1.    Consider the market for cigarettes. Suppose that the tobacco industry (i.e. the big tobacco          companies) conducts research on the health benefits of smoking, and to their surprise they find that smoking actually causes cancer.

a.    Using a generic model of supply and demand, illustrate and discuss how the market for cigarettes wo uld likely be affected if this information became widely available.

This information would affect the tastes and preferences of consumers and their expectations of the costs and benefits of smoking. It would make individual consumers less willing to smoke because of  the negative health consequences and, if able, consumers would start reducing their demand. We     would see a leftward shift in the demand curve.

This would result in lower prices and a lower number of cigarettes being sold, as illustrated in the following graph:

Q

b.   Would the tobacco industry want to conceal or reveal that information? Explain your answer.

Absolutely they would want to conceal this information. Recall that profits are given by: Profits = Price * Quantity – Costs = (Price – average total cost)*Quantity

The tobacco companies would likely see revenue falling much faster than costs, and as such their  profits would hurt (lower profit per unit and fewer units sold). They would have every incentive to conceal this information, even though it would be quite relevant to consumers.

2.    Emily faces a gamble, namely, her risky income next year, which may be $0, $100, or $289, where all three possibilities are equally likely. She has (expected) utility function u (m) = √m  where m is the actual income that is realized.

a.    Calculate her expected utility from this gamble.

Because there are three possible outcomes, one and only one of those outcomes will occur (i.e. it is a collectively exhaustive and mutually exclusive set), we know that the sum of the                 probabilities must equal 1, implying the possibility of each outcome equals 1.

EU(Job) = (  ) √0 + (  ) √100 + (  ) √289 =  +  = 9

b.    Calculate her certainty equivalent and risk premium for this gamble.                                      

The Certainty Equivalent of the Job is such that U(CE) = EU(Job). This implies that √CE = 9, or rather that CE = 81.

The Risk Premium from the Job is defined as rpB = Ew(Job)  − CE  where Ew(Job) = ( ) 0 +

c.    Characterize her preferences towards risk using one of the approac hes discussed in lecture last week.

We can characterize our consumer’s preferences towards risk with several equivalent arguments. First, we could recognize that the risk premium is positive, meaning that our consumer would be willing to    pay 48.7 of her expected wealth to eliminate the uncertainty. Therefore, she prefers certainty to            uncertainty and is risk averse.

Alternatively, we could argue that the utility of her expected wealth, (389/3 = 11.4, is greater than   her expected utility, 9. Therefore, she would prefer to have a certain asset over an uncertain asset that generates the same expected wealth.

3.    Jack has two options next year, Job A and Job B. The earnings in the two jobs differ and are         random. In Job A, earnings will be $25 with probability 1/2, $36 with probability 1/6, or $81 with probability 1/3. In Job B, earnings will be $36 with probability 2/3 or $64 with probability 1/3.

a.    If Jack has preferences over earnings given by u(e) = √e, where e denotes actual earnings, which job should he take and why?

We compare his expected utilities from the two jobs:

EU(Job A) = (  ) √25 + (  ) √36 + (  ) √81 = 6.5

and

EU(Job B) = (  ) √36 + (  ) √64 = 6.66

So, the expected utility from Job B is higher so he should take Job B.

b.    Calculate the Certainty Equivalents and risk premiums for Jack for both Job A and Job B.

The Certainty Equivalent of Job A is such that U(CEA) = EU(Job A). This implies that (CEA = 6.5, or rather that CEA = 42.25. The expected wealth is given by: Ew(Job A) = () 25 +      () 36 + () 81 = 45.5, so we have the risk premium of Job A equal to 45.5-42.25=3.25.

The Risk Premium from Job B is defined as rpB = Ew(Job B) − CEB where CEB is calculated as U(CEB) = EU(Job B), implying that the CE is 6.66^2=44.36. As the Ew(Job B) = () 36 +     () 64 = 408/9, so we have the risk premium of Job B equal to 0.97.

4.   You are the manager of a local band that has developed a following around the country. You are   planning the band’s next tour. Some of the band members want to keep the tour to a US tour as   you’ve done in the past, while others want to take the tour international (which you’ve never       done before). Because of past experience, you know that if you go on the US tour you will profit   $50,000. However, if you go international, you are unsure about the size of the crowd so you are  uncertain about your profits: with probability one half you yield high profits of $100,000, yet with equal probabilities you get either middling profits of $40,000 or profits of $0 because a very low   turnout means you just barely break even.

a)    If all that you care about is the expected profits from the tour, what advice to you give the band and why? Should they take the tour international?

Expected profits should be $60K for the international tour, so you should go on the tour because this is more than the expected profits of the US tour which are $50K!

E冗(international) =  ∗ $100K +  ∗ $40K +  ∗ $0K = $60K

b)   Suppose now that you have the option of paying a market research company for a perfect          forecast of the profitability of an international tour prior to deciding upon which type of tour to undertake. How much would you be willing to pay for such a forecast, i.e. what is the value of   that additional information? A good way to start is with a decision tree!

To see this, recognize that your first decision is whether or not to buy the information. If you don’t,         you’ll behave according to part (a) above and earn an expected profit of $60K. If you do buy the               information, it tells you with probabilities ½, ¼, and ¼ whether you would make profits of $100K, $40K,  or $0K. Knowing what the international tour would bring in allows you to make a better decision in each possible state of the world – once you have the information you’ve eliminated the uncertainty about      profits from an international tour! So, if you know you’ll make $100K you’ll go on the international tour. But, if you know you’ll only make $40K or $0K you’ll decide to stay in the US and earn the $50K. So,         expected profits with the information is given by:

E冗(with info) =  ∗ $100K +  ∗ $50K +  ∗ $50K = $75K

So, what is the value of the information? The difference of expected profits with and without info ::

Value of Info = E几(with info) − E几(without info) = $75K − $60K = $15K

No Info

$100K

Info

5.   Johnathan has $1 million in savings and owns a house. With a 0.1% probability, the house burns

down and costs $800,000 to rebuild. He can buy insurance at a cost of $1.20 per $1,000 of

coverage (in other words, he pays $0.0012 for each dollar of insurance). Johnathan’s preferences

over realized wealth are given by u(m) = √m . (Hint: this implies that Jonathan’s marginal utility

is given by: MU(m) =  m− 1/2)

a)   What is Jonathan’s expected utility?

EU(Job) = (0.001)(m with fire + (0.999)(m without fire

Now, in the absence of any fire, we have:

m with fire = 1,000,000 − 800,000 = 200,000

m without fire = 1,000,000

So

EU(Job) = (0.001)(200,000 + (0.999)(1,000,000 = 0.4472 + 999 = 999.4472

b)   What is Jonathan’s certainty equivalent?

The Certainty Equivalent of the Job is such that U(CE) = EU(Job). This implies that √CE = 999.4472, or rather that CE = 998,894.7.

c)    What is Jonathan’s optimal amount of insurance to purchase?

Well, the solution here is similar to the problem above. A good place to start is to write out the                  conceptual expected utility function that you are trying to maximize. Start with the source of uncertainty – whether or not there is a fire. If there is a 0.1% probability of a fire, then there must be a 99.9% probability that there is not since the two states of the world form a mutually exclusive and collectively exhaustive set. The expected utility is then the weighted sum of the utility Jonathan would receive in    those two states of the world, where the weights are the probabilities of the states occurring:

EU = 0.999U(wealth with no fiTe) + 0.001U(wealth with fiTe) = 0.999(mnf+ 0.001(mf

With our objective function clear in mind, the next piece is to write an expression fo r wealth in those       two states of the world. When conducting decision-making under uncertainty, we have a choice to make if we can trade off our wealth between the two potential future states. Insurance provides just such a     mechanism! If we denote K as the number of units of insurance we purchase, then we have:

mnf = 1,000,000 − .0012K

mf = 1,000,000 − 800,000 − .0012K + K = 200,000 + 0.9988K

Recall that each unit of insurance costs $1.2 to purchase, so the premium of 1.2K has to be paid no matter what. And in the event of the fire, the insurance company pays out $1,000 for each unit of   insurance purchased, so the agent receives an extra 1,000K.

We will use the expression derived in class to solve this problem. We showed in class that the result of a properly specified expected utility maximization problem is the following condition for choosing an         optimal level of insurance:

MU(mD)     (1 − 几)y

MU(mL)    几 (1 − y)

What are these components? Well, 几 is the probability that the accident occurs and the house burns   and (1 − 几) is the probability it doesn’t, as discussed previously (these states of the world occur with  probability 0.1% and 99.9%, respectively). The MU is the marginal utility, which is the derivative of the utility function. For this particular utility function, it is given by:

MU =  =  m − 1/2

This utility function will be evaluated at the different income levels derived above, mD and mL, and these income levels depend upon our c hoice of level of insurance! Lastly, y, is the cost per dollar of

insurance. If $1.2 purchases $1,000 worth of insurance, then y =  = 0.0012.

Putting this all together we have:

1                               − 1

2