Research

Works in Progress

• Global Comparative Statics via the Implicit Function Theorem

Brief Description. Comparative statics based on the implicit function theorem are powerful but inherently local, describing how equilibria respond to infinitesimal parameter changes. In empirical and quantitative applications, however, researchers often study finite changes in taxes, costs, policies, or other parameters. This paper develops a framework that identifies when local IFT-based comparative statics can be extended to such finite changes. 

Abstract. Comparative statics in smooth equilibrium models is typically characterized using the implicit function theorem, which yields local predictions based on derivatives of the equilibrium system. This paper develops a general framework for extending such local results to finite parameter changes. The analysis proceeds in two steps. First, we establish conditions under which the equilibrium system admits a globally defined, continuously differentiable selection, using either a properness-based global inversion argument or injectivity conditions applied slice-by-slice. Second, we show that global comparative statics can be obtained by integrating local responses along parameter paths. The key requirement is a cone invariance condition: parameter changes must generate shocks to the equilibrium system that lie in an admissible shock cone, and the propagation operator must map those shocks into an admissible cone of outcome changes. Under this condition, finite equilibrium changes inherit the qualitative properties of local comparative statics. A complementary result establishes that, under a strengthened finite-change hypothesis, such global behavior implies corresponding pointwise restrictions on the Jacobian. Together, these results provide a general link between local derivative-based comparative statics and global predictions in smooth equilibrium systems.

• Comparative Statics for Difference-in-Differences

Abstract. In a model with spillovers, consider the difference in the impact of a positive shock on average outcomes among the treated and untreated, or the marginal average effect of treatment among the treated with spillovers (MATTS). MATTS is positive if, and only if, the negated Jacobian inverse of the equilibrium system is a B-matrix by columns. This condition is also sufficient to answer traditional comparative statics questions. I also give several sufficient, and sometimes necessary, conditions on the noninverted Jacobian–which correspond to common modeling assumptions–under which its inverse is a B-matrix by columns. Sign restrictions on MATTS are testable because the sample difference-in-differences is an unbiased estimator for MATTS from a superpopulation perspective. I also show that MATTS generalizes the ATT and the ATE when interference, or spillovers, are present. I apply the results to oligopoly and contests.

• The Role of Learning in Search with Statistical Discrimination