{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

AppendixC

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

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
©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
©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 + >= \ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
©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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 35

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online