1.You can choose between two dierent portfolios, A and B, that produce the following uncertain returns in the two possible states of the world

Good Bad

Portfolio A 100 0

Portfolio B 60 80

Before making your choice you receive one of two messages, m₁ or m₂, that convey some information about the future realization of the state of the world. Messages are costless to receive.

Consider the following specifications of the information structure (each row gives you the conditional probabilities of the two states of the world, given the message you receive):

Perfect Information No Information Noisy Information

Good Bad Good Bad Good Bad

m1 1 0 0.5 0.5 0.8 0.2

m2 0 1 0.5 0.5 0.2 0.8

(i)Draw the decision tree corresponding to each of the above spec- ifications of the information structure.

Suppose you receive message m1.What would your optimal choice be in any of the above scenarios?[8]

(ii) Let a = {A, B}, e = {Good, Bad}, m = {m₁, m₂} and denote the conditional probability of e given m by p(e | m) and the return to portfolio a in state of the world e by U(a, e). The value of the information is defined as:

X X

p(m) max p(e | m)U(a, e)m(a,e)

m a e

Suppose you are equally likely to receive any of the two mes- sages.

What is the value of the information in each of the above sce- narios?

Would you be led to conclude that the Ecient Market Hypoth- esis holds in this setting?[8]

(i)Now consider a market that is efficient, but not perfect (in that information is costly), where a large number of traders can choose whether to acquire Lots of Information at a high cost or Little Information at a low cost. After having decided what package of information to buy, traders are randomly matched in pairs and the payoffs from the interaction are summarized in the following payoffs matrix

Trade Column

Lots of Info Little Info

Lots of Info 4 , 4 8 , 2

Little Info 2 , 8 4 , 4

a) Denote by p the fraction of traders in the market that have bought Lots of Information.If you choose to buy Lots of Information, what do you expect your payoff to be?

If you choose to buy Little Information, what do you expect your payoff to be?

b) Show that for p = 2/3 the market is in an equilibrium