In: Economics
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?