Extension of the Cournot Model

Collusion

You may ask yourself why firms don't agree to work together to maximize profits for all rather than competing amongst themselves. In fact, we will show that firms do benefit when cooperating to maximize profits.

Assume both Firm 1 and Firm 2 face the same total market demand curve:

Q = 90 - P

where P is the market price and Q is the total output from both Firm 1 and Firm 2. Furthermore, assume that all marginal costs are zero, that is:

MC = MC1 = MC2 = 0

Verify that the reaction curves according to the Cournot model can be described as:

Q1* = 45 - Q2/2

Q2* = 45 - Q1/2

Solving the system of equations, we find:

Cournot Equilibrium: Q1* = Q2* = 30

Each firm produces 30 units for a total of 60 units in the market place. P is then 30 (recall P = 90 - Q). Because MC = 0 for both firms, the profit for each firm is simply 900 for a total profit of 1,800 in the market.

However, if the two firms were to collude and act as a monopoly, they would act differently. The demand curve and the marginal costs remain the same. They would act together to solve for the total profit maximizing quantity Q. Revenues in this market can be described as:

Total Revenue = P * Q = (90 - Q) * Q = 90 * Q - Q^2

Marginal Revenue is therefore:

MR = 90 - 2 * Q

Imposing the profit maximizing condition (MR = MC), we conclude: Q = 45

Each firm now produces 22.5 units for a total of 45 in the market. The market price P is therefore 45. Each firm makes a profit of 1,012.5 for a total profit of 2,025.

Notice that the Cournot equilibrium is much better for the firms than perfect competition (under which no one makes any profits) but worse than the collusive outcome. Also, the total quantity supplied is lowest for the collusive outcome and highest for the perfectly competitive case. Because the collusive outcome is more socially inefficient than the competitive oligopoly outcome, the government restricts collusion through anti-trust laws.

We now extend the Cournot Model of duopolies to an oligopoly where n firms exist. Assume the following:

Each firm chooses a quantity to produce.

All firms make this choice simultaneously.

The model is restricted to a one-stage game. Firms choose their quantities only once.

All information is public.

Recall that in the Cournot model, the strategic variable is the output quantity. Each firm decides how much of a good to produce. All firms know the market demand curve, and each firm knows the cost structures of the other firms. The essence of the model: each firm takes the other firms' choice of output level as fixed and then sets its own production quantities.

Let's walk through an example. Assume all firms face a single market demand curve as follows:

Q = 100 - P

where P is the single market price and Q is the total quantity of output in the market. For simplicity's sake, let's assume that all firms face the same cost structure as follows:

MC_i = 10 for all firms I

Given this market demand curve and cost structure, we want to find the reaction curve for Firm 1. In the Cournot model, we assume Qi is fixed for all firms i not equal to 1. Firm 1's reaction curve will satisfy its profit maximizing condition, MR1 = MC1. In order to find Firm 1's marginal revenue, we first determine its total revenue, which can be described as follows

Total Revenue = P * Q1 = (100 - Q) * Q1 = (100 - (Q1 + Q2 +...+ Qn)) * Q1 = 100 * Q1 - Q1 ^ 2 - (Q2 +...+ Qn)* Q1

The marginal revenue is simply the first derivative of the total revenue with respect to Q1 (recall that we assume Qi for i not equal to 1 is fixed). The marginal revenue for firm 1 is thus:

MR1 = 100 - 2 * Q1 - (Q2 +...+ Qn)

Imposing the profit maximizing condition of MR = MC, we conclude that Firm 1's reaction curve is:

100 - 2 * Q1* - (Q2 +...+ Qn) = 10 => Q1* = 45 - (Q2 +...+ Qn)/2

Q1* is Firm 1's optimal choice of output for all choices of Q2 to Qn. We can perform analogous analysis for Firms 2 through n (which are identical to firm 1) to determine their reaction curves. Because the firms are identical and because no firm has a strategic advantage over the others (as in the Stackelberg model), we can safely assume all would output the same quantity. Set Q1* = Q2* =... = Qn*. Substituting, we can solve for Q1*.

Q1* = 45 - (Q1*)*(n-1)/2 => Q1* ((2 + n - 1)/2) = 45 => Q1* = 90/(1+n)

By symmetry, we conclude:

Qi* = 90/(1+n) for all firms I

In our model of perfect competition, we know that the total market output Q = 90, the zero profit quantity. In the n firm case, Q is simply the sum of all Qi*. Because all Qi* are equal due to symmetry:

Q = n * 90/(1+n)

As n gets larger, Q gets closer to 90, the perfect competition output. The limit of Q as n approaches infinity is 90 as expected. Extending the Cournot model to the n firm case gives us some confidence in our model of perfect competition. As the number of firms grow, the total market quantity supplied approaches the socially optimal quantity.

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