Machina Math Handout

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Unformatted text preview: , the method of Lagrangians is nothing more than a roundabout way of generating our condition that the normal vector of ƒ(x1,..., xn) be λ times the normal vector of g(x1,..., xn), and the constraint g(x1,..., xn) = c be satisfied. We will not do second order conditions for constrained optimization (they are a royal pain). Econ 100A 11 Mathematical Handout F. OPTIMIZATION #2: COMPARATIVE STATICS OF SOLUTION FUNCTIONS Having obtained the first order conditions for a constrained or unconstrained optimization problem, we can now ask how the optimal values of the control variables change when the parameters change (for example, how the optimal quantity of a commodity will be affected by a price change or an income change). Consider a simple maximization problem with a single control variable x and single parameter α max ƒ( x ;α ) x For a given value of α, recall that the solution x* is the value that satisfies the first order condition ∂ ƒ( x*;α ) =0 ∂x Since the values of economic parameters can (and do) change, we have defined the solution function x*(α) as the formula that specifies the optimal value x* for each value of α. Thus, for each value of α, the value of x*(α) satisfies the first order condition for that value of α. So we can basically plug the solution function x*(α) into the first order condition to obtain the identity ∂ ƒ( x*(α );α ) ≡0 α ∂x We refer to this as the identity version of the first order condition. Comparative statics is the study of how changes in a parameter affect the optimal value of a control variable. For example, is x*(α) an increasing or decreasing function of α ? How sensitive is x*(α) to changes in α? To learn this about x*(α), we need to derive its derivative d x*(α)/dα. The easiest way to get d x*(α)/dα would be to solve the first order condition to get the formula for x*(α) itself, then differentiate it with respect to α to get the formula for d x*(α)/dα. But sometimes first order conditions are too complicated to solve. Are we up a creek? No: there is another approach, implicit differentiation, which always gets the formula for the derivative d x*(α)/dα. In fact, it can get the formula for d x*(α)/dα even when we can’t get the formula for the solution function x*(α) itself ! Implicit differentiation is straightforward. Since the solution function x*(α) satisfies the identity ∂ ƒ( x*(α );α ) ∂x ≡0 α we can just totally differentiate this identity with respect to α, to get ∂ 2 ƒ( x*(α );α ) d x*(α ) ⋅ dα ∂x 2 and solve to get Econ 100A d x*(α ) dα ≡ α + ∂ 2 ƒ( x*(α );α ) ∂x ∂α ∂ 2 ƒ( x*(α );α ) − ∂x ∂α 12 ≡0 α ∂ 2 ƒ( x*(α );α ) ∂x 2 Mathematical Handout For example, let’s go back to that troublesome problem max α⋅x2 – ex, with first order condition 2⋅α⋅x* – ex* = 0. Its solution function x* = x*(α) satisfies the first order condition identity 2 ⋅α ⋅ x*(α ) − e x*(α ) ≡ α 0 So to get th...
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This document was uploaded on 09/18/2013.

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