Question

Show how two Cournot mineral water producers use their cost and demand curves to determine their outputs given each others output.

Show how this determines their best response curves.

Answer 1

Let there be two mineral water producers with costs C1(q1)= c1q1 and C2(q2) = c2q2. First firm has a marginal cost of c1 and second firm has a marginal cost of c2. The newly created firms compete in quantities that is Cournot competition

The market demand function is P = a – bQ Where Q is the sum of each firm’s output q₁ and q_2.

Each firm will maximize its profit. This is done when each of them equates marginal revenue function and marginal cost function

Revenue for firm 1

R₁ = P*q₁ = (a– b(q₁ + q₂))*q₁ = aq₁ – bq₁² – bq₁q₂.

Firm 1 has the following marginal revenue and marginal cost functions:

MR₁ = a – 2bq₁ – bq₂

MC₁ = c1

Profit maximization implies:

MR₁ = MC₁

a – 2bq₁ – bq₂ = c1

which gives the best response function:

q₁ = (a – bq₂ – c₁)/2b

The same is done for firm 2

R₂ = P*q₂ = (a– b(q₁ + q₂))*q₂ = aq₂ – bq₂² – bq₁q₂.

Firm 1 has the following marginal revenue and marginal cost functions:

MR₂ = a – 2bq₂ – bq₁

MC₂ = c2

Profit maximization implies:

MR₂ = MC₂

a – 2bq₂ – bq₁ = c2

which gives the best response function:

q₂ = (a – bq₁ – c₂)/2b

Cournot equilibrium is determined at the intersection of these two best response functions:

This implies that the final ouput by each firm is determined as

q1* = (a - 2c1 + c2)/3b and q2* = (a - 2c2 + c1)/3b

The case with similar cost is easier. Both firms have same marginal cost = c. Hence the results are slightly changed. The best response functions are now q₁ = (a – bq₂ – c)/2b and q₂ = (a – bq₁ – c)/2b. These are drawn in the graph below. For the best response function for firm1, if q2 = 0, we have q1 = (a - c)/2b. If q1 is 0, we have q2 = (a - c)/b. This shapes the BRF for firm 1 as a red line. Similarly the BRF for firm 2 is a green line. Simultaneous equilibrium results when both BRFs meet at A. This gives Cournot equilibrium as

q1 = q2 = (a - c)/3b