Question D1. There is a two firm oligopoly where each firm is selling differentiated products. Firm 1 faces a demand curve q, = 10 – p1 + 0.75p2 and Firm 2 q2 = 10 – p2 + 1/3p1. Each firm has fixed marginal costs of 5. a) If firm 1 sets a price of p, what is firm 2's best response? b) Work out the Nash Equilibrium for the game detailed above. c) Does this model still suffer from the Bertrand Paradox? Explain your answer.
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- 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…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 ThConsider 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?
- Two firms are considering simultaneously developing a new product for a market. The costs of developing the product are $10m but there will only be revenue in the market of $40m if only one of the firms develops the product. If both firms develop the product then earnings/revenues will be competed away. a) Capture this entry game in a payoff matrix. b) What is the Nash equilibrium and why? Please provide explanation. c) Does either firm have a dominant strategy? Please provide explanation.Consider an oligopoly market with 3 firms. They face a demand curve given by p(Q) = 300 – 5Q, where Pmin refers to the lowest price offered by any firm. The marginal costs are the following: C₁ = : 80, C₂ 210. The fixed costs for each firm are zero and firms compete a la Bertrand. Which of the following is consistent with a Nash equilibrium of this game? € refers to the minimal price change possible. = (a) Pmin (b) Pmin (c) Pmin (d) Pmin (e) Pmin = 80 and only firm 1 sells. 200, C3 - 190 and only firm 1 sells. = = 200 - € and only firm 1 sells. 200 and firms 1 and 2 sell. - = = 210 and all firms sell.Problem 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 =
- QUESTION 13 Consider a market where two firms (1 and 2) produce differentiated goods and compete in prices. The demand for firm 1 is given by D₁(P₁, P2) = 140 - 2p1 + P2 and demand for firm 2's product is D2 (P1, P2) 140 - 2p2 + P1 Both firms have a constant marginal cost of 20. What is the Nash equilibrium price of firm 1? (Only give a full number; if necessary, round to the lower integer; no dollar sign.)Two identical firms each have a cost function TC = 2y2 and the market demand for their output is P = -4Q+192a) Write the âbest responseâ function for each firm.b) Find the Nash equilibrium in this model c) Show that if each firm produces 1 fewer units than the result in (b), both firms make more profit. Use this information to construct a normal form game. Explain why this game is a prisonerâs dilemma.Table 2 below represents the payoff matrix for two firms, X and Y, who compete with each other. Payoffs are in millions of pounds (£) profit. Each firm may choose one of two strategies i.e. set a high price for its output or set a low price for its output. Neither firm knows what strategy the other will adopt. Table 2 11. Firm Y High Price Low Price Firm X High Price X-£7mn, Y-£7mn X=£1mn, Y=£15mn Low Price X-£15mn, Y-£1mn X=£4mn, Y=£4mn In the absence of collusion, which combination of strategies is most likely to occur? a) X sets a low price and Y sets a low price. b) X sets a high price and Y sets a low price. c) X sets a low price and Y sets a high price. d) X sets a high price and Y sets a high price.
- In a repeated game of Bertrand competition, suppose two firms are playing the grim trigger strategy. They discount future payoffs at B. A firm will play the monopoly price and earn 8 every period, so long as the other firm does the same. However, if a firm deviates from playing the monopoly price, the other firm plays the one-shot Bertrand outcome forever. Collusion is a Nash equilibrium if B is Select J [ Select) Space 1 options: less than or equal to, equal to, greater than or equal to Space 2 options: 0, 0.5, 1, 8, 16Game Theory. Consider a Stackelberg competition game with three firms. Firm 1 chooses q1 first. Firm 2 observes q1 and chooses q2. Firm 3 observes both and chooses q3. These three firms are the only firms in the market, so the sum of their outputs is equal to total market supply, i.e. q1+q2+q3=Q. Suppose demand is given by P=12-Q. For simplicity of calculation, suppose each firm has marginal costs of 0, i.e. c1(q1)=0, c2(q2)=0 and c3(q3)=0. (1) What quantity does Firm 1 produce in the SPNE of the game? (2) What quantity does Firm 2 produce in the SPNE of the game? (3) What quantity does Firm 3 produce in the SPNE of the game?Perrier and Apollinaris. Perrier and Apollinaris are two companies that sell mineral water in Tampa, FL. Each company has a fixed cost of $5,000 per period, regardless whether they sell anything or not. The two companies are competing for the same market and each firm must choose a high price ($2 per bottle) or a low price ($1 per bottle). Here are the rules of the game: At a price of $2, 5,000 bottles can be sold for total revenue of $10,000. At a price of $1, 10,000 bottles can be sold for total revenue of $10,000. If both companies charge the same price, they split the sales evenly between them. If one company charges a higher price, the company with the lower price sells the whole amount and the company with the higher price sells nothing. Payoffs are total profits. In this case, Apollinaris has: no dominant strategy. Perrier has a dominant strategy of P=$1. a dominant strategy of P=$1. Perrier also has a dominant strategy of P=$2. a dominant strategy…