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1. A consumer faces a utility function given by: U(X,Y) = XY2, where income is denoted M and the prices of X and Y are given by Px; Py respectively. (a) Assuming that this consumer maximises utility, derive the Marshall demand functions. (5 marks) (b) Now assume that the price of good X increases to Px. Find the Hicksian demands and explain what they show, using a diagram to support your answer. (10 marks) (c) Given your answer to part (b), write out the expenditure function and explain what it shows and its properties. (5 marks)
. 2. Consider the following one-shot game between two players. 'D' 'C' 'D' 1,1 -2,6 'C' 6,-2. 3,3 (a) Find the Nash equilibrium/equilibria of this game. Are any of the Nash equilibria Pareto optimal? (4 marks) (b) Is every rationalizable strategy in this game also dominant? (2 marks) Now assume that the game is repeated every period and that both players discount future payoffs with discount factor 8 where 0 <8 <1. (c) Define the 'Grim Trigger' strategy and find the condition on which permits the surplus maximising outcome to arise as a subgame perfect equilibrium (SPE). (7 marks) (d) Explain whether both players playing 'Grim Trigger' is the unique SPE of this repeated game. (2 marks) (e) What insight does this repeated game model and the concept of 'coordination failure' give us about the prospect of cooperation in prisoners' dilemmas? (5 marks)
3. Consider the intertemporal utility function U = x/x2 where xı represents consumption today and X2 represents consumption tomorrow. The consumer has income of M, and M, in each respective time period and can lend/borrow at the rate of interest, r, which is constant across both time periods. Assume that prices are p = 1 and that the price of commodities remains constant. (a) Give the equation for the intertemporal budget constraint (x2 as a function of the other variables) and identify the slope. If there was in increase in future income, what would happen to the budget constraint? (5 marks) (b) Set up the utility maximising problem and solve to find an expression for the optimal quantities of X, and xz. (7 marks) 2 (Question 3 continued overleaf) EC1090 (c) Assume that M = £10.00 and M2 = £8.00 and r = 5%. How much should this consumer spend on current consumption and how much will the consumer be able to spend in the future? (3 marks) (d) If the rate of interest increased, what would happen to the budget constraint and the consumer's decision about being a lender or borrower? Does your answer change if the interest rate decreases? (5 marks)
113 4. Suppose that you are given a cost function (w,r,x) = 2wzrāxż where w is the wage rate for labour, r is the rental rate of capital and x is the output level for a price taking firm (a) By deriving an expression for the marginal cost function, explain whether the production process that gives rise to the total cost function has increasing, decreasing or constant returns to scale. (3 marks) (b) Given your answer to part (a), what do we know about the relationship between the marginal cost, average cost and supply curves? Illustrate these curves in a diagram. (5 marks) (c) Derive the supply function for the firm and explain what it tells us. (Hint you can use the answer to part (b) to help you think about where the supply curve lies and how to derive it). How does output by the firm change as input and output prices change? Explain your answer. (7 marks) 1 1 1 (d) If the cost function had been c(w,r, x) = 2wärzxz instead, how would your answer to part (a) change? What would the average and marginal cost curves now look like? What does this mean about the profit maximizing output of the firm and does it make sense? (5 marks)
5. Consider a firm producing an output, Q, using capital, K and labour, L as the two inputs. (a) Assume that the production technology to produce the output is given by: Q = 2K + L. Illustrate the isoquants for this technology (Labour measured on the horizontal axis) and explain how the firm would determine the optimal use of inputs, when the price of labour is w and the price of capital is r. (4 marks) 3 (Question 5 continued overleaf) EC1090 (b) If the production technology now becomes: Q = min{2K, L}, what will the isoquants now look like? Where will the optimal production point be? (3 marks) Now consider a firm with the following production technology: Q = LK with the price of labour denoted by w and the price of capital denoted by r. (c) Find an expression for the conditional input demands for labour and capital in the long run. Suppose that w = 16;r = 9; Q = 144. How many units of labour and capital should this firm use? (8 marks) (d) If the firm was now producing in the short run, with capital fixed at R, would this firm's optimal input demands result in a tangency? Use a diagram to explain and justify your answer, commenting on how short run costs typically compare with long run costs. (5 marks)
6. Anita and Ben want to watch a movie and must choose from W, X, Y and Z. Utilities from each movie are given below: Utility for B 1 Movie w х Y z Utility for A 4 4 7 2 1 -2 9 (a) Which outcomes are Pareto optimal? (4 marks) (b) Suppose we were to square Anita's utility from each movie (i.e. U (W) becomes 16). 0 Does the set of Pareto optimal outcomes change? Why/why not? (3 marks) (ii) Does the surplus maximising outcome change? Why/why not? (3 marks) (c) Now suppose that a third person, Caroline, enters and wants to watch the movie. Her utility from each movie is uc(W) = 5, uc(X) = 2, 4c(Y) = 0 and uc(Z) = 0. If Anita and Ben retain the utilities from part (a), which outcomes are Pareto optimal? (5 marks) 4 (Question 6 continued overleaf) EC1090 (d) What are the pros and cons of using Pareto optimality as an efficiency criterion? Is total surplus a better measure of efficiency? (5 marks)
7. A monopolist has a demand function where quantity demanded depends on the price they set (p) and the amount of advertising they do (A). The demand function is given by: Q = (60 - p)(1+0.2A - 0.032A?) Their total cost is given by: TCCQ, A) = 50 + 100 + 5A (a) Assume that there is no advertising (A = 0). What will be the optimal price for the monopolist? What is the corresponding quantity and profit? (5 marks) (b) If we fix the price at its optimum from part (a) and allow for advertising then what will be the optimal level of advertising? How do quantity and profit compare to the case of part (a)? (7 marks) (c) Suppose the monopolist knows that if their profit levels exceed 675 then other firms will be encouraged to enter the market (competing away all profits to zero). Assume that they set the same price as in part (a), how would the monopolist optimally choose the level of advertising in this case? (Hint: Notice that 0.032 = 4/12) (8 marks)
8. Two producers of bottled water i and j are competing on quantities in a market. The producers have no fixed cost and a constant marginal cost c = 10. The inverse demand function is given by PQ) = 100 - 5Q where Q = 91 +9j. (a) Do the firms have a dominant strategy in this game? (4 marks) (b) Find the unique Nash equilibrium in quantities. What are equilibrium profits for each firm? (6 marks) (c) Using diagrams of the best response functions explain how the market might dynamically adjust to a new equilibrium if the inverse demand function changed to: PQ) = 210 - 100 (5 marks) (d) To what extent is the Cournot model a realistic model of competition? Are realistic assumptions an important consideration when working with theoretical models? (5 marks)