1. [Monopoly and Duopoly] Consider a market of one good with demand P = A -...

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1. [Monopoly and Duopoly] Consider a market of one good with demand P = A - BQ where A and B are positive constants and Q is the industry output. We assume throughout that there are no costs of production for any firm operating in the market. (a) Suppose there is a monopolist serving the entire market. Find the monopolist's optimal quantity and optimal profit. (b) Now suppose the market is a duopoly market with two firms. Firm 1 chooses its quantity of output q and firm 2 chooses q2. Write down the profit functions of the two firms, find the best response functions of the firms, and calculate the equilibrium quantities for firms 1 and 2 in this market. Is the total equilibrium quantity of the two firms higher than the monopolist's optimal quantity in part [a]? (c) Now suppose the firms 1 and 2 could agree that for every unit of output produced by firm 1, the firm must pay a tax t to firm 2, and that firm 2 must similarly pay a tax t2 to firm 1 for every unit of output produce by firm 2. Explain how these taxes can increase the profits of both firms. Assuming that t =t2t, find the equilibrium quantities with the taxes. What is the tax level that maximizes the firms' profits. 1. [Monopoly and Duopoly] Consider a market of one good with demand P = A - BQ where A and B are positive constants and Q is the industry output. We assume throughout that there are no costs of production for any firm operating in the market. (a) Suppose there is a monopolist serving the entire market. Find the monopolist's optimal quantity and optimal profit. (b) Now suppose the market is a duopoly market with two firms. Firm 1 chooses its quantity of output q and firm 2 chooses q2. Write down the profit functions of the two firms, find the best response functions of the firms, and calculate the equilibrium quantities for firms 1 and 2 in this market. Is the total equilibrium quantity of the two firms higher than the monopolist's optimal quantity in part [a]? (c) Now suppose the firms 1 and 2 could agree that for every unit of output produced by firm 1, the firm must pay a tax t to firm 2, and that firm 2 must similarly pay a tax t2 to firm 1 for every unit of output produce by firm 2. Explain how these taxes can increase the profits of both firms. Assuming that t =t2t, find the equilibrium quantities with the taxes. What is the tax level that maximizes the firms' profits.