Bertrand Price Competition with Homogeneous Goods

Introduction

In quantity competition, firms choose how much to produce, and the market price follows from total output. In the basic Bertrand model, firms instead choose prices. That change in strategic variable matters because, when products are identical and buyers can switch freely, even a small price difference can redirect the market’s demand.

The model’s assumptions must be kept distinct. Consumers’ preferences and the tie-breaking rule determine who gets demand; sufficient capacity lets the lower-priced seller serve that demand; and marginal cost is used to calculate profit. In particular, cost does not decide which identical product a consumer prefers.

Assumptions of the Basic Bertrand Model

Definition: In the basic homogeneous-good Bertrand model, a set of firms simultaneously chooses prices for products consumers regard as identical. Consumers observe the prices and buy from the lowest-priced firm, subject to a specified rule for ties. Each firm has sufficient capacity to meet the entire market demand at its price. For profit calculations, each firm is assumed to have a constant marginal cost and no fixed cost unless one is explicitly included.

The components of this definition have separate roles:

Simultaneous price setting: each firm chooses its price without first observing the other firms’ current choices. The resulting allocation can be analyzed conditional on the prices, but a firm choosing its price does not have advance knowledge of its rivals’ simultaneous choices.
Homogeneous goods: buyers view the products as interchangeable. There are no differences in quality, brand preference, location, or service that would make a buyer prefer one firm at the same price.
Buyer awareness and freedom to switch: consumers can compare prices and purchase from the lower-priced firm without search, switching, or transaction costs that would prevent them from doing so.
Sufficient capacity: a firm charging the lowest price can supply all market demand at that price. Without this condition, the preferred seller might be unable to serve everyone who wants to buy from it.
Constant marginal cost: producing each additional unit costs the same amount, denoted $c_i$ for firm $i$. This simplifies the profit calculation; it is not what makes consumers choose one firm over another.

Let $D(p)$ denote total market demand when the common price paid by buyers is $p$. A standard model treats $D(p)$ as nonnegative and weakly decreasing in price over the relevant range. If firm $i$ charges $p_i$, the firms’ prices determine which price consumers face and how that demand is allocated.

Allocation rule: If one firm charges a lower price than every rival, consumers who buy choose that firm’s product. With sufficient capacity, it serves $D(p_i)$ units at its price $p_i$, and the higher-priced firms sell zero. If firms tie for the lowest price, demand is divided according to the stated tie-breaking rule. In a symmetric example, an equal division is often assumed.

Why Buyers Choose the Lower Price

The price-choice rule follows directly from identical products. Consider a consumer who values one unit of the product at $v$. If firm A charges $p_A$, the consumer’s net benefit from buying from A is $v-p_A$; from firm B, it is $v-p_B$. When $p_A<p_B$, the difference in net benefits is

$$ (v-p_A)-(v-p_B)=p_B-p_A>0. $$

The consumer therefore prefers A whenever buying is worthwhile. Consumers whose value is at least $p_A$ are willing to buy at that price, so aggregate demand at the lowest price is $D(p_A)$. Sufficient capacity allows A to fulfill those purchases. If the prices are equal, the consumer receives the same net benefit from either firm; the model needs a tie-breaking assumption to determine how the market is divided.

This reasoning identifies the role of each assumption. Product identity and the buyer-choice rule explain the preference for the lower price. The capacity assumption makes it feasible for that firm to serve the demand. Neither of these conclusions requires constant marginal cost.

Calculating Sales and Profit

Once the allocation has been determined, constant marginal cost makes profit straightforward to calculate. If firm $i$ sells $q_i$ units at price $p_i$, has constant marginal cost $c_i$, and has no fixed cost, then revenue is $p_iq_i$, variable cost is $c_iq_i$, and profit is

\pi_i=p_iq_i-c_iq_i=(p_i-c_i)q_i.

This formula is conditional on the sales allocation. It does not determine that allocation: first apply consumer choice and the tie rule to find $q_i$, then use costs and prices to calculate profit. If a fixed cost $F_i$ is included, profit instead equals $(p_i-c_i)q_i-F_i$.

Worked Example: A Lower Price Captures Market Demand

Two firms sell an identical product. Market demand is $D(p)=180-12p$ units for prices from 0 to 15 dollars and zero above 15 dollars. Firm A charges $6$ dollars and has constant marginal cost $2.50$ dollars. Firm B charges $8$ dollars and has constant marginal cost $3$ dollars. Both have sufficient capacity and no fixed costs. Find each firm’s sales and profit.

Method. Apply the price-based allocation rule first, then calculate each firm’s profit. This avoids using marginal cost to decide which firm consumers choose.

1. Identify the lowest price. Since $6<8$, consumers who buy choose firm A. A has capacity to serve all demand at its price.

2. Calculate total demand at A’s price.

$$ D(6)=180-12(6)=180-72=108. $$

Thus $q_A=108$ and $q_B=0$.

3. Calculate profit. A’s margin per unit is $6-2.50=3.50$ dollars, so $\pi_A=3.50(108)=378$ dollars. B sells zero units, so its variable revenue and variable cost are both zero and $\pi_B=(8-3)(0)=0$.

Check. A’s revenue is $6(108)=648$ dollars and its variable cost is $2.50(108)=270$ dollars; $648-270=378$ dollars. The sales add to $108$, exactly the total demand at the lowest price.

Worked Example: Dividing Demand at a Price Tie

Two firms charge $7$ dollars for an identical product. Total demand is $D(p)=240-20p$ units for prices from 0 to 12 dollars. Assume tied firms split demand equally. Each firm has constant marginal cost of $3.20$ dollars, sufficient capacity, and no fixed cost. Find total demand, each firm’s sales, and each firm’s profit.

Method. Use the tie rule to divide demand before applying the profit formula. This avoids assigning all demand to both firms.

1. Calculate total demand at the common price.

$$ D(7)=240-20(7)=240-140=100. $$

2. Divide demand equally. Each firm sells $100/2=50$ units.

3. Calculate each firm’s margin and profit. The margin is $7-3.20=3.80$ dollars per unit. Each firm’s profit is $3.80(50)=190$ dollars.

Check. Each firm’s revenue is $7(50)=350$ dollars, and its variable cost is $3.20(50)=160$ dollars. The difference is $190$ dollars. The two firms together sell $50+50=100$ units, not 200.

Conditional Price Comparisons

The allocation rule can also be used to compare possible price choices against a given rival price. In a simultaneous-choice setting, those comparisons are conditional calculations: they show what sales and profit would result from each price pair, not what a firm knows about a rival’s choice when selecting its own price.

Worked Example: Comparing Prices Against a Rival’s Price

Consider two firms selling an identical product with demand $D(p)=300-15p$ units for prices from 0 to 20 dollars. Firm A has constant marginal cost $4$ dollars, and firm B has constant marginal cost $5$ dollars. Both have sufficient capacity and no fixed cost. Evaluate A’s sales and profit if B charges $10$ dollars and A charges, respectively, $9.50$, $10$, or $10.50$ dollars. Use an equal split at a tie.

Method. For each price pair, determine A’s sales from the consumer-choice and tie rules, then multiply by A’s margin. This keeps the allocation decision separate from the cost calculation.

1. A charges $9.50$ dollars. A is cheaper, so it serves all demand at $9.50$:

$$ D(9.50)=300-15(9.50)=300-142.50=157.50. $$

A’s margin is $9.50-4=5.50$ dollars, giving profit $5.50(157.50)=866.25$ dollars. A’s sales are $157.50$ units.

2. A charges $10$ dollars. The firms tie. Total demand is $D(10)=300-150=150$ units, so A sells $150/2=75$ units. Its margin is $10-4=6$ dollars, and its profit is $6(75)=450$ dollars.

3. A charges $10.50$ dollars. B is cheaper, so A sells zero units. Thus A’s profit is $(10.50-4)(0)=0$ dollars.

Check. A’s sales are $157.50$, $75$, and $0$ units in the three cases. The tie case allocates $75$ units to each firm, totaling $150$, which equals demand at the common price. These calculations compare outcomes conditional on prices; they do not assume A observes B’s simultaneous choice in advance.

What Marginal Cost Does—and Does Not—Determine

The distinction between customer allocation and profit is essential. For identical products, a consumer compares the prices paid, not the firms’ production costs. A firm with lower marginal cost does not automatically attract demand if it charges more. Its cost affects the margin it earns on sales, but consumer allocation follows the model’s price and tie-breaking rules.

The allocation formula and profit formula can be summarized separately:

Question	Relevant assumption or rule	What it determines
Which firm do buyers choose?	Identical goods, observed prices, and the buyer-choice rule	Allocation toward the lowest-priced firm
Can the lowest-priced firm serve all buyers?	Sufficient capacity	Whether it can supply all demand at its price
How is a lowest-price tie divided?	Specified tie-breaking rule	Each tied firm’s quantity sold
What profit does a firm earn?	Price, allocated quantity, and cost assumptions	$(p_i-c_i)q_i$ when marginal cost is constant and there is no fixed cost

Common Mistakes and Model Limits

Using cost to allocate demand. A lower-cost firm does not receive customers simply because it can produce more cheaply. In this model, consumers choose by comparing prices for identical products.
Giving each tied firm the full market demand. If tied firms split demand equally, each gets half. Their sales together must equal $D(p)$, not twice $D(p)$.
Forgetting to evaluate demand at the lowest price. When one firm charges less, its quantity is $D(p_i)$ at its own price, not demand evaluated at the rival’s higher price.
Applying the profit formula before finding sales. The margin $p_i-c_i$ alone is not profit. Multiply it by the quantity allocated to the firm, and subtract any fixed cost if the problem includes one.
Ignoring capacity. If the lowest-priced firm cannot supply all demand, the sufficient-capacity assumption fails. The basic allocation rule alone does not specify how the shortfall is rationed or how remaining buyers are served.
Treating conditional comparisons as observed information. Calculating what would happen at a given pair of prices does not mean a firm sees its rival’s simultaneous choice before choosing its own.

Key takeaway: In homogeneous-good Bertrand competition, the lowest price determines which firm consumers prefer, the tie rule determines how equal lowest prices divide demand, and sufficient capacity makes the stated allocation feasible. Constant marginal cost enters afterward, when calculating profit from price and sales.

Check Your Understanding

For each problem, identify the allocation before calculating profit. Unless stated otherwise, assume no fixed costs and sufficient capacity.

Demand is $D(p)=160-8p$ units for prices from 0 to 20 dollars. Firm A charges $5$ dollars and has marginal cost $2$ dollars; firm B charges $6$ dollars. Find each firm’s sales and profit. (Answer: A sells 120 units and earns $360; B sells 0 units and earns $0.)
Two firms tie at $8$ dollars, split demand equally, and each has marginal cost $3$ dollars. If $D(p)=210-15p$, find each firm’s sales and profit. (Answer: Total demand is 90 units; each sells 45 units and earns $225.)
Demand is $D(p)=260-10p$. Firm A charges $11$ dollars and has marginal cost $6$ dollars; firm B charges $10$ dollars and has marginal cost $4$ dollars. Find each firm’s sales and profit. (Answer: B sells 160 units and earns $960; A sells 0 units and earns $0.)
Demand is $D(p)=180-12p$. Firm B charges $7$ dollars. Firm A has marginal cost $2$ dollars. Compare A’s sales and profit at prices $6.50$, $7$, and $7.50$, assuming an equal split at a tie. (Answer: At $6.50, A sells 102 and earns $459; at $7, A sells 48 and earns $240; at $7.50, A sells 0 and earns $0.)
At a price of $5$ dollars, demand is 100 units. Firm A is the only lowest-priced firm, but its capacity is only 40 units. What can the basic sufficient-capacity Bertrand allocation rule conclude, and what additional information is needed to specify actual sales? (Answer: The basic rule’s sufficient-capacity condition is violated; it does not specify actual allocation beyond A being consumers’ preferred seller. A rationing or residual-demand rule is needed to determine sales.)

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