Homework Problems: Value of Information; Cost of Misinformation

1. The following sequence of questions is based on the example presented in class.

• Smith is deciding whether to invest $50,000 in a small local company that may go public within the year. If that happens S will double [quadruple] her money, but if it doesn't she'll lose 5%[20%].
• The other option is to take out a certificate of deposit that pays 10%. The buzz around town is that the chances are 50/50 that it will go public.

a) Calculate the expected utility (EU) of each option (i.e., stock vs. CD) in each scenario (i.e., double vs. quadruple).

b) Let the probability that the stock goes public be P. (In both the above scenarios P is 0.5.) For each scenario (double and quadruple), use algebra to find the value of P such that the EUs of the two options in that scenario are equal.

Suppose now a new rumor floats around town that the decision has already been made and that for a price one can get a hold of the insider information. Consider the metadecision of whether to pay for info or not. Let us assume (unrealistically) that if we get inside info it will be perfectly accurate.

c) For each of the two original scenarios calculate the EU of the option of buying insider information.

d) Now change the probability of going public from 0.5 to 0.1. Using this new probability calculate the new EU of the option of buying insider information for each scenario.

2. The following problem is also about the value of additional information, but it is presented in a more artificial terms so that our intuitions furnish less guidance.

A closet contains 800 type A urns and 200 type B urns. The urns appear identical but A-type urns contain six blue balls and four red ones; B-type urns contain one blue ball and nine red ones. An urn is drawn at random from the closet and you must bet on the type of the urn. If you bet on type A and it is an A-type, you win $20; otherwise you lose $20. If you bet on type B and it is a B-type, you win $80; otherwise you lose $10. Assume that you maximize expected monetary values.

e) Set up a decision matrix for the choice between the EUs of the two bets. Which should you choose?

f) Prior to actually making your bet, what is the maximum amount you should pay to learn the type of the urn?

g) Suppose that in the process of choosing the urn at random from the closet a ball accidentally tumbles out and you see that it is blue before it is restored to the urn. How would this bit of information affect the amount that you would pay to learn the type of the urn prior to making you bet, increase it, decrease it, make no difference? (If you say increase or decrease you need not calculate how much more or less.)

h) What difference would it make, if any, if the ball which tumbled out were red?