Question

A person spends all his/her $400 weekly income on goods, X and Y. His/her utility function...

A person spends all his/her $400 weekly income on goods, X and Y. His/her utility function is U(X,Y)=XY.

a) What is the marginal rate of substitution (MRS) for consuming 4 units of X and 8 units of Y?

b) How much of each good would be purchased if the price of X is 4 and the price of Y is 8?

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Answer #1

ANswer
a)

MRS=MUx/MUy
MU is the change in utility, and that is found by partial differentiation with respective the specific good
MUx=dU/dX=Y and MUy=dU/dY=X
MRS=-(Y/X)

X=4 and Y=8 as given

MRS=8/4=-2, the MRS is shown in neagative terms

b)
A consumer maximize consumption at
MUx/MUy=Px/Py
the MRS=price ratio
equating both
Y/X=4/8
Y=0.5X

the budget line is the form of
XPx+YPy=I
X=quantity of X, Y=quantity of Y
Px= price of X, Py= price of Y
I=income
4X+8Y=400
Y=0.5X
4X+8*(0.5X)=400
4X+4X=400
8X=400
X=400/8
X=50
Y=0.5*50
Y=25
a consumer will purchase 50 units of X and 25 units of Y for the given utility function and income with the prices.

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