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Econ 600 Schroeter Fall 2007 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 the wording. a. If is differentiable with ℜ→ℜnF :()vxF22∂∂ negative definite for all , then is strictly concave. nv ℜ∈()⋅F b. If is differentiable and quasi-concave, then ℜ→ℜnF :()( )0≥−∂∂vuvxF for all nvu ℜ∈,. c. Given differentiable and such that ℜ→ℜnF :nx ℜ∈*()0*=∂∂xxF and (*22x)xF∂∂ is negative definite. Then is a strict global maximum of *x()⋅F. d. Any sequence contained in a closed subset of has a convergent subsequence. nℜ e. Given , differentiable, assume that the problem ℜ→ℜ2:F()axFxtrw;max... has, for each value of a, a strict global solution given by the differentiable function . Define the value function: ()ax*() ()()aaxFaF ;**≡ . Then, for every value of 0aa=, () ()()00*0*;aaxaFadadF∂∂= and () ()()00*2202*2;aaxaFadaFd∂∂≥ . 2. Let {} be a sequence in . Prove that ∞=1nnxmℜ{}∞=1nnx is a Cauchy sequence in if and only if every one of its component sequences is a Cauchy sequence in . mℜ1ℜ23. A consumer has utility ()()321321lnlnln,, xcxbaxxxxU+++=, where and are non-negative consumption levels of goods 1, 2, and 3; and a, b, and c are positive parameters. The consumer purchases goods at fixed prices; and ; and total expenditure can be no larger than money income, I. ,,21xx3x,,21pp3p a. Write down the consumer's problem of maximizing utility subject to an inequality and non-negativity constraints. Write down the Kuhn-Tucker conditions for this problem. b. Use the Kuhn-Tucker conditions to show that this problem can have solutions of only two types; where, by a solution "type," we mean a given classification of the inequality and non-negativity constraints as "binding" or "not binding." c. For each solution type: (i.) Find a restriction on the parameters ()Ipppcba ,,,,,,321 that must hold if a solution of that type is going to occur. (ii.) Solve for the demand equations. 4. Given differentiable functions: ()ℜ→ℜℜ xxxf221:;,α and ()ℜ→ℜ221:, xxg . Consider the following problem: ()α;,max21,...21xxfxxtrw subject to (),,21bxxg= where b is a constant. Note that and are choice variables and 1x2xα is a scalar parameter. Assume that the problem has a strict global regular solution for each value of α, with the optimal values of the choice variables given by differentiable functions of α: and . (Recall that a "regular" solution is one at which the second order sufficient conditions are satisfied.) For a given value of ()α*1x()α*2xα, 0α, assume that () ()()0;,00*20*122=∂∂∂ααααxxxf and () ()()0,0*20*12≠∂∂ααxxxg. Show that (0*1αα∂∂x) has the same sign as () ()()00*20*112;,ααααxxxf∂∂∂. (Hint: Write down the first-order conditions and differentiate with respect to


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ISU ECON 600 - Exam #2

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