Econ 600 Schroeter Fall 2011 Exam #1 Do all three problems. Weights: #1 - 30%, #2 - 35%, #3 - 35%. Closed book, closed 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. Let U be a convex subset of nℜ . If ℜ→UF : is differentiable then ()⋅F is concave if and only if () () ()( )vuvxFvFuF −∂∂+≤ for all Uvu ∈,. b. Let U be a convex subset of nℜ . If ℜ→UF : is differentiable then ()⋅F is strictly concave if and only if ()vxF22∂∂ is negative definite for all Uv ∈ . c. Let ℜ→ℜnF : be differentiable. If nx ℜ∈* is such that ()0*=∂∂xxF and ()*22xxF∂∂ is positive semi-definite then *x is a local minimum of ()⋅F . d. Let ℜ→ℜ2:F and ℜ→ℜ2:g be differentiable and let ℜ∈b. Let 2*ℜ∈x be a point at which ()0*≠∂∂xxg and a local solution to ()xFxtrw ...max such that ()bxg=. Then there exists ℜ∈*λ such that()()***xxgλxxF∂∂=∂∂. e. ℜ→ℜℜ xF : is differentiable. For each value of the scalar parameter a, ()axFxtrw;min... has a regular solution at ()ax*. Define: ()()()aaxFaF ;**≡ . Then, for each value of the parameter 0a : ()()()0*2200*22; aFdadaaxFa≥∂∂.22. First recall the following definition of a quasi-convex function: Let U be a convex subset of nℜ . ℜ→UF : is quasi-convex if, for all Uvu ∈, , and for all []1,0∈h , ()()()(){}vFuFvhhuF ,max1≤−+. Now, for U a convex subset of nℜ , let ℜ→ℜ UxFm: be such that ()axF ; is quasi-convex in a for every mx ℜ∈ . Consider the problem ()axFxtrw;max... and assume that it has a global solution ()ax* for each Ua∈. Define the value function in the usual way: () ()()aaxFaF ;**≡ . (Note that ()aF* is well-defined even if ()ax* is not unique.) Prove that ℜ→UF :* is quasi-convex. 3. Consider the problem: 2211,...21min xaxaxxtrw+ such that ()cxxg=21, , where c is a constant and 1a and 2a are positive constants. The constraint function, ℜ→ℜ2:g is differentiable with first- and second-order partial derivatives (denoted using subscripts) that satisfy the following sign restrictions throughout 2ℜ : ;0;0,;0,12221121><> ggggg and 02122211>− ggg . a. Write down the Lagrangian and the first order necessary conditions for this problem. The assumptions made above insure that the second-order sufficient conditions for a strict local minimum are satisfied at any point that satisfies the first-order necessary conditions (although you don’t have to show this). b. Use ()(),,,,21*221*1aaxaax and ()21*, aaλ to denote the optimal values of 1x and 2x , and the solution value for the Lagrange multiplier, as functions of 1a and 2a . Show that 01*>∂∂aλ and