1. (32 total points) Suppose there are two consumers, A and B. The utility funct

Question

1. (32 total points) Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = X*Y2 Therefore: For consumer A: MUX = Y; MUY = X For consumer B: MUX = Y2; MUY = 2XY The initial endowments are: A: X = 200; Y = 46 B: X = 40; Y = 26 a) (20 points) Suppose the price of Y, PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) (8 points) How much of each good does each consumer demand in equilibrium? Consumer A's Demand for X: Consumer A's Demand for Y Consumer B's demand for X Consumer B's demand for Y c) (4 points) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

Explanation / Answer

The MRS = -MUX/MUY. After substituting from the marginal utilities given above and simplifying, the marginal rate of substitution is given by:MRS=-Y/X.

In order to maximize utility subject to the budget constraint, the consumer must choose a bundle on the budget line where the MRS = MRT.   As discussed in the previous section, the MRT is just the slope of the budget constraint and is equal to –PX/PY. So the consumer will choose a bundle where, -Y/X=-Px/Py.So,Px=Y*Py/X or,Px=.23

For consumer B,MRS=MUx/MUy=Y2/2XY=-Y/2X.Here also consumer will choose bundle where,-Y/2X=-Px/Py.Putting valueof Py,we have Px=Y/2X=26/80=.33

b)We need to find out consumer A's demand for X and consumer A's demand for Y.

For consumer A,Y/X=Px/Py.

Or,Y=XPx/Py.

The budget constraint is,M=PxX+PyY

Substituting Y in budget constraint to solve X we get,

X*=M/2Px.

Now for consumer A demand for Y is,Y*=XPx/Py=M/2Py.

For consumer B we have,

Y/2X=Px/Py.Or,Y=2PxX/Py.

If the budget constraint isPxX+PyY=M

then,3PxX=M

Or,X*=M/3Px.

Putting value of X* We have Y*=2M/3Py.

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