Bertrand Competition

Bertrand Competition Definition

What is the definition of Bertrand competition? Bertrand's competition model is an oligopoly model where firms producing homogeneous products compete in price. Let's take a look at an overview of some of this model's characteristics.

• The Bertrand model has the following characteristics:
  1. It is a single-period model;
  2. It describes the behavior of firms in an oligopoly;
  3. It is a static model.

These characteristics will become more apparent later when we explore an example of a duopoly. But before we move further, it would be helpful to look at some of the assumptions of Bertrand competition.

Bertrand Competition Assumptions

Let's look at this overview of the assumptions of Bertrand competition.

• There are several assumptions in the Bertrand model:
  1. Firms are rational, and their objective is to maximize their profits;
  2. Firms produce homogeneous products;
  3. Firms compete by setting prices;
  4. Firms make decisions simultaneously;
  5. Each firm treats the price set by its competitor as a given;
  6. There is no cooperation between the firms;
  7. Each firm can meet the entire market demand;
  8. The firm that sets the lowest price captures the entire market.

Hold onto these, as everything will become more transparent in the next section, where we will look at the model in terms of equations and graphs!

Bertrand Competition Example

Let's take a look at an example of Bertrand competition.We, economists, love to name things, so let's give the two companies that will be featured in this example some names: 'The Smart Firm' and 'The Witty firm.'We will consider a duopoly case where both firms move simultaneously to set their prices. To decide on these prices, they have to treat the quantities set by their competitor as a given. The two firms produce homogeneous products.The Smart firm is contemplating how it would set its price. It is taking a look at the market demand curve:\(P=300-Q=300-(Q_1+Q_2)\)

Where:

\(Q=Q_1+Q_2\)\(Q_1 - \hbox{Smart Firm's output}\)\(Q_2 - \hbox{Witty Firm's output}\)\(Q - \hbox{Total output of both firms}\)Both firms know that their marginal costs are quite similar and are equal to:\(MC_1=MC_2=\$30\)The Smart Firm knows the Cournot duopoly model. It has gone through the exercise of finding the equilibrium outputs if the two firms set their quantities simultaneously.It found them to be:\(Q_1=Q_2=90\)And the market price in the Cournot case was found to be:\(P=\$120\)And each firm would make a profit of:\(\pi_1=\pi_2=TR_1-TC_1=TR_2-TC_2=\)\(= P \times Q_i - MC_i \times Q_i = \$120 \times 90 - \$30 \times 90 = \$8,100\)

While the Smart Firm was doing all these calculations, the Witty Firm figured out that if they set the prices simultaneously, the lowest-price firm would capture the entire market! Indeed, it will make the most profits then! The firm selecting a higher price would simply lose all its customers. Why? Because they would choose to purchase an identical product at a lower price. But if the two firms set their prices equally, they would split the market in half. Therefore, the price that both firms would settle on would be a Nash equilibrium.

The Witty Firm decided to establish the price that it would set ahead of its competitor. It thinks of a reasonable price but is afraid that the Smart Firm will undercut it.

The Witty Firm continues to go through this exercise. Still, at all the reasonable prices, it is afraid that the Smart Firm will set its price just below and win the entire market. The Witty firm then bids its price down to where it can no longer lower it -- at its marginal cost.By this point, the Smart Firm has also figured it out and decided to set the price at their marginal cost.In this case:\(P_1=P_2=\$30\)From the demand equation:\(P=300-Q=300-(Q_1+Q_2)\)The total market output is:\(Q=300-P=300-30=270\)But what happened to all these profits that both firms had hoped for? Due to the price war, the firms make zero profits as they sell their products at a price equal to their marginal cost. This is what the Bertrand competition model predicts! This is the same equilibrium as in the perfect competition model.

Bertrand Competition Diagram

There is no diagram for the Bertrand competition. However, if we assume that the firms sell differentiated rather than homogeneous products. In that case, we can have some mathematical and diagrammatical analysis!

Let's use the Smart Firm and the Witty Firm again but in a different scenario.

Now, they each have no variable costs but have a fixed cost of:\(FC_1=FC_2=\$20\)

The demand curves for the Smart Firm and the Witty Firm are respectively:\(Q_1=12-2 \times P_1 + P_2\)\(Q_2=12-2 \times P_2 + P_1\)The duopolists will set their prices simultaneously while treating their competitor's price as given.The Smart Firm solves for its profit:

\(\pi_1=TR_1-TC_1= (P_1 \times Q_1) - FC_1=\)\(= P_1 \times (12-2 P_1 + P_2) - 20 = \)\(= 12 P_1 - 2 P_1^2 + P_1P_2 - 20 \)The Smart Firm treats the price that the Witty Firm will set as given and maximizes with respect to its own price:

\( \frac{\Delta \pi_1}{\Delta P_1}=12-4P_1+P_2=0\)Rearranging for P1, the Smart Firm finds its reaction function to be:\(P_1=3+\frac{1}{4}P_2\)The Witty Firm decided not to bother with the calculations. It secretly asked the accountant from the Smart Firm to share their findings and figured that their The Witty Firm's reaction function is symmetric:\(P_2=3+\frac{1}{4}P_1\)

Let's plot these reaction functions below:

Fig. 1 - Reaction functions example

Figure 1 shows the reaction functions of the Smart Firm and the Witty Firm. Where the two reaction functions intersect lies a Nash equilibrium. This equilibrium is stable as neither firm wants to deviate.But what would the prices the two firms charge at this equilibrium be? Take a look at Figure 2 below:

Fig. 2 - Price competition competitive equilibrium

Figure 2 shows that the two firms will charge $4 at the equilibrium. Each firm will earn a profit of $12.What would happen if the two firms colluded to fix prices? They would set the price that would maximize their joint profits. What would this price be? Take a look at Figure 3 below.

Fig. 3 - Price competition collusive equilibrium

Figure 3 above shows the price the Smart and Witty Firms would charge if they colluded. They would charge $6, each earning a profit of $16. The firms had finally figured out that a better outcome is possible when the two brilliant minds come together!

Bertrand Competition vs. Cournot Model

What is the difference between Bertrand competition vs. the Cournot model? There is a significant difference, and it is how the firms compete with each other. In Bertrand competition, firms compete by setting prices. In contrast, in the Cournot model, firms compete by setting quantities. These differences are at the core of different conclusions from the two models.Bertrand contended that an outcome of a duopoly could be the same result that the perfect competition model predicts. Prices will be as low as the firms' marginal costs. Conversely, Cournot thought that a colluding duopoly would set quantities and prices just like a monopolist does.

In Bertrand competition, firms engage in price wars as the primary assumption is that they compete by setting prices. On the other hand, Cournot assumed that the firms compete by setting quantities. He predicted that there would be no price wars, as there would be an equilibrium from which no firm would want to deviate.

Both models predict a final outcome that results in a Nash equilibrium. No firm can be better off in such an equilibrium by defecting from the status quo. In Bertrand competition, firms cannot lower their prices below marginal costs as they would be making losses. Similarly, they cannot raise the prices above marginal costs as they would lose all their customers to their competitors. In the Cournot model, no firm would be better off reducing or increasing its output. This is because the output was already chosen to maximize profits, given what their competitor is doing.