Aggregate comparative statics

In aggregative games, each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players. Many common games in industrial organization, political economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in aggregative games. We focus on two classes of games: (1) aggregative games with strategic substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies. We provide simple sufficient conditions under which positive shocks to individual players increase their own actions and have monotone effects on the aggregate. The results are illustrated with applications to public good provision, contests, Cournot competition and technology choices in oligopoly.     

Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach

We analyze comparative risk aversion in a new way, through a comparative statics problem in which, for a cost, agents can shift from an initial probability distribution toward a preferred distribution. The Ross characterization arises when the original distribution is riskier than the preferred distribution and the cost is monetary, and the Arrow-Pratt characterization arises when the original distribution differs from the preferred distribution by a simple mean-preserving spread and the cost is a utility cost. Higher-order increases in risk lead to higher-order generalizations, and the comparative statics method yields a unified approach to the problem of comparative risk attitudes.     

Rationalizing multiple consumption-saving puzzles in a unified framework

Empirical evidence suggests that it may cost time, effort, and resources to implement an optimal consumption-saving plan, although the cost may differ across individuals. This paper explores the implications of such friction. We begin by documenting a series of facts on consumption and savings over the life cycle, each of which goes against the prediction of the standard life-cycle model. While the existing studies demonstrate how various permutations of the benchmark model may help resolve one or another of these puzzles, it appears difficult to jointly rationalize even a small subset of these facts within the existing theories. We then incorporate a feature of costly implementation into the benchmark model and show that the addition of this one feature moves in the right direction in jointly resolving all these puzzles. This friction is the sole and common mechanism in our model for rationalizing this series of facts, as we intentionally abstract from the mechanisms explored in the existing literature that are known to help explain one or another of these facts, in order to isolate the role of costly implementation. The implementation costs in the model are small, yet our results show that the mechanism can be important in complementing the existing theories to help account for these consumption and saving facts in a unified framework.     

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.     

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.     

A note on comparative statics and dynamics of stationary states

Certain results are communicated regarding effects of parameter changes, when the stationary state is stable, on the stationary state and an optimal paths in its neighbourhood in discrete time multi-sectoral optimal growth models where period utilities are discounted.     

Weak and strong monotone comparative statics

Summary This paper develops necessary and sufficient conditions for the set of solutions to an optimization problem to be nondecreasing in a weak sense still strong enough to guarantee the existence of an increasing selection, and thus strong enough to guarantee monotonicity when the solution is unique, as well as necessary and sufficient conditions for the set of optimizers to be nondecreasing in a strong sense which is strong enough to rule out the possibility of a decreasing selection. These necessary and sufficient conditions are variations of quasisupermodularity and the single crossing property introduced in Milgrom-Shannon [13]. Moreover, to determine when an objective function satisfies these conditions, this paper develops several characterizations of quasisupermodularity and the single crossing property and their variants, both in terms of differential conditions and in terms of restrictions on the structure of the level sets of these functions. Several examples are given to choice theory under loss aversion and to an auction problem.I am grateful to Don Brown and Paul Milgrom for numerous helpful conversations concerning earlier versions of this paper. This paper is a revised and expanded version of the paper An Ordinal Theory of Games with Strategic Complementarities (Shannon [15]). I am pleased to acknowledge the financial support of the National Science Foundation, and an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship.     

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.     

Necessary conditions for comparative statics under uncertainty

We investigate the comparative static properties of three classes of CDF changes: first and second degree stochastic dominant shifts, and mean preserving contractions. For each class, we provide conditions that are necessary and sufficient for a dominating shift to cause an unambiguous change in the choice variable. This allows us to see the trade-offs one must make between restrictions on preferences and CDF changes to obtain interesting comparative statics results. We then investigate the implications of our results for the two-period consumption-savings models and for distinguishing agents that do and do not obey the expected utility hypothesis.     

Nash equilibrium vs. maximin : A comparative game statics analysis

This paper derives comparative game statics results for both maximin and Nash strategies, thereby allowing a more systematic comparison between these two approaches.     

A distribution dynamics approach to regional GDP convergence in unified Germany

This paper uses nonparametric techniques to study GDP convergence across German labor market regions and counties during the period 1992–2004. The main result is that regional convergence in unified Germany has been substantial. In the first years after German unification the distribution of GDP has been characterized by a pronounced bimodality. The dispersion of the GDP distribution has become substantially smaller over time. Although some bimodality remains in most recent years, this bimodality is weak in comparison to previous years. Nevertheless, disparities among regions located in the Eastern and Western part of the country are still apparent.     

Welfare variations and the comparative statics of demand

By employing order-theoretic comparative statics, we provide necessary and sufficient conditions on the direct utility function (alternatively, on the indirect utility function) to guarantee that the compensating and equivalent variations are monotonically increasing in income. We also show that these conditions are sufficient to guarantee normal demand. Our results do not depend on the smoothness or strict quasiconcavity of the utility function, and can be extended to a setting with non-linear prices.     

Further results on comparative statics under uncertainty. A comment on Machnes

This paper re-examines the results in Machnes (1993). When a risk-averse firm faces both uncertain demand and uncertain fixed costs, the Arrow-Pratt theory of decreasing absolute risk aversion may be too weak to yield unambiguous comparative statics. Herein, it is shown that the stronger notions of risk behavior proposed by Ross (1981) and Kimball (1993) are useful in providing intuitive results in this context.     

Core structure and comparative statics in a hybrid matching market

Flexible firms compete by means of wages in the Assignment market while rigid firms have no flexibility over terms of appointment in the Marriage market. Workers trade with both kinds of firms in the hybrid market.Examples show that standard results that characterize the core of the Marriage market (respectively, Assignment market) are not robust to the entrance of flexible (respectively, rigid) firms to this market. A new algebraic structure provides a different characterization for the core of the hybrid model and reflects a sort of robustness to the exit of rigid (respectively, flexible) firms from this market. Meaningful comparative static results are derived.     

Dynamics and comparative statics in the theory of residential location

This paper introduces a dynamic adjustment mechanism of residential boundaries among a finite number of household classes in a circular city model. First we derive some meaningful conditions for all the residential boundaries to be locally stable. Then, assuming that the stability conditions are satisfied, we examine the comparative static properties of equilibrium boundary positions, e.g., the effects of changes in a land tax, an income subsidy, the number of households, etc.     

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.     

A unified approach to incorporating demographic or other effects into demand systems

"A general method of introducing demographic effects into any demand system, using modifying functions, is described which permits complicated interactions of demographic variables with prices and expeditures. Theorems give properties the modifying functions must have to ensure integrability of the resulting system. Demand equations of the new system are given explicitly as functions of the original demand equations and the modifying functions. The procedure is interpreted as altering a household's technology, and is shown to encompass adult equivalent scales and related methods. Examples of modifying functions are derived and applications of the technique to uses other than demographic variation are considered."