ASHE analysis

Antoine Cournot was a French mathematician and philosopher. In the early to mid-1880s, Cournot used his understanding of mathematics to formulate and publish a significant model of what oligopolies look like. The model, known as the Cournot Duopoly Model (or the Cournot Model), places weight on the quantity of goods and services produced, stating that it is what shapes the competition between the two firms in a duopoly. In Cournot’s model, the key players in the duopoly make an arrangement to essentially divide the market in half and share it.

Cournot’s model speculates that in a duopoly, each company receives price values on goods and services based on the quantity or availability of the goods and services. The two companies maintain a reactionary relationship in regard to market prices, where each company changes and makes adjustments to their respective production, ending when an equilibrium is reached in the form of equal halves of the market for each firm.

Cournot duopoly, also called Cournot competition, is a model of imperfect competition in which two firms with identical cost functions compete with homogeneous products in a static setting. Cournot’s duopoly represented the creation of the study of oligopolies, more particularly duopolies, and expanded the analysis of market structures which, until then, had concentrated on the extremes: perfect competition and monopolies.

Cournot really invented the concept of game theory almost 100 years before John Nash, when he looked at the case of how businesses might behave in a duopoly. There are two firms operating in a limited market. Market production is: 𝑃(𝑄) = 𝑎 − 𝑏𝑄, where 𝑄 = 𝑞₁+ 𝑞₂ for two firms. Both companies will receive profits derived from a simultaneous decision made by both on how much to produce, and also based on their cost functions:

𝑇𝐶_𝑖 = 𝐶 − 𝑞_𝑖. So, algebraically:

𝑀𝑎𝑥 𝜋_𝑖(𝑄) = [𝑎 − 𝑏(𝑞_𝑖 + 𝑞_𝑗) − 𝑐]𝑞_𝑖 In order to maximise, the first order condition will be:

𝑞_𝑖 =𝑎 − 𝑏𝑞_𝑗 − 𝑐 2𝑏 And, if 𝑞_𝑖 = 𝑞_𝑗, then both equal:

𝑎 − 𝑐 3𝑏

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Therefore, the reaction functions (blue lines), where the key variable is the quantity set by the other firm, will take the following form:

𝑅𝐹_𝑖(𝑞_𝑗) = 𝑎 − 𝑐 2𝑏 −1

2𝑞_𝑗

What all this explains is a very basic principle. Both companies are vying for maximum benefits. These benefits are derived from both maximum sales volume (a larger share of the market) and higher prices (higher profitability). The problem stems from the fact that increasing profitability through higher prices can damage revenue by losing market share. What Cournot’s approach does is maximise both market share and profitability by defining optimum prices. This price will be the same for both companies, as otherwise the one with the lower price will obtain full market share, which makes this a Nash equilibrium, also known for this model the Cournot-Nash equilibrium.

If we consider isoprofit curves (those which show the combinations of quantities that will render the same profit to the firm, red curves) we can see that the equilibrium of the game is not Pareto efficient, since isoprofit curves are not tangent. The outcome is below that of perfect competition and therefore is not socially optimal, but it is better than the monopoly outcome.

Extending the model to more than two firms, we can observe that the equilibrium of the game gets closer to the perfect competition outcome as the number of firms increases, decreasing market concentration.

Illustration

Let the inverse demand function and the cost function be given by 𝑃 = 50 − 2𝑄 𝑎𝑛𝑑 𝐶 = 10 + 2𝑞

respectively, where Q is total industry output and q is the firm’s output.

First consider first the case of uniform-pricing monopoly, as a benchmark. Then in this case Q = q and the profit function is

𝜋(𝑄) = (50 − 2𝑄)𝑄 − 10 − 2𝑄 = 48𝑄 − 2𝑄² − 10.

Solving 𝑑𝜋

𝑑𝑄 = 0 𝑤𝑒 𝑔𝑒𝑡 𝑄 = 12, 𝑃 = 26, 𝜋 = 278, 𝐶𝑆 = 12(50 − 26)

2 = 144, 𝑇𝑆

= 278 + 144 = 422.

MONOPOLY

Q P π CS TS

12 26 278 144 422

Now let us consider the case of two firms, or duopoly. Let 𝑞₁ be the output of firm 1 and 𝑞₂ the output of firm 2. Then 𝑄 = 𝑞₁ + 𝑞₂ and the profit functions are:

𝜋₁(𝑞₁, 𝑞₂) = 𝑞₁ [50 − 2(𝑞₁ + 𝑞₂)] − 10 − 2𝑞₁ 𝜋₂(𝑞₁, 𝑞₂) = 𝑞₂[50 − 2(𝑞₁ + 𝑞₂)] − 10 − 2𝑞₂

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A Nash equilibrium is a pair of output levels (𝑞₁^∗, 𝑞₂^∗) such that:

𝜋₁(𝑞₁^∗, 𝑞₂^∗) ≥ 𝜋₁(𝑞₁, 𝑞₂^∗) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑞₁ ≥ 0 and

𝜋₁(𝑞₁^∗, 𝑞₂^∗) ≥ 𝜋₁(𝑞₁^∗, 𝑞₂) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑞₂ ≥ 0

This means that, fixing 𝑞₂ at the value 𝑞₂^∗ and considering 𝜋₁ as a function of 𝑞₁ alone, this function is maximized at 𝑞₁=𝑞₁^∗. But a necessary condition for this to be true is that

𝜕𝜋₁

𝜕𝑞₁ (𝑞₁^∗, 𝑞₂^∗) = 0.

Similarly, fixing 𝑞₁ at the value 𝑞₁^∗ and considering 𝜋₂ as a function of 𝑞₂ alone, this function is maximized at 𝑞₂=𝑞₂^∗. But a necessary condition for this to be true is that

𝜕𝜋₂

𝜕𝑞₂ (𝑞₁^∗, 𝑞₂^∗) = 0.

Thus the Nash equilibrium is found by solving the following system of two equations in the two unknowns 𝑞₁ 𝑎𝑛𝑑 𝑞₂:

{

𝜕𝜋₁

𝜕𝑞₁ (𝑞₁^∗, 𝑞₂^∗) = 50 − 4𝑞₁− 2𝑞₂− 2 = 0

𝜕𝜋₂

𝜕𝑞₂ (𝑞₁^∗, 𝑞₂^∗) = 50 − 2𝑞₁− 4𝑞₂− 2 = 0

The solution is 𝑞₁^∗= 𝑞₂^∗ = 8, 𝑄 = 16, 𝑃 = 18, 𝜋₁ = 𝜋₂ = 118, 𝐶𝑆 = ^16(50−18)

2 = 256, 𝑇𝑆 = 118 + 118 + 256 = 492.

Let us compare the two.

MONOPOLY

Q P π CS TS 12 26 278 144 422

DUOPOLY q1 q

2 Q P π

1 π

2 tot π CS TS 8 8 16 18 118 118 236 256 492

Thus competition leads to an increase not only in consumer surplus but in total surplus:

the gain in consumer surplus (256 − 144 = 112) exceeds the loss in total profits (278 − 236 = 42).

In the above example we assumed that the two firms had the same cost function (C = 10 + 2q). However, there is no reason why this should be true. The same reasoning applies to the case where the firms have different costs. Example: demand function as before (P = 50 − 2Q) but now

𝑐𝑜𝑠𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑖𝑟𝑚 1: 𝐶₁ = 10 + 2𝑞₁ 𝑐𝑜𝑠𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑖𝑟𝑚 2: 𝐶₂ = 12 + 8𝑞₂. Then the profit functions are:

𝜋₁(𝑞₁, 𝑞₂) = 𝑞₁ [ 50 − 2(𝑞₁ + 𝑞₂)] − 10 − 2𝑞₁

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𝜋₂(𝑞₁, 𝑞₂) = 𝑞₂ [ 50 − 2(𝑞₁ + 𝑞₂)] − 12 − 8𝑞₂ The Nash equilibrium is found by solving:

{

𝜕𝜋₁

𝜕𝑞₁ (𝑞₁^∗, 𝑞₂^∗) = 50 − 4𝑞₁− 2𝑞₂− 2 = 0

𝜕𝜋₂

𝜕𝑞₂ (𝑞₁^∗, 𝑞₂^∗) = 50 − 2𝑞₁− 4𝑞₂− 8 = 0

The solution is 𝑞₁^∗= 9, 𝑞₂^∗ = 6, 𝑄 = 15, 𝑃 = 20, 𝜋₁ = 152, 𝜋₂ = 60. Since firms have different costs, they choose different output levels: the low-cost firm (firm 1) produces more and makes higher profits than the high-cost firm (firm 2).

SELF-ASSESSMENT EXERCISE

Consider the Cournot model when firm 1 has a cost advantage over firm 2, where c1 < c2. All other conditions remain the same so that firm profits are

𝜋₁ = 𝑎𝑞₁ – 𝑏𝑞₁² – 𝑏𝑞₁𝑞₂ – 𝑐₁𝑞₁ 𝜋₂ = 𝑎𝑞₂ – 𝑏𝑞₂² – 𝑏𝑞₁𝑞₂ – 𝑐₂𝑞₂ Obtain the Cournot equilibrium 𝑞₁^∗, 𝑞₂^∗, 𝑃^∗, 𝜋₁^∗ 𝑎𝑛𝑑 𝜋₂^∗,

HINT: solve the first-order conditions of profit maximization and solve them simultaneously for output, and plug these optimal values into the demand and profit functions to obtain the Cournot equilibrium.