Microeconomics, IB and IBP Regular EXAM, December 2011 Open book, 4 hours There are two pages in this exam. In total, there are six questions in the exam. The questions are organized into four sections. Each section is worth one quarter of your grade. Your answers should be clear and concise, and your hand writing should be legible. Section 1 1. Consider a market with the following demand function: P = 78 2Q (a) If there is a monopolist supplying the market, with a total cost function: T C = 10 + 2Q then what is the profit-maximizing quantity sold and how much profit does the firm earn? (b) Now suppose another firm enters the market, and the first firm is a Stackleberg leader. If both firms have the same total cost function, T C = 10 + 2Q, then what is the new market price? How much profit does the Stackleberg leader earn and how much profit does the follower earn? (a) The monopolist s marginal cost is MC = 2 and her marginal revenue is MR = 78 4Q. The quantity produced is where marginal cost is equal to marginal revenue. MC = MR 2 = 78 4Q Q = 19 Price is found from the demand curve: P = 78 2Q = 78 2 19 = 40 Profit is total revenue minus total cost. Π = T R T C = 40 19 10 2 19 = 712
(b) Firm 1 is the Stackleberg leader. Firm 2 is the follower. First find Firm 2 s reaction function. Firm 2 chooses output so that MC = MR. Marginal revenue for Firm 2 is: MR 2 = 78 2Q 1 4Q 2 MC = MR 2 = 78 2Q 1 4Q 2 Q 2 = 19 1 2 Q 1 To find Firm one s marginal revenue, substitute Firm two s reaction function into Firm one s demand curve. P = 78 2Q 1 2Q ( 2 P = 78 2Q 1 2 19 1 ) 2 Q 1 P = 40 Q 1 MR 1 = 40 2Q 1 The Stackleberg leader chooses output such that MR = MC MC = 40 2Q 1 2 = 40 2Q 1 Q 1 = 19 Firm two s output can be found from its reaction function: Q 2 = 19 1 19 = 9.5 2 Market price comes from the market demand curve evaluated at total supply P 2 (Q 1 + Q 2 ) = 78 2 (19 + 9.5) = 21 = 78 Firm one s profit: Π 1 = 21 19 10 2 19 = 351 Firm two s profit: Π 2 = 21 9.5 10 2 9.5 = 170.5 Section 2 2. Explain in your own words, what assumption distinguishes a Cournot duopolist from a Bertrand duopolist. Bertrand duopolists choose their prices assuming that the other firm s prices will not respond. Cournot duopolists assume that the other firm s quantity does not respond when it chooses its own quantity. Page 2
3. There are only two consumers in a market for umbrellas: Consumer A and Consumer B. Their demand functions are P = 20 4Q A and P = 16 2Q B (a) The market demand function will have a kink in it. At what price will the kink occur? (b) What is the market demand function for umbrellas at prices above the kink? (c) What is the price elasticity of demand in the market when the price is 4? (a) The kink will be at P= 16. (b) At prices above P = 16, market demand is P = 20 4Q. (c) To find the price elasticity of demand, first find market demand. P = 20 4Q A Q A = 5 1 4 P P = 16 2Q B Q B = 8 1 2 P Market demand is found by horizontal summing the individual demand: Q = Q A + Q B = 5 1 4 P + 8 1 2 P Q = 13 3 4 P When price is 4 market demand is: Q = 13 3 4 = 10 (1) 4 Elasticity is found using the point-slope formula 1 P ɛ = slope Q ɛ = 3 4 4 10 = 3 10 Remember the slope in this equation refers to the slope when P is on the left hand side. Page 3
Section 3 4. (a) Which assumption about rational preferences implies that indifference curves are downward sloping? (b) Explain why an upward sloping indifference curve would violate this assumption. (a) Monotonicity or more is better implies downward sloping indifference curves. (b) Some bundles on an upward sloping indifference curve would contain more of both commodities which violates monotonicity. A rational individual can not be indifferent between two bundles when one bundle contains more of both commodities. 5. Margrethe consumes only apples and bananas and her utility function is: U = 0.4 ln(a) + 0.6 ln(b) where A is the number of apples and B is the number of bananas she is consuming. Bananas are twice as expensive as apples. (a) If Margrethe had 9 bananas and 3 apples, how many bananas would she be willing to give up to get one more apple? (b) If Margrethe consumes 8 apples when she is maximizing her utility, subject to her constraints, how many bananas does she consume? (a) The marginal rate of substitution indicates the rate a person is willing exchange one good for another. MU A = 0.4 A MU B = 0.6 B MRS = MU A MU B = MRS = 2 9 3 3 = 2 0.4 A 0.6 B She is willing to give up 2 bananas to get one more apple. = 2 B 3 A You might have also gotten a solution of 1.57 if you calculated the total utility when she had 9 bananas and 3 apples, then figured out how many bananas would give her exactly that utility when she had 4 apples. Either approach is fine. Page 4
U = 0.4 ln(a) + 0.6 ln(b) U = 0.4 ln(3) + 0.6 ln(9) = 1.76 U = 0.4 ln(4) + 0.6 ln(b) = 1.76 B = 7.43 Therefor the number of Bananas she will trade for one apple is 9 7.43 = 1.57 (b) Margrethe is maximizing her utility subject to her constraints if her consumption bundle is the point where the MRS is equal to the price ratio. MRS = P A P B 2 B 3 A = 1 2 2 B 3 8 = 1 2 B = 6 If she is consuming 8 apples and she is maximizing her utility then she must be consuming 6 bananas. Section 4 6. Henrik has 180 Danish kroner which he must invest in either a risk-free government bond or a risky stock. There is a 80 per cent chance that the risky stock will pay an interest rate of 25 per cent on his investment and a 20 percent chance that the risky stock will double his investment. (a) What is the expected value of investing the 180 kroner in the risky stock? (b) If Henrick s utility is U (M) = M 2 what is the expected utility of investing the 180 kroner in the risky stock? (c) Now assume Henrick is risk averse and he has decided to invest in the risky stock. Given that he has invested in the risky stock, the interest rate on the risk-free government bond must be less than what number? (a) If the interest rate is 0.25 then the pay-off is 180 1.25 = 225. If the money is doubled then the pay-off is 180 2 = 360 The expected value of the risky investment option is: EV = 0.8 225 + 0.2 360 = 252 (b) The expected utility of the risky investment option is EU = 0.8 225 2 + 0.2 360 2 = 66420 (c) If Henrick is risk averse and he invests in the risky stock then we know that the return on the risky stock must be more than fair relative to the risk-free investment. If investing at the bank paid out at least as much as the expected value of the risky stock, then a risk averse person Page 5
would never take the risky stock. This means that the interest rate can not be higher than the interest rate that solves this equation: EV = (1+R)*180, where R is the interest rate on the government bond. Since EV = 252, this equation is, 252=(1+R)*180. Therefore, R = 0.4. If the government bond pays 0.4 or more, then a risk averse person would have never invested in the risky stock. Page 6