| Bertrand | Cournot | |
|---|---|---|
| Firms choose | Prices | Quantities |
| Equilibrium \(P\) | \(= c\) | Between \(c\) and \(P^M\) |
| Market power | None | Positive |
| Best responses | Strategic complements | Strategic substitutes |
| # firms for \(P = c\) | 2 | \(\infty\) |
Same industry, same firms, but drastically different predictions. Which model is right?
Bertrand: A small price cut steals the entire market from your rival. Huge incentive to undercut.
Cournot: A small increase in your output has only a marginal effect on price and on your rival’s revenue. Less aggressive competition.
Key insight: The “sharpness” of competition depends on how sensitive one firm’s demand is to the other firm’s action.
Kreps & Scheinkman (1983): If firms first choose capacity, then compete in prices, the outcome is the Cournot equilibrium.
Intuition:
Takeaway: Cournot is a better model when capacity is costly and chosen before pricing. Bertrand is better when firms can flexibly adjust output at their chosen price.
When products are not identical, undercutting doesn’t capture the entire market.
Demand: \[q_1 = a - bp_1 + dp_2 \qquad q_2 = a - bp_2 + dp_1\]
where \(b > d > 0\) (own-price effect dominates cross-price effect).
Firm 1 maximizes: \[\pi_1 = (p_1 - c)(a - bp_1 + dp_2)\]
FOC: \[a - 2bp_1 + dp_2 + bc = 0\]
Best response: \[p_1^*(p_2) = \frac{a + bc + dp_2}{2b}\]
Symmetric equilibrium (\(p_1 = p_2 = p^*\)):
\[p^* = \frac{a + bc}{2b - d}\]
As long as \(c < a/(b-d)\), which is the highest price that yields positive demand, we have \(p^* > c\). So firm makes positive profits even with two firms. The Bertrand paradox is resolved!
Best responses are upward sloping: if your rival raises their price, you raise yours. Prices are strategic complements.
In a one-shot Bertrand game: \(p = c\) is the only equilibrium.
But firms interact repeatedly. Can the threat of future punishment sustain collusion?
Tacit collusion: Firms maintain high prices without an explicit agreement, enforced by the threat of a price war.
Key distinction:
Grim trigger strategy: Both firms charge the monopoly price \(P^M\). If either firm deviates, both revert to \(p = c\) forever.
Collusion profit stream (each firm gets half the monopoly profit each period):
\[V_{\text{collude}} = \frac{\pi^M/2}{1 - \delta}\]
Deviation profit: Undercut slightly, capture all monopoly profit for one period, then earn zero forever:
\[V_{\text{deviate}} = \pi^M + 0 + 0 + \cdots = \pi^M\]
Collusion is sustainable when \(V_{\text{collude}} \geq V_{\text{deviate}}\):
\[\frac{\pi^M/2}{1 - \delta} \geq \pi^M \implies \delta \geq \frac{1}{2}\]
If firms value the future enough (\(\delta \geq 1/2\)), they can sustain monopoly prices.
What makes collusion easier?
With \(n\) firms splitting monopoly profit equally:
\[V_{\text{collude}} = \frac{\pi^M / n}{1 - \delta}, \qquad V_{\text{deviate}} = \pi^M\]
Collusion requires: \[\delta \geq 1 - \frac{1}{n} = \frac{n-1}{n}\]
| \(n\) | Min \(\delta\) for collusion |
|---|---|
| 2 | 0.50 |
| 3 | 0.67 |
| 10 | 0.90 |
OPEC is a real-world example of attempted collusion:
This is exactly the repeated game logic: collusion sustained by the threat of a price war.
What if firms move sequentially rather than simultaneously?
Stackelberg model: Firm 1 (leader) chooses \(q_1\) first. Firm 2 (follower) observes \(q_1\) and then chooses \(q_2\).
Solve by backward induction:
Step 1: Firm 2’s best response (same as Cournot): \[q_2^*(q_1) = \frac{a - c}{2b} - \frac{q_1}{2}\]
Step 2: Firm 1 anticipates this and substitutes into its profit function.
Firm 1 maximizes: \[\pi_1 = \left[a - b\left(q_1 + \frac{a-c}{2b} - \frac{q_1}{2}\right)\right]q_1 - cq_1\]
Simplifying: \[\pi_1 = \left[\frac{a - c}{2} - \frac{b q_1}{2}\right]q_1\]
FOC: \[\frac{a - c}{2} - bq_1 = 0 \implies q_1^S = \frac{a - c}{2b}\]
Follower’s response: \(q_2^S = \frac{a-c}{4b}\)
Using \(P = 100 - Q\) and \(c = 20\):
| Cournot | Stackelberg Leader | Stackelberg Follower | |
|---|---|---|---|
| Output | 26.67 | 40 | 20 |
| Price | 46.67 | 40 | 40 |
| Profit | 711 | 800 | 400 |
The leader produces more and earns more. By committing to a large quantity first, the leader forces the follower to scale back, a “top dog” strategy.
Total output is higher and price is lower than Cournot → closer to competition.
Consider Bertrand game from before with \(a = 1, b = 1, d = 0.5, c = 0\).
\[ q_i = 1 - p_i + 0.5 p_j \]
Price leadership: Firm 1 (leader) chooses \(p_1\) first. Firm 2 (follower) observes \(p_1\) and chooses \(p_2\).
Follower’s best response: \[p_2^*(p_1) = \frac{1 + 0.5 p_1}{2}\]
Leader’s problem: \[\max_{p_1} p_1 \left(1 - p_1 + 0.5 \left( \frac{1 + 0.5 p_1}{2} \right) \right)\]
FOC: \[1 - 2p_1 + 0.25 p_1 + 0.25 = 0 \implies p_1 = 0.714\]
Follower’s response: \[p_2 = \frac{1 + 0.5 (0.714)}{2} = 0.679\]
| Bertrand | Price Leader | Price Follower | |
|---|---|---|---|
| Price | 0.667 | 0.714 | 0.679 |
| Quantity | 0.666 | 0.625 | 0.678 |
| Profit | 0.444 | 0.446 | 0.460 |
Quantity game (Stackelberg): Leader overproduces → “top dog” strategy
Price game (price leadership): Leader raises price → “puppy dog” strategy
Key insight: Whether the first mover is aggressive or accommodating depends on the slope of the follower’s best response function.
In the long run, positive profits attract entry. How many firms will enter?
Setup: Entry requires a sunk cost \(K\). After entry, firms compete à la Cournot.
Equilibrium number of firms \(n^*\): the greatest integer such that:
\[g(n^*) = \frac{(a-c)^2}{b(n^* + 1)^2} \geq K\]
The equilibrium number of firms is decreasing in sunk costs \(K\) and increasing in market size \((a - c)\).
An incumbent may try to prevent entry rather than accommodate it.
Methods:
Trade-off: Entry deterrence is costly as the incumbent distorts its behavior. Worth it only if the benefit of being alone in the market exceeds the cost of deterrence.
Demand function: \(P = 100 - Q\), MC = 0, sunk entry cost \(K\).
To deter entry, incumbent firm 1 will produce a quantity such that even if firm 2 enters and produces its best response, it will not earn enough to cover the entry cost.
Firm 2’s best response to firm 1’s quantity \(q_1\) is: \[q_2^*(q_1) = \frac{100 - q_1}{2}\]
Profit for firm 2 if it enters: \[\pi_2 = q_2^* (100 - q_1 - q_2^*) = \frac{(100 - q_1)^2}{4}\]
To deter entry, we need \(\pi_2 < K\): \[\frac{(100 - q_1)^2}{4} < K \implies q^{det}_1 = 100 - 2\sqrt{K}\]
Firm 1’s profit if it deters entry: \[\pi_1^{det} = (100-2\sqrt{K}) \cdot 2\sqrt{K}\]
If firm 1 accommodates entry, it produces \(q_1^{acc} = 33.33\) and earns \(\pi_1^{acc} = 1111\).
Firm 1 will deter entry if \(\pi_1^{det} > \pi_1^{acc}\).
We need to solve the following equation for \(K\): \[(100-2\sqrt{K}) \cdot 2\sqrt{K} = 1111 \]
To simplify, let \(x = 2\sqrt{K}\): \[100x - x^2 = 1111 \implies x^2 - 100x + 1111 = 0\]
Using the quadratic formula: \[x = \frac{100 \pm \sqrt{100^2 - 4 \cdot 1111}}{2} = \frac{100 \pm 74.54}{2}\]
Using the smaller root (\(x =12.73\)): \(K = 40.5\). If entry costs are below this, deterrence is not profitable.
An important subtlety for applied work:
Greater sunk costs constrain entry even in the long run, so prices tend to be higher in industries with large sunk costs.
Models of oligopoly:
Resolving the Bertrand paradox:
Strategic concepts: