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When firms sell identical goods, the Bertrand competition model results in the same equilibrium that we find in a perfectly competitive market, where price equals marginal cost. Because consumers care only about the price of the good (the product is identical across firms), each firm faces a demand that is perfectly elastic. Any increase in a firm’s price results in it losing all of its market share. The demand for the product will go to the firm offering the lowest price.

But what if firms face capacity constraints, and thus a limit on how much demand they can fill in the short run? With this restriction, if a firm undercuts another’s price, it can only steal as many customers as it has available capacity, and this capacity is probably not the size of the whole market.

In this model, there won’t be as much pressure for a firm to respond to price cuts because each firm will not lose all of its customers even if it keeps its price higher than that of a competitor. In fact, if the capacity of the low-price company is small enough, its competitor may not feel the need to cut prices much at all. This avoids the price-cutting spiral we saw in the Bertrand model.

In this situation, the critical issue is for a firm to determine how much capacity it has and thus what quantity it can produce.

We raise the idea of capacity constraints to motivate another major oligopoly model, Cournot competition (named after its first modeler, Augustin Cournot—yet another nineteenth-century French mathematician and economist).

In Cournot competition, firms produce identical goods and choose a quantity to produce rather than a price at which to sell the good. Individual firms do not control the price of their goods as they do in the Bertrand model. First, all firms in the industry decide how much they will produce; then based on the quantity produced by all firms, the market demand curve determines the price at which all firms’ output will sell. In Chapter 9, we learned that when dealing with a monopolist, the price-quantity outcome is the same whether a firm sets the price of its product or the number of units of output it produces. In an oligopoly, however, the market outcome differs depending on whether the firm chooses to set its price or its quantity.

To be more specific, let’s say there are two firms in a Cournot oligopoly, Firm 1 and Firm 2. (There can be more; we keep it at two to make it easy.) Each has a constant marginal cost of c, and both firms independently and simultaneously choose their production quantities q₁ and q₂. The good’s inverse demand curve is

P = a – bQ

where Q is the total quantity produced in the market: Q = q₁ + q₂.

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Firm 1’s profit π₁ is the quantity q₁ it produces times the difference between the market price P and its production costs c, or

π₁ = q₁ × (P – c )

Substituting the inverse demand equation for P, we find that

π₁ = q₁ × [a – b (q₁ + q₂ ) – c ]

Similarly, Firm 2’s profits are given by the equation

π₂ = q₂ × [a – b ( q₁ + q₂ ) – c ]

These two profit equations make clear that the firms in this oligopoly strategically interact. Firm 1’s profit is not just a function of its own quantity choice q₁, but also of its competitor’s quantity q₂. Likewise, Firm 2’s profit is affected by Firm 1’s output choice. The logic is that each firm’s production choice, through its influence on the market price P, affects the other firm’s profit.

An example of an industry that is like the Cournot model is the crude oil industry. Crude oil is a commodity; consumers are indifferent about oil from different sources. The price of oil is set on a worldwide market, and it depends on the total amount of oil supplied at a given time. Therefore, it’s realistic to assume that oil producers, even those such as Saudi Arabia or Iran with large oil reserves, do not choose the price of their outputs. They just choose how much to produce. (As we discussed earlier, OPEC chooses production quantity targets, not prices.) Oil traders observe these production decisions for all oil producers, and they bid oil’s market price up or down depending on how the total quantity produced (the market supply) compares to current demand. This price-setting process derives from the demand curve that connects total output to a market price.

Finding the equilibrium for a Cournot oligopoly will be easier to follow using an example. Suppose for simplicity that only two countries pump oil, Saudi Arabia and Iran. Both have a marginal cost of production of $20 per barrel. Also assume that the inverse demand curve for oil is P = 200 – 3Q, where P is in dollars per barrel and Q is in millions of barrels per day.

Finding the equilibrium for the Cournot model is similar to doing so for a monopoly, but with the one change noted above: The market quantity Q is the sum of the quantities produced in Saudi Arabia q_S and Iran q_I , rather than just the monopolist’s output: Q = q_S + q_I . After recognizing this difference, we follow the same steps used to solve for a monopoly’s profit-maximizing output. That is, we find each country’s marginal revenue curve, and then find the quantity at which marginal revenue equals marginal cost.

Let’s examine Saudi Arabia’s profit maximization first. As we learned in Section 9.2, we can more easily find a firm’s marginal revenue curve by starting with its inverse demand curve. Therefore, we start by writing the inverse demand curve equation in terms of the quantity choices of each country:

P = 200 – 3Q = 200 – 3(q_S + q_I) = 200 – 3q_S – 3q_I

Because the slope of the marginal revenue curve is twice the slope of the inverse demand function, Saudi Arabia’s marginal revenue curve is8

MR_S = 200 – 6 q_S – 3q_I

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Saudi Arabia maximizes profit when it produces the quantity at which its marginal revenue equals its marginal cost:

200 – 6q_S – 3q_I = 20

We can solve this equation for Saudi Arabia’s profit-maximizing output:

q_S = 30 – 0.5q_I

This outcome differs from the monopoly outcome: If Saudi Arabia were a monopoly, setting marginal revenue equal to marginal cost would result in a single quantity Q because its quantity supplied q_S would be the market quantity supplied Q. In this example, however, Saudi Arabia’s profit-maximizing output depends on the competitor’s output q_I. Similarly, Iran’s profit-maximizing q_I depends on q_S because it faces the same market demand curve and has the same marginal cost:

q_I = 30 – 0.5q_S

This result shows that one country’s output choice effectively decreases the demand for the other country’s output. That is, the demand curve for one country’s output is shifted in by the amount of the other country’s output. If the Saudis expect Iran to produce, say, 10 million barrels per day (bpd), then Saudi Arabia would effectively be facing the demand curve

P = 200 – 3q_S – 3q_I = 200 – 3q_S – 3(10) = 170 – 3q_S

If it expected Iran to pump out 20 million bpd, Saudi Arabia would face the demand curve

P = 200 – 3q_S – 3(20) = 140 – 3q_S

This demand that is left over to one country (or more generally, firm), taking the other country’s output choice as given, is called the residual demand curve. We just derived Saudi Arabia’s residual demand curves for two of Iran’s different production choices, 10 and 20 million bpd.

In effect, a firm in a Cournot oligopoly acts like a monopolist, but one that faces its residual demand curve rather than the market demand curve. The residual demand curve, like any regular demand curve, has a corresponding marginal revenue curve (it’s called . . . wait for it . . . the residual marginal revenue curve). The firm produces the quantity at which its residual marginal revenue equals its marginal cost. That’s why Saudi Arabia’s optimal quantity is the one that sets 200 – 6q_S – 3q_I = 20. The left-hand side of this equation is Saudi Arabia’s residual marginal revenue (expressed in terms of any expected Iranian output level q_I ). The right-hand side is its marginal cost.

How does the profit-maximizing output of one country change with the other country’s expected production? In other words, what role do strategic interactions play in a Cournot oligopoly? This can be seen in Figure 11.3, which shows Saudi Arabia’s residual demand, residual marginal revenue, and marginal cost curves. (The Iranian case would be the same, just with the two countries’ labels switched.) The residual demand RD_S¹ and residual marginal revenue RMR_S¹ curves correspond to an Iranian output level of 10 million bpd. In other words, if Saudi Arabia expects Iran to produce 10 million bpd, Saudi Arabia’s optimal output quantity is 25 million bpd. If it expects Iran to produce 30 million bpd, Saudi Arabia’s residual demand and marginal revenue curves shift in to RD_s² and RMR_s² to P = 110 – 3q_S and MR = 110 – 6q_S , respectively. The Saudis’ optimal quantity then falls to 15 million bpd. At Iranian output levels higher than 30 million bpd, Saudi Arabia’s residual demand and marginal revenue curves would shift in further and its optimal quantity would fall.

Each competitor’s profit-maximizing output depends on the other’s and in an opposite direction: If a firm expects its competitor to produce more, it will reduce its production. Although this kind of interaction seems to create a hopeless chicken-and-egg problem, we can still pin down the specific production quantities for each country if we return to the concept of Nash equilibrium: Each producer does the best it can, taking the other producer’s action as given.

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To see what that implies for a Cournot oligopoly, note that the equation for each country’s profit-maximizing output is described given the particular output choice of the other country. The equation for q_S gives Saudi Arabia’s best response to any production level q_I that Iran might choose. Likewise, the q_I equation gives Iran’s best response to any Saudi production decision. In other words, when both equations hold simultaneously, each country is doing the best it can given the other country’s action. So, the Nash equilibrium is the combination of outputs that make both equations hold.

Cournot Equilibrium: A Graphical Approach We can show this graphically in Figure 11.4. Saudi Arabia’s output is on the vertical axis and Iran’s output is on the horizontal axis. The curves illustrated are reaction curves. A reaction curve shows the best production response a country (or firm, in a more general oligopoly context) can make given the other country’s/firm’s action. Because both reaction curves are downward-sloping, a firm’s optimal output falls as the other producer’s output rises.

Reaction curve SA shows Saudi Arabia’s best response to any production choice of Iran—it shows the points at which q_S = 30 – 0.5q_I . If it expects Iran to produce no oil (q_I = 0), for example, the profit-maximizing Saudi response is to produce q_S = 30 million bpd. This combination occurs at point A. The optimal q_S falls as Iranian production rises. If Iran produces q_I = 10, then Saudi Arabia maximizes its profit by producing q_S = 30 – 0.5(10) = 25 million bpd (point B). If q_I = 30, then the optimal q_S is 15 million bpd (point C ). Saudi Arabia’s optimal production continues to fall as Iran’s production rises until it hits zero at q_I = 60 million bpd (point D). At any q_I greater than 60 million bpd, the market price is below $20 per barrel (see the demand curve). Because this price is below the marginal cost of production, it wouldn’t be profitable for Saudi Arabia to pump any oil if Iran produced 60 million bpd.

Line I is the corresponding reaction curve for Iran’s profit-maximizing quantity q_I = 30 – 0.5q_S . It’s essentially the same as SA, except with the axes flipped. Just as Saudi Arabia’s profit-maximizing output falls with Iran’s production choice, Iran’s optimal production decreases with expected Saudi production q_S . The optimal q_I is 30 million bpd if q_S = 0, and it falls toward 0 as q_S rises toward 60 million bpd.

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Each country realizes that its actions affect the desired actions of its competitor, which in turn affect its own optimal action, and so on. This back-and-forth strategic interaction is captured in firms’ reaction curves and is why the equilibrium is found where reaction curves intersect. The intersection of the two reaction curves at point E shows the quantities at which both competitors are simultaneously producing optimally given the other’s actions. That is, point E is the Nash equilibrium of the Cournot oligopoly—the mutual best response. If one country is producing at point E, the other country would only reduce its profits by unilaterally producing at some other point. At this equilibrium, each country produces 20 million bpd, and total output is 40 million bpd.

Cournot Equilibrium: A Mathematical Approach In addition to finding the Cournot equilibrium graphically, we can solve for it algebraically by solving for the output levels that equate the two reaction curves. One way to do this is to substitute one equation into the other to get rid of one quantity variable and solve for the remaining one. For example, if we substitute Iran’s reaction curve into Saudi Arabia’s reaction curve for q_I , we find

q_S = 30 – 0.5q_I = 30 – 0.5(30 – 0.5q_S )

= 30 – 15 + 0.25q_S

0.75q_S = 15

q_S = 20

Thus, the equilibrium output for Saudi Arabia is 20 million bpd. If we substitute this value back into Iran’s reaction curve, we find that q_I = 30 – 0.5q_S = 30 – 0.5(20) = 20. Iran’s optimal production is also 20 million bpd. Equilibrium point E in Figure 11.4 has the coordinates (20, 20), and total industry output is 40 million bpd.

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The equilibrium price of oil at this point can be found by plugging these production decisions into the inverse market demand curve. Doing so gives P = 200 – 3(q_S + q_I ) = 200 – 3(20 + 20) = $80 per barrel. Each country’s profit is 20 million bpd × ($80 – $20) = $1,200 million = $1.2 billion per day, so the industry’s total profit is $2.4 billion per day.

Let’s compare this equilibrium in a Cournot oligopoly (Q = 40 million bpd at P = $80) and profit ($2.4 billion per day) to the outcomes in other oligopoly models we’ve analyzed. These results are described in Table 11.2.

Collusion Let’s first suppose Saudi Arabia and Iran can actually get their acts together and collude to act like a monopolist. In that case, they would treat their separate production decisions q_I and q_S as a single total output Q = q_S + q_I . Following the normal marginal-revenue-equals-marginal-cost procedure, we would find that Q = 30 million bpd. Presumably, the two countries would split this output evenly at 15 million bpd because they have the same marginal costs. This is less than the total Cournot oligopoly production of 40 million bpd that we just derived. Furthermore, because monopoly production is lower, the price is higher, too: Plugging this monopoly quantity into the demand curve, price becomes P = 200 – 3(30) = $110 per barrel. We also know that total industry profit must be higher in the collusive monopoly outcome. In this case, it’s 30 million bpd × ($110 – $20) = $2.7 billion per day (or $1.35 billion for each country). This total is $300 million per day higher than the Cournot competition outcome. At the collusive monopoly equilibrium, output is lower than at the Cournot equilibrium, and price and profit are higher.

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Bertrand Oligopoly with Identical Products Next, let’s consider the Nash equilibrium if the two countries competed as in the Bertrand model with identical products. In this case, we know that price will equal marginal cost, so P = $20. Total demand at this price is determined by plugging $20 into the demand curve: P = 20 = 200 – 3Q, or Q = 60 million bpd. The two countries would split this demand equally, with each selling 30 million bpd. Because both countries sell at a price equal to their marginal cost, each earns zero profit. At the Bertrand equilibrium, output quantity is higher than at the Cournot equilibrium, price is lower, and there is no profit.

Summary To summarize, then, in terms of total industry output, the lowest is the collusive monopoly outcome, followed by Cournot, then Bertrand:

Q_m < Q_c < Q_b

The order is the opposite for prices, with Bertrand prices the lowest and the collusive price the highest:

P_b < P_c < P_m

Similarly, profit is lowest in the Bertrand case (at zero), highest under collusion, with Cournot in the middle:

π_b = 0 < π_c < π_m

Therefore, the Cournot oligopoly outcome is something between those for monopoly and Bertrand oligopoly (for which the outcome is equivalent to perfect competition). And, unlike the collusive and Bertrand outcomes, the price and output in the Cournot equilibrium depend on the number of firms in the industry.

These intermediate outcomes are for a market with two firms. If there are more than two firms in a Cournot oligopoly, the total quantity, profits, and price remain between the monopoly and perfectly competitive extremes. However, the more firms there are, the closer these outcomes get to the perfectly competitive case, with price equaling marginal cost and economic profits being zero. Having more competitors means that any single firm’s supply decision becomes a smaller and smaller part of the total market. Its output choice therefore affects the market price less and less. With a very large number of firms in the market, a producer essentially becomes a price taker. It therefore behaves like a firm in a perfectly competitive industry, producing where the market price equals its marginal cost. Most Cournot markets are not at this limit, so price is usually above marginal cost, but for intermediate cases, more firms in a Cournot oligopoly lead to lower prices, higher total output, and lower average firm profits.

The fact that the intensity of competition changes with the number of firms in the market is a nice feature of the Cournot model. This prediction is more in line with many people’s intuitive view of oligopoly than the Bertrand model’s prediction that anything more than a single firm leads to a perfectly competitive outcome. The downside of the Cournot framework is that it’s a bit more of a stretch than usual to assume that companies can only compete in their quantity choices and have no ability to charge different prices. How many oligopolies could that describe? Oil seems a very special case.

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Economists David Kreps and José A. Scheinkman examined this assumption in more detail. They proved an important result (though it’s too mathematically advanced to detail here) that helps expand the applicability of the Cournot model.9 Kreps and Scheinkman showed that under certain conditions, even if firms actually set their prices instead of quantities, the industry equilibrium can look like the Cournot outcome. The key added element in Kreps’s and Scheinkman’s Cournot story is that firms must first choose their production capacity before they set their prices. The firms are then constrained to produce at or below that capacity level once they make their price decisions.

As an example of a market described by the Cournot model, imagine that a few real estate developers in a college town build student apartments that are identical in quality and size. Once these developers build their apartment buildings, they can charge whatever price the market will bear for the apartments, but their choice of prices will be constrained by the number of apartments they have all built. If, for some reason, the developers want to charge a ridiculously low rent of, say, $50 per month, they would probably not be able to satisfy all of the quantity demanded at that low price because they only have a fixed number of apartments to rent. If the developers first choose the number of apartments in their buildings and then sell their fixed capacity at whatever prices they choose, Kreps and Scheinkman show that the equilibrium price and quantity (which, as it turns out, will equal the developers’ capacity choice in this case) will be like a Cournot oligopoly.

This result means that in industries in which there are large costs of investing in capacity so that firms don’t change their capacity very often, the Cournot model will probably be a good predictor of market outcomes even if firms choose their prices in the short run. (In the long run, the firms could both change their capacity by building more apartment buildings and change the prices they choose.)