Suppose we have two goods, whose quantities are denoted by A and B , each being...

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?

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Rahul Sunny answered 1 week ago

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