Suppose two firms engage in simultaneous quantity competition. Both firms have 0 marginal cost. Firm A : P(Q)= 24-Q Firm B: P(Q)= 24-2Q a) Find the Nash Equilibrium quantities q^NE and profits.
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Suppose two firms engage in simultaneous quantity competition. Both firms have 0
marginal cost.
Firm A : P(Q)= 24-Q
Firm B: P(Q)= 24-2Q
a) Find the Nash
(b) Find the
(c) Now suppose the game is repeated infinitely and each firm has a common discount
factor δ. Find the required discount factor to sustain the following grim trigger
strategy as a SPNE: Play Q^M /2 if this has been played in every previous period,
otherwise play q^NE.
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- Inverse elasticity rule Use the first-order condition (Equation 15.2 ) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm's residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule: pMCp=sieQ,p , where si=qi/Q is firm i's market share and eQp is the elasticity of market demand. Compare this version of the inverse elasticity rule with that for a monopolist from the previous chapter.Two firms produce a homogeneous good and compete in price. Prices can only take integer values. The demand curve is Q = 6 p, where p denotes the lower of the two prices. The lower - priced firm meets all the market demand. If the two firms post the same price p, each one gets half the market demand at that price, i. e., each gets (6p)/2. Production cost is zero.a) Show that the best response to your rival posting a price of 6 is to post the monopoly price of 3. What is the best response against a rival's price of 4? of 5?Market demand is given by p= 12-Q. There are two firms; the incumbent firm (1) and the entrant firm (E). Incumbent moves first by choosing quantity q, from the interval [0.1]. The entrant observes q and decides whether or not to enter and how much to produce if he enters (qe). There is no fixed cost of entry. If the entrant decides to stay out, his profit is zero and the incumbent enjoys a monopoly position. Suppose that both incumbent and entrant have identical marginal costs equal to c= 8 a) What is the subgame perfect equilibrium of this game? What are the quantities produced by the incumbent and entrant? What are their profits? b) What is the minimum quantity that must be produced by the incumbent to deter entry (to make entry unprofitable)? In this game, will the incumbent ever try to deter entry by increasing quantity?
- Gamma and Zeta are the only two widget manufacturers in the world. Each firm has a cost function given by: C(q) = 10+20q + q^2, where q is number of widgets produced. The market demand for widgets is represented by the inverse demand equation: P = 200 - 2Q where Q = q1 + q2 is total output. Suppose that each firm maximizes its profits taking its rival's output as given (i.e. the firms behave as Cournot oligopolists). a) What will be the equilibrium quantity selected by each firm? What is the market price? What is the profit level for each firm? Equilibrium quantity for each firm__ price__ profit__ b) It occurs to the managers of Gamma and Zeta that they could do a lot better by colluding. If the two firms were to collude in a symmetric equilibrium, what would be the profit-maximizing choice of output for each firm? What is the industry price? What is the profit for each firm in this case? Equilibrium quantity for each firm__ price__ profit__ c) What minimum discount factor is required…Suppose two firms face market demand of P=150-Q, where Q=q1+q2 . Both firms have the same unit cost of 26. Assume the firms compete a la Stackelberg. Firm 1 is the leader and Firm 2 is the follower in this market. （a）What is the follower’s total revenue function? （b）Determine the equilibrium output level for both the leader and the follower. （c）Determine the equilibrium market price. （d）Determine the profits of the leader and the follower (Please write clearly, thank you）Consider Cournot competition with n identical firms. Suppose that the inverse demand function is linear with P(X) = a - bX, where X is total industry output, a; b > 0. Each firm has a linear cost function of the form C(x) = cx, where x stands for per firm output. It is assumed that a > c. a. At the symmetric equilibrium, what are the industry output and price levels? What are the equilibrium per firm output and profit levels? What is the equilibrium social welfare (defined as the difference between the area under the demand function and total cost)? b. Now let m out of n firms merge. Show that the merger is profitable for the m merged firms if and only if it involves a pre-merger market share of 80 percent. c. Show that each of the (n – m) non merged firms is better off after the merger. d. Show that the m-firm merger increases industry price and also lowers consumer welfare.
- Suppose a certain city has a monopoly cable-television company. This company has total costs TC = 0.25Q2 + 30Q + 70. (Hint: using calculus, this means MC = 0.5Q+ 30since MC is the derivative of TC with respect to output.) The demand in the community is approximated by the equationQd = 60- P/2(alternatively, you can write the demand equation as Qd = 60–0.5P). Graphically depict the demand curve as well as the marginal cost (MC) curve. If the cable company is free to choose its own pricePm and quantityQm, graphically depictthe monopoly equilibrium price and quantity. Add any other curve(s) to your diagram that may be required to obtain this outcome. Compute and state the exact monopolist equilibrium pricePm and quantityQm that you depicted graphically.Suppose two firms face market demand of P=150-Q, where q=q1+q2 . Both firms have the same unit cost of C=20. Assume the firms compete a la Stackelberg. Firm 1 is the leader and Firm 2 is the follower in this market. a) What is the follower’s total revenue function? b) Determine the equilibrium output level for both the leader and the follower. c) Determine the equilibrium market price. d) Determine the profits of the leader and the follower.Suppose two firms face market demand of P=150-Q, where . Both firms have the same unit cost of C, C= 22. Assume the firms compete a la Stackelberg. Firm 1 is the leader and Firm 2 is the follower in this market. What is the follower’s total revenue function? Determine the equilibrium output level for both the leader and the follower. Determine the equilibrium market price. Determine the profits of the leader and the follower.
- Suppose two brothers own identical skydiving companies but have decided to experiment with different pricing structures. The older brother’s company, Air Adventures, charges everyone the same price, while the younger brother’s company, Sky Warriors, sets its prices using a twotiered, price-discrimination model. Assuming that both companies face the same market demand curves, marginal costs, and costs of production, and wield significant market power for their service area, which of the following is most likely to occur? a. Air Adventures will generate a similar net revenue to Sky Warriors. b. Sky Warriors will generate a higher net revenue than Air Adventures. c. Sky Warriors will generate a lower net revenue than Air Adventures. d. Air Adventures will generate a higher net revenue than Sky Warriors. e. Sky Warriors will eventually switch to the Air Adventures model.Two firms, 1 and 2, compete in price Market demand in period t is given by D(t) = AtD(p) with A > 0 The common discount factor is ? ? (0, 1) Suppose the firms use trigger strategies to collude at the monopoly price pm = arg max(p ? c)(A)tD(p) ? (A)t?m (note that pm does not depend on A and t due to the function form) Suppose the punishment after deviation is returning to marginal cost pricing forever If the firms collude, they set the same prices and evenly split the profits What are firms’ collusive profits in period t? If a firm undercuts below pm in period t, what are the (optimal) deviating price and deviating profit Write down the no-deviating condition in period t? Simplify the no-deviating condition and derive the critical discount factor ? Compared to when the market is shrinking (A 1) make collusion easier? Explain in words your finding in [e]Consider two firms, i = 1; 2, producing differentiated products and engaged in Cournot a. Given the market demands, what are the best-response functions of the two firms? b. Draw the best-response functions both for complements (d 0). c. Compute the Cournot equilibrium quantities and prices in this market. d. Compare the outcome between substitutes and complements goods. e. What are the profit-maximizing quantities and prices if firm i is a monopolist in this market? Compare with part c.
