Question

Sara likes consume Bananas (X) and Mangoes (Y). The cost of banana is birr 2 and that of Mango is bírr 4. Sara's monthly budget for these fruit is birr 100 and her utility function is U(X,Y) = X ^ 2 * Y ^ 1/2,Then calculate Utility maximizing units of Bananas and Mangoes.

Answer 1

U(_xy)=X$^2$ Y^0.5

Budget line,where p_x=2,p_y=4

P_xX+P_yY=100

2X+4Y=100

Utility is maximized when

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

MUx=$\frac{du}{dy}$ X$^2$ Y^0.5

MU_x=2XY^0.5

MU_y=0.5X$^2$Y^-0.5

^therefore,

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

$\frac{2XY^\frac{1}{2}}{0.5X^2Y^\frac{-1}{2}}=\frac{2}{4}$

$\frac{2y}{0.5x}=\frac{1}{2}$ ,

0.5x=4y

x=8y

substitute in budget line function

2x+4y=100

2(8y)+4y=100

16y+4y=100

20y=100

y=5

2x+4y=100

2x+(4$\times5)$ =100

2x+20=100

2x=100-20

2x=80

x=40

maximum utility Bananas(x)=40,Mangoes(y)=5