Econ 600 Schroeter Fall 2010 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 wording. a. Let U be a convex subset of nℜ. If ℜ→UF : is quasi-concave then ()()()(){}vFuFvhhuF ,max1 ≥−+ for all Uvu∈, and for all []1,0∈h. b. Given ℜ→ℜ:f differentiable and ℜ∈*x . If ()0*=′xf and ()0*<′′xf then ()⋅f has a strict local maximum at *x . c. Given differentiable functions ℜ→ℜℜxFn: and ℜ→ℜng : . Consider the problem: ()axFxtrw;max... subject to ()bxg =, where x is an n-dimensional vector of choice variables, a is a scalar parameter that enters the objective function but not the constraint function, and b is a scalar constant. Assume that for 1aa =, the problem has a regular solution at ()1*ax . Define the value function: ()()()aaxFaF ;**≡. Then ()⋅*F is differentiable at 1a and ( ) ( )( )11*1*; aaxaFadadF∂∂=. d. Given ℜ→ℜnF : differentiable. If nxℜ∈* is a local solution to ()xFxtrw...max subject to nx+ℜ∈, then, for ()0,,,2,1*≥∂∂= xxFniiK . e. Cauchy sequences in nℜ are bounded.22. Consider three sequences in {}{},,:111∞=∞=ℜnnnnyx and {}∞=1nnz. For each nnnzyxn ≤≤= ,,3,2,1K. Also, both {}∞=1nnx and {}∞=1nnz converge and have the same limit: {}axnn→∞=1 and {}aznn→∞=1. Prove that {}aynn→∞=1. 3. 12: ℜ→ℜg is differentiable with 0,21>∂∂∂∂xgxg and 22xg∂∂ negative definite throughout 2ℜ . 1a and 2a are positive constants. Consider the problem: 2211,...21min xaxaxxtrw+ subject to ()0,21=xxg. i. Write down the Lagrangian and the first-order necessary conditions for this problem. Show that the second-order sufficient conditions for a strict local minimum will be satisfied at any solution to the first-order conditions. ii. Let ()()21*221*1,,, aaxaax , and ()21*, aaF denote the solution values of the choice variables and the value function, respectively. Show that 1*aF∂∂ has the same algebraic sign as *1x . Sketch and explain a couple of graphs that illustrate the intuition for this result. iii. Find and sign expressions for 1*1ax∂∂ and 1*2ax∂∂. 4. Consider the following utility maximization problem: ()baxxUxxtrw,;,max21,...21 subject to 0,0,212211≥≥≤+ xxIxpxp, where 1x and 2x are quantities of the two goods, ()⋅U is the utility function, a and b are positive parameters of the utility function, 1p and 2p are positive prices, and I is positive income. For each of the following utility functions, find restrictions on prices, income, and utility function parameters that are implied by zero consumption of good 1 at the optimum. i. ()()()bxaxbaxxU +⋅+=2121,;,. ii. ()()axxaxxU