Question

2. A consumer has the utility function U ( X₁, X₂ ) = X₁ + X₂ + X₁X₂ and the budget constraint P₁X₁ + P₂X₂ = M ,

where M is income, and P₁ and P₂ are the prices of the two goods. .

a. Find the consumer’s marginal rate of substitution (MRS) between the two goods.

b. Use the condition (MRS = price ratio) and the budget constraint to find the demand functions for the two goods.

c. Are the goods complements or substitutes for each other? How do you know?

Answer 1

(a)

MU1 = U/X1 = 1 + X2

MU2 = U/X2 = 1 + X1

MRS = MU1/MU2 = (1 + X2) / (1 + X1)

(b)

Utility is maximized when MRS = P1/P2

(1 + X2) / (1 + X1) = P1/P2

P2 + P2.X2 = P1 + P1.X1

P2.X2 = P1 + P1.X1 - P2

Substituting in budget line,

M = P1.X1 + P2.X2

M = P1.X1 + P1 + P1.X1 - P2

M = 2P1.X1 + P1 - P2

2P1.X1 = M - P1 + P2

X1 = (M - P1 + P2) / 2P1

Again,

P1.X1 = P2.X2 - P1 + P2

Substituting in budget line,

M = P2.X2 - P1 + P2 + P2.X2

M = 2P2.X2 - P1 + P2

2P2.X2 = M + P1 - P2

X2 = (M + P1 - P2) / 2P2

(c)

From demand function of X1, as P2 increases (decreases), X1 increases (decreases). From demand function of X2, as P1 increases (decreases), X2 increases (decreases). So X1 and X2 are substitutes.