Capacity Constraints in Bertrand Competition

Why Capacity Changes Bertrand Competition

In the standard homogeneous-good Bertrand model, a firm that slightly undercuts its rival can attract the market’s demand. That conclusion depends on the firm being able to supply all the customers who switch to it. If the firm has a production limit, an undercut may instead fill its capacity while leaving some customers for the higher-priced rival. The higher-priced firm can then earn positive sales even though its product is otherwise identical.

The central calculation is therefore not just total market demand at a price. It is the demand remaining after the low-priced firm has made its sales. The result depends on assumptions about how customers are allocated when a firm is capacity constrained. This tutorial uses efficient rationing: when a low-priced firm cannot serve everyone, it serves customers with the highest willingness to pay first. Other rationing rules can produce different residual demand and different strategic conclusions.

Definition: In a capacity-constrained Bertrand model, firms selling a homogeneous product choose prices simultaneously, but firm $i$ cannot sell more than its capacity $K_i$. Let $Q(p)$ be market demand at price $p$, and let $c_i$ be firm $i$'s constant marginal cost. With no fixed costs, firm $i$'s profit is $\pi_i=(p_i-c_i)q_i$, where $q_i$ is its actual sales and $q_i\leq K_i$. Under efficient rationing, a lower-priced firm serves the highest-willingness-to-pay customers first.

Assume prices do not exceed consumers’ reservation prices, firms have no other binding production limits, and customers buy from the lowest-priced firm they can access. If the two firms charge different prices, write $p_L<p_H$, with capacities $K_L$ and $K_H$. The low-priced firm sells up to its capacity, while the high-priced firm faces demand at its own price minus the units already supplied to the customers willing to pay that price.

q_L=\min\{K_L,Q(p_L)\},\qquad q_H=\min\{K_H,\max[Q(p_H)-q_L,0]\}.

The subtraction uses $Q(p_H)$, not $Q(p_L)$. A customer willing to buy at the higher price is also willing to buy from the cheaper firm. Under efficient rationing, the low-priced firm serves the customers with the highest willingness to pay, so those customers are removed first from the demand available to the high-priced firm. The remaining eligible demand is $Q(p_H)-q_L$, if positive.

Allocating Sales and Calculating Profit

When one firm charges less, its capacity may bind even though total market demand is much larger. The high-priced firm’s sales then depend on both its own capacity and the residual demand. Profit uses actual quantity sold, not the quantity the firm could have sold without the capacity restriction.

Worked Example: A Lower Price Does Not Capture the Whole Market

Two firms sell the same product. Market demand is $Q(p)=150-5p$. Firm L has capacity 40 and charges $p_L=\$12$; Firm H has capacity 70 and charges $p_H=\$16$. Their marginal costs are $c_L=\$6$ and $c_H=\$6$. Under efficient rationing, calculate each firm’s sales and profit.

Method. First find demand at each firm’s price. Then apply the low-priced firm’s capacity limit and subtract its actual sales from demand at the higher price.

At Firm L’s price, $Q(12)=150-5(12)=150-60=90$. Its capacity is 40, so its actual sales are $q_L=\min(40,90)=40$. Firm L cannot supply the other 50 units of demand at its price.

At Firm H’s price, $Q(16)=150-5(16)=150-80=70$. After the low-priced firm’s 40 sales, residual demand for Firm H is $\max(70-40,0)=30$. Since Firm H’s capacity is 70, its actual sales are $q_H=\min(70,30)=30$.

Firm L’s profit is $(12-6)(40)=6(40)=\$240$. Firm H’s profit is $(16-6)(30)=10(30)=\$300$. The check is that total sales are $40+30=70$, no greater than demand of 70 at the higher price, and neither firm exceeds its capacity. The common mistake is to assign Firm H all 70 units of demand at \$16. Those 70 units include customers already served by Firm L; Firm H has only 30 residual units.

The sales rule also clarifies the role of a capacity limit in a price cut. If a firm has enough capacity to serve all market demand, undercutting can redirect many customers to it. If it has little capacity, a price cut may increase its sales only up to that limit. The effect on the rival depends on the residual demand left at the rival’s price.

Why the Residual-Demand Formula Holds

The formula follows from consumers’ willingness to pay and the priority given to the cheaper product. At the high price $p_H$, $Q(p_H)$ customers in total are willing to buy. Under efficient rationing, the low-priced firm serves $q_L$ of these customers first whenever its capacity binds. Subtracting those customers leaves $Q(p_H)-q_L$ possible buyers for the high-priced firm. If that difference is negative, residual demand is zero; and if it exceeds the high firm’s capacity, sales stop at $K_H$.

\text{Residual demand at }p_H =\max\{Q(p_H)-q_L,0\},

so applying the high-priced firm’s own capacity gives $q_H=\min\{K_H,\max[Q(p_H)-q_L,0]\}$. This result requires the stated rationing assumption. With proportional rationing, for example, customers may be assigned differently and the higher-priced firm’s residual demand need not equal this expression. A numerical answer should therefore follow the allocation rule specified in the problem rather than silently assuming one.

Find demand at each price.
Evaluate $Q(p_L)$ and $Q(p_H)$ separately. Demand is generally lower at the higher price.

Apply the low-price capacity limit.
Set $q_L=\min\{K_L,Q(p_L)\}$. The low-priced firm cannot sell beyond capacity.

Calculate residual demand.
Under efficient rationing, subtract $q_L$ from $Q(p_H)$, then replace a negative result with zero.

Apply the high-price capacity and calculate profits.
Limit the high firm’s sales to $K_H$, then multiply actual sales by the relevant price-minus-cost margin.

Comparing Price Choices

A firm considers not only the demand associated with its own price but also the rival’s price and capacity. The following example makes the price choices explicit and shows how capacity affects best responses. To make the equilibrium conclusion precise, each firm is restricted to the listed price menu; it is not a claim about every continuous-price Bertrand model.

Worked Example: Finding an Equilibrium on a Price Menu

Two firms face demand $Q(p)=100-4p$, have marginal cost $c_A=c_B=\$4$, and have capacities $K_A=30$ and $K_B=50$. Each may choose only a price of \$10, \$12, or \$14. At a tie, assume demand is divided in proportion to capacities, unless total demand exceeds combined capacity, in which case each firm sells its full capacity. Find the pure-strategy Nash equilibrium on this menu.

Method. Compute profits for each price pair. Under a tie, divide demand in the ratio $30:50$; when prices differ, use efficient rationing. Then check whether either firm can gain by changing its own price.

At a tie of \$10, demand is $Q(10)=100-4(10)=60$. The capacity ratio is $30:50$, so A sells $60(30/80)=22.5$ and B sells $60(50/80)=37.5$. Their profits are $(10-4)(22.5)=\$135$ and $(10-4)(37.5)=\$225$.

At a tie of \$12, demand is $Q(12)=100-48=52$. A sells $52(30/80)=19.5$, and B sells $52(50/80)=32.5$. Their profits are $8(19.5)=\$156$ and $8(32.5)=\$260$.

At a tie of \$14, demand is $Q(14)=100-56=44$. A sells $44(30/80)=16.5$, and B sells $44(50/80)=27.5$. Their profits are $10(16.5)=\$165$ and $10(27.5)=\$275$.

For unequal prices, the efficient-rationing rule applies. If A charges \$10 and B charges \$12, A sells 30; demand at B’s price is 52, leaving B $52-30=22$ sales. Profits are $(10-4)(30)=\$180$ for A and $(12-4)(22)=\$176$ for B. If B charges \$14 instead, its residual demand is $44-30=14$, giving profits of \$180 for A and $10(14)=\$140$ for B.

If B charges \$10 and A charges \$12, B sells 50, leaving A $52-50=2$ sales. Their profits are $\$300$ for B and $\$16$ for A. If A charges \$14 instead, its residual demand is $\max(44-50,0)=0$, so A earns zero while B earns $(10-4)(50)=\$300$.

If A charges \$12 and B charges \$14, A sells 30 and B’s residual demand is $44-30=14$. Profits are $\$240$ for A and $\$140$ for B. If A charges \$14 and B charges \$12, B sells 50 and A has zero residual demand at \$14; profits are zero for A and $(12-4)(50)=\$400$ for B.

Now check the candidate where both charge \$10. A earns \$135. Raising its price to \$12 while B stays at \$10 would leave A only $52-50=2$ sales, for a profit of \$16; raising to \$14 would leave A no sales. A has no lower price on the menu. B earns \$225 at the tie. Raising to \$12 against A at \$10 would leave B $52-30=22$ sales, for a profit of \$176; raising to \$14 would leave B 14 sales, for a profit of \$140. B also has no lower price on the menu. Neither firm can profitably deviate, so $(\$10,\$10)$ is a pure-strategy Nash equilibrium on the specified price menu. The calculation checks the capacity constraints and all available unilateral deviations; it does not establish an equilibrium for prices outside that menu.

Best Responses When One Firm Is Capacity Constrained

At a given rival price, a firm compares the profit from undercutting, matching, or charging more. Matching can matter because the tie rule may allocate customers differently from the unequal-price rule. A price increase can also reduce demand enough that a firm earns less despite a wider margin. The next example compares these options directly.

Worked Example: Comparing an Undercut, a Tie, and a Higher Price

Demand is $Q(p)=180-6p$. Firm A has capacity 35 and marginal cost \$8; Firm B has capacity 50 and marginal cost \$8. Firm B charges \$18. Firm A is considering \$17, \$18, or \$20. At a tie, demand is allocated in proportion to capacities. Find A’s profit under each choice and identify its best response among these three prices.

Undercut at \$17. Demand is $Q(17)=180-102=78$, so A sells its full capacity of 35. At B’s price, demand is $Q(18)=180-108=72$; after A’s 35 sales, B has residual demand $72-35=37$, below its capacity of 50. A’s profit is $(17-8)(35)=9(35)=\$315$.

Match at \$18. Demand is 72, below combined capacity $35+50=85$. Proportional allocation gives A $72(35/85)=29.6470588\ldots$ sales. Its profit before rounding is $(18-8)(29.6470588\ldots)=\$296.470588\ldots$. To the nearest cent, A earns $\$296.47$.

Charge \$20. B remains the lower-priced firm and sells $\min(50,Q(18))=\min(50,72)=50$. Demand at A’s price is $Q(20)=180-120=60$, so A’s residual demand is $60-50=10$, below its capacity. Its profit is $(20-8)(10)=12(10)=\$120$.

Among the three choices, A’s best response is to undercut at \$17, earning \$315. The result is not simply “the lower price gets the whole market”: A sells only 35 units, and B still sells 37. The common mistake is to use $Q(17)=78$ as A’s sales. Capacity limits A to 35 units.

Common Traps and Economic Significance

Capacity constraints change the link between price and quantity. A firm may sell its full capacity at a price below its rival’s and still leave the rival with customers willing to pay the higher price. Conversely, a higher-priced firm may receive no sales if the low-priced rival has enough capacity to serve all demand at the higher price. These outcomes help explain why the standard intuition that any undercut captures the entire market cannot be applied without checking capacity and rationing.

Using demand at the wrong price. The high-priced firm’s residual demand starts with $Q(p_H)$, not $Q(p_L)$.
Ignoring capacity. A firm’s actual sales cannot exceed $K_i$, even if demand at its price is larger.
Subtracting a rival’s capacity instead of its sales. The relevant subtraction is $q_L$. If the low-priced firm does not fill its capacity, its capacity is not the number of customers already served.
Forgetting the allocation rule at a tie. Proportional allocation, priority rules, or other specified rules can imply different sales and profits when prices are equal.
Claiming a universal equilibrium result. The existence and price of equilibrium depend on the demand curve, capacities, costs, feasible prices, tie-breaking, and rationing assumptions. A result for a discrete price menu need not hold when firms may choose any price.

For an actuarial or business application, the same logic can be used in a hypothetical market for a product with limited production capacity: a price change matters through both the margin per unit and the number of units that can actually be supplied. Capacity can therefore affect the value of undercutting, the sales available to a higher-priced competitor, and the profit comparison among feasible prices. The model is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Key takeaway: With efficient rationing and unequal prices, calculate the low-priced firm’s actual sales first, then subtract those sales from demand at the higher price and apply the high-priced firm’s capacity. Use the stated tie rule when prices are equal, and base profit on actual sales.

Check Your Understanding

For each problem, apply the stated capacity and allocation assumptions before calculating profit.

Market demand is $Q(p)=200-5p$. Firm A charges \$20, has capacity 45, and marginal cost \$7. Firm B charges \$24, has capacity 60, and marginal cost \$9. Under efficient rationing, calculate each firm’s sales and profit. (Answer: A sells 45 and earns \$585; B sells 35 and earns \$525.)
Demand is $Q(p)=140-4p$. A low-priced firm charges \$15, has capacity 30, and marginal cost \$6. A rival charges \$18, has capacity 45, and marginal cost \$10. Under efficient rationing, how many units does the rival sell, and what is its profit? (Answer: 38 units and \$304.)
Demand is $Q(p)=90-3p$. Firms A and B have capacities 20 and 40, respectively, and charge \$10 each. Demand is allocated in proportion to capacity; marginal costs are \$4 for A and \$7 for B. Find their sales and profits. (Answer: A sells 20 and earns \$120; B sells 40 and earns \$120.)
Demand is $Q(p)=160-5p$. Firm X has capacity 30 and marginal cost \$6. Its rival charges \$18, has capacity 50, and marginal cost \$6. Under efficient rationing, compare X’s profits if it charges \$16, \$18, or \$20. Which choice is best among these options? (Answer: \$300 at \$16, \$315 at \$18, and \$140 at \$20; the best choice is \$18.)
Demand is $Q(p)=120-4p$. A low-priced firm charges \$14 and has capacity 25; a rival charges \$17 and has capacity 50. Both have marginal cost \$5. Under efficient rationing, find both firms’ sales and profits, and verify neither capacity is exceeded. (Answer: The low-priced firm sells 25 and earns \$225; the rival sells 27 and earns \$324; neither capacity is exceeded.)

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