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Problem 1(25pts) A decision maker (DM) has a Bernoulli utility of wealth that is given

by the function

u(w) = (

w if w ≤ 2

3

2w − 1 if w > 2.

This DM is considering three lotteries:

Lottery A results in wealth that is distributed according to the pdf

f(w) = (

32

w3

if w ≥ 4

0 if w < 4.

.

Lottery B results in wealth that is distributed according to the pdf

g(w) = (

2e

−2w if w ≥ 0

0 if < 0.

.

Lottery C results in wealth w = 1 with probability 0.5 and wealth w = 5 with probability

0.5

a) (5pts) Is the DM risk averse? risk neutral? or neither? Justify your answer

b) (20pts) Which lottery (of the above three lotteries) does the DM prefer?

Problem (25pts) Consider a system consisting of two urns (labelled urn 1 and urn 2),

three white balls and two red balls. At t = 0, two white balls are placed in urn 1. From the

remaining balls, one more ball is randomly selected and placed in urn 1. The remaining two

balls are now placed in urn 2. For each t = 1, 2, 3, · · · , we randomly draw one ball from each

urn and exchange them (the ball that was in urn 1 goes to urn 2 and the one that was in

urn 2 goes to urn 1).

For any t ≥ 0, we say that our system is in state i -for i = 1, 2, 3- if the number of white

balls in urn 1 is i. Let X(t) be the random variable representing the state of the system at

time t.

a)(5pts) Find the transition probability matrix for this process.

b)(5pts) Find the initial probability distribution of the states at t = 0.

c)(5pts) What is the probability that the state will be i = 2 at t = 3?

d) (5pts) Find E[X(3)]

e) (5pts) Suppose you get 10 dollars every time the system is at state 1 or 2, and you loose

10 dollars every time the system is at state 3. Suppose you let the system run for a very

long time (say until t=10,000). What is your expected payoff?

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