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AppendixC

AppendixC - John Riley 15 August 2011 APPENDIX C...

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John Riley 15 August 2011 APPENDIX C: OPTIMIZATION C.1 TWO VARIABLES 1 C.2 UNCONSTRAINED OPTIMIZATION 5 C.3 IMPLICIT FUNCTION THEOREM 9 C.4 CONSTRAINED MAXIMIZATION 16 C.5 SUPPORTING HYPERPLANE 24 C.6 TAYLOR EXPANSION 27 33 pages 11 figures
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©John Riley Appendix C page 2 C.1 MAXIMIZATION WITH 2 VARIABLES Key Ideas: Necessary and sufficient conditions Rather than leap directly to the analysis of multi-variable optimization problems we begin by examining the two variable case. Suppose that the function f is differentiable at 00 0 12 (,) x xx = . If f takes on its maximum over 2 \ at 0 x , then taking one variable at a time, we know that each of the first partial derivatives must be zero. Proposition C.1-1: First-Order Conditions (FOC) for a Maximum If (, ) fxx takes on its maximum over 2 \ at 0 x , then 0 ()0 , 1 , 2 i f xi x == . We also know that each of the second partial derivatives must be negative. To obtain a further necessary condition we consider the change in f as x changes from 0 x in the direction of some other vector 1 x .We do this by considering the weighted average x λ of 0 x and 1 x , that is 01 0 1 0 (1 ) ( ) x x x x λλ =− + = + , and define 0 () ( ( ) gf x x x =+ . This function is depicted below. Fig. C.1-1: Cross-section of f 2 x bb 1 x 0 x 1 x O x f g
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©John Riley Appendix C page 3 The mapping, ( ) g λ , depicted in the cross-section is a function from \ into \ . Then we can appeal to the necessary conditions for one-variable maximization. In particular, for a maximum at 0 x , the second derivative of () g must be negative at 0 = . The first derivative of g is 12 ( ) ff gx z x z xx λλ ∂∂ =+ , where 10 zx x . For a maximum this must be zero for all z , hence the partial derivatives must both be zero. Differentiating again, 22 2 11 2 2 ( ) 2 ( ) f gz x z z x z x ′′ + [] 1 2 2 2 2 1 zz z x z x ⎡⎤ = ⎢⎥ ⎣⎦ Note that the right-hand side of the last equation is a quadratic form. Appealing to Proposition B.2-2 we have the following result. Proposition C.1-2: Second-Order Conditions (SOC) for a Maximum If (, ) fxx takes on its maximum over 2 \ at 0 x , then 2 2 00 0 0 2 2 1 2 ( ) ()0 , 1 , 2 a n d ( ) ()( ) 0 i f f ix i i ixx x x x x ≤= . Exercise C.1-1: Consumer Choice Bev faces prices 1 p and 2 p and her income is I . Her utility function Ux x satisfies 2 ( ) 0, 1,2, j U xj x x + >= \ .
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©John Riley Appendix C page 4 (a) Argue that the budget constraint must be binding and hence reduce Bev’s optimization problem to a 1 variable problem. Hence obtain first order necessary conditions for a corner solution. (b) Suppose that 0 lim ( ) , 1,2 j x j U xj x =∞ = . That is, the marginal utility of each commodity increases without bound as consumption of the commodity declines to zero. Show that the necessary conditions for a maximum cannot be satisfied at a corner.
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AppendixC - John Riley 15 August 2011 APPENDIX C...

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