2.3. 2Indirect Utility Function Therefore, as a result of the above lemma, under the local nonsatiation assumption, utility-maximizing problem can be restated as w(p, y)=max u(x) s.t. px=y The function v(p, y) that gives the answer to the consumer's problem is called indirect utility function. The value of x that solves this problem is called the consumer's demandable bundle: it gives how much of each good the consumer desires at a given level of prices and income Proposition: The indirect utility function has the following properties (a)Ip, y) is homogeneous of degree 0 in(p, y) (b)wp, y) is nonincreasing in p and increasing in y (c)Ip, y)is quasi-convex with respect to p, that is, the set (p: v(p, y)s) is convex for very y>0 and c. Proof: Parts(a)and (b) are straightforward. Let us prove(c). For any real number c, suppose v(pl, y)scand v(p, y)sc. Then for any t: 0<I<l, let p=tpl +(1-1)p2. Define Ci=(x: prx sy) for i I and 2, C=x: px syl. We claim that CC CI U C2. It suffices to show that C S(CI U C2) This is true since x∈(C1∪C2)→px> y and p2x>y →px=|px+(l-D)p2x>y→x∈C Now, 1(p, y)=max(u(x):xE C). Let x* be an optimal solution, then x∈C1∪C2→(x*)≤(p1,y)or以(x*)≤v(p2,y) Hence p=tpI +(l-Op2E (p: v(p, y)sc. max((pl, y),v(pl,y))sc →v(p,y)=l(x*)≤ Example 1( Cobb-Douglas Utility Function): a1>0,i=1, Then the corresponding indirect utility function is v(p, y)=max subject to px=y Then using the first-order conditions(n= Lagrange multiplier) ax∏x 2p;=0, Multiplying xleadsto ∏x-2x1=0,i Summing these equations over i and letting a= a.. we get 0→A which leads to6 2.3.2Indirect Utility Function Therefore, as a result of the above lemma, under the local nonsatiation assumption, a utility-maximizing problem can be restated as: v(p, y) = max u(x) s.t. px = y The function v(p, y) that gives the answer to the consumer's problem is called indirect utility function. The value of x that solves this problem is called the consumer's demandable bundle: it gives how much of each good the consumer desires at a given level of prices and income. Proposition: The indirect utility function has the following properties: (a) v(p, y) is homogeneous of degree 0 in (p, y); (b) v(p, y) is nonincreasing in p and increasing in y. (c) v(p, y) is quasi-convex with respect to p, that is, the set {p: v(p, y)  c} is convex for very y > 0 and c. Proof: Parts (a) and (b) are straightforward. Let us prove (c). For any real number c, suppose v(p1, y)  c and v(p2, y)  c. Then for any t: 0 < t < 1, let p = tp1 + (1-t)p2. Define Ci ={x: pix  y} for i = 1 and 2, C ={x: px  y}. We claim that C  C1  C2. It suffices to show that C c  (C1  C2) c . This is true since x  (C1  C2) c  p1x > y and p2x > y  px = tp1x + (1-t)p2x > y  x  C c Now, v(p, y) = max{u(x): x  C}. Let x* be an optimal solution, then x*  C1  C2  u(x*)  v(p1, y) or u(x*)  v(p2, y)  v(p, y) = u(x*)  max(v(p1, y), v(p1, y))  c. Hence p  tp1 + (1-t)p2  {p: v(p, y)  c}.  Example 1 (Cobb-Douglas Utility Function): ( ) , 0, 1,..., . 1 u x x i i m m i i i =   = =   Then the corresponding indirect utility function is y v y x m i i i  = = = p x p subject to ( , ) max 1  Then using the first-order conditions ( = Lagrange multiplier): 0, 1,..., . 1 x x pi i m j i i i j j i  − = =  −     Multiplying xi leads to 0, 1,..., . 1 x pi xi i m m j i j j  − = = =    Summing these equations over i and letting = = m i i 1   , we get   = = −  =  = m j j m j j j j x y x 1 1 0     p x  which leads to