an:04114349
Zbl 0679.90014
Becker, Robert A.; Boyd, John H. III; Sung, Bom Yong
Recursive utility and optimal capital accumulation. I: Existence
EN
J. Econ. Theory 47, No. 1, 76-100 (1989).
00169791
1989
j
91B62 91B28 49J45
recursive utility; existence of optimal capital accumulation paths; Weierstrass theorem; upper semicontinuity
This paper demonstrates existence of optimal capital accumulation paths when the planner's preferences are represented by a recursive objective functional. Time preference is flexible. Mathematically, the problem is as follows:
\[
\max imize\quad \int^{\infty}_{0}L(t,k,\dot k)\exp (\int^{\infty}_{0}R(s,k,\dot k)ds)dt,
\]
s.t. k: \({\mathbb{R}}_+\to {\mathbb{R}}^ m\) is an absolutely continuous function; \(\dot k\in G(t,k)\) a.e.; \(0\leq k(0)\leq x\), where L: \(\Omega \to {\mathbb{R}}_+\) and R: \(\Omega\) \(\to {\mathbb{R}}\) are continuous on \(\Omega\) and convex in k, which are the felicity function and the discounting function of the economy respectively; \(\Omega \subset {\mathbb{R}}\times {\mathbb{R}}^ m\times {\mathbb{R}}^ m\) is the technology set of the economy and \(G(t,k)=\{y: (t,k,y)\in \Omega \}\) is the investment correspondence, which is compact-convex- valued and upper semicontinuous in t. Existence of optimal paths is addressed via the classical Weierstrass theorem. An improved version of a lemma due to Varaiya proves compactness of the feasible set for the compact-open topology. A monotonicity argument is combined with a powerful theorem of Cesari to demonstrate upper semicontinuity.
S.Shi