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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
Chapter 15 Risk and Information Solutions to Review Questions 1. Why must the probabilities of the possible outcomes of a lottery add up to 1? As a general rule, the sum of the probabilities of all possible outcomes is equal to one. This rule ensures that all possible outcomes are accounted for. If the probabilities for the possible outcomes summed to a number less than one, it would imply there were other possible outcomes that had not been included. 2. What is the expected value of a lottery? What is the variance? The expected value of a lottery is a measure of the average payoff the lottery will generate. The variance of a lottery characterizes the average squared deviation between the possible outcomes of the lottery and the expected value of the lottery. The variance measures the risk associated with the lottery; a smaller variance implies greater certainty associated with the expected payoff. 3. What is the difference between the expected value of a lottery and the expected utility of a lottery? The expected value of a lottery measures the average payoff of the lottery in monetary terms. The expected utility of a lottery takes into account how much the decision maker values the expected payoff, particularly in terms of the risk associated with the expected payoff. For example, while the expected payoff from one lottery may exceed the expected payoff from a second lottery, if the second lottery has less risk associated with it, a decision maker might prefer the second lottery to the first. 4. Explain why diminishing marginal utility implies that a decision maker will be risk averse. A utility function that exhibits diminishing marginal utility will imply the decision maker is risk averse. This is because with a utility function with diminishing marginal utility a decision maker will prefer a sure thing to a lottery with the same expected value. By preferring the sure thing, the decision maker prefers less risk, implying the decision maker is risk averse. 5. Suppose that a risk-averse decision maker faces a choice of two lotteries, 1 and 2. The lotteries have the same expected value, but Lottery 1 has a higher variance than Lottery 2. What lottery would a risk-averse decision maker prefer? A risk-averse decision maker, when comparing lotteries with the same expected value, will prefer the lottery with lower risk. In this instance, since Lottery 1 has a higher variance, Lottery 1 will have more risk associated with it. Thus, the decision maker will prefer Lottery 2.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
6. What is a risk premium? What determines the magnitude of the risk premium? A risk premium is the difference between the expected value of a lottery and the payoff from a sure thing so that the decision maker is indifferent between the lottery and the sure thing. A key determinant of the risk premium is the variance associated with the lottery. If two lotteries have the same expected value but one has a higher variance, the risk premium associated with the lottery that has a higher variance will be larger. The decision maker is requiring a greater premium to take on greater risk. 7. What is fair insurance? Why will a risk-averse consumer always be willing to buy full insurance that is fair? A fair insurance policy is one in which the insurance premium is equal to the expected value of the damage being covered. A risk-averse individual will always prefer to purchase a fair insurance policy that provides full insurance against a loss in order to eliminate risk. 8. What is the difference between a chance node and a decision node in a decision tree? In a decision tree, a decision node indicates a particular decision that the decision maker faces. Each branch from a decision node corresponds to a possible alternative that the decision maker might choose. A chance node indicates a particular lottery that the decision maker faces. Each branch from a chance node corresponds to a possible outcome of the lottery. 9. Why does perfect information have value, even for a risk-neutral decision maker? Perfect information has value because it allows the decision maker to tailor its decisions to the underlying circumstances it faces. By knowing the outcome of the lottery, the decision maker can select the decision alternative best suited to the outcome of the lottery. 10. What is the difference between an auction in which bidders have private values and one in which they have common values? In an auction in which bidders have private values, bidders have their own personal valuation of the object. The bidder knows how he values the object but is unsure how other potential bidders value the object. In an auction with common values, the value of the object is the same for all buyers, but no buyer knows exactly what that value is. 11. What is the winner’s curse? Why can the winner’s curse arise in a common-values auction but not in a private-values auction? The winner’s curse arises in auctions in which bidders have common values. The bidder who wins the auction must have submitted the highest bid and therefore must have had the most optimistic estimate of the value of the object. The winning bidder will have almost surely overestimated the value of the object he or she is bidding on. Thus, the winner may suffer from the winner’s curse – bidding more for the object than the item’s intrinsic value.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
12. Why is it wise to bid conservatively in a common values auction? A bidder should anticipate that if she wins a common values auction it is because she had the highest estimate of the object’s value. To avoid the winner’s curse, paying more for the object than the object’s intrinsic value, the bidder should act as if her value is something less than her initial estimate. This new estimate then becomes the starting point when trying to determine the bid; this is the starting point because the bidder may still want to scale down her bid even more, recognizing that other bidders may scale down their bids in a similar way.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
Solutions to Problems 15.1. Consider a lottery with three possible outcomes: a payoff of -10, a payoff of 0, and a payoff of +20. The probability of each outcome is 0.2, 0.5, and 0.3, respectively. a) Sketch the probability distribution of this lottery. b) Compute the expected value of the lottery. c) Compute the variance and the standard deviation of the lottery. a) 0.6 Probability
0.5 0.4 0.3 0.2 0.1 0 -30
-20
-10
0
10
20
30
Payoff
b)
EV 0.2(10) 0.5(0) 0.3(20) EV 4.0
c) Variance 0.2(10 4) 2 0.5(0 4) 2 0.3(20 4) 2 Variance 124 Standard Deviation Variance Standard Deviation 124 Standard Deviation 11.14 15.2. Suppose that you flip a coin. If it comes up heads, you win $10; if it comes up tails, you lose $10. a) Compute the expected value and variance of this lottery. b) Now consider a modification of this lottery: You flip two fair coins. If both coins come up heads, you win $10. If one coin comes up heads and the other comes up tails, you neither win nor lose—your payoff is $0. If both coins come up tails, you lose $10. Verify that this lottery has the same expected value but a smaller variance than the lottery with a single Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
coin flip. (Hint: The probability that two fair coins both come up heads is 0.25, and the probability that two fair coins both come up tails is 0.25.) Why does the second lottery have a smaller variance? a)
EV 0.5(10) 0.5( 10) EV 0.0 Variance 0.5(10 0) 2 0.5(10 0) 2 Variance 100
b)
EV 0.25(10) 0.5(0) 0.25(10) EV 0 Variance 0.25(10 0) 2 0.5(0 0) 2 0.25(10 0) 2 Variance 50
This second lottery has a smaller variance because the probabilities associated with winning or losing $10, which are 0.25, are smaller than the probabilities associated with winning or losing $10 in the first lottery, which are 0.50. In the second lottery there is a 50% chance of a $0 payoff and that reduces the overall variance of the lottery. 15.3. Consider two lotteries. The outcome of each lottery is the same: 1, 2, 3, 4, 5, or 6. In the first lottery each outcome is equally likely. In the second lottery, there is a 0.40 probability that the outcome is 3, and a 0.40 probability that the outcome is 4. Each of the other outcomes has a probability 0.05. Which lottery has the higher variance? The first lottery has the higher variance. This can be verified by direct calculation. It can also be seen in the fact in the second lottery, it is much more certain that the outcome will be confined to one of two numbers, 3 and 4 (each of which lies in the middle of the distribution), whereas in the first lottery any number is equally likely. 15.4. Consider a lottery in which there are five possible payoffs: $9, $16, $25, $36, and $49, each occurring with equal probability. Suppose that a decision maker has a utility function given by the formula U = √I. What is the expected utility of this lottery? The expected utility is:
0.20 9 0.20 16 0.20 25 0.20 36 0.20 49 5.
15.5. Suppose that you have a utility function given by the equation U = √50I. Consider a lottery that provides a payoff of $0 with probability 0.75 and $200 with probability 0.25.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
a) Sketch a graph of this utility function, letting I vary over the range 0 to 200. b) Verify that the expected value of this lottery is $50. c) What is the expected utility of this lottery? d) What is your utility if you receive a sure payoff of $50? Is it bigger or smaller than your expected utility from the lottery? Based on your answers to these questions, are you risk averse? a) 120
Utility
100 80 60 40 20 0 0
50
100
150
200
I
b)
EV 0.75(0) 0.25(200) EV 50
c) Expected Utility 0.75 50(0) 0.25 50(200) Expected Utility 25 d) Utility 50(50) Utility 50 The utility associated with the certain payoff of 50 is higher than the expected utility of the lottery with the same expected payoff. Thus, with this utility function the decision maker is risk averse since the decision maker prefers the sure thing to a lottery with the same expected payoff. 15.6. You have a utility function given by U = 2I + 10√I. You are considering two job opportunities. The first pays a salary of $40,000 for sure. The other pays a base salary of $20,000, but offers the possibility of a $40,000 bonus on top of your base salary. You believe that there is a 0.50 probability that you will earn the bonus.
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
a) What is the expected salary under each offer? b) Which offer gives you the higher expected utility? c) Based on your answer to (a) and (b), are you risk averse? a) The expected salary under the first job offer is $40,000. The expected salary under the second job offer is also $40,000: 0.5($20,000) + 0.5($60,000) = $40,000. b) Your expected utility under the first offer is U 2( 40000) 10 40000 82,000
Your expected utility under the second offer is
U .5 2( 20000) 10 20000 .5 2(60000) 10 60000 81,932.
The first offer gives you the higher expected utility. c) The two offers have the same expected value. Since you prefer the certain salary to the risky salary, it follows that you are risk averse. 15.7. Consider two lotteries, A and B. With lottery A, there is a 0.90 chance that you receive a payoff of $0 and a 0.10 chance that you receive a payoff of $400. With lottery B, there is a 0.50 chance that you receive a payoff of $30 and a 0.50 chance that you receive a payoff of $50. a) Verify that these two lotteries have the same expected value but that lottery A has a bigger variance than lottery B. b) Suppose that your utility function is U = √I + 500. Compute the expected utility of each lottery. Which lottery has the higher expected utility? Why? c) Suppose that your utility function is U = I √ 500. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral, or risk loving? d) Suppose that your utility function is U = (I √ 500)2. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral, or risk loving? a)
EVA 0.90(0) 0.10(400) EVA 40 EVB 0.50(30) 0.50(50) EVB 40
Thus, both lotteries have the same expected value. VarianceA 0.90(0 40) 2 0.10(400 40) 2 VarianceA 14, 400 VarianceB 0.50(30 40) 2 0.50(50 40) 2 VarianceB 100
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
Thus, Lottery B has a smaller variance than Lottery A. b) Expected Utility A 0.90 0 500 0.10 400 500 Expected Utility A 23.13 Expected Utility B 0.50 30 500 0.50 50 500 Expected Utility B 23.24 Thus, Lottery B has the higher expected utility. In general, when two lotteries have the same expected value but different variance, a risk-averse decision maker will have a higher expected utility from the lottery with the lower variance. c)
Expected Utility A 0.90(0 500) 0.10(400 500) Expected Utility A 540 Expected Utility B 0.50(30 500) 0.50(50 500) Expected Utility B 540
With this utility function both lotteries have the same expected value and same expected utility. In general, when two lotteries have the same expected value and different variances, a riskneutral decision maker will be indifferent between the two lotteries, i.e., will have the same expected utility for both lotteries. Thus, this utility function corresponds with a risk-neutral decision maker. d) Expected Utility A 0.90(0 500) 2 0.10(400 500) 2 Expected Utility A 306, 000 Expected Utility B 0.50(30 500) 2 0.50(50 500) 2 Expected Utility B 291, 700 With this utility function the decision maker has a higher expected utility for Lottery A than for Lottery B. In general, when two lotteries have the same expected value but different variances, a risk-loving decision maker will prefer the lottery with the higher variance, Lottery A in this case. 15.8. Consider two lotteries A and B. With Lottery A, there is a 0.8 probability that you receive a payoff of $10,000 and a 0.2 chance that you receive a payoff of $4,000. With Lottery B, you will receive a payoff of $8,800 for certain. You should verify for yourself that
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
these two lotteries have the same expected value, but that Lottery A has a higher variance. For each of the utility functions below, please fill in the table below:
Utility function
Expected utility lottery A
Expected utility lottery B
Which lottery gives the highest expected utility?
9,264.91 8,800
9,380.83 8,800
8,320
7,744
Lottery A Both give the same expected utility Lottery B
Does the utility function exhibit risk aversion, risk neutrality, or risk loving? Risk aversion Risk neutrality Risk loving
15.9. Sketch the graphs of the following utility functions as I varies over the range $0 to $100. Based on these graphs, indicate whether the decision maker is risk averse, risk neutral, or risk loving: a) U = 10I – (1/8)I2 b) U = (1/8)I2 c) U = ln (I + 1) d) U = 5I a)
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
300
Utility
200 100 0 -100 0
20
40
60
80
100
-200 -300 I
Since this utility function increases at a decreasing rate, the decision maker will prefer a sure thing to a lottery with the same expected value. Thus, this utility function corresponds to a riskaverse decision maker.
Utility
b) 1400 1200 1000 800 600 400 200 0 0
20
40
60
80
100
I
Since this utility function increases at an increasing rate, the decision maker will prefer a lottery to a sure thing with the same payoff. Thus, this utility function corresponds to a risk-loving decision maker. c)
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
3
Utility
2 2 1 1 0 0
20
40
60
80
100
I
Since this utility function increases at a decreasing rate, the decision maker will prefer a sure thing to a lottery with the same expected value. Thus, this utility function corresponds to a riskaverse decision maker. d) 600
Utility
500 400 300 200 100 0 0
20
40
60
80
100
I
Since this utility function increases at a constant rate, the decision maker will be indifferent between a sure thing and a lottery with the same expected value. Thus, this utility function corresponds to a risk-neutral decision maker. 15.10. a) Write down the equation of a utility function that corresponds to a risk-neutral decision maker. (Note: there are many possible answers to this part and the next two parts.) b) Write down the equation of a utility function that corresponds
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
to a risk-averse decision maker. c) Write down the equation of a utility function that corresponds to a risk-loving decision maker. a) Examples of a risk-neutral utility function would be: U = 10I, U = 20 + 5I, or U = I/50. The key point is that the marginal utility is constant in income b) Examples of a risk-averse utility function would be U I , U = log I, or U = 1 – I-2. In all of these cases, marginal utility decreases in I. As an example, consider U = 1 – I-2. Marginal utility for this utility function is MU = 2I-3. If you graph this function, you will see that marginal utility decreases in I. c) Examples of a risk-loving utility function would be U = I2, U = 5I3, U = 2I + 3I2. In all of these cases, marginal utility increases in I. As an example, consider U = 2I + 3I2. Marginal utility for this utility function is MU = 2 + 6I. This is clearly increasing in I. 15.11. Suppose that I represents income. Your utility function is given by the formula U = 10I as long as I is less than or equal to 300. If I is greater than 300, your utility is a constant equal to 3,000. Suppose you have a choice between having an income of 300 with certainty and a lottery that makes your income equal to 400 with probability 0.5 and equal to 200 with probability 0.5. a) Sketch this utility function. b) What is the expected value of each lottery? c) Which lottery do you prefer? d) Are you risk averse, risk neutral, or risk loving? a) U
3,000
0
I 300
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
b) The expected value of each option is 300. c) Your (certain) utility under the first option is 3,000. Your expected utility under the second option is 0.5U(200) + 0.5U(400) = 0.5(2,000) + 0.5(3,000) = 2,500. Thus, you prefer the sure thing to the lottery. d) Since the sure thing and the lottery have the same expected value, but since you prefer the sure thing to the lottery, it follows that you are risk averse. 15.12. Suppose that your utility function is U = √ I. Compute the risk premium of the two lotteries described in Problem 15.7. If your utility function were U I , then the risk premium associated with Lottery A would be 0.90 0 0.10 400 40 RPA 40 RPA 2 40 RPA 4 RPA 36 The risk premium associated with Lottery B would be 0.50 30 0.50 50 40 RPB 40 RPB 6.27 40 RPB 39.36 RPB 0.64 Lottery A has a risk premium of 36 and Lottery B has a risk premium of 0.64. 15.13. Suppose you are a risk-averse decision maker with a utility function given by U(I) = 1 – 10I-2, where I denotes your monetary payoff from an investment in thousands. You are considering an investment that will give you a payoff of $10,000 (thus, I = 10) with probability 0.6 and a payoff of $5,000 (I = 5) with probability 0.4. It will cost you $8,000 to make the investment. Should you make the investment? Why or why not? If you do not make the investment, your utility is: 1 – 10(8)-2 = 0.84375 If you make the investment, your utility is: (0.6)(1 – 10(10)-2) + (0.4)(1-10(5)-2) = (0.6)(0.9) + (0.4)(0.6) = 0.78
Copyright © 2014 John Wiley & Sons, Inc.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
Since the expected utility from the investment is less than the utility from not making the investment, you should not make the investment. 15.14. You have a utility function given by U = 10 lnI where I represents the monetary payoff from an investment. You are considering making an investment which, if it pays off, will give you a payoff of $100,000, but if it fails, it will give you a payoff of $20,000. Each outcome is equally likely. What is the risk premium for this lottery? The expected payoff of this lottery is given by 0.5(100,000) + 0.5(20,000) = 60,000. The risk premium RP of for this lottery is the solution to the equation 0.5[10*ln(100,000)] + 0.5[10*ln(20,000)] = 10*ln(60,000 – RP) which is equivalent to 107.08 = 10*ln(60,000 – RP) Solving this equation tells us that RP = $15,278.64. 15.15. In the upcoming year, the income from your current job will be $90,000. There is a 0.8 chance that you will keep your job and earn this income. However, there is 0.2 chance that you will be laid off, putting you out of work for a time and forcing you to accept a lower paying job. In this case, your income is $10,000. The expected value of your income is thus $74,000. a) If your utility function has the formula 100I - 0.0001I2, determine the risk premium associated with this lottery. b) Provide an interpretation of the risk premium in this particular example. (a) The risk premium RP solves the following equation 0.8*U(90,000) + 0.2U(10,000) = U(74,000 – RP) Now: U(90,000) = 8,190,000 U(10,000) = 990,000 0.8*U(90,000) + 0.2U(10,000) = 6.750,000 U(74,000 – RP) = 100(74000 – RP) -0.0001(74,000 – RP)2 Thus, RP is the solution to: 6,750,000 = 100(74000 – RP) -0.0001(74,000 – RP)2 This is a quadratic equation that has two solutions. It can be verified that one solution is negative and one is positive. We can ignore the negative solution (which doesn’t make economic sense). The positive solution is (approximately), RP = 1,200 (b) In this problem, the RP of $1,200 can be interpreted as the maximum amount that you would be willing to pay to receive unemployment insurance that fully replaces your income loss if you are laid off.
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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual
15.16. Consider a household that possesses $100,000 worth of valuables (computers, stereo equipment, jewelry, and so forth). This household faces a 0.10 probability of a burglary. If a burglary were to occur, the household would have to spend $20,000 to replace the stolen items. Suppose it can buy an insurance policy for $500 that would fully reimburse it for the amount of the loss. a) Should the household buy this insurance policy? b) Should it buy the insurance policy if it cost $1,500? $3,000? c) What is the most the household would be willing to pay for this insurance policy? How does your answer relate to the concept of risk premium discussed in the text? a) If you remain uninsured, you face a lottery in which you have 10% chance of $80,000 in valuables and a 90% chance of $100,000 in valuables. The expected value of valuables is thus $98,000. If you purchase the insurance policy for $500, then with no burglary you have 100, 000 500 $99,500 and with a burglary you have 100, 000 500 20, 000 20, 000 $99,500 . The expected value if you purchase the policy is therefore $99,500. Since the expected value at year end with insurance exceeds the expected value at year end without insurance, you should purchase the insurance policy for $500. b) We can set up a table that shows the possible outcomes. The values in the table represent the value of valuables at year end depending on the corresponding row and column situations. Here is the $1500 case. Burglary No Burglary Expected Value No Insurance $80,000 $100,000 $98,000 Insurance $98,500 $98,500 $98,500 Probability 0.10 0.90 If the policy costs $1,500, you are $500 better off with the policy. Here is the $3,000 case. No Insurance Insurance Probability
Burglary $80,000 $97,000 0.10
No Burglary $100,000 $97,000 0.90
Expected Value $98,000 $97,000
If the policy costs $3,000, you are $1,000 better off without the policy. c) Since without the policy you would have an expected value $2,000 less than the value of the valuables, the most you would be willing to pay for an insurance policy that fully reimburses for loss is $2,000. 15.17. If you remain healthy, you expect to earn an income of $100,000. If, by contrast, you become disabled, you will only be able to work part time, and your average income will drop to $20,000. Suppose that you believe that there is a 5 percent chance that you could Copyright © 2014 John Wiley & Sons, Inc.
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become disabled. Furthermore, your utility function is U = √ I. What is the most that you would be willing to pay for an insurance policy that fully insures you in the event that you are disabled? If you do not purchase insurance, your expected utility is .95 100,000 0.05
20,000 307.49
If you do purchase insurance at a price P, your expected utility is 100,000 P . The highest price that you would be willing to pay is P such that: 100,000 P 307.49, or 100,000 – P = 94,548, which implies that P = $5,451. Thus, the most you’d be willing to pay for this insurance policy is $5,451. 15.18. You are a risk-averse decision maker with a utility function U(I) = 1-3200I-2, where I denotes your income expressed in thousands. Your income is $100,000 (thus, I =100). However, there is a 0.2 chance that you will have an accident that results in a loss of $20,000. Now, suppose you have the opportunity to purchase an insurance policy that fully insures you against this loss (i.e., that pays you $20,000 in the event that you incur the loss). What is the highest premium that you would be willing to pay for this insurance policy? If you do not buy insurance, your expected utility is: 0.2[1 – 3200(100 – 20)-2] + 0.8[1 – 3200(100)-2] = (0.2)(0.5) + (0.68)(0.5) = 0.644 If you do buy insurance at price P, you have no risk and your utility is 1 – 3200(100 – P)-2 The most you would be willing to pay for insurance would be just a shade less than the P that makes your utility with insurance equal to your expected utility with no insurance. In terms of equations: 1 – 3200(100 – P)-2 = 0.644 3200(100 – P)-2 = 0.356 (100 – P)-2 = 0.356/3200 (100 – P)2 = 3200/0.356 100 – P = [3200/0.356](1/2) 100 – P = 94.81
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P = 5.19. Thus, the most that you would be willing to pay for the insurance, would be a shade less than $5.19 per thousand dollars of coverage. 15.19. You are a relatively safe driver. The probability that you will have an accident is only 1 percent. If you do have an accident, the cost of repairs and alternative transportation would reduce your disposable income from $120,000 to $60,000. Auto collision insurance that will fully insure you against your loss is being sold at a price of $0.10 for every $1 of coverage. Finally, suppose that your utility function is U = √I. You are considering two alternatives: buying a policy with a $1,000 deductible that essentially provides just $59,000 worth of coverage, or buying a policy that fully insures you against damage. The price of the first policy is $5,900. The price of the second policy is $6,000. Which policy do you prefer? Your expected utility if you buy the first policy is 0.01 120,000 60,000 59,000 5900 0.99 120,000 5900 337.77 .
If you buy the second policy, your expected utility is 0.01 120,000 60,000 60,000 6000 0.99 120,000 6000 337.64
Your expected utility is higher when you buy the first policy. 15.20. Consider a market of risk-averse decision makers, each with a utility function U = √I. Each decision maker has an income of $90,000, but faces the possibility of a catastrophic loss of $50,000 in income. Each decision maker can purchase an insurance policy that fully compensates her for her loss. This insurance policy has a cost of $5,900. Suppose each decision maker potentially has a different probability q of experiencing the loss. a) What is the smallest value of q so that a decision maker purchases insurance? b) What would happen to this smallest value of q if the insurance company were to raise the insurance premium from $5,900 to $27,500? a)
If an individual purchases insurance, her (certain) utility is 90,000 5,900 290.
If an individual does not purchase insurance, her expected utility is q 90,000 50,000 (1 q ) 90,000 200q 300(1 q ) 300 100q
An individual will purchase insurance if
290 300 100q, or q 0.10.
In other words, individuals that are 90 percent or more certain that they will not experience the loss will not purchase insurance. b) If the insurance premium is increased to $27,500, an individual who purchases insurance will achieve a certain utility of
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90,000 27,500 250.
The individual still receives an expected utility of 300 – 100q if it does not purchase insurance. Thus, an individual will purchase insurance if 250 300 100 q, or q 0.50.
15.21. An insurance company is considering offering a policy to railroads that will insure a railroad against damage or deaths due to the spillage of hazardous chemicals from freight cars. Different railroads face difference risks from hazardous spills. For example, railroads operating on relatively new tracks face less risk than railroads with relatively older right of ways. (This is because a key cause of chemical spills is derailment of the train, and derailments are more likely on older, poorer tracks.) Discuss the difficulties that the insurance company might face in offering this type of policy; that is, why might it be difficult for the insurance company to make a profit from this type of policy? There are two potential problems that might make it difficult for the insurance company to make a profit. The first is the adverse selection problem. As noted in the problem, not all railroads are equally risky, but the insurance company may find it difficult to discern a railroad’s risk characteristics. For example, the railroad will probably be much better informed about the condition of its track than the insurance company. This makes it difficult to tailor the terms of the insurance policy to the risk characteristics of the railroad. This is a problem because railroads whose track is in good conditions may choose to go without insurance. They may choose to “self insure” by taking precautions against derailment (which might not be very costly, since derailment risk for these railroads is low anyway), or they may simply do without insurance altogether. This means that the market for this insurance coverage might be primarily made up of railroads whose track is in poor condition and whose risk of derailment is correspondingly higher. The insurance company might not be able to make much profit from this pool of high-risk railroads. To make matters worse, if the insurance company tries to raise price to increase profit, the railroad that are most likely to “drop out” of the market in response to a now-more expensive insurance policy, are those which (among the pool of high-risk railroads) have the lowest risks. Hence, the insurance company makes things worse by trying to raise price! The second problem is moral hazard. Once a railroad is insured against the risk of a chemical spill, it may become less careful in preventing such skills. Train operators may operate the train faster, maybe even violating speed limits, thus increasing the risk of a derailment. The company may skimp on investments in new track, new track ties, or ballast, also making a derailment more likely. With less care exerted by the railroad and the risk of derailment increased, the insurance company may find it difficult to make a profit on this particular insurance product. 15.22. A firm is considering launching a new product. Launching the product will require an investment of $10 million (including marketing expenses and the costs of new facilities). The launch is risky because demand could either turn out to be low or high. If the firm does
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not launch the product, its payoff is 0. Here are its possible payoffs if it launches the product.
a) Draw a decision tree showing the decisions that the company can make and the payoffs from following those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the firm acts as a risk-neutral decision maker, what action should it choose? What is the expected payoff associated with this action? Demand is 0.5high
$20 million
Launch 0.5
Do not launch
Demand is low –$10 million
$0
If the firm launches the product, its expected payoff is 0.5(20) + 0.5(–10) = $5 million. Because this number is bigger than the payoff from not launching, a risk-neutral firm should launch the new product. 15.23. A large defense contractor is considering making a specialized investment in a facility to make helicopters. The firm currently has a contract with the government, which, over the lifetime of the contract, is worth $100 million to the firm. It is considering building a new production plant for these helicopters; doing so will reduce the production costs to the company, increasing the value of the contract from $100 million to $200 million. The cost of the plant will be $60 million. However, there is the possibility that the government will cancel the contract. If that happens, the value of the contract will fall to zero. The problem (from the company’s point of view) is that it will only find out about the cancellation after it completes the new plant. At this point, it appears that the probability that the government will cancel the contract is 0.45. a) Draw a decision tree reflecting the decisions the firm can make and the payoffs from those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the firm is a risk-neutral decision maker, should the firm build a new plant? What is the expected value associated with the optimal decision?
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c) Suppose instead of finding out about contract cancellation after it builds the plant, the firm finds out about cancellation before it builds the plant. Draw a new decision tree corresponding to this new sequence of decisions and events. Again assuming that the firm is a risk-neutral decision maker, should the firm build the new plant? a & b) The decision tree for this situation is shown below. The chance nodes are circles, and the decision node is square.
Build
Do not build
Contract is cancelled (probability = 0.45)
$0 - $60 million = -$60 million
Contract is not cancelled (probability = 0.55)
$200 - $60 million = $140 million
Contract is cancelled (probability = 0.45) Contract is not cancelled (probability = 0.55)
$0
$100 million
Since the decision maker is risk neutral, we can evaluate payoffs using expected values. The expected value if you build the plant is: (0.45)(-$60 million) + (0.55)($140 million) = $50 million The expected value if you do not build the plant: (0.45)($0) + (0.55)($100 million) = $55 million. Not building the plant is the best course of action. c) The answer to question of whether the firm should build the plant is: it depends! The decision tree for the revised sequence of decisions and event is shown below.
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Build
$0 - $60 million = -$60 million
Contract is cancelled (probability = 0.45)
Do not build Build
Contract is not cancelled (probability = 0.55)
$0
$200 - $60 million = $140 million
Do not build
$100 million
We see from the tree that if the contract is cancelled, the best action is not to build the plant, which results in a payoff of $0. However, if the contract is not cancelled, it’s better to build the plant than to not build the plant. Hence, the decision to build the plant depends on the circumstances the firm faces, in particular whether the contract is cancelled. To finish the decision tree analysis, the picture below shows the folded back tree. There are no further decisions to be evaluated so all that needs to be done is to compute the expected vaue associated with this situation. That expected value is: (0.45)($0) + (0.55)($140) = $77 million. Note that the firm’s expected value when the plant-building decision is made after the status of the contract is know is bigger ($77 million versus $55 million) than its value when the plant decision must be made when the contract status is still uncertain. This difference reflects the value of having perfect information about the status of the contract in the second situation. Contract is cancelled (probability = 0.45)
Do not build
$0
Contract is not cancelled (probability = 0.55) $200 - $60 million = $140 million
Folded back decision tree
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15.24. A small biotechnology company has developed a burn treatment that has commercial potential. The company has to decide whether to produce the new compound itself or sell the rights to the compound to a large drug company. The payoffs from each of these courses of action depend on whether the treatment is approved by the Food and Drug Administration (FDA), the regulatory body in the United States that approves all new drug treatments. (The FDA bases its decision on the outcome of tests of the drug’s effectiveness on human subjects.) The company must make its decision before the FDA decides. Here are the payoffs the drug company can expect to get under the two options it faces:
a) Draw a decision tree showing the decisions that the company can make and the payoffs from following those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the biotechnology company acts as a risk-neutral decision maker, what action should it choose? What is the expected payoff associated with this action? a)
A is a decision node; B and C are chance nodes. Approves Sell rights
B Does not approve Approves
A Produce Self
C Does not approve
10 2 50 –10
b) The expected payoff for “Sell rights” is 0.20(10) + 0.80(2) = 3.60. The expected payoff for “Produce yourself” is 0.20(50) + 0.80(–10) = 2.0. Therefore, the risk-neutral company should sell the rights, for an expected payoff of 3.60. 15.25. Consider the same problem as in Problem 15.24, but suppose that the biotech company can conduct its own test—at no cost—that will reveal whether the new drug will be approved by the FDA. What is the biotech company’s VPI? If the firm does not conduct a test, it should launch the new product, resulting in an expected value of $5 million. If the outcome of the test marketing is that demand is high, the firm should launch the new product, and in so doing it receives a payoff of $20 million.
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If the outcome of the test marketing is that demand is low, the firm should not launch the new product, and in so doing, it receives a payoff of $0. The expected value from conducting the test is thus: 0.5($20 million) + 0.5(0) = $10 million. VPI = expected value from conducting costless test – expected value from not conducting the test = $10 million – $5 million = $5 million. If the test indicates the FDA will approve, the company will choose “Produce yourself” and earn a payoff of 50. If the test indicates the FDA will not approve, then the company will choose “Sell the rights” and earn 2. The expected payoff from conducting the costless test is therefore 0.20(50) + 0.80(2) = 11.60. The VPI is the difference between the expected payoff with the test and the expected payoff without the test. Thus, VPI = 11.60 – 3.60 = 8.00. 15.26. You are bidding against one other bidder in a first-price sealed-bid auction with private values. You believe that the other bidder’s valuation is equally likely to lie anywhere in the interval between $0 and $500. Your own valuation is $200. Suppose you expect your rival to submit a bid that is exactly one half of its valuation. Thus, you believe that your rival’s bids are equally likely to fall anywhere between 0 and $250. Given this, if you submit a bid of Q, the probability that you win the auction is the probability that your bid Q will exceed your rival’s bid. It turns out that this probability is equal to Q/250. (Don’t worry about where this formula comes from, but you probably should plug in several different values of Q to convince yourself that this makes sense.) Your profit from winning the auction is profit = (200 - bid) x probability of winning. Show that your profit maximizing strategy is bidding half of your valuation. From the given information, the profit from winning the auction is (assuming you bid Q ) Q (200 Q) 250 (0.80 0.004Q)Q At the optimal bid marginal profit equals zero. Thus, at the optimum, the bid must satisfy 0.80 0.008Q 0 0.008Q 0.80 Q 100 The optimal bid is therefore 100, which is equal to one-half of your true valuation, 200. Thus, a strategy of bidding one-half of your valuation is a Nash equilibrium; it is the best you can do given the other player’s strategy.
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