A method for implementing counterfactual experiments in models with multiple equilibria

Abstract

This paper proposes a method for implementing counterfactual experiments in estimated models that have multiple equilibria. The method assumes that the researcher does not know the equilibrium selection mechanism and wants to impose minimum restrictions on it. Our key assumption is that the equilibrium selection function does not jump discontinuously between equilibria as we change marginally the structural parameters of the model. Under this assumption, we show that, although the equilibrium selection function is unknown, the researcher can obtain an approximation of this function in a neighborhood of the estimated values of the structural parameters. Under the additional assumption that the counterfactual equilibrium is stable, this approximation can be combined with iterations in the equilibrium mapping to obtain the exact counterfactual equilibrium. We illustrate the differences between our approach and other methods, such as the selection of a counterfactual equilibrium that is closer to the equilibrium in the data, and equilibrium mapping iterations using the equilibrium in the data as the initial value. We show that, in general, these alternative methods are not consistent with the assumption that the equilibrium selection mechanism is continuous with respect to the structural parameters.

Recent Developments in Empirical IO: Dynamic Demand and Dynamic Games

“…Therefore, when a consumer makes a purchase, she is influenced by her current holdings of the good and by her expectations about future prices and attributes of available products. For 1 See, for example, Bresnahan (1981Bresnahan ( , 1987, Porter (1983), Hausman, Leonard and Zona (1994), Berry, Levinsohn and Pakes (1995), Goldberg (1995), Nevo (2001). 2 As we noted in the Introduction, our goal is not to provide a complete survey, so we will not offer a comprehnsive discussion of this wide litterature.…”

“…Here we describe a simple homotopy method that has been proposed in Aguirregabiria (2009) and applied in Aguirregabiria and Ho (2009). Under the assumption that the equilibrium selection mechanism, which is unknown to the researcher, is a smooth function of the structural parameters, we show how to obtain a Taylor approximation to the counterfactual equilibrium.…”

“…Once cost has been recovered, the estimated demand and cost can be used to study the form of competition, understand firm behavior, generate counterfactuals (e.g., the likely effect of a merger) or quantify welfare gains (for example, from the introduction of new products.) 1 Much of the literature has relied on static demand models for this type of exercise. However, in many markets demand is dynamic in the sense that (a) consumers current decisions affect their future utility, or (b) consumers' current decisions depend on expectations about the evolution of future states.…”

Natural Monopoly and Distorted Competition: Evidence from Unbundling Fiber-Optic Networks

“…This paper applies the methodologies used in Aguirregabiria (2012) and Aguirregabiria and Ho (2012). We calculate the counterfactual conditional choice probabilities using the following Taylor-series approximation:…”

A dynamic game of airline network competition: Hub-and-spoke networks and entry deterrence

“…sequential game of entry between an incumbent hub-and-spoke carrier and a point-to-point regional carrier. 3 The profit function of a hub-and-spoke airline is supermodular with respect to its entry decisions at different city-pairs. This complementarity implies that a hub-andspoke airline may be willing to operate non-stop flights in a city-pair even when profits are negative because operation in that city-pair can generate positive profits in other routes.…”