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Comparative statics
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In economics, comparative statics is the comparison of two different equilibrium states, before and after a change in some underlying exogenous parameter. As a study of statics it compares two different unchanging points, after they have changed. 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.
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In economics, comparative statics is the comparison of two different equilibrium states, before and after a change in some underlying exogenous parameter. As a study of statics it compares two different unchanging points, after they have changed. 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. The term 'comparative statics' itself is more commonly used in relation to microeconomics (including general equilibrium analysis) than to macroeconomics. Comparative statics was formalized by John R. Hicks (1939) and Paul A. Samuelson (1947) (Kehoe, 1987, p. 517).
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:
.
The factor of proportionality is sometimes called the multiplier of a on x.
Stability
The stability assumption is important for two reasons. On one hand, if the equilibrium were unstable, a small parameter change might cause a large jump in the value of , invalidating the use of a linear approximation. On the other hand, Paul A. Samuelson's correspondence principle states that stability of equilibrium has testable implications about the comparative static effects. In other words, knowing that the equilibrium is stable may help us predict whether the coefficient is positive or negative.
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 change in the endogenous variables can be approximated arbitrarily well by . In this case, represents the -by- matrix of partial derivatives of the equations with respect to the variables , and represents the -by- matrix of partial derivatives of the equations with respect to the parameters . (The derivatives in and are evaluated at the initial values of and .)
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