Infinite-horizon choice functions

We analyze infinite-horizon choice functions within the setting of a simple technology. Efficiency and time consistency are characterized by stationary consumption and inheritance functions, as well as a transversality condition. In addition, we consider the equity axioms Suppes–Sen, Pigou–Dalton, and resource monotonicity. We show that Suppes–Sen and Pigou–Dalton imply that the consumption and inheritance functions are monotone with respect to time—thus justifying sustainability—while resource monotonicity implies that the consumption and inheritance functions are monotone with respect to the resource. Examples illustrate the characterization results.     

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.     

Strong comparative statics of equilibria

Some results in the monotone comparative statics literature tell us that if a parameter increases, some old equilibria are smaller than some new equilibria. We give a sufficient condition such that at a new parameter value every old equilibrium is smaller than every new equilibrium. We also adapt a standard algorithm to compute a minimal such newer parameter value and apply this algorithm to a game of network externalities. Our results are independent of a theory of equilibrium selection and are valid for games of strategic complementarities.     

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 sufficiently 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 fixed interval (as long as the price caps are binding for some values in that interval).     

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.     

Regular Quantal Response Equilibrium

The structural Quantal Response Equilibrium (QRE) generalizes the Nash equilibrium by augmenting payoffs with random elements that are not removed in some limit. This approach has been widely used both as a theoretical framework to study comparative statics of games and as an econometric framework to analyze experimental and field data. The framework of structural QRE is flexible: it can be applied to arbitrary finite games and incorporate very general error structures. Restrictions on the error structure are needed, however, to place testable restrictions on the data (Haile et al., 2004). This paper proposes a reduced-form approach, based on quantal response functions that replace the best-response functions underlying the Nash equilibrium. We define a regular QRE as a fixed point of quantal response functions that satisfies four axioms: continuity, interiority, responsiveness, and monotonicity. We show that these conditions are not vacuous and demonstrate with an example that they imply economically sensible restrictions on data consistent with laboratory observations. The reduced-form approach allows for a richer set of regular quantal response functions, which has proven useful for estimation purposes. JEL Classification: D62, C73     

An additively separable representation in the Savage framework

This paper proposes necessary and sufficient conditions for an additively separable representation of preferences in the Savage framework (where the objects of choice are acts: measurable functions from an infinite set of states to a potentially finite set of consequences). A preference relation over acts is represented by the integral over the subset of the product of the state space and the consequence space which corresponds to the act, where this integral is calculated with respect to an evaluation measure on this space. The result requires neither Savage's P3 (monotonicity) nor his P4 (weak comparative probability). Nevertheless, the representation it provides is as useful as Savage's for many economic applications.     

Topics in nonparametric comparative statics and stability

This paper presents the analysis of the Jacobian matrix of a small, macroeconomic model to determine the robustness of the model's comparative statics. The values of the model's coefficients are assessed with respect to seven alternative estimation strategies such as ordinary least squares or two-stage least squares. For each alternative version, the invertibility and stability of the Jacobian matrix is studied. The model is robust to the degree that these characteristics are shared by any matrix with the same sign pattern or other similar nonparametric conditions on its entries. One way to address the relative success of the different estimation strategies is through the robustness of the comparative statics of the resulting model.     

Stochastic dominance and thick-tailed wealth distributions

Right-skewed and thick-tailed wealth distributions have been documented as an empirical regularity across space and time. A key mechanism for explaining these distributional features is proportional random growth. We investigate the comparative statics of a well-defined class of random growth models when allowing for stochastically ordered shifts in the wealth return process. An order-contingent monotone comparative statics property is identified, according to which pure increases in risk (e.g. higher volatility of capital returns) foster top wealth concentration whereas first-order stochastically dominated shifts in the return process (induced by e.g. proportional capital income taxation) rather lower inequality at the upper end of the distribution. Our analysis points to the potentially ambiguous effects on top wealth inequality of introducing or modifying capital income tax treatments in the presence of stochastic returns.     

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.     

A unified approach to comparative statics puzzles in experiments

The paper shows that several game-theoretic solution concepts provide similar comparative statics predictions over a wide class of games. I start from the observation that, in many experiments, behavior is affected by parameter shifts that leave the Nash equilibrium unchanged. I explain the direction of change with a heuristic structural approach, using properties such as strategic complementarities and increasing differences. I show that the approach is consistent with general comparative statics results for (i) the Nash equilibrium of a game with perturbed payoff functions, (ii) the quantal response equilibrium, (iii) level-k reasoning. I also relate the structural approach to equilibrium selection concepts.     

A program for algebraic and qualitative comparative statics and stability analysis

This paper presents a computer program designed by the author which determines algebraically and qualitatively both short- and long-run comparative statics multipliers for standard linear or linearized economic models. When the model is dynamic, i.e., when it involves linear of linearized differential equations, the program also provides the algebraic expressions and the qualitative signs for the coefficients of the characteristic polynomial.     

Asymmetric Monotone Comparative Statics for the Industry Compositions

Within a standard model of international trade with heterogenous firms and two asymmetric countries, we derive sufficient conditions for monotone comparative statics (MCS) for the industry composition. This model outcome is defined as first‐order stochastic dominance shifts in the equilibrium distributions of all activities across active firms. MCS for the industry composition occurs in a country that experiences a decline in its costs of serving the foreign market and meanwhile experiences an increase in its level of competition. In the other country, the industry‐level implications are exactly opposite. These clear industry‐level results hold while firms respond asymmetrically to the trade shock.     

A Graphical Approach to the Standard Principal-Agent Model

An obstacle to the teaching of principal-agent theory is the technical complexity and intractability of the general model. Even in academic studies strong assumptions are often imposed so as to derive an analytical solution. The author describes a graphical approach to the standard principal-agent model. Characterizing equilibrium in the contract space defined by the incentive parameter and insurance component of pay under a linear contract, this approach provides a simple and intuitive method for analyzing the principal-agent problem, which can be easily understood by students of economics with basic knowledge of algebra and differentiation. The approach has shown to be convenient and rich for comparative statics analyses.     

A qualitative approach to Markovian equilibrium in infinite horizon economies with capital

Using lattice programming and order theoretic fixpoint theory, we develop a new class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium decision processes for a large class of infinite horizon economies with capital. The class of economies includes models with public policy, valued fiat money, monopolistic competition, production externalities, and various other nonconvexities in the production sets. The results can be adapted to construct symmetric Markov equilibrium in models with many agents and market incompleteness. As the methods are constructive, they provide the foundations for a rigorous analysis of numerical approximation schemes that study extremal Markovian equilibrium. Equilibrium comparative statics results relative to the space of economies are available. Of independent interest, we provide new conditions for preserving complementarity under maximization, and new generalized envelope theorems for nonconcave dynamic programming problems. Our fixed point algorithms are sharp, and are able to distinguish sufficient conditions under which Markovian equilibrium form a complete lattice of Lipschitz continuous, uniformly continuous and semicontinuous monotone functions as well as unique continuously differentiable equilibrium.     

Welfare and comparative statics implications of fair social security : A steady-state analysis

This paper examines the steady-state welfare and comparative statics implications of fair social security in an overlapping generations model with uncertain lifetime. The model is designed to capture the effects of lifetime uncertainty extending over the individual's entire lifespan, the effects of intentional and unintentional bequests, and the effects of the earning profile. He shows that the welfare effects depend on the size of the benefits and the structure of the social security tax. In particular, even fair social security may have negative welfare implications if the underlying tax structure is ill-designed. Some comparative statics implications are analyzed.     

The purity of impure public goods

In this paper, we provide a new perspective on the links between the analysis of the voluntary provision of pure and impure public goods. In particular, it is shown that the impure public good model can be transformed into a pure public good one. This innovative method not only leads to new comparative statics results, but also provides new insights on the impure public good model, for example, on causes of the nonneutrality of income transfers with regard to Nash equilibria in the impure public good case.     

Empirical Analysis of Limit Order Markets

We provide empirical restrictions of a model of optimal order submissions in a limit order market. A trader's optimal order submission depends on the trader's valuation for the asset and the trade-offs between order prices, execution probabilities and picking off risks. The optimal order submission strategy is a monotone function of a trader's valuation for the asset. We test the monotonicity restriction in a sample of order submissions and their realized outcomes from the Stockholm Stock Exchange. We do not reject the monotonicity restriction for buy orders or sell orders considered separately, but reject the monotonicity restriction for buy and sell orders considered jointly.     

COMPARATIVE STATICS OF A MONOPOLISTIC FIRM FACING RATE‐OF‐RETURN AND COMMAND‐AND‐CONTROL POLLUTION CONSTRAINTS

The intrinsic comparative statics properties of a general rate‐of‐return regulated, profit‐maximizing model of a monopolist facing a command‐and‐control pollution constraint are derived. Recent advances in the theory of comparative statics are used to derive the basic comparative statics of the model, which are contained in an observable negative semi‐definite matrix and possess the form of Slutsky‐like expressions. We consider several command‐and‐control pollution constraints that are commonly implemented in practice, and conclude that the intrinsic comparative statics properties of the model are qualitatively invariant to the type of command‐and‐control pollution constraint imposed. We compare our results with those extant, and find that several basic results from the standard A–J model no longer hold in our model.