Question

In the Bertrand duopoly, market demand is Q = a-Bp, and firms have no fixed costs and identical marginal cost. Find a Bertrand equilibrium pair of prices, (p1 , p2 ), and quantities, (q1, q2), when the following hold. a. Firm1 has fixed costs F>0. b. Both firms have fixed costs F > 0. c. Fixed costs are zero, but firm 1 has lower marginal cost than firm 2, so c2 > c1 > 0. (For this one, assume the low-cost firm captures the entire market demand whenever the firms charge equal prices.).

Answer 1

As per the question:

Bertand model is equivalent with Cournot model, while the choice variable is price in the Bertrand model. These models produce different examples of a Nash equilibrium and this is different as according to monopoly case. If the firm whether uses output or price as its strategic variable.They mostly used Bertand model.

market demand is Q = a-Bp, and firms have no fixed costs and identical marginal cost. Find a Bertrand equilibrium pair of prices, (p1 , p2 ), and quantities, (q1, q2), when the following hold. a. Firm1 has fixed costs F>0. b. Both firms have fixed costs F > 0. c. Fixed costs are zero, but firm 1 has lower marginal cost than firm 2, so c2 > c1 > 0

a linear inverse demand function, p = a – Bp, where Q = q1 + q2 and the parameters a and b are positive.1 Each owner then sets output to maximize its profits, and the equilibrium price clears the market (p*).

TheBertandt problem is to determine the optimal values of our variables of interest: q1, q2, p1, and p2. Notice that because the products are homogeneous, however, that p1 =

p2 = p* in equilibrium if both firms are to participate.Recall that for this to be a game, we must define the players, their choice variables, their payoffs, and the information set. For the remainder of this chapter, we will assume that assume that costs are positive and the firm i’s total cost equation is TCi = cqi, c > 0. In terms of

notation, c is the unit cost of production, subscript i signifies firm 1 or 2, and subscript j signifies

the other firm. Each firm’s goal is to choose the level of output that maximizes profits, given the

output of the other firm. The relevant characteristics of the game are:

· Players: Firms or owners 1 and 2.

Let p1 = p(q1) and p2 = p(q2) be

the prices at the optimal quantities. Then, by optimization:

p1q1 ? c1(q1) _ p2q2 ? c1(q2),

because profits at (p1, q1) must be larger than any other price/quantity combo including

(p2, q2). Similarly:

p1q1 ? c2(q1) _ p2q2 ? c2(q2).

Subtracting these equations, we get:

c2(q1) ? c1(q1) _ c2(q2) ? c1(q2).

Or,

c2(q1) ? c2(q2) ? (c1(q1) ? c1(q2)) _ 0.

Which can be written: Z q1

q2

c02(x)dx ? Z q1 q2

c01(x)dx _ 0.

Z q1 q2

[c02(x) ? c01(x)]

| _0 b{yzass. }

dx _ 0.

Hence if q2 > q1, the area would have to be negative which would violate this last

condition. Thus q2 _ q1 as required.