Quantity vs. price competition (Cournot vs. Bertrand)
Learning goals
After this page, you can:
- Explain quantity (capacity) competition in words, compute its price with the given formulas, and show that more rivals push the price toward cost.
- Explain why price competition with an identical product drives the price down to marginal cost, even with two firms — and why differentiation keeps it above cost.
- Classify a real industry as closer to quantity competition or to price competition, with one piece of evidence.
The idea
Two cruise lines decide this year how many ships to sail on a route two years from now. Once the ships are built, they fill them at whatever price sells the cabins. Two gas stations change the number on the sign any morning they like. Both are “two rivals”. But their price fights look very different. The difference is what they compete with.
Take one market, one demand line and one cost. There are three ways the rivals can end up.
1. The cartel. The rivals act like one owner. They hold volume down and share the monopoly profit. This is the highest price. (The cartels page shows why it rarely lasts.)
2. Quantity competition. In quantity competition (also called Cournot competition), each firm chooses how much to sell — how many ships, seats, or gallons. The price then falls to whatever level sells the total. Each firm does the best it can given what the others sell. Here is the key point. When one firm adds a unit, it gains the margin on that unit. It also pushes the price down a little — on its own units and on its rivals’ units. The firm counts the damage to its own units. It ignores the damage to its rivals. So each firm sells more than its cartel quota. Together they sell more than a cartel, and the price lands below the cartel price but above cost.
More rivals, lower price. With more firms, each one is a smaller part of the market. It feels less of the price damage from its own extra units, so it pushes volume harder. With two firms the price is well above cost. With three it is lower. With many, it is close to cost.
3. Price competition with an identical product. In price competition (also called Bertrand competition), each firm chooses its price, and customers buy from the cheapest. If the product is identical — same gasoline, same charger — a price one cent lower takes every customer. So any price above cost invites a rival to undercut it by a cent. The undercutting stops only when the price equals marginal cost. Then no one can cut further without losing money on every sale. The surprise: this happens even with just two firms. Two can be enough for a full price war.
Why real prices stay above cost: differentiation. Real gas stations do not sell at cost. Why? Because their products are not quite identical. One is on your side of the road. One has a car wash. One gives points in your loyalty app. Differentiation means customers see a real difference between sellers. Then a small price gap moves some customers, not all of them. Each seller keeps its loyal customers, and the price stays above cost. In the class price-war game, the stations have partly loyal drivers and a cost of $3.00. When each station does the best it can given the others’ prices, the price ends up at $5.00, not $3.00.
So what decides it? Not only how many rivals there are, but how they compete:
- Quantity-like: capacity is committed in advance and is hard to change. Airlines, cruise lines, chip plants, hotels. The price then adjusts to fill the capacity. Prices stay above cost. The danger is a rival adding capacity.
- Price-like: prices change daily, on products that customers compare in seconds. Gas, delivery fees, generic goods sold online. Without differentiation, prices sink toward cost. What keeps them up is location, brand, loyalty programs, and switching costs.
For your memo. Look for evidence of which world your company is in. A rival announcing a new factory or new planes points to quantity competition. Rivals’ prices for the same item, checked on the same day, sitting within a few cents and moving within hours of each other, point to price competition. One piece of real evidence beats a general claim.
Market demand P = a − b·Q, the same marginal cost c for every firm, n firms.
Cartel: Q = (a − c) ÷ (2b), P = a − b·Q
Quantity competition: each firm q = (a − c) ÷ ((n + 1)·b) — with two firms, (a − c) ÷ (3b). Total Q = n·q, P = a − b·Q. (Check: P = (a + n·c) ÷ (n + 1).)
Price competition, identical product: P = c
Gross profit per firm = (P − c) × q (× 1,000 when q is in thousands). With identical costs: cartel > quantity competition > price competition = c.
Try it
The simulator opens on the Exit 41 example: two gas stations, demand P = 6.90 − 0.30·Q, cost $3.30 a gallon.
- Read the three bars. Which price is highest? Which one sits right on the marginal-cost line?
- Change the number of stations from 2 to 3, then to 4. Watch the “add a rival” strip. Which price moves, and which ones stay put?
- Raise the wholesale cost by $0.50. Does every price rise by $0.50?
- Press Use my numbers. Pick how many sellers really compete for your customers. Which line is today’s price closest to?
Worked example
Problem. Exit 41 (fictional): P = 6.90 − 0.30·Q, Q in thousand gallons a day, c = $3.30, so a − c = 3.60. Find the price under each way of competing. Then add rivals.
Step 1. Cartel. Q = 3.60 ÷ 0.60 = 6, P = $5.10. Each station earns (5.10 − 3.30) × 3 × 1,000 = $5,400 a day.
Step 2. Quantity competition, two stations. Each sells q = 3.60 ÷ (3 × 0.30) = 4. Total 8. P = 6.90 − 0.30 × 8 = $4.50. Each earns 1.20 × 4 × 1,000 = $4,800.
Step 3. Price competition, identical gasoline. P = c = $3.30. Gross profit is $0. Each station still pays its rent and wages, so it actually loses money.
Step 4. Add rivals (quantity competition). Three stations: q = 3.60 ÷ (4 × 0.30) = 3, total 9, P = 6.90 − 2.70 = $4.20, each (4.20 − 3.30) × 3 × 1,000 = $2,700. Four stations: q = 3.60 ÷ (5 × 0.30) = 2.4, total 9.6, P = 6.90 − 2.88 = $4.02, each 0.72 × 2.4 × 1,000 = $1,728. The price walks toward $3.30 as rivals are added.
Step 5. Differentiation. In the price-war game, four stations have partly loyal drivers (customers = 90 − 20 × own price + 10 × the average of the others’ prices) and a cost of $3.00. When each does the best it can given the others’ prices, they end up at $5.00, not $3.00.
Step 6. Classify. Cruise lines and airlines commit capacity first: quantity-like. Gas, delivery fees and phone chargers sold online are compared in seconds: price-like, softened by brand, location and loyalty programs.
Check yourself
Two stations each choose how many gallons to sell. Demand is P = 6.90 − 0.30·Q (Q = total thousand gallons a day), cost is $3.30 a gallon, and each sells q = (a − c) ÷ (3b) = 4 thousand gallons. What is the pump price?
Volume competition: use total quantity in the demand line; the price lands between cartel and cost.
Same exit, but a third station opens and each of the three still chooses its volume: q = (a − c) ÷ (4b) = 3 thousand gallons each. What happens to the price?
With volume competition, each added rival lowers the price toward marginal cost.
Which industry is closest to quantity (capacity) competition?
Capacity chosen before the price is set → quantity-like; prices changed quickly on comparable products → price-like.
Two identical stations are both selling at their $3.30 cost. Which change is most likely to let one of them charge above cost without losing every driver?
Differentiation keeps price competition from driving price all the way to cost.
Which piece of evidence best supports calling your company’s market price-competition-like in memo Section 4?
Same-day prices for the same item, matched quickly, are the best evidence of price competition.
Video
Video coming soon.