A dynamic model of input demand for agriculture in the Southeastern United States

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Title:
A dynamic model of input demand for agriculture in the Southeastern United States
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vii, 148 leaves : ill. ; 28 cm.
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English
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Monson, Michael James, 1956-
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Subjects / Keywords:
Agriculture -- Economic aspects -- Southern States   ( lcsh )
Duality theory (Mathematics)   ( lcsh )
Food and Resource Economics thesis Ph. D
Dissertations, Academic -- Food and Resource Economics -- UF
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 141-147.
Statement of Responsibility:
by Michael James Monson.
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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oclc - 15464499
notis - AEH5769
sobekcm - AA00004868_00001
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Full Text












A DYNAMIC MODEL OF INPUT DEMAND
FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES






By

MICHAEL JAMES MONSON


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


1986















































Copyright 1986

by

Michael James Monson













ACKNOWLEDGEMENTS


Custom alone does not dictate the need for me to

express my gratitude to the members of my committee, as

each member has provided valuable assistance in the

completion of this dissertation. I thank Dr. Boggess

for enabling me to pursue a variety of topics and

enhancing the breadth of my graduate research

experience, as well as keeping a down-to-earth

perspective. Dr. Taylor has served me well as motivator

and mentor of my dissertational research. I also

express my appreciation to Dr. Majthay for the classroom

instruction in optimal control theory and tolerance of

an agricultural economist's employment of the theory,

Dr. Langham for some timely advice, and Dr. Emerson for

restoring my faith in nonlinear models at a crucial

moment.

Additionally, thanks to the staff of the FARM lab,

particularly Rom Alderman, for technical support in

preparing the manuscript, and to Pat Smart for doing her

best to see that I turned the required forms in on time.

Finally, I thank my wife, Sandra, son, Jeffery, and

my parents. Sorry it took so long.


iii












TABLE OF CONTENTS


ACKNOWLEDGEMENTS......................

ABSTRACT.................................


CHAPTER

I




II







III




IV




V

APPENDIX

A
B

C

D


INTRODUCTION..........................

Background.. ........................
Objectives. ...........................

THEORETICAL MODEL....................

Dynamic Models Using Static
Optimization.......................
Dynamic Optimization.................
Theoretical Model.....................
The Flexible Accelerator..............

EMPIRICAL MODEL AND DATA...............

Empirical Model.......................
Data Construction..................

RESULTS............................. .

Theoretical Consistency...............
Quasi-fixed Input Adjustment...........

SUMMARY AND CONCLUSIONS...............



INPUT DEMAND EQUATIONS................
REGIONAL EXPENDITURE, PRICE,
AND INPUT DATA.........................
EVALUATION OF CONVEXITY OF
THE VALUE FUNCTION.....................
ANNUAL SHORT- AND LONG-RUN
PRICE ELASTICITY ESTIMATES............


REFERENCES......................................

BIOGRAPHICAL SKETCH...............................

iv


PAGE

iii

v


1

3
15

18


20
27
35
42

46

46
60

75

75
99

110



119

123

130

132

141

148











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 DYNAMIC MODEL OF INPUT DEMAND
FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES




By

Michael James Monson

August 1986


Chairman: William G. Boggess
Cochairman: Timothy G. Taylor
Major Department: Food and Resource Economics


The current crisis in U.S. agriculture has

focused attention on the need to adjust to lower

output prices as a result of a variety of factors.

The ability of agriculture to adjust is linked to the

adjustment of inputs used in production. Static

models of input demand ignore dynamic processes of

adjustment. This analysis utilizes a model based on

dynamic optimization to specify and estimate a system

of input demands for Southeastern U.S. agriculture.

Dynamic duality theory is used to derive a system

of variable input demand and net investment equations.








The duality between a value function representing the

maximized present value of future profits of the firm

and the technology are reflected in a set of

regularity conditions appropriate to the value

function.

Labor and materials are taken to be variable

inputs, while land and capital are treated as

potentially quasi-fixed. The equations for optimal

net investment yield short- and long-run demand

equations for capital and land. Thus, estimates of

input demand price elasticities for both the short run

and long run are readily obtained.

The estimated model indicates that the data for

the Southeast support the assumption of dynamic

optimizing behavior as the regularity conditions of

the value function are generally satisfied. The

model indicates that labor and capital, and land and

capital are short-run substitutes. Land and labor,

and materials and capital are short-run complements.

Similar relationships are obtained in the long run as

well.

Estimated adjustment rates of capital and land

indicate that both are slow to adjust to changes in

equilibrium levels in response to relative price

changes. The estimated own rate of adjustment in the

difference between actual and equilibrium levels of








capital within one time period is 54 percent. The

corresponding adjustment rate for land is 16 percent.

However, the rate of adjustment for capital is

dependent on the difference in actual and equilibrium

levels of land.


vii












CHAPTER I
INTRODUCTION


If one were to characterize the current situation

in U.S. agriculture, one might say that agriculture

faces a period of adjustment. Attention has focused

on.the need to adjust to lower output prices as a

result of a variety of factors including increasing

foreign competition, trade regulations, a declining

share of the world market, strength of the U.S.

dollar, changes in consumer demand, high real rates of

interest and governmental policies. The adjustments

facing agriculture as a result of low farm incomes

have become an emotional and political issue as

farmers face bankruptcy or foreclosure.

The ability of the agricultural sector to respond

to these changing economic conditions is ultimately

linked to the sector's ability to adjust the inputs

used in production. Traditionally the analysis of

input use in agriculture has been through the

derivation of input demands based on static or single-

period optimizing behavior. While static models

maintain some inputs as fixed in the short-run yet

variable in the long-run, there is nothing in the

theory of static optimizing behavior to explain a

1










divergence in the short- and long-run levels of such

inputs. Previous models based on static optimization

recognize that some inputs are slow to adjust, but

they lack a theoretical foundation for less-than-

instantaneous input adjustment.

A theoretical model of dynamic optimization has

been recently proposed by Epstein (1981). Epstein has

established a full dynamic duality between a dual

function representing the present value of the firm

and the firm's technology. This value function can be

used in conjunction with a generalized version of

Hotelling's Lemma to obtain expressions for variable

input demands and optimal net investment in less-than-

variable inputs consistent with dynamic optimizing

behavior.

While the theoretical implications of dynamic

duality are substantial, there are a limited number of

empirical applications upon which an evaluation of the

methodology can be based. An empirical application to

agriculture presenting a complete system of factor

demands is absent in the literature. Chambers and

Vasavada (1983b) present only net investment equations

for the factors of land and capital in U.S.

agriculture, focusing on the rates of adjustment.

This same approach is used by Karp, Fawson, and

Shumway (1984). The rates of adjustment in less-than-








3

variable inputs are important, but the methodology is

richer than current literature would indicate.

The purpose of this study is to utilize dynamic

duality to specify and estimate a dynamic model of

aggregate Southeastern U.S. agriculture. The model

enables a clear distinction between the short- and

long-run. Thus, input demands and elasticities for

the short- and long-run may be determined. A key part

of the methodology based on the explicit assumption of

dynamic optimizing behavior is whether the data

support such an assumption. This analysis intends to

explore the regularity conditions necessary for a

duality between the dynamic behavioral objective

function and the underlying technology.

Background

The term "period of adjustment" is an appropriate

recognition of the fact that agriculture does not

always adjust instantaneously to changes in its

environment. Agriculture has faced and endured

periods of adjustment in the past. The Depression and

Dust Bowl eras and world wars produced dramatic

changes in the agricultural sector (Cochrane, 1979).

In more recent history, agricultural economists have

noted other periods of adjustment in response to less

spectacular stimuli. Examples include the exodus of

labor from the agricultural sector in the 1950s that








4

stemmed from the substitution of other inputs for

labor (Tweeten, 1969), and the introduction of hybrid

seed (Griliches, 1957) and low-cost fertilizers

(Huffman, 1974). Two points are crucial. First, the

period of adjustment in each case is less than

instantaneous if not protracted, and second, the

adjustments are ultimately reflected in the demand for

inputs underlying production.

Quasi-fixed Inputs

The assumption of short-run input fixity is

questionable. In a dynamic or long-run setting, input

fixity is, of course, inconsistent with the definition

of the long run as the time period in which all inputs

are variable. The recognition that some inputs are

neither completely variable or fixed dictates using

the term "quasi-fixed" to describe the pattern of

change in the levels of such inputs. Given a change

in relative prices, net investment propels the level

of the input towards the long-run optimum. The input

does not remain fixed in the short-run, nor is there

an immediate adjustment to the long-run optimum.

Alternatively, the cost of changing the level of

an input does not necessarily preclude a change as is

the case for fixed inputs, nor is the cost solely the

marginal cost of additional units in the variable

input case. No inputs are absolutely fixed, but










rather fixed at a cost per unit time. The quasi-

fixity of an input is limited by the cost of adjusting

that input.

Short-run Input Fixity

In light of the notion of quasi-fixity, input

fixity in short-run static models is based on the

assumption that the cost of adjusting a fixed input

exceeds the returns in the current period. In the

static model, the choice is either no adjustment in

the short-run initial level or a complete adjustment

to a long-run equilibrium position. The static model

is unable to evaluate this adjustment as a gradual or

partial transition from one equilibrium state to

another. Given such a restricted horizon, it is

understandable that static models of the short run

often treat some inputs as fixed.

The assumption of input fixity in the short run

casts agriculture in the framework of a putty-clay

technology, where firms have complete freedom to

choose input combinations ex ante but once a choice is

made, the technology becomes one of fixed coefficients

(Bischoff, 1971). Chambers and Vasavda (1983a) tested

the assumption of the putty-clay hypothesis for








6

aggregate U.S. agriculture with respect to capital,

labor, and an intermediate materials input. Land was

maintained as fixed and not tested.

The methodology, developed by Fuss (1978),

presumes that uncertainty with respect to relative

prices determines the fixed behavior of some inputs.

The underlying foundation for Chambers and Vasavada's

test of asset fixity is a trade-off between

flexibility of input combinations in response to

relative input price changes and short-run efficiency

with respect to output. Essentially, the farmer is

faced with choosing input combinations in anticipation

of uncertain future prices. If the prices in the

future differ from expectations and the input ratios

do not adjust, the technology is putty-clay. If the

input ratios can be adjusted such that static economic

conditions for allocative efficiency are fulfilled

(marginal rate of technical substitution equal to the

factor price ratio), the technology is putty-putty.

Finally, if the input ratios do adjust, but not to the

point of maximum efficient output, the technology is

said to be putty-tin.

Chambers and Vasavda concluded that the data did

not support the assumption of short-run fixity for

capital, labor, or materials. However, the effects of

the maintained assumptions on the conclusions is








7

unclear. The assumption of short-run fixed stocks of

land makes it difficult to determine if the degree of

fixity exhibited by the other inputs is inherent to

each input or a manifestation of the fixity of land.

In addition to the assumption that land is fixed, the

test performed by Chambers and Vasavada also relied on

the assumption of constant returns to scale, an

assumption used in previous studies at the aggregate

level (Binswanger, 1974; Brown, 1978). The measure of

efficiency, output, is dependent on the constant

returns to scale assumption. If there were in fact

increasing returns to scale presumably inefficient

input combinations may actually be efficient.

The Cost of Adjustment Hypothesis

The cost of adjustment hypothesis put forth by

Penrose (1959) provides an intuitive understanding of

why some inputs are quasi-fixed. Simply stated, the

firm must incur a cost in order to change or adjust

the level of some inputs. The assumption of short-run

fixity is predicated on the a priori notion that these

costs preclude a change.

The cost of search model by Stigler (1961) and

the transactions cost models of Barro (1969) can be

interpreted as specialized cases of adjustment costs

characterized by a "bang-bang" investment policy.

Such models lead to discrete jumps rather than a








8

continuous or gradual pattern of investment. For a

model based on static optimization, this appears to be

the only adjustment mechanism available. Inputs

remain at the short-run fixed level until the returns

to investment justify a complete jump to the long-run

equilibrium.

Land is most often considered fixed in the short-

run (Hathaway, 1963; Tweeten, 1969; Brown, 1978;

Chambers and Vasavada, 1983a). Transaction costs

appear to be the basis for short-run fixity of land

and capital in static models. Galbraith and Black

(1938) reasoned that high fixed costs prohibit

substitution or investment in the short run. G. L.

Johnson (1956) and Edwards (1959) hypothesized that a

divergence in acquisition costs and salvage value

could effectively limit the movement of capital inputs

in agriculture.

While labor and materials are usually considered

as variable inputs in the short run, a search or

information cost approach may be used to rationalize

fixity in the short run. Tweeten (1969) has suggested

that labor may be trapped in agriculture as farmers

and farm laborers are prohibited from leaving by the

costs of relocation, retraining, or simply finding an

alternative job. However, the treatment of










agriculture as a residual employer of unskilled labor

seems inappropriate given the technical knowledge and

skills required by modern practices.

These same technical requirements can be extended

to material inputs such as fertilizers, chemicals, and

feedstuffs, in order to justify short-run fixity. The

cost of obtaining information on new material inputs

could exceed the benefits in the short run.

Alternatively, some material inputs may be employed by

force of habit, such that the demand for these inputs

is analagous to the habit persistence models of

consumer demand (Pope, Green, and Eales, 1980).

Recognizing the problems of such "bang-bang"

investment policies, the adjustment cost hypothesis

has been extended to incorporate a wider variety of

potential costs. The adjustment costs in the first

dynamic models may be considered as external (Eisner

and Strotz, 1963; Lucas, 1967; Gould, 1968). External

costs of adjustment are based on rising supply prices

of some inputs to the individual firm and are

inconsistent with the notion of competitive markets.

Imperfect credit markets and wealth constraints may

also be classified as external costs of adjustment.

Internal costs of adjustment (Treadway 1969,

1974) reflect some foregone output by the firm in the

present in order to invest in or acquire additional








10

units of a factor for future production. The

assumption of increasing adjustment costs, where the

marginal increment of output foregone increases for an

incremental increase in a quasi-fixed factor, leads to

a continuous or smooth form of investment behavior.

Quasi-fixed inputs adjust to the point that the

present value of future changes in output are equal to

the present value of acquisition and foregone output.

The adjustment cost hypothesis, particularly

internal adjustment costs, is important in a model of

dynamic optimizing behavior. The costs of adjustment

can be reflected by including net investment as an

argument of the underlying production function. The

constraint of input fixity in a short-run static model

is relaxed to permit at least a partial adjustment of

input levels in the current period. Assuming that

these adjustment costs are increasing and convex in

the level of net investment in the current period

precludes the instantaneous adjustment of inputs in

the long-run static models. Therefore, a theoretical

foundation for quasi-fixed inputs can be established.

Models of Quasi-fixed Input Demand

Econometric models which allow quasi-fixity of

inputs provide a compromise between maintaining some

inputs as completely fixed or freely variable in the

short-run. The empirical attraction of such models is








11

evident in the significant body of research in factor

demand analysis consistent with quasi-fixity surveyed

in the following chapter. Unfortunately, a

theoretical foundation based on dynamic optimizing

behavior is generally absent in these models.

By far the most common means of incorporating

dynamic elements in the analysis of input demand has

been through the use of the partial adjustment model

(Nerlove, 1956) or other distributed lag

specifications. Such models typically focus on a

single input. The coefficient of adjustment then is a

statistical estimate of the change in the actual level

of the input as a proportion of the complete

adjustment that would be expected if the input was

freely variable.

The principle shortcoming of the single-equation

partial adjustment model is that such a specification

ignores the effect of, and potential for, quasi-fixity

in the demands for the remaining inputs. The recent

payment-in-kind (PIK) program is an excellent example

of the significance of the interrelationships among

factors of production. The program attempted to

reduce the amount of land in crops as a means of

reducing commodity surpluses. The effect of the

reduction in one input, land, reduced the demand for

other inputs, such as machinery, fertilizer and








12

chemicals, and the labor for operation of equipment

and application of materials. The consequences of a

reduction in this single input extended beyond the

farm gate into the industries supporting agriculture

as well.

The partial adjustment model has been the

foundation for many so-called dynamic optimization

models cast in the framework of a series of static

problems with the imposition of an adjustment

coefficient as the linkage between the individual

production periods (e.g., Day, 1962; Langham, 1968;

Zinser, Miranowski, Shortle, and Monson, 1985). This

effectively ignores the potential for instantaneous

adjustment. Adjustments and relationships among

inputs are determined arbitrarily. Others maintain

potentially quasi-fixed inputs as variable (e.g.,

McConnell, 1983). The actual rate of adjustment in

the system may be slower than the model would indicate

as the adjustment of the maintained quasi-fixed inputs

depends on the adjustments in supposedly variable

inputs.

The multivariate flexible accelerator (Eisner and

Strotz, 1963) is an extension of the partial

adjustment model to a system of input demands rather

than a single equation. Lucas (1967), Treadway (1969,

1974), and Mortensen (1973) have demonstrated that








13

under certain restrictive assumptions concerning the

production technology and adjustment cost structure, a

flexible accelerator mechanism of input adjustment can

be derived from the solution of a dynamic optimization

problem. The empirical usefulness of this approach is

limited, however, as the underlying input demand

equations are expressed as derivatives of the

production function. Thus, any restrictions inherent

in the production function employed in the

specification of the dynamic objective function are

manifest in the demand equations.

Berndt, Fuss, and Waverman (1979), and Denny,

Fuss, and Waverman (1979) derived systems of input

demand equations consistent with the flexible

accelerator by introducing static duality concepts

into the dynamic problem. The derivation of input

demands through the use of a static dual function

reduces the restrictions imposed by the primal

approach utilizing a production function. Given the

assumption of quadratic costs of adjustment as an

approximation of the true underlying cost structure,

these analyses obtained systems of variable input

demand functions and net investment equations by

solving the Euler equation corresponding to a dynamic

objective function. This methodology, however, is








14

generally tractable for only one quasi-fixed input and

critically relies on the quadratic adjustment cost

structure.

Duality is a convenience in models of static

optimizing behavior. However, the application of

static duality concepts to a dynamic objective

function is somewhat limited. McClaren and Cooper

(1980) first explored a dynamic duality between the

firm's technology and a value function representing

the maximum value of the integral of discounted future

profits. Epstein (1981) established a full

characterization of this dynamic duality using the

Bellman equation corresponding to the dynamic problem.

The optimal control theory underlying the

solution to a dynamic optimization problem is

consistent with quasi-fixity and the cost of

adjustment hypothesis. The initial state is

characterized by those inputs assumed fixed in the

short run. Net investment in these inputs serves as

the control (optimal in that the marginal benefit

equals the cost of investment) that adjusts these

input levels towards a desired or optimal state. This

optimal state corresponds to the optimal levels of a

long-run static optimization in which all inputs are

variable.








15

Objectives

The objective of this analysis is to utilize

dynamic duality to specify and estimate a system of

variable input demands and net investment equations

for aggregate southeastern U.S. agriculture. Inherent

to this effort is a recognition of the empirical

applicability of dynamic duality theory to a small

portion of U.S. agriculture. While the study has no

pretense of determining the acceptance or rejection of

the methodology for aggregate economic analysis, a

presentation of the methods employed and difficulties

encountered may provide some basis for further

research.

In addition to obtaining estimates of the optimal

rates of net investment in land and capital,

appropriate regularity conditions are evaluated. The

clear distinction between the short- and long-run

permits derivation of short- and long-run price

elasticities for all inputs. Furthermore, the

specificiation used for the value function permits the

testing of hypotheses concerning the degree of fixity

of land and capital and the degree of interdependence

in the rates of net investment in these inputs. The

potential significance of this interdependence with

respect to policy is briefly explored.










Scope

The dynamic objective function is expressed in

terms of quasi-fixed factor stocks, net investment,

the discount rate, and relative input prices.

Endogenizing the factors conjectured as responsible

for lower output prices in the introduction is no less

difficult in a dynamic setting than in a static model.

This study is content to explore the effects of

relative price changes on factor demands and

adjustment.

Labor and materials are taken to be variable

inputs while land and capital are considered as

potentially quasi-fixed. This treatment is dictated

by the available data at the regional level consistent

with measurement of variable inputs rather than factor

stocks. The development of appropriate stock measures

for labor should involve a measure of human capital

(Ball, 1985). In fact, the incorporation of human

capital in a model of dynamic factor demands is a

logical extension of the methodology as a means of not

only determining but explaining estimated rates of

adjustment. However, such a model exceeds the scope

of this analysis.

Additionally, a method of incorporating policy

measures in the theory remains for the future. Noting

this limitation, the Southeast is perhaps best suited








17

for an initial exploration of the methodology. The

diversity of product mix in the components of total

output in the Southeast reduces the influence of

governmental policies directed at specific commodities

or commodity groups. In 1980, the revenue share of

cash receipts for the commodity groups typically

subject to governmental price support in the U.S.,

namely dairy, feed grains, food grains, cotton,

tobacco, and peanuts, was nearly 36 percent of total

cash receipts, while the share of those commodities in

the Southeast was 18 percent.

Overview

A review of previous models incorporating dynamic

elements in the analysis of factor demands leads

naturally to the theoretical model developed in

Chapter II. An empirical model potentially consistent

with dynamic duality theory and construction of the

data measures follows in Chapter III. The estimation

results and their consistency with the regularity

conditions, the measures of short- and long-run factor

demands and price elasticities, and hypotheses tests

are presented in Chapter IV. The final chapter

discusses the implications of the results and the

methodology.













CHAPTER II
THEORETICAL DEVELOPMENT


The primary objective of this study is to specify

and estimate a system of dynamic input demands for

southeastern U.S. agriculture. In order to explore the

adjustment process of agricultural input use, the model

should be consistent with dynamic optimizing behavior,

quasi-fixity, and the adjustment cost hypothesis. This

entails an exploration of the empirical applicability of

a theory of dynamic optimization capable of yielding

such a system. Yet models of input adjustment, hence

dynamic input demands, based on static optimization

generally lack a theoretical foundation. Treadway

summarizes the incorporation of theory in these models.

A footnote is often included on the
adjustment-cost literature as if that
literature had fully rationalized the
econometric specification. And other
adjustment mechanisms continue to appear with
no discernible anxiety about optimality
exhibited by their users. Furthermore, it is
still common for economists to publish studies
of production functions separately from
studies of dynamic factor demand without so
much as mentioning that the two are
theoretically linked. (Treadway, 1974, p. 18)

In retrospect, the search for a theoretical foundation

for empirical models rationalized on the notions of

quasi-fixity and adjustment costs culminate in the very








19

theory to be empirically explored. While these prior

empirical models of input demand and investment are not

necessarily consistent with the theoretical model

finally developed, they are important elements in its

history.

This chapter includes a review of dynamic input

demand models, dynamic in the sense that changes in

input levels are characterized by an adjustment process

of some form. The alternative models are evaluated with

respect to empirical tractability and adherence to

economic theory. In the first section, an adjustment

mechanism is imposed on an input demand derived from

static optimization. These models, whether the

adjustment mechanism is a single coefficient or a matrix

of coefficients, are empirically attractive but lack a

firm theoretical foundation.

The models in the second group are based on dynamic

optimization. These models are theoretically

consistent, yet limited by the form of the underlying

production function or the number of inputs which may be

quasi-fixed, even though static duality concepts are

incorporated.

The advantages presented by static duality lead to

the development of dynamic duality theory. The

application of dynamic duality theory to a problem of

dynamic optimizing behavior permits the derivation of a








20

system of dynamic input demands explicitly related to

the underlying production technology and offers a means

of empirically analyzing the adjustment process in the

demand for agricultural inputs in a manner consistent

with optimizing behavior.

Dynamic Models Using Static Optimization

The input demand equations derived from a static

objective function with at least one input held constant

provide limited information relative to input

adjustment. The derived demands are conditional on the

level of the fixed input(s). The strict fixity of some

inputs makes such models inappropriate for a dynamic

analysis.

The demands derived from the static approach

without constraints on factor levels characterize

equilibrium or optimal demands if the factors are in

fact freely variable. However, a full adjustment to a

new equilibrium given a change in prices is inconsistent

with the observed demand for some inputs. Input

demand equations derived from a static optimization

problem, whether cost minimization or profit

maximization, characterize input demand for a single

period and permit either no adjustment in some inputs or

instantaneous adjustment of all inputs. There is

nothing in static theory to reflect adjustment in input








21

demands over time. Models of dynamic input demand based

on static optimization attempt to mimic rather than

explain this adjustment process.

Distributed Lags and Investment

The investigation of capital investment through

distributed lag models seems to represent a much greater

contribution to econometric modeling and estimation

techniques than to a dynamic theory of the demand for

inputs. However, models of capital investment

characterize an early empirical approach to quasi-fixity

of inputs, recognizing that net investment in capital is

actually an adjustment in the dynamic demand for capital

as a factor of production. Additionally, these models

developed an implicit rental price or user cost of a

quasi-fixed input as a function of depreciation, the

discount rate, and tax rate as the appropriate measure

of the quasi-fixed input price (e.g., Hall and

Jorgenson, 1967).

Particular lag structures identified and employed

in the analysis of capital investment include the

geometric lag (Koyck, 1954), inverted V lag (DeLeeuw,

1962), polynomial lag (Almon, 1965), and rational lag

(Jorgenson, 1966). The statistical methods and problems








22

of estimating these lag forms has been addressed

extensively. Lag models inherently recognize the

dynamic process, as current levels of capital are

assumed to be related to previous stocks. A problem

with these models is that lag structures are arbitrarily

imposed rather than derived on the basis of some theory.

Griliches (1967, p. 42) deems such methods "theoretical

ad-hockery."

The basic approach in distributed lag models is to

derive input demand equations from static optimization

with all inputs freely variable. The demand equations

obtained characterize the long-run equilibrium position

of the firm. If an input is quasi-fixed, it will be

slow to adjust to a new equilibrium position. The

amount of adjustment, net investment, depends on the

difference in the equilibrium demand level and the

current level of the input.

The underlying rationale of a distributed lag is

that the current level of a quasi-fixed input is a

result of the partial adjustments to previous

equilibrium positions. The various forms of the

distributed-lag determine how important these past

adjustments are in determining the current response.

The finite lag distributions proposed by Fisher (1937),



1 Maddala (1977, pg. 355-76) presents econometric
estimation methods and problems.








23

DeLeeuw (1962), and Almon (1965) limit the number of

prior adjustments that determine the current response.

The infinite lag distributions (Koyck, 1954; Jorgenson,

1966) are consistent with the notion that the current

adjustment depends on all prior adjustments.

While such models may characterize the adjustment

process of a single factor, there is little economic

information to be gained. There is no underlying

foundation for a less-than-complete adjustment or

existence of a divergence of the observed and optimal

level of the factor. The individual factor demand is a

component of a system of demands derived from static

optimization, yet its relationship to this system is

often ignored.

The Partial-Adjustment Model

The partial adjustment model put forth by Nerlove

(1956) provides an empirical recognition of input demand

consistent with quasi-fixity. The partial adjustment

model as a dynamic model has been widely employed

(Askari and Cummings (1977) cite over 600 studies), and

continues to be applied in agricultural input demand

analyses (e.g., Kolajo and Adrian, 1984).

The partial adjustment model recognizes that some

inputs are neither fixed nor variable, but rather quasi-








24

fixed in that they are slow to adjust to equilibrium

levels. In its simplest form, the partial adjustment

model may be represented by


(2.1) xt Xt-l = a ( x*t xt-1)


where xt is the observed level of some input x in period

t, and x* t is the equilibrium input level in period t

defined as a function of exogenous factors. The

observed change in the input level represented by xt -

xt-l in (2.1) is consistent with a model of net

investment demand for the input. The observed change in

the input is proportional to the difference in the

actual and equilibrium input levels. Assuming the firm

seeks to maximize profit, x t becomes a function of

input prices and output price such that


(2.2) x*t = f(Py, w1 *.... wn).


where output price is denoted by py, and the wi, i = 1

to n, are the input prices. Equation (2.2) allows the

unobserved variable, x*t, in (2.1) to be expressed as

observations in the current and the prior period. The

parameter a in (2.1) represents the coefficient of

adjustment of observed input demand to the equilibrium

level.

The model is a departure from static optimization

theory in that x*t is no longer derived from the first-







25

order conditions of an optimal solution for a static

objective function. While static theory does not

directly determine the form of the adjustment process,

there is an explicit recognition that certain inputs are

slow to adjust to long-run equilibrium levels.

Interrelated Factor Demands

Coen and Hickman (1970) extended the approach of

distributed-lag models to a system of demand equations

for each input of the production function employed in

the static optimization. Essentially, the input demand

equations derived from the production function under

static maximization conditions are taken as a system of

long-run or equilibrium demand equations. A geometric

lag is arbitrarily imposed on the differences in actual

and equilibrium input levels. The shared parameters

from the underlying production function are restricted

to be identical across equations.

Coen and Hickman apply this model to labor and

capital demands derived from a Cobb-Douglas production

function. However, this method becomes untractable when

applied to a more complex functional form or a much

greater number of inputs. Additionally, the adjustment

rate or lag structure for each input is not only

arbitrary but remains independent of the disequilibrium

in the other factors.








26

Nadiri and Rosen (1969) formulate an alternative

approach to a system of interrelated factor demands

where the adjustment in one factor depends explicitly on

the degree of disequilibrium in other factors. The

model is a generalization of the partial adjustment

model to n inputs such that


(2.6) xt xt-l = B [x*t Xt-1].


This specification is similar to that in (2.1), except

that xt and x*t are n x 1 vectors of actual and

equilibrium levels of inputs, and the adjustment

coefficent becomes an n x n matrix, B. Individual input

demand equations in (2.6) are of the form

n
(2.7) xit = z bij*(x*j,t xj,t-1) +
jfi

bi'*(x*i,t xi't-l) + xit-1*

This representation permits disequilibrium in one input

to affect demand for another input. This

interdependence allows inputs to "overshoot" equilibrium

levels in the short run. For example, assume an input

is initially below its equilibrium level. Depending on

the sign and magnitude of the coefficients in (2.7), the

adjustment produced by disequilibria in other inputs may

drive the observed level of the input beyond the long-

run level before falling back to the optimal level.










Nadiri and Rosen (1969) derived expressions for

equilibrium factor levels using a Cobb-Douglas

production function in a manner similar to Coen and

Hickman (1968). However, they failed to consider the

implied cross-equation restrictions on the parameters

implied by the production function. Additionally,

stability of such a system requires that the

characteristic roots of B should be within the unit

circle, yet appropriately restricting each bij is

impossible.

Neither Coen and Hickman, or Nadiri and Rosen

provide a distinction between variable and quasi-fixed

inputs. All inputs are treated as quasi-fixed and the

adjustment mechanism is extended to all inputs in the

system. They do provide key elements to a model of

dynamic factor demands in that Coen and Hickman

recognize the relationship between the underlying

technology and the derived demands and Nadiri and Rosen

incorporate an interdependence of input adjustment.

Dynamic Optimization

Recently, there has been a renewed interest in

optimal control theory and its application to dynamic

economic behavior. As Dorfman (1969, pg. 817) notes,

although economists in the past have employed the

calculus of variations in studies of investment

(Hotelling, 1938; Ramsey, 1942), the modern version of








28

the calculus of variations, optimal control theory, has

been able to address numerous practical and theoretical

issues that previously could not even be formulated in

static theory.

Primal Approach

Eisner and Strotz (1963) developed a theoretical

model of input demand consistent with dynamic optimizing

behavior and a single quasi-fixed input. Lucas (1967)

and Gould (1968) extended this model to an arbitrary

number of quasi-fixed inputs. However, these extensions

are limited by the nature of adjustment costs external

to the firm. Thus, the potential interdependence of

adjustment among quasi-fixed inputs is ignored.

Treadway (1969) introduced interdependence of

quasi-fixed inputs by internalizing adjustment costs in

the production function of a representative firm. The

firm foregoes output in order to invest in or adjust

quasi-fixed inputs. Assuming all inputs are quasi-

fixed, the underlying structure of this model is shown

by


(2.8) V = max f e-rt(f(x,x) p'x) dt,



where V represents the present value of current and

future profits, x is a vector of quasi-fixed inputs, and

x denotes net investment in these inputs. The vector p

represents the user costs or implicit rental prices of








29

the quasi-fixed inputs normalized by output price. The

current levels of the quasi-fixed inputs serve as the

initial conditions for the dynamic problem.

Assuming a constant real rate of discount, r, and

static price expectations, the Euler equations

corresponding to an optimal adjustment path2 for the

quasi-fixed inputs are given by


(2.9) [fkk]i + [f.xJk = [f (x,*)] + r[f.(x,k)] p.


Treadway assumed the existence of an equilibrium

solution to (2.9), where x = x = 0, in order to derive a

system of long-run or equilibrium demand equations for

the quasi-fixed inputs. However, derivation of a demand

equation for net investment is more complicated, in that

an explicit solution for x exists for only restricted

forms of the production function (e.g., Treadway, 1974).

Yet, net investment demand is the key to characterizing

the dynamic adjustment of quasi-fixed inputs.

The difficulty of deriving input demand equations

from a primal dynamic optimization problem are apparent.

While input demand equations derived from a primal

static optimization involve first-order derivatives of



2 Additionally, the optimality of x depends on a
system of transversality conditions where
-rt
lim e [f] = 0, and a Legendre condition that

[f..] negative semi-definite (Treadway, 1971 p. 847).
xxfK








30

the production function, derivation of input demands

from (2.9) involve second-order derivatives as well.

Treadway (1974) shows that the introduction of variable

inputs further increases the difficulty in deriving

input demands from a primal dynamic optimization.

This derivation is necessarily in terms of a

general production function, owing to the primal

specification of the objective function. In terms of

empirical interest, estimation of such a system is

nonexistent. However, the establishment of necessary

conditions for an optimal solution to the dynamic

problem in terms of the technology provides the

foundation for the use of duality that follows.

Application of Static Duality Concepts

The primary difficulty in estimating a system of

dynamic factor demands from the direct or primal

approach is that the characteristic equations underlying

the dynamic optimization problem in (2.8) are

necessarily expressed in terms of first- and second-

order derivatives of the dynamic production function.

Thus, unless a truly flexible functional form of the

production function (e.g., Christensen, Jorgenson, and

Lau, 1973) is employed, restrictions on the underlying

technology are imposed a priori.

In a static model, a behavioral function such as

the profit or cost function with well-defined properties








31

can serve as a dual representation of the underlying

technology (Fuss and McFadden, 1978). A system of

factor demands is readily derived from the profit

function by Hotelling's Lemma (7r (p,py)/apj =

-x* (p,py)) or from the cost function by Shephard's

Lemma (aC(p,y)/apj = xj(p,y)). However, all inputs are

necessarily variable. These static models had been

extended to the restricted variable profit and cost

functions that hold some factors as fixed (e.g., Lau,

1976).

Berndt, Fuss, and Waverman (1979) incorporated a

restricted variable profit function into the primal

dynamic problem in order to simplify the dynamic

objective function. Berndt, Morrison-White, and Watkins

(1979) derive an alternative method employing the

restricted variable cost function as a component of the

dynamic problem of minimizing the present value of

current and future costs. The advantages of static

duality reduce the explicit dependence on the form of

the production function and facilitate the incorporation

of variable inputs in the dynamic problem.

Berndt, Fuss, and Waverman specified a normalized

restricted variable profit function presumably dual to








32

the technology in (2.8) based on the conditions for such

a static duality as presented by Lau (1976). This

function may be written as


(2.10) 7r(w,x,x) = max f(L,x,x) w'L
L > 0


Assuming that the level of net investment is optimal for

the problem in (2.8), the remaining short-run problem as

reflected in (2.10) is to determine the optimal level of

the variable factor L dependent on its price, w, and on

the quasi-fixed factor stock and net investment.

Current revenues are 7(w,x,x) p'x, which can be

substituted directly into (2.8). The use of static

duality in the dynamic problem allows the production

function to be replaced by the restricted variable

profit function. A general functional form for 7,

quadratic in (x,x), can be hypothesized as


(2.11) IL(w,x,k)= a0 + a'x + b'x + l/2[x' x'] CA x


where ao, the vectors a, b, and matrices A, B, and C

will be dependent on w in a manner determined by an

exact specification of 7(w,x,k). The Euler equation for

the dynamic problem in (2.8) after substitution of

(2.11) is


(2.12) Bx + (C'- C rB)x (A + rC')x = rb p + a


Note that this solution is now expressed in terms of the








33

parameters of the restricted variable profit function

instead of the production function. A steady state or

equilibrium for the quasi-fixed factors denoted as

x(p,w,r) can be computed from (2.12) evaluated at x =
x = 0, such that


(2.13) x (p,w,r) = -[A + rC']-1(rb p + a).


Applying Hotelling's Lemma to the profit function yields

a system of optimal variable input demands, L*,

conditional on the quasi-fixed factors and net

investment of the form


(2.14) L*(x,p,w,r) = -!w(w,x,x).


The system of optimal net investment equations can

expressed as


(2.15) x(x,p,w,r) = M(w,r)[x x(p,w,r)].


The exact form of the matrix M is uniquely determined by

the specification of the profit function in the solution

of (2.10). Only in the case of one quasi-fixed factor

has this matrix been expressed explicitly in terms of

the parameters of the profit function, where M = r/2 -

(r2/4 + (A + rC)/B)1/2, where A, B, and C are scalars.

In order to generalize this methodology to more

than one quasi-fixed factor, LeBlanc and Hrubovcak

(1984) specified a quadratic form such that the optimal








34

levels of variable inputs depend only on factor stocks

and are independent of investment. Therefore, they rely

on external adjustment costs reflected by rising supply

prices of the factors. In addition, the adjustment

mechanism for each input is assumed independent of the

degree of disequilibrium in other quasi-fixed factors

such that the implied adjustment matrix is diagonal.

This facilitates expression of net investment demand

equations for more than one quasi-fixed input in terms

of parameters of the profit function, but at

considerable expense to the generality of their

approach.

Dynamic Duality

The use of static duality in these models of

dynamic factor demands leads naturally to the

development of a dual relationship of dynamic optimizing

behavior and an underlying technology. Such a general

dynamic duality was conjectured by McLaren and Cooper

(1980). Epstein (1981) establishes the duality of a

technology and a behavioral function consistent with

maximizing the present value of an infinite stream of

future profits termed the value function.








35

Theoretical Model

The firm's problem of maximizing the present value

of current and future profits3 may be written as


(2.16) JO(K,p,w) = max.f e [F(L,K,K) w'L p'K] dt
0
L > 0, K 0

subject to K(O) = KO > 0.


The production function F(L,K,K) yields the maximum

amount of output that can be produced from the vectors

of variable inputs, L, and quasi-fixed inputs, K, given

that net investment K is taking place. The vectors w

and p are the rental prices or user costs corresponding

to L and K respectively, normalized with respect to

output price. Additionally, r > 0 is the constant real

rate of discount, and KO is the initial quasi-fixed

input stock. J(Ko,p,w,r) then characterizes a value

function reflecting current and discounted future

profits of the firm.

The following regularity conditions are imposed on

the technology represented by F(L,K,K) in (2.16):

T.l. F maps variable and quasi-fixed inputs and
net investment in the positive orthant; F,
FL, and FK are continuously differentiable.
T.2. FL, FK > 0, Fk > 0 as K < 0.
T.3. F is strongly concave in (L, K).



3 The exposition of dynamic duality draws heavily
on the theory developed by Epstein (1981).








36

T.4. For each combination of K, p, and w in the
domain of J, a unique solution for (2.16)
exists. The functions of optimal net
investment, K *(K,p,w), variable input demand,
L* (K,p,w), and supply, y*(K,p,w) are
continuously differentiable in prices, the
shadow price function for the quasi-fixed
inputs, (K,p,w), is twice continuously
differentiable.
T.5. *p(K,p,w) is nonsingular for each
combination of quasi-fixed inputs and input
prices.
T.6. For each combination of inputs and net
investment, there exists a corresponding set
of input prices such that the levels of the
inputs and net investment are optimal.
T.7. The problem in (2.16) has a unique steady
state solution for the quasi-fixed inputs
that is globally stable.

Condition T.1 requires that output be positive for

positive levels of inputs. Declining marginal products

of the inputs characterize the first requirements of

T.2. Internal adjustment costs are reflected in the

requirements of Fk. The extension allowing for positive

and negative levels of investment requires that the

adjustment process be symmetric in the sense that when

net investment is positive some current output is

foregone but when investment is negative current output

is augmented. Consistent with assumed optimizing

behavior, points that violate T.6 would never be

observed.








37

Assuming price expectations are static, inputs

adjust to "fixed" rather than "moving" targets of long-

run or equilibrium values. However, prices are not

treated as fixed. In each subsequent period a new set

of prices is observed which redefine the equilibrium.

As the decision period changes, expectations are altered

and previous decisions are no longer optimal. Only that

part of the decision optimal under the initial price

expectations is actually implemented.

Given the assumption of static price expectations

and a constant real discount rate, the value function in

(2.16) can be viewed as resulting from the static

optimization of a dynamic objective function. Under

these assumptions and the regularity conditions imposed

on F(L,K,K), the value function J(K,p,w) is at a maximum

in any period t if it satisfies the Bellman (Hamilton-

Jacobi) equation for an optimal control (e.g.,

Intriligator, 1971, p. 329) problem such that


(2.17) rJ*(K,p,w) = max {F(L,K,K) w'L p'K +

JK(K,p,w) K*},

where JK(K,p,w) denotes the vector of shadow values

corresponding to the quasi-fixed inputs, and K*

represents the optimal rate of net investment.








38

Through the Bellman equation in (2.17), the dynamic

optimization problem in (2.16) may be transformed into a

static optimization problem. In particular, (2.17)

implies that the value function may be defined as the

maximized value of current profit plus the discounted

present value of the marginal benefit stream of an

optimal adjustment in net investment. Thus, through the

Maximum Principle (e.g., Intriligator, 1971, p. 344) the

maximizing values of L and K in (2.17) when K = KO are

precisely the optimal values of L and K in (2.16) at t =

0.

Utilizing (2.17), Epstein (1981) has demonstrated

that the value function is dual to F(L,K,K) in the

dynamic optimization problem of (2.16) in that,

conditional on the hypothesized optimizing behavior,

properties of F(L,K,K) are manifest in the properties of

J(K,p,w). Conversely, specific properties of J(K,p,w)

may be related to properties on F(L,K,K). Thus, a full

dynamic duality can be shown to exist between J(K,p,w)

and F(L,K,K) in the sense that each function is

theoretically obtainable from the other by solving the

appropriate static optimization problem as expressed in

(2.17). The dual problem can be represented by


(2.18) F*(L,K,K) = min {rJ(K,p,w) + w'L + p'K -
p,w

JK-K).








39

The static representation of the value function in

(2.17) also permits derivation of demand functions for

variable inputs and net investment in quasi-fixed

factors. Application of the envelope theorem by

differentiating (2.17) with respect to w yields the

system of variable factor demand equations


(2.19) L*(K,p,w) = -rJw' + JwKK'*,


and differentiation with respect to p yields a system of

optimal net investment equations for the quasi-fixed

inputs,


(2.20) K*(K,p,w) = JpK-1.(rJp'+ K)


This generalized version of Hotelling's Lemma permits

the direct derivation of a complete system of input

demand equations theoretically consistent with dynamic

optimizing behavior. The ability to derive an equation

for net investment is crucial to understanding the

short-run changes or adjustments in the demand for

quasi-fixed inputs. The system is simultaneous in that








40

the optimal variable input demands depend on the optimal

levels of net investment, K*. In the short-run, when

K* f 0, the demand for variable inputs is conditional on

net investment and the stock of quasi-fixed factors.

In addition, a supply function for output is

endogenous to the system. The optimal supply equation

derived by solving (2.17) for F(L,K,K) where

(L,K)=(L*,K*) may be expressed as

(2.21) y*(K,p,w) = rJ + w'L* + p'K JKK*.


As for the variable input equation, optimal supply

depends on the optimal level of net investment. This is

consistent with internal adjustment costs as the cost of

adjusting quasi-fixed factors through net investment is

reflected in foregone output.

The regularity conditions implied by the properties

of the production function are manifested in (2.19)-

(2.21) and provide an empirically verifiable set of

conditions on which to evaluate the theoretical

consistency of the model. Consistency with the notion

of duality dictates that the previously noted properties

of the technology be reflected in the value function.

The properties (V) manifest in J from the technology are

V.1. J is a real-valued, bounded-from-below
function defined in prices and quasi-fixed
inputs. J and JK are twice-continously
differentiable.








41

V.2. rJK + P JKK(k*) JK > o, J0O as k*>0.
V.3. For each element in the domain of J, y*0;
for such K in the domain of J, (L*, K, K*)
maps the domain of q onto the domain of F.
V.4. The dynamic system K*, K(0) = KO, in
the domain of J defines a profile K(t) such
that (K(t),p,w) is in the domain of J for all
t and K(t) approaches R(p,w), a globally
steady state also in the domain of J.
V.5. JpK is nonsingular.
V.6. For the element (K,p,w) in the domain of J, a
minimum in (2.18) is attained at ( if
(K,L)=(K*,L*).
V.7. The matrix Kw Lp is nonsingular for


each element, (K,p,w), in the domain of J.

These regularity conditions are essential in

establishing the dynamic duality between the technology

and the value function. In fact, the properties of J

are a reflection of the properties of F. The definition

of the domain of F implies V.1. Condition V.2 reflects

in (p,w) the restrictions imposed on the marginal
products of the inputs, FL and Fk, and net investment,

FK, in T.2. The conditions in V.3 with respect to an
optimal solution in price space, (p,w), are dual to the

conditions for an optimal solution in input space,

(L,K), maintained in T.6. V.4 is the assumption of the
global steady state solution as in T.7. Given JK =X "

noted earlier, V.5 is the dual of T.5. V.7 is a

reflection of the concavity requirement of T.3.








42

Condition V.6 may be interpreted as a curvature

restriction requiring that first-order conditions are

sufficient for a global minimum in (2.18). Epstein

(1981) has demonstrated that if JK is linear in (p,w),

V.6 is equivalent to the convexity of J in (p,w).

An advantage of dynamic duality is that these

conditions can be readily evaluated using the parameters

of the empirically specified value function. The

specification of a functional form for J must be

potentially consistent with these properties.

The Flexible Accelerator

Dynamic duality in conjunction with the value

function permits the theoretical derivation of input

demand systems consistent with dynamic optimizing

behavior. Such a theoretical foundation establishes the

relationship between quasi-fixed and variable input

demand and an adjustment process in the levels of quasi-

fixed inputs as a consequence of the underlying

production technology.

One may note that the net investment demand

equation for a single quasi-fixed input derived from the

incorporation of the restricted variable profit function

in the primal dynamic problem yields a coefficient of

adjustment as a function of the discount rate and the

parameters of the profit function similar to the

constant adjustment coefficient employed in the partial








43

adjustment model. However, an explicit solution of the

system of net investment equations with two or more

quasi-fixed inputs in terms of an adjustment matrix is

difficult. Nadiri and Rosen (1969) considered their

model as an approximate representation of an adjustment

matrix derived from dynamic optimization.

Dynamic duality provides a theoretical means of

deriving a wide variety of adjustment mechanisms. The

difficulties in relating a specific functional form of

the production function to the adjustment mechanism in

the direct or primal approach and the limited

applicability of the adjustment mechanism derived from

incorporating the restricted variable profit function in

the dynamic objective function are alleviated

considerably. However, the functional form of the value

function is critical in determining the adjustment

mechanism.

The adjustment mechanism of interest in this

analysis is the multivariate flexible accelerator.

Although the theoretical model relies on a constant real

discount rate, it is not unreasonable to hypothesize

that this constant rate of discount is partially

responsible for the rates of adjustment in quasi-fixed

inputs to their equilibrium levels. Noting the

potential number of parameters and non-linearities in

the demand equations, an adjustment matrix of








44

coefficents as a function of the discount rate and the

parameters of the value function may be the desired form

of the adjustment process for empirical purposes.

Epstein (1981) establishes a general form of the

value function from which a number of globally optimal

adjustment mechanisms may be derived. The adjustment

mechanism of constant coefficents is a special case.4

The flexible accelerator [r + G] is globally optimal if

the value function takes the general form

(2.22) J(K,p,w) = g(K,w) + h(p,w) + p'G-lK.

This form yields JpK = G-1 and Jp = hp(p,w) + G-lK.

Substituting in (2.20) yields the optimal net investment

equations of the form


(2.23) K*(K,p,w) = G[rhp(p,w)] + [r + G]K.


Solving (2.23) for X(p,w) at K*=0,


(2.24) K(p,w) = -[r + G]-1G[rhp(p,w)].


Multiplying (2.24) by [r + G] and substituting directly

in (2.23) yields


(2.25) K*(K,p,w) = -[r + G]R(p,w) + [r + G]K =

[r + G] [K K(p,w)].



4 The derivation and proof of global optimality of
a general flexible accelerator is provided by Epstein,
1981, p. 92.








45

Thus, the flexible accelerator derived in (2.25) is

globally optimal given a value function of the form

specified in (2.25). While the accelerator is dependent

on the real discount rate, the assumption that this rate

is constant implies a flexible accelerator of constants.

The linearity of JK in (p,w), which implies the

convexity of J in (p,w), is crucial in the derivation of

a globally optimal flexible accelerator of fixed

coefficients.













CHAPTER III
EMPIRICAL MODEL AND DATA


Empirical Model

The specification of the value function J is taken

to be log-quadratic in normalized prices and quadratic

in the quasi-fixed inputs. The specific form of the

value function J(K,p,w) is thus given by


(3.1) J(K,p,w) = ag + a'K + b'log p + c'log w +

1/2(K'AK + log p'B log p + log w'C log w) +

+ log p'D log w + p'G'-lK + w'NK + p'G'-VKT -

W'VLT


where K = [K, A], a vector of the quasi-fixed inputs,

capital and land, p = [Pk, Pa] denotes the vector of

normalized (with respect to output price) prices for

the quasi-fixed inputs, and w = [pw, Pm], the vector

of normalized variable input prices for labor and

materials respectively. Thus, log p = [log pk, log

Pa] and log w = [log py, log pm]. T denotes a time
trend variable.

Parameter vectors are defined by a = [aK, aA], b

= [bk, ba, c = [cw cm), VK = IVK, VA], and VL =

[vL, vM]. The vectors VK and VL are technical change
parameters for the quasi-fixed inputs and variable

46








47

inputs. The variable input vector is defined by L =

[L, M), where L denotes labor and M denotes materials.
Parameter matrices are defined as:

A = aKK aKA B = [bkk bka C = c wm ,
aAK aAA bak baa cmw CmJ
D = dkw dkm N = nKw nKm and G-1= gKK KA.
daw dam nAw nAmj AK AA
Let G = [gKK gKA The matrices A, B, and C are
19AK 9AA
symmetric.

The incorporation of some measure of technical

change is perhaps as much a theoretical as empirical

issue. The assumption of static expectations applies

not only to relative prices but the technology as

well. The literature contains two approaches to the

problem of technical change in dynamic analysis:

detrending the data (Epstein and Denny, 1983) or

incorporating an unrestricted time trend (Chambers and

Vasavada, 1983b; Karp, Fawson, and Shumway, 1984). An

argument for the former (Sargent, 1978, p. 1027) is

that the dynamic model should explain the

indeterminate component of the data series--that which

is not simply explained by the passage of time.

However, as Karp, Fawson, and Shumway (1984, p. 3)

note, the restrictions of dynamic model reflected in

the investment equations involve real rather than

detrended economic variables so the restrictions may

not be appropriate for detrended values.








48

The latter approach is adopted in the above

specification of the value function in (3.1). Thus,

investment and variable input demand equations derived

from the value function in (3.1) include an

unrestricted time trend. This form allows the

technical change parameters to measure in part the

relative effect of technical change with respect to

factor use or savings over time. Note that the

presence of G-1 in the interaction of p, VK, and T in

the interaction of p, VK, and T in (3.1) ensures that

the technical change parameters enter the investment

demand equations without restriction.

The incorporation of technical change in the

value function serves as an illustration of the

difficulty in incorporating policy, human capital, and

other variables besides prices into the value

function. In static optimization, the input demand

equations are determined by first-order derivatives of

the objective function. Therefore, the interpretation

of parameters in terms of their effects on the

objective function is straightforward. The demand

equations derived from dynamic optimization contain

first- and second-order derivatives of the value

function. The value function can be specified to

permit a direct interpretation of the parameters in

terms of the underlying demand equations. However,








49

relating these parameters to the dynamic objective

function becomes difficult. Without estimating the

value function directly, one must rely on the

regularity conditions implied by dynamic duality to

ensure consistency of the empirical specification and

underlying theory.

Input Demand Eauations

Utilizing the generalized version of Hotelling's

Lemma in (2.20), the demand equations for optimal net

investment in the quasi-fixed inputs are given by

A
(3.2) K*(K,p,w) = G[rp"1(b + B log p + D log w) +

rVKT] + [r + G]K,


where K*(K,p,w) = [K (K,p,w), A*(K,p,w)] signifies

that optimal net investment in capital and land, is a

function of factor stocks and input prices. r is a

diagonal matrix of the discount rate, and p is a

diagonal matrix of the quasi-fixed factor prices. The

specification of G-1 in (3.1) permits direct

estimation of the parameters of G in the net

investment equations.

Net investment, or the rate of change in the

quasi-fixed factors, is determined by the relative

input prices and the initial levels of the quasi-fixed

factors, as evidenced by the presence of K in (3.2).

The premultiplication by G (G = JpK-1 from (2.20))








50

yields a system of net investment demand equations

that are nonlinear in parameters. The technical

change component for the quasi-fixed inputs in the

value function (3.1) enters the net investment demand

equations in a manner consistent with the assumption

of disembodied technical change.

The optimal short-run demand equations for the

variable inputs are derived using (2.19), and are

given by


(3.3) L*(K,p,w) = -rw'1(c + D log p + C log w) + rVwT

rNK + NK*(K,p,w),


where L*(K,p,w) = [L* (K,p,w), M*(K,p,w)], the

optimal short-run input demands for the variable

inputs, labor and materials. r is again a diagonal

matrix of the discount rate, and w is a diagonal

matrix of the variable input prices.

The short-run variable input demand equations

depend not only on the initial quasi-fixed input

stocks but the optimal rate of net investment in these

inputs as well. While variable inputs adjust

instantaneously, the adjustments are conditioned by

both K and K*. The presence of k*(K,p,w) in the

variable input demand equations dictates that net

investment and variable input demands are determined

jointly, requiring a simultaneous equations approach.








51

The derivation of optimal net investment and

variable input demands in (3.2) and (3.3) are

presented as systems in matrix notation. The precise

forms of the individual net investment and variable

input demands used in estimation are presented in

Appendix Table A-1.

Long-run Demand Equations

In the dynamic model, the quasi-fixed inputs

gradually adjust toward an equilibrium or steady

state. The long-run level of demand for an input is

defined by this steady state, such that there are no

more adjustments in the input level. In other words,

net investment is zero.

The long-run or steady state demands for the

quasi-fixed inputs are derived by solving (3.2) for K

when K*(K,p,w) = 0. The long-run demand equation for

the quasi-fixed factors is thus given by

A
(3.4) K(p,w) = [I + rG1]-1[rp-1(b + B log p +

D log w) + rVKT],


where K(p,w) =[K(p,w), A(p,w)]. Note that these long-

run demand equations are functions of input prices

alone.

Noting that the short-run demand equations for

the variable inputs in (3.3) are conditional on K and










K*, substitution of K(p,w) for K and K*(K,p,w) = 0 in

the short-run equations yields


(3.5) L(K,p,w) = -rw 1(c + D log p + C log w) + rVLT

rNK(p,w),


where L(K,p,w)=[L(K,p,w), M(K,p,w)]. The long-run

variable input demands are no longer conditional on

net investment, but are determined by the long-run

levels of the quasi-fixed inputs. The individual

long-run demand equations for all inputs are presented

in Appendix Table A-2.

Short-run Demands

The short-run variable input demands were

presented in (3.3). The variable input demands are

conditional on the initial levels of the quasi-fixed

inputs and optimal net investments. The short-run

demand for the quasi-fixed inputs requires the

explicit introduction of time subscripts in order to

define optimal net investment in discrete form as


(3.6) Kt*(K,p,w) = K*t Kt-11


where K*t = [K*t, A*t], the vector of quasi-fixed

inputs at the end of period t. Therefore, the short-

run demand for capital at the beginning of period t is


(3.7) K*t(K,p,w) = Kt-1 + K*t(K,p,w),








53

where K*t(K,p,w) is the optimal demand for the quasi-

fixed inputs in period t, Kt-1 is the initial stock at

the beginning of the period, and K*t is net investment

during the previous period. The short-run demand

equations for the quasi-fixed inputs are optimal in

the sense that the level of the quasi-fixed input,

K *t, is the sum of the previous quasi-fixed input

level and optimal net investment during the prior

period.

Returning to (3.2), the short-run demand equation

for the quasi-fixed input vector can be written


(3.8) K*t(K,p,w) = G[rp-l(b + B log p + D log w)] +

rVKT + [I + r + G]Kt-l,


where the time subscripts are added to clarify the

distinction between short-run demand and initial

stocks of the quasi-fixed inputs. The individual

short-run demand equations for the quasi-fixed inputs

are presented in Appendix Table A-3.

The Flexible Accelerator

The flexible accelerator matrix M = [r + G] was

shown to be consistent with the general form of the

value function in (3.1) in the previous chapter.








Rewriting (3.2) and multiplying both sides of the

equation by G-1 yields


(3.9) G-lk*(K,p,w) = rp"1(b + B log p + D log w) +

rVKT + [I + rG'1]K.


Multiplying both sides by [I + rG-1]-1 and noting that

[I + rG-1]-1 = [r + G]-1G, then (3.9) can be written
as


(3.10) [r + G]-1k*(K,p,w) = [I + rG--1]-lr-1^(b +

B log p + D log w) + rVKT] + K.

The first term on the right hand side of (3.10) is

identical to the negative of the long-run quasi-fixed

input demand equation in (3.4). Substituting -K(p,w)

in (3.10) and solving for K*(K,p,w) yields

(3.11) K *(K,p,w) = [r + G] [K K(p,w].


As may be noted, this is precisely the form of the

multivariate flexible accelerator.

Solving (3.11) for the individual equations, the

optimal net investment in capital is


(3.12) K* = (r + gKK)(K K) + gKA(A A),


and optimal net investment in land may be written


(3.13) A* = gAK(K X) + (r + gAA)(A A).








55

Thus, gKA and the parameters associated with land in

the value function appear in the net investment

equation for capital. Likewise, gAK and the

parameters associated with capital in the value

function appear in the net investment equation for

land.

Hypotheses Tests

The form of the flexible accelerator in (3.11)

permits direct testing of hypotheses on the adjustment

matrix in terms of nested parameter restrictions. The

appropriateness of these tests are based on Chambers

and Vasavada (1983b). Of particular interest is the

hypothesis of independent rates of adjustment for

capital and land which can be tested via the

restrictions gKA = gAK = 0. Independent rates of

adjustment indicate that the rate of adjustment to

long-run equilibrium for one quasi-fixed factor is

independent of the level of the other quasi-fixed

factors.

The hypothesis of an instantaneous rate of

adjustment for the quasi-fixed inputs relys on

independent rates of adjustment. Thus, a sequential

testing procedure is dictated. Given that the

hypothesis of independent rates of adjustment is not

rejected, instantaneous adjustment for land and

capital requires r + gKK = r + gAA = -1, in addition








56

to gKA = =AK = 0. If both inputs adjust
instantaneously, the adjustment matrix takes the form

of a negative identity matrix. Capital and land would

adjust immediately to long-run equilibrium levels in

each time period.

Regularity Conditions

An attractive feature of the theoretical model is

the regularity conditions that establish the duality

of the value function and technology. Even so, little

focus has been given to these conditions in previous

empirical studies beyond the recognition of the

existence of steady states for the quasi-fixed factors

and a stable adjustment matrix required by condition

V.4.

Without estimating the supply function or value

function directly it is impossible to verify the

regularity conditions stated in V.1., V.2 and Y*>0,

the first part of condition V.3. One can note with

slight satisfaction, however, that these conditions

are likely to be satisfied if ag, aK, and aA are

sufficiently large positive (Epstein, 1980, pg 88).

The differentiability of J and JK are, of course,

implicitly maintained in the choice of the value

function. The conditions in V.4 are readily verified

by determining if the long-run or equilibrium factor

demands at each data point are positive to ensure the








57

existence and uniqueness of the steady states.

Furthermore, the stability of these long-run demands

is ensured if the implied adjustment matrix is

nonsingular and negative definite. The nonsingularity

of the adjustment matrix is related to condition V.5,

the nonsingularity of JpK, as JpK = [M r]-l

demonstrated in the previous chapter. Regularity

condition V.7 is easily verified by the calculation of

demand price elasticities for the inputs.

Condition V.6 may be viewed as a curvature

restriction ensuring a global minimum to the dual

problem. Since JK is linear in prices, this condition

is equivalent to the convexity of the value function J

in input prices. The appropriate Hessian of second-

order derivatives is required to be positive definite.

Elasticities

One particularly attractive aspect of dynamic

optimization is the clear distinction between the

short run, where quasi-fixed inputs only partially

adjust to relative price changes along the optimal

investment paths, and the long run, where quasi-fixed

inputs fully adjust to their equilibrium levels.

However, expectations with respect to the signs of

price elasticities based on static theory are not

necessarily valid in a dynamic framework.








58

Treadway (1970) and Mortensen (1973) have shown

that positive own-price elasticities are an indication

that some inputs contribute not only to production but

to the adjustment activities of the firm. Thus in the

short-run, the firm may employ more of the input in

response to a relative price increase in order to

facilitate adjustment towards a long-run equilibrium.

However, this does not justify a positive own-price

elasticity in the long-run. This same contribution to

the adjustment process may also indicate short-run

effects which exhibit greater elasticity than the long

run. The firm may utilize more of an input in the

short-run in order to enhance adjustment than in the

long-run in response to a given price change.

Short-run variable input demand elasticities may

be calculated from (3.3). For example, the elasticity

of labor demand with respect to the price of the jth

input, e L,pji is



(3.15) E: + + A.
L,p *
3 Pj 9K apj 9A a p. L


The short-run elasticity of demand for a variable

input depends not only on the direct effect of a price

change, but the also on the indirect effects of a

price change on the short-run demand for the quasi-

fixed inputs.








59

The short-run price elasticity for a quasi-fixed

input is obtained from (3.7). The short-run demand

elasticity for capital with respect to a change in the

price of the jth input, sK,pj is


(3.16) E s K pj.
pj ap K


The short-run elasticity of demand for a quasi-fixed

input depends only on the direct price effect in the

short-run demand equation.

The long-run elasticity of demand for a variable

input can be obtained from (3.5). In the long run,

all inputs are at equilibrium levels. Thus, the long-

run elasticity of demand for labor with respect to the

price of input j is


1 L +DL 3 K + L D A pj
(3.17) gLlp = + + .
Spj DK a p. 3 A D p. L


The long-run elasticity of a variable input is

conditional on the effect of a price change in the

equilibrium levels of the quasi-fixed inputs.

The long-run demand elasticity for a quasi-fixed

input is determined from (3.4). The long-run








60

elasticity of demand for capital with respect to the

jth input price is



(3.18) = _
a Pj


In contrast to the short-run demand for a quasi-fixed

input, where the short-run demand for one quasi-fixed

input is determined in part by the level of the other

quasi-fixed input, the long-run demand for a quasi-

fixed input is solely an argument of prices.

Data Construction

The data requirements for the model consist of

stock levels and net investment in land and capital,

quantities of the variable inputs, labor and

materials, as well as normalized (with respect to

output price) rental prices for the inputs for the

Southeast region. This region corresponds to the

states of Alabama, Florida, Georgia, and South

Carolina. The appropriate data are constructed for

the period from 1949 through 1981.

Data Sources

Indices of output and input categories for the

the Southeast are provided in Production and

Efficiency Statistics (USDA, 1982). The inputs

consist of farm power and machinery, farm labor, feed,

seed, and livestock purchases, agricultural chemicals,








61

and a miscellaneous category. These indices provide a

comprehensive coverage of output and input items used

in agriculture for the respective categories.

Annual expenditures for livestock, seed, feed,

fertilizer, hired labor, depreciation, repairs and

operations, and miscellaneous inputs for each state

were obtained from the State Income and Balance Sheet

Statistics (USDA) series. The expenditures for each

of the Southeastern states are summed to form regional

expenditures corresponding to the appropriate regional

input indices cited above. This same series also

contains revenue data for each state in the categories

of cash receipts from farm marketing, value of home

consumption, government payments, and net change in

farm inventories. These data are aggregated across

states to form a regional measure of total receipts.

These sources provide the data for the

construction of capital, materials, and labor quantity

indices and capital and materials price indices. A

GNP deflator is used to convert all expenditures and

receipts to 1977 dollars. Additional data is drawn

from Farm Labor (USDA) in order to construct a labor

price index. Farm Real Estate Market Developments

(USDA) provides quantity and price data for land. The

undeflated regional expenditure and input data are

provided in Appendix B.










Capital

Capital equipment stocks and investment data are

not available below the national level prior to 1970.

Therefore, the mechanical power and machinery index

was taken as a measure of capital stocks. As Ball

(1985) points out, this index is intended to measure

the service flow derived from capital rather than the

actual capital stock. The validity of the mechanical

power and machinery index as a measure of capital

stock rests on the assumption that the service flow is

proportional to the underlying capital stock.

It is possible that the service flow from capital

could increase temporarily without an increase in the

capital stock if farmers used existing machinery more

intensely without replacing worn-out equipment.

Eventually, worn-out capital would have to replaced.

Ball relies on a similar assumption of proportionality

in employing the perpetual inventory method

(Jorgenson, 1974) in deriving capital stocks. This

method relies on the assumption of a constant rate of

replacement in using gross investment to determine

capital stocks such that


(3.19) Ait = lit + (1 6i)Ai,t-1,


where Ait is capital stock i in period t, lit is gross

investment, and 6i is the rate of replacement. Even








63

the regional level, the perpetual inventory approach

appears to share the potential weakness of the

mechanical power and machinery index.

Determining the appropriate price of capital

presents additional difficulty. Hall and Jorgenson

(1967) and Jorgenson (1967) define the user cost or

implicit rental price of unit of capital as the cost

of the capital service internally supplied by the

firm. This actual cost is complicated by the discount

rate, service life of the asset, marginal tax rate,

allowable depreciation, interest deductions, and

degree of equity financing.

An alternate measure of user cost is provided by

expenditure data representing actual depreciation or

consumption of capital in terms of replacement cost

and repairs and operation of capital items (Appendix

Table B-l). By combining these expenditure categories

in each time period to represent the user cost of the

capital stock in place during the period, these

expenditures and the machinery index can be used to

construct an implicit price index for the region.








64

An implicit price index for capital is

constructed using Fisher's weak factor reversal test

(Diewert, 1976). The implicit price index may be

calculated by


(3.20) Pit = (Eit/Eib) / Qit,


where Qit and Pit denote the quantity and price

indices corresponding to the ith input in period t,

and expenditures on the ith input in the same time

period are denoted by Eit, and b denotes the index

base period. Fisher's weak factor reversal test for

price and quantity indices is satisfied if the ratio

of expenditures in the current time period to the base

is equal to the product of the price and quantity

indices in the current time period. Since the

machinery index and expenditure data are based in

1977, the resulting implicit price index for capital

is also based in 1977.

Land

The land index represents the total acres in

farms in the Southeast. The regional total is the sum

of the total in each state. Hence, farmland is

assumed homogeneous in quality within each state. An

adjustment in these totals is necessary for the years








65

after 1975 as the USDA definition of a farm changed.5

Observations after 1975 are adjusted by the ratio of

total acres under the old definition to total acres

using the new definition.

A regional land price index is constructed by

weighting the deflated index of the average per acre

value of farmland in each state by that state's share

of total acres in the region. Unlike most price

indices, the published index of farmland prices is not

expressed in constant dollars. As rental prices are

not available for the region, the use of an index of

price per acre implicitly assumes that the rental rate

is proportional to this price. The regional acreage

total, quantity index, and price index may be found in

Appendix Table B-2.

Labor

Beyond the additional parameters needed in the

empirical model to treat labor as quasi-fixed, the

farm labor index reflects the quantity employed, not

necessarily the stock or quality of labor available.

Hence, the regional labor index by definition

represents a variable input. The USDA index of labor

weights all hours equally, regardless of the human


5 Prior to 1975, a farm was defined as any unit
with annual sales of at least $250 of agricultural
products or at least 10 acres with annual sales of at
least $50. After 1975, a farm is defined as any unit
with annual sales of at least $1000.








66

capital characteristics of the workers. Additionally,

this quantity index is not determined by a survey of

hours worked but calculated based on estimated

quantities required for various production activities.

This presents some difficulties.

The USDA farm labor quantity index includes

owner-operator and unpaid family labor as well as

hired labor, while the corresponding expenditures

include wages and perquisites paid to hired labor, and

social security taxes for hired labor and the owner-

operator. Derivation of a price index as in (3.20)

using these quantity and expenditure data treats

owner-operator and family labor as if they were free.

Instead, the USDA expenditures on hired labor and

a regional quantity index of hired labor for the

region calculated from Farm Labor (USDA) are used to

calculate a labor price index. This assumes that

owner-operators value their own time as they would

hired labor. While this may seem inappropriate, the

relative magnitude of hired labor to owner-operator

labor in the Southeast reduces the impact of such an

assumption. The regional total for expenditures on

hired labor, the hired labor quantity index, and labor

price index are presented in Appendix B, Table B-3.










Materials

Expenditure data on feed, livestock, seed,

fertilizer, and miscellaneous inputs are used to

construct budget shares that provide the appropriate

weights for each input in constructing an aggregate

index. The indices represent quantities used rather

than stocks, so the materials index characterizes a

variable input. Some part of the livestock

expenditure goes toward breeding stock, which is

potentially quasi-fixed. The impact of investment in

breeding stock is minimal, as the relative share of

expenditures on livestock in the region is quite

small.

Again, Fisher's weak factor reversal test as

shown in (3.20) can be readily applied to derive an

implicit price index for materials. The expenditures

on each of the inputs are aggregated and deflated.

The ratio of aggregate materials expenditures in each

time period to expenditures in 1977 is divided by the

corresponding ratio of the aggregate materials input

index. The regional expenditures for material inputs,

aggregate materials index, and materials price index

are presented in Appendix Table B-4.

Output Price

Equation (3.20) can also be used to construct an

implicit output price index for the Southeast region








68

in order to normalize input prices. By combining the

value of cash receipts, government payments, net

inventory change, and the value of home consumption as

a measure of output value for each region, this value

and the aggregate output quantity index can be used to

derive an implicit output price index. The output

price of the prior year is used to normalize input

prices to reflect that current price is not generally

observed by producers when production and investment

decisions are made. Regional total receipts, output

quantity index, and output price index are found in

Appendix Table B-5.

Net Investment

The observations on the USDA input indices

correspond to quantities used during the production

period. This is satisfactory for the variable inputs,

labor and materials. However, the mechanical power

and machinery index in effect reflects stock in place

at the end of the production period. Therefore, this

index is lagged one time period to reflect an initial

level of available capital stock. The same procedure

applies to the index of total acres in farms for the

Southeast, as total acres are measured at year's end.

As noted earlier, it is not possible to obtain

estimates of gross investment in capital for the

Southeast region over the entire data period. A








69

measure of net investment in capital and land for each

time period can be defined for each of the inputs by


(3.21) Ki,t = Ki,t Ki,t-l,


where kit is net investment in the quasi-fixed input

i during period t, Ki,t is the level of the input

stock in place at the end of period t, and Kit-1 is

the level of input stock in place at the beginning of

period t.

By developing the model in terms of net

investment, the need for gross investment and

depreciation rate data in the determination of quasi-

fixed factor stocks via (3.19) is eliminated.

Since the estimated variable is actual net investment,

it has been common practice in previous studies

(Chambers and Vasavada, 1983b; Karp, Fawson, and

Shumway, 1984) to assume constant rates of actual

depreciation in order to calculate net investment from

gross investment data. However, it is possible that

the rate of depreciation could vary over observations.

By using the difference of a quasi-fixed input index

between two time periods as a measure of net

investment, this problem can be at least partially

avoided.










Data Summary

Before proceeding to the estimation results of

the empirical model, a brief examination of input use

in the Southeast is in order. The quantity indices

for capital, land, labor, and materials inputs used in

the Southeast region for the years 1949 through 1981

are presented in Appendix Table B-6. The base year

for the quantity and price indices is 1977.

Figure 3-1 depicts the quantity indices for the

1949-1981 period. During the early years of the data

period, agricultural production in the Southeast was

characterized by a substantial reliance on labor and

land relative to materials and capital. The quantity

index of labor in 1949 was over three-and-one-half

times the quantity index in 1981. Except for a short

period of increase from 1949 to 1952, the quantity of

land in farms has gradually declined from a high of

774 million acres in 1952 to 517 million in 1981, a

decrease of nearly 35 percent. On the other hand,

capital stocks nearly doubled, from 52 to 105, and the

use of aggregate materials rose 250 percent, 44 to

112, from 1949 to 1981.










Quantity
360

340-

320

300-

280-

260-

240-

220-

200-

180-

160-

140-

120

100

80-

60-

4 0 1 1 1 1 1 1 1 1 ,1 1 1 1 1 1 1 1 1 1 ,1 1 1 1,-I T -
50 55 60 65 70 75 80
Year

Capital + Land Labor A Materials

Figure 3-1. Observed Input Demand for Southeastern
Agriculture, 1949-1981.








72

Turning to the normalized input prices, Figure

3-2 charts these input prices over the period of

analysis. Not surprisingly, the same inputs whose

quantities have dropped the most, labor and land,

correspond to the inputs whose normalized prices have

increased dramatically, labor increasing seventeen-

fold, from 0.10 to 1.71, and land six-fold, 0.21 to

1.35, over the data period. The most dramatic

increase in the labor price index begins in 1968, such

that nearly eighty percent of the increase in the

labor price index occurs from 1967 to 1981, jumping

from 0.39 to 1.71. The increase in the normalized

land price index is more gradual, such that 50 percent

of the increase occurs prior, 0.21 to 0.66, and 50

percent, 0.66 to 1.35, after 1966, the midpoint of the

data period. The normalized price of capital doubled

between 1949 and 1981, from 0.68 to 1.21, while the

materials price increased only 10 percent, from 0.913

to 1.04.

Interpretation of these changes in the normalized

price indices should be tempered by recognizing that

the indices are normalized with respect to output

price. A drop in the output price would produce an

increase in the normalized input price, everything

else constant. However, examination of the actual

output price index in Appendix Table B-5 shows only a











Capital, Land
Index
1.8-1--


50 55 60 65 70 75 80
Year


D Capital
Price


Land
Price


Labor
Price


Materials
A Price


Figure 3-2. Normalized Input Prices for the
Southeast, 1949-1981.








74

12 percent change in the output price index from

endpoint, 1.08 in 1949, to endpoint, 0.94 in 1981.

The rapid increase in output price of nearly 25

percent from 1972 to 1973, 0.89 to 1.112, produced a

substantial drop in the normalized price indices for

capital, land, and labor. The materials price index,

however, rose even faster than the output price index,

so the normalized price of materials increased.

These data indicate that the Southeast has

undergone some substantial changes from 1949 to 1981.6

The normalized price of labor has risen as

dramatically as the quantity index has fallen. The

Southeast has come to rely substantially more on

materials and capital than in the past. The quantity

of land in farms has gradually declined. It remains

for the next chapter to see what light a dynamic model

of factor demands can shed on these changes.
















6. McPherson and Langham (1983) provide a
historical perspective of southern agriculture.












CHAPTER IV
RESULTS

Theoretical Consistency

This chapter presents the results of estimating

net investment demand equations for capital and land

and variable input demand equations for labor and

materials. The consistency of the data with the

assumption of dynamic optimizing behavior is

considered by evaluating the regularity conditions of

the value function. Estimated short- and long-run

levels of demand are obtained from the parameters of

the estimated equations and compared to observed input

demand. Estimates of short- and long-run price

elasticities are computed in order to identify gross

substitute/complement relationships among the inputs.

Method of Estimation

The system of equations presented in the previous

chapter were estimated using iterated nonlinear three-

stage least squares.7 For purposes of estimation, a

disturbance term was appended to the net investment

and variable input demand equations to reflect errors

in optimizing behavior. This convention is consistent


7. The model was estimated using the LSQ option
of the Time Series Processor (TSP) Version 4.0 as
coded by Hall and Hall, 1983.

75








76

with other empirical applications (Chambers and

Vasavada, 1983b; Karp, Fawson, and Shumway, 1984),

although Epstein and Denny assume a first order

autoregressive process in the error term for the

quasi-fixed input demand equations.8

The iterated nonlinear three stage least squares

estimation technique is a minimum distance estimator

with the distance function D expressed as


(4.1) D = f(y,b)'[S- 1 H (H'H)-1H'] f(y,b)

where f(y,b) is the stacked vector of residuals from

the nonlinear system, S is the residual covariance

matrix, and H is the Kronecker product of an identity

matrix dimensioned by the number of equations and a

matrix of instrumental variables. For this system,

the instruments consist of the normalized prices and

their logarithms, quasi-fixed factor levels, and the

time trend. Although the system is nonlinear in

parameters, it is linear in variables. Hence, the

minimum distance estimator is asympotically equivalent

to full information maximum likelihood (Hausman, 1975)

and provides consistent and asymptotically efficient

parameter estimates.


8. Such an assumption necessitates estimation of
a matrix of autocorrelation parameters. For two
quasi-fixed inputs, this would require estimation of
four additional parameters.








77

A constant real discount rate of five percent was

employed in the estimation. This rate is consistent

with the estimates derived by Hoffman and Gustafson

(1983) of 4.4 percent reflecting the average twenty

year current return to farm assets, 4.3 percent

obtained by Tweeten (1981), and 4.25 percent by

Melichar (1979).9

The parameter estimates and associated standard

errors are presented in Table 4-1. Thirteen of the

twenty-six parameters are at least twice their

asymptotic standard errors. Given the nonlinear and

simultaneous nature of the system, it is difficult to

evaluate the theoretical and economic consistency of

the model solely on the structural parameters. Thus,

one must consider the underlying regularity conditions

and the consistency of the derived input demand

equations with observed behavior in order to assess

the empirical model.

Regularity Conditions

An important feature of the dual approach,

whether applied to static or dynamic optimization, is

that the relevant conditions (V in Chapter II) are

easy to check. Lau (1976) notes the difficulty of

statistically testing the conditions for a static


9. The parameter estimates are fairly
insensitive but not invariant to the choice of
discount rates.











Table 4-1. Parameter Estimates Treating Materials
and Labor as Variable Inputs, Capital and Land
as Quasi-Fixed.


Parameter Standard
Parameter Estimate Errora


1913.160
-423.746

-677.077
-343.822

2472.608
-243.214
-123.888

-105.878
-233.275
406.767

-49.625
98.263
152.451
-111.878

1.900
-1.879
0.826
-0.159

18.639
4.155
17.267
23.045

-0.588
0.490
-0.023
-0.213


bkk
bka
baa

cww
cwm
Cmm

dwk
dwa
dmk
dma

nwK
nwA
nmK
nmA

VK
VA
VW
VM

9KK
gKA
gAK
gAA


496.327*
229.533

212.572*
658.473

478.233*
153.757
98.104

123.786
72.084*
636.244

110.041
102.696
79.123
74.397

0.731*
2.283
0.309*
0.530

5.808*
0.707*
1.507*
8.312*

0.160*
0.242*
0.015
0.056*


times its


a indicates parameter estimate two
standard error.








79

duality, concluding that such tests are limited to

dual functions linear in parameters. Statistical

testing of the regularity conditions underlying

dynamic duality is even more difficult. However,

these conditions can be numerically evaluated.

Since one of the objectives of this study is to

obtain estimates of the adjustment rates of the quasi-

fixed inputs and since the elements of the adjustment

matrix M=[r+G] can be determined readily from the

parameter estimates, the regularity conditions of

nonsingularity of JpK-1 and stability of M are

examined first. The nonsingularity of JpK-1 is

determined from the estimates of the elements of G, as

JpK-l = G. The determinant of G is -0.334, thus
satisfying the nonsingularity of Jp-l1. The stability

of the adjustment matrix requires that the eigenvalues

of M have negative real parts and lie within the unit

circle. The eigenvalues of G are -0.196 and -0.505,

which satisfy the necessary stability criteria. The

equilibrium demand levels for capital and land are

positive at all data points. The existence and

uniqueness of equilibrium or steady state levels of

capital, K(p,w), and land, A(p,w)' as a theoretical

requirement are also established.








80

It was shown in Chapter II that convexity of the

value function in normalized input prices is

sufficient to verify the necessary curvature

properties of the underlying technology when JK is

linear in prices, as is the empirical specification

used to derive the current estimates. In fact, the

linearity of JK in prices is necessary to generate an

accelerator matrix consistent with net investment

equations of the form K* = M (K-K). The elements of

the matrix of second-order derivatives of the value

function with respect to prices in this model are

dependent upon the exogenous variables (prices) in the

system. Thus, the Hessian must be evaluated for

positive definiteness at each data point. This

regularity condition was satisfied at thirty-one of

the thirty-three observations (See Appendix C-1 for

numerical results).

The only exceptions were the years 1949 and 1950.

Given that these observations immediately follow the

removal of World War II agricultural policies, the

return of a large number of the potential agricultural

work force, and rapidly changing production practices

incorporating newly available materials, it is perhaps

not surprising that the data are inconsistent with

dynamic optimizing behavior at these points.










Technical Change

The parameters representing technical change in

the system of equations indicate that technical change

has stimulated the demand for all inputs in the

Southeast. Incorporation of these parameters as a

linear function of time implicitly assumes technical

change is disembodied. The relative magnitutude of

these estimates indicates that technical change has

been material-using relative to labor, and capital-

using relative to land. While some studies of U.S.

agriculture have found technical change to be labor-

and land-saving (Chambers and Vasavada, 1983b), the

estimated positive values for these inputs is not

surprising given the rebirth of agriculture in the

Southeast over the past quarter century. At least

some portion of technical change has aided in

maintaining the demand for labor in the face of rising

labor prices by increasing productivity for many crops

in the Southeast that rely on hand-harvesting, such as

fresh vegetables and citrus.

Consistency with Observed Behavior

Evaluation of the empirical model relies on more

than the theoretical consistency of the parameter

estimates with respect to the regularity conditions.

In addition, the economic consistency of the model is

determined by the correspondence of observed net








82

investment and input use with the estimates or

predicted values obtained from the derived demand

equations. Satisfaction of the regularity conditions

alone is not verification that dynamic optimizing

behavior is an appropriate assumption.

The observed and estimated values of K* in Table

4-2 show that the Southeast has been characterized by

a steady increase in net capital investment, with only

a few periods of net disinvestment. The estimated

values of K* correspond fairly closely to observed net

investment. Observed capital stocks and the estimated

short-run demand for the stock of capital correspond

closely with never more than a two percent difference.

However, there is a notable divergence of observed and

equilibrium capital stock demand from 1949 to 1973.

Contrary to the concerns of overcapitalization

today, the Southeast only initially exhibited an

excess of capital. However, the equilibrium level of

capital rises in response to changing relative prices

such that by 1974 observed and equilibrium levels are

in close correspondence. The observed capital use and

short- and long-run demand levels for capital

presented in Figure 4-1 further illustrate this

convergence. However, one should note the adjustment










Table 4-2. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Capital.


Net Investment Capital Demand

K K* K K*(K,p,w) R(p,w)
observed optimal observed short- long-
Year run run


49 8.00


5.00
6.00
5.00
2.00
1.00
2.00
2.00
-1.00
0.00
2.00

-2.00
-1.00
-1.00
1.00
0.00
1.00
1.00
3.00
1.00
0.00

-1.00
4.00
-2.00
3.00
4.00
2.00
2.00
1.00
6.00
6.00

-5.00
-2.00


7.13

5.22
5.69
5.53
4.65
0.29
0.94
0.68
-1.67
-0.42
1.27

-0.92
0.33
0.44
0.49
0.44
0.85
0.92
1.22
0.22
-0.23

0.58
4.80
2.52
3.37
3.75
0.38
1.67
0.96
2.07
-0.12

-2.54
2.58


52.00

60.00
65.00
71.00
76.00
78.00
79.00
81.00
83.00
82.00
82.00

84.00
82.00
81.00
80.00
81.00
81.00
82.00
83.00
86.00
87.00

87.00
86.00
90.00
88.00
91.00
95.00
97.00
99.00
100.00
106.00

112.00
107.00


53.81

60.34
65.60
71.56
75.98
75.13
76.56
78.59
79.24
79.32
80.69

81.48
80.43
79.68
78.90
80.03
80.36
81.44
82.83
85.19
86.05

86.70
88.55
91.06
89.82
93.28
95.09
97.86
99.44
101.22
105.88

110.29
108.57


16.58

17.33
22.46
29.16
32.54
26.29
28.64
33.48
34.22
40.18
47.80

49.57
52.08
54.34
56.71
60.29
62.11
63.89
68.77
70.21
72.96

76.50
84.92
84.73
87.85
95.12
93.03
96.18
97.72
101.59
105.04

106.43
109.84






84

of an excess capital stock to equilibrium levels is

not achieved by a disinvestment in capital, but by an

increase in the equilibrium level of capital demand.

An examination of net investment and demand

levels for land in Table 4-3 reveals a situation

completely opposite from that of capital. Apart from

a short period initially, the Southeast has exhibited

a gradual reduction in the stock of land in farms.

While estimated long-run demand exceeds observed and

short-run demand for land stocks until 1961, the

equilibrium level of demand falls at a faster rate

than the short-run and observed levels. After 1961,

the Southeast was marked by a slight degree of

overinvestment in land stocks, owing primarily to an

increase in the relative price of land.

Observed stocks of land and estimates of short-

run demand correspond closely over the data range.

While the equilibrium level of capital increased in

response to the increasing relative price of labor,

the long-run demand for land has declined in response

to an increase in the relative price of land as well

as increase in the relative price of labor. Observed

and estimated short- and long-run demand levels for

land are presented graphically in Figure 4-2.









Capital
Index
120-


110-


100-


90-


80-


70-


60-


50


40-


30-


20-


10-


50 55 60 65 70 75 80
Year


Observed
u Demand


Estimated Estimated
+ Short-run 0 Long-run
Demand Demand


Figure 4-1. Comparison of Observed and Estimated Demand
for Capital in Southeastern Agriculture, 1949-1981.











Table 4-3. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Land.


Net Investment Demand for Land

A A* A A* (K,p,w) A(p,w)
observed optimal observed short- long-
Year run run


3.45

1.14
0.76
0.19
-0.95
-0.57
-2.86
-3.05
-3.63
-3.82
-3.44

-2.48
-2.86
-2.86
-3.63
-1.72
-1.53
-1.53
-1.91
-2.29
-1.91

-1.34
-0.95
-0.76
-1.15
-0.57
0.19
-1.23
-1.04
-1.23
0.87

0.22
-1.62


2.64 142.32


2.11
1.01
-0.16
-0.92
-1.85
-2.68
-3.18
-3.52
-3.02
-2.70

-2.83
-2.77
-2.65
-2.24
-2.21
-2.21
-2.34
-1.48
-2.19
-1.74

-1.45
-1.63
-1.56
-0.87
-0.38
-0.28
-0.61
-0.58
-0.48
-0.38

-0.48
-0.48


145.77
146.91
147.68
147.87
146.91
146.34
143.48
140.43
136.80
132.99

129.55
127.07
124.21
121.35
117.72
116.00
114.48
112.95
111.04
108.75

106.85
105.51
104.56
103.79
102.65
102.08
102.27
101.04
100.00
98.77

99.63
99.85


146.78

150.01
149.98
149.35
148.79
147.27
145.87
142.29
138.86
135.44
131.66

128.00
125.47
122.61
120.04
116.35
114.60
112.94
112.14
109.49
107.59

105.87
104.11
103.31
103.09
102.27
101.90
101.76
100.54
99.52
98.37

99.20
99.32


193.99

198.58
189.99
177.78
172.26
170.28
163.14
152.68
146.27
141.64
135.40

129.24
125.47
121.53
119.86
114.83
112.55
109.86
112.78
104.95
105.01

103.60
96.77
97.74
99.82
99.84
101.64
99.29
97.83
96.33
95.52

96.70
95.36










Land
Index
200


50 55 60 65 70 75 80
Year


Observed
D Demand


Estimated Estimated
+ Short-run 0 Long-run
Demand Demand


Figure 4-2. Comparison of Observed and Estimated Demand
for Land in Southeastern Agriculture, 1949-1981.








88

The observed use of labor and estimated short-

and long-run demands for labor as presented in Table

4-4 indicate almost complete adjustment of observed

labor demand to the estimated equilibrium within one

time period. This is consistent with the assumption

that labor is a variable input. Returning to the

parameter estimates in Table 4-1 and the short-run

demand equation for labor in Table A-1, capital stocks

slightly reduce the short-run demand for labor.

Capital investment increases the short-run demand for

labor. This indicates that labor facilitates

adjustment in capital. The effect of land stocks on

the short-demand for labor indicates an increase in

land increases the short-run demand for labor. The

effect of net investment in land decreases the short-

run demand for labor. Labor appears to have a

negative effect on the adjustment of land.

The long-run demand for labor depends on the

equilibrium levels of capital and land to the same

degree that short-run labor demand depends on capital

and land stocks. An increase in the equilibrium level

of capital decreases the long-run demand for labor.

Conversely, an increase in the equilibrium level of

land increases the long-run demand for labor.

Disequilibrium in the quasi-fixed inputs could

potentially cause a divergence in the short- and long-











Table 4-4. Comparison of Observed and Estimated
Short- and Long-Run Demands for Labor.


Labor Demand

Labor L* (K,p,w) L(K,p,w)
Year observed short-run long-run


49 351.195


320.518
336.653
311.155
295.817
267.331
265.139
239.841
205.578
192.231
196.813

189.442
181.873
179.681
175.299
161.355
146.813
136.454
138.048
128.685
129.283

122.908
120.319
114.542
113.147
109.761
106.375
103.785
100.000
96.016
92.430

95.817
91.434


351.812

336.175
320.176
306.751
294.246
270.240
253.921
232.346
217.464
211.425
204.972

189.117
181.008
173.914
169.490
155.314
148.204
138.309
143.208
126.861
124.008

121.029
115.411
108.599
113.642
116.091
107.527
105.591
101.159
99.784
92.872

89.070
95.881


351.592

339.369
319.446
302.778
290.103
273.312
253.412
230.377
219.113
210.896
200.875

188.722
177.777
170.290
166.351
151.941
143.835
133.380
139.392
123.177
122.116

117.851
102.439
100.668
105.218
107.571
106.405
101.054
98.056
94.424
92.140

93.235
89.346








90

run demand for labor. However, the magnitudes of the

parameter estimates associated with the dependence of

labor demand on the quasi-fixed factors are small.

Thus, the short-and long-run demands for labor are

similar.

This is also true for the materials input as

shown in Table 4-5. The degree of correspondence of

observed and short- and long-run demands for materials

is even greater than for labor. The short- and long-

run demands for materials depend on quasi-fixed input

stocks and equilibrium levels only slightly.

Materials appear to facilitate adjustment in capital

and slow adjustment in land.

The substantial disequilibrium in the Southeast

with respect to capital and land during the first part

of the sample period may be interpreted from at least

two viewpoints, one empirical and one intuitive.

Empirically, the specification of the adjustment

mechanism in the model is only indirectly dependent on

factor prices through the determination of equilibrium

levels of the quasi-fixed inputs. The accelerator

itself is a matrix of constants. Yet the degree of

adjustment in each factor level depends on the

disequilibrium between actual and equilibrium input

levels, which in turn are a function of the input

prices. Changes in relative prices of the inputs,











Table 4-5. Comparison of Observed and Estimated
Short- and Long-Run Demands for Materials.


Materials Demand

Materials M*(K,p,w) M(K,p,w)
Year observed short-run long-run


44.237

42.623
49.666
50.194
51.917
51.345
54.943
55.006
54.630
57.657
63.332

63.839
66.081
68.409
69.281
76.910
78.817
79.390
81.210
82.814
87.717

94.124
98.432
101.150
101.801
97.131
86.274
96.985
100.000
105.649
114.375

105.367
112.323


44.877

44.431
47.821
50.111
52.350
49.255
53.015
55.332
54.572
58.220
64.191

63.245
67.067
69.178
70.373
75.303
78.569
79.980
83.445
85.197
88.000

93.462
97.382
98.821
94.462
96.437
92.328
100.289
101.008
107.951
109.014

103.573
114.331


41.464

43.123
45.440
47.166
49.974
51.308
53.908
56.009
57.477
59.693
63.653

64.828
67.159
69.021
70.176
75.007
77.764
78.993
82.368
84.826
88.212

92.798
91.829
95.828
90.681
92.264
91.928
98.369
99.833
105.538
108.972

106.169
111.323








92

especially labor, have caused the equilibrium level of

capital to rise more rapidly than observed or short

run capital demand. A complementary relationship

between land and labor and substitute relation between

capital and labor contribute to a high demand for land

in the long-run and a low equilibrium for capital

given the low initial prices of land and labor.

Intuitively, these low prices may be attributed

to a share-cropper economy, itself a vestige of the

old plantations. While the relative prices of labor

and land in 1949 reflect this notion, the observed

levels of land and capital do not. It thus appears

very plausible that during the initial postwar period,

agriculture in the Southeast anticipated a change in

this system and had already begun investing in capital

and reducing land stocks.

Elasticity Measures

Given the inability to estimate the supply

equation, only Marshallian (uncompensated) input

demand elasticities were estimated. The explicit

recognition of dynamic optimization provides a clear

distinction between the short run, where quasi-fixed

inputs adjust only partially to relative price changes

along an optimal investment path, and the long run,

where quasi-fixed input stocks are fully adjusted to

equilibrium levels.








93

Average short-run gross elasticities for capital,

land, labor and materials for selected periods are

presented in Table 4-6. The short run own-price

elasticities were negative at each data point for all

inputs. Short-run elasticity estimates for each year

are presented in Appendix Tables D-1 D-4. As noted

earlier, positive short-run own-price elasticities are

not inconsistent with the adjustment cost model.

However, such a result did not occur in this analysis.

The signs of the elasticity measures indicate

that land and labor are short-run complements. This

is not surprising given the labor-intensive crops that

characterize production in the Southeast. Labor and

capital and labor and materials are short-run

substitutes. Materials and capital are short-run

complements while materials and land are substitutes.

Finally, capital and land are short-run substitutes.

The short-run own-price elasticities for all four

inputs are inelastic. Given capital and land are

quasi-fixed inputs and the short-run demands for labor

and materials are conditional on these inputs,

inelastic short-run demands should be expected. The

short-run own price elasticity of land is the most

inelastic of the four inputs, ranging from an average

of -0.08 in 1949-1955 to -0.02 in 1976-1981. Capital

generally has next lowest own-price elasticity. The