Bertrand–Edgeworth model

Model homogeneous-product price competition among capacity-constrained sellers under an explicit rationing rule, so residual demand can prevent the pure marginal-cost equilibrium of unconstrained Bertrand competition.

Version

v2 · 2026-09-19 · History

Domain-specific #

1371

Domain group

Social Sciences

Origin domain

Economics & Finance

Subdomain

Capacity Constrained Price Competition → Economics & Finance

Aliases

Bertrand–Edgeworth competition, Capacity-constrained Bertrand model

Core Idea

The Bertrand–Edgeworth model is a price-setting oligopoly game for a homogeneous good in which each firm has limited supply capacity or an output constraint. Buyers first seek the lowest offered price, but a low-price firm may be unable to serve all demand. A declared rationing rule allocates unmet demand to higher-price firms. Capacity therefore creates residual demand and changes best responses relative to the unconstrained Bertrand model.

Each firm chooses a price and sells up to capacity against market demand, production cost, rivals' prices, and the rationing rule. Undercutting can attract demand but may sacrifice margin once capacity binds. Raising price can remain profitable because rivals cannot serve the whole market.

How would you explain it like I'm…

Stands That Run Out

Two lemonade stands sell the same lemonade, and kids go to the cheaper one first. But each stand only has so many cups. If the cheap stand runs out, the other kids have to buy from the pricier stand. So being the cheapest does not always win every customer, and a stand can sometimes charge more and still sell.

Price Wars with Limited Supply

In the Bertrand–Edgeworth model, shops sell the same thing and compete by picking prices, but each shop can only make or hold a limited amount. Shoppers try the cheapest shop first. If that shop runs out, the leftover shoppers go to the more expensive shops, following a set rule about who gets served. Because of this, cutting your price to be cheapest may not help much once you are sold out, and charging more can still work because your rival can't serve everyone. Depending on the details, the prices may settle down, or they may never settle into one steady choice.

Capacity-Constrained Price Competition

The Bertrand–Edgeworth model is a price-competition model for identical goods where each firm has a capacity limit. Customers try the lowest price first, but if the cheapest firm cannot supply everyone, a stated rationing rule decides which unserved customers go to higher-priced firms. That leftover demand is called residual demand, and it changes each firm's best response compared with ordinary Bertrand competition. Undercutting a rival can win customers but may just lower your margin once you are at capacity, while raising your price can still pay off because rivals cannot serve the whole market. Depending on capacities, demand, costs and the rationing rule, there may be a stable single-price equilibrium, none in pure strategies, or an equilibrium where firms randomize prices.

 

The Bertrand–Edgeworth model is a price-setting oligopoly game for a homogeneous good in which each firm faces a capacity or output constraint. Consumers seek the lowest price, but a low-price firm may be unable to meet all demand, so a declared rationing rule (e.g., efficient or proportional rationing, which yield different residual demands) allocates unserved demand to higher-priced firms. Each firm chooses a price and sells up to capacity given market demand, cost, rivals' prices and the rationing rule. Capacity makes undercutting less attractive once it binds and makes a higher price potentially profitable because rivals cannot serve the whole market. Outcomes are parameter-dependent: a pure-strategy equilibrium may exist at a competitive or other price, may fail to exist in an intermediate regime, or be replaced by a mixed-strategy equilibrium. The model is distinct from unconstrained Bertrand competition and from quantity-setting Cournot competition. The Edgeworth paradox, the possible nonexistence of a pure equilibrium, is one outcome within some specifications, not the model itself. Variants such as endogenous capacity chosen in a prior stage, product differentiation, demand uncertainty, discrete prices or convex costs are distinct extensions that should be labeled.

Scope of Application

The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Bertrand–Edgeworth model itself, not metaphors based only on resemblance.

• Industrial organization. Analyzing how finite capacity alters price-setting equilibrium.
• Market design. Testing rationing and residual-demand rules in constrained supply settings.
• Electricity economics. Using carefully adapted capacity-constrained price games without assuming the baseline captures network constraints.
• Oligopoly theory. Comparing price and quantity competition under capacity limits.
• Equilibrium existence. Locating parameter regions with pure versus mixed strategies.
• Large-market limits. Studying convergence toward competitive outcomes as firms become smaller and more numerous.

Clarity

A clear account of Bertrand–Edgeworth model must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. State firms, demand, costs, capacities, price action set, tie rule, and rationing rule. Distinguish exogenous capacity from a prior capacity-choice stage. Report equilibrium existence as conditional on the whole specification. Separate a model implication from an empirical claim about an actual market.

Manages Complexity

Bertrand–Edgeworth model manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: price-setting firms supplies two or more strategic sellers choose posted prices.; homogeneous product supplies buyers rank offers primarily by price under the baseline model.; market demand supplies a price-dependent demand function supplies potential sales.; capacity constraints supplies each firm faces an upper bound on quantity served.; cost structure supplies marginal or convex costs determine profit from realized output..

Abstract Reasoning

1. Define the market demand and each firm's feasible output and cost.
2. Choose efficient, proportional, or another explicit rationing rule.
3. Derive realized sales and profit for every price profile, including ties.
4. Calculate best responses in each capacity and residual-demand regime.
5. Test candidate pure equilibria against upward and downward deviations.
6. If no pure equilibrium exists, establish conditions for a mixed or approximate solution rather than asserting one automatically.

Knowledge Transfer

The strict upward abstraction is Competition. Bertrand–Edgeworth Model instantiates Competition because capacity-constrained sellers choose rival prices for the same demand, and one firm's captured sales reduce the residual prize available to rivals. Within capacity constrained price competition, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Bertrand–Edgeworth model after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.

Relationships to Other Abstractions

Current abstraction Bertrand–Edgeworth model Domain-specific

Parents (1) — outgoing typed relations

• Bertrand–Edgeworth model is a kind of Competition Prime

  Bertrand–Edgeworth Model instantiates Competition because capacity-constrained sellers choose rival prices for the same demand, and one firm's captured sales reduce the residual prize available to rivals.

Hierarchy path (1) — routes to 1 parentless root

• Bertrand–Edgeworth model → is a kind of Competition

Neighborhood in Abstraction Space

Bertrand–Edgeworth model sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Edgeworth Paradox — 0.85
• Bertrand competition — 0.83
• Differentiated Bertrand competition — 0.81
• Oligopoly — 0.81
• Demand Shaping — 0.79

Computed from structural-signature embeddings · 2026-10-25