Mathematical Methods for Economics Mock Exam 4 - Amsterdam University College PDF

Mathematical Methods for Economics 900124ACCY MOCK Exam 4 NA, NA Duration of the exam is 90 minutes The exam was made by Marina Aguiar Palma and peer-re...

Mathematical Methods for Economics 900124ACCY MOCK Exam 4 NA, NA Duration of the exam is 90 minutes The exam was made by Marina Aguiar Palma and peer-reviewed by Jaap de Jonge Course examiner: Marina Aguiar Palma Student name Instructions: ˆ Please write your answers in the space provided. Use a separate sheet if you need more space, with your name clearly written on it. ˆ Scrap paper for additional computations is provided. ˆ All your answers must be clearly motivated. In particular, please show all your work- ing. ˆ No calculators, textbook, personal notes, etc. allowed. These questions are worth 20 points jointly, 2 points for each part, of which 2 you are given to start with, and will count for 10% of your final grade You have 90 minutes to complete the test (115 minutes if granted extra time). Amsterdam University College DATE Mathematical Methods for Economics, Mock exam 4 NA Question 1 √ √ Let Y = F (K, L) = 144 K 3 L be the output of a production process when K > 0 units of capital and L > 0 units of labour are used as inputs. (a) Compute FL′ (100, 64). What is the economic interpretation of this derivative? [3 pts] (b) If capital input increases from K = 100 to K = 101 and labour input is fixed at 64, what is the approximate increase in output? [4 pts] Question 2 A function f of two variables is given for all x and y by f (x, y) = 5x2 − 2xy + 2y 2 − 4x − 10y + 5 (a) Find the first- and second-order partial derivatives of f. [5 pts] (b) Find the critical points of f , and classify them as maxima or minima, or as saddle points. [5 pts] Question 3 A consumer spends a positive amount m in order to buy x units of one good at the price of 6 per unit, and y units of a different good at the price of 10 per unit. The consumer chooses x and y to maximize the utility function U (x, y) = (x + y)(y + 2). (a) Suppose that 8 ≤ m ≤ 40. Find the optimal quantities x∗ and y ∗ , as well as the Lagrange multiplier, all as functions of m. [14 pts] (b) Find the maximum utility value as a function of m. What is its derivative at m = 20 ? Interpret it. [8 pts] (c) Assuming that the consumer is restricted to choose nonnegative values of x and y, what are the solutions for x∗ and y ∗ if: (i) m < 8; (ii) m > 40? [8 pts] Question 4 [6 pts] A firm produces three commodities A, B and C. When it sells x, y, and z units of these commodities, respectively, its weekly profits are given by P (x, y, z) = 50x + 30y + 10z + 2xy + xz + 3yz − 8000. Because of a transport bottleneck, however, it can only ship a total of 100 units per week of the three commodities, so that x + y + z = 100. What values of x, y, and z maximise this firm’s profits? Question 5 [6 pts] Consider the LP problem:  x + 2y ≤ 4  maximise x + 2y subject to −x + y ≤ 1 , x, y ≥ 0.  2x − y ≤ 3  Solve the problem graphically.