A perfectly competitive firm has a Cobb-Douglas production function f (X1, X2) = X1X2. Suppose that input prices are w1 = 1 and W2 = 1. The firm wants to find the cheapest way of producing y = 32. a. Suppose that in the short run the quantity of input factor 2 is fixed at X2 = 8. Solve the firm's short-run cost minimization problem to derive the optimal input quantity x1. b. Derive the corresponding costs Cs of producing y = 32 in the short run. c. Now consider the long run, in which both input factors are variable. Set up the Lagrangian function for this firm's long-run cost minimization problem. d. Derive the first-order conditions of the above long-run cost minimization problem. e. Solve the above first-order conditions to derive the optimal input quantities x1 and X2. f. Derive the corresponding costs c* of producing y = 32 in the long run.2. A set of risk averse consumers, each with identical preferences and identical wealth w >0, faces identical independent risks of experiencing boss w > L > 0. True, false, or uncertain and why: There are no potential gains from trade in risk among these consumers.5. When hiring young lawyers, law firms attempt to screen out applicants who plan to have children during the first ten years of their career (call these applicants "family types"). This is because lawyers tend to reduce their billable hours and thus generate less profit for their law firms once they have children. True, false, uncertain or why: The full disclosure principle says that job applicants will explicitly or implicitly reveal their type (family type, non family type) during the interview process, and hence screening for this characteristic is not necessary.6. State Lotteries sold $53 billion of lottery tickets in 2010. Suppose the cost of playing is $1 and the expected value of playing is 50 cents (i.e. E;p;W, = $0.50 where W, is payout i, and p; is probability of that payout). Mr. Jo, a VNM expected utility maximizer, is indifferent between playing the lottery or not. Is Mr. Jo risk-averse, risk-loving, neither, or is there not enough information to say? Explain.county received police cameras. 2. Does TRD provide an unbiased estimate of the causal effect of receiving police cameras on police brutality? Explain why or why not, along with any assumptions you need to make. (6 points)3. Suppose that you find TRD = -5. President Obama thinks that this is strong enough evidence to support the expansion of the policy to the remaining counties in the U.S. Do you think that Tao is an unbiased estimate of the causal effect for these remaining counties with H,