A nonparametric analysis of the Cournot model

An observer makes a number of observations of an industry producing a homogeneous good. Each observation consists of the market price, the output of individual firms and perhaps information on each firm's production cost. We provide various tests (typically, linear programs) with which the observer can determine if the data set is consistent with the hypothesis that firms in this industry are playing a Cournot game at each observation. When cost information is wholly or partially unavailable, these tests could potentially be used to derive cost information on the firms. This paper is a contribution to the literature that aims to characterize (in various contexts) the restrictions that a data set must satisfy for it to be consistent with Nash outcomes in a game. It is also inspired by the seminal result of Afriat (and the subsequent literature) which addresses similar issues in the context of consumer demand, though one important technical difference from most of these results is that the objective functions of firms in a Cournot game are not necessarily quasiconcave

Discounting and Patience in Optimal Stopping and Control Problems

This paper establishes that the optimal stopping time of virtually any optimal stopping problem is increasing in "patience," understood as a particular partial order on discount rate functions. With Markov dynamics, the result holds in a continuation- domain sense even if stopping is combined with an optimal control problem. Under intuitive additional assumptions, we obtain comparative statics on both the optimal control and optimal stopping time for one-dimensional diusions. We provide a simple example where, without these assumptions, increased patience can precipitate stopping. We also show that, with optimal stopping and control, a project's expected value is decreasing in the interest rate, generalizing analogous results in a deterministic context. All our results are robust to the presence of a salvage value. As an application we show that the internal rate of return of any endogenously-interrupted project is essentially unique, even if the project also involves a management problem until its interruption. We also apply our results to the theory of optimal growth and capital deepening and to optimal bankruptcy decisions.capital growth, comparative statics, discounting, internal rate of return, optimal control, optimal stopping, patience, present value, project valuation

Comparative Statics, Informativeness, and the Interval Dominance Order

We identify a natural way of ordering functions, which we call the interval dominance order and develop a theory of monotone comparative statics based on this order. This way of ordering functions is weaker then the standard one based on the single crossing property (Milgrom and Shannon, 1994) and so our results apply in some settings where the single crossing property does not hold. For example, they are useful when examining the comparative statics of optimal stopping time problems. We also show that certain basic results in statistical decision theory which are important in economics – specifically, the complete class theorem of Karlin and Rubin (1956) and the results connected with Lehmann’s (1988) concept of informativeness – generalize to payoff functions obeying the interval dominance order.single crossing property, interval dominance order, supermodularity, comparative statics, optimal stopping time, complete class theorem, statistical decision theory, informativeness

Law of Demand

We formulate several laws of individual and market demand and describe their relationship to neoclassical demand theory. The laws have implications for comparative statics and stability of competitive equilibrium. We survey results that offer interpretable sufficient conditions for the laws to hold and we refer to related empirical evidence. The laws for market demand are more likely to be satisfied if commodities are more substitutable. Certain kinds of heterogeneity across individuals make the laws more likely to hold in the aggregate even if they are violated by individuals.

A Nonparametric Analysis of the Cournot Model

An observer makes a number of observations of an industry producing a homogeneous good. Each observation consists of the market price, the output of individual rms and perhaps information on each rm's production cost. We provide various tests (typically, linear programs) with which the observer can determine if the data set is consistent with the hypothesis that rms in this industry are playing a Cournot game at each observation. When cost information is wholly or partially unavailable, these tests could potentially be used to derive cost information on the rms. This paper is a contribution to the literature that aims to characterize (in various contexts) the restrictions that a data set must satisfy for it to be consistent with Nash outcomes in a game. It is also inspired by the seminal result of Afriat (and the subsequent literature) which addresses similar issues in the context of consumer demand, though one important technical di erence from most of these results is that the objective functions of rms in a Cournot game are not necessarily quasiconcave. Keywords:

The Weak Axiom and Comparative Statics

This paper examines conditions which guarantee that the excess demand function of an exchange economy will satisfy the weak axiom in an open neighborhood of a given equilibrium price. This property ensures that the equilibrium is locally stable with respect to Walras' tatonnement. A related issue is the possibility of local comparative statics; in particular, the paper examines conditions which guarantee that when an economy's endowment is perturbed, the equilibrium price will move in a direction opposite to that of the perturbation. A distinguishing feature of this paper's approach is the heavy use of the indirect utility function, though we also provide results that allow for the translation of conditions imposed on indirect utility functions to conditions imposed on direct utility functions. Indeed we apply this to the special case of exchange economies where all agents have directly additive utilities - essentially a complete markets finance model with agents having von Neumann-Morgenstern utility functions. We show that the structural properties of demand near an equilibrium price depend on variations in the coefficient of relative risk aversion.

Comparative Statics with Concave and Supermodular Functions

Certain problems in comparative statics, including (but not exclusively) certain problems in consumer theory, cannot be easily addressed by the methods of lattice programming. One reason for this is that there is no order on the choice space which orders choices in a way which conforms with the comparison desired, and which also orders constraint sets in the strong set order it induces. The objective of this paper is to show how lattice progamming theory can be extended to deal with situations like these. We show that the interaction of concavity and supermodularity in objective or constraint functions yield a structure that is very useful for comparative statics.lattices, concavity, supermodularity, comparative statics, demand, normality

Risk Aversion over Incomes and Risk Aversion over Commodities

This note determines the precise connection between an agent's attitude towards income risks and his attitude over risks in the underlying consumption space. Our results follow a general mathematical theory connecting the curvature properties of an objective function with the ray-curvature properties of its dual.risk aversion, concavity, duality

Comparative Statics with Concave and Supermodular Functions

Certain problems in comparative statics, including (but not exclusively) certain problems in consumer theory, cannot be easily addressed by the methods of lattice programming. One reason for this is that there is no order on the choice space which orders choices in a way which conforms with the comparison desired, and which also orders constraint sets in the strong set order it induces. The objective of this paper is to show how lattice programming theory can be extended to deal with situations like these. We show that the interaction of concavity and supermodularity in objective or constraint functions yield a structure that is very useful for comparative statics.

Revealed Price Preference: Theory and Empirical Analysis

With the aim of determining the welfare implications of price change in consumption data, we introduce a revealed preference relation over prices. We show that an absence of cycles in this preference relation characterizes a model of demand where consumers trade-off the utility of consumption against the disutility of expenditure. This model is appropriate whenever a consumer's demand over a {\em strict} subset of all available goods is being analyzed. For the random utility extension of the model, we devise nonparametric statistical procedures for testing and welfare comparisons. The latter requires the development of novel tests of linear hypotheses for partially identified parameters. In doing so, we provide new algorithms for the calculation and statistical inference in nonparametric counterfactual analysis for a general partially identified model. Our applications on national household expenditure data provide support for the model and yield informative bounds concerning welfare rankings across different prices.Comment: 53 page