JOURNAL

OF ECONOMIC

THEORY

Vintage

55,

323-339 (1991)

Capital,

Investment,

and Growth*

JESS BENHABIB Department

of Economics, New York, New

New York

York Universit)). 10003

AND ALDO RUSTICHINI Department

of Economics, Northwestern Evanston, Illinois 60208

University,

Received June 29, 1990; revised April 19, 1991

We study the dynamics of growth and investment in a continuous time model with vintage capital. Vintage capital models may be characterized by non-exponential rates of depreciation and technical change and can incorporate “gestation lags” as well as “learning by using.” We investigate the effect of such features on the dynamics of investment and growth and show how they can contribute to explaining the volatile nature of investment time-series. Journal of Economic Literature Classification Numbers: 110, 130, 020. 0 1991 Academic PXSS, hc.

1. INTRODUCTION In models of vintage capital, equipment of different vintages may differ in productivity due to technical progress or due to the effects of variable depreciation rates. In particular, such models allow for the possibility of non-uniform or non-exponential rates of depreciation and technical progress. Indeed, in certain plausible situations, equipment may, because of “learning by using,” i become progressively more productive during an initial phase of its lifetime before it depreciates later on. A limiting example of such a structure is “gestation lags,” where initially new equipment is *We thank the C. V. Starr technical assistance. ’ Strictly speaking, “learning with cumulative output. We improvements that result from based on usage and experience. change. See footnote 3.

Center at New York University for research facilities and by doing” refers to productivity improvements that increase are loosely thinking of productivity and management increased familarity with the capabilities of new equipment, This is particularly relevant in the case of embodied technical

323 0022-0531/91 $3.00 642’55

2-7

Copyright Q 1991 by Academic Press, Inc All rights of reproduction in any form reserved

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totally unproductive and costly to maintain. In this paper we analyze the implications of non-exponential depreciation, and of “learning by using,” for the dynamics of investment in an optimal growth model with vintage capital. Essentially, in such a model one must keep track of equipment of different vintages to describe the investment dynamics. As we show below, non-exponential depreciation structures, with or without “learning by using,” may help explain the highly volatile nature of investment timeseries. (For an empirical investigation in an aggregative stochastic model in discrete time, see Benhabib and Rustichini [3].) To introduce the discussion, it may be helpful to begin with the classical view of capital. If we denote by K(t) the capital stock at time t, and by k(t) the investment (according to a notation which will be consistently used in the rest of this paper), then the standard model of exponential depreciation, k(f)=k(t)-yK(f),

(1.1)

has a solution (provided K(t) e-” + 0 as t + co) given by

Note that in this formulation the efticiency of an investment good of vintage t has its efficiency reduced by a factor eP”‘. The dynamics of the optimal path in the exponential depreciation case are well known. We can take the case of a linear utility as an easy reference. It is known that if the initial capital stock is lower than the steady state value, then investment grows (with the optimal level of consumption equal to zero) until the steady state is reached. After that, both the capital stock and the investment are constant. One of the points that we shall argue in this paper is that the assumption of exponential depreciation suffers by virtue of its own simplicity (that is, by dramatically reducing the possible dynamics that an optimal growth model can describe). Our model is closely related to the work of Mitra and Wan [7, S] and Mitra, Ray, and Roy [9]. Mitra and Wan [7, S] study the dynamics of a forestry model in discrete time. The primary input, iand, is used either alone, to produce saplings, or in combination with standing timber of some age to produce, according to a fixed proportions production function, a consumable good. The latter may be used either as the final good (harvested wood) or as standing timber of any age. In our model the primary input, labor, is used in variable proportions with various vintages of capital to produce a final good, which is divided between consumption and capital of a new vintage. Section 2 sets out some notation and definitions. Section 3 describes our model of vintage capital and some properties of its solution. Section 4,

VINTAGE

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CAPITAL

which contains the main results, describes the different investment dynamics that emerge under various assumptions on depreciation schemes, including those with “learning by using” and “gestation lags.” The standard “exponential depreciation” case turns out to be a particularly special case. Finally, Section 5 shows how persistent and robust oscillations in investment can occur in a model with “learning by using” and a strictly concave utility function. 2.

NOTATION

AND

DEFINITIONS

Let R, denote [O, + co) and I% denote ( - w, O]. We let D be the family of positive finite measures on the real line with the Bore1 o-field; for p, YED we say ~2 y if p(A)>?(A) for every Bore1 set A. The cone D* c D of measure of increasing depreciation is the set of measures which satisfy the condition that for every 6 > 0, the function t * p( [r, t + 6)), defined on ( - co, -61, is increasing. To any PEED, we associate the measure ,U*E D* defined by ~*([a, 6))~ suplsa ,u( [t, t + b -a)). M defines the space of (Lebesgue) measurable functions defined on ( - co, 01; &? the analogous space defined over R. M+ and IV’ denote the positive cones of these spaces. For a function k: ( - co, t] -+ R, with t > 0, we denote by k,(s) = k(t + s) the function k,: (-co, 0] + R. Also, for any interval [a, b] the function kCa,61 denotes the restriction of k to [a, b]. To every ke M, we associate the quantity K(t) E Jtcu k(s) dp(s - t). Finally, xA denotes the characteristic function of the set A : xA(t) E { 1 if tcA, 0 if t#A}. 3.

AN

OPTIMAL

GROWTH

MODEL

WITH

VINTAGES

The technology in our economy is given by a neoclassical production function of the capital stock K(t), denoted byf: We assume: (i) f is strictly concave, Cz in the positive real line; and (ii) lim, _ + o. f’(x) = 0.

(3.1)

To a given REM (the initial capital profile) we associate the set of admissible pairs of capital and consumption paths A(4), namely the set of pairs (c, k) E M+ x ii?’ which satisfy (C is some positive constant) k. = 4;

f 1’

-co

k(s) dp(s - t) - c(t) = k(t)

> F~c(t)20,

(Lebesgue a.e. t)

k(t)aO.

W)(i) (3.2)(ii)

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Let r be the positive discount rate. The optimal defined by 2

+z e -“c( s

sup (C,k)EA 0

growth problem is then

t) dt.

(G)

Remark. We ignore at first, for simplicity of exposition, the case of a non-linear utility function. This extension is easily accomplished, however, and will be considered later (see Section 5). Another easy extension, that we shall not consider explictly, is to a production technology which includes labor and embodied technical progress of the labor augmenting type. 3 Before discussing the characterization of the optimal path, we establish its existence in Lemmas 3.1 and 3.2. The conditions on f give, for every E> 0, a real number A, such that f(x) 6 A, + EX. Then we have: LEMMA

fCj’m

3.1. The path of maximal accumulation, 4s) 4s - t)), satisfies the inequality

defined by k(t) =

where C denotes a positive constant. Proof The existence of such a path is given by a standard contraction argument which gives the existence of a solution to the integral equation. By replacing /J by p* if necessary we may assume without loss of generality that LED*. Then for .any he [0, 11, and t>O, k(t+h)
which implies the result.

1

a Two recent papers, Mehdin [S, 61, study a control process governed by an integral equation with integral constraints. The author studies existence and optimality conditions with relaxed controls and derives conditions for which relaxed controls reduce to ordinary controls in the context of a finite horizon and a non-linear integral kernel. By contrast, in our framework the integral kernel is linear and we have an infinite horizon. 3 If utility is linear or homogeneous, for example if U(c)=?, CXE(0, 11. and L-y
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VINTAGE CAPITAL

Now it is easy to prove: LEMMA

3.2.

The optimal growth problem has a solution for any k, E M, .

Proof. Since the problem is concave, this follows immediately, standard arguments, from the estimate in Lemma 3.1. i

by

We can now proceed to characterize the optimal path. We first need some conditions on the measure p. More specifically, the following assumption (Al) will stand for the remainder of the paper: (i)

dp(s)=m(s)d(s),

(ii)

m(0) # 0.

mEL’(K;

[O, 11) (Al)

We shall see that optimal interior which we immediately introduce,

paths satisfy an integral

equation

0 s

-m

k( t + s) d/i(s) = C,

(1)

ko=i,

where the constant, dependent only on p andf, is given by -1

C=C(p,f)-f’-’

KJ

O e’“dp(s) -130

)

I

(3.3)

We say that a measurable function k: R -+ R which satisfies (3.2) for almost every t 3 0 is a solution of the integral equation. If, in addition, 0

c>f

-cc

k(t+s)dp(s)

>

-k(t)>0

and k(t) 2 0, then the solution is said to be feasible; it is interior if the inequalities hold strictly. We now discuss the relationship between the solutions of the optimal growth problem and the integral equation. THEOREM 3.1. Let (Al) be satisfied. Then an interior optimal solution to the growth problem (G) satisfies the integral equation (I).

Prooj

See Appendix 1.

In the following sections we study the dynamic properties of the optimal solution by studying the solutions of the integral equation (I). To justify this we have to take care of a few technical points. Theorem 3.1 states that an interior optimal solution satisfies (I). We prove an existence and uniqueness result for solutions of (I) under conditions on the depreciation

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BENHABIB AND RUSTICHINI

measure which are general enough to cover the case analyzed in Section 4. The result is technical, and is presented in Appendix A2; see in particular Lemma A.2.4.

4. THE DYNAMIC

IMPLICATIONS

OF THE DEPRECIATION

PROFILE

In view of Section 3 and Appendix 2, we can use the integral equation (I) to describe the solutions to the optimal investment problem (G) in Section 3. We associate with the integral equation (I) the characteristic equation 0

s -cc

ezsdp( s) = 0.

(4.1)

The left hand side of the equation is the Laplace transform of the function s H m( --s), denoted &( - . )(z). When m has compact support, fi( - . ) is an analytic function defined on @; otherwise it is defined on a proper subset of C. The spectrum associated with Eq. (4.1) is the set S~{zEC:ti(Any linear combination

~)(z)=o}u{o}.

(4.2)

of the set of eigenvalues of the form

k(t) = C Aieqf,

with

z,=o,

A,=C,

A,EC,

satisfies the integral equation (I). The analysis of the asymptotic distribution of the elements of the spectrum can be reduced to the analysis of the distribution diagram of the pairs (pi, m,) of the exponential polynomial of the form n

p(z) = C p,z”Je@( 1 + E(Z)),

(4.3)

j=O

where O=p,
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VINTAGE CAPITAL

special case, corresponds to the standard case of smooth exponential depreciation. For all other examples, the spectrum S has complex elements and therefore the dynamics of investment are necessarily oscillatory. EXAMPLE

1.

Let where

T> 0.

(4.4)

This case corresponds to a linear depreciation schedule, where the lifetime of a machine is T. The characteristic equation is zT+e-‘T-

1 =O.

(4.5)

PROPOSITION 4.1. Any element in the spectrum associated with m in (4.4) has non-positive real parts. The only element with zero real part is z,, = 0. Consequently for any initial capital profile 4, the solution k+: R + If3 of the integral equation satisfies

lim k$(t) = C. f--t +‘x

(4.6)

Proof From the characteristic equation (4.5) and Hayes’ theorem (see Bellman and Cooke [l, p. 444]), if ZE S, then Re zd0.

A similar situation arises in the following case with an exponential depreciation schedule where a machine still lasts T periods. Note, however, that there is a discontinuity at time T. EXAMPLE

2. Let y>O, T>O.

4s) =eYS~(--T,ol,

(4.7)

The characteristic equation is 1 -e-(Y+‘)T=o

9

(4.8)

and therefore S= (0) u {-y+ (2ni) kTP1 Ik=O, fl, ...}. The non-zero elements of the spectrum have negative real parts and the eigenfunctions have arbitrarily small periods. The solution from any initial condition converges to the constant solution at exponential rate y. This last fact is also immediate from the fact that any solution of the integral equation satisfies k(t) = yC + k( t - T) ePy*, and therefore k(t+nT)=yCi

e-jyT+ q5(t - T) ePnyT,

and so k(t) + yC/( I- ePYT) uniformly

as t + + 00.

tE CO, T)

(4.9)

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BENHABIB AND RUSTICHINI

As T+ + co in Example 2, we get to the classical case of exponential decay, which we discuss next. EXAMPLE

3. Let m(s) = P,

SElK.

(4.10)

Differentiating and integrating by parts easily show that this is the case of a standard accumulation equation for aggregate capital, given by k(t) = k(t) - Yat). Now +z( - . ) is only defined on {z: Re z > -7 }, and in this region the characteristic equation is (z + y)- ’ = 0, so that S = (0). Any solution of the integral equation is characterized as the solution of k(t) - y j” m k(t + s) &(s) = 0, so k(t) = yC. In other words, as soon as the set 7 is reached the investment becomes a constant. This is as expected, since it is the standard situation corresponding to an optimal growth model with a linear utility function. The following proposition sums this up: PROPOSITION 4.2. Any optimal path km satisfies the equality k”(t) = yC for any t > 0 such that kf E 7.

Note that the convergence to the steady state value is exponentially fast: indeed the convergence is estimated by the eigenvalue with the largest (negative) real part. This is the picture of a classical turnpike theorem. As we shall see in a moment, however, this is not the only possible case. EXAMPLE

4 (One-Hoss Shay).

Let (4.11)

in this case a machine does not depreciate but has a lifetime of T. The characteristic equation is I- eezT = 0, and therefore S = (0) u (2xik: k = ) 1, +2, ...}. All the eigenfunctions are purely periodic: there is no dampening of an initial perturbation. In fact, any solution of the integral equation satisfies k(t) = k(t - T); so it is periodic with period T, and not necessarily continuous. PROPOSITION 4.3. Any feasible solution of the integral equation, kf E .?, satisfies kf(s) = kt(s + T) for any s > 0.

The above understanding limit situation. regular pace,

with

case of one-hoss shay is an extreme example. To get a better of its dynamic behavior it may be useful to consider it as a In the following example, depreciation takes place at a more and has the one-hoss shay case as its limit.

331

VINTAGE CAPITAL EXAMPLE

5. Let m(s) = m,( I- 0) + m,(8 + s)

SE [-t&O]

m(s)=m,(l

SE c-1,

+s)

-e-j,

(4.12)

where 8 E [0, 11, ml( 1 - 6) + m,B = 1. We are mostly interested in the case m,>m,20. The characteristic equation is -zeZ+m,e’-mm,+(m,-m2)e”‘-0)=0.

(4.13)

PROPOSITION 4.4. The spectrum associated with (4.13) is asymptotically distributed in the strip (z: /Re(z + log z)l < C, > for some positive constant C, ; so the real part of the eigenvalues is eventually negative. As m2 + 0, m, --* + co, 0 + 1, the ‘spectrum tends pointwise on compact subsets of the complex plane to the spectrum associated with the case of “one-boss shay” (4.11).

Proof. This is a consequence of Theorem 12.9, Bellman and Cooke [ 11. Note that by setting m, =m,= 1, (4.13) reduces to (4.5), the case of Example 1. We also remark that the characteristic equation given by (4.13) for m2 < 0 corresponds to the case for which the efficiency of the investment goods increases over an initial period. Let us now consider the possibility that new investment goods do not reach the peak of their efficiency when they are introduced, but actually see their efficiency increase with time, at least for some time. We first consider a simple example. EXAMPLE

6. Let

4s) = (a + bs) xc- 1,~~

a > 0, b < a.

;

Then the characteristic equation is -Z

b

be-’

~+(b-a)~-z2+-i-

= 0. Z

By Theorem 12.9 of Bellman and Cooke [I], the spectrum associated with this characteristic equation is that of an equation of neutral type, that is, with a spectrum asymptotically distributed on a strip in the complex plane of the form {z: 1Re zI ,( C,} for some constant C,. Indeed, integration by parts gives that differentiable solutions of the integral equation are solutions of the equation of the neutral type ak(t)=

-b(k(t)-k(t-l))+(a-b)k(t-1).

(4.14)

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BENHABIB

AND

RUSTICHINI

The zeros of the characteristic equation are asymptotically the zeros of the equation

distributed

e’- 1-ba =O, ( 1

like

(4.15)

so that asymptotically the real parts of the roots of (4.14) have Re z= log( 1 -b/a) and so have positive (or negative) real parts if b < 0 (b > 0, respectively). In other words, if the relative efficiency of investment goods increases with time, the steady state 7 becomes unstable. Intuitively, current investment decreases the incentive to invest in the nearby future not only because of diminishing returns, but also because the stock of aggregate capital tends to “increase” simply by virtue of the passage of time. We also note that the sudden depreciation to zero for s < - 1 is not the cause of the instability. If investment goods are fully efficient for a period after the time they reach their peak, as in the case of the depreciation scheme m(s)=(a-bb)~[-~,-~]+(a+b~)~[~~.~~,

(4.16)

the result does not change. In fact the characteristic equation is (4.17) which is again of neutral type; the zeros are asymptotically distributed the zeros of e*’ - (1 - b/a) = 0, and the above analysis is unchanged.

like

For a final example, we turn to the case of pure “gestation lags.” EXAMPLE 7. We consider the case in which investment has a fixed, possibly negative, contribution to the aggregate capital stock during an initial period, and then has a positive contribution which decays at an exponential rate. This case may represent an initial gestation lag for investment which has non-negative maintenance costs. Formally we define m(s)=eY’“+R)X,s~-T)+cX,~Tcs~O)

(4.18)

In this case the characteristic equation is given by (4.19) Again by Theorem 12.9 of Bellman and Cooke [ 1] the spectrum associated with this characteristic equation is of the neutral type. The roots

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VINTAGE CAPITAL

are asymptotically distributed as the roots of 1 -c + ce’= 0, and therefore the limit of the real part of the roots is log11 - l/cl. There are two important values of c. As c + 0, the asymptotic limit of the real part of the roots tend to infinity. So for c E (-co, 0) u (0, 1) the roots have asymptotically a positive real part, and the constant solution of the integral equations is unstable. The case c = 0 is very special. In this case the spectrum degenerates to a single point, S= (0). The other important value of c is c = 1. As c crosses this value the limit of the real part changes from positive to negative, and we have a critical Hopf bifurcation. The intuition for the results of this example are very similar to that of the previous example in which the productivity of an investment outlay increases for a while before depreciation takes over.

5. THE NON-LINEAR

UTILITY

CASE

An interesting application of the analysis of the linear utility case developed above can now be given for the case of non-linear utility. We note first that in a similar way to the derivation of Theorem 3.1 one can prove that interior optimal paths are characterized as solutions of the equation -e p”U’(c(t))+

J’” I

eCrsU’(c(s))f’(K(s))

m(t -as) ds= 0,

(5.1)

where c(s) =f(K(s)) -k(s), for s > 0. We have seen in the analysis of the linear utility case that as the slope, b, of the depreciation function m(s) = (a + bs) x1- 1,0,(s) passes from positive to negative values, the eigenvalues of the characteristic equation cross the imaginary axis. The Hopf bifurcation is “critical” and therefore degenerate because the equation is linear. A similar conclusion holds for the case of gestation lags or “time to build” in Example 7. In this section we analyze this transition in the non-linear model (5.1) above. We first linearize Eq. (5.1) at the steady state function (c*, k*), and compute the associated characteristic equation. This is done by differentiating the first order condition (5.1) above with respect to the vintage produced at time U, k(u), and then integrating over R the product of this derivative with the function eZU.When (as in the case we are interested in) the support of the function m is bounded, all the computations formally performed are justified. Also, it is clearly sufficient (by the stationarity of the problem) to consider Eq. (5.1) above at t = 0. The computations outlined above will give a characteristic equation

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BENHABIB AND RUSTICHINI

T(z) =O, where T is the sum of the different terms T,, T,, T, defined below: T,(z) = U”(c*)[ 1 -f’(K*)

fi( - . )(z)]

and T*(z) = U”(c*)f’(K*)

jo+ r; e’“‘“[f’(K*)

C(p)-m(

-u)]

du.

The integral in T, will converge on the region {Re z < r }. Since the asymptotic behavior of the roots of fi( - . )(z) satisfies Re z = log( 1 - b/a), this restriction creates no problems for small enough b. Recall that C(p) = j”no e%(s) ds. Finally the third term is given by T, = tii( - . )(z) lo+ z e(;--,‘Um(-u)f”(K*)du.

Note that the first two terms can be made, in compact subsets of the complex plane, arbitrarily small. The critical Hopf bifurcation in the case of linear utility becomes a non-critical bifurcation here. (For an analysis of Hopf bifurcation for integral equations, see Diekmann and van Gils [4]; see also Rustichini [ 111.) We conclude therefore that persistent oscillations in investment that are robust can occur with non-linear utility functions when we allow for some of “learning by using.” Such persistent oscillations in continuous time are different in nature from the multisector cycles obtained by Benhabib and Nishimura [2], which arise from factor intensity relationships in production. They represent another departure from the classical turnpike results studied by McKenzie [lo].

APPENDIX

1: PROOF OF THEOREM 3.1

Given the assumption of absolute continuity of p (i.e., dp(s) = m(s) d(s)), the proof is immediate by differentiation. More precisely, if we define as a perturbation of the Iro,h = X[to-h, ro+h,W and k,(t)=k(t)+~l~~,,, optimal path k, and then differentiate with respect to E the function of E given by fO+O”e -rs{f(s”,, k,(s+u)dp(u))-k,(s)} ds, we derive, at E=O, +*

r

epr”f’(K(.s))tj(t,h,s)ds-e-“(ep’h-e’h)=O s 0

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VINTAGE CAPITAL

for every t > 0, where we have defined 0

$(t,h,s)=

A-s+Ct-h,sl) i p(-s+[t-hh,

t+h])

s
or, after dividing by h and taking the limit as h + 0,

s+*

C’“f’(K(s))m(t-s)ds--ep”=O,

I

for every

t

2 0.

Note that if K(s) = C, a constant, then the above equation is satisfied if and only if CE~‘~‘(C(~)-‘) and C(P)=~“~ e’“&(s).

APPENDIX

2. THE DYNAMICS OF STOCKS AND FLOWS

Theorem 3.1 showed the connection between interior optimal paths and solutions to the integral equation (I). In this section, we establish conditions under which a unique solution to the integral equation (I) exists and provide a characterization of the solution (see Lemma A2.3 below). The characterization of the solution of the integral equation will allow us in Section 4 to study the dynamics of the optimal investment path under different assumptions of schedules for depreciation and under “learning by using.” DEFINITION.

3= 8EMf:J0 fj(s)dp(s)=C. -CL i I We now define the family of operators T,: M -+ M for every t 2 0, and s,: f+3: T,: it4 + M

for

every t > 0 is the optimal

solution

operator; S,: f + 7 for every t 2 0 is the integral equation operator. We assume for the moment that {S,] is well defined, i.e., for every 4 E 3 there exists a k E ii? such that j” o. k,(s) &L(s) = 0. We shall discuss later the conditions which ensure that {S,} is well defined for every t. Note that T, is well defined because a solution to the optimization problem exists and is unique iff is strictly concave (which we assume).

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Theorem 3.1 established that an interior optimal solution solves the integral equation (I). Lemma A2.1 will establish that a unique solution to the integral equation (I), along which consumption and (gross) investment remains non-negative, is an optimal solution.

LEMMAA2.1. If S,d satisfies (S,d)(O) b0 and f(j”oo S,&(O) 3 0 for t b 0, then T, = S, for every t > 0.

S,&s)

dp(s))-

The proof is immediate. If k is the optimal path with initial condition 4, then we define T,d = k,. Note that since the problem is stationary, T is a semigroup, that is, T,( T,$) = T,, ,$ for every t, s > 0. Also note that T is continuous in the Ls norm. Analogous statements hold for S,. We now introduce a subset of J: DEFINITION.

Note that j is non-empty because the constant function C (resealed by a scalar) is in it. Note f2 1. (Indeed j= {C} is a possibility, as we see in the “learning by using” Example 6 of Section 4. The existence of (continuous) solutions to the integral equation is easy to derive if we assume m(s) is an absolutely continuous Define now the problem

function.

C.42)

(D) by

k(t) = m(O)-’

j-”

,x

k( t + s) m’(s) ds,

t>O

k,=#EM.

(Dl) 032)

Note that the function k 1co,+ =,) which satisfies the condition automatically continuous.

(Dl ) is

LEMMA A2.2 Assume Al and A2. Then a function k E I%? which solves (D) with f”oc 4(s) dp(s) = C, also solves the integral equation (I).

Proof:

Let k E ii? be such that

1. kl to, + ooj is continuous 2.

k(t)=m(0)p’fO,

k(t+s)m’(s)ds

t>O;

k(O)=C.

If k(t)mJ”, k(t+s)m(s)d s we want to prove K(t) = C for t 2 0. Fix any T > 0 and consider a family { k”}E, o of functions in M such that

VINTAGE

331

CAPITAL

ki = $, k:,,, =, + k c0,T1 uniformly, k&, + ccI,E C “( R!+ ). These define K”(r) E j” m kE(t + S) m(s) ds. It suffices to show lim, _ 0 K”(t) = C for every t E [0, T]. Using condition 2 above, we have 3. k”(t)=m(O)-‘J”,k”(t+s)m’(s)ds+A(E,

t)

with A(&, .) +O as E -0, uniformly in t E [0, T]. (Note that A(&, t)= {k”(t)-k(t)+jO, [k(t+s)--k”(t+s)]m’(s)ds}m(O))’ satisfies that condition.) But, integrating by parts, we have -$-‘(t)=k’-m(0))ll’O

kE(t + s) m’(s) ds = A(&, t),

~%8

and our claim follows.

tE co, n

a

The system D in fact characterizes the solution of the integral equation, when m’ is bounded. In fact, LEMMA A2.3. Let k E I@, let k,E L”(K) satisfy I for every t, and let m’ E L”( lR’-). Then for every t 2 0, Dl and D2 hold. Also, if d(s) = 0 for s < - T, T < + co, then k is continuous at any t B 0.

Proof

From the integral equation, for any t 2 0 and h > 0, we have t+h

sf

k(s)m(s-t-h)ds 0

-I -50 k(s)[m(s

- t) - m(s - t - h)] ds = 0.

The first statement now follows from the Lebesgue differentiation theorem and the dominated convergence theorem. The continuity of k now follows from the fact that k satisfies (D), and continuity in Lp. l LEMMA A2.4. Assume Al, A2, m’ E L( [wp ). Then S,: J+ J” is well defined, i.e., for any #EM there exists a unique solution of the integral equation. Furthermore, S,d 1Co,r, is a continuous function.

Proof: From the equivalence condition of Lemma A2.3, it suffices to prove that there is a unique solution, continuous on [0, + co), to (D). The proof is now a standard application of a contraction mapping argument on 4 the space of continuous functions on R, In the rest of the paper we shall sometimes be interested in examples where the condition A2 is not satisfied. It is interesting therefore to record results similar to Lemmata A2.2 and A2.3 above in a somewhat weaker situation.

338

BENHABIB AND RUSTICHINI

Define the (A2’) condition

on M as:

m is piecewise absolutely continuous,

,.f,)’ ti+l <

m=CmjxCt,

ti

. <

m, is absolutely continuous

to

i.e.

= 0, where

(AZ’)

on [ tj+ 1, ti).

Then define k(t)=m(O)-’

fk(r+r (

,+ICmj+l(ti+l-)-mj(?i+,+)]

i

-k(t+t,)m,,(f,,+)-Jo

- Ix’

k(t+s)m’(s)ds),

CD’11

where the arguments t + (t - ) denote as usual limit from the right (left), and

The analogues of Lemmata A2.2 and A2.3 hold: LEMMA

A2.2’.

Let (Al ), (A2’) hold. Then a solution of (D’) is a solution

A2.3’.

A solution of

of I. LEMMA

(D’)

exists for

an initial

condition

4EC((--oO,Ol, W). Remark. A solution of the integral equation is not continuous, even for positive times, as the example m(s) = x~-,,~,(s), k(s) = {s-n: for se [n, n+ l)} shows.

It may be of interest to note that, due to the linearity of the problem, the sets 7 and j have a very simple form. We can describe it in the following lemma. LEMMA

2.

A25

1.

2, j are convex sets;

The restrictions of T, to 3 satisfies:

(and therefore so does S,, by the lemma.)

VINTAGE CAPITAL Proof:

If tii~.?, i= 1,2, then by definition

339

of j,

and so f3T,$, = (1 - 0) T,I)~ also satisfies the above inequality for every r. It follows that f3T,tj, + (1 - 0) TtlCIZ l j. By the uniqueness of the solution of the integral equation (I), the claim follows. i

REFERENCES BELLMAN AND K. L. COOKE, “Differential-Difference Equations,” Academic Press, New York/London, 1963. J. BENHABIB AND K. NISHIMURA,The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth, J. Econ. Theory 21 (1979), 421+&U. J. BENHABIB AND A. RUSTICHINI, A vintage capital model of investment and growth: Theory and evidence, in “Growth, General Equilibrium and International Trade II: The Legacy of Lionel McKenzie” (R. Becker, M. Boldrin, R. Jones, and W. Thompson, Eds.), Academic Press, San Diego, in press. 0. DIEKMANN AND S. A. VAN GILS, Invariant manifolds for Volterra integral equations of convolution type, J. Lhfferenfial Equafions 54 (1984), 139-180. N. MEHDIN, Optimal processes governed by integral equations, J. Math. Anal. Appl. 120 (1986), 1-12. N. MEHDIN, Optimal processes governed by integral equation with integral constraints, J. Math. Anal. Appl. 129 (1988), 269-283. T. MITRA AND H. WAN, JR., Some theoretical results on the economics of forestry, Rev. Econ. Bud. 52 (1985), 2633282. T. MITRA AND H. WAN, JR., On the Faustmann solution to the forest management problem, J. Econ. Theory 40 (1986), 229-249. T. MITRA, D. RAY, AND R. ROY, The economics of orchards: An exercise in point-intput, flow-output capital theory, J. Econ. Theory 53 (1991), 12-50. L. MCKENZIE, Optimal economic growth and turnpike theorems, in “Handbook of Mathematical Economics,” Vol. 11 (K. Arrow and M. Intriligator, Eds.), North-Holland, New York, 1986. A. RUSTICHINI,Hopf bifurcation for functional differential equations of the mixed type, J. Dynamics Differential Equations 1 (1989). 145-177.

1. R.

2. 3.

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