Focusing on the testable revealed preference restrictions on the equilibrium manifold, we show that the rationalizability problem is NP-complete. Subsequently, we present a mixed integer programming (MIP) approach to characterize the testable implications of general equilibrium models. Attractively, this MIP approach naturally applies to settings with any number of observations and any number of agents. This is in contrast with existing approaches in the literature. We also demonstrate the versatility of our MIP approach in terms of dealing with alternative types of assignable information. Finally, we illustrate our methodology on a data set drawn from the US economy. In this application, an important focus is on the discriminatory power of the rationalizability tests under study.
Highlights
► Empirical application of general equilibrium models. ► Focus on the revealed preference restrictions on the equilibrium manifold. ► Verifying whether the data is a competitive equilibrium is an NP-complete problem. ► Applications of mixed integer programming techniques.
We introduced a mixed integer programming (MIP) approach for verifying testable implications on the equilibrium manifold. A core motivation for our MIP approach is that the rationalizability problem for general equilibrium models is View the MathML sourceNP-complete, which suggests using easy-to-implement non-polynomial time algorithms (such as MIP algorithms) for checking rationalizability. Interestingly, our MIP approach naturally allows for dealing with any number of agents and observations. This contrast with existing approaches, and is particularly convenient in view of empirical applications. We further demonstrated the versatility of our MIP approach by showing that it naturally deals with alternative types of assignable information. Finally, we illustrated the practical usefulness of the MIP methodology by means of an empirical application to data drawn from the US economy. In this application, an important focus was on the power of the rationalizability conditions under consideration. We conclude that, for our data set, assignable quantity information is crucial for meaningful (= powerful) tests. More generally, our application suggests using additional prior structure for the rationalizability condition in order to obtain tests with reasonable power. The MIP approach set out in this paper may provide a useful basis for imposing such a structure.
To focus our discussion, we have concentrated on testing consistency with alternative specifications of the general equilibrium model. If the behavior observed is consistent with a particular specification (i.e. can be rationalized), then the natural question that follows pertains to recovering/identifying the structural features of the model under consideration. For example, such a recovery analysis can focus on the individual preferences and/or individual private quantities. Chiappori et al. (1999) derived identifiability/recoverability results for general equilibrium models, which enable recovery by starting from some a priori (parametric) specification of the individual utility functions. Because the approach discussed in this paper does not require a (usually non-verifiable) prior specification for the utility functions, it address recovery questions by ‘letting the data speak for themselves’ (i.e., it only uses the information that is directly revealed by the data). See, for example, Afriat (1967) and Varian, 1982 and Varian, 2005 for detailed discussions of revealed preference recoverability; these authors consider the basic setting with a single consumer, and thus start from the rationalizability concept in Definition 1. As for the general equilibrium setting considered in this paper, recovery can proceed along the lines of Cherchye et al. (2011), who focused on the MIP revealed preference restrictions of collective household consumption models.
Finally, the rationalizability tests discussed above are ‘sharp’ tests; they only tell us whether observations are exactly consistent with the rationalizability condition that is under evaluation. However, as argued by Varian (1990), the exact consistency may not be a very interesting hypothesis. Rather, one may be interested whether the behavioral model under study provides a reasonable way to describe the behavior observed. For most purposes, ‘nearly optimizing behavior’ is just as good as ‘optimizing’ behavior. This pleads for using measures that quantify the goodness-of-fit of the model under study. In our application, all data passed the rationalizability tests. Thus, the data perfectly fit the rationalizability conditions, which made the goodness-of-fit concern redundant in this case. Still, it is worth indicating that, based on the methodology of Varian, 1985 and Varian, 1990, our MIP approach allows for taking such goodness-of-fit concerns into account for data sets that do reject rationalizability.
