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The Existence of Steady States in Multisector Capital Accumulation Models with Recursive Preferences
Journal of Economic Theory ET2190 journal of economic theory 71, 289297 (1996) article no. 0118
NOTES, COMMENTS, AND LETTERS TO THE EDITOR
The Existence of Steady States in Multisector Capital Accumulation Models with Recursive Preferences* John H. Boyd III Department of Economics, University of Rochester, Rochester, New York 14627 Received December 31, 1994; revised October 24, 1995
This paper proves the existence of a non-trivial stationary optimal path in a multisectoral capital accumulation model with recursive preferences. The reduced-form recursive preferences are represented by an aggregator funtion. I introduce a new form of $-normality that is appropriate for use with recursive preferences. Under some mild conditions on the aggregator, non-trivial steady states exist when the technology is bounded and $-normal. Journal of Economic Literature Classification Numbers: D90, C61, C62. 1996 Academic Press, Inc.
1. INTRODUCTION Steady states play an important role in the analysis of optimal growth models. The Euler equations may be linearized about the steady states. Their stability properties inform us about the stability of the optimal paths. Moreover, the linearization is often useful in numerical approximations to the optimal paths. In models with additively separable utility, the issue of the existence of steady states has long been settled. It was first broached in multisector models by Sutherland [15]. The first real existence results were obtained by Hansen and Koopmans [8] in a type of von Neumann model, and by Peleg and Ryder [14] in a consumption-based model. Cass and Shell [3] handled the continuous-time case. Flynn [7] and McKenzie [12, 13] independently established the current theory for reduced-form models. This analysis has been further enriched by Kahn and Mitra [9], using * I thank Robert Becker, Lionel McKenzie, Karl Shell, and Tomoichi Shinotsuka for helpful discussions concerning this project and their comments on earlier drafts of this paper. I also thank the referees for their comments.
289 0022-053196 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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purely primal methods, and by Becker and Foias [1], who use Ky Fan's Inequality instead of the Kakutani Theorem. Models with more general recursive preferences are a different story. If there is one sector, the existence of steady states follows from appropriate Inada conditions and the intermediate value theorem. If a steady state exists, a local analysis of the optimal path is sometimes possible (e.g., Epstein [4, 5]). Unfortunately, conditions that guarantee existence of steady states in general multisector models have previously been unknown. This paper proves the existence of a non-trivial stationary optimal path in a reduced-form multisectoral capital accumulation model with recursive preferences. The key to the proof is a new form of $-normality that is appropriate for use with recursive preferences. Under some mild conditions on the aggregator, non-trivial steady states exist when the technology in bounded and $-normal. The actual proof is akin to those in McKenzie [12, 13] and Kahn and Mitra [9]. Section 2 sets up the basic model, and Section 3 proves the existence of steady states.
2. MULTI-SECTOR CAPITAL ACCUMULATION MODELS The reduced-form model is described by a technology set T/R 2m + obeying (T1)(T5) and an aggregator function W: T_R R + obeying (A1)(A5) below. Given a technology set T and initial capital stock k 0 , a path k=[k t ] t=0 is feasible from k 0 if (k t&1 , k t ) # T for t=1, 2, ... . Define the shift operator S by S k=(k 1 , k 2 , . ..). The utility function is defined as the unique function U that obeys the recursion U(k)=W(k 0 , k 1 , U(Sk)) for all feasible k. The function W is referred to as the aggregator. 1 The aggregator allows us to express lifetime utility as a function of current inputs, current outputs, and future utility. By recursively applying W, we may generate the lifetime utility function. An optimal path from k 0 is a feasible path that maximizes utility over all feasible paths from k 0 . A stock k is a steady state if (k, k, . . .) is optimal from k. Technology Assumptions. A set T/R 2m + is a bounded Malinvaud technology set if: (T1)
T is a closed set. (closure)
(T2)
T is a convex set. (convexity)
1 See Koopmans [10] for a discussion of the properties of preferences that yield such a utility function. The preferences considered here are somewhat more general. The converse problem of deriving a utility function U from the aggregator W is discussed in Lucas and Stokey [11] and Boyd [2].
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(T3) (T4) disposal)
There is (a, b) # T with br0. ( productivity) If (a, b) # T and a$ a, 0 b$ b, then (a$, b$) # T. ( free
(T5) There is a ;>0 such that &b&<&a& whenever (a, b) # T with &a&>;. (bounded paths) When (a, b) # T, we interpret a as the input stock and b as the output stock, once consumption has been subtraced. These assumptions on technology: closure, convexity, productivity, free disposal, and bounded paths, are all fairly standard. Neither the inaction postulate (0 # T) nor the no free lunch postulate ((0, b) # T implies b=0) are needed. Condition (T5) ensures that all feasible paths are bounded. It implies that any steady state k has &k& ;, and any feasible path starting at k 0 with &k 0 & ; obeys &k t & ;. Let U be a closed interval in R _ [& ]. Aggregator Assumptions.
The aggregator W: T_U U obeys:
(A1) W(a, b, u) is finite whenever (a, b) # int T and u # U is finite. If & # U, then W(a, b, & )=& for all (a, b) # T. W is both upper semicontinuous and lower semicontinuous on T_U. 2 (A2) W 3 #dWdu exists and is continuous in (a, b, u) whenever W(a, b, u)>& . Moreover, 0<$ & W 3 $ + <1 for some $ & and $ +. (A3) Let u # U. Each W nu defined inductivily (Sk)) and W 0u(k)=u is concave in k. W(k 0 , k 1 , W n&1 u (A4)
by
W nu(k)=
If a$ a, 0 b$ b, and u$ u then W(a$, b$, u$) W(a, b, u).
(A5) For all (a, b), (a$, b$) # T and finite u, u$ # U, W obeys W(a, b, u)> W(a$, b$, u) if and only if W(a, b, u$)>W(a$, b$, u$). Assumption (A1) is meant to include commonly used cases, such as the additively separable aggregator W(a, b, u)=log( f (a)&b)+$u in the case m=1. Here the technology set is T=[(a, b) # R 2+ : b f (a)] with f a smooth, increasing, concave production function with f (0)=0 and f $(a);. This example also illustrates that continuity of W with respect to (a, b) does not follow from (A2). Assumption (A3) amounts to assuming that the derived utility function U is concave. Monotonicity (A4) is standard in reduced-form models. Finally, (A5) is Koopman's [10] limited independence postulate in the reduced-form context. There are also non-separable examples that satisfy (A1)(A5). Again take the one-sector case with production function f above. Let W(a, b, u)=&1+e &v( f (a)&b)u with u # U=R & , where v is a smooth, 2
This means W is continuous as an extended real-valued function.
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increasing, concave function with v(0)>0. Properties (A1), (A4), and (A5) are obvious. Set $ & =e &v( ;) and $ + =e &v(0) to satisfy (A2). To see that (A3) holds, examine n
\
t&1
+
W nu(k)=& : exp & : v( f (k {&1 )&k { ) +u t=1
\
n
{=1
+
_exp & : v( f (k t&1 )&k t ) . t=1
Now W nu is concave because the mapping x [ &e &x is concave and each of the sums t&1 {=1 v( f (k {&1 &k { ) is concave. To derive the utility function, let u=sup U. 3 Then W(k 0 , k 1 , u) u, i.e., (k) for all n. W 1u(k) W 0u(k). An easy induction shows W nu(k) W n&1 u Define the utility function U(k)=inf n W nu(k). The Lipschitz condition implicit in (A2) implies that |W nv &W nu | ($ + ) n |u&v| for any finite u, v # U. As a result, the infimum of the W nu does not depend on u. The (Sk))= function U is recursive as U(k)=inf W nu(k)=infW(k 0 , k 1 , W n&1 u (S k))=W(k , k , U(S k)). The infimum is upper semiW(k 0 , k 1 , inf W n&1 0 1 u continuous in the product topology. Condition (A3) guarantees that U is concave. When (k, k) # T, define 8(k)=U(k, k. . .). Note that if W(k, k, u)>& for some u # U then 8(k)>& . Of course 8 inherits upper semicontinuity from U. We also need a joint assumption on W and T. Even in the additive case, productivity must be sufficient to overcome discounting if there is to be a steady state. Define $ k =W 3(k, k, 8(k)). This is the discount factor on the constant path (k, k, . . .). Let $=inf $ k $ & >0. $-Normality Assumption. The reduced-form model (T.W) is $-normal if there is a pair (a, b ) # T such that 0Ra $b and W(a, b, u)>8(0) for some u # U, where we interpret 8(0) as & if (0, 0) T. There is a finite u # U with W(a, b, u ) u because of (A2). By (A4), W(b, b, u ) u. This implies W nu (b, b, . . .) is non-decreasing, and so 8(b ) u.
3. THE EXISTENCE OF STEADY STATES In this section, we will assume (T, W), obeys (T1)(T5), (A1)(A5), and $-normality. Define T 1 =[a # R m +: (a, b) # T for some b and &a& ;] and K=[k # R m +: (k, k) # T and 8(k) u ]. We will obtain a steady state in K. 3
This construction is based on the ``partial summation'' method of Boyd [2].
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Of course, &k& ; whenever k # K, so K/T 1 . Note that K and T 1 are compact and convex and contain b. For k # T 1 , denote the closest point in K to k by g(k). This defines a continuous function from T 1 to K. The mappings k [ $ g(k) and 9(k)= 8(g(k)) extend $ k and 8 k . Abusing notation, we also denote the extension of $ k by $ k . For k # T 1 , define ,(k)=[(a, b) # T: a (1&$ k )+$ k b]. Because (a, b ) # T defines a correspondence. ,(k), ,(k) is non-empty and ,: T 1 If (a, b) # ,(k) with &a&>;, &a& (1&$ k ) &k&+$ k &b& (1&$ k ) &k&+ $ k &a& by (T5). But then, &a& &k& ;. It follows that &a& ; for all (a, b) # ,(k). By (T5), &b& ;. It follows that ,(k) is compact and convex for every k # T 1 . A stock k # K is a recursive golden rule stock if (k, k) # T and W(k, k, 8(k)) W(a, b, 8(k)) for all (a, b) # ,(k). When W is additively separable, with W(a, b, u)=v(a, b)+$u, this definition reduces to the usual definition of a discounted golden rule. A golden rule is non-trivial if 8(k)>8(0). Theorem 1. Suppose (A1)(A4), (T1)(T5), and $-normality hold. If k is a recursive golden rule, the stationary path (k, k, . . .) is optimal from k. Proof. By $-normality, ,(k) has an interior. Now (a, b) [ W(a, b, 8(k)) is concave on ,(k) and has a maximum at (k, k). Thus there is a vector ( p, q) which supports both W and ,(k). In other words, W(a, b, 8(k)) W(k, k, 8(k))+p(a&k)+q(b&k) for all (a, b) # T, and ( p, q)=*( p$, q$)+ m i=1 * i (e i &$ k e i ) for some *, * i 0, where ( p$, q$) supports T. 4 That is, p$a+q$b pk+qk for all (a, b) # T. Let x be feasible from k and let k=(k, k, . . .). Then W n8(k)(k)=8(k)= U(k). Because W n8(k) is concave, and W is continuously differentiable and non-decreasing in the third argument, we have W n8(k)(x)&W n8(k)(k) n&1
(x t &k)+$ nk q(x n &k). p(x 0 &k)+ : ($ k p+q) $ t&1 k t=1
Recalling that x 0 =k, and taking the limit as n yields U(x) t&1 (x t &k). U(k)+ t=1 ($ k p+q) $ k Now $ k p+q=*$ k p$+*q$+ m i=1 * i ($ k e i &$ k e i )=*$ k p$+*$ k q$. t&1 (x t&1 , x t ) # T. 5 Consider the convex combination (1&$ k ) &1 t=1 $ k 4
Here e i denotes the ith standard basis vector of R m. This construction has been used in a primal context by Dechert and Nishimura [4] and Khan and Mitra [9]. 5
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t&1 By the support property of ( p$, q$), ( p$x t&1 + q$x t ) t=1 $ k t&1 t=1 $ k ( p$k+q$k). Since x is feasible from x 0 =k 0 , we can rewrite t&1 ($ k p$+q)(x t &k) 0. But $ k p+q=$ k p$+q$, so to obtain t=1 $ k U(x) U(k). Thus k is optimal from k. K
Because recursive golden rules are steady states, it is enough to show that recursive golden rules exist. Given any stock k # T 1 , the implicit programming problem is to solve J(k)=sup[W(a, b, 9(k)): (a, b) # ,(k)]. This problem has a solution because ,(k) is compact and W(a, b, 9(k)) is upper semicontinuous in (a, b). By $-normality, (a, b ) # ,(k) for all k # K and J(k) W(a, u, 9(k)). Since 9(k) u, it follows that J(k) W(a, b, u ) u > & by free disposal and monotonicity. The maximizer correspondence +(k) is defined by +(k)=[(a, b) # ,(k): W(a, b, 9(k))=J(k)]. Note that +(k) is also non-empty, compact, and convex for each k # K. Thus a stock k # K is a recursive golden rule if and only if (k, k) # +(k). We will only show the existence of non-trivial golden rules when the technology is both bounded and $-normal. As is well known, discounted golden rules may not exist if the technology is not bounded, as in models that generate balanced growth. If the technology is bounded but not sufficiently productive, all optimal paths may converge to 0. The $-normality condition rules out cases where all optimal paths converge to 0 and is a generalization of the requirement that $f $(0+ )>1 in the onesector model. The proof of existence of golden rules proceeds as follows. We start with a preliminary lemma that shows that 9 and $ k are continuous on T 1 . We next show , has a lower semicontinuity property in Lemma 2. We then use that fact to show +(k) is a closed correspondence in Lemma 3. 6 Finally, we project + onto the input space T 1 . This new correspondence on T 1 is closed with compact and convex values. The Kakutani Fixed Point Theorem yields a fixed point of , and that fixed point is a recursive golden rule. Lemma 1.
The function 9 and the mapping k [ $ k are continuous on T 1 .
Proof. Because g is continuous, it is enough to show $ k and 8(k) are continuous on K. Let =>0 and k # K. An induction shows that for 6 A correspondence , is closed if whenever x n x and y n y with y n # ,(x n ), we have y # ,(x).
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k$ # K, |W n8(k)(k$)&8(k)| (1&$ + ) &1 |W(k$, k$, 8(k))&W(k, k, 8(k))|, where k$=(k$, k$, . . .). Because W(k, k, 8(k))> & , (A1) implies we may pick '>0 with |W n8(k)(k$)&8(k)| <=2 for &k$&k&<'. Letting n shows |8(k$)&8(k)| <= for &k$&k&<'. Now $ k =W 3(k, k, 8(k)). By (A2) and the continuity of 8, $ k is also continuous. K Lemma 2. Suppose (x, y) # ,(k) with (x, y){(a, b ). If =>0, there is : with 0<:0 such that ((1&:) x+:a, (1&:) y+:b ) # ,(k$) whenever &k$&k&<'. Proof. Let z=(1&:) x+:a and w=(1&:) y+b. By convexity of T 1 , (z, w) # T. Then define q(k$)=(1&:)($ k$ &$ k ) y+[(1&$ k$ ) k$&(1&:)(1&$ k ) k]+:($ k$ &$) b. Note that q is continuous in k$ and that q(k)=:(1&$ k ) k+:($ k &$) b 0. There are now two cases to consider. First, if $ k >$, q(k)r0. If follows that q(k$)r0 for k$ near k. Second, if $ k =$, the first and third terms of q(k$) are always non-negative, and the middle term in non-negative when k$ is sufficiently near k. Either way, we may choose '>0 with q(k$) 0 for &k$&k&<'. Using : to form a convex combination of x (1&$ k ) k+$ k y and a $b, we find z (1&$ k$ ) k$+$ k$ w&q(k$) (1&$ k$ ) k$+$ k$ w for &k$&k&<'. It follows that (z, w) # ,(k$) for &k$&k&<'. K Lemma 3.
T is a closed correspondence. The correspondence +: T 1
Proof. As in Lemma 1, we need only to show + is closed on K. Suppose (a n , b n ) # +(k n ) with (a n , b n ) (a, b) and k n k. Then (a, b) # T by closure and a (1&$ k ) k+$ k b, so (a, b) # ,(k). Since W and 8 are upper semicontinuous and W is non-decreasing in u, J(k) W(a, b, 9(k))= lim sup W(a n , b n , 8(k n ))=lim sup J(k n ). We need only show W(a, b, 8(k)) =lim sup W(a n , b n , 8(k n )) J(k) to complete the proof. Note that W(a, b, 8(k)) lim sup J(k n ) W(a, b, 8(k)). Suppose J(k)>W(a, b, 8(k)). Then J(k)>W(a, b, 8(k)). Take (x, y) # ,(k) with W(x, y, 8(k))=J(k). By Lemma 2, we can construct a subsequence of the k n (also labeled k n ) and : n <1n with (x n , y n ) # ,(k n ), where x n = (1&: n ) x+: n a and y n =(1&: n ) y+: n b. Now W(x n , y n , 8(k n )) J(k n ) because (x n , y n ) # ,(k n ). Furthermore, W(x n , y n , 8(k n )) (1&: n ) W(x, y, 8(k n ))+: n W(a, b, 8(k n ))
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by concavity. Rearranging and using J(k n ) W(x n , y n , 8(k n )), we obtain : n[W(x, y, 8(k n ))&W(a, b, 8(k n ))] W(x, y, 8(k n ))&J(k n ).
(1)
Let =>0. Taking n large enough, (1) shows =>[W(x, y, 8(k n ))&J(k n )]. But W(x, y, 8(k n )) J(k), so =2+J(k n )>J(k) for n large. It follows that J(k) lim sup J(k n ) lim inf J(k n ) J(k), and so W(a, b, 8(k))=J(k). This shows + is closed. K Theorem 2. Suppose (A1)(A5), (T1)(T5), and $-normality hold. A non-trivial recursive golden rule exists. T 1 by (k)=[a # T 1: (a, b) # Proof. Define a correspondence : T 1 +(k) for some b]. We know maps into T 1 because , and + are bounded by ;. It is clear that has convex, compact, non-empty values. It is also closed. Let k n k and a n a with a n # (k n ). Take b n with (a n , b n ) # +(k n ). Now &a n & ; and &b n & ;. By passing to a subsequence, we may assume b n b. But then (a, b) # +(k) because + is closed. Hence is closed. Apply the Kakutani Fixed Point Theorem to obtain k with k # (k). There is a b with (k, b) # +(k). Then k (1&$ k ) k+$ k b, which implies k b. By free disposal (k, k) # T, and W(k, k, 9(k)) W(k, b, 9(k))=J(k) by monotonicity. Now J(k)=W(k, k, 9(k)) W(a, b, 9(k)). Using (A5), we find W(k, k, u ) W(a, b, u )=u. Induction then shows W nu(k, k, . . .) u. Letting n shows 8(k) u, so k # K. Therefore K is is a recursive golden rule. From non-triviality, use (A5) to see that 8(k)=W(k, k, 8(k)) W(a, b, 8(k))>W(0, 0, 8(k)). If follows that W n8(k)(0, 0, . . .) is non-increasing. Taking the limit shows 8(k)>8(0). K Remark. If there is a u with 8(a) u whenever (a, a) # T and if W(a, b, u)>W(0, 0, u) for all u # U with u u, we can dispense with (A5) in Theorem 2. Note that (A5) is used twice, both times in the proof of Theorem 2. Under these alternative assumptions, the first use of (A5) is unnecessary. 8(k) u automatically. To show k is non-trivial, use Theorem 1 and monotonicity to find W(k, k, 8(k))=8(k) U(k, 0, . . .)= W(k, 0, 8(0)) W(k, k, 8(0)), implying 8(k) 8(0). Then W(k, k, 8(k)) W(a, b, 8(k))>W(0, 0, 8(k)) W(0, 0, 8(0))=8(0) establishes nontriviality. REFERENCES 1. R. A. Becker and C. Foias, A minimax approach to the implicit programming problem, Econ. Lett. 20 (1986), 171175. 2. J. H. Boyd III, Recursive utility and the Ramsey problem, J. Econ. Theory 50 (1990), 326345.
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3. D. Cass and K. Shell, The structure and stability of competitive dynamical systems, J. Econ. Theory 12 (1976), 3170. 4. W. D. Dechert and K. Nishimura, A complete characterization of optimal growth paths in an aggregated model with a non-concave production function, J. Econ. Theory 31 (1983), 332354. 5. L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica 55 (1987), 329355. 6. L. G. Epstein, A simple dynamic general equilibrium model, J. Econ. Theory 41 (1987), 6895. 7. J. Flynn, The existence of optimal invariant stocks in a multi-sector economy, Rev. Econ. Stud. 47 (1980), 809811. 8. T. Hansen and T. C. Koopmans, On the definition and computation of a capital stock invariant under optimization, J. Econ. Theory 5 (1972), 487523. 9. M. A. Khan and T. Mitra, On the existence of a stationary optimal stock for a multi-sector economy: A primal approach, J. Econ. Theory 40 (1986), 319328. 10. T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica 28 (1960), 287301. 11. R. E. Lucas, Jr. and N. L. Stokey, Optimal growth with many consumers, J. Econ. Theory 32 (1984), 139171. 12. L. W. McKenzie, A primal route to the turnpike and Liapounov stability, J. Econ. Theory 26 (1982), 194209. 13. L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in ``Handbook of Mathematical Economics, III'' (K. J. Arrow and M. D. Intrilligator, Eds.), North-Holland, New York, 1986. 14. B. Peleg and H. E. Ryder, Jr., The modified golden rule of a multi-sector economy, J. Math. Econ. 1 (1974), 193198. 15. W. R. S. Sutherland, On optimal development in a multi-sectoral economy: The discounted case, Rev. Econ. Stud. 37 (1970), 585588.
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NOTES, COMMENTS, AND LETTERS TO THE EDITOR
The Existence of Steady States in Multisector Capital Accumulation Models with Recursive Preferences* John H. Boyd III Department of Economics, University of Rochester, Rochester, New York 14627 Received December 31, 1994; revised October 24, 1995
This paper proves the existence of a non-trivial stationary optimal path in a multisectoral capital accumulation model with recursive preferences. The reduced-form recursive preferences are represented by an aggregator funtion. I introduce a new form of $-normality that is appropriate for use with recursive preferences. Under some mild conditions on the aggregator, non-trivial steady states exist when the technology is bounded and $-normal. Journal of Economic Literature Classification Numbers: D90, C61, C62. 1996 Academic Press, Inc.
1. INTRODUCTION Steady states play an important role in the analysis of optimal growth models. The Euler equations may be linearized about the steady states. Their stability properties inform us about the stability of the optimal paths. Moreover, the linearization is often useful in numerical approximations to the optimal paths. In models with additively separable utility, the issue of the existence of steady states has long been settled. It was first broached in multisector models by Sutherland [15]. The first real existence results were obtained by Hansen and Koopmans [8] in a type of von Neumann model, and by Peleg and Ryder [14] in a consumption-based model. Cass and Shell [3] handled the continuous-time case. Flynn [7] and McKenzie [12, 13] independently established the current theory for reduced-form models. This analysis has been further enriched by Kahn and Mitra [9], using * I thank Robert Becker, Lionel McKenzie, Karl Shell, and Tomoichi Shinotsuka for helpful discussions concerning this project and their comments on earlier drafts of this paper. I also thank the referees for their comments.
289 0022-053196 18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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purely primal methods, and by Becker and Foias [1], who use Ky Fan's Inequality instead of the Kakutani Theorem. Models with more general recursive preferences are a different story. If there is one sector, the existence of steady states follows from appropriate Inada conditions and the intermediate value theorem. If a steady state exists, a local analysis of the optimal path is sometimes possible (e.g., Epstein [4, 5]). Unfortunately, conditions that guarantee existence of steady states in general multisector models have previously been unknown. This paper proves the existence of a non-trivial stationary optimal path in a reduced-form multisectoral capital accumulation model with recursive preferences. The key to the proof is a new form of $-normality that is appropriate for use with recursive preferences. Under some mild conditions on the aggregator, non-trivial steady states exist when the technology in bounded and $-normal. The actual proof is akin to those in McKenzie [12, 13] and Kahn and Mitra [9]. Section 2 sets up the basic model, and Section 3 proves the existence of steady states.
2. MULTI-SECTOR CAPITAL ACCUMULATION MODELS The reduced-form model is described by a technology set T/R 2m + obeying (T1)(T5) and an aggregator function W: T_R R + obeying (A1)(A5) below. Given a technology set T and initial capital stock k 0 , a path k=[k t ] t=0 is feasible from k 0 if (k t&1 , k t ) # T for t=1, 2, ... . Define the shift operator S by S k=(k 1 , k 2 , . ..). The utility function is defined as the unique function U that obeys the recursion U(k)=W(k 0 , k 1 , U(Sk)) for all feasible k. The function W is referred to as the aggregator. 1 The aggregator allows us to express lifetime utility as a function of current inputs, current outputs, and future utility. By recursively applying W, we may generate the lifetime utility function. An optimal path from k 0 is a feasible path that maximizes utility over all feasible paths from k 0 . A stock k is a steady state if (k, k, . . .) is optimal from k. Technology Assumptions. A set T/R 2m + is a bounded Malinvaud technology set if: (T1)
T is a closed set. (closure)
(T2)
T is a convex set. (convexity)
1 See Koopmans [10] for a discussion of the properties of preferences that yield such a utility function. The preferences considered here are somewhat more general. The converse problem of deriving a utility function U from the aggregator W is discussed in Lucas and Stokey [11] and Boyd [2].
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(T3) (T4) disposal)
There is (a, b) # T with br0. ( productivity) If (a, b) # T and a$ a, 0 b$ b, then (a$, b$) # T. ( free
(T5) There is a ;>0 such that &b&<&a& whenever (a, b) # T with &a&>;. (bounded paths) When (a, b) # T, we interpret a as the input stock and b as the output stock, once consumption has been subtraced. These assumptions on technology: closure, convexity, productivity, free disposal, and bounded paths, are all fairly standard. Neither the inaction postulate (0 # T) nor the no free lunch postulate ((0, b) # T implies b=0) are needed. Condition (T5) ensures that all feasible paths are bounded. It implies that any steady state k has &k& ;, and any feasible path starting at k 0 with &k 0 & ; obeys &k t & ;. Let U be a closed interval in R _ [& ]. Aggregator Assumptions.
The aggregator W: T_U U obeys:
(A1) W(a, b, u) is finite whenever (a, b) # int T and u # U is finite. If & # U, then W(a, b, & )=& for all (a, b) # T. W is both upper semicontinuous and lower semicontinuous on T_U. 2 (A2) W 3 #dWdu exists and is continuous in (a, b, u) whenever W(a, b, u)>& . Moreover, 0<$ & W 3 $ + <1 for some $ & and $ +. (A3) Let u # U. Each W nu defined inductivily (Sk)) and W 0u(k)=u is concave in k. W(k 0 , k 1 , W n&1 u (A4)
by
W nu(k)=
If a$ a, 0 b$ b, and u$ u then W(a$, b$, u$) W(a, b, u).
(A5) For all (a, b), (a$, b$) # T and finite u, u$ # U, W obeys W(a, b, u)> W(a$, b$, u) if and only if W(a, b, u$)>W(a$, b$, u$). Assumption (A1) is meant to include commonly used cases, such as the additively separable aggregator W(a, b, u)=log( f (a)&b)+$u in the case m=1. Here the technology set is T=[(a, b) # R 2+ : b f (a)] with f a smooth, increasing, concave production function with f (0)=0 and f $(a);. This example also illustrates that continuity of W with respect to (a, b) does not follow from (A2). Assumption (A3) amounts to assuming that the derived utility function U is concave. Monotonicity (A4) is standard in reduced-form models. Finally, (A5) is Koopman's [10] limited independence postulate in the reduced-form context. There are also non-separable examples that satisfy (A1)(A5). Again take the one-sector case with production function f above. Let W(a, b, u)=&1+e &v( f (a)&b)u with u # U=R & , where v is a smooth, 2
This means W is continuous as an extended real-valued function.
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increasing, concave function with v(0)>0. Properties (A1), (A4), and (A5) are obvious. Set $ & =e &v( ;) and $ + =e &v(0) to satisfy (A2). To see that (A3) holds, examine n
\
t&1
+
W nu(k)=& : exp & : v( f (k {&1 )&k { ) +u t=1
\
n
{=1
+
_exp & : v( f (k t&1 )&k t ) . t=1
Now W nu is concave because the mapping x [ &e &x is concave and each of the sums t&1 {=1 v( f (k {&1 &k { ) is concave. To derive the utility function, let u=sup U. 3 Then W(k 0 , k 1 , u) u, i.e., (k) for all n. W 1u(k) W 0u(k). An easy induction shows W nu(k) W n&1 u Define the utility function U(k)=inf n W nu(k). The Lipschitz condition implicit in (A2) implies that |W nv &W nu | ($ + ) n |u&v| for any finite u, v # U. As a result, the infimum of the W nu does not depend on u. The (Sk))= function U is recursive as U(k)=inf W nu(k)=infW(k 0 , k 1 , W n&1 u (S k))=W(k , k , U(S k)). The infimum is upper semiW(k 0 , k 1 , inf W n&1 0 1 u continuous in the product topology. Condition (A3) guarantees that U is concave. When (k, k) # T, define 8(k)=U(k, k. . .). Note that if W(k, k, u)>& for some u # U then 8(k)>& . Of course 8 inherits upper semicontinuity from U. We also need a joint assumption on W and T. Even in the additive case, productivity must be sufficient to overcome discounting if there is to be a steady state. Define $ k =W 3(k, k, 8(k)). This is the discount factor on the constant path (k, k, . . .). Let $=inf $ k $ & >0. $-Normality Assumption. The reduced-form model (T.W) is $-normal if there is a pair (a, b ) # T such that 0Ra $b and W(a, b, u)>8(0) for some u # U, where we interpret 8(0) as & if (0, 0) T. There is a finite u # U with W(a, b, u ) u because of (A2). By (A4), W(b, b, u ) u. This implies W nu (b, b, . . .) is non-decreasing, and so 8(b ) u.
3. THE EXISTENCE OF STEADY STATES In this section, we will assume (T, W), obeys (T1)(T5), (A1)(A5), and $-normality. Define T 1 =[a # R m +: (a, b) # T for some b and &a& ;] and K=[k # R m +: (k, k) # T and 8(k) u ]. We will obtain a steady state in K. 3
This construction is based on the ``partial summation'' method of Boyd [2].
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Of course, &k& ; whenever k # K, so K/T 1 . Note that K and T 1 are compact and convex and contain b. For k # T 1 , denote the closest point in K to k by g(k). This defines a continuous function from T 1 to K. The mappings k [ $ g(k) and 9(k)= 8(g(k)) extend $ k and 8 k . Abusing notation, we also denote the extension of $ k by $ k . For k # T 1 , define ,(k)=[(a, b) # T: a (1&$ k )+$ k b]. Because (a, b ) # T defines a correspondence. ,(k), ,(k) is non-empty and ,: T 1 If (a, b) # ,(k) with &a&>;, &a& (1&$ k ) &k&+$ k &b& (1&$ k ) &k&+ $ k &a& by (T5). But then, &a& &k& ;. It follows that &a& ; for all (a, b) # ,(k). By (T5), &b& ;. It follows that ,(k) is compact and convex for every k # T 1 . A stock k # K is a recursive golden rule stock if (k, k) # T and W(k, k, 8(k)) W(a, b, 8(k)) for all (a, b) # ,(k). When W is additively separable, with W(a, b, u)=v(a, b)+$u, this definition reduces to the usual definition of a discounted golden rule. A golden rule is non-trivial if 8(k)>8(0). Theorem 1. Suppose (A1)(A4), (T1)(T5), and $-normality hold. If k is a recursive golden rule, the stationary path (k, k, . . .) is optimal from k. Proof. By $-normality, ,(k) has an interior. Now (a, b) [ W(a, b, 8(k)) is concave on ,(k) and has a maximum at (k, k). Thus there is a vector ( p, q) which supports both W and ,(k). In other words, W(a, b, 8(k)) W(k, k, 8(k))+p(a&k)+q(b&k) for all (a, b) # T, and ( p, q)=*( p$, q$)+ m i=1 * i (e i &$ k e i ) for some *, * i 0, where ( p$, q$) supports T. 4 That is, p$a+q$b pk+qk for all (a, b) # T. Let x be feasible from k and let k=(k, k, . . .). Then W n8(k)(k)=8(k)= U(k). Because W n8(k) is concave, and W is continuously differentiable and non-decreasing in the third argument, we have W n8(k)(x)&W n8(k)(k) n&1
(x t &k)+$ nk q(x n &k). p(x 0 &k)+ : ($ k p+q) $ t&1 k t=1
Recalling that x 0 =k, and taking the limit as n yields U(x) t&1 (x t &k). U(k)+ t=1 ($ k p+q) $ k Now $ k p+q=*$ k p$+*q$+ m i=1 * i ($ k e i &$ k e i )=*$ k p$+*$ k q$. t&1 (x t&1 , x t ) # T. 5 Consider the convex combination (1&$ k ) &1 t=1 $ k 4
Here e i denotes the ith standard basis vector of R m. This construction has been used in a primal context by Dechert and Nishimura [4] and Khan and Mitra [9]. 5
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t&1 By the support property of ( p$, q$), ( p$x t&1 + q$x t ) t=1 $ k t&1 t=1 $ k ( p$k+q$k). Since x is feasible from x 0 =k 0 , we can rewrite t&1 ($ k p$+q)(x t &k) 0. But $ k p+q=$ k p$+q$, so to obtain t=1 $ k U(x) U(k). Thus k is optimal from k. K
Because recursive golden rules are steady states, it is enough to show that recursive golden rules exist. Given any stock k # T 1 , the implicit programming problem is to solve J(k)=sup[W(a, b, 9(k)): (a, b) # ,(k)]. This problem has a solution because ,(k) is compact and W(a, b, 9(k)) is upper semicontinuous in (a, b). By $-normality, (a, b ) # ,(k) for all k # K and J(k) W(a, u, 9(k)). Since 9(k) u, it follows that J(k) W(a, b, u ) u > & by free disposal and monotonicity. The maximizer correspondence +(k) is defined by +(k)=[(a, b) # ,(k): W(a, b, 9(k))=J(k)]. Note that +(k) is also non-empty, compact, and convex for each k # K. Thus a stock k # K is a recursive golden rule if and only if (k, k) # +(k). We will only show the existence of non-trivial golden rules when the technology is both bounded and $-normal. As is well known, discounted golden rules may not exist if the technology is not bounded, as in models that generate balanced growth. If the technology is bounded but not sufficiently productive, all optimal paths may converge to 0. The $-normality condition rules out cases where all optimal paths converge to 0 and is a generalization of the requirement that $f $(0+ )>1 in the onesector model. The proof of existence of golden rules proceeds as follows. We start with a preliminary lemma that shows that 9 and $ k are continuous on T 1 . We next show , has a lower semicontinuity property in Lemma 2. We then use that fact to show +(k) is a closed correspondence in Lemma 3. 6 Finally, we project + onto the input space T 1 . This new correspondence on T 1 is closed with compact and convex values. The Kakutani Fixed Point Theorem yields a fixed point of , and that fixed point is a recursive golden rule. Lemma 1.
The function 9 and the mapping k [ $ k are continuous on T 1 .
Proof. Because g is continuous, it is enough to show $ k and 8(k) are continuous on K. Let =>0 and k # K. An induction shows that for 6 A correspondence , is closed if whenever x n x and y n y with y n # ,(x n ), we have y # ,(x).
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k$ # K, |W n8(k)(k$)&8(k)| (1&$ + ) &1 |W(k$, k$, 8(k))&W(k, k, 8(k))|, where k$=(k$, k$, . . .). Because W(k, k, 8(k))> & , (A1) implies we may pick '>0 with |W n8(k)(k$)&8(k)| <=2 for &k$&k&<'. Letting n shows |8(k$)&8(k)| <= for &k$&k&<'. Now $ k =W 3(k, k, 8(k)). By (A2) and the continuity of 8, $ k is also continuous. K Lemma 2. Suppose (x, y) # ,(k) with (x, y){(a, b ). If =>0, there is : with 0<:
T is a closed correspondence. The correspondence +: T 1
Proof. As in Lemma 1, we need only to show + is closed on K. Suppose (a n , b n ) # +(k n ) with (a n , b n ) (a, b) and k n k. Then (a, b) # T by closure and a (1&$ k ) k+$ k b, so (a, b) # ,(k). Since W and 8 are upper semicontinuous and W is non-decreasing in u, J(k) W(a, b, 9(k))= lim sup W(a n , b n , 8(k n ))=lim sup J(k n ). We need only show W(a, b, 8(k)) =lim sup W(a n , b n , 8(k n )) J(k) to complete the proof. Note that W(a, b, 8(k)) lim sup J(k n ) W(a, b, 8(k)). Suppose J(k)>W(a, b, 8(k)). Then J(k)>W(a, b, 8(k)). Take (x, y) # ,(k) with W(x, y, 8(k))=J(k). By Lemma 2, we can construct a subsequence of the k n (also labeled k n ) and : n <1n with (x n , y n ) # ,(k n ), where x n = (1&: n ) x+: n a and y n =(1&: n ) y+: n b. Now W(x n , y n , 8(k n )) J(k n ) because (x n , y n ) # ,(k n ). Furthermore, W(x n , y n , 8(k n )) (1&: n ) W(x, y, 8(k n ))+: n W(a, b, 8(k n ))
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by concavity. Rearranging and using J(k n ) W(x n , y n , 8(k n )), we obtain : n[W(x, y, 8(k n ))&W(a, b, 8(k n ))] W(x, y, 8(k n ))&J(k n ).
(1)
Let =>0. Taking n large enough, (1) shows =>[W(x, y, 8(k n ))&J(k n )]. But W(x, y, 8(k n )) J(k), so =2+J(k n )>J(k) for n large. It follows that J(k) lim sup J(k n ) lim inf J(k n ) J(k), and so W(a, b, 8(k))=J(k). This shows + is closed. K Theorem 2. Suppose (A1)(A5), (T1)(T5), and $-normality hold. A non-trivial recursive golden rule exists. T 1 by (k)=[a # T 1: (a, b) # Proof. Define a correspondence : T 1 +(k) for some b]. We know maps into T 1 because , and + are bounded by ;. It is clear that has convex, compact, non-empty values. It is also closed. Let k n k and a n a with a n # (k n ). Take b n with (a n , b n ) # +(k n ). Now &a n & ; and &b n & ;. By passing to a subsequence, we may assume b n b. But then (a, b) # +(k) because + is closed. Hence is closed. Apply the Kakutani Fixed Point Theorem to obtain k with k # (k). There is a b with (k, b) # +(k). Then k (1&$ k ) k+$ k b, which implies k b. By free disposal (k, k) # T, and W(k, k, 9(k)) W(k, b, 9(k))=J(k) by monotonicity. Now J(k)=W(k, k, 9(k)) W(a, b, 9(k)). Using (A5), we find W(k, k, u ) W(a, b, u )=u. Induction then shows W nu(k, k, . . .) u. Letting n shows 8(k) u, so k # K. Therefore K is is a recursive golden rule. From non-triviality, use (A5) to see that 8(k)=W(k, k, 8(k)) W(a, b, 8(k))>W(0, 0, 8(k)). If follows that W n8(k)(0, 0, . . .) is non-increasing. Taking the limit shows 8(k)>8(0). K Remark. If there is a u with 8(a) u whenever (a, a) # T and if W(a, b, u)>W(0, 0, u) for all u # U with u u, we can dispense with (A5) in Theorem 2. Note that (A5) is used twice, both times in the proof of Theorem 2. Under these alternative assumptions, the first use of (A5) is unnecessary. 8(k) u automatically. To show k is non-trivial, use Theorem 1 and monotonicity to find W(k, k, 8(k))=8(k) U(k, 0, . . .)= W(k, 0, 8(0)) W(k, k, 8(0)), implying 8(k) 8(0). Then W(k, k, 8(k)) W(a, b, 8(k))>W(0, 0, 8(k)) W(0, 0, 8(0))=8(0) establishes nontriviality. REFERENCES 1. R. A. Becker and C. Foias, A minimax approach to the implicit programming problem, Econ. Lett. 20 (1986), 171175. 2. J. H. Boyd III, Recursive utility and the Ramsey problem, J. Econ. Theory 50 (1990), 326345.
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3. D. Cass and K. Shell, The structure and stability of competitive dynamical systems, J. Econ. Theory 12 (1976), 3170. 4. W. D. Dechert and K. Nishimura, A complete characterization of optimal growth paths in an aggregated model with a non-concave production function, J. Econ. Theory 31 (1983), 332354. 5. L. G. Epstein, The global stability of efficient intertemporal allocations, Econometrica 55 (1987), 329355. 6. L. G. Epstein, A simple dynamic general equilibrium model, J. Econ. Theory 41 (1987), 6895. 7. J. Flynn, The existence of optimal invariant stocks in a multi-sector economy, Rev. Econ. Stud. 47 (1980), 809811. 8. T. Hansen and T. C. Koopmans, On the definition and computation of a capital stock invariant under optimization, J. Econ. Theory 5 (1972), 487523. 9. M. A. Khan and T. Mitra, On the existence of a stationary optimal stock for a multi-sector economy: A primal approach, J. Econ. Theory 40 (1986), 319328. 10. T. C. Koopmans, Stationary ordinal utility and impatience, Econometrica 28 (1960), 287301. 11. R. E. Lucas, Jr. and N. L. Stokey, Optimal growth with many consumers, J. Econ. Theory 32 (1984), 139171. 12. L. W. McKenzie, A primal route to the turnpike and Liapounov stability, J. Econ. Theory 26 (1982), 194209. 13. L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in ``Handbook of Mathematical Economics, III'' (K. J. Arrow and M. D. Intrilligator, Eds.), North-Holland, New York, 1986. 14. B. Peleg and H. E. Ryder, Jr., The modified golden rule of a multi-sector economy, J. Math. Econ. 1 (1974), 193198. 15. W. R. S. Sutherland, On optimal development in a multi-sectoral economy: The discounted case, Rev. Econ. Stud. 37 (1970), 585588.
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