Bertrand duopoly equilibrium calculator

The Bertrand model is the workhorse of simultaneous price competition: two firms sell a homogeneous good, post prices at the same moment, and the cheaper seller takes the entire market — they split demand only when the two prices match. With constant marginal cost and no capacity constraint, each firm wants to undercut a rival who is pricing above cost, so the unique Nash outcome drives both prices down to marginal cost. That is the Bertrand paradox: two sellers are enough for the competitive price. This Bertrand duopoly equilibrium calculator takes linear inverse demand P = a − bQ together with each firm's constant marginal cost and returns both posted prices, the quantities each firm sells, total output, and profits. When costs are equal the market splits at p = c with zero economic profit; when they differ the low-cost firm either posts its monopoly price or limit-prices at the rival's cost. Students, tutors, and industrial-organization notebooks get a deterministic key instead of a spreadsheet that mishandles the split-versus-winner branch.

What a homogeneous Bertrand equilibrium means

In the homogeneous-goods Bertrand duopoly, two firms choose prices simultaneously. Buyers treat the products as perfect substitutes, so they purchase only from the cheaper firm. If the posted prices are equal, the standard textbook convention is that the two firms split market demand in half.

Inverse demand is written P = a − bQ, where a is the choke price on the vertical axis and b is a positive slope, so quantity demanded at a posted price P is Q = (a − P) / b whenever P sits below a. Each firm has a constant marginal cost and no binding capacity, so the profit of a firm that wins the market at price p is (p − c) times residual demand.

The strategic logic is undercutting: if the rival posts a price above your cost, you can steal the whole market by posting a slightly lower price and still earn a positive margin. That incentive continues until neither firm can profitably go lower.

When the two costs are equal, the unique Nash equilibrium is both prices equal to that common cost, each firm serving half of residual demand, and both economic profits equal to zero — the famous result that price competition with homogeneous goods drives the market to marginal cost even with only two sellers.

When costs differ, the low-cost firm can still undercut; it either charges its unconstrained monopoly price if that price already sits at or below the rival's cost, or it limit-prices at the high-cost firm's cost and serves the entire market. This calculator implements that closed-form equilibrium, not a numerical best-response iteration and not a Cournot quantity game.

Inputs, closed-form branches, validation, and the response

Both marginal costs must be non-negative, and the choke price must strictly exceed the lower cost so that some units are demanded at cost. On the symmetric branch, when the two costs are equal to c, both firms post p = c, total quantity is (a − c) / b, each firm sells half of that total, and both profits are zero. On the asymmetric branch the low-cost firm's monopoly price is (a + c_L) / 2.

If that monopoly price is at most the rival's cost, the low-cost firm posts the monopoly price, the high-cost firm posts its own cost and sells nothing, and the winner earns the monopoly profit. If the monopoly price is above the rival's cost, the low-cost firm limit-prices at the high-cost firm's cost, takes the entire residual demand at that price, and earns (c_H − c_L) times that quantity. Numbers are rounded to twelve decimal places for stable goldens.

Where a Bertrand duopoly calculator earns its keep

Introductory industrial-organization and intermediate micro courses assign Bertrand drills because the paradox is the first sharp contrast with Cournot: the same two firms, the same linear demand, but prices rather than quantities as the strategic variable, and a completely different equilibrium price. Tutoring platforms and LMS auto-graders need a deterministic oracle that returns the split-at-cost outcome when costs match and the correct winner-takes-all outcome when they do not, instead of a student spreadsheet that always halves quantity.

Policy and strategy notebooks that compare Cournot and Bertrand on the same demand and cost numbers can pin the Bertrand side here and keep Cournot in a separate quantity solver. Pin the rounded prices and the regime string in tests, keep intercepts and costs finite, and you get reproducible Bertrand equilibria for homework, exam generators, and lightweight IO notebooks without standing up a numerical game solver.