EXAM 2 SPRING 2021 SOLUTIONS without solutions for #5.pdf

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OTM 732: Economics for Managers EXAM 2 1) (20 points) This is the “Rational Pigs Game.” Two pigs, one big and the other small, are placed in a large box. There is a lever at one end of the box which, when pressed, dispenses food at the other end. This means that the pig that presses the lever must run to the other end to get any food; and by the time it gets there, the other pig will have eaten some, and perhaps all, of the food. The big pig can prevent the little pig from getting any of the food when both are at the food. If the pigs can reason like game theorists, which pig will press the lever? For concreteness, let’s attach some hypothetical numbers to the game. Suppose that 12 kernels of corn are dispensed if the lever is pressed (by either pig). If the small pig presses the lever, then the big pig eats all 12 kernels before the small pig can run to the other side of the box where the food appeared. However, if it was the big pig who pressed the lever, then the small pig has time to eat 10 of the 12 kernels before the big pig pushes it away and eats the remaining two kernels. Suppose that in the unlikely event that they press the lever together, the small pig, because it can run faster, gets 4 kernels before the big pig pushes it away. And once that happens, the big pig will eat the remaining 8 kernels. Finally, suppose that pressing the lever and running to the other end requires effort equal to one kernel of corn. This leads to the following game: a) (5 points) Does the SMALL pig have a strictly dominant strategy? If so, identify it. YES, “DON’T PRESS” is a SDS for the small pig b) (5 points) Does the BIG pig have a strictly dominant strategy? If so, identify it. NO c) (5 points) What is the Nash equilibrium to this game? Which pig does better in the NE? NE is (DON’T PRESS by Small, PRESS by Big) for a payoff of (10,1). The Small Pig does better! d) (5 points) Is this game a Prisoner’s Dilemma game? Why or why not? No, this is not a PD because the Big Pig does not have a SDS….and there is not a non-equilibrium outcome that is strictly better for both of them.

***** This may seem like this is a silly game, but studies have shown that pigs can learn to behave in the ways described by this NE. 2) (20 points) You are playing a game with someone (not a friend and not someone you can communicate with). The game starts with both of you being awarded $3. The game’s rules are: • You announce whether you want to add $0, $1, $2 or $3 to your initial $3 award. Doing so reduces your opponent’s award by $0, $2, $4, or $6, respectively. (In words, they automatically lose twice what you gain.) • Your opponent simultaneously and independently has the same choices: add $0, $1, $2, or $3 to their initial $3 award, which automatically decreases your award by $0, $2, $4, or $6, respectively. Hi M So, for each of you, your final award equals: (your initial $3) + (your choice of $0, $1, $2, or $3) – 2 x (opponent’s choice of $0, $1, $2, or $3). If the game ends with a negative award for a player, that’s OK and you must pay that amount. a) (15 points) Create the game matrix for this game, showing players, actions, and payoffs. Identify the game’s Nash equilibrium strategies and payoffs for the players. 

 Each player (simultaneously and independently of the other) chooses to add to their award $0, $1, $2, $3, $4, ……$999,999, or $1,000,000. Doing so decreases the other player’s award by twice that amount. The game matrix for this expanded game has 1,000,001 rows and 1000,001 columns. DO NOT DRAW IT! But building on what you learned in part (a), answer the following two questions: i) What is the Nash equilibrium of this game? ii) What are the Nash equilibrium payoffs for the players in this game? 3)a) (10 points) A monopolist has the demand and MC (marginal cost) curves below. Assuming the monopolist must offer the same price to all potential buyers, draw the MR line, identify the profitmaximizing P* and Q*, shade the areas representing PS, CS, and any inefficiency. By calculating areas of triangles, report the numerical sizes of PS, CS, and any inefficiency

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separate the potential buyers into two segments: 1) those on the top half of the demand curve; and 2) those on the bottom half. Suppose also that the monopolist can third degree price discriminate by charging different prices in the two segments. Solve for the optimal 3 rd degree price discrimination prices. (Do this graphically or analytically – you choose. But I need to see your work.) What are the optimal price and quantity in each segment? Calculate total CS and PS. Compare these values with PS and CS from part (a). You can solve this problem on one graph, but now that we’re studying 3 rd degree price discrimination, we can study each segment in isolation. So it’s possible (and easier, I think) to give each segment its own graph. This is my approach below. Once we do that, the LOW case is easy and very standard, with optimal choices of P LO =6 and Q LO =1 for a PS LO =2. For HIGH demand, the challenge is that the demand curve stops at Q=4. This follows because the maximum demand is for 4 units, and lowering the price to 3, 2, 1 or even 0 does not add sales (in segment HIGH). Visually, our downward-sloping curve becomes vertical at Q=4. This means that MR is only relevant for 4 units. MC=MR at 3 units, so the optimal price is 10, PS is 18, and CS is 9. The curious shaded area is the inefficiency, which is all about the unrealized GTT of unit 4. (AN ASIDE: In part (a), we didn’t think about the HI and LO segments; we just solved for the overall profit-maximizing price. Looking back now, notice that the optimal solution in (a) was to sell ONLY in the HI segment. Given that, it’s no surprise that when we focus on the HI segment, we do exactly what we did in part (a). The idea is: if we can make choices from sets A and B, and it’s optimal to choose only

from set A, then if we redo our optimization with set B eliminated, then we should do exactly what we did before. If B doesn’t matter when it’s available, then it shouldn’t make a difference to eliminate it as candidate.) Summing up, 3 rd degree P.D. increased overall profit from 18 to 20. It increased overall CS from 9 to 10, and overall inefficiency December decreased from 9 to 6. Notice that what really happened in this example is that LOWER segment was totally ignored in part (a), but 3 rd degree P.D. allowed the seller to serve the LOWER segment without undermining its profits in HIGH. Now let’s superimpose all of this on the top graph. See next page!

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4) (20 points) You produce WORD and EXCEL. You have two customers, Fabio and Hart. Willingness to pay for each item for each person appears below. Fabio Hart WORD 8 14 EXCEL 23 14 In parts (a)-(c) below, suppose marginal cost is zero for both WORD and EXCEL. a) (4 points) If you sell EXCEL and WORD separately, what price for each maximizes your profit? What is your total profit? P* word = 8 for two sales and profit of 16 P* excel = 14 for two sales and profit of 28 Total profit is 44. b) (4 points) If you bundle EXCEL and WORD together, what bundle price maximizes your profit? What is your profit? P* bundle = 28 for the sale of two bundles and profit of 2x28 or 56 c) (4 points) What is the optimal mixed bundling strategy? Is there an advantage to mixed bundling in this example? Why or why not? No. The total potential value to offer is 8+23+14+14 or 59, and cost is always zero. So the potential GTT are 59-0 or 59. The bundle gets us to 56. We can’t do better with Hart, since the bundle is already realizing 100% of his WTP. Splitting the bundle into separate units to cater to Fabio will temput Hart to buy a single WORD rather than the bundle. And splittling the bundle into separate units to cater to Hart will tempt Fabio to buy the a single Excel rather than the bundle. So we cannot do better with mixed bundling. d) (8 points) Now suppose your MC has increased from zero to 10 for each unit of EXCEL and each unit of WORD. Repeat parts (a)-(c) with this change in costs. d/a) P* word = 14, one sale to Hart, and profit of 14-10 = 4. P* excel = 23, for one sale to Fabio, and profit of 23-10 = 13 Total profit is 17 d/b) P* bundle = 28, sales to both Fabio and Hart, and profit of 2x(28-20) = 16. d/c) Consider P* bundle = 28, P* excel = 20, and not selling WORD separately. Hart will buy the bundle. Fabio has equal CS of 3 with both offers, so let’s suppose he buys EXCEL. The resulting profit is 28-20 or 8 from Hart, and 20-10 or 10 from Fabio, for a total profit of 18.

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