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On optimal growth under uncertainty
JOURNAL
OF ECONOMIC
THEORY
On Optimal
11, 329-339 (1975)
Growth
under
Uncertainty*
J. MIRMAN+
LEONARD
University of Illinois, Urbana, Illinois AND ITZHAK
ZILCHA
The Hebrew University of Jerusalem, Jerusalem, Israel, and the University of INinois, Urbana, Illinois Received April 30, 1975
In a recent paper Brock and Mirman showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process. This paper considers a similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. These conditionslead, in an example, to an explicit optimal policy function, which is used to display the steady-state solution for the capital stock under an optimal policy. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved by Brock and Mirman are satisfied even when the production functions are not ordered.
1. INTRODUCTION
In a recent paper Brock and Mirman [l] showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process.This paper considersa similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. The sufficient conditions for linear models were proved by Levhari and Srinivasan [2] (also see [5]). These conditions lead, in an example, to an explicit optimal policy function, which is used to display the steady-state * The authors would like to thank Prof. B. Peleg for his helpful comments. + Partially supported by the NSF Grant GS-05317.
329 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
330
MIRMAN
AND
ZILCHA
distribution for the capital stock under an optimal policy. Although the existence and uniqueness of this stationary distribution, is known, no explicit stationary distribution for optimal policies has been reported in the literature. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved in [l] are satisfied even when the production functions are not ordered. Economically, this allows the study of economic systems for which production processes that are superior at one level of capital (or capitallabor ratio) are inferior at other levels. For example, consider an agricultural economy for which the random variable is rain. For low capital stock, more rain is better than less rain. For high capital stocks, it is possible to put in an irrigation system, in which case more rain might be inferior to less rain. In any case, even in this regime there will exist a unique steady state under an optimal policy function. However, as is shown by example, it might turn out that the steady state is a degenerate distribution, i.e., all randomness disappears in the long run.
2. THE
MODEL
The model of economic growth under uncertainty in this paper is similar to the model of Brock and Mirman [l]; hence, only the essential features of the model will be discussed. Consider a central planner who has complete information about the utility function of society extending into the infinite future. Moreover, the central planner knows all technological possibilities, in the sense that if productive relationships are random, then the distribution of possibilities is known. The problem of the planner is to divide the available good at the beginning of each period t, between consumption ct and investment xt for that period. The decision is made in the light of future uncertainty, where the cirterion is the maximization of the expected value of the discounted sum of utilities with an indefinite future. It is assumed that the past values of the random production process as well as the present or past capital stock do not affect the present or future probabilities. Hence given the structure of the random process, i.e., the utility function, the production function, and the probability distributions, only the current output is necessary for an optimal decision. Let (a, 9, P) be the probability space on which the random variable r is defined. Formally, let s > 0 be the inititial stock and f(x, r) be the production function.
ON OPTIMAL
GROWTH UNDER UNCERTAINTY
331
Then cg + xg = s, ct + xt = f(xt-,
3rt-1)
t = 1, 2,....
(1)
Here ct and xt are consumption and capital stock at period t, respectively, and rt is the random variable affecting the production processat time t. The properties of the function f(x, r) are the usual neoclassical properties. Namely, (I) f(* , r) is an increasing, concave differentiable function for all r withf(0, r) = 0 for all r. For simplicity the Tnada conditions are assumed,namely (II)
f’(0, r) = co,f’(co, r) = 0 for all r. (Heref’(x,
r) is Zf(x, r)/h).
The utility function u(c) will also be endowed with the usual properties, i.e., (III) II is an increasing concave and differentiable function with u(0) = 0. (IV)
u’(0) = co.
DEFINITION. Let x = (x,, , x1 ,... ), c = (co, c1 ,... ). Here xt and ct satisfy (1) for t = 0, I,.. .. The pair (x, c) is said to be a feasible program from initial stock S. The mapping r: Q + [0, co) generates a measure on the Bore1 subsets .9? of the real line given by v(B) = P{r E B} for all BE 97’. Moreover it is also assumed that the set (7 /f(x, 7) is strictly concave} has positive measure.
3. PROPERTIES OF THE MODEL
Define U(s) = sup 1 %Eu(c,). c t=n where the supremum is taken over all c for which (x, c) is a feasible program from initial stock s and 6 is the discount factor with 0 < 6 < 1. By assumptions(I) and (II) it is easy to verify that U(s) is well-defined, i.e., 03 > U(s) > 0 for all s > 0 , and is concave Using a dynamic programming argument,
U(s) = oyfys . . {u(c) + ~EU[f(s - c, r)lI.
(2)
MIRMAN
332
AND ZILCHA
The maximum is obtained at a unique point c = g(s) since the maximand is strictly concave, moreover 0 < g(s) < s for s > 0 (since u’(0) = Cc). The optimal consumption policy function g(s) is defined on (0, cc) by equation (2). It is easy to verify that if (E, E) is determined by g(s), i.e., co = g(s), x0 = s - 2, , Xt = f(Ftel , rtel) - Ct where Ct = g(f(& , rdpl)) then U(s) = f FEu(i;t). t=o
The optimal LEMMA
program (E, P) from initial stock is a feasible program. 1.
U’(s) exists for all s > 0 and U’(s) = u’(g(s)).
Proof. Let [ > 0 and E the optimal consumption plan from initial stock 5 determined by g(s), the optimal policy function (O.P.F.). Define a feasible consumption program from [ + At where de > 0 as follows: co = Z. + 05, ct = Ct for t > 1. By definition of U it follows that
U(t + &?I - Wt) 3 F @El44 - 4G)l = 4co + 06) - 4.4 t=o
= u’(Co)A[ + o(Af). Therefore U+‘(t) > u’(Eo) = u’( g(t)). Let 0 < Of < E, . Define a feasible program from f - 05 as follows: co = co - 06 and Ct = ct for t 3 1. Using the same argument as above ut1(5- A0 -
u(E) 2 -u’(c,)
A5 + oW>.
Hence, U-‘(e) < u’(c,) = u’(g(5)). Therefore, U’(!9 = +m~ Denote by Ft+dd = df(xt
for all [ > 0.
, 7)) = (Et+1I rt = d
THEOREM 1. Let (E, E) be a feasible program determined by a function g(x). Necessary and suficient conditions that g(x) be an optimal policy function are:
(a) u’(G) = s .ff’(% ,v> u’(G+drl)) 4dd t = 0, I,... E{SW(Z,) Xt} --f 0 as t + co.
(b)
Sketch of the Proof.
The proof of sufficiency is similar to that in [2].
ON
OPTIMAL
GROWTH
UNDER
333
UNCERTAINTY
To prove the necessity of (a), differentiate the maximand in (2) with respect to c which is possible since CJis differentiable. Using Lemma 1,
u’(gW> = 6 s-fYx - g(x), 4 u’W(x - g(x), ?))I Gd
(3)
which yields (b). To show that the transversality condition (b) is true when (Z, E) is optimal, consider the case where the Ct are not bounded away from 0 (otherwise the assertion is trivial since Xt are bounded from above). For Xt sufficiently small f(Zt, 7) > Xt for all 71. U(x) is concave hence U(x)/x > U’(x) or U(x) 2 u’(g(x)) x, therefore K8 3 StU(XJ Xt > &i’(g(x,))
X6 .
Hence SW(&) xt -++m 0. Note that the sufficiency does not depend on the assumption u(O) = 0. EXAMPLE A. Let f(x, p) = apxp, O
a,>0
for
Define ,5 = Jt pv(dp). Then, the solution is h = 1 - 6p. Hence (1 - 6p) x is an O.P.F. since, as we will show, condition (b) is satisfied. It is worthwhile to remark that the O.P.F., (1 - 6p) x is independent of a, , CL< p < /3. It is not difficult to see that U(x) = (l/(1 -
6p)) In x.
To show that Zt are bounded away from 0, let us assume for simplicity that a0 = a > 0, for all p. Since on [0, 11, ax0 3 ax6 for all p, 01 < p d /3 it is enough to show that Xt are bounded away from 0 when rt = /3, t = 1, 2 ,.... In this case, x,= (1 - h)s x1 = (1 - h)f(Z, ) fi) = (1 - h)‘+“f(s, jq xt = (1 - h)f(X,_, ) /I) = (1 - X)1+~+~2+..~+qqs, /3) (ft(s, /3) means applyingf(*
, p) on s, t times).
334
MIRMAN AND ZILCHA
Hence Et -ftern [(l - A) ~]ll(l-~) > 0. Therefore, {CJ are uniformly bounded from 0 for all realizations of r, and hence, condition (b) is satisfied.
4. ON THE STOCHASTIC STEADY STATE
In this section we shall see that the results of Brock-Mirman [I] about the existence and uniqueness of the steady state remains true in the model in which the production function f(x, 7) is not a monotone function of 7. The assumption that the f(x, *) functions are ordered seems to be very restrictive, since it is possible that production processes that are superior at one level of the capital-labor ratio are inferior at other levels (e.g., the agricultural example stated in the Introduction). Define
Let h(x) = x -g(x) and N(x, r) = h(f(x, r)). Let H,,(x) = min, H(x, q), and HM(x) = max, H(x, q). Since h(x) is monotone increasing and continuous we have
Knw = wiLw>
and
HIM(x) = Kh4w
Define -~?n= max{x I H,(x)
= x},
X~ = min{x j H,(x)
= x}.
We first show that x,,, < x~, hence the interval [x, , x~] is the stable interval (see [3]) of the stochastic process Xt+, = H(Xt , r,). LEMMA
2. Let a = H,(a) and b = H&b) then a < b.
Proof. Define d(x, p) = u’[g(f(x, p))]. The decreasing for all p. Since H(x, v) is continuous (in both arguments),
function
d( , p) is
a = H,(a) = H(a, pm) for some pnLin [01,/?I, and b = H&b) = H(b, pM) for some p,+, in (cx,/3]. By (3),
ON OPTIMAL
GROWTH UNDER UNCERTAINTY
335
Hence
and 46 PM) = 6 Jm
7) a,
7) ddd.
(6)
Since f(a, pill) < f(a, v), for all 77, we see that d(a, pm) 3 n(u, T), for all 7. Similarly d(b, p&l < @, 7) for all 7. Therefore from (5), 1 < 6 J f’(a, 7) v(dr)) and from (6), 1 > 6 J f’(b, 7) u(d~). Since is a J”fk 77)477) is . a strictly concave function of X, J f’(x, 7) I strictly decreasingfunction; hence, b > a. Lemma 2 rules out the situation in Fig. 1 and all other similar cases.We will see that generally, the same proof given by Brock and Mirman in [l] proves the existence and uniqueness of the stochastic modified golden-rule without the assumption that the functions f(x, .) are ordered. We prove in the Appendix that Lemma 4.2 in [l] remains true even when f(x, 7) is not an increasing function of v. Obviousiy the proofs of Lemma 4.3 (see remark in the Appendix) and [l, Theorem 4.11 do not use this assumption. Therefore, by making the proper changes in the proof of Lemma 4.2, we come to the sameresults. Denote by F,(x) the distribution functions of the random variables Xt t = 0, I,....
FIGURE
1
THEOREM 2. There exists a distribution function F(x) such that F,(x) + F(x) uniformly for all x. Furthermore, F(x) does not depend on the initial stock s.
In the next two examples we compute the stochastic steady state using the result of Example A. EXAMPLE B. Suppose the random variable r assumes two values, say 1 and 2, with probability 4 and +, respectively. Let f(x, 1) = 2x112
336
MIRMAN
AND
ZILCHA
and f(x, 2) = 4xY3 and 6 = 6. Further, assume u(c) = In c. It was proven in Example A that the O.P.F. g(x) is linear and is given by g(x) = { 1 - $[i * g + + * $I} x = $x. Therefore h(x) = 3x, hence H(x, 1) =$x1/z and H(x, 2) = &G/3. Since f(x, 1) = f(x, 2) at x = 26, then H(x, 1) = H(x, 2) at x = 26 (see Fig. 2). In this case all uncertainty disappears in the long-run steady state, i.e., F,(x) + F(x) where
F(x)= I; This is illustrated
x < 26 x326 .
in Fig. 2.
f,H
x
FIGURE
2
EXAMPLE C. Let the random variable Y assume two values, say 1 and 2, with probability 4 each. Let u(c) = In c and 6 = &. The production functions are f(x, 1) = x112 and f(x, 2) = 2x1i2. Using the result of Example A, we find that the O.P.F. is g(x) = Ix and h(x) = ix. Hence H(x 1) = $x1l2 and H(x, 2) = +x1i2. Their inverse functions are g(x,‘l) = 16x2 and q(x, 2) = 4x2. The fixed points of H(x, 1) and H(x, 2) are IL6 and 2. Let Fk(x) be the probability distribution function of Xl,. Denote by 0~~and 01~the probabilities that r may assume the values 1 and 2, respectively. Then
I’$(x) = P{xk < x} = i
P{H(:,-,
, i) < x> ai
i=l
= i i=l
Pcz&-l < q(x, i)} OIi = 5 E;;cel(q(x, i)) cY(. i=l
ON
OPTIMAL
GROWTH
UNDER
UNCERTAINTY
337
Hence, Fdx) = ; po((4~)~)
+ Fo (; (4d2)]
F,(x) = $ [Fo(42(4x)4) + F,,(4(4~)3
By induction,
+ Fo ((4~)~) + Fo ($ (4~)~)] .
we see that F,(x)
= & ‘zl F,[42”-i(4x)2”]. a=2
We will prove that .F,(x) converges to a distribution function F(x). Let b = log,(s/4), then 42”-i(4x) 2n > 4b if 2 + log, x 3 (i + b)/2”. Hence . = F(x) =
0 2 + log, x
= 11
x < 1% I.. < x d t x>*
F(x) is the modified stochastic golden-rule for this example.
APPENDIX
In [l], Brock and Mirman assumed that H(x, a) has some positive fixed points (finite or infinite in number) ignoring the possibility, as shown in [4], that H(x, CX)may have no positive fixed point. It is not difficult to observe that their proof remains true even when H(x, CX)< x for all x > 0. The proof that for any 01 -=cq < /3 there is an E, > 0 such that for all x E (0, E,,), H(x, q) > x is correct. Now take a sequence x, decreasing to 0. Define A, = [x, , xMB] (see [l]). The same proof given in [I, Lemma 4.21 remains true for the sets A, , i.e., A, are transient sets with respect to the “inverse” process, q(x, v). Since for all 7 > 01, q(x, 7) < x near 0, for each E > 0 there exists an N such that for all t > T, and all n > N, Prob{E, E An} < E. The set [xm , x~] (see Fig. 3) is the stable interval of the stochastic process Et+i = H(Xt , rt) (see [3]). Hence in the case that x, = X~ , it is not difficult to see that this process “converges” to the point x,,, = X~ . (Mirman [3] shows that the support of the stationary measure is the stable interval). Therefore Theorem 2 is clear for this case. Let us assume that x,,, -=cx,+, . Define ynz = min{x 1H,(x)
= x}
and
Let us prove [I, Lemma 4.21 for our model.
yM = max{x 1HM(x) = x}.
338
MIRMAN
AND ZILCHA
q,(x)
xM
FIGURE
Y
3
LEMMA. Let A = [ y,m, y,]. Then A is a transient setfor the stochastic process 32’,= q(f,-, , rt), 9, = X. (This processis defined in [l].)
Proof. Let a = (x, + xJ/2, (see Fig. 3), X, < a < X~ . q&x) the inverse of HM(x) (which does exist). Define El =
min Y,
is
x - 4W(x) 2 .
Let B = (7 I dx, 7) < x - q, yrn G x < 4. Since q(. , 7) is continuous and qM(x) < x on (0, x,) we see that v(B) = A, > 0 (it has been assumedthat v([p, fl]) > 0 and Y([(Y.,p]) > 0 for all 01< p < 13). Making the same arguments as in [I, Lemma 4.21, there is a constant integer N such that for all n 3 N, q”(x, q) < ym for allTEBandxE[y,, a]. Hence for all x in [ ym , a] with probability AIN, at least, the process leaves [ ynL, a] and never returns, i.e., [ y, , a] is a transient set. Let qm(x) be the inverse of H,(x). Let E2 =
min q7R(x) U
x
.
Define c = {rl I 4(x, 7) 2 x +
E2
3
a < x d YMM).
By continuity of q(* , 9) we seethat V(C) = A, > 0. Therefore, there is a constant integer K such that for all n > K, qn(x, q) > y,+, for all 7 E C and x E [a, ~~1. Hence for all x in [a, yw] with probability X2K,at least, the process leaves the interval [a, y,,,] and never returns. Therefore the expected number of visits to A of any x in A is finite. This proves the lemma.
ON
OPTIMAL
GROWTH
UNDER
UNCERTAINTY
339
REFERENCES 1. W. A. BROCK AND L. J. MIRMAN, Optimal economic growth and uncertainty: The discounted case, J. Econ. Theory 4 (1972). 2. D. LEVHARI AND T. N. SRINIVASAN, Optimal saving under uncertainty, Rev. Econ. Srud. 36 (1969). 3. L. J. MIRMAN, The steady state behavior of a class of one-sector growth models with uncertain technology, J. Econ. Theory 5 (1973). 4. L. J. MIRMAN AND I. ZILCHA, Unbounded shadow prices for optimal stochastic growth models, Znt. Econ. Rev., to be printed. 5. J. A. MIRRLEES, Optimal growth and uncertainty, Nuffield College, Oxford, 1971.
OF ECONOMIC
THEORY
On Optimal
11, 329-339 (1975)
Growth
under
Uncertainty*
J. MIRMAN+
LEONARD
University of Illinois, Urbana, Illinois AND ITZHAK
ZILCHA
The Hebrew University of Jerusalem, Jerusalem, Israel, and the University of INinois, Urbana, Illinois Received April 30, 1975
In a recent paper Brock and Mirman showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process. This paper considers a similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. These conditionslead, in an example, to an explicit optimal policy function, which is used to display the steady-state solution for the capital stock under an optimal policy. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved by Brock and Mirman are satisfied even when the production functions are not ordered.
1. INTRODUCTION
In a recent paper Brock and Mirman [l] showed that in a one-sector model of economic growth under uncertainty the long-run behavior of the optimal capital stock is governed by the basic properties of an acyclic ergodic Markov process.This paper considersa similar model and has two purposes. First, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted. The sufficient conditions for linear models were proved by Levhari and Srinivasan [2] (also see [5]). These conditions lead, in an example, to an explicit optimal policy function, which is used to display the steady-state * The authors would like to thank Prof. B. Peleg for his helpful comments. + Partially supported by the NSF Grant GS-05317.
329 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
330
MIRMAN
AND
ZILCHA
distribution for the capital stock under an optimal policy. Although the existence and uniqueness of this stationary distribution, is known, no explicit stationary distribution for optimal policies has been reported in the literature. Secondly, in the Brock and Mirman paper it was assumed that the production functions are ordered. We show that all the properties proved in [l] are satisfied even when the production functions are not ordered. Economically, this allows the study of economic systems for which production processes that are superior at one level of capital (or capitallabor ratio) are inferior at other levels. For example, consider an agricultural economy for which the random variable is rain. For low capital stock, more rain is better than less rain. For high capital stocks, it is possible to put in an irrigation system, in which case more rain might be inferior to less rain. In any case, even in this regime there will exist a unique steady state under an optimal policy function. However, as is shown by example, it might turn out that the steady state is a degenerate distribution, i.e., all randomness disappears in the long run.
2. THE
MODEL
The model of economic growth under uncertainty in this paper is similar to the model of Brock and Mirman [l]; hence, only the essential features of the model will be discussed. Consider a central planner who has complete information about the utility function of society extending into the infinite future. Moreover, the central planner knows all technological possibilities, in the sense that if productive relationships are random, then the distribution of possibilities is known. The problem of the planner is to divide the available good at the beginning of each period t, between consumption ct and investment xt for that period. The decision is made in the light of future uncertainty, where the cirterion is the maximization of the expected value of the discounted sum of utilities with an indefinite future. It is assumed that the past values of the random production process as well as the present or past capital stock do not affect the present or future probabilities. Hence given the structure of the random process, i.e., the utility function, the production function, and the probability distributions, only the current output is necessary for an optimal decision. Let (a, 9, P) be the probability space on which the random variable r is defined. Formally, let s > 0 be the inititial stock and f(x, r) be the production function.
ON OPTIMAL
GROWTH UNDER UNCERTAINTY
331
Then cg + xg = s, ct + xt = f(xt-,
3rt-1)
t = 1, 2,....
(1)
Here ct and xt are consumption and capital stock at period t, respectively, and rt is the random variable affecting the production processat time t. The properties of the function f(x, r) are the usual neoclassical properties. Namely, (I) f(* , r) is an increasing, concave differentiable function for all r withf(0, r) = 0 for all r. For simplicity the Tnada conditions are assumed,namely (II)
f’(0, r) = co,f’(co, r) = 0 for all r. (Heref’(x,
r) is Zf(x, r)/h).
The utility function u(c) will also be endowed with the usual properties, i.e., (III) II is an increasing concave and differentiable function with u(0) = 0. (IV)
u’(0) = co.
DEFINITION. Let x = (x,, , x1 ,... ), c = (co, c1 ,... ). Here xt and ct satisfy (1) for t = 0, I,.. .. The pair (x, c) is said to be a feasible program from initial stock S. The mapping r: Q + [0, co) generates a measure on the Bore1 subsets .9? of the real line given by v(B) = P{r E B} for all BE 97’. Moreover it is also assumed that the set (7 /f(x, 7) is strictly concave} has positive measure.
3. PROPERTIES OF THE MODEL
Define U(s) = sup 1 %Eu(c,). c t=n where the supremum is taken over all c for which (x, c) is a feasible program from initial stock s and 6 is the discount factor with 0 < 6 < 1. By assumptions(I) and (II) it is easy to verify that U(s) is well-defined, i.e., 03 > U(s) > 0 for all s > 0 , and is concave Using a dynamic programming argument,
U(s) = oyfys . . {u(c) + ~EU[f(s - c, r)lI.
(2)
MIRMAN
332
AND ZILCHA
The maximum is obtained at a unique point c = g(s) since the maximand is strictly concave, moreover 0 < g(s) < s for s > 0 (since u’(0) = Cc). The optimal consumption policy function g(s) is defined on (0, cc) by equation (2). It is easy to verify that if (E, E) is determined by g(s), i.e., co = g(s), x0 = s - 2, , Xt = f(Ftel , rtel) - Ct where Ct = g(f(& , rdpl)) then U(s) = f FEu(i;t). t=o
The optimal LEMMA
program (E, P) from initial stock is a feasible program. 1.
U’(s) exists for all s > 0 and U’(s) = u’(g(s)).
Proof. Let [ > 0 and E the optimal consumption plan from initial stock 5 determined by g(s), the optimal policy function (O.P.F.). Define a feasible consumption program from [ + At where de > 0 as follows: co = Z. + 05, ct = Ct for t > 1. By definition of U it follows that
U(t + &?I - Wt) 3 F @El44 - 4G)l = 4co + 06) - 4.4 t=o
= u’(Co)A[ + o(Af). Therefore U+‘(t) > u’(Eo) = u’( g(t)). Let 0 < Of < E, . Define a feasible program from f - 05 as follows: co = co - 06 and Ct = ct for t 3 1. Using the same argument as above ut1(5- A0 -
u(E) 2 -u’(c,)
A5 + oW>.
Hence, U-‘(e) < u’(c,) = u’(g(5)). Therefore, U’(!9 = +m~ Denote by Ft+dd = df(xt
for all [ > 0.
, 7)) = (Et+1I rt = d
THEOREM 1. Let (E, E) be a feasible program determined by a function g(x). Necessary and suficient conditions that g(x) be an optimal policy function are:
(a) u’(G) = s .ff’(% ,v> u’(G+drl)) 4dd t = 0, I,... E{SW(Z,) Xt} --f 0 as t + co.
(b)
Sketch of the Proof.
The proof of sufficiency is similar to that in [2].
ON
OPTIMAL
GROWTH
UNDER
333
UNCERTAINTY
To prove the necessity of (a), differentiate the maximand in (2) with respect to c which is possible since CJis differentiable. Using Lemma 1,
u’(gW> = 6 s-fYx - g(x), 4 u’W(x - g(x), ?))I Gd
(3)
which yields (b). To show that the transversality condition (b) is true when (Z, E) is optimal, consider the case where the Ct are not bounded away from 0 (otherwise the assertion is trivial since Xt are bounded from above). For Xt sufficiently small f(Zt, 7) > Xt for all 71. U(x) is concave hence U(x)/x > U’(x) or U(x) 2 u’(g(x)) x, therefore K8 3 StU(XJ Xt > &i’(g(x,))
X6 .
Hence SW(&) xt -++m 0. Note that the sufficiency does not depend on the assumption u(O) = 0. EXAMPLE A. Let f(x, p) = apxp, O
a,>0
for
Define ,5 = Jt pv(dp). Then, the solution is h = 1 - 6p. Hence (1 - 6p) x is an O.P.F. since, as we will show, condition (b) is satisfied. It is worthwhile to remark that the O.P.F., (1 - 6p) x is independent of a, , CL< p < /3. It is not difficult to see that U(x) = (l/(1 -
6p)) In x.
To show that Zt are bounded away from 0, let us assume for simplicity that a0 = a > 0, for all p. Since on [0, 11, ax0 3 ax6 for all p, 01 < p d /3 it is enough to show that Xt are bounded away from 0 when rt = /3, t = 1, 2 ,.... In this case, x,= (1 - h)s x1 = (1 - h)f(Z, ) fi) = (1 - h)‘+“f(s, jq xt = (1 - h)f(X,_, ) /I) = (1 - X)1+~+~2+..~+qqs, /3) (ft(s, /3) means applyingf(*
, p) on s, t times).
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Hence Et -ftern [(l - A) ~]ll(l-~) > 0. Therefore, {CJ are uniformly bounded from 0 for all realizations of r, and hence, condition (b) is satisfied.
4. ON THE STOCHASTIC STEADY STATE
In this section we shall see that the results of Brock-Mirman [I] about the existence and uniqueness of the steady state remains true in the model in which the production function f(x, 7) is not a monotone function of 7. The assumption that the f(x, *) functions are ordered seems to be very restrictive, since it is possible that production processes that are superior at one level of the capital-labor ratio are inferior at other levels (e.g., the agricultural example stated in the Introduction). Define
Let h(x) = x -g(x) and N(x, r) = h(f(x, r)). Let H,,(x) = min, H(x, q), and HM(x) = max, H(x, q). Since h(x) is monotone increasing and continuous we have
Knw = wiLw>
and
HIM(x) = Kh4w
Define -~?n= max{x I H,(x)
= x},
X~ = min{x j H,(x)
= x}.
We first show that x,,, < x~, hence the interval [x, , x~] is the stable interval (see [3]) of the stochastic process Xt+, = H(Xt , r,). LEMMA
2. Let a = H,(a) and b = H&b) then a < b.
Proof. Define d(x, p) = u’[g(f(x, p))]. The decreasing for all p. Since H(x, v) is continuous (in both arguments),
function
d( , p) is
a = H,(a) = H(a, pm) for some pnLin [01,/?I, and b = H&b) = H(b, pM) for some p,+, in (cx,/3]. By (3),
ON OPTIMAL
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335
Hence
and 46 PM) = 6 Jm
7) a,
7) ddd.
(6)
Since f(a, pill) < f(a, v), for all 77, we see that d(a, pm) 3 n(u, T), for all 7. Similarly d(b, p&l < @, 7) for all 7. Therefore from (5), 1 < 6 J f’(a, 7) v(dr)) and from (6), 1 > 6 J f’(b, 7) u(d~). Since is a J”fk 77)477) is . a strictly concave function of X, J f’(x, 7) I strictly decreasingfunction; hence, b > a. Lemma 2 rules out the situation in Fig. 1 and all other similar cases.We will see that generally, the same proof given by Brock and Mirman in [l] proves the existence and uniqueness of the stochastic modified golden-rule without the assumption that the functions f(x, .) are ordered. We prove in the Appendix that Lemma 4.2 in [l] remains true even when f(x, 7) is not an increasing function of v. Obviousiy the proofs of Lemma 4.3 (see remark in the Appendix) and [l, Theorem 4.11 do not use this assumption. Therefore, by making the proper changes in the proof of Lemma 4.2, we come to the sameresults. Denote by F,(x) the distribution functions of the random variables Xt t = 0, I,....
FIGURE
1
THEOREM 2. There exists a distribution function F(x) such that F,(x) + F(x) uniformly for all x. Furthermore, F(x) does not depend on the initial stock s.
In the next two examples we compute the stochastic steady state using the result of Example A. EXAMPLE B. Suppose the random variable r assumes two values, say 1 and 2, with probability 4 and +, respectively. Let f(x, 1) = 2x112
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AND
ZILCHA
and f(x, 2) = 4xY3 and 6 = 6. Further, assume u(c) = In c. It was proven in Example A that the O.P.F. g(x) is linear and is given by g(x) = { 1 - $[i * g + + * $I} x = $x. Therefore h(x) = 3x, hence H(x, 1) =$x1/z and H(x, 2) = &G/3. Since f(x, 1) = f(x, 2) at x = 26, then H(x, 1) = H(x, 2) at x = 26 (see Fig. 2). In this case all uncertainty disappears in the long-run steady state, i.e., F,(x) + F(x) where
F(x)= I; This is illustrated
x < 26 x326 .
in Fig. 2.
f,H
x
FIGURE
2
EXAMPLE C. Let the random variable Y assume two values, say 1 and 2, with probability 4 each. Let u(c) = In c and 6 = &. The production functions are f(x, 1) = x112 and f(x, 2) = 2x1i2. Using the result of Example A, we find that the O.P.F. is g(x) = Ix and h(x) = ix. Hence H(x 1) = $x1l2 and H(x, 2) = +x1i2. Their inverse functions are g(x,‘l) = 16x2 and q(x, 2) = 4x2. The fixed points of H(x, 1) and H(x, 2) are IL6 and 2. Let Fk(x) be the probability distribution function of Xl,. Denote by 0~~and 01~the probabilities that r may assume the values 1 and 2, respectively. Then
I’$(x) = P{xk < x} = i
P{H(:,-,
, i) < x> ai
i=l
= i i=l
Pcz&-l < q(x, i)} OIi = 5 E;;cel(q(x, i)) cY(. i=l
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Hence, Fdx) = ; po((4~)~)
+ Fo (; (4d2)]
F,(x) = $ [Fo(42(4x)4) + F,,(4(4~)3
By induction,
+ Fo ((4~)~) + Fo ($ (4~)~)] .
we see that F,(x)
= & ‘zl F,[42”-i(4x)2”]. a=2
We will prove that .F,(x) converges to a distribution function F(x). Let b = log,(s/4), then 42”-i(4x) 2n > 4b if 2 + log, x 3 (i + b)/2”. Hence . = F(x) =
0 2 + log, x
= 11
x < 1% I.. < x d t x>*
F(x) is the modified stochastic golden-rule for this example.
APPENDIX
In [l], Brock and Mirman assumed that H(x, a) has some positive fixed points (finite or infinite in number) ignoring the possibility, as shown in [4], that H(x, CX)may have no positive fixed point. It is not difficult to observe that their proof remains true even when H(x, CX)< x for all x > 0. The proof that for any 01 -=cq < /3 there is an E, > 0 such that for all x E (0, E,,), H(x, q) > x is correct. Now take a sequence x, decreasing to 0. Define A, = [x, , xMB] (see [l]). The same proof given in [I, Lemma 4.21 remains true for the sets A, , i.e., A, are transient sets with respect to the “inverse” process, q(x, v). Since for all 7 > 01, q(x, 7) < x near 0, for each E > 0 there exists an N such that for all t > T, and all n > N, Prob{E, E An} < E. The set [xm , x~] (see Fig. 3) is the stable interval of the stochastic process Et+i = H(Xt , rt) (see [3]). Hence in the case that x, = X~ , it is not difficult to see that this process “converges” to the point x,,, = X~ . (Mirman [3] shows that the support of the stationary measure is the stable interval). Therefore Theorem 2 is clear for this case. Let us assume that x,,, -=cx,+, . Define ynz = min{x 1H,(x)
= x}
and
Let us prove [I, Lemma 4.21 for our model.
yM = max{x 1HM(x) = x}.
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AND ZILCHA
q,(x)
xM
FIGURE
Y
3
LEMMA. Let A = [ y,m, y,]. Then A is a transient setfor the stochastic process 32’,= q(f,-, , rt), 9, = X. (This processis defined in [l].)
Proof. Let a = (x, + xJ/2, (see Fig. 3), X, < a < X~ . q&x) the inverse of HM(x) (which does exist). Define El =
min Y,
is
x - 4W(x) 2 .
Let B = (7 I dx, 7) < x - q, yrn G x < 4. Since q(. , 7) is continuous and qM(x) < x on (0, x,) we see that v(B) = A, > 0 (it has been assumedthat v([p, fl]) > 0 and Y([(Y.,p]) > 0 for all 01< p < 13). Making the same arguments as in [I, Lemma 4.21, there is a constant integer N such that for all n 3 N, q”(x, q) < ym for allTEBandxE[y,, a]. Hence for all x in [ ym , a] with probability AIN, at least, the process leaves [ ynL, a] and never returns, i.e., [ y, , a] is a transient set. Let qm(x) be the inverse of H,(x). Let E2 =
min q7R(x) U
x
.
Define c = {rl I 4(x, 7) 2 x +
E2
3
a < x d YMM).
By continuity of q(* , 9) we seethat V(C) = A, > 0. Therefore, there is a constant integer K such that for all n > K, qn(x, q) > y,+, for all 7 E C and x E [a, ~~1. Hence for all x in [a, yw] with probability X2K,at least, the process leaves the interval [a, y,,,] and never returns. Therefore the expected number of visits to A of any x in A is finite. This proves the lemma.
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REFERENCES 1. W. A. BROCK AND L. J. MIRMAN, Optimal economic growth and uncertainty: The discounted case, J. Econ. Theory 4 (1972). 2. D. LEVHARI AND T. N. SRINIVASAN, Optimal saving under uncertainty, Rev. Econ. Srud. 36 (1969). 3. L. J. MIRMAN, The steady state behavior of a class of one-sector growth models with uncertain technology, J. Econ. Theory 5 (1973). 4. L. J. MIRMAN AND I. ZILCHA, Unbounded shadow prices for optimal stochastic growth models, Znt. Econ. Rev., to be printed. 5. J. A. MIRRLEES, Optimal growth and uncertainty, Nuffield College, Oxford, 1971.