Global Comparative Statics via an Implicit Function Theorem

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.