4. Suppose a subjective learning DM has prior belief that assigns equal weight, 1/3, to the three states, wi, w2, and w3, of the world (i.e., good, bad, and ugly weather). Suppose the DM expects to receive information about the true state. In particular, let be the DM's subjective information structure which puts equal weight on each posterior belief, p = (2/3, 1/6, 1/6) and q = (1/6,2/3, 1/6) and r = (1/6, 1/6,2/3), but zero weight on any other posterior belief. Let f = (12,6,0) and g = (0,12,6) and h = (6,0,12) be three acts available to the DM and suppose the DM receives payoff from each outcome a by the function u(a) = a. Answer the following questions. (a) Show that the DM strictly prefers (f, g} over 1/2{g} + 1/2{h} (b) Show that the DM strictly prefers (f, g} over {f, 1/2g + 1/2h} (c) Show that the DM strictly prefers {f, g, h} over {f, 1/2g+ 1/2h, h}

5. Suppose that the government must undertake an irreversible policy decision regarding the extent of air pollution regulation. The government is making this decision in a situation of uncertainty. however. In particular, there is some probability p that the benefits will remain the same as they are this year for all future years, but there is some probability 1-p that benefits will be less in all future years. If we take into consideration the multiperiod aspects, should we err on the side of overregulation or underregulation, compared to what we would do in a single-period choice?

8. Suppose that a sales force has found 20 qualified buyers and has begun the sales process. The sales manager estimates that 10% eventually proceeds to make a purchase. Assume that a professional company offers three services, priced at $2,000, $7,000 and $20,000, respectively. Based on past results or the sales manager’s estimates, you project that 60% of first-time buyers will choose the cheapest option, 30% will choose the middle option and 10% will choose the most expensive option. a. Calculate the size of a likely sale for any prospect that makes a purchase.  b. How many qualified buyers must be found in order for the company to generate $80,000 in sales in a given period? Your final answer must be rounded to the nearest whole number.

[Adverse Selection] Each of the two players receives an envelope, in which there is anamount of money that is equally distributed from $0, $1, $2, ..., $100. The amounts in twoenvelopes are independent. After receiving the envelope, each individual can check exactlyhow much money is put in his/her own envelope. Then each player has the option to exchangehis/her envelope for the other individual's prize. The decisions are made simultaneously. Ifboth individuals agree to exchange, then the envelopes are exchanged; otherwise, if at leastone player chooses not to exchange, each individual keeps his/her own envelope and receivesits attached sum of money.a. Model this game as a static Bayesian game (write the normal formrepresentation) and find the Bayesian Nash equilibrium.b. Consider a new game where the probability distribution of money in eachenvelope is changed. The amount is equal to $100 with probability 90%, and is equalto each number in $0, $1, $2, ... ,$99 with probability 0.1%.…

Insurance Example o Initial income = $15,000 o Risk averse consumer faces 20% chance of losing $10,000 o Actuarially fair insurance premium (AFP) for covering this risk equals the expected value of the loss (.2x$10,000 = $2000) %3D Question 1) Suppose Al has an income of $25,000 and faces a 20% chance of having a serious medical problem that requires $15,000 worth of medical care. Al's utility function is U(I) = 10.5 [or square root of income]. There are two possible health insurance plans available. The first has 25% coinsurance and has a premium of $2500. The second plan has a $1500 deductible then 25% coinsurance and has a premium of $2200. Which plan would Al choose, if any? What is the AFP for each plan? What is the loading factor for each plan? (loading factor is % by which premium exceeds the AFP)

13. The government wishes to model the number of calls made to a particular citizen's advice helpline. Over the last month they estimate that there have been on average 1.8 calls/hour. What distribution would be best to model this random variable? O Normal Distribution O Poisson Distribution O Binomial Distribution O Uniform Distribution O T-Distribution

The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica- tion of preferences in economics that captures risk aversion and intertemporal consump- tion smoothing. The CRRA utility function has the desirable property that the degree of risk aversion is constant and independent of the level of consumption. This means that as a household's consumption grows, its willingness to take risks remains the same. The coefficient of relative risk aversion (σ) measures the extent to which households are risk- averse and prefer a smooth consumption path over time. A higher value of σ indicates a greater degree of risk aversion and a stronger preference for consumption smoothing. Consider a two-period endowment economy with a large number of identical house- holds. Each household has the following lifetime utility function: U(j) = C+(j) 1-0 - 1 1-σ +ẞ C++1(j) 1-0 - 1 1-σ where C₁(j) and C++1(j) are consumption in periods t and t + 1 for household j, re- spectively, ẞ is…

Suppose the market for auto insurance is made of up two types of buyers: high-risk and low-risk. Buyers’ willingness to pay (WTP) for auto insurance plans, and sellers’ willingness to accept (WTA) when selling plans to each type of buyer, are outlined in a photo    Assume now that there is asymmetric information and that insurance companies do not knowhow risky an individual buyer is. In the face of this uncertainty, they determine that the probability that a “walk-in” is high-risk is 0.75.   What is the minimum price sellers are willing to accept when selling aninsurance plan? At this price, will low- and high-risk buyers both be willing to purchase this insurance plan? Explain. Be sure the mention adverse selection in your answer.   Returning to the conditions outlined in Q1, suppose that buyers of auto insurance (high- and low-risk) were offered a $1,000 subsidy to purchase coverage. This would raise their WTP by $1,000. Would the market for both insurance plans clear after the…

Why does dependent variables must covariate in the multivariate analyses like MANOVA/MANCOVA? What happens if they don't covariate, will you be able to proceed in multivariate analyses?

1. Imagine a person who wants to find a job within two months. Define X corresponding to the outcome, such that X = 1 if the person finds a job and X = 0 otherwise. (a) Explain why X can be viewed as a random variable. What is the support of X? (b) Suppose the probability of finding a job is 0.75. What is the probability distribution of X? Be specific.