Jack and Mike harvest timber and sell it to local sawmills. Harvesting timber requires a special government permit. Jack and Mike have the only two permits and are therefore the only two producers of timber in this market. Harvested timber is a homogenous product. The inverse market demand is estimated to be P = 40,000 0.2Q where Q is total market output in tons of timber and P is the price in $/ton. Since Jack and Mike are the only two suppliers in this market, Q = Qs + Qwm, with Qy and Qum representing Jack's and Mike's outputs, respectively. Thus, for given output choices Qs and Qum, the market-clearing price is P = 40,000 0.2(Qs + Qwm), and total revenues for Jack and Mike can be expressed as follows: TR; = PQ; = [40,000 0.2(Q; + Qu)]Q; = 40,000Q; 0.2Q7 0.2QxQ: TRy = PQy = [40,000 0.2(Q; + Q4)]Qy = 40,000Q,, 0.2Q% 0.2Q,,Q,. Jack's and Mike's total costs can be expressed as functions of their output as TC; = 200Q; and TCy = 200Qy. Jack and Mike each choose their output to maximize their profit, I, = TR, T(; a) Suppose that Jack and Mike select their outputs simultaneously. Determine the Nash equilibrium in output choices Qs and Qm. What is the resulting market-clearing price? What are Jack's and Mike's profits? b) Suppose that the government grants Jack the legal right to harvest timber before Mike. Calculate the equilibrium solution in which Jack makes his production decision before Mike. What is the resulting market-clearing price? What are Jack's and Mike's profits? ) Suppose now that there are N producers in the industry (including Jack and Mike), all with the same total cost TCi =200Qi for 1= 1,2,...,N. Thus Firm i's profit can be expressed as: II; = TR; TC; = [40,000 0.2(Q; + @, + -+ Qy)]Q; 200Q;. All firms choose output simultaneously. Find the Nash equilibrium quantities and the market clearing price. (Hint: Use the fact that this is a symmetric game where in equilibrium all firms choose the same output.)