Two agents A and B have the following indirect utility functions

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

 

answer:

We can get demand funciton form indirect utility function of the consumers A and B by Roy’s identity. It is written as x(p,I) = (∂v/∂pi)/(∂v/∂I). Now we calculate 6 partial differentiation ∂vA/∂P1 = -a/P1 , ∂VA/∂P2=-(1-a)/P2 and ∂vA/∂I=1/IA. Similarly we get ∂vB/∂P1=-b/P1, ∂vB/∂P2=-(1-b)/P2 and ∂vB/∂I = 1/IB. Now we calculate the demand function as X1A=aIA/P1 and X2A=(1-a)IA/P2 and for agent B we have X1B= bIB/P1 and X2A=(1-b)IB/P2.

The initial endowments are eA1 and eA2 for agent A and eB1 and eB2 for agent B. Then X1A+X2A=2 and X1B+X2B=2. Now by using the demand function we have aIA/P1+bIB/P1=2 so P1= (aIA+bIB)/2 and similarly we get P2=[(1-a)IA+(1-b)IB]/2. So we can calculate P1/P2 and thses prices would be the market clearing price.

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