Comparative Statics, Stability, and Uniqueness

Consider an economic model whose equilibrium can be represented as the fixed point of a system of differentiable equations. Using the theory of B-matrices, I show that comparative statics are well-behaved if the interactions between the equations are not too large, and the negative interactions are not too varied. When there are only positive interactions, for example when strategic complements prevail in a strategic setting, I prove a version of Samuleson's (1947) Correspondence Principle in that equilibrium is nondecreasing for any positive parameter shock if and only if equilibrium is exponentially stable under discrete time best reply dynamics . If there are only negative interactions, like when strategic substitutes prevail in a game theoretic context, I use the theory of inverse M-matrices to significantly relax Dixit's (1986) conditions under which comparative statics are well-behaved. For every comparative statics result I show that if the conditions apply globally then equilibrium is unique. Applications are provided to differentiated products Cournot oligopoly, market demand with interdependent preferences, and games on fixed networks.