Intermediate Microeconomic Theory | Perloff 5th Ed., Chapter 14 (Sections 14.4–14.5)
Tags: Stackelberg model, Bertrand model, first-mover advantage, leader-follower, price competition, Nash-Bertrand equilibrium, differentiated products, strategic trade policy, government subsidy
Difficulty: Intermediate to Advanced | Prerequisites: Part 2 notes (Cournot Model), Chapter 11 (Monopoly), Chapter 13 (Game Theory, credible threats)
These two sections complete the trio of classic oligopoly models. The Stackelberg model modifies Cournot by letting one firm move first (sequential rather than simultaneous). The Bertrand model changes the strategic variable from quantity to price. Each model gives strikingly different predictions about equilibrium outcomes, which is why understanding when each applies matters for exam success and for real-world analysis.
In Stackelberg, the first mover (leader) produces more and earns higher profit than the follower, because it commits to a large output and forces the follower to accommodate. In Bertrand with identical products, price competition drives the price all the way down to marginal cost (the competitive outcome), even with just two firms. With differentiated products, Bertrand prices sit above marginal cost and respond to demand conditions, making it more realistic for most markets.
Stackelberg model / Stackelberg equilibrium
An oligopoly model in which one firm (the leader) chooses its output first, and the other firm(s) (followers) choose after observing the leader's choice. The leader produces more and earns more than in the symmetric Cournot equilibrium.
First-mover advantage
The benefit a firm gains by committing to its output (or other strategic variable) before rivals. In Stackelberg, moving first is valuable because it lets the leader commit credibly to a large output, forcing followers to produce less.
Credible threat
A threat that a firm would actually carry out if called upon. In Stackelberg, the leader's large output is credible because it has already committed to producing it. In simultaneous play, the same threat is not credible because the firm would prefer to reduce output if the rival did not back down.
Bertrand equilibrium / Nash-Bertrand equilibrium / Nash-in-prices equilibrium
A set of prices such that no firm can obtain a higher profit by choosing a different price, given the other firms' prices. The strategic variable is price, not quantity.
Strategic trade policy
A government policy (typically a subsidy) designed to give a domestic firm a strategic advantage in international competition, effectively turning it into a Stackelberg leader.
The leader knows that once it sets its output, the follower will choose output using its Cournot best-response function. The leader substitutes the follower's best-response function into its own profit function, so that its profit depends only on its own output. It then maximises this profit.
For inverse demand p = a - bQ, with both firms having marginal cost m:
The follower's best-response (from the Cournot model) is: q_2 = (a - m)/(2b) - (1/2)q_1
The leader substitutes this into its profit function:
π_1 = [(a - m - bq_1) / 2] × q_1
Maximising with respect to q_1 gives:
Leader's output: q_1 = (a - m) / (2b)
Follower's output: q_2 = (a - m) / (4b)
The leader produces twice as much as the follower. In the airline example (a = 339, b = 1, m = 147): q_leader = 96, q_follower = 48, Q = 144, p = $195.
If firms move simultaneously, a threat to produce the Stackelberg leader quantity is not credible. Each firm knows the other would be worse off producing that much if the rival also produced the Cournot quantity. Both end up producing the Cournot output instead. Sequential timing makes the threat credible because the leader has already committed to the large output before the follower acts.
A government can effectively turn its domestic firm into a Stackelberg leader by giving it a per-unit subsidy. The subsidy lowers the firm's marginal cost, shifting its best-response curve outward. In the airline example, a $48-per-passenger subsidy to United drops its MC from $147 to $99, causing United to produce 96 (the Stackelberg leader quantity) and American to produce 48.
Key results of the subsidy:
Price falls from $211 to $195 (consumers gain)
Subsidised firm's profit rises from $4.1M to $9.2M
Rival's profit falls from $4.1M to $2.3M
The government's welfare (firm's profit minus subsidy cost) is maximised at the $48 subsidy
Four conditions for strategic trade policy to work:
The other government must not retaliate (if both subsidise, both may lose)
The government must commit before firms choose output
The subsidy must be credible (the foreign firm must believe it will happen)
The government must know the demand and cost functions well enough to set the right subsidy
Competition | Stackelberg | Cournot | Collusion | |
|---|---|---|---|---|
Total output Q (thousands) | 192 | 144 | 128 | 96 |
Price p ($) | 147 | 195 | 211 | 243 |
Consumer surplus ($M) | 18.4 | 10.4 | 8.2 | 4.6 |
Total profit π ($M) | 0 | 6.9 | 8.2 | 9.2 |
The ordering from most to least output: Competition > Stackelberg > Cournot > Collusion. The price ordering is the reverse.
With identical products and constant marginal cost, the Nash-Bertrand equilibrium price equals marginal cost, the same as perfect competition. The logic: if Firm 2 charges $10 and MC = $5, Firm 1's best response is to charge $9.99 and capture the entire market. This undercutting continues until both charge $5 (= MC). At that point, neither firm can profitably undercut further.
This result is striking because just two firms produce the competitive outcome, compared to Cournot where two firms produce a price well above marginal cost.
The Bertrand model with identical goods has two implausibility issues:
The competitive price with only two firms seems unrealistic. Real-world oligopolies typically earn positive profits.
The Bertrand price depends only on cost, not on demand conditions or the number of firms. The Cournot price responds to both, which matches observed market behaviour better.
With differentiated products, the Bertrand model becomes much more plausible and is widely used. The key difference: one firm can charge more than another without losing all its sales. A Coke loyalist will not switch to Pepsi over a small price difference.
Coke-Pepsi example (from Gasmi, Laffont, and Vuong, 1992):
Coke demand: q_C = 58 - 4p_C + 2p_P
Pepsi demand: q_P = 63.2 - 4p_P + 1.6p_C
With MC = $5 for both:
Coke's best-response: p_C = 9.75 + 0.25p_P
Pepsi's best-response: p_P = 10.4 + 0.2p_C
Nash-Bertrand equilibrium: p_C = p_P = $13 per unit (well above the $5 marginal cost).
Cournot best-response curves slope downward (in quantity space): a firm produces less when its rival produces more.
Bertrand best-response curves slope upward (in price space): a firm charges more when its rival charges more.
This difference is intuitive. In Cournot, more rival output floods the market and pushes your optimal output down. In Bertrand, a higher rival price shifts demand toward you, letting you charge more too.
Differentiation leads to higher prices, which harms consumers. But differentiation is also valuable in its own right because consumers prefer variety. The net welfare effect is ambiguous: the price increase hurts, but the value of new varieties helps. The welfare impact depends on how much consumers value the differentiation.
"The Stackelberg leader always earns the monopoly profit." In the linear case, the leader earns more than the Cournot profit but less than the monopoly profit (because the follower still produces positive output, which depresses the price).
"First-mover advantage always exists." It exists in the Stackelberg quantity game, but in other settings (e.g., pricing with network effects) moving first can be a disadvantage.
"Bertrand competition always drives price to marginal cost." Only with identical products and constant marginal costs. With differentiated products, Bertrand prices are well above marginal cost.
"Cournot and Bertrand should give the same answer." They do not. Cournot models quantity competition, Bertrand models price competition. The appropriate model depends on the market: Cournot fits better when firms commit to capacity or production levels; Bertrand fits better when firms set prices and let consumers choose quantities.
⚠️ Be able to derive the Stackelberg equilibrium by substituting the follower's best-response function into the leader's profit function. This is a standard exam problem.
⚠️ Know the ordering: Competition > Stackelberg > Cournot > Collusion (for output), and the reverse for price.
⚠️ Understand why the Stackelberg threat is credible only with sequential timing, not simultaneous timing.
⚠️ For Bertrand with identical goods, know that p = MC (competitive outcome). Be prepared to explain why this is unrealistic and when Cournot is more appropriate.
⚠️ For Bertrand with differentiated goods, know that prices are above MC and best-response curves slope upward. Be able to derive the equilibrium from demand functions.
⚠️ The government subsidy problem (Solved Problem 14.3) is a classic exam question. Know how the subsidy shifts the best-response curve and what happens when both governments subsidise (both firms produce more, price falls, both countries may be worse off).
True or False: In a Stackelberg model, the leader and follower produce equal quantities. (False, the leader produces more, specifically twice as much in the linear case.)
Fill in the blank: The Stackelberg leader's advantage comes from its ability to __________ to a large output before the follower acts. (Commit / credibly commit.)
True or False: With identical goods and constant MC, the Bertrand equilibrium price equals marginal cost. (True.)
True or False: Bertrand best-response curves slope downward. (False, they slope upward, in price space.)
Fill in the blank: When both governments subsidise their firms, both firms produce __________ and both countries may be __________. (More; worse off.)
Q: Duopoly firms face demand p = 150 - q_1 - q_2. Each has MC = $60. What is the Stackelberg equilibrium when Firm 1 moves first?
A: Follower's best-response: q_2 = (150 - 60)/(2) - (1/2)q_1 = 45 - (1/2)q_1. Leader maximises π_1 = [(150 - 60 - q_1)/2] × q_1 = [(90 - q_1)/2] × q_1. FOC: (90 - 2q_1)/2 = 0, so q_1 = 45. Then q_2 = 45 - 22.5 = 22.5. Price p = 150 - 45 - 22.5 = $82.50.
Q: Why do most economists prefer the Cournot model over Bertrand for identical-good oligopolies?
A: The Bertrand model with identical goods predicts the competitive outcome (p = MC, zero profit) with just two firms, which is implausible. It also predicts that price does not respond to demand conditions or the number of firms. The Cournot model predicts prices between competitive and monopoly levels, responding to both demand and market structure, which better matches observed market behaviour.
Q: In the Coke-Pepsi example, what happens if Coke's marginal cost rises to $14.50 while Pepsi's stays at $5?
A: Coke's best-response function shifts up (Coke charges more at every Pepsi price). In the new equilibrium, Coke charges $18 and Pepsi charges $14. The higher-cost firm charges a higher price, and the rival also raises its price because Bertrand best-response curves slope upward.
Q: Why does a government subsidy to one firm in a Cournot duopoly lead to the Stackelberg outcome?
A: The subsidy lowers the domestic firm's marginal cost, shifting its best-response curve outward. The domestic firm now produces more at every level of rival output. In the new Nash-Cournot equilibrium, the subsidised firm produces the Stackelberg leader quantity and the rival produces the Stackelberg follower quantity. The subsidy makes the domestic firm's "threat" of large output credible because its lower cost makes it genuinely profitable.
The Stackelberg model connects directly to the concept of credible threats from Chapter 13: the leader's large output is credible only because of the sequential timing. The strategic trade policy application connects to international economics and the Prisoners' Dilemma: when both governments subsidise, both may be worse off, which is why the WTO has forbidden explicit export subsidies. The Bertrand model with differentiated products connects to monopolistic competition (Section 14.6, Part 4), where free entry drives profit to zero.
Stackelberg model, leader-follower, first-mover advantage, credible threat, Bertrand model, Nash-Bertrand equilibrium, price competition, quantity competition, strategic trade policy, government subsidy, Coke Pepsi duopoly, differentiated products, identical products, best-response function, contract curve, competitive equilibrium, collusive equilibrium, airline duopoly, Boeing Airbus subsidy