Sensitivity Analysis for Applied General Equilibrium Models in the Presence of Multiple Equilibria

Pagan and Shannon's (1985) widely used approach employs local linearizations of a system of non-linear equations to obtain asymptotic distributions for the endogenous parameters (such as prices) from distributions over the exogenous parameters (such as estimates of taste, technology, or policy variables, for example). However, this approach ignores both the possibility of multiple equilibria as well as the problem (related to that of multiplicity) that critical points might be contained in the confidence interval of an exogenous parameter. Critical equilibria occur for parameter values that generate a singular excess demand Jacobian at the equilibrium prices. At such points, the equilibrium correspondence might not be lower hemi-continuous and the selection of equilibria made by a computation algorithm (or by a tatonnement process) can jump. From a statistical viewpoint, the presence of critical economies means that statistical error in the parameter estimates can have a large and discontinuous impact on error in the endogenous variables, such as prices. We generalize Pagan and Shannon's approach to account for critical economies and multiple equilibria by assuming that the choice of equilibrium is described by a continuous random selection. We develop an asymptotic theory regarding equilibrium prices, which establishes that their probability density function is multimodal and that it converges to a weighted sum of normal density functions. An important insight is that if multiple equilibria exist but multiplicity is ignored, the computed solution will be an inconsistent estimator of the true equilibrium, even if the computation algorithm tracks the same equilibrium as the economy's tatonnement.