Econ 600 Schroeter Practice Problem Set #1 1. Let be defined by ℜ→ℜ2:F()22221312122, xxxxxxF ++= . Let and . Solve for the value of ()′= 2,1*x()′=Δ 1,1xθ implicitly defined by ()()() ().*21***22xxxxFxxxxFxFxxF Δ⋅Δ+∂∂⋅′Δ+Δ⋅∂∂+=Δ+θ 2. Let X be a convex subset of . Given nℜℜ→XF : and ℜ→XG :, define by ℜ→XH :() ()()uGuFuH +=. Prove or provide a counterexample: a. If and are concave on ()⋅F()⋅GX, then ()⋅H is concave on X. b. If and ()⋅F()⋅G are quasi-concave on X, then ()⋅H is quasi-concave on X. c. If is concave and is quasi-concave on ()⋅F()⋅GX, then ()⋅H is concave on X. d. If is concave and is quasi-concave on ()⋅F()⋅GX, then ()⋅H is quasi-concave on X. (Hint: If you think that a claim is false, try to find a counterexample for the case n = 1. A single counterexample may suffice to disprove more than one of the claims.) 3. a. Consider defined by ℜ→ℜ++2:u()βα2121, xxxxu = where α and 0>β. (Note: denotes the strictly positive quadrant of ; that is, 2++ℜ2ℜ(){}.0,:,212212>ℜ∈=ℜ++xxxx ) Show that ()⋅u is concave if 1≤+βα but not if 1>+βα. (Hint: Remember the characterization of concavity in terms of the Hessian matrix.) b. Now consider defined by ℜ→ℜ++2:v()().log,2121βαxxxxv = Show that is concave for all ()⋅vα and 0>β.24. a. is quasi-concave and is convex. Define ℜ→ℜnF :nX ℜ⊂ () (){}.: XxallforxFxFXxS∈≥∈= That is, S is the set of solutions to the problem ()xFxtrw ...max such that Xx∈. Assume that S is not empty. Prove that S is convex. b. is strictly quasi-concave and is convex. Define S as above. ℜ→ℜnF :nX ℜ⊂Assume that S is not empty. Prove that S contains a single point. 5. (Simon and Blume 21.5) Let U be a convex subset of and consider a function . Define nℜℜ→UF : ()(){}.:,1xFyandUxyxBn≤∈ℜ∈=+ (Note: B can be described as the set "below the graph of ()⋅F.") Prove that is concave if and only if B is convex. ()⋅F 6. Consider and the following two properties of : ()ℜ→ℜℜmnxaxF :;()⋅F A. is concave in (axF ;)()mnax+ℜ∈;. B. (axF ;) is concave in for each nx ℜ∈ma ℜ∈. (axF ;) is concave in for each ma ℜ∈nx ℜ∈. Prove that A implies B. Give a counterexample to show that B does not imply A. (Hint: Take n = m = 1 and look at s that are linear (hence concave) in x for each fixed value of a, and linear in a for each fixed value of x.) ()⋅F 7. (part of Simon and Blume 21.19). Prove the equivalence of the following two alternative definitions of quasi-concavity: I. U convex . nℜ⊂ℜ→UF : is quasi-concave if (){}axFUx ≥∈: is convex for all . ℜ∈a II. U convex . nℜ⊂ℜ→UF : is quasi-concave if for all Uyx∈, such that , and for all () ()yFxF ≥[]()()()yFyhhxFh