Question

Suppose a consumer consuming two commodities X and Y has the following utility function 0.4 0.6 U  X Y .

If price of good X and Y are 2 and 3 respectively and income constraint is Birr 50.

a. Find the quantities of X and Y which maximize utility.

b. Show how a rise in income to Birr 100 will affect the quantity of X and Y.

c. Calculate the maximum utility for case “b”.

Answer 1

Solution:

Derive MRS_XY = $\frac{MU_{X} }{MU_{Y}} = \frac{P_{X} }{P_{Y}}$

U= 10X^0.4Y^0.6

MUx = $\frac{\partial U} {\partial X}$ = 4X^-0.6Y^0.6

MUy = $\frac{\partial U} {\partial Y}$ = 6X^0.6Y^-0.4

 

Set $\frac{MU_{X} }{MU_{Y}} = \frac{P_{X} }{P_{Y}}$

$\frac{4X^{-0.6}Y^{0.6} }{6X^{0.6}Y^{-0.4} } = \frac{2 }{3}$

Y = X^1.2

Budget constraint: M = PxX + PyY

50 = 2X + 3Y

Substitute the value of X in the budget constraint to derive Y:

50 = 2X + 3X^1.2

X = 7.7

Y = X^1.2 = 7.7^1.2 = 11.6

The quantities of X and Y which maximize utility (Uxy) = (7.7, 11.6)

 

New budget constraint: 100 = 2X + 3Y

100 = 2X + 3X^1.2

X = 14.1

Y = X^1.2 = 14.1^1.2 = 23.9.

The new quantities of X and Y which maximize utility (Uxy) = (14.1, 23.9).

 

The rise in income to Birr 100, will allow the consumer to double the quantities of X and Y since they will have more funds to spare.