(6) For each situation, solve for the Cournot-Nash equilibrium (Capacity Constraints) 6a) Suppose Sarah’s constant marginal cost is $5 but Joe's is $8. Recall that in a Cournot model, products are homogeneous. So each supplier faces the same aggregate demand function: P(Q) = 20 – 0.1 Q where Q = Qjoe + Qsarah 6b) Suppose Joe and Sarah have the same cost functions as earlier (constant MC of $5) but market inverse demand is now P(Q) = 30 – 0.2Q
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- The marginal cost of a product is fixed at MC = 20. The demand for the product is Q = 100 - 2P. (a) Now consider a Cournot model with two firms that are choosing quantities simultaneously. What is the best reply (best response) function for each firm? What is theNash equilibrium? What is the total surplus? (b)What do you expect the total surplus would be with three firms? Why? (You do not need to calculate an exact value. You can say ”total surplus is at least 100”, or ”total surplus is at most 80”)An industry consists of two firms, firm 1 and firm 2. The demand function for the product of each firmsis given by q1 = 720 - 3p1 + 2p2 and q2 = 720 - 3p2 + 2p1. We assume for simplicity that the totalcost of production is zero:TC1 =TC2 =0. a) Are these two products substitutes or complements? Assuming that firms compete over prices, find theprice best-response functions for firm 1 and fin 2. Draw a diagram that shows the BRFs and theequilibrium. Are prices strategic substitutes or complements? Find the Bertrand Nash equilibrium inprices and outputs. Obtain the profits of each firm. b) Show that the duopolists have incentives to collude. Find their joint profit-maximizing price, output, andprofit: find each firm's price. output and protit. Is collusion a Nash equilibrium? If not, what is the optimaldefection for each firm? Show this game in a 2X2 matrix form. What does this imply about the Nashequilibrium or the stability of their collusive agreement? Is it a Prisoner's Dilemma…Suppose that there are two firms producing a homogenous product and let the market demand besiven by Q(P) = 120 -P/2 . For simplicity assume that each fir operates with zero total cost. a) Assuming that firms compete over quantities, find the price best-response functions of firms 1 and2. Draw a diagram that shows the BRFs and the equilibrium, Are outputs strategic substitutes orcomplements? Find each firm's Cournot equilibrium output, price, profit, and total surplus. DefineNash equilibrium and argue that it is indeed a Nash equilibrium. b) Show that the duopolists have incentives to collude, Find their joint profit-maximizing price, output,and profit: find each firm's output and profit. Is collusion a Nash equilibrium? If not, what is theoptimal defection for each firm? Show this game in a 2X2 matrix form. What does this imply aboutthe Nash equilibrium or the stability of their collusive agreement? Is it a Prisoner's Dilemma Type? c) Suppose now that fims play the above game in…
- The inverse market demand for fax paper is given by P=100-Q. There are two firms who produce fax paper. Firm 1 has a cost of production of C1= 15*Q1 and firm 2 has a cost of production of C2=20*Q2 a) Suppose firm 1 and firm 2 compute simultaneously in quantities. What are the Cournot quantities and prices?What are the profits of firm 1 and 2?b) Suppose firm 1 and firm 2 compete simultaneously in prices. What are the Bertrand quantities and prices?What are the profits of firm 1 and 2?(Cournot competition with different marginal costs) Our best estimate for total marketdemand in a given market is P 1000-2Q. Two firms (1 and 2) are competing in this market in quantities, choosing Q1 and Q2 simultaneously. Firm 1 has marginalcost equal to c1 = 100 and Firm 2 produces at marginal cost c2 = 200. (a) Write down the profits of both firms and and their best response functions. (b) Find the Cournot - Nash equilibrium in quantities, and calculate equilibrium profits for both firms. (c) Suppose that each firm has the option, at a previous stage, to invest in an R&D project that will reduce its marginal cost of production by 50% if successful. What is the value of this innovation to each firm? Given that R&D costs and successprobabilities are equal, which one has greater incentives to invest in R&D ? You can think in terms of per - period profits to set aside timing issues.Consider a coastal economy served by two (2) fishing firms. There is only one type of fish that is harvested. All firms send their boats out at the same time and they compete on quantity. Demand given by P = 200 – Q and each firm has a constant marginal cost of $20. There are no fixed costs. a. Using the appropriate model, what will be an individual firm’s quantities and profits? b. Suppose that Firm 1 adopts a new production technology that reduces his marginal cost of production from $20 to $10. Firm 2 does not adopt the technology and continues to have a marginal cost of $20. What will be the new quantity and profit outcomes? How has the technology affected the relative position of Firm 1 and Firm 2 in the market?
- Consider the differentiated goods Bertrand price competition model where firms A and B produce similar goods and sell them at pA and pB. The demand for each firm’s product is given byqA =60−2pA +pB andqB =60–2pB +pA ,and there are NO costs of producing either good (all costs are 0). (a)Calculate the (Bertrand) equilibrium prices and the net profits of each firm (b) Now suppose that -instead of competing to maximize their own individual profits- the firms decide to “collude” and set prices pA and pB to maximize their joint profits (sum of their profits). What would be each firm’s optimal price and net profits? Compare these prices and profits with what you found in (a) (greater/smaller/the same?).Two firms produce a homogeneous product. Let p denote the product’s price. The output levelof firm 1 is denoted by q1, and the output level of firm 2 by q2. The aggregate industry output isdenoted by Q, Q = q1 + q2. The aggregate industry demand curve for this product is given byp = 70 − Q.Assume that the unit cost of firm 1 is c1 = 10 and the unit cost of firm 2 is c2 = 20.a. Suppose the firms move simultaneously and compete on quantities. Derive the firms’ bestresponse functions, and find the Cournot-Nash equilibrium. What are the profits of thefirms?b. Suppose the firms move sequentially with firm 1 setting its level of output before firm 2.Find the Stackelberg-Cournot equilibrium. What are the profits of firm 1 and firm 2?c. Now assume that firm 2 sets its level of output before firm 1. Find the Stackelberg-Cournotequilibrium. What are the profits of firm 1 and firm 2? Is there a difference in your findingsbetween part b and c? Explain why.Question 3:Suppose the inverse demand for a good is given by P = 50 – 4Q, where Q is the totalquantity supplied by all firms in the market. Suppose each firm in the market has a constantmarginal cost of 18.Q3 a) Assume the market consists of two firms that set their quantities simultaneously.Calculate the duopoly levels of production and the equilibrium price. Q3 b) Now assume firm 1 chooses its production level before firm 2 does. What will be theequilibrium quantities, price and profits in this case?Q3 c) Now instead suppose that the two firms compete over prices rather than quantities.What will be the equilibrium price and profits of firms 1 and 2 in this case? Finally, if firm 1manages to lower its marginal cost to 14, what will be the new equilibrium price, quantitiesand profits?
- Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand function is Qd = 10,000 – 100P, where P is the price of a cubic yard of concrete and Qd is the number of cubic yards demanded per year. Marginal cost is $25 per cubic yard. Suppose that Joe and Rebecca compete in quantities and competition in this market is described by Cournot model. What are Joe and Rebecca’s Nash equilibrium outputs? What is the resulting price? What do they each earn as profit? How does the price compare to the marginal cost? Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand function is Qd = 10,000 – 100P, where P is the price of a cubic yard of concrete and Qd is the number of cubic yards demanded per year. Marginal cost is $25 per cubic yard. Suppose that Joe and Rebecca compete in quantities and competition in this market is described by Cournot model. What are Joe and Rebecca’s Nash equilibrium outputs? What is the resulting price? What do they each…Consider a "Betrand price competition model" between two profit maximizing widget producers say A and B. The marginal cost of producing a widget is 4 for each producer. Each widget producer has a capacity constraint to produce only 5 widgets. There are 8 identical individuals who demand 1 widget only, and individuals value each widget at 6. If the firms are maximizing profits, then which of the following statement is true: a) Firm A and Firm B will charge 4 b) Firm A and Firm B will charge 6 c) Firm A and Firm B will charge greater than or equal to 5 d) None of the options are correct. Explain clearly.There are two firms selling differentiated products. Firm A faces the following demand for his product: QA=20-1/2PA+1/4PB Firm B faces the following demand: QB=220-1/2PB+1/4PA PA represents the price set by firm A. PB represents the price set by firm B.Assume that the marginal cost is zero both for firm A and firm B.What are the equilibrium prices of a simultaneous price competition?What would the equilibrium prices be if A is the leader and B is the follower?