Bertrand Competition

The model

Joseph Bertrand wrote a 1883 review of Antoine Augustin Cournot's Recherches sur les principes mathématiques de la théorie des richesses (1838), arguing that Cournot's choice of quantity as the strategic variable was the wrong call. Real firms, Bertrand wrote, post prices and let demand sort itself out. If they did, what would happen?

Bertrand's setup is austere. Two firms, indexed 1 and 2, produce a homogeneous good at constant marginal cost c. They simultaneously post prices p_1 and p_2. Market demand at the lower price is D(p), and all consumers buy from whichever firm is cheaper; if both firms tie, they split demand. Each firm sets price to maximise profit, taking the other's price as given. The equilibrium concept is Nash.

demand for firm 1:
    Q_1(p_1, p_2) = D(p_1)        if p_1 < p_2
                  = D(p_1) / 2     if p_1 = p_2
                  = 0              if p_1 > p_2

profit:  π_i(p_1, p_2) = (p_i − c) × Q_i(p_1, p_2)

That is the entire model. No capacity, no differentiation, no dynamics. Find the prices at which neither firm has a profitable unilateral deviation.

Why the equilibrium is p = MC

Suppose, hypothetically, that both firms posted the same price p > c. Each earns roughly (p − c) D(p) / 2. Either firm can then shave its price by an arbitrarily small ε > 0, become strictly cheaper, capture all demand, and earn approximately (p − ε − c) D(p) — almost double the previous profit. So no symmetric price above marginal cost is stable; whichever firm goes first wants to undercut.

By symmetry the same argument applies to any asymmetric pair with both prices above c. The higher-price firm earns zero (it has been undercut), and the lower-price firm always has a profitable infinitesimal undercut up to c + ε. The chain of best responses ends only when both firms charge exactly c. At that point neither firm can undercut profitably (pricing below MC means losses on every unit) and neither wants to charge above (it would sell zero). So p_1 = p_2 = c is the unique pure-strategy Nash equilibrium. Each firm earns zero economic profit.

Note the discontinuity. Demand for firm 1 jumps from D(p) to zero as p_1 crosses p_2; the profit function is discontinuous in p_i. That is what makes the undercutting incentive so strong, and it is what distinguishes Bertrand from Cournot, where the profit function is everywhere differentiable in quantities.

The Bertrand paradox

Two firms — just two — and price equals marginal cost? The same outcome perfect competition delivers with infinitely many firms? That clashes badly with what we see in the world. Duopolies and tight oligopolies earn positive margins everywhere: Coke and Pepsi, Boeing and Airbus, Visa and Mastercard. Economists call this clash the Bertrand paradox.

The paradox is what the model gets right, not what it gets wrong. The model says: if your assumptions about constant-MC, identical goods, perfect price observability, infinite capacity, and one-shot interaction hold exactly, two firms suffice to compete the margin away. Real oligopolies preserve margins by violating at least one of those assumptions. Each of the resolutions below identifies which one. The interesting question is never "why is the paradox wrong" — it is "which assumption fails first in your market".

Five canonical resolutions

Resolution	Author	Mechanism	Equilibrium prediction
Capacity constraints	Edgeworth (1897); Kreps-Scheinkman (1983)	Rationed customers spill to higher-priced rival	No pure-strategy NE; cycles, or Cournot quantities
Product differentiation	Hotelling (1929); Spence-Dixit-Stiglitz	Smooth, non-collapsing residual demand	p > c, markup ∝ differentiation
Repeated interaction	Friedman (1971); folk theorem	Trigger strategies sustain tacit collusion	Any price ∈ [c, monopoly] is SPE
Search costs	Diamond (1971); Stahl (1989)	Search frictions protect markup	Even tiny ε > 0 can collapse to monopoly
Consumer heterogeneity	Salop-Stiglitz; Varian (1980)	Informed and uninformed buyers coexist	Mixed-strategy price dispersion equilibrium

Capacity constraints — the Edgeworth resolution

Francis Ysidro Edgeworth, two papers later (1897), pointed out the flaw in Bertrand's argument. If firm 1 has capacity k_1 < D(c) — not enough to serve the whole market at marginal cost — then undercutting captures only k_1 customers, not all of D(p). The rationed consumers turn to firm 2, which now faces residual demand D(p_2) − k_1 and can profitably price above c. So firm 2 will not match the undercut; it will instead charge close to its residual-monopoly price.

Now firm 1 reasons the same way. If firm 2 is committed to a high price, firm 1's best response may itself be to raise price — and the cycle is on. Edgeworth showed there is no pure-strategy Nash equilibrium in this game; prices cycle between low (undercutting) and high (monopoly on the residual) regions. The mixed-strategy equilibrium exists but is messy.

Kreps and Scheinkman (1983) gave the cleanest patch. Allow firms to choose capacity in a first stage and then play Bertrand in the second; the equilibrium capacity is exactly the Cournot quantity, and the resulting prices reproduce Cournot's prediction. This is why Cournot and Bertrand are not really rival theories of the same situation — they describe what happens when capacity binds and when it does not.

Product differentiation

If the two firms' goods are imperfect substitutes — Coke versus Pepsi, an iPhone versus a Galaxy — then a firm pricing slightly above its rival does not lose all demand, only a fraction proportional to a differentiation parameter. The profit function becomes smooth in p_i, the discontinuity vanishes, and the equilibrium is interior. A standard linear-demand model with substitutability γ ∈ [0, 1] gives

p_i* = (a + c) / (2 − γ)  − c  + c
     → p* = c + (a − c) / (2 − γ)

γ → 1 (perfect substitutes):  p* → c   (Bertrand limit)
γ → 0 (independent goods):    p* → monopoly price each

Even modest differentiation — γ < 0.95, say — lifts equilibrium price several percent above MC and yields positive profit. The Hotelling linear-city model gives the same answer geometrically: two firms located at opposite ends of a [0, 1] consumer line each sell to the consumers closer to them and earn margins proportional to transport cost.

Repeated interaction

If the same two firms meet day after day, they can sustain prices above marginal cost using trigger strategies: charge the monopoly price; if the rival ever undercuts, revert to Bertrand pricing forever. The one-shot gain from undercutting is finite; the punishment payoff is zero (Bertrand profit) for the rest of time. With discount factor δ close enough to 1, the deviation is unprofitable and the cooperative price is sustained as a subgame-perfect equilibrium.

Formally, the folk theorem says any price between marginal cost and monopoly can be sustained in equilibrium of the infinitely repeated game, given a high enough discount factor. This is the canonical model of tacit collusion, and it is the reason competition authorities scrutinise repeated price-matching announcements and price-leadership patterns in oligopolies.

Search costs

The Diamond paradox (Peter Diamond, 1971) shows that even an arbitrarily small consumer search cost ε > 0 can collapse the Bertrand equilibrium to the monopoly outcome. A consumer who has paid ε to learn one firm's price faces a choice between buying immediately or paying another ε to learn the next firm's price. If all firms are charging the same price, the consumer never benefits from extra search; firms know this and post the monopoly price. The result feels paradoxical for the opposite reason — search costs deliver monopoly even with many firms — but it captures something real about why local retailers can sustain markups despite competition five minutes down the road.

Consumer heterogeneity

Hal Varian's "Model of Sales" (1980) populates the market with two types of consumers: informed shoppers who know all prices and buy from the cheapest, and uninformed shoppers who pick a firm randomly. Pure-strategy Bertrand fails; the equilibrium is in mixed strategies, with each firm randomising its price over an interval. Some days the firm posts a low "sale" price to attract informed shoppers; other days a high "regular" price to milk the uninformed. This is the textbook explanation for observed price dispersion in retail markets where products are identical (gasoline, grocery staples, electronics).

Cournot versus Bertrand — why the pricing variable matters

The most striking lesson of Bertrand is comparative. Hold everything else fixed — same firms, same costs, same products, same demand — and just change whether they pick quantities or prices. The predicted equilibrium moves discretely.

Model	Strategic variable	Best-response shape	Equilibrium price	Equilibrium profit
Cournot (1838)	Quantity q_i	Downward-sloping in q_j (strategic substitutes)	p* > c, decreases in n	π* > 0
Bertrand (1883)	Price p_i	Step function (undercut just below p_j)	p* = c for n ≥ 2	π* = 0
Stackelberg (1934)	Quantity, sequential	Leader internalises follower BR	p* between Cournot and PC	Asymmetric, leader earns more

So which model is right? Empirically, neither one nor the other; the right model depends on what firms can commit to. A market where capacity is built in advance and is hard to expand quickly (steel, chemicals, semiconductors) looks Cournot — capacity is the lasting decision, prices then clear residual demand. A market where capacity is elastic and price posts are revisable in minutes (online retail, airline yield management, gasoline) looks Bertrand — prices move first and quantities follow. The Kreps-Scheinkman result formalises this: capacity-then-price collapses to Cournot quantities; price-only with infinite capacity collapses to MC.

Where Bertrand competition shows up in the wild

• Airline ticket prices. When two airlines offer competing flights on a route at similar times, online tariffs are observably matched within hours and undercutting cycles are visible in revenue-management traces. Yield-management algorithms read each other's posted prices in real time and react. Margins are sustained mainly by capacity constraints (Edgeworth) and product differentiation across times/classes.
• Retail gasoline. Stations on the same intersection post nearly identical prices because consumers compare visually before turning in. Empirical work (Noel 2007 on Edmonton) finds explicit Edgeworth-like cycles: stations undercut for days, then jump back up in coordinated fashion.
• Online retail (commodity electronics, books). Algorithms scrape competitor prices and adjust within minutes. Amazon's Buy Box mechanism is essentially a real-time Bertrand auction among sellers of the same SKU.
• Digital platforms with near-zero marginal cost. Cloud storage, messaging, search, basic API access. Marginal cost is essentially zero; if the service is undifferentiated, Bertrand predicts a zero-price floor. Hence "free" tiers of Gmail, WhatsApp, Dropbox, and consumer search — these are not loss leaders; they are the Bertrand equilibrium. Revenue migrates to differentiated paid tiers or to the other side of the market (ads).
• Generic pharmaceuticals after patent expiry. Multiple manufacturers produce chemically identical molecules; FDA inspections cap differentiation. Prices collapse to roughly marginal cost within five generic entrants — the empirical literature finds 80%+ price reduction by the fifth entrant. The convergence is essentially Bertrand.
• Spot electricity markets. Generators bid prices into a day-ahead pool; cheapest bids clear first. With surplus capacity, prices are pinned near marginal fuel cost. When capacity binds (cold snap, heat wave), Edgeworth's mechanism kicks in: marginal generators set very high clearing prices on residual demand. This is exactly the Edgeworth cycle replayed in megawatt-hours.

Worked example — duopoly with linear demand

Two firms; market demand D(p) = 100 − p; constant marginal cost c = 10.

Step 1. Compute monopoly price as a benchmark. π(p) = (p − 10)(100 − p), maximised at p_m = 55; monopoly profit = 45 × 45 = 2025.

Step 2. Bertrand pure-strategy equilibrium: p_1 = p_2 = 10. Total quantity = 90, split 45 each. Profit = 0 each. Consumer surplus = (1/2)(90)(90) = 4050.

Step 3. Cournot equilibrium for comparison. q_i* = (a − c) / 3 = 30; total Q = 60; price p_c = 40; profit per firm = (40 − 10) × 30 = 900. Consumer surplus = (1/2)(60)(60) = 1800.

Step 4. Compare welfare:

Regime	Price	Quantity	Firm profit (each)	Consumer surplus	Total welfare
Monopoly	55	45	2025 (single firm)	1012.5	3037.5
Cournot duopoly	40	60	900	1800	3600
Bertrand duopoly	10	90	0	4050	4050
Perfect competition	10	90	0	4050	4050

Two observations. First, Bertrand and perfect competition deliver identical welfare. Second, the welfare gap between Cournot and Bertrand for the same two firms is 12.5% of total surplus, illustrating how much the choice of strategic variable matters. If a regulator can structurally force firms to compete in prices rather than quantities — for instance by forbidding capacity commitments or by requiring price-posting transparency — the move from Cournot to Bertrand is a 12.5% welfare gain at zero cost.

Extensions and active research

• Bertrand-Edgeworth with stochastic demand. Adds demand uncertainty on top of capacity constraints. Generates more realistic-looking price distributions and predicts when undercutting is rational versus suicidal.
• Bertrand with asymmetric costs. If c_1 < c_2, the equilibrium is p_1 = p_2 = c_2 − ε (the low-cost firm just barely undercuts and earns positive margin). This is the Bertrand explanation for why cost leadership is so valuable.
• Search-and-bargain models (Burdett-Judd 1983). Endogenises the distribution of informed and uninformed consumers; predicts that the equilibrium price distribution depends on the share of comparison shoppers.
• Algorithmic Bertrand and tacit collusion. Active 2020s literature on whether pricing algorithms (Q-learning, deep reinforcement learning) trained on each other's behaviour converge to supra-competitive prices without explicit collusion. Calvano et al. (2020) found that even simple Q-learners can sustain prices above the Bertrand-Nash floor — a regulatory concern for online marketplaces.
• Two-sided markets with zero MC. Tirole-Rochet pricing for platforms where marginal cost is zero on the consumer side, revenue comes from the seller/advertiser side, and Bertrand-floor pricing on the free side is endogenous.

Common pitfalls

• Forgetting the tie-breaking rule. If consumers split 50/50 at ties, p_1 = p_2 = c is the equilibrium. Under other tie-breaking rules (e.g. firm 1 wins all ties), only p_1 = p_2 = c remains an equilibrium if both firms break even at c, but some variants give no pure-strategy equilibrium at all. The convention matters; always state it explicitly.
• Assuming Bertrand requires identical costs. It does not. Asymmetric MC gives p* = c_high − ε, with the low-cost firm capturing the market and earning (c_high − c_low) × D(c_high). The model still has a clean equilibrium and an interesting profit prediction.
• Conflating Bertrand and perfect competition. Bertrand has two firms and ends at marginal cost; perfect competition has infinitely many firms and ends at marginal cost. They share the price prediction but differ in strategic structure: Bertrand has best-reply discontinuities, perfect competition does not. Empirically distinguishing them requires watching strategic responses, not just observing prices.
• Treating the paradox as a refutation. The Bertrand paradox is a feature, not a bug — it forces the modeller to identify which Bertrand assumption fails in the real market and to model the failure explicitly. Skipping that step and reaching for Cournot by default is intellectually lazy.
• Ignoring the no-pure-strategy case. With capacity constraints in a one-shot game, the equilibrium is mixed. Computing expected profits then requires integrating over the price distribution; a common error is to use the average price as if it were the equilibrium price.