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Intertemporal price indices for the firm
Journal of Economic Dynamics and Control 6 (1983) 109-126. North-Holland
INTERTEMPORAL
PRICE
INDICES
FOR THE FIRM*
Larry G. EPSTEIN University
of Toronto,
Toronto.
Canada
A45S IA1
Received December 1981, final version received November 1982 A notion of intertemporal price aggregation is analysed within the adjustment-cost model of the firm, and is shown to be feasible precisely for those technologies which imply an accelerator adjustment rule. A unique feature of the common quadratic production function specification is established. The analysis is shown to be relevant to the possibility of decentralization within the firm, to functional form specification for dynamic demand functions, and to comparative dynamics analysis.
1. Introduction Within the framework of the adjustment-cost model of the firm, this paper focuses on non-autonomous control problems solved by firms facing time varying expected prices. The following question is addressed: When can the entire path of expected factor rental prices be aggregated into a single (vector) index in such a way that the firm’s factor decisions may be made by assuming that prices will remain constant forever at the level given by the index? Such an index defines, for each given vector price path, a ‘constantequivalent price vector’ where ‘equivalence’ refers to equivalence from the point of view of the firm’s input and output decisions. There are several reasons for being interested in the existence of such an index. The principal application of this paper is to the literature on empirical factor demand systems [Schramm (1970), Berndt and Morrison (1981), Meese (1980), Hansen and Sargent (1980)], and in particular to the problem of functional form specification for dynamic factor demand functions. A direct approach to deriving such functional forms requires the explicit solution of the Euler-Lagrange equations, which is rarely possible even for an autonomous problem. In practice this approach is limited to the specification of a quadratic production function. (See all the empirical studies cited above.) Since there is no reason to expect most or indeed any data sets. to conform to the latter specification, some scope for alternative functional forms in empirical work is clearly desirable. Epstein (1981) has described an *Financial support from the University of Toronto and the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. 0165-1889/83/%3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
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alternative approach based on a duality between the production function and value function which gives the maximum value of the intergral of discounted future profits. A generalized Hotelling’s Lemma may be applied to the value function to generate a large class of functional forms for factor demands consistent with the adjustment cost model, under the assumption of static price expectations or [see Epstein and Denny (1983)] when expectations are generated by first-order differential or difference equations. But if a general path of expected prices possesses a constant-equivalent price the scope of the value function approach is extended and it may be used to generate functional forms for dynamic factor demands given a general model of expectations. A second application may be noted. Treadway (1970), Mortensen (1973) and Epstein (1982) have carried out qualitative comparative dynamics analyses of the adjustment-cost model under the assumption of static price expectations. When an intertemporal price index exists these analyses are readily extended to non-static expectations. By the Chain Rule of Calculus, the effect on input demand of an arbitrary (marginal) change in a price profile may be decomposed into two components: (a) the change induced in the constant equivalent prices, and (b) the effect on input demand induced by the change in (a). Component (b) may be derived from the cited analyses of models with static expectations, while information about the other component is provided in this paper. A final justification for the ensuing analysis is the following: the intertemporal profit maximizing firm must in general solve a complicated nonautonomous control problem. When an intertemporal index may be constructed, this diflicult task may be decomposed into two stages and decentralized. One department within the tirm may be charged with forecasting future prices and using them to construct a constant-equivalent price vector. The latter may be relayed to the department in charge of input and output decisions. This department need only solve a simpler autonomous control problem the solution of which frequently converges to a steady state. [The existence of a steady state facilitates the computation of optimal plans and permits the calculation of approximately optimal control laws that are informationally and computationally less demanding. See Heal (1973), Infante and Stein (1973).] The two-stage decomposition and decentralization is akin to that permitted by notions of separability and price aggregation in standard consumer and producer models [Blackorby, Brimont and Russell (1978)]. The paper and major results may be outlined as follows: In section 2 the model is defined and the notion of an intertemporal price index is made precise. The heart of the paper is section 3 which characterizes (Theorem 1) the class of technologies consistent with the construction’ of constantequivalent prices. The analysis is local in the sense of applying to a proper
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subset of all conceivable quantity and price variables, e.g. the subset defined by a body of data in a particular empirical exercise. The essential characterizing feature, given some mild regularity conditions, is that the technology imply a (flexible) accelerator adjustment rule for quasi-fixed factors. One commonly specified member of this class of technologies is the quadratic production function. Indeed under some additional occasionally applicable hypotheses, the latter is shown (Theorem 2) to be the unique specification which generates an accelerator adjustment rule. In light of the importance of the accelerator rule and the popularity of the quadratic specification, this result is of independent interest. Proofs are collected in an appendix. The following notation is adopted throughout: Ek is k-dimensional Euclidean space, G!“ c Ek is the open positive orthant; x E Qk is written x>O and x 2 0 means that x E ak, the closure of Bk; all vectors are column vectors; the superscript r denotes transposition; 1x1 denotes the Euclidean norm of XE Ek; a dot over a function denotes differentiation with respect to time; if h(x)=h(x,,..., xk) iS a real-valued fUUCtiOn, h,, iS the matrix (h,l,,)r,j; if h(x) = (h’(x), . . . ,/r’(x)), then h, is the matrix (/I;,),,,. If P maps [0, co) into Ek, P denotes the function on [0, co) defined by P+T(t)=P(T+t). If r is a sc+afar, then r^ denotes the scalar multiple T of the identity matrix of suitable dimension. 2. The model and basic deftitions Consider first the following maximization:
autonomous
problem
J(K,,p)~max~e-“[F(K,z)-p’K]dt,
of intertemporal
profit
(1)
0
subject to
K=z,
K(O)=Ko,
z(t)~@(K(t))
and
(K(t),p)~@
for all t,
where F(K,z) is a production function giving the maximum amount of the scalar output y that can be produced given the stock of quasi-fixed factors K E 52” and given that net investment z is taking place. Properties of F will be described below. PE s1” is the rental price corresponding to K, normalized with respect to the output price and expected to stay constant throughout the horizon. r>O is the real rate of discount which is fixed throughout. K, is the vector of initial stocks. J is the value of the problem (1) assuming that a solution exists. 0 cG!‘” is a bounded open set which is the domain of J.
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Bhe imposed boundedness of optimal stock profiles is not very restrictive (Brock and Scheinkman (1977, pp. 400-401).] @CR” x E”, bounded and open, is the domain of definition of F. @(K)={z:(K,z)E a}. The constraints z(~)E @(K(t)) and (K(r),p)~ 0 are taken to be never binding. For future reference, O(K) = {p:(K, p) E O}. Denote by &K,,p) the policy function corresponding to (1) describing optimal net investment at t=O. A(K,, p) denotes the optimal shadow price at t=O.
For many environments static expectations on the part of the firm are ‘so irrational’ that the model (1) is unsatisfactory. Thus it will be assumed that firms solve the following more complicated problem: V(K,,P)=max
Tee-“[F(K,z)-P(t)K]dt,
(2)
0
subject to
K=z,
fW=Ko,
z(t) E @(K(t))
and
(K(t), P,,) E S for all
t,
where P:[O, 03)*52” represents the expected rental price profile. r and @ are as before. Once again the constraints z E Q(K) and (K(t), P,,) ES are assumed to be non-binding. V is the value functional which extends J of problem (1) to non-constant price profiles. S is the domain of definition of I! Define S(K)E{P:(K,P)ES} and gru{S(K):K~51”}. Denote by K*(K,,P), for (K,,P)ES, optimal net investment in (2) at t = 0. The notion defined:
of price aggregation
Deenition 1. The functional I :S+s1” and satisfies (K, P) E S*(K,
the policy function
that describes
which will be of interest may now be
I aggregates rental price profiles (over S) if
Z(P)) E 0,
(3)
and x*(K, P) = <(K, Z(P)) for all
(K, P) ES.
(4)
Eq. (4) expresses the requirement that the optimal t =0 behaviour dictated by (2) is identical to that derived from the static expectations problem (1)
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when p=Z(P).' (3) ensures the consistency of domains necessary for (4) to be meaningful. Note that the behavioural equivalence extends also to output supply functionals since output in (2) = F(K, K*(K, P)) = F(K, r(K, I(P))) = output in (1) when p = Z(P). The technologies which permit the aggregation of rental price profiles are characterized in the next section. Those characterizations are derived while maintaining the following regularity conditions on the production function F: Conditions (T)
(T.l) F maps @c.R” x E” into a ‘; F is twice continuously differentiable, F,>O for each i, Fzi 30 as Zi ~0. (T.2) F is concave in (K,z) and F,,(K;) is negative definite throughout @. (T.3) For each (I&P) E@ there exists a unique solution to problem (1); moreover, the corresponding policy function 5 is continuously differentiable. (T.4) 5,(&p) is non-singular for all (K,p) E 0 where < is the policy function for problem (1). (T.5) For every (K, z) E @ there exists (K, p) E 0 such that z = r(K,p). A detailed discussion of Conditions (T) appears in Epstein (1981). Thus only a few brief comments are offered here. (T.lHT.3) are standard and selfexplanatory. (T.5) requires that every point in the domain of F is conceivably optimal for some (K,p). Thus it requires that 0 be sufficiently large. Any point (K,z) which violated (T.5) would never be observed empirically if 8 describes the set of possible environments facing the firm. Finally, the non-singularity in (T.4) is reminiscent of the requirement in the standard static production model that the Jacobian of factor demand functions with respect to factor prices have full rank. The latter is true if and only if the production function is strongly concave. The characterization of (T.4) is more difficult but the following observation may be helpful: the nonsingularity of 5, at any finite number of points in 0 could not be refuted empirically since singularity could be eliminated by arbitrarily small perturbations of the data. 3. Rental price aggegation Some further assumptions, formulated:
required
for the characterizations,
are now
‘If I were allowed to depend on K, the notation of aggregation would lose most of its force since the equation K*(K,P)=c[K,I(K,P)] could frequently be used to solve for I(K,P). Note that if n= 1, (4) requires that P we weakly separable from K in each functional @. Also if each I’ is independent of P, for j#i, then (4) requires that in each functional kf, P, be weakly separable from K (and from P,, . . . , P, _ 1, Pi + 1,. . . , P.). In general, the restriction imposed by (4) is’not readily expressable in terms of separability notions. [See, however, the general functional structure in Gorman (1976, section 8).]
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Assumption I. If (K,P) ES, then P:[O, co)+sZ” is continuously and supt le-“P(t)1 and supt /e-rfP(t)l are both finite. Assumption 2. For each (K,, P) ES, a unique convergent integrals) exists for problem (2).
solution
differentiable
(in the sense of
The following assumptions relate to a given index functional I: Assumption 3.
@={(K,Z(P)):(K,P)ES}.
Given S and the aggregator I, Assumption 3 may be used to define 0 the domain of the value function J. 0 so defined is the largest set over which we can infer anything about the value function J or policy function <‘from the solution of (2). Assumption 4.
continuously
For each (K,P) ES, the function differentiable.
h(t) =Z(P+,),
t E [0, co), is
The following is a sufficient condition for Assumption 4: topologize the set of profiles P with the norm ll~ll~=sup~ le-“P(t)/ and let Z have a continuous Frtchet derivative on each S(K). [See Luenberger (1969, ch. 7).] Use the fact that P~S(K)+sup,le-“P(t)(coo. Assumption 5. For each (K, P) ES and u E E” there exist UE E’, a #O, and P’ E S(K) such that Z(P’) = Z(P) and P1(0) -Z(P) = au.
Assumptions 14 do not rule out the possibility that each S(K) contains only constant profiles in which case rental price aggregation is trivial; Z(p) = p is a valid aggregator of constant profiles. For the problem of aggregation to be meaningful and interesting there must be a ‘sufficient’ number of nonconstant profiles. Assumption 5 is a mild condition to ensure the latter. It states that any given value of the index Z(P) may be achieved by a profile P’ whose value at 0, P1(0), lies in any desired direction v when measured from Z(P)=Z(P’) as the origin. Consider an intuitive ‘derivation’ of Assumption 5 from other mild and loosely specified hypotheses. Let BC’[O, co) denote the set of funtions Q:[O, co)+E” with sup, le-“Q(t)\O [i.e., Z(Q) f Q(0) follows from (ii)] suggests that the
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equality should be achievable. Thus Assumption 5 may be interpreted as placing mild restrictions on the ‘size’ of each S(K) and the continuity of I. [Since (i) and (ii) are not necessary for Assumption 5 the weaker assumption is made.] Finally, it may help to formulate Assumption 5 for the case n= l:V(K, P) ES 3P’ ES(K) such that Pi(O) #Z(Pi) =Z(P). With the help of these assumptions the following characterization may be derived: Theorem 1. . Let F satisfy Conditions (Tl)-(i’Y). Under Assumptions 1-5, if.a rental price aggregator Z exists over the set S, then the policy function for problem (I) has the form
(a)
V(K,p)E@, for an n xn matrix M(p) all of whose eigenvalues have non-positive real parts.
(5)
(b)
Conversely, (5) implies the existence of a rental price aggregator Z and a set S consistent with Assumptions I-5.
The theorem produces the rather striking result that rental price aggregation is feasible precisely when the policy function has an adjustment matrix M(p) = &(K, p) that is independent of K. Refer to (5) as an accelerator adjustment rule even though that term is generally reserved for the case where M(p) is stable. In fact (5) is also consistent with limit cycles around R(p). Since a stable steady state is commonly assumed in adjustment-cost models the subsequent discussion will restrict itself to the case of a stable matrix M(p). The corresponding policy function K* for problem (2) with a non-constant price profile is K*(K, P) = M(Z(P))[K
- &Z(P))],
V(K, P) ES.
(6)
This is also an accelrator adjustment rule but towards the continually moving target K(Z(P)). As time proceeds from t =0 to t = t,, the relevant price profile and steady state change to P+,O and K(Z(P+J), respectively. (Of course, this is in the absence of expectations revisions.) The necessity part of the theorem is the more remarkable because of the weakness in particular of Assumption 5. However, the sufficiency part is stronger the larger is the class of profiles which may be aggregated. At the conclusion of the proof of Theorem 1 (and in the examples below) an attempt is made to give some indication of the size of this class. Given conditions sufficient to ensure the existence of a rental price aggregator a logical question is whether the aggregator is unique. This is
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necessarily the case if the function <(K, a) is one-to-one over each O(K), for then ((K, Z(P)) = QK, Z’(P))*Z(P) =Z’(P). Similarly the univalence of c(K, a) implies that Z(p) =p for any PE O(K), i.e., the index value associated with a constant price path Z’(t) =p, Vt, is necessarily the constant level p. The technologies consistent with (5) may be characterized by making use of the duality between production functions F and value functions .Z. Epstein (1981) has shown that .Z and F provide equivalent representations of the technology, and that the use of .Z is advantageous for some purposes. For instance, it is demonstrated that all technologies which generate policy functions satisfying (5) have value functions J of the form JK
(7)
P)= a(p) + B(K) + Y’(PW.
The corresponding
adjustment matrix is (8)
M(p)=F+J,k’(K,p)=i+(y;‘(p))‘.
(Recall that i denotes the n x n identity matrix multiplied
by the scalar r.)
Epstein (1981) provides some examples of functional forms of this sort that are dual to production functions F that satisfy (T) over a set @CD”‘, i.e., (K, z) E @*z 2 0. In the same way for numerous appropriate specifications of a( a) and /I( *) the following functional forms can be shown to be wellbehaved in a region 0 (dual to @) in which negative investment is never optimal2 The functional forms differ in the specification of y and hence M, which determines the nature of the price aggregator. (The specific rental price aggregators given below [eqs. (10) and (13)] are derived by applying eq. (23) from the proof of Theorem 1.) Example 1.
Let .Z in (7) have
Y(P)= A’P,
(9)
with A a non-singular n x n matrix such that r^+ A-’ is stable. Then M(p) = ?+A-‘, a constant. Consider price profiles such that supr le-rfP(t)l < co. Then Z(P) =(i-M’)
7 e-(p-w)rP(t)
dt.
0
(10)
Z(P) is a weighted sum of the values P(t).
‘One could equally well have 2,
arise when for some i, zi can
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A common special case is where a and B are quadratic.3 The value function is then dual to F quadratic, the standard specification. [See Epstein (1981, example l).]
Example2. Let A be n x n non-singular, A - ’ = (a,;‘). Define .Z by (7) where i=
l,**.
2 n*
(11)
Then (12)
Y,(P) = - A - ‘W4
where D(p) is diagonal with ith diagonal element d,(p). M’(p) =i+r;‘(p) which depends on p and is stable if r-d,: ‘(p) < 0 over 0, i = 1,. . . , n. (M is diagonal itI A is diagonal.) Consider price profiles P such that inf, di(P(t))>0, Vi. The corresponding price aggregator is given by
I(P)=/'
)
(13)
Because the adjustment matrix depends on rental prices Z is not a simple weighted sum as in Example 1. In fact it is not even linearly homogeneous in P. That the quadratic specification for the technology permits the aggregation of rental price profiles is well-known and is one reason for its popularity. Rental price aggregation is generally consistent with a large class of technologies as Theorem 1 and Examples 1 and 2 demonstrate. But in fact, under some additional occasionally applicable assumptions, the quadratic specification is the only one consistent with an intertemporal price index. This result is the content of Theorem 2. The theorem thus describes circumstances under which the quadratic specification is not only convenient, but also necessary for the specification of functional forms for dynamic factor demand functions in the presence of a general model of price expectations. (See the discussion in the Introduction of existing approaches to functional form specification.) Of course interest in the accelerator adjustment rule extends beyond the ‘Quadratic
refers to a polynomial
of degree 2; constant and linear terms may be present.
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specific concern of this paper, namely the aggregation of rental price profiles. The accelerator adjustment rule is the most popular specification in dynamic factor demand theory. Thus Theorem 2 is similarly of somewhat broader interest since it provides an explicit characterization of the underlying production functions which is applicable in an empirically relevant class of circumstances. Theorem 2. Let F satisfy conditions (T) and generate the policy function (5). Adopt the following additional assumptions:
(a) M = tR(K, p) is independent of p and is stable. (b) For every i=-1,2,. . . , n, 3jf i such that mij #O, where M=(mij); particular, n > 1). (c) For euery (K, p) E 0,3(K, @)E 0 such that J,,(K,
(d) (K, z) E Q,for some z*(K,
(in
@) is negative dejnite.
0) E Sp.
Then F is a quadratic production function of the form F(K, z) = a, + a’K +$K’AK
+$z’Bz,
(14)
where A and B are (symmetric and) negative definite n x n matrices and B is diagonal.
Hypothesis (a) restricts the adjustment rule (5) so that M is independent of rental prices, which is the common specification. The constancy of M guarantees that the corresponding intertemporal price aggregator Z can be derived explicitly; in fact, it is given by the formula (10). An explicit representation for Z is required to generate explicit functional forms for the dynamic factor demand functions corresponding to the non-autonomous problem (2). Hypothesis (b) rules out the existence of a factor whose optimal adjustment is independent of the levels of all other factors. This hypothesis could not be refuted empirically as long as n> 1. .Z is concave in K since F is concave. Assumption (c) strengthens the concavity of .Z to strong concavity over a subset of its domain. Finally, (d) requires that the domain of F include all steady state positions (K,O). This assumption may be forced onto the empirical analyst by the nature of the data set which he is attempting to explain by means of model (2). Alternatively, the assumption may be required for counterfactual experiments which extrapolate from the data set to a larger domain. Its significance is that, in conjunction with Condition (T.l), it implies that, for each i=l,...,n, F,,(K,z)=O,
V(K,Z)E@,
such that
zi=O.
(15)
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Note that the functional form (14) is the most general quadratic specification which satisfies (15) over a domain @ consistent with (d). Theorem 2 shows clearly the unique feature of the common quadratic specification. At the same time, however, it does not vitiate interest in the sufficiency part of Theorem 1. The latter shows that even for non-quadratic technologies rental price aggregation is possible over some domains. From the point of view of functional form specification for empirical estimation, aggregate data sets occasionally involve everywhere positive net investment in all stocks. In that case the domain hypothesis (d) of Theorem 2 need not apply and’ Theorem 1 provides additional scope for generating functional forms for investment consistent with the adjustment-cost model and a general model of expectations. The case for Theorem 1 may be strengthened considerably by noting that natural modifications of the theorems apply if adjustment costs depend on gross rather than net investment. Then Theorem 2 is not applicable if, as is often the case, gross investment in all stocks is positive over the relevant domain. [See Epstein and Denny (1983) for an empirical analysis with such a data set.] Then Theorem 1 may be of value in generating alternative functional forms.
Appendix Proof of Theorem I
Suppose an aggregator I exists and prove (5). Let {K(t)},” be optimal in (2). By hypothesis &t)=t(K(t),I(P+,))
for all
(16)
th0,
which implies
K(t) must solve the Euler-Lagrange equation after substitution of (16) and (17), becomes
corresponding
1
~zz(K(thO [ ~w(K(t),I(P+,))r(K(t),I(P+,))+C,~I(P.,) vtzo. (18) t> 0. Consider problem (1) with initial stock= K(t)and p=Z(P+,).
+~Zd(K(O~ Z(P+,)) =fi+rF: -P(t),
Fix
to (2) which,
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Since < is the policy function for (l), the corresponding equation (evaluated at the initial position) yields
~,,um
Euler-Lagrange
ocsrdm~ Z(P+,))mm Z(P+dl (19
+F,,r(K(t),z(P+,))-FS,+rF:-z(P+,). Substitute (19) into (18) to obtain
Moreover, this equation is valid for all t 20 since t was chosen arbitrarily. The equation F,(K iw, PI) = - JKUC PI is a well-known first-order respect to p to obtain F,,(K
condition
(21)
associated with (1). Differentiate
5(K PM, = - J,,W, PI-
with
(22)
Thus (20) may be rewritten in the form
-J,,(K(t)~~(P+,))$CI(P+,)I=~(P+*)-P(r),
vtzo.
(23)
Evaluate (23) at t=O to obtain
(24) Differentiate with respect to K,, (the first component
of K,) to derive
By Assumption 5 there exist P' ,.. .,P" such that Z(P')=*..=Z(P")=Z(P) and {Z(P)-Pp'(O),...,Z(P)-P"(O)} is a linearly independent set. The appropriate versions of (24) and the non-singularity of JK,, [Conditions (T.2) (T.4) and
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(22)] now imply that
{
$c~(P:~)I,=o,...,$r~(P”+,)lr=a
is a linearly independent
set. Precisely as (25) was derived one can deduce i=l,...,n.
~c~~,(K,,z(P))l$C~(P:~)l~=,=O¶ 01
But the n vectors in these equations are linearly independent. 0 = &Z,,(K,, Z(P))/X,,. Similarly AZ,(K,, Z(P))/Xoi =0, Vi. Thus JpK is independent Jacobi equation for (l),
Therefore,
of K. To prove (5), consider the Hamiltonian-
rJ(K,p)=max{F(K,z)-p’K+J~K,p)z}.
(27)
I
Differentiate that
with respect to p and apply the Envelope Theorem to deduce
rJ; + K = J,&
(28)
from which (5) follows. For the reverse implication, (B.i)-(B.2)} where:
suppose (5) is valid. Define f={4:3
(B.1) 4:[0, co)-+UK O(K) is twice continuously
satisfies
differentiable.
(B.2) h(t)=$(t)-[f-M’(S(t))]-‘J(r) is such that h(t)~62”, Vt, inf,h’(t)>O, Vi, supt (e-“h(t)1 and sup, le-“h(;(t)l are both finite. Let - be an equivalence relation on f defined by Y1 -4’ iff 3T > 0 such that ~‘(t)-[i-~(91(t))]-‘~‘(t)=92(t)-[i-M’(92(t))]-’~2(t), Vthll: Denate by Z the set of equivalence classes p/and denote a typical equivalence class by f, a representative element of the equivalence class. Introduce some terminology: a path (K( .), P( e)) is regular if there exists a compact set Cc 0 such that (K(t), P(t)) E C, Vt. Finally let (K,, P) ES iff there exists 9 E Z (necessarily unique) such that P(t)=9(t)-[i-M’(9(t))]-‘~(t),
vtzo,
(29)
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(a(.),$(.))
is regular,
where K:(s) solves
K(0) = Ko.
(30)
The aggregator Z is defined by Z(P)s9(0)
where
PES=U
{S(K):KEQ”},
(31)
and 9 is the unique element of Z satisfying (29) and (30). It remains to verify that Z is a valid aggregator and that Assumptions l-5 are satisfied by I, S and 0 so defined. Assumption 1 is a direct consequence of (B.l). Turn to Assumption 2. First note that because of the construction of Z, if Y corresponds to P as in (29), then $+t corresponds to P,, . Thus (29) may be rewritten in the form (32)
which is precisely eq. (23). It suffices to show that K( .) defined in (30) with Y(t) =Z(P+,), uniquely solves problem (2). Note that g(t) =
(4
P(0) =f(O),
(‘-9
p(t)
=a[?-M’(9(0))]ll, 1=0
Vt. It
L.G. Epstein, Intertemporal price indices for theli*n,
(4
(K’(t), S’(t)) E 0,
Vt,
123
where K’ solves
K(O) = Ko.
It is possible to choose J1 or satisfying (a), (b), sup~[J(t)-9’(t)l
INTERTEMPORAL
PRICE
INDICES
FOR THE FIRM*
Larry G. EPSTEIN University
of Toronto,
Toronto.
Canada
A45S IA1
Received December 1981, final version received November 1982 A notion of intertemporal price aggregation is analysed within the adjustment-cost model of the firm, and is shown to be feasible precisely for those technologies which imply an accelerator adjustment rule. A unique feature of the common quadratic production function specification is established. The analysis is shown to be relevant to the possibility of decentralization within the firm, to functional form specification for dynamic demand functions, and to comparative dynamics analysis.
1. Introduction Within the framework of the adjustment-cost model of the firm, this paper focuses on non-autonomous control problems solved by firms facing time varying expected prices. The following question is addressed: When can the entire path of expected factor rental prices be aggregated into a single (vector) index in such a way that the firm’s factor decisions may be made by assuming that prices will remain constant forever at the level given by the index? Such an index defines, for each given vector price path, a ‘constantequivalent price vector’ where ‘equivalence’ refers to equivalence from the point of view of the firm’s input and output decisions. There are several reasons for being interested in the existence of such an index. The principal application of this paper is to the literature on empirical factor demand systems [Schramm (1970), Berndt and Morrison (1981), Meese (1980), Hansen and Sargent (1980)], and in particular to the problem of functional form specification for dynamic factor demand functions. A direct approach to deriving such functional forms requires the explicit solution of the Euler-Lagrange equations, which is rarely possible even for an autonomous problem. In practice this approach is limited to the specification of a quadratic production function. (See all the empirical studies cited above.) Since there is no reason to expect most or indeed any data sets. to conform to the latter specification, some scope for alternative functional forms in empirical work is clearly desirable. Epstein (1981) has described an *Financial support from the University of Toronto and the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. 0165-1889/83/%3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
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alternative approach based on a duality between the production function and value function which gives the maximum value of the intergral of discounted future profits. A generalized Hotelling’s Lemma may be applied to the value function to generate a large class of functional forms for factor demands consistent with the adjustment cost model, under the assumption of static price expectations or [see Epstein and Denny (1983)] when expectations are generated by first-order differential or difference equations. But if a general path of expected prices possesses a constant-equivalent price the scope of the value function approach is extended and it may be used to generate functional forms for dynamic factor demands given a general model of expectations. A second application may be noted. Treadway (1970), Mortensen (1973) and Epstein (1982) have carried out qualitative comparative dynamics analyses of the adjustment-cost model under the assumption of static price expectations. When an intertemporal price index exists these analyses are readily extended to non-static expectations. By the Chain Rule of Calculus, the effect on input demand of an arbitrary (marginal) change in a price profile may be decomposed into two components: (a) the change induced in the constant equivalent prices, and (b) the effect on input demand induced by the change in (a). Component (b) may be derived from the cited analyses of models with static expectations, while information about the other component is provided in this paper. A final justification for the ensuing analysis is the following: the intertemporal profit maximizing firm must in general solve a complicated nonautonomous control problem. When an intertemporal index may be constructed, this diflicult task may be decomposed into two stages and decentralized. One department within the tirm may be charged with forecasting future prices and using them to construct a constant-equivalent price vector. The latter may be relayed to the department in charge of input and output decisions. This department need only solve a simpler autonomous control problem the solution of which frequently converges to a steady state. [The existence of a steady state facilitates the computation of optimal plans and permits the calculation of approximately optimal control laws that are informationally and computationally less demanding. See Heal (1973), Infante and Stein (1973).] The two-stage decomposition and decentralization is akin to that permitted by notions of separability and price aggregation in standard consumer and producer models [Blackorby, Brimont and Russell (1978)]. The paper and major results may be outlined as follows: In section 2 the model is defined and the notion of an intertemporal price index is made precise. The heart of the paper is section 3 which characterizes (Theorem 1) the class of technologies consistent with the construction’ of constantequivalent prices. The analysis is local in the sense of applying to a proper
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subset of all conceivable quantity and price variables, e.g. the subset defined by a body of data in a particular empirical exercise. The essential characterizing feature, given some mild regularity conditions, is that the technology imply a (flexible) accelerator adjustment rule for quasi-fixed factors. One commonly specified member of this class of technologies is the quadratic production function. Indeed under some additional occasionally applicable hypotheses, the latter is shown (Theorem 2) to be the unique specification which generates an accelerator adjustment rule. In light of the importance of the accelerator rule and the popularity of the quadratic specification, this result is of independent interest. Proofs are collected in an appendix. The following notation is adopted throughout: Ek is k-dimensional Euclidean space, G!“ c Ek is the open positive orthant; x E Qk is written x>O and x 2 0 means that x E ak, the closure of Bk; all vectors are column vectors; the superscript r denotes transposition; 1x1 denotes the Euclidean norm of XE Ek; a dot over a function denotes differentiation with respect to time; if h(x)=h(x,,..., xk) iS a real-valued fUUCtiOn, h,, iS the matrix (h,l,,)r,j; if h(x) = (h’(x), . . . ,/r’(x)), then h, is the matrix (/I;,),,,. If P maps [0, co) into Ek, P denotes the function on [0, co) defined by P+T(t)=P(T+t). If r is a sc+afar, then r^ denotes the scalar multiple T of the identity matrix of suitable dimension. 2. The model and basic deftitions Consider first the following maximization:
autonomous
problem
J(K,,p)~max~e-“[F(K,z)-p’K]dt,
of intertemporal
profit
(1)
0
subject to
K=z,
K(O)=Ko,
z(t)~@(K(t))
and
(K(t),p)~@
for all t,
where F(K,z) is a production function giving the maximum amount of the scalar output y that can be produced given the stock of quasi-fixed factors K E 52” and given that net investment z is taking place. Properties of F will be described below. PE s1” is the rental price corresponding to K, normalized with respect to the output price and expected to stay constant throughout the horizon. r>O is the real rate of discount which is fixed throughout. K, is the vector of initial stocks. J is the value of the problem (1) assuming that a solution exists. 0 cG!‘” is a bounded open set which is the domain of J.
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Bhe imposed boundedness of optimal stock profiles is not very restrictive (Brock and Scheinkman (1977, pp. 400-401).] @CR” x E”, bounded and open, is the domain of definition of F. @(K)={z:(K,z)E a}. The constraints z(~)E @(K(t)) and (K(r),p)~ 0 are taken to be never binding. For future reference, O(K) = {p:(K, p) E O}. Denote by &K,,p) the policy function corresponding to (1) describing optimal net investment at t=O. A(K,, p) denotes the optimal shadow price at t=O.
For many environments static expectations on the part of the firm are ‘so irrational’ that the model (1) is unsatisfactory. Thus it will be assumed that firms solve the following more complicated problem: V(K,,P)=max
Tee-“[F(K,z)-P(t)K]dt,
(2)
0
subject to
K=z,
fW=Ko,
z(t) E @(K(t))
and
(K(t), P,,) E S for all
t,
where P:[O, 03)*52” represents the expected rental price profile. r and @ are as before. Once again the constraints z E Q(K) and (K(t), P,,) ES are assumed to be non-binding. V is the value functional which extends J of problem (1) to non-constant price profiles. S is the domain of definition of I! Define S(K)E{P:(K,P)ES} and gru{S(K):K~51”}. Denote by K*(K,,P), for (K,,P)ES, optimal net investment in (2) at t = 0. The notion defined:
of price aggregation
Deenition 1. The functional I :S+s1” and satisfies (K, P) E S*(K,
the policy function
that describes
which will be of interest may now be
I aggregates rental price profiles (over S) if
Z(P)) E 0,
(3)
and x*(K, P) = <(K, Z(P)) for all
(K, P) ES.
(4)
Eq. (4) expresses the requirement that the optimal t =0 behaviour dictated by (2) is identical to that derived from the static expectations problem (1)
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when p=Z(P).' (3) ensures the consistency of domains necessary for (4) to be meaningful. Note that the behavioural equivalence extends also to output supply functionals since output in (2) = F(K, K*(K, P)) = F(K, r(K, I(P))) = output in (1) when p = Z(P). The technologies which permit the aggregation of rental price profiles are characterized in the next section. Those characterizations are derived while maintaining the following regularity conditions on the production function F: Conditions (T)
(T.l) F maps @c.R” x E” into a ‘; F is twice continuously differentiable, F,>O for each i, Fzi 30 as Zi ~0. (T.2) F is concave in (K,z) and F,,(K;) is negative definite throughout @. (T.3) For each (I&P) E@ there exists a unique solution to problem (1); moreover, the corresponding policy function 5 is continuously differentiable. (T.4) 5,(&p) is non-singular for all (K,p) E 0 where < is the policy function for problem (1). (T.5) For every (K, z) E @ there exists (K, p) E 0 such that z = r(K,p). A detailed discussion of Conditions (T) appears in Epstein (1981). Thus only a few brief comments are offered here. (T.lHT.3) are standard and selfexplanatory. (T.5) requires that every point in the domain of F is conceivably optimal for some (K,p). Thus it requires that 0 be sufficiently large. Any point (K,z) which violated (T.5) would never be observed empirically if 8 describes the set of possible environments facing the firm. Finally, the non-singularity in (T.4) is reminiscent of the requirement in the standard static production model that the Jacobian of factor demand functions with respect to factor prices have full rank. The latter is true if and only if the production function is strongly concave. The characterization of (T.4) is more difficult but the following observation may be helpful: the nonsingularity of 5, at any finite number of points in 0 could not be refuted empirically since singularity could be eliminated by arbitrarily small perturbations of the data. 3. Rental price aggegation Some further assumptions, formulated:
required
for the characterizations,
are now
‘If I were allowed to depend on K, the notation of aggregation would lose most of its force since the equation K*(K,P)=c[K,I(K,P)] could frequently be used to solve for I(K,P). Note that if n= 1, (4) requires that P we weakly separable from K in each functional @. Also if each I’ is independent of P, for j#i, then (4) requires that in each functional kf, P, be weakly separable from K (and from P,, . . . , P, _ 1, Pi + 1,. . . , P.). In general, the restriction imposed by (4) is’not readily expressable in terms of separability notions. [See, however, the general functional structure in Gorman (1976, section 8).]
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Assumption I. If (K,P) ES, then P:[O, co)+sZ” is continuously and supt le-“P(t)1 and supt /e-rfP(t)l are both finite. Assumption 2. For each (K,, P) ES, a unique convergent integrals) exists for problem (2).
solution
differentiable
(in the sense of
The following assumptions relate to a given index functional I: Assumption 3.
@={(K,Z(P)):(K,P)ES}.
Given S and the aggregator I, Assumption 3 may be used to define 0 the domain of the value function J. 0 so defined is the largest set over which we can infer anything about the value function J or policy function <‘from the solution of (2). Assumption 4.
continuously
For each (K,P) ES, the function differentiable.
h(t) =Z(P+,),
t E [0, co), is
The following is a sufficient condition for Assumption 4: topologize the set of profiles P with the norm ll~ll~=sup~ le-“P(t)/ and let Z have a continuous Frtchet derivative on each S(K). [See Luenberger (1969, ch. 7).] Use the fact that P~S(K)+sup,le-“P(t)(coo. Assumption 5. For each (K, P) ES and u E E” there exist UE E’, a #O, and P’ E S(K) such that Z(P’) = Z(P) and P1(0) -Z(P) = au.
Assumptions 14 do not rule out the possibility that each S(K) contains only constant profiles in which case rental price aggregation is trivial; Z(p) = p is a valid aggregator of constant profiles. For the problem of aggregation to be meaningful and interesting there must be a ‘sufficient’ number of nonconstant profiles. Assumption 5 is a mild condition to ensure the latter. It states that any given value of the index Z(P) may be achieved by a profile P’ whose value at 0, P1(0), lies in any desired direction v when measured from Z(P)=Z(P’) as the origin. Consider an intuitive ‘derivation’ of Assumption 5 from other mild and loosely specified hypotheses. Let BC’[O, co) denote the set of funtions Q:[O, co)+E” with sup, le-“Q(t)\
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equality should be achievable. Thus Assumption 5 may be interpreted as placing mild restrictions on the ‘size’ of each S(K) and the continuity of I. [Since (i) and (ii) are not necessary for Assumption 5 the weaker assumption is made.] Finally, it may help to formulate Assumption 5 for the case n= l:V(K, P) ES 3P’ ES(K) such that Pi(O) #Z(Pi) =Z(P). With the help of these assumptions the following characterization may be derived: Theorem 1. . Let F satisfy Conditions (Tl)-(i’Y). Under Assumptions 1-5, if.a rental price aggregator Z exists over the set S, then the policy function for problem (I) has the form
(a)
V(K,p)E@, for an n xn matrix M(p) all of whose eigenvalues have non-positive real parts.
(5)
(b)
Conversely, (5) implies the existence of a rental price aggregator Z and a set S consistent with Assumptions I-5.
The theorem produces the rather striking result that rental price aggregation is feasible precisely when the policy function has an adjustment matrix M(p) = &(K, p) that is independent of K. Refer to (5) as an accelerator adjustment rule even though that term is generally reserved for the case where M(p) is stable. In fact (5) is also consistent with limit cycles around R(p). Since a stable steady state is commonly assumed in adjustment-cost models the subsequent discussion will restrict itself to the case of a stable matrix M(p). The corresponding policy function K* for problem (2) with a non-constant price profile is K*(K, P) = M(Z(P))[K
- &Z(P))],
V(K, P) ES.
(6)
This is also an accelrator adjustment rule but towards the continually moving target K(Z(P)). As time proceeds from t =0 to t = t,, the relevant price profile and steady state change to P+,O and K(Z(P+J), respectively. (Of course, this is in the absence of expectations revisions.) The necessity part of the theorem is the more remarkable because of the weakness in particular of Assumption 5. However, the sufficiency part is stronger the larger is the class of profiles which may be aggregated. At the conclusion of the proof of Theorem 1 (and in the examples below) an attempt is made to give some indication of the size of this class. Given conditions sufficient to ensure the existence of a rental price aggregator a logical question is whether the aggregator is unique. This is
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necessarily the case if the function <(K, a) is one-to-one over each O(K), for then ((K, Z(P)) = QK, Z’(P))*Z(P) =Z’(P). Similarly the univalence of c(K, a) implies that Z(p) =p for any PE O(K), i.e., the index value associated with a constant price path Z’(t) =p, Vt, is necessarily the constant level p. The technologies consistent with (5) may be characterized by making use of the duality between production functions F and value functions .Z. Epstein (1981) has shown that .Z and F provide equivalent representations of the technology, and that the use of .Z is advantageous for some purposes. For instance, it is demonstrated that all technologies which generate policy functions satisfying (5) have value functions J of the form JK
(7)
P)= a(p) + B(K) + Y’(PW.
The corresponding
adjustment matrix is (8)
M(p)=F+J,k’(K,p)=i+(y;‘(p))‘.
(Recall that i denotes the n x n identity matrix multiplied
by the scalar r.)
Epstein (1981) provides some examples of functional forms of this sort that are dual to production functions F that satisfy (T) over a set @CD”‘, i.e., (K, z) E @*z 2 0. In the same way for numerous appropriate specifications of a( a) and /I( *) the following functional forms can be shown to be wellbehaved in a region 0 (dual to @) in which negative investment is never optimal2 The functional forms differ in the specification of y and hence M, which determines the nature of the price aggregator. (The specific rental price aggregators given below [eqs. (10) and (13)] are derived by applying eq. (23) from the proof of Theorem 1.) Example 1.
Let .Z in (7) have
Y(P)= A’P,
(9)
with A a non-singular n x n matrix such that r^+ A-’ is stable. Then M(p) = ?+A-‘, a constant. Consider price profiles such that supr le-rfP(t)l < co. Then Z(P) =(i-M’)
7 e-(p-w)rP(t)
dt.
0
(10)
Z(P) is a weighted sum of the values P(t).
‘One could equally well have 2,
arise when for some i, zi can
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A common special case is where a and B are quadratic.3 The value function is then dual to F quadratic, the standard specification. [See Epstein (1981, example l).]
Example2. Let A be n x n non-singular, A - ’ = (a,;‘). Define .Z by (7) where i=
l,**.
2 n*
(11)
Then (12)
Y,(P) = - A - ‘W4
where D(p) is diagonal with ith diagonal element d,(p). M’(p) =i+r;‘(p) which depends on p and is stable if r-d,: ‘(p) < 0 over 0, i = 1,. . . , n. (M is diagonal itI A is diagonal.) Consider price profiles P such that inf, di(P(t))>0, Vi. The corresponding price aggregator is given by
I(P)=/'
)
(13)
Because the adjustment matrix depends on rental prices Z is not a simple weighted sum as in Example 1. In fact it is not even linearly homogeneous in P. That the quadratic specification for the technology permits the aggregation of rental price profiles is well-known and is one reason for its popularity. Rental price aggregation is generally consistent with a large class of technologies as Theorem 1 and Examples 1 and 2 demonstrate. But in fact, under some additional occasionally applicable assumptions, the quadratic specification is the only one consistent with an intertemporal price index. This result is the content of Theorem 2. The theorem thus describes circumstances under which the quadratic specification is not only convenient, but also necessary for the specification of functional forms for dynamic factor demand functions in the presence of a general model of price expectations. (See the discussion in the Introduction of existing approaches to functional form specification.) Of course interest in the accelerator adjustment rule extends beyond the ‘Quadratic
refers to a polynomial
of degree 2; constant and linear terms may be present.
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specific concern of this paper, namely the aggregation of rental price profiles. The accelerator adjustment rule is the most popular specification in dynamic factor demand theory. Thus Theorem 2 is similarly of somewhat broader interest since it provides an explicit characterization of the underlying production functions which is applicable in an empirically relevant class of circumstances. Theorem 2. Let F satisfy conditions (T) and generate the policy function (5). Adopt the following additional assumptions:
(a) M = tR(K, p) is independent of p and is stable. (b) For every i=-1,2,. . . , n, 3jf i such that mij #O, where M=(mij); particular, n > 1). (c) For euery (K, p) E 0,3(K, @)E 0 such that J,,(K,
(d) (K, z) E Q,for some z*(K,
(in
@) is negative dejnite.
0) E Sp.
Then F is a quadratic production function of the form F(K, z) = a, + a’K +$K’AK
+$z’Bz,
(14)
where A and B are (symmetric and) negative definite n x n matrices and B is diagonal.
Hypothesis (a) restricts the adjustment rule (5) so that M is independent of rental prices, which is the common specification. The constancy of M guarantees that the corresponding intertemporal price aggregator Z can be derived explicitly; in fact, it is given by the formula (10). An explicit representation for Z is required to generate explicit functional forms for the dynamic factor demand functions corresponding to the non-autonomous problem (2). Hypothesis (b) rules out the existence of a factor whose optimal adjustment is independent of the levels of all other factors. This hypothesis could not be refuted empirically as long as n> 1. .Z is concave in K since F is concave. Assumption (c) strengthens the concavity of .Z to strong concavity over a subset of its domain. Finally, (d) requires that the domain of F include all steady state positions (K,O). This assumption may be forced onto the empirical analyst by the nature of the data set which he is attempting to explain by means of model (2). Alternatively, the assumption may be required for counterfactual experiments which extrapolate from the data set to a larger domain. Its significance is that, in conjunction with Condition (T.l), it implies that, for each i=l,...,n, F,,(K,z)=O,
V(K,Z)E@,
such that
zi=O.
(15)
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Note that the functional form (14) is the most general quadratic specification which satisfies (15) over a domain @ consistent with (d). Theorem 2 shows clearly the unique feature of the common quadratic specification. At the same time, however, it does not vitiate interest in the sufficiency part of Theorem 1. The latter shows that even for non-quadratic technologies rental price aggregation is possible over some domains. From the point of view of functional form specification for empirical estimation, aggregate data sets occasionally involve everywhere positive net investment in all stocks. In that case the domain hypothesis (d) of Theorem 2 need not apply and’ Theorem 1 provides additional scope for generating functional forms for investment consistent with the adjustment-cost model and a general model of expectations. The case for Theorem 1 may be strengthened considerably by noting that natural modifications of the theorems apply if adjustment costs depend on gross rather than net investment. Then Theorem 2 is not applicable if, as is often the case, gross investment in all stocks is positive over the relevant domain. [See Epstein and Denny (1983) for an empirical analysis with such a data set.] Then Theorem 1 may be of value in generating alternative functional forms.
Appendix Proof of Theorem I
Suppose an aggregator I exists and prove (5). Let {K(t)},” be optimal in (2). By hypothesis &t)=t(K(t),I(P+,))
for all
(16)
th0,
which implies
K(t) must solve the Euler-Lagrange equation after substitution of (16) and (17), becomes
corresponding
1
~zz(K(thO [ ~w(K(t),I(P+,))r(K(t),I(P+,))+C,~I(P.,) vtzo. (18) t> 0. Consider problem (1) with initial stock= K(t)and p=Z(P+,).
+~Zd(K(O~ Z(P+,)) =fi+rF: -P(t),
Fix
to (2) which,
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Since < is the policy function for (l), the corresponding equation (evaluated at the initial position) yields
~,,um
Euler-Lagrange
ocsrdm~ Z(P+,))mm Z(P+dl (19
+F,,r(K(t),z(P+,))-FS,+rF:-z(P+,). Substitute (19) into (18) to obtain
Moreover, this equation is valid for all t 20 since t was chosen arbitrarily. The equation F,(K iw, PI) = - JKUC PI is a well-known first-order respect to p to obtain F,,(K
condition
(21)
associated with (1). Differentiate
5(K PM, = - J,,W, PI-
with
(22)
Thus (20) may be rewritten in the form
-J,,(K(t)~~(P+,))$CI(P+,)I=~(P+*)-P(r),
vtzo.
(23)
Evaluate (23) at t=O to obtain
(24) Differentiate with respect to K,, (the first component
of K,) to derive
By Assumption 5 there exist P' ,.. .,P" such that Z(P')=*..=Z(P")=Z(P) and {Z(P)-Pp'(O),...,Z(P)-P"(O)} is a linearly independent set. The appropriate versions of (24) and the non-singularity of JK,, [Conditions (T.2) (T.4) and
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(22)] now imply that
{
$c~(P:~)I,=o,...,$r~(P”+,)lr=a
is a linearly independent
set. Precisely as (25) was derived one can deduce i=l,...,n.
~c~~,(K,,z(P))l$C~(P:~)l~=,=O¶ 01
But the n vectors in these equations are linearly independent. 0 = &Z,,(K,, Z(P))/X,,. Similarly AZ,(K,, Z(P))/Xoi =0, Vi. Thus JpK is independent Jacobi equation for (l),
Therefore,
of K. To prove (5), consider the Hamiltonian-
rJ(K,p)=max{F(K,z)-p’K+J~K,p)z}.
(27)
I
Differentiate that
with respect to p and apply the Envelope Theorem to deduce
rJ; + K = J,&
(28)
from which (5) follows. For the reverse implication, (B.i)-(B.2)} where:
suppose (5) is valid. Define f={4:3
(B.1) 4:[0, co)-+UK O(K) is twice continuously
satisfies
differentiable.
(B.2) h(t)=$(t)-[f-M’(S(t))]-‘J(r) is such that h(t)~62”, Vt, inf,h’(t)>O, Vi, supt (e-“h(t)1 and sup, le-“h(;(t)l are both finite. Let - be an equivalence relation on f defined by Y1 -4’ iff 3T > 0 such that ~‘(t)-[i-~(91(t))]-‘~‘(t)=92(t)-[i-M’(92(t))]-’~2(t), Vthll: Denate by Z the set of equivalence classes p/and denote a typical equivalence class by f, a representative element of the equivalence class. Introduce some terminology: a path (K( .), P( e)) is regular if there exists a compact set Cc 0 such that (K(t), P(t)) E C, Vt. Finally let (K,, P) ES iff there exists 9 E Z (necessarily unique) such that P(t)=9(t)-[i-M’(9(t))]-‘~(t),
vtzo,
(29)
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(a(.),$(.))
is regular,
where K:(s) solves
K(0) = Ko.
(30)
The aggregator Z is defined by Z(P)s9(0)
where
PES=U
{S(K):KEQ”},
(31)
and 9 is the unique element of Z satisfying (29) and (30). It remains to verify that Z is a valid aggregator and that Assumptions l-5 are satisfied by I, S and 0 so defined. Assumption 1 is a direct consequence of (B.l). Turn to Assumption 2. First note that because of the construction of Z, if Y corresponds to P as in (29), then $+t corresponds to P,, . Thus (29) may be rewritten in the form (32)
which is precisely eq. (23). It suffices to show that K( .) defined in (30) with Y(t) =Z(P+,), uniquely solves problem (2). Note that g(t) =
(4
P(0) =f(O),
(‘-9
p(t)
=a[?-M’(9(0))]ll, 1=0
Vt. It
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(4
(K’(t), S’(t)) E 0,
Vt,
123
where K’ solves
K(O) = Ko.
It is possible to choose J1 or satisfying (a), (b), sup~[J(t)-9’(t)l
0. Define P’ for arbitrarily prespecified [r-M’(~‘(t))]-‘S’(t). Then (&,,P’)ES iff (P’(e), S’(e)) is regular where K’( -) sohes ti(t)=M(Sl(t))[K(t)-K(fl(t))], K(O)=&. Let C be the compact set containing the regular path (&( *), 9( *)) and let N,(C) c 8 be a closed A >O neighborhood of C. The regularity of (K( *, 9( *)) implies that there exists a closed neighborhood D c UK O(K), which contains Y(t), Vt. By appropriate choice of ,u, S’(t) ED, Vt, also. M( *) and R( *) are uniformly continuous and bounded on D and real parts of eigenvalues of M(p) are bounded away from zero as p varies over D. Use Hahn (1967, pp. 291-293) to deduce that supt (&(t)-&l(Q1sB(p) for a function B which tends to zero as p+O. Thus for a suitably small p< A, (K’(t),9l(t))~N~(C)c8, i.e., (K’( *),S’( *)) is regular and Assumption 5 is proven. The argument has proceeded from the set f and has defined only indirectly the rental price profiles P which may be aggregated. One could also begin by taking S to be the set of pairs (K,,P) for which eq. (23) admits a solution 9 consistent with (30). Existence theorems from differential equations theory would be applied to (23). [For example, consider a profile P such that P(t) = 0, Vt > T and the corresponding version of (23),
Apply Hartman (1973, theorem 1.1) to deduce existence of a solution 4 under further assumptions on bounds for M(m) and P( *).I This approach would demonstrate explicitly some profiles P which may be aggregated. The approach adopted above has the advantage of proving that aggregation is valid for all profiles P which generate suitable solutions 9 via (23). 0 Proof of Theorem 2
It has been noted that the policy function (5) corresponds to a value function having the form (7). But if M is constant, then (8) implies that
J(K r-4= 44 + B(K)-I-P’YK
(33)
124
L.G.
Epstein, Intertemporal
price
indices for
thejirm
and M=i+y-‘,
(34)
where y is a constant n x n matrix. There is no loss of generality in assuming that, V(K,p) E 0, (R(p),O) EQ) where R(p) is defined in (5). [Otherwise, we could restrict attention to points (K,p) consistent with this condition. By (d) and (T.5) there are ‘sufficiently many’ such points for our purposes.] Thus @K(p), p) = 0 for all (K, p) E 0. By (15) and (21), J,(R(p),p)=O, V(K,p)e@. Therefore (33)~/?,(K(p))+p’y=O, and so
J&G PI= BKW)- PlmPN. Condition
(35)
(T.l) and the linear adjustment rule (5) imply that, for all i,
Bq (K) Z BQ (R(P))
according as 1 m,jKj g 1 m, jRj(p), i i
where, of course, M = (mij). Hypothesis (d) and Condition all i, /YK,(K) g /?K,(IQ
.
mijmiizO
and
(37)
lie in @.
An immediate implication of (37) is that for each a strictly increasing function hi such that
Moreover, t_he derivative = O=@K,x, W) = 0, W-Jx Differentiate (38) to derive
(T.5) imply that, for
according as 1 mij Kj g C mijI?j, i j
for any K and K such that (K,O) and ($0)
(36)
i=
1,2,. . . , n, there exists
(38)
hi >,O throughott the relevant domain. [hi(K) , ,t,(K,p) =0, V(K,p) E 0, which violates (dj.] flKiK,(K)=h~mii =hjlmji =BK,+(K). Thus, mij=Oomji=O.
mii z0a.J K,K,(K,p) =BKIKi(K)=O,
(3%
V(K,p) E 0, which contradicts (c). Thus, for
all i, lllii
#O.
(40)
L.G. Epstein,
Intertemporal
price
indices for
125
the firm
Integrate (38), for the 1st and ith stocks and equate the results to obtain BW)=f’K,...,
(41) for suitable functions f’, f’, H’ and Hi. Upon differentiating BK,Ki
=fkiKi
BK,fc,
=mliH:l=milH1l*
one obtains (42)
+m~iH~llmll=miiH’ll~
and
(Hi,
(43)
denotes the second-order derivative of Hi, and so on.)
Pick i# 1 so that ml, # 0. [Such an i exists by hypothesis (b).] Then mi, #O also because of (39). Substitute (43) into (42) to obtain
The square bracket on the right-hand
side is non-zero because of(c).
Thus neither side in (44) can vanish.] Differentiate (44) with respect to K, to deduce that m,,H~,, =0 or If;,, =O. Now (41)=@, K (K) is independent of K, Vj. Similar arguments show that P&K) is a con&ant matrix. By (c) it is non-singular. By Hotelling’s Lemma for value functions [Epstein (1981)], [(K,p) = J;i(,‘(K,p)[rJ;(K,p) +K]. Thus, r(R(p),p)=Oora~(p)+(ry + l)R(p)=O* R(p)= -t-y-‘M-‘a;(p).
(45)
On the other hand, I?(p) also satisfies J,(&p),p) =0, which implies that R = -/3&p. In combination with (45) this implies that a; = My/?&p/r, i.e., ap is linear in p and a is quadratic in p. Thus J is quadratic in (K,p). By Epstein (1981, example 1) the production function F(K,k) is quadratic. The parameter restrictions contained in the JEDC-
E
126
L.G.
Epstein,
Intertemporal
specific quadratic (14) are readily Condition (T.l) and (d). 0
price
indices/or
established,
the firm
primarily
by
applying
References Bemdt, E. and C. Morrison, 1981, Short-run labour productivity in a dynamic model, Journal of Econometrics 16,339-36X Blackorby, C., D. Primont and R. Russell, 1978, Duality, separability and functional structure: Theory and economic applications (North-Holland, Amsterdam). Brock, W. and J. Scheinkman, 1977, On the long-run behaviour of a competitive firm, in: G. Schwodiauer, ed., Equilibrium and disequilibrium in economic theory (D. Reidel, Dordrecht). Epstein, L.G., 1981, Duality theory and functional forms for dynamic factor demands, Review of Economic Studies 48,81-9X Epstein, L.G., 1982, Comparative dynamics in the adjustment-cost model of the firm, Journal of Economic Theory 27, 77-100. Epstein, L.G. and M. Denny, 1983, The multivariate llexible accelerator model: Its empirical restrictions and an application to U.S. manufacturing, Econometrica 51, 647-674. Gorman, W.M., 1976, Tricks with utility functions, in: M. Artis and R. Nobay, eds., Essays in economic analysis (Cambridge University Press, Cambridge). Hahn, W., 1967, Stability of motion (Springer-Verlag, New York). Hansen, L. and T. Sargent, 1980, Formulating and estimating dynamic linear rational expectations models, Journal of Economic Dynamics and Control 2, 7-46. Hartman, P., 1973, Ordinary differential equations (Johns Hopkins University Press, Baltimore, MD). Heal, G., 1973, The theory of economic planning (North-Holland, Amsterdam). Infante, E. and J. Stein, 1973, Optimal growth with robust feedback control, Review of Economic Studies 40,47-60. Luenberger, D., 1969, Optimization by vector space methods (Wiley, New York). Meese, R., 1980, Dynamic factor demand schedules for labour and capital under rational expectations, Journal of Econometrics 14, 141-158. Mortensen, D., 1973, Generalized costs of adjustment and dynamic factor demand theory, Econometrica 4L657-665. Schramm, R., 1970, The influence of relative prices, production conditions and adjustment costs on investment behaviour, Review of Economic Studies 37, 361-375. Treadway, A., 1970, Adjustment costs and variable inputs in the theory of the competitive firm, Journal of Economic Theory 2, 329-347.