SUMMER TERM 2016 ECON2601: ECONOMICS II (Combined Studies)

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1 SUMMER TERM 2016 ECON2601: ECONOMICS II (Combined Studies) TIME ALLOWANCE: 3 hours Answer ALL questions from Part A, ONE question from Part B, and 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 B and 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. A1 You spend your weekly allowance of m = 20 (in ) on either units of coffee (denoted c) or units of biscuits (denoted b). If the prices of the two goods are p c = 2 (in /unit) and p b = 1 (in /unit), then what is your budget constraint? Suppose that your café offers you the tenth unit of coffee for free if you purchase nine units of coffee that week at the full price. What is your budget constraint with this incremental-unit discount? Assume that this discount holds only for the tenth unit of coffee, i.e., the eleventh unit in the week will have to be purchased at the full price. A2 One property of a utility function is that a positive monotonic transformation does not change the shapes of the underlying indifference curves. Using the transformation f(u) = ln u and by referring to the marginal rate of transformation, show that the original utility function, u(x 1, x 2 ), has indifference curves with the same shapes as those of the transformed utility function, v(x 1, x 2 ), where x i, i = 1, 2 represents the quantity of good i that is consumed. A3 Suppose that your utility function is u(x 1, x 2 ) = a 1 x 1 + a 2 x 2, where x i, i = 1, 2 represents the quantity of good i that is consumed. Here, a i > 0, i = 1, 2, are constants and the price of good i is p i > 0. If p 1 p 2 < a 1 a 2 and the target utility level is ū > 0, then what is your Hicksian quantity demanded for good 1, h 1 (p 1, p 2, ū)? What is the corresponding expenditure function, e(p 1, p 2, ū)? Verify that Shephard s lemma holds here. Although it is not possible to solve the expenditure-minimisation problem via calculus to obtain the Lagrange multiplier, it is still possible to determine the Lagrange multiplier by partially differentiating the expenditure function with respect to the target utility level. Based on this observation, what is the Lagrange multiplier, θ (p 1, p 2, ū)? Obtain the Marshallian demand function, x 1 (p 1, p 2, m), from the Hicksian ECON TURN OVER

2 demand function and the indirect utility function, v(p 1, p 2, m), by applying the correct identity. Note that the exogenous income is m > 0. A4 A consumer has a strictly positive endowment for two goods, (ω 1, ω 2 ), that sell at prices p 1 > 0 and p 2 > 0. Let (x 1, x 2 ) denote the final consumption of the two goods. Suppose that this consumer s utility function is u(x 1, x 2 ) = min {a 1 x 1, a 2 x 2 }, where a i > 0, i = 1, 2, is a constant. Given a ceteris paribus increase in the price of good 1 to p 1, determine the substitution effect, ordinary income effect, and endowment income effect associated with the price change. A5 A casino offers a gamble to a risk-averse consumer in which the payoff is either Z 1 = 4 (with probability p) or Z 2 = 16 (with probability 1 p). There are, thus, two mutually exclusive states of the world associated with the uncertain payoff, Z. Suppose that the consumer tells you that her certainty equivalent is 8. If you know that her utility function is u(y ) = Y (where Y represents her wealth) and her initial wealth, Y 0, is zero, then what is the value of p? A6 A monopoly can produce output y at a cost given by the function c(y). It faces an inverse market demand curve p(y) where p is the price per unit. Assuming that the monopoly is profitmaximising, find the price markup it charges over the marginal cost. A7 Suppose there are two types of consumers i = 1, 2 each possessing a quasi-linear utility function of the form U i (x, y) = u i (x) + y for x units consumed of a particular good and remaining income y. Here u i (0) = 0 for both consumers, and for all positive values of x, u 1 (x) > u 2 (x) and u 1 (x) > u 2 (x). Good x is produced by a monopoly that wishes to undertake second-degree price discrimination by designing a take-it or leave-it contract for each consumer type where r i is paid by the consumer to receive a quantity x i of the monopolist s good. Show that the willingnessto-pay constraint for consumer type 1 does not bind in the monopoly s profit-maximisation problem. A8 The table below shows the payoffs for two players, A and B, in a one-shot simultaneous move game where each player can play either L, M or R. The first number in each cell is the payoff to player A and the second to player B. Player B L M R L (10, 10) (2, 12) (0, 14) Player A M (12, 2) (5, 5) (0, 0) R (14, 0) (0, 0) (1, 1) What is/are the pure Nash equilibrium/equilibria of this game? A9 A lower rate of saving over a long period of time will cause a permanent drop in the rate of growth of output per worker. By considering the Solow model of economic growth with technological progress, explain carefully using a graph and words whether this statement is true, false or uncertain. ECON CONTINUED

3 A10 Using the Solow growth model with technological progress and a Cobb-Douglas aggregate production function, derive the Solow Residual and briefly explain how it enables technological progress in an economy to be measured. ECON TURN OVER

4 PART B Answer ONE question from this section. B1 A consumer has the Cobb-Douglas utility function, u(x 1, x 2 ) = x α 1 xβ 2, where 0 < α < 1 and 0 < β < 1 are constants. The prices of the two goods are p 1 > 0 and p 2 > 0, and the exogenous income is m > 0. (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 solve for x 1 (p 1, p 2, m), x 2 (p 1, p 2, m), and λ (p 1, p 2, m). (b) The second-order sufficiency condition for this constrained maximisation problem is that the bordered-hessian matrix has border-preserving principal minor submatrices with determinants that alternate in sign starting with negative. What is the bordered-hessian matrix for this problem? Verify that the second-order sufficiency condition is satisfied. (c) Find the indirect utility function, v(p 1, p 2, m), for this problem. Verify that the marginal utility of wealth is equal to λ (p 1, p 2, m). (d) Invert v(p 1, p 2, m) to determine the expenditure function, e(p 1, p 2, ū), where ū > 0 is the target utility level. Apply Shephard s lemma to find the Hicksian demand function, h 1 (p 1, p 2, ū). Verify the identity h 1 (p 1, p 2, ū) x 1 (p 1, p 2, e(p 1, p 2, ū)). (e) Suppose that the price of good 1 increases to p 1. Calculate the compensating and equivalent variations for this consumer. ECON CONTINUED

5 B2 You run a factory with production function f(x 1, x 2 ) = x α 1 xβ 2 in a perfectly competitive industry. Here, x i represents the quantity of input factor i, where i = 1, 2, and α > 0 and β > 0 are constants such that α + β < 1. The price for input factor i is w i > 0, while the output from your factory is sold at price p > 0. (a) Formulate your long-run profit-maximisation problem taking care to indicate the decision variables. What are the first-order necessary conditions for this problem? (b) For this unconstrained maximisation problem, the second-order sufficiency condition is that the Hessian matrix must be negative definite. This may be verified by checking that the determinants of the Hessian s principal minor submatrices alternate in sign starting with negative. Show that the second-order sufficiency condition is satisfied here. (c) Use the first-order necessary conditions in part (a) to solve for the unconditional input factor demand functions, x 1 (p, w 1, w 2 ) and x 2 (p, w 1, w 2 ). (d) What is the supply function, y (p, w 1, w 2 )? What is the profit function, π (p, w 1, w 2 )? π (e) Show that Shephard s lemma holds, i.e., w 1 Explain intuitively what these results mean. = x 1 (p, w 1, w 2 ) and π p = y (p, w 1, w 2 ). ECON TURN OVER

6 PART C Answer ONE question from this section. C1 Consider an exchange economy consisting of two agents, A and B, and two goods, x and y. Agent A has preferences U A (x, y) = x 2/3 y 1/3 and starts with an endowment consisting of zero units of good x and e y units of good y. Agent B has preferences given by U B (x, y) = min(x, 2y) and starts with an endowment consisting of four units of good x and one unit of good y. (a) Define the Walrasian equilibrium of an exchange economy. conditions a Walrasian equilibrium is Pareto-efficient. Briefly explain under what (b) Find the Walrasian-equilibrium price and allocation for the above economy in terms of e y. (c) Draw an Edgeworth box showing the position of the initial endowment allocation, the position of the final allocation, and the indifference curves of the agents passing through both initial and final allocations for the case e y = 1. (d) For what range of values of e y does a Walrasian equilibrium exist? (e) Show that the utility of agent A is maximised after trading with agent B for the case e y = 1/2. For what value of e y is agent B s utility maximised after trading? ECON CONTINUED

7 C2 An industry consists of two competing firms and the inverse market demand function is of the form p = a b(y 1 + y 2 ) where y i is the output produced and sold by firm i in each period and p is the price per unit paid by consumers. Suppose firm 1 has constant marginal cost c 1 and firm 2 has constant marginal cost c 2. (a) Assume the firms play a one-shot simultaneous move game where each firm decides, without consultation, the level of output to produce. Determine the Cournot-Nash equilibrium where no firm wishes to deviate ex-post. (b) What is the equilibrium price and output produced when firm 1 chooses and announces their output first followed by firm 2? (c) Assume now that c 1 = c 2 = c and that the two firms sign an agreement to restrict supply and raise the market price by each firm only producing half the output of a single monopoly. (i) Calculate the payoff each firm would get by cooperating in this way, π coop. (ii) Show that the payoff a firm would receive if it were to stop cooperating and to deviate onto their best-response curve would be (9/8)π coop. (d) Propose a trigger strategy that could enforce this agreement in an infinitely repeated game and determine the values of the discount rate δ that would ensure continued cooperation. ECON END OF PAPER