Demand Shifts and Second Degree Price discimination - the Impact of DVDs on the Motion Pictures Industry

This paper applies models of price discrimination to the motion picture industry. Movies are durable goods with no resale market. Therefore, price discrimination using time can be used. The distributors release the movie in two different periods: theaters and video. The first is a high quality product and the second is a low quality product issued in a later point in time. The quality gap between the two versions of the product has shrunk as the DVD technology penetrated the market. This paper compares two years: 1995 and 2000. Initial results show a difference between the two years. The most evident one is a shrinkage of the time between the theatrical release and the video releaseprice discrimination, motion picture industry, time consistency

The Dynamics of Retail Oligopolies

Dynamic Models, Retail Indusry, Markov perfect equilibrium, Oligopoly

Asymptotic Properties for a Class of Partially Identified Models

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include sample means and best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs, and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the identification region of the model with respect to the Hausdorff distance. We then show that the Hausdorff distance between the estimator and the population identification region, when properly normalized by square-root-of-n, converges in distribution to the supremum of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate this limiting distribution. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and to calculate the power of the test. We show that these results can be used to construct a confidence collection, that is a collection of sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test.

Asymptotic properties for a class of partially identified models

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include sample means and best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs, and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the identification region of the model with respect to the Hausdorff distance. We then show that the Hausdorff distance between the estimator and the population identification region, when properly normalized by vn, converges in distribution to the supremum of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate this limiting distribution. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and to calculate the power of the test. We show that these results can be used to construct a confidence collection, that is a collection of sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test.Partial Identification, Confidence Collections, Set-Valued Random Variables.

Sharp identification regions in models with convex moment predictions

We provide a tractable characterization of the sharp identification region of the parameters θ in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ 1, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ 1. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method. This paper is a revised version of CWP27/09.

Sharp identification regions in games

We study identification in static, simultaneous move finite games of complete information, where the presence of multiple Nash equilibria may lead to partial identification of the model parameters. The identification regions for these parameters proposed in the related literature are known not to be sharp. Using the theory of random sets, we show that the sharp identification region can be obtained as the set of minimizers of the distance from the conditional distribution of game's outcomes given covariates, to the conditional Aumann expectation given covariates of a properly defined random set. This is the random set of probability distributions over action profiles given profit shifters implied by mixed strategy Nash equilibria. The sharp identification region can be approximated arbitrarily accurately through a finite number of moment inequalities based on the support function of the conditional Aumann expectation. When only pure strategy Nash equilibria are played, the sharp identification region is exactly determined by a finite number of moment inequalities. We discuss how our results can be extended to other solution concepts, such as for example correlated equilibrium or rationality and rationalizability. We show that calculating the sharp identification region using our characterization is computationally feasible. We also provide a simple algorithm which finds the set of inequalities that need to be checked in order to insure sharpness. We use examples analyzed in the literature to illustrate the gains in identification afforded by our method.Identification, Random Sets, Aumann Expectation, Support Function, Capacity Functional, Normal Form Games, Inequality Constraints.

Partial identification using random set theory

This paper illustrates how the use of random set theory can benefit partial identification analysis. We revisit the origins of Manski's work in partial identification (e.g., Manski (1989, 1990)), focusing our discussion on identification of probability distributions and conditional expectations in the presence of selectively observed data, statistical independence and mean independence assumptions, and shape restrictions. We show that the use of the Choquet capacity functional and of the Aumann expectation of a properly defined random set can simplify and extend previous results in the literature. We pay special attention to explaining how the relevant random set needs to be constructed, depending on the econometric framework at hand. We also discuss limitations in the applicability of specific tools of random set theory to partial identification analysis.