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

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u(w) =

{ w if w ≤ 2 3 2 w − 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?

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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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Problem 3(25pts) Consider the following EOQ inventory model with a constant demand rate of 4 units per unit time. Suppose that the cost of ordering y units is given by

C(y) =

{ 1 + 7y if w ≤ 10 1 + 5y if y > 10.

.

Assume further that The holding cost is 10 cents per unit stored per unit time. Finally, assume there is no time lag between the time the order is placed and the time it is delivered. a)(17pts) What is the optimal inventory management policy? In other words, when do we order, how much should we order every time we place our order, and how long do we have to wait between two consecutive orders? b)(8pts) what will change in your answer in part (b), if it actually takes one unit of time for an order to arrive?

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Problem 4(20pts) Suppose you want to invite one of three friends to a party. We shall refer to these friends as number 1,2, and 3. You are going to proceed with your invitations sequentially. Once you invite a person and that person accepts, the process stops and you go to the party with that person. Otherwise, you move on and invite another friend. However, you cannot re-invite someone who already said no to you. Your utilities of going to the party with friends 1, 2, and 3 are 16, 8, and 12 respectively. If no one accepts the invitation, you will not go to the party and your utility is zero. Each friend will know when you invite someone else, and the probability of a particular friend accepting your invitation varies based on whether you invited them first, second, or third. Let pij be the conditional probability that friend j accepts the invitation, given that he/she is the ith person invited by you. More specifically, For friend 1: p11 = 0.5, p21 = 0.2, p31 = 0.1. For friend 2: p12 = 0.8, p22 = 0.3, p32 = 0.3. For friend 3 : p13 = 0.6, p23 = 0, p33 = 0. Note that, for j = 1, 2, 3, p1j, p1j ,p3j are conditional probabilities and need not add up to 1. a)(8) Write a Bellman equation that can be solved recursively to find the optimal order of invitations you should attempt in order to maximize your expected utility. b)(12) Find the optimal order of invitations you should attempt in order to maximize your expected utility. You can find the optimal order by solving the Bellman equation you found in part (a) or by using any other method you can come up with.

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Problem 5(25pts) a) Solve Problem 1 Section 23.2A in Chapter 23 of the text (posted on Canvas) for an infinite number of periods using the exhaustive enumeration method (EEM). Clearly show your work for every step of the EEM method.

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