Hi, please help me with the following question. Thanks in advance! Suppose that

Question

Hi, please help me with the following question. Thanks in advance!

Suppose that a consumer=s preferences over two goods, x and y, are represented by a Cobb-Douglas utility function: 

U(x,y)   =    x1- y   

a. Find the expenditure function of the consumer, and the compensated demand functions. 

b. Suppose that = 1/4 and that initially a consumer is in equilibrium with px = 1, py = 1, I = 100.  What are the demands for x and y. c. Starting from the position in part b, the price of good y rises to 2.  What is the increase in the "consumer price index" (CPI) for this consumer?   

d. What is the minimum increase in income necessary for the consumer to be as well off under price px = 1, py = 2, as she/he was at prices px = 1, py = 1?   Explain why the percentage increase is smaller than the increase in the CPI for the consumer that you derived in part c.   

Explanation / Answer

A. We want to minimize Px*X + Py*Y subject to X^(1-B)*Y^B = U The Lagrange function is: L = Px*X + Py*Y - v(X^(1-B)*Y^B - U), where v is a Lagrange multiplier Take the first order conditions: X: Px = v*(1-B)*(Y/X)^B Y: Py = v*B*(Y/X)^(B-1) Let's divide the top equation by the bottom. (Px/Py) = (1-B)/B * (X/Y) So, X = Y * (Px/Py) * B/(1-B) And Y = X * (Py/Px) * (1-B)/B Now substitute X into the utility function. X^(1-B)*Y^B = U [Y * (Px/Py) * B/(1-B)]^(1-B)*Y^B = U Solve for Y. [(Px/Py) * B/(1-B)]^(1-B) * Y = U Y = U * [(Px/Py) * B/(1-B)]^(B-1) And substitute Y into the utility function: X^(1-B)*Y^B = U X^(1-B)*[X * (Py/Px) * (1-B)/B]^B = U [(Py/Px) * (1-B)/B]^B * X = U X = U * [(Py/Px) * (1-B)/B]^(-B) X = U * [(Px/Py) * B/(1-B)]^B These are the compensated demands. You get the expenditure function by substituting these into the budget constraint. E = Px*X + Py*Y E = Px*[U * [(Px/Py) * B/(1-B)]^B] + Py*[U * [(Px/Py) * B/(1-B)]^(B-1)] B. The ordinary demands can be found by maximizing the utility function subject to the budget constraint. Max X^(1-B)*Y^B s.t. Px*X + Py*Y = I X: (1-B)*(Y/X)^B = vPx Y: B*(X/Y)^(1-B) = vPy (Px/Py) = (1-B)/B * (Y/X) Y = Px/Py * B/(1-B) * X X = Py/Px * (1-B)/B * Y I = Px*X + Py*Y I = Px*X + Py*[Px/Py * B/(1-B) * X] I = Px*X + Px * B/(1-B) * X I/Px = X * (1 + B/(1-B)) X = I/Px * (1-B) Y = I/Py * B These are the ordinary demands. Substitute you values X = I/Px * (1-B) X = 100/1 * (1-(1/4)) X = 75 Y = I/Py * B Y = 100/1 * (1/4) Y = 25 If Py increases to 2: Y = I/Py * B Y = 100/2 * (1/4) Y = 12.5 The old consumer bundle cost $100. That same bundle would now cost 75*1 + 25*2 = 125. So, the CPI increases by 25. D. The initial utility is: X^(1-B)*Y^B = U 75^(1-(1/4))*25^(1/4) = U 56.99 = U After the price increase, it is: X^(1-B)*Y^B = U 75^(1-(1/4))*12.5^(1/4) = U 47.92 = U The income needed for the initial utility is 100. But the income increase needed to keep utility the same is less than 25 because the consumer can substitute away from the higher-priced good. Min 1*X + 2*Y s.t. X^(3/4)*Y^(1/4) = 56.99 X: 1 = v(3/4)*(Y/X)^(1/4) Y: 2 = v(1/4)*(X/Y)^(3/4) (1/2) = 3 * (Y/X) Y/X = 1/6 X = 6Y X^(3/4)*Y^(1/4) = 56.99 6Y^(3/4)*Y^(1/4) = 56.99 Y = 56.99/6 Y = 9.5 X = 6*9.5 X = 57 The amount needed to purchase this bundle is: 57*1 + 9.5*2 = 76 So, the amount needed to compensate is only 100 - 76 = 24

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