[EC7086] Mock Examination 2010 No. of Pages: [7] No. of Questions: [6] Subject [Economics] Title of Paper [EC7086: Microeconomic Theory] Time Allowed [Two (2) hours] Instructions to candidates Please answer any three (3) questions. Please remember that if you answer more than three, I will mark three of them based entirely on my discretion. Important: Please note that this mock exam is meant to be an indication of the format, rather than the content, of the actual exam. In particular, do not take these questions as hints (covert or overt) as to the questions, or the topics on which the questions will be based, in the actual exam. Every topic we have covered is fair game as far as the actual exam questions are concerned. The actual exam will also contain 6 questions, each carrying equal weight, and you will be expected to do any 3 of them.
Page 2 of 7 EC 7086 EC 7086: Microeconomics Mock Exam Good luck Question 1: (100 points) (A) State the Le Chatelier s principle, and discuss briefly the intuition behind this principle. Explain also why the result is trivially true when there are two inputs. (B) Is the following input requirement set monotonic, convex? V (y) = {x 1, x 2 : 4x 1 + x 1 x 2 + 3x 2 y}; y > 0 (C) The CES production function is given by: y = ( ) δ 1 x ρ 1 + δ 2 x ρ 1 ρ 2 ; δ 1 + δ 2 = 1; ρ > 0; (i) Find MP i, the marginal product of factor i. Hence, calculate MRTS 12, the marginal rate of technical substitution between factors 1 and 2. (ii) How does MRTS 12 vary with y and with x 1 /x 2? (iii) As we have discussed in the lectures, the elasticity of substitution, σ 12 is given by; σ 12 d ln(x 1/x 2 ) d ln(mp 1 /MP 2 ) Show that in this case (iv) Find the profit function. σ 12 = 1 1 + ρ
Page 3 of 7 EC 7086 Question 2 (100 points): (A) Discuss briefly the differences between Compensating Variation CV, Equivalent Variation EV, and change in consumer s surplus, CS. Explain, very briefly as to why these three measures coincide when the utility function is quasilinear. (B) Consider the 4-good Cobb-Douglas utility function: u(x 1, x 2 ) = (x 1 ) α 1 (x 2 ) α 2 (x 3 ) α 3 (x 4 ) α 4 where α i > 0 for all i = 1, 2, 3, 4 and 4 i=1 α i = 1 (i) Derive the Marshallian demand functions. (ii) Derive the indirect utility functions. (iii) Derive the expenditure function (iv) Use Sheppard s Lemma to obtain the Hicksian demand function for good 1 (v) Derive the Hicksian demand function for good 1 directly (i.e. by doing the cost minimization subject to utility constraint) and verify that the answer is the same as the one you got in part (iv).
Page 4 of 7 EC 7086 Question 3 (100 points): (A) The cost function of a monopolist is given by C(q, θ) where q is output and θ is quality. The (inverse) demand is represented by P (q, θ). Assume both C and P are increasing in θ. Assume also that the profit function is concave in q and θ. (i) Suppose the monopolist s objective is to maximize profit by choosing q and θ. Write the first order conditions. (ii) Let the social welfare be the sum of consumer s surplus and profit. Show that it is possible for the monopolist to produce a higher quality than the one that maximizes the social welfare. (iii) Explain intuitively your result in part (ii).[a diagram might be useful]. (B) A monopolist produces output at a constant average and marginal cost of 20. There are two groups of buyers (designated groups 1 and 2) for the product with the inverse demands given by p 1 = 100 q 1 p 2 = 180 2q 2 (i) If the monopolist can carry out (third degree) price discrimination, what are the profit maximizing prices in the two markets? What are the quantities sold? (ii) If the monopolist is forced to charge a uniform price for both groups, what is the profit maximizing price? (iii) If welfare is measured by the sum of consumer surpluses and profit, is the welfare higher under price discrimination or under uniform pricing. (iv) Answer parts (i), (ii) and (iii) again but now assume that the inverse demand for the first group is given by: p 1 = 40 q 1 The second group s inverse demand is as before.
Page 5 of 7 EC 7086 Question 4 (100 points): (A) State the two welfare theorems and briefly discuss the assumptions behind them. (B) The two individuals, A and B, in an exchange economy have the utility functions: U A = ln(x A 1 ) + 2 ln(x A 2 ) The endowment are given by U B = 2 ln(x B 1 ) + ln(x B 2 ) ω A = (9, 3); ω B = (12, 6) (i) Find the excess demand for each good and verify that Walras law holds. (ii) Find the equilibrium price ratios. (C) In an exchange economy consisting of person A and person B and two goods, 1 and 2, suppose the utility functions and endowments are given by: U A = x A 1 x A 2 ; ω A = (8, 2) U B = x B 1 x B 2 ; ω B = (2, 8) (i) Show that the allocation x A = (4, 4) and x B = (6, 6) is in the core of the economy. (ii) Consider the 2-fold replica of the economy so now there are two A type and two B type people. Is the allocation given in part (i) still in the core? Explain.
Page 6 of 7 EC 7086 Question 5: (100 points) (A) Write the expressions for the coefficients of absolute and relative risk aversion and explain briefly what they measure. (B) State the independence axiom of the von Neumann Morgenstern axioms. Discuss briefly the rationale behind this axiom. Hence, discuss why such an axiom would be an extremely restrictive axiom when modelling consumer behavior under certainty. (C) An individual who has utility of wealth function given by u(w) = w1 R 1 R ; 0 < R < 1 assigns subjective probability p to Manchester United winning their next match. He decides to bet y on them winning so that if they win he wins y and if they fail to win (including drawing), he loses y. Suppose you know that his initial wealth is W and you observe the amount y that he bets (it is assumed that you know the utility function). Show that you can determine his subjective odds p/1 p. (D) Consider the choice between gambles A and B where in A he gets 300 with probability 0.25 and gets 0 with probability 0.75, whereas in B he gets 450 with probability 0.2 and and 0 with probability 0.8. Now consider two other gambles. In gamble C, first, with probability 0.5, the game will end and the decision maker gets nothing. With probability 0.5 he now faces another gamble where he gets 300 with probability 0.5 and 0 with probability 0.5. In gamble D, first with probability 1/3 the game ends and he gets nothing. With probability 2/3 he is faced with another gamble in which he gets 450 with probability 3/10 and 0 with probability 7/10. Explain what is the relation between the two situations. If a DM is an expected utility maximizer then show that if you know his preference over one pair (A versus B, or C versus D) then you can deduce his preferences over the other pair.
Page 7 of 7 EC 7086 Question 6 (100 points): (A) Explain briefly what the terms adverse selection and moral hazard mean. Explain in particular how they affect efficient outcome in the insurance market. (B) Explain why second-degree price discrimination leads to inefficient outcome. (C) Suppose that a buyer values a good at kv s where k is some positive number and v s is the seller s valuation of the good. The seller knows his valuation but the buyer doesn t. The buyer knows however that v s is distributed over the interval [0, 1] with cumulative distribution given by F (v s ) = (v s ) 2. The buyer can make a take-it-or-leave it offer of p to the seller if he wishes. If the seller accepts they trade at the price p and otherwise there is no trade. (a) Argue that as long as k > 1 the efficient outcome is for trade to take place irrespective of the value of v s. (b) Show that there is a cut-off value of k, say k, such that the outcome is not efficient for k < k. END OF EXAMINATION