Question 2.3. Consider an industry with two firms, each having marginal costs equal to zero. The industry demand is P(Y)=100-Y, where Y=y1+y2 is total output (a) What is the competitive equilibrium output? (b) If each firm behaves as a Cournot competitor, what is firm 1s optimal output given firm 2s output? (c)Calculate the Cournot equilibrium output for each firm (d)Calculate the cartel output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm Question 2.4. Consider a Cournot industry in which the firms'outputs are denoted yn, aggregate output is denoted by Y=2ia yi, the industry demand curve is denoted by P(Y), and the cost function of each firm is given by ci(yi)= cyi.For simplicity, assume P(Y<0. Suppose that each firm is required to pay a specific tax ti on output (a) Write down the first-order conditions for firm i (b) Show that the industry output and price only depend on the sum of tax rates (c) Consider a change in each firms tax rate that doesn't change the tax burden on the industry. Letting At i denote the change in firm i's tax rate, we require that Ci Ati=0. Assuming that no firm leaves the industry, calculate the change in firm i's equilibrium output Ay;. Hint: use the equations from the derivations of (a) nd( b)I Question 2.5.(Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let where a>0 and c>0 are two constants Stage 1: All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K>0 Stage 2: All firms that have entered play a Bertrand gameQuestion 2.3. Consider an industry with two firms, each having marginal costs equal to zero. The industry demand is P(Y ) = 100 − Y, where Y = y1 + y2 is total output. (a) What is the competitive equilibrium output? (b) If each firm behaves as a Cournot competitor, what is firm 1’s optimal output given firm 2’s output? (c) Calculate the Cournot equilibrium output for each firm. (d) Calculate the cartel output for the industry. (e) If firm 1 behaves as a follower and firm 2 behaves as a leader, calculate the Stackelberg equilibrium output of each firm. Question 2.4. Consider a Cournot industry in which the firms’ outputs are denoted by y1,...,yn, aggregate output is denoted by Y = Sn i=1 yi, the industry demand curve is denoted by P(Y ), and the cost function of each firm is given by ci(yi) = cyi. For simplicity, assume P00(Y ) < 0. Suppose that each firm is required to pay a specific tax of ti on output. (a) Write down the first-order conditions for firm i. (b) Show that the industry output and price only depend on the sum of tax rates Sn i=1 ti. (c) Consider a change in each firm’s tax rate that doesn’t change the tax burden on the industry. Letting ∆ti denote the change in firm i’s tax rate, we require that Sn i=1 ∆ti = 0. Assuming that no firm leaves the industry, calculate the change in firm i’s equilibrium output ∆yi. [Hint: use the equations from the derivations of (a) and (b)]. Question 2.5. (Entry Cost in a Bertrand Model). Consider an industry with an entry cost. Let ci(y) = cy, pd (y) = a − y, where a > 0 and c ≥ 0 are two constants. Stage 1: All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K > 0. Stage 2: All firms that have entered play a Bertrand game. 2—2