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^{1/2}Y^{1/2}

(1)

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

here

MUx(marginal utility of X) =dU/dX

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

MUy(marginal utility of Y) =dU/dY

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

So

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

MRS=Y/X

(2)

d(MRS)/dX=-(Y/X^{2})

(3)

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

let z is a constant variable

U(zX,zY)=(zX)^{1/2}(zY)^{1/2}

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

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