In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter. As a type of static analysis it compares two different equilibrium states, after the process of adjustment (if any). It does not study the motion towards equilibrium, nor the process of the change itself. Comparative statics is commonly used to study changes in supply and demand when analyzing a single market, and to study changes in monetary or fiscal policy when analyzing the whole economy. Comparative statics is a tool of analysis in microeconomics (including general equilibrium analysis) and macroeconomics. Comparative statics was formalized by John R. Hicks (1939) and Paul A. Samuelson (1947) (Kehoe, 1987, p. 517) but was presented graphically from at least the 1870s. For models of stable equilibrium rates of change, such as the neoclassical growth model, comparative dynamics is the counterpart of comparative statics (Eatwell, 1987). Linear approximation Comparative statics results are usually derived by using the implicit function theorem to calculate a linear approximation to the system of equations that defines the equilibrium, under the assumption that the equilibrium is stable. That is, if we consider a sufficiently small change in some exogenous parameter, we can calculate how each endogenous variable changes using only the first derivatives of the terms that appear in the equilibrium equations. For example, suppose the equilibrium value of some endogenous variable is determined by the following equation: where is an exogenous parameter. Then, to a first-order approximation, the change in caused by a small change in must satisfy: Here and represent the changes in and , respectively, while and are the partial derivatives of with respect to and (evaluated at the initial values of and ), respectively. Equivalently, we can write the change in as: Dividing through the last equation by da gives the comparative static derivative of x with respect to a, also called the multiplier of a on x: Many equations and unknowns All the equations above remain true in the case of a system of equations in unknowns. In other words, suppose represents a system of equations involving the vector of unknowns , and the vector of given parameters . If we make a sufficiently small change in the parameters, then the resulting changes in the endogenous variables can be approximated arbitrarily well by . In this case, represents the × matrix of partial