Question

Suppose a consumer consuming two commodities X and Y has the following utility function X0.4 Y0.6. If price of good X and Y are 2 and 3 respectively and income constraints birr 50.

A/ Find the quantities of X and Y which Maximize utility

B/ Find the MRSxy

Answer 1

Solution:

A.). Quantities of X and Y which maximize utility:

Utility maximizing condition is where $\frac{MUx} {MUy} = \frac{Px} {Py}$

U(x,y) = 10X^0.4Y^0.6

Determine MUx and MUy:

MUx = $\frac{\partial U} {\partial X}$ = 4X^0.4-1Y^0.6 = 4X^{– 0.6}Y^0.6

 

MUy = $\frac{\partial U} {\partial Y}$ = 6X^0.4 Y^0.6-1 = 6X^0.4Y^-0.4

 

Set $\frac{MUx} {MUy} = \frac{Px} {Py}$ :

Px = 2

Py = 3

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

Simplify:

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

Y = X

Derive the budget constraint:

Budget constraint: I = PxX + PyY

50 = 2X + 3Y

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

50 = 2X + 3X

50 = 5X

X = 10

Since Y = X, then Y = 10

The quantities of X and Y that maximizes utility (Ux,y) are = (10,10)

B.). Find the MRSxy:

MRSxy = MUx/MUy = $\frac{4X^{-0.6}Y^{0.6} } {6X^{0.4}Y^{-0.4}} = \frac{2Y} {3X} = \frac{2(10)} {3(10)} = \frac{20} {30} = \frac{2} {3}$ =20/30 = 2/3

MRSxy = $\frac{2} {3}$ or 0.67