Question

Suppose a consumer consuming two commodities X and Y has the following unity fiction
U=X04 Y06 if price of good X and Y are 2 and 3 respectively and income constraint is Birr 50.
1. Find the quantities of X and Y which maximize utility.
2. Show how a rise in income to Birr 100 will affect the quantity of X and Y.

Answer 1

Solution:

1.). U = X^0.4Y^0.6

Budget constraint = 2X + 3Y = 50

Utility Maximization: $\frac{MU_{X} }{MU_{Y}} = \frac{P_{X} }{P_{Y}}$

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

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

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

$\frac{0.4X^{-0.6}Y^{0.6} }{P_{X} } = \frac{0.6X^{0.4}Y^{-0.4} }{P_{Y}}$

Simplify:

X = Y

Substitute in the budget constraint:

50 = 2Y + 3Y

50 = 2Y + 3Y

50 = 5Y

Y = 10

X = 10

 

2.). New budget constraint: 100 = 2X + 3Y

100 = 2Y + 3Y

100 = 5Y

Y = 20

The new quantity of X = 20

The new quantity of Y = 20