Consider the market of homogenous beer with N production plants and a linear aggregate demand given by p = 30 - Q. Let qi denote the output level of plant i (measured in billions of bottles) and the cost function of plant i is given by TC i (qi) = 9 4(qi) 2 . a. Suppose that a cartel (NBPA, National Beer Producer Association) organizes all the N plants and decides how much each plant should produce. The objective of NBPA is to maximize their joint profit. Find the optimal total quantity and the corresponding market price of beer. Remember that NBPA is an association, so it cannot force any plant to shut down (N is fixed). b. Suppose that a monopolist owns all the plants in this market, so it can shut down some plants if it wants to (N is variable). Find the optimal number of plants for the monopolist. Note that shutting down a plant does not incur any scrap cost/value