3. Consider an environment with a firm that has two projects, project 1 and project 2, in which it can invest. There are two possible states of the world, state 1 and state 2, which the firm initially views as equally likely. If the firm were to invest I; > 0 in project 1, then the expected profit that would be generated by project 1 in state 1 is /I; I; and the expected profit that would be generated by project 1 in state 2 is I;. Similarly, if the firm were to invest I, > 0 in project 2, then the expected profit that would be generated by project 2 in state 1 is I; and the expected profit that would be generated by project 2 in state 2 is v/Is I. Throughout this problem, suppose that the firm's choice of investment levels is not capped and take the firm's objective to be maximizing its expected profit. (a) What are the investment levels that would maximize the firm's expected profit? (b) Now, suppose that, before choosing its investment levels, the firm can purchase an information structure o that, conditional state 1, results in signal a occurring with probability % and signal b occurring with probability %, and, conditional on state 2, results in signal a occurring with probability % and signal b occurring with probability %. What would be the highest amount that the firm would be willing to pay for this information structure? (c) Instead, suppose that the firm can select the information structure it purchases before choosing its investment levels. In particular, for arbitrary precision % 0 in project 1, then, regardless of the state, the expected profit that would be generated by project 1 is +/I; I, and, if the firm were to invest I, > 0 in project 2, then the expected profit that would be generated by project 2 in state 1 is I and the expected profit that would be generated by project 2 in state 2 iS\\/[Q+11I2. (d) Answer the questions given in 3(b) and 3(c) above for this alternate environment. 4.1 Example Consider the following modification to the example given above concerning an agent considering purchasing 1 and the an umbrella. The set of states of the world is Q = {r,d}, the ex-ante probability of r is p, = 3, ex-ante probability of d is pq = % Moreover, X = {u,n}. The agent's Bernoulli utility u satisfies u(w, u,t) =101 and 22t fw=r, u(w,n,t) = t ifw=d for all (w,t). In the absence of an informative information structure, the agent would choose to purchase an umbrella, since doing so would result in an expected utility of 10, which is strictly greater than 11, the expected utility the agent would obtain from not purchasing an umbrella. Thus, here V,,, info = 10. Suppose that the agent can purchase at price 7 R, an information structure (o, S), in which S = {a, b}, ola|r] = o[bld] = 2, and o[b|r] = o[a|d] = ;. Note that, with such an information structure, the probability of signal a occurring is P, 5 = % and the probability of signal b occurring is Pp 5,5 = % Moreover, under such an information structure, the conditional probability of the state being r given signal a is profalr] P, o = 3 P ,S[Tla] J>j:.r,ar._.'jv'[ifl] and the conditional probability of the state being d given signal a is paclald] Pp,a.51a)' Py os(rla] = b3 [=S so the conditional probability distribution over states given signal a is %ck + %503. Similarly, the conditional probability distribution over states given signal b is ir + %511' Given signal a, purchasing an umbrella would induce probability distribution over outcomes %J(r,u__,{) + %J(d__u, and thus give the agent expected utility 1 Epurchase|u(w, z,t)|a] = %fu(r', u, ) + Zu(d' u,m) = 10 7, while not purchasing an umbrella would induce probability distribution over outcomes %6(,\"_\"__,,,) + %6({1__,@,\")