A simple dynamic equilibrium model with adjustment costs

Journal of Economic Dynamics and Control 6 (1983) 79-98. North-Holland

4 SIMPLE DYNAMIC EQUILIBRIUM MODEL WITH ADJUSTMENT COSTS* Robert A. BECKER Indiana

University,

Bloomington,

IN

47405,

USA

Received November 1981, final version received October 1982 The equivalence of a perfect foresight equilibrium to a Fisher competitive equilibrium and an optimal planning model is demonstrated for a model with an adjustment cost technology and a representative household. Adjustment costs lead to a non-trivial dynamic optimization problem for the producer. The results depend on the necessity of a transversality condition for the household, producer and planner in their respective optimization problems. The equivalence theorems are used to discuss conditions for multiple steady state equilibrium solutions and the corresponding stability properties.

1. Introduction A major problem in the theory of dynamic economic models is to characterize the competitive equilibrium configuration. A rational expectations equilibrium which coordinates the intertemporal maximizing behavior of many agents provides a mechanism to characterize competitive equilibrium in dynamic models. This approach was followed by Lucas and Prescott (1971), Brock (1972, 1974) and Scheinkman (1977). A common theme in each of these papers is that the proposed model’s equilibrium path solved a maximization problem. In Brock (1974) firms were static profit maximizers, whereas in Lucas and Prescott (1971) consumers’ demands had no intertemporal features. Scheinkman (1977) formulated a model with intertemporal maximization problems for a representative consumer and producer; this model did not provide an analysis of capital accumulation. The model formulated here was stimulated by Scheinkman. Our formulation features a representative consumer and representative producer that solve an intertemporal allocation problem. The model incorporates capital accumulation within the context of a private ownership economy. *This is a revision of ‘A Simple Dynamic Equilibrium Model with Stable and Unstable LongRun Equilibria’, which was Chapter I of my Ph.D. thesis at the University of Rochester. I have greatly benefited from the comments and support given by my advisors, Lionel W. McKenzie and Lawrence Benveniste. William Brock and Jose Scheinkman made numerous suggestions during the development of this project. Conversations with Michael J.P. Magill were also helpful. I have also benelited from many suggestions by the knowledgeable referees at this and another journal. B165-1889/83/%3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

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Becker,

A

simple dynamic

equilibrium model

Uncertainty is ignored, so the rational expectations equilibrium concept reduces to long-run perfect foresight. Analysis of the perfect foresight equilibrium problem reduces to an analysis of a growth model formally identical to the one sector joint production model of Liviatan and Samuelson (1969) by Theorems 1 and 2. Thus characterizing the equilibrium path reduces to their analysis. This implies that a perfect foresight equilibrium path may not have the turnpike property: there may be multiple long-run equilibria, alternately stable and unstable. Because our model is built up from costs of adjustment, another interpretation of the Liviatan and Samuelson model as representing a competitive equilibrium may be of interest. The main result of the paper is to show that a perfect foresight equilibrium for the model is equivalent to a Fisher competitive equilibrium representing Fisher’s (1930) ‘Second Approximation’ as well as to the optimal growth path obtained from the Liviatan and Samuelson model. The proof of the equivalence theorems depends on establishing the transversality condition as a necessary condition for optimality in a class of infinite horizon problems in addition to the usual Euler equations. Here, the arguments rely on the results of Benveniste and Scheinkman (1982) for supporting the necessity of the transversality condition in a class of models. This means that the competitive economy (or Fisherian economy) imposes on itself the transversality condition; this limits the choice of initial prices consistent with equilibrium at each instant [see Burmeister (1980) and Becker (1981) for further discussion on this point]. The adjustment cost model is also restricted so that shadow prices are always non-negative. This might not hold for other models with adjustment costs. The equivalence of an optimal growth solution to either a perfect foresight and Fisher equilibrium was developed in Becker (1981, 1982) for a multisector capital accumulation model. In those papers, the producer’s technology was described by a point-input point-output transformation function. This resulted in a myopic producer’s problem in equilibrium. This myopic problem resulted from the fact that capital goods prices equalled the discounted value of the future rental stream. Thus, with efficient markets, producer’s only examined their instantaneous profit; all capital accumulation decisions were borne by the representative consumer. The model proposed below alters this picture by introducing internal adjustment costs in the producer’s problem. This results in a non-trivial dynamic problem for the producer. Given adjustment costs, at each instant the producer must balance the marginal return from holding an additional unit of capital with the foregone rate of return obtained by ‘investing’ in additional capital. This is the content of the arbitrage condition in part B in the proof of Theorem 1. The marginal return is the discounted difference between the marginal product of capital

R.A.

Becker,

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and the instantaneous rental rate; the foregone rate of return is a marginal cost arising from adjustment costs. If adjustment costs are not present, then the marginal product of capital and rental rate are equated at each instant. This is Arrow’s (1964) myopic rule; this is the type of result obtained in Becker (1981, 1982). The presence of adjustment costs shows this rule is only valid at steady states. Section 2 contains the formulation of the perfect foresight equilibrium model and’ the corresponding optimal growth model. Section 3 includes Theorems 1 and 2 giving the equivalence between perfect foresight equilibrium paths and optimal growth paths. Section 4 contains the Fisher equilibrium model and demonstration of the equivalence between a Fisher competitive equilibrium and a perfect foresight equilibrium. Section 5 contains a summary of the results and concluding comments. 2. The model 2.1. Consumer’s problem

There is a single representative consumer. This consumer takes as given a path R = {r(t); t E [0, co)} for the rental rate on capital services yielded by the single capital good he owns and rents to firms; ZZ= {x(t); TV [0, co)} for the path of dividends paid out to the consumer from the ownership of firms. The path 17 may be negative at some instants of time. An alternative interpretation is to regard ZZ as the wage rate with one unit of labor supplied inelastically at each time.’ At each time t the consumer decides how much he wants to invest, Z(t), and how much to consume, c(t). Capital does not depreciate, hence Z(t) =k(t) for each time t. Formally, the consumer wishes to solve the problem P,(k,,, R,n): max i e-” u(c(t)) dt,

(c.k) o

subject to c(t)+i(t)=r(t)k(t)+a(t), c(t) 2 0, k(0) = k,,

k(t) 2 0

(1)

for all

k, given.

t E [0, co),

(4

(3)

‘Scheinkman (1977) treats the case of a variable labor supply derived from a labor-leisure choice; his II path may be interpreted as a residual paid to a fxed capital factor.

R.A. Becker,

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The firm pays out as dividends at time t it’s net cash flow, n(t). If n(t) is negative, then the owners of the firm must make good the deficit as this is a private ownership economy. The owners of the firm face a perfect market for capital services at each time and discount future cash flows according to the time path of interest rates R. The owners of the firm wish to choose (k,i) to maximize the present value of cash flows. Formally the firm’s problem is stated by P,(k,, R): max 7 emR(‘)n(t) dc, (k-i)

0

subject to

4t) = f(k(% k’(t))-r(wdt),

(4)

k(t) 2 0 for all

(5)

k(0) = k,,

t E [0, co),

k0 given,

and where R(t) =j r(s) ds.

(6)

0

Let

Y” = {y”(t); t E [0, co)} and Kd = {kd(t); t E [O, co)} denote P,(k,, R) where ys=f(kd, id) at each t.

a solution

to

Remark 2. If kd is a solution to P,(k,, R), then condition (c) of Assumption 2 implies kd(t) > 0 for all t E [0, co). 2.3.

Equilibrium

In an abstract economy where consumers solve P, and firms solve P, given continuous functions r:[O, co)+ R + and 7~:[0, co)+ R corresponding to R and I7 we may solve for the consumer’s consumption demand and investments supply, cd(t) and k(t), respectively, and capital stock (services) supplies, k”(t), assuming consumers perfectly predict R and 17. On the other hand, given R we may solve for the firm’s capital services demands, kd(t), goods supply, y”(t), and ‘investment’ demands, kd(t), as well as determine n(t) by (4) provided firms perfectly predict R. The following definition of equilibrium is used: Definition. Given k, >O, a pair of paths R and ii are a perfect foresight competitive equilibrium (P.F.C.E.) if the solution Rd, Y” of P,(k,,R) is such

R.A.

Becker.

A simple

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The firm pays out as dividends at time t it’s net cash flow, n(t). If z(t) is negative, then the owners of the firm must make good the deficit as this is a private ownership economy. The owners of the firm face a perfect market for capital services at each time and discount future cash flows according to the time path of interest rates R. The owners of the firm wish to choose (k,# to maximize the present value of cash flows. Formally the firm’s problem is stated by PF(kO,R): ;y

$ e-R(” x(t) dt,

subject to (4) k(t) 2 0 k(0) = k,,

for all

t E [0, co),

(5)

k, given,

and where R(t) = j r(s) ds.

(6)

Y” = {y’(t); t E [0, co)} and Kd= {kd(t); t E [0, co)} denote Pdk,, R) where ~7= f(kd, k”) at each t.

Let

Remark 2. If kd is a solution to Pdk,, R), then condition 2 implies kd(t) > 0 for all t E [O, co). 2.3.

a solution

to

(c) of Assumption

Equilibrium

In an abstract economy where consumers solve Pc and firms solve P, given continuous functions r:[O, co)+&!+ and K: [0, co)+R corresponding to R and n we may solve for the consumer’s consumption demand and investments supply, cd(t) and p(t), respectively, and capital stock (services) supplies, k”(t), assuming consumers perfectly predict R and n. On the other hand, given R we may solve for the firm’s capital services demands, kd(t), goods supply, y”(t), and ‘investment’ demands, id(t), as well as determine n(t) by (4) provided firms perfectly predict R. The following definition of equilibrium is used: Definition. Given k, >O, a pair of paths i7 and it are a perfect foresight competitive equilibrium (P.F.C.E.) if the solution Ed, P of Pdk,,iT) is such

R.A. Becker, A simple dynamic equilibrium model

84

and cd(t), p(t) that if. 7?(t)=f(Ed(r), i(t))-f(t)iGd(t), Ed(r)+ R(r) = a(r) and IEd(t) = p(t) for all t E [0, co). Equilibrium

solve P,(k,, R, m

then

requires that, for each time t E [0, co), ‘stock market equilibrium’,

(El) (E2.a)

‘flow market equilibrium’. (E2.b) In this abstract economy, a form of Walras’ Law obtains: Given (El) and I;“(O)=I;“(O)=k,, then (E2a) holds, and with the budget constraint (1) then (E2b) also obtains.

W)

--LIn a P.F.C.E., let C, K, K, Y denote the given equilibrium solutions to P, and P,. Combining Remarks 1 and 2 for equilibrium paths yields Remark 3.

Given a P.F.C.E. path (C, R), then E(t) >O and K(r) >O for all

tE L-0,00). 2.4. The growth problem

As mentioned in the introduction the equilibrium functions of our model are also solutions to an optimization problem. This results in a considerable simplification for deducing properties of equilibrium. Consider the following growth problem: P,(kJ: 7”;: $ eMdru(c(t)) dt,

c,

subject to (7) c(t) 2 0, k(t) 2 0 for all k(O)=k,,

t E [O, co),

k, given.

(8) (9)

Let C and K be solution paths for P&k,). Remark 4. If (C,K) is a solution for P&k,), then Assumptions c(t)>0 and k(t)>0 for all t~[O,co).

1 and 2 imply

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R.A. Becker, A simple dynamic equilibrium model

The problem P&-J is formally identical to the one-sector optimal growth problem with joint production introduced by Liviatan and Samuelson (1969) when we define a transformation function F(k, k) by c= F(k, I$ =f(k, k) -f. Then observe u(F(k,]t)) is concave in (k,k) with Fi 50 and a strict inequality whenever fk ~0. In section 3 we will demonstrate the equivalence between solutions to P&k,) and equilibrium solutions to the economy described by P, and P,. 3. The equivalence theorems The equivalence between solutions to P, and equilibrium solutions to the economy described by P, and P, is given in Theorems 1 and 2. The proofs of each theorem utilize a common strategy: a candidate solution of either PB or P, and P, is given by knowledge that it solves P, and & or P,; the candidate solution is demonstrated to be a solution if it can be shown to satisfy the sufficiency conditions for variational problems with concavity assumptions. We shall show the candidates are competitive, i.e., satisfy the necessary conditions of Pontriagin’s Maximum Principle for an appropriate path of dual variables and that the transversality condition also holds along our suspected solution. The particular version of the Maximum Principle we employ was developed in Benveniste and Scheinkman (1982). They demonstrate that a transversality condition for a class of concave variational problems over an infinite horizon obtains as a necessary condition of maximization.2 We may now state: Theorem I.

Suppose R solves P&k,), and paths R and I7 are defined by

F(t) = 6 -(d/dt)

In u/(?(t)),

(10)

(11) - then the paths I? and Ii constitute a P.F.C.E. where R solves P,(k,, R, lI) and P,(ko, @. Proof.

(A)

After some preliminaries, - R solves P,(k,, R, l7),

we will divide the analysis into two parts:

(B)

R solves Pdk,,R).

*Halkin (1974) gives an example where the transversality condition for infinite horizon programs does not hold as a necessary condition of optimization. Thus, we are restricting our attention to the class of concave variational problems covered by Benveniste and Scheinkman (1982).

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model

Since i? solves P&k,), it follows from Pontriagin’s Maximum Principle [see Benveniste and Scheinkman (1982, theorem 3)] that there is a C(l) function of present value prices or duals, satisfying the competitive

Preliminaries:

conditions C&h G(O)= -e-d’Dg(l;(O,

(12)

RS),

where g(k, r;) = u(j-(k, k) - rt> and

Ddk @ = (a, gA,

and the transversality condition lim q(ict)fF(t) =o. 1-m

(13)

We may write (12) as i(C) = -eedt fi’(t)j”k(t),

(14)

l)ii’(t),

q((t)= -eed’(fk(t)-

(15)

with ii’(t)=u’(C((t)), &(t)=j@(t), i(C)), etc., note ij(t) 20 by AIId. -(A) R solves P,(k,, R,l7): Define L(k, f, t) =u[E(t) +f(t)k(t) -i(t)] e-“; note that L is strictly concave in (k,];) for each t provided- -c(t) defined by (1) is non-negative. A path K* is competitive for P,(k,, R,l7) if there is a C(l) function l*:[O, co)+R such that (l*(t), A*(t)) = -(L:(t), L;(t)). We want to show K is a competitive path with duals X[O, co)-tR and that the transversality condition lim,, m ;T(t)k(t) = 0 obtains. Now define I(C)= -L@(t),C(t),

t)=e-d’ii’(t).

(16)

Show T(t) = -Jr&;(t),

IQ), t).

From (16), we calculate f(t)= -6e-d’zi’(t)+e-dr(d/dt)ti’(t) =e -dr C’(t)[(d/dt) In ii’(t) -S]

= -X(t)f((t)

by (10)

thus I? is competitive with duals ii =

{X(C); t E [0, co)}.

R.A. Becker, A simple dynamic equilibrium model

87

From (16), we have

=4WW3 -.&WI by WI. Since l-Tk(t)zl,

(13) implies lim,,,J((t)k(t)=O.

Therefore - - R satisfies the transversality P,@,, R, n).

condition

and thus is a solution to

Let Y be given from (10); a path K* is competitive C(l) function p*:[O, co)+R such that

for P,(k,, R) if there is a

(B)

I? solves P,(k,,R):

Define

G(k,k,t)=e-“(‘)[f(k(t),k(t(t))-F(t)k(t)], where

($*(t),p*(t))= - D(,,~)G(k*(t), k*(t), t), or P*(t)= -em”(‘) [f:(t)-f(t)],

p*(t) = -eeR(‘) f:(t).

We show R is a competitive path for P,(k,,iT) and that the transversality condition lim ,+,P((t)fE(t)=O, where P={p((t);t~[O,c~)} is the path of duals associated with R. Define &(t)=-DrG(li(t),

@t),t)=-e’(‘)Jlk(t)lO

for all

t~[O,co).

Show ,@I = -D,G(l;(t),t(t),

t).

Notice that (15) implies jT((t)= -e-“(‘)

[l -edf g(t)/ti’(t)],

hence (17)

R.A. Becker, A simple dynamic equilibrium model

88

Differentiation of (17) with respect to t yields the correct calculation below demonstrates,

j(t)

as the

we break up this calculation into d -e-““‘=jz(t)e-““‘, dt

W

g {e[dr-R(f)l ij(t)/ii’(t)}.

(1%

and

Differentiation

(19) is found by putting

=e[d’-R(r)l i Cs-4t)]~(r)i&‘(t)+~(~(t)/u’(t)) I q(t)/ii’(t)

= erdf -ml

by (10)

&)/g(t)

=e[af-R(r)l(-e-arU’(t)Tk(t))/U’(t)

=-e

+-$(ij(t)/ii’(t))

by (14)

-*q(t).

Combining (18) and (19) yields j(t)=-e-““’

Cfk(t)- ~tt)l= W(IE(tLh,> 0,

hence I? is competitive for PAk,,R)

with associated dual path P.

R.A. Becker, A simple dynamic equilibrium model

Furthermore,

a9

we have

p(t)@;(t)= -em”(‘) &(t)&(t) c-e

-& W) mfk(t)k(t)

by

by (15)

=--&(At)-e-drti’(r))k(t)

This implies lim, _ m p-(t)E(t) =O; P,(k,, 17). Q.E.D.

R(t)=&-lnti’(t)+lnti’(O)

hence

R

is an optimal

solution

to

We can also prove: Theorem 2. If paths i? and li are a P.F.C.E. for a given k,?O solution to- P,(k,,R) with - - C, defined for each t by E(t) =f(lt(t),I((t)), to P,(k,, R, II), then (C, K) solves P,(k,).

and R is a a solution

Prooj: Let g(k, k) = u[f(k, k) - k]; we shall show R is competitive and satisfies an appropriate transversality condition. Define

for P&k,)

by AIM.

q(t)= -e-“g&t),@t))zO Show i(t)=

-e-“g&(t),@t))

and

lim g((t)@t)=O. r+Q)

Because K is a P.F.C.E., there are C (‘I functions p(t), I(t) such that: (A) - (K, P) are competitive paths for P,(k,,R) and a- transversality condition obtains; (B) (R, ?i) are competitive paths for P,(k,, R, n) and a transversality condition obtains. Formally, Benveniste and Scheinkman’s (1982) theorem 3 implies for (A) and (B), respectively, p(t) = -e-“(“h(t), $(
[J.(t)-j;(t)], (20)

lim I?(t)k( t) = 0;

R.A. Becker, A simple dynamic

90

equilibrium

model

X(t)=eedr ii’(t), i(t)=

(21)

--I(t)

lim I((t)lE(t)=O. t+co

I claim the relations (22) and (23) follow from the competitive conditions of a P.F.C.E.:

flt)=+la’(t).

(23)

Note that (22) and (23) are the Euler equations for P, and P,, respectively. Defining G= G(k, $, t) and E=L(E, c, t) as in Theorem 1, we observe the Euler equations for PF and P, are (d/dt)( - Gk) = -G, and (d/dt)( -Lk) = - Lk, respectively. Proof(22):

The Euler equation for P, may be written out as

-$-e-‘(f)fr(t)]=

-e-“(‘)[fk(t)-F(t)],

or emR(‘)i;(t)fk(t)-ee-“(‘)

-&i(t)=

-e-@)

[j&t)-F(t)],

Cancelling e -‘(‘) and rearranging yields (22). Proof (23): The Euler equation for P, may be written (X(t), X(r)) to read (d/dr)[eedf $(t)] = -X(t)?(t), hence

out in terms of

Cancelling i?(t) emdr yields (23). We use (22) and (23) to show l? is a competitive path for P&k,) as follows by defining a path Q by q(t)= -e-dfgk(t)=

-em” [h(t)-l]zi’(t).

R.A. Becker, A simple dynamic equilibrium model

Using this definition

of g(t), we calculate

-eedf[jk(t)

d - l] zr7(t)

=ii’(t) e -dl 6[~(t)-11-$~(r)-Cf,(t)-l1$Ina.(t) I =ii’(t)e

-“‘{(a-$l.s.(t))lX(t)-ll-$fr(~)~

= -f?(t) emdrf,(t) c-e

I

by (22)

-a’&.

Therefore e, R is a competitive

path for the problem P&J.

We have $t)E(t) = -ebb’ [z(t)=e ‘w-”

l]ti’(t)&E(t)

ii’(t)p(t)k(t) +X(t)E(t)

by (20) and (21)

=ti’(O)p(t)E(t)+X(t)lt(t),

since we have R(t)-&=

This implies hm,,,

-lnI’(t)+lniZ(O)

by (23).

g(t)lE(t)=O by (20) and (21).

Therefore (c,I?) is a solution to P&k,).

Q.E.D.

Remark. Existence of a P.F.C.E. path follows from the existence of solutions to P,(kJ.

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4. A Fisher competitive equilibrium model This section is devoted to a demonstration that an equilibrium solution of the abstract economy (P,,P,,k,) is equivalent to a Fisher Competitive Equilibrium that is an infinite horizon analog of the well-known Fisher Separation Principle. The Fisherian model in this section determines in equilibrium a time path for the rate of interest by combining principles of investment opportunity and impatience with a market clearing principle.3 In the Fisherian problem, a representative consumer faces a given path of interest rates, R = {r(t); t E [0, co)}, a given initial present value of wealth, W,(k,, R) + k,,, and wishes to solve PF(k,, R, W,(k,, R)): yy

7 emdt u(c(t)) dt, 0

subject to $ ebRtt) c(t) dt = W,(k,, R) + k,

and

c(t) 20

for all t.

(24)

The constraint (24) requires the present value of any consumption stream capitalized according to R to be equal to the given present value of initial wealth. This present value of initial wealth consists of two components, W,(k,, R) and k,; the former component may be interpreted as arising from an optimization process based upon investment opportunities while k, represents the initial endowment of capital. Investment opportunities

are modeled by the problem

PF(k,, R): W,(k,, R) zmax 7 eeR(‘) n(t) dt,

(k,i)

0

subject to n(t)=f(k(t),k(t))--r(t)k(t),

k(t)20 k(0) = k,

for all t, given,

(25)

and where R is a given time path of the interest rate. Problem PF(k,, R) states that a Fisherian household faces investment opportunities according to the constraint (25) given an initial capital stock. This consumer will seek to ‘See Fisher (1930, pp. 148-149) regarding his ‘Second Approximation’.

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c(t)+ i(t) = f(W), @I)

(26)

holds for each time t. The Market Principle takes the form of imposing (26) as a requirement to be met by candidate solutions of Pf and PF. This leads to the following: Definition. A path R is a Fisher Competitive Equilibrium (F.C.E.) if there is a path C = {E(t); t E [!I, co)},. R = {k(t); t E [0, co)} such that if R solves Pf(k,, R) and E(t)=j(li(t),&t))-k(t) at each t E [O, co), then C is a solution to P:(k,, R, W,(k,, RN.

The result of this section is that a F.C.E. is equivalent to a P.F.C.E., and, therefore, from Theorems 1 and 2 a F.C.E. is equivalent to an optimal growth path for P&k,-,). Theorem 3. Let k, >O be given; Zet R = {f(t); t E [0, m)}, n= {$t); t E [0, co)} be a P.F.C.E. path with corresponding P.F.C.E. paths R, C as a solution to - - P,(k,,R,l7) and P,(k,,@, then R is a F.C.E. and K, C solve P,F(k,,, W) and P,F(k,, R, W,(k,, R)), respectively. Proof Since R is a solution for P,(k,,R), obviously R is also a solution for P:(k,,R). Therefore, we have only to show that c is also a solution for

P%,, R N,(k,, RN. Since R and D are a P.F.C.E., we can use the results in the proof of Theorem 2. By (23), the Euler equation for P,, we have

= ~(Olt(W’(O) by (B), in Theorem 2. This implies lim eeRtr) E(t) =0 r-m JEDC

D

since

lim X(t)L(t) =O. r+m

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Hence we have ““‘(it(t)+~~)l;(t)--(t)dt

$e-‘(‘)qt)dt=~e0

= W,(k,, R) + k, - lim e-R’f’ k(t) 1’03 = W,(k,, R) + k,.

This proves that C is a feasible path for P,F(k,, R, W,(k,, R)). Now suppose that C is not a solution for P,F(ko,R, W,(k,,R)). is a path Co={co(t);t~[O, co)} such that

Then, there

$ e-” u(co(t)) dt> $ eVdr u(F((t))dt, and $ e-“(I) c,(t) dt= W,(k,, R) + ko. Define functions F: [0, l] -P R and G: [O, l] + R by F(O)= 7 emat u(@c,(t) +(l -@)2(t)) dt

for each

0 Eco,11,

for each

OE[O, 11.

0

and G(O) = 7 e-R(‘)(Oco(t) +( 1 -O)?(t)) dt 0

The concavity of u together with the assumption F(0) < F( 1) implies F’(0) > 0, where F’(0)=~e-d’u’(E(t))(co(t)-~(t))dt. 0

Also, G can be shown to be a constant function implying G’(0) =O. However, the Euler equation (23) of Theorem 2 implies F’(0) = 7 e-“(‘) ii’(O)(c,(t) -F(t)) dt, 0

=ti’(O)C(O).

U’(0) = zqE(O)),

R.A.

This is a contradiction.

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model

Thus, c is a solution to Pf;(k,,R, W&e,&).

Q.E.D.

We may also show: Theorem- -4. Given k,>O, suppose a path of interest rates R is a F.C.E. with paths C, K as. solutions to PF(kO,R, W,(k,,R)) and P,Fk,,R),- -respectiuely. If m=ff(l;(d, I;(w-WlE(t)- is- used to define the path if, then R,Il is a P.F.C.E. where %, R solve P,(k,, R, II) and PdkO, K), respectively. Proof: Since R is a solution for PF(k,, W), obviously R is a solution PF(k,,,R). - - Therefore we have only to show that c is a solution

for for

Pcbo, R n).

Since i7 is a F.C.E., we have for each t, z(t)+i(t)=qt)E(t)+ii(t). - -

This implies that the path (C,R) is feasible for P,(k,, R,l7). Also, this implies (27)

since I? solves P,FkO, R) and C solves P,F(k,, R, WO(k,, W)). Hence, integrating the left-hand side of (27) yields $e-R”)(f(t)li(t)-k’o)dt

= -1im e-R(r)&t)+k(0)=k,,

t4co

and hence lim e-“(‘)E(t)=O.

,+a

Furthermore, since PF(k,,,IT, W,(k,,R)) is a typical problem of the calculus of variations with integral constraints, a well-known necessary condition is that for a constant b > 0, e-dtiZ(~(t))=be-@r)

Therefore F(t) = 6 -(d/dt)

for all t.4

In ii’(t) obtains for all t.

%ee Hadley and Kemp (1971, theorem 3.2.1, p. 178), for a statement of this necessary condition for isoperimetric problems. In the problem P,’ the Hamiltonian takes the form H(t, c, f) = b, evd’u(c(t))

+ be-%(t),

where b, and b are constants not both of which are zero. The Euler condition is (d/dr)Zf, = H, which .reduces to H,(t,L$),E(t))=O in this example. Assumption 1 asserts that u’(c)>0 implying both b, and b are non-zero, so that we may assume b is positive without loss of generality.

R.A. Becker,

96

A simple

dynamic

equilibrium

model

- - Now we are ready to prove that (C, K) solves P,(k,, R, L’). Define L(k,k,t)=u(Z(t)+?(t)k(t)-L(t))e-”.

Also define

Then we have i(t) =$

Lewd’ 12(t)]

=e -afiT

-8+$ln3(t) [

= -L,JE(;(t), E(r), t)

1

by the above result.

Thus R is a competitive path. Also, we have ;T(t)E(t)=e-” =be-‘(I)

ti’(t)k(t)

E(t) by the above result.

Therefore lim e-Rcr)K(t)=O implies lim X($(t) =O. t-m 1-m - - Therefore, (C, K) is a solution to P,(ko, R, II). Q.E.D. Applying Theorems 1 to 4 yields the following: Corollary. A F.C.E. - - path i? yields a solution (e,R) to the problem P&k,), and a solution (C,K) to the problem P&k,) yields a F.C.E. path. 5. Concluding

comments

and summary

The equivalence theorems permit analysis of either equilibrium models by appealing to the analysis of the optimal growth problems. This problem is formally identical to the joint-production model of Liviatan and Samuelson (1969). Also, the analysis of Magi11 (1977) may be used for this problem. In particular, the one capital good structure permits use of Magill’s potential function which completely characterizes steady state equilibria. His analysis implies that a steady state solution, kd, for P,(k,) is unstable if and only if

R.A.

Becker,

A simple

dynamic

equilibrium

model

97

&.(kd, 6) 2 0, where

g(k,li)=u(j-(k&I;).

Thus, &.(kd, 6) 2 0 can occur only when &(k’, 0) > 0. In this case, there are typically multiple steady states. If & > 0 at ok’, then kd unstable means that there are smaller losses of foregone output from large changes in capital than small changes. Thus, for k, sufficiently near k’, the optimal path results in large changes in the capital stock leading the optimal program to escape any sufficiently small neighborhood of k*. The ‘large’ adjustment cost of foregone output from keeping k close to zero and returning to k* along the optimal path can be overcome by altering capital to approach a new steady state capital stock. This study has exhibited an example of a model of intertemporal competitive equilibrium that may exhibit multiple steady states; in particular global asymptotic stability of a P.F.C.E. path of capital stocks may not occur in this model. Thus many of the stability difficulties one would expect in a multi-sector problem representing dynamic equilibrium are shown to arise in a one-sector model. The perfect foresight equilibrium model has also been shown to be equivalent to a representation of Fisher’s Second Approximation. This equivalence produces a demonstration of the Separation Principle for an infinite horizon model where investment projects are not necessarily of the point-input point-output variety. Generalizing the results in Becker (1981, 1982) along these lines would be one possible problem for future analysis.

References Arrow, Kenneth, 1964, Optimal capital policy, the cost of capital and myopic decision rules, Annals of the Institute of Statistical Mathematics, 21-30. Becker, Robert, 1981, On the duality of a dynamic model of equilibrium and an optimal growth model: The heterogeneous capital goods case, Quarterly Journal of Economics 96,271-300. Becker, Robert, 1982, The equivalence of a Fisher competitive equilibrium and perfect foresight competitive equilibrium in a multi-sectoral model of capital accumulation, International Economic Revtew 23, 19-34. Benveniste, Lawrence and Jose Scheinkman, 1982, Duality theory for dynamic optimization models of economics: The continuous time case, Journal of Economic Theory 27, 1-19. Brock, William, 1972, On models of expectations that arise for maximizing behavior of economic agents over time, Journal of Economic Theory 5.348-376. Brock, William, 1974, Money and growth: The case of long-run perfect foresight, International Economic Review 15, 75Q-777. Burmeister, Edwin, 1980, Capital theory and dynamics (Cambridge University Press, Cambridge).

R.A. Becker, A simple dynamic equilibrium model

98

Fisher, Irving, 1930, Theory of interest, Kelley reprint edition 1970 (Augustus M. Kelley, New York). Hadley, George and Murray Kemp, 1971, Variational methods in economics (North-Holland, Amsterdam). Halkin, Hubert, 1974, Necessary conditions for optimal control problems with infinite horizons, Econometrica 42,267-272. Liviatan, Nissan and Paul Samuelson, 1969, Notes on turnpikes: Stable and unstable, Journal of Economic Theory L454-475. Lucas, Robert and Edward Prescott, 1971, Investment under uncertainty, Econometrica 39, 659 682.

Magill, Michael, 1977, Some new results on local stability of the process of capital accumulation, Journal of Economic Theory 15, 174-210. Scheinkman, Jose, 1977, A simple dynamic equilibrium model, Unpublished notes (dated around February 1977).