Question

Consider a firm facing market demand qa p with a > 0; its cost of production c0 (a)(2pt] Find the optimal price p for this firm. In the questions below, consider only pure strategies Assume next for the questions that follow below that there are two firms, each with zero cost of production, who together face the market demand q 1 p. Firm l supplies the quantity q1 to the market. After observing this quantity, Firm 2 sets the price and supplies the rest of the market not covered by Firn 1 (which is 1-11-p if 1-11-p > 0 and 0 otherwise), while Firn l accepts the price set by Firin 2 and sells either q1 if q1 < 1-p, or 1-p, otherwise. In other words, Firm 1 gets the highest-valuation customers, which is analogous to what the lower-pricing firm in the Bertrand model with capacity constraints gets, and Firm 2 gets the rest of customers, which is analogous to what the higher-pricing firm in the same model get:s (b)(2pt] Write the solution to Firm 2's problem, given Firm 1's choice (c)[2pt] Write the maximisation problem of Firm 1 (d)(2pt Find the optimal quantity of Firm 1 (e)[2pt Find one subgame perfect Nash equilibriuof this game. Suppose now that firms play this game simultaneously. The game is the same otherwise: Firm 1 chooses the quantity 1, Firm 2 chooses p (f)(2pt Write the maximisation problem of Firm 1 (g)2pt Solve the maximisation problem of Fir (h)[2pt] Write the maximisation problem of Firm 2 (g)[2pt Solve the maximisation problem of Firm 2 (i)2pt Find one pure-strategy Nash equilibrium or explain why such an equilibrium does not exist

Answer 1

Answer:

(a) Market demand function: cost of production (c) 0 Firm maximize its profit Profit = p*q-c Derivate profit w.r.t to p ES Therefore, optimal price for this firm is a/2

where qq1+q2 given firm 1 choice q1, firm 2 demand is q2= 1-q1-p firm 2 maximization problem is profit 2 p q2-c firm 2 maximize w.r.t q2 aprofit/aq2-11-22-0 42 q1/2 p 1-g1-42 -1-91-1/2 -1-3q1/2

(c) firm 1 maximize its profit Therefore its maximization problem is profit1 p qi-c -(1-q1-q2 1- substituting value of q2

to calculate optimal quantity of firm 2 profit (1- 3q1 /2 91- aprofit/oqi-1-3q1-0 p 1- 3 (1/3)/2 therefore optimal quantity for firm 1 is 1/3

firm 1 produce 1/3 quantity at price 1/2 and firm 2 produce 1/6 at price 1/2

(f) when the firms play simultaneously The profit (objective) function of firm 1 is: Profit 1-p qi- as c 0 Profit1 1- q1 -q2)"q To solve firm 1 maximization problem, derivate firm 1 profit w.r.t to its quantity therefore q (1-q2)/2 and p 1- q2 - (1-q2)/2 1/2- 92/2

The profit (objective) function of firm 2 is: Profit 1-p 42- (1-q1-92)"92-c as c= 0 To solve firm 1 maximization problem, derivate firm 2 profit w.r.t to its quantity therefore q2 = (1-q1)/2 and p 1-q(1-q1)/2

0) The pure nash equilibrium is substituting value of q2 in q1 q1(1-q2)/2 q1(1-(1-q1)/2)/2 (1+q14 3q1/4 1/4 Therefore q1 1/3 Similarly a2 1/3 and p- 1-1/3-1/3