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.
Abstract: We develop 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 nonsmooth objectives. Our methods emphasize the role of the Strict Mangasarian–Fromovitz Constraint Qualification (SMFCQ), and include envelope theorems for both the convex and nonconvex case, allow for noninterior solutions as well as equality and inequality constraints. We give new sufficient conditions for the value function to be directionally differentiable, as well as continuously differentiable. We apply our results to stochastic growth models with Markov shocks and constrained lattice programming problems.
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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.
Abstract: We examine the effect of a change in interest rates on an agent’s consumption and savings decisions when her income is fluctuating In each period, a long-lived agent decides how much to save (ie, invest in a risky bond) and how much to consume while her income and the rate of return on her savings are uncertain and depend on the state of the economy We show that under the concavity of the consumption function, a condition that ensures that the substitution effect dominates the income effect, lower interest rates encourage the agent’s consumption across all states
10 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
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.
6 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
Posted Content•
TL;DR: In this paper, the authors established the argmin of a random objective function to be unique almost surely, without convexity of the objective function, and applied it to a variety of applications in statistics, including uniqueness of M estimators, both classical likelihood and penalized likelihood estimators.
Abstract: This paper establishes the argmin of a random objective function to be unique almost surely. This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed, including uniqueness of M-estimators, both classical likelihood and penalized likelihood estimators, and two applications of the argmin theorem, threshold regression and weak identification.
3 citations
References
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1,952 citations
TL;DR: The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative as mentioned in this paper.
Abstract: The standard envelope theorems apply to choice sets with convex and topological structure, providing sufficient conditions for the value function to be differentiable in a parameter and characterizing its derivative. This paper studies optimization with arbitrary choice sets and shows that the traditional envelope formula holds at any differentiability point of the value function. We also provide conditions for the value function to be, variously, absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to mechanism design, convex programming, continuous optimization problems, saddle-point problems, problems with parameterized constraints, and optimal stopping problems.
1,059 citations
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