Everything about Comparative Statics totally explained
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've changed. It doesn't 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's 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).
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
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:
. 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
.)
Further Information
Get more info on 'Comparative Statics'.
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