Comparative Statics - Linear Approximation

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 elements of are sometimes called the multipliers of the elements of a on the elements of x.