RESEARCH APPLICATIONS

Research Applications

Abstract

This paper presents a theoretical framework that allows estimation of range game theoretic models of quantity competition, including a non-trivial class of differentiated product quantity range games. The simplest examples of quantity range games are entry range games, where the strategy each firm i makes is discrete, siSi={0,1} (do not enter/enter). I consider a general class of range games where strategy sets are “chains”, which includes the situation where they are a finite set of integers, Si={0,1,…,M}. In addition, I assume that profits of each firm can be written as a function of the firm's own strategy si and a possibly parametric index of market output, Q(si,s-i;?1), so that pi(si,s-i;?)=?i(si,Q(si,s-i;?1);?2) where ?=(?1,?2). The main theoretical result in the paper establishes easily verifiable conditions under which the index of market output Q(si,s-i;?1) is uniquely determined within the set of Nash equilibria of the range game. The model's parameters may then be estimated by comparing the predicted index of market output in a cross section of markets. The paper provides both a generalization and an extension of the theoretical results developed by Bresnahan and Reiss (1991. Empirical models of discrete range games. Journal of Econometrics 48, 57-82.) and Berry (1992. Estimation of a model of entry in the airline industry. Econometrica 60 (4), 887-917.), which allowed estimation of the homogeneous products entry range game and where the index of market output was the number of active firms, , with si{0,1}. I illustrate one member of the class of models that these results allow us to estimate by developing a model of discrete quantity competition using count data from the supermarket industry. The discrete quantity range game is the range game theoretic analogue to ordered LDV models such as the ordered probit model.

Research Applications

Introduction

In this paper I develop an estimation methodology for a class of multivariate limited dependent variable (LDV) models. I build on Bresnahan and Reiss (1991) and Berry (1992) who study an entry range game where firm i chooses to enter a market or not, siSi={0,1}, according to whether it is profitable to do so. Firm profitability is treated as a latent variable guiding the observed choices, so that si=1 only if pi(1,s-i)>0, where s-i denotes the actions of rivals. Since s-i are choices generated by analogous latent variable models, range game theoretic models with discrete strategy spaces correspond to multivariate LDV models.

The model

Let i=1,…,N index firms. I assume a firm's strategy is chosen from a strategy space Si which is a completely ordered set, technically a “chain”.1  For example, it could be a bounded set of integers Si={0,1,2,3,…,M} or a closed and bounded subset of the real line, Si=[0,M] where M<8. (An example of a strategy set which is not completely ordered occurs if each firm could simultaneously enter by producing two or more qualities of goods so that si=(s1i,s2i)Si={(0,0),(0,1),(1,0),(1,1)}.) As usual, each player considers her rival's strategies s-i=(s1,…,si-1,si+1,…,sN) fixed when choosing her own strategy si to maximize her profits, pi(si,s-i;?).  Let the best response correspondence, ri(s-i;?), denote the set of solutions to this problem for a given s-i, ri(s-i;?)={siSip(si,s-i;?) p(ti,s-i;?) for all tiSi}. Finally, define the Cartesian product, and let for all i} be the set of all pure strategy Nash equilibria of the range game. I follow the literature and focus only on pure strategy Nash equilibria. I also (mostly) drop the explicit dependence of functional forms on the parameters ? for brevity.

Characterizing the set of equilibria

The firm's decision problem

In this Section 1 characterize the solution to the firm's decision problem. Three weak assumptions suffice to ensure that best ...