(1) In each of the following constrained optimization problems, use the Lagrangian method to solve for a solution where possible. When the Lagrangian method fails to give the unique correct answer carefully explain why.

???? 1x2 ifx0,

(a) Let f(x) = 2 Solve maxx f(x) s.t. x3 1.

(b) Letf(x,y)=x+y.Solvemax(x,y)f(x,y)s.t.x2 0andy2 1.

(c) Letf(x,y)=2(lnx+lny).Solvemax(x,y)f(x,y)s.t.x+y2andx2+y2 2.

What would happen if the first constraint became x + y 3?

(2) Prove the Lagrange sufficiency theorem in Rn with k inequality constraints. That is,letL = f(x)+????ki=1i[bi gi(x)],wherex U Rn. Showthatif,forsome = (1,...,k), x solves maxx L(x,), and (i) i[bi gi(x)] = 0, (ii) i 0, and (iii) gi(x) bi for all i, then x solves maxx f(x) s.t. gi(x) bi for all i.

(3) Show that if f : U ???? R is a C1 quasiconcave function on an open convex set U Rn and Df(x) = 0 for all x U, then

Df(y)(xy)0 f(x)f(y).

(4) LetU Rn beopenandconvex. Letf : U ???? RbeC1 quasiconcavewithDf(x) = 0 for all x U. Let g1,...,gk : U ???? R be C1 quasiconvex. Suppose that the NDCQ holds. Let L(x, ) = f(x) + ????ki=1i[bi gi(x)]. Using the result stated in question (3) above, show that if there exists x and = (1, . . . , k) such that

L(x,)=0 forall j={1,...,n}, xj

i 0,gi(x)bi,andi[bi gi(x)]=0foralli={1,...,k},then x = argmaxf(x) s.t. g1(x) b1,...,gk(x) bk.

x

(5) A consumer's preferences are represented by utility u(x,y) = xy, where x and y are the quantities consumed of two goods. Let > 0 and > 0. Suppose the consumerfacespricespX >0andpY >0,hasincomem>0,andchoosesx0 and y 0 to maximize utility. Solve the associated Lagrangian problem.

(a) For what values of and is this utility function concave?

(b) For what values of and is this utility function quasiconcave?

(c) For what values of and are the solutions to the Lagrangian problem also the solution to the consumer's constrained maximization problem? Why?