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'a, _

where all the partial dérivaiives are evaluated at the point

and so are given numbers. Given that the determinant | F\ of the left-hand matrix is nonzero, we can solve for dx*/daj by using Cramer's rule. This involves forming the determinant | Fn | by replacing the /th column of | with the column r -fl 1

Jctj

and evaluating daj |F|

When the model is one of constrained optimization, the functions fk will be the first-order partial derivatives of the Lagrange function and the equilibrium conditions are the first-order conditions. In that case, the determinant | F| is the Hessian determinant of the system. Recall that the sufficient second-order conditions are expressed in terms of the sign of |F|. These conditions can be used to help sign the comparative statics. We see how this works out in some examples.

The Slutsky Equation

In chapter 13 we considered the problem of a consumer's optimal choice of consumption quantities. The solution to this problem takes the form of a set of demand functions. one for each good. A major aim of the analysis is to show how the consumer's demand for a good varies with prices, and so this is a problem in comparative-statics analysis. In this example, we consider this problem for the case of two goods, A | and x2. The consumer's demands for Lhese goods are given by the solution to the problem max mU'i . X2) s.t. p 1 x 1 -+■ p2x2 = 111

Here the endogenous variables are the demands ,V| and vj, and the exogenous variables are the prices p\ and pi. and income/«. Applying ihe Lagrangean method gives the first-order conditions

" 1 (jt* , x£) — A* pi — 0 U2(X*,X2) - I* P2 = 0 <11 — p\x* — p\x2 = 0

We now look at the effects of changes in prices and income on the demands. Applying the standard method leads to