Stackelberg duopoly

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.

Bertrand Duopoly

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.