Consider a duopoly market, where two firms sell differentiated prod-ucts, which are imperfect substitutes. The market can be modelledas a static price competition game, similar to a linear city model.The two firms choose prices p1 and p2 simultaneously. The deriveddemand functions for the two firms are: D1 (P1, P2) = ; +and D2 (P1, P2) =+ 2, where S > 0 and the parameter t > 0measures the degree of product differentiation. Both firms haveconstant marginal cost c > 0 for production.SP2-P12tS(a) Derive the Nash equilibrium of this game, including the prices,outputs and profits of the two firms.Pj-Pi derive(b) From the demand functions, q; = D; (pi, Pj) =the residual inverse demand functions: p; = P;(qi, Pi) (work outP:(qi, Pi)). Show that for t > 0, P:(q;, P;) is downward-sloping,aP:(gi-Pj)+2ti.e.,< 0. Argue that, taking p; > 0 as given, firm iis like a monopolist facing a residual inverse demand, and theoptimal q; (which equates marginal revenue and marginal cost)or pi makes P;(qi, P¡) = Pi > c, i.e., firm i has market power.

Demand Function for Firm 1Q1 = S2+P2-P12tDemand Function for Firm 2Q2 = S2+P1-P22tBoth Firm Have…

Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices pi and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = ; + and D2 (p1, P2) =+2, where S> 0 and the parameter t >0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. P2-P1 2t S (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms. (b) From the demand functions, q; = D; (pi, P;) = + ", derive the residual inverse demand functions: p; = P;(qi, P¡) (work out P:(qi, P;)). Show that for t > 0, P;(q;, P;) is downward-sloping, < 0. Argue that, taking p; 20 as given, firm i is like a monopolist facing a residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or p; makes P:(qi, p;) = pi > c, i.e., firm i has market power. Pj-Pi i.e., !be

Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices pi and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = ; + and D2 (p1, P2) =+2, where S> 0 and the parameter t >0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. P2-P1 2t S (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms. (b) From the demand functions, q; = D; (pi, P;) = + ", derive the residual inverse demand functions: p; = P;(qi, P¡) (work out P:(qi, P;)). Show that for t > 0, P;(q;, P;) is downward-sloping, < 0. Argue that, taking p; 20 as given, firm i is like a monopolist facing a residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or p; makes P:(qi, p;) = pi > c, i.e., firm i has market power. Pj-Pi i.e., !be

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 2E

See similar textbooks

Similar questions

Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices p1 and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = + P2P1 and D2 (P1, P2) =+ 2, where S > 0 and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. S 2t S (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms. Pj-Pi derive (b) From the demand functions, q; = D; (pi, Pj) = the residual inverse demand functions: p; = P:(qi, P¡) (work out P;(qi, P;)). Show that for t > 0, P;(4;, P;) is downward-sloping, aP:(gi-Pj) + 2t i.e., c, i.e., firm i has market power. (c) Calculate the limits of the equilibrium prices and profits as t → 0. What is P;(qi, p;) as t → 0? Is it downward sloping? Ar- gue that…
Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices pi and p2 simultaneously. The derived demand functions for the two firms are: D1 (p1, P2) and D2 (p1, P2) = + 2, where S > 0 and the parameter t > 0 을 + S + P2-P1 2t 2t measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms.
Consider a duopoly market, where two firms sell differentiated products, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices p, and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = SG+P1)and D2 (P1, P2)= S(;+-P2), where S > 0 and the parameter t > 0 measures the 2t 2t degree of product differentiation. Both firms have constant marginal cost c> 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profit of the two firms. (b) From the demand functions, qi= D¡ (p; , p¡ )= SG+P), derive the residual inverse demand functions: p; = P; (qi , p¡) (work out: P; (qi , P;)). Show that for t > 0, P;(qi , P;) is downward 2t aPi (qi ,Pj) . sloping, i.e., c, i.e., firm į has market power. %3D (c) Calculate the limits of the equilibrium prices and profit as t → 0 ? What is P; (qi , p;) as t → 0? Is it downward sloping?…
Consider a market that only includes two large firms. The (inverse) market demand is P = 100 – Q. 3q2. Firm 1 has a cost function of C, = 2q1, and firm 2 has a cost function of C2 Use a Cournot model to calculate the Nash equilibrium outputs q, and q2 of the two firms. and 92 (a) Give each firm's profit as a function of (b) Compute the Nash equilibrium q, and q2.
Two firms who make identical products engage in Cournot competition. The inverse market demand curve is: P(Q) = 60 - 10Q, where Q=9₁ +92. The cost functions are C₁(9₁) 1091 and C₂ (92) : 592 for firms 1 and 2, respectively. Find the Nash equilibrium quantities for each firm. (a) 9₁ = = 1.5, 92 2 (b) q₁ = 92 = 2 (c) q₁ = 92 = 1.5 (d) 9₁ = 1.5, 92 = 3 (e) None of the above options is correct.
Consider Hotelling's model (a street of length one, consumers uniformly distributed along the street, each consumer has a transportation cost equal to 2t, where t is the distance traveled). Suppose there are two gas stations, one located at 1/4 and the other located at 1. (a) Calculate the demand functions for the two firms. (b) If the two gas stations compete in prices and settle at a Nash equilibrium, will they charge the same price for gasoline? (assume that production costs are zero, that is, firms maximize revenue).
Two identical firms are engaged in Cournot competition, with cost functionsTCA(QA) = 10 QA and TCB(QB) = 10 QB. The market demand is given by P = 610 –2Q.a) Plot the best response functions and report the Cournot-Nash equilibrium quantities, price and profits.b) What are the prices, quantities, and profits for the firms if they decide to collude and share profits equally? c) Show that firms have an incentive the deviate from the collusive outcome.d) Find the Stackelberg equilibrium if A leads and B follows.e) Show the equilibria in the previous parts on the inverse demand function. Calculate and identify consumer surplus and deadweight loss in each equilibrium..
OLIGOPOLY 1.- Each of two firms, firms 1 and 2, has a cost function C(q) = 30q; the inverse demand function for the firms' output is p = 120-Q, where Q is the total output. Firms simultaneously choose their output and the market price is that at which demand exactly absorbs the total output (Cournot model).(a) Obtain the reaction function of a firm.(b) Map the function obtained in (a), and graphically represent the Cournot equilibrium in this market.(c) Repeat (b), this time analytically.(d) Now suppose that firm 1's cost function is C(q) = 45q instead, but firm 2's cost is unchanged. Analyze the new solution in the market.(e) Obtain the total surplus, consumer surplus, and industry profits in both cases, and compare. What is the effect of the worsening in firm 1's cost?
Consider a Stackelberg duopoly:There are two firms in an industry with demand Q = 1 − Pd.The “leader” chooses a quantity qL to produce. The “follower” observes qL and chooses a quantity qF.Suppose now that the cost function is Ci(qi) = qi2 for i = L, F. (a) Find the subgame perfect equilibrium. (b) Compare the equilibrium you found with the Nash equilibrium if the game was simultaneous (i.e., Cournot competition). Is the Nash equilibrium of the Cournot game also a Nash equilibrium of the sequential game? Why or why not?
consider a market with inverse demand P(Q) = 10 − Q and two firms with cost curves C1(q1) = 2q1 and C2(q2) = 2q2 (that is, they have the same marginal costs and no fixed costs). They compete by choosing quantities. Now consider a modified game, which goes as follows: First, Firm 1 decides whether to enter the market or not. As in the previous question, there is no fixed cost, even if the firm decides to enter. Next, Firm 2 observes Firm 1’s entry choice and decides whether to enter or not.Firm 2 has no fixed cost as well. If no firm enters, the game ends. If only one firm enters, that firm chooses quantity, operating as a monopolist. If both firms enter, then Firm 1 chooses quantity q1. Then, Firm 2 observes Firm 1’s choice of q1 and then chooses q2 (like in the previous question). If a firm does not enter, it gets a payoff of zero. Which of the following statements is consistent with the SPNE of this game? Hint: you don’t need complicated math to solve this problem.(a) Neither firm…
Cournot model: linear demand; identical firms. Q(P)=D-P TC(C)=cQ, where D>c a) Suppose that there are 2 firms. They can either choose to produce the Cournot quantity, or choose to produce one half of the monopoly quantity. Write down the 2X2 “payoff matrix” for this game. b) If D= 6 and c = 2, suppose that the game is repeated infinitely often with a discount factor of beta. For what values of beta will it be possible to sustain collusion? c) Now consider the same game with 3 firms. Compute the profits in the static Cournot- Nash equilibrium, and the profits when the 3 firms each produce one third of the monopoly quantity. For what values of beta will it be possible to sustain collusion in this case?
Consider a Cournot duopoly. The inverse demand function of the market is given by p = 10-Q, where p is the market price, and Q = 91 +92 is the aggregate output. The marginal costs of the two firms are C₁ 1 and C₂ = 4. = (a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits. (b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities 9₁ and 92, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices p₁ and P2. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for 10 Piāj if Pi > Pj firm ij, is Di (Pi, Pj) = 10-Pi 2 = if pipi. (Note, if Pi < Pj 10 - Pi here we assume that the efficient/parallel rationing applies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing…