The Review of Economic Studies, Ltd.
Asymptotic Growth under Uncertainty: Existence and Uniqueness Author(s): Fwu-Ranq Chang and A. G. Malliaris Reviewed work(s): Source: The Review of Economic Studies, Vol. 54, No. 1 (Jan., 1987), pp. 169-174 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/2297452 . Accessed: 25/06/2012 13:21 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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Review of Economic Studies (1987) LIV, 169-174 ? 1987 The Society for Economic Analysis Limited
Asymptotic
0034-6527/87/00110169$02.00
Growth
under
Uncertainty:
Existence
and
Uniqueness
FWU-RANQ CHANG Indiana University and A. G. MALLIARIS Loyola University of Chicago First version received July 1985; final version accepted June 1986 (eds.) This paper demonstrates, using the Reflection Principle, the existence and uniqueness of the solution to the classic Solow equation under continuous time uncertainty for the class of strictly concave production functions which are continuously differentiable on the nonnegative real numbers. This class contains all CES functions with elasticity of substitution less than unity. A steady state distribution also exists for this class of production functions which have a bounded slope at the origin. A condition on the drift-variance ratio of the stochastic differential equation alone, independent of technology and the savings ratio, is found to be necessary for the existence of a steady state.
1. INTRODUCTION Stochastic calculus techniques have been used to study continuous time economic growth under uncertainty. For example, Bourguignon (1974) and Merton (1975) have, extended the neoclassical model of economic growth developed by Solow (1956) to incorporate uncertainty in a continuous time framework by extensive use of Ito's lemma. However, the existence and uniqueness of the solution to the stochastic equation of evolution in the proposed models have been assumed without proof. In addition, Merton proves the existence of a steady state distribution for the class of production functions satisfying the Inada conditions. While it is important to include Cobb-Douglas production functions in the analysis, the Inada conditions unfortunately rule out all other CES production functions (Wan (1971, p. 39)). This paper fills the gap in the literature by proving the existence and uniqueness of the solution to the stochastic Solow equation for the class of strictly concave production functions which are continuously differentiable on the semi-closed interval of non-negative real numbers. This class of functions contains all CES production functions with elasticity of substitution less than unity. A steady state distribution also exists for this class of production functions which have a bounded slope at the origin. This result complements Merton's theorem on the existence of a steady state with Inada-type production functions. The existence and uniqueness gap in the literature is partly attributed to the fact that Ito's theorem on the existence and uniqueness of the solution to a given stochastic differential equation cannot be directly applied because the stochastic Solow equation has a non-negativity constraint on the capital-labour ratio. The theorem only applies to 169
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stochastic processes with state space the whole real line. The technique presented in this paper is to extend the original equation from the non-negative real numbers to the whole real line by "reflection". Such an extension is akin to the Reflection Principle in complex analysis, see, for example, Ahlfors (1966). The production function is extended to a new function which is defined on the whole real line by assigning the value of the mirror image as the mirror image of the value, where the mirror is the y-axis. The advantage of this reflection is that both the drift and the variance are odd functions of the capital-labour ratio. Then, the differentiability of the original function is extended if the production function is continuously differentiable at the origin. Hence, the existence and uniqueness theorems for equations with continuously differentiable coefficients can be applied. Note that the condition f'(O) < oo is needed for reflection. Then we proceed to verify that this solution is the desired one by showing that capitallabour ratios never change signs. Specifically, if the initial condition is positive, then the solution to the equation is always positive. The conditions we impose on the production function f(k) are rather standard: f is strictly concave, continuously differentiable on the nonnegative real line, f(O) = 0 and f'(oc) = 0. It is known that the solution is uniquely defined up to a Brownian stopping time (i.e. explosion time). Based on the nature of the derived equation, we show that, with probability one, the explosion time is +oo. The solution technique employed in this paper rules out the possibility of k, = 0 as an accessible barrier. In the proof of our main theorem, a condition on the drift-variance ratio of the stochastic differential equation alone, independent of technology and the savings ratio, is found to be necessary for the existence of a steady state distribution. This result further complements Merton's finding. This paper is organized as follows. We summarize mathematical results needed for this analysis in Section 2. Then we restate Merton's derived stochastic Solow equation, describe our reflection technique and state and prove our main theorem in Section 3. Concluding remarks are given in Section 4. 2. MATHEMATICAL PRELIMINARIES For convenience, we shall list the basic definitions and theorems from stochastic calculus employed in this paper. Given a probability space (Q, F, P), let {zj} be a Wiener process and let {FJ} be the filtration generated by the Wiener process {zj}. That is, F, is the smallest or-field with respect to which z, is measurable, {FJ} is an increasing family, and F, is independent of the a-field generated by the process z(t + s) - z(t), s_ 0. Assume f and g are measurable in (t, cl). Consider a stochastic differential equation of the form dXt =f(t,
Xt)dzt
Xt)dt+g(t,
(1)
XO = C
defined on t e [0, T]. In integral form this equation is written as ot
Xt= c+ ,ff(s, o
rt
g(s, Xs)dzs.
Xs)ds +
(2)
o
Definition. An equation of the form of (1) is called an Ito stochastic differential equation. A stochastic process Xt is called a solution of an Ito stochastic differential equation in [0, T] if it satisfies the following properties: (a) Xt is F,-measurable,
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171
(b) the functions f(t,X (w)) and g(t, X (c(o)) are such that, with probability 1, Jj'If(s, XS(&))Ids < oo and J[ O. Notice that equation (5) is defined on k ?0. Therefore, the theorems stated in the previous section cannot be directly applied. Merton (1975) assumes that a unique solution of (5) exists without further elaboration. The technique presented in this paper is to extend the original equation from the non-negative real numbers to the whole real line by "reflection". We first extend the production function f(k,) to a new function defined on the whole real line as if kt-?, ff(kt) t-f(- kt) if kt < . Thus, f is an odd function defined on R. Next, we extend the saving function s(k) to the whole real line by setting s(k) = s(-k), for k < 0, and s(k) = s(k) for k' 0. In other words, s(k) is an even function. Let F(kt) =(kt)f(kt)-(n )kt and G(kt)=-o-kt. Clearly F and G are odd functions on R. The following stochastic differential equation dkt = F(kt)dt+ G(kt)dzt,
(6)
ko= c>O,
is an extension of equation (5) in the sense that any solution to (6) with kt:- 0 is a solution to equation (5) and that any solution to (5) solves (6). In what follows we shall firstdemonstrate the existence and uniqueness of the solution to equation (6). Then we show that the solution thus found is everywhere positive if the initial condition is positive, ko= c > 0. Therefore, equation (5) has a unique solution. Now we state and prove the Existence and Uniqueness Theorem of the Stochastic Solow Equation. Theorem. If the productionfunction f is strictly concave, continuously differentiable on [0, oo), f(O) = 0, and limk+,,f'(k) = 0, then there is a unique solution to (5). Proof First, we extend (5) to (6). By assumption, f(k) is strictly concave and f(O) = 0. The boundedness of savings ratio, 0 < s(k) 1, implies that there exists a constant Bo> 0 such that
Is(k)f(k)I?_k* Notice that, if k* does not exist, then (8) is automatically satisfied. Choose B = Max {6B2, 6(n - o-2)2, 3c2}. Clearly, F2+ G2 [s(k )f(kt) - (n - o2)kt]2+ o-2kt '-2[(s(k-)f(kt))2
+ (n
-
o2)2k2]+ a2kt B(1
2 kt).
Thus, the solution exists and is unique by the Nonexplosive Theorem,
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Next, notice that the stochastic process that satisfies the following equation dkt = -(n
-a
2)ktdt - oktdzt with ko> 0,
(9)
is lognormal whose solution is given by kt = koexp{
- (n Q2/2)ds -
odzs
>
0.
(10)
The addition of g(k&)f(kt)dt, when kt > 0, to (9) will simply shift the stochastic process (10) upward. Consequently, a zero capital-labour ratio is an inaccessible boundary, hence, the solution to (6) is everywhere positive provided ko= c > 0. 11 Corollary. There is a unique solution to (5) for all CES productionfunctions with elasticity of substitution less than unity. Remark. If ko= c < 0, the kt of (10) is everywhere negative. The addition of g(k&)f(kt) to (10) will forever lower the value of kt. Note that we can also obtain this by looking at -k, since both the drift and the variance are odd functions. An advantage of our approach is that the solution is decomposed into two parts s(k)f(k)dt and a lognormal equation (10). To show the existence and uniqueness of the solution to equation (5), no restrictions on parameters n and orare required as we can see from the proof. Since the explosion time is +oo, we know that kt = oo is an inaccessible boundary. From the proof of the theorem, we see kt = 0 is also inaccessible. Thus, the theorem demonstrates that the solution to (5) will neither explode nor degenerate to zero in finite time, a useful result in studying growth paths. Equation (10) also helps us see that n - cr2/2>0,
(11)
is necessary for the existence of a steady state distribution; otherwise, the drift grows exponentially with time. In fact, condition (11) without equality, is one of a set of sufficient conditions for the existence of a steady state as pointed out by Merton (1975, footnote 1, p. 379). This observation is particularly useful because it is a restriction only on the source of uncertainty, population size in this case. It states that the drift-variance ratio cannot be less than one-half. Note especially that the necessity part is independent of conditions on the production function and the savings ratio, as long as they are not negative. Finally, it can be shown that a steady state distribution exists for this class of production functions if the following bounded slope condition s(O)f'(0) > n - cr2/2,
(12)
is satisfied. This is because we assume f'(0) < oo instead of f'(0) = oo. The rest of the proof follows Merton's argument. Thus, in addition to Cobb-Douglas production functions, a steady state distribution also exists for some CES production functi -ns. 4. REMARKS AND CONCLUSION The method of reflection presented in this paper is fairly general. The key is to make sure that both the drift and the variance have a finite slope at the origin so that the extended functions after reflection maintain nice properties of the original functions on the entire real line. In this regard, the method applies to sources of uncertainty other
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than population growth as well. For example, technical change is often argued as a more important cause of changes in the aggregate behaviour of the economy. As long as technical progress can be shown to follow a diffusion process, the above argument for the existence and uniqueness of the corresponding stochastic differential equation is still valid. In summary, the demonstration of the existence and uniqueness of the solution to the stochastic Solow equation places the stochastic Solow model of economic growth under uncertainty on firm mathematical foundations. It also opens the possibility that the corresponding optimal stochastic control problem has a non-empty set of admissible feedback controls. It is hoped that the reflection technique introduced in this paper would have applications in the development of existence theorems for various stochastic control problems. Acknowledgement. The authors are grateful to William Brock for suggesting this problem, and to Robert Becker, John Boyd, Michael Magill and two anonymous referees for their valuable comments. None of the above are responsible for errors that remain. An earlier version of this paper, entitled "Properties of Continuous Time Stochastic Economic Growth," was presented at the 1981 Winter Meeting of the Econometric Society. REFERENCES ARNOLD, L. (1974) Stochastic Differential Equations: Theory and Applications (New York: John Wiley and Sons). AHLFORS, L. (1966) Complex Analysis 2nd ed. (New York: McGraw-Hill). BOURGUIGNON, F. (1974), "A Particular Class of Continuous Time Stochastic Growth Models", Journal of Economic Theory,9, 141-158. MCKEAN, H. P., JR. (1969) Stochastic Integrals (New York: Academic Press). MERTON, R. (1975), "An Asymptotic Theory of Growth Under Uncertainty", Review of Economic Studies, 42, 375-393. SOLOW, R. (1956), "A Contribution to the Theory of Economic Growth", QuaterlyJournal of Economics, 70, 65-94. WAN, H. Y., JR. (1971) Economic Growth (New York: Harcourt Brace Jovanovich, Inc).
