, UNIVERSITY COLLEGE LONDON EXAMI~ATION FOR JNTERNAL STUDENTS MODULE CODE ECON2601 ASSESSMENT ECON2601D PATTERN MODULE NAME Economics 2 (Combined Studies) DATE· Friday 27 April 2018 TIME 10:00 TIME ALLOWED 3 hrs This paper is suitable for- candidates who attended classes for this module in the following academic year(s): Year 2015/2016, 2016/2017 and 2017/2018 EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE 2016/17-ECON2601 D-001-EXAM-Statistical Science 114 2016 University College London TURN OVER SUMMER TERM 2018 ECON2601: Economics II (Combined Studies) TIME ALLOWANCE: 3 hours Answer ALL questions from Part A, ONE question from Part B, ana ONE question from Part C. Correct but unexplained answers will not receive high marks. Questions in Part A carry five percent of the total mark each, and questions in Parts Band C carry twenty-five percent of the total mark each. In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers). All remaining answers will be ignored. PART A Answer ALL questions from this section. A.l Your next holiday will be spent at Mayworld, a fantasy-themed amusement park that offers rides on unicorns to its customers. It has a two-part pricing structure that charges you a fixed fee of f = 50 (in shillings) to enter and subsequently PI = 1 (in shillings per unit) to go on each unicorn ride, where Xl denotes the number of unicorn rides you consume. Let X2 be your expenditUre on all other goods, and you may assume that both Xl and X2 are continuous, non- negative variables. Suppose that your utility function for unicorn rides and all other expenditure is U(XI, X2) = Xl + 2JX2. If your exogenous income is m = 100 (in shillings), then how many unicorn rides are you willing to consume What is the highest fixed fee that you will pay and still be willing to consume a strictly positive number of unicorn rides Call this value the reservation fee, fR, and find the corresponding number of rides, xr. Note that if you consume zero unicorn rides, Le., Xl = 0, then you do not pay the fixed fee and have X2 = m. A.2 The absolute value of the marginal rate of substitution (MRS) between goods i and j is the rati!J of their marginal utilities, MUi and MUj, respectively. Suppose that a consumer’s utility function for commuting is u(w, t, c) = aw+ ,6t+,c, where w, t, and c represent the walking time (in hours), total travel time (in hours), and the cost (in ￡) of commuting, respectively. What do the constants a, ,6, and, mean in this context How much is the consumer willing to pay in order to reduce her trip by one hour How much of an increase in the total travel time is the consumer willing to endure in order to reduce her walking time by one hour ECON2601 1 TURN OVER A.3 Suppose e(PI,P2, u) = UVPIP2. Find the Hicksian and Marshallian demand functions for good 1. Verify that the identity hi(PI,P2,U) == Xi(PI,P2,e(PI,P2,U)) is satisfied. A.4 You have the utility function U(XI’ X2) = xi1 xi1 and face prices PI = P2 = 1 with exogenous income m= 100. Determine the compensating and equivalent variations when there is. a ceteris paribus increase in PI from 1 to 2. Explain intuitively which income adjustment is greater here. A.S A gambler with initial wealth Yo > 0 goes to a casino that offers only one type of game: with probability 0 ::;; P ::;; 1, the gambler ends up with wealth Yo + ZI, and with probability 1 – P, she ends up with wealth Yo + Z2. Here, ZI > 0 and Z2 < a such that YO + Z2 ::::: O. Find the gambler's expected terminal wealth from playing a single game at the casino. Next, calculate her utility of expected wealth, expected utility of wealth, certainty equivalent, and risk premium if her utility function is u(Y) = .;y. A.6 Calculate the profit-maximising solution for a monopoly and show how the output price P is related to the elasticity of demand with respect to that output price, and defined here as c. Assuming a constant marginal cost, show that dp/dc is positive and sketch a graph of the P against c. . A.7 Two countries A and B agree to cooperate by imposing quotas on fishing in shared waters. Their respective payoffs each year are as follows: Country B overfish keep quota Country A overfish keep quota (60,60) (40,110) (110,40) (80,80) where the first number is the payoff to Country A and the second is the payoff to Country B. Assuming a one-shot simultaneous move game, what is the pure Nash equilibrium If the game is repeated infinitely many times and the countries have the same annual discount factor, 8 = 0.7, check whether an equilibrium in trigger strategies exists and examine if this equilibrium induces a different outcome in each period compared to the static Nash equilibrium of the game. A.8 Two firms compete in a duopolistic industry. Firm 1 produces output qi and has a constant marginal cost k > O. Firm 2 produces output q2 and has cost function C2(q2) = kq2 + f3q~. The market demand schedule.is given by P = a – (qi + q2) for price per unit P and with a given positive real parameter a > k. Determine the output levels of each firm (in terms of a, k, and (3) in a one-shot simultaneous move game where each firm decides, without consultation, the level of output to produce. Explain briefly how 13 affects the resulting Cournot-Nash equilibrium. ECON2601 2 CONTINUED A.9 “A lower rate of population growth over a long period of time will cause a permanent rise in the rate of growth of output per worker.” By considering the Solow Growth Model with technological progress, explain carefully using a graph and words whether this statement is true, false, or uncertain. A.IO During the past twenty years the standard of living (output per head) in Country A has grown at the same rate on average as the standard of living in Country B. However, the growth rate . of technological progress has been lower in Country A over the same period. How would you explain this phenomenon using the Solow Growth Model ECON2601 3 TURN OVER PARTB Answer ONE question from this section. 2 H.l A consumer has a generic utility function, U(XI, xz), where aau > 0 (for i = 1,2), pau < 0 (for Xt Xi i = 1,2), and aa 2 u = O. The two goods have prices PI and pz, and the consumer has anXl aX2 endowment (WI, wz) instead of exogenous income m. (a) Formulate the Lagrangian for the utility-maximisation problem. Make sure you indicate the three decision variables. Find the three first-order necessary conditions and indicate the solutions as Xi(PI,PZ,WI,WZ), X;(PI,PZ,WI,WZ), and >‘*(PI,PZ,WI,WZ). (b) What is the second-order sufficiency condition for this constrained maximisation problem Show that it is satisfied here. (c) Find ~ by substituting the optimal solutions from part (a) into the three first-order nec- ~ . essary conditions and partially differentiating them with respect to Pl. Can you determine ax’the sign of -a17Pi (d) Find ~:~ by substituting the optimal solutions from part (a) into the three first-order necessary conditions and partially differentiating them with respect to WI. What is the sign f~ o aWl. (e) How would the sign of aaxi in part (c) differ if income were exogenous Explain your answer Pi intuitively. ECON2601 4 CONTINUED B.2 You are the Chief Operating Officer of GlobalHypoMega Corporation which has two U.K. pro- duction plants, one in Birmingham and one in Manchester. All output from these two plants belonging to GlobalHypoMega is sold in a perfectly competitive industry, and the cost functions at the plants are CB(YB) = !Y~ and CM(YlvI) = Yfu., where B and M denote Birmingham and Manchester, respectively, and Yi is the output of plant i, where i = B, M. (a) YoUr objective is to produce a total of Y ~ YB + YM units of output from the two U.K. plants at minimum overall cost. Formulate your constrained cost-minimisation problem. What is the Lagrangian Indicate the decision variables clearly. (b) Find the three first-order necessary conditions to the problem in part (a). Show that the second-order sufficiency condition, which is that the determinants of all principal minor submatrices of the bordered Hessian matrix must be negative, is satisfied. (c) Solve the three first-order necessary conditions in part (b) simultaneously to obtain the optimal production at each plant as a function of Y, Le., find Y’B(Y) and YNI(Y)’ You should also be able to solve for the Lagrange multiplier, >.*(y). Substitute Y’B(Y) and YNI(Y) into CB(YB) and CM(YlvI), respectively, and add them to obtain the minimised overall cost function, c(y). (d) What is the economic interpretation of the Lagrange multiplier in this context Prove your assertion mathematically. (e) Recall that given the output price, P, a plant’s marginal cost curve is also its inverse supply curve. From this, the supply curves may be found and added to determine the overall U.K. supply curve for GlobalHypoMega Corporation. Verify that GlobalHypoMega’s overall U.K. supply curve calculated in this way is simply the marginal cost curve from part (c). ECON2601 5 TURN OVER PART C Answer ONE question from this section. C.l A monopolist produces a single good x and sells it to two consumer types, A and B, whose preferences are given by the utility functions where x is the quantity consumed of the monopolist’s good and y is the consumer’s disposable income. There is an equal number (say N) of both types of consumer in the population and the monopolist has a constant marginal cost equal to 4. (a) Determine each consumer type’s demand function for the monopolist’s good based on utility maximisation. (b) Suppose that the monopoly can accurately identify consumer type and wishes to design specific “take it or leave it” contracts to perfectly price discriminate and maximise profits. These contracts take the form of a specific quantity qi sold for an overall price of ri for each consumer type i = A, B. (i) Write down the profit-maximisation problem and the two constraints faced by the monopolist. (ii) Determine the specific quantity and overall price for each contract, (qA,rA) and (qB,rB). (c) Suppose now that the monopoly cannot accurately identify consumer type and so the contracts must be tailored in such a way that the consumers of each type have the incentive to self select a given contract. (i) Write down the willingness-to-pay and self-selection con- straints for the monopolist’s profit-maximisation problem and highlight which constraints are slack and which constraints are binding. (ii) Graphically or otherwise determine the revised contracts (qA, rA) and (qB’ rB) for this second-degree price discrimination problem that maximise the monopolist’s profit. ECON2601 6 CONTINUED C.2 Consider an exchange economy consisting of two agents, A and B, and two goods. Agent A has preferences UA(:XA,yA) = x A + alnyA and starts with an endowment consisting of a units of good x and 1 unit of good y. Agent B has preferences UB(xB, yB) = xB + 2ln yB and starts with an endowment consisting of 2 units of good x and 2 units of good y. (a) Define the Walrasian equilibrium of an exchange economy. (b) In an exchange economy, can one agent be made strictly better off if we are already at a Pareto-efficient allocation If we are not at a Pareto-efficient allocation, can we make all agents strictly better off by trading to a Pareto-efficient allocation Explain your answers. (c) Determine agent A’s and agent B’s demand functions for each good. (d) Find the Walrasian-equilibrium price and allocation. (e) What would be the Walrasian-equilibrium price and allocation if agent B only has one unit of good x and zero units of good y (f) Are the Walrasian-equilbrium allocations obtained above Pareto efficient Explain your answer. ECON2601 7 END OF PAPER
