Answer to Question #288828 in Microeconomics for Ali

Question

Donald derives utility from only two goods, carrots (X) and donuts (Y). His utility function

is as follows: U(X, Y) = XY. Donald has an income (M) of $120 and the price of carrots

(PX) and donuts (PY) are both $1.

a) Find the marginal utility that Donald receives from carrots (MUX) and from

donuts(MUY) (4marks)

b) Determine the marginal rate of substitution of X for Y (MRSXY)(4 marks)

c) How does MRSXY change as the firm uses more X, holding utility constant.(3 marks)

d) What is Donald's budget line and relative price (PX/ PY )(4 marks)

e) What quantities of carrots and donuts will maximize Donald's utility? (5 marks)

Answer 1

Solution:

a.). MUx = $\frac{\partial U} {\partial X}$ = Y

MUy = $\frac{\partial U} {\partial Y}$ = X

 

b.). MRS_XY = $\frac{\partial Y} {\partial X} = \frac{MU_{X} } {MU_{X} = } = \frac{px} {Py}$ =

 

c.). MRSXY will reduce as the firm uses more X, since the marginal utility derived from each additional unit declines.

 

d.). Budget line: M = PxX + PyY

120 = X + Y

 

PX/PY = 1/1 = 1                          

 

e.). Utility Maximization:

MUx/MUy = Px/Py

Y/X = 1/1

Y = X

 

Substitute in the budget constraint:

120 = X + Y

120 = X + X

120 = 2X

X = 60

Y = X = 60

The quantities of carrots and donuts that will maximize Donald’s utility are = 60 carrots and 60 donuts.