Econ 600 Schroeter Fall 2006 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 the wording. a. Given differentiable. If there exists and ℜ→ℜnF :nx ℜ∈*0>ε such that for all then () (xFxF ≥*)()*xBxε∈ ()*22xxF∂∂ is negative semi-definite. b. Given differentiable. ℜ→ℜnF :()⋅F is strictly concave if and only if () ( ) ()(*** xxx)xFxFxF −∂∂+< for all .*,*, xxxxn≠ℜ∈ c. Given differentiable; , a 1 x n vector with ℜ→ℜnF :ng ℜ∈0≠g; and b a scalar constant. Consider the problem: ()xFxtrw ...max subject to ()*,bxg=⋅ and define the Lagrangian: ()()().; xgbxFxL⋅−+≡λλ If is quasi-concave and and ()⋅Fnx ℜ∈*0*≠λ are such that ()**;λx is a stationary point of , then is a global solution to problem (*). ()⋅L*x d. Given differentiable and with nonzero gradient vector, ℜ→ℜ2:F2* ℜ∈x()*xxF∂∂. Then (*x)xF∂∂ is tangent to the level curve of ()⋅F through . *x e. Define . At the point ()2221211, xxxxF −−=()()0,0,21=xx, the second order sufficient condition for a strict local maximum is not satisfied.22. is strictly quasi-concave. Supposed that is a local solution to ℜ→ℜnF :nx ℜ∈* ().max...xFxtrw Prove that is a strict global solution. *x 3. A firm uses n inputs in quantities to produce output via the differentiable production function . The firm faces parametric prices for output (p) and for each of the n inputs (. Assume that, for every vector of strictly positive prices, profit maximizing employment levels of the n inputs exist as differentiable functions of prices: nxxx .,..,,21(nxxxFq .,..,,21=))nwww .,..,,21 ()niwwwpx .,..,,;*21 for ..,..,2,1 ni= Show that, for all ,.,..,2,1 nji =≠ .**ijjiwxwx∂∂=∂∂ (Hint: Use the envelope theorem and Young's theorem. Young's theorem says that the second-order cross-partial derivatives of a differentiable function are equal, regardless of the order in which the derivatives are