Why the Number of Firms Matters
In Cournot competition, firms choose quantities rather than prices. Each firm recognizes that its output contributes to total market supply and therefore affects the price earned by every unit sold. With more firms making these decisions, each one accounts for a smaller share of total output. The equilibrium price and total quantity consequently move toward the competitive benchmark, but they do not generally equal it for any finite number of firms.
This tutorial derives that result for identical firms with constant marginal cost and linear inverse demand. The focus is the connection between firm count, each firm's output, and the market's price-cost margin—not a change in demand or costs.
The assumptions matter. A homogeneous good has one market price, so a firm's output affects the price received by its rivals as well as its own. Constant marginal cost makes the cost of producing \(q_i\) equal to \(cq_i\). The model does not describe firms setting prices, differentiated products, or firms with different costs.
From Best Response to Symmetric Equilibrium
Let \(Q_{-i}=\sum_{j\ne i}q_j\) denote the output of all firms other than firm \(i\). Given \(Q_{-i}\), firm \(i\)'s profit is its revenue less its production cost. Its choice of quantity is therefore a best response to the quantities chosen by its rivals.
To find an interior best response, differentiate profit with respect to \(q_i\), holding \(Q_{-i}\) fixed. The first-order condition equates marginal revenue to marginal cost. Because the second derivative is \(-2b<0\), profit is strictly concave in \(q_i\), so a positive solution to the first-order condition is the firm's profit-maximizing response.
Output cannot be negative. If the expression on the right is zero or below, the firm's best response is zero instead. Thus the general best response is \(q_i=\max\{0,(a-c-bQ_{-i})/(2b)\}\). In a symmetric equilibrium, every firm selects the same positive output \(q\), so \(Q_{-i}=(N-1)q\). Substituting that condition into the best response gives the symmetric solution.
Set \(q_i=q\) for each firm, so the other \(N-1\) firms together produce \((N-1)q\).
\(a-c-b(N-1)q-2bq=0\), so \(a-c=b(N+1)q\).
\(q=(a-c)/[b(N+1)]\), which is positive because \(a>c\) and \(b>0\).
Multiply by \(N\) for total output, then substitute that total into inverse demand to obtain price.
The price expression follows by substituting the total quantity into demand: \(P^*=a-bN(a-c)/[b(N+1)]\). Combining the terms over the common denominator \(N+1\) gives \(P^*=[a+Nc]/(N+1)=c+(a-c)/(N+1)\). Since each firm's margin per unit is \(P^*-c\), its profit is \((P^*-c)q_i^*\), which yields the stated profit formula.
Price-Cost Margins and the Competitive Limit
The dollar price-cost margin is \(P^*-c\). It measures the amount by which price exceeds marginal cost for each unit sold. The Lerner index is the proportional margin, \((P^*-c)/P^*\), when price is positive. Keeping these measures distinct avoids confusing a dollar amount with a percentage of price.
The first expression falls as \(N\) increases: its numerator is fixed and its denominator grows. For the second expression, when \(c\ge 0\), the denominator \(a+Nc\) is nondecreasing in \(N\), so the proportional margin does not rise. In the usual case \(c>0\), it also approaches zero. The dollar margin approaches zero whether \(c\) is positive or zero.
To compare the oligopoly outcome with the competitive benchmark, hold demand and cost fixed and let the number of firms grow. At price equal to marginal cost, demand is \(Q_c=(a-c)/b\). The Cournot total output can be written as that benchmark multiplied by a fraction:
At the same time, \(q_i^*\) tends to zero for any one firm, while \(P^*\) tends to \(c\). Thus, as the number of firms becomes very large, total market output approaches the quantity demanded at marginal cost and the price approaches marginal cost. This is a limit result: the expressions approach those values as \(N\) grows; it does not say that a finite-\(N\) Cournot equilibrium is exactly competitive.
Worked Example: Four Firms in a Linear-Demand Market
Suppose inverse demand is \(P=120-2Q\), marginal cost is \(c=20\), and four identical firms compete in quantities. Find each firm's output, total output, price, dollar margin, Lerner index, and profit. The method is to use the symmetric Cournot formulas; the check is whether the calculated price agrees with demand at total output.
Step 1: Identify the parameters. Here \(a=120\), \(b=2\), \(c=20\), and \(N=4\), so \(a-c=100\) and \(N+1=5\).
Step 2: Find individual and total output. \(q_i^*=100/[2(5)]=100/10=10\). Therefore \(Q^*=4(10)=40\).
Step 3: Find price and the dollar margin. Demand gives \(P^*=120-2(40)=120-80=40\). The price-cost margin is \(40-20=20\). The formula gives the same result: \(100/5=20\).
Step 4: Find the Lerner index and profit. The Lerner index is \(20/40=0.5\), or 50%. Each firm's profit is \((40-20)(10)=200\). Equivalently, the profit formula gives \(100^2/[2(5^2)]=10{,}000/50=200\).
Check. Total output of 40 produces price 40 on the demand curve, and the calculated margin is exactly \(P^*-c\). The common mistake avoided here is multiplying the individual output by the wrong firm count when calculating market price.
How Outcomes Change as Firm Count Grows
The formulas reveal the direction of change while keeping demand and cost fixed. Since \(q_i^*=(a-c)/[b(N+1)]\), output per firm decreases as \(N\) rises. Total output, however, increases toward \((a-c)/b\), because the increase in the number of firms outweighs the decline in each firm's quantity. The market price falls toward \(c\), and the dollar price-cost margin shrinks.
| Equilibrium measure | Expression | Behavior as \(N\) grows |
|---|---|---|
| Output per firm | \((a-c)/[b(N+1)]\) | Falls toward zero |
| Total output | \(N(a-c)/[b(N+1)]\) | Rises toward \((a-c)/b\) |
| Price | \(c+(a-c)/(N+1)\) | Falls toward \(c\) |
| Dollar price-cost margin | \((a-c)/(N+1)\) | Falls toward zero |
| Profit per firm | \((a-c)^2/[b(N+1)^2]\) | Falls toward zero |
Worked Example: Comparing Two Firm Counts
A market has inverse demand \(P=90-1.5Q\) and constant marginal cost \(c=30\). Compare the symmetric equilibrium with \(N=2\) firms to the equilibrium with \(N=5\). For each case, calculate individual output, total output, price, and each firm's profit. This calculation isolates the effect of firm count by holding the demand and cost parameters fixed.
With two firms: \(a-c=90-30=60\), \(b=1.5\), and \(N+1=3\). Thus \(q_i^*=60/[1.5(3)]=60/4.5=13\frac{1}{3}\), and \(Q^*=2(13\frac{1}{3})=26\frac{2}{3}\). Price is \(90-1.5(26\frac{2}{3})=90-40=50\). The margin is \(50-30=20\), so each firm's profit is \(20(13\frac{1}{3})=266\frac{2}{3}\).
With five firms: \(N+1=6\), so \(q_i^*=60/[1.5(6)]=60/9=6\frac{2}{3}\). Total output is \(5(6\frac{2}{3})=33\frac{1}{3}\). Price is \(90-1.5(33\frac{1}{3})=90-50=40\). The margin is \(40-30=10\), and each firm's profit is \(10(6\frac{2}{3})=66\frac{2}{3}\).
Check. Moving from two to five firms reduces each firm's output and profit, but increases total output and lowers price. The competitive benchmark is \(Q_c=(90-30)/1.5=40\) at price 30; the five-firm outcome is closer to both benchmark values than the two-firm outcome.
Worked Example: Inferring Firm Count from an Observed Margin
Suppose inverse demand is \(P=150-3Q\), marginal cost is \(c=30\), and identical firms are at a symmetric Cournot equilibrium. An analyst reports a market price of 50. Infer the number of firms, then calculate the implied output per firm, total output, and profit per firm. The method is to use the price formula to solve for \(N\), then verify the implied quantities against demand.
Step 1: Use the price-cost margin. The reported margin is \(50-30=20\). Since \(a-c=150-30=120\), the formula \(P^*-c=(a-c)/(N+1)\) gives \(20=120/(N+1)\). Therefore \(N+1=120/20=6\), so \(N=5\).
Step 2: Calculate outputs. Here \(b=3\), so each firm produces \(q_i^*=120/[3(6)]=120/18=6\frac{2}{3}\). Total output is \(Q^*=5(6\frac{2}{3})=33\frac{1}{3}\).
Step 3: Verify price and calculate profit. Demand gives \(P=150-3(33\frac{1}{3})=150-100=50\), as reported. Each firm's profit is \((50-30)(6\frac{2}{3})=133\frac{1}{3}\).
Check. The inferred firm count is a positive integer, and the implied total output reproduces the observed market price. The common mistake avoided is using the markup as a percentage of price in the equation for the dollar margin.
Common Mistake: Confusing Individual and Total Output
A frequent error is to treat \((a-c)/[b(N+1)]\) as total market output. It is the output of one firm. Total output requires multiplying by \(N\). For example, with \(a=100\), \(b=2\), \(c=20\), and \(N=4\), the individual quantity is \(80/[2(5)]=8\), not 32. The total is \(4(8)=32\), and price is \(100-2(32)=36\). Substituting 8 as total output would instead give a price of 84, which is inconsistent with four firms each producing 8 units.
A related trap is to claim that price equals marginal cost as soon as a market has several firms. In this model, \(P^*-c=(a-c)/(N+1)>0\) for every finite \(N\) when \(a>c\). Price approaches marginal cost only in the limit as \(N\) grows without bound. This distinction between a finite-market result and a limiting result is central to interpreting the model.
The analysis is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Its central discipline is to distinguish each firm's quantity from total market quantity, then apply inverse demand to the total.
Check Your Understanding
Use the symmetric Cournot formulas and show enough intermediate steps to verify each result.
- Inverse demand is \(P=100-2Q\), marginal cost is 20, and three identical firms compete in quantities. Find individual output, total output, price, and profit per firm. (Answer: \(q_i^*=10\), \(Q^*=30\), \(P^*=40\), and \(\pi_i^*=200\).)
- Inverse demand is \(P=200-4Q\), marginal cost is 40, and there are four firms. Calculate individual output, total output, price, and the dollar price-cost margin. (Answer: \(q_i^*=8\), \(Q^*=32\), \(P^*=72\), and \(P^*-c=32\).)
- Inverse demand is \(P=75-0.5Q\), marginal cost is 15, and five firms compete. Find the Cournot total output and compare it with output at the competitive benchmark \(P=c\). (Answer: \(Q^*=100\); competitive output is \(120\), so Cournot output is 20 units below it.)
- Inverse demand is \(P=140-2Q\), marginal cost is 20, and the symmetric equilibrium price is 40. Infer the number of firms, then calculate output per firm and total output. (Answer: \(N=5\), \(q_i^*=10\), and \(Q^*=50\).)
- Inverse demand is \(P=110-3Q\) and marginal cost is 20. What is the smallest integer number of firms for which the dollar price-cost margin is no more than 6? At that firm count, calculate total output and price. (Answer: \(N=14\); \(Q^*=28\) and \(P^*=26\), giving a margin of 6.)