Universidad Carlos III Microeconomics CONSUMER THEORY:UNCERTAINTY
1. An individual su¤ers a severe illness. The options are to follow a medication or to have surgery. The medication permits to eliminate reasonably well the majority of symptoms but he will have a probability of 2/3 to live 20 more years and a probability of 1/3 to live only 10 more years. Surgery eliminates with probability 0:7 the illness and allows him to live 30 more years. However, with probability 0:3 he will die during the surgery. The patient’s preferences are represented by the Bernoulli utility function u(x) =p
x, where x represents the additional years the patient is expected to live. Which decision will he take?
2. A student has just graduated. He has just received an inheritance of 4 millions of euros and is considering whether to invest 2 millions of euros in a start up business. If the business is successful, he expects a gross pro…t of 6 millions of euros, but if it fails he will lose the investment. The probability of success is p = 1=2.
a. Supposing that the student’s preferences are represented by the Bernoulli utility function u(x) = x, would he try this investment? What if u(x) = x2? And if u(x) =p
x?
b. Suppose now the student’s preferences are represented by the utility function u(x) = x2: A study that costs a millions of euros predicts with certainty if the investment will be lucky or not. Should the student buy the mentioned study if a = 1? And if a = 0:5?
c. Suppose …nally that the student’s preferences are represented by the utility function u(x) = px: Let’s suppose that, to a this student; it is o¤ered the possibility to implement the the previous investment if he receives a tax break of b millions of euros. Should the student accept this tax break if b = 1? And if the tax break is b 6= 1?
3. The NBA team Memphis Grizzlies has the objective of playing the playo¤s this season. The team’s managers are considering alternative strategies to achieve this goal. As the club is in a …nancially stable situation –its estimated value is $2500 millions –they can a¤ord to hire Kobe Bryant. If they do so, the probability of classifying for the playo¤s is 0.6, while without Kobe this probability is only 0.1. The cost of hiring Kobe is $196 millions. If the team hires Kobe and makes it to the playo¤s, then it expect to have a chance to hire Shaquille O’Neil for
$228 millions. With Shaq, the Grizzlies would have a probability of winning the …nal of 0.9, while without him this probability is only of 0.3. If the team did not hire Kobe but classi…ed for the playo¤s nevertheless, then they do not they will be able to hire anybody before playing the matches, and the probability of winning the …nal under these circumstances is only 0.01.
Classifying for the playo¤s would increase the teams value by $525 millions for the club, and winning the …nal would further increase the value by an additional $420 millions. The teams preferences are represented by the Bernoulli utility function u(x) =p
x:
a. Describe the problem using a decision tree.
b. Suppose that the team have hired Kobe and has classify for the playo¤s, and is now considering whether to hire Shaq or not. What should they do it?
c. Determine whether the Grizzlies should contract Kobe, or Kobe-and-Shaq, or none.
3. Pedro Banderas has a wealth of 100 thousand euros and is considering whether to produce a movie whose budget is 250 thousand euros. A …lm company is willing to …nance the movie but wants Pedro to share some of the risk (and pro…ts); speci…cally it is willing to …nance 80% of the budget. Assuming that the distributors like the movie, Pedro expects the movie to generate box o¢ ce revenue of 250 thousand euros if the reviews are bad, and as much as 1,5 million euros it the reviews are good. It is known that distributors like 8 out of 10 movies that are produced, and that 1 out of 10 movies that are distributed get good reviews. Pedro’s preferences are represented by the Bernoulli utility function u(x) =p
x.
a. Represent the decision problem and determine whether or not Pedro should produce the movie.
b. Determine whether Pedro may be willing to …nance 40% (instead of 20%) of the movies’
budget.
4. You have to guess the result of tossing a coin (heads or tails). If you win (you guess right), you get 10 euros and you have the choice of playing again for a maximum of three times (in total). If you loose, you give all your earnings back and you cannot play again. At the end of the game you have to pay 2 euros for each bet. Represent this decision problem using a decision tree. Determine the decision that maximizes expected utility for u(x) = x.
5. An investment may lead to the following pro…ts (in millions of euros) with the following corresponding probabilities:
Pro…ts -20 -10 0 20 Pr 0.2 0.2 0.4 0.2 :
If the result of the investment is 20 millions, then the investor has the possibility of making a second investment that can lead to the following pro…ts with the following corresponding probabilities:
Pro…ts 50 -10 Pr 0.8 0.2 :
The investor preferences are represented by a Bernoulli utility function u(x) = p x that satis…es the following:
x -20 -10 0 10 20 30 40 45 50 60 70
u (x) 0 0.3 0.5 0.65 0.75 0.825 0.9 0.93 0.95 0.975 1
a. Draw the decision tree corresponding to this problem taking into account that the pro…ts are 0 if the …rst investment is not made.
b. Determine whether, in accordance with the criterion of maximizing the expected utility, the investor should make each of the two investments.
c. What is the certainty equivalent and the risk premium for the second investment?
6. The oil company Tibitrol has bought some deserted land in Monegros. The company’s geol- ogist estimates that the probability that they will …nd oil in this land is 0.2. The drilling of the land in order to check whether or not it really has oil costs 100 millions of euros. If they
…nd oil, then the company will make revenues of 300 millions of euros. If they do not …nd oil, then the drilling will be completely useless. The company has to decide whether or not it will do the drilling. If the company is risk neutral what will it decide to do in order to maximize its expected utility? And if the company is risk averse?
7. A consumer has a house with a 250,000 euros value and her preferences are represented by the utility function u (x) =p
x; where x is the wealth of the consumer at the end of the year.
The probability that the house will be totally destroyed by an accidental …re (in which case it will loose all its value) is 0.01.
a. Would she accept to pay 3,000 euros for a full insurance of his house? What is the maximum premium she is willing to pay for this insurance? What is the relationship between these premium, the certainty equivalent and the risk premium of the lottery she faces?
b. Assuming that the risk of …re is the same for all the consumers (and it is independent among them), is 3000 euros an acceptable insurance premium for a risk neutral insurance company? What is the minimum premium the company is willing to o¤er?
8. The owner of a shop (which is worth 64 million euros) thinks that his shop will be destroyed by a …re during the year with probability 1%. The preferences of the shop’s owner are represented by the utility function u (x) = p
x;where x represents his wealth at the end of the year.
a. Calculate the expected utility of the lottery faced by the shop’s owner and its certainty equivalent. Would he agree to sell his shop for 60 millions of euros? What about for 63 millions?
b. An insurance company o¤ers an annual contract that covers all the risk for 1 million of euros. Should the shop’s owner accept this contract?
c. Assume now that the shop’s owner has (in addition to his shop) 1 million of euros in cash. A …rm proposes him to rent a …re team that will reduce the probability of …re to 0.5%. Should the shop’s owner pay 50,000 euros for renting this …re team? What is the maximum annual amount that he should pay in order to rent this team?
9. An individual whose preferences are represented by the Bernoulli utility function u(x) = p x has just insured his new motorcycle — whose value is 2,500 euro— against theft. Motorcycle theft is very common in the town where he lives, and when a motorcycle is stolen, it never appears afterwards. Knowing that if the price of the insurance policy increases then the individual will not purchase insurance, and that its actual price is 99 euro, you are asked to:
a. Calculate the certainty equivalent of the lottery faced by the individual.
b. If this individual received a prize of 3,600 euro, would this individual still buy insurance?
10. A participant in a television show who is risk neutral has answered correctly to all the questions asked so far. In the last question there remain two possible answers but he is completely indecisive (that is, he thinks that the probabilities of each answer of being correct are equal).
Until now his accumulated gains are 361 euros. If he abandons he wins these gains. If he decides to answer and is correct he obtains in addition 315 euros, …nishing with a total of 676 euros. But if his answer is wrong, he loses 261 euros, leaving with only 100 euros.
a. Should he continue playing if he is risk neutral? Should he continue playing when his utility function is given by u(x) =p
x?
b. What is the certainty equivalent of the lottery when his Bernoulli utility function is u(x) = px? What is the risk premium?
11. In the market of car insurance, there are two kinds of drivers, the good ones (they have one accident per year with probability 0.1 and no accident with probability 0.9) and the bad ones (they have one accident per year with probability 0.1, two accident with probability 0.05 and no accident with probability 0.85). The costs of repairing a car are on average 2,000 euros.
The proportion of good and to bad drivers is 2 to 1.
a. Assume that the insurance companies are risk neutral and they cannot distinguish between good and bad drivers. What is the minimum price that these companies would be willing to o¤er in order to cover the risk of an accident?
b. Imagine that the preferences of the drivers are represented by the utility function u (x) = px;and that their initial wealth is 5,000 euros. Which type of drivers (good and/or bad) will subscribe to an insurance policy with the minimum price determined in part (a)?
12. A salesman preferences are represented by the Bernoulli utility function u(x) = x. He makes sales by phone, and received a phone list of potential customers. Each day he can make a limited number of phone calls. Each phone call costs one euro and for each successful sale he receives 20 euros for commission. According to his experience, he manages to speak to the right person in 3 out of 10 phone calls. Moreover, when he manages to speak to the right person, he succeeds in making the sale in 2 out of 10 cases.
a. Draw the decision tree. Which is the expected monetary value of each phone call?
b. The phone company also o¤ers a service called “person-to-person”. With this service you only pay the cost of the phone call p only if you reach the person you want. Which is the maximum price p that the salesman is willing to pay per phone call in this new service?
13. A risk neutral person needs to put a mortgage on one of his buildings in order to get 200,000 euros. He has to pay back this amount in 2 annual payments of 100,000 euros, each one with the corresponding interest rate. The mortgage credits among which he could choose are:
Fixed interest rate: 10% per year.
Interest rate 9% in the …rst year which can increase to 14%, decrease to 8% or remain the same in the second year.
Interest rate 7% in the …rst year which can increase to 20%, decrease to 6% or remain the same in the second year.
a. Determine the decision which maximizes the expected pro…ts knowing that the interest rate increases with probability 0.6 and decreases with probability 0.2.
b. How much is this person willing to pay in order to learn whether the interest rate will increase, decrease or remain the same?
15. A consumer must choose between buying an apartment in Madrid or a house in the suburb.
Both choices would cost him 120,000 euros. He is indi¤erent between the two options, except for his expectation regarding revaluation. If the housing prices keep on increasing (event E1), the price of the apartment will reach 140,000 euros, while the price of the house will reach 340,000 euros. The probability that this will happen is 0.3. If the opposite thing (decrease in the housing prices) happens (event E2), the price of the apartment will be 70,000 euros and the price of the house 20,000. The preferences of the consumer are represented by the utility function u(x) = p
x, where x is the wealth expressed in euros. The initial wealth of the consumer is 140,000 euros.
a. Represent the decision problem and determine whether the consumer should buy the house or the apartment.
b. Should he pay 20,000 euros in order to learn whether the housing prices will decrease or increase?
16. The introduction of a new product in the market takes includes three stages: Design, Exper- imentation, and Production. 7 out of 10 products do not pass the design stage. From those that do pass it, only 10% pass the experimentation stage and are being produced. Only 1 out of 5 products produced has success in the market. For each new product the costs of each stage are 100,000, 20,000, and 200,000 euros, respectively. The expected pro…ts from a product that passed successfully the three stages are 60 millions of euros.
a. Which is the expected value of constructing a new product?
b. For 15,000 euros a consultant can predetermine (without any uncertainty) whether or not a product that has already passed the design stage will pass the experimentation stage.
What is the value of the consultant’s services, assuming the entrepreneur is risk neutral?
17. The marketing chief of a big computer producer has to decide whether to launch a new campaign before (d1) or after the month of May (d2). If he does before, he will manage to obtain 100 millions Euros of sales. If he does after, there is a risk that its competitor launches its own campaign before (C), which will occur with probability 0.4. Moreover, the sales also depend on the predictions of the state of the economy, which can be good (A) with probability 0.5, stable (E) with probability 0.3, and bad (R). If the economy is good, and the competitor has not launched its campaign, sales can reach 150 millions Euros, and if its competitor did launch its campaign, sales would reach a value of 120 millions Euros. If the economy is stable, sales would reach 90 millions of Euros if the competitor launches its campaign and 110 millions if it does not. Finally, when the economy is bad, and if the competitor launches its campaign, sales will reach 70 millions Euros while they would go up to 80 millions Euros if the competitor does not. Assuming that the producer is risk-neutral, what is the best decision? How much would be the marketing chief ready to pay in order to know with certainty all the uncertain variables of the problem? How much would he be willing to give to an industrial spy who would tell him with certainty whether the competitive …rm will launch its own campaign or not?
18. A professional has an annual wage of 250; 000 and his income tax rate is 50%. He considers whether he should declare his full income, declaring half of his income, or declaring nothing
at all. It is known that the probability of a Hacienda inspection is 0:1. If the inspection detects that he misdeclared his income, he will have to pay the amount of the missing tax plus the same amount as a fee. His preferences are represented by the Bernoulli utility function u(r) =p
r, where r is his income. (For negative amounts of r it is u(r) = 2p r:) a. Draw the decision tree that corresponds to this problem.
b. Suppose now that he decides not to declare anything and that after doing so he gets afraid of a possible inspection and asks a friend to help him. In such a situation, he has to pay m euros in order to be sure that he will not have any problems with the inspection. How much is he willing to pay for the service of his friend (m)?
c. Would your answer in part (a) change if the utility function was given by u(r) =p
r? And if it was given by u(r) = 2r?
d. Suppose now that Hacienda has already decided (before the professional makes the dec- laration) the list of persons that will be inspected. His friend o¤ers to check whether his name is on the list for 20,000 euros. Will the professional accept? Draw the condi- tions for …nding out the maximum amount that the professional is willing to pay for this information.
19. In a region there are …ve risk neutral contractors who regularly go to auctions of electricity projects. The contract for the project is given to the one that o¤ers the lowest price. Suppose that you are the owner of one of companies that operates as an electricity contractor. The name of your company is Los Muhonestos. Today you have a work lunch with the other contractors of your region. One knows that in only 10% of these meetings people talk about the “allocation” of the electricity contracts - that is, about which contractor will make the lowest o¤er for the future contracts (There are only 4 more contractors in the region and the decision about who is going to be the contractor that will make the lowest o¤er is decided through a lottery - all the contractors have the same probability of winning this lottery).
Winning the contract (i.e. being the contractor which makes the lowest o¤er) corresponds to winning 100,000 euros in average. Alternatively, you can decide not to go to the work lunch and to spend the afternoon playing golf in the local golf club. One knows that there is a 50%
probability of meeting someone important in the club. The average bene…ts of such type of contacts are equal to 6,000 euros.
a. Determine which act (going to the work lunch or playing golf) will maximize your expected pro…ts.
b. Which is the value of knowing in advance whether or not they will talk about the “alloca- tion” of the electricity contracts during the lunch?
c. Ralph Sonrisas (a business friend) has o¤ered to provide you with precise information about the topics that will be discussed during the work lunch (this way you will be able to decide whether or not to go). Moreover, he guarantees that, in the case that they will talk about the contract, you will be the one wining the contract. How much are you willing to pay Ralph for his services?
