Econ 600 Schroeter Fall 2009 Exam #2 Do all four problems. They will be equally weighted. Closed book, open notes. Be sure your answers are presented in a neat and well-organized manner. 1. Answer True or False for each of the following statements. If the statement is false, indicate how it could be changed to a true statement with a small change in wording. a. Consider , where U is a convex subset of . ℜ→UF :nℜ()⋅F is concave if and only if { is convex for all ()axF ≥}Ux ∈ :ℜ∈a . b. Given , differentiable. Let be a local solution to: ℜ→ℜnF :nx ℜ∈* ()xFxtrw ...max subject to for 0≥ix ni ,,2,1 K=. Then, for each , ni ,,2,1 K=()0*≤∂∂xxFi and ()0**=∂∂xxFxii. c. Given , differentiable, assume that the problem ℜ→ℜ2:F()axFxtrw;min...(ax*, has, for each value of a, a strict global solution given by the differentiable function . Define the value function: )() ()()aaxF ;*≡aF*. Then, for every value of 0aa=, () ()()00*0*;aaxaFadadF∂∂= and () ()()00*2202*2;aaxaFadaFd∂∂≤ . d. If a Cauchy sequence, , has a convergent subsequence, then {}∞=1nnx{}∞=1nnx converges. e. If is differentiable and has a strict local maximum at , then ℜ→ℜnF :nx ℜ∈*()*x22xF∂∂ is negative definite.22. A consumer purchases and consumes two goods, in non-negative quantities x and y, to maximize strictly quasi-concave utility, ()yxU , , subject to the constraint that eon the goods be no greater than income 0>I . xpenditure ()⋅U tric positive marginal utilthroughout 2+ℜ . Good y is purchased at price 0>yp . Purchases of good x are of two types: 21xxx += . 1x units are purcha d at a subsidized price of 0>− spx, where units are purchased at the "regular" price of 0>xp . The consu limitedin the amount of good x that can be purchased at the subsidized price: has s tly ities r is se0>s . 2x mexx ≤1, where 0>x is a cons ch that ()tant su Ixspx<−. a. Write down the consumer's utility maximization problem subject to inequality and non-negativity constraints. b. Write down the Lagrangian for the problem and the Kuhn-Tucker conditions. c. Denoting the consumer's optimal purchases by ()***,, yxx21, use the Kuhn-Tucker conditions to establish the following claims. (Note: A simple budget-line-indifference-curve graph will help with the intuition of these results. In each case, the claim is pretty obvious. The problem is to formally derive each one from the Kuhn-Tucker conditions.) i. xxx =⇒>*1*20. ii. xx =*1 and ()()yxyxyxppyxxUyxxUpspx ≤++≤−⇒=**2*1**2*1*2,,0. 3. Recall the following definitions of the underlined terms: nC ℜ⊂ is open if, for all , there exists such that Cx ∈ 0>ε()CxBε⊂ . A function is mnf ℜ→ℜ: continuous if, for every convergent sequence in we have {},nnnxℜ∞=,1nx ℜ∈→(){}()mnnxfxf ℜ∈→∞=1. Use these definitions to prove the following proposition: Let be continuous and let be open. Then mnf ℜ→ℜ:()mC ℜ⊂(){}Cxff ∈−xCnℜ∈≡ :1 is open.34. Consider the problem: ()21,...,max21xxFxxtrw subject to (),,21bxxg= where is differentiable and strictly concave, is differentiable and strictly convex, and b is a scalar parameter. Assume that the problem has a regular solution for , insuring that optimal values of the choice variables exist as differentiable functions of b, and ℜ→ℜ2:F0bb =ℜ→ℜ2:g()bx*1()bx*2, in some neighborhood of . Assume that and have strictly positive first partial derivatives at 0b()⋅F()⋅g()()()()0*20, bx*1bx0*bx ≡ . Define the value function: ()()()bx*FbF*≡ . Show that: ()00*>bdbdF and