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*Show that the menotone likelihood ratio condition (f(x)/g(x) increasing in x, where x is a real number) implies firstorder stochastic dominance (F(x) 5 0(a) for all it). Let B = B] or 9; denote the state of nature in a two state world- Let pa = Pr(9 = 81)- Let f(p) be an arbitrary distribution on the unit interval such that lplip)dp = [39. Show that f(p) can be construed as a distribution over posterior probabilities of p, stemming from an experiment with outcomes y, a prior probability p = po and two likelihood functions p(y|9.) and p(y|92). [It may be easier to solve the problem if you assume that f(p) consists of two mass points so that the experiment will have only two relevant outcomes, say, yL and y".] 3. An agent has to decide between two actions a, and a2, uncertain of the prevailing state of nature s, which can be either s, or $2. The agent's payoff as a function of the action and the state of nature is as follows: u(a1,$1) = 10, u(a1,$2) = 7 u(a2,$1) = 5, u(a2,$2) = 11 The prior probability of state s, is p = Pr($1). a. Give a graphical representation of the agent's decision problem as a function of the probability of state s1. If p = (.4), what is the agent's optimal decision? b. Let the agent have access to a signal y before taking an action. Assume y has two possible outcomes with the following likelihoods Pr(y1|$1) = 21 Pr(y1/$2) = 12 How valuable is this information system if 21 = 22 = 1/2? How valuable is it if 21 = 1/2 and 22 = 0? c. Show that, irrespectively of the prior, the information system 21 = 1/2 and 12 = 0 is preferred to the information system 21 = 1/20 + 1/2B and 22 = B, where a and B are any numbers between 0 and 1. (Hint: show that latter system is garbling of former).4. *An agent can work hard (e = e") or be lazy (e = eL), where e\" > eL > 0. There are two prot levels, in and x;; it.