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Question

Two agents A and B have the following indirect utility functions.....

Two agents A and B have the following indirect utility functions: u^A = ln I^A - a ln P_1 - (1 - a) ln P_2 v_B = ln I^B - b ln P_1 - (1 - b) ln P_2 The initial endowments are e^I = (e^i_1, e^i_2), i = A, B respectively What is the price ratio P_1/P_2 in competitive market equilibrium? What are the competitive equilibrium allocations?

Solution

xx can xxx xxxxxx xxxxxxxx form xxxxxxxx utility xxxxxxxx xx xxx consumers x and x xy xxy'x identity. xx is xxxxxxx xx x(x,x) = (x/xx)/(x/x). Now xx xxxxxxxxx x partial xxxxxxxxxxxxxxx vA/P1 = -x/xx , VA/P2=-(1-a)/P2 xxx vA/I=1/IA. xxxxxxxxy xx xxx vB/P1=-b/P1, xx/xx=-(x-x)/xx and xx/x = x/xx. Now xx calculate xxx xxxxxx xxxxxxxx as Xxx=xxx/xx and Xxx=(x-x)xx/xx xxx xxx agent x we xxxx Xxx= xxx/xx and Xxx=(x-x)xx/xx.

The xxxxxxx xxxxxxxxxx xxx eA1 xxx eA2 xxx xxxxx x and xxx and xxx xxx xxxxx B. xxxx X1A+X2A=2 xxx Xxx+Xxx=x. xxx by xxxxx the xxxxxx xxxxxxxx xx have xxx/xx+xxx/xx=x so xx= (xxx+xxx)/x xxx similarly xx get xx=[(x-x)xx+(x-x)xx]/x. xx xx can xxxxxxxxx P1/P2 xxx xxxxx xxxxxx would xx the xxxxxx xxxxxxxx xxxxx.

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