Question

Consider a consumer with the following utility function: U(X, Y ) = X1/2Y 1/2

(a) Derive the consumer’s marginal rate of substitution

(b) Calculate the derivative of the MRS with respect to X.

(c) Is the utility function homogenous in X?

(d) Re-write the regular budget constraint as a function of PX , X, PY , &I. In other words, solve the equation for Y .

(e) State the optimality condition that relates the marginal rate of substi- tution to the prices.

(f) Use the optimality condition from part (e) and the budget constraint

from part (d) to solve for this consumer’s demand functions X and Y .

(g) Now suppose there are 25 consumers with this utility function and each consumer has I = $10. Denote the market quantity demanded for good Y as QY . Find QY .

(h) Suppose the market supply for good Y is given by the function: QY =5PY

Solve for the equilibrium price and quantity in the market for good Y .

Answer 1

Solution

U(X, Y)=X^1/2Y^1/2

(1)

The marginal rate of substitution(MRS)=MUx/MUy

here

MUx(marginal utility of X) =dU/dX

MUx=1/2(X^-1/2Y^1/2)

MUy(marginal utility of Y) =dU/dY

MUy=1/2(X^1/2Y^-1/2)

So

MRS= [1/2(X^-1/2Y^1/2)] / [1/2(X^1/2Y^-1/2)]

MRS=Y/X

(2)

d(MRS)/dX=-(Y/X²)

(3)

U(X, Y)=X^1/2Y^1/2  

let z is a constant variable

U(zX,zY)=(zX)^1/2(zY)^1/2  

U(zX,zY)=z^1/2z^1/2(X^1/2Y^1/2 )

U(zX,zY)=zU(X, Y)

so the variable completely factor out so utility function is homogenous

(4)

Budget line

I=X(Px)+Y(Py)

Y(Py)=I-X(Px)

Y=(I/Py)-X(Px/Py)