ITI-420 Information Economics. LARGE PROBLEM SET

Due at my office, in paper version, by the close of business on December 12th, 2002

There are 5 problems. Please do all parts of all problems.

Problem 1. Nash equilibrium and broadcasting. Suppose that each of 4 television stations is deciding what to broadcast in the evening. Each has a choice of broadcasting at 9pm or at 10pm. Each has the choice of broadcasting a program that appeals to a young audience (Y) or an older audience (M). So the 4 choices are 9Y, 9M, 10Y, and 10M. They can't afford to produce more than one show for the season, so in the other time slot they will broadcast public service announcements, which produce no audience share that they can sell to advertisers. Audience surveys have shown that the potential audience for each of these four possibilities is as shown in the following table

A. Enumerate all the possible profiles of strategic choices or "outcomes", such as "all 4 at 9Y", etc. Be systematic to be sure that you list them all. [4 points]

B. Figure out the size of the audiences that each station will get, in each of the outcomes. [4 points]

C. Figure out how many "real world alternatives" each of the profiles corresponds to. [For example, a profile with numbers 3 at 9Y and 1 at 10M could be realized in 4 different ways, according to which station is respresented by the "1".

D. For each of the profiles, list at least one profile to which one of the stations would switch, to improve its audience, if such an improvement is possible. For all other cases, label the Nash Equilibria.

E. Decide whether the Nash equilibrium maximizes the Social Welfare, and explain your answer.

Problem 2. A complicated utility function. Suppose that differing students have different judgments about the relative importance of different aspects of the educational experience. They place some value t on having a class hour free; they place some value e on learning all the material; they have some estimate f of how much they get by relying on a friend to attend the class; they have some value g that they attach to the grade that they earn. So, having decided to take a course, they then must decide whether to attend it themselves, or join up with a friend, and attend only half the classes themselves. Having taken 420 they know how to write formulas for the utility of each strategy.

Attend All: U(All) =eL +gL That is, they get full value of however much they can learn (L) and they get a grade that is proportional to how much they learn.

Attend Half: U(Half)= e(0.5L + 0.5fL') +0.5t + g(0.5L + 0.5fL') That is, there is no direct penalty for not attending. But they only get a fraction (f) of the amount (L') that their friend manages to absorb while attending class.

For 5 individual students, the actual values of the coefficients in their utility functions are as shown in the table below.

A. Figure out the utility, for each student, of attending class all the time

B. Find at least one way for them to pair up which increases the utility for every one of the students.

C. Calculate the total increase in utility that they get by pairing up this way.

D. If none of them think the free time is worth anything, will anyone want to team up with Miss Piggy? Will she want to team up with them?

E. (One paragraph) Do you think this kind of model of the decision to team up on class attendance is realistic? If not, what features would you add to it?

Problem 3. Probabilities and Statistical Significance. Recall that we say an occurrence is "statistically significant" if the chance that it would occur by having the teams toss a fair coin, rather than play the games, is less than 5%. [Specifically, we then say that we are 95% confident that it did not occur by chance.] We saw that a "four-game sweep" in the World Series does not establish that the winning team is better, at the 95% confidence level. Apply the same kind of reasoning to answer each of these specific questions.

A. If the World Series were 9 games long, so that a sweep requires winning the first 5 games, would a sweep be significant at the 95% confidence level?

B. How long must the World Series be for a sweep to establish that the winning team did not win simply by chance (at the 95% confidence level).

C. If the World Series, under present rules, lasts all 7 games, what is our confidence that the winning team is actually better? Give an explanation of your reasoning, please.

D. Suppose that we knew, as the series approached, that one of the teams seemed to be the underdog, and people pretty much agreed that this team should only win about 40% of the time. If they actually do win the first 4 games, could this have occurred by chance (at 95% confidence level.) Be careful!

E. If this same "underdog" managed to win not in 4 games, but in 5 games, then would it be statistically significant at the 96% confidence level? What is the chance that this would occur if they tend to win 50% of the games?

Problem 4. A non-linear utility function . Extensive research in the Department of Underworked Researchers at Enormous State University has revealed the following general theory of where students sit in class. There is a utility function, which varies from student to student, and each student tries to maximize his or her utiltiy.

The variable that students can adjust, x, is distance from the door, in columns of seats. The classroom is 20 columns wide. Each student is characterized by a "Sociability" represented by the letter S, and a "laziness" represented by the letter L.

A. Show that students who are more lazy (that is, have larger values of L will prefer to sit nearer to the door.

B. Where will each of these students prefer to sit (that is, how many columns from the door)?

C. Why does preference seem to depend so much more on laziness than on sociability? (This is hard).

D. If the utility function were not a product, as shown here, but were a sum,

where would each of the students prefer to sit?

E. In your experience, is the actual behavior of students who arrive early more like that predicted by the product formula, or the sum formula? In your experience, do students exhibit the same kind of behavior in every class they take, or does it vary from class to class?

Problem 5. Explain the idea of economic modeling. We know that different people behave in different ways, and are not always rational in their behavior. So how can it make sense to talk about an "average customer" or to talk about the average preference that people have? Illustrate your argument by making up some example data where some people think a CD is worth $12, some think it is worth $15, and some think it is worth $18. In this situation, what is the apparent "average value" of the CD? Suppose you run a retail store. If you sell the CD at above the average value, how many people will buy it? Does it makes sense to sell it at $14 or $ 13 (hint: why not?). How about cutting the price all the way down to $12? Does that decision depend at all on how much it costs you to buy the CD from the wholesaler? In answering this question you can make up numbers but I am looking to see that you reason clearly, based on the numbers that you chose. In particular, what use could you make, if any, of the information about the "average value" of the CD to your customers?