Monopolies & Oligopolies: Duopolies and Oligopolies

Previous Next

The solution to the Cournot model lies at the intersection of the two reaction curves. We solve now for Q₁^*. Note that we substitute Q₂^* for Q₂ because we are looking for a point which lies on Firm 2's reaction curve as well.

Q1* = 45 - Q2*/2 = 45 - (44 - Q1*/2)/2
= 45 - 22 + Q1*/4
= 23 + Q1*/4
=> Q1* = 92/3

By the same logic, we find:

Q2* = 86/3

Again, we leave the actual computation of Q₂^* as an exercise for the reader. Note that Q₁^* and Q₂^* differ due to the difference in marginal costs. In a perfectly competitive market, only firms with the lowest marginal cost would survive. In this case, however, Firm 2 still produces a significant quantity of goods, even though its marginal cost is 20% higher than Firm 1's.

An equilibrium cannot occur at a point not in the intersection of the two reaction curves. If such an equilibrium existed, at least one firm would not be on its reaction curve and would therefore not be playing its optimal strategy. It has incentive to move elsewhere, thus invalidating the equilibrium.

The Cournot equilibrium is a best response made in reaction to a best response and, by definition, is therefore a Nash equilibrium. Unfortunately, the Cournot model does not describe the dynamics behind reaching equilibrium from a non-equilibrium state. If the two firms began out of equilibrium, at least one would have an incentive to move, thus violating our assumption that the quantities chosen are fixed. Rest assured that for the examples we have seen, the firms would tend towards equilibrium. However, we would require more advanced mathematics to adequately model this movement.

The Stackelberg duopoly model of duopolies is very similar to the Cournot model. Like the Cournot model, the firms choose the quantities they produce. In the Stackelberg model, however, the firms do not move simultaneously. One firm holds the privilege to choose production quantities before the other. The assumptions underlying the Stackelberg model are as follows:

Each firm chooses a quantity to produce.
A firm chooses before the other in an observable manner.
The model is restricted to a one-stage game. Firms choose their quantities only once.

To illustrate the Stackelberg model, let's walk through an example. Assume Firm 1 is the first mover with Firm 2 reacting to Firm 1's decision. We assume a market demand curve of:

Q = 90 - P

Furthermore, we assume all marginal costs are zero, that is:

MC = MC1 = MC2 = 0

We calculate Firm 2's reaction curve in the same way we did for the Cournot Model. Verify that Firm 2's reaction curve is:

Q2* = 45 - Q1/2

To calculate Firm 1's optimal quantity, we look at Firm 1's total revenues.

Firm 1's Total Revenue = P * Q1 = (90 - Q1 - Q2) * Q1
= 90 * Q1 - Q1 ^ 2 - Q2 * Q1

However, Firm 1 is not forced to assume Firm 2's quantity is fixed. In fact, Firm 1 knows that Firm 2 will act along its reaction curve which varies with Q₁. Firm 2's quantity very much relies on Firm 1's choice of quantity. Firm 1's Total Revenue can thus be rewritten as a function of Q₁:

R1 = 90 * Q1 - Q1 ^2 - Q1 * (45 - Q1/2)

Marginal revenue for firm 1 is thus:

MR1 = 90 - 2 * Q1 - 45 + Q1
= 45 - Q1

When we impose the profit maximizing condition (MR = MC), we find:

Q1 = 45

Solving for Q₂, we find:

Q2 = 22.5

Although much of the logic behind the Stackelberg model is used in the Cournot model, the two outcomes are radically different: being the first to announce creates a credible threat. In the Cournot model, both firms make their choices simultaneously and have no communication beforehand. In the Stackelberg model, Firm 1 not only announces first, but Firm 2 knows that when Firm 1 announces, Firm 1's actions are credible and fixed. This demonstrates how a slight change in the flow of information can drastically impact the outcome of a market.

The Bertrand duopoly Model, developed in the late nineteenth century by French economist Joseph Bertrand, changes the choice of strategic variables. In the Bertrand model, rather than choosing how much to produce, each firm chooses the price at which to sell its goods.

Rather than choosing quantities, the firms choose the price at which they sell the good.
All firms make this choice simultaneously.
Firms have identical cost structures.
The model is restricted to a one-stage game. Firms choose their prices only once.

Although the setup of the Bertrand Model differs from the Cournot model only in the strategic variable, the two models yield surprisingly different results. Whereas the Cournot model yields equilibriums that fall somewhere in between the monopolistic outcome and the free market outcome, the Bertrand model simply reduces to the competitive equilibrium, where profits are zero. Rather than take you through a series of convoluted equations to derive this result, we will simply show there could be no other outcome.

The Bertrand equilibrium is simply the no profit equilibrium. First, we will demonstrate that the Bertrand outcome is indeed an equilibrium. Imagine a market in which two identical firms sell at market price P, the competitive price at which neither firm earns profits. Implicit in our argument is our assumption that at equal price, each firm will sell to half the market. If Firm 1 were to raise its price above the market price P, Firm 1 would lose all its sales to Firm 2 and would have to exit the market. If Firm 1 were to lower its price below P, it would be operating below cost and therefore at a loss overall. At the competitive outcome, Firm 1 cannot increase profits by changing its price in either direction. By the same logic, Firm 2 has no incentive to change prices. Therefore, the no profit outcome is an equilibrium, in fact a Nash equilibrium, in the Bertrand model.

We now demonstrate uniqueness of the Bertrand equilibrium. Naturally, there can be no equilibrium where profits are negative. In this case, all firms would operate at a loss and exit the market. It remains to be shown that there is no equilibrium where profits are positive. Imagine a market in which two identical firms sell at market price P, which is greater than cost. If Firm 1 were to raise its price above the market price P, Firm 1 would lose all its sales to Firm 2. However, if Firm 1 were to lower its price ever so slightly below P (while still remaining above MC), it would capture the entire market at a profit. Firm 2 is faced with the same incentives, so Firm 1 and Firm 2 would undercut each other until profits are driven to zero. Therefore no equilibrium exists when profits are positive in the Bertrand model.

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 Q_i 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 Q_i 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 Q_n. 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^* = ... = Q_n^*. 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 Q_i^*. Because all Q_i^* 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.