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

 

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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 2 0 and B > 4. His utility function is: U (A, B) = In(A + 5) + In(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. ) Solve the problem by Lagrange's method, ignoring the constraints A2 0, B > 4. Show that the solutions for A and B that you obtain are valid demand functions if and only if i 2 5p + 49. () Suppose I 2 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 49 < I s 5p + 4q. What happens if I s 4q? In each of the following parts, consider the above cases (1) and (i1) separately. (ii) 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? () Find the algebraic expressions for the own price derivatives aA/ap, ƏB/aq. Which, if either, of the goods is a Giffen good? 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 2 0 and B > 4. His utility function is: U (A, B) = In(A + 5) + In(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. ) Solve the problem by Lagrange's method, ignoring the constraints A2 0, B > 4. Show that the solutions for A and B that you obtain are valid demand functions if and only if i 2 5p + 49. () Suppose I 2 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 49 < I s 5p + 4q. What happens if I s 4q? In each of the following parts, consider the above cases (1) and (i1) separately. (ii) 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? () Find the algebraic expressions for the own price derivatives aA/ap, ƏB/aq. Which, if either, of the goods is a Giffen good?