BEPP 250 Managerial Economics Final Exam Name: Penn ID #: Section Number: Professor’s Name: Spring 2015 My signature below certifies that I have complied with the University of Pennsylvania’s Code of Academic Integrity in completing this examination. (SIGNATURE) INSTRUCTIONS 1. SIGN IN. Your proctor has the sign-­‐‑in sheet. 2. Do NOT take the exam apart. 3. Write your legal name and your Penn Student ID number on this page and the back of the exam. 4. Calculators are provided. No other electronics are allowed. 5. Notes and/or “cheat sheets” are NOT permitted. 6. Show your work. Clearly explain your solutions. No credit will be given if work is not shown or explanations are missing. 7. Write in the spaces provided after each question. Or, if you need more space, use the overflow spaces provided. 8. Write legibly. 9. SIGN OUT and turn in your exam BEFORE you leave. DO NOT START UNTIL 6:00 PM DO NOT LEAVE THE ROOM WITH THIS EXAM EVEN IF YOU ARE PLANNING TO WITHDRAW FROM THE COURSE Page 1 of 32 Page 2 of 32 1. 25 POINTS Consider a market with two firms that offer homogeneous products. Firm 1 has a constant marginal cost of 4, and firm 2 has a constant marginal cost of 2. The market demand function is 𝐷 𝑝 = 10 − 𝑝 Firms compete by simultaneously choosing price. Given that firms’ products are identical, all consumers buy from the firm with the lowest price. For example, if firm 1’s price is 𝑝' and firm 2’s price is 𝑝( and 𝑝' < 𝑝( then firm 1’s demand is 𝐷(𝑝' ) and firm 2 has zero demand. If both firms offer an identical price 𝑝 then they split the market demand equally so each firm’s demand is 𝐷 𝑝 /2. As just described, this is the Bertrand price game. However, contrary to what was specified in class, each firm is restricted to choose its price from the set: {2,3, … ,10}. That is, price must be an integer between 2 and 10. It is assumed that each firm is interested in maximizing its profit. a. 3 POINTS Is it a Nash equilibrium for both firms to price at their marginal cost? (Show your work) Answer: No. Given firm 2 prices at 2, firm 1’s profit is zero from pricing at 4 and it cannot do any better by pricing differently. As long as its price exceeds 2 then its demand and profit are zero. If it prices at 2 then it has negative profit. Thus, firm 1’s price of 4 is a best response to firm 2’s price of 2. However, firm 2’s price is not a best response. Pricing at 2 produces zero profit, while pricing at 3 yields positive profit of 10 − 3 3 − 2 = 7 so a price of 2 is not optimal for firm 2. Thus, both firms pricing at marginal cost is not a Nash equilibrium. Page 3 of 32 b. 3 POINTS Is it a Nash equilibrium for both firms to price at 4? (Show your work) Answer: No. While firm 1’s price is a best response, firm 2’s price is not. By pricing at 4, firm 2 earns profit of 10 − 4 1 2 4 − 2 = 6 but does better by pricing at 3 and earning profit of 7. Page 4 of 32 c. 6 POINTS Find all Nash equilibria. Answer: As an initial step to answering this question, the best response function for firm 2 will be derived. It can be shown that firm 2’s monopoly price is 6. Thus, if firm 1’s price is 7, 8, 9, or 10 then firm 2 will have a monopoly if it prices below firm 1 in which case firm 2’s best response is 6. If firm 1’s price is 6 then firm 2 could price at 6 and earn profit of 10 − 6 1/2 6 − 2 =
8 (as it shares demand with firm 1), price at 5 and earn profit of 10 − 5 5 − 2 = 15 (as it has all of market demand), price at 4 and earn profit of 10 − 4 4 − 2 = 12, or price at 3 and earn profit of 10 − 3 3 − 2 = 7. (It clearly does not want to price at 2 and earn zero profit.) Thus, firm 2’s best response is 5 when firm 1 prices at 6. If firm 1 prices at 5 then firm 2’s best response is 4 as that delivers profit of 12 while pricing at 5 yields profit of 7.5 (and we already showed that a price lower than 4 has lower profit). If firm 1 prices at 4 then firm 2’s best response is 3 as that delivers profit of 7 while pricing at 4 yields profit of 6. If firm 1 prices at 3 then firm 2’s best response is 3 as that delivers profit of 3.5 while pricing at 2 yields profit of 0. In sum, if firm 1’s price is 7 or higher than firm 2’s best response is 6. If firm 1’s price lies in {4,5,6} then firm 2’s best response is to undercut firm 1’s price by 1. There are two Nash equilibria. One Nash equilibrium has firm 1 price at 4 and firm 2 price at 3. Firm 1’s profit (and demand) is zero for any price exceeding 3. If it prices at 3 or less, its demand is positive but, as its price is less than its cost, its profit is negative. 3 is firm 2’s best response to firm 1 pricing at 4. A second Nash equilibrium has firm 1 price at 5 and firm 2 price at 4. Firm 1’s profit (and demand) is zero for any price exceeding 4. If it prices at 4 then, though it has positive demand, its profit is still zero as it is pricing at marginal cost. If it prices at 3 or less, its demand is positive but, as its price is less than its cost, its profit is negative. A price of 4 is firm 2’s best response to firm 1 pricing at 5. Page 5 of 32 Now suppose the firms play this game for an infinite number of periods. Each firm seeks to maximize the sum of its discounted profits (or, equivalently stated, the present value of its profit stream) where a firm’s discount factor is 𝛿 where 0 < 𝛿 < 1. Consider the following strategy pair. Firm 1’s strategy: In period 1, firm 1 prices at 6. In any future period, firm 1 prices at 6 if both firms priced at 6 in all past periods, and prices at 4 if one or more firms priced differently from 6 in some past period. Firm 2’s strategy: In period 1, firm 2 prices at 6. In any future period, firm 2 prices at 6 if both firms priced at 6 in all past periods, and prices at 3 if one or more firms priced differently from 6 in some past period. d. 3 POINTS Find each firm’s payoff (starting with profit received in period 1) if the firms uses these strategies. Answer: Firm 1 will earn 0.5 10 − 6 6 − 4 = 4 each period so its payoff is Firm 2 will earn 0.5 10 − 6 6 − 2 = 8 each period so its payoff is Page 6 of 32 ;
'<=
>
'<=
. . e. 4 POINTS In period 1, each firm’s strategy calls for it to price at 6. Derive the conditions under which this decision to price at 6 in period 1 is optimal for both firms. Answer: If a firm prices differently from 6 then the future price path is the same: firm 1 prices at 4 and firm 2 prices at 3. Thus, if a firm is to deviate price from 6, it ought to do so as to maximize current profit given that the future profit stream is the same. The optimal price in that case is to just undercut the collusive price by pricing at 5. Firm 1: 4
𝛿
≥ 10 − 5 5 − 4 +
×0 1−𝛿
1−𝛿
→
4
≥ 5 1−𝛿
→ 𝛿 ≥ 1/5 Firm 2: 8
𝛿
≥ 10 − 5 5 − 2 +
(10 − 3)(3 − 2) 1−𝛿
1−𝛿
→
8
𝛿
≥ 15 +
7 1−𝛿
1−𝛿
→ 𝛿 ≥ 7/8 Page 7 of 32 f. 3 POINTS Derive the conditions for this strategy pair to be a subgame perfect Nash equilibrium. Answer: We’ve already shown that the strategy is optimal for histories that call for it to price at 6 as long as 𝛿 ≥ 1/5 and 𝛿 ≥ 7/8. For histories which call for firm 1 to price at 4 and for firm 2 to price at 3, the prescribed prices are optimal because they form a static Nash equilibrium, as shown in part c. Page 8 of 32 Consider the following strategy pair. Firm 1’s strategy: In period 1, firm 1 prices at 6. In any future period, firm 1 prices at 6 if both firms priced at 6 in all past periods, and prices at 4 if one or more firms priced differently from 6 in some past period. Firm 2’s strategy: In period 1, firm 2 prices at 6. In any future period, firm 2 prices at 6 if both firms priced at 6 in all past periods, and prices at 2 if one or more firms priced differently from 6 in some past period. g. 3 POINTS Show that this strategy pair is NOT a subgame perfect Nash equilibrium. Answer: In the event that some firm deviated from 6, both firms are supposed to price at marginal cost forever. However, firm 2’s behavior is not optimal with such behavior. Given that firm 1 is always going to price at 4 then firm 2 can earn profit of 7 in every period by pricing at 3. That yields a payoff of 7 1 − 𝛿 which exceeds the zero payoff from pricing at 2. Page 9 of 32 OVERFLOW SPACE FOR QUESTION 1 Page 10 of 32 2. 25 POINTS Consider a market with two firms that offer identical products. The inverse market demand function is 𝑃 𝑞' + 𝑞( = 210 − 4 𝑞' + 𝑞( where 𝑞' is firm 1’s quantity and 𝑞( is firm 2’s quantity. Firms make simultaneous quantity decisions and the price each firm receives is 𝑃 𝑞' + 𝑞( . Firm 1’s cost function is 𝐶' 𝑞' = 20𝑞' and firm 2’s cost function is 𝐶( 𝑞( = 2 𝑞( ( a. 2 POINTS Derive each firm’s marginal cost. Answer: Firm 1’s marginal cost is 20, and firm 2’s marginal cost is 4𝑞( . Page 11 of 32 b. 3 POINTS Derive each firm’s marginal revenue. Answer: Firm 1’s revenue is 𝑅' 𝑞' , 𝑞( = 𝑃 𝑞' + 𝑞( 𝑞' = 210 − 4 𝑞' + 𝑞( 𝑞' so its marginal revenue is 𝜕𝑅' 𝑞' , 𝑞(
= 210 − 8𝑞' − 4𝑞( 𝜕𝑞'
Firm 2’s revenue is 𝑅( 𝑞' , 𝑞( = 𝑃 𝑞' + 𝑞( 𝑞( = 210 − 4 𝑞' + 𝑞( 𝑞( so its marginal revenue is 𝜕𝑅( 𝑞' , 𝑞(
= 210 − 4𝑞' − 8𝑞( 𝜕𝑞(
Page 12 of 32 c. 6 POINTS Derive each firm’s best response function. Answer: Firm 1’s profit function is 𝜋' 𝑞' , 𝑞( = 𝑃 𝑞' + 𝑞( 𝑞' − 𝐶' 𝑞'
= 210 − 4 𝑞' + 𝑞( 𝑞' − 20𝑞' Firm 1’s best response function is the quantity that maximizes its profit which is found by setting marginal profit equal to zero and solving for 𝑞' : 𝜕𝜋' 𝑞' , 𝑞(
= 210 − 8𝑞' − 4𝑞( − 20 = 0 𝜕𝑞'
Firm 1’s best response function: 𝑞' 𝑞( =
190 1
− 𝑞( = 23.75 − 0.5𝑞( 8
2
Firm 2’s profit function is 𝜋( 𝑞' , 𝑞( = 𝑃 𝑞' + 𝑞( 𝑞( − 𝐶( 𝑞( = 210 − 4 𝑞' + 𝑞( 𝑞( − 2𝑞(( 𝜕𝜋( 𝑞' , 𝑞(
= 210 − 4𝑞' − 8𝑞( − 4𝑞( = 0 𝜕𝑞'
Firm 2’s best response function: 𝑞' 𝑞( =
210 1
− 𝑞 = 17.5 − 0.33𝑞' 12 3 (
Page 13 of 32 d. 3 POINTS Is a firm’s best response function increasing or decreasing in its rival’s quantity? Explain your answer in economic terms (that is, dealing with such concepts as demand and cost). Answer: Each firm’s best response function is decreasing in its rival’s quantity: 𝑑𝑞' 𝑞(
1
= − 𝑑𝑞(
2
𝑑𝑞( 𝑞'
1
= − 𝑑𝑞'
3
The higher is the rival’s quantity, the lower is the price that a firm receives for a given quantity that it produces. In other words, a higher rival quantity shifts in a firm’s demand curve. As a result of weaker firm demand, a firm’s optimal quantity is smaller. Page 14 of 32 e. 6 POINTS Derive the Nash equilibrium quantities. Answer: We need to find a pair of quantities, 𝑞'∗ and 𝑞(∗ , such that both firms are producing according to their best response functions: 𝑞'∗ =
190 1 ∗
− 𝑞( 8
2
𝑞(∗ =
210 1 ∗ 210 1 190 1 ∗
− 𝑞 =
−
− 𝑞( 12 3 '
12 3 8
2
→ 𝑞(∗ = 11.5 𝑞'∗ =
190 1 ∗ 190 1
− 𝑞( =
− ×11.5 = 18 8
2
8
2
Page 15 of 32 f. 5 POINTS Now suppose firms have hired a cartel manager who will choose 𝑞' and 𝑞( in order to maximize total industry profit (which is the sum of the two firms’ profits). The cartel manager has determined that the total quantity that maximizes joint profit is 23.75. How should those 23.75 units be allocated among the two firms in order to maximize joint profit? Answer: If both firms produce positive amounts at the joint profit maximum then they must have equal marginal cost. If that wasn’t the case so that, for example, firm 1’s marginal cost exceeded firm 2’s marginal cost then total profit could be increased by maintaining the same total quantity and shifting a little bit of output from firm 1 to firm 2. Given total quantity is unchanged, revenue is the same; but total cost is lower as some output has been shifted to the firm with the lower marginal cost. Given that firm 1’s marginal cost is 20 and firm 2’s is 4𝑞( , we want to find 𝑞( such that: 20 = 4𝑞( → 𝑞( = 5. Given total quantity is 23.75 then the joint profit maximum has 𝑞' = 18.75 and 𝑞( = 5. Page 16 of 32 OVERFLOW SPACE FOR QUESTION 2 Page 17 of 32 3. 25 POINTS Montse just bought a TV worth $1000 and is now deciding whether to buy an insurance contract that covers the value of the TV. With probability 0.05, the TV completely breaks and with probability 0.95, the TV remains fully functional. The insurance contract costs $𝑃 and fully covers the value of the TV if it breaks. Assume Montse has utility 𝑈(𝑊) = 𝑊, where W is total wealth, which includes the value of the TV ($1000) plus total savings of $999,000. That is, without insurance, total wealth is 𝑊 = $1,000,000 with probability 0.95, while total wealth goes down to 𝑊 =
$1,000,000 − $1000 = $999,000 with probability 0.05. a. 2 POINTS Is Montse risk averse, risk neutral or risk loving? Explain your answer in terms of a condition on 𝑈(𝑊). Answer: Risk averse since P
1
𝑈 OO = − 𝑊 <( < 0 4
Page 18 of 32 b. 5 POINTS What is Montse’s expected utility if she does NOT buy insurance? Answer: 𝐸𝑈RS = 0.05 999,000 + 0.95 1,000,000 = 999.975 Page 19 of 32 c. 3 POINTS Up to how much is Montse willing to pay for insurance? Answer: 𝐸 𝑈S = 1,000,000 – 𝑃. Thus 𝐸 𝑈S ≥ 𝐸𝑈RS if 𝑃 ≤ 50.0119. Page 20 of 32 Suppose another consumer, Lily, buys the same TV and is thinking about buying the insurance contract. Lily owns a dog at home, which raises the probability that her TV fails from 0.05 to 0.20. Assume that Lily has the same utility 𝑈(𝑊) = 𝑊 and total savings of $999,000 (the same as Montse). d. 5 POINTS Up to how much is Lily willing to pay for insurance? Answer: 𝐸𝑈RS = 0.20 999,000 + 0.80 1,000,000 = 999.9 𝐸𝑈S =
1,000,000 − 𝑃 Thus, 𝐸 𝑈S ≥ 𝐸𝑈RS if 𝑃 ≤ 200.04. Page 21 of 32 e. 3 POINTS If the insurer prices the insurance contract at $150 and the two possible buyers are Montse and Lily, who will buy insurance? Answer: Only Lily will buy insurance because $150 < $200.04 but $150 >
$50.01. Page 22 of 32 f. 7 POINTS Derive the insurer’s expected profits as a function of price 𝑃 and find the price that maximizes expected profits. Answer: If 𝑃 < $50.01, then expected profit is 2𝑃 − 0.05 + 0.20 1000 = 2𝑃 − 250 If $50.01 < 𝑃 ≤ $200.04, then expected profit is 𝑃 − 0.20 1000 = 𝑃 − 200 If 𝑃 > $200.04, then expected profit is zero. Therefore, the retailer maximizes expected profit by setting 𝑃 = $200.04 Page 23 of 32 OVERFLOW SPACE FOR QUESTION 3 Page 24 of 32 4. 25 POINTS There are two bidders 𝐴 and 𝐵, each with independent private valuations 𝑣Z and 𝑣[ . With probability 1/2, a bidder values the object for $50; and with probability 1/2, a bidder values the object for $100. a. 3 POINTS Suppose the object is sold through a second-­‐‑price sealed-­‐‑bid auction. Is there a bidding strategy that is optimal regardless of how the rival is bidding? Explain the intuition behind your answer in five sentences or less. Answer: Bidding your valuation is optimal regardless of how one’s rival is bidding. Your own bid only affects the probability that you win and not your surplus and thus you want to increase your bid as much as possible. However, you don’t want to go over your valuation because it’s only worth outbidding your opponent if he bids below your valuation, otherwise you’ll get negative surplus. Page 25 of 32 In what follows, suppose the object is sold through a first-­‐‑price sealed-­‐‑bid auction and bidders can only bid $30, $60 or $70. If the two bidders submit the same bid, then a coin is flipped to determine the winner. So, in that event, each bidder has probability 1/2 of winning the auction. Consider the following strategy profile 𝑆 ∗ : For each bidder 𝑖 = 𝐴, 𝐵, 𝑏_∗ =
30 𝑖𝑓 𝑣_ = 50
70 𝑖𝑓 𝑣_ = 100
b. 2 POINTS If bidder 𝐵 bids according to 𝑆 ∗ , show that bidder 𝐴 wins with probability 0.25 if he bids $30. Answer: Let 𝟏{𝑋} be an indicator function such that 𝟏{𝑋} = 1 if 𝑋 is true, otherwise 𝟏{𝑋} = 0. The probability that bidder 𝑖 wins if he bids 𝑏_ and the rival bids according to 𝑆 ∗ is 𝑃𝑟(𝑖 𝑤𝑖𝑛𝑠) = 0.5(𝟏{𝑏_ > 70} + 0.5 ⋅ 𝟏{𝑏_ = 70} ) + 0.5(𝟏{𝑏_
> 30} + 0.5 ⋅ 𝟏{𝑏_ = 30}) Thus if 𝑏_ = 30, then 𝑃𝑟(𝑖 𝑤𝑖𝑛𝑠) = 0.25. Page 26 of 32 c. 2 POINTS If bidder 𝐵 bids according to 𝑆 ∗ , show that bidder 𝐴 wins with probability 0.5 if he bids $60. Answer: Applying the same formula as above gives 𝑃𝑟(𝑖 𝑤𝑖𝑛𝑠) = 0.5 if 𝑏_ =
60. Page 27 of 32 d. 4 POINTS If bidder 𝐵 bids according to 𝑆 ∗ , what is the expected payoff of bidder 𝐴 if he bids $70 and 𝑣Z = 100? Answer: 𝑃𝑟(𝑖 𝑤𝑖𝑛𝑠) = 0.25 + 0.5 = 0.75. If 𝑣Z = 100, then surplus is 100 − 70 = 30. Thus expected payoff is 22.5. Page 28 of 32 e. 7 POINTS Show that 𝑆 ∗ is a symmetric Nash Equilibrium. Answer: Consider some bidder 𝑖 . It suffices to show that bidder 𝑖 ’s best response is to bid according to 𝑆 ∗ given that the rival bidder bids according to 𝑆 ∗ . We have to check this for 𝑣_ = 50 and for 𝑣_ = 100. The following table shows the best response of 𝑣_ = 50 to a rival bidding according to 𝑆 ∗ is to bid 30 while the best response of 𝑣_ = 100 to a rival bidding according to 𝑆 ∗ is to bid 70. Thus, 𝑆 ∗ is a symmetric NE. 𝑣_ = 50 𝑏_ 𝑣_ − 𝑏_ 𝑃𝑟(𝑤𝑖𝑛) Expected Payoff 30 20 0.25 5 60 -­‐‑10 0.50 -­‐‑5 70 -­‐‑20 0.75 -­‐‑15 𝑣_ = 100 𝑏_ 𝑣_ − 𝑏_ 𝑃𝑟(𝑤𝑖𝑛) Expected Payoff 30 70 0.25 17.5 60 40 0.50 20 70 30 0.75 -­‐‑22.5 Page 29 of 32 f. 7 POINTS Continue assuming that that the auction is still first-­‐‑price sealed-­‐‑bid and that bidders can only bid $30, $60 or $70. However, it is now assumed to be a common value setting. 𝑣Z and 𝑣[ are not valuations, but instead private signals about this common value. It is assumed the true common value of the object is the minimum of the signals received: 𝑉 = min{𝑣Z , 𝑣[ } Private signals are still independently drawn from the same distribution as before, i.e. $100 and $50 with equal probability. To be clear, a bidder receives a signal of the value of the object up for auction and the true value of the object equals the minimum of the two bidders’ signals. Show that 𝑆 ∗ is no longer a symmetric Nash Equilibrium. Answer: It suffices to show that bidder 𝑖 has a profitable deviation from bidding according to 𝑆 ∗ given the rival plays 𝑆 ∗ . Consider 𝑣_ = 100. 𝑆 ∗ prescribes 𝑖 bidding 70. Expected payoff under this strategy is (0.5)(0.5) 100 − 70 + (0.5)(50 − 70) = −2.5 Consider the deviation 𝑏_ = $30 for 𝑣_ = 100. Expected payoff under this deviation is (0.5)(0) + (0.5)(0.5)(50 − 30) = 5 > −2.5, and therefore is a profitable deviation from 𝑆 ∗ . (As an aside, notice that the deviation 𝑏_ = $60 gives expected payoffs of 0.5 0 + (0.5)(50 − 60) = −5 gives you even worse than 𝑆 ∗ , so this is not the right deviation to consider.) Page 30 of 32 OVERFLOW SPACE FOR QUESTION 4 Page 31 of 32 Name: Penn ID: DO NOT WRITE BELOW Question Max Points 1a 3 1b 3 1c 6 1d 3 1e 4 1f 3 1g 2a 3 2 2b 3 2c 6 2d 3 2e 6 2f 3a 5 2 3b 5 3c 3 3d 5 3e 3 3f 4a 7 3 4b 2 4c 2 4d 4 4e 7 4f 7 Score Total EXAM TOTAL Page 32 of 32