Question

1. Two firms produce a homogeneous good. Each unit that a firm produces costs it c, where 0 < c < 1. Market demand at a price of p equals 1−p for prices between 0 and 1.

1. (a) What is the maximal level of welfare? [50%]

2. (b) What is the level of welfare in equilibrium if the two firms compete by simultaneously choosing how much to produce, with price adjusting to equate market demand to total supply? [50%]

Answer 1

Solution:

Given that marginal cost, MC = c, and market demand function : q = 1 - p, where q = q1 + q2 with q1 = quantity supplied by firm 1, q2 = quantity supplied by firm 2.

Also, 0 < c, p < 1

a) Maximal welfare is attained when the market structure is perfectly competitive (or in case of Bertrand structure). Basically, when there is Marginal cost pricing, welfare is maximized. So, if the two firms follow marginal cost pricing, that is the price charged is same as the marginal cost. So, p = c (for both firms).

p = 1 - q

With p = c, c = 1 - q*

q* = (1 - c)

For firms, profits = total revenue - total cost

where total revenue = p*qi, total cost = c*qi, i = {1, 2}

With p = c, profits for each firm = 0 or producer surplus = 0 + 0 (for two firms) = 0

Consumer surplus (area of triangle for linear demand curve) = (1/2)*(q*)*(p(q=0) - c)

Now, p(q) = 1 - q, so p(q=0) = 1 - 0 = 1 (maximum willingness to pay when quantity = 0)

Consumer surplus = (1/2)*(1 - c)*(1 - c) = (1-c)²/2

Total welfare = producer surplus + consumer surplus

TW = 0 +  (1-c)²/2

So, maximal level of welfare =  (1-c)²/2

b) When deciding how much to produce, the two firms compete in Cournot manner.

Profits (as already mentioned) = p*qi - c*qi for firm i

For firm 1 then, profits = (1 - q1 - q2)*q1 - c*q1 = (1 - q2 - c)q1 - q1²

Differentiating this with respect to q1 and equating to 0 (first order condition for maximization), we get (1 - q2 - c) - 2q1 = 0

2*q1 + q2 = (1 - c) ... (1)

Similarly, for firm 2, profits = (1 - q1 - q2)*q2 - c*q2 = (1 - q1 - c)q2 - q2²

Differentiating this with respect to q2 and equating to 0 (first order condition for maximization), we get (1 - q1 - c) - 2q2 = 0

q1 + 2*q2 = (1 - c) ... (2)

Solving (1) and (2) simultaneously, we get

q1 = q2 = (1 - c)/3

So, q = q1 + q2 = 2(1-c)/3

price, p = 1 - q

p = 1 - 2(1 - c)/3 = (1 + 2c)/3

So, firm 1's profit = firm 2's profit = ((1 + 2c)/3)((1-c)/3) - c*((1-c)/3) = (1-c)²/9

So, producer surplus = 2*[(1-c)²/9] = 2*(1-c)²/9

Consumer surplus = (1/2)*(2(1-c)/3)*(1 - (1+2c)/3) = 2*(1-c)²/9

So, total welfare = PS + CS

TW =  2*(1-c)²/9 + 2*(1-c)²/9 = 4(1-c)²/9

Clearly, this welfare is less than maximal welfare ((1/2)(1-c)² > (4/9)(1-c)²)