Question

There are two goods, Good 1 and Good 2, with positive prices p₁ and p₂. A consumer has the utility function U(x₁, x₂) = min{2x₁, 5x₂}, where “min” is the minimum function, and x₁ and x₂ are the amounts she consumes of Good 1 and Good 2. Her income is M > 0.

(a) What condition must be true of x₁ and x₂, in any utility-maximising bundle the consumer chooses? Your answer should be an equation involving (at least) these two variables.

(b) Use the answer to (a), along with the budget constraint, to calculate the consumer’s demand functions for both goods.

(c) Is Good 2 a normal good? Explain your answer.

(d) Are the goods complements, substitutes, or neither? Explain your answer.

Answer 1

U(x₁, x₂) = min{2x₁, 5x₂}

(a)

at equilibrium, it must be true that 2x1 = 5x2

x2 = (2/5)*x1

(b)

Budget Constraint:

M = p1X1 + p2X2

Put x2 = (2/5)*x1 in above equation

p1X1 + p2*(2/5)*x1 = M

X1(P1 + 0.4P2) = M

x1 = M / (P1 + 0.4P2) -----demand function for commodity 1

we know that,

x2 = (2/5)*x1

x2 = (0.4*M) / (P1 + 0.4P2) ----demand function for commodity 2

(c)

x1 = M / (P1 + 0.4P2)

dx1/dM > 0. As income increases, quantity of x1 increases. So, X1 is a normal good.

Similarly,

x2 = (0.4*M) / (P1 + 0.4P2)

dx2/dM > 0. So, x2 is also a Normal Good.

(d)

Since, the utility function is in Min form, both the goods are said to complements of each other. Such preferences are called Leontief Preferences. Goods have to be consumed in equal proportions.

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