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Let ci be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by P=1-Q.
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Calculate the Nash
equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry prof- its, consumer surplus, and total welfare. -
Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1’s cost would change the equilibrium. Draw a representative isoprofit for firm 1.
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- Three firms compete in the style of Cournot. All firms have a constant returns to scale technology: There are no fixed cost and each firm's marginal cost is constant. The market demand is given by Q(P) = 9 - P. Firm 1's marginal cost is MC1 = 1, firm 2's marginal cost is MC2 = 2. Let MC3 be the marginal cost of Firm 3. Which of the below is a necessary condition so that q > 0 for all three firms in a Nash equilibrium? a. MC3 < 1 b. MC3 < 4 c. MC3 < 3 d. MC3 > 1 e. MC3 < 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.The market demand function is Q=10,000-1,000p. Each firm has a marginal cost of m=$0.28. Firm 1, the leader, acts before Firm 2, the follower. Solve for the Stackelberg-Nash equilibrium quantities, prices, and profits. Compare your solution to the Cournot-Nash equilibrium. The Stackelberg-Nash equilibrium quantities are: q1=___________ units and q2=____________units The Stackelberg-Nash equilibrium price is: p=$_____________ Profits for the firms are profit1=$_______________ and profit2=$_______________ The Cournot-Nash equilibrium quantities are: q1=______________units and q2=______________units The Cournot-Nash equilibrium price is: p=$______________ Profits for the firms are profit1=$_____________ and profit2=$_______________
- Assume a Nash-Cournot equilibrium. How much output does firm 1 produce? Assume a Nash-Cournot equilibrium and no fixed cost. How much profit does firm 2 make? Now assume a collusive equilibrium. What is firm 1's output?How would the Cournot equilibrium change in the airline example if American's marginal cost were $90 and United's were $180? The demand the duopoly quantity-setting firms face is Q=339−p with an inverse demand function of p=339−1qA −1qU where qA is the quantity produced by American and qU is the quantity produced by United. The Cournot-Nash equilibrium occurs where qA equals ? enter your response here and qU equals? enter your response here. (enter numeric responses using integers) Furthermore, the equilibrium occurs at a price of ? (round your answer to the nearest penny)Consider a Cournot Oligopoly. One firm has costs C1(Q1) = 12Q1 while the other firm’s cost function is C2(Q2) = 10Q2. The demand for both firms’ products Q=Q1 +Q2 isQD(P)=200−2P. (a) Determine the equilibrium price P, the market shares s1, s2, and the quantities Q1, Q2 produced by both firms. (b) Suppose more firms with the lower cost technology, i.e., with cost function Ci(Qi) = 10Qi enter the market. How many firms with this technology must be in the market such that firm 1’s profit becomes negative. In other words, suppose there is one firm with the high costs, and n firms with the low costs. At what level n will profits of the high-cost firm be negative?
- Consider a homogeneous goods industry where two firms operate and the linear demand is given by p(y1 + y2 ) = a - b(y1 + y2 ), where p is the market price, and y1 (y2) is the output produced by firm 1 (2). There are no costs for firm 1 or firm 2. Derive the best responses (reaction curve) for firm 1 and firm 2. Explain the term best response (reaction curve). Illustrate the best responses in a diagram. b) For the case in (a) determine the Cournot equilibrium (Nash equilibrium in quantities) when firm 1 and firm 2 compete simultaneously in quantities. How large are firm 1’s and firm 2’s profits? What is the industry output? Suppose the inverse demand curve in a market is D(p) =a-bp, where D(p) is the quantity demanded and p is the market price. Firm 1 is the leader and has a cost function c1(y1)=cy1 while firm 2 is the follower with a cost function c2(y2 )=. Firm 1 sets its price to maximise its profit. Firm 1 correctly forecasts that the follower takes the price leader’s chosen…Consider a homogeneous goods industry where two firms operate and the linear demand is given by p(y1 + y2 ) = a - b(y1 + y2 ), where p is the market price, and y1 (y2) is the output produced by firm 1 (2). There are no costs for firm 1 or firm 2. Derive the best responses (reaction curve) for firm 1 and firm 2. Explain the term best response (reaction curve). Illustrate the best responses in a diagram. b) For the case in (a) determine the Cournot equilibrium (Nash equilibrium in quantities) when firm 1 and firm 2 compete simultaneously in quantities. How large are firm 1’s and firm 2’s profits? What is the industry output? c) Suppose the inverse demand curve in a market is D(p) =a-bp, where D(p) is the quantity demanded and p is the market price. Firm 1 is the leader and has a cost function c1(y1)=cy1 while firm 2 is the follower with a cost function c2(y2 )=y22 /2. Firm 1 sets its price to maximise its profit. Firm 1 correctly forecasts that the follower takes the price…Consider the following statements about the Stackelberg game from the slides, assuming both firms are identical: (I) Denote by qC the Cournot equilibrium quantity produced by each firm, and by qPC the competitive quantity defined by P(qPC) = c (price equals marginal cost). Let s2 denote a strategy where firm 2 plays q2 = qC if it observes q 1 = qC, and plays q2 = qPC otherwise. Let s 1 denote a strategy where firm 1 plays q1 = qC. Then, (s1,s2) is a Nash equilibrium of the Stackelberg game, but it’s not a subgame perfect Nash equilibrium. (II) The first firm is allowed to change its quantity after observing firm 2’s quantity chosen at the second stage. (III) Consumers are worse off in the Stackelberg game compared with the Cournot outcome given the same parameters. Group of answer choices: a. Only II is correct b. Only I is correct. c. All options are incorrect. d. Only III is correct e. More than one option is correct.
- Suppose that two firms produce mountain spring water and the market demand for mountain spring water is given as follows: P= 254 - 91 - 92 Firm 1 and Firm 2 have a MC = 50 a) Find the Cournot-Nash equilibrium price and quantity of each firm. b) Assume now that firm 1 becomes the Stackelberg leader. What will be the market price, output by each firm? Compared to part a, who gains? c) If Firm 1 chooses a quantity, then Firm 2 chooses a quantity (having observed Firm 1's quantity), then Firm 1 has an opportunity to revise its quantity (having observed Firm 2's quantity), then payoffs are determined, does either firm stand to gain relative to the case of simultaneous quantity choice? Why or why not? (hint: there is no need to do any calculation here).Three electricity generating firms are competing in the market with the inverse demand given by P(Q) = 20 – Q. All firms have constant marginal costs. Firm 1’s marginal cost is MC = 5; it has a capacity constraint of K1 = 5 units. Firm 2’s marginal cost is MC = 8; it has a capacity constraint of K2 = 2.5 units. Firm 3’s marginal cost is MC = 10; it has a capacity constraint of K3 = 2.5 units. A. The three firms compete in the style of Cournot. Please compute the Nash equilibrium quantities. Also compute the price in the Nash equilibrium. B. Which of these firms would have produced a larger quantity if it had a larger capacity?Two firms are competing in a Bertrand setting. The demand and costs equations are: Q1 = 88–4P1+2P2, Q2 = 88–4P2+2P1; MC1 = 9; and MC2 = 10. Instructions: Use no decimals. Do not round values if used for other calculations. d. Profits Firm 1 = $ and profits Firm 2 = $ e. If Firm 1 instead produces P1 = 16, the optimal P2 = . f. When one of the firms set a P < P-Duopoly, the best strategy for the other firm is to set: A. a P-BRF, and continue with this strategy afterward with the risk of economic profits = 0. B. a P-BRF, and after this one time, then continue with P-Duopoly. C. a P > P-Duopoly from now on, until the market reaches P-Monopoly. D. also P < P-Duopoly one time, then set P-Monopoly.
