Machina Math Handout

# x1 2 x1 x2 x2

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Unformatted text preview: ) ∂ 2 ƒ ( x1 (α ), x2 (α );α ) ⋅ + ⋅ + 2 ∂ x1 ∂ x2 ∂α ∂ x2 ∂α ∂ x2 ∂α ≡0 α α This is a set of two linear equations in the two derivatives ∂x1*(α)/∂α and ∂x2*(α)/∂α, and we can solve for ∂x1*(α)/∂α and ∂x2*(α)/∂α by substitution, or by Cramer’s Rule, or however. Econ 100A 14 Mathematical Handout G. OPTIMIZATION #3: COMPARATIVE STATICS OF EQUILIBRIA Implicit differentiation isn’t restricted to optimization problems. It also allows us to derive how changes in the parameters affect the equilibrium values in an economic system. Consider a simple market system, with supply and demand and supply functions QD = D(P,I) and QS = S(P,w) where P is market price, and the parameters are income I and the wage rate w. Naturally, the equilibrium price is the value Pe solves the equilibrium condition D(Pe,I) = S(Pe,w) It is clear that the equilibrium price function, namely Pe = Pe(I,w), must satisfy the identity D( Pe(I,w) , I ) I≡ S( Pe(I,w) , w ) ,w So if we want to determine how a rise in income affects equilibrium price, totally differentiate the above identity with respect to I, to get ∂D ( P e ( I , w), I ) ∂P e ( I , w) ∂D ( P e ( I , w), I ) ⋅ + ∂P e ∂I ∂I then solve to get ∂P e ( I , w) ∂I = ≡ w, I ∂S ( P e ( I , w), w ) ∂P e ( I , w) ⋅ ∂P e ∂I ∂D ( Pe ( I ,w), I ) ∂I ∂S ( Pe ( I ,w),w) ∂D ( Pe ( I ,w), I ) − ∂Pe ∂Pe In class, we’ll analyze this formula to see what it implies about the effect of changes in income upon equilibrium price in a market. For practice, see if you can derive the formula for the effect of changes in the wage rate upon the equilibrium price. Summary of How to use Implicit Differentiation to obtain Comparative Statics Results The approach of implicit differentiation is straightforward, yet robust and powerful. It is used extensively in economic analysis, and as seen above, consists of the following four steps: STEP 1: Obtain the first order conditions for the optimization problem, or the equilibrium conditions of the system. STEP 2: Substitute the solution functions into these first order conditions, or equilibrium conditions. STEP 3: Totally differentiate these equations with respect to the parameter that’s changing. STEP 4: Solve for the derivatives with respect to that parameter. Econ 100A 15 Mathematical Handout H. OPTIMIZATION #4: COMPARATIVE STATICS OF OPTIMAL VALUES The final question we can ask is how the optimal attainable value of the objective function varies when we change the parameters. This has a surprising aspect to it. In the unconstrained maximization problem: max ƒ( x1 ,..., xn ;α ) x1 ,..., xn * recall that we get the optimal value function φ(α) by substituting the solutions x1*(α),..., xn (α) back into the objective function, i.e.: * * φ (α ) ≡ ƒ( x1 (α ),..., xn (α );α ) α Thus, we could simply differentiate with respect to α to get: = * * * ∂ ƒ( x1 (α ),..., xn (α );α ) dx1 (α ) ⋅ ∂x1 d...
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## This document was uploaded on 09/18/2013.

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