Question

Suppose a consumer consuming two commodities X and Y has the following utility function U= 10X0.4Y0.6. If price of good X and Y are 2 and 3 respectively and income constraint is Birr 50, then find

Find quantities of X and Y which maximize utility

Show how the rise in income to Birr 100 will affect the quantities of X and Y

Answer 1

Solution:

To maximize utility: MRSxy =$\frac{MUx}{MUy} = \frac{Px}{Py}$

Derive MUx:

U = 10X^0.4Y^0.6

MUx = $\frac{\partial U} {\partial X}$ = 4X^-0.6Y^0.4

Derive MUy:

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

MRSxy = $\frac{MUx}{MUy}$ = $\frac{4X^{-0.6}Y^{0.4} }{6X^{0.4}Y^{-0.4} }$

MRSxy = $\frac{Px}{Py}$

$\frac{2Y^{0.8} }{3X }$ = $\frac{2 }{3 }$

Y = X^1.25

Budget constraint: I = PxX + PyY

50 = 2X + 3Y

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

50 = 2X + 3(X^1.25)

50 = 2X + 3X^1.25

X = 7

Y = X^1.25 = 7^1.25 = 12

Y = 12

The quantities of X and Y which maximize utility (Uxy) = (7, 12)

 

New budget constraint: 100 = 2X + 3Y

Y = X^1.25

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

100 = 2X + 3(X^1.25)

100 = 2X + 3X^1.25

X = 13

Y = X^1.25 = 13^1.25 = 25

Y = 25

The new quantities of X and Y which maximize utility (Uxy) = (13, 25)

The rise in income to Birr 100, will enable the consumer to double the quantities of X and Y since the consumers will be having more funds to spare.