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.

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.

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