Final Exam Solutions - Intermediate Microeconomic Theory | ECO 202, Exams of Microeconomics

Download Final Exam Solutions - Intermediate Microeconomic Theory | ECO 202 and more Exams Microeconomics in PDF only on Docsity! Name Davidson College Mark C. Foley Department of Economics Aug – Dec 2003 Intermediate Microeconomic Theory Review #4 Selected Suggested Solutions Directions: This review is a closed-book, closed-notes exam to be taken in one sitting, no time limit. You may use a calculator. There are 100 points on the exam. Each short answer question is worth 5 points. The problems in Section II are worth 20, 15, and 10 points, respectively. The problems in Section III are worth 15 points each. You must show all your work to receive full credit. Any assumptions you make and intermediate steps should be clearly indicated. Do not simply write down a final answer to the problems without an explanation. Read the questions carefully, answering what is asked. Think clearly and work efficiently. Good luck. Label the optimal number of workers to hire as L*. Why does the MRP L curve slope downward after some point? MRPL slopes downward because MPL falls as labor increases (diminishing marginal productivity of labor). 5. Using a properly labeled graph, draw two reaction functions (no need to specify equations) for which Cournot duopolists would not reach an equilibrium. Start in an initial disequilibrium situation, label it point “N,” and explain why no equilibrium would be reached. The reaction function of the firm on the y-axis must be steeper. Then the responses of the duopolists to an initial disequilibrium situation will take them further and further away from the equilibrium point (the intersection of the reaction functions). For example, if firm 1 starts a little above what would be its equilibrium output level, say at q1a, then the arrows trace out the reactions and we can see that this is moving away from the Cournot-Nash equilibrium. q 2 q 1 Firm 2’s reaction function Firm 1’s reaction function q 1a N Section II: Problems Do these 3 questions (No choice ). 1. Consider a monopolist with production function Q = L½K½ and facing demand curve P = 56 - 2Q, and competitive input prices w = $4 and r = $16. (a) Find the conditional factor demand equations for L and K in terms of w,r, & Q. 2 1 2 1 ..min LKQtsrKwLTC  ][ 2 1 2 1 LKQrKwL   FOCs:  2 1 2 1 2 1 2 1 2 00 2 1      LK w LKw L    2 1 2 1 2 1 2 1 2 00 2 1      KL r KLr K    2 1 2 1 0 LKQ       and  imply  r wL K L K r w KL r LK w   2 1 2 1 2 1 2 1 22 substitute into  to obtain 2 1 2 1 2 12 1 *1                    w r QLL w r QL r wL Q Substituting into , 2 1 2 1 * * *              r w Q w r Q r w K r wL K (b) What is the maximum level of profits? First set MR = MC. MR is found from the demand curve and doubling the slope (when written as P = f(Q)). Or you can write down TR = PQ and take the derivative with respect to Q. MR = 56 – 4Q. MC is the derivative of TC. Qwr r w Qr w r QwrKwLTC 2 12 1 2 1 )(2             So 2 1 )(2 wrMC  Setting MR = MC we get 10 4 )(256 )(2456 * 2 1 2 1    QQ wr wrQ So P = 56 – 2Q = 56 – 20 = $36. Profit = TR – TC = ($36*10) – (2*2*4)(10) = $360 – 160 = $200 50.1$0320 *  PP dP dTR per hour. So the optimal fee is $5.0625 per week or $263.25 per year. Weekly profits are 2*5.0625 + (1.50)*(9- (3*1.5)/2) = $20.25 - $5 = $15.25. 3. A college dean wants to know whether her school’s alcohol ban reduces drinking by freshman. All first-year students at her college live on campus in either Dorm A or Dorm B (which are the same size) and each is allowed to choose his or her dorm during the first two weeks of the fall semester. The college is located in a dry (i.e., alcohol-free) town, so there is no off-campus drinking. To estimate the impact of the ban, the dean runs an experiment over two years. In year 1, she hires student informants to monitor the drinking habits of each dorm’s residents (regardless of where they do their drinking). She finds that students from Dorm A drink 3 beers per week on average while those from Dorm B drink 2.8 beers per week on average. In year 2, the dean announces that during the two weeks in which students can choose their dorm, that drinking will be allowed in Dorm B this year, but not in Dorm A. Her informants again monitor drinking and find that Dorm A residents drink 2.1 beers per week on average while Dorm B residents drink 3.7 beers per week on average. The dean uses the difference-in-differences methodology to estimate the impact of the drinking ban on the drinking behavior of the average student. She finds a statistically significant difference. (a) What is the difference-in-difference estimate of the impact of the alcohol ban on drinking by first-year college students? A B Year 1 3.0 2.8 Year 2 2.1 3.7 -.9 .9 Impact of ban in dorm A is -.9 - .9 = -1.8 drinks per week (b) Does the dean’s design constitute a valid natural experiment? Explain why or why not, indicating any assumptions you make. The experiment is not really a valid natural experiment because the choice of dorm is not random (it is endogenous, or determined within the set-up, not outside of it) as students can choose based on their preferences and a dorm’s drinking policy. Thus, assignment to the treatment group (dorm A) versus the control group (dorm B) was not random. (c) The college’s trustees are considering lifting the drinking ban and ask the dean to report on the impact this will have on student drinking. What is her best estimate? Average drinking is 2.9 in both years, suggesting the ban had no effect. It appears that this poorly designed experiment encouraged students to sort themselves by dorm according to their preferences, without actually impacting overall drinking. Section III: Problems Do 2 of the following problems (Choice ). If you start more than two, be sure to indicate (by circling the question #) which you want graded. 4. (a) Assume there are two airlines which each have constant marginal cost of $147 (per passenger per flight), and face a market demand curve of Q = 339 - P, where Q is total quantity of both airlines combined (Q = q1 + q2) and P is the price per flight. Fill in the following table. Model Firm 1’s quantit y (q1) Firm 2’s quantit y (q2) Total Output (Q) Market Price (P) Firm 1’s profit (p1) Firm 2’s profit (p2) Cartel (splitting p equally) 48 48 96 $243 $4608 $4608 Cournot duopoly 64 64 128 $211 $4096 $4096  Let’s start with the Cournot model. Each firm equates its marginal cost and marginal revenue, treating the output of the other firm as a constant. The reaction function for each firm is found by solving MRi = MCi, i = {1,2}. TR1 = Pq1 = (339 - q1 - q2)q1 = 339q1 - q12 - q2q1  MR1 = 339 – 2q1 - q2 MR1 = MC1  339 – 2q1 - q2 = 147  2q1 = 192 – q2  q1*= 96 – q2/2. This is firm 1’s reaction function. Similarly, we can derive q2*= 96 – q1/2. Solving for firm 1, we get q1= 96 – [96- q1/2]/2  q1 = 48 + q1/4  3q1/4 = 48  q1 = 64 so q2 = 64. Total output is Q = q1 + q2 = 128, so P = 339 – Q = 339 – 128 = = $211. p1 = (AR-AC)q1 = Pq1 = ($211 - $147)(64) = 4096 = p2 also.  If they collude, they should treat their output as identical since they have the same cost curves. They will split the monopoly output and profit. Let Q= q1 + q2. Then the two firms maximize joint profit by setting MR = MC. TR = PQ = (a - bQ)Q, so MR = a – 2bQ, which in this case is MR = 339 – 2Q. Thus, 339 – 2Q = 147  Q* = 96. So each should make 96/2 = 48 . P = 339 – Q = 339 – 96 = $243. Total p = (AR – AC)Q = (243 - 147)(96) = 9216, so each earns 4608. 5. (a) Assume there are two airlines which each have constant marginal cost of $147 (per passenger per flight), and face a market demand curve of Q = 339 - P, where Q is total quantity of both airlines combined and P is price per flight. Fill in the following table. Model Firm 1’s quantit y (q1) Firm 2’s quantit y (q2) Total Output (Q) Market Price (P) Firm 1’s profit (p1) Firm 2’s profit (p2) Cournot duopoly 64 64 128 $211 $4096 $4096 Stackelberg leader (firm 1) with Cournot follower 96 48 144 195 $4608 $2304  Cournot work is above.  Under Stackelberg model, one firm is a leader, choosing output first and the other firm reacts in a naïve Cournot fashion. With firm 1 as the leader, they will set MR = MC. TRStack = Pq1 = (a - bq1 -bq2)q1 = (a - bq1 –b[(a – c – bq1)/2b])q1 = aq1 - bq12 – aq1/2+ cq1/2 +bq12/2 = aq1/2 + cq1/2 – bq12/2  MR1, Stack = (a+c)/2 – bq1 MR1, Stack = MC  (339+147)/2 – 1q1 = 147  q1, Stack = (339-147)/2 = 96, so q2, Cournot Follower = [(a – c – bq1)/2b] = (a – c – b((a-c)/2b))/2b = (a-c)/4b = (339-147)/4 = 48. Total output = Q = (a-c)/2b + (a-c)/4b = 3(a-c)/4b = 144 P = a –b(3(a-c)/4b) = (a+3c)/4 = (339+3*147)/4 = 195 p1, Stack = (AR-AC)q1, Stack = (195 – 147)96 = 4608. p2, Cournot Follower = (AR-AC)q2, Cournot Follower = (195 – 147)48 = 2304