A nonsmooth approach to envelope theorems
TL;DR: In this paper, a nonsmooth approach to envelope theorems applicable to a broad class of parameterized constrained nonlinear optimization problems that arise typically in economic applications with nonconvexities and/or non-smooth objectives was developed.
About: This article is published in Journal of Mathematical Economics.The article was published on 2015-12-01. It has received 10 citations till now. The article focuses on the topics: Differentiable function & Nonlinear programming.
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TL;DR: This paper examined the effect of a change in interest rates on an agent's consumption and savings decisions when her income is fluctuating and showed that lower interest rates encourage the agent to consume more.
11 citations
TL;DR: It is shown 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.
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.
8 citations
12 Aug 2018
TL;DR: In this paper, the authors survey how the methods of dynamic and stochastic games have been applied in macroeconomic research, focusing on strategic dynamic programming, which has found extensive application for solving macroeconomic models.
Abstract: In this chapter, we survey how the methods of dynamic and stochastic games have been applied in macroeconomic research. In our discussion of methods for constructing dynamic equilibria in such models, we focus on strategic dynamic programming, which has found extensive application for solving macroeconomic models. We first start by presenting some prototypes of dynamic and stochastic games that have arisen in macroeconomics and their main challenges related to both their theoretical and numerical analysis. Then, we discuss the strategic dynamic programming method with states, which is useful for proving existence of sequential or subgame perfect equilibrium of a dynamic game. We then discuss how these methods have been applied to some canonical examples in macroeconomics, varying from sequential equilibria of dynamic nonoptimal economies to time-consistent policies or policy games. We conclude with a brief discussion and survey of alternative methods that are useful for some macroeconomic problems.
8 citations
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TL;DR: In this paper, the converse envelope theorem with a converse was shown to be equivalent to a first-order condition for mechanism design, and was used to extend the canonical result in mechanism design that any increasing allocation is implementable.
Abstract: I prove an envelope theorem with a converse: the envelope formula is equivalent to a first-order condition. Like Milgrom and Segal's (2002) envelope theorem, my result requires no structure on the choice set. I use the converse envelope theorem to extend to abstract outcomes the canonical result in mechanism design that any increasing allocation is implementable, and apply this to selling information.
5 citations
TL;DR: In this article, a general framework is developed for studying screening in many-agent discrete type environments where each agent's preferences depend endogenously on the allocations received by the other agents, and the solution to the principal's problem is analyzed by decomposing it, a la Rothschild and Scheuer (2013, 2014b), into an inner problem with fixed preferences and an outer problem with varying preferences.
Abstract: A general framework is developed for studying screening in many-agent discrete type environments wherein each agent’s preferences depend endogenously on the allocations received by the other agents. Applications include optimal income taxation, performance contracting with across-worker externalities, and insurance with endogenous risks. The solution to the principal’s problem is analyzed by decomposing it, a la Rothschild and Scheuer (2013, 2014b), into an inner problem with fixed preferences and an outer problem with varying preferences. The outer problem is typically discontinuous at points where the preferences of two or more types endogenously coincide. As a result, the principal will frequently find it optimal to select allocations which involve two or more types with endogenously coinciding preferences, even though such allocations may appear, ex-ante, to be highly unusual. Assuming that types are strictly ordered by their single-crossing preferences is, therefore, not innocuous in endogenous preference environments.
4 citations
References
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01 Jan 1982
TL;DR: In this article, the marginal function of a mathematical program neither assumed convex in its variables or in its parameters is studied. But the bounds for the Dini directional derivatives and estimates for the Clarke generalized gradient are obtained.
Abstract: This paper consists in a study of the differential properties of the marginal or perturbation function of a mathematical programming problem where a parameter or perturbation vector is present. Bounds for the Dini directional derivatives and estimates for the Clarke generalized gradient are obtained for the marginal function of the mathematical program neither assumed convex in its variables or in its parameters. This study generalizes some previously published results on this subject for the special case of right-hand side parameters or perturbations.
180 citations
TL;DR: In this paper, necessary and sufficient conditions for optimal policy functions are derived in a regime in which future utilities are discounted, leading to an explicit optimal policy function, which is used to display the steady-state solution for the capital stock under an optimal policy.
161 citations
TL;DR: In this article, the problem of maximizing a real-valued function when the objective function is constrained to lie in some subset of R l is studied, and a natural way to order the constraint sets C and find the corresponding restrictions on the objective functions that guarantee that optimal solutions increase with the constraint set is developed.
Abstract: This paper develops and applies some new results in the theory of monotone comparative statics. Let f be a real-valued function defined on R l and consider the problem of maximizing f(x) when x is constrained to lie in some subset C of R'. We develop a natural way to order the constraint sets C and find the corresponding restrictions on the objective function f that guarantee that optimal solutions increase with the constraint set. We apply our techniques to problems in consumer, producer, and portfolio theory. We also use them to generalize Rybcsynski's theorem and the LeChatelier principle.
127 citations
TL;DR: It is shown that this new constraint qualification is a necessary and sufficient condition for the uniqueness of Kuhn—Tucker multipliers and implies the satisfaction of second order necessary optimality conditions at a local minimum.
Abstract: Recently Fujiwara, Han and Mangasarian introduced a new constraint qualification which is a slight tightening of the well-known Mangasarian—Fromovitz constraint qualification. We show that this new qualification is a necessary and sufficient condition for the uniqueness of Kuhn—Tucker multipliers. We also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum.
125 citations
01 Jan 1982
TL;DR: In this article, a Lagrange multiplier rule is derived for finite-dimensional optimization problems with locally Lipschitzian equality and inequality constraints and also an abstract constraint described by a closed set.
Abstract: For finite-dimensional optimization problems with locally Lipschitzian equality and inequality constraints and also an abstract constraint described by a closed set, a Lagrange multiplier rule is derived that is sharper is in some respects than the ones of Clarke and Hiriart-Urruty. The multiplier vectors provided by this rule are given meaning in terms of the generalized subgradient set of the optimal value function in the problem with respect to perturbational parameters. Bounds on subderivatives of the optimal value function are thereby obtained and in certain cases the existence of ordinary directional derivatives.
104 citations