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- 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) = ; + 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 P2-P1 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, Pi) (work out P:(qi, Pi)). Show that for t > 0, P:(q;, P;) is downward-sloping, aP:(gi-Pj) + 2t i.e., 0 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 pi makes P;(qi, P¡) =…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 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 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).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?The total cost for a product-testing firm is C(q)=70 + 20q2 q= number of products tested Price of a product = average cost Each corporation purchases one product test per year from a product-testing firm in the same city. All other inputs are ubiquitous. Suppose five corporations are initially distributed uniformly, with one corporation in each city (A,B,C,D,E). Is the initial distribution a Nash Equilibrium? Demonstrate it is not by finding how much one corporation would pay if they deviate and move to another city? What is the average price of having two tests conducted? (Which is the price that the corporation would pay if they "live" in a city where two tests are conducted) The average price of moving and thus, having two tests is: $_____
- Two firms, Firm 1 and Firm 2, compete by simultaneously choosing prices. Both firms sell an identical product for which each of 100 consumers has a maximum willingness to pay of $40. Each consumer will buy at most 1 unit, and will buy it from whichever firm charges the lowest price. If both firms set the same price, they share the market equally. Costs are given by ci(qi) = 16q₁. Because of government regulation, firms can only choose prices which are integer numbers, and they cannot price above $40. Answer the following: a) If Firm 1 chooses p₁ = 25, Firm 2's best response is to set what price? 24 b) If Firm 2 chooses the price determined in the previous question, Firm 1's best response is to choose what price? 23 c) If Firm 1 chooses p₁ = 12, Firm 2's best response is a range of prices. What is the lowest price in this range? 16 d) Now suppose both firms are capacity-constrained: Firm 1 can produce at most 34 units, and Firm 2 can produce at most 42 units. If firms set different…Three firms share a market. The demand function is P(q1, 92, 93) = 10 – q1 – 92 – 93, where q; is the output of firm i, Player i. The marginal cost per unit for each firm is zero. Suppose firm 1 is a market leader, and that firm 2, and firm 3 choose q2 and q3 simultaneously after observing q1 Find the SPE of this game. How much is produced by firm 2? Numerical answer How much is produced by firm 1, 2, and 3?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.
- Consider two firms producing homogeneous goods. Firm 1 and firm 2 simultaneously set outputs q and q2. The inverse demand is P=20-3(q₁+9₂) and both firms have marginal costs of 2. In a Nash equilibrium, the firms produce a. (qq) = (2,2) b. (992) = (1.5,3) Ⓒc. Each of the other suggestions might occur in a Nash equilibrium d. (9₁.92 )=(3,1.5)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..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?
