Answers-B_even_-1

Answers-B_even_-1 - John Riley 24 July 2011 ANSWERS TO EVEN...

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© John Riley 24 July 2011 Answers to Exercises in Appendix B page 1 ANSWERS TO EVEN NUMBERED EXERCISES IN APPENDIX B Section B.2. FUNCTIONS OF VECTORS Exercise B.2-2: Positive definite quadratic form What are the necessary and sufficient conditions for 22 11 () ij i j ij qx axx == = ∑∑ to be strictly positive for all 0 x ? ANSWER If 0 > for all 0 x , then ( ) 0 −< for all 0 x . This is the case where 11 0 a −> and 11 22 12 21 0 aa −− > , which is the same as 11 0 a < and 11 22 12 21 0 . Exercise B.2-4: Convex lower contour sets (a) Show that the lower contour sets of a function are convex if and only if the function is quasi-convex. (b) The output of commodity j has labor input requirements j j f q , 1,. .., j n = where j f is convex. Show that the total labor requirements function 1 ( ) n j j j Lq f q = = is quasi-convex. (c) Hence show that if the supply of labor is fixed, the set of feasible outputs is convex. ANSWER (a) The proof is almost identical to the proof for quasi-concave functions. We demonstrate necessity (only if). Suppose the lower contour sets of f are convex and consider any 0 x and 1 x such that 10 () () f xf x . Then both vectors are in the set (){ |( ) () } L Cx xfx fx =≤ . By hypothesis, this set is convex. Thus all convex combinations x λ lie in 1 L . It follows immediately that 1 f x .
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© John Riley 24 July 2011 Answers to Exercises in Appendix B page 2 (b) 1 () n jj j Lq f q λ = = . Since j f is convex, 01 ( 1) ( ) ( ) j j j j f qf q f q λλ ≤− + . Hence 1 ( 1 )() n j fq = + 0 1 11 (1 ) ( ) ( ) ) ( ) ( ) nn f q L q L q == =− + + ∑∑ Thus is convex and so quasi-convex. It follows that the lower contour sets of L are convex. (c) Let L be the fixed supply of labor and choose q so that L = . Then the set of feasible outputs must satisfy the constraint . This is a lower contour set and is therefore convex.
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This note was uploaded on 01/11/2012 for the course ECON 200 taught by Professor Riley during the Fall '10 term at UCLA.

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Answers-B_even_-1 - John Riley 24 July 2011 ANSWERS TO EVEN...

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