Econ 600 Schroeter Fall 2004 Homework #1 1. For , define the set of lower bounds for ℜ⊂A(),, ALA as follows: (){}.: AaallforallAL∈≤ℜ∈= If φ≠)(AL (A has at least one lower bound; that is, A is bounded below), define the infimum of A (or the greatest lower bound of A), denoted , as follows: Ainf *inf lA = such that ()ALl∈* and for all ll ≥*().ALl∈ It is a fundamental property of the real numbers that any set of real numbers that is bounded below has an infimum and it is well-defined by the statement above. In other words, for any A, bounded below, there is a unique l* that satisfies the stated conditions. Let {}{}{}IicCIibBIiaAiii∈=∈=∈= :,:,: and {}IidDi∈=: be finite or infinite sets of real numbers indexed by the same set, I. (I might be the set of positive integers, for example, in which case, A, B, C, and D would each contain countably infinitely many elements.) a.) Suppose that both A and B are bounded below and that for all Prove iiba ≥ .Ii ∈ .infinf BA ≥ b.) Suppose that both C and D are bounded below and let α and β be positive real numbers. Prove {}.infinf:inf DCIidciiβαβα+≥∈+ c.) Let for , be a finite or infinite collection of concave functions. ℜ→ℜnif :Ii ∈Define as follows: ℜ→ℜnf : For ()(){}.:inf, Iixfxfxin∈=ℜ∈ (Assume that the set that appears on the right-hand-side of this equation is bounded below for all .) Use the results of a and b to prove that nx ℜ∈()⋅f is concave.22. Let be a concave function. Let A be an n x m matrix and let Consider the function defined by: ℜ→ℜnf :.nb ℜ∈ℜ→ℜmg : For ()()., bAxfxgxm+=ℜ∈ Prove that is concave. ()⋅g 3. In lecture we showed that defined by the formula ℜ→ℜ+2:f()2121, xxxxf ⋅= is quasi-concave. Use the same formula to define a function on domain instead of . Is that function also quasi-concave? Explain. 2ℜ2+ℜ 4. Let U be a convex subset of and suppose that the function is both concave and strictly quasi-concave. Is nℜ ℜ→Uf :()⋅f strictly concave? Either prove or provide a counterexample. 5. Given U, a convex subset of we say that a function nℜℜ→UF : is quasi-convex if, for all , and for all , Uvu ∈,[]1,0∈h ()()()(){}.,max1 vFuFvhhuF≤−+ Prove that is quasi-convex if and only if ()⋅F()⋅−F is