Consider an economy consisting of two firms, labeled 1 and 2. Each firm i = 1,2 chooses x¡ € [0, 1], which yields a profit of 0₁ (xj)² πi (x₁, xj) = Xi +ti, 2 where t; indicates a transfer from the government to firm i and 0¡ € [1,2] indicates the firm i's type. Assume that 0₁ and 02 are independently drawn from a common distribution F over the interval [1,2]. (a) Suppose that the firms choose the x simultaneously and independently. Assume there are no transfers: t₁ = t2 = 0. Compute a Bayesian Nash equilibrium of this game. (b) Suppose that the government can observe the firms' types (0₁,02), and that it seeks to max- imize 7₁ + 7₂. Setting transfers equal to zero, compute the first-best allocation (x,x) as a function of the types. (c) Now suppose that the government cannot observe the firms' types. Design a direct reve- lation mechanism satisfying the following two properties: (i) the mechanism implements the first-best allocation in dominant strategies, and (ii) the mechanism always results in a balanced budget, i.e., t₁ (0₁, 02) + t2(01,02) = 0.