Why Cournot Calculations Go Wrong
Cournot calculations are short, but several quantities can look deceptively similar: a firm’s output, the rival’s output, and total market output. The central discipline is to define each quantity before substituting it. In particular, a firm chooses its own quantity while treating its rivals’ quantities as fixed; the market price then depends on the sum of all firms’ quantities.
The examples below use a homogeneous product and linear inverse demand. They focus on calculation and interpretation errors rather than on extending the model. The treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Keep Firm Output Separate From Market Output
The notation \(q_i\) means the quantity produced by firm \(i\), while \(Q\) means the sum of output across all firms. With two firms, \(Q=q_1+q_2\); with three firms, \(Q=q_1+q_2+q_3\). The market demand equation uses \(Q\), not one firm’s quantity alone.
A firm’s profit calculation uses its own quantity, not total market output. Once the market price has been found, firm \(i\)’s profit is \((P-c_i)q_i\). Multiplying the margin by \(Q\) would calculate neither that firm’s profit nor, in general, total industry profit.
Worked Example: Firm Quantities, Market Output, and Price
Two firms face \(P=90-Q\). Firm 1 has marginal cost \(c_1=18\), and firm 2 has marginal cost \(c_2=30\). Find the simultaneous Cournot equilibrium, market output and price, and each firm’s profit. The method is to derive each best response and solve the two equations together. This avoids treating one firm’s quantity as if it were the market total.
For firm 1, \(Q=q_1+q_2\), so its profit is \(\pi_1=[90-q_1-q_2-18]q_1=(72-q_1-q_2)q_1\). Differentiating with respect to \(q_1\), while holding \(q_2\) fixed, gives \(72-2q_1-q_2=0\). Therefore its interior best response is \(q_1=36-\frac{1}{2}q_2\).
For firm 2, profit is \(\pi_2=[90-q_1-q_2-30]q_2=(60-q_1-q_2)q_2\). Differentiating with respect to \(q_2\), while holding \(q_1\) fixed, gives \(60-q_1-2q_2=0\), so its interior best response is \(q_2=30-\frac{1}{2}q_1\).
Substitute firm 2’s response into firm 1’s: \(q_1=36-\frac{1}{2}(30-\frac{1}{2}q_1)=21+\frac{1}{4}q_1\). Thus \(\frac{3}{4}q_1=21\), giving \(q_1=28\). Then \(q_2=30-\frac{1}{2}(28)=16\).
Market output is \(Q=28+16=44\), so market price is \(P=90-44=46\). Firm 1’s profit is \((46-18)(28)=28(28)=784\). Firm 2’s profit is \((46-30)(16)=16(16)=256\).
Check. At the proposed quantities, firm 1’s marginal revenue is \(90-q_2-2q_1=90-16-56=18=c_1\). Firm 2’s is \(90-q_1-2q_2=90-28-32=30=c_2\). Both first-order conditions hold, and the quantities add to the total used to calculate price.
Residual Demand and Marginal Revenue
To find a firm’s best response, first hold the rivals’ quantities fixed and express the price as a function of the firm’s own quantity. This is that firm’s residual demand. For linear inverse market demand \(P=a-bQ\), firm \(i\)’s residual inverse demand is \(P=a-b(q_i+Q_{-i})\), where \(Q_{-i}\) is the sum of rival quantities.
The residual demand slope is not itself the marginal-revenue slope. Because revenue is price times the firm’s own quantity, differentiating revenue doubles the coefficient on that own quantity. The rival-output term remains fixed during this derivative.
The expression follows directly from profit maximization. With rivals’ output fixed, firm \(i\)’s profit is \(\pi_i=[a-b(q_i+Q_{-i})-c_i]q_i\). Expanding and differentiating with respect to \(q_i\) gives \(\frac{d\pi_i}{dq_i}=a-c_i-bQ_{-i}-2bq_i\). Setting this derivative equal to zero gives the interior condition. The second derivative is \(-2b<0\), so the profit function is strictly concave and the interior condition identifies a maximum. If its implied quantity is negative, the firm cannot choose that quantity; the nonnegative constraint makes zero the best response.
Worked Example: Residual Demand Is Not Market Demand
Market inverse demand is \(P=84-2Q\). A firm considers its output while its rival produces 9 units, and the firm’s marginal cost is 18. Find its best-response quantity, the resulting market price, and its profit at that response. The method is to substitute the fixed rival output into demand, then differentiate the firm’s revenue. This avoids using the market demand slope as if it were marginal revenue.
With \(Q=q_i+9\), residual inverse demand is \(P=84-2(q_i+9)=66-2q_i\). The firm’s revenue is \(R_i=(66-2q_i)q_i=66q_i-2q_i^2\), so its marginal revenue is \(MR_i=66-4q_i\).
Set marginal revenue equal to marginal cost: \(66-4q_i=18\). Hence \(4q_i=48\) and \(q_i=12\), which is positive and therefore feasible. Total market output is \(Q=12+9=21\), and market price is \(P=84-2(21)=42\). The firm’s profit is \((42-18)(12)=24(12)=288\).
Check. At \(q_i=12\), marginal revenue is \(66-4(12)=18\), equal to marginal cost. The original market demand gives the same price as residual demand: \(84-2(21)=42\) and \(66-2(12)=42\).
Substitute Best Responses Simultaneously
A best response answers a conditional question: what quantity maximizes this firm’s profit for a specified set of rival quantities? A Cournot equilibrium is a set of quantities in which each firm’s choice is a best response to the quantities actually chosen by all rivals. Therefore, the response equations must hold at the same time.
An easy mistake is to calculate one firm’s response to a particular rival output and then treat that response as an equilibrium without checking whether the rival is responding optimally in turn. The reliable method is to solve the system of response equations, then substitute the resulting quantities back into every equation and into the market demand equation.
State \(Q\) as the sum of the firms’ quantities. Keep each firm’s own quantity distinct from the total.
For each firm, hold rival quantities fixed, form profit, differentiate with respect to that firm’s quantity, and apply the nonnegative-output constraint.
Use substitution or simultaneous-equation methods; do not assume a response calculated at an arbitrary rival quantity is an equilibrium.
Check every best-response equation, sum quantities to obtain \(Q\), calculate price from market demand, and calculate each firm’s profit using its own output.
Worked Example: A Three-Firm Equilibrium With Two Similar Rivals
Three firms face \(P=120-Q\). Firm 1 has marginal cost 20; firms 2 and 3 each have marginal cost 30. Find the Cournot quantities, total output, price, and profits. The method is to write all three best responses and use the symmetry between firms 2 and 3 only after stating it. This avoids assuming that all three firms have identical costs.
Firm 1’s best response is \(q_1=(120-20-q_2-q_3)/2=50-\frac{1}{2}(q_2+q_3)\). Firm 2’s response is \(q_2=(120-30-q_1-q_3)/2=45-\frac{1}{2}(q_1+q_3)\); firm 3 has the corresponding response \(q_3=45-\frac{1}{2}(q_1+q_2)\).
Because firms 2 and 3 have the same cost and face the same demand environment, their equilibrium quantities are equal. Let \(q_2=q_3=x\) and \(q_1=y\). Then firm 1’s equation gives \(y=50-x\). Firm 2’s equation gives \(x=45-\frac{1}{2}(y+x)\), or \(3x=90-y\). Substituting \(y=50-x\) yields \(3x=40+x\), so \(x=20\) and \(y=30\).
Thus \(q_1=30\), \(q_2=20\), and \(q_3=20\). Total output is \(Q=30+20+20=70\), and price is \(P=120-70=50\). Profits are \(\pi_1=(50-20)(30)=900\), \(\pi_2=(50-30)(20)=400\), and \(\pi_3=(50-30)(20)=400\).
Check. Firm 1’s response to the other firms’ total output of 40 is \((100-40)/2=30\). Firm 2’s response to rival output \(30+20=50\) is \((90-50)/2=20\); firm 3’s check is identical. All quantities satisfy their response equations, and the price follows from total output 70.
Corner Solutions and a Common Calculation Trap
The interior best-response formula must not be used to justify a negative quantity. A firm with a sufficiently high marginal cost may optimally produce zero. When one firm’s response is zero, solve the remaining firms’ responses with that zero quantity included, then check that the inactive firm still has no profitable positive response.
Worked Example: Checking a Zero-Output Best Response
Two firms face \(P=50-Q\), with marginal costs \(c_1=20\) and \(c_2=45\). Determine whether firm 2 produces, and find the equilibrium quantities and price. The method is to apply the nonnegative constraint to the best responses and verify the inactive firm’s incentive. This avoids reporting a negative quantity as an equilibrium output.
The unconstrained responses are \(q_1=(30-q_2)/2\) and \(q_2=(5-q_1)/2\), with each response replaced by zero if its numerator implies a negative value. Suppose firm 2 produces zero. Firm 1’s response is then \(q_1=30/2=15\). Given \(q_1=15\), firm 2’s unconstrained response is \((5-15)/2=-5\), so its feasible best response is \(q_2=0\).
Therefore \(Q=15\) and \(P=50-15=35\). Firm 1’s profit is \((35-20)(15)=225\); firm 2’s profit is zero. The check confirms both firms’ choices are best responses: firm 1’s response to zero is 15, and firm 2’s response to 15 is zero.
A frequent error is to divide the demand intercept net of cost by two and call that a firm’s equilibrium quantity, without accounting for the rival’s equilibrium output. For example, with two identical firms, \(P=90-Q\), and marginal cost 18, that shortcut gives \((90-18)/2=36\) per firm. But if the rival produces 36, the actual best response is \((72-36)/2=18\), not 36. The proposed quantity fails its own best-response check.
The correct symmetric equilibrium sets the two best responses equal: \(q=(72-q)/2\), so \(3q=72\), \(q=24\), \(Q=48\), and \(P=42\). Each firm’s marginal revenue at that outcome is \(90-24-2(24)=18\), equal to marginal cost. The error was not merely a wrong total: it omitted the rival quantity when constructing the firm’s residual demand.
| Common error | Why it fails | Correction |
|---|---|---|
| Using \(q_i\) where demand requires \(Q\) | Market price depends on every firm’s output. | Write \(Q=\sum_j q_j\) before calculating price. |
| Using residual-demand slope as marginal-revenue slope | Revenue multiplies price by the firm’s own quantity, so differentiation changes that coefficient. | Form \(R_i=P(q_i)q_i\), then differentiate. |
| Substituting a rival’s quantity only once | A one-sided response need not satisfy the rival’s best response. | Solve all response equations simultaneously and check each one. |
| Keeping a negative response quantity | Firms cannot choose negative output. | Use the nonnegative constraint and check the zero-output response. |
| Using total output to calculate one firm’s profit | Each firm earns its margin on its own sales. | Calculate \((P-c_i)q_i\) for firm \(i\). |
Check Your Understanding
Work through the steps in each problem and use the final answer to check your result.
- Two identical firms face \(P=70-Q\) and each has constant marginal cost 10. Find each firm’s Cournot quantity, market output, price, and each firm’s profit. (Answer: \(q_1=q_2=20,\ Q=40,\ P=30,\ \pi_1=\pi_2=\$400\).)
- Market demand is \(P=100-2Q\). A firm has marginal cost 14, and its rival currently produces 19 units. Find the firm’s residual inverse demand, marginal revenue, best-response quantity, and the resulting market price. (Answer: \(P=62-2q_i,\ MR_i=62-4q_i,\ q_i=12,\ Q=31,\ P=38\).)
- Two firms face \(P=96-2Q\), with marginal costs \(c_1=16\) and \(c_2=28\). Find both simultaneous equilibrium quantities and the market price. (Answer: \(q_1=46/3,\ q_2=28/3,\ Q=74/3,\ P=140/3\).)
- Three firms face \(P=90-Q\), with marginal costs 18, 24, and 24. Find each equilibrium quantity and price, using symmetry only for the two firms with equal costs. (Answer: \(q_1=21,\ q_2=q_3=15,\ Q=51,\ P=39\).)
- Two firms face \(P=50-Q\), with marginal costs 20 and 45. Check whether an interior solution for both firms is feasible; if not, find the corner equilibrium and verify the inactive firm’s response. (Answer: \(q_1=15,\ q_2=0,\ Q=15,\ P=35\); firm 2’s unconstrained response is \(-5\), so its feasible response is zero.)