4. Consider a market with two upstream firms (Ul and U2) selling an intermediate good to two downstream firms (DI and D2). The downstream firms then use the intermediate good as an input to produce a final good that is sold to consumers. It costs the upstream firm $20 to make each unit of the intermediate good and it costs the downstream firm $20 per unit (excluding the cost of the intermediate good) to convert one unit of the intermediate good into one unit of the final good. The two upstream firms compete in quantities and sell the intermediate good to the downstream firms at price PY. The two downstream firms use the intermediate good to produce the final good and they compete in quantities which results in a market clearing price of price PD for consumers. Consumer demand is given by the function P = 200 - 2Q. a. Derive the profit maximizing output level for each downstream firm and the resulting price PD given the price of the intermediate good PU. (5 pts) b. Derive the profit function for each of the upstream firms. Solve for the profit maximizing price PU set by the upstream firms. Use this to derive total sales in this market and calculate the profits of all four firms (10 pts) Suppose that the firms Ul and DI merge in order to form a new vertically integrated firm /. We will assume that firm / will not supply the downstream firm D2. c. Write down the profit function of the integrated firm (1) and the independent downstream firm (D2), Using this, derive the profit maximizing output level for I and D2 and the resulting price P given the price of the intermediate good PU (5 pts) d. Derive the profit function of the upstream firm U2. Solve for the profit maximizing price P set by U2. Use this to derive total sales in this market and calculate the profits of all three firms (1, U2 and D2). Was the merger between Ul and DI profitable? (10 pts)