How to Calculate Cournot Equilibrium

The Cournot equilibrium comes from Cournot's competition model, which shows how two companies in a duopoly can successfully compete without price fixing or colluding on their output. The model was developed in the 19th century by French mathematician Augustin Cournot while analyzing two companies selling spring water.

Cournot's model has had some modifications over the past century, most notably due to the work of John Nash, which is why the Cournot equilibrium is also called the Cournot-Nash equilibrium.

Cournot competition theory is an economic model that describes how two rival companies can compete when they are offering the same product in the same market without colluding. When there is only one company competing in a market, it's called a monopoly. When there is more than one companies competing, it's called an oligopoly, and when there are only two companies, it's called a duopoly.

Cournot's model works with the law of supply and demand, which means that producing more goods drives down prices, while producing fewer goods increases prices. A company with a monopoly only has to monitor its own production levels, but if you're in an oligopoly, the prices you charge will be determined by what you and your competitors produce. So, you should try to determine how much your competitor is going to produce so that you will be able to maximize your profits.

For Cournot's model to work, you need to assume:

• The companies should produce identical products or standardized products.
• The companies do not collude or form a cartel.
• The companies have the same insights on market demand.
• The market determines the sales price.

On the surface, it would seem to make sense that if there are two companies competing in the same market, then colluding on their pricing and output strategies would be mutually beneficial, assuming they didn't get caught. In the United States, the federal government and most state governments have severe penalties for this type of unethical business practice.

The fact that collusion is illegal is just one problem the companies face. The second problem with collusion is that there is nothing to stop either company from breaking the agreement to increase its own profit to the detriment of the other.

If two companies agree to collude and one decides to break that agreement, the other company could quickly go out of business, leaving the market with only one company: a monopoly. The principles behind this dilemma of colluding or not colluding can be easily understood by looking at the classic duopoly analysis game: the prisoner's dilemma.

Imagine that two criminals have been jointly accused of a serious crime and are being interviewed by the police. Evidence is sparse, so if neither prisoner confesses, they will both be charged with misdemeanors and will each serve one year in prison. If one confesses and the other does not, then the confessor will go free, and the other will serve 20 years in prison. If they both confess, they each will serve 10 years in prison.

If you were prisoner one, your choice is to confess or not regardless of what prisoner two does. The consequences of your decision would be:

• Confession: freedom or 10 years in prison
• No confession: one year or 20 years in prison

It should be obvious that your best choice is to confess since the minimum and maximum penalties for this decision are better than the minimum and maximum penalties of the other. Prisoner two, of course, is in the same predicament, and his best choice would be to confess as well. This is called a Nash equilibrium, named after John Nash, in which each person takes the best strategy for himself regardless of what the other person chooses.

For two companies in a duopoly, the dilemma to collude with each other or not to collude is exactly the same as the prisoner's dilemma. In this case, however, the best decision for both companies is not to abide to any collusion agreement.

Suppose two companies have agreed to collude on their production output and pricing so that they each make $10,000 profit. However, if firm A breaks the agreement and firm B does not, it will make $12,000 profit while firm B makes only $8,000 profit. If they both break the agreement, they would both make $9,000 profit.

• Collude: $10,000 or $8,000 profit
• Don't collude: $12,000 or $9,000 profit

Just as in the prisoner's dilemma, a decision to not collude gives a better worst-case and best-case reward.

Before calculating the Cournot equilibrium point, you must first know the demand curve for your market. In a demand curve, the quantity demanded (Q) is a function of price (P), which is Q = f(P).

Typically, as the price goes up, demand goes down, but this varies with every market. To calculate the demand curve, you would need data showing how sales were affected by changes in price, which you could then plot on a graph to show the curve.

Mathematically, the demand curve is expressed as P = a - b(Q), where (a) represents the intercept where price is 0, and (b) is the slope of the demand curve. Because there are two companies competing in a duopoly, the total quantity (Q) demanded is expressed as the quantity (q) for each company: Q = q₁ + q_2. Assuming both companies are producing the same quantities, this can also be expressed as Q = 2 q.

The marginal revenue curve of the two companies working in collusion can be determined by calculating the total change in revenue in the market for each one-unit change in output: MR = a - b(Q), where (a) represents the intercept where marginal revenue is 0, and (b) is the slope of the demand curve.

Once you know the optimal demand and optimal revenues for the market as a whole, you can now calculate the point of equilibrium for either company's production, disregarding any collusion between the two using this formula: π = P(Q) q − C(q).

In this formula:

• π is the individual company's profit.
• Q is the level of total market output.
• q is the individual company's output.
• P is the price of the product.
• C(q) is a function of the company's total costs associated with each level of its output. 

Using this formula, each company or either company can adjust its own production quantity (q) so that its individual profit (π) is at its maximum.

Because the Cournot model requires that a market has two companies competing in the same market and providing identical or nearly identical goods, there are few real examples where this could be found. Unlike Cournot's era, there are now many companies selling spring water.

Perhaps the closest example today would be fuel costs at gas stations. It's become nearly routine that gas stations in the same area often have similar or identical prices. Many people automatically assume that the gas companies are colluding on their pricing. However, this is probably not the case.

If you have two gas stations in the same neighborhood, their demand curves would be nearly identical, as would their costs (like rent), and thus, their marginal revenue costs would be nearly identical. If each company independently uses the formula for Cournot equilibrium, prices would be the same or nearly the same. Gas stations in a second town would have different demand curves and different costs, and thus, they would arrive at slightly different prices than the first town, but prices in that second town would also be nearly identical to each other.