated from
(20) since all inputs x^t i 4 r,s are unaffected by productivity enhancement.
Graphically, the total cost function derived from (20) is plotted on Figure
5-9 and labeled TC(20) while the total cost function for (29) is plotted and
labeled TC(29) .
75
-------
Let us now examine scenario (25) where the productivity enhancement does
not affect the constrained input. In this case output does not expand beyond
Q* and the optimal demands are
/N
x - aQ* for i * r (30)
5arQ* for r 4 s (3D
which implies that the total cost is equal to
Pr(6arQ*) (32)
The total cost function is plotted on Figure 5-9 and labeled TC(32) .
Finally, if we consider scenario (26) where no productivity enhancement
takes place then output does not increase beyond Q* and the optimal factor
demands are
^ = aiQ* for all i (33)
and therefore the implied total cost function is
and is graphed on Figure 5~9 and labeled 10(3*0.
76
-------
Cost
_. ._ _ TC(29)
Jl'C(20)
-------
Cost
MC(34)
MC(32)
CO
MC(29)
MC(20)
0
Q*
Figure 5-10,
Marginal cost functions under alternative scenarios regarding the differential impact
of ozone concentration changes on factors of production
-------
Each marginal cost function corresponding to the four total cost func-
tions are displayed on Figure 5-10 and labeled in a manner analogous to
Figure 5-9. Since we will be integrating under these marginal cost functions
to obtain welfare estimates, the importance of differential productivity
enhancements embodied in the cost curves is fairly important.
It is reasonable to assume that land is a quasi-fixed factor in agricul-
ture (a constraining input in the terminology of our analysis above) and all
other inputs freely variable. Thus, we shall be concerned with a model
similar to the total and marginal cost functions described by Equations 14
and 15- Further, NCLAN biological evidence suggest that yields are inversely
related to ozone concentrations and therefore lower concentrations will
elicit higher per acre yields. If we believe that it costs more to harvest a
bumper crop than a normal crop then we would be inclined to adopt a model of
cost similar to the total cost function (29). In such a model a distinction
is made between harvest and nonharvest cost and the productivity of factors
allocated to the two categories is permitted to be differentially impacted by
changes in ambient ozone concentrations. This dichotomized cost model forms
•
the basis for the RMF welfare calculations.
5.4. WELFARE CALCULATIONS
Before we discuss the specifics of the RMF welfare calculations we
briefly review the interplay between production supply and consumer demand
functions in the calculation of welfare changes. As we have previously
stated the aggregate supply functions derived from the RMF will be upward
sloping step functions. For ease of exposition let us consider them linear
functions with positive slopes. In this instance all we will be concerned
79
-------
with is the elasticity of linear demand functions. Perfectly elastic demand
functions will not be considered since the assumption is totally untenable
given the huge inventory of unsold agricultural products that currently
exists.
In the case of perfectly inelastic demand welfare estimates are based
only on the resource cost savings obtained in production. Thus, the impact
of governmental price support programs will not affect these welfare calcu-
lations. When a less than perfectly inelastic demand is assumed an unknown
effect may be present.
We first consider a perfectly inelastic demand relation as depicted in
Figure 5-11. Before a reduction in ozone the producer supply curve is given
by SQ and the market clearing price is PQ. Consumer surplus is the infinite
area E and producer surplus is the area C + D. After a reduction in ozone
the producer supply curve shifts to S1 and market price falls to P1. Con-
sumer surplus now expands to E + D + B and producer surplus is C + A. The
net gain in consumer and producer surplus is therefore A + B which is equal
to the resource savings concept discussed previously.
If we now consider a demand function which is not perfectly inelastic
such as that depicted in Figure 5~12 the benefit calculation is somewhat
different. Before the ozone change the producer supply curve is S- market
price is PQ, quantity demanded and supplied is QQ, consumer surplus is E and
producer surplus is D + C. After a reduction in ozone the producer supply
curve shifts to S1 , market price falls to P^, quantity expands to Q1,
consumer surplus isE+D+B+F and the producer surplus is equal to C + A
• •*
+ G. The net gain in consumer and producer surplus isA + B + F+G where A
+ B is the resources savings and G + F is the value of the difference between
80
-------
po
p, -
Figure 5-11. Perfectly inelastic demand.
81
-------
Figure 5-12. Demand Not 'perfectly inelastic.
82
-------
the marginal benefit of consumption and the marginal cost of production for
the extra quantity CL - QQ. While A + B+G + Fis the total welfare gain
the division of this gain between consumers and producers is dependent upon
the elasticities of supply and demand.
The difference between the two welfare calculations described above is
the area G + F. The more elastic the demand curve the greater this area.
Without explicit knowledge of regional demand functions by crop and by region
directly calculating the area G 4- F is impossible. To ascertain the poten-
tial magnitude of G + F we intend to construct a linear demand curve for a
particular crop and assign it alternative arc elasticities. We then
calculate the area G + F under these elasticity alternatives as part of a
benefit calculation sensitivity study.
5.5. OPERATIONALIZING THE WELFARE CALCULATION
We describe below the steps necessary to perform the actual welfare cal-
culations. Bear in mind the NCLAN experimental work provides the basis for
the biological dose-response functions, the FEDS provides the cost structure
of the RMF and assumptions regarding demand elasticity are employed to calcu-
late the value of additional output.
The first step in the process is to determine the intersection of crop
types for which NCLAN dose-response functions and FEDS budgets exist. At the
time the research described in the report was being conducted NCLAN had
published a limited number of dose-response functions. For the most part
these functions were linear or quadratic and in our opinion required further
refinement. The decision was made to employ published NCLAN experimental
results and reestimate the dose-response equations using a flexible
functional specification. The published data limited our efforts to five
83
-------
crops: soybeans, corn, wheat, cotton and peanuts; thus, the majority of the
discussions in this report concerns these five crops and is based upon the
dose-response equations estimated by the authors. Recently, NCLAN has
published a new set of dose-response equations based upon a flexible
functional specification. These new equations are available for the five
crops referenced above plus sorghum and barley. In Chapter 9 we examine the
impact which these new NCLAN equations have had on the welfare calculations
and also supply welfare estimates for sorghum and barley based on these new
functions.
We next examine the NCLAN data for a particular crop and define the
period over which the dose-response function is calculated (specifics of the
actual dose-response function estimation are contained in Chapter 6). We
then proceed to the EPA supplied data base containing county level
concentrations of ozone for the year 1978. For each county this data set
contains monthly averages of seven hour daily maximums for the months
April-October. In the case of soybeans we select the data for July, August
and September and average it to a three-month value consistent with the
experimental data.
The third step is to select from the FEDS file those budgets for a
particular crop, in this present example soybeans. For each budget we
determine the counties contained in the budget's region and once again
average the three-month county ozone data to the level of the appropriate
FEDS area.
The fourth step is to match relevant NCLAN dose-response functions to
the appropriate FEDS area. To do this we first map all FEDS areas into the
specified NCLAN regions. Then, if there are three dose-response functions
for soybeans, each derived from a different NCLAN lab, we apply the individ-
84
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ual dose-response functions to those FEDS areas contained in the NCLAN region
which developed the function. If a soybean producing area does not lie in a
NCLAN region that has a soybean dose-response function we use the function
from the geographically closest NCLAN region. Naturally, if we have only one
dose-response function it is applied to all producing regions. When multiple
dose-response functions exist for alternative crop varietals we employ the
method of Frontier Tidwell discussed in Chapter 6, section 6.7.
Once we have identified the appropriate dose-response function we pass
to it the area-wide ozone concentration, before any regulatory change in the
standard, and calculate the value of the yield variable. Using scenarios
supplied by EPA we next pass to the function post regulation values for the
area-wide ozone levels and recompute the yield variable. Using the formula
below we compute the increase in yield to be expected in the FEDS area.
Y*-Y
AYIELD = -—i (35)
Y
where: Y is the yield before regulatory action
Y* is the yield after regulatory action
Having selected the budgets for a particular crop, we order these
budgets by their marginal costs of production and assemble the aggregate
supply functions as displayed on Figure 5-2. To calculate the marginal cost
of production after the regulated change we recast Equation 35 as below.
6 =» 1/0+ AYIELD) (36)
85
-------
where <5 is the same as that employed in the analytical discussions above.
After calculating 6, which varies by FEDS area, we are in a position to
construct the new aggregate supply function.
To capture the differential impacts of ozone changes on nonharvest and
harvest cost we aggregate all factors of production for each region into
these two components of total cost and employ the following formula to
compute the marginal cost for a specific area.
MC = (1/(UAYIELD))(MARNONHRV) + (1/0 +YAYIELD)) (MARHRV) (37)
where: MARNONHRV = marginal nonharvest cost
Y = differential harvest effects 0 < Y £ 1
MARHRV = marginal harvest cost
If Y = 1 then the productivity of factors of production employed in harvest-
ing is enhanced by an amount equal to the nonharvest factors. If Y = 0 then
harvest factors are unaffected. Varying Y between 0 and 1 allows for a range
of impacts.
We are now in a position to calculate the welfare changes under the
assumption of perfectly inelastic demand. We first integrate under the
preregulation supply function from zero to the aggregate output level con-
tained in the FEDS. We then integrate under the post regulation supply curve
from zero again to the FEDS output figure. The difference between the value
of these two integrals is the net consumer and producer welfare gains. When
86
-------
the demand elasticity is not equal to zero the calculation is somewhat more
complex.
The procedures outlined above are repeated for each of the five crops we
shall be considering in this analysis. The sum of the welfare gains for each
crop represents the total social welfare gain occasioned by regulatory
action.
5-6. CONCLUSION
The regional model farm approach to agricultural benefits estimation is
admittedly simplistic. A particular weakness of the RMF is its static nature
and therefore its inability to capture the adjustment decisions of farm mana-
gers and to present a dynamic perception of agricultural responses to changes
in pollutant levels. On the positive side of the ledger the RMF easily
incorporates experimental data in a consistent fashion; and most importantly,
provides the ability to calculate regional benefits at a high level of reso-
lution. This regional disaggregation of the RMF is depicted on Maps 1-10
where the ten production regions of the RMF are displayed. If we consider
for the moment Region 01 "Northeast" we can see that it is composed of some
20 subregions. The total agricultural benefits occurring to the northeast
will be sum of the subregion benefits. Thus we will be able to determine,
for example, how the benefits will be distributed between upstate New York
and Western New York.
Regional distribution of benefits and the associated equity considera-
tions have been highlighted as crucial issues in the latest NCLAN 1981 Annual
Report. Since the sensitivity of crop types to ozone varies across crops and
since these crops are planted in geographically distinct areas, some farmers
will stand to gain more than others if ozone concentrations are reduced.
87
-------
Moreover, even for the same crop grown in a contiguous geographic area,
ambient ozone concentrations differ. Even in the unlikely event that a
change in the regulatory standard changes ozone concentrations by a constant
proportional amount in all areas, the differences in the absolute level will
imply different yield effects when the biological dose-response mechanism is
nonlinear-
-------
FOOTNOTES FOR CHAPTER 5
1 . In personal conversations with Boyce Thompson Institute Staff it has
become apparent that a focus on yield alone may not be adequate to
characterize effects of environmental pollution. In the case of acid-rain,
for example, yield is apparently unaffected but the insecticide retention
capability of various plant species is greatly lessened. Given the high cost
of insecticide, welfare gains can be expected if acid rain is abated even
though yields may not increase.
2. The concept of resource savings is analogous to the notion of
technical efficiency developed by Farrell (1957) and the coefficient of
resource utilization introduced by Debreu (1951).
3. Note the resource savings results from the production of the same
quantity Q* with fewer inputs due to the decline in ozone concentrations and
thus increase in productivity of factor inputs.
89
-------
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-------
CHAPTER 6
THE ESTIMATION OF DOSE-RESPONSE FUNCTIONS
6.1. INTRODUCTION
The purpose of this chapter is to discuss the empirical dose-response
functions estimated from published NCLAN experimental results. Very recent
NCLAN estimated dose-response functions are discussed in Chapter 9. The RMF
requires four types of information to estimate the welfare loss/gain which
may accrue to society in the event of a rise/decline in ambient ozone
concentrations. The major informational component of the RMF is a detailed
account of the cost structure for the production of specific agricultural
commodities for specific areas of the continental U.S.. The second component
is an estimate of county level ozone concentrations for rural agricultural
areas while the third component is an estimate of the demand elasticity for
specific agricultural commodities. The final component is a mathematical
expression which relates a measure of a particular crop/variety yield in a
specific region to a measure of ambient ozone concentration. This functional
expression is employed in the RMF to adjust the marginal costs of crop/region
specific agricultural production to changes in ambient ozone.
Conceptually, there are two approaches to the identification of the
relationship between ambient ozone and crop specific yield. The first is
statistical in nature and employs actual measures of ozone concentration,
measures of yield and measures of all the variables relevant to yield. Such
99
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an approach might involve the estimation of an agricultural yield equation
over regional subareas where ozone concentrations are known to differ.
Alternatively, one might estimate a crop specific cost function over the same
subareas with ozone as an argument in the function (see Chapter 3 for a dis-
cussion of the microtheoretic econometric approach). Regardless of the
method employed, primal or dual, the reliability of the estimated relation-
ship between ozone and yield or cost is dependent upon the accuracy of the
ozone data, the variation in ozone, yield and cost data across the sample,
and the ability to control for all factors other than ozone which may affect
yield or cost.
The second approach to the dose-response problem is experimental and
involves subjecting particular crop varieties to alternative levels of ozone
under conditions of experimental control. The variation in yield resulting
from these experiments can then be directly linked to ozone concentrations
and a simple two variable equation (yield and ozone) estimated to describe
the relationship. This experimental approach is pursued by NCLAN (National
Crop Loss Assessment Network).
The reliability of the experimental approach is a function of several
factors. First, crops in farmers' fields must respond to ozone in the same
manner as those in the controlled experimental plots. Second, for the large
part, experimental control is maintained by holding all factors other than
ozone constant. If factors such as pathogen concentrations affect the
relationship between ozone and yield and cannot be controlled for in the
experiment, then the simple two variable dose-response equation is inade-
quate. Third, identical crops grown on different plots must respond iden-
tically to ozone concentrations. This is, of course, necessary if one
100
-------
intends to generalize the experimental results to a regional basis as NCLAN
intends. Finally, the correct mathematical specification of the dose-
response relationship must be specified in the estimation. Choosing a
quadratic or plateau linear form when the true relation is logistic can
result in serious distortions to the relationship.
After reviewing the statistical and experimental approaches we have
decided to utilize the NCLAN experimental results for the following reasons.
First, given the limited budget of this project, using available experimental
results is very cost effective and therefore attractive. Second, it is not
at all clear that the detailed agricultural, climatological and soil data at
sufficient level of regional disaggregation can be obtained in order to
statistically control for all factors affecting yield. Third, the database
containing county level ozone concentrations is generated by an interpolation
technique using monitoring sites in primarily urban areas to estimate ozone
concentrations in rural counties. The accuracy of this data is unknown.
Finally, county level concentrations of other pollutants such as SO- <}O not
currently exist and thus the effects of SO- on yield could not be adequately
controlled for in a statistical sense. At the present time we believe it is
advisable to use the experimentally derived results in preference to an
estimated statistical function.
The design of the NCLAN experiments and their execution are well docu-
mented in Heck _et _al. (1981) and (1982) and will not be discus-sed in this
chapter. In Section 6.2 we present a fairly detailed investigation of the
problems involved in the estimation of a dose-response relationship from
experimental data. 'Since it is not at all clear that yield changes calcu-
101
-------
lated from an estimated dose-response relation are robust with respect to
functional form and estimation technique, Section 6.2 begins an investigation
of these problems. Section 6.3 presents the published NCLAN dose-response
functions for the crops included in the RMF (soybeans, wheat, cotton, corn
and peanuts). Section 6.4 discusses the Box-Tidwell functional specification
and the estimation procedure used to re-estimate the NCLAN dose-response
functions. Section 6.4 also presents the data employed in the estimates
drawn from Heck et al. (1981), (1982) and Heagle (1979). Section 6.5
presents the RFF estimates of Box-Tidwell dose-response functions for
soybeans, wheat, cotton, corn and peanuts. Section 6.6 discusses the method
of frontier Box-Tidwell employed to handle the variety averting behavior
problem. Finally, Section 6.7 presents some concluding remarks.
6.2. STATISTICAL CONSIDERATIONS IN FITTING DOSE-RESPONSE FUNCTIONS
/
Response Surfaces
"The reported results of the NCLAN experiments to date have involved the
estimation of linear dose-response functions based on experimental data. The
scope of the experiments does not yet allow the empirical study of the crop
yield response surface; that is, the empirical modeling of the nature and
strength of all of the disparate influences on yield, including weather, soil
type, and farming practice, along with the concentration of all influential
pollutants, including but not limited to ozone.
Response surface methodology (Box et al., 1978, Ch. 15; Biles and Swain,
1980, Ch. 3) can answer a number of interesting questions:
How is a yield response affected by a given set of explanatory
variables (rainfall, temperature, ozone, sulfur dioxide, etc.)
over some specified region of interest?
102
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What combination of levels, if any, of the explanatory variables
will produce maximum (local or global) yields, and what does the
response surface look like around the maximum (or maxima)?
Up to now, data to fit a crop yield response surface have been lacking.
Such data may be forthcoming from the work of NCLAN. But, at present, data
limitations have important implications for the transferability of single
equation dose-response models estimated from any particular site where the
data generating experiments were conducted to any other producing area. As
we know, weather conditions, soil type, and farming practices can vary widely
across the country and even between two adjacent farms. So, empirical ozone
dose-yield response models of the narrow sort based on local experiments
require certain assumptions to be consistently applied elsewhere. Specific-
ally, the partial derivatives of the log of the true yield response surface
(which is unknown) with respect to ozone concentrations must be independent
of the levels of each and every other important variable affecting yield in
the model. If this is not the case, response surfaces are needed.
6.3. CROP YIELD-OZONE DOSE MODEL SPECIFICATION: THE SINGLE VARIABLE CASE
Lack of experimental information necessary to estimate a response sur-
face confines us to very simple empirical models for estimating the influence
of ozone concentrations over the growing season on crop yield, all else
having been held constant in the design of the NCLAN experiments. The fact
that published NCLAN data for any particular experiment represent average
yields over a number of plots does not help, for such averaging reduces the
amount of information in the data, inflates measures of goodness of model
fit, and introduces the possibility of heteroskedasticity in the error term
103
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(Kmenta, 1971, pp. 322-336; Haitovsky 1973). Assuming away the potential for
heteroskedasticity by assuming an equal number of observations (plants) per
plot whence the averages came still leaves the problem of very few observa-
tions (generally 7-10) per experiment. This in turn means any models to be
estimated using the averaged NCLAN data must be parsimonious in terms of
parameters.
Even so, the appropriate mathematical form of the dose-response rela-
tionship in the single explanatory variable model must be determined. Two
approaches are open.
The first, and certainly the most convenient approach is to be able to
say with some certainty on a priori theoretical grounds (plant pathology in
this case) that one particular functional form is best. Ex post, the
reliability of the theoretical model could be exposed to statistical
specification error tests (Ramsey, 1969, 1974; Thursby and Schmidt, 1977;
Thursby, 1979; Harvey, 1981).2 If these tests reveal a serious problem, a
rethinking of the prior nonsample information forming the basis of the
unreliable model would be in order.'
Plant pathologists, if anyone, may be in a position to advocate one par-
ticular form over all others. But unfortunately there seems to be no consen-
sus on functional form among the experts, based on a Delphi survey conducted
by General Research Corporation (Carriere _e_t al. 1982). A review of the
literature shows that a preponderance of biologists have in practice fitted
linear functions to experimental data, whatever their theoretical preconcep-
tions (Heck et al., 1982). Thus the extant dose-response literature gives us
a menu of functional forms, without recommending any particular selection.
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The second avenue Is wider and much less well defined but basically it
involves letting the data indicate which form is best. Here "best" takes on
a wide range of meanings, each with a different level of sophistication,
ranging from eyeball inspection of data plots, R2 comparisons across alterna-
tive forms where the yield data is measured commensurately in the regres-
sions, power transforms evaluated in the maximum likelihood context, to tests
of nonnested hypotheses. Some of these approaches are briefly reviewed
below.
Ordinary Least Squares: Piecewise Linear Approximations, Polynomial Approxi-
mations and Simple Tests for Nonlinearity
One simple way to handle potential nonlinearity in an ordinary least
squares (OLS) context is to approximate a nonlinear function with linear line
segments. In the dose-response literature, one example of this approach is
the plateau linear model. (Heck et al. , 1982, for an example; Kmenta, 1971,
p. 469, and Judge et al. , 1980, p. 388, for a discussion of the more general
case ) .
In the single variable case, a sample plateau model could be written
as: ^
- X))
where b_, 5, are parameters to be estimated and the usual assumptions of the
classical normal linear regression model are presumed to be satisfied. Here,
Y£ represents yield over the i = 1, ..., n observations, Xi represents ozone
dose, D represents a* dummy variable which takes on a value of one if dose is
equal to or greater than some known critical level, X* and e. is N(0,o2).
105
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In this model, b represents a constant yield between Xi = 0 and the
break point defined by X*. Thereafter, the yield function has an augmented
intercept and the change in yield per unit change in X is given by b1 for all
doses greater than X*. The null hypothesis of a linear relationship of con-
stant slope and intercept across all values of X in this simple case (i.e.,
*i - bg + b^Xi + e^) can be tested by an F test (Kmenta, 1971, p. 469). The
problem with this approach is that we usually do not know, a priori, the best
way to approximate the nonlinear function with linear segments — i.e., where
to place the break point' (or points) X*. Said otherwise, we do not know how
to best break up the overall sample into separate subsamples, each of which
behaves according to its own (approximately linear) regime. Positing a
number of break points where slopes and intercepts change can quickly exhaust
degrees of freedom."
Another legitimate method which can be wasteful of degrees of freedom is
the fitting of an approximating polynomial to a nonlinear function based on
the Taylor's series expansion of any function" f(X) around some arbitrary
point (Kmenta, pp. 452-454). Essentially this method involves estimating a
function in the powers of X. Linearity can be tested via the F test of the
null hypothesis that the parameters attached to the higher order terms are
zero.
A practical problem with this second approach is that the columns of the
matrix of explanatory variables X, X^, X^, etc. tend to be highly .correlated,
leading to inflated estimated variances of the parameter estimates.
A theoretical objection to both of the tests for linearity mentioned
above is that such tests lead to pretest estimators which can have undesir-
able sampling properties if the null hypothesis is incorrect (Judge _et al.,
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1980, Ch. 3). This objection can almost always be made, however, whatever
course of model selection is pursued.
Ordinary Least Squares; Linearizing Transformations
There are any number of models which are nonlinear with respect to the
variables but linear with respect to the parameters to be estimated. After
appropriate transformation of the independent and/or dependent variables,
such functional forms can be estimated by ordinary least squares (OLS) . For
example, in the one explanatory variable case, where e. is N(0,o2)» we can
write:
Untransformed Model Transformed Model
Y. = b^i 1
b1Xi+£i
Y. , b0e InYi - b0
' "0 +
There are many examples of such implicitly linear forms in standard
econometrics texts — Daniel and Wood (1980), for example, present a large
selection. The problem with this approach is that, a priori, we often do not
know which nonlinear model is appropriate — unless theoretical considera-
tions lead us to one particular model. (Biles and Swain, pp. 156-57). The
107
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procedure of preselecting one convenient nonlinear form that is implicitly
linear can, in practice, be rather ad hoc.
Further, if we are interested in a family of such candidates there may
be no simple way to choose among them using standard OLS regression packages.
This is particularly the case if the alternative models are nonnested in the
sense that one model cannot be obtained from the other as a limiting process
i e<
(for example, Yj_ = bQXt e versus Y^ = i/(bQ + ^e ) or Y£ = bQ +
* £j_ versus Yi = bQ + b-jX^ + &^ ) . In many instances we read applied work
stating that one functional form was chosen over all others because it "fit
the data better", was "more satisfactory". What is usually meant in these
cases is that the specification with the smallest residual variance (or high-
est R2) was selected as best, a criterion suggested by Theil (1971, p. 543).
Theil claims that on average, the residual- variance estimator of the in-
correct specification will exceed that of the correct specification given
that one of the alternative models is indeed the "true" model. So, seme
practitioners using OLS compare alternative models (when the dependent
variable is measured the same way in each) on the basis of goodness of fit
(R2, or A ) or, if the dependent variable is not measured the same way (e.g.,
InY, 1/Y and Y) on the basis of "pseudo" R2 or pseudo mean square error
measures based on transformed residuals. This is a rather unpersuasive
procedure which has been criticized by Pesaran (1974). If used with two
models where the dependent variable is measured differently. Theil 's residual
variance criterion may give contradictory results. Suppose a linear and a
log linear model are fitted to the data using OLS. Then, we have four mean
square error measures to compare. The two actual measures (MSB.. for the
linear model and M3E, for the double log) are based on the residuals from
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the fitted models. The pseudo measures are based on the sum of the squared
differences between the log of the actual dependent variable minus the log of
the linear model's predictions (PMSElin) Or the sum of the squared differ-
ences between the actual dependent variable minus the antilog of the log
model's predictions (PMSElQg). We compare MSElin * PMSEiog and PMSElin *
MSE1 . But, there is no reason to expect these two comparisons to give con-
sistent results; that is both comparisons favoring one of the forms over the
other.
Nonlinear Modeling: Sophisticated Methods
The preselection of a set of transformations to permit ordinary least
squares analysis has become less common with the advent of efficient nonlin-
ear modeling programs which permit the direct estimation of the transforming
parameters, either by nonlinear least squares or maximum likelihood.
One example is the Box-Cox (1964) class of transformations on the
dependent and independent variables. The general model for the single
independent variable case is of the form
Ui) (A2)
Yi =bO + Mi +£i
Ui ) Ai (A2) . *2
where ^ = (YA - DA.,, and Xj. =
The model is intrinsically nonlinear. The functional form resultant from the
Box-Cox transformation depends on the values of the X^, which are estimated
along with the b's (See Spitzer, 1982, for a complete discussion of the
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maximum-likelihood, nonlinear least squares and iterated OLS methods for
estimating the parameters of the Box-Cox transformation.)
With the Box-Cox transformation, if AI = o and A2 = 1 tne raodel is semi-
log; if A.J = x2 = 0 the model is double-log; and if X1 = A2 = 1 tne model is
linear. Other intermediate cases are, of course, possible.
The Box-Cox procedure is an attractive way to allow the data to dictate
functional form, and linearity can easily be tested by a likelihood ratio
test. But there are at least three problems with the method. The first is
that the true model may not be included in the general form, so the inad-*-
quacy of a false maintained hypothesis cannot be tested (Aneuryn-Evans and
Deaton, 1980). The second difficulty is that the conventional Box-Cox maxi-
mum likelihood estimator does not, as usually performed, separate out the
decision of whether Y (and hence the error term) should be treated as
homoskedastic or heteroskedastic from the decision regarding the correct
functional form. Specifically, there is bias in estimating A toward a
transformation of Y which reduces heteroskedasticity (Zarembka, 1974; Judge
_et al., 1980). This problem might be remedied by using Lahiri and Egy's
(1981) amended version of the Box-Cox maximum likelihood estimator, but even
so another more critical problem remains. This third problem is that the
transformed dependent variable — and hence the error term — will be
truncated for all values other than AI = o. With Y assumed to be greater
*1
than or equal to zero, (Y - 1)^ wm be greater than or equal to -1/A-, for
A1 greater than or equal to zero, and conversely for A., less than-or equal to
zero (Amemiya and Powell, 1982). Thus the transformed variable Y cannot
strictly be normally distributed unless A = 0. • This leads Amerniya and Powell
to note that the Box-Cox maximum likelihood estimator is not, strictly
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speaking, a statistical model. It is merely a method of estimating the
parameters which potentially can produce inconsistent parameter estimates.
Amemiya and Powell propose a nonlinear two-stage least squares estimator
(NL2S) of the Box-Cox model, instead of the customary methods outlined in
Spitzer (1982).
In our application, the data simply do not merit the expense and diffi-
culty of constructing a program to implement the NL2S Box-Cox estimator. We
follow a simpler but not unsophisticated course, which is explained in detail
in subsequent discussions. The following discussion presumes nonlinear esti-
mation procedures are employed for all models discussed.
Fitting Logistic and Box-Tidwell Functions
Briefly, with sufficient data the essence of our approach would be to
posit a theoretically appealing model — the logit function — and compare it
with a variable transformation model of the Box-Tidwell (1962) type. The
method of comparison could be based on Sargan's unmodified likelihood ratio
for model discrimination. Ideally, such a comparison should also involve
tests of nonnested hypotheses. The Davidson-MacKinnon (1981) "C" and "P"
tests are attractive candidates, but unfortunately cannot be used with much
confidence in the face of very small samples. Yet because future dose-
response research might profit from the application of such tests, their
outlines are sketched in a subsequent section. But first the form of the
models which could be estimated from the available NCLAN data is explained.
The inherently nonlinear logistic model in three parameters (Maddala,
1977, p. 7) is given by:
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Yi
The logistic takes on the value b /(-j + b1) when X is zero and as X goes to
infinity the value of Y approaches zero.^ Physical considerations based on
threshold values provide a common sense justification for using a logistic
model to represent the dose-response relationship (Cox, 1970). The logistic
model has been extensively used in biological work, and in the case of crop
yield it makes sense (see Carriere et al., 1982). Negative predicted yields
are impossible however high the ozone dose, and the logistic model reflects
this aspect of physical reality that a simple linear model does not.
An even more general logistic with a lower threshold equal to some posi-
tive constant greater than zero is the four parameter model:
As X approaches infinity, Y. approaches bQ. Parameter restrictions on this
general model produce other logistic models often seen in the literature such
as:
bo +
112
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1+e
However, biologists may not be satisfied with any of the above logistic
representations, so an alternative is to let the data dictate functional
form.
So, in contrast to choosing a theoretical model a priori, we can write a
nonlinear model and consider estimating a transforming parameter in the
explanatory variable dose as part of a nonlinear estimation procedure. The
Box-Tidwell (1962) method is based on such an approximating transformation.
' The Box-Tidwell transformation is just a. special case of the generalized
Box-Cox transformation. Its advantage is that since no transformation of the
dependent variable is involved, neither of the two problems mentioned in con-
nection with the Box-Cox transform (het er os ke das ti city, truncated error)
arises.
With one explanatory variable, the Box-Tidwell model requires three
parameters to be estimated, b 5 and A:
0 + ^X* + ei , et - N(0,o2)
X
where X. is exponentiated to the A power and is not equal to the transformation
of the X variable on page 108.
In the positive quadrant, the sign of the second derivative of the Box-
Tidwell function cannot change. Therefore, it has no inflection points,
.# "*
which means that the model is more restrictive than the logistic, which does
113
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have an inflection point (i.e., the first derivative of the function has an
extreme value). Said simply, the Box-Tidwell model permits no change in
curvature (i.e., from convex to concave) while the logistic model does.9
Obviously, since both of these models (logistic, Box-Tidwell) are intrin-
sically nonlinear, nonlinear least squares estimation is required (see Draper
and Smith, 1966; Judge jet ^l._, 1980).
Practically speaking, the Box-Tidwell model is just a "graduating" func-
tion which is expected to represent the true function over a limited region
of the _X space reasonably well. It cannot capture the "S" shape of the
logistic. But, the data may not show a logistic pattern simply because the
complete domain of dose was not represented in the experiments, and thus the
point of inflection not revealed. If it were revealed, the logistic model
should better represent the data, where the criterion of better is given by
the class of nonnested hypothesis tests mentioned above.
To illustrate, Figure 6-1 shows three extreme possibilities assuming the
true dose-response function is logistic. In panel A the full shape of the
logistic is revealed by the data, with the point of inflection (X*) posi-
tioned near the mean of the observed dose data. In panel B, the logistic's
inflection point is positioned near a dose of zero, giving the function the
appearance of being convex to the origin over the observed range of dose. In
panel C the inflection point shows up near the uppermost dose measurement
giving the impression of a concave function in the positive quadrant.
Because it has no inflection point, the Box-Tidwell approximation to the
true logistic is visibly better for the cases represented in panels B and C
of Figure 6-1 than it is for the situation shown in panel A. In the two
former situations (B,C) it may in fact be difficult to statistically distin-
114
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guish between the true logistic and the Box-Tidwell approximation in small to
moderate sized samples. Of course, this discussion assumes that the logistic
function is in fact the true model. But, a less restrictive point of view
would admit that either, both, or neither of the posited models could have
generated the observed data.
Model Discrimination. Nested and Nonnested Hypothesis Tests
When testing nested hypotheses, it is not possible to simultaneously
reject the null and alternative hypotheses and conclude that neither is
correct.
For example, the linear model is just a special (nested) case of the
Box-Tidwell model with the restriction that X = 1 . Thus it is simple to test
for linearity (a test of the null hypothesis that X = 1) by constructing a
A
confidence interval around X at the chosen level of type one error to see if
it encompasses the value of one given by the null hypothesis of linearity.
But of course, neither model may be correct.
Distinguishing between two intrinsically nonlinear functions such as the
logistic and Box-Tidwell is not so straightforward. Properly speaking, non-
nested hypothesis tests should be used, since the models are nonnested
(Pesaran and Deaton, 1978; Aneuryn-Evans and Deaton, 1980; Davidson and
MacKinnon, 1981).
In lieu of such tests, there is the simpler alternative of the unmodi-
fied likelihood ratio. This model discrimination criterion is the nonlinear
estimation analogue of Theil's R2 criterion in OLS. Like Theil's criterion,
the likelihood ratio (variously labelled the Sargan test or Akaike's Informa-
tion Criterion) is not really a statistical test with known statistical
115
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KEY
Logistic
Linear
! Box-Tidwell
Dose
Panel A: Full Logistic with Concave and
Convex Regions
Dose
Panel B: Logistic Principally Convex
to Origin
X* Dose
Panel C: Logistic Principally Concave
to Origin
Figure 6-1. Alternative locations of the observed point of inflection, X*
of the logistic function, with linear and Box-Tidwell approximations.
-------
properties. Instead it is just a method of model discrimination which is
easy to calculate and should be successful "on average" presuming one of the
models in the comparison set is the true model. No significance level can be
set for such a comparison: one just chooses the model with the higher like-
lihood (Aneuryn-Evans and Deaton, 1980; Harvey, 1981).
Specifically, suppose the null hypothesis HQ is represented by the three
parameter logistic function with parameter vector 6. The alternative hypoth-
esis H.| is represented by the three parameter Box-Tidwell approximation func-
tion with parameter vector g.
In general matrix notation we can write the models compactly as:
Logistic
HQ : Y = f(X,9) + EO
Box-Tidwell
H, : Y - g(X,S) + e1
where Y is an n x 1 column vector of observations, X is an n x k matrix of
explanatory variables, e e. are n x 1 column vectors of disturbances, and
f,g represent the logistic and Box-Tidwell functions respectively with
parameter vectors 8 and g.
With the assumption of normally distributed errors the two log likeli-
hood functions (log L) in the three parameter case can be written, with the
observation index i = 1, ..., n, as:
117
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Logistic
log L(9,o§) = -(n/2) log 2ir - (n/2) log OQ - Eeoi/2°0
where from our earlier notation:
S ' (Y ~
-------
last terms on the rhs of both log likelihood functions are equal to n/2.
Therefore, the estimate of the difference between the two log likelihood
functions, LR, simplifies to:
LR = (-n/21no2) - (-
or
LR
All this means is that if LR is positive, accept the model specification
°f HQ, otherwise accept H^ . Even more simply, when the intrinsically nonlin-
ear models are estimated by either nonlinear least squares or maximum likeli-
hood techniques (and no transformation of the dependent variable is under-
taken) the criterion tells us to accept the model with the lesser mean square
error (or higher R2). This is just Theil's model discrimination criterion
applied to intrinsically nonlinear models.
The Monte Carlo evidence presented in Aneuryn-Evans and Deaton (1980)
suggests the unmodified likelihood ratio (which they call the Sargan test but
is also known as the Akaike Information Criterion (AIC)) is a useful discrim-
inator between two alternative models provided one can be certain- in advance
that either HQ or H1 is in fact true.11 When both HQ and H., are false the
likelihood ratio discriminator is misleading for it forces a decision when in
fact indecision is possible — both models should be rejected.
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The advantage' of the Sargan method is computational simplicity, a
feature not shared with the Cox-Pesaran type nonnested test procedures for
functional form specification set out in Pesaran and Deaton (1978) and
Aneuryn-Evans and Deaton (1980). Fortunately, a family of nonnested
hypothesis testing procedures has recently been developed by Davidson and
MacKinnon (1981)(DM) which are simple to compute — the C and P tests.
With a sufficiently large sample, the idea of the class of DM tests
would be to test the logistic model HQ against the Box-Tidwell model H1 ,
conditional on the truth of HQ. Reversing roles, the Box-Tidwell model, HQ,
would be tested against the logistic model, conditional on the truth of the
new .HO. Obviously, the following outcomes are all possible under the
nonnested hypothesis testing scheme:
Logistic
Box-Tidwell
Hn : Y = f(X,9)
H
g(X,8)
Accept
Rej ect
Accept
Reject
The possibility of rejecting both models with these tests is rather
unsettling. Such a nihilistic outcome would not satisfy an investigator
seeking an immediate solution to a problem, since it can only elicit the
familiar call for more research. But, in dismissing the idea that relative
superiority of model fit is a useful way to compare models Pesaran and Deaton
(1978, p. 678) state their position quite strongly:
It is important that notions of the absolute fit or performance play
no part in the analysis. Indeed it should be clear ... that, apart
from the nested case, we regard such indicators as meaningless. In
considering whether an alternative hypothesis, together with the
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data contains sufficient information to reject the currently main-
tained hypothesis, the question of whether that alternative 'fits'
well or badly, even if meaningful, is certainly irrelevant. An
hypothesis, which one would not wish to consider seriously in its
own right, can be a perfectly effective tool for disproving an
alternative, even if that alternative may in some respects seem much
more promising. rt _is_ thus important that tests between nonnested
hypotheses _or models should encompass the possibility of_ rejecting
both, _as does the Cox procedure. [ Ital ics added]
This position may involve a bit of intentional overstatement, since
later in the same paper the authors merely suggest that their tests be used
as a supplement to, but not a replacement for, "current practices", which
reasonably could be taken to mean model discrimination on the basis of fit.
Such issues aside, the set-up for the family of Davidson-MacKinnon tests
is quite simple.12 These tests are closely related but not identical to the
Pesaran-Deaton (1978) tests. As before, we have the two competing (non-
nested) nonlinear models:
HQ : Y = f(X,eO + e0
H! : Y = g(X,g) + EI
Both error terms are assumed to be normally independently distributed with
zero mean and respective variances OQ and of.
Define the maximum likelihood predictions (*) of each observation of the
Y. vector given the maximum likelihood (ML) estimates of 9 and g as:
f!
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g!
The C (conditional) test of the truth of H involves a linear regression
to estimate the test parameter a, conditional on the 0M, vector:
- Ct)f* + QLg
*
or
Yi - f* = a(f* - gf) + 6i
The validity of HQ can be tested by using a conventional t test of the
null hypothesis that a*, the estimate of a, equals zero. However, the t
statistic for a* is not distributed asymptotically as N(0,1) if HQ is true.
Rather, the estimate of the variance of the distribution of the t statistic
for the C test is asymptotically biased below 1 when H is true. Practically
speaking, this means that the nominal level of significance chosen for the
test will overstate the true asymptotic level of significance, or otherwise
said, the true probability of Type I error (probability of rejecting a true
HQ) will be less than the nominal level chosen. The C test is therefore
conservative in the sense that it is less likely to reject a true H- than one
wishes it to be.
To produce a test statistic which is asymptotically distributed as
N(0,1) the authors suggest the J (joint) test which estimates a and g jointly
in the nonlinear regression:
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Yi = (1 -
g
However, a simpler computational test procedure when HQ is nonlinear,
which shares the same asymptotic properties of the J test, is the P test. The
P test involves a linearization of the J test, around the g* vector:
- fj = a(g* - f*)
where f denotes 3f/3Bk|8kML for k = 1, .... K parameters in the nonlinear
model under HQ ancj 5^ 4>k are parameters to be estimated along with a in
the P regression. To complete either the C or P procedures, the roles of HQ
and H.J are reversed and the tests repeated. It should also be noted that
several models can be simultaneously compared using an extension of the J or
P procedures.
Unfortunately for our purposes, the aggregate data contained in the
NCLAN annual reports are not sufficiently large to merit the indulgence in
such sophisticated hypothesis testing as that described above. The small
sample performance of these tests is largely unknown, but their application
to samples of even twenty observations would appear unwise (Pesaran, 1982;
Davidson and MacKinnon, 1982).
In fact, the aggregate data sets available in the annual reports are so
small as to preclude statistical tests of functional form. Yet 'differences
in functional form of the dose-response relationship obviously could have
significant impacts on the economic benefits produced by models relying upon
them. In the same vein, it does not seem unreasonable to presume that
natural-world relationships are likely to be nonlinear. It is germane to
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raise this question, although our logit and Box-Tidwell approximating func-
tions are poor answers to it, given the data at hand.
6.4. NCLAN REPORTED DOSE-RESPONSE FUNCTIONS
The Firm Enterprise Data System (FEDS) contains production cost informa-
i
tion on twenty-nine major crops grown in the continental U.S.. At the time
the research described in this report was conducted the intersection of FEDS
crops and NCLAN experimental information contained soybeans, wheat, corn,
cotton, and peanuts. Since the FEDS data underlies the Regional Model Farm
benefit estimation method, we are constrained by FEDS in the crops that can
be examined. This constraint makes it impossible to employ information on
crops such as tomatoes and beans.
We note at the outset and caution the reader that in all of the RFF
estimates dose-response functions the ambient air plots were included in the
estimation data base. This was done to increase the number of observations
in our data sets but may impart some bias to our results if these ambient air
plots lead to a systematic bias in crop yields.
In the 1980 NCLAN annual report (Heck et al. (1981)) dose-response func-
tions are reported for the Corsoy variety of soybeans and NC-6 peanuts, with
the experiments conducted at Argonne National Laboratory and North Carolina
State University respectively. Both experiments were of the open-top-chamber
variety. Linear functions were used to describe the experimental results and
related a measure of yield to an experimentally maintained level of ozone
over the 7 hr. period 0900-1600. In the case of soybeans the ozone fumiga-
tion began on August 6 and ended on October 9, while for peanuts the period
of fumigation extended from June 16 to October 6.
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Table 6-1. NCLAN ESTIMATED DOSE-RESPONSE FUNCTIONS DEVELOPED IN 1980
AND PUBLISHED IN THE 1980 NCLAN ANNUAL REPORT
Crop: SOYBEAN (CORSOY)
NCLAN Region: Central States (ARGONNE)
Interactions: NONE
Y* = 23.14 - 123.20(0zone)
Crop: PEANUTS (NC-6)
NCLAN Region: Southeast (N.C. STATE)
Interactions: NONE
Y** = 173.20 - 1045.6(0zone)
Notes: Y* = seed weight per plant, Y** = weight of pods. Ozone is
measured in part per million. Corsoy and NC-6 are varieties of soybeans
and peanuts respectively. Ozone concentrations were added during the
same 7-hour period each day: 0900-1600 hr std time.
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The NCLAN estimated dose-response functions for soybeans and peanuts are
presented in Table 6-1. The measures of yield incorporated in the functions
are seed weight per plant for soybeans and weight of pods for peanuts.
In the case of soybeans it is not clear from Heck et al._ (1980) whether
other functional specifications were estimated. While not stated, apparently
twenty or more observations were available for use in the estimation permit-
ting more complex association between yield and ozone than that portrayed by
the linear function. As outsiders to NCLAN with access to only the
summarized NCLAN results, it is impossible for us to make an objective
assessment of the statistical reliability of the estimated dose-response
functions. Based on the preliminary NCLAN reports any conclusions we might
draw would be of dubious value and unfair to the NCLAN researchers.
Therefore, we merely present the remainder of the published NCLAN
dose-response functions without comment.
In Heck et al._ (1982) (the 1981 NCLAN Annual Report) dose-response func-
tions are reported for .corn, soybeans, and cotton. In the case of soybeans
and corn alternative varieties were examined. This variety analysis provides
us with dose-response functions for two major corn varieties, and four types
of soybeans, Hodgson, Davis, Williams, and Essex. Only a single variety of
cotton was examined, Acala SJ2. For the two corn varieties stepped linear
functions (termed "plateau linear" by NCLAN) were estimated. The soybean
functions are predominantly quadratic with two exceptions which' are linear
and the cotton functions are linear. All of the dose-response functions
published in the 1981 NCLAN Annual Report are displayed on Table 6-2.
126
-------
TABLE 6-2. NCLAN ESTIMATED DOSE-RESPONSE FUNCTIONS DEVELOPED IN 1981
AND PUBLISHED IN THE 1981 NCLAN ANNUAL REPORT
Crop: CORN (PIONEER 3780)
NCLAN Region: Central States (ARGONNE)
Interactions: NONE
Y = 10836 + D(-78993(OZONE - 0.071))
where: D = 0 if OZONE < 0.071
D = 1 otherwise
Crop: CORN (PAG 397)
NCLAN Region: Central States (ARGONNE)
Interactions: NONE
Y = 12221 + D(-105751(OZONE - 0.090))
where: D = 0 if OZONE < 0.090
D = 1 otherwise
Crop: SOYBEAN (HODGSON)
NCLAN Region: Northeast (BOYCE-THOMPSON)
Interactions: NONE
Y = 2628 - 9875(OZONE)
127
-------
Table 6-2 (continued)
Crop: SOYBEAN (DAVIS)
NCLAN Region: Southeast (N.C. STATE)
Interactions: NONE
Y = 53^5 - 39886(OZONE) + 109600(OZONE)2
Crop: SOYBEAN (WILLIAMS)
NCLAN Region: Southeast (BELTSVILLE)
Interactions: NONE
4426 - 110429(OZONE)
Crop: SOYBEAN (ESSEX)
NCLAN Region: Southeast (BELTSVILLE)
Interactions: NONE
Y = 3901 - 5038(OZONE)
Crop: COTTON (ACALA SJ2)
NCLAN Region: Southwest (SHAFTER)
Interactions: MOISTURE (NORMAL)
Y = 2036 - 6884(OZONE)
Crop: COTTON (ACALA SJ2)
NCLAN Region: Southwest (SHAFTER)
Interactions: MOISTURE (STRESSED)
Y - 1301 - 2784(OZONE)
]28
-------
Table 6-2 (continued)
Crop: SOYBEAN (DAVIS)
NCLAN Region: Southeast (N.C. STATE)
Interactions: S02(So2 = 0.026 ppm)
Y = 5220 - 39194(OZONE) + 109600(OZONE)2
Crop: SOYBEAN (DAVIS)
NCLAN Region: Southeast (N.C. STATE)
Interactions: S02(so2 = 0.085 ppm)
Y = 4937 - 37624(OZONE) +• 1 09600(OZONE)2
Crop: SOYBEAN (DAVIS)
NCLAN Region: Southeast (N.C. STATE)
Interactions: S02(so2 = 0.367 ppm)
Y = 3585 - 30120(OZONE) + 1 09600(OZONE)2
Crop: SOYBEAN (WILLIAMS and ESSEX)
NCLAN Region: Southeast (BELTSVILLE)
Interactions: S02(S02 = 0.071 ppm)
Y = 4503 - 37798(OZONE) + 164897(OZONE)2
Crop: SOYBEAN (WILLIAMS and ESSEX)
NCLAN Region: Southeast (BELTSVILLE)
Interactions: S02(so2 = 0.148 ppm)
Y = 4212 - 25322(OZONE) + 103541(OZONE)2
129
-------
Table 6-2 (continued)
Crop: SOYBEAN (WILLIAMS and ESSEX)
NCLAN Region: Southeast (BELTSVILLE)
Interactions: S0(So = 0.334
Y = 3863 - 26153(OZONE) + 92033COZONE)2
Notes: All yields are in KG/HA, ozone is measured in parts per million
of 7 hr average concentrations. Names in parentheses following crop
identifications are variety identifiers. Ozone concentrations were
added during the same 7-hour period each day: 0900-1600 hr std time.
130
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6.5. RE-ESTIMATING THE NCLAN DOSE-RESPONSE FUNCTIONS
In Section 6.2 of this report we discussed the merits of a logistic
specification but conclude, given the range of ozone concentrations employed
in the experiments, that a Box-Tidwell form is more appropriate. All of the
dose-response functions reported in Table 6-1, 6-2 and several additional
functions were estimated with a common Box-Tidwell specification. In the
following subsection we discuss in detail the estimation procedure and
present the computer estimation code.
Estimating the Box-Tidwell Dose Response Function
Recall that the Box-Tidwell (BT) model can be written as
(D
where Y, X, and e are nx1 and bQ> 5^ and A are scalar parameters. The
objective is estimation of the parameters b b., and X.
Several approaches are possible. If one were to assume that e was a
normally distributed random vector, with E(e)-0 and Var(e)=o2I, i.e. the e
were i.i.d. normal, then the two most appealing approaches — maximum likeli-
hood estimation (MLE) and nonlinear least squares (NLLS) — are identical.
The problem with the normality assumption is that it admits the possibility
of negative yields. This, in fact, is a weakness of any BT specification
which allows both for E(e)=0 and nondegenerate variances.
Without making an explicit assumption about the distribution of the e,
save that E(e)=0, (1.) can be estimated by NLLS. Given certain regularity
131
-------
conditions, which may in fact be violated here, the asymptotic distribution
of the NLLS estimator for 6 = [b 5. xj is
—> N(0,o2 plim(n-1F(5)'F(5)r1)
where F(<5) is nxk and F = Ofj/a6j), where i indexes observations and j
indexes parameters, and f=b_ + biX* * e- F(<5) and o2 are typically estimated
at the NLLS estimates with o2 estimated as (n-k)~1SSR (see Judge, et al., p.
723 for a more detailed discussion of this asymptotic distribution and its
derivation).
.Nonlinear regression algorithms converge most quickly to the optimal
parameter estimates when provided with parameter starting values "close" to
those that satisfy the criterion function, in this case, the minimum of the
sum of squared residuals. Indeed, without proximate starting values, it is
possible that the solution algorithms will take quite a long time to converge
even if the second-order conditions for unique minima are satisfied. There
is thus a premium to be put on obtaining good starting values.
Box and Tidwell suggest an iterative method for obtaining the parameter
starting values. Their method will approximately converge to the first
moment of the NLLS parameter estimates if the relevant second order condi-
tions are satisfied.
Box and Tidwell proceed as follows. Considering only the univariate
model specified in (1), and given observations on y an(j x > u=1,...,n,
assume E(yu). ^ and E(yu-nu)(yv-nu)=o2 for u=v and =0 for u^v. Further, it
is assumed that n=f(£.8) where £ is a vector of the transformed X vector such
132
-------
that C=g(X,A), A being, in general, a parameter vector of the transformation,
but in the case of (1) a scalar parameter. Thus, the BT formulation is
yu = f(g(xu,A),3) * e (2)
For present purposes, it is the BT treatment of the power transformation that
is of moment. Here, define for the ith round transformation (i.e. the trans-
formation made on the 1th iteration) 5. sucft that
xu for
for
Of interest, of course, is the estimation of the parameters A and 0 of
(2). Assume 1 as a starting value for A, i.e. A. =1 . Expanding f(£,g)
around A =1 in a Taylor series gives:
f(xu,6) + (A-1)Of(5u,B)/3A) + R (3)
Evaluating (3f(O/3A) at AI =1 gives
f(xu,8) + (A-1)Of(xu,S)/3xu)(xuln(xu))
A first round estimate of Of (xu>g)/aXu) can tie produced from the esti
» T.
mated slope coefficient of a linear regression of Y on a constant term and X
A
Denote this slope estimate as Y. . Using this, fit the OLS equation
133
-------
7U = f(xu,8) + (A-DY^ulnCxu) (5)
or
f(xu,B) + e^ulnCxu) (6)
From the estimate of e^ , e1 , obtained in (6), one can back out a second-round
estimate of A as \2 = (e^/y-,) + 1. Using this, one retransforms the X vector
\2 A2
as £ = x > regresses Y on a constant term and X , obtains the slope coef-
SI
ficient Y2, and fits the equation
Yu = fCXy.S) * (A-1)Y2xuln(xu) (51)
or
yu = f(xu,B) + 92xuln(xu) (6')
From this, the third-round estimate of A, A,, is derived as Ao = (92/Y2) + 1,
and the process continues until convergence.
The reason that the BT parameter estimates obtained from the iterative
OLS method must be treated as starting values for a NLLS algorithm rather
than as the parameter estimates themselves is that the estimates of the
moments of bQ and b1 at any iteration are conditional on the value of A. At
each iteration, A is treated parametrically, with optimization carried out
only with respect to bQ and b-,. Because of this, there will be no estimate
-------
of the standard error of X and the OLS estimates of the variances and covar -
ances of bQ and t^ will be biased. However, by using the BT values as start-
ing values in a NLLS algorithm, one avoids this problem because X, b , and b.
are treated as parameters to be estimated simultaneously.
The attached SAS program documents the method used to obtain the BT
starting values. Initial values for X, b , and b are obtained, respec-
tively, as the value of L1 and the parameter estimates for the intercept and
slope of the regression of Y on XNORM in the final iteration. Four to six
iterations are all that are typically required to obtain "correct" starting
values. Using these values, the SAS PROCs MODEL, SYSNLIN, and NLIN are used
to calculate the parameter estimates of the dose-response functions and their
standard errors.
The Experimental Data
The data which underlie our re-estimation of the NCLAN dose-response
functions are drawn from three sources: Heck et al. (1981), (1982) and
Heagle et al. (1979). All data reported in these documents were derived from
experiments conducted in approximate accordance with NCLAN protocols. The
experiments are of the open-top-chamber and zonal types and thus exclude all
closed control chamber and green house studies. With the exception of a set
of experiments conducted on four red winter wheat varieties (Heagle (1979))
all experiments and resulting data are described in NCLAN annual reports.
The lack of availability of the disaggregate experimental data, that is,
data pertaining to each chamber of a multi-chamber experimental design,
significantly limits our estimation of dose reponse form. Rather, we employ
average information across all chambers which were intended to receive
135
-------
TABLE 6-3. INDEX OF DOSE-RESPONSE VARIABLES
DRAWN FROM 1980 NCLAN ANNUAL REPORT
SOYBEAN (CORSOY), Central States NCLAN Region
OZ3MO Ozone cone. (PPM) 7/1 - 9/30
OZ2MO Ozone Cone. (PPM) 8/6 - 9/30
NS Number of Seeds
SW . Seed Weight
SWP Seed Weight per Plant
SWHP Seed Weight per Healthy Plant
WS Weight per Seed
OIL Percentage Oil
PROT Percentage Protein
PEANUTS (NC-6), Southeast NCLAN Region
OZ5MO Ozone Cone. 6/1 7 - 10/6
SHTW Fresh Shoot Weight
RTW Fresh Root Weight
PODW Total Pod Weight
MPODW Marketable Pod Weight
MPODN Marketable Pod Number
136
-------
TABLE 6-4. RAW EXPERIMENTAL DATA
SOYBEAN (CORSOY), NCLAN Central States Region
OZ3MO
.037
.050
.050
.064
.079
.094
OZ2MO
.022
.042
.042
.064
.089
.115
NS
718
742
784
694
612
508
SW
T08
105
112
95
77
58
SWP
13-9
13.8
14.1
11.5
9.4
7.0
SWHP
20.4
19.0
18.4
14.9
11.7
9.4
WS
.149
.141
.143
.137
.125
.115
OIL
19.5
19.2
19.2
18.9
19.0
18.2
PROT
38.9
38.9
38.3
39.6
39.5
40.7
137
-------
TABLE 6-5. RAW EXPERIMENTAL DATA
PEANUTS (NC-6), NCLAN Southeast Region
OZ5MO
.056
.025
.056
.076
.101
.125
SHTW
893
1008
'761
483
402
219
RTW
20
21
16
12
9
5
PODW
204
187
145
110
77
^3
MPODW
158
142
122
92
69
40
MPODN
77
70
58
45
34
22
138
-------
TABLE 6-6. INDEX OF DOSE-RESPONSE VARIABLES
DRAWN FROM 1981 NCLAN ANNUAL REPORT
CORN (2 Varieties), NCLAN Central States Region
OZ4MO Ozone Cone. 6/20 - 9/10
KGHAPI Yield KG/HA - PIONEER 3780
SDWPI Weight of 100 Seeds - PIONEER 3780
PTKPI Percent Kerneled - PIONEER 3780
KGHAPA Yield KG/HA - PAG 397
SDWPA Weight of 100 seeds - PAG 397
PTKPA Percent Kerneled - PAG 397
SOYBEAN (HODGSON), NCLAN Northeast Region
OZ3MO Ozone Cone. 7/23 - 9/30
NOS Number of Seeds
SDW Seed Weight
SOYBEAN (DAVIS 0, AND S02), NCLAN Southeast Region
OZ Ozone Cone.
S02 S02 cone.
SD100 Weight of 100 Seeds
SDW Weight of Seeds per Meter of Row
139
-------
Table 6-6 (continued)
SOYBEAN (ESSEX and WILLIAMS, 0 and S02), NCLAN Southeast Region
OZFM Ozone Cone. During Fumigation
OZSEA Ozone Cone. During Season
PLTSE Plants/M Row ESSEX
PLTSW Plants/M Row WILLIAMS
YIELDE Yield G/M Row ESSEX
YIELDW Yield G/M Row WILLIAMS
SDSIZE Seed Size ESSEX
SDSIZW Seed Size WILLIAMS
SEEDSE Seed Numbers ESSEX
SEEDSW Seed Numbers WILLIAMS
COTTON (ACALA SJ2), NCLAN Southwest Region
OZ Ozone cone.
LD Percent Leaf Damage
YLD Mean Gross Yield
140
-------
TABLE 6-7. RAW EXPERIMENTAL DATA
CORN (2 VARIETIES), NCLAN Central States Region
OZ4MO
.044
.015
.044
.073
.100
.129
.156
KGHAPI
10474
10991
10743
10909
8237
6101
4232
SDWPI
24.2
25.7
24.3
24.7
20.0
17.6
15.4
PTKPI
89.5
88.3
87.4
88.2
87.5
88.5
82.1
KGHAPA
11387
11832
12911
11461
11044
8319
5040
SDWPA
23-7
25.8
25.6
25.3
24.0
18.6
15.9
PTKPA
91 .0
89.8
93.0
89.4
91 .0
89.0
84.2
141
-------
TABLE 6-8. RAW EXPERIMENTAL DATA
SOYBEAN (HODGSON), NCLAN Northeast Region
OZ3MO NOS SOW
.017 76.3 12.1
.035 73-5 11.5
.035 73-3 11.1
.060 69.3 9.7
.084 66.7 8.4
.122 60.6 7.1
142
-------
TABLE 6-9. RAW EXPERIMENTAL DATA
SOYBEAN (DAVIS), NCLAN Southeast Region
oz
.0245
.0553
.0687
.0858
.1058
.1247
.0531
.0245
.0553
.0687
.0858
.1058
.1247
.0531
.0245
.0553
.0687
.0858
.1058
.' 1 247
.0531
.0245
S02
0
0
0
0
0
0
0
.026
.026
.026
.026
.026
.026
.026
.085
.085
.085
.085
.085
.085
.085
.367
SD100
17.6
17.0
15.9
14.2
13-4
13-3
16.0
18.2
15.9
15.0
13-3
13.3
13.2
18.1
15.4
13.6
13.7
13.2
12.3
15.3
SDW
412
381
318
273
246
222
379
438
318
313
238
250
190
426
329
294
233
198
193
286
143
-------
Table 6-9 (continued)
OZ S02 SD100 SOW
.0553
.0687
.0858
.1058
.1247
.0531
.367
.367
.367
.367
-367
-367
14.2
13.3
13.1
13.0
12.8
237
192
189
154
164
144
-------
TABLE 6-10. RAW EXPERIMENTAL DATA
SOYBEAN (ESSEX AND WILLIAMS), NCLAN Southeast Region
OZFM
.014
.039
.071
.096
OZSEA
.014
.039
.060
.077
PLTSE
18.7
19.4
18.9
20.1
PLTSW
20.6
19.4
19.5
20.2
YIELDE
343
289
259
242
YIELDW
363
340
268
262
SDSIZE
13.6
13-0
12.2
12.0
SDSIZW
19.1
17.7
16.7
16.1
SEEDS E
2553
2235
2219
1959
SEEDS W
1805
1970
1656
1579
Note: The yield variables were averaged across alternative sulfur dioxide
concentrations.
145
-------
TABLE 6-11. COTTON (ACALA SJ2), NCLAN Southwest Region
OZ LD YLD
.018 0 1123.8
.045 0 1356.0
.071 6 1109.3
.111 29 859.5
.143 55 864.3
.185 61 592.5
.077 5 1194.0
146
-------
TABLE 6-12. INDEX OF DOSE-RESPONSE VARIABLES
DRAWN FROM HEAGLE ET AL. (CANADIAN JOURNAL OF BOTANY)
WHEAT (4 Varieties RED WINTER), Experiments conducted
in Southeastern U.S.
OZ2MO Ozone Cone. 4/9 - 5/31
SDWBB Seed Weight per Plant - BLUEBOY II
SDWCOK Seed Weight per Plant - COKE, 4?-27
SDWHOL Seed Weight per Plant - HOLLY
SDWOA Seed Weight per Plant - OASIS
Note: Experiments conducted prior to NCLAN formation.
147
-------
TABLE 6-13. RAW EXPERIMENTAL DATA
WHEAT (4 VARIETIES RED WINTER), Southeast U.S.
OZ2MO
.06
• 03
.06
.10
.13
SDWBB
4.79
5.84
5.74
4.97
4.02
SDWCOK
4.01
5.09
4.55
3.82
2.91
SDWHOL
4.16
4.95
4.91
4.43
3-30
SDWOA
4.06
4.45
4.41
3.89
3.28
Note: Experiments conducted prior to NCLAN formation.
148
-------
the same ozone concentrations. The result of this averaging is a reduction
in the degrees of freedom (number of observations) available for our
re-estimation of the dose-response functions.
Table 6-3 presents the variable index for the data sets drawn from the
1980 NCLAN Annual Report. Tables 6-4 and 6-5 display the accompanying raw
data used in our estimation programs. It is readily apparent from an
examination of Tables 6-4 and 6-5 that only six observations exist for the
estimation of the soybean and peanut functions.
Table 6-6 displays the variable index for data sets drawn from the 1981
NCLAN Annual Report while Tables 6-7 - 6-11 display the associated raw data
sets. Finally, Tables 6-12 and 6-13 display the variable index and raw data
pertaining to the wheat experiments reported in Heagle et al. (1979).
6.6. RFF BOX-TIDWELL DOSE-RESPONSE FUNCTION ESTIMATES
In the following sequence of tables we present our estimates of dose-
response 'functions based upon the data sets displayed in subsection 6.4.
Each estimated function is based on common specification which we have termed
a Box-Tidwell in recognition of its developers. Recall the form of the
Box-Tidwell as given below in the case of single independent variable x.
(7)
where b is an intercept term, b. a slope parameter and X a curvature param-
eter. In the event that X = 1 the expression (7) reduces to a linear func-
tion of x. If X > 1 then (7) becomes a concave function and if \ < 1 (7) is
a convex function. Thus, for example, if the true relationship underlying
149
-------
the estimation of the Corsoy soybean dose-response function (Table 6-1) is
indeed linear, as suggested by the NCLAN choice of functional form, then we
would expect X to be very close to unity. Similarly, if the plateau function
employed in the case of Pioneer 3780 corn (Table 6-2) is a reasonable
approximation to the true relation we will expect X to be greater than unity.
In the tables to follow we present estimated functions for specific
crop/varieties which vary by choice of yield variable and in some instances
by ozone averaging times.
We note that all the data points presented in Tables 4-2, 4-3 and 4-5 -
4-12 were used in the estimation. These data points include control plots
exposed to ambient air without chambers. If a chamber bias exists in the
experiments then our estimated functions will be impacted by the inclusion of
the control plot.
Table 6-14 displays the functions estimated for Corsoy soybeans. These
functions are comparable to the NCLAN relationship depicted in Table 6-1.
The equations displayed in Table 6-14 range over five alternative yield
variables and two averaging times. The NCLAN function depicts seed weight
per healthy plant as a function of a two month averaging time. Our curvature
parameter suggests that the linear form employed by NCLAN is a reasonable
approximation to the data.
It is not the purpose of this report to evaluate each of our functions
with respect to their NCLAN counterparts. The purpose of the above discus-
sion is merely to highlight the sensitivity of the relationship to the choice
of functional form, yield variable and averaging time.
Tables 6-15 - 6-21 display our estimated functions for peanuts, corn
hybrids, wheat varieties, corn varieties Pioneer 3780, and PAG 397, Hodgson
150
-------
soybeans, Essex and Williams soybeans, Acala SJ2 cotton and Davis soybeans.
In all but one instance the nonlinear Box-Tidwell estimation approach per-
formed quite well. In the case of function SE1 (Table 6-20) the estimation
algorithm did not converge to a satisfactory level of confidence. At the
present time we have not identified the source of the problem.
We draw attention to a series of experiments conducted on Davis soybeans
by NCLAN researchers at North Carolina State University. In this set of
experiments concentrations of sulfur dioxide were administered to
open-top-chambers in addition to ozone concentrations. Three alternatives
$®2 regimes including no SC^ were administered. In the absence of any S02
the relationship between yields of Davis soybeans and ozone concentrations is
remarkably linear. As SO. concentrations are increased in steps to a maximum
of .367 ppm the functions become more nonlinear. Finally, it is possible to
pool all the data across SO regimes and estimate a Box-Tidwell function of
the following form.
1
Y = o + SO 1 + YS02
The RFF estimate of this function is given below.
Y = 933.347 - 1109.74(0 )'2089 - 297.483(S02)1'175 R2 = .9310
151
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TABLE 6-14. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: SOYBEAN (CORSOY)
NCLAN Region: Central States
Function ID
SC1
SC2
SC3
SC4
SC5
SC6
SC7
SC8
SC9
SC10
Measure
of
Output
NS
SW
SWP
SWHP
WS
NS
SW
SWP
SWHP
WS
Dose
OZ3MO
OZ3MO
OZ3MO
OZ3MO
OZ3MO
OZ2MO
OZ2MO
OZ2MO
OZ2MO
OZ2MO
#OBS
6
6
6
6
6
6
6
6
6
6
a
760.
115.
15.
28.
•
755.
113-
14.
23.
a
4634
749
43259
06625
156729
5221
3243
8961
998
152759
A
b
-2145473-
-44348.
-2025.
-220.
-2.
-76216.
-4261
-318.
-113.
•
2
42
277
57102
6
12
683
573
66197
3
2
2
1
1
2
1
1
1
A
A
.8167
.8017
.3099
.0364
.7415
.6383
.99905
.6995
.93716
.3227
R2
.9254
.9728
.9734
.9853
.9912
.9278
.9738
.9739
.9841
.9914
152
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TABLE 6-15. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: PEANUTS (NC-6)
NCLAN Region: Southeast U.S.
Function ID
Measure
of
Output
Dose #OBS
PS1
PS2
PS3
PS4
SHTW OZ5MO 6 1264.378 -7654.69
RTW OZ5MO 6 24.1078 -265.083
PODW OZ5MO 6 211.2775 -4158.15
MPODW OZ5MO 6 156.9733 -4995.8
.94902 .9350
1.25525 .9332
1.52339 .8611
1.78762 .8747
153
-------
TABLE 6-16. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: WHEAT (RED WINTER, BLUEBOY II, COKER 47-27, HOLLY, OASIS)
NCLAN Region: Southeast U.S.
Function ID
Measure
of
Output
Dose #OBS
WB1
WC1
WH1
W01
SDWBB
SDWCOK
SDWHOL
SDWOA
OZ2MO
OZ2MO
OZ2MO
OZ2MO
5
5
5
5
5.8993
5.8657
4.6979
4.4423
-48.888
-15.6303
-149335.
-172.732
1.6222 .7497
.83509 .9117
5.6777 .7858
2.4582 .9247
154
-------
TABLE 6-17. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: CORN (PIONEER 3780, PAG 397)
NCLAN Region: Central States
Measure
of
Function ID Output Dose #OBS a b A R2
CPI1 KGHAPI OZ4MO 7 11163.32 -515292. 2.3004 .9669
CPG1 KGHAPA OZ4MO 7 12075.55 -6960261. 3-707221 .9664
155
-------
TABLE 6-18. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: SOYBEAN (HODGSON)
NCLAN Region: Northeast U.S.
Function ID
SH1
SH2
SH3
Measure
of
Output
NOS
SOW
NOSSDW
"Dose
OZ3MO
OZ3MO
OZ3MO
#OBS
6
6
6 1
a
77.892
13.252
061 .823
b
-221 .175
-49.467
-3938.49
A
X
1 .1738
.9465
.8335
R2
.9922
.9922
.9952
156
-------
TABLE 6-19. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: SOYBEAN (ESSEX and WILLIAMS)
NCLAN Region: Southeast U.S.
Function ID
SE1
SW1
SE2
SW2
Measure
of
Output
YIELDE
YIELDW
YIELDE
YIELDW
Dose
OZFM
OZFM
OZSEA
OZSEA
#OBS
4
4
4
4
A
a
A
b
A R2
NONCONVERGENCE
397.132
491 .162
383-508
-93^.339
-547.452
-2748.27
.8008 .9342
.3063 .9998
1.1906 .9167
157
-------
TABLE 6-20. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTION
Crop: Cotton (ACALA SJ2)
NCLAN Region: Southwest U.S.
Measure
°f
Function ID Output Dose #OBS a b A
CA1 YLD OZ 7 1561.073 -45^0.42 .9193 .9543
158
-------
TABLE 6-21 . RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
Crop: SOYBEANS (DAVIS)
NCLAN Region: Southeast U.S.
Measure
Function of
ID
SD1
SD2
SD3
SD4
Output
SOW
SDW
SOW
SDW
Dose
OZ
OZ
OZ
OZ
#OBS
7
6
6
6
so2
0.000
0.026
0.085
0.367
469
1126
807
1467
A
a
.553
.782
• 73
.863
b
-2283.
-1
-1
-1
353.
199.
509.
45
00
88
07
y\
A
1 .0453
0.1817
0.3108
0.0664
R2
.9510
.9588
.98816
.9447
159
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6.7. AVERTING BEHAVIOR AS EMBODIED IN VARIETY SWITCHING
The dose-response function evidence provided in the previous section
demonstrates clearly that the sensitivity of a particular crop to concentra-
tions of ozone varies with the variety of that particular crop. If the dose-
response relationship across varieties is merely a neutral displacement of a
common relation then for our benefit estimates the differing varieties are
not a problem. However, if a nonneutral displacement is found then the
benefit estimates will vary with variety.
Ideally, we would like to know exactly which varieties were planted in
what quantities in which areas at each point in time. Our contacts of the
Economic Research Service (ERS) of USDA are inclined to believe that this
data at the level of resolution required by the RMF is not available. We
shall continue to pursue our efforts with ERS but must have a fallback
position which is acceptable from an economic standpoint and within the time
and budget limitations of the project.
We propose the following based on the simple assumptions that farmers
choose crop varieties in an effort to maximize yields in their respective
regions presuming that all varieties receive identical applications of
fertilizer and other inputs. Under these assumptions farmers choose
varieties which maximize yields given their ambient ozone concentrations,
climate, soil type, etc.. Consider the four varieties of wheat study by
Heagle et al. (1979). If we were to form the uppermost envelope 'of this set
of functions we would have defined what we shall term the "frontier
dose-response function." Under our assumption of producer behavior the
variety Blueboy will be chosen by all wheat farmers which experience ambient
ozone concentrations from 0.0-.24 ppm. Since this variety provides the
160
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greatest yield where ozone concentrations are less than .24. If
concentrations exceed this terminal value then the farmers are induced to
switch to Coker.
In each instance where we have multiple, variety specific dose-response
functions we form the frontier of these functions and use that frontier as
the relevant dose-response curve ' reflecting the variety choice of the
producers.
6.8. CONCLUDING REMARKS
It is fair to say that the NCLAN experiments conducted over the last two
years have added to the evidence suggesting the existence of harmful ozone
effects on plants, both in terms of leaf injury and yield. However, the
design and results of these recent experiments, though extremely useful, do
not provide all of the desired information for theoretically and empirically
sound national benefit estimates.
When we speak of an agricultural yield response-ozone dose function in
the narrowest sense, we presume that all other factors affecting yield of the
particular crop under consideration — climate, soil type, farming practices,
concentrations of other pollutants and the like — are held constant in the
design of the experiments which generate the data. In these circumstances,
an' attempt to empirically relate crop yield and ozone dose, say in a
regression context, could be made. But for either of two reasons, the
influence of other candidate variables on yield cannot be accounted for
because of data limitations. Specifically the experiments could either have
been designed to hold these variables constant or, improperly, could have
inadvertently allowed them to vary but failed to obtain their measurements.
161
-------
If the crop yield response to ozone is in fact independent of the levels
of all other potential explanatory variables, this method, labeled the
"classical one-variable-at-a-time strategy" (Box _et al., 1978) is, at least
mathematically, benign. However, even if independence holds, relating yield
to each separate variable (such as ozone, rainfall, soil type) in a sequence
of one-variable ordinary least squares (OLS) regressions can have serious and
quite undesirable consequences if all theoretically important explanatory
variables apart from the one of interest in the one-variable regressions were
not actually held constant in the experiments. Allowing omitted variables to
vary in the experiments but failing to measure their levels means that, to
the extent that such omitted explanatory variables are correlated with the
included explanatory variable, the parameter estimates of any single variable
yield regression will be biased and inconsistent. Even in the absence of
such correlation, the intercept, parameter estimate will be biased, as will
the estimated variance of the slope (Kraenta, 1971, Ch. 10).
We presume that, because the NCLAN dose-response experiments were care-
fully designed, all omitted variables in any particular experiment were in
fact held constant, so that omitted variable bias is not a problem. The
results presented in this report are conditioned on the assumption that crop
yield can be legitimately estimated as a function of ozone dose alone to
accurately represent what happened at _a particular experiment station. To
predict what could happen across the nation on the basis of this Information
is another question altogether.
162
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CHAPTER 6 FOOTNOTES
Some experiments investigated the simultaneous effects of ozone and
sulfur dioxide.
2
Specification error tests are designed to discriminate between random
(white noise) variation in the residuals and systematic variation which can
be related to other variables. Misspecified models produce the latter, but
sometimes the net effect of several simultaneous specification errors may
lead to apparent white noise residuals, defeating the tests.
^For an example of the application of the Ramsey-type tests to linear
epidemiological dose-response models relating human morbidity and pollution
see Smith (1975). The RESET test is a cousin to the method of using the
higher powers of the explanatory variables as a test for nonlinearity
discussed in the next section (see Thursby and Schmidt, 1977, p. 637).
4
The same GRC report also produced a wide range of opinion over the
appropriate way to measure ozone dosage, particularly the common assumption
of equivalent yield reductions from mathematically equivalent doses (e.g.,
0.06 ppm for 100 hours versus 12 ppm for 50 hours).
It is easy to visualize the plateau model and its representation in the
regression context. Graphically, let b2 be the Y axis intercept of the down-
ward sloping segment of the function (with slope b1) and b0 be the plateau
level for X < X*.
163
-------
X*
Algebraically we can write the model without error as:
Y = (1 - D)bQ + D(b2 - b-,X) where D = 1 if X > X*
To force the horizontal line segment and the downward sloping line
segments of the plateau model to join at X* the following restriction must be
imposed:
- b.,X* = b0 or b2 = b-,X* + bQ
Substituting for b given by the restriction we get:
- D)b
Q + DC^X* + b0 -
164
-------
which can be simplified to the equation in the text:
A more sophisticated approximation method is the cubic spline function.
Cubic splines are cubic polynomials in a single independent variable joined
together smoothly at known points.
If the break points (changes in regression regimes) are unknown a
priori, an attempt to locate them empirically can be made using a variety of
methods which are frequently applied in time series analysis. (See Hackl,
1980, for an exhaustive survey, and Harvey, 1981 , for a lucid discussion of
the cumulative sum of recursive residuals (CUSUM) test for structural mis-
specification) .
7
A similar sort of model specification test was performed for alterna-
tive functional forms of the travel cost model in Smith (1975). Also, see
Aneuryn-Evans and Deaton (1980) for a theoretical treatment and some Monte
Carlo evidence on the performance of Cox-Pesaran test.
Q
Note that the observed values of yield for any given value of dose in
this formulation theoretically can extend from minus infinity to plus
infinity because we have assumed a normal distribution for the error term.
Put otherwise, there always exists a finite probability that observed yield
will be nonpositive. To get around this problem, we can either truncate the
distribution of the error term or assert that the expected value of the
dependent variable will always be, say, five standard errors above zero.
Practically speaking, • the latter means that the probability of observing a
nonpositive Y is so close to zero that it can be ignored. Without using this
165
-------
dodge or arbitrarily- truncating the error term we must be willing to use the
natural log of yield as the dependent variable in all the models considered
(i.e., the linear model is rejected outright). Then, the logit model would
be:
b X
V(1+b1e > £i
Y. = e e
We do not entertain this possibility here.
9
Box-Tidwell: First and Second Derivatives with Respect to X:
Y = f(X) = b0 + bl
Logistic: First and Second Derivatives with Respect to X:
b?X
Y = g(X) = b + (
p ;> -
g'(X) = -b2bie /(1+l^e )2 < 0 iff b1f b2 > 0
or b-
b X b X b.X b.X b,X
-(1+b e ' ) bjb.e * *2b5b.e ^ (1+^e ^ Xb.b.e ^
g (X) = 1 LI LL- 1 fJ
2 U
(1+5 * r
To find the point of inflection given b.> b2 set:
166
-------
b X b X b X
equal to:
b X b X
* ) be *
Cancelling terms and simplifying:
V
b.je = 1 or X = -Inb /b
For testing nested hypotheses, -2(LR) has, for large samples, a chi-
square distribution with degrees of freedom equal to the number of parameters
restricted to specific values under HQ. in the nonnested case, it is only a
measure of plausibility with no such distributional properties.
The discussion assumes both models have an identical number of param-
eters to be estimated.
12
See Davidson and MacKinnon (1981) for a theoretical derivation of the
asymptotic properties of their tests which they show are similar to the
asymptotic properties of the Pesaran and Deaton (1978) tests.
167
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CHAPTER 7
YIELD CHANGES USING EPA OZONE SCENARIOS
In this chapter we exploit the dose-response functions described in
Chapter 6 in conjunction with the air quality and crop yield information con-
tained in the RMF to examine the impact of alternative ozone exposure
scenarios on the yields of selected crops. We note at the outset that these
calculations assume no economic adjustments on the part of agricultural
producers to changing crop yields. We simply employ the RFF re-estimated
NCLAN dose-response functions to calculate the change in yield associated
with a particular change in ozone concentrations and then multiply this
change in yield by the 1978 yields contained in the FEDS.
The actual calculations are described below.
1. Actual 1978 ozone concentrations by FEDS areas contained in the RMF
data base are associated with the quantity of soybeans, wheat, corn, cotton
and peanuts produced by the respective areas in 1978.
2. The actual 1978 ozone concentrations for each FEDS area are located
on the appropriate NCLAN dose-response function and the value of the yield
proxy variable (the response variable) recorded.
3. Using EPA/OAQPS supplied ozone exposure scenarios (see Table 7-1) we
bring all FEDS areas to the same ozone concentration as specified by the
scenario.
4. The scenario concentrations are then located on the dose-response
functions and the new level of the proxy yield variable recorded.
168
-------
5. For each FEDS area for each crop the following formula (reproduced
from Chapter 5) is calculated.
*
AYIELD => -y- - 1
where Y is the yield at the 1978 ozone concentration
Y* is the yield associated with each ozone scenario.
6. (AYIELD + 1) is multiplied by the 1978 quantities of each crop
produced in each FEDS area and then summed by crop across areas.
.The relevant dose-response functions used in this exercise along with
their pictorial representations are displayed on Tables 7-2 - 7-8. The
specific dose-response functions (highlighted by rectangles in the tables)
were chosen to be regionally consistent with FEDS areas. In the cases of
wheat and corn we employ the method of frontier Tidwell discussed in
Chapter 6.
Tables 7-9 - 7-15 report the changes in biological yield for the five
crops examined in this study across the seven ozone concentrations displayed
in Table 7-1. Soybeans, wheat and corn are dimensioned in bushels, peanuts
in pounds and cotton in bales.
169
-------
TABLE 7-1
EPA/OAQPS OZONE CONCENTRATION SCENARIOS
Scenario No.
1
2
3
4
5
6
7
8
9
10
Concentration in PPM
.01
.02
• 03
.04
.05
.06
.07
.08
.09
.10
Note: Ozone concentrations are measured
as seasonal 7 hour daily means.
170
-------
TABLE 7-2. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: SOYBEANS
Crop: SOYBEAN (CORSOY)
NCLAN Region: Central States
Function ID
SCI
SC2
SC3
SC4
SC5
SC6
SC7
SC8
SC9
SC10
Measure
of
Output
NS
SW
SWP
SWHP
WS
NS
SW
SWP
SWHP
WS
Dose #OBS
OZ3MO
OZ3MO
OZ3MO
OZ3MO
OZ3MO
OZ2MO
OZ2MO
OZ2MO
OZ2MO
OZ2MO
6
6
6
6
6
6
6
6
6
6
760.
115.
15.
28.
^ t
755.
113.
14.
23.
.
a
4634
749
43259
06625
156729
5221
3243
8961
998
152759
>%
b
-2145473-
-44348.
• -2025.
-220.
-2.
-76216.
-4261 .
-318.
-113-
•
2
42
277
57102
6
12
683
573
66197
3
2
2
1
1
2
1
1
1
A
\
.8167
.8017
.3099
.0364
.7415
.6383
.99905
.6995
.93716
.3227
R2
.9254
.9728
.9734
.9853
.9912
.9278
.9738
.9739
.9841
.9914
171
-------
TABLE 7-3. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: SOYBEANS
Crop: SOYBEAN (HODGSON)
NCLAN Region: Northeast
Function ID
SH1
SH2
SH3
Measur e
of
Output
NOS
SOW
NOSSDW
Dose
OZ3MO
OZ3MO
OZ3MO
#OBS
6
6
6 1
A
a
77.892
13.252
061 .823
b
-221 .175
-49.467
-3938.49
A
1 .1738
.9465
.8335
R2
.9922
.9922
.9952
172
-------
TABLE 7-4. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: SOYBEANS
Crop: SOYBEAN (ESSEX and WILLIAMS)
NCLAN Region: Southeast
Function ID
Measure •
of
Output Dose #OBS a b
A R2
SET
SW1
SE2
YIELDE OZFM 4 NONCONVERGENCE
YIELDW OZFM 4 397.132 -934.339
YIELDE OZSEA 4 491.162 -547.452
.8008 .9342
.3063 .9998
SW2
YIELDW
OZSEA
4
383.508
-2748.27
1 .1906
.9167
173
-------
TABLE 7-5. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: COTTON
Crop: Cotton (ACALA SJ2)
NCLAN Region: Southwest
Measure
of
Function ID Output Dose #OBS
CA1
YLD
OZ
7
1561
.073
-45^0.
42
.91
93
.9543
174
-------
TABLE 7-6. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: CORN
Crop: CORN (PIONEER 3780, PAG 397)
NCLAN Region: Central States
Measure
of
Function ID Output Dose #OBS
CPU KGHAPI OZ4MO 711163-32 -515292. 2.3004 .9669
CPG1 KGHAPA OZUMO 712075.55 -6960261. 3-707221 .9664
175
-------
TABLE 7-7. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NON-NCLAN EXPERIMENTAL DATA: WHEAT
Crop: WHEAT (RED WINTER, BLUEBOY II, COKER 47-27, HOLLY, OASIS)
NCLAN Region: Southeast
Function ID
Measure
of
/V A
Output Dose #OBS a b
A R2
WB1
WC1
WH1
W01
SDWBB
SDWCOK
SDWHOL
SDWOA
OZ2MO
OZ2MO
OZ2MO
OZ2MO
5
5
5
5
5.
5.
4.
4.
8993
8657
6979
4423
-48.
-15.
-149335.
-172.
888
6303
732
1
5
2
.6222
.83509
.6777
.4582
.7497
.9117
.7858
.9247
176
-------
TABLE 7-8. RESOURCES FOR THE FUTURE DOSE-RESPONSE FUNCTIONS
ESTIMATED FROM NCLAN EXPERIMENTAL DATA: PEANUTS
Crop: PEANUTS (NC-6)
NCLAN Region: Southeast
Function ID
Measure
of
Output Dose #OBS
PS1
PS2
SHTW OZ5MO 6 1264.378 -7654.69
RTW OZ5MO 6 24.1078 -265.083
.94902 .9350
1.25525 -9332
PS3
PODW
OZ5MO
fa
211
.2775
-4158.
15
1
.52339
.8611 |
i
PS4
MPODW
OZ5MO 6 156.9733
-4995.8
1.78762 .8747
177
-------
TABLE 7-9. OUTPUT CHANGES
Crop: SOYBEANS
NCLAN Region: Central States
Ozone concentrations
Output
Output change
from .04 ppra
Change
by increment
.04 ppm
.05 ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
. 10 ppm
1,541,393,410
1.454,137,600
1,340,680,450
1,199,492,100
1,029,211,650
828,609,792
596,553,984
-87,255,810
-200,712,960
-341,901,310
-512,181,760
-712,783,618
-944,839,426
-87,255,810
-113,457,150
-141 ,188,350
-170,280,450
-200,601,858
-232,055,808
178
-------
TABLE 7-10. OUTPUT CHANGES
Crop: SOYBEANS
NCLAN Region: Northeast
Ozone concentrations
Output
Output change
from .04 ppm
Change
by increment
. 04 ppm
.05 ppm
.06 ppm
'.07 ppm
.08 ppm
.09 ppm
. 1 0 ppm
29,081 ,344
27,061,936
25,109,168
23,210,080
21,355,808
19,539,840
17,757,232
-2,019,408
-3,972,176
-5,871,264
-7,725,536
-9,541,504
-11,324,112
-2,019,408
-1,952,768
-1,899,088
-1,854,272
-1,815,968
-1,782,608
179
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TABLE 7-11. OUTPUT CHANGES
Crop: SOYBEANS
NCLAN Region: Southeast
Ozone concentrations
Output
Output change
from .04 ppm
Change
by increment
.04 ppm
.05 ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
.10 ppm
837,101,056
790,301,952
741,672,704
691 ,467,264
639,872,512
587,029,248
533,053,440
-46,799,104
-95,428,352
-145,633,792
-197,228,544
-250,071 ,808
-304,047,616
-46,799,104
-48,629,248
-50,205,390
-51,594,752
-52,843,264
-53,975,808
180
-------
Crop: COTTON
Region: U.S.
TABLE 7-12. OUTPUT CHANGES
Ozone concentrations
Output
Output change
from .04 ppm
Change
by increment
.04 ppm
.05 ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
.10 ppm
.7,837.458,430
7,520,448,510
7,208,534,020
6,900,822,020
6,596,636,670
6,295,523,330
5,997,109,250
-317,009,920
-628,924,410
-936,636,410
•1 ,240,821 ,760
•1,541,935,100
•1,840,349,180
-317,009,920
-311 ,914,490
-307,712,000
-304,185,350
-301,113,340
-298,414,080
181
-------
Crop: CORN
Region: U.S.
TABLE 7-13- OUTPUT CHANGES
Ozone concentrations
Output
Output change
from .04 ppra
Change
by increment
.04 ppra
.05- ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
.10 ppm
7,029,059,580
6,994,669,570
6,935,658,500
6,843,064,320
6,706,827,260
6,515,740,670
6,257,717,250
-34,390,010
-93,401,080
-185,995,260
-322,232,320
-513,318,910
-771,342,330
-34,390,010
-59,011,070
-92,594,180
-136,237,060
-191,086,590
-258,023,420
182
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Crop: WHEAT
Region: U.S.
TABLE 7-14. OUTPUT CHANGES
Ozone concentrations
Output
Output change
from .04 ppm
Change
by increment
.04 ppm
.05 ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
.10 ppm
2,135, 484, 420
2,127,833,340
2,077,558,780
2,021,774,590
1,960,794,620
1,894,871,550
1,824,221,440
-7,651,080
-57,925,640
-113,709,830
-174,689,800
-240,612,870
-311,262,980
-7,651,080
-50,274,560
-55,784,190
-60,979,970
-65,923,070
-70,650,110
183
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Crop: PEANUTS
Region: U.S.
TABLE 7-15. OUTPUT CHANGES
Ozone concentrations
Output
Output change
from .04 ppm
Change
by increment
.04 ppm
.05 ppm
.06 ppm
.07 ppm
.08 ppm
.09 ppm
. 10 ppm
4,060,651,260
3,779,529,220
3,467,225,860
3,126,350,590
2,758,942,210
2,366,644,220
1,950,814,980
-281 ,122,040
-593,425,400
-934,300,670
-1,301,709,050
-281,122,040
-312,303,360
-340,875,270
-367,408,380
-1,694,007,040 -392,297,990
-2,109,836,280 -415,829,240
184
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CHAPTER 8
SOME WELFARE EXERCISES USING THE REGIONAL MODEL FARM
8.1. INTRODUCTION
The purpose of this chapter is to demonstrate the capabilities of the
RMF as a tool for the analysis of societal welfare effects forthcoming fron
the agricultural production sector in response to changes in rural ozone con-
centrations. We note at the outset that the estimates of net producer and
consumer surplus reported in this chapter are solely illustrative. These EPA
supplied ozone scenarios treat the standard as a strict equality, not as a
less than or equal to inequality constraint. If, for example, the standard
is tightened to .10 ppm, which might translate to average rural concentra-
tions of .05 ppm, all counties below .05 ppm are assumed to pollute up to .05
ppm. In a Regulatory Impact Analysis (RIA) proposed standards would be
translated into expected ozone monitor readings at the actual monitor sites
(primarily urban areas). These expected readings would then serve as data to
an interpolation procedure which would predict expected ozone concentrations
in rural areas. Finally, these interpolated ozone concentrations would serve
as data for the RMF.
In the illustrations to follow simple ozone scenarios are employed to
obtain the area specific ozone exposures. Specifically, we assume that the
»
ozone concentrations in all rural counties (indeed all counties rural or
urban) attain uniform levels as specified by the EPA ozone scenarios
185
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displayed in Table 8-1. Welfare estimates are then based upon the difference
between the sum of producer and consumer surplus calculated at 1978 ambient
county level ozone concentrations (these ambient concentrations vary county
to county) and the sum of producer and consumer surplus calculated at each
EPA scenario ozone concentration. Thus, if we are examining the .05 ppm
scenario some county concentrations will rise to the .05 ppm level from 1978
ambient while others will fall. This information is then used to calculate
the increment benefits between alternative ozone scenarios.
8.2. MAINTAINED ASSUMPTIONS USED IN THE ILLUSTRATIVE WELFARE EXERCISES
The process by which the RMF calculates net producer and consumer sur-
plus (welfare) estimates is discussed in section 5-3 of this report and will
not be repeated. The purpose of this section is to identify those assump-
tions which underlie the illustrative results reported below.
The first assumption concerns the differential effect which ozone has on
the productivity of preharvest and harvest factors of production. The
results reported in this chapter assume that the preharvest production func-
tion is neutrally displaced in input-output space in accordance with the
NCLAN dose-response functions discussed and reported in Chapter 6. We
further assume that harvest production function is unaffected by charges in
ozone concentration and is therefore "ozone stationary". This set of assump-
tions is manifested in the parameter Y (Equation 37, Chapter 5), where Y = 0
for all our illustrations. The sensitivity of the RMF welfare estimates to
this set of maintained assumptions is examined in the following chapter.
The second set of maintained assumptions concerns the dose-response
functions used in the welfare calculations. In the case of soybeans we
186
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TABLE 8-1. EPA/OAQPS OZONE CONCENTRATION SCENARIOS
Scenario No.
1
2
3
4
5
6
7
8
9
10
Concentration in ppra
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
187
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employ region-specific dose-response functions. But for wheat, corn, cotton
and peanuts we must use a single function for all regions. Further, each
dose-response function employs a common nonlinear form referred to as a Box-
Tidwell. The specific dose-response functions utilized are displayed in
Chapter 7.
In the case of wheat and corn we have been able to estimate dose-
response functions for alternative varieties of each crop. Since a priori we
do not know which variety farmers are planting or would plant under differing
ozone regimes we have adopted the behavioral rule that farmers plant that
variety which produces the greatest yield. This rule allows us to envelope
the uppermost portions of a set of varietal specific dose-response functions
and employ that envelope as a function .which in some sense incorporates
varietal switching behavior- The sensitivity of our results to this partic-
ular assumption is examined in the following chapter.
The third assumption concerns the assumed elasticity of demand assigned
to each crop. The elasticities employed in this study are drawn from the
USDA model entitled "A Mathematical Programming Model for Agricultural Sector
Policy Analysis" and are displayed in Table 8-2. The sensitivity of our
results to the elasticity estimates is examined in the following chapter.
8.3- BENEFIT CALCULATIONS WITH ELASTIC DEMAND
In what follows we describe the methods employed to compute net producer
and consumer surplus when the aggregate demand for agricultural crops pos-
sesses some elasticity. Table 8-2 below displays the point estimates of
demand elasticities for the five crops covered in this study.
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TABLE 8-2: PRICE ELASTICITIES OF DEMAND
FOR SELECTED AGRICULTURAL CROPS
Crop
Cotton
Corn
Soybeans
Wheat
Peanuts
Demand Elasticity
-.22
-.33
-.80
-.35
-.80
*These estimates were drawn from "A Mathematical
Programming Model for Agricultural Sector Policy
Analysis," Robert House, Oct. 20, 1982 United States
Dept. of Agriculture, Economic Research Service.
189
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Figures 8-1 and 8-2 display the heuristics of the benefit calculation
under alternative scenarios concerning the level of ambient ozone. In
Figure 8-1 ozone concentrations are reduced below current ambient. This has
the effect of shifting the agricultural supply function from S° to S* and
thus increasing output from Q^ to Q*. The shaded area represents the net
gain in consumer and produce surplus. Area S°ABS* is obtained by suitable
integration of the appropriate marginal cost curves. The area ABC is
calculated with knowledge of the elasticity given in Table 8-2 and thus the
slope of DD' and the change in output given by Q* - Q^.
Figure 8-2 displays a case in which ambient ozone concentrations rise
reducing crop yields and forcing the supply function upward as indicated by
the shift from S° to S1 . To evaluate the welfare loss we must determine the
area S°S1ABCD. We first determine S°S1AB by suitable integration under the
supply curves from 0 to Q1 and then determine DABC with knowledge of Q^ - Q1
and the slope of DD'.
8.-4. WELFARE ESTIMATES UNDER EPA/OAQPS SUPPLIED OZONE SCENARIOS
Tables 8-3 - 8-9 display the net producer and consumer surplus estimates
generated by the RMF under the EPA/OAQPS specified ozone scenarios and the
maintained assumptions discussed in section 8.2. We remind the reader that
each welfare estimate represents the difference in the sum of producer and
consumer surplus, based on the production of a specific crop between the base
ozone regime and the scenario regime. The base regime represents an estimate
of the actual 1978 ambient ozone concentrations prevailing in each FEDS area
and the consumer and producer surplus calculated on the basis of 1978 factor
prices and yields. The ten alternative scenario regimes assume that the
190
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Figure 8-1.
191
-------
Figure 8-2.
192
-------
ambient levels either rise or fall to the concentration given by the
scenario. Thus, for any particular scenario, the actual percentage change in
1978 ambient ozone will vary across FEDS regions.
As an example let us consider the results reported in Table 8-3 where we
examine the welfare gains and losses associated with the EPA scenarios for
the production of soybeans in the northeast United States. In this area of
the country the estimated mean growing season ambient ozone concentration is
approximately .055 ppm. Thus, if ozone concentrations in all FEDS areas in
this region rose to a uniform level of .06 ppm, one would expect economic
loss which is reflected in Table 8-3 as net welfare loss of $3,525,134*
Decreasing ozone concentrations from 1978 ambient to a uniform level of .05
ppm results in a net increase in welfare of $1,236,760. In the extreme
scenarios reductions to .01 ppm would yield welfare gains of $18,366,336 and
increases in ozone to a uniform .10 ppm would result in losses of
$30,367,024.
Tables 8-5 and 8-6 round out these illustrative soybean examples by
reporting results for soybean production in the Southeast and Midwest.
Tables 8-6 - 8-9 represent national estimates for the crops corn, wheat,
cotton and peanuts respectively.
8.5. CONCLUDING REMARKS
For the purposes of regulatory impact and other analyses welfare
estimates based on alternative ozone standards would be constructed in a
manner quite different from these estimates reported in the previous section.
In Chapter 10 we address some of the issues and particularly the need for
additional rural monitoring sites. One should also bear in mind that the
i
welfare estimates will vary dramatically from one portion of the country to
193
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another, even from one portion of a state to another. Thus, while we have
not done so in these illustrations, estimates made for actual policy
simulations should be regionally disaggregated at least to the state level.
194
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TABLE 8-3. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE NORTHEAST REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare Incremental welfare
gain/loss gain/loss
$ 18,366,336
13,932,556
9,690,281
5,052,414
136,379
-3,525,134
-8,986,080
-15,409,551.
-22,730,768
-30,367,024
4,433,780
4,242,275
4,637,867
4,916,035
3,661,513
5,460,946
6,423,471
7,321,217
7,636,256
195
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TABLE 8-4. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE SOUTHEAST REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS
Net welfare Incremental welfare
Concentration gain/loss gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
$ 651,690,496
570,665,472
481,455,360
343,926,272
189,834,000
9,038,215
-191,245,908
-367,553,280
-547,632,640
-742,565,632
81,025,024
89,210,112
137,529,088
154,092,272
180,795,785
200,284,123
176,307,372
180,079,360
194,932,992
196
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TABLE 8-5. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE CENTRAL STATES REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS
Net welfare
Concentration gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
$ 428,407,712
399,954,944
341,329,152
245,927,920
77,812,576
-198,836,368
-552,760,064
-1,086,211,330
-1,901,005,570
-3,074,742,020
Incremental welfare
gain/loss
28,452,768
58,625,792
95,401,232
168,115,344
276,648,944
353,923,696
533,451,266
814,794,240
1,173,736,450
197
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TABLE 8-6. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR CORN PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
Net welfare Incremental welfare
Concentration gain/loss gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
•$ 141,439,728
138,554,752
125,264,480
91,308,864
34,874,448
-68,029,264
-221,512,768
-447,547,392
-792,965,376
-1,315,634,690
2,884,976
13,290,272
33,955,616
56,434,416
102,903,712
153,483,504
226,034,624
345,417,984
522,669,314
198
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TABLE 8-7. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR WHEAT PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
Net welfare Incremental welfare
Concentration gain/loss gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
$ 262,120,464
224,526,304
165,511,312
79,262,624
-17,772,240
-132,422,384
-257,741,504
-401,955,840
-563,645,184
-751,795,712
37,594,160
59,014,992
86,248,688
97,034,864
114,650,144
125,319,120
144,214,336
161,689,344
188,150,528
199
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TABLE 8-8. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR COTTON PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
Net welfare Incremental welfare
Concentration gain/loss gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
• $ 634,018,304
512,547,584
389,435,392
253,104,528
94,547,872
-76,303,344
. -290,614,272
-540,368,384
-831,184,128
-1,172,176,380
121,470,720
123,112,192
136,330,864
158,556,656
170,851,216
214,310,928
249,754,112
290,815,744
340,992,252
200
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TABLE 8-9. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR PEANUT PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
Net welfare Incremental welfare
Concentration gain/loss gain/loss
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
$ 111,490,240
94,479,968
82,811,520
60,723,424
22,996,576
-35,673,600
-78,029,184
-127,927,056
-187,841,216
-263,253,008
17,010,272
11,668,448
22,088,096
37,726,848
58,670,176
42,355,584
49,897,872
59,914,160
75,411,792
201
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CHAPTER 9
SENSITIVITY STUDIES
9.1. INTRODUCTION
The purpose of this chapter is to discuss the results of three sensitiv-
ity studies designed to examine the impact which particular characteristics
of our data and set of maintained assumptions has had on the producer and
consumer surplus estimates discussed in the previous chapter. Specifically,
we shall examine: 1) the nature of the harvest-nonharvest cost differential
discussed in Chapter 4, 2) the choice of the frontier Tidwell approach to
varietal switching, and 3) the USDA estimates of crop demand elasticity.
9.2 HARVEST-NONHARVEST COST DIFFERENTIAL
If we think of the agricultural production process for field crops as a
sequence of production activities we may logically draw a boundary between
those activities which- are associated with harvesting the crop and those
which are not. The nonharvest or preharvest activities involve all of the
land preparation activities, the dispersement of herbicides, fertilizer, seed
and pesticides and the general maintenance of the crop until harvest. The
biological experiments forming the basis for the dose-response functions
reported in Chapter 6 are concerned with this first stage of production since
it is during this stage that ozone is expected to have an impact on crops.
202
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If the ozone impact is such that it neutrally displaces the preharvest pro-
duction function (as biological evidence suggests and as we have assumed in
our analysis) then one can think of the ozone effect as displacing the
productivity of each input by equal proportions.
Since we are unaware of any impact which ozone might have on the produc-
tion activity of harvesting, we assume that the harvest production function
is unaffected by ozone and remains stationary with respect to changes in
ambient concentrations. Since the RMF explicitly recognizes the stages of
production we are able to adjust the productivity of preharvest factors of
production without changing the productivity of the harvest factors. Eco-
nomic assessment models which do not explicitly recognize the sequence of
production activities must assume that both the preharvest and harvest
production functions are impacted (shifted) equally by changes in ozone
concentrations and will therefore lend to over/under estimates of welfare
gains associated with decreases/increases in ambient ozone.
To determine the possible magnitude of these errors in the measurement
of welfare changes we report in the table to follow changes in net consumer
and producer surplus for all crops in our study when ozone concentrations
fall from estimated 1978 ambient levels to a uniform level of .04 ppm across
all FEDS areas. We calculate the welfare changes under the assumption that
the productivity of all inputs is enhanced equally and then under the
assumption that only the preharvest factors are affected.
Recalling from our discussion in Chapter 4, we reproduce below the
sequenced formula for the marginal cost of production.
MC = (1/1+AYIELD)(MAHNONHRV) + (1 /1+YAYIELD) (MARHRV)
203
-------
where Y = differential harvest effect parameter 0 * Y 1
MARNONHRV = marginal nonharvest cost
MARHRV = marginal harvest cost.
We note that if Y = 1 then the productivity of factors employed in harvesting
is enhanced by the same proportion as the nonharvest factors. However, if Y
= 0 the harvest factors are unaffected.
Table 9-1 displays the change in net producer and consumer surplus
brought about by a reduction in ozone from 1978 ambient levels to .04 ppm.
The first column of estimates assumes that the productivity of nonharvest
factors is impacted positively by the reduction in ozone but that the
productivity of nonharvest input remains unaffected (Y =0). The last column
assumes that all factors, harvest and nonharvest, have their productivities
enhanced in equal proportions by ozone reductions (Y = 1). The middle column
allows for some productivity enhancement of harvest factors due solely to
economies of scale in harvesting bumper crops (Y = 0.2).
It is readily apparent from Table 9-1 that a failure to dichotomize the
stages of production and to explicitly recognize differential productivity
affects leads to wild overstatements of benefits.
9.3. THE PROBLEM OF VARIETAL SWITCHING
The agricultural producer of field crops may choose from several differ-
ent varieties of particular crops. These varieties differ in their growing
characteristics with regard to soil and moisture requirements and to air
pollutants. If, for 'example, concentrations of ambient ozone increase, pro-
duction managers will choose in subsequent planting seasons varieties of
204
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TABLE 9-1. ESTIMATES OF NET CONSUMER AND PRODUCER SURPLUS FORTHCOMING
FROM A DECLINE IN AMBIENT OZONE TO .04 PPM UNDER ALTERNATIVE ASSUMPTIONS
REGARDING HARVEST PRODUCTIVITY EFFECTS
Crop/region
Harvest productivity parameters
Y = 0.0 Y = 0.2 Y - t.0
Soybeans (NERCLAP)
Soybeans (SERCLAP)
Soybeans (CSRCLAP)
Corn
Wheat
Cotton
Peanuts
5,052,414
343,834,000
245,927,920
91,308,864
79,262,124
253,104,528
60,723,424
7,283,587
404,077,312
348,942,848
106,100,544
96,374,688
302,895,184
72,485,632
16,245,349
590,561,024
714,143,744
164,707,424
163,819,824
488,226,048
117,834,480
205
-------
corn, wheat, soybeans, etc. with a higher tolerance to ozone concentrations.
If the price and cost of alternative varieties is equal then the manager will
choose that variety which produces the greatest yield.
The data which we have available for this study does not permit us to
identify the particular variety planted in each FEDS area. Thus, we have
assumed that the price and cost of each variety is uniform and therefore the
variety producing the greatest yield under alternative ozone regimes is the
variety chosen by agricultural producers.
The results reported in Chapter 8 are based upon the varietal choice
principle stated above and therefore this principle determines the specific
dose-response function to be used under alternative ozone regimes. Over the
ozone range 0.00 ppm to .24 ppm the variety BLUEBOY produces the greatest yield
and is the variety whose dose-response function we employ in Chapter 8.
To determine the sensitivity of our Chapter 8 results to our choice of
dose-response function we report below a set of welfare estimates for wheat
comparable to those presented in Chapter 8 but based on the extreme assumption
that farm managers choose that variety which produces the poorest yield. We
realize that such an assumption is unreasonable but we take this extreme posi-
tion in order to place bounds on our estimates.
Table 9-2 displays the net producer and consumer surplus estimates for
wheat production that can be expected to obtain if the ambient ozone concentra-
tion of all FEDS areas attained a uniform concentration as specified in the
first column. The column of estimates labeled "Full Frontier" corresponds to
the estimates presented in Chapter 8 and utilizes the notion of frontier dose-
response functions discussed in Chapter 6. The last column entitled
"Antifrontier" provides estimates on net producer and consumer sur-
206
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TABLE 9-2. NET PRODUCER AND CONSUMER SURPLUS DERIVED FROM WHEAT
PRODUCTION UNDER VARYING OZONE CONCENTRATION REGIMES
DIFFERENTIATED BY ASSUMED VARIETAL SWITCHING BEHAVIOR
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Full frontier
262,120,464
224,526,304
165,511,312
79,262,624
-17,772,240
-132,422,384
-257,741 ,504
-401 ,955,840
-563,645,184
-751,795,712
Ant if rentier
105,558,640
95,482,992
75,290,832
39,960,240
-9,544,298
-77,965,520
-36,460,208
-85,183,440
-173,051 ,008
-319,173,632
207
-------
plus under the assumption that a lower envelop of the varietal dose-response
functions is consistent with the choice behavior of Agricultural producers.
Table 9-2 highlights the importance of the varietal problem and suggests
that errors of as much as 50% can be made in the estimation of benefits if the
wrong varietal dose-response function is employed. The reliable estimation of
welfare benefits requires the knowledge of varietals currently being planted,
varietals within a specific region's choice set and finally a battery of dose-
response functions for these varietals. Unfortunately; such information does
not exist and one must resort to fairly ad hoc rules such as the full frontier
approach advocated in this study.
In the case of corn the frontier function is set by PAG 397 and is the
function used for the estimates presented in Chapter 8. The antifrontier
function is PIONEER 3780. Using our _ad hoc rule that agricultural managers
plant that crop variety which ceteris paribus maximizes yield, leads to the
column of net producer and consumer surplus estimates given on Table 9-3
labeled "Full Frontier". If managers had chosen to plant PIONEER 3780 which
produces a lower yield the welfare estimates would be those displayed under the
heading antifrontier.
9.4. ALTERNATIVE ESTIMATES OF CROP DEMAND ELASTICITY
The elasticity of demand estimates embedded in the RMF are reasonably
close to the estimates one will find in USDA's model entitled "A Mathematical
Programming Model for Agriculture Sector Policy Analysis." While one may
acknowledge that these estimates are generally reliable one may still be
concerned with the sensitivity of producer and consumer surplus estimates to
tiae magnitudes of these elasticities. In this section we shall specifically
208
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TABLE 9-3. NET PRODUCER AND CONSUMER SURPLUS DERIVED FROM CORN PRODUCTION
UNDER VARYING OZONE CONCENTRATION REGIMES
DIFFERENTIATED BY ASSUMED VARIETAL SWITCHING BEHAVIOR
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Full frontier
141,439,728
138,554,752
125,264,480
91,308,864
37,874,448
-68,029,264
-221,512,768
-447,547,392
-792,965,376
-1,315,634,690
Antifrontier
614,787,584
574,511,104
462,406,400
233,101,312
103,655,280
-200,046,256
-578,447,616
-1,094,658,050
-1,812,615,680
-2,797,287,680
209
-------
examine this issue by forming an interval around the USDA estimates. Our low
elasticity estimate is 75% of the USDA figure and our high estimate is 125* of
the figure. As an extreme case we employ a perfectly inelastic demand function
and calculate welfare estimates under the assumption that any shortfalls in
supply are made up by imports at a price equal to the marginal cost of the last
domestically produced crop unit. Table 9-4 below presents the alternative
elasticity estimates used in this sensitivity analysis.
For each crop under consideration we vary ozone concentrations from 1978
ambient to .04 ppm and then calculate the net producer and consumer surplus
gain under the three elasticity estimates. Naturally, the more elastic the
estim'ates the larger will be the gain. We then vary the concentration from
ambient to .08 ppm and calculate the welfare loss. Tables 9-5 - 9-9 display
the results of this analysis for soybeans, wheat, corn, cotton and peanuts
respectively.
Examining Table 9-5 - 9-9 one quickly sees that the sensitivity of the
estimates to alternative elasticity assumptions is considerably less than the
sensitivity to varietal choice. If one were to attempt a refinement of the
estimates reported in Chapter 8 it would seem that further work on elasticity
refinement would be unwarranted.
9.5. ALTERNATIVE DOSE-RESPONSE EQUATIONS
Upon completion of the research described in this report two papers (Heck
et al. (1984a, I984b)) authored by members of NCLAN presented dose-response
equations for a wide variety of agricultural crops based on a Wybul functional
specification. The intersection of the crops covered by these new functions
and the crops found in FEDS contains soybeans, corn, wheat, cotton, peanuts,
210
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TABLE 9-4. DEMAND ELASTICITIES EMPLOYED
IN THE SENSITIVITY ANALYSIS
Crop
Soybeans
Wheat
Corn
Cotton
Peanuts
Alternative
USDA
-.80
-.35
-.33
-.22
-.80
elasticity
High
-1.0
-.44
-.41
-.28
-1.0
estimates
Low
-.60
-.26
-.25
-.17
-.60
TABLE 9-5. NET PRODUCER AND CONSUMER SURPLUS ESTIMATES
UNDER ALTERNATIVE ASSUMPTIONS REGARDING THE DEMAND ELASTICITY
OF SOYBEANS
Elasticity Ranges
Concentration High USDA Low Inelastic
.04 612,691,572 594,906,606 589,417,321 413,250,880
.08 -1,380,893,693 -1,469,174,161 -1,622,899,744 1,662,523,472
211
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TABLE 9-6. NET PRODUCER AND CONSUMER SURPLUS ESTIMATES
UNDER ALTERNATIVE ASSUMPTIONS REGARDING THE DEMAND ELASTICITY
OF CORN
Elasticity Ranges
Concentration High USDA Low Inelastic
.04
.08
91 ,524,592
-437,176,064
91,308,864
-447,547,392
91 ,088,944
-464,832,512
86,81 4,720
-501 ,420,032
TABLE 9-7. NET PRODUCER AND CONSUMER SURPLUS ESTIMATES
UNDER ALTERNATIVE ASSUMPTIONS REGARDING THE DEMAND ELASTICITY
OF WHEAT
Elasticity Ranges
Concentration High USDA Low Inelastic
.04 79,802,688 79,262,624 81,381,347 76,537,856
.08 -387,188,736 -401,955,840 -426,716,672 -434,364,,416
212
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TABLE 9-8. NET PRODUCER AND CONSUMER SURPLUS ESTIMATES
UNDER ALTERNATIVE ASSUMPTIONS REGARDING THE DEMAND ELASTICITY
OF COTTON
Elasticity Ranges
Concentration High USDA Low Inelastic
.04
.08
253,
-482,
373,
269,
824
952
253,
-540,
104
368
,528
,389
253
-550
,077,
,005,
440
504
251
-601
,437,056
,348,957
TABLE 9-9. NET PRODUCER AND CONSUMER SURPLUS ESTIMATES
UNDER ALTERNATIVE ASSUMPTIONS REGARDING THE DEMAND ELASTICITY
OF PEANUTS
Elasticity Ranges
Concentration High USDA Low Inelastic
.04 62,531,984 60,723,424 58,921,760 31,847.424
.08 -122,173,936 -127,927,056 -137.475,728 -204,983,040
213
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sorghum, and barley. For each of these seven crops we have modified the RMF by
replacing the Box-Tidwell dose-response equation with the Wybul equations found
in Heck et al. (1984a, 1984b) and adding production cost information for
sorghum and barley.
The Wybul functional form may be written
Y = a exp[-(x/b)°]
where: Y = a measure of yield
x = ozone
a, b, c parameters to be estimated
The results presented in this section are based on nine Wybul dose-response
equations given in Table 9-10.
The standard set of maintained assumptions (see Chapter 8, Section 8.2)
are employed in the model runs described below. We have arbitrarily set the
demand elasticities for sorghum and barley equal to -.5 due to the lack of
alternative estimates. Given the sensitivity results of Section 9.4 we believe
such assumed values will not greatly distort our welfare estimates. For each
of the seven crops and the three distinct regions for soybean production we
have calculated welfare estimates using the EPA supplied scenarios displayed in
Table 9-11. These scenarios are identical to those used in Chapter 8.
The welfare calculations made from the RMF using the NCLAN Wybul dose-
response equations are reported in Tables 9-12 through 9-20. To provide a
comparison of the Wybul and Box-Tidwell results we have calculated the ratio of
the welfare estimates made using the Box-Tidwell dose-response equations to
214
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TABLE 9-10: NCLAN DOSE-RESPONSE EQUATIONS BASED ON THE WYBUL
FUNCTIONAL SEPCIFICATION
Species
Cultivar
Date, Location
Estimated
Parameters
Barley
'Poco1
1982 - Shafter, Calif.
a
b
c
1 .988
0.205
4.278
lean, Kidney
'Calif. Light Red'
(Full Plots - FP)
1982 - Ithaca, NY
a
b
c
2878.
0.120
1 .171
Corn
'PAG 397', 1981
•- Argonne, 111.
a
b
c
13953.
0.160
4.280
Cotton
'Acala SJ-2'
1981 - Shafter, Calif,
(Irrigated - I)
a = 5546.
b = 0.199
c = 1.288
Peanut
'NC-61
1980 - Raleigh, NC
a
b
c
7485.
0.111
2.249
Sorghum
•DeKalb - 28'
1982 - Argonne, 111.
a
b
c
8137-
0.296
2.217
215
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Table 9-10 continued
Species
Cultivar Estimated
Date, Location Parameters
Soybean
'Corsoy' a = 2785.
1980 - Argonne, 111. b = 0.133
c - 1.952
'Williams' a = 4992.
1981 - Beltsville, Md. b = 0.211
c = 1.100
'Hodgson1 a = 2590.
1981 - Ithaca, NY b = 0.138
(Full Plots - FP) c = 1 .000
Tomato
•Murrieta' a - 32.9
1981 - Tracy, Calif. b - 0.142
c - 3.807
Wheat, Winter
•Abe', 1982 a = 5363-
- Argonne, 111. b = 0.143
c = 2.423
216
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TABLE 9-11. EPA/OAQPS OZONE CONCENTRATION SCENARIOS
Scenario No.
1
2
3
4
5
6
7
8
9
10
Concentration in ppra
.01
.02
.03
.04
.05
.06
.07
.03
.09
.10
217
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TABLE 9-12. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE NORTHEAST REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/ loss
15,657,280
12,273,369
8,735,396
5,024,708
1,121,977
-3,399,020
-8,390,933
-13,900,266
-19,774,416
-25,757,584
218
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TABLE 9-13. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE SOUTHWEST REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/loss
498,4^3,776
423,408,640
339,527,680
244,486,720
141., 413, 632
43,799,744
-135,775,952
-249,254,592
-356,041,728
-458,332 928
219
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TABLE 9-14. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SOYBEAN PRODUCTION IN THE CENTRAL STATES REGION OF NCLAN:
ESTIMATES IN 1978 DOLLARS BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/ loss
432,759,040
390,963,456
307,099,648
210,175,344
-45,044,720
-158,656,384
-391,853,568
-672,811,776
-1 ,007,974,400
-1,401,567,230
220
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TABLE 9-15. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR CORN PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/ loss
170,826,736
168,713,408
158,458,672
126,192,016
53,624,192
-98,725,904
-314,127,872
-593,981,184
-1,021 ,463,810
-1,693,116,160
221
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TABLE 9-16. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR WHEAT PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Net welfare
Concentration gain/loss
.01 395,600,640
.02 368,051,456
.03 308,114,944
.04 201,541,696
.05 -78,489,184
.06 -317,720,832
.07 -586,202,368
.08 -927,098,368
.09 -1,380,957,700
.10 -1,954,862,080
222
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TABLE 9-17. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR COTTON PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Net welfare
Concentration gain/loss
.01 634,127,104
.02 529,935,360
.03 410,888,704
.04 274 312 960
.05 72,711,248
.06 -83,150,640
.07 -321,918,464
.08 -599,386,624
.09 -926,649,344
.10 -1,304,902,660
223
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TABLE 9-18. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR PEANUT PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/ loss
82,988,832
77,972,496
68,426,672
52,822,080
22,302,848
-35,117,504
-77,737,584
-127,325,984
-184,010,272
-249,357,152
224
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TABLE 9-19. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR SORGHUM PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
Net welfart
gain/loss
58,697,168
53,528,944
43,798,384
28,930,272
1,659,738
-23,320,848
-52,791 ,424
-81 ,747,600
-110,017,696
-141 ,185,088
225
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TABLE 9-20. WELFARE ESTIMATES UNDER EPA/OAQPS OZONE SCENARIOS
FOR BARLEY PRODUCTION IN THE U.S.: ESTIMATES IN 1978 DOLLARS
BASED ON NCLAN WYBUL DOSE-RESPONSE EQUATIONS
Concentration
.01
.02
• 03
.04
.05
.06
.07
.08
.09
.10
Net welfare
gain/loss
1,968,748
1,924,958
1,707,037
792,822
-396,877
-3,178,408
-7,960,204
-15,519,651
-25,280,720
-35,859,824
226
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TABLE 9-21. RATIO OF WELFARE ESTIMATES CALCULATED
USING BOX-TIDWELL DOSE-RESPONSE FUNCTION TO NCLAN WYBUL FUNCTIONS
Ozone concentration
Crop
Soybeans
Corn
Wheat
Cotton
Peanuts
All*
.03 ppm
1.27
0.79
0.54
0.95
1 .21
1 .00
.08 ppra
1.57
0.75
0.43
0.90
1 .00
0.94
*This ratio is calculated by aggregating across the welfare estimates and
then computing the ratios; it is not a simple average of the ratios in the
table.
227
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those estimates using the NCLAN Wybul equations for two ozone concentrations
.03 ppm and .08 ppm. These ratios are displayed in Table 9-21.
An examination of these ratios, crop by crop, reveals a substantial
difference in the welfare estimates. For example, in the case of soybeans the
Box-Tidwell equation leads to welfare estimates of gains and losses in excess
of 27% and 57% respectively over the Wybul equations. On the other hand,
calculations made for the wheat crop show that the Tidwell form leads to gain
and loss estimates much smaller than the Wybul form. However, if one
aggregates across all crops, the resulting national welfare estimate is
remarkably similar.
Unfortunately, the above analysis is not sufficient to discriminate
between the Box-Tidwell and Wybul forms for the Regulatory Impact Analysis.
While the differences in welfare estimates are disturbingly large, one cannot
attribute the differences to functional form alone. Recall from Chapter 6 the
Box-Tidwell equations were estimated from published, aggregated NCLAN
experimental results, while the Wybul functions were estimated by NCLAN
researchers from the unpublished, disaggrregate experimental results. Given
this disparity in data sets, conclusions as to the correct functional form, or
statements regarding the differences in welfare estimates due to alternative
forms, cannot be made on the basis of the above results.
If one were to proceed directly to a Regulatory Impact Analysis without
the ability to research the functional form issue further it would be our
recommendation that the NCLAN Wybul dose-response functions be employed, solely
upon the criterion that they were estimated from the original disaggregate
data.
228
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9.6. CONCLUDING REMARKS
This chapter has reported the results on four sensitivity studies dealing
with differential productivity effects, the varietal choice problem, estimates
of crop demand elasticity, and the choice of dose-response equation function
specification studies. Of the four studies the choice of dose-response
functional specification leads to the greatest sensitivity in welfare
estimates. The problem of harvest/nonharvest differential productivity is
substantive in the sense that a failure to recognize the distinction seriously
distorts the perceived welfare impacts, but since we feel that a model of
differentiable productivity is the only defendable approach we believe little
concern should be directed toward this .problem. The impact of alternative
elasticity estimates on the welfare calculations is minor and probably not
worth pursuing further-
While there are many other issues one could have pursued in an expanded
sensitivity analysis, the four issues cited above seem the most important to
examine with a limited budget. If we were to expand the effort we would
concentrate on those aspects of the study concerned directly with the dose-
response functions. Our experience has led us to believe that minor variations
in these functions can have marked impacts on welfare estimates; and unfortu-
nately, these functions are the weakest link in the sequence of analysis that
has led to'the welfare estimates of this chapter and of Chapter 8.
229
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CHAPTER 10
CONCLUSIONS AND AGENDA FOR FUTURE RESEARCH
10.1. INTRODUCTION
We have organized this discussion of future research around two topics:
1) further analysis of ozone's impact using biologically determined dose-
response functions and microtheoretic economic assessment models, and 2)
further analysis of ozone using statistically determined dose-response
relations and microtheoretic assessment models. We are led to believe that
the second topic is important to consider in future air pollution studies
since it addresses the problems we have encountered in using the biological
evidence amassed by NCLAN, and the difficulty of using yield experiments to
learn about production activities which may be nonneutrally impacted by
pollutants other than ozone.
For the purposes of the eventual RIA for ozone the hypothesized
neutrality of ozone on agricultural production activities justifies the use
of biologically driven economic assessment models. The biologically driven
Regional Model Farm assessment model discussed in this report provides for
broad crop coverage and significant regional disaggregation in a sound
microtheoretic structure and hence possesses the qualities necessary to
provide benefit estimates to an RIA.
230
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10.2. FURTHER ANALYSIS OF OZONE USING BIOLOGICAL DOSE-RESPONSE FUNCTIONS
This report has described an economic assessment model capable of esti-
mating the welfare gains or losses emanating from the agricultural production
sector in response to changes in rural ozone concentrations. The assessment
model is comprised of four major components which may be improved to lead to
more reliable welfare estimates. These components are: 1) the biological
information contained in the dose-response functions, 2) the air quality data
supplied by EPA for both baseline and alternative exposure scenarios, 3) the
economic information on agricultural cost and production contained in the
RMF, and 4) crop specific demand functions. In the paragraphs below we shall
discuss some areas of future research which could lead to improved components
of the assessment model without changing the basic structure of the
assessment framework.
10.2.1. Improvements in Biological Dose-Response Functions
Improved biological dose-response functions will require a greater
emphasis on the selection of crops, the selection of particular varieties and
hybrids, and the specification of dose-response relationship functional form.
Certainly, the development of full dose-response surfaces would also lead to
greatly improved functions. However, such surfaces may take more time and be
more costly to develop. Therefore, we confine our remarks to the three areas
noted above.
To appropriately assess the economic impact of a change in ozone
concentrations one must be able to model the reactions of agricultural
producers to their awareness of decreased or increased yields. The ozone
neutrality property referred to in Chapter 5 only holds for input demands and
suggests that agricultural managers will not adjust the mix of their inputs
231
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in response to changes in ozone. However, ozone neutrality does not extend
to output mix considerations of farm managers. In particular, if ozone
differentially affects corn and wheat, then in areas of the country where
both crops are feasible production choices farmers will adjust the mix of
such crops in response to ozone. This output mix nonneutrality suggests that
the appropriate methodology for choosing crops to study would be to choose
crops which comprise feasible output choices in given areas. The failure to
do so prevents the economic modeling of the output choice and therefore leads
to an understatement of benefits and an overstatement of losses associated
with changes in ozone concentrations.
In addition to the problem of crop coverage, the companion problem of
variety choice within a single crop type must also be addressed. Again we
recommend a choice methodology which will provide the basis for economic
assessments. Research should not be focussed on varieties which are believed
to be ozone sensitive. Rather it should examine those feasible varieties
within the choice set of agricultural producers. The rationale for such a
methodology again rests on the ability of farm managers to choose varieties
in response to yield changes. If one excludes the possibility of varietal
switching (averting behavior) one will, ceteris paribus, always understate
benefits and overstate losses.
The correct functional specification of the dose-response relationship
is vital to biologically driven assessments models since it in large measure
determines the magnitude of supply function shifts. In this report RFF
proposed a specific functional form (Box-Tidwell) only because it was
impossible using the ,aggregate summary NCLAN data to undertake rigorous
statistical tests of functional specification. Without strong a priori
232
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theoretical justification for a particular specification such statistical
analysis seems the most prudent path to pursue when one is choosing
alternative specification for the RIA benefits analysis.
10.2.2. Air Quality Data
Rural ozone concentrations are required by the assessment model for two
different purposes. First, concentrations determine the relative position of
current crop yields on the biological dose-response functions. Any deviation
between the actual concentration and the concentration supplied to the model
will falsely position the baseline yield. If the dose-response function were
linear this false positioning would not affect the welfare estimates since
the change in yield relative to the change in ozone is constant at all ozone
concentrations. However, the dose-response functions are for the most part
decidedly nonlinear, and thus false positioning can over or understate yield
changes given a change in concentrations.
The relative difference in ozone concentrations across areas of the
country is important in modeling the range of regulatory alternatives to the
current standard. For example, if a concentration of .06 ppm mean 7 hour
growing season concentration recorded in Iowa is consistent with the .12 ppm
hourly, one expected exceedence per year, standard, then regulatory scenarios
which tighten the standard to say .10 ppm hourly, one expected exceedence per
year, would lead to reductions in Iowa concentrations of .05 ppm, mean 7 hour
growing season concentration which the model would reflect in increased
yields and positive welfare benefits. However, if there exist errors in the
ozone data such that a baseline rural ambient value of .07 ppm was passed to
the model, then benefits larger than actual would be reported by the model.
233
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Similarly, a false baseline ambient value of .04 pprn would lead to no
benefits at all.
Unfortunately, few ozone monitors exist in rural areas and as a
consequence county level ozone concentrations used in the assessment model
described in this report are interpolated values based primarily on metro-
politan monitors. It is believed that in the future these interpolated data
will be supplanted with concentrations derived from a more detailed air
model. However, the data will still represent extensions of urban air
modeling.
Given the strong biological evidence supporting the hypothesis that
ozone seriously reduces important crop (grains) yields it seems only natural
to begin monitoring ozone concentrations in crop growing areas. Even a
handful of monitoring sites in the Great Basin, would increase the relia-
bility of interpolated or model generated air quality data.
10.2.3. The Economic Modeling Component
Under the ozone neutrality assumption implicit in the NCLAN experiments
and maintained in the structure of the assessment model described in this
report, there exists only one area in which refinement of the RMF would lead
to more reliable benefit estimates. This area of research concerns the
output choices of agricultural managers in response to changing relative crop
yields brought about by changes in ambient ozone concentrations. See Kopp et
al. (1984) for a discussion of such a model.
10.2.4. Crop Demand Functions
The final area for improved economic assessment modeling using
biological dose-response functions concerns the estimates of consumer demand.
234
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In the present study we have employed USDA crop specific demand elasticities
in conjunction with the assumption of linear demand functions to determine
probable equilibrium prices and quantities. In preparation for the RIA one
would want to investigate the possibility of using region specific demand
equations rather than the national estimates employed in this study.
Moreover, to the extent possible demand equations which possess cross-price
responses are again more desirable than those employed in the current study.
10.3. NON DOSE-RESPONSE FUNCTION APPROACHES TO THE AGRICULTURAL IMPACTS OF
OZONE
In section 2 of this chapter we have discussed improvements which might
be made to the assessment methodology described in this report. In this
section we briefly discuss an alternative methodology for assessing the
economic impact of air pollutants on agriculture and society which does not
employ biologically based dose-response functions. Rather, the methodology
we shall discuss employs a statistical dose-response function estimated
jointly with the agricultural supply function within the context of a
microtheoretic econometric economic assessment model (see Chapter 3 for
details).
The statistically identified dose-response relationship has two impor-
tant advantages over experimentally derived relationships. First, the
statistical relations do not assume ozone neutrality but leave the assumption
as a hypothesis which may be subjected to rigorous statistical test. Second,
the statistical functions incorporate the reactions of farm managers to
changing crop yields; reactions which may manifest themselves in varietal
switches, crop mix changes and changes in input composition. If the
geographic area over which the statistical relations are estimated is
235
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sufficiently small or explanatory variables such as soil characteristics,
weather patterns and the like added to the model, the statistical functions
become more characteristic of specific areas than experimental functions
which must often be applied far from the original experimental site.
Techniques for implementing this methodology and the benefits to be
gained are described fully in Chapter 3 along with the methodology's draw-
backs. The two greatest stumbling blocks are informational requirements.
The first requirement is a set of detailed U.S. production and cost informa-
tion at a fine level of regional disaggregation. While many researchers such
as Crocker et^ al^ (1981) have been unsuccessful in developing such a national
data set, researchers at RFF have assembled and are employing such a data set
at this time in the analysis of acid rain impacts. Thus, the extraordinarily
detailed economic information required by the nonbiological statistical
approach is readily available for a significant set of crops.
The second piece of information is reliable estimates of rural ozone
concentrations. As stated above, the current source of such information is
an interpolated data set for 1978. In the past months this data set has been
revised and improved and now a second data series for 1980 exists. Further-
more, advances continue to be made in the development of air models for ozone
which will also be able to provide rural estimates of ozone concentrations.
While neither of these two approaches can be as reliable as actual monitoring
information we believe it is prudent to develop a statistical dose-response
econometric model based on such data for comparison with the biological dose-
response model described in this report. Such an approach will enable us to
better bound the benefit estimates.
236
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-45Q/5-84-Q03
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
Agricultural Sector Benefits Analysis for Ozone
Methods Evaluation and Demonstration
i. REPORT DATE (Data nf
June 15. 1984'™' gt1fln)
6. PERFORMING ORGANIZATION CODE'
7. AUTHOR(S)
Raymond J. Kopp, William J. Vaughan, and Michael
Hazilla
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Resources for the Future
1755 Massachusetts Ave., N.W.
Washington, DC 20036
10. PROGRAM ELEMENT NO.
12A2A
11. CONTRACT/GRANT NO.
68-02-3583
12. SPONSORING AGENCY NAME AND ADPRESS
u.S. Environmental Protection Agency
Office of Air Quality Planning and Standards (MD-12)
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final Report
en
INC
14. SPONSORING AGENCY CODE
OAQPS
15. SUPPLEMENTARY NOTES
Project Officer: Thomas 6. Walton
16. ABSTRACT
Th'is report describes the development of an applied model capable of using
exogenously supplied agricultural sector dose response information, agricultural
cost of production data, and air quality information to estimate changes in
producer and consumer welfare due to changes in ozone exposures for agriculture.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI I'ield/Group
Benefit Analysis
Air Pollution, 03
Agricultural Economic Models
18. DISTRIBUTION STATEMENT
Release Unlimited
19. SECURITY CLASS (ThisReport)
20. SECURITY CLASS f This page)
21. NO. OF PAGES
_2J5J
22. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION is OBSOLETE
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