Question

Suppose a consumer’s utility function is given a U = 100X^0.25Y^0.75.The prices of the two commodities X and Y are Birr 2 and Birr 5 per unit respectively. If the consumer’s income isBirr 280, how many units of each commodity should the consumer buy to maximize his/her utility? 

Answer 1

Solution:

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

MRS = $\frac{MUx}{MUy} = \frac{Px}{Py}$

First, derive MRS:

MRS = $\frac{MUx}{MUy} = \frac{Px}{Py}$

MUx =  $\frac{\partial U} {\partial X}$ = 25Y^0.75

MUy = $\frac{\partial U} {\partial Y} =$ 75X^0.25

MRS = $\frac{MUx}{MUy}$ = $\frac{25Y^{0.75} }{75X^{0.25} }$

Set MRS equal to Px/Py to derive the utility maximizing bundle:

Px = 2

Py = 5

$\frac{25Y^{0.75} }{75X^{0.25} }$ = $\frac{2}{5}$

X = $\frac{625Y^{3} }{1296 }$

Plug X into the budget constraint to derive Y:

Budget constraint: M = PxX + PyY

280 = 2X + 5Y

280 = 2(625Y³/1296) + 5Y

Y = 6.4

Plug this into X equation:

X = 625Y³/1296 = 625(6.4³)/1296 = 163840/1296 = 126.4

X = 126.4

Utility maximizing bundle (Ux,y) = (126.4, 6.4)