A strong correspondence principle for smooth, monotone environments

In discrete time dynamic systems that are locally monotone, we show that comparative statics are well-behaved at a regular equilibrium if and only if equilibrium is exponentially stable.  We apply the result to show: (i) if excess demand satisfies the gross substitutes property then the competitive exchange equilibrium is stable iff intuitive comparative statics hold, (ii) the market demand curve for bandwagon (or network) goods slopes upwards only if equilibrium is unstable, and (iii) equilibria of supermodular games are stable iff they exhibit monotone comparative statics. We also provide a non-local version of the result for linear models, and apply the insights to the input-output model and network games with linear best replies and strategic complementarities.