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Signaling Game Problems

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Problem 1, p 348 QualityProbabilityValue to SellerValue to Buyer Good Carq10,00012,000 Lemon1-q6,0007,000 their Expected Value of a random car is 12000q+7000(1-q)=7,000+5,000q If Buyers believe that the fraction of good cars on market is q, In this case, we can expect all used cars to sell for about PU=7,000+5,000q. If q>3/5, then PU=7000+5000q> 10,000 and so owners of lemons and of good cars and of will be willing to sell at price PU. Thus the belief that the fraction q of all used cars are good Is confirmed. We have a pooling equilibrium.

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There is also a separating equilibrium QualityProbabilityValue to SellerValue to Buyer Good Carq10,00012,000 Lemon1-q6,0007,000 Suppose that buyers all believe that the only used cars on the market. Then they all believe that a used car is only worth $7000. The price will not be higher than $7000. At this price, nobody would sell his good car, since good used cars are worth $10,000 to their current owners. Buyers beliefs are confirmed by experience. This is a separating equilibrium. Good used car owners act differently from lemon owners.

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Problem 3, page 348 Suppose that buyers believe that product with no warranty is low quality and that with warranty is high quality. High quality items work with probability H and low quality items work with probability L. Consumer values a working item at V. Buyers are willing to pay up to LV an item that works with probability L. Buyers are willing to pay up to V for any item with a money back guarantee. (If it works, their net gain is V-P and if it fails they get their money back so their net gain is 0. Therefore they will buy if P<V.)

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Equilibrium If the item with warranty sells for just under V and that with no warranty sells for just under LV, buyers will take either one. Given these consumer beliefs, V is the highest price that sellers can get for high quality with warranty and LV is the highest price for the low quality without warranty. Sellers profits from high quality sales with guarantee are hV-c and profits from low quality without guaranty are LV-c. If seller put a guarantee on low quality items and sold them for V, his profit would be LV-c, which is no better than he does without a guarantee on these.

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Equilibrium If buyers believe that only the good items have guarantees, the Nash equilibrium outcome confirms this belief. If fraction of items sold that are of high quality is r, then retailers average profit per unit sold Is rHV+(1-r)LV. Retailer can not do better with a pooling equilbrium in which he guaranteed nothing, or in one in which he guaranteed everything. Can you show this?

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Problem 5, page 350 George Bush and Saddam Hussein

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The story Bush believes that probability Hussein has WMDs is w<3/5. When is there a perfect Bayes-Nash equilibrium with strategies? Hussein: If WMD, Dont allow, if no WMD allow with probability h. Bush: If allow and WMD, Invade. If allow and no WMD, Dont invade, If dont allow, invade with probability b.

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Payoffs for Hussein if he has no WMDs Payoff from not allow is 2b+8(1-b)=8-6b Payoff from allow is 4, since if he allows Bush will not invade. Hussein is indifferent if 4=8-6b or equivalently b=2/3. So he would be willing to use a mixed strategy if he thought that Bush would invade with probability 2/3 if Hussein doesnt allow inspections.

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Probability that Hussein has WMDs if he uses mixed strategy If Hussein does not allow inspections, what is probability that he has WMDs? Apply Bayes law. P(WMD|no inspect)= P(WMD and no inspect)/P(no inspect)= w/(w+(1-w)(1-h))

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Bushs payoffs if Hussein refuses inspections If Bush does not invade: 1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h))) If Bush invades: 3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h)) Bush will use a mixed strategy only if these two payoffs are equal. We need to solve the equation 1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h))) =3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h)) for h.

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Solution Solving equation on previous slide, we see that if Saddam refuses inspections, Bush is indifferent between invading and not if h=3- 5w/3(1-w). (Remember we assumed w<3/5) so 0<h<1) If Saddam has no WMDs, he is indifferent between allowing and not allowing inspections Bush would invade with probability 4/5 if there are no inspections.

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Describing equilibrium strategies Saddam: Do not allow inspections if he has WMD. Allow inspections with probability h=3-5w/3(1-w) if he has no WMD. (e.g. if w=1/2, h=1/3. If w=1/3, h=2/3.) Bush: Invade if Saddam has WMD and allows inspections, Dont invade if Saddam has no WMD and allows inspections. Invade with probability 4/5 if Saddam does not allow inspections.

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Problem 5, p 350 Students are of 3 types, High, medium, and low. Cost of getting a college degree to a student is 2 if high, 4 if medium, and 6 if low. 1/6 of students are of high type, ½ of medium type, 1/3 are of low type. Salaries for managers are 15, and 10 for clerks. An employer has one clerks job to fill and one managers job to fill. Employers profits (net of wages) are 7 from hiring anyone as a clerk, 4 from hiring a low type as a manager, 6 from hiring a medium type as manager, 14 from hiring a high type as manager.

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Equilibrium where high and medium types go to college, low does not. If high and medium types go to college, what is the expected profit from hiring a college grad as a manager? Find probability p that someone is of high type given college: P(H|C)=P(H and C)/P( C)=(1/6) / (1/6+1/2)=1/4 Expected profit is 1/4x14+3/4x6=8. If you hire a college grad as clerk, expected profit is 7. So better off to hire her as manager.

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Equilibrium for workers. High types get paid 15 as manager have college costs of 2. So net wage is 13. Thats better than the 10 that nondegree people get as clerks. Medium type get paid 15 as manager have college costs 4, net wage of 11, so they prefer college and managing to no college and clerk. Low types would get 15 as manager with college costs of 6. Net pay of 15-6=9 is less than they would get with no college and being a clerk.

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Professor Drywalls Lectures

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A fable Imagine that the labor force consists of two types of workers: Able and Middling with equal proportions of each. Employers are not able to tell which type they are when they hire them. A worker is worth $1500 a month to his boss if he is Able and $1000 a month if he is Middling. Average worker is worth $ ½ 1500 + ½ 1000=$1250 per month.

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Competitive labor market The labor market is competitive and since employers cant tell the Able from the Middling, all laborers are paid a wage of $1250 per month.

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One employer believes that Drywalls lectures are useful and requires its workers attend 10 monthly lectures by Professor Drywall and pays wages of $100 per month above the average wage. – Middling workers find Drywalls lectures excruciatingly dull. Each lecture is as bad as losing $20. – Able workers find them only a little dull. To them, each lecture is as bad as losing $5. Which laborers stay with the firm? What happens to the average productivity of laborers?

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Other firms see what happened Professor Drywall shows the results of his lectures for productivity at the first firm. Firms decide to pay wages of about $1500 for people who have taken Drywalls course. Now who will take Drywalls course? What will be the average productivity of workers who take his course? Do we have an equilibrium now?

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Professor Drywall responds Professor Drywall is not discouraged. He claims that the problem is that people have not heard enough lectures to learn his material. Firms believe him and Drywall now makes his course last for 30 hours a month. Firms pay almost $1500 wages for those who take his course and $1000 for those who do not.

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Separating Equilibrium Able workers will prefer attending lectures and getting a wage of $1500, since to them the cost of attending the lectures is $5x30=$150 per month. Middling workers will prefer not attending lectures since they can get $1000 if they dont attend. Their cost of attending the lectures would be $20x30=$600, leaving them with a net of $900.

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So there we are.

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