1. Consider the Hotelling model of horizontal product differentiation. Suppose that two products (noted 1 and 2) are located at the extreme locations of the [0, 1] interval. (Assume firm 1 is located at 0 and firm 2 is located at 1.) Firm 1 and firm 2 have constant marginal costs of c₁ and c₂ respectively for production and maximize profits (c₁> C₂ > 0). Consumers are uniformly distributed on the unit interval and incur a disutility from traveling to the lo- cation of the product, which is linear in distance. i) Are the price charged by both the firms the same in the equilibrium?

1. Consider the Hotelling model of horizontal product differentiation. Suppose that two products (noted 1 and 2) are located at the extreme locations of the [0, 1] interval. (Assume firm 1 is located at 0 and firm 2 is located at 1.) Firm 1 and firm 2 have constant marginal costs of c₁ and c₂ respectively for production and maximize profits (c₁> C₂ > 0). Consumers are uniformly distributed on the unit interval and incur a disutility from traveling to the lo- cation of the product, which is linear in distance. i) Are the price charged by both the firms the same in the equilibrium?

Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)

14th Edition

ISBN:9781305506381

Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Publisher:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Chapter12: Price And Output Determination: Oligopoly

Section: Chapter Questions

Problem 1E

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2. A homogenous good industry consists of two firms (firm 1 and firm 2). Their cost functions are cq and cq2, respectively, where c<2. The market demand function is p=10-Q, where Q=q₁+q₂. (a) Assume that the two firms play the Bertrand price game. Find the firms' choices in the Bertrand-Nash equilibrium. (b) Assume that the two firms play the Cournot quantity game. Find the firms' choices in the Cournot-Nash equilibrium. (c) Assume the two firms play the Stackelberg game with firm 1 as the leader. Find the firms' equilibrium choices in the Stackelberg equilibrium.
We now consider a duopoly model where the firms offer products with different qualitiesand consumers differ from each other in how much they care for quality. Suppose the firms offer their products with quality si ∈ [0, 1] and consumers’ ‘location’ in terms ofhow much they care about quality is given by a parameter θ ∈ [0, 1] and consumers are uniformlydistributed over this interval. Suppose the firms first choose their quality s1 and s2 and then setprices p1 and p2. A consumer of type θ derives utilityvi = r − pi + θsifrom consuming a unit from firm i and where reservation price r is high enough that everyonepurchases from one of the two firms. Suppose the marginal cost of producing increases with quality.There are no fixed costs and the total variable cost isC(qi, si) = csiqiso that marginal cost is csi and increases with quality. Further, let c = 1 so marginal cost is siConsider the second stage where given a choice of qualities s1 and s2 with s1 < s2, the firmssimultaneously…
Two firms produce identical products at zero cost, and theycompete by setting prices. If each firm charges a low price,then both firms earn profits of zero. If each firm charges ahigh price, then each firm earns profits of £30. If one firmcharges a high price and the other firm charges a low price,the firm that charges the lower price earns profits of £50, andthe firm charging the higher price earns profits of zero. (a) Which oligopoly model best describes this situation?(b) Write this game in normal form.(c) Suppose the game is infinitely repeated. Can theplayers sustain the "collusive outcome" as a Nashequilibrium if the interest rate is 50 percent? Explain. Please answer the a, b and c parts.
1. Two firms (A and B) play a competition game (i.e. Cournot) in which they can choose any Qi from 0 to ¥. The firms have the same cost functions C(Qi) = 10Qi + 0.5Qi2, and thus MCi = 10 + Qi. They face a market demand curve of P = 220 – (QA + QB). a. Assume firm A chooses quantity first. Frim B observes this choice and then chooses its own quantity. What is Frim B's profit as a function of QA and QB? b. Firm B has MRB = 220 – 2QB – QA. What is firm B’s best response to an arbitrary QA selected by firm A? c. Given that firm A expects firm B’s best response, what is firm A’s profit as a function of QA? (Hint: the only unknown variable in the profit function should be QA) d. Firm A has MRA = 150 – 4QA/3. What are the equilibrium QA and QB selected in this game? e. What is the equilibrium price, and how much profit does each firm collect?
2. Suppose that identical duopoly firms have constant marginal costs of $10 per unit. Firm 1 faces a demand function of q₁ - 100 - 2p₁ + P2, where q₁ is Firm 1's output, p₁ is Firm 1's price, and p2 is Firm 2's price. Similarly, the demand function Firm 2 faces is 92 100 2p2 + P₁. Solve for the Nash-Bertrand equilibrium. =
5. N - Σ Consider a Cournot model in which N firms compete with each other by setting quantities. The market inverse demand function is P = a i=1 qi, where a > 0 and q; is the quantity of firm i. Firm i's cost function is quadratic: q, where c₂ > 0. (a) Suppose N 2. Find the Nash equilibrium. Show which firm produces more in the equilibrium and explain your result. (b) = Suppose N≥ 2 and ci = c for all i. Find the Nash equilibrium. Show whether the firms produce more or less than the constant marginal cost case where the cost function is cqi, with a>c>0.
Problem 3 Consider a market with two firms. Each firm is located at one end of a line with lenght one. There is a mass one of consumers. The location of each consumer is given by 0 < x < 1 which is uniformly distributed (with density 1). Firms have no cost of production and set price simultaneously. a) Derive the demand for each firm by identifying the location of the indifferent con-sumer for each price pair. Assume that all consumers know about both products. b) Consider again that consumers can only buy after receiving an ad. Suppose there is an avdertising company that offers the firms to coordinate the targeting of their ads. The company suggests to inform all consumers with a location between 0 and 0.4 the product of the firm at location 0 and to all consumers between 0.6 and 1 the product of the firm at location 1. Determine the optimal prices for both firms if they accept this offer. What are the resulting profits? ( please solve question b only)
Two firms produce and sell differentiated products that are substitutes for each other. Their dernand curves are Firm 1:Q, 40-3P,+ P, Firm 2: 0, - 40 - 3P,+ P, Bolh firms have constant marginal costs of $5.00 per unit. Both fims set their own price and take their competitor's price as fixed. Use the Nash equilibrium concept to determine the equiltbirium set of prices. Since Ihe firms ane identical, they wil set the same prices and produce the same quantities. In equibrium, each firm will charge a price of 8 and produce units of output. (Enter your responses rounded to wo decimal places)
Consider a duopoly with a demand curve given by P = a –bQ, where a and b are positive constants and Q is the total production by the two firms. Firms sell identical goods and have an identical constant marginal cost of production c. Fixed costs are equal to zero. We assume firms choose quantities simultaneously (Cournot competition). a. Obtain the first order condition of profit maximization for each firm. Use graphical analysis and economic intuition to explain what they represent. [30%] b. Obtain the profit maximizing quantity for each firm. Explain what they represent using game theory concepts. [20%] c. Demonstrate using relevant graphical analysis and economic intuition that the results obtained in b are not a Pareto Optimum for the firms involved. [20%] d. How would the graphical analysis in part a change if Firm A had a fixed cost of production?
Initially there are six firms producing differentiated products. The demand function for the good produced by firm i, i=1,2..,6, is given by qi = 10-2pi+0.3 summation pj where the sum is taken over the five prices other than firm i. Each firm has the same marginal cost c. The firms choose prices simultaneously; that is, they are differentiated products Bertrand competitors. (a) Solve for the symmetric Nash equilibrium prices. (b) Suppose that you observe each firm to set a price of 4.8. What must c be? (c) Suppose that two of the six firms merge to become a single firm. The firm continues to produce both goods. Using the marginal cost you found in (b), derive the new post-merger Nash equilibrium prices.
demand is y = 500 1. The Corleone and Chung families are the only providers of a good in the U.S. The market 20p. The costs of production for each of them are represented by the cost functions C₁(y₁) = 10∙y₁ and C₂(y₂) = 7 y2, respectively. Suppose both families must choose their output levels simultaneously. a) Derive their reaction functions. b) Calculate the Cournot equilibrium in this market. Indicate output levels, market price, and individual profits. (Answer: y₁ = 80 units, P = 14 USD) -
1. Best responses in a Cournot Oligopoly Firm A and Firm B sell identical goods Total market demand for the good is: The inverse demand function is therefore 1 P(QM) = 780 -Q=780 -0.02222QM 45 QM is total market production (i.e., combined production of firm's A and B. That is: Q(P) = 35, 100- 45P 2M = A +QB As a result, the inverse demand curve for each firm is: P(QA, QB) = 780- -1/32₁-752 45 Unlike the example in class, the two firms have different costs. = 4000A TCA (QA) TCB (QB) = 260QB = 780 -0.022220A -0.02222QB a. Using the demand function and the cost functions above, what is firm A's profit function. b. Using the profit function above and assuming that firm B produces Qg, calculate what firm A's best response is to firm B’s decision to produce QB- Note: Firm A's best response should be a function of B