Question

2. A consumer has the utility function U ( X1, X2 ) = X1 + X2...

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?

Homework Answers

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.

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