A framework to analyze comparative dynamics in a continuous time stochastic growth model

Economic Modelling 15 Ž1998. 125]134

A framework to analyze comparative dynamics in a continuous time stochastic growth model Sumit Joshi Department of Economics, The George Washington Uni¨ersity, 2201 G Street NW., Washington DC 20052, USA

Accepted July 1996

Abstract This paper develops a general framework to analyze the comparative dynamic properties of optimal growth paths with respect to any parameter in a continuous time model of uncertainty. A distinguishing feature of the analysis is that the entire dynamic time path of the capital accumulation process can be characterized rather than restricting attention to a comparison of steady states as in conventional analysis. Since any policy variable enters as a parameter in a growth model, the paradigm developed immediately yields the dynamic impact of any policy tool on the time paths of the optimal capital accumulation program. The powerful Ito’s ˆ Lemma of stochastic calculus then determines the impact of the policy parameter on other variables of interest such as the wage rate and the return to capital. Q 1998 Elsevier Science B.V. Keywords: Comparative dynamics; Continuous time; Stochastic; Growth; Monotonicity JEL classification: D90

1. Introduction This paper constructs a general growth model of continuous time uncertainty in which the comparative dynamic properties of the capital accumulation process with respect to any policy parameter can be analyzed. In growth models, policy instruments such as the Žinitial. stock of money or the rate of capital income taxation 0264-9993r98r$19.00 Q 1998 Elsevier Science B.V. All rights reserved P I I S 0 2 6 4 - 9 9 9 3 Ž 9 7 .0 1 0 4 6 - 2

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enter into the analysis as parameters. A perturbation in these parameters influences the dynamic behaviour of the economy. A significant portion of the literature in economic dynamics has concentrated attention exclusively on the long-run impact on capital accumulation Žor, the effect on the steady state of capital. of a change in some policy parameter. This literature is vulnerable to the criticism that, in confining attention to the long-run or steady-state behavior of capital, it distracts attention from the adjustment process of the economy in response to a policy change and the behavior of the economy over finite horizons. A more realistic analysis requires a characterization of the entire dynamic time path of the captial accumulation process. This is the major motivation behind the paper. The basic framework that is utilized is the continuous time growth model of Solow]Merton. Although the techniques for analyzing comparative dynamics are relatively harder, the effect of a policy parameter on the behavior of the entire capital accumulation path can be determined. In particular, the time path of the capital]labour ratio is shown to satisfy a monotonicity or non-crossing property with respect to any policy parameter if the response of the savings function to the policy instrument is given. The powerful Ito’s ˆ Lemma } a cornerstone of continuous time stochastic calculus } then allows an explicit determination of the time paths of all major variables such as the return to capital and the wage rate. To correspond with conventional analysis, the framework is then extended to incorporate steady-state comparisons. While in the deterministic case, the steady state corresponds to a capital stock that is invariant with respect to time, in the stochastic case, the steady state corresponds to a probability distribution on the set of capital stocks that is invariant to time. It is shown that the distribution functions corresponding to the stochastic steady state also display monotonicity with respect to the policy parameter in the sense of first-order stochastic dominance. A continuous time framework, of course, has the advantage of corresponding more closely with the temporal character of the real world, as well as allowing a sharp distinction between flow variables such as investment and stock variables such as capital. However, it has a more crucial role in the current paper } it permits the analysis to be conducted within the fixed-sa¨ings function paradigm. Similar fixed-savings function models have also been the object of study in Atkinson and Stiglitz Ž1980., Boadway Ž1979., Chang and Malliaris Ž1987. and Merton Ž1975.. Generally, fixed-savings function models are subject to the criticism that savings behavior is ad hoc and not derived from intertemporal utility maximization. However, the work of Chang Ž1988. on the inverse optimal problem in continuous time mitigates much of the force of this criticism by indicating that the lack of an explicit utility-maximizing framework may not be an omission. Chang has demonstrated that, given the Solow growth equation and any consumption Žor savings. function, cŽ k ., and a production function, f Ž k ., satisfying standard restrictions, there exists a utility function, U, and a discount factor, b , such that cŽ k . is the solution to a representative consumer or planner maximizing the objective function EH0`ey b t UŽ c .dt over the set of consumption and capital accumulation processes that are subject to the growth constraint imposed by the stochastic Solow equation. In this sense, the fixed-savings function being used can be construed as

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being generated by a suitable optimization exercise. Therefore, the underlying utility-maximizing framework is implicit in the analysis rather than explicit. It is noted in Chang Ž1988. that this facility obtains only in continuous time models and not discrete time models. Since it is considerably simpler to operate with a fixed-savings function that depends on a given policy parameter, the continuous time paradigm is maintained precisely to exploit the analytical convenience that can be derived from the work of Chang Ž1988..

2. The Solow–Merton model This section extends the continuous time stochastic neoclassical model of Solow]Merton. The aggregate production function is given by Y s F Ž K, L. where F is linearly homogeneous. K Ž t . and LŽ t . denote the capital stock and labour input in period t, respectively. In per capita terms, y s f Ž k . where y s YrL, k s KrL and f Ž k . ' F Ž KrL, 1.. The per capita production function f is assumed to be twice continuously differentiable, increasing and strictly concave on Rq with f Ž0. s 0. The proportion of Y that is saved Žand invested. is given by the savings ratio sŽ k, u .. The savings ratio is allowed to depend on k. More importantly, the savings ratio is allowed to depend on any policy parameter u under consideration. This is because any policy parameter will influence the savings]consumption decision, and thereby the dynamic behavior of the economy, through the savings ratio. Given s, capital accumulation is determined as follows: K˙ '

dK dt

s s Ž k, u . Y .

Ž1.

It follows in particular that: K˙ K

s

s Ž k, u . f Ž k . k

.

Ž2.

Following Merton Ž1975., the source of uncertainty is the size of the population. Let Ž V, C, n . denote a probability space and W s Wt , CtW ; 0 F t - `4 denote the Brownian motion on this space.  CtW 4 denotes the filtration generated by Wt 4 ; that is, CtW is the smallest s-algebra with respect to which Wt is measurable, the sequence  CtW 4 is increasing, and CtW is independent of the s-algebra generated by Wtqiy Wt , i G 0. The size of the population follows a diffusion process given by: L t s L0 q

t

H0 nL

s

ds q

t

H0 s L

s

dWs ,

Ž3.

where n represents the expected rate of growth of the population per unit of time, s is the instantaneous variance, and L0 is the initial size of the population. Using the tools of Ito ˆ stochastic calculus Že.g. Merton, 1975, Appendix A, or Chang, 1988,

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Appendix A., the stochastic Solow equation can be written for 0 F t - ` as: d k t s s Ž k t , u . f Ž k t . y Ž n y s 2 . k t dt y s k t dWt ,

Ž4.

or, in integral form as: kt s k0 q

t

H0

sŽ ks , u . f Ž ks . y Ž n y s 2 . ks d s y

t

H0 s k

s

dWs ,

Ž5.

where k 0 s K 0rL0 , K 0 being the initial stock of capital. Note that the second integral on the right-hand side of Ž5. has the interpretation of a stochastic integral. The existence of a solution  k tu 4 , t g Rq, to the stochastic differential equation Ž5. follows from the Reflection Principle techniques of Chang and Malliaris Ž1987..

3. Monotonicity with respect to the policy parameter This section analyzes the impact of any policy parameter on the time path of the capital]labour ratio. In any continuous time stochastic model, the form of the function, I Ž k, u . s sŽ k, u . f Ž k ., representing total savingsrinvestment associated with k, assumes critical important Že.g. Merton, 1975.. One standard assumption to impose following the mathematical literature is the following:1 Lipschitz condition. There exists 0 - h - ` such that for any k, kX g Rq: < s Ž k, u . f Ž k . y s Ž kX , u . f Ž kX . < F h < k y kX < .

Ž6.

The main result of this paper is true under the Lipschitz condition. Unfortunately, the Lipschitz condition is rather limiting in the sense that sufficient conditions ensuring that Ž6. will hold are rather stringent. First, the requirement on the production function is that it must have a bounded slope at the origin, i.e. f X Ž0. - `. Secondly, the savings ratio must satisfy ­ sr­ k F 0. Recall that sŽ k, u . g Ž0, 1. for all k and u . Then, from the concavity of f, for k ) kX ) 0: s Ž k, u . f Ž k . y s Ž kX , u . f Ž kX . F s Ž kX , u .w f Ž k . y f Ž kX .x F f Ž k . y f Ž kX . F f X Ž kX .w k y kX x F f X Ž 0 .w k y kX x . However, if ­ sr­ k ) 0, an additional boundedness condition is required. To see this, note from the Mean Value theorem that: I Ž k, u . y I Ž kX , u . s I X Ž h .Ž k y kX . ,

kX F h F k,

where I X Ž h. s ­ I Ž h, u .r­ k. The Lipschitz condition will now hold if I X Ž h. is 1

This assumption essentially implies that the investment function is uniformly continuous. That is, for X any « ) 0 there exists a n Ž « . ) 0 such that for any capital stocks k and k that are within X n Ž « .-distance of each other, the investment levels I Ž k, u . and I Ž k , u . are within «-distance of each other.

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uniformly bounded from above. But note that IX Ž h. s

­ s Ž h, u . ­h

f Ž h . q s Ž h, u . f X Ž h . F

­ s Ž h, u . ­h

f Ž h . q f X Ž0. .

The problem now is the uniform boundedness of f Ž h., which is hard to ensure in a stochastic model. In Merton Ž1975., it was shown that k s ` is an inaccessible boundary and hence the capital]labour ratio will not explode. However, this does not imply that it will be uniformly bounded. Hence, Ž6. may not hold for this case. The more serious restriction, however, is the condition f X Ž0. - `. This assumption accommodates the class of Constant Elasticty of Substitution production functions with elasticity of substitution less than unity. However, it rules out the important parametric class of Cobb]Douglas production functions. Therefore, to allow the widest possible class of production functions, this paper does not impose a Lipschitz condition on the savings function but uses the alternative assumption on the savings ratio that ­ sr­u / 0. This assumption specifically requires the following: if g and d are two alternative values of the policy parameter u where g - d , then for all k ) 0: s Ž k, g . f Ž k . - Ž ). s Ž k, d . f Ž k . ,

­s

if

­u

) Ž - . 0.

With this implication, a result in Karatzas and Shreve Ž1991, Exercise 2.19, p. 294. can be used to dispense with the Lipschitz requirement on I Ž k, u .. This result shows the existence of a real-valued function Z:Rqª Rq, which satisfies Ž6. and is such that s Ž k, g . f Ž k . F Z Ž k . F s Ž k, d . f Ž k . ,

if

s Ž k, g . f Ž k . G Z Ž k . G s Ž k, d . f Ž k . ,

if

­s ­u ­s ­u

) 0, - 0.

Letting su s ­ sr­u , the main result of this section can now be proved. Theorem 1. Suppose g and d are two ¨alues of the parameter u with g - d . If su ) Ž-. 0 for all k ) 0, than k td G ŽF. k gt n-a.s. for all t ) 0. Proof. See Appendix. This result is significant on the following three counts: Ži. In a general model, the theorem determines the dynamic impact of a change in any policy parameter given that the sign of su is uniform for all k ) 0. Such a uniformity of sign restriction is common in conventional models when analyzing the effect of a policy parameter on the steady state Žfor instance, the steady-state analysis of capital income taxation in Atkinson and Stiglitz, 1980, p. 238..

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Žii. The theorem generalizes the existing literature, which has confined itself to long-run or steady-state comparisons of a change in the policy parameters. The above result relates the sign of su to the entire dynamic path of the capital]labour ratio. It shows that the time path of the capital]labour ratio enjoys the following monotonicity or non-crossing property with respect to the policy parameter u : if the savings function depends positively Žnegatively. on u for all k ) 0, then for all realizations of the stochastic environment Žexcept possibly those belonging to a set of probability measure zero., the capital]labour ratio for an economy with parameter value d always lies above Žbelow. the capital]labour ratio for an economy with parameter value g in each time period. Žiii. The comparative dynamics of all other variables of interest that are functions of the capital]labour ratio can be worked out as a corollary of the above result. Two instances are the following. 3.1. The return to capital The return to capital is given by R ut s f X Ž k tu .. Ito’s ˆ Lemma allows an explicit determination of the time path of R ut as: d R ut s a Ž k tu . dt q b Ž k tu . dWt ,

Ž7.

where the drift and variance terms are given by:

a Ž k . ' f Y sf y Ž n y s 2 . k q Ž 1r2. f Zs 2 k 2 , b Ž k . ' ys kf Y . Since f is concave, it follows from Theorem 1 that if su ) Ž-. 0 for all k ) 0, then R td F ŽG. R gt n-a.s. for all t ) 0. That is, the return to capital also exhibits a monotonicity property with respect to u : if the savings function depends positively Žnegatively. on the policy parameter u , then Žalmost surely. the return to capital for an economy with parameter value d lies below Žabove. that for an economy with parameter value g for every time period. The comparative dynamic property of the entire time path of the return to capital is thus determined. 3.2. The wage rate The competiti¨e wage rate, wtu s f Ž k tu . y k tu f X Ž k tu ., follows the dynamic path given by dwtu s lŽ k tu . dt q m Ž k tu . dWt , where the drift and variance terms are:

lŽ k . ' ykf Y sf q Ž n y s 2 . k 2 f Y y Ž 1r2.w f Y y kf Z x s 2 k 2 , mŽ k . ' s k2 f Y .

Ž8.

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The monotonicity properties of the wage rate can once again be determined by exploiting the concavity of f. For instance, consider the case where su ) 0 for all k ) 0. From Theorem 1, k td G k gt n-a.s. It now follows that with probability one: f Ž k td . y f Ž k gt . G f X Ž k td . k td y k gt G k td f X Ž k td . y k gt f X Ž k td . G k td f X Ž k td . y k gt f X Ž k gt . , and rearranging the above gives wtd G wtg n-a.s. In general, if su ) Ž-. 0, then wtd G ŽF. wtg n-a.s. for all t ) 0. Therefore, if savings depend positively Žnegatively. on u , then the competitive wage rate for an economy with policy parameter value d lies almost surely above Žbelow. that for the economy with parameter value g in each time period.

4. Steady-state analysis The conventional results on steady-state behavior can be derived here as a corollary of Theorem 1. Define the distribution functions  Ftu 4 , t g Rq, associated with the time path of the capital]labour ratio  k tu 4 as follows: Ftu Ž k . s Pr  k tu F k 4 ,

t G 0.

The corollary now shows that the distribution functions also exhibit monotonicity with respect to the policy parameter. In particular, if su is positive Žnegative. for all k ) 0, then the distribution functions  Ftd 4 dominate Ž are dominated by . in the first-order stochastic sense the sequence  Ftg 4 for each t G 0. Corollary 1. If su ) Ž-.0 for all k ) 0, then Ftd F ŽG. Ftg for all t G 0. Proof. From Theorem 1, k td G ŽF. k gt when su ) Ž-. 0. Therefore, for all t G 0: Ftd Ž k . s Pr  k td F k 4 F Ž G. Pr  k gt F k 4 s Ftg Ž k . . I A stochastic steady state corresponds to the limit of distribution functions, if it exists. Sufficient conditions for the existence of a steady-state distribution are given in Merton Ž1975.. Assuming a stochastic steady state exists, let F u s lim t ª` Ftu . Since Corollary 1 shows that the inequalities hold for all t, letting t ª ` yields the result that F d F ŽG. F g if su ) Ž-. 0 for all k ) 0. In other words, the stochastic steady state of an economy with policy parameter value d dominates Žis dominated by. in the first-order stochastic sense the stochastic steady state of the economy with the parameter value g if the savings ratio depends positively Žnegatively. on the policy parameter.

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5. Conclusion In a continuous time uncertainty model, using a fixed-savings function and implicit intertemporal utility maximization, this paper was able to characterize the impact of any policy parameter on the time path of the capital accumulation process Žand any function of the capital]labour ratio. as well as the stochastic steady states. An application of the tools developed in this paper to specific policy instruments in more specialized frameworks is an objective of future research.

Acknowledgements This research was started in Summer 1993 and was made possible by a Junior Scholar Incentive Award granted by the George Washington University for Summer 1993.

Appendix Proof of Theorem 1. The proof uses techniques developed in Karatzas and Shreve Ž1991, Chapter 5. to study the properties of solutions of stochastic differential equations. The proof is given for su ) 0 and is identical for the other case. Given any k, kX g Rq, let x s < k y kX <. Note that: lim « ª 0 HxŽ0 , « . Ž x .Ž 1rx 2 . d x s `, where xŽ0, « . is the indicator function of the set Ž0, « ..2 This implies there exists a sequence  sn4 : Ž0, 1x , n s 1, 2, . . . , such that sn G snq1 , s0 s 1, lim n ª ` sn s 0 and HxŽ s n , s ny 1 . Ž x .Ž 1rx 2 . d x s n,

; n G 1.

Let rnŽ x . s 1rnx 2 and note that it is continuous for x ) 0. Further, by construction, it has support on the interval Ž sn , sny1 .. Now let D t s k gt y k td and consider the functions: Cn Ž D t . s

<

t<

u

H0 D H0 r Ž x . d x du, n

Fn Ž D t . s Cn Ž D t . xŽ0 , `. Ž D t . .

The following properties of these functions will be employed in the proof. First, cn is twice continuously differentiable on Rq. Second, since the interval, Ž sn , sny1 . is shrinking, for a fixed u and all sufficiently large n, H0u rnŽ x . d x s 1. Therefore, using the Lebesgue Dominated Convergence theorem, lim nª`FnŽ D t . s xŽ0, `. lim n ª ` cnŽ D t . s xŽ0, `. < D t < s X Ž . sup0, D t 4 ' Dq t . Third, for D t F 0, Fn D t s 0 and for D t ) 0: FnX Ž D t . s CnX Ž D t . s

2

Dt

H0

rn Ž x . d x F 1,

It is defined as follows: xŽ0, « .Ž x . s 1 if x g Ž0, « . and xŽ0, « .Ž x . s 0, otherwise.

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using a result from Rudin Ž1976, Theorem 6.20..3 Another application of this result gives FnY Ž D t . s rnŽ D t .. From Ž5. and Ž6., it follows that: dD t s w s Ž k gt , g . f Ž k gt . y s Ž k td , d . f Ž k td .x dt q Ž n y s 2 .Ž k td y k gt . dt q s Ž k td y k gt . dWt . From Ito’s ˆ Lemma: dFn Ž D t . s FnX Ž D t . dD t q 12 FnY Ž D t .Ž dD t . . 2

The operational rules of the Ito ˆ calculus for a Brownian motion process requires Ždt . 2 s ŽdWt .Ždt . s 0 and ŽdWt . 2 s dt. Using these rules, it follows that dFn Ž D t . s FnX Ž D t .w s Ž k gt , g . f Ž k gt . y s Ž k td , d . f Ž k td .x dt q FnX Ž D t .Ž n y s 2 .Ž k td y k gt . dt q 12 FnY Ž D t . s 2 Ž k td y k gt . dt q FnX Ž D t . s Ž k td y k gt . dWt .

Ž A1.

Recall the existence of a function ZŽ k . satisfying Ž6. such that s Ž k gt , g . f Ž k gt . F Z Ž k gt . ,

s Ž k td , d , d . f Ž k td . G Z Ž k td . .

Consider the first term after the equality sign on the right-hand side of ŽA1.. It follows that FnX Ž D t .w s Ž k gt , g . f Ž k gt . y s Ž k td . f Ž k td .x dt F FnX Ž D t .w Z Ž k gt . y Z Ž k td .x dt. q Therefore, the first two terms on the right-hand side of ŽA1. are F j FnX Ž D t . Dq t F jD t , X 2. Ž Ž . Ž . where j s h q n y s and 0 F Fn D t F 1. Also, note that E dWt s 0, ; t G 0. Hence, rewriting ŽA1. in integral form and taking expectations

t

EFn Ž D t . F j

q. s

H0 E Ž D

d s q 12 E

t

Y n

H0 F s

2Ž

2

D s . d s.

Ž A2.

Recall that FnY Ž D t . s rnŽ D t .. Hence, the second term on the RHS of ŽA2. is less than or equal to t

H0 s r Ž D .Ž D . 2

n

s

s

2

ds s

t

H0

s2 n

ds s

ts 2 n

.

Therefore, the inequality in ŽA2. now becomes: EFn Ž D t . F j

t

q. s

H0 E Ž D

ds q

ts 2 n

.

Ž A3.

Since the above holds for all n, let n ª `. Using the Lebesgue Dominated Convergence 2 . Ž . theorem, lim nª ` EFnŽ D t . ª EŽ Dq t . Also, t s rn ª 0 and hence A3 becomes 3

This result states that if F Ž x . s H0x f Ž y . d y and f is continuous at x, then F is differentiable at x and X F Ž x . s f Ž x ..

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.Fj E Ž Dq t

t

q. s

H0 E Ž D

d s.

5 q q . Now, using the Gronwall inequality,4 it follows that EŽ Dq t s 0. Since D t G 0, D t s 0 q n-a.s. Recalling that D t s sup0, D t 4, this implies that D t F 0 n-a.s., giving the result. I

References Atkinson, A.B. and J.E. Stiglitz, 1980, Lectures on public economics ŽMcGraw Hill Inc., New York.. Boadway, R., 1979, Long-run tax incidence: a comparative dynamic approach, Review of Economic Studies 46, 505]511. Chang, F.R., 1988, The inverse optimal problem: a dynamic programming approach, Econometrica 56 Ž1., 147]172. Chang, F.R. and A.G. Malliaris, 1987, Asymptotic growth under uncertainty: existence and uniqueness, Review of Economic Studies, LIV, 169]174. Karatzas, I. and S.E. Shreve, 1991, Brownian motion and stochastic calculus ŽSpringer-Verlag, Berlin.. Merton, R., 1975, An asymptotic theory of growth under uncertainty, Review of Economic Studies 42, 375]393. Rudin, W., 1976, Principles of mathematical analysis ŽMcGraw-Hill Inc., New York.. 4

This inequality states the following: consider a continuous function g Ž t . where 0 F g Ž s . F bH0t g Ž s .d s for some integrable real values function a Ž t .. Then, g Ž t . F a Ž t . F bH0t a Ž s .e b Ž tys. d s. 5 . and a Ž t . ' 0, for all t G 0. In the Gronwall inequality, take g Ž t . ' EŽ Dq t