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-Fromowitz 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.
11 citations
11 Jun 2018
TL;DR: In this paper, the authors consider a bipartite network where potential buyers and sellers are divided into types, and they represent the compatibility across these types using a graph of nodes on one side correspond to buyer types and nodes on the other side to seller types.
Abstract: Platforms facilitating the exchange of goods and services between individuals are prevalent: one can purchase goods from others on eBay, arrange accommodation through Airbnb, and find temporary projects/workers on online labor markets such as Upwork. The majority of these markets exhibit three key features. First, the platforms do not dictate the transaction prices, i.e., buyers and sellers determine at which price the goods/services will be exchanged. Second, not all buyers or sellers on a platform are compatible. This may be due to taste differences (a buyer may be interested only in the types of goods/services a subset of the sellers offer), geographical or import/export restrictions (e.g., being able to provide services only regionally), or other sources of mismatch (e.g., a mismatch in the desired and available skills in online labor markets). Finally, buyers/sellers are heterogeneous in their valuations for goods or services they receive/provide.In exchange for facilitating trade, these platforms commonly obtain a commission from each transaction and/or charge subscription fees to sellers and buyers who access the platform. As such, their revenues depend both on the chosen commission/subscription fees and on the prices at which buyers/sellers transact. In settings that exhibit the aforementioned features, how should a platform design commission/subscription fees with the objective of maximizing its revenues? Is it sufficient to charge these to only one side of the market or is it necessary to charge them to both sides? What is the role of the underlying compatibility structure, and which structures are more conducive to higher revenues? In order to answer these questions, we consider a model where potential buyers and sellers are divided into types. Not all buyer and seller types are compatible with each other, and we represent the compatibility across these types using a bipartite network: nodes on one side correspond to buyer types, nodes on the other side correspond to seller types, and edges capture compatibility between different types of buyers and sellers. Agents within each type differ in their valuation for the goods they buy/sell. The platform chooses subscription fees, which must be paid by all agents who participate in the market. In addition, it chooses commissions, and a buyer/seller who exchanges goods/services pays a fraction of the transaction price to the platform as indicated by these commissions. The value distributions, subscription fees, and commissions are all possibly type-specific. Given the commissions/subscriptions chosen by the platform, the transaction prices and equilibrium trades are formed endogenously at a competitive equilibrium.We establish that, in order to maximize its revenues, the platform may need to charge different commissions/subscriptions to different types, depending on their network position. In fact, we show that if the same commissions/subscriptions are employed for all agents on the same side, the revenue loss can be unbounded. We complement this worst-case result by providing a bound on the revenue loss in terms of the supply/demand imbalance across the network under homogeneous value distributions. Surprisingly, we also show that, in general, charging commissions/subscriptions to only one side of the market (i.e., only to buyers or only to sellers) leads to lower revenues than optimal, even when different types on the same side can be charged different fees. Furthermore, we characterize the impact of the network structure on the revenues of the platform. Finally, we investigate how the commissions/subscriptions chosen by the platform impact social welfare. We establish that, under mild convexity assumptions on the value distributions, the revenue-maximizing commissions/subscriptions induce at least 2/3 of the maximum achievable social welfare.
10 citations
TL;DR: In this paper, the issue of measuring segregation in a population of small units, considering establishments in their application, is considered, where each establishment may have a different probability of hiring an individual from the minority group.
Abstract: We consider the issue of measuring segregation in a population of small units, considering establishments in our application. Each establishment may have a different probability of hiring an individual from the minority group. We define segregation indices as inequality indices on these unobserved, random probabilities. Because these probabilities are measured with error by proportions, standard estimators are inconsistent. We model this problem as a nonparametric binomial mixture. Under this testable assumption and conditions satisfied by standard segregation indices, such indices are partially identified and sharp bounds can be easily obtained by an optimization over a low dimensional space. We also develop bootstrap confidence intervals and a test of the binomial mixture model. Finally, we apply our method to measure the segregation of foreigners in small French firms.
Segregation small units partial identification C13 C14 J71
9 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
Cites background from "A Nonsmooth Approach to Envelope Th..."
...…given this fact, an immediate complication for characterizing incentive-constrained solutions is that value functions associated with recursive reformulations of these problems are generally not differentiable (e.g., see Rincón-Zapatero and Santos (2009) and Morand et al. (2015) for discussion)....
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References
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TL;DR: The Calculus of Variations as discussed by the authors is a generalization of the calculus of variations, which is used in many aspects of analysis, such as generalized gradient descent and optimal control.
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"A Nonsmooth Approach to Envelope Th..." refers background or methods in this paper
...2 in Clarke [7]....
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...The stability bound results follow from the application of Corollary 4 in Clarke [7] to the modified program, as the MFCQ for the original program is equivalent to the MFCQ for the above program, which implies (in Clarke’s terminology) that the set of abnormal multipliers reduces to {0}....
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...See also Clarke [7] and Bonisseau and LeVan [5]....
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..., q where I = {i : gi(a, s) = 0} 2 If A is closed, then the abstract constraint a ∈ A induces an additional term in the Lagrangian (see, for instance, Clarke [7], Chapter 6)....
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...The first inclusion is demonstrated in Clarke ([7], Corollary 1, p 242)....
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TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
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TL;DR: In this article, the authors introduce the concept of lattices, supermodular functions, and optimal decision models for cooperative games and non-cooperative games, and present a review of the literature.
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1,981 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.
1,183 citations