Final2021_A_solutions

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Page 1 of 12 Name: Recitation teacher’s name: Day and time of recitation: Microeconomics Spring 2021 Final Examination There are two sections to the exam. Part A contains 12 short-answer questions worth 5 points each (60 points total) and Part B contains two long-answer questions (40 points total). Instructions: - Upload your answers to the “Assignment” tab on NYU Classes. You may use the answer template provided or write your answers on scratch paper. - Submit your answers only when you're ready. You may only submit once. - You are not permitted to work with other students or communicate electronically (cell phone, email, etc.). - Read the questions carefully. Please write legibly and provide concise but justified answers. - Show your work . Include labels on axes/curves in all diagrams you draw. - You have 1 hour and 50 minutes. Set an alarm for yourself. We will allow a 10 min grace period after the deadline for scanning and uploading. Exams submitted past this grace period will not be accepted. Good luck!

Page 2 of 12 I. Short Answer Problems (60 points total) 1. [5 pts] Stu visits a casino in Las Vegas with $8 in his pocket. His utility function is U=I 2 where I represents the amount of money in Stu’s pocket. If Stu spends his time gambling, odds are he’ll end up with $14 in his pocket with probability 0.5 or $2 with probability 0.5. Stu’s friend Doug, who thinks gambling is bad, offers to pay Stu not to play. What is the minimum amount Doug needs to offer Stu to convince him not to gamble? Explain. Answer: The question is looking for us to compute Stu’s expected utility from gambling vs. his utility from not gambling. We’ll then find the risk premium by looking for the amount x that we need to pay Stu to equate the two values. Here, Stu is risk loving; we’ll find Stu needs to be paid not to gamble given the numbers in the question. If Stu plays the game, his expected utility is: EU = 0.5*u(2) + 0.5*u(14) =0.5*(2 2 ) + 0.5*(14 2 ) =2 + 98 = 100. If Stu doesn’t play, his utility is u(8) = 8 2 =64. Doug needs to offer Stu x, where u(8+x)=100. Thus, (8+x) 2 =100, or 8+x=10. Doug needs to pay Stu $2 to make him indifferent between gambling or not gambling. 2. [5 pts] The table below shows the payoffs in a simultaneous move game with two players. Find all pure strategy Nash Equilibria of this game. Are there any dominant strategies? Are there any dominated strategies? Explain. Player 2: Left Center Right Player 1: High 1, 4 2, 5 4, 6 Medium 6, 8 5, 9 3, 7 Low 7, 6 3, 7 1, 1 Answer: There are two pure strategy Nash Equilbria of this game: (High, Right) and (Medium, Center). In both of those outcomes, neither player has an incentive to deviate unilaterally; each would receive a lower payoff from deviating. No strategy is dominant here. Depending on Player 1’s play, Player 2 will play either Center or Right. Depending on Player 2’s play, Player 1 will play High, Medium, or Low. For Player 2, playing Left is a dominated strategy: no matter what Player 1 chooses, Player 2 will never best respond by playing Left. There are no dominated strategies for Player 1.

Page 3 of 12 3. [5 pts] In the figure below, what is consumer surplus if a monopolist serves the market? How does this amount compare to the consumer surplus under perfect competition? Provide a number and explain. Answer: Here, we can see that a monopolist, setting MR=MC, will choose a quantity equal to 4. At a quantity of 4 on the demand curve, the monopolist chooses a price equal to 12. Consumer surplus is then (1/2)*(20-12)*4 = 16. Under perfect competition, we find P=MC, so in that setting P=4, Q=8. CS under perfect competition equals (1/2)(20- 4)*8 = 64. Thus CS is larger under perfect competition. The difference is 64-16 = 48. 4. [5 pts] The retailer Zappas is the monopoly seller of autographed Frank Zappa memorabilia. The company faces an inverse demand curve given by P=100-10Q and has a marginal cost of 20 and no fixed costs. Zappas is approached by Friendbook, a company that claims to have so much information about Frank Zappa fans that Zappas would be able to implement first degree price discrimination. What is the highest fee Zappas would be willing to pay Friendbook for access to this information? Answer: Our goal here is to calculate the profits to Zappas as a monopolist that (a) sets a single price or (b) acts as a first degree price discriminating monopolist. Zappas would be willing to pay Friendbook up to the difference in profits to be able to first degree price disciminate. If Zappas must set a single price, we can find its profits by solving the standard monopoly problem: TR = P(Q)*Q = (100-10Q)*Q 20 8 4 4 MC D MR 12

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Page 4 of 12 =100Q – 10Q 2 . Thus, MR = dTR/dQ = 100 – 20Q. Setting MR = MC, we find 100-20Q = 20, or Q=4. At Q=4, P = 100-10(4) = 60. Profits are thus equal to TR-TC = 60*4 – 20*4 = 160. Under first degree price discrimination, Zappas would serve the perfectly competitive quantity where P=MC. Here, that means 100-10Q = 20, or Q=8. Under first degree price discrimination, the monopolist charges a different price to each buyer equal to her willingness to pay; the firm’s total revenue would equal the area under the demand curve up to a quantity of 8. Here, that equals TR= (1/2)(100-20)*8 + 20*8 = 320 + 160 = 480. The total costs, when fixed costs = 0, are TC = 8*20 = 160. Thus, profits to Zappas here are TR- TC = 480-160 = 320. Comparing the profits under the two scenarios, Zappas earns an additional profit of 320-160 =160. Zappas would be willing to pay up to 160 to gain access to Friendbook’s information. 5. [5 pts] Both Anton and Tina produce identical boxes of steel cut oatmeal. Both want to sell their oatmeal in Argentina. Inverse demand in Argentina is given by P = 10-2Q. In Argentina, Anton has a marginal cost of $2/box while Tina faces a higher marginal cost of $8/box. If the firms compete simultaneously in price (Bertrand), what is the equilibrium price in the market? How much oatmeal does Anton sell? How much oatmeal does Tina sell? Explain. Answer: To find the equilibrium, we want to think about each firm’s best response function. Here, we can write Anton’s best response function as: ࠵? ! (࠵? " ) = % ࠵? ! # ࠵?࠵? ࠵? " > ࠵? ! # ࠵? " − 0.01 ࠵?࠵? ࠵?࠵? ! < ࠵? " ≤ ࠵? ! # ࠵?࠵? ! ࠵?࠵? ࠵? " ≤ ࠵?࠵? ! Where ࠵? ! # is Anton’s monopoly price and MC A =2. Tina’s best response looks similar. Because we see that Anton has a lower marginal cost than Tina, we know that in the Bertrand Equilibrium, Anton will acquire the whole market. Tina will sell no oatmeal in Argentina. What price and quantity will Anton achieve in equilibrium? A natural guess might be $7.99, slightly undercutting Tina’s marginal cost and serving the entire market. But given the demand curve, we can find Anton’s monopoly price. Here. TR = (10-2Q)Q = 10Q – 2Q 2 . Thus, MR = 10-4Q. Setting MR=MC, we find 10-4Q =2, or Q=2. Then, P=10- 2(2)=6. Since this is less than Tina’s MC, Anton chooses P=6 and sells Q=2. As a note, if Tina were a monopolist here, she would set MR=10-4Q=8, choose Q=.5 and P=9. But that is not the Bertrand Equilibrium, since Anton would best respond to Tina setting P=9 by setting P=6.

Page 5 of 12 6. [5 pts] The table below shows the payoffs in a simultaneous move game with two players. What is the mixed strategy Nash Equilibrium of this game? Player 2: L R Player 1: A 3, 3 2, 2 B 2, 2 3, 3 Answer: Let p be the probability that P1 chooses A; (1-p) is the probability that P1 chooses B. We find p by setting the payoffs for P2 from playing L or R equal to each other: Payoff to P2 from L: 3*p + 2*(1-p) = 3p + 2 – 2p =p+2 Payoff to P2 from R: 2*p + 3*(1-p) = 2p +3 – 3p = 3-p. Setting the payoffs equal, we find p+2 = 3-p, or 2p=1, p=1/2. Let q be the probability that P2 chooses L; (1-q) is the probability that P2 chooses R. We find q by setting the payoffs for P1 from playing A or B equal to each other: Payoff to P1 from A: 3*q + 2*(1-q) = 3q + 2 – 2q= q+2 Payoff to P1 from B: 2*q + 3*(1-q) = 2q + 3 – 3q = 3-q Setting the payoffs equal, we find q+2 = 3-q, or 2q =1 , or q=1/2. The mixed strategy NE here is for P1 to play A 50% and B 50%, and for P2 to play L 50% and R 50%. 7. [5 pts] The US government is planning to auction off the rights to harvest timber in an area of the Pacific Northwest. Regulators must decide between using an English or a Dutch auction (you can ignore the possibility that firms might use bid information to collude). True/False/Uncertain: The English auction will generate more revenue than the Dutch auction. Explain. Answer: True. The auction of timber is best thought of as a common values auction. In the case of common values, bidders in the auction have an incentive to shade their bids below their values for fear of the winner’s curse. By having public information on the bids in the English auction format, a bidder can learn about its competitors’ estimates of the timber’s value from the level of other bids. This may lead bidders to shade less, since they gain information on the likely value of the timber. Revenue will likely go up relative to the Dutch auction (which is similar to a sealed bid first price auction).

Page 6 of 12 8. [5 pts] In the market for used iPhones, there are two types of phones: ‘good’ used phones and ‘duds’. Buyers value good used phones at $400 and duds at $0. Sellers value good used phones at $300 and duds at $0. Let q be the fraction of duds in the market. What is the highest fraction of duds that can be in the market for all used iPhones to be sold? Explain. Answer: For all iPhones to sell, we need the buyer’s expected value of a phone to be greater than or equal to the seller’s value of the good used iPhone. Here, that means we need: ࠵? ∗ (0) + (1 − ࠵?) ∗ 400 ≥ 300 This is true when (1-q) ³ (3/4), or (1/4) ³ q. So, at most 25% of phones can be duds for all used iPhones to sell. 9. [5 pts] Suppose a monopolist faces the following demand curve for its product: Q=6 – (1/2)P. Find the elasticity of demand at a price of P = 4. Starting at P=4, if the monopolist raises its price a little, will revenue increase or decrease? With the information given, determine whether P=4 is the monopoly price, lower than the monopoly price, higher than the monopoly price, or whether the answer is uncertain. Explain. Answer: The elasticity of demand is E D =(dQ/dP)*(P/Q). Here, dQ/dP = -(1/2) and P=4. Plugging P=4 into the demand curve, we find Q=4. Thus, E D =(-1/2)*(4/4) = -.5. Demand is thus inelastic. We know that firms earn greater revenue by raising price with inelastic demand. We also know that the equilibrium price for the monopolist must be greater than 4. We can’t determine the exact monopoly price without knowing the marginal cost, but here we know that it must be larger than 4. For example, even if MC=0, we’d set MR=MC and we would find MR = 12-4Q =MC=0, or Q=3. At Q=3, P = 12 – 2(3) = 6, which is greater than 4. 10. [5 pts] The figure below illustrates the supply and demand in the market for copper wire. Suppose the government imposes a new quantity restriction such that copper wire manufacturers cannot sell more than 4,000 units of wire to prevent excess extraction of copper deposits. What is the equilibrium price in the market following the imposition of the quantity restriction? What is the amount of deadweight loss under the quantity restriction?

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Page 7 of 12 Answer: The competitive equilibrium quantity and price in the diagram above are P=5 and Q = 3,000. The quantity restriction of Q=4000 is above the competitive equilibrium, so has no effect on the market equilibrium here. Thus P=5 and Q=3000 after the quantity restriction. There is no deadweight loss at the competitive outcome. 11. [5 pts] Leo’s lumber yard produces wood planks using the production function Q = LK 2 , where L is labor hours and K is capital units. Leo pays wages (w) equal to $10/hour and the cost of capital (r) is $20/unit. If Leo minimize costs by hiring L=4 hours of labor, how much capital does he use? What quantity Q of wood planks does Leo produce? Explain. Answer: Here, we want to find the MRTS and set it equal to the ratio of wages to the cost of capital. To find MRTS = MP L /MP K , we first find the marginal products: MP L =K 2 MP K =2KL Thus, MRTS = K/2L. Here, w/r = 10/20 = 1/2. Thus, (K/2L) = 1/2, or K=L. If L=4, K=4. Here, Q = LK 2 =4*(4 2 ) = 64. 8 0 1 2 3 4 5 6 7 9 0 1 2 3 4 5 6 7 8 Quantity, in 000s Price, in $

Page 8 of 12 12. [5 pts] Joshua’s Bakery is famous for selling rainbow-colored bagels. The shop sells these bagels to both college students and professors. Student demand for the bagels is given by Q S = 20 – 2P S while professor demand is given by Q P = 20 – P P . Suppose Joshua has only 20 rainbow bagels to sell and no marginal costs. If he can tell the two groups apart, how many rainbow bagels will he sell to students? How many to professors? Explain. Answer: Here, we want to find the marginal revenue Joshua earns from selling bagels to students and professors, respectively. For students, Q S = 20 – 2P S translates to P S =10-(1/2)Q S . To find MR S , we compute TR S =(10-(1/2)Q S )Q S =10Q S -(1/2)Q S 2 . MR S =dTR S /dQ S =10 – Q S . For Professors, P P =20 – Q P . TR P =(20-Q P )Q P , so MR P =20-2Q P . If we set MR P =MR S and substitute the constraint that Q S +Q P =20, we find: 10-Q S =20-2Q P 10-(20-Q P )=20-2Q P Thus, 3Q p =30, or Q P =10. Then Q S =10. At those quantities, MR P =MR S =0, which is greater than or equal to MC=0 in this setting; Joshua will sell all 20 bagels.

Page 9 of 12 II. Long Answer Problems (40 points) 1. [25 points] Competition in the electric car market Tesla got an early start in the market for electric cars, but now faces potential competition from VW. Suppose that the demand for electric cars is given by P=120-5Q. Tesla’s marginal cost of production is $40/unit and the company pays a fixed cost of $50. (a) [5 pts] Suppose Tesla is the only company in the electric car market. What price should Tesla charge? What quantity does it produce? How much profit does it make each year? Answer: With demand given by P=120-5Q, Tesla’s revenue function is: TR = P(Q)*Q=120Q-5Q 2 and marginal revenue is MR=dTR/dQ =120-10Q. Setting this MR equal to the MC of 40 and solving, we get Q=8. Plugging this quantity back into the demand expression gives P=120-5*8=80. So Tesla produces Q=8 and charges P=80. Profit is given by π=(P-AVC)*Q-FC and since the MC is constant, AVC=MC=40. So we have π=(80-40)*8-50 which yields π=270. (b) [7 pts] VW is considering entering the electric car market. VW can produce cars with a marginal cost of $20, but to enter the market they would need to pay a fixed cost of $250 per year. Suppose Tesla cannot anticipate whether VW will enter, so that if VW does enter the two firms compete by making simultaneous quantity choices. What quantity does VW expect to be able to sell if it enters the market? What is the market price? Answer: Since firms will make simultaneous quantity choices if VW enters, we are solving a Cournot problem. We begin by finding each firm’s best response function. With two firms in the market, demand is given by P=120-5Q T -5Q VW where Q T is the quantity produced by Tesla and Q VW is the quantity produced by VW. The revenue function for Tesla is R T =120Q T -5Q T 2 -5Q VW Q T . Taking the derivative with respect to Q T gives the marginal revenue for Tesla: MR T =120-10Q T -5Q VW . Setting MR T equal to Tesla’s marginal cost of 40 we have 120-10Q T -5Q VW = 40 which can be reorganized to yield: Q T =8-(1/2)Q VW The revenue function for VW is R VW =120Q VW -5Q T Q VW -5Q VW 2 and marginal revenue for VW is MR VW =120-10Q VW -5Q T . Setting this equal to the VW marginal cost of 20, we have 120-10Q VW -5Q T =20 which reorganizes to yield the VW best response function: Q VW =10-(1/2)Q T We can substitute Tesla’s best response function into the VW best response function: Q VW =10-(1/2)[8-(1/2)Q VW ]=10-4+(1/4)Q VW (3/4) Q VW = 6 Q VW =8 Plugging this into the Tesla best response function gives Q T = 8-(1/2)Q VW = 4.

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Page 10 of 12 Now we know that the total quantity produced if VW enters will be 8+4=12. Plugging this into the demand equation gives P=120-5*(12)=60. (c) [3 pts] Does it make sense for VW to enter the market and compete with Tesla simultaneously in quantities? Why or why not? Answer: To answer this question, we need to calculate the profits that VW earns if it enters the market. With a constant marginal cost of 20 (so AVC=MC=20) and a fixed cost of 250, we can write down VW’s profit as: π VW = (P-AVC)*Q-FC = (60-20)*8-250=70 Since profits are positive, it makes sense for VW to enter the market. (d) [7 pts] Suppose that Tesla hears about VW’s plans for entering the electric vehicle market. This allows Tesla to commit in advance to a level of production (Stackelberg leader). If VW decides to enter the market as a follower, what quantity can it expect to sell? At what price? Answer: Now we are solving a Stackelberg problem. We can start by using the follower’s (VW’s) best response function from the Cournot problem, which is Q VW =10-(1/2)Q T . We want to substitute this into the leader’s revenue function before taking the derivative to find marginal revenue. From part (b), the Tesla’s revenue function is: R T =120Q T -5Q T 2 -5Q VW Q T Plugging in VW’s best response function gives: R T =120Q T -5Q T 2 -5Q T *[10-(1/2)Q T ] R T = 120Q T -5Q T 2 -50Q T +(5/2) Q T 2 = 70Q T – (5/2)Q T 2 The marginal revenue for Tesla is then: MR T =70-5Q T Setting MR T equal to Tesla’s marginal cost, we have 40=70-5Q T which we solve to get Q T =6. Plugging this back into the VW best response function gives us Q VW =10-(1/2)*6 = 10-3 so Q VW =7. To find price, we plug the sum of these quantities back into the demand expression: P=120-5*(6+7)=55 (e) [3pts] Should VW enter the market if it knows that Tesla will be able to choose quantity first? Why or why not? Answer: To answer this, we need to find VW’s profit with the price and quantity from part (d). This is given by π VW = (55-20)*7-250 = -5. Since profit is negative, VW should not enter the market if it knows that Tesla will be able to choose quantity first.

Page 11 of 12 2. [15 points]. Adverse selection and moral hazard in hiring. Google is looking to expand its workforce in San Francisco. Google knows that among the set of workers looking for a new job, half of the potential applicants will be “Hard Workers” and half will be Slackers. Hard Workers have an outside option of 100,000 while Slackers have an outside option of 50,000. (a) [3 pts] Suppose that Google cannot tell if an applicant is a Hard Worker or a Slacker, and that the company gets a value of 60,000 from hiring Slackers. Suppose that Google gets a value V HW from hiring Hard Workers. Write down Google’s expected value from hiring a worker, as a function of V HW , when 50% of applicants are Hard Workers and 50% are Slackers. Answer: EV = ½ * 60,000 + ½* V HW (b) [4 pts] What is the lowest value V HW that Google must get from hiring a Hard Worker to make it possible for at least some Hard Workers to end up working for Google? Explain. Answer: If half of the workers are Hard Workers and half are Slackers, then Google’s expected value from hiring a worker must be greater than the outside option of the Hard Workers. So if V HW is the value Google gets from hiring Hard Workers, then this must satisfy ½ * 60,000 + ½ * V HW ≥ 100,000. This means V HW ≥ 140,000. (c) [5 pts] Suppose that once Google hires a Hard Worker, the worker can choose either to work hard or take it easy. If they work hard, they generate a good outcome of $1,000,000 in value with probability 0.9 and a bad outcome of $200,000 with probability 0.1. If they take it easy, they generate the good outcome of $1,000,000 with probability 0.1 and the bad outcome of $200,000 with probability 0.9. Effort costs the worker $100,000 and workers are risk neutral. Google is committed to paying a fixed wage of $150,000, but it can also pay a bonus in the event of a good outcome. What level of bonus, along with the wage, does Google need to pay to incentivize the worker to choose effort? Explain. Answer: To incentivize the worker to work hard, the payoff to choosing effort must be higher than the payoff to choosing no effort. That means, 0.9*(150,000+b) + 0.1*150,000 – 100,000 ≥ 0.1 (150,000+b) + 0.9*150,000 This means 0.8*b ≥ 100,000 or b≥125,000.

Page 12 of 12 (d) [3 pts] What is Google’s profit if it pays the hard worker a wage of $150,000 and a bonus that is just large enough to incentivize the worker to exert effort? Is this compensation scheme worth it for Google relative to paying only the fixed wage? Explain. Answer: The company’s payoff if the bonus is just large enough to incentivize the worker to exert effort (so b=125,000) is: 0.9*(1,000,000-125,000) + 0.1*200,000-150,000 = 657,500 If the company does not pay a bonus and therefore the worker does not exert effort, the company’s payoff is: 0.1*1,000,000 + 0.9*200,000 – 150,000 = 130,000 So Google would prefer to pay the bonus in order to incentivize the worker to exert effort.

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