Question

Given two commodities A and B, where price of A is ₦50 and the price of B is ₦60. If the consumer’s income is ₦1200 and the utility function U = AB, Find: i. the equilibrium values of the two commodities. ii. Mention four properties of the demand function derived in (i) above

Answer 1

Solution:

i). Equilibrium values of the two commodities:

P_A = 50

P_B = 60

I = 1200

U (_{A, B}) = AB

Derive Budget line:

I = P_AA + P_BB

1200 = 50A + 60B

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

U (_{A, B}) = AB

MU_A = $\frac{\partial _{u} }{\partial _{A}}$ = B

MU_B = $\frac{\partial _{u} }{\partial _{B}}$ = A

$\frac{MU_{A} }{MU_{B}}$ = $\frac{B }{A}$

Therefore: $\frac{MU_{A} }{MU_{B}}$ = $\frac{P_{A} }{P_{B}}$

$\frac{B }{A}$ = $\frac{50}{60}$

B = $\frac{5 }{6} A$

Solve for A:

I = P_AA + P_BB

1200 = 50A + 60B

1200 = 50A + 60 $(\frac{5 }{6} A)$

1200 = 50A + 50A

1200 = 100A

A = 12

Derive B:

B = $\frac{5 }{6} A$

B =$(\frac{5 }{6}) 12$

B = 10

U (_{A, B}) = AB

U (_{12, 10}) = $12\times 10 = 1,200$

The equilibrium value of commodity A = 12

The equilibrium value of commodity B = 10

ii). The four properties of the demand function derived above include the following:

·        The demand function is single-valued.

·        The demand function is insensitive to proportional increases in price and income.

·        The demand function exists.

·        The demand function exhausts the consumer’s budget.