Monotone Comparative Statics Research Articles

Monotone Comparative Statics for Games with Strategic Substitutes

Under some conditions, parameterized games with strategic substitutes exhibit monotone comparative statics of equilibria. These conditions relate to a tradeoff between a direct parameter effect and an opposing, indirect strategic substitute effect. If the indirect effect does not dominate the direct effect, monotone comparative statics of equilibria are guaranteed. These conditions are available for best-response functions, differentiable payoff functions, and general payoff functions. Results are extended to correspondences, the analysis applies to asymmetric equilibria, and several examples are provided.

The Neglect of Monotone Comparative Statics Methods

Monotone methods enable comparative static analysis without the restrictive assumptions of the implicit-function theorem. Ease of use and flexibility in solving comparative static and game-theory problems have made monotone methods popular in the economics literature and in graduate courses, but they are still absent from undergraduate mathematical economics courses and textbooks. In this article, the authors illustrate the generality of monotone comparative statics relative to the implicit function approach. For example, to sign the effect of a discrete policy shift on a choice variable, the marginal returns will increase with the policy parameter. They also apply monotone methods in game theory settings. As mathematical economics courses and majors gain popularity, incorporating monotone methods into curriculum and textbooks would provide a modern treatment of comparative static analysis.

Monotone Comparative Statics with Separable Objective Functions

The sum of a supermodular function, assumed nondecreasing in the choice variable, and of a 'concavely supermodularizable' function, assumed nonincreasing in the parameter variable, satisfies the Milgrom- Shannon (1994, Monotone comparative statics, Econometrica 62, 157-180) single crossing condition. As an application, I prove existence of a pure strategy Nash equilibrium in a Cournot duopoly with logconcave demand, affiliated types, and nondecreasing costs.

Monotone Comparative Statics of Characteristic Demand

This study deals with comparative statics of the consumer’s demand. According to Lancaster [1966, Journal of Political Economy], a utility function should be defined on the set of all characteristics that affect well-being of the consumer and these characteristics should be derived by consuming commodities. We show the sufficient condition for monotone comparative statics of the demand for characteristics and investigate the properties of the demand for commodities by considering the relationship between characteristics and commodities. We do not restrict the domain of a utility function and price systems, that is, the set of characteristics, to the Euclidean space. In particular, we allow discrete choice and nonlinear price systems. This theory enables us to predict the properties of the demand for the commodities that are not present in the market, such as new commodities. Our sufficient condition on a direct utility function can be characterized by the properties of the welfare variations for the change in the level of characteristics, which are more transparent and easier to verify. Further, the results of this study are derived by employing a new mathematical technique, which can be regarded as the generalization of the lattice theoretical comparative statics.

A uniqueness proof for monetary steady state

The framework in Lagos and Wright (2005) [20] combining decentralized and centralized markets is used extensively in monetary economics. Much is known about that model, but there is a loose end: only under special assumptions about bargaining power or decentralized market preferences has it been shown that the monetary steady state is unique. For general decentralized market utility and bargaining, I prove uniqueness for generic parameters with fiat money, and for all parameters with commodity money. As a corollary, I get monotone comparative statics.

Lemons and Money Market

This paper identifies simple conditions for monotone comparative statics of a unique equilibrium in the Akerlof-Wilson model. Separate conditions apply to trade volume and price. Trade volume increases when supply becomes both stronger and more elastic. In contrast, price decreases when supply becomes both stronger and less elastic. An application to the interbank market suggests surprisingly specific measures to address elevated term rates and market breakdown.

Generalized monotonicity analysis

Complex economic models often lack the structure for the application of standard techniques in monotone comparative statics. Generalized monotonicity analysis (GMA) extends the available methods in several directions. First, it provides a way of finding parameter moves that yield monotonicity of model solutions. Second, it allows studying the monotonicity of functions or subsets of variables. Third, GMA naturally provides bounds on the sensitivity of variables to parameter changes. Fourth, GMA may be used to derive conditions under which monotonicity obtains with respect to functions of parameters, corresponding to imposed parameter moves. Fifth, GMA contributes insights into the theory of comparative statics, for example, with respect to dealing with constraints or exploiting additional information about the model structure. Several applications of GMA are presented, including constrained optimization, nonsupermodular games, aggregation, robust inference, and monotone comparative dynamics.

Testing Models With Multiple Equilibria by Quantile Methods

This paper proposes a method for testing complementarities between explanatory and dependent variables in a large class of economic models. The proposed test is based on the monotone comparative statics (MCS) property of equilibria. Our main result is that MCS produces testable implications on the (small and large) quantiles of the dependent variable, despite the presence of multiple equilibria. The key features of our approach are that (i) we work with a nonparametric structural model of a continuous dependent variable in which the unobservable is allowed to be correlated with the explanatory variable in a reasonably general way; (ii) we do not require the structural function to be known or estimable; (iii) we remain fairly agnostic on how an equilibrium is selected. We illustrate the usefulness of our result for policy evaluation within Berry, Levinsohn, and Pakes's (1999) model.

Comparative Statics, Informativeness, and the Interval Dominance Order

We identify a new way to order functions, called the interval dominance order, that generalizes both the single crossing property and a standard condition used in statistical decision theory. This allows us to provide a unified treatment of the major theorems on monotone comparative statics with and without uncertainty, the comparison of signal informativeness, and a non-Bayesian theorem on the completeness of increasing decision rules. We illustrate the concept and results with various applications, including an application to optimal stopping time problems where the single crossing property is typically violated.

Matching and Inequality in the World Economy

This paper develops tools and techniques to analyze the determinants of factor allocation and factor prices in economies with a large number of goods and factors. The main results of our paper characterize sufficient conditions for robust monotone comparative statics predictions in a Roy-like assignment model. These general results are then used to generate new insights about the consequences of globalization.

The global LeChatelier Principle and multimarket equilibria

LeChatelier (Comptes Rendus 99:786, 1884) and LeChatelier (Ann Mines 13(2):157, 1888) proposed that the adjustment of a physical system to a disturbance in its equilibrium was smaller as constraints are added to the adjustment process. Samuelson (Econometrica 28:368–379, 1960) showed that this Principle applied to a stable multimarket equilibrium for the case that all commodities are gross substitutes. We note that the Principle also applies to the stable equilibria of markets for which commodities can also be gross complements as formulated by Morishima (Osaka Econ Pap 1:101–113, 1952). Further, the Morishima and gross substitute cases are the only cases to which the Principle can apply, based only upon the stability hypothesis and a specification of the gross substitute/complement relations among commodities. We show that the Milgrom and Shannon (Econometrica 62:157–180, 1994) single crossing conditions specified for monotone comparative statics can be configured appropriate to the Morishima case to give global conditions for the LeChatelier Principle to apply to a multimarket equilibrium.

The Neglect of Monotone Comparative Static Methods

Monotone methods enable comparative static analysis without the restrictive assumptions of the implicit function theorem. Ease of use and flexibility in solving comparative static and game theory problems have made monotone methods popular in the economics literature and in graduate courses, but they are still absent from undergraduate mathematical economics courses and textbooks. We illustrate the generality of monotone comparative statics relative to the implicit function approach. For example, to sign the effect of a discrete policy shift on an optimal control or choice variable, one need only check that marginal returns increase with the policy parameter. We also apply monotone methods in game theory settings. As mathematical economics courses and majors gain popularity, incorporating monotone methods into curriculum and textbooks would provide a modern treatment of comparative static analysis.

Ordinal notions of submodularity

We consider several ordinal formulations of submodularity, defined for arbitrary binary relations on lattices. Two of these formulations are essentially due to Kreps [Kreps, D.M., 1979. A representation theorem for “Preference for Flexibility”. Econometrica 47 (3), 565–578] and one is a weakening of a notion due to Milgrom and Shannon [Milgrom, P., Shannon, C., 1994. Monotone comparative statics. Econometrica 62 (1), 157–180]. We show that any reflexive binary relation satisfying either of Kreps’s definitions also satisfies Milgrom and Shannon’s definition, and that any transitive and monotonic binary relation satisfying the Milgrom and Shannon’s condition satisfies both of Kreps’s conditions.

Monotone Comparative Statics: Geometric Approach

We consider the comparative statics of solutions to parameterized optimization problems. A geometric method is developed for finding a vector field that, at each point in the parameter space, indicates a direction in which monotone comparative statics obtains. Given such a vector field, we provide sufficient conditions under which the problem can be reparameterized on the parameter space (or a subset thereof) in a way that guarantees monotone comparative statics. A key feature of our method is that it does not require the feasible set to be a lattice and works in the absence of the standard quasi-supermodularity and single-crossing assumptions on the objective function. We illustrate our approach with a variety of applications.

The Comparative Statics of Constrained Optimization Problems

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.

Screening contracts in the presence of positive network effects

Based on the critical assumption of complementarity, this paper builds a general model to describe and solve the screening problem faced by a monopolist seller of a network good. By applying strict monotone comparative statics, we demonstrate that the joint presence of asymmetric information and positive network effects leads to a strict downward distortion from the welfare-maximizing allocation for all consumers. We also show that the equilibrium allocation is an increasing function of network effect intensity.

Monotone Comparative Statics in Ordered Vector Spaces

This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the Henstock-Kurzweil integral, we generalize existing results on increasing differences and supermodularity for C1 or C2 functions. None of the results are based on the assumption that the order is Euclidean. As applications we consider a teamwork game and a monopoly union model.

On Price Caps Under Uncertainty

This paper shows how standard arguments supporting the impo- sition of price caps break down in the presence of demand uncertainty. In particular, though in the deterministic case the introduction or lowering of a price cap (above marginal cost) results in increased production, increased total welfare, decreased prices, and increased consumer welfare, we show that all of the above comparative statics predictions fail for generic uncertain demand functions. For example, for price caps su-ciently close to marginal cost, a decrease in the price cap always leads to a decrease in production and total welfare under certain mild conditions. Under stronger regularity assumptions, all of the monotone comparative statics predictions from the deterministic case also do not hold for a generic uncertain demand if we restrict attention to price caps in an arbitrary flxed interval (as long as the price caps are binding for some values in that interval).

On Price Caps Under Uncertainty

This paper shows how standard arguments supporting the imposition of price caps break down in the presence of demand uncertainty. In particular, though in the deterministic case the introduction or lowering of a price cap (above marginal cost) results in increased production, increased total welfare, decreased prices, and increased consumer welfare, we show that all of the above comparative statics predictions fail for generic uncertain demand functions. For example, for price caps su ciently close to marginal cost, a decrease in the price cap always leads to a decrease in production and total welfare under certain mild conditions. Under stronger regularity assumptions, all of the monotone comparative statics predictions from the deterministic case also do not hold or a generic uncertain demand if we restrict attention to price caps in an arbitrary fixed interval (as long as the price caps are binding for some values in that interval).

Monotone Comparative Statics for Models of Politics

We elucidate a powerful yet simple method for deriving comparative statics conclusions for a wide variety of models: Monotone Comparative Statics (Milgrom and Shannon 1994). Monotone comparative static methods allow researchers to extract robust, substantive empirical implications from formal models that can be tested using ordinal data and simple nonparametric tests. When these methods apply, they can replace a diverse range of more technically difficult mathematics (facilitating richer, more realistic models), assumptions that are hard to understand or justify substantively (highlighting the political intuitions underlying a model's results), and a complicated set of methods for extracting implications from models. We present an accessible introduction to the central monotone comparative statics results and a series of practical tools for using these techniques in applied models (with reference to original sources, when relevant). Throughout we demonstrate the techniques with examples drawn from political science.