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
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.
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
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