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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
Creator:
Monson, Michael James, 1956-
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English
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vii, 148 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Agricultural land ( jstor )
Agriculture ( jstor )
Capital shortages ( jstor )
Capital stocks ( jstor )
Elasticity of demand ( jstor )
Financial investments ( jstor )
Labor demand ( jstor )
Mathematical variables ( jstor )
Net investment ( jstor )
Price elasticity ( jstor )
Agriculture -- Economic aspects -- Southern States ( lcsh )
Dissertations, Academic -- Food and Resource Economics -- UF
Duality theory (Mathematics) ( lcsh )
Food and Resource Economics thesis Ph. D
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 141-147.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Michael James Monson.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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15464499 ( OCLC )
AEH5769 ( NOTIS )
AA00004868_00001 ( sobekcm )

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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.
in


TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iii
ABSTRACT V
CHAPTER
I INTRODUCTION 1
Background 3
Objectives 15
II THEORETICAL MODEL 18
Dynamic Models Using Static
Optimization 20
Dynamic Optimization 27
Theoretical Model 35
The Flexible Accelerator 42
III EMPIRICAL MODEL AND DATA 46
Empirical Model 46
Data Construction 60
IV RESULTS 75
Theoretical Consistency 75
Quasi-fixed Input Adjustment 99
V SUMMARY AND CONCLUSIONS 110
APPENDIX
A INPUT DEMAND EQUATIONS 119
B REGIONAL EXPENDITURE, PRICE,
AND INPUT DATA 12 3
C EVALUATION OF CONVEXITY OF
THE VALUE FUNCTION 130
D ANNUAL SHORT- AND LONG-RUN
PRICE ELASTICITY ESTIMATES 132
REFERENCES 141
BIOGRAPHICAL SKETCH 148
iv


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.
v


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
vi


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


2
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


5
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
continous 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


9
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 persistance 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 continous 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.


16
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
18


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
extensively1. 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_x = a ( x*t xt_x),
where x^ 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 x^ -
x-(-_i 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*-(- becomes a function of
input prices and output price such that
(2.2) x t f (Py, w^ , w^j) .
where output price is denoted by Py, and the w, i = 1
to n, are the input prices. Equation (2.2) allows the
unobserved variable, x*^, 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*^ 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) x.(- = B [x t -
This specification is similar to that in (2.1), except
that x-t 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) xi,t = ^ bij*(x j,t xj,t-l) +
bfi* (x*j^t x-^t-l) + xi t-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.


27
Nadiri and Rosen (1969) derived expressions for
equilibrium factor levels using a Cobb-Douglas
production function in a manner similiar 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 bj 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
CO
(2.8) V = max / e-rt(f(x,x) p'x) dt,
X 0
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
intitial 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) [fxx]^ + [fxx]* = + 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
v*+* ,
lim e [f.] = 0, and a Legendre condition that
t-> OO
[f..] negative semi-definite (Treadway, 1971 p. 847).


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 (Sir (p,Py)/3Pj =
-x*j(p,Py)) or from the cost function by Shephard's
Lemma (9C(p,y)/9pj = 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)it(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 tt(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 it,
quadratic in (x,x), can be hypothesized as
(2.11) tl(w,x,x)= aQ + a'x + b'x + l/2[x' x']
A
C
X
C
B
J

LXJ
where ag, the vectors a, b, and matrices A, B, and C
will be dependent on w in a manner determined by an
exact specification of tt(w,x,x). 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) = -jrw(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 + rCJ/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
OO
(2.16) J0(K,p,w) = max./ ert [F (L, K, K) w'L p'K] dt
L > 0, K < 0
subject to K(0) = K0 > 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 K0 is the initial quasi-fixed
input stock. J(K0,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 Fj< are continuously differentiable.
T.2. Fl, Fk > 0, F < 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, A*(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.l 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 F£# 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 funtion. 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 = K0 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' + Jwk'K*,
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.


V. 2 .
V. 3 .
41
V. 4 .
V. 5.
V. 6.
V. 7 .
rJK + P JKK(K*), Jk > JK< as K*<0.
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 $ onto the domain of F.
The dynamic system K*, K(0) = K0, 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 K(p,w), a globally
steady state also in the domain of J.
JpK is nonsingular.
For the element (K,p,w) in the domain of J, a
minimum in (2.18) is attained at (p,w) if
(K,L) = (K*,L*) .
The matrix
Lp
is nonsingular for
Kw K£
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.l. Condition V.2 reflects
in (p,w) the restrictions imposed on the marginal
products of the inputs, FL and F, 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 =A *
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 Jj< 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-1K.
This form yields JpK = G_1 and Jp = hp(p,w) + G-1K.
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 K(p,w) at K*=0,
(2.24) K(p,w) = -[r + Grl-Gtrhp^w) ] .
Multiplying (2.24) by [r + G] and substituting directly
in (2.23) yields
(2.25) K*(K,p,w) = -[r + G]K(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 Jj^ 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) = a0 + a'K + b'log p + c'log w +
1/2(K1AK + log p'B log p + log w'C log w) +
+ log p'D log w + p' G-1K + 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 pw, log pm]. T denotes a time
trend variable.
Parameter vectors are defined by a = [aK, aA], b
= [bk, ba], c = [cw, cm], VK = [vK, 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
aAK &AA_
...
bkk bka~
bak baa
cww cwm
cmw cmm
!
D =
dkw dkm
daw dam
...
nKw nKm
nAw nAm
, and G
-1 =
[gKK gKA "
gAK gAA
Let G = gKK ^kaT The matrices A, B, and C are
_9AK 9aaJ
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 seriesthat 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
unrestricited 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 Equations
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), *(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-l.
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 + rG-1]-1[rp-1(b + B log p +
D log w) + rVj^T] ,
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


52
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_1#
where K*-(- = [K*^, A*tl / 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_x + K*t(K,p,w),


53
where K*t(K,p,w) is the optimal demand for the quasi-
fixed inputs in period t, K^-i is the initial stock at
the beginning of the period, and K*-j- 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-1(b + B log p + D log w)] +
rVKT + [I + r + G]Kt_!,
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.


54
Rewriting (3.2) and multiplying both sides of the
equation by G-1 yields
(3.9) G-1K*(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]-1[rp-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) * = gAK(K K) + (r + g^) (A A) .


55
Thus, gj^ 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 gj^ = gAK = o. 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 + gK^ = r + g^ = -1, in addition


56
to 9KA = 9ak = 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.I., 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 a0, 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]-1
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,pj' is
. *
9 L 3 L

*
* *
3K + 3 L
*
*
3 A
pj.
*
3 Pj 3 K
3Pj 3 A
3Pj
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, £K,pj i-s
*
<3-16> e5,Pj
9K Pj
*
9Pj 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
(3.17)
9 L 3 L
3p_. 3 K
3 K 3 L
3 Pj 3 A
3_^_ _£j
9 Pj 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)
8 K
3-pT
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.


62
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 = Iit + (1 6i)Ai/t-l,
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) / it'
where 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 E-^, 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.


67
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 = Kift Ki t.lf
where is net investment in the quasi-fixed input
i during period t, is the level of the input
stock in place at the end of period t, and 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 elimimated.
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.


70
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.


71
Quantity
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 suprisingly, 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


73
Capital, Land
Index
Year
n Capital Land Labor Materials
U Price T Price v Price 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-1S 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.


78
Table 4-1. Parameter Estimates Treating Materials
and Labor as Variable Inputs, Capital and Land
as Quasi-Fixed.
Parameter
Standard
Parameter
Estimate
Errora
bK
1913.160
496.327*
bA
-423.746
229.533
CW
-677.077
212.572*
cM
-343.822
658.473
bkk
2472.608
478.233*
bka
-243.214
153.757
baa
-123.888
98.104
cww
-105.878
123.786
cwm
-233.275
72.084*
cmm
406.767
636.244
dwk
-49.625
110.041
dwa
98.263
102.696
dmk
152.451
79.123
dma
-111.878
74.397
nwK
1.900
0.731*
nwA
-1.879
2.283
nmK
0.826
0.309*
nmA
-0.159
0.530
VK
18.639
5.808*
VA
4.155
0.707*
VW
17.267
1.507*
VM
23.045
8.312*
-0.588
0.160*
9KA
0.490
0.242*
-0.023
0.015
9aa
-0.213
0.056*
a indicates parameter estimate two times its
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-1 = G. The determinant of G is -0.334, thus
satisfying the nonsingularity of JpK-1* 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-l 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.


81
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
JL ,
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


83
Table 4-2. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Capital.
Net Investment
+
K K
observed optimal
Year
Capital Demand
K K*(K,p, w)
observed short-
run
K(p,w)
long-
run
49
8.00
7.13
52.00
53.81
16.
58
50
5.00
5.22
60.00
60.34
17.
33
51
6.00
5.69
65.00
65.60
22.
46
52
5.00
5.53
71.00
71.56
29.
16
53
2.00
4.65
76.00
75.98
32.
54
54
1.00
0.29
78.00
75.13
26.
29
55
2.00
0.94
79.00
76.56
28 .
64
56
2.00
0.68
81.00
78.59
33.
48
57
-1.00
-1.67
83.00
79.24
34.
22
58
0.00
-0.42
82.00
79.32
40.
18
59
2.00
1.27
82.00
80.69
47.
80
60
-2.00
-0.92
84.00
81.48
49.
57
61
-1.00
0.33
82.00
80.43
52 .
08
62
-1.00
0.44
81.00
79.68
54 .
34
63
1.00
0.49
80.00
78.90
56.
71
64
0.00
0.44
81.00
80.03
60.
29
65
1.00
0.85
81.00
80.36
62.
11
66
1.00
0.92
82.00
81.44
63 .
89
67
3.00
1.22
83.00
82.83
68 .
77
68
1.00
0.22
86.00
85.19
70.
21
69
0.00
-0.23
87.00
86.05
72 .
96
70
-1.00
0.58
87.00
86.70
76.
50
71
4.00
4.80
86.00
88.55
84.
92
72
-2.00
2.52
90.00
91.06
84.
73
73
3.00
3.37
88.00
89.82
87.
85
74
4.00
3.75
91.00
93.28
95.
12
75
2.00
0.38
95.00
95.09
93.
03
76
2.00
1.67
97.00
97.86
96.
18
77
1.00
0.96
99.00
99.44
97.
72
78
6.00
2.07
100.00
101.22
101.
59
79
6.00
-0.12
106.00
105.88
105.
04
80
-5.00
-2.54
112.00
110.29
106.
43
81
-2.00
2.58
107.00
108.57
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.


85
Capital
Index
Observed Estimated Estimated
c Demand ^ Short-run ^ Long-run
Demand Demand
Figure 4-1. Comparison of Observed and Estimated Demand
for Capital in Southeastern Agriculture, 1949-1981.


86
Table 4-3. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Land.
Net Investment Demand for Land
Year

A
observed
A*
optimal
A
observed
A*(K,p,w)
short-
run
A(p,w)
long-
run
49
3.45
2.64
142.32
146.78
193.99
50
1.14
2.11
145.77
150.01
198.58
51
0.76
1.01
146.91
149.98
189.99
52
0.19
-0.16
147.68
149.35
177.78
53
-0.95
-0.92
147.87
148.79
172.26
54
-0.57
-1.85
146.91
147.27
170.28
55
-2.86
-2.68
146.34
145.87
163.14
56
-3.05
-3.18
143.48
142.29
152.68
57
-3.63
-3.52
140.43
138.86
146.27
58
-3.82
-3.02
136.80
135.44
141.64
59
-3.44
-2.70
132.99
131.66
135.40
60
-2.48
-2.83
129.55
128.00
129.24
61
-2.86
-2.77
127.07
125.47
125.47
62
-2.86
-2.65
124.21
122.61
121.53
63
-3.63
-2.24
121.35
120.04
119.86
64
-1.72
-2.21
117.72
116.35
114.83
65
-1.53
-2.21
116.00
114.60
112.55
66
-1.53
-2.34
114.48
112.94
109.86
67
-1.91
-1.48
112.95
112.14
112.78
68
-2.29
-2.19
111.04
109.49
104.95
69
-1.91
-1.74
108.75
107.59
105.01
70
-1.34
-1.45
106.85
105.87
103.60
71
-0.95
-1.63
105.51
104.11
96.77
72
-0.76
-1.56
104.56
103.31
97.74
73
-1.15
-0.87
103.79
103.09
99.82
74
-0.57
-0.38
102.65
102.27
99.84
75
0.19
-0.28
102.08
101.90
101.64
76
-1.23
-0.61
102.27
101.76
99.29
77
-1.04
-0.58
101.04
100.54
97.83
78
-1.23
-0.48
100.00
99.52
96.33
79
0.87
-0.38
98.77
98.37
95.52
80
0.22
-0.48
99.63
99.20
96.70
81
-1.62
-0.48
99.85
99.32
95.36


87
Land
Index
Observed Estimated Estimated
^ Demand + Short-run *' 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-l, 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-


89
Table 4-4. Comparison of Observed and Estimated
Short- and Long-Run Demands for Labor.
Year
Labor
observed
Labor Demand
L*(K,p,w)
short-run
L (K, p, w)
long-run
49
351.195
351.812
351.592
50
320.518
336.175
339.369
51
336.653
320.176
319.446
52
311.155
306.751
302.778
53
295.817
294.246
290.103
54
267.331
270.240
273.312
55
265.139
253.921
253.412
56
239.841
232.346
230.377
57
205.578
217.464
219.113
58
192.231
211.425
210.896
59
196.813
204.972
200.875
60
189.442
189.117
188.722
61
181.873
181.008
177.777
62
179.681
173.914
170.290
63
175.299
169.490
166.351
64
161.355
155.314
151.941
65
146.813
148.204
143.835
66
136.454
138.309
133.380
67
138.048
143.208
139.392
68
128.685
126.861
123.177
69
129.283
124.008
122.116
70
122.908
121.029
117.851
71
120.319
115.411
102.439
72
114.542
108.599
100.668
73
113.147
113.642
105.218
74
109.761
116.091
107.571
75
106.375
107.527
106.405
76
103.785
105.591
101.054
77
100.000
101.159
98.056
78
96.016
99.784
94.424
79
92.430
92.872
92.140
80
95.817
89.070
93.235
81
91.434
95.881
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
similiar.
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,


91
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
49
44.237
44.877
41.464
50
42.623
44.431
43.123
51
49.666
47.821
45.440
52
50.194
50.111
47.166
53
51.917
52.350
49.974
54
51.345
49.255
51.308
55
54.943
53.015
53.908
56
55.006
55.332
56.009
57
54.630
54.572
57.477
58
57.657
58.220
59.693
59
63.332
64.191
63.653
60
63.839
63.245
64.828
61
66.081
67.067
67.159
62
68.409
69.178
69.021
63
69.281
70.373
70.176
64
76.910
75.303
75.007
65
78.817
78.569
77.764
66
79.390
79.980
78.993
67
81.210
83.445
82.368
68
82.814
85.197
84.826
69
87.717
88.000
88.212
70
94.124
93.462
92.798
71
98.432
97.382
91.829
72
101.150
98.821
95.828
73
101.801
94.462
90.681
74
97.131
96.437
92.264
75
86.274
92.328
91.928
76
96.985
100.289
98.369
77
100.000
101.008
99.833
78
105.649
107.951
105.538
79
114.375
109.014
108.972
80
105.367
103.573
106.169
81
112.323
114.331
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.


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.
in

TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iii
ABSTRACT V
CHAPTER
I INTRODUCTION 1
Background 3
Objectives 15
II THEORETICAL MODEL 18
Dynamic Models Using Static
Optimization 20
Dynamic Optimization 27
Theoretical Model 35
The Flexible Accelerator 42
III EMPIRICAL MODEL AND DATA 46
Empirical Model 46
Data Construction 60
IV RESULTS 75
Theoretical Consistency 75
Quasi-fixed Input Adjustment 99
V SUMMARY AND CONCLUSIONS 110
APPENDIX
A INPUT DEMAND EQUATIONS 119
B REGIONAL EXPENDITURE, PRICE,
AND INPUT DATA 12 3
C EVALUATION OF CONVEXITY OF
THE VALUE FUNCTION 130
D ANNUAL SHORT- AND LONG-RUN
PRICE ELASTICITY ESTIMATES 132
REFERENCES 141
BIOGRAPHICAL SKETCH 148
iv

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.
v

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
vi

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

2
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

5
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
continous 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

9
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 persistance 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 continous 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.

16
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
18

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
extensively1. 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_! = a( x*t - xt_x),
where x-¡- 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 x^ -
x-)-_i 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*^ becomes a function of
input prices and output price such that
(2.2) x t — f (Py, w^ , , w^j) .
where output price is denoted by Py, and the w¿, i = 1
to n, are the input prices. Equation (2.2) allows the
unobserved variable, x*^, 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*^ 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) x.(- — = B [x t -
This specification is similar to that in (2.1), except
that x-t 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) xi,t = ^ bij-(x j,t “ xj,t-l) +
bfi* (x*j^t - x-^t-l) + xi,t-l*
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.

27
Nadiri and Rosen (1969) derived expressions for
equilibrium factor levels using a Cobb-Douglas
production function in a manner similiar 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 b¿j 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
00
(2.8) V = max / e-rt(f(x,x) - p'x) dt,
X 0
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
intitial 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) [fxx]^ + [fxx]* = + _ 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
— V“t* ,
lim e [f.] = 0, and a Legendre condition that
t-*- oo
[f..] negative semi-definite (Treadway, 1971 p. 847).

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 (Sir (p,Py)/3Pj =
-x*j(p,Py)) or from the cost function by Shephard's
Lemma (9C(p,y)/9pj = 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)it(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 tt(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 it,
quadratic in (x,x), can be hypothesized as
(2.11) tl(w,x,x)= aQ + a'x + b'x + l/2[x' x']
A
C
X
C
B
J
•
LXJ
where ag, the vectors a, b, and matrices A, B, and C
will be dependent on w in a manner determined by an
exact specification of tt(w,x,x). 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) = -ttw(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 + rCJ/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
OO
(2.16) J0(K,p,w) = max./ e“rt [F (L, K, K) - w'L - p'K] dt
L > 0, K < 0
subject to K(0) = K0 > 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 K0 is the initial quasi-fixed
input stock. J(K0,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 Fj< are continuously differentiable.
T.2. Fl, Fk > 0, F¿ < 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, A*(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.l 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 F£# 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 funtion. 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,u) 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 = K0 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' + JwK*K*,
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.l. J is a real-valued, bounded-from-below
function defined in prices and quasi-fixed
inputs. J and JK are twice-continously
differentiable.

V. 2 .
V. 3 .
41
V. 4 .
V. 5.
V. 6.
V. 7 .
rJK + P “ JKk(K*), Jk > JK<° as K*<0.
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 $ onto the domain of F.
The dynamic system K*, K(0) = K0, 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 K(p,w), a globally
steady state also in the domain of J.
JpK is nonsingular.
For the element (K,p,w) in the domain of J, a
minimum in (2.18) is attained at (p,w) if
(K,L) = (K*,L*) .
The matrix
Lp
is nonsingular for
Kw K£
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.l. Condition V.2 reflects
in (p,w) the restrictions imposed on the marginal
products of the inputs, FL and F¿, 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 =A *
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 Jj< 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-1K.
This form yields JpK = G_1 and Jp = hp(p,w) + G-1K.
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 K(p,w) at K*=0,
(2.24) K(p,w) = -[r + Gr^-Gtrhp^w) ] .
Multiplying (2.24) by [r + G] and substituting directly
in (2.23) yields
(2.25) K*(K,p,w) = -[r + G]K(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 Jj^ 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) = a0 + a'K + b'log p + c'log w +
1/2(K1AK + log p'B log p + log w'C log w) +
+ log p'D log w + p' G-1K + 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 pw, log pm]. T denotes a time
trend variable.
Parameter vectors are defined by a = [aK, aA], b
= [bk, ba], c = [cw, cm], VK = [vK, 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
aAK &AA_
...
’bkk bka~
bak baa
cww cwm
cmw cmm
r
D =
dkw dkm
daw dam
...
nKw nKm
nAw nAm
, and G
-1 =
[gKK gKA "
gAK gAA
Let G = gKK 9kaT The matrices A, B, and C are
_9AK 9aaJ
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
unrestricited 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 Equations
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), Á*(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-l.
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 + rG-1]-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

52
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_1#
where K*-(- = [K*^, A*tl < 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_! + K*t(K,p,w),

53
where K*t(K,p,w) is the optimal demand for the quasi-
fixed inputs in period t, K^-i is the initial stock at
the beginning of the period, and K*-j- 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-1(b + B log p + D log w)] +
rVKT + [I + r + G]Kt_!,
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.

54
Rewriting (3.2) and multiplying both sides of the
equation by G-1 yields
(3.9) G-1K*(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]-1[rp-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) Á* = gAK(K - K) + (r + g^) (A - A) .

55
Thus, gj^ 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 gj^ = gAK = o. 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 + gK^ = r + g^ = -1, in addition

56
to 9KA = 9ak = 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.I., 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 a0, 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]-1
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, z l,pj' is
. * *
3 L 3 L
•
*
* *
9K + 9 L
*
*
9 A
pj.
*
9 Pj 9 K
9Pj 9 A
3Pj
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, £K,pj i-s
*
<3-16> e5,Pj
9K _ Pj
*
9Pj 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
(3.17)
9 L 9 L
3p_. 3 K
3 K 3 L
3 Pj 3 A
3_^_ . _£j
9 Pj 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)
8 K
3-pT
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.

62
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 = Iit + (1 - 6i)Ai/t_1,
where A-j^ is capital stock i in period t, Ij^ is gross
investment, and 6j_ 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) / ®it'
where 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 E-^, 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.

67
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) K¿/t = Kift -
where is net investment in the quasi-fixed input
i during period t, is the level of the input
stock in place at the end of period t, and 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 elimimated.
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.

70
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.

71
Quantity
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 suprisingly, 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

73
Capital, Land
Index
Year
n Capital , Land , Labor Materials
U Price T Price v Price 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-1S 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.

78
Table 4-1. Parameter Estimates Treating Materials
and Labor as Variable Inputs, Capital and Land
as Quasi-Fixed.
Parameter
Standard
Parameter
Estimate
Errora
bK
1913.160
496.327*
bA
-423.746
229.533
CW
-677.077
212.572*
cM
-343.822
658.473
bkk
2472.608
478.233*
bka
-243.214
153.757
baa
-123.888
98.104
cww
-105.878
123.786
cwm
-233.275
72.084*
cmm
406.767
636.244
dwk
-49.625
110.041
dwa
98.263
102.696
dmk
152.451
79.123
dma
-111.878
74.397
nwK
1.900
0.731*
nwA
-1.879
2.283
nmK
0.826
0.309*
nmA
-0.159
0.530
VK
18.639
5.808*
VA
4.155
0.707*
VW
17.267
1.507*
VM
23.045
8.312*
-0.588
0.160*
9KA
0.490
0.242*
-0.023
0.015
9aa
-0.213
0.056*
a * indicates parameter estimate two times its
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-1 = G. The determinant of G is -0.334, thus
satisfying the nonsingularity of JpK-1* 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-l 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.

81
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
• JL ,
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

83
Table 4-2. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Capital.
Net Investment
• • +
K K
observed optimal
Year
Capital Demand
K K* (K, p, w)
observed short-
run
K(p,w)
long-
run
49
8.00
7.13
52.00
53.81
16.
58
50
5.00
5.22
60.00
60.34
17.
33
51
6.00
5.69
65.00
65.60
22.
46
52
5.00
5.53
71.00
71.56
29.
16
53
2.00
4.65
76.00
75.98
32.
54
54
1.00
0.29
78.00
75.13
26.
29
55
2.00
0.94
79.00
76.56
28.
64
56
2.00
0.68
81.00
78.59
33.
48
57
-1.00
-1.67
83.00
79.24
34.
22
58
0.00
-0.42
82.00
79.32
40.
18
59
2.00
1.27
82.00
80.69
47.
80
60
-2.00
-0.92
84.00
81.48
49.
57
61
-1.00
0.33
82.00
80.43
52 .
08
62
-1.00
0.44
81.00
79.68
54 .
34
63
1.00
0.49
80.00
78.90
56.
71
64
0.00
0.44
81.00
80.03
60.
29
65
1.00
0.85
81.00
80.36
62.
11
66
1.00
0.92
82.00
81.44
63 .
89
67
3.00
1.22
83.00
82.83
68 .
77
68
1.00
0.22
86.00
85.19
70.
21
69
0.00
-0.23
87.00
86.05
72 .
96
70
-1.00
0.58
87.00
86.70
76.
50
71
4.00
4.80
86.00
88.55
84.
92
72
-2.00
2.52
90.00
91.06
84.
73
73
3.00
3.37
88.00
89.82
87.
85
74
4.00
3.75
91.00
93.28
95.
12
75
2.00
0.38
95.00
95.09
93.
03
76
2.00
1.67
97.00
97.86
96.
18
77
1.00
0.96
99.00
99.44
97.
72
78
6.00
2.07
100.00
101.22
101.
59
79
6.00
-0.12
106.00
105.88
105.
04
80
-5.00
-2.54
112.00
110.29
106.
43
81
-2.00
2.58
107.00
108.57
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.

85
Capital
Index
Observed
Demand
Estimated
Short-run
Demand
Estimated
Long-run
Demand
Figure 4-1. Comparison of Observed and Estimated Demand
for Capital in Southeastern Agriculture, 1949-1981.

86
Table 4-3. Comparison of Observed and Estimated
Levels of Net Investment and Demand for Land.
Net Investment Demand for Land
Year
•
A
observed
A*
optimal
A
observed
A*(K,p,w)
short-
run
A(p,w)
long-
run
49
3.45
2.64
142.32
146.78
193.99
50
1.14
2.11
145.77
150.01
198.58
51
0.76
1.01
146.91
149.98
189.99
52
0.19
-0.16
147.68
149.35
177.78
53
-0.95
-0.92
147.87
148.79
172.26
54
-0.57
-1.85
146.91
147.27
170.28
55
-2.86
-2.68
146.34
145.87
163.14
56
-3.05
-3.18
143.48
142.29
152.68
57
-3.63
-3.52
140.43
138.86
146.27
58
-3.82
-3.02
136.80
135.44
141.64
59
-3.44
-2.70
132.99
131.66
135.40
60
-2.48
-2.83
129.55
128.00
129.24
61
-2.86
-2.77
127.07
125.47
125.47
62
-2.86
-2.65
124.21
122.61
121.53
63
-3.63
-2.24
121.35
120.04
119.86
64
-1.72
-2.21
117.72
116.35
114.83
65
-1.53
-2.21
116.00
114.60
112.55
66
-1.53
-2.34
114.48
112.94
109.86
67
-1.91
-1.48
112.95
112.14
112.78
68
-2.29
-2.19
111.04
109.49
104.95
69
-1.91
-1.74
108.75
107.59
105.01
70
-1.34
-1.45
106.85
105.87
103.60
71
-0.95
-1.63
105.51
104.11
96.77
72
-0.76
-1.56
104.56
103.31
97.74
73
-1.15
-0.87
103.79
103.09
99.82
74
-0.57
-0.38
102.65
102.27
99.84
75
0.19
-0.28
102.08
101.90
101.64
76
-1.23
-0.61
102.27
101.76
99.29
77
-1.04
-0.58
101.04
100.54
97.83
78
-1.23
-0.48
100.00
99.52
96.33
79
0.87
-0.38
98.77
98.37
95.52
80
0.22
-0.48
99.63
99.20
96.70
81
-1.62
-0.48
99.85
99.32
95.36

87
Land
Index
Observed Estimated Estimated
^ Demand + Short-run ^ 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-l, 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-

89
Table 4-4. Comparison of Observed and Estimated
Short- and Long-Run Demands for Labor.
Year
Labor
observed
Labor Demand
L*(K,p,w)
short-run
L (K, p, w)
long-run
49
351.195
351.812
351.592
50
320.518
336.175
339.369
51
336.653
320.176
319.446
52
311.155
306.751
302.778
53
295.817
294.246
290.103
54
267.331
270.240
273.312
55
265.139
253.921
253.412
56
239.841
232.346
230.377
57
205.578
217.464
219.113
58
192.231
211.425
210.896
59
196.813
204.972
200.875
60
189.442
189.117
188.722
61
181.873
181.008
177.777
62
179.681
173.914
170.290
63
175.299
169.490
166.351
64
161.355
155.314
151.941
65
146.813
148.204
143.835
66
136.454
138.309
133.380
67
138.048
143.208
139.392
68
128.685
126.861
123.177
69
129.283
124.008
122.116
70
122.908
121.029
117.851
71
120.319
115.411
102.439
72
114.542
108.599
100.668
73
113.147
113.642
105.218
74
109.761
116.091
107.571
75
106.375
107.527
106.405
76
103.785
105.591
101.054
77
100.000
101.159
98.056
78
96.016
99.784
94.424
79
92.430
92.872
92.140
80
95.817
89.070
93.235
81
91.434
95.881
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
similiar.
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,

91
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
49
44.237
44.877
41.464
50
42.623
44.431
43.123
51
49.666
47.821
45.440
52
50.194
50.111
47.166
53
51.917
52.350
49.974
54
51.345
49.255
51.308
55
54.943
53.015
53.908
56
55.006
55.332
56.009
57
54.630
54.572
57.477
58
57.657
58.220
59.693
59
63.332
64.191
63.653
60
63.839
63.245
64.828
61
66.081
67.067
67.159
62
68.409
69.178
69.021
63
69.281
70.373
70.176
64
76.910
75.303
75.007
65
78.817
78.569
77.764
66
79.390
79.980
78.993
67
81.210
83.445
82.368
68
82.814
85.197
84.826
69
87.717
88.000
88.212
70
94.124
93.462
92.798
71
98.432
97.382
91.829
72
101.150
98.821
95.828
73
101.801
94.462
90.681
74
97.131
96.437
92.264
75
86.274
92.328
91.928
76
96.985
100.289
98.369
77
100.000
101.008
99.833
78
105.649
107.951
105.538
79
114.375
109.014
108.972
80
105.367
103.573
106.169
81
112.323
114.331
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-l - 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

94
Table 4-6. Estimated Short-run Average Uncompensated
Input Demand Elasticities for Southeastern U.S.
Agriculture for Selected Subperiods, 1949-1981.
Elasticity with Respect to the
Input Period price of:
Labor Materials Capital Land
Labor 1949-55
1956-60
1961-65
1966-70
1971-75
1976-81
Materials 1949-55
1956-60
1961-65
1966-70
1971-75
1976-81
Capital 1949-55
1956-60
1961-65
1966-70
1971-75
1976-81
Land 1949-55
1956-60
1961-65
1966-70
1971-75
1976-81
-0.703
0.311
-0.629
0.272
-0.542
0.242
-0.436
0.196
-0.335
0.134
-0.248
0.100
0.280
-0.170
0.195
-0.234
0.166
-0.299
0.148
-0.367
0.112
-0.325
0.104
-0.358
0.062
-0.071
0.046
-0.052
0.043
-0.049
0.035
-0.040
0.034
-0.039
0.026
-0.029
-0.041
0.029
-0.028
0.020
-0.022
0.016
-0.018
0.013
-0.015
0.011
-0.012
0.009
0.073
-0.207
0.065
-0.181
0.059
-0.161
0.046
-0.131
0.039
-0.090
0.025
-0.068
-0.098
0.130
-0.073
0.091
-0.064
0.078
-0.060
0.070
-0.043
0.053
-0.043
0.050
-0.344
0.066
-0.191
0.048
-0.154
0.045
-0.074
0.037
-0.126
0.036
-0.041
0.027
0.023
-0.082
0.016
-0.057
0.012
-0.044
0.010
-0.033
0.009
-0.024
0.007
-0.018

95
own-price elasticity of demand for capital declines in
absolute value throughout, with the exception of an
increase from 1966 to 1970. The demand for labor is
more elastic than the demand for materials until the
1976-81 time period, when the own-price elasticity of
labor demand is -0.25 and the corresponding elasticity
for materials is -0.36.
The estimated long-run uncompensated demand
elasticities for selected periods are presented in
Table 4-7 (See Appendix Tables D-5 through D-8 for
annual estimates). All long-run own-price demand
elasticities are negative. The substitute/complement
relationships of the long run are identical to those
of the short run. Labor and capital and land and
capital are long-run substitutes. Land and labor are
long-run complements, as well as land and materials.
Finally, materials and labor are long-run substitutes.
With the exception of labor, the own-price
elasticities are more elastic in the long run than the
short run, consistent with the Le Chatilier principle
which states that long-run own price elasticities
should be at least a large as the corresponding short-
run elasticities. This difference is especially
obvious in the short- and long-run elasticities for
the quasi-fixed inputs.

96
Table 4-7. Estimated Long-run Average Uncompensated
Input Demand Elasticities for Southeastern U.S.
Agriculture for Selected Subperiods, 1949-1981.
Elasticity with Respect to the
Input Period price of:
Labor
Materials
Capital
Land
Labor
Materials
Capital
Land
1949-55
-0.636
0.325
0.074
-0.166
1956-60
-0.577
0.270
0.066
-0.140
1961-65
-0.505
0.235
0.062
-0.120
1966-70
-0.410
0.184
0.046
-0.096
1971-75
-0.318
0.120
0.049
-0.062
1976-81
-0.239
0.087
0.026
-0.046
1949-55
0.275
-0.196
-0.115
0.129
1956-60
0.195
-0.245
-0.091
0.095
1961-65
0.168
-0.306
-0.084
0.084
1966-70
0.150
-0.369
-0.085
0.075
1971-75
0.115
-0.330
-0.057
0.059
1976-81
0.107
-0.360
-0.062
0.055
1949-55
0.044
-0.215
-3.105
0.804
1956-60
0.024
-0.118
-1.370
0.441
1961-65
0.016
-0.079
-0.844
0.294
1966-70
0.011
-0.056
-0.413
0.209
1971-75
0.009
-0.047
-0.468
0.177
1976-81
0.007
-0.035
-0.197
0.131
1949-55
-0.229
0.169
0.398
-0.555
1956-60
-0.187
0.138
0.326
-0.474
1961-65
-0.154
0.114
0.268
-0.391
1966-70
-0.127
0.094
0.222
-0.316
1971-75
-0.112
0.083
0.195
-0.236
1976-81
-0.087
0.064
0.152
-0.189

97
For the variable inputs, the direct effect of a
change in the own price of the input is identical in
the short-run and long-run. But as noted earlier, the
elasticity of a variable input is also conditional on
the levels of the quasi-fixed inputs and net
investment in the short-run. The greater short-run
elasticity of labor arises as labor enhances the
adjustment of capital. This is evident in the
elasticity of capital with respect to a change in the
price of labor. The estimated short-run cross-price
elasticity is more elastic than the long-run.
The absolute value of the long-run own-price
elasticity of demand for capital is greater than one
through 1960. This may be attributed to the
relatively small value of equilibrium capital demand
during that time period. The other factor demands are
inelastic throughout the time span of the data. After
1965 the long-run demand for land is the most own
price inelastic, and is consistently less elastic than
the long-run demand for capital. The own-price
elasticity of materials is the least elastic of the
four inputs until 1966. However, by 1981, materials
demand is the most elastic.
The own- and cross-price elasticities for the
short- and long-run trend downward with the exception
of the own-price elasticity of materials which becomes

98
more elastic over time. One may also note that the
normalized prices of labor, capital, and land trend
upward over time which increase the numerators in the
various elasticity formulae. However, Epstein and
Denny also noted a trending in elasticity measures in
their study of U.S. manufacturing using detrended
data. While the rise in the normalized price of
materials over the data period is not as great as that
of the other inputs, the price does increase so that
attributing the downward trend in elasticity measures
to an upward trend in normalized prices would be
somewhat inconsistent with the increasing own- price
elasticity of materials demand.
The Multivariate Flexible Accelerator
The adjustment matrix M may be obtained almost
directly from the estimated parameters by adding the
discount rate to the diagonal elements of G. The
implied rate of adjustment for capital, r + gKK, is “
-0.538. This implies that nearly 54 percent of the
optimal net investment in capital occurs in the first
year following a change in relative prices given an
equilibrium level of land. The adjustment rate for
land, r + g^A' i-s “0*163. Thus, net investment in
land reduces the difference in the equilibrium and
actual level of land by 16 percent in the first year
in response to a change in relative prices. The rate

99
of adjustment for one quasi-fixed input also depends
on the degree of disequilibrium in the other input.
Capital is especially dependent on the disequilibrium
in land stocks given the estimated magnitude of g^ at
0.49. An overinvestment in land would significantly
reduce the net adjustment of an excess capital stock.
Net investment in land is much less dependent on
an equilibrium in capital stocks. This is reflected
by gAK = -0.0213 not significantly different from
zero. The negative value of gAK implies that
overinvestment in capital stocks speeds the reduction
of land stocks given an overinvestment in land.
The accelerator mechanism derived from the value
function is amenable to a sequential test of
independent rates of adjustment and instantaneous
rates of adjustment, as both hypotheses are nested
within the unrestricted model. The test of
independent rates of adjustment is relevant given the
number of studies previously cited that assume the
rate of adjustment of one quasi-fixed factor is
unaffected by the degree of disequilibrium in other
quasi-fixed input levels. The hypothesis of
instantaneous adjustment for both quasi-fixed inputs
is conditional on independent rates of adjustment.
Instantaneous rates of adjustment indicate that the
quasi-fixed inputs, capital and land, adjust

100
completely to equilibrium levels within one time
period. This does not imply that the quasi-fixed
factors are in fact variable inputs, as the demands
for labor and materials are still conditional on net
investment and the levels of capital and land.
The results of the sequential hypotheses tests
are presented in Table 4-8. The hypothesis of
independent rates of adjustment, H1, imposed by
restricting the adjustment matrix to be diagonal, is
rejected at a significance level of 0.05. Rejection
of this hypothesis should terminate the testing
procedure at this point, implicitly rejecting the
hypothesis of instantaneous rates of adjustment. This
explains the large magnitude of the test statistic for
H2, where the adjustment matrix is restricted to the
negative of an identity matrix. These results support
the empirical specification of the multivariate
flexible accelerator treating capital and land as
quasi-fixed factors interdependent on the degree of
disequilibrium in their respective levels.
Beyond the result that capital and land do appear
to be quasi-fixed, exhibiting a less-than-complete
adjustment to equilibrium levels given a change in
relative prices, the interdependence of adjustment has
some significant implications. Were the adjustments
of land and capital to equilibrium levels independent

101
Table 4-8. Sequential Hypothesis Tests of Independent
and Instantaneous Rates of Adjustment for Capital and
Land.
Hypothesis
Test
Statistic3
Critical
Value
:
Independent rates of
adjustment
<%A= 9aK “ °)
9.444
X2,0.025= 7-378
«1 =
Unrestricted model
Ho :
Instantaneous
adjustment
<9KK+ r = §AA+ r = 1
/
¿KA = ®AK = 0)
405.459
X3,0.025= 9'348
H1 :
Independent rates of
adjustment
a The test statistic utilized is T° = n(S° - S) where
S° denotes the minimized distance of the residual
vector under the null hypothesis. S represents this
same value for the unrestricted model and n is the
sample size. Under the null hypothesis, T° - X2 with
degrees of freedom equal to the number of independent
restrictions (Gallant and Jorgenson, 1979).

102
or if land was at its equilibrium level, capital would
achieve 75 percent of the necessary adjustment in two
years, while land would require nearly nine years to
achieve the same level of adjustment if the level of
capital was fully adjusted to its equilibrium level.
It may be noted in Figures 4-1 and 4-2 that changes in
the equilibrium level of capital produce a much
steeper change in the short-run demand for capital
than appears in the short-run demand for land with
respect to a corresponding change in the equilibrium
level of land. In 1978, the observed level of capital
was quite close to the equilibrium level. The index
of short-run demand for land exceeded equilibrium
demand by 3.3 percent. The predicted disinvestment in
land was 15 percent of this disequilibrium, quite
close to the own rate of adjustment for land.
Recognition of the own rate of response in these
figures, especially for capital, is clouded by the
interdependence of factor adjustments.
Disequilibrium in the stock of land also
substantially affects the adjustment of capital.
While a scenario of fixed relative prices is somewhat
unrealistic, it may be used to illustrate the
importance of this interdependent relationship and the
hazards of single-equation partial adjustment models.
Figure 4-3 and 4-4 illustrate the adjustment of land

103
and capital to equilibrium levels reflecting two
alternative starting points. Figure 4-3 corresponds
to the observed and equilibrium levels of land and
capital based on 1980 relative prices. Initially,
observed values of land and capital exceed equilibrium
levels. While the own rates of adjustment indicate
that capital adjusts over three times as quickly as
land, the overinvestment in land significantly reduces
the net adjustment in capital so that after five
years, 72 percent of the adjustment in capital and
nearly 70 percent of the adjustment in land has been
achieved.
Figure 4-4 corresponds to the relative prices in
1981 and respective observed and equilibrium levels of
capital and land. In this case, capital is initially
below equilibrium and land stocks exceed equilibrium
levels. The overinvestment in land speeds net
investment in capital to the point that simulated
short-run capital "overshoots" the equilibrium level.
Stability of the adjustment matrix ensures that
capital levels will eventually approach equilibrium.
However, the original disequilibrium was one of
underinvestment in capital. With the overinvestment
in land present in 1981, it would require three years
for capital stocks to peak and then begin to return to
an equilibrium level.

104
Capital, Land
Index
n Initial , Equilibrium. Initial Equilibrium
Capital Capital Land ü Land
Figure 4-3. Adjustment Paths of Capital and Land Given
1980 Relative Prices and Initial Capital and Land Stocks.

105
Capital, Land
Index
Year
g Initial
Capital
, Equilibrium Initial
Capital Land
Equilibrium
Land
Figure 4-4. Adjustment Paths of Capital and Land Given
1981 Relative Prices and Initial Capital and Land Stocks.

106
An alternative policy to fixing relative prices
is one that artificially maintains an input above its
equilibrium level. While such a policy is likely to
affect relative prices, identifying that effect is
difficult. Figures 4-5 and 4-6 represent the same
time periods as the previous graphs with identical
initial disequilibria and relative prices. However,
no adjustment in land stocks is permitted. For the
1980 conditions, capital initially moves toward its
equilibrium level but levels out. In effect, a policy
maintaining land stocks also maintains capital above
its equilibrium level. Similarly, if capital
initially is below equilibrium as shown in the 1981
graph, maintaining land stocks not only causes capital
to overshoot its equilibrium but to remain at a excess
level. Rather than peaking and returning to the long-
run level of demand, capital levels continue to climb
above the equilibrium level.
The difference in simulated short-run and
equilibrium levels of capital has a small effect on
the adjustment process of land stocks unless the
capital disequilibrium is extremely large. The
relative independence of the adjustment of land would
indicate that policies directed at land stocks appear
to work, yet create undesirable effects in the level
of other factors. When such a policy is removed, the

107
Capital, Land
Index
Year
Initial Equilibrium Initial Equilibrium
U Capital + Capital 0 Land A Land
(fixed)
Figure 4-5. Adjustment Path of Capital with 1980
Relative Prices and Initial Capital and Land Fixed.

114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
] I:
c
are
Land
0—9—9—0 0 0 0 » 0 9 9—OOOOOOO 0
A- ■ ft- A- - A - -A A -A- A--A--A—A--A—ft A ft A ft ft -A
85 90 95
Year
.tial
)ital
+
Equilibrium^
Capital
Initial
Land
(fixed)
. Equilibrium
4 Lind
4-6. Adjustment Path of Capital with 1981
: Prices and Initial Capital and Land Fixed.

109
new disequilibrium could potentially be worse than the
original. Finally, ignoring the relative price
effects of a policy not directed specifically at
factor stocks may be worse than a failure to recognize
the interdependence of the adjustment process.

CHAPTER V
SUMMARY AND CONCLUSIONS
The application of duality theory in static models
of economic analysis is somewhat of a convenience.
Although primal formulations may be difficult to solve,
analytic solutions can be found for a wide variety of
objective functions. Within the realm of dynamic
economic analysis, however, the use of duality in order
to generate theoretically consistent systems of demand
equations is a necessity rather than a convenience in
all but the simplest of model specifications. In spite
of the importance of dynamic duality in generating
systems of demand equations, its use in empirical
analysis has been limited.
The purpose of this study was to specify and
estimate a complete system of dynamic input demand
equations for southeastern U.S. agriculture by drawing
upon the recent developments in the theory of dynamic
duality. The results of this study not only provide
information on the dynamic adjustment and short- and
long-run interrelationships of inputs, but also provide
some further insight and evidence concerning the
empirical usefulness of dynamic duality theory.
110

Ill
The empirical model treated capital and land as
quasi-fixed inputs and labor and materials as variable
inputs. The estimated system was generally consistent
with the regularity conditions of the theory.
Equilibrium levels of capital and land are positive at
each data point and the matrix that characterizes
adjustment of capital and land toward equilibrium levels
is stable. Furthermore, the required convexity of the
value function in input prices is satisfied by the data
at all but two observations.
As noted in the analysis, explicit use of a dynamic
objective function permits a clear distinction between
the short run and long run. The substitute/complement
relationships indicated by the short-run price
elasticity estimates indicate capital and land are
short-run substitutes, as are capital and labor.
Capital and materials are estimated to be complements in
the short run, while labor and materials are short-run
substitutes.
The long-run elasticity estimates indicate the same
substitute/complement relationships as in the short run.
The demand for land is generally more inelastic than the
other input demands in the short and long run. Long-run
demands are generally more elastic than short-run.
Labor appears to facilitate the adjustment of capital
and thus labor demand is more elastic in the short-run

112
With the exception of the own-price elasticity of
materials, elasticity estimates in the short and long
run become less elastic over time.
The ability to estimate both short-run and long-run
demands permits estimation of the relative amount of
disequilibria in inputs, or just how far the short-run
observed levels are from the long-run levels. In the
initial years of the data period, the model indicates a
substantial degree of disequilibrium in capital and land
inputs. However, from 1965 the differences in long-run
equilibrium levels and observed demands are much less.
The model indicates that there is a slight
overinvestment in land in the Southeast from 1960 to
1981. Contrary to the notion of "backward"
agricultural production practices in the South, in the
sense that the substitution of capital for labor
indicates technical progress, the data indicate that the
Southeast was overcapitalized from 1949 to 1978. The
adjustment of capital to near long-run equilibrium was
not acheived through disinvestment in capital stocks,
however, but was a consequence of the rising relative
price of labor, which increased the long-run demand for
capital. The parameter estimates of technical change
indicate that technical change has stimulated the demand
for all inputs in the Southeast. This is consistent
with the general growth of agriculture in the South.

113
The differences in short-run and long-run demand
for capital and land from 1949 to 1965 are perhaps a
result of the maintained assumption that price
expectations are stationary. In retrospect, it is easy
to recognize the trend of rising prices for land and
labor. As noted earlier, perhaps producers in the
Southeast perceived this trend and had already invested
in capital as a substitute for labor and land.
Alternatively, the relatively large difference in long-
and short-run levels of demand for capital and land may
be a lingering effect of agricultural policies initiated
prior and during World War II which were not removed
until 1948.
The empirical results indicate that capital and
land are slow to adjust to changes in equilibrium
levels. Given an equilibrium level of land, the model
indicates an adjustment of nearly 54 percent of the
difference in actual and long-run level within one time
period. Land appears to adjust less rapidly, reducing
the difference in actual and equilibrium levels by 15
percent in one time period. However, the rate of
adjustment for capital appears to be particularly
dependent on the degree of disequilibrium in land
stocks.

114
The relatively slow adjustment rate of land
provides an indication of why some static optimization
models maintained land as fixed in the short run.
However, the dependence of adjustment of capital on the
land is shown to prevent capital from attaining a long-
run equilibrium if a disequilibrium in land was fixed.
In fact, whether capital is initially above or below its
equilibrium position, maintaining land stocks above
equilibrium also drives the capital stock above its
equilibrium position. Although it may be unrealistic to
assume relative prices remain unchanged for an extended
period, the simulated adjustments of land and capital
based on relative prices and existing disequilibrium in
1980 and 1981 illustrates the importance of
interdependent rates of adjustment. The problems of
"overshooting" and fixing the level of one input are
heightened if relative price changes increase the degree
of disequilibrium. The appropriateness of policy
directed at alleviating the disequilibrium of a single
input is questionable at best, and may be
counterproductive to desired goals.
It should be noted that the adjustment rates
estimated for the Southeast differ considerable from
estimates derived from a similar model of U.S
agriculture. Chambers and Vasavada (1983b) and Karp,
Fawson, and Shumway (1984), conclude that at the

115
national level land adjusts almost instaneously to
changes in its long-run or equilibrium position. It is
not surprising that they reach the same conclusion as
they appear to be using the same data set. However,
whether the difference in this analysis and the prior
analyses is a result of definitional differences or a
real difference in adjustment in land in the Southeast
from the U.S. is unclear. If one considers the
significance of perennial crops in the Southeast, such
as fruits and tree nuts, and the acreage allotment
policies for peanuts and tobacco that serve to keep land
in agriculture, perhaps it is not surprising that the
adjustment of land in the Southeast is slow.
At the present time, the application of dynamic
duality in economic theory is relatively unexplored. The
opportunities for its application extend far beyond the
area of input demands. However, there are several
issues pertinent to this analysis that deserve mention.
Even though a wide variety of functional forms of
the value function potentially consistent with the
underlying regularity conditions may be specified, there
appears to be a trade-off between simplicity of the
specification and satisfaction of the regularity
conditions in empirical application. Initial efforts
using a strictly quadratic form of the value function
for the Southeast, other regions, and the U.S. proved to

116
be unsatisfactory. Limitations on time and money
prevented application of the log-quadratic model to
other regions for this dissertation.
Given the number of parameters and nonlinearities
inherent in the estimation equations, the multivariate
flexible accelerator as a matrix of constants may be the
only practical adjustment mechanism for empirical work.
Ideally, a general form of input demand equations, for
which the flexible accelerator as a matrix of constants
is a special case imposed by restricting parameters,
could be tested. Even then, the application is limited
by the data in that each additional quasi-fixed input
considered expands the parameters of the adjustment
mechanism. Finally, the test for instantaneous
adjustment of an input depends on independent rates of
adjustment. Thus, appropriately testing quasi-fixity
for a single input does not appear possible.
The notion of rational expectations is often
considered in determining the appropriate price for
decision-making in static models (see Fisher, 1982).
Taylor (1984) noted that the derivation of input demands
through dynamic duality depends on the certainty of
price expectations and incorporation of risk in the
dynamic model implies a different set of conditions for
an optimal solution. However, Karp, Fawson, and Shumway
(1984) estimate a dynamic model under several price

117
expectation hypotheses and conclude that the adjustment
rate is fairly insensitive to the underlying price
expectation. Finally, Chambers and Lopez (1984) argue
that stationary price expectations may be rational given
the costs of information and difficulties in formulating
alternative expectations in a dynamic setting.
Additionally, the treatment of relative input
prices as exogenous variables may be not be appropriate.
It would seem that the adjustment rates and prices would
be related. If a large degree of overinvestment and a
"fast" rate of adjustment were indicated for a quasi-
fixed input, then a large quantity of the input is
potentially for sale, in turn depressing the price of
the input. Likewise, if an input was below its long-run
equilbrium, it would seem that the price of the input
would be bid up as firms adjust toward equilibrium.
The treatment of prices as exogenous also has
implications for policy based strictly on the rates of
adjustment. Policy directed at physical quantities of
inputs, such as set-asides, buy-outs, and land
retirment, inevitably affect input price. In turn,
these price changes determine a new equilibrium
position, and the adjustment process starts again.
Recent work on joint determination of technolgoy and
price expectations by Epstein and Yatchew (1985) may
provide some insight to this problem.

118
Another potential issue is the symmetry of
adjustment costs. The empirical model assumes that net
disinvestment augments output by the same amount that
net investment decreases output. It is entirely
possible that adjustment costs are not symmetric, in
that it may be more or less costly to invest rather than
disinvest in a quasi-fixed input. In its current state,
dynamic duality theory has little to say about this
issue, but appears to be most likely method for
exploration.
In closing, it should be noted that the application
of optimal control theory in the first manned mission to
the moon indicated that the Apollo spacecraft was "off
course" 90 percent of the time (Garfield, 1986). Yet
the theory was vital to the conceptualization of man
actually landing on the moon. Perhaps an analogy can be
drawn for the application of dynamic duality theory in
economic analyses. For the present, emphasis should not
be on how "on course" such analyses are, but recognize
instead the ability to conceptualize and address issues
that were previously outside the possibilities of
economic theory.

APPENDIX A
INPUT DEMAND EQUATIONS

Table A-l. Estimation Equations for Optimal Net
Investment in Quasi-fixed Inputs and Short-run
Variable Input Demands.
Optimal Net Investment in Capital:
K*(K,p,w) = (r/pK) • [ (bk-gKK + b^g^) + (gKK*bkk +
gKA*bka)‘l0(3 PK + (9KK*bka + gKA*baa)'log Pa +
(9KK-dkw + gKA'daw)’log PL + (9KK*dkm +
^KA’dam)’lo9 Pm] + r'vK*T + (r + gKK)*K + 9ka'a
Optimal Net Investment in Land:
Á*(K,p,w) = (r/pA) • [ (bk*gAK + b^g^) + (gAK‘bkk +
gAA‘bka)'log PK + (*?AK"bka + gAA,baa)'lo ( 9AA* dam) *lo Short-run Demand for Labor:
L* (K, p, w) = (-r/pL) • (cw + c^-log pL + c^-log pM +
dkw'log PK + daw‘log Pa)~r"(nKw* ^ + nAw'A) +
(nKw‘K*(K,p,w) + nAw* Á*(K,p,w)) + r-vL*T
Short-run Demand for Materials:
M*(K,p,w) = (-r/pM)-(cm + c^* log pL + cmm*log pM +
dkm*log PK + dam’log PA)-r,(nKm‘K + nAm‘A) +
(nKm'K*(K,p,w) + nAm-A*(K,p,w)) + r*vM*T
120

121
Table A-2. Long-run Input Demand Equations.
Long-run Demand for Capital:
K(p,w) = (-r/pk) • (c0 + cx * log pk + c2-log pa +
c3’l°g Pw + c4•log pm) - r•c5•T
Long-run Demand for Land:
A(p,w) = (-r/pa) • (d0 + dx * log pk + d2*log pa +
d3 * log pw + d4’log pm) - r-d5-T
Long-run Demand for Labor:
L(K;p,w) = (-r/pw) • (cw + cw'log pw + c^-log pm +
dkw‘lo Long-run Demand for Materials:
M(K,p,w) = (-r/pm) ' (cm + c^-log pw + c^-log pm +
dkm’log Pk + dam‘lo where
c0 = mxl•a0 + m12*a0 d0 =
C1 = mll’al + m12’al dl =
c2 = mn" a2 + m12'a2 d2 =
Ci = m-i-i'a-i + mi/ai do =
c4 = m11,a4 + itii2' a4 d4 =
c5 = in-vK + m12-vA d5 =
mll = (r + gj^/Cir + gKK) ' (r +
m2 2 = (r + gKK)/[(r + gKK)'(r +
™12 = “9ka/ [ (r + 9kk) ' (r + ro21 = -9AK/[(r + 9kk)‘(r + ?Aa)
and
m2i'b0 + m22•b0
mo i•bn + moo•bi
m21 * b2 + m22'b2
mii-bo + moo *bo
m2l'b4 + m22*b4
m21’vk + ™22*VA ,
9aa) “ ^KA’gAK)
9aa) " 9KA‘9akJ
- 9ka‘9ak)
_ 9ka"9ak) #
a0 “ bk* ^KK + ba,(?KA
al = bkk’9KK + bka * 9ka
a2 = bka’9kk + baa * 9ka
a3 “ dkw**?KK + daw’9KA
a4 = dkm* *?KK + dam‘ 9kA
b0 = bk,(?AK + ba * 9aA
bl = bkk’9aK + bka* b2 = bka * 9aK + haa'^AA
b3 “ dkw‘^AK + daw’9AA
b4 = dkm,
122
Table A-3. Short-run Demand Equations for the Quasi-
fixed Inputs.
Short-run Demand for Capital:
K*(K,p,w) = (r/pK) * [ (bk*gKK + ba-g^) + (gKK*bkk +
gKA*bka)'log PK + (gKK‘bka + gKA-baa)‘log Pa +
(9KK*dkw + gKA,daw)’log Pl + (9KK*dkm +
gKA'dam)*log Pm] + r>vK'T + (1 + r + gKK)-K +
Short-run Demand for Land:
A*(K,p,w) = (r/pA)•[(bk-gAK + ba-gAA) + (gAK‘bkk +
gAA'bka)‘log PK + ( (dkw + gAA,daw)’log PL + (9AK'dkm +
gAA*dam)’log Pm] + r'vA*T + gAK*K +
(1 + r + gAA)-A

APPENDIX B
REGIONAL EXPENDITURE, PRICE, AND INPUT DATA

Table B-l. Expenditures on Capital Use and Capital
Price Index for the Southeast.
Repairs and
Operations
Depreciation
and Consumption
Capital
Price
Index
Year
Million
dollars
49
139.2
118.3
0.721
50
148.0
136.6
0.744
51
166.7
162.7
0.775
52
181.8
172.3
0.731
53
190.3
183.6
0.735
54
198.0
201.9
0.764
55
203.8
207.4
0.765
56
214.6
204.2
0.734
57
224.1
214.0
0.760
58
228.2
223.5
0.760
59
237.7
236.6
0.760
60
227.4
229.5
0.738
61
222.0
227.6
0.725
62
227.8
237.2
0.746
63
229.9
249.1
0.748
64
227.2
268.5
0.757
65
247.8
292.9
0.807
66
259.2
311.5
0.826
67
278.5
342.8
0.848
68
292.3
369.4
0.847
69
304.5
408.4
0.887
70
312.1
422.3
0.874
71
298.8
441.9
0.740
72
301.0
476.1
0.819
73
323.5
541.8
0.861
74
411.5
637.5
0.959
75
454.1
744.1
0.978
76
533.6
805.6
0.957
77
600.5
868.7
1.000
78
687.1
1123.5
1.081
79
834.9
1274.0
1.060
80
992.8
1367.4
1.165
81
1091.7
1522.4
1.144
124

125
Table B-2. Acres and Acreage Index of Land in Farms
and Value of Farm Real Estate Index for the
Southeast.
Acres in Acreage Value
Farms Index Index
Year Million acres
48
749
142.909
-
49
767
146.343
0.226
50
770
146.916
0.220
51
773
147.488
0.241
52
774
147.679
0.256
53
771
147.106
0.266
54
770
146.916
0.268
55
752
143.481
0.278
56
736
140.428
0.296
57
717
136.803
0.323
58
697
132.987
0.347
59
679
129.553
0.379
60
666
127.072
0.399
61
651
124.210
0.408
62
636
121.348
0.445
63
617
117.723
0.462
64
608
116.006
0.493
65
600
114.480
0.535
66
592
112.953
0.558
67
582
111.045
0.570
68
570
108.756
0.632
69
560
106.848
0.665
70
553
105.512
0.681
71
548
104.558
0.705
72
544
103.795
0.730
73
538
102.650
0.795
74
535
102.078
0.960
75
536
102.269
0.911
76
530
101.038
0.924
77
524
100.000
1.000
78
518
98.769
1.138
79
522
99.643
1.163
80
523
99.859
1.249
81
517
98.628
1.272

126
Table B-3. Expenditures on Hired Labor and Quantity
and Price Index for the Southeast.
Hired
Labor
Hired Labor
Index
Labor Price
Index
Year
Million dollars
49
189.9
674.98
0.097
50
190.6
582.38
0.110
51
214.8
549.97
0.123
52
215.4
531.82
0.126
53
215.6
516.43
0.128
54
215.6
481.92
0.135
55
225.6
458.74
0.146
56
235.7
410.47
0.165
57
246.4
397.40
0.172
58
249.4
361.86
0.188
59
261.7
347.13
0.201
60
258.3
322.37
0.210
61
265.8
309.04
0.223
62
268.1
287.96
0.237
63
268.9
274.99
0.246
64
277.5
246.10
0.279
65
278.5
212.91
0.317
66
278.8
179.36
0.370
67
298.5
182.95
0.372
68
318.0
166.68
0.416
69
348.7
165.41
0.437
70
365.6
153.28
0.470
71
453.3
124.43
0.683
72
484.8
116.52
0.749
73
568.8
126.42
0.766
74
632.2
113.11
0.875
75
689.6
110.21
0.896
76
711.7
100.93
0.960
77
777.6
100.00
1.000
78
925.3
88.74
1.249
79
1047.5
84.21
1.371
80
1185.2
87.71
1.362
81
1417.3
80.91
1.614

127
Table B-4. Expenditures on Materials and Materials
Price Index for the Southeast.
Material:
Feed
Live¬
stock
Seed
Ferti¬
lizer
Miscell¬
aneous
Price
Index
Year
Million
dollars
49
123.4
22.5
37.1
187.2
103.1
0.969
50
136.3
26.6
36.5
188.3
107.9
1.053
51
182.6
41.4
41.1
205.9
129.8
1.054
52
208.9
42.3
43.7
214.5
132.3
1.037
53
196.8
40.3
43.9
203.1
129.2
0.943
54
222.1
52.4
39.4
202.0
126.8
0.983
55
225.4
59.8
43.9
191.4
136.6
0.937
56
259.2
66.8
40.7
193.5
144.1
0.963
57
283.4
70.3
40.6
190.9
133.7
0.993
58
335.8
87.9
40.4
187.7
141.7
0.990
59
358.5
74.0
36.5
196.8
159.8
0.905
60
365.8
89.7
37.7
201.6
165.7
0.910
61
382.1
82.3
37.8
206.6
174.4
0.878
62
412.4
89.1
36.8
211.9
187.1
0.883
63
457.5
88.9
38.1
207.7
196.1
0.899
64
464.5
83.9
40.4
219.5
214.3
0.814
65
486.6
97.3
47.3
217.2
230.9
0.820
66
544.0
115.6
45.6
225.2
245.6
0.853
67
568.8
116.0
52.4
234.7
254.0
0.854
68
514.9
116.4
48.6
237.9
279.6
0.784
69
562.4
138.0
52.0
231.0
308.5
0.768
70
629.5
137.4
52.4
227.9
339.2
0.721
71
776.1
127.1
84.7
304.4
533.6
0.831
72
776.5
137.1
87.6
329.0
587.9
0.799
73
1260.1
161.5
110.0
412.2
606.4
1.050
74
1410.5
151.4
132.7
680.1
720.5
1.247
75
1213.0
151.0
154.1
699.1
794.6
1.275
76
1307.6
169.6
178.7
586.9
912.1
1.039
77
1268.3
220.4
215.7
612.6
952.9
1.000
78
1274.7
270.2
192.8
633.3
1118.6
1.007
79
1626.0
310.7
216.3
685.2
1364.3
0.968
80
1659.7
319.0
244.8
839.1
1520.9
1.063
81
1637.5
357.4
289.0
843.4
1900.4
0.982

128
Table B-5. Total Gross Receipts, Output Index and
Output Price Index for the Southeast.
Total
Receipts
Output
Index
Output
Price
Index
Year
Million dollars
48
2057.7
62.80
1.062
49
1948.7
58.76
1.085
50
2103.0
58.66
1.149
51
2525.5
68.57
1.108
52
2400.9
64.16
1.109
53
2439.4
71.26
0.999
54
2137.0
63.54
0.970
55
2537.8
76.49
0.936
56
2402.1
73.84
0.890
57
2343.1
65.62
0.944
58
2642.4
68.18
1.008
59
2664.8
72.63
0.932
60
2659.0
74.35
0.894
61
2788.1
78.08
0.884
62
2842.2
79.81
0.866
63
3008.4
83.45
0.864
64
3017.4
81.08
0.878
65
3163.9
86.77
0.842
66
3313.9
78.19
0.961
67
3427.4
91.14
0.817
68
3433.0
83.83
0.852
69
3857.1
89.31
0.854
70
3871.2
89.66
0.811
71
4686.8
101.18
0.828
72
5144.9
98.65
0.895
73
6897.3
100.74
1.112
74
7252.6
105.97
1.021
75
7725.2
108.26
0.974
76
7798.2
112.13
0.902
77
8156.8
100.00
1.000
78
9629.2
111.26
0.988
79
11008.2
119.80
0.965
80
10543.1
107.31
0.944

129
Table B-6. Prices of Inputs Normalized with Respect
to Output Price Used in Estimations.
Normalized
Land
Price of:
Capital
Labor
Materials
Year
49
0.213
0.679
0.091
0.913
50
0.203
0.686
0.101
0.971
51
0.210
0.674
0.107
0.917
52
0.231
0.660
0.114
0.936
53
0.240
0.663
0.115
0.850
54
0.268
0.764
0.135
0.984
55
0.287
0.789
0.150
0.966
56
0.316
0.784
0.176
1.028
57
0.363
0.854
0.193
1.116
58
0.368
0.805
0.199
1.049
59
0.376
0.754
0.199
0.898
60
0.428
0.792
0.225
0.977
61
0.457
0.811
0.250
0.982
62
0.503
0.844
0.269
0.999
63
0.533
0.864
0.284
1.038
64
0.571
0.876
0.323
0.942
65
0.609
0.919
0.361
0.934
66
0.663
0.981
0.439
1.013
67
0.593
0.883
0.387
0.889
68
0.774
1.037
0.510
0.960
69
0.781
1.041
0.514
0.902
70
0.797
1.023
0.550
0.844
71
0.870
0.913
0.843
1.025
72
0.881
0.988
0.905
0.964
73
0.888
0.961
0.856
1.173
74
0.864
0.863
0.787
1.122
75
0.892
0.958
0.877
1.249
76
0.949
0.983
0.985
1.067
77
1.108
1.108
1.108
1.108
78
1.138
1.081
1.249
1.007
79
1.177
1.073
0.388
0.980
80
1.294
1.207
1.411
1.101
81
1.347
1.212
1.709
1.040

APPENDIX C
EVALUATION OF CONVEXITY OF THE VALUE FUNCTION

Table C-l. Results for Convexity Test of the Value
Function with Respect to Prices.
Jkk
Jkk Jka
Jak Jaa
Jkk Jka Jkw
Jak Jaa Jaw
Jwk Jwa Jww
Jkk Jka Jkw Jkm
Jak Jaa Jaw Jam
Jwk Jwa Jww Jwm
Jmk Jma Jmw Jmm
Year
49
2.246E+03
1.376E+08
7.041E+12
-1.490E+13 *
50
2.113E+03
1.420E+08
6.325E+12
-1.060E+13*
51
2.328E+03
1.458E+08
5.641E+12
1.340E+13
52
2.601E+03
1.349E+08
4.611E+12
1.990E+13
53
2.607E+03
1.250E+08
3.963E+12
3.810E+13
54
1.384E+03
5.375E+07
1.374E+12
8.140E+12
55
1.210E+03
4.094E+07
8.506E+11
7.980E+12
56
1.286E+03
3.598E+07
5.683E+11
6.060E+12
57
8.293E+02
1.7 68E+07
2.411E+11
2.190E+12
58
1.181E+03
2.447E+07
3.048E+11
4.090E+12
59
1.682E+03
3.320E+07
3.790E+11
9.020E+12
60
1.370E+03
2.101E+07
1.968E+11
4.310E+12
61
1.248E+03
1.680E+07
1.293E+11
3.100E+12
62
1.048E+03
1.168E+07
7.83 6E+10
1.980E+12
63
9.373E+02
9.325E+06
5.727E+10
1.370E+12
64
9.168E+02
7.916E+06
3.644E+10
1.23 OE+12
65
7.194E+02
5.449E+06
2.031E+10
7.260E+11
66
4.823E+02
3.067E+06
8.265E+09
2.430E+11
67
9.117E+02
7.255E+06
2.338E+10
9.440E+11
68
3.535E+02
1.640E+06
3.211E+09
1.080E+11
69
3.531E+02
1.606E+06
3.009E+09
1.160E+11
70
4.245E+02
1.858E+06
2.975E+09
1.410E+11
71
8.858E+02
3.273E+06
2.590E+09
9.23 0E+10
72
5.728E+02
2.050E+06
1.395E+09
5.220E+10
73
6.4 63E+02
2.300E+06
1.8 69E+09
5.040E+10
74
1.153E+03
4.340E+06
4.025E+09
1.22 0E+11
75
6.509E+02
2.299E+06
1.829E+09
4.330E+10
76
5.987E+02
1.856E+06
1.115E+09
3.630E+10
77
2.609E+02
5.845E+05
2.818E+08
6.770E+09
78
3.494E+02
7.461E+05
2.773E+08
9.040E+09
79
3.309E+02
6.853E+05
2.029E+09
8.020E+09
80
1.094E+02
1.685E+05
5.066E+07
2.433E+10
81
1.206E+02
1.726E+05
3.564E+07
2.201E+10
* indicates determinant inconsistent with convexity.
131

APPENDIX D
ANNUAL SHORT- AND LONG-RUN PRICE ELASTICITY ESTIMATES

Table D-l. Short-run Uncompensated Price Elasticities
for Capital.
Short-run Elasticity of Capital Demand with respect to
the price of:
Labor Materials Capital Land
Year
49
0.080
-0.091
-0.453
0.086
50
0.070
-0.080
-0.388
0.077
51
0.066
-0.075
-0.393
0.070
52
0.062
-0.071
-0.399
0.065
53
0.058
-0.066
-0.379
0.060
54
0.051
-0.058
-0.213
0.055
55
0.048
-0.055
-0.186
0.050
56
0.047
-0.054
-0.196
0.049
57
0.043
-0.049
-0.121
0.046
58
0.046
-0.052
-0.180
0.048
59
0.048
-0.055
-0.253
0.050
60
0.045
-0.052
-0.207
0.048
61
0.045
-0.051
-0.194
0.047
62
0.044
-0.050
-0.166
0.046
63
0.043
-0.049
-0.149
0.045
64
0.042
-0.048
-0.147
0.045
65
0.040
-0.045
-0.115
0.041
66
0.037
-0.042
-0.071
0.038
67
0.040
-0.046
-0.147
0.042
68
0.033
-0.038
-0.045
0.035
69
0.033
-0.037
-0.045
0.035
70
0.033
-0.038
-0.061
0.035
71
0.036
-0.041
-0.149
0.038
72
0.033
-0.037
-0.094
0.034
73
0.034
-0.039
-0.106
0.036
74
0.036
-0.042
-0.179
0.039
75
0.032
-0.037
-0.101
0.034
76
0.030
-0.035
-0.093
0.033
77
0.027
-0.030
-0.032
0.028
78
0.027
-0.031
-0.051
0.028
79
0.026
-0.029
-0.056
0.027
80
0.022
-0.025
-0.003
0.023
81
0.022
-0.025
-0.008
0.024
133

134
Table D-2. Short-run Uncompensated Price Elasticities
for Land.
Short-run Elasticity of Land Demand with respect to
the price of:
Labor Materials Capital Land
Year
49
-0.045
0.032
0.025
-0.095
50
-0.046
0.033
0.026
-0.093
51
-0.045
0.032
0.025
-0.087
52
-0.041
0.029
0.023
-0.080
53
-0.039
0.028
0.022
-0.076
54
-0.036
0.025
0.020
-0.073
55
-0.033
0.024
0.019
-0.068
56
-0.031
0.022
0.017
-0.062
57
-0.028
0.020
0.016
-0.060
58
-0.028
0.020
0.016
-0.058
59
-0.028
0.020
0.016
-0.054
60
-0.026
0.018
0.014
-0.051
61
-0.024
0.017
0.014
-0.048
62
-0.023
0.016
0.013
-0.046
63
-0.022
0.016
0.012
-0.045
64
-0.021
0.015
0.012
-0.041
65
-0.020
0.014
0.011
-0.038
66
-0.019
0.013
0.010
-0.035
67
-0.021
0.015
0.012
-0.037
68
-0.017
0.012
0.009
-0.031
69
-0.017
0.012
0.009
-0.031
70
-0.017
0.012
0.009
-0.029
71
-0.015
0.011
0.009
-0.023
72
-0.015
0.011
0.009
-0.022
73
-0.015
0.011
0.009
-0.024
74
-0.016
0.011
0.009
-0.025
75
-0.015
0.011
0.009
-0.025
76
-0.015
0.010
0.008
-0.021
77
-0.013
0.009
0.007
-0.020
78
-0.012
0.009
0.007
-0.017
79
-0.012
0.009
0.007
-0.016
80
-0.011
0.008
0.006
-0.017
81
-0.010
0.007
0.006
-0.014

135
Table D-3. Short-run Uncompensated Price Elasticities
for Labor.
Short-run Elasticity of Labor with respect to the
price of:
Labor Materials Capital Land
Year
49
-0.724
0.330
0.077
-0.219
50
-0.726
0.311
0.073
-0.207
51
-0.714
0.314
0.074
-0.208
52
-0.705
0.312
0.075
-0.207
53
-0.692
0.323
0.078
-0.214
54
-0.688
0.297
0.069
-0.197
55
-0.672
0.291
0.067
-0.194
56
-0.656
0.276
0.065
-0.184
57
-0.646
0.266
0.061
-0.177
58
-0.633
0.268
0.064
-0.179
59
-0.613
0.282
0.069
-0.187
60
-0.598
0.267
0.065
-0.178
61
-0.579
0.257
0.063
-0.171
62
-0.563
0.250
0.061
-0.167
63
-0.554
0.243
0.058
-0.162
64
-0.519
0.236
0.058
-0.157
65
-0.496
0.225
0.054
-0.150
66
-0.468
0.201
0.047
-0.134
67
-0.477
0.217
0.055
-0.145
68
-0.426
0.189
0.043
-0.126
69
-0.415
0.189
0.043
-0.126
70
-0.393
0.184
0.043
-0.123
71
-0.334
0.138
0.041
-0.093
72
-0.317
0.132
0.036
-0.089
73
-0.339
0.132
0.037
-0.089
74
-0.347
0.140
0.044
-0.094
75
-0.337
0.128
0.036
-0.086
76
-0.300
0.121
0.034
-0.081
77
-0.277
0.111
0.027
-0.075
78
-0.249
0.103
0.027
-0.069
79
-0.231
0.093
0.026
-0.064
80
-0.233
0.093
0.020
-0.063
81
-0.199
0.080
0.018
-0.054

136
Table D-4 . Short-run Uncompensated Price
Elasticities for Materials.
Short-run Elasticity of Materials Demand with
respect to the price of:
Labor Materials Capital Land
Year
49
0.327
-0.128
-0.114
0.152
50
0.296
-0.106
-0.102
0.137
51
0.297
-0.161
-0.102
0.138
52
0.276
-0.173
-0.091
0.128
53
0.290
-0.239
-0.098
0.135
54
0.242
-0.175
-0.090
0.113
55
0.233
-0.206
-0.088
0.109
56
0.208
-0.204
-0.077
0.097
57
0.186
-0.179
-0.073
0.087
58
0.189
-0.215
-0.071
0.089
59
0.207
-0.300
-0.075
0.097
60
0.186
-0.272
-0.068
0.087
61
0.177
-0.281
-0.066
0.083
62
0.169
-0.281
-0.064
0.079
63
0.160
-0.266
-0.061
0.075
64
0.165
-0.328
-0.064
0.078
65
0.160
-0.341
-0.063
0.075
66
0.145
-0.318
-0.059
0.068
67
0.159
-0.364
-0.062
0.075
68
0.142
-0.354
-0.059
0.067
69
0.146
-0.381
-0.060
0.069
70
0.148
-0.416
-0.061
0.070
71
0.121
-0.362
-0.046
0.057
72
0.124
-0.389
-0.049
0.058
73
0.107
-0.297
-0.042
0.051
74
0.110
-0.308
-0.040
0.052
75
0.100
-0.269
-0.039
0.047
76
0.109
-0.338
-0.043
0.052
77
0.104
-0.329
-0.043
0.049
78
0.108
-0.375
-0.044
0.051
79
0.107
-0.392
-0.043
0.051
80
0.099
-0.342
-0.042
0.047
81
0.099
-0.374
-0.042
0.047

137
Table D-5. Long-run Uncompensated Price Elasticities
for Capital.
Long-run Elasticity of Capital Demand with respect to
the price of:
Labor Materials Capital Land
Year
49
0.060
-0.294
-4.407
1.101
50
0.058
-0.285
-4.107
1.065
51
0.047
-0.230
-3.499
0.861
52
0.038
-0.187
-3.035
0.699
53
0.034
-0.167
-2.734
0.625
54
0.036
-0.180
-2.145
0.674
55
0.033
-0.162
-1.805
0.605
56
0.029
-0.142
-1.658
0.531
57
0.026
-0.129
-1.196
0.481
58
0.024
-0.117
-1.329
0.437
59
0.021
-0.105
-1.446
0.392
60
0.020
-0.097
-1.221
0.363
61
0.018
-0.091
-1.090
0.339
62
0.017
-0.084
-0.930
0.313
63
0.016
-0.079
-0.826
0.294
64
0.015
-0.073
-0.760
0.273
65
0.014
-0.068
-0.616
0.253
66
0.013
-0.062
-0.445
0.232
67
0.013
-0.064
-0.662
0.238
68
0.011
-0.053
-0.321
0.200
69
0.010
-0.051
-0.306
0.191
70
0.010
-0.050
-0.333
0.186
71
0.010
-0.051
-0.543
0.190
72
0.009
-0.047
-0.384
0.175
73
0.009
-0.047
-0.418
0.174
74
0.010
-0.048
-0.598
0.179
75
0.009
-0.044
-0.399
0.165
76
0.008
-0.042
-0.357
0.156
77
0.007
-0.036
-0.187
0.136
78
0.007
-0.036
-0.222
0.134
79
0.007
-0.035
-0.231
0.131
80
0.006
-0.031
-0.091
0.115
81
0.006
-0.030
-0.091
0.111

138
Table D-6. Long-run Uncompensated Price Elasticities
for Land.
Long-run Elasticity of Land Demand with respect to the
price of:
Labor Materials Capital Land
Year
49
-0.233
0.172
0.405
-0.586
50
-0.239
0.177
0.416
-0.582
51
-0.242
0.178
0.421
-0.564
52
-0.235
0.174
0.409
-0.547
53
-0.234
0.173
0.407
-0.534
54
-0.212
0.156
0.368
-0.543
55
-0.207
0.152
0.359
-0.528
56
-0.201
0.149
0.350
-0.505
57
-0.183
0.135
0.318
-0.499
58
-0.186
0.137
0.324
-0.478
59
-0.191
0.141
0.332
-0.450
60
-0.176
0.130
0.305
-0.436
61
-0.170
0.125
0.295
-0.420
62
-0.159
0.117
0.276
-0.407
63
-0.152
0.112
0.265
-0.399
64
-0.149
0.110
0.258
-0.372
65
-0.142
0.105
0.247
-0.358
66
-0.134
0.099
0.233
-0.343
67
-0.146
0.108
0.254
-0.339
68
-0.120
0.088
0.209
-0.311
69
-0.119
0.088
0.207
-0.303
70
-0.118
0.087
0.205
-0.286
71
-0.116
0.086
0.202
-0.234
72
-0.114
0.084
0.198
-0.232
73
-0.110
0.081
0.191
-0.242
74
-0.113
0.083
0.196
-0.231
75
-0.107
0.079
0.187
-0.239
76
-0.104
0.076
0.180
-0.216
77
-0.090
0.066
0.156
-0.209
78
-0.089
0.066
0.155
-0.188
79
-0.087
0.064
0.151
-0.173
80
-0.078
0.057
0.135
-0.182
81
-0.076
0.056
0.132
-0.163

139
Table D-7. Long-run Uncompensated Price Elasticities
for Labor.
Long-run Elasticity of Labor Demand with respect to
the price of:
Labor Materials Capital Land
Year
49
-0.656
0.353
0.079
-0.178
50
-0.655
0.328
0.072
-0.169
51
-0.643
0.330
0.075
-0.169
52
-0.639
0.327
0.080
-0.166
53
-0.624
0.339
0.083
-0.170
54
-0.625
0.304
0.067
-0.158
55
-0.610
0.296
0.064
-0.155
56
-0.598
0.277
0.064
-0.145
57
-0.593
0.264
0.057
-0.139
58
-0.581
0.266
0.063
-0.139
59
-0.560
0.281
0.075
-0.142
60
-0.551
0.263
0.070
-0.134
61
-0.535
0.251
0.067
-0.129
62
-0.524
0.244
0.064
-0.125
63
-0.518
0.236
0.062
-0.121
64
-0.485
0.227
0.061
-0.116
65
-0.464
0.215
0.056
-0.111
66
-0.439
0.189
0.045
-0.100
67
-0.444
0.206
0.059
-0.106
68
-0.403
0.176
0.041
-0.092
69
-0.392
0.177
0.041
-0.092
70
-0.371
0.171
0.042
-0.088
71
-0.318
0.124
0.054
-0.063
72
-0.300
0.117
0.040
-0.061
73
-0.323
0.118
0.044
-0.062
74
-0.331
0.126
0.062
-0.063
75
-0.320
0.114
0.043
-0.060
76
-0.286
0.106
0.041
-0.055
77
-0.267
0.097
0.026
-0.052
78
-0.240
0.089
0.029
-0.046
79
-0.221
0.081
0.030
-0.041
80
-0.224
0.079
0.015
-0.043
81
-0.193
0.067
0.014
-0.036

140
Table D-8. Long-run Uncompensated Price Elasticities
for Materials.
Long-run Elasticity of Materials Demand with respect
to the price of:
Labor Materials Capital Land
Year
49
0.320
-0.164
-0.131
0.148
50
0.289
-0.141
-0.119
0.133
51
0.290
-0.193
-0.117
0.136
52
0.271
-0.202
-0.100
0.129
53
0.283
-0.264
-0.111
0.136
54
0.239
-0.192
-0.112
0.112
55
0.231
-0.219
-0.113
0.109
56
0.207
-0.216
-0.096
0.100
57
0.186
-0.188
-0.094
0.089
58
0.190
-0.225
-0.089
0.093
59
0.207
-0.312
-0.092
0.102
60
0.186
-0.282
-0.085
0.093
61
0.179
-0.290
-0.084
0.089
62
0.171
-0.288
-0.083
0.085
63
0.162
-0.273
-0.080
0.081
64
0.166
-0.334
-0.085
0.083
65
0.162
-0.345
-0.087
0.081
66
0.147
-0.321
-0.082
0.074
67
0.161
-0.370
-0.083
0.081
68
0.144
-0.355
-0.084
0.072
69
0.148
-0.382
-0.087
0.074
70
0.150
-0.418
-0.088
0.075
71
0.123
-0.368
-0.061
0.064
72
0.126
-0.393
-0.069
0.064
73
0.110
-0.302
-0.056
0.057
74
0.113
-0.315
-0.049
0.059
75
0.103
-0.274
-0.051
0.053
76
0.112
-0.342
-0.059
0.058
77
0.106
-0.331
-0.062
0.054
78
0.110
-0.377
-0.064
0.056
79
0.109
-0.394
-0.063
0.056
80
0.101
-0.341
-0.063
0.051
81
0.101
-0.374
-0.063
0.052

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BIOGRAPHICAL SKETCH
Michael James Monson was born August 15, 1956, the
first of James and Glenda Monson*s three children. He
grew up on a farm near Stratford, Iowa.
After graduating from Stratford Community High
School in 1974, he entered Iowa State University. In
1978, he received a Bachelor of Science degree in
agricultural business, graduating with distinction. He
continued his education at Iowa State as a research
assistant in agricultural economics from 1978 to 1979,
research associate in 1980 and 1981, and taught
undergraduate Agricultural Policy in 1981 and 1982.
Michael earned a Master of Science degree from the
Department of Agricultural Economics specializing in
natural resource economics in May, 1982.
In August of 1982 he entered the graduate program
of the Department of Food and Resource Economics as a
research assistant in order to pursue a doctoral degree.
On July 2, 1983, he married Sandra Lynn Johnson. On
August 16, 1984, Mike and Sandra were blessed with their
first child, Jeffery James.
Michael James Monson is a member of the Phi Kappa
Phi and Phi Eta Sigma honorary societies and the
American Agricultural Economics Association.
148

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
JQ-
William G. Boggese, Chairman
Associate Professor of Food and
Resource Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Timothy Taylor, Cdchairman
Assistant Professor of Food and
Resource Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the detjree^>f Doc tot: of Philosophy.
' ' ' * ' •
V
¿bbert D. Emerson
Associate Professor of Food and
Resource Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Max R. Langham
Professor of Food
Economics
nd Resource

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Antal Majthay
Associate Professor of
Management
This dissertation was submitted to the Graduate Faculty of
the College of Agriculture and to the Graduate School and
was accepted as partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
August 1986
Dean, Graduate School

UNIVERSITY OF FLORIDA
II ill ill mu mu
3 1262 08554 1521








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