A mathematical programming model of the U.S. beef sector

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Title:
A mathematical programming model of the U.S. beef sector
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vi, 251 leaves : ill. ; 28 cm.
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English
Creator:
Peters, Mark A., 1956-
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Subjects / Keywords:
Food and Resource Economics thesis Ph. D
Dissertations, Academic -- Food and Resource Economics -- UF
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Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 246-250).
Statement of Responsibility:
by Mark A. Peters.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text











A MATHEMATICAL PROGRAMMING MODEL
OF THE U.S. BEEF SECTOR


















By

MARK A. PETERS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990














ACKNOWLEDGMENTS


It has been a very rewarding learning experience to

work with Dr. Thomas H. Spreen, chairman of my dissertation

committee. Dr. Spreen provided many helpful insights. I

appreciate his efforts in making the completion of this

dissertation possible and in finding me employment.

I would also like to thank Dr. John S. Shonkwiler, Dr.

William G. Boggess, Dr. Timothy G. Taylor, and Dr. Douglas

G. Waldo for serving on my dissertation committee. My

appreciation is also extended to Dr. Rodney R. Martin at

Auburn University and Dr. Kenneth E. Nelson at the Economic

Research Service for providing me with much needed

information.

Financially speaking, I am grateful to the Food and

Resource Economics Department of the University of Florida

for funding much of my stay here as well as to the USDA

which provided me with a fellowship for three years.

Emotionally speaking, I am indebted to the friends I

have made in Gainesville and to my family who have supported

me during this endeavor. I am especially indebted to May

Mercado who gave me her unconditional love.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . ii

ABSTRACT . . .

CHAPTERS

1. CHANGES OCCURRING IN THE STRUCTURE OF THE
U.S. BEEF INDUSTRY .. .. 1

Objectives . 11
Overview of Study . .. 12

2. MATHEMATICAL PROGRAMMING AND SECTOR LEVEL
ANALYSIS .......... .. .. 13

Incorporating Demand Systems into
Mathematical Programming Models 15
The Integrability Problem 18

3. A MODEL OF THE U.S. BEEF SECTOR 27

Organization of Beef Sector 27
Supply Response . .. 46
The Model . ... 47

4. SPECIFICATION OF AN INVERSE DEMAND SYSTEM
FOR FRESH MEATS . .. 76

Criteria for Selecting a Demand System
to be Used in the Programming
Model . 76
The Two-Stage Budgeting Process and the
Representation of Consumer
Preferences . ... .80
The Derivation of the Inverse Almost
Ideal Demand System . 88
The Data . .90
Estimation Procedures .. 97
The Results . .. 98


iii










Adjustments Made to Demand Equations

5. DESCRIPTION OF ACTIVITY ANALYSIS MODEL OF
BEEF PRODUCTION SYSTEM .

6. BASE RESULTS AND SIMULATIONS .

The Base Run ... .
Scenarios .. .. .

7. SUMMARY AND CONCLUSIONS . .


APPENDICES


A. TRANSFORMATIONS OF EXPENDITURE AND PRICE
DATA ................

B. TRANSFORMATIONS TO COST OF PRODUCTION
DATA . .

C. GAMS PROGRAM ..............

D. THE EMPIRICAL COMPENSATED DEMAND SYSTEM


REFERENCES . .. .

BIOGRAPHICAL SKETCH . .


S 210


S 213

S 223

S 244

S 246

S 251


117


123

148

148
157

198














Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


A MATHEMATICAL PROGRAMMING MODEL OF THE U.S.
BEEF SECTOR

By

Mark A. Peters

December, 1990

Chairman: Thomas H. Spreen
Major Department: Food and Resource Economics


The main purpose of this study was to build an econom-

ic model of the U.S. beef sector for use in policy analysis.

The model developed is a price endogenous spatial equilib-

rium model. It integrates a system of demand equations for

beef with an activity analysis model of the U.S. beef pro-

duction system. The representation of the beef production

system partitions the U.S. into five supply regions and four

production stages. The representation of demand for beef

partitions the U.S. into four demand regions.

The use of this methodology has been limited by

restrictions placed on model formulation by integrability

conditions. A new approach was used to solve the integra-

bility problem. In this approach a compensated demand








system is used in place of the more generally used uncom-

pensated demand system.

An inverse Almost Ideal Demand System (AIDS) contain-

ing eight meat commodities was derived and estimated for

each of the four demand regions. The demand systems were

estimated by pooling the data on household expenditures on

meat obtained from the Consumer Expenditure Survey, 1982-

1986.

The model was used to analyze the impact of an in-

crease in poultry consumption, an increase in beef imports

and an increase on beef imports on participants in the beef

sector. Increased levels of poultry consumption and

increased levels of beef exports caused a decline in the

quantity of beef produced and a decline in the size of fin-

ished cattle. Increased exports of beef caused both the

quantity of beef produced and the size of finished cattle to

increase.

The impact of these adjustments on participants in the

sector depended on the location of the supply region and the

stage of production. The quantity of beef produced declined

significantly in the Southeast due to an increase in poultry

consumption or beef imports. The decline in the size of

finished cattle caused the quantity of cattle finished to

decline in the Southwest and to increase in the Plains. The

increase in cattle size due to an increase in exports of

beef caused the quantity of cattle finished in the Southwest

to increase and to decline in the Plains.















CHAPTER 1
CHANGES OCCURRING IN THE STRUCTURE OF THE U.S.
BEEF INDUSTRY


The beef industry is one of the most important com-

ponents of the U.S. agricultural sector. In 1985 consumer

expenditures on beef totaled $45.6 billion (USDA, 1988)

while on farm cash receipts for beef totaled $29.1 billion.

This amount represented 20% of total on-farm cash receipts

in the entire agricultural sector for that year (USDA,

1988). In addition, the industry is the major consumer of

grains, and as a consequence, changes in the beef industry

which affect the supply and demand for beef reverberate

throughout the agricultural sector.

The beef industry is in a state of flux with major

changes occurring in its structure in both supply and

demand. On the supply side the geographic location of

production operations is moving westward and sizes of

operations are increasing. Specialization in production is

intensifying as large commercial feedlots and single species

meatpacking plants increase in numbers. Vertical integra-

tion is also increasing as meatpacking firms continue to

integrate the functions carried out by independent whole-

salers into their operations. On the demand side, there has

been a dramatic shift in the pattern of meat consumption.

1










Poultry's share of household expenditures on meat has

increased, while beef's share of meat expenditures has

declined. The effect of this shift in the pattern of meat

consumption is reflected in the decline in the level of per

capital consumption of beef throughout the eighties.

The changes presently occurring in the structure of

the supply side of the beef industry are largely attributed.

to two events: the advent of irrigation in the high plains

states and the introduction of boxed beef in the meatpacking

industry in the mid-sixties. The introduction of irrigation

in the high plains has been responsible for the geographic

shift in the location of feeding and slaughtering operations

while boxed beef is responsible for the increasing size of

meatpacking plants and increased vertical integration in the

processing and distribution of fresh beef.

Historically, cattle feeding has been located in areas

of highly concentrated feed grain production known as the

cornbelt, but starting in 1960 a gradual shift occurred in

its location as cattle feeding moved away from the cornbelt

to the western cornbelt and high plains states of Texas,

Nebraska, Kansas, Iowa and Colorado. This shift is primar-

ily a result of the greater availability of feed grain

supplies in the plains states and economies of scale result-

ing from large commercial feedlots (McCoy and Sarhan, 1988)

made possible by the introduction of irrigation and the low

opportunity cost for land compared to crop production on











nonirrigated lands in this region. Both the greater avail-

ability of feed grains and economies of scale led to lower

average costs of production, giving these states a compara-

tive advantage in cattle feeding over the central cornbelt

states.

The movement of the meatpacking industry has followed

that observed for feedlots. Most cattle are slaughtered

within 100 miles of where they are finished. This is not

surprising given the nature of meatpacking which is essen-

tially a process of reducing the size of the primary input.

Thus, the closer the meatpacking facility is located to

where the live animal is produced, the less weight that

needs to be shipped long distances. When live animals were

produced by small atomized producers widely dispersed geo-

graphically, the high cost of assembly and transporting the

animals to slaughter made it economically infeasible to

locate meatpacking facilities at any other location than at

the terminal points of transportation routes. However, with

the development of large scale feedlots assembly costs were

greatly reduced. Thus, the meatpacking facilities were

located closer to the areas where the large feedlots were

located.

The introduction of boxed beef in 1969 is the most

significant technological innovation to occur in the meat-

packing industry during the past thirty years. Both the

increase in size of meatpacking plants and the increase in









4
level of concentration in the meatpacking industry have been

attributed to the introduction of boxed beef and the econ-

omies of scale associated with this technology. It has also

transformed the distribution sector since it eliminated the

need for specialty wholesalers to service large retail out-

lets such as grocery chains (Duewer, 1984).

Boxed beef is the end product of a process by which

the carcass of the slaughtered cow is fabricated (cut) into

primals, subprimals or both, vacuum wrapped and shipped to

wholesalers or retailers. Primal cuts consist of the major

divisions of the carcass such as rounds, loins, and chucks,

while subprimals cuts include the smaller cuts obtained from

these divisions, such as chuck steak, and chuck roasts.

Since its introduction the amount of beef marketed as

boxed beef has increased steadily and by 1983 accounted for

nearly 90% of fresh beef marketing (Duewer, 1984). There

are large economies of scale associated with boxed beef.

Before the introduction of boxed beef the efficient size

packing plant had a capacity of 250,000 head while after its

introduction the efficient size plant had a capacity of

1,000,000 head (Marion). In addition to economies of size,

boxed beef was widely adopted by the meatpacking industry

because it permitted less fat and bone to be shipped,

allowed the buyer to order specific cuts, reduced shrinkage

during shipping, increased the shelf life of the product,

and required less space in shipping (Nelson, 1987a).










As important as the changes occurring in the

production of beef are, the most important change is

occurring on the demand side where a dramatic shift has

occurred in the pattern of meat consumption highlighted by

poultry's supplanting of beef as the major meat product

consumed (Table 1.1). For thirty years prior to 1977, per

capital levels of beef consumption had increased steadily at

approximately the same rate as the rate of growth in

personal income, reaching a high of 94.0 lbs. in 1976. In

1977, the level of beef consumption dropped to 91.0 Ibs.

marking the beginning of a period of sharp decline in per

capital levels of beef consumption. The decline continued

until 1980 when beef consumption dropped to a level of

76.4 lbs. per person. From 1980-1986 per capital beef

consumption leveled off, but began to decline again in 1987

when it dropped to 73.4 Ibs. During the 1977-1987 time

period the level of beef consumption declined by 18.0 lbs.

per person. In contrast, the per capital level of

consumption of chicken increased steadily during this same

time period and in 1987 surpassed that of beef. In 1975 the

level of poultry consumption stood at 48 lbs. per person and

climbed to a level of 77 lbs. per person in 1987 (Table

1.1). This represents an increase in the level of poultry

consumption of 29 lbs. per person.

In addition to the change which has occurred in the

consumption pattern among the major meat species, a change












TABLE 1.1


COMPARISON OF PER CAPITAL CONSUMPTION OF BEEF AND
POULTRY TO THE RELATIVE PRICE OF POULTRY TO
BEEF. 1975-87.


Relative Price
Consumption Consumption of Poultry in
Year of Beef of Poultry Terms of Beef


lbs. $

1975 88.0 48.3 .955
1976 94.2 51.6 .947
1977 91.4 52.9 .958
1978 87.2 55.5 .860
1979 78.0 60.1 .708
1980 76.4 60.3 .706
1981 77.1 62.0 .729
1982 76.8 63.4 .706
1983 78.2 64.7 .725
1984 78.1 66.5 .793
1985 78.8 69.7 .791
1986 78.4 72.0
1987 73.4 77.8


Source: USDA, 1987.









7
has also occurred in the pattern of consumption found within

the beef category itself. In 1965, steak ranked as the

number one beef cut in terms of quantity consumed with

roasts ranked second and ground beef third. However, by

1984 the picture had changed dramatically as hamburger was

now ranked number one, steak had dropped to number two and

roasts came in third.

Among agricultural economists there has been con-

siderable debate over the major factors contributing to the

changes occurring in the pattern of meat consumption and to

the decline in levels of per capital beef consumption. Some

have attributed these changes to a fundamental shift in the

structure of the demand for meats, due either to increased

health concerns on the part of consumers (Chavas, 1983;

Braschler, 1983; and Buse, 1986) or an increased desire for

convenience in food preparation (Carnes, 1984; Eales and

Unnevehr, 1988; and Duewer, 1984). Others assert that the

changes which have occurred in the pattern of meat consump-

tion are easily explained by changes which have taken place

in such traditional economic variables as the relative price

of beef to poultry, income, and the demographic composition

of the U.S. population (Haidecker et al., 1982; Chalfant and

Alston, 1988; Hager, 1985; Dahlgran, 1987; Heien and Pom-

pelli, 1988).

With regard to the argument that there has been a

shift in consumer preferences due to increased health











concerns and need for greater convenience in food

preparation, two explanations are usually given to explain

how these concerns cause consumers to eat less beef. The

first explanation is that the U.S. population has become

increasingly concerned about the fat content of their diets

due to reports that over consumption of fat leads to heart

disease and other health problems. This has led them to try

to reduce the amount of fat consumed, especially animal

fats. Beef products contain more fat than poultry products.

Consequently consumers desire to consume more poultry and

less beef in order to reduce the amount of fat in their

diet. The second explanation is that the number of two-

income households has increased rapidly during the 1975-1987

time period. As a consequence Americans have less time to

spend on meal preparation and desire greater convenience or

ease of use in food products. The traditional beef

products, roasts in particular, require considerably more

preparation time and come in larger portions than poultry

products. Thus consumers desire more poultry and less beef.

With regard to the argument that it is changes in

economic variables such as prices and income which have

caused the decline in beef consumption and not a shift in

preferences, its supporters point out that the relative

price of beef compared to poultry increased significantly

during the period in which the changes in the consumption

pattern occurred. During the 1975-1985 time period the











price of poultry relative to the price of beef declined by

17% (see Table 1.1). The primary cause of the decline in

the relative price of poultry to beef being increased effi-

ciency in the production of poultry which has not been

matched by beef producers. Thus as beef becomes more expen-

sive relative to poultry consumers demand more poultry and

less beef.

The debate over the cause of declining beef consump-

tion is not just of esoteric interest to econometricians,

but has important implications for the beef industry as

well. Already, the beef industry is putting its energy into

efforts to increase the demand for beef. These efforts

include the generic promotion of beef, development of a new

grading system by the USDA, private labeling, and the devel-

opment of new products such as lean beef. However, if the

major cause of the decline in beef consumption is due to the

increase in the relative price of beef to poultry then the

current efforts by the industry to increase demand will be

futile. In this case beef producers should have greater

success in regaining market share if they concentrate on

reducing the cost of producing beef.

All other things being equal, one of the major impacts

of the decline in beef consumption will be a reduction in

the overall size of the nation's beef herd. It is also

likely that the decline in beef consumption and the con-

sequent reduction in the beef herd will accelerate the










current trends in the beef sector with regard to the shift

in location of the beef herd, the increase in the scale of

operations and the increase in the level of concentration in

meatpacking as higher cost participants are squeezed out.

Thus one question which needs to be answered is to

what extent these trends will continue. If so, who will be

the winners and the losers?

Other areas of concern raised by the current trends

focus on their impact on the efficiency of beef production.

For example, will increased concentration in meatpacking

allow meatpackers to exercise a degree of monopsony power

and reduce returns to cattle producers?

Also, the industry's efforts to increase the demand

for beef through the introduction of new products such as

lean beef and restructured beef cuts has raised many ques-

tions concerning the impact of these new products on the

sector. How successful will a new product need to be in

order to prevent further reductions in the herd size? How

will the type of production systems used to produce the new

beef products affect the structure of the sector? Will the

new production systems change the location of beef produc-

tion in the U.S.? If meatpacking firms integrate backwards

into the production of beef cattle in order to ensure proper

quality of their product, how will this affect cow-calf

producers?









11
Policy makers have expressed a need for an integrated

model of the U.S. livestock sector which would enable them

to assess the impact of the changes in the structure of the

U.S. livestock sector on the different sets of producers in

the sector (Nelson et al., 1988). The changes that are

occurring in the consumption of beef only serve to highlight

the need for the development of a model of the beef sector-

for policy analysis. The purpose of this dissertation is to

develop a model for analyzing the impact of changes in the

pattern of consumption of meat on the beef industry.


Objectives
The main objective of this research is to develop an

integrated model of the U.S livestock sector to be used for

policy analysis. The primary use of the model in this study

will be to determine the long run impact of the decline in

the consumption of beef on the size and location of the beef

herd. To this end the following subobjectives are outlined:

1. Determine the technological coefficients on pro-

duction activities occurring in the beef industry.

2. Estimate a system of regional inverse demand

equations for meats to determine the interdepend-

encies among the various meat products.

3. Develop a price endogenous spatial equilibrium

programming model of the livestock sector which

incorporates a detailed representation of produc-

tion activities occurring on the supply side and a









12

set of inverse demand equations for meats to rep-

resent the demand side.

4. Use GAMS (General Algebraic Modeling System) as a

matrix generator and report writer in order to

facilitate the flexibility of the model and to

promote its continued usage in the future.


Overview of Study

In the second chapter a discussion of the issues

surrounding the use of mathematical programming to conduct

economic analysis at the sectoral level is presented. A

detailed discussion of the problems associated with incor-

porating a system of demand equations into a mathematical

programming model and the representation of supply curves

with an activity analysis model is provided. In Chapter 3

the mathematical formulation of the mathematical programming

model of the U.S. beef sector is developed. Chapter 4 con-

tains the specification and estimation of a system of demand

equations for fresh meats. Chapter 5 contains the empirical

model used for the analysis, and in Chapter 6 the model is

used to analyze the impact of the decline in beef consump-

tion on the beef sector.














CHAPTER 2
MATHEMATICAL PROGRAMMING AND
SECTOR LEVEL ANALYSIS

Mathematical programming models have been used exten-

sively by agricultural economists to model the livestock

industry (Nelson, 1987b). Samuelson (1952) was the first to

demonstrate that the spatial equilibrium problem could be

cast as a constrained maximization problem. Since then many

extensions of the model have been formulated. Takayama and

Judge (1971) demonstrated how a spatial equilibrium problem,

which incorporates linear supply and demand equations, could

be solved as a quadratic programming problem. The applica-

tion of this formulation, however, has been limited by com-

putational difficulties caused when nonlinear demand and

supply equations are introduced. Separable programming

techniques developed by Duloy and Norton (1975) broadened

the scope of problems which could be solved using this type

of analysis by approximating the nonlinear model in such a

way as to allow the simplex algorithm to be used to generate

solutions. They accomplished this by approximating the non-

linear demand and supply equations with linear line seg-

ments. Hazell and Scandizzo (1977) further extended the

applicability of the spatial equilibrium analysis by incorp-

orating risk behavior into the formulation. McCarl and

13









14
Spreen (1980) discussed price equilibrium models which could

be formulated with implicit supply relationships. They have

shown that a sectoral level analysis of the type being con-

sidered here may be effectively conducted using a price

endogenous mathematical programming model. McCarl and

Spreen also provide a good summary of the use of these types

of models by agricultural economists.

The multicommodity price endogenous programming prob-

lem seeks to determine the vectors of prices and quantities

which establish a price equilibrium in the markets of sever-

al related markets. It takes as data the technological

coefficients on production activities, levels of fixed

resources, demand relationships of final products, and sup-

ply relationships for purchased inputs and generates a solu-

tion which gives the equilibrium prices and quantities of

final goods, the usage pattern for the factors of produc-

tion, prices of purchased factors, and imputed prices for

owned resources and production activities. The equilibrium

is partial because such factors as consumer income and the

prices of commodities not endogenous to the system are

treated as exogenous variables.

There are several advantages to using a programming

model over other techniques given the goals of the study.

First, the model's explicit representation of producer

behavior allows each production unit to adjust endogenously

its supply of products and its use of production inputs.










Thus, the model is able to simulate the response of pro-

ducers to changes in the economic environment, making it

possible to identify not only increases or decreases in

supply caused by changes in exogenous variables, but to also

identify the pattern of production activities used. Second,

the model allows for the introduction of new production

activities. Thus, it is possible to simulate the impact of

these activities on the profitability of the activity and

the effects they will have on the rest of the sector.

Third, it does not require knowledge of derived demand and

supply curves at each production level of the sector, but

only input supply and final product demand curves.

In addition, a price endogenous mathematical program-

ming model theoretically allows the introduction of changes

in the demand structure for goods, whether they are due to a

shift in consumer preferences or the introduction of new

products. Thus, it is possible to determine the impact of

the introduction of new technologies, new products, and

changes in the demand structure on the industry.


Incorporating Demand Systems
into Mathematical Programming Models

The Spatial Equilibrium Model

In order to mathematically formulate the price

endogenous programming problem, let













Pi = di(Q1,Q21* IQj I)


denote the inverse demand equation for commodity i, Pi is

the demand price of commodity i, Q1, i=l,---,n is the

quantity demanded of commodity i, and I is consumer income.

Let


pJ = sJ (Q1,Q2,...,QIlZ)



be the inverse supply equation for commodity j, PJ is the

supply price of commodity j, QJ, j = 1,---,n is the quantity

supplied of commodity j and Z is a vector of supply

shifters. The constrained optimization problem can be

written as

n
Max NSB = f f f***' di(QI,Q21,--,Q) dQ1dQ2".'dQ.
i=l
(2.1)

n
-_ f 'f... f-s(Q1,Q2,...,Q") dQ'dQ2-- dQn
j=1


s.t. Qi < Qi i=l,--* ,n (2.2)



Qi, Qi 2 0 (2.3)










The objective function (2.1) maximizes the sum of

areas under each demand function less the sum of the areas

under each supply function. The inequality (2.2) insures

that the quantity demanded is less than or equal to quantity

supplied (no excess demand). Expression (2.3) imposes the

nonnegativity conditions.

Expression (2.1) is a simplification of the integra-

tion that must be performed to properly describe consumer's

plus producer's surplus in a multicommodity framework. Fol-

lowing Hazell and Norton (1986, p. 168), a series of line

integrals are performed in which the first term is




Qi
f Di(Q1, Q2, ,Qn) dtl
0



but all succeeding terms are



Qi
f Di(Q1,Q2,7 I,Qn I Q1=0,Q2=2,*.,Qi-1=0)di
0



and similar expressions are formed when integrating the

supply functions. For further explanation see Hazell and

Norton.

If the demand and supply relationships are linear,


e.g.









18


n
Pi = g, E hi, Qk i=l,--,n
k=l
and
n
p = el + E fik Qk j=l,-",n
k=1



then (2.1) (2.2) can be written as a quadratic programming

problem

n n n
Max E giQi 1/2 : E QiQkhik (2.4)
i=l i=l k=l


n n n
M eJQj 1/2 E : QjQkfjk
j=1 j=1 k=1


s.t. Qi s Qi, i=l,---n (2.5)


Qi,Q' e 0. (2.6)



The Integrability Problem

Two important assumptions are made to ensure that the

solution of (2.1) (2.3) is unique. The first assumption

is that the income generated from the commodities under

study does not affect consumer demand. If the sector under

study is small relative to entire economy, this assumption

should not prove to be restrictive. Otherwise a general

equilibrium framework must be employed. The second

assumption is that the demand and supply functions are

integrable.











Integrability requires that the Jacobian matrix of

both the demand system and the supply system be symmetric.

It also requires that the Jacobian matrix of the demand

system be negative definite and the Jacobian matrix of the

supply system to be positive definite. Symmetry requires

that



adi ad

aQk 8Qi
and
asj 8sk

aQk aQo



This ensures that the optimal solution to the constrained

maximization problem does not depend on the order of

integration. If this requirement is not satisfied there

will be as many optimal solutions as there are possible

orderings for integration.

Of the two integrability conditions the symmetry

requirement for demand systems is believed to be the most

difficult to fulfill. The negativity requirement is

generally believed to be satisfied if the demand equations

are downward sloping and the supply equations are upward

sloping.

In the case of supply functions, symmetry is not a

stringent requirement following Zusman (1989), ". in the










case of supply functions the classical assumptions of the

theory of production, in fact, yield the symmetry con-

ditions" (p. 55). However, in the case of consumer demand

functions, symmetry is a very stringent requirement. The

demand relationship consists of a symmetric substitution

effect plus an income effect. The income effect is not gen-

erally symmetric. Thus, the assumptions of demand theory do

not yield the symmetry conditions.


Approaches to Handling the Integrability Problem

Several approaches have been used to deal with the

integrability problem posed by demand systems with non-

symmetric cross-price effects. An ad hoc approach is to

simplify the demand system so that each demand function

includes only own price and own quantity (Hazell and Norton,

1986, p. 168). In this case, all cross-price effects are

zero and hence the integrability condition is satisfied.

Another approach is to reformulate the problem by

incorporating both price and quantity variables into the

primal form of the model (Plessner and Heady). Thus both

price and quantity equilibrium conditions are imposed in the

primal as opposed to (2.1) (2.3) in which quantity equil-

ibrium conditions are imposed in the primal and price equil-

ibrium conditions are imposed implicitly through the dual.

In the case of linear supply and demand equations, the

primal-dual formulation is












n n n n
Max Z (gi hikQk)Q (ej + E fkQk)QJ
i=1 k=l j=1 k=1


s.t. Qi Q1i 0

Pi(Qi Q') = 0

n
(g, E hikQk) Pi 0
k=1


n
Qi(gi E hkQ, pi) = 0
k=l


n
(e + E fikQk) + Pi 0
k=l


n
Qi(e+ + 2
k=l


i=1,-**,n
i=l,-*-n


i=l,--*n




i=l,*--n


fikQk + Pi) = 0


QQiQpi s 0



The objective function no longer represents the area between

the demand and supply functions but represents net social

monetary gain (Takayama and Judge, 1971).

This problem can be solved using linear complemen-

tarity programming (LCP) (Takayama and Judge, 1971;

Stoecker, 1974). The computer code LINDO (Schrage, 1984)

has an option which uses LCP. To be solved by LINDO, the

demand system must be linear. For many problems, this










approach is theoretically sound and computationally

tractable. For large problems, however, it may pose a

problem of size. For example, a problem with 10 commod-

ities, 1000 other primal variables, 10 market clearing

inequalities and 500 resource constraints would result in an

LCP with 1520 variables and 3,040 constraints.

A third approach to the integrability problem is to

transform the demand system so that the Jacobian matrix is

symmetric. This is accomplished by averaging the cross-

price effects and entering them in the off diagonal posi-

tions. The problem with this solution is that the first

order conditions are altered so that price no longer needs

to equal marginal cost. Consequently, the new optimal

solution no longer satisfies the conditions for a competi-

tive equilibrium.

A fourth approach is to use the compensated (Hicksian)

demand system rather than the uncompensated (Marshallian)

demand system. The Marshallian demand systems do not, in

general, satisfy the integrability conditions because the

assumptions of demand theory do not imply that the system's

Jacobian matrix will be either negative definite or sym-

metric. However, economic theory does imply that the

Jacobian matrix of the compensated demand system will be

negative semi-definite and symmetric. Thus it is

unnecessary to reformulate the problem.










Following Silberberg (1978, pp. 232-40), let Q, =

di"(p,, 21, ,p,m) i=1,-**,n represent the uncompensated

demand system and Qi = dh(pi,p,21..,pi,uo) i=l,--*,n

represent the corresponding compensated system of demand

equations. The Slutsky decomposition of the uncompensated

system can be written


adi"(p,m) adh(p,uo) adi"
S-- Ql" (2.7)
a pj a pj Cm


It shows that the change in the quantity of commodity i

demanded due to a change in the price of commodity j can be

split into two parts: a substitution effect and an income

effect. The substitution matrix is negative definite and

symmetric. In equation (2.7) it is represented by the

cross-price effect of the Hicksian demand system. Thus, the

Jacobian matrix of the compensated demand system is both

negative definite and symmetric,



aQi(p,u) aQj(p,uO)

8pj api


thereby satisfying the integrability conditions of the price

endogenous mathematical programming model.

Is the use of compensated demand functions in price

endogenous mathematical programming models appropriate? The

answer depends on the difference in the equilibrium position









24
arrived at when a system of compensated demand equations is

used instead of the corresponding system of ordinary demand

equations.

As shown in Figure 2.1, equilibrium price and quantity

(point E) generated by the simultaneous solution of the

uncompensated demand equation (DO) and the supply equation

is identical to the price-quantity pair resulting from the

simultaneous solution of the compensated equation (Db) and

the supply equation. But if supply shifts from S to S1, the

equilibrium established by the uncompensated demand equation

is at point A, while the equilibrium suggested by the

compensated demand equation is point B. The difference in

quantity demanded is qh-ql" and the difference in price is

plh -pl"

The difference between quantity demanded and the

difference in price will be determined by three factors:

(1) the magnitude of the movement away from the original

equilibrium, (2) the magnitude of the income elasticity of

the commodity for which the price changed, and (3) the share

of consumer's income spent on the commodity. Peters and

Spreen (1989) examined at the difference in the solutions

found by using the compensated and uncompensated demand

systems. They used the demand system for meat estimated by

Eales and Unnevehr (1988) to simulate the equilibrium estab-

lished by the uncompensated and compensated demand system.

They found that for many agricultural products, such as



















p


q q




Figure 2.1. Change in equilibrium along
compensated and uncompensated demand curves due to
an exogenous shift in supply.










beef, there will be little difference between the two

solutions.

This fourth approach is the one that will be used to

formulate the programming model of the U.S. beef sector. It

has been selected because it permits the integrability con-

ditions to be satisfied without reformulating the problem

as required in complementary programming. This preserves

the economic meaning of the objective function, reduces the

size of problem to be solved, and permits nonlinear demand

systems to be used with little cost with respect to the

accuracy of the solution.















CHAPTER 3
A MODEL OF THE U.S. BEEF SECTOR


The purpose of this chapter is to develop a price

endogenous sectoral level programming model of the U.S. beef

sector. It is important that the model accurately portray

the activities occurring in the sector. In the first

section the production activities which are found in the

sector are described. This has the additional benefit of

providing a foundation for validating the base model. The

material contained in this section has been drawn from four

major sources: Marion (1986), McCoy and Sarhan (1988),

Simpson and Farris (1982), and Nelson (1987a). In the

second section the cattle cycle, sector supply response and

the usefulness of static versus dynamic models of the sector

are discussed. In the final section the mathematical formu-

lation of the model is laid out and described.


Organization of the Beef Sector

The organization of the beef sector is complex and the

task of coordinating production activities in the industry

is difficult. The time frame for producing beef is long.

It takes about 2 1/2 years from the time of breeding to the

slaughter of a mature animal. Also, a relatively large

proportion of cattle and calves change ownership as they

27










move through the stages of production, except at the

distribution stage where processors and retailers have taken

over wholesale activities. Coordination is made even more

complex by the distance between the major areas of beef

production and the main demand centers.

Given the complexity of its organization it is con-

venient to arrange production activities occurring in the

sector into a vertical system (Figure 3.1). There are five

major stages of production in the vertical system: cow-

calf, growing, finishing, slaughtering and processing, and

distribution. The initial products (beef calves) enter the

system at the cow-calf stage and are passed sequentially

through the next four stages of production until they reach

their final form (fresh beef products). At this point they

are sold to consumers and exit the system.


The Cow-Calf Stage

The primary activities occurring at the cow-calf stage

are the maintenance and breeding of the cow herd and the

production of stocker or feeder calves. This includes

feeding, breeding and culling of the cow herd and the

production of calves. The primary inputs required at this

stage of production are land for grazing, breeding stock,

and harvested roughage. The level of investment for

operators is high. Consequently, operations are affected by

changes in land values and interest rates.






















COOROINATIONN
EXCHANGE


Intemal or Market *



Internal or market ..



Market ............




L
Market or Formula -
Price Agreement -


Contracts *


FUNCTIONAL STAGES


TYPICAL
C0ua 6


cull
cows






I]


and


I HRI


Market


I Consumer i


Source: Marion, 1986.


Figure 3.1. Organization of the beef sector.









30
Calves are weaned at the age of six months. They are

then either retained for replacement of culled breeding

stock or to expand the cow-herd, sent to the growing stage,

or sent directly to the finishing stage. The decision to

carry the weaned calf into the growing stage is determined

by the availability of forage. In some operations calves

are placed on feed for a short time after they are weaned

and then sold as vealers.

The decision to retain the weaned calf for the cow

herd is based on the size of herd the cow-calf operator

wants to maintain. If the cow-calf operator feels the cow

herd needs to be increased then more calves need to be

retained than the number required for replacement of culled

breeding stock. Likewise, if the operator wants to decrease

the size of the cow herd, then less calves will be retained

than the amount needed to replace culled breeding stock.


The Growing Stage

In the growing stage weaned calves are placed on

forage and roughage for a period of 6 to 12 months for the

purpose of increasing the development of the body frame.

This stage in the production process is often referred to as

the stocker or backgrounding stage. From the growing stage

stockered cattle are either sent directly to the slaughter

plant for processing as nonfed beef or sent to the finishing

stage to be fattened for slaughter. The most common route

used is from backgrounding to the finishing stage then to










the processing stage to be slaughtered. While the market-

ings of nonfed beef are significant the far greater amount

of cattle are marketed as fed beef. For example, in 1988,

fed beef comprised 78% of the total of beef cattle

slaughtered (USDA, 1989). In addition, most of the nonfed

beef slaughter comes from beef and dairy culls.


The Finishing Stage

At the finishing stage the cattle are confined in a

feedlot and placed on a high concentrate ration. The major

production inputs used at this stage are feeder cattle,

feed, feeding facilities, feed storage facilities, and feed

processing and delivery equipment. Feeder cattle enter the

feedlot from either the cow-calf or stocker stages. The

animals are kept on the feedlot for varying lengths of time

depending on their placement weight and the slaughter cattle

to corn price ratio. The lower the ratio the shorter the

period of time the cattle are retained on the feed lot.

Cattle placed in feedlots immediately after weaning are

fattened to a light slaughter weight (900-1100 lbs.).

Yearlings are either short-fed to a light weight or long-fed

to a heavy slaughter weight (1200-1300 lbs.). Older place-

ment cattle are finished heavy. The usual time period for

finishing is six months. From the finishing stage the

cattle are sent to the meatpacking plant for slaughter.

The three production stages described above are not

totally separate or distinct from each other. A large









32
portion of enterprises found at these levels of the vertical

system have integrated more than one of the production

stages into their operations. However, it is uncommon for

all three stages to be completely integrated under the

umbrella of a single enterprise. The stocker stage is the

least distinct of the production stages as it is often

integrated into either the cow-calf or finishing stages.


The Slaughtering and Processing Stage

The slaughtering and processing stage encompasses all

activities involved in the slaughter of beef cattle and the

cutting of carcasses into smaller units for sale to inde-

pendent wholesalers or retail outlets. Primary inputs used

at this stage are slaughter cattle, facilities, labor, and

containers. Live animals can enter the meat-packing plant

from the cow-calf stage, the stocker stage, the finishing

stages or the dairy herd. They are killed, halved, dressed

and their carcasses chilled. The chilled carcasses are then

either sold or cut up further into boxed beef. The calves

coming from the cow-calf stage or the dairy herd are pro-

cessed as vealers, while cattle coming from the stocker

stage or the culled dairy and beef herd are slaughtered as

nonfed beef.


The Distribution Stage

The distribution system for fresh beef is complex and

contains many components (Figure 3.2). The wholesaler























a;

*4



4
a









*4



04


4.



0
Go
c


















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0



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classification includes both the activities of processors

and retail enterprises which perform the wholesale function

as well as independent wholesale organizations. At present

the marketing of beef products is dominated by processors

that sell directly to retail outlets. It is estimated that

over 80% of total beef production is marketed as boxed beef

(Duewer, 1984).

In the distributing stage fresh beef is moved from the

processing stage to the retail outlets. Retail outlets can

be broken into two groups: retail supermarkets and the

hotel, restaurant, and institutional (HRI) trade. Retail

supermarkets are the most significant outlet for fresh beef

although the importance of the HRI trade is growing. Large

retail supermarkets maintain central warehouse facilities

for handling boxed beef purchased from the meat packing

plant. Some of the supermarket chains also maintain a

central cutting facility where carcasses are fabricated into

boxed beef.

Many of the smaller food stores and HRI outlets

require the services of independent wholesalers. Brokers do

not take ownership of the beef products but execute sales on

a commission basis. Jobbers buy and sell to retail custom-

ers. Purveyors also sell on their own account, but

specialize in providing beef to a special set of clientele,

such as expensive hotels.










Regional Structure of the Industry

Cattle raising' is widely dispersed throughout the

U.S. Some 31 states individually account for at least 1% of

total beef cattle production (Table 3.1). However, cattle

raising is not evenly distributed geographically. Of the 19

states producing less than 1% of the total amount of beef

cattle, 15 are located in the heavily populated Northeast.

On the other hand the top 6 producing states account for 41%

of production. All are located west of the Mississippi

River and east of the Rocky Mountains. In all, 57% of beef

cattle production is located between the Mississippi River

and the Rocky Mountains.

Most beef cattle operations are small, with an average

herd size of 34 head. Seventy percent of calves produced

come from cow herds with less than 200 head (Nelson, 1987a).

Boykin et al. (1980) has identified four production

systems which characterize the type of enterprises involved

in cattle raising: cow-calf-feeder, cow-calf-slaughter,

stocker purchase-slaughter sales, and stocker purchase-

feeder sales. The cow-calf-feeder system includes both the

cow-calf and cow-calf-yearling operations. In the cow-calf

operation calves are sold after weaning, whereas in the

cow-calf-yearling operation, calves are carried over into

the growing stage and then sold to a feedlot for finishing.



'Cattle raising includes both the cow-calf and stocker
stages of production.












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In the cow-calf-slaughter system weaned calves are carried

over into the stocker stage and sold as slaughter calves or

kept in the stocker stage longer and sold as nonfed beef.

In some instances operators using this system will place

stocker cattle into feedlots and then sell them as fed

cattle. In the stocker purchase-slaughter sales system

weaned calves are purchased and placed on small grasses,

field stocks, and other feed sources through the growing

stage, then placed in a feedlot for finishing and then sold

for slaughter. Finally, in the stocker-purchase-feeder

sales system, operators purchase weaned calves and place

them on range or pasture during the growing stage. Feeder

cattle are then sold to feedlots for finishing.

The type of beef production system found in a region

is determined by availability of forage, feed, and alter-

natives to cattle production. Of the four systems described

the most prevalent is the cow-calf-feeder system. The cow-

calf system predominates in the southeast while cow-calf-

stocker operations are common in the great plains states.

In the midwest the stocker purchase-slaughter sale system

predominates.

Given the current conversion rate of feed into gain

for beef cattle the finishing of beef cattle occurs pri-

marily in the regions where feed grains are produced (see

Table 3.2). Finishing operations are more highly concen-

trated geographically than cattle production. In 1988, the














TABLE 3.2.


NUMBER OF CATTLE FEEDLOTS AND FED CATTLE MARKETED BY
SIZE OF FEEDLOT CAPACITY, 13 STATES, 1987.


Cattle Feedlot Capacity (head)

< 1.000 1.000-2,000 2,000-3.999 4,000-7,999
Marketed Marketed Marketed Marketed
State Lots (000 hd) Lots (000 hd) Lots (000 hd) Lots (000 hd)


ARIZ 9 17 0 0 -
CALIF 10 3 5 7 7 15 10 51
COLO 140 45 50 90 55 200 30 265
IDAHO 35 10 15 10 15 30 6 50
ILL 8,750 725 40 60 10 40 0 0
IOWA 9,655 1,215 250 321 95 214 -
KANS 1,627 70 92 71 57 190 34 267
MINN 5,931 432 53 65 16 43 0 0
NEBR 8,950 1,340 175 340 135 570 77 690
OKLA 206 30 4 17 3 8
S DAK 4,146 269 24 67 17 77 13 237
TEX 849 90 9 20 12 35 21 170
WASH 62 6 7 35 -

TOTAL 40,353 4,226 723 1,061 409 1,320 211 1,690


Source: USDA, 1989.
















TABLE 3.2--Extended


Cattle Feedlot Capacity (head) Total
of all
8,000-15.999 16.000-31.999 > 32.000 Feedlots
Marketed Marketed Marketed Marketed
Lots (000 hd) Lots (000 hd) Lots (000 hd) Lots (000 hd)


6 177 5 266 20 4
11 98 11 266 6 365 60 7
16 310 11 425 8 895 310 22
6 100 4 260 81 4
0 0 0 0 8,800 8
0 0 0 0 10,000 17
49 1,014 26 1,065 15 1,353 1,900 40
0 0 0 0 0 0 6,000 5
45 920 13 540 5 500 9,400 49
8 83 4 142 5 410 230 6
0 0 4,200 6
36 625 40 1,385 33 2,930 1,000 52
5 33 6 342 80 4

185 3,285 119 4,347 81 7,042 42,081 229










top 13 states with respect to number of cattle on feed

account for 85% of fed beef marketed and the top 6 states

account for 67% of fed beef marketed (see Table 3.3).

Finishing operations can be grouped into farmer and

commercial classifications. The typical farmer operated

feedlot maintains a one time carrying capacity of less than

1,000 head. The feedlot is often just one of several

enterprises operated on the farm, and cattle are on feed

only part of the year. In contrast, the commercial feedlot

has a one-time carrying capacity greater than 1,000 head are

operated as single enterprises and feed cattle the year

round. Regionally, the majority of the farmer feedlots are

found in the midwestern states of Iowa, Illinois, and

Nebraska, while the commercial feedlots are found in the

southwest and great plains states. While a majority of

feedlots are farmer feedlots the majority of fed slaughter

cattle come from the commercial feedlots. Feedlots with a

capacity of less than 1,000 head accounted for 18% of cattle

marketed in 1987, whereas feedlots with capacity of greater

than 8,000 head accounted for 64% of cattle marketed in that

year (see Table 3.3).

The location of slaughter and processing plants mir-

rors the location of feedlots (see Table 3.4). Most cattle

are slaughtered less than 100 miles from where they were

fed.














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Due to the substantial economies of scale associated

with boxed beef two types of enterprises have evolved. The

efficient size of slaughter-processing plant which produces

boxed beef as an output is believed to be between 500,000 to

1 million head annual capacity (Marion, 1986). One type is

the large slaughter-processor with plant capacity often

exceeding 500,000 head who specialize in fed beef and sell

their output as boxed beef. The other type of enterprise

operates a smaller capacity plant and caters to smaller,

specialized customers. The number of firms at this stage is

small. Concentration measures indicate that the top four

firms involved in slaughter and processing of beef account

for 82% of fed beef slaughtered (Marion, 1986).


Dairy Herd

The culls from the dairy herd are a significant source

of beef, typically accounting for 20% of total cattle

slaughter. As a consequence, the amount of dairy culls

slaughtered do have an important effect on price received by

producers in the beef sector. However, the quantity of

culls coming from the dairy sector is not determined by con-

ditions found in the beef system, but by conditions prevail-

ing in the dairy sector. Thus the supply of dairy culls is

exogenous to the beef system. The slaughter of cull cows is

more dispersed than the slaughter of fed beef reflecting the

distribution of cow-calf operations and the dairy herd.










Foreign Trade

Like the dairy herd, foreign trade is significant but

does not play a major role in the decisions made by beef

producers. Thus, level of imports and exports can be treat-

ed as exogenous to the sector. It must noted here, the role

of foreign trade is increasing and that domestic beef

producers hope that increase in beef exports to foreign mar-

kets will offset the decline in beef consumption in the U.S.


Supply Response
Given the long lag in time from when the decision to

produce a calf is made to the time that the live animal is

slaughtered, the decisions made at the cow-calf stage ultim-

ately determine the supply of beef at any point in time.

Thus, the sectors supply response of the sector is deter-

mined by the ability of cow-calf operators to increase or

decrease the number of feeder calves produced.

Cow-calf operators cannot increase production rapidly

in response to favorable market conditions due to two

related factors. First is the length of time it takes to

produce new breeding stock. It takes two years to produce a

heifer for breeding purposes. Second, there are no alter-

native uses for breeding stock other than for the production

of calves. As a consequence, breeding stock in excess of

the amount needed to meet calf production needs are not kept

but sold for slaughter.









47
The slowness of the supply response is believed to be

the principal cause of the cattle cycle. The cattle cycle

has been observed to last 10 years starting with a 3 to 4

year period of contraction in the number of beef cattle

produced followed by a 6 to 7 year period of expansion.

The discussion on the sector's supply response raises

the issue of time and the appropriateness of casting the

model of the beef sector in a static or dynamic framework.

A resolution to the time problem depends on the type of

information one is interested in gaining. If one is inter-

ested in forecasting the prices received by producers or

cattle numbers at any particular point in time then a

dynamic framework is appropriate. However, if one is inter-

ested in long-run trends then the dynamic framework is a

hindrance. The seasonal fluctuations and production cycles

which are incorporated in a dynamic model only serve to hide

the trend. In this case a static framework is appropriate.


The Model
The description of the organization of the beef sector

serves as a conceptual framework with which to mathematic-

ally formulate the programming model. The system for beef

production is represented by a linear activity analysis

model. Regional differences in the production of beef are

accounted for by building an activity analysis model for

each supply region. The demand for beef is represented by a

system of demand equations for meat at the retail level.









48
Transportation activities permit the transportation of live

animals between supply regions and boxed beef from supply

regions to demand regions. The size of the breeding herd,

the level of beef imports and the level of dairy culls

slaughtered are exogenously determined.

Conceptually the model is not complicated. It is a

spatial equilibrium model where the supply functions are

being implicitly represented have been replaced by a linear

activity analysis model of beef production. The mathe-

matical representation of the model is complicated by the

number of dimensions to each variable. However, the

production activities in one region are simply replicated

for the other supply regions. The supply models differ by

the values of their parameters. Variables and their

dimensions are defined in Tables 3.5 and 3.6.

The beef production system as represented by the

following model draws heavily from earlier versions of the

model developed by Nelson et al. (1982), Kennedy (1982), and

Disney (1989). The structure of the model and variable

definitions are as defined in Nelson et al. and Kennedy.

The parameter values were updated by Disney.












TABLE 3.5. VARIABLE DIMENSION DEFINITIONS


Description


Regions
Supply regions are identified by:
(i,i')=1,2,3,4,5






Demand regions are identified by:
j=1,2,3,4


Production Stages
Production stages are identified
by:
s=1,2,3,4,5


The Cow-calf Stage
Production processes used are
identified by:
kl=1,2


i=1 Southeast
i=2 Midwest
i=3 Southwest
i=4 Plains
i=5 West



j=l= Northeast
j=2= South
j=3= Midwest
j=4= West


s=l = cow-calf stage
s=2 = stocker or growing
stage
s=3 = finishing stage
s=4 = slaughtering stage
s=5 = hamburger processing
stage


kl=1 = producing a weaned
calf
kl=2 = culling breeding stock


Subject











TABLE 3.5.--Continued


Description


Types of cattle produced are
identified by:
ul=1,2



The Stocker Stage
Type of animal used in production
activities are identified by:
v2=l

Production processes used are
identified by:
k2=1,2




Types of cattle produced are
identified by:
u2=1,2



The Finishing Stage
Type of animal used in production
activities are identified by:
v3=1,2,3




Production processes used are
identified by:
k3=1,2,3,4,5,6


ul=l = weaned calf
ul=2 = cull





v2 = weaned calf




k2=1 = producing a yearling
k2=2 = producing a 1 1/2 year
old




u2=1 = yearling
u2=2 = 1 1/2 year old





v3=1 = weaned calf
v3=2 = yearling
v3=3 = 1 1/2 year old




k3=l = producing 900 lb.
animal from a weaned
calf


Subject











TABLE 3.5.--Continued


Description


k3=2 = producing 1100 lb.
animal from a weaned
calf
k3=3 = producing 1200 lb.
animal from yearling
k3=4 = producing 1300 lb.
animal from 1 1/2 year
old
k3=5 = producing 900 lb.
animal from yearling
k3=6 = producing 1100 lb.
from 1 1/2 year old


Types of cattle produced are
identified by:
u3=1,2,3,4


u3=1
u3=2
u3=3
u3=4


The Slaughtering Stage
Types of meatpacking plants used
are identified by:
1=1,2,3,4,5






Type of animal used in production
activities are identified by:
v4=1,2,3,4,5,6,7


= 900 lb. animal
= 1100 lb. animal
= 1200 lb. animal
= 1300 lb. animal


1=1 = plant
1=2 = plant
1=3 = plant
1=4 = plant
1=5 = plant


v4=1 = 900 lb. fed animal
v4=2 = 1100 lb. fed animal


Subject











TABLE 3.5.--Continued


Description


v4=3
v4=4
v4=5
v4=6
v4=7


= 1200 lb. fed animal
= 1300 lb. fed animal
= yearling
= 1 1/2 year old
= cull


Production activities used
are identified by:
k4=1,2,3,4,5,6,7
















Product forms produced are
identified by:
u4=1,2,3,4,5


k4=1 = fabricate 900
animal
k4=2 = fabricate 1100
animal
k4=3 = fabricate 1200
animal
k4=4 = fabricate 1300
animal
k4=5 = fabricate year
k4=6 = fabricate 1 1/
old
k4=7 = fabricate cull


u4=1
u4=2
u4=3
u4=4
u4=5


lb. fed


lb.

lb.


lb. fed


ling
2 year


= roast
= steak
= lean trim
= medium trim
= fat and bone


v5=l = roast
v5=2 = steak


Subject


fed

fed


The Hamburger Processing Stage
Product forms used are identified
by:
v5=1,2,3,4











TABLE 3.5.--Continued


Description


Production activities used are
identified by:
k5=1,2,3,4,5,6,7


























Product forms sold are
identified by:
u=1,2,3


v5=3 = lean trim
v5=4 = medium trim


k5=1 = make hamburger from
primal cuts obtained
from 900 lb. fed
animal
k5=2 = make hamburger from
primal cuts obtained
from 1100 lb. fed
animal
k5=3 = make hamburger from
primal cuts obtained
from 1200 lb. fed
animal
k5=4 = make hamburger from
primal cuts obtained
from 1300 lb. fed
animal
k5=5 = make hamburger from
primal cuts obtained
from yearling
k5=6 = make hamburger from
primal cuts obtained
from 1 1/2 year old
k5=7 = make hamburger from
primal cuts obtained
from culls


u=l = hamburger
u=2 = roast
u=3 = steak


Subject












TABLE 3.6 VARIABLE DEFINITIONS

Variables Description


fj(Y) represents the market level inverse demand
system for beef in the jth demand region.

PjU represents the price received for beef
product u consumed away from the home (the
hotel, restaurant, and institutional trade)
in demand region j.

AYj,, the quantity (cwt/year) of beef product u
consumed away from the home in demand region
j.
P. represents the price received for beef
product u exported to other countries.
EYU represents the quantity (cwt) of beef
product u exported on an annual basis.
rdi the price of dairy culls in the ith supply
region.
ZDi the quantity (head) of dairy culls in the
ith supply region utilized in the beef
production system.

rlia1 the price of weaned beef calves (ul=l) or
beef culls (ul=2) in supply region i.

ql,i,kl the level (head) of breeding herd activity
kl utilized in region i.

Z1,i,um quantity (head) of weaned beef calves and
beef culls in region i utilized in the beef
production system.

r2,i,k2 the cost ($/hd) of stockering activity k2 in
supply region i.
q2,i,k2 the level (head of cattle) of stockering
activity k2 utilized in region i.

r3,i,k3 the cost ($/head) of feeding activity k3 in
region i.











q3,1,k3


r4,i,



q4,i,l,k4






qs,ilk5


1,1'



X3,i'''U1-1


X3,i,i',u2


X4,1i,*,1,u2,v4


T,.i',uD


X4,i,i',1,ul-2,v4-7


the level (head of cattle) of feeding
activity k3 utilized in region i.

the cost ($/head) of slaughtering and
producing boxed beef in plant 1 in supply
region i.

the level (head of cattle) of slaughtering
activity k4 utilized in plant 1 in supply
region i.

the cost ($/cwt.) of manufacturing hamburger
in plant 1 in region i.

the level (cwt.) of the k5 hamburger pro-
duction activity utilized in plant 1 in
supply region i.

cost of transporting weaned calves(ul=1) or
beef culls (ul=2) from supply region i to
supply region i'.

number of weaned calves (ul=1) in supply
region i transported to region i' to be
utilized in stockering activities.

number of weaned calves (ul=1) transported
from region i to region i' to be utilized in
feeding activities.

cost of transporting stockered cattle
(u2=yearling or 1-1/2 year old) from supply
region i to supply region i'.

number of stockered cattle,(u2=yearling or
1-1/2 year old) transported from region i to
region i' to be utilized in feeding
activities.

head of stockered cattle (u2=yearling or 1-
1/2 year old) transported from region i to
be slaughtered in plant 1 in region i'.

cost of transporting culls (uD=dairy) from
supply region i to supply region i'.

number of beef culls (ul=2) transported from
region i to be slaughtered in plant 1 in
region i'.










X4,i,iI,,,.-,iv4-7 number of dairy culls (uD=l) shipped for
slaughter from region i to plant 1 in region
i'.

Ti,r,u3 cost of transporting fed beef (u3=900 lbs.,
1100 lbs., 1200 Ibs., or 1300 Ibs.) from
supply region i to supply region i'.

X4,ii',1,,u3,v4 head fed beef (u3=900 lbs., 1100 lbs., 1200
lbs., or 1300 lbs.) transported from region
i to plant 1 in region i'.

TIj,u.- cost of transporting imported beef
(u=hamburger) to demand region j.

XIj,u.I quantity of hamburger (u=l) imported to
demand region j.

TEj,u cost of transporting exported beef (u=roast
or steak) from supply region i to export
markets.

XEn,u quantity of beef products (u=steak or roast)
exported from supply region i.

Tij,u cost of transporting beef products
(u=hamburger, roast or steak) from supply
region i to demand region j.

Xiju quantity of beef products (u=hamburger,
roast or steak) shipped from supply region i
to demand region j.

Mj,u marketing margin ($/cwt.) for beef product
u=hamburger, roast or steak) in demand
region j.

Yj,u quantity (lbs./month) of beef product u
consumed at home in demand region j.

Ij quantity (cwt.) of hamburger available for
importation.
RICJ,n quantity of beef product u consumed away
from home in demand region j.

Zi,, quantity (cwt.) of beef product u sold in
supply region i.
Z4,i,1,u4,14 quantity (cwt.) of primal cut u4 produced in
plant 1 in supply region i from production
activity k4.











Z4,,1,,u4


5s,1,U1,,v5


quantity (cwt.) of primal cuts (u4=roast or
steak) in region i allocated to sales
activity.

quantity (cwt.) of primal cuts (u4=roast,
steak, lean trim or medium trim) in region i
allocated to hamburger activity.

the percentage of fat found in primal cut v5
used in the hamburger activity in supply
region i.

quantity (cwt.) of primal cut v5 used by one
unit of hamburger production activity k5 in
plant 1.

the maximum percentage of fat permitted in
hamburger in supply region i.

head of cattle slaughtered in plant 1 in
supply region i.

capacity (head of cattle) of slaughter plant
1 in supply region i.

quantity (cwt.) of primal cut u4 produced by
one unit of the k4th slaughter activity of
plant 1 in supply region i.

quantity (head) of cattle used by one unit
of production activity k4 in plant 1 in
supply region i.

adjustment to live animal numbers due to
shrinkage during transit from supply region
i to supply region i'.

adjustment to final beef product weights due
to shrinkage during transit from supply
region i to demand region j.

quantity (head) of live animal v4
slaughtered in plant 1 in region i.

head of fed cattle u3 produced in supply
region i.

head of live animal v3 placed on feedlots in
supply region i.


Cs,,1,vS,k5


FATi


TCAPI,i


S,j


Z4,1,l,v4


23,i,u3


Z3,l,v3


c4,i,l,v4,k4










d3,1,3, k3 quantity (head) of fed cattle produced by
one unit of feeding activity k3 in supply
region i.

c3,,v3,k3 quantity (head) of live animals v3 used by
one unit of feeding activity k3 in supply
region i.

Zz,i,u2 head of stockered cattle u2 produced in
supply region i.
Z2,i,v2 head of live animal v2 used in stockering
activities in supply region i.

d21,i,2.k2 quantity (head) of stockered cattle produced
by one unit of stockering activity k2 in
supply region i.

c2,iv2,2 quantity (head) of live animals v2 used by
one unit of stockering activity k2 in supply
region i.

Z1,iu1 head of weaned calves or culls (ul=1 or 2)
produced by breeding and maintenance
activities in supply region i.

Zo,i,vl head of breeding stock used in breeding and
maintenance activities in supply region i.

dl,i,ul,kl quantity (head) of weaned calves (ul=l) or
culls (ul=2) produced by one unit of
breeding (kl=1) and maintenance (kl=2)
activities in supply region i.

c1,i,vl,kl quantity (head) of breeding stock vl used by
one unit of breeding (kl=l) or maintenance
(kl=2) activity in supply region i.
ZDi quantity of dairy culls from supply region i
utilized.


dairy cull supply in region i.


DAIRYi










The Oblective Function


4 4 3
max NSB = M J f(Y)dy, dy3 + 2 2 Pj.Ayj,
j=i L j=l u=l



3 5
+ E P, EY, E rdi ZDi
u=2 i=1



5 2
E. rl,iUl Z,i,l
i=l ul=l



5 2
r2, i,k2 q2,i,k2
i=l k2=1



5 6
~- r3,i,ik3 q3,i,k3
i=l k3=1



5 5 7
r4,i,l q4,i,l,k4
i=l k3=1 k4=1



5 5 4
E rs5,,1 q5,i,1,k5
i=l 1=1 k5=1



5 5
T=1i,i ,,l-1 X ,ii',ul1-
i=1 i'=1













Ti,i',ul-1 X3,ii,u11-


2

u2=1


5 5
- i'=
i=l i'=l


5
- 2
i=1


5

i'=1


5 5
- i'
i=l i'=l


5
- 2
i=l



4
J-
j=l


5

i'=l


5

1=1



5
2
1=1



5

1=1 i


5

1=1


(TIJu-l+ Mj,u.1) XIJ,u-1


5 3
-= u3=2
i=l u3=2


5
- 2
i=l 1


4

j=1


TEi,u XEi,u





T,,u X.j.u


5

i=l


5
-1
i=l


5
E
i'=l


5

i'=l


2

u2=1



4

13=1


6

v4=5



4
v4=
v4=1


Ti,i',u2 X3,i,1',u2


Ti',1,,,u12 X4,ii',1,ul-2,v4-7


Tii',u2 X4,i,i',l,u2,v4





i,i',u3 X4,i,i', ,u3,v4


Ti,i',uD X4,i,i',l,uD,v4-7











The objective function is maximizing the area under-

neath the demand curves in all the demand regions minus the

total cost of activities at each stage of production and

total transportation costs. It also takes account of expen-

ditures on away from home consumption and the cost of

importing beef.

Constraints

The maximization of the objective function is subject

to a number of constraints. The constraints embody the

production activities, transportation activities and the

equilibrium conditions for a competitive market. They are

expressed as follows:

1. Dairy cull supply constraint.



ZDi DAIRYi : 0; i=1,*--,5;



This constraint ensures that the number of dairy culls

utilized in region i is no greater than the number of

dairy culls available in region i.

2. Dairy cull transfer constraint.



5 5
-ZDi + 2 2 X4,,i,,l, s 0; i=l,-' ,5;
i'=l 1=1 uD=l;


This constraint guarantees that the number of dairy

culls transferred from region i (X4,i,,, ,1,) is less than











or equal to the number of dairy culls utilized in

region i (-ZD,).

3. Maintenance of breeding herd constraint.



2
-Zo,i,vi + Z Cl,,vlkl ql,i,kl 0;
kl=l
i=l,**,5;
vl=l;
kl=1,2;


This constraint ensures that the number of cows

utilized in the replacement and breeding activities

(cl,ivlkl ql1,i,kl) in region i is less than or equal to

the number of cows available in region i (Zo,i,vj).

4. Weaned calf and cull production constraint.



2
Zi,,u dli,ul,kl ql,i,k < 0; i=l, ,5;
kl=l ul=1,2;



This constraint ensures that the number of weaned

calves or culls produced in region i (dl,i,ul,k ql,i,ki)

is greater than or equal to the available supply of

weaned calves or culls in region i (Z1i,,,a).











5. Beef cull transfer constraint.



5 5
-Zi,1,ui-2 + E 2 X4,i,,,i,2.2 0; i=1,*'',5;
i'=1 1=1



In this constraint the cattle culled from the breeding

herd are transferred to the slaughter-processing stage.

It ensures that the number of beef culls transferred

from region i to the slaughtering stage (X4,,1,i.1*2) is

less than or equal to the available supply of beef

culls in region i (Z1,,i1.2).

6. Weaned calf transfer constraint.



5 5
-Z1,1,.1-1 + X2,i,i',aul- + X3,ii,ul. < 0;
i'=1 i'=l
i=1,** ,5;



In this constraint weaned calves in region i are

transferred to either the stocker or finishing stages.

It guarantees that the number of weaned calves

transported from region i to the stocker (X2,i,i'.1) and

finishing stages (X3,i,i',u-1i) is less than or equal to

the supply of weaned calves in region i (Z1,1,ul.1).









64

7. The receipt of weaned calves into the stockering stage

constraint.



5
Si,i, X2,iiui'l + Z2,i,v2 -' 0; i=1,'* ,5;
i'=l v2=1;



This constraint assembles weaned calves for use in

stockering activities in region i. It ensures that the

number of weaned calves assembled for use in stockering

activities (Z2,1,12) is no greater than the number of

weaned calves shipped to the stockering stage in region

i times a shrinkage parameter which adjusts the number

of weaned calves shipped for injury and death in

transit (si,, X2,ii,ul=1 ).

8. The utilization of calves in the stockering stage

constraint.



2
-Z2,i,v2 + 2 C2,i,v2.k2 q2,i,k2 < 0;
k2=1
i=1,- ,5;
v2=1;
k2=1,2;


This constraint ensures that the number of weaned

calves utilized in the stockering activity in region i

(c2,i,v2,k2 q2,i,k2) is no greater than the supply of
stocker calves in region i.











9. The stocker output constraint.



2
Z2,1,-2 2 d2,i.u2,k2 q2,ik2 < 0;
k2=1
i=1, 5;
u2=1,2;



This constraint ensures that the number of stockered

cattle produced in region i (d2,i,. k2 q2,i,k2) is greater

than or equal to the available supply of stockered

cattle in region i (Z2,i.u2).

10. The transfer of stockered cattle constraint.



5 5 5
Z2,i.u + E X3,ii',u2 + X4,i,i'lu2 < 0;
i'=1 i'=1 1=1
i=1," ,5;
u2=1,2;
k2=1,2;


This constraint transfers stockered cattle in region i

to either the finishing or slaughter stages. It

guarantees that the number of stockered cattle trans-

ported from region i to the finishing (X3,i,,.,,3) and

slaughter stages (X4,i,1.,,3) is less than or equal to the

available supply of stockered cattle in region i

(Z2,1,u2)











11. The receipt of weaned calves into the finishing stage

constraint.



5
tS i, X ,i,i',ui.i + Z3,i,v3.1 : 0;
i'=1
i=l,---,5;



This constraint collects weaned calves for use in the

finishing stage in region i. It ensures that the

number of weaned calves collected for use in feeding

activities (Z3,1,,.1) is no greater than the number of

weaned calves shipped to the finishing stage in region

i times a shrinkage parameter which adjusts the number

of weaned calves shipped for injury and death in

transit (sl,i, X3,iiU-.1).

12. The receipt of stockered cattle into the finishing

stage constraint.



5
2 X3,i,i',u2 + Z3,i,3 0; i=l,' ,5
i'=l u2=l, v3=2;
u2=2, v3=3;



This constraint collects stockered cattle for use in

the finishing stage in region i. It ensures that the

number of stockered cattle collected for use in feeding

activities (Z3,1,3) is no greater than the number of

stockered cattle shipped to the finishing stage in









67
region i times a shrinkage parameter which adjusts the

number of stockered cattle shipped to region i for

injury and death in transit (s,i, X3,1,1,,,3).

13. The utilization of cattle in the finishing stage

constraint.



4
Z3,,v3 + E C3,1, q3,, s 0; i=1, --,5;
k3=1 v3=1,---,3;


This constraint ensures that the head of cattle

utilized in feeding activities in region i (c3,,v3,3 *

q3,i,3) is no greater than the supply of cattle for

finishing in region i (Z3,1,,3).

14. The finishing stage output constraint.



4
Z3,i,u3 ~ d3,,u3,k3 q3,i,k3 0; i=1, ,5;
k3=1 u3=1,---,4;


This constraint ensures that the number of finished

cattle produced in region i (d3,i,.3,k3 q,,,,,3) is greater

than or equal to the available supply of finished cat-

tle in region i (Z3,,,,3).

15. The transfer of finished or fed cattle constraint.



5 5
Z3u,u3 + 0 2 X4,i,i,,l,u3 0; i=1, 0;,5;
i'=l 1=1 u3=1,-' ,4;











This constraint transfers finished cattle in region i

to the slaughter stage. It guarantees that the number

of fed cattle transported from region i to the

slaughter stage (X4,i,i,1,u3) is less than or equal to the

supply of finished cattle in region i (Z3,,3).

16. The receipt of culls into the slaughter stage con-

straint.



5
Z4,i,l,v4-7 3 Si,i, X4,i,i',l,ul-2
i=l



si,, X4, i,i,,1,U 0; i=1, '1,5;
1=1,-**,5;



This constraint assembles culls for slaughter in region

i. It ensures that the number of culls assembled for

use in slaughtering activities (Z4,1,,,,4.7) is no greater

than the number of beef culls shipped to the slaughter-

ing stage in region i times a shrinkage parameter which

adjusts the number of weaned calves shipped for injury

and death while in transit (si, X4,i,i,,,-,,2) and

the number of dairy culls shipped to the slaughtering

stage in region i times a shrinkage parameter

(Sii, X4,i,i',I,ul-2) *











17. The receipt of stockered cattle into the slaughter

stage constraint.



5
Z4,iv4 si, X4,1, ,u2 0; i=1,- ,5;
i'=l 1=1,***,5;
v4=5,6;
u2=1,2;


This constraint assembles stockered cattle for

slaughter in region i. It ensures that the number of

stockered cattle assembled for use in slaughter

activities (Z4,1,1,,4) is no greater than the number of

stockered cattle shipped to the slaughter stage in

region i times a shrinkage parameter which adjusts the

number of stockered cattle shipped to region i for

injury and death in transit (si,i, X4,i,i',1,u2).

18. The receipt of finished cattle into the slaughter stage

constraint.



5
Z4,i,l,v4 si,i, X4, i,,,1,u3 0; i=1,-- ,5;
i'=1 1=1,-- ,5;
v4=1,-* ,4;
u3=1,-**,4;


This constraint assembles finished cattle for slaughter

in region i. It ensures that the number of fed cattle

assembled for use in slaughter activities (Z4,i,.,v4) is

no greater than the number of fed cattle shipped to the









70
slaughter stage in region i times a shrinkage parameter

which adjusts the number of fed cattle shipped to

region i for injury and death in transit (s,, *



19. The utilization of cattle in the slaughter stage

constraint.



7
Z4,1,1,v4 E2 C4,i,i,,v4,k4 q4,,l,lk4 0;
k4=1 i=1,-",5;
1=1,-- ,5;
v4=1,"--,7;


This constraint ensures that the quantity of cattle

slaughtered in plant 1 (c4,i,l,v4,k4 q4,i.ik4) is no

greater than plant i's supply of slaughter cattle



20. The slaughter activity output constraint.



7
Z 4,1,.1 4, 2: d4,i,l,u4,k4 q4,i,l,k4 < 0; i=1 5 ;
k4=1 1=1,---,5;
u4=1,"**,5;


This constraint guarantees that the number of primal

cuts produced by plant 1 (d4,,1,4 q4,,1.,k4) is greater

than or equal to the available supply of primal cuts in

plant 1 (Z4,i,1,u4).











21. The slaughter of cattle in meatpacking plant

constraint.



7
2 q4,1,i,kM QA,i.I = 0; i=1,* 0,5;
k4=1 1=1,**-,5;



This constraint accounts for the head of cattle

slaughtered in plant 1 in region i.

22. The slaughter capacity constraint.



QAi,i TCAPi, 0; i=1,'*',5;
1=1,-** ,5;


This constraint requires that the quantity of cattle

slaughtered does not exceed plant capacity.

23. The transfer of primal cuts to sale and hamburger

processing activities constraint.



Z4,ilu4 + 6Z4,1,l,.4 + 8'Z5,i,1,.15 0;

i=4,...,5; 1=1,***,5;
u4=1,'--,4; v5=1,-'',4;
a = 1 if u4=1,2
0 if u4=3,4,5
a' = 1 if v5=1,-**,4;
0 if v5=5;


This constraint allocates the quantity of primal cuts

produced in plant 1 to sales or hamburger processing

activities. It ensures that the quantity of primal











cuts allocated to sales and hamburger processing

(Z4,1,1,v4 + Zs,i,1,vs) is less than or equal to the quantity

of primal cuts available in plant 1 (Z4,i,1,u4,k4).

24. Regional supply of steaks and roasts constraint.



5
S Z4,1,,4 + Z,u 0; i=1,--,5;
1=1 u4=1, u=2;
u4=2,u=3;


This constraint assembles all the steaks and roasts al-

located to the sales activity in the meatpacking plants

in region i (Z4,,,1,,4) at a single distribution center.

It guarantees that the quantity of steaks and roasts

available for distribution in region i (Zi,,) is no

greater than the quantity of steaks and roasts al-

located to sale by the meatpacking plants in region i.

25. Utilization of primal cuts in the production of

hamburger constraint.



4
Z5,i,1,vs + Z C5,i.l,v5,k5 qs,i,l,k5 0;
k5=1
i=1,-**,5;
l=1l,--*5;
v5=1,-**,4;


This constraint ensures that the quantity of primal

cuts used to produce hamburger in plant 1 (cs5,i,,v5,ks

qs,i.l,ks) is less than or equal to the quantity of primal









73
cuts allocated to the hamburger processing activity in

plant 1 (Zsi,ls ).

26. The hamburger processing constraint.


4 5
2 Z
k5=1 1=1


qs,1,1,ks + Zi,-,,, 0;


This constraint ensures that the quantity of hamburger

in region i (Z,,,,) is no greater than the quantity of

hamburger produced in the meatpacking plants in region

2 (ksHam qf,i,1,k5) t

27. Hamburger fat content constraint.


4 5
k2 E
k5=1 1=1


a5,i,k5 q5,i,1,k5 FATi Zi,u.1 s 0;


This constraint ensures that percentage of fat by

weight contained in hamburger produced in region i does

not exceed 27%.

28. The transfer of boxed beef constraint.


4

j=1


Xi,j,u + XEi,u Z,., 0;


This constraint transfers box beef cuts from supply

region i to domestic and foreign markets. It ensures


i=1,... 5;
u=1,---,3;










that the quantity of boxed beef cuts transported from

supply region i (X,,j,. + XEi,u) is less than or equal to

the supply of boxed beef cuts in region i (Zi,u).

29. Hamburger import constraint.




j Xj,u.i I < 0; u=l;



This constraint ensures that the quantity of hamburger

imported is less than or equal to an exogenously

determined supply of imported hamburger.

30. Beef consumption constraint.



8XIju-1 Ei sij Xi,j,u

+ (.12) Yj,u + AYJ,U < 0; u=1,2,3;
j=1,- -,4;
8= 1 if u=l
0 if u=2,3;


This constraint ensures that the quantity of fresh beef

consumed at and away from the home (Yj,u + AYJ,U) in

demand region j is less than or equal to the quantity

of foreign and domestically produced transported to

region j (X,,j, + XFj,u,-).

31. Away from home consumption constraint.



AYJ,u + HRIj,u = 0; j=l,'" ,4;
u=l,** ,3;








75

This constraint requires that away from home

consumption of beef in region j equal the away from

home demand for beef.















CHAPTER 4
SPECIFICATION OF AN INVERSE DEMAND
SYSTEM FOR FRESH MEATS


The purpose of this chapter is to specify a system of

demand equations for fresh meats to be used in the pro-

gramming model. The demand system needs to satisfy the

restrictions placed on it by both economic theory and the

programming model.

The first section of the chapter lays out the con-

ditions the demand system needs to satisfy in order for it

to be incorporated into the programming model (Table 4.1).

In the second section of the chapter a demand system for

fresh meats which is consistent with the conditions

established is derived. Characteristics of the data, the

estimation procedure used and the estimated system are

discussed in the third section. In the final section the

demand system to be used in the programming model and a

description of the steps taken to incorporate the estimated

demand system into the programming model are discussed.


Criteria for Selectinq a Demand System to be
Used in the Programming Model

The first condition which the demand system must meet

in order to be used in the programming model is that it be

specified in price-dependent or inverse form. This

76









77

TABLE 4.1. THE CRITERIA USED FOR SELECTING AN APPROPRIATE
SYSTEM OF DEMAND EQUATIONS.

1. They must be specified in price-dependent form.

2. They must satisfy the integrability conditions of the
price endogenous mathematical programming model.

a. Negative definite Jacobian matrix
b. Symmetric cross-quantity effects

3. They must satisfy nonseparability of preferences among
the various meat products.










condition is imposed by the formulation of the programming

model being used. By specifying the demand system in price-

dependent form it is much easier to formulate the program-

ming model and to make economic interpretations.

In demand theory, the demand system is usually derived

by assuming that prices and income are given and that it is

the quantity consumed which is adjusted by consumers in

order to maximize their utility. As a result the consumer's

demand system is specified in quantity-dependent form.

Fortunately, one of the basic results obtained through the

use of duality theory is that the consumer's preferences can

be represented by either the quantity-dependent demand

system or the associated inverse demand system. Anderson

(1980) has further shown that the inverse demand systems

have properties analogous to the properties of the quantity-

dependent demand systems.

In many cases the use of an inverse demand system is

just a matter of expedience with regards to the modeling

effort. There being no reason to represent the demand

relationship in inverse form other than to make the for-

mulation of the programming model simpler and easier to

interpret in an economic sense (see Takayama and Judge,

1971). However, for a product such as beef, the production

decisions are made long before the final product is avail-

able to consumers. This implies that at any given point in

time the supply of beef is fixed, and it is the price of










beef which must adjust in order to achieve market equilib-

rium. Given this it can then be argued that the demand for

beef should be specified in its inverse form, not only

because it is more convenient for modeling purposes, but

because it properly represents the manner in which

equilibrium in the market for beef is determined.

The second condition which the demand system must

fulfill is that it satisfy the integrability conditions of

the programming model. The integrability conditions require

that the Jacobian matrix of the demand equations be negative

definite and have symmetric cross-quantity effects. As dis-

cussed in Chapter 2 the integrability conditions ensure that

a unique solution exists.

A third condition which must be satisfied by the de-

mand system is that it reflect the nonseparability of pref-

erences for meat products by species. The reason for this

assumption about the separability of consumers' preferences

for fresh meat is discussed in greater detail in the next

section.

In addition to the conditions just discussed it is

important that the demand system reflect regional differ-

ences in consumer preferences for meat. This is based on

the assumption that differences in the levels of fresh meat

consumed between regions is determined not only by differ-

ences in population and level of income between regions, but

also by differences in their customs and traditions.









80
Finally, given the controversy over whether a shift in

consumer preferences for meat has occurred the data used to

estimate the demand system should come from the period after

the change in the structure in the demand for meat is

believed to have occurred. This will ensure that the estim-

ated parameters will reflect the current demand structure

for fresh meat.


The Two-Stage Budgeting Process and the
Representation of Consumer Preferences

According to demand theory the fundamental decision

facing consumers is how to allocate their income among the

available goods so as to maximize their utility. This

decision is determined by the consumer's preferences, the

prevailing commodity prices and the consumer's income.

While the consumer's preferences are unobservable they can

be represented by a system of demand equations which relates

the utility maximizing level of consumption to prices and

income.

Demand theory assumes that consumers know and take

into consideration all commodities available when making

their consumption decisions. This implies that the system

of demand equations should contain all commodities. How-

ever, empirical demand equations only contain the commod-

ities of interest and a small subset of related commodities.

One reason for this is that it is impractical to estimate a

demand system which contains all commodities. A second










reason for this is the realization that consumers do not

take into consideration all goods when they make their

expenditure decisions, but only a subset of related goods.

Thus, a demand system, if appropriately defined, only needs

to contain the commodities of interest in order to appropri-

ately represent the consumer's behavior.

The two-stage budgeting process is often used to

represent the consumer's decision-making process and to

define the set of commodities to be included in the demand

system. In the two-stage budgeting process the consumer

first decides how to allocate his or her income over a broad

category of goods, then in the second stage how to allocate

each category's expenditures over the goods contained in

them.

The two-stage budgeting process is used to construct a

utility tree which shows which goods consumers take into

consideration when making their expenditure decisions. The

utility tree depicted in Figure 4.1 demonstrates how the

two-stage budgeting process is used to select the commod-

ities to be included in a demand system. The branches of

the tree represent the budget or expenditure categories and

nodes or leaves on the branches represent the commodities

contained in each of the budget categories. By following

along the branches of this tree one can follow the process

used by the consumer when making a decision to purchase a

particular good. The first leaf on the tree represents all

































































*4

:>4

f-I

4a
41









r1

*^









83
available goods. At this stage the consumer decides how to

allocate expenditures among housing, transportation, food,

and entertainment located at the nodes at the end of each

branch. Then in the next stage the consumer then decides

how to allocate food expenditures among the items in the

food category, such as meats, dairy products, grains, and

fruit.

In order for a particular decision tree to be an

appropriate representation of the consumer's budgeting pro-

cess, the consumer's preferences must be weakly separable

according to the branches of the decision tree. Weak sepa-

rability implies that each branch of the decision tree can

be defined by a separate sub-utility function. Thus, the

two-stage budgeting process provides a theoretical founda-

tion for demand systems which only contain a portion of the

goods available by allowing a demand system to be derived

from the sub-utility function containing the goods of

interest.

There are any number of decision trees which could be

used to represent consumers' decisions to purchase fresh

meats. Two likely candidates are represented in the two

panels found in Figures 4.2 and 4.3.

The utility tree in Figure 4.2 represents a two-stage

budgeting process for fresh meat based on the animal species

from which the meat comes. At the first level of
























































Figure 4.2. Utility tree A.



























































Figure 4.3. Utility tree B.










decision-making the consumer has a given level of expendi-

tures to allocate to food. The consumer then decides how to

allocate the given level of expenditure between meat and

nonmeat categories. At the second level the consumer then

decides the amount of expenditures for meat to allocate to

the purchase of beef, pork, and poultry. At the final level

of the decision tree the consumer then decides the amount of

beef expenditures to spend on hamburger, steak, and roasts.

The utility tree in Figure 4.2 represents the two-

stage budgeting process underlying most empirical studies of

the structure of the consumer's demand for meat. To a large

extent this has been simply a matter of expedience. For the

most part the data on consumption of meat has been based on

disappearance and not on knowledge of the amount actually

purchased by consumers at the retail level. The information

available is published by species. It is also argued that

the assumption that consumers determine how much to spend on

beef, pork and poultry before selecting the particular beef

pork or poultry product they purchase reflects the way con-

sumers budget their meat expenditures. One only need to

look at the meat case at the supermarket where meat is sepa-

rated by species to find corroboration of this observation.

The decision tree in Figure 4.3 represents an

alternative view of the consumer's budgeting process for

meats. In this case, once the consumer determines the

allocation for meat expenditures then the consumer does not










make a further distinction between the different meat

products based on origin of species alone. The consumer

decides how much of food expenditures to allocate towards

the purchase of meat, then at the next level determines the

amount to spend on the various meat products represented in

the diagram. This utility tree implies that consumers'

preferences for meat are weakly separable while their

preferences for hamburger, steaks, roasts, pork, whole

chickens, and chicken parts are not.

Recent empirical evidence indicates that consumers do

not make a distinction between meat products based on

species of origin alone. It indicates that consumers

determine how much of their food expenditures to spend on

meat and then decide how much of their meat expenditures to

spend on steaks, roasts, hamburger, pork, whole chickens,

chicken parts and other meat products (Eales and Unnevehr,

1988) as represented by the utility tree in Figure 4.3.

Thus, in order for a system of demand equations to be

consistent with the decision making process of the consumer

it must be specified at a level which encompasses all meat

cuts, not just those of a particular species. It is a

maintained assumption of this study that consumers'

preferences for fresh meat are weakly separable from all

other goods but not separable among the various meat

products.










The Derivation of the Inverse
Almost Ideal Demand System

Consider the Almost Ideal Demand System (AIDS) pro-

posed by Deaton and Muellbauer (1980). The AIDS model was

developed to test the restrictions on consumer demand

derived from demand theory. As a result it is possible to

impose symmetry of the substitution matrix during estim-

ation. Furthermore, the AIDS model is specified in level

form. This gives it an advantage over the Rotterdam model

which is specified in first difference form because this

permits it to be readily incorporated into a static pro-

gramming model. The AIDS model also has the advantage that,

as long as consumer's preferences are of the price independ-

ent generalized logarithmic (PIGLOG) form, it permits per-

fect aggregation over consumers without assuming that pref-

erences are additive. Thus, with this particular demand

system there is a theoretical justification for imposing

demand restrictions based on individual behavior at the

market level.

Deaton and Muellbauer (1980) derive the AIDS model by

applying Shepherd's Lemma to an expenditure function. An

inverse demand system analogous to the AIDS model can be

derived by specifying a distance function which is dual to

the expenditure function used by Deaton and Muellbauer

(1980) and applying the Shepherd-Hancock Lemma. By using

the distance function similar in form to the AIDS










expenditure function the properties possessed by the AIDS

model are carried over into the inverse demand system.

The distance function is

In D(U,q)= a(q) +U*b(q) (4.1)
where
a(q)= a. + E ai*lnqi + 1/2 EE Yij Inqglnqj (4.2)
i ij
and

b(q)=p)lqPi (4.3)

U represents the level of utility and q is the vector of

commodities consumed. D(U,q) is homogenous in q if Ea=I,,


EiYij = jYij = pi = 0, and yj = Yji.

The compensated share equations in quantity dependent

form are derived by taking the partial derivatives of the

distance function with respect to In q,. Resulting in


wi = ai + E Yij In qj + UpiP.Iqg, (4.4)


The equations are functions of quantity and utility and

represent the compensated inverse demand system in share

equation form. If it is assumed that preferences are homo-

thetic then it is possible to substitute real expenditures

(X/P) for utility. An estimable system which does not rely

on the assumption that preferences are homothetic is found

by deriving an expression for utility from the distance









90
function, 4.1-4.3, and then substituting this expression for

U in 4.4.

Utility maximization implies that D(U,q)=1 and along

with 4.1-4.3 that


U = -(a, + E ai*lnqi + 1/2 EE Yij Inqilnqj)/PJIq (4.5)
i ij

Substituting the result from 4.5 into 4.4 and simplifying

results in



Wi = ai + E Ylnqj Piln(Q) (4.6)


where

ln(Q) = a(q), E ai = 1, yij = yl, Ej yij = 0 E yij = 0

and

E, Pi = 0.
The system of share equations represented by 4.6 are

functions of q alone and represent the uncompensated inverse

demand system. It is linear in parameters if ln(Q) is

approximated by Ekwklnqk.


The Data
Data on the level of household expenditures on fresh

meat products were obtained from the Bureau of Labor Sta-

tistics (BLS) Consumer Expenditure Survey (CES), 1982-1986.

The survey is a representative sample of the U.S. popula-

tion. It records the weekly expenditures of a household











over a two week period. The U.S. is divided into four

regions: Northeast, South, Midwest and West (Figure 4.4).

As with all household surveys a zero level of expenditures

was reported for any particular commodity for many of the

households. Consequently, the data were aggregated from the

household level to the regional level in order to eliminate

the possibility of having zero as an observation.

Expenditure levels for 22 meat products are reported

in the survey which were in turn aggregated into eight

categories for estimation purposes (see Table 4.2). This

aggregation process used the formulas provided in the CES

documentation (see Appendix A for the formulas). The new

data set contains the level of average monthly household

expenditure on the eight aggregate meat commodities by

region.

Data on monthly prices of meat products in the four

demand regions were also obtained from the BLS. The data

contain the U.S. average price as well as separate regional

prices for 24 meat products. The prices reported were

roughly consistent with the commodity categories obtained

from the expenditure survey (see Table 4.2). In the several

instances where multiple prices were reported for a single

expenditure category in the CES data the prices were

averaged together. There were three exceptions to this

procedure. In the steak category the price for porterhouse

steak was not used in calculating the average price for















TABLE 4.2


EXPENDITURES AND PRICES REPORTED FOR MEAT COM-
MODITIES BY THE BUREAU OF LABOR STATISTICS (BLS)
AND THE COMMODITY CATEGORY USED IN THE DEMAND
SYSTEM.


Prices Expenditures Aggregate
Reported Reported Commodities


Turkey


Chicken Legs
Chicken Breast

Whole Chicken

Beef Liver
Bologna
Franks

Ham, canned
Sausage
Other Ham
Pork Chop
Bacon
Pork Shoulder
Pork Roast

Porterhouse Steak
T-Bone Steak
Chuck Steak
Sirloin Steak
Round Steak

Rib Roast
Round Roast
Chuck Roast


Ground Chuck
Ground Beef


Other Poultry

Chicken Parts


Whole Chickens

Other Meat
Other Meat
Other Meat


Canned Ham
Pork Sausage
Ham (excluding canned)
Pork Chops
Bacon
Other Pork
Other Pork

Other Steak
Other Steak
Other Steak
Sirloin Steak
Round Steak

Other Roast
Round Roast
Chuck Roast


Ground Beef
Ground Beef


Other Poultry

Chicken Parts


Whole Chickens


Other Meat


Pork








Steak





Roast


Ground Beef










other steak. This was done because its price was reported

for only two of the demand regions. In the ground beef

category only the price for ground chuck was used because

the ground beef price was not reported prior to 1984. In

the pork category only the prices for sausage, canned ham,

pork chop, and bacon were used. Pork shoulder, pork roast

and other ham were not used because their prices were not

consistently reported in any of the demand regions.

The eight price indexes were calculated by using a

weighted average of the prices of the individual elements of

a meat category. The weights used were the individual

commodities' share of consumer's expenditures on the

aggregate commodity categories.

As alluded to above there was a problem with missing

observations in the price series provided by the BLS. In

some cases the entire price series for a commodity was not

reported in one of the demand regions. This was particu-

larly true for steak prices in the West region (see Table

4.3).

In the instances where this occurred three different

approaches were used to impute the missing prices. The

first approach was to use a price of a substitute commodity

to represent the missing price series. This was done in the

case of pork where the canned ham price was used as a proxy

for the fresh ham prices. It was also done in the case of

other steak where the porterhouse price was used as a proxy