Econ 600 Schroeter Fall 2008 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. If is differentiable and is a strict global maximum of , then ℜ→ℜnF :nx ℜ∈*()⋅F ()0*=∂∂xxF and (*22x)xF∂∂ is negative definite. b. If is strictly quasi-concave then it is both strictly concave and quasi-concave. ℜ→ℜnF : c. If is differentiable and convex, and is such that ℜ→ℜnF :nx ℜ∈*()0*=∂∂xxF, then is a global minimum of *x()⋅F . d. Let and be differentiable. Consider the problem: ℜ→ℜnF : ℜ→ℜng : ()xFxtrw ...max subject to ()bxg= ()* where b is a scalar constant. If is a local solution to nx ℜ∈*()* at which ()0*≠∂∂xxg, then there exists such that ℜ∈*λ()()***xxgxxF∂∂=∂∂λ.2e. Let and be differentiable. Consider the problem: ℜ→ℜ2:F ℜ→ℜ2:g ()xFxtrw ...max subject to ()bxg= ()* where b is a scalar constant. Define ()()()()xgbxFxL−+≡λλ; . If is a local solution to ( at which nx ℜ∈*)*()0*≠∂∂xxg then () ()() () ()() () ()⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂**222**212*2**212**212*1*2*1;;;;0λλλλxxLxxxLxxgxxxLxxLxxgxxgxxg is negative semi-definite. 2. Let be defined by ℜ→ℜnF :()AxxxF′= where A is a symmetric n x n matrix and x is an n x 1 vector of arguments. Use the definition of concavity to show that is concave if and only if A is negative semi-definite. ()⋅F Note: An alternative approach to this problem would use the second-order calculus criterion for differentiable concave functions. Here I’m asking you to prove the claim by using the definition of concavity directly. Hint: For and , express nxx ℜ∈21,[1,0∈h]()()()( )()[]212111 xFhxhFxhhxF −+−−+ in terms of a single quadratic form. Also remember: ()ABAB′′=′. 3. A firm uses n inputs to produce a single output via a differentiable production function. The firm faces parametric prices for output and for each of the inputs. Assume that the firm’s profit maximization problem has a regular solution for each vector of strictly positive prices, so that input demand functions exist as differentiable functions of prices. Set up the firm’s profit-maximization problem and use the envelope theorem and Young’s theorem to show that an increase in output price raises the demand for a given input if and only if an increase in the price of the input reduces optimal output. Hint: 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