An Introduction to Monotone Comparative Statics

1. Overview Comparative statics (called sensitivity analysis in engineering fields) means characterizing how the endogenous outcomes of a model change with its exogenous parameters. The " statics " in the name is because one is simply comparing two versions of the model, for two different values of exogenous parameters, rather than tracing out a dynamic process of change. One can do comparative statics in a dynamic model. For example, one can characterize how the time-paths of investments differ for different market conditions, for different investment costs, or for different initial values of R&D. There are many different techniques for comparative statics. There are some techniques of general use, but each comparative statics exercise can involve also some ad hoc techniques. Two of the " general-purpose " tools are monotone comparative statics and the implicit function theorem. The purpose of these notes is to convey the main ideas of monotone comparative statics. A common setting in which one may do comparative statics is in optimization problems. How does the solution to an optimization problem depend on parameters? How does demand depend on prices? How does a firm's optimal price depend on the prices of competing firms? What type of monopolist would choose a higher level of investment in cost reduction: a profit-maximizer or a social-welfare maximizer? The parameters that affect decisions in optimization problems may do so either because they affect preferences (e.g., this is the case of all the examples above except for the demand example) or because they affect the feasible set (e.g., prices affect demand not because the person cares directly about prices but because prices determine the budget set). Monotone comparative statics are methods for characterizing how parameters that affect preferences thereby affect choices, keeping fixed the constraint set. They are methods for unconstrained maximization. The term " monotone " in " monotone comparative statics " is for two reasons. First, these are methods for characterizing whether an increase in a parameter causes the decision to increase or decrease. Second, historically the implicit function theorem was used for this purpose and the implicit function theorem not only tells you whether the decision increases or decreases but also the rate of change. In contrast, monotone comparative statics tells you only " up " or " down " , i.e., it gives an ordinal rather than cardinal answer. However, compared to the implicit function theorem, monotone comparative statics is …