Generalized Envelope Theorems: Applications to Dynamic Programming

Olivier Morand (), Kevin Reffett () and Suchismita Tarafdar ()
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Olivier Morand: University of Connecticut
Kevin Reffett: Arizona State University
Suchismita Tarafdar: Shiv Nadar University

Journal of Optimization Theory and Applications, 2018, vol. 176, issue 3, No 8, 650-687

Abstract: Abstract We show in this paper that the class of Lipschitz functions provides a suitable framework for the generalization of classical envelope theorems for a broad class of constrained programs relevant to economic models, in which nonconvexities play a key role, and where the primitives may not be continuously differentiable. We give sufficient conditions for the value function of a Lipschitz program to inherit the Lipschitz property and obtain bounds for its upper and lower directional Dini derivatives. With strengthened assumptions we derive sufficient conditions for the directional differentiability, Clarke regularity, and differentiability of the value function, thus obtaining a collection of generalized envelope theorems encompassing many existing results in the literature. Some of our findings are then applied to decision models with discrete choices, to dynamic programming with and without concavity, to the problem of existence and characterization of Markov equilibrium in dynamic economies with nonconvexities, and to show the existence of monotone controls in constrained lattice programming problems.

Keywords: Envelope theorem; Lipschitz function; Clarke regularity; Supermodularity; Lipschitz dynamic programming; Recursive dynamic program (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10957-018-1241-5

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