Question

5. Suppose a consumer’s utility function is given a U = 100X0.25Y0.75.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

Utility function given :

$U=100X^{0.25}Y^{0.75}$

Budget line:$I = PxX+PyY$

$280 = 2X+5Y......... (1)$

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

From utility function: 

$U=100X^{0.25}Y^{0.75}\\ MU_x= 100_x 0.25 (X^{0.25−1}) (Y^{0.75})\\MU_x= 25\frac{ Y^{0.75}}{X^{0.75}}\\ MU_y=100 _x ( X^{0.25}) x 0.75 (Y^{0.75−1})\\MU_y =75 x \frac{X^0.25}{Y^0.25} \\ \frac{MU_x}{MU_y}=\frac{1}{3}x \frac{Y}{X}$

Equating Marginal utilities and prices we get, 

$\frac{ MU_x}{MU_y}=\frac{ Px}{Py}\\\frac{1}{3}x\frac{Y}{X}=\frac{2}{5}\\Y=\frac{6}{5}X$

Putting the value of Y in , I =2X+5Y that is 280= 2X+5Y we get

$280=2X+5.\frac{6}{5}X\\280=8X\\X=35\\ Y=\frac{6}{5}.X\\Y=\frac{6}{5}.35\\Y=42$

Answer: X=35 and Y=42