Answer to Question #235546 in Microeconomics for Johdrick

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