Question

A household consumes only apples (A) and bananas (B). The preference of the house hold is 

given by the utility function U(A, B) = A0.8B

0.2. If the income of the household is $20 and 

the price of apple (A) and banana (B) is $4 and $2, respectively, then 

a) Find the optimal consumption of A and B

b) Show the equilibrium condition graphically

Answer 1

$U(A,B)= A^{0.8}B^{0.2}$

I= 20

P$_A= 4$

P$_B= 2$

$\frac{Mu_A}{Mu_B}=\frac{P_A}{P_B}$

$=\frac{0.8A^{-0.2}B^{0.2}}{0.2A^{0.8}B^{-0.8}}= \frac{4}{2}$

$=\frac{0.8B^{0.2}B^{0.8}}{0.2A^{0.8}A^{0.2}}= \frac{4}{2}$

$=\frac{0.8A}{0.2A}=\frac{4}{2}$

A=2B

B= 0.5A

From the Budget Equation

4A+2B= 20

Replace A= 2B

4(2B)+2B= 20

10B= 20

B$^*$ = 2

Replace B= 0.5B

4A+2(0.5)= 20

4A+A=20

A$^*$ = 4

The equilibrium condition can be shown in the diagram below.