Question

Question 1 Suppose we have two goods, whose quantities are denoted by A and B, each being a real number. A consumer’s consumption set consists of all (A; B) such that A ≥ 0 and B > 4.

His utility function is: U (A, B) = ln(A + 5) + ln(B - 4). The price of A is p and that of B is q; total income is I. You have to find the consumer’s demand functions and examine their properties. You need not worry about second-order conditions for now.

(i) Solve the problem by Lagrange’s method, ignoring the constraints A ≥ 0, B > 4. Show that the solutions for A and B that you obtain are valid demand functions if and only if I ≥ 5p + 4q. (ii) Suppose I ≥ 5p + 4q. Solve the utility maximization problem subject to the budget constraint and an additional constraint A ≥ 0, using Kuhn-Tucker theory (Bear in mind that the Kuhn-Tucker conditions coincide with the ordinary first-order Lagrangian conditions). Show that the solutions for A and B you get here are valid demand functions if and only if 4q < I ≤ 5p + 4q. What happens if I ≤ 4q?

In each of the following parts, consider the above cases (i) and (ii) separately.

(iii) Find the algebraic expressions for the income elasticities of demand for A; B. Which, if either, of the goods is a luxury?

(iv) Find the marginal tendencies to spend on the two goods. Which, if either, of the goods is inferior?

(v) Find the algebraic expressions for the own price derivatives ?A/?p, ?B/?q. Which, if either, of the goods is a Giffen good?

Answer 1

(i)

(ii) If I < 5p + 4q, we see from (i) that unconstrained demand for A would be negative. If we impose that A 0, we know from Kuhn-Tucker theory that the consumer will optimize demand by setting A = 0 exactly.

Thus, he will reach his best utility outcome by purchasing good B only. With the income I, he can purchase I/q units of good B.

So his total demand is: (A; B ) = (0; I/q )

Note that this demand function holds only for 4q < I 5p + 4q. If I > 5p + 4q, we have the demand function as in (i).

If I 2q, then the consumer will not have enough money to buy even 2 units of B, thus not allowing him to achieve any utility above negative infinity. In this case, we say that utility is not clearly defined (since ln(B-2) does not exist), so demand is not defined either.

(iii)  

(iv)

(v)