Abduls utility is U(X A, Y A)= min{X A, Y A}, where X A and Y A are his consumptions of goods X and Y respectively.Babettes utility function is U(X B, Y B)= X B Y B, where X B and Y B are her consumptions of goods X and Y.Abduls initial endowment is no units of Y and 5 units of X.Babettes initial endowment is no units of X and 11 units of Y.If X is the numeraire good and p is the price of good Y, then supply will equal demand in the market for Y if
A)11/(p + 1)+ 2.50 = 11.
B)5/(p + 1)+ 5.50 = 11.
C)11/5 = p.
D)min{5, 11} + 5/2p = 11.
E)min{5, 11} + 11/2p = 11.

Question

Quizplus · Accepted Answer

Since X is the numeraire good, its price is normalized to 1. Thus, Abdul's initial income is 5 and Babette's initial income is 11/p. 
To find the equilibrium condition in the market for good Y, we need to equate the quantity demanded and quantity supplied of good Y. 
Abdul's demand for good Y is given by the marginal rate of substitution: 
MRS(A) = dYA/dXA = -MUx/MUy = p 
where MUx and MUy are the marginal utilities of goods X and Y respectively. 
Substituting Abdul's utility function, we get: 
p = -d(U(A))/d(YA) / d(U(A))/d(XA) = -10YAXA^-2 
Similarly, Babette's demand for good Y is given by: 
p = -d(U(B))/d(YB) / d(U(B))/d(XB) = XB / YB 
At the equilibrium, total demand for good Y = total supply of good Y 
That is, p(QA + QB) = 11, where QA and QB are the quantities of good Y demanded by Abdul and Babette respectively. 
Substituting the demand functions derived above, we get: 
p(-10XAA^-2 + QB / YB) = 11 
Substituting Abdul's initial endowment (XA = 5, YA = 0) and Babette's initial endowment (XB = 0, YB = 11/p), we can solve for the equilibrium price of good Y: 
p = 11 / (5/1 + 11/p) = 5.5 
Substituting p = 5.5 in either Abdul's or Babette's demand function for good Y, we can solve for the quantities demanded by Abdul and Babette: 
QA = 5 / (5.5) = 10/11 
QB = (11/p) / (1 + (XB / YB)) = 11 / (1 + (5.5 / 11)) = 6/11 
Therefore, supply will equal demand in the market for Y if Abdul demands 10/11 units of Y and Babette demands 6/11 units of Y. The only answer choice that matches this is B.

Abduls Utility Is U(X A, Y A)= Min{X A, Y