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ISU ECON 600 - Exam-1

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Econ 600 Schroeter Fall 2004 Exam #1 Do all three problems. Weights: #1 - 25%, #2 - 35%, #3 - 40%. 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 the wording. a. Given differentiable. ℜ→ℜnF :()⋅F is concave if and only if () () ()( ).,, vuvuallforvuvxFvFuFn≠ℜ∈−∂∂+< b. For , if is either strictly increasing or strictly decreasing, then ℜ→ℜ:F()⋅F()⋅F is strictly quasi-concave. c. is differentiable and . If ℜ→ℜnF :nx ℜ∈*()0*=∂∂xxF and ()*22xxF∂∂ is negative definite then is a strict local maximum of *x()⋅F. d. In an equality-constrained optimization problem with 3 choice variables (n = 3) and 2 constraints (m = 2), the bordered Hessian is a 5 x 5 matrix and the second order necessary condition is a single sign restriction on the determinant of the entire bordered Hessian. e. Let and be differentiable and let be a point such that ℜ→ℜnF :ℜ→ℜng :nx ℜ∈*()0*≠∂∂xxg. Then is a local solution to *x ()()bxgthatsuchxFxtrw=...max if and only if there exists such that ℜ∈*λ()**,λx is a stationary point of ()()()[]xgbxFxL−+≡λλ,.22. Consider the problem: ,min...Uxthatsuchxaxtrw∈⋅ where x is an n x 1 vector of choice variables, a is a 1 x n vector of parameters, and Suppose that the problem has a global solution, .nU ℜ⊂(),*ax for all , and define the value function na ℜ∈()().**axaaF ⋅≡ (Some facts that you don't need to prove: A sufficient condition for the problem to have a global solution for all is that U is closed and bounded. While the optimal value of x, need not be unique for a given a, the value function is uniquely defined. That is, if, for a given a, there are multiple optimal they all yield the same value of the objective function.) na ℜ∈(),*ax,*sx Prove that is concave. ℜ→ℜnF :* 3. Consider the following equality-constrained maximization problem: ()().)(,,,,max321321,,...321ibxxxgthatsuchxxxfxxxtrw= where and are differentiable functions and b is a constant. Also consider the unconstrained optimization problem: ℜ→ℜ3:f ℜ→ℜ3:g ().)(,max21,...21iixxFxxtrw where is differentiable. ℜ→ℜ2:F a. Write down the Lagrangian and the first-order conditions for problem (i.). b. Write down the first-order conditions for problem (ii.). Now suppose that ()()213321,,, xxhxxxxg−= where is differentiable, and that ℜ→ℜ2:h() ()(bxxhxxfxxF)+≡212121,,,,. For this case, show that c. if ()*3*2*1,, xxx satisfies the first-order conditions for problem (i.), then ()*2*1, xx satisfies the first-order conditions for problem (ii.). d. if ()*2*1, xx satisfies the first-order conditions for problem (ii.), then ()()bxxhxx +*2*1*2*1,,, satisfies the first-order conditions for problem


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ISU ECON 600 - Exam-1

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