Comparative statics
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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.
It is one of the primary analytical methods used in economics, where it is very commonly used in the study of changes in supply and demand when analyzing a market and changes in monetary and fiscal policy when analyzing the 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).
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[edit] Linear approximation
In practical terms, comparative statics results are usually derived by studying a linear approximation to the system of equations that define 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 x is determined by the following equation:
f(x,a) = 0
where a is an exogenous parameter. Then, to a first-order approximation, the change in x caused by a small change in a must satisfy:
Bdx + Cda = 0
Here dx and da represent the changes in x and a, respectively, while B and C are the partial derivatives of f with respect to x and a (evaluated at the initial values of x and a), respectively. Equivalently, we can write the change in x as:
dx = − B − 1Cda.
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 x, 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 B − 1C is positive or negative.
All the equations above remain true in the case of a system of n equations in n unknowns. In other words, suppose f(x,a) = 0 represents a system of n equations involving the vector of n unknowns x, and the vector of m given parameters a. If we make a sufficiently small change da in the parameters, then the resulting change in the endogenous variables can be approximated arbitrarily well by dx = − B − 1Cda. In this case, B represents the n-by-n matrix of partial derivatives of the equations f with respect to the variables x, and C represents the n-by-m matrix of partial derivatives of the equations f with respect to the parameters a. (The derivatives in B and C are evaluated at the initial values of x and a.)
[edit] See also
[edit] References
- John Eatwell et al., ed. (1987). "Comparative dynamics," The New Palgrave: A Dictionary of Economics, v. 1, p. 517.
- John R. Hicks (1939). Value and Capital.
- Timothy J. Kehoe, 1987. "Comparative statics," The New Palgrave: A Dictionary of Economics, v. 1, pp. 517-20.
- Paul A. Samuelson (1947). Foundations of Economic Analysis.
