There are two firms (Firm A and Firm B) competing in a Cournot game. The market demand is P=400-2Qa- 2Qb. The marginal cost for both firms is constant and is equal to $40. a) Derive the reaction functions for these firms. b) Draw the reaction functions on the graph, as done in class. Label the axes clearly and indicate which curve belongs to each firm. c) Solve for the Cournot equilibrium quantities for each firm and the market price.
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- 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 the following normal form representation of the standard competition between firm A and firm B. Each firm can choose either standard A or standard B. Their payoffs are given as follows: Firm B A В A Firm A В 1 1 3 1 (1) (10 points) What's Nash equilibrium (NE) in this game? If there are more than one, find them all. But there is no NE, state that there is no NE. (2) (10 points) If you find a NE (or multiple Nash equilibria), is it (or are they) Pareto efficient?Duopoly: 1. Consider a duopoly game with 2 firms. The market inverse demand curve is given by P(Q) = 120-Q, where Q = 9₁ +9₂ and q; is the quantity produced by firm i. The firm's long run total costs are given by C₁(9₁)=2q₁ and C₂(92)=92, respectively. a. Determine the Nash Equilibrium for Cournot competition, in which firms compete based on quantity. What is each firm's best response as a function of the other firm's output? Graph these best response functions in on the same graph. Compute the associated payoffs for each firm. b. Determine the Nash Equilibrium for Bertrand competition, in which firms compete based on price and stand ready to meet market demand at that price. What is each firm's best response as a function of the other firm's price? Graph these best response functions in on the same graph. Compute the associated payoffs for each firm. Game Th
- In a given market the demand for a homogenous product is given by p(q) = 120 – 5Q. The market has two firms, firm 1 has a marginal cost cı 5 and firm 2 has a marginal cost c2 = : 10. (i) Assume that the firms compete in a Cournot game. Compute the price in equilibrium, quantity produced by each firm and deadweight loss generated in this market.2. An industry contains two firms that have identical cost functions C(q)=10+2q. The inverse demand function for the market is P=50-2Q where Q is the total industry output. Assuming the firms compete in quantities: Find the firms' best response functions. b. Solve for the Cournot Nash Equilibrium of the game. What is the total industry output in equilibrium? What is the equilibrium price? с. i. If both firms could collude, what would the industry output and price be? Suppose they decide that each firm produces half of the industry output found in part (i). Is this agreement self-enforcing? Explain. ii. a.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..
- Consider a Cournot duopoly with the inverse demand P = 200 − 2Q. Firm 1 and 2 compete by simultaneously choosing their quantities. Both firms have constant marginal and average cost MC = AC = 20. A) Find each firm’s best response function. Plot the best response functions (label the x-axes as ?1 and y-axes as ?2 ). B) Find the Cournot-Nash equilibrium quantities, profits and market price. Illustrate the equilibrium point on your graph in part (A). C) Suppose instead that firm 1 had MC = AC = 20, but firm 2’s MC = 8. What is the Cournot-Nash equilibrium outputs and profits now? How would this affect your answers to part (B)? ExplainConsider a Cournot duopoly with the inverse demand P = 200−2Q. Firm 1 and 2 compete by simultaneously choosing their quantities. Both firms have constant marginal and average cost MC = AC = 20. A) Find each firm’s best response function. Plot the best response functions (label the x-axes as ?1 and y-axes as ?2 ). B) Find the Cournot-Nash equilibrium quantities, profits and market price. Illustrate the equilibrium point on your graph in part AProblem 5.1. The inverse market demand for printer paper is given by P = 400 – 2Q. There are two firms who compete to produce this paper, each with a marginal cost of production equal to c = 40 over a large range of output (ie, assume constant marginal cost). The two firms compete in quantities, in other words they each simultaneously choose a quantity to produce (Cournot competition). Derive the Cournot-Nash equilibrium of this game. Please write final answers in the boxes, showing work in blank areas. (a) The reaction function for each firm. 91 (92): 92 (91) (b) Optimal output q for each firm. 92 = р = = π1 = (c) Market price (from demand curve). (d) Firm profits. 92 = π2 =
- 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…I have constructed a Bertrand game (competition in prices) and presented you with the reaction functions of each firm. p1 = 12.5 + p2/4 p2 = 5/2 + p1/2 a) Use excel to draw the reaction functions. b) Solve for the Nash equilibriumConsider a market in which there are two firms: A and B. Each firm produces a differentiated product and chooses its price. Assume that each firm can set price equal to $60 or $70. The payoffs associated with each set of prices are shown. If the firms choose price simultaneously, then the Nash equilibrium price for firm A is chooses price first and can commit to that price, then firm A will set its price equal to If firm A ○ A. $70; $60 B. $70; $70 ○ C. $60; $70 ○ D. $60; $60 Q Firm B's Price ✓ $60 $70 $1800 $1650 $60 $1800 $2250 Firm A's Price $2250 $2200 $70 $1650 $2200