7. (15 points)(Vertical Monopolies) Suppose there are upstream and downstream markets, and each market is served by one firm (so, there are upstream and downstream monopolists). The downstream market has the market demand function of D(p) = 1000 p, where p is the price of the final output produced by the downstream firm. The downstream monopolist has a production function y(z) = 3x, where x is the output of the upstream monopolist. The unit price of x is denoted k, and the downstream monopolist only uses = as an input, and has no other costs (so, the total cost of the firm is kz). The upstream monopolist has a cost function c(x) = zx, where z is a positive constant. (a) Write down the equation for the downstream monopolist's profit maximization problcmE] (b) Solve the above profit maximization problem and derive the downstream firm's reaction function (i.e., express y in terms of k). (c) Solve the upstream firm's profit maximization problem given the downstream firm's solution, and find the equilibrium level of z*, k*, y* and p*.] (d) Now suppose the downstream and upstream monopolies merge. Find the equilibrium level of y* and p*. Is the market price cheaper than (c)? Explain why