Adv. Math. Econ. 14, 39–67 (2011)

An overview of turnpike theory: towards the discounted deterministic case ^{3}

M. Ali Khan^{1} and Adriana Piazza ^{2}

1 Department of Economics, The Johns Hopkins University, Baltimore, MD 21218, USA (e-mail: akhan@jhu.edu)

2 Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. Espa.na 1680, Casilla 110-V, Valparaso, Chile (e-mail: adriana.piazza@usm.cl)

Received: July 22, 2010 Revised: July 30, 2010

JEL classification: C62, D90, Q23

Mathematical Subject Classification (2010): 91B62, 49J45, 49O2

Abstract. In the last 5 years, there has been extensive work on the existence and characterization of solutions of undiscounted optimal programs in simple discretetime models of the ‘choice of technique’ in development planning and of lumber extraction in the economics of forestry. In this expository essay, we present a unified treatment of the characterization results in two of these models. Furthermore, with an eye towards extensions to the discounted setting, we present the general theory, both with or without “smoothness hypotheses” on the felicity function, and in continuous-time and discrete-time taking special care to distinguish asymptotic convergence of optimal programs from their classical turnpike properties. We show how the general results do not translate directly to the particular toy models examined here, and thereby suggest open problems for a more universal theory.

Key words: Intertemporal resource allocation, discount factor, optimal programs, asymptotic convergence, classical turnpike theory, neighborhood turnpike theorem, development planning, forest management

3A preliminary version of this work was presented at the Workshop on Mathematical Economics held in Keio University, Tokyo during November 13–15, 2009. The authors are grateful to Professor Toru Maruyama for his invitation, and to him and Professors Alexander Ioffe, Alejandro Jofre, Boris Mordukhovich and Haru Takahashi for discussion and encouragement. Adriana Piazza gratefully acknowledges the financial support of Programa Basal PFB 03, Centro de Modelamiento Matematico (CMM), Universidad de Chile and that of FONDECYT under project 11090254. Ali Khan would like to acknowledge, in addition, the hospitality of the Departamento de Matemática, Universidad Técnica Federico Santa María, of the CMM and Facultad de Economía y Negocios at the Universidad de Chile during his visit in July 2010, when the final version of this work was completed.