Math for Econ II - Homework Assignment 3 New York University Due in Recitation, Friday, September 30th Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work you will not be given any credit. Do not forget to write your name and Recitation section number on your work and to staple your work, if you have multiple pages. 1. (4 pts) The figures below show the optimal point (marked with a dot) in three optimization problems with the same constraint. Arrange the corresponding values of from smallest to largest. Assume > 0 in each case. Briefly justify your answer. (In diagram (II), the bottom is not labeled \"fx = 1.\" It is the curve f = 1, with the x-axis labeled below it) (II) (I) (III) 2. (6 pts) An individual purchases quantities a, b, and, c of three different commodities whose prices are p, q, and r, respectively. The consumer spends m dollars, where m > 2p, and the utility function is given by U (a, b, c) = a + ln(bc) Show that the optimal expenditure on the second and third good is equal to p (that is, show that when utility is maximized, the expenditure on each of the second and the third good is equal to p). Find a, b, and c when utility is maximized (you may assume Lagrange multipliers gives the max solution) . 3. (6 pts) A company needs three different materials to manufacture its product. Given x units of the first material, y units of the second material, and z units of the third material, the company produces f (x, y, z) = 25x2/5 y 1/5 z 1/5 tons of its product. Each unit of the first material costs $80 to buy, each unit of the second material costs $12, and each unit of the third material costs $10. The company's budget for materials is $24, 000. What values of x, y, and z will maximize production? (You may assume Lagrange multipliers gives the maximum) 4. A firm that uses two inputs x and y has production given by U (x, y) = 5x1/3 y 1/3 , for x, y 0. The cost of input x is $20 and the cost of input y is $40 and the firm earns $96 for each unit of output. Futhermore, the firm is contrained by the government to use exactly 100 units of x. (a) (4 pts) How much of y does it use to maximize profit? (b) (3 pts) What is the most that the firm is willing to bribe the government inspector to use a little more x? 5. (7 pts) Find the minimum and maximum of f (x, y) = exy in the elliptical region x2 + 2y 2 1. Extra problems-not to be handed in: 1. Find the maximum of f (x, y) = 24x x2 + 16y 2y 2 when x2 + 2y 2 = 44. What is the approximate change in the maximum value of f if the constraint is changed to x2 + 2y 2 = 45? 2. (a) (5 pts) Find the minimum and maximum of f (x, y, z) = x2 + y 2 + z subject to the constraint x2 + 2y 2 + 4z 2 = 1. (b) (2 pts) Suppose the constraint was changed to x2 + 2y 2 + 4z 2 = 1.5. Without re-solving the problem under the new constraint, estimate the maximum value of f under the new constraint