Why Homogeneous Bertrand Competition Drives Price to Cost

Why a Small Price Difference Can Change the Outcome

When firms choose quantities, a firm’s sales depend partly on what its rivals produce. In homogeneous-good Bertrand competition, firms instead choose prices, and consumers buy from the lowest-priced seller. If a firm can supply the whole market, charging slightly less than a rival can therefore change its sales sharply: it may go from a share of demand to the entire market. This is the force behind the result that price competition can push the market price down to marginal cost.

The result is not a claim that every price above cost can always be profitably undercut by an arbitrarily small amount. At some prices, demand may be zero. The argument must account for that possibility and state the conditions under which a small undercut works.

Definition: In the homogeneous-good Bertrand model, $n\geq 2$ firms simultaneously choose prices for identical products. Each firm has the same constant marginal cost $c$, no fixed costs, and enough capacity to serve the entire market. Consumers buy from the firm or firms charging the lowest price. If firms tie for the lowest price, they split market demand equally. Market demand at price $p$ is $D(p)$, which is nonnegative, nonincreasing, continuous, and positive at prices from $c$ through some interval above $c$. A firm’s profit is its price-cost margin multiplied by its sales.

For the argument, assume prices and marginal cost are nonnegative. If one firm charges strictly less than all rivals, it serves all market demand at its price. If $k$ firms tie at the lowest price $p$, each serves $D(p)/k$. Thus, a firm in that tie earns:

\pi_{\text{tie}}=\frac{(p-c)D(p)}{k}.

The assumptions matter. With a homogeneous product, buyers have no product-based reason to choose a higher-priced seller. Sufficient capacity lets a low-priced firm meet the resulting demand. Equal marginal costs make each firm’s price-cost margin comparable. Without these conditions, the undercutting argument may change; capacity constraints and differentiated products require separate analysis.

Deriving the Undercutting Incentive

Suppose $k\geq 2$ firms tie at a lowest price $p>c$, and demand at that price is positive. A firm in the tie can lower its price to $p-\varepsilon$, where $\varepsilon>0$ is small enough that $p-\varepsilon>c$. It then becomes the unique lowest-priced seller and serves the whole market.

Its profit from the undercut is:

\pi_{\text{undercut}}(\varepsilon)=(p-\varepsilon-c)D(p-\varepsilon).

By continuity of demand, as $\varepsilon$ approaches zero, this profit approaches $(p-c)D(p)$. The firm’s original tied profit is only $(p-c)D(p)/k$. Since $k\geq 2$ and $(p-c)D(p)>0$, the limiting undercut profit is strictly greater than the original profit. The strict difference means a sufficiently small positive $\varepsilon$ also gives a profitable deviation.

\lim_{\varepsilon\to 0^+}\pi_{\text{undercut}}(\varepsilon) =(p-c)D(p) > \frac{(p-c)D(p)}{k} =\pi_{\text{tie}}.

This is the central derivation: a firm that previously served only a fraction of positive demand can capture nearly the same total demand while keeping nearly the same margin. Continuity makes the small price reduction’s effect on demand small, while the firm’s share of demand jumps from $1/k$ to all of it.

Start with a tie above cost.
Each tied firm earns a positive margin on only its equal share of positive demand.

Lower one firm’s price slightly.
Choose the reduction small enough that the new price remains above marginal cost.

Compare profits.
The deviating firm captures the market. By continuity, its profit for a sufficiently small reduction exceeds its former tied profit.

What If Demand at the Common Price Is Zero?

A common-price profile above cost does not necessarily have positive sales. For example, if demand has a choke price, a common price at or above that price can leave every firm with zero profit. At such a price, the continuity argument for a small undercut does not establish a gain: a slightly lower price might still leave demand at zero.

The positive-demand-near-cost assumption supplies a different deviation. If all firms charge $p>c$, choose a price $p'$ strictly between $c$ and $p$, close enough to $c$ that $D(p')>0$. The deviating firm is then the unique low-priced seller, and its profit $(p'-c)D(p')$ is positive, while its original profit was zero. More generally, if demand is positive at the common price, the small-undercut derivation applies; if demand is zero there, the lower price near cost rules out that profile instead.

Worked Example: A Profitable Undercut When Demand Is Positive

Two hypothetical firms sell an identical replacement part. Each has constant marginal cost $c=\$20$, enough capacity to serve all buyers, and no fixed costs. Market demand is $D(p)=120-2p$ for prices in the relevant range. Both firms charge $\$35$. Find each firm’s current profit and compare it with the profit from charging $\$34.90$. Use the method of comparing tied profit with whole-market profit; this avoids treating the undercut as if the firm still received only half of demand.

Current demand. At $p=\$35$,

$$ D(35)=120-2(35)=120-70=50. $$

Current sales and profit. The firms tie, so each sells $50/2=25$ parts. The margin is $\$35-\$20=\$15$, giving profit $25(\$15)=\$375$ per firm.

Profit after the undercut. At $\$34.90$, the deviating firm is the unique low-priced seller. Demand is $120-2(34.90)=120-69.80=50.20$ parts. Its margin is $\$34.90-\$20=\$14.90$, so its profit is $50.20(\$14.90)=\$747.98$.

The deviation raises profit from $\$375.00$ to $\$747.98$. The small margin reduction is more than offset by serving the whole market rather than half of it.

Worked Example: A Common Price with Zero Demand

Two hypothetical sellers have marginal cost $c=\$10$, sufficient capacity, and no fixed costs. Demand is $D(p)=\max(90-3p,0)$. Both charge $\$40$. Calculate their profits, then test a deviation to $\$12$. The method is to check demand at the original price before applying the small-undercut argument; this avoids assuming positive sales where none exist.

Original price. At $\$40$, the expression $90-3(40)=90-120=-30$, so demand is $\max(-30,0)=0$. Each firm sells zero and earns zero profit.

Deviation. At $\$12$, demand is $\max(90-3(12),0)=\max(54,0)=54$. The deviating firm is cheaper than its rival and serves all 54 units. Its margin is $\$12-\$10=\$2$, giving profit $54(\$2)=\$108$.

The deviation increases profit from $\$0$ to $\$108$. Here, an arbitrarily small undercut of $\$40$ would not be shown to work; the profitable deviation is a lower price near cost where demand is positive.

Why Marginal Cost Is the Symmetric Equilibrium Price

A symmetric price profile has every firm charging the same price. If that common price is above $c$, it cannot be an equilibrium. When demand there is positive, any firm can make the profitable small undercut just derived. When demand there is zero, it can instead choose a price between cost and the common price where demand is positive. In either case, at least one firm has a profitable deviation.

At a common price of $c$, each firm’s margin and profit are zero. A firm that charges less than $c$ has a negative margin if it makes sales. A firm that charges more than $c$ loses all sales to the firms still charging $c$, so it earns zero. Neither deviation gives a higher profit than zero. Therefore, charging marginal cost is a symmetric Nash equilibrium: no firm can increase its profit by changing its price alone.

Worked Example: Checking Deviations at Marginal Cost

Three hypothetical firms sell the same component, each with constant marginal cost $c=\$15$, sufficient capacity, and no fixed costs. Demand is positive at $\$15$. All three charge $\$15$. Check the profit from a price of $\$14.99$ and from a price of $\$15.01$. The method is to compare each deviation with the other firms’ unchanged prices; this avoids assuming that a price increase retains sales.

At the common price. The margin is $\$15-\$15=\$0$, so each firm earns zero profit, regardless of its share of demand.

Price below cost. At $\$14.99$, the deviating firm is the unique low-priced seller and serves positive demand. Its margin is $\$14.99-\$15=-\$0.01$, so its profit is negative. The exact demand quantity is not needed to determine the sign: a negative margin multiplied by positive sales gives a loss.

Price above cost. At $\$15.01$, the firm is more expensive than the other two firms, which remain at $\$15$. It makes no sales and earns zero profit, the same as before rather than more.

Neither deviation raises profit. The same reasoning applies to any price below or above $\$15$, not just these one-cent changes.

Scope of the Result and Common Traps

The result concerns the market price, not necessarily every firm’s posted price in every possible equilibrium profile. In the symmetric equilibrium, all firms charge $c$. In some non-symmetric profiles, multiple firms may charge $c$ while other firms quote higher prices and make no sales. Those higher quotes do not change the lowest price paid by consumers. A single firm that is the only seller at $c$, with all rivals quoting more, could generally raise its price a little while remaining the lowest-priced seller; that is why such a profile need not be an equilibrium.

The claim also depends on the model assumptions. If a firm cannot supply the whole market, an undercut may not capture all demand. If products differ, some consumers may prefer a higher-priced product. If marginal costs differ, the same cost-price comparison does not apply to every firm. In each case, the simple homogeneous-good, identical-cost argument must be modified.

Assuming positive demand at every above-cost price. A demand curve may reach zero. Check demand at the common price; if it is zero, use a price near cost with positive demand.
Keeping the tied market share after undercutting. With sufficient capacity and a strict lowest price, the deviator serves the whole market, not its former fraction.
Ignoring the margin. An undercut below cost can attract demand but lose money. The profitable-deviation argument uses a new price still above $c$.
Claiming the result without its assumptions. Homogeneity, identical costs, sufficient capacity, and the tie rule support the standard argument.

Key takeaway: With identical constant marginal costs, homogeneous products, sufficient capacity, and positive demand near cost, a common price above marginal cost can be undermined by a profitable deviation. At a common price equal to marginal cost, no firm can gain by changing its price alone. Thus marginal cost is the symmetric Bertrand equilibrium price, aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, identify whether the relevant deviation captures the market or makes no sales, then compare profits.

Two firms have marginal cost $\$12$, sufficient capacity, and no fixed costs. Demand is $D(p)=100-2p$ in the relevant range. Both charge $\$22$ and split demand equally. Find each firm’s current profit and the deviator’s profit if it charges $\$21.90$. (Answer: Current profit is $\$280.00$ per firm; the undercutting firm earns $\$556.38$.)
Two firms have marginal cost $\$8$, sufficient capacity, and no fixed costs. Demand is $D(p)=\max(70-2p,0)$. Both charge $\$40$. Find current profit per firm and the profit to one firm from charging $\$10$. (Answer: Current profit is $\$0$; the deviation earns $\$100$.)
Four firms have marginal cost $\$5$, sufficient capacity, and no fixed costs. Demand is $D(p)=200-5p$ in the relevant range. At a common price of $\$20$, calculate each firm’s tied profit. Then find one firm’s profit at $\$19.99$. (Answer: Tied profit is $\$375.00$ per firm; the undercutting firm earns $\$1,499.75$.)
Two firms have marginal cost $\$10$, sufficient capacity, and no fixed costs. Demand is $D(p)=120-4p$ in the relevant range. Firm 1 charges $\$18$ and firm 2 charges $\$27$. Find firm 2’s profit before and after it changes its price to $\$17.99$. (Answer: Firm 2 initially earns $\$0$; after undercutting it earns $\$383.84$.)
Two firms have marginal cost $\$14$, sufficient capacity, and no fixed costs. Demand is $D(p)=100-p$ in the relevant range. Both charge $\$14$. Find each firm’s profit, then calculate the profit of a unilateral deviation to $\$13.99$ and explain the result of a deviation to $\$14.01$. (Answer: Each earns $\$0$ at $\$14$; a deviation to $\$13.99$ earns $-\$0.86$, while a deviation to $\$14.01$ earns $\$0$ because the firm makes no sales.)

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