Question

Suppose a consumer consuming two commodities x and y has the following utility function u=x⁰⁴y⁰⁶. If price of good x and good y are birr 4 and birr 6 respectively, and income constraint is birr 100.a.find marginal rate of substitution of x for y and y for x. B.find the quantities of x and y which maximize utility. C.show how a rise in income to birr 200 will affect the quantities of x and y at equilibrium.

Answer 1

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

Derive MUx:

U = 10X^0.4Y^0.6

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

Derive MUy:

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

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

MRSxy = $\frac{Px}{Py}$

$\frac{0.2Y^{0.8} }{0.3X }$ = $\frac{0.2 }{0.3 }$

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)