J O U R N A L OF ECONOMIC THEORY
.ner's behaviour, Oxford Econ.
3. W. E. ARMSTRONG,
A note on the theory of I
Papers 2 (1950), 119-1 22.
rransfinite Numbers," Dover,
4. G. CANTOR,''Contributions to the Foundir.
. . ..
New York, 1915.
. . 1'.
5. J. S. CHIPMAN,
Consumption theory without t n , , ive indifference, in "Preference,
Utility and Demand: A Minnesota Symposii;. '(J. S. Chipman, L. Hurwicz,
-court, Brace and Jovanovich,
M. K. Richter and H. Sonnenschein, Eds.).
. , .. . ,
New York, to be published.
6. S. EILENBERG,
Ordered topological spaces, Amer. .Math. 63 (1941), 39-45.
Threshold in choice an!, he theory of demand, Econo7. N. GEORGESCU-ROEGEN,
metrica 26 (1958), 157-168.
8. H. S. HOUTHAKKER,
Revealed preference and the I:., ity function, Economica N.S.
17 (1950), 159-174.
9. H. S. HOUTHAKKER,
The present state of consumplaon theory, Econometrica
29 (1961), 704-740.
10. L. H u ~ w l c zA N D M. K. RICHTER,
Revealed preference without demand continuity
assumptions, in "Preference, Utility and Demand: A Minnesota Symposium"
(J. S. Chipman, L. Hurwicz, M. K. Richter and H. Sonnenschein, Eds.). Harcourt,
Brace and Jovanovich, New York, to be published.
11. R. D. LUCE,Semiorders and a theory of utility discrimination, Econometrica
24 (1956), 178-191.
12. E. J. MCSHANEAND T. BOTTS,"Real Analysis,'' Van Nostrand, Princeton, New
Jersey, 1952.
13. M. K. RICHTER,Revealed preference theory, Econometrica 34 (1966), 635-645.
A note on the pure theory of consumer's behavior, Ecortomica
14. P. A. SAMUELSON,
N.S. 5 (1938), 61-71 and 353-354.
15. P. A. SAMUELSON,
Consumption theory in terms of revealed preference, Economica
N.S. 15 (1948), 243-253.
16. E. SZP~LRAJN,
Sur I'extension de I'ordre partiel, Fundamento Mathematicae
16 (1930), 386-389.
17. H. P. THIELMAN,
"Theory of Functions of Real Variables," Prentice-Hall, Englewood
Cliffs, New Jersey, 1953.
18. J. VON NEUMANN
AND 0 . MORGENSTERN,
"Theory of Games and Economic Behavior," Princeton Univ. Press, Princeton, New Jersey, 1947.
.
2, 329-347 (1970)
A d j u s t m ~ Josts and Variable Inputs
in the The / of the Competitive Firm
Deparrnlenr of' Econotnics.
)Ilege of Arrs and Sciences, Northwestern University,
Evansron, Illinois
deceived November 7, 1969
.I'
In this paper we examine a number of familiar propositions in the theory
of the profit-maximizing competitive firm within a dynamic optimization
model incorporating costs of adjustment.' Though the theory of the
profit-maximizing firm has been established o n a reasonably logical basis
for several decades and has served as the chief a priori foundation for
much econometric work o n factor-demand and product-supply, it is based
on a static o r atemporal optimization concept. This has made it necessary
to adopt essentially ad hoc adjustment mechanisms such as the flexible
accelerator for use in many econometric contexts in which partial adjustment was obviously operative. Attempts t o provide a consistent optimal
dynamic theory of the firm to either justify o r replace these a d hoc
adjustment mechanisms have recently been made; the adjustment cost
literature began with attempts t o rationalize extant adjustment processes.
However, the existing adjustment cost models have been "disciplined" t o
reproduce the inferences of the static theory in those situations in which
the latter theory is capable of generating predictions. Our objective in the
present paper is to formulate and analyze a somewhat more general
dynamic model in a n effort to see how far one can go without vitiating
the familiar theorems of static production theory.
A primary role is played in this paper by perfectly variable factors of
production, a familiar concept t o be defined more precisely below. In
particular, the possibility that such factors facilitate o r inhibit the process
of adjustment of quasifixed factors, by reducing o r raising marginal
adjustment costs, will be shown to cast into doubt many of the most
central inferences of the received static theory.
' The static theory is presented more or less r~gorouslyin Samuelson [Ill, particularly
Chaps. I11 and IV. Adjustment cost models of firm behavior have been analyzed by
Eisner and Strotz [2], pp. 63-86, Lucas [7], [8],Could [4], and the author [12], [13].
330
THEORY OF THE COLIPETITlVE FIRM
TREADWAY
For example, it is well-known that the static profit-maximization theory
rules out "falling" long-run product-supply curves and "rising" long-run
factor-demand curves. It will be shown below that circumstances may
exist a priori in which such theorems are false in a dynamic model
involving adjustment costs. That is, we shall call into question certain
canonized theorems on optimal long-run behavor.
Until the adjustment cost literature arose, a standard analysis of
short-run behavior involved applicatioil of the Le Chatelier principle.*
That is, the short-run was regarded as a period in which certain factor
inputs, called quasifixed, were not susceptible to variation whereas such
inputs could be varied in the long-run, which was some period of greater
length. The primary inferences based on this comparative statics methodology are that long-run product-supply and factor-demand curves must be
more elastic than the corresponding short-run curves. Professor Mundlak
has applied these results in an effort to provide theoretical restrictions on
distributed lag coefficients, arguing that distributed lag adjustment
processes should not violate theorems of optimal comparative statics.3
This would seem a reasonable requirement to apply; however, when we
formulate an adjustment cost model to yield not only comparative statics
theorems but also the lag process itself, we find that possible a priori
conditions exist under which the standard comparative statics theorems
themselves are not valid. This stems from the fact that there is no necessity
for a static production function consrraint it7 a dynamic optinlization theory.
For example, if a certain variable factor tends to facilitate the adjustment
of the quasifixed factors, by lowering marginal adjustment costs, as well as
being directly productive, the optimizing firm may hire more of it in the
short-run than in the long-run in response to a fall in its wage, i. e., the
short-run factor-demand curve may be tnore elastic than the long-run
curve in contradiction to the static proposition.
The inferences of the static profit-maximization theory that we shall
refer to most frequently below are: (a) the nonpositivity of own-price
effects on factor demands, both in the long-run and in the short-run,
(b) the symmetry of cross-price effects on factor demands, (c) the nonnegativity of the effect of product price on product supply, both in the
long-run and in the short-run, and (d) the relative elasticity of long-run
factor-demand or product-supply curves in comparison t o short-run
curve^.^
In Section I1 we present our model of the firm, relate it to existing
"ee Sarnuelson [I 11, pp. 3 6 4 6 .
See Mundlak 191, [lo].
All of these theorems are presented and proved in Samuelson [ I l l . Other inferences
are available from this theory, but are not central to it or germane to our efforts.
33 1
adjustment cost models and provide calculus of variations necessary
conditions along with economic interpretations of them. In Section 111
the analysis of optimal behavior is carried out. Section IV is a summary.
A. Notation and the Objective Fuir,?c~io~lal
The firm is assumed to sell Q units per period of a single product at
a unit price P. It is assumed to produce this product instantaneously by
the application of two factors of production, a "variable" factor called
labor services which is hired in a q~iantityL at a wage Wand the services
of a "quasi-fixed" factor called physical capital, a stock K of which is
owned by the firm. The services of capital are assumed proportional to
the stock. The gross rate of physical investment is denoted by I and the
market price of capital goods is C. The instantaneous net cash flow of the
firm is thus PQ - W L - GI. We assume that the firm can borrow (or
lend) at a rate of interest r. Furthermore. this firm will be assumed to be
competitive ill the sense that its product, the services of labor, investment
goods and funds are traded at prices (P, W , G, r ) which are invariant to
the actions of the firm.
We postulate that the firm acts so as to maximize its (expected) present
value, which amounts to maxi~nizingprof ts under circumstances identified
below with the static case.j The plan is made at time t = 0. We shall
assume that price expectations are stationary, i.e., current ( t = 0) prices
are expected to obtain forever. This expectations hypothesis has many
drawbacks, but it is ideal for our purposes. We shall attempt to compare
the results of an optimal dynamic adjustment theory with those of the
traditional comparative statics theory; the stationary expectations hypothesis allows us to abstract from many phenomena that might otherwise
clutter up our results. Since our primary results are critical of the static
theory, this is even more for'ceful when we recognize that these negative
results are not due to expectatiotzal effects but rather to a particular kind
of transition from an atemporal theory to an intertemporal theory.
Under the assumptions described above the present value of the firm
is a functional, depending on the time paths of product, labor services and
gross real investment:
[
[ P I
Q L C )I t ) ] =
0
-
L C )- G t ) ] e - d t .
(I)
This criterion is consistently derivable from stockholder utility maximization in
the tradition of Fisher [3] given the expectations concept used here.
THEORY OF THE COMPETITIVE FIRM
TREADWAY
332
is used in [I21 and [13]; in this case F,, = F,, = 0 in terms of the notation
of (4). In [8] Lucas dealt with arbitrary finite numbers of inputs, both
variable and quasifixed, but assumed a structure similar to (4') with
separability between the levels of all inputs and the rates of investment.9
In [4] and [7] a weaker separability assumption is maintained:
We will assume that depreciation, If it occurs at all, is proportional to
the stock of capital:
where K is net physical investmenL6 This depreciation hypothesis is not
particularly defensible, but it has the advantage of simplicity. Condition (2)
constrains the maximization of (I), because a fixed stock KOof physical
capital is inherited from the past at the time of the plan.
333
I
Q = F'(K, L) - C(K, K),
+
(4")
though Gould [4] further restricts C to the form C(K pK) = C(1) and
Lucas [7] restricts C to the form C(K/K).~O
The restriction (4") requires
F2, = 0,but not F,, = 0,using the terminology of (4).
The critical constraint for the optimization of (1) is, however, the
generalization of the production function that we shall use. We assume
that the quantity of product generated by the firm depends not only on the
capital and labor services, Kand L, but also on the rate of net investment K:
The latter dependence flows from the assumption that the magnitude of
capital stock can be changed only by incurring adjustment costs. The
inclusion of K in F is one means of specifying the notion of the quasifixity of the capital input. Such quasifixity may arise as a consequence of
planning costs, installation or break-in costs or other frictions in the
growth process that are inlert~alto the firm. There is then
- -a trade-off for
the firm given the levels of the two factor inputs, say (IS,L), as in Fig. 1;
the firm can produce and sell more today if it is willing to give up some
current growth (and, hence, future sales), or it can grow faster and get
higher future sales at the expense of current production and revenue.
Figure 1 describes an iso-input set for F, the locus of (Q, K)combinations
consistent with given levels of K and L.s
The specification (4) is similar to adjustment-cost specifications found
in the recent literature on optimal investment theory. The main difference
between (4) and the existing formulations is that no separability assumptions
are here made on F whereas some separability postulates characterize all
previous adjustment-cost models. For example, the form
A dot over a variable denotes a total time derivative.
Our depreciation hypothesis makes it irrelevant whether the third argument of Fis
net or gross real investment.
8 Figure 1 is drawn with a negative slope and concave; neither property is assumed
a priori in what follows.
6
It will become apparent below that some restriction on Ft3 is needed if
we are to salvage any of the short-run theorems of the static case when
we consider the dynamic optimization theory. The separability postulate
F,,
0 will help, though it is as unjustified a priori as is any such
' Lucas [81 did not distinguish very explicitly between output-reducing adjustment
costs, which must be conceived of as internal costs, and adjustment costs arising from
monopsony in the market for investment goods. The latter, of course, involve a natural
separability of the net-cash-flow function, since the argument of a monopsony-type
adjustment cost should be I and this cost should be independent of K or L because of
its market origin.
Eisner and Strotz 121, pp. 63-86, consider only one input and utilize a separability
aSsumption like (4') without the argument L in F'.
'O Gould [4] actually requires C to be quadratic in I which is inessential for the
results he obtains.