Question

Bread and Butter are the two complementary goods that people often prefer to consume for breakfast in the weekend. The utility function for the two goods are:

U (B, M) =   min (4B, 16M)

•  Find the demand function for Bread (B) and Butter (M) using the above utility function. 
• What would be the quantities of Bread (B) and Butter (M) if an individual is willing to spend $20 on Bread (B) which costs $2, and Butter (M) which costs $4.

Answer 1

Solution:

1.). Find the demand functions:

Derive the marginal utility functions:

MU_B = $\frac{\partial U} {\partial B} = \frac{64} {B}$

MU_M = $\frac{\partial U} {\partial M} = \frac{64} {M}$

$\frac{MU_{B} }{P_{B}} =\frac{MU_{M} }{P_{M}}$

P_B = 2

P_M = 4

Solve for B:

$\frac{64}{B} \div 2 = \frac{64}{M} \div 4$

B = 2M

The demand function for Bread (B): B = 2M

Solve for M:

$\frac{64}{B} \div 2 = \frac{64}{M} \div 4$

$M = \frac{B}{2}$

The demand function for Butter (M): M = $\frac{B}{2}$

 

2.). Find the quantities of Bread (B) and Butter (M) if an individual is willing to spend $20:

Derive the budget constraint:

I = PbB + PmM

Where: I = Income = 20

           Pb = Price of B = 2

           Pm = Price of B = 2

20 = 2B + 4M

B = 2M

Substitute in the budget constraint:

20 = 2(2M) + 4M

20 = 4M + 4M

20 = 8M

M = 2.5

B = 2M = 2 $\times$2.5 = 5

U_BM = (5, 2.5)

The quantities of Bread (B) and Butter (M) if an individual is willing to spend $20 are 5 and 2.5