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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? 2 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? 3 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?

4

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.

5

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.

Compelling correspondence is essential to the achievement all things considered but since of the changing idea of the present working environments, successful correspondence turns out to be more troublesome, and because of the numerous impediments that will permit beneficiaries to acknowledge the plan of the sender It is restricted. Misguided judgments.In spite of the fact that correspondence inside the association is rarely completely open, numerous straightforward arrangements can be executed to advance the effect of these hindrances.

Concerning specific contextual analysis, two significant correspondence standards, correspondence channel determination and commotion are self-evident. This course presents the standards of correspondence, the act of general correspondence, and different speculations to all the more likely comprehend the correspondence exchanges experienced in regular daily existence. The standards and practices that you learn in this course give the premise to additionally learning and correspondence.

This course starts with an outline of the correspondence cycle, the method of reasoning and hypothesis. In resulting modules of the course, we will look at explicit use of relational connections in close to home and expert life. These incorporate relational correspondence, bunch correspondence and dynamic, authoritative correspondence in the work environment or relational correspondence. Rule of Business Communication In request to make correspondence viable, it is important to follow a few rules and standards. Seven of them are fundamental and applicable, and these are clear, finished, brief, obliging, right, thought to be, concrete. These standards are frequently called 7C for business correspondence. The subtleties of these correspondence standards are examined underneath: Politeness Principle: When conveying, we should build up a cordial relationship with every individual who sends data to us.

To be inviting and polite is indistinguishable, and politeness requires an insightful and amicable activity against others. Axioms are notable that gracious “pay of graciousness is the main thing to win everything”. Correspondence staff ought to consistently remember this. The accompanying standards may assist with improving courtesy:Preliminary considering correspondence with family All glad families have the mystery of progress. This achievement originates from a strong establishment of closeness and closeness. Indeed, through private correspondence these cozy family connections become all the more intently. Correspondence is the foundation of different affiliations, building solid partners of obedient devotion, improving family way of life, and assisting with accomplishing satisfaction (Gosche, p. 1). In any case, so as to keep up an amicable relationship, a few families experienced tumultuous encounters. Correspondence in the family is an intricate and alluring marvel. Correspondence between families isn’t restricted to single messages between families or verbal correspondence.

It is a unique cycle that oversees force, closeness and limits, cohesiveness and flexibility of route frameworks, and makes pictures, topics, stories, ceremonies, rules, jobs, making implications, making a feeling of family life An intelligent cycle that makes a model. This model has passed ages. Notwithstanding the view as a family and family automatic framework, one of the greatest exploration establishments in between family correspondence centers around a family correspondence model. Family correspondence model (FCP) hypothesis clarifies why families impart in their own specific manner dependent on one another ‘s psychological direction. Early FCP research established in media research is keen on how families handle broad communications data. Family correspondence was perceived as an exceptional scholastic exploration field by the National Communications Association in 1989. Family correspondence researchers were at first impacted by family research, social brain science, and relational hypothesis, before long built up the hypothesis and began research in a family framework zeroed in on a significant job. Until 2001, the primary issue of the Family Communication Research Journal, Family Communication Magazine, was given. Family correspondence is more than the field of correspondence analysts in the family. Examination on family correspondence is normally done by individuals in brain science, humanism, and family research, to give some examples models. However, as the popular family correspondence researcher Leslie Baxter stated, it is the focal point of this intelligent semantic creation measure making the grant of family correspondence special. In the field of in-home correspondence, correspondence is normally not founded on autonomous messages from one sender to one beneficiary, yet dependent on the dynamic interdependency of data shared among families It is conceptualized. The focal point of this methodology is on the shared trait of semantic development inside family frameworks. As such, producing doesn’t happen in vacuum, however it happens in a wide scope of ages and social exchange.

Standards are rules end up being followed when performing work to agree to a given objective. Hierarchical achievement relies significantly upon compelling correspondence. So as to successfully impart, it is important to follow a few standards and rules. Coming up next are rules to guarantee powerful correspondence: clearness: lucidity of data is a significant guideline of correspondence. For beneficiaries to know the message plainly, the messages ought to be sorted out in a basic language. To guarantee that beneficiaries can without much of a stretch comprehend the importance of the message, the sender needs to impart unmistakably and unhesitatingly so the beneficiary can plainly and unquestionably comprehend the data.>

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