(a) it is homogeneous of degree I in p: (b) it is nondecreasing in p and increasing in u (c)it is concave in p Proof: Straightforward Proposition: u is continuous, X=R"+. Consider Jv(p, y)=max u(x) s.pX≤y yelp, u)=min p-x t.u(x)≥u (i)Assume that the preference relation satisfies local nonsatiation. If x solves(A), then x solves B)with u=v(, y) (ii) Assume that px*>0. Ifx* solves(B), then x*also solves(A)with y =e(p, u) (i) Suppose x*solves(A), but not(B) →彐 x such that px<px*andl(x)≥u=(p,y) Local nonsatiation→彐xl, near x, such that px'spx*≤ yand t(x)>M(x)≥u =x' is feasible to(A)and u(x)>v(p, y), contradicting to the optimality of x (ii) Suppose x*solves(B). Let y=px*>0. We want to show: if pxs y, then ux)su(x*) Consider such a x and let x'=ax,0<a< 1. It is clear that p x'<y. Therefore, x' is infeasible fo (B), which implies that u(x)<u(x*). Then by continuity of the utility function u, it follows that imal(ax)=l(x)≤l(x*) as required Two Important Identities(Direct Results from the Above Proposition) (a) e(p, v(p, y)) (b)(p,e(p,u)=l. ply that the indirect utility function and the expenditure functi 2.4 Duality in Consumer Theory Duality Theorem: Assume that u is continuous, quasi-concave, and strongly monotone. Let X (a) For p>0 I(p, 1)=max u(x) t.px≤1 X (b)For x>0 u(x)=min v(p, 1) ≤1 p8 (a) it is homogeneous of degree 1 in p; (b) it is nondecreasing in p and increasing in u; (c) it is concave in p. Proof: Straightforward. Proposition: u is continuous, X = Rm +. Consider      = st y v y u p x p x x . . ( , ) max ( ) (A)     =  st u u e u . . ( ) ( , ) min x p p x x (B) Then, (i) Assume that the preference relation satisfies local nonsatiation. If x * solves (A), then x * solves (B) with u = v(p, y). (ii) Assume that px* > 0. If x* solves (B), then x* also solves (A) with y = e(p, u). Proof: (i) Suppose x* solves (A), but not (B):   x such that p x < p x* and u(x)  u  v(p, y). Local nonsatiation   x 1 , near x, such that p x1  p x*  y and u(x 1 ) > u(x)  u.  x 1 is feasible to (A) and u(x 1 ) > v(p, y), contradicting to the optimality of x * . (ii) Suppose x* solves (B). Let y = p x* > 0. We want to show: if p x  y, then u(x)  u(x*). Consider such a x and let x' = x, 0 <  < 1. It is clear that p x' < y. Therefore, x' is infeasible for (B), which implies that u(x') < u(x*). Then by continuity of the utility function u, it follows that lim ( ) ( ) ( *) →1 u x = u x  u x as required.  Two Important Identities (Direct Results from the Above Proposition) (a) e(p, v(p, y)) = y. (b) v(p, e(p, u)) = u. • They together imply that the indirect utility function and the expenditure function are somehow "equivalent". 2.4 Duality in Consumer Theory Duality Theorem: Assume that u is continuous, quasi-concave, and strongly monotone. Let X = R m +. Then (a) For p > 0, v(p, 1) = max u(x) s.t. px  1 (C) x  X. (b) For x > 0, u(x) = min v(p, 1) s.t. px  1 (D) p  0