Intermediate Microeconomic Theory | Perloff 5th Ed., Chapter 14 (Section 14.3)
Tags: Cournot model, Nash-Cournot equilibrium, best-response function, reaction function, residual demand, duopoly, oligopoly, quantity competition
Difficulty: Intermediate to Advanced | Prerequisites: Part 1 notes (Market Structures and Cartels), Chapter 11 (Monopoly), Chapter 13 (Nash Equilibrium)
The Cournot model is the first of three non-collusive oligopoly models in this chapter, and it is the workhorse model for quantity-setting oligopolies. It answers the question: what happens when firms independently choose how much to produce, simultaneously, without coordinating? The key insight is that each firm treats its rival's output as given and chooses its own profit-maximising output in response. The equilibrium lies between the competitive and monopoly outcomes.
In a Cournot oligopoly, firms simultaneously choose quantities. Each firm's profit-maximising output depends on what it believes rivals will produce (the best-response function). The Nash-Cournot equilibrium is where all best-response functions intersect: no firm wants to change its output given what the others are doing. As the number of firms increases, the Cournot equilibrium approaches the competitive outcome.
Cournot equilibrium / Nash-Cournot equilibrium / Nash-in-quantities equilibrium
A set of quantities chosen by firms such that, holding the quantities of all other firms constant, no firm can obtain a higher profit by choosing a different quantity. This is a Nash equilibrium where the action is choosing output.
Best-response function (reaction function)
A function that gives a firm's profit-maximising output as a function of its rival's output. In the linear duopoly: q_A = 96 - (1/2)q_U. The slope is negative, meaning a firm produces less the more it expects its rival to produce.
Residual demand curve
The market demand that is not met by other sellers at any given price. If the market demand function is D(p) and other firms supply S^o(p), then the residual demand is D^r(p) = D(p) - S^o(p). Think of it as the leftover demand a firm can serve after its rivals have taken their share.
Duopoly
An oligopoly with exactly two firms. The simplest case for illustrating the Cournot, Stackelberg, and Bertrand models.
Spurious differentiation
When a firm convinces consumers its product is different from rivals' products even though the products are physically or chemically identical. Bayer aspirin and Clorox bleach are classic examples.
The basic Cournot model makes four assumptions:
The market lasts one period (each firm chooses quantity only once)
Firms act simultaneously (no firm moves before the other)
Firms are identical (same cost functions, undifferentiated products)
Start with a duopoly, then generalise to n firms
The textbook uses American Airlines and United Airlines competing on the Chicago to Los Angeles route.
Market demand: Q = 339 - p (linear), where Q is thousands of passengers per quarter.
Costs: Each airline has constant MC = AC = $147 per passenger.
Step 1: Find the residual demand. If United flies q_U passengers, American's residual demand is q_A = (339 - p) - q_U.
Step 2: Derive the best-response function. American acts as a monopolist on its residual demand. Its residual inverse demand is p = 339 - q_A - q_U. Revenue is R = (339 - q_A - q_U) × q_A. Marginal revenue from the residual demand is MR^r = 339 - 2q_A - q_U.
Setting MR^r = MC: 339 - 2q_A - q_U = 147
Solving: q_A = 96 - (1/2)q_U (American's best-response function)
By symmetry: q_U = 96 - (1/2)q_A (United's best-response function)
Step 3: Solve simultaneously. The Nash-Cournot equilibrium is where both best-response functions intersect: q_A = q_U = 64 (thousands of passengers per quarter). Total output Q = 128, price p = $211.
The best-response curve has a negative slope, which means a firm produces less the more it expects its rival to produce. Key points:
If q_U = 0, American produces the monopoly quantity: q_A = 96
If q_U = 64, American produces q_A = 64 (the Nash-Cournot equilibrium)
If q_U = 192, American shuts down: q_A = 0
Any pair of quantities other than (64, 64) is not a Nash-Cournot equilibrium. For example, if q_A = 96 and q_U = 0, American is on its best-response curve (happy) but United is not. United would want to produce q_U = 48, not zero. Only at the intersection do both firms simultaneously sit on their best-response curves.
For n identical firms with inverse demand p = a - bQ and constant marginal cost m:
Best-response function (firm i):
q_i = (a - m) / (2b) - [(n - 1) / 2] × q
where q is the common output of each other identical firm.
Equilibrium output per firm:
q = (a - m) / [(n + 1)b]
Total market output:
Q = n(a - m) / [(n + 1)b]
Equilibrium price:
p = (a + nm) / (n + 1)
Lerner Index:
(p - MC) / p = (a - m) / (a + nm) = -1 / (nε)
where ε is the market demand elasticity.
Airline example values: a = 339, b = 1, m = 147. With n = 2: q = 64, Q = 128, p = $211, Lerner Index = 0.30.
As n increases, each firm's output falls toward zero, but total output approaches (a - m)/b (the competitive quantity). Price approaches marginal cost m. The Lerner Index approaches zero. With n = 1 you get monopoly; as n approaches infinity you get perfect competition. The Cournot model neatly bridges these two extremes.
Key data from the airline example (a = 339, b = 1, m = 147):
n = 1: p = $243, Lerner = 0.40
n = 2: p = $211, Lerner = 0.30
n = 3: p = $195, Lerner = 0.25
n = 4: p = $185, Lerner = 0.21
n = 10: p = $164, Lerner = 0.11
n = 100: p = $149, Lerner = 0.01
When firms have different marginal costs, the low-cost firm produces more and earns higher profit. If Firm 1 has MC = m and Firm 2 has MC = m + x (where x > 0):
q_1 = (a - m + x) / (3b) (produces more)
q_2 = (a - m - 2x) / (3b) (produces less)
π_1 = (a - m + x)² / (9b) > π_2 = (a - m - 2x)² / (9b)
Both firms charge the same price (undifferentiated products), but the low-cost firm captures more of the market and earns higher profit.
If consumers view the products as different, each firm faces its own inverse demand function (e.g., Intel vs AMD CPUs). The slopes of the demand functions differ: a firm's own-price coefficient is larger than the cross-price coefficient, reflecting that its own output affects its price more than the rival's output does. In this case, Nash-Cournot quantities and prices differ across firms.
"The Cournot equilibrium is the same as the competitive equilibrium." It is not. The Cournot price is above marginal cost (but below the monopoly price). Only as n approaches infinity does Cournot approach perfect competition.
"Firms actually choose quantities in real life." This is a modelling choice, not a claim about literal behaviour. The Cournot model works well for markets where firms commit to capacity or production levels (airlines choosing flight schedules, manufacturers setting production runs).
"The best-response function is the same as the demand curve." The best-response function maps the rival's output to the firm's profit-maximising output. The demand curve maps price to quantity demanded. They are different objects.
"A firm with lower costs always drives the rival out of the market." In Cournot, the high-cost firm still produces positive output (as long as a - m - 2x > 0). It just produces less.
⚠️ Be able to derive the best-response function from the residual demand curve and MR = MC. This is a core exam skill.
⚠️ Know that the Nash-Cournot equilibrium is at the intersection of the best-response curves and be able to explain why any other point is not an equilibrium.
⚠️ Memorise (or be able to derive) the general formulas for the linear model: q = (a - m)/[(n+1)b] and p = (a + nm)/(n + 1).
⚠️ The Lerner Index expression (p - MC)/p = -1/(nε) connects market structure to market power and is commonly tested.
⚠️ Know how to handle the unequal-cost case: different best-response functions, different equilibrium quantities, but the same equilibrium price (for undifferentiated goods).
True or False: In a Cournot duopoly, each firm chooses its price. (False, each firm chooses its quantity.)
Fill in the blank: The Nash-Cournot equilibrium occurs at the __________ of the best-response curves. (Intersection.)
True or False: A negative slope of the best-response curve means a firm produces more when its rival produces more. (False, it produces less.)
Fill in the blank: As the number of Cournot firms approaches infinity, the equilibrium price approaches __________. (Marginal cost / the competitive price.)
True or False: In a Cournot duopoly with unequal costs, the low-cost firm earns higher profit. (True.)
Q: In the airline duopoly, if American believes United will fly 96,000 passengers, what is American's best response?
A: Using American's best-response function q_A = 96 - (1/2)q_U = 96 - (1/2)(96) = 96 - 48 = 48 thousand passengers.
Q: Why does the Cournot model predict a price between the monopoly price and the competitive price?
A: Each firm has some market power because it is large relative to the market, but it cannot ignore the output of its rival. The rival's output limits each firm's ability to raise price. With one firm (n = 1), you get monopoly pricing. With many firms, each firm's market share shrinks, its residual demand curve becomes more elastic, and the price falls toward marginal cost.
Q: Suppose the market demand is Q = 1000 - 1000p and each firm has MC = $0.28. What is the Nash-Cournot equilibrium for a duopoly?
A: First rewrite in inverse demand form: p = 1 - Q/1000, so a = 1, b = 1/1000, m = 0.28. Each firm's output: q = (1 - 0.28)/[(2+1)(1/1000)] = 0.72/0.003 = 240. Total output Q = 480. Price p = 1 - 480/1000 = $0.52.
Q: How does product differentiation affect the Cournot equilibrium?
A: When products are differentiated, each firm faces its own demand curve. The cross-price effects are weaker than own-price effects. This allows firms to charge different prices and typically leads to higher equilibrium prices and profits compared to the identical-products case, because differentiation makes the residual demand less elastic.
The Cournot model builds directly on the monopoly model from Chapter 11: each firm acts as a monopolist on its residual demand. The best-response function concept comes from game theory (Chapter 13). The Cournot equilibrium is a special case of Nash equilibrium where the strategic variable is quantity. Part 3 covers what happens when one firm moves first (Stackelberg) or when firms choose prices instead of quantities (Bertrand).
Cournot model, Nash-Cournot equilibrium, best-response function, reaction function, residual demand, duopoly, oligopoly, quantity competition, Lerner Index, market power, linear demand, marginal revenue, best response curve, identical firms, unequal costs, differentiated products, spurious differentiation, Intel AMD duopoly, airline competition