5.5 Cournot, Bertrand, and Stackelberg models

Cournot, Bertrand, and Stackelberg Models

Cournot, Bertrand, and Stackelberg models explain how firms compete in oligopolies. Each model captures a different strategic scenario: firms choosing quantities simultaneously, firms choosing prices simultaneously, or one firm moving before the other. The key variable in each case is what firms optimize over (price vs. quantity) and whether decisions happen at the same time or sequentially.

These models matter because they produce strikingly different predictions about prices, output, and profits, even when describing the same market structure. Comparing them helps you understand why the type of competition matters just as much as the number of competitors.

Cournot Model of Quantity Competition

Key Assumptions and Equilibrium Concept

In the Cournot model, firms choose how much to produce at the same time, without knowing what their rivals picked. The market price then adjusts based on total industry output through the inverse demand function.

• Each firm picks the quantity that maximizes its own profit, taking the other firm's output as given
• The equilibrium is a Nash equilibrium: no firm can do better by unilaterally changing its quantity
• Each firm's optimal choice is described by a reaction function (or best response function), which maps the rival's output to the firm's profit-maximizing output
• The Cournot equilibrium sits where the reaction functions intersect
• Prices end up above marginal cost but below the monopoly price
• Total industry output falls between the competitive and monopoly levels

Mathematical Representation

Firm $i$'s profit is:

$\pi_i = P(Q)q_i - C_i(q_i)$

where $P(Q)$ is the inverse demand function, $Q$ is total industry output, $q_i$ is firm $i$'s output, and $C_i(q_i)$ is its cost function.

To find the profit-maximizing quantity, take the first-order condition:

$\frac{\partial \pi_i}{\partial q_i} = P(Q) + P'(Q)q_i - C'_i(q_i) = 0$

Notice the term $P'(Q)q_i$. This captures the fact that when firm $i$ produces more, it pushes the market price down on all of its units. That's the key difference from perfect competition, where firms are price-takers and this term vanishes.

Solving the system of first-order conditions for all firms simultaneously gives you the Cournot-Nash equilibrium quantities.

Worked example (symmetric duopoly): Suppose inverse demand is $P = a - bQ$, both firms have constant marginal cost $c$, and $Q = q_1 + q_2$.

1. Write firm 1's profit: $\pi_1 = (a - b(q_1 + q_2))q_1 - cq_1$

2. Take the FOC with respect to $q_1$: $a - 2bq_1 - bq_2 - c = 0$

3. Solve for firm 1's reaction function: $q_1^* = \frac{a - c}{2b} - \frac{q_2}{2}$

4. By symmetry, firm 2's reaction function is: $q_2^* = \frac{a - c}{2b} - \frac{q_1}{2}$

5. Substitute one into the other. In the symmetric equilibrium: $q_1^* = q_2^* = \frac{a - c}{3b}$

6. Total output: $Q^* = \frac{2(a-c)}{3b}$, and market price: $P^* = \frac{a + 2c}{3}$

Each firm's equilibrium profit is $\pi^* = \frac{(a-c)^2}{9b}$. You can verify this by plugging $P^*$ and $q^*$ back into the profit function.

Model Extensions and Market Dynamics

• More firms push the outcome toward perfect competition. With $n$ symmetric firms, each produces $\frac{a-c}{(n+1)b}$, total output is $\frac{n(a-c)}{(n+1)b}$, and as $n \to \infty$, price approaches marginal cost $c$.
• The model extends to asymmetric firms with different cost structures. Lower-cost firms produce more and earn higher profits in equilibrium.
• It can also accommodate product differentiation and capacity constraints, though these change the math considerably.

Bertrand Model of Price Competition

Core Principles and the Bertrand Paradox

In the Bertrand model, firms choose prices simultaneously rather than quantities. Consumers buy from whichever firm offers the lowest price (splitting demand equally if prices are tied).

The Bertrand paradox is the surprising result that emerges with homogeneous products and identical marginal costs: the equilibrium price equals marginal cost, and both firms earn zero economic profit. With just two firms, you get the same outcome as perfect competition.

Why? If either firm charges above marginal cost, the other can undercut by a tiny amount and steal the entire market. This undercutting logic continues until price is driven all the way down to $c$. At $p = c$, neither firm wants to deviate: cutting price means losses, and raising price means losing all customers.

This is called a "paradox" because it seems implausible that two firms alone could replicate the competitive outcome. Much of the literature on Bertrand competition focuses on what assumptions need to break for more realistic predictions to emerge.

Mathematical Framework

With differentiated products, firm $i$ faces a demand function that depends on all firms' prices:

$q_i = D_i(p_i, p_{-i})$

where $p_{-i}$ is the vector of rival prices. Firm $i$'s profit is:

$\pi_i = (p_i - c_i)D_i(p_i, p_{-i})$

The first-order condition is:

$\frac{\partial \pi_i}{\partial p_i} = D_i(p_i, p_{-i}) + (p_i - c_i)\frac{\partial D_i}{\partial p_i} = 0$

The first term is the gain from raising price on existing units sold. The second term is the loss from reduced demand as some consumers switch away. At the optimum, these balance out.

With differentiation, $\frac{\partial D_i}{\partial p_i} < 0$ but is finite (not $-\infty$ as in the homogeneous case), so the equilibrium price exceeds marginal cost. The degree of differentiation determines how much markup each firm can sustain.

Model Variations and Extensions

• Product differentiation resolves the paradox. When products are imperfect substitutes, each firm faces a downward-sloping demand curve and can charge above marginal cost. Think of Coke vs. Pepsi: a small price difference won't cause all consumers to switch.
• Capacity constraints also break the paradox. If a firm can't serve the whole market at a low price, undercutting doesn't capture all demand. This reintroduces an element of quantity competition (this connects to the Edgeworth critique of Bertrand, and more formally to the Kreps-Scheinkman result showing that capacity choice followed by price competition yields Cournot outcomes).
• Repeated interactions can sustain prices above marginal cost through tacit collusion, where firms maintain high prices under the threat of future price wars.

Stackelberg Model of Sequential Competition

Sequential Decision-Making Structure

The Stackelberg model modifies Cournot by making decisions sequential: one firm (the leader) commits to a quantity first, and the other firm (the follower) observes this and then chooses its quantity.

The leader gains a first-mover advantage because it can commit to a large output level, effectively forcing the follower to produce less. The crucial element here is credible commitment: the leader's quantity choice must be observable and irreversible. If the leader could secretly change its output after the follower decides, the model collapses back to Cournot.

Solving the Model: Backward Induction

You solve Stackelberg models using backward induction, working from the last decision back to the first:

1. Start with the follower. Given the leader's output $q_L$, the follower maximizes its own profit. This gives the follower's reaction function $q_F^*(q_L)$, which is the same as its Cournot reaction function.
2. Move to the leader. The leader anticipates the follower's reaction function and substitutes it into its own profit function. The leader then chooses $q_L$ to maximize profit, knowing exactly how the follower will respond.
3. Compute equilibrium. Plug the leader's optimal $q_L^*$ back into the follower's reaction function to get $q_F^*$.

Using the same linear example ($P = a - bQ$, equal marginal cost $c$):

1. Follower's reaction function (same as Cournot): $q_F^* = \frac{a - c}{2b} - \frac{q_L}{2}$

2. Leader substitutes this into its profit: $\pi_L = \left(a - b\left(q_L + \frac{a-c}{2b} - \frac{q_L}{2}\right)\right)q_L - cq_L$

which simplifies to $\pi_L = \left(\frac{a - c}{2} - \frac{b}{2}q_L\right)q_L$

1. Taking the FOC: $\frac{a-c}{2} - bq_L = 0$, so $q_L^* = \frac{a - c}{2b}$

2. Follower produces: $q_F^* = \frac{a - c}{4b}$

3. Total output: $Q^* = \frac{3(a-c)}{4b}$, and price: $P^* = \frac{a + 3c}{4}$

The leader produces exactly the monopoly quantity, and the follower produces half of what the leader does. The leader's profit is $\frac{(a-c)^2}{8b}$ and the follower's is $\frac{(a-c)^2}{16b}$.

Equilibrium Characteristics

• The Stackelberg equilibrium is a subgame perfect Nash equilibrium (SPNE), meaning it's a Nash equilibrium in every subgame. The follower's decision is optimal given the leader's choice, and the leader's choice is optimal given the follower's anticipated response.
• The leader produces more than its Cournot quantity; the follower produces less.
• Total industry output exceeds Cournot output, so the market price is lower.
• The leader earns higher profit than the follower, and higher profit than either firm would earn in Cournot. The follower earns less than it would under Cournot.

Model Extensions and Applications

• The model extends to multiple leaders, multiple followers, or multiple periods.
• It's most relevant for markets where one firm has a clear timing advantage, such as an incumbent facing a new entrant.
• Real-world examples include dominant firms in technology or natural resource sectors that set capacity before competitors respond.
• A Stackelberg price leadership variant also exists, where the leader sets price first and the follower responds. The welfare implications differ from the quantity version.

Cournot vs. Bertrand vs. Stackelberg Outcomes

Comparative Market Outcomes

For the standard case (homogeneous products, identical constant marginal costs, linear demand):

• Prices follow: Bertrand $<$ Stackelberg $<$ Cournot $<$ Monopoly
• Total output follows the reverse order
• Bertrand competition yields zero economic profits, while Cournot and Stackelberg allow positive profits
• The Stackelberg leader earns the highest individual firm profit among the oligopoly models, but the follower earns less than a Cournot duopolist

Model Applicability and Industry Characteristics

Choosing the right model depends on how competition actually works in a given market:

• Bertrand fits markets with price competition and low switching costs (retail, online marketplaces, airline tickets on the same route)
• Cournot fits markets where firms set production quantities or capacity in advance (oil production, cement, agriculture). It also serves as a good approximation when firms first choose capacity and then compete on price (per the Kreps-Scheinkman result).
• Stackelberg fits markets with a clear first-mover or dominant incumbent (a tech giant entering a new product category before rivals can respond)

The key factors to consider:

• Nature of the strategic variable (price vs. quantity)
• Degree of product homogeneity
• Timing of decisions (simultaneous vs. sequential)
• Whether capacity is chosen before prices

Efficiency and Welfare Implications

• Bertrand yields the most efficient oligopoly outcome: price equals marginal cost (with homogeneous goods), consumer surplus is maximized, and there's no deadweight loss.
• Cournot and Stackelberg both produce deadweight loss because prices exceed marginal cost. Stackelberg generates more total output and less deadweight loss than Cournot.
• Monopoly is the least efficient: highest price, lowest output, greatest deadweight loss.

From a consumer welfare perspective, more aggressive competition (Bertrand > Stackelberg > Cournot) consistently benefits consumers through lower prices and higher output. From a total welfare perspective, the ranking is the same, since producer surplus losses from lower prices are more than offset by consumer surplus gains (up to the competitive/Bertrand benchmark where total surplus is maximized).