Question

2. A consumer has the utility function U ( X_{1},
X_{2} ) = X_{1} + X_{2} +
X_{1}X_{2} and the budget constraint
P_{1}X_{1} + P_{2}X_{2} = M ,

where M is income, and P_{1} and P_{2} 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.**

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