Problem 1 Two natural gas producing rms face the inverse demand curve P =5 0 -5Q, where Q =Q, +Q2. The rms' cost functions are C1(Qi) = 20 + IOQ: and C2(Q:) = 10 + 12Q2. Suppose both rms have entered the natural gas industry. What is the joint prot maximizing level of output? How much will each rm produce? How would your answer change if the firms have not yet entered the industry? b. What is each firm's equilibrium output of natural gas and prot if they behave non cooperatively? Use the Coumot model. Draw the rms' reaction curves and show the equilibrium. Problem 2 A monopolist in the natural gas market can produce at a constant average (and marginal) cost of AC = MC = $5. It faces a market demand curve given by Q = 53 - P. a. b. Find the prot-maximizing price and quantity for this monopolist. What are its prots? Suppose a second rm enters the market. Let Q; be the output of the first rm and Q2 be the output of the second. Market demand is now given by Q: + Q2 = 53 P. Assuming that this second lm has the same costs as the rst, write the prots ofeach rm as functions of Q1 and Q2. Suppose (as in the Cournot model) that each rm chooses its profit-maximizing level of natural gas output based on the assumption that its competitor's output is xed. Find each rm's \"reaction curve" (i.e., the rule that gives its desired output in terms ofits competitor's output). Calculate the Coumot equilibrium (i.e., the values onl and 02 for which each rm is doing as well as it can given its competitor's output). What are the resulting market price and prots of each rm? Problem 3 Assume two identical energy rms produce natural gas and that they are the only rms in the market. Their costs are given by C: = 60Qr and C2 = 6012.). where Q] is the output of Firm 1 and Q: the output of Firm 2. The inverse demand curve is P = 300 - Q, where Q = Q1 + Q2. a. Find the Cournot-Nash equilibrium. Calculate the prot of each rm at this equilibrium. b. Suppose the two rms form a cartel to maximize joint prots. What will be the output of two rms? Calculate each rm's prot. c. Suppose Firm 1 were the only rm in the industry. How would market output and Firm ['3 prot differ from that found in part b above? d. Returning to the duopoly of part b, suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. What will be the output of Firm 2? What will be each rm's prots? Problem 4 Suppose the natural gas industry consisted ofonly two firms. Let these rms have identical cost functions. C(qr) = 40:]. Assume the demand curve for the industry is given by P = 100 , Q and that each rm expects the other to behave as a Coumot competitor. 3. Calculate the CoumotNash equilibrium for each rm, assuming that each chooses the output level that maximizes its profits when taking its rival's output as given. What are the prots of each rm? What would be the equilibrium quantity if Firm 2 had constant marginal and average costs of$25 and Firm 1 had constant marginal and average costs of $40? Assuming that both rms have the original cost function, C(q) = 40g, how much should Firm 2 be willing to invest to lower its marginal cost from 40 to 25, assuming that Firm 1 will not follow suit? How much should Firm I be willing to spend to reduce its marginal cost to 25, assuming that Firm 2 will have marginal costs of 25 regardless of Firm 1's actions
