Cournot Comparative Statics: Demand, Costs, and Firm Count

Why Cournot Comparative Statics Matter

A Cournot equilibrium describes firms that choose quantities while taking their rivals’ quantities as given. The equilibrium formulas are useful not only for calculating one market outcome, but also for asking how that outcome changes when demand, production costs, or the number of firms changes. Those comparisons reveal an important distinction: a change can move each firm’s output in one direction while moving total market output in another.

The analysis below holds the basic Cournot structure fixed and changes one parameter at a time. This makes the source of each result clear. The conclusions are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Consider $n$ identical firms choosing quantities simultaneously. Inverse market demand is $P=a-bQ$, where $Q=\sum_{i=1}^{n}q_i$, $a$ is the demand intercept, and $b>0$ is the absolute slope. Each firm has constant marginal cost $c$, no fixed cost, and chooses $q_i\geq 0$. In the interior symmetric equilibrium, where $a>c$, each firm chooses the same quantity. The formulas below assume the market price is given by this demand curve and that firms have no capacity constraints.

The Symmetric Equilibrium and Its Comparative Statics

Firm $i$'s profit is its sales revenue minus variable cost. Given the quantities of its rivals, its profit is $\pi_i=[a-b(q_i+Q_{-i})-c]q_i$, where $Q_{-i}$ is the total quantity produced by all firms other than firm $i$. Differentiating with respect to $q_i$ gives the first-order condition $a-c-bQ_{-i}-2bq_i=0$. In a symmetric equilibrium, $Q_{-i}=(n-1)q_i$. Substitution yields $(a-c)-b(n+1)q_i=0$, which gives the equilibrium quantity for each firm.

q_i^*=\frac{a-c}{b(n+1)},\qquad Q^*=\frac{n(a-c)}{b(n+1)},\qquad P^*=\frac{a+nc}{n+1},\qquad \pi_i^*=\frac{(a-c)^2}{b(n+1)^2}.

The market quantity follows by multiplying each firm's quantity by $n$. To obtain the price, substitute $Q^*$ into inverse demand: $P^*=a-bQ^*=a-n(a-c)/(n+1)=(a+nc)/(n+1)$. Each firm's profit equals its equilibrium margin, $P^*-c=(a-c)/(n+1)$, multiplied by its output. This gives the stated profit formula.

This derivation also shows why the equilibrium is an interior one only when $a>c$. If $a\leq c$, the positive-output formulas do not describe an equilibrium with active production: at a nonpositive margin, firms do not choose positive quantities under these assumptions. The comparative statics below concern the active case $a>c$.

Comparative-static result: With $b$, $c$, and $n$ fixed, a higher $a$ raises each firm's and the market's output, the market price, and each firm's profit. With $a$, $b$, and $n$ fixed, a higher $c$ lowers output and profit but raises price. With $a$, $b$, and $c$ fixed, adding firms lowers each firm's output, price, and profit, but raises total output.

For the demand intercept and marginal cost, the direction of change can be checked from the formulas. For example, $\partial Q^*/\partial a=n/[b(n+1)]>0$, whereas $\partial Q^*/\partial c=-n/[b(n+1)]<0$. The price formula gives $\partial P^*/\partial a=1/(n+1)>0$ and $\partial P^*/\partial c=n/(n+1)>0$. Thus, more willingness to pay shifts the market price upward as well as increasing output; higher production cost reduces output but is partly reflected in a higher price.

The number of firms is an integer, so it is best to compare equilibrium outcomes at $n$ and $n+1$, rather than interpret a derivative with respect to firm count literally. The total-output difference is $Q^*_{n+1}-Q^*_n=(a-c)/[b(n+1)(n+2)]>0$. Meanwhile, the price difference is $P^*_{n+1}-P^*_n=-(a-c)/[(n+1)(n+2)]<0$. Each firm's quantity is $(a-c)/[b(n+1)]$, so it falls as $n$ rises. The market gains output because the increase in the number of producing firms more than offsets the lower output per firm.

As $n$ becomes large, $Q^*$ approaches $(a-c)/b$, the quantity at which price equals marginal cost under the linear demand curve. At the same time, $P^*$ approaches $c$ and each firm's profit approaches zero. These limiting statements describe the model's implication as the number of identical firms grows; they do not mean that any finite number of firms produces exactly the competitive outcome.

Worked Example: A Higher Demand Intercept

Suppose three identical firms face inverse demand $P=120-2Q$, and each has marginal cost $c=20$. Calculate the symmetric equilibrium. Then suppose the demand intercept rises to $136$, with all other parameters unchanged. Find the new outcomes and compare them. State quantities in units and prices and profits in dollars per unit and dollars, respectively.

Method. Here $n=3$, $b=2$, and the initial intercept is $a=120$. The equilibrium denominator for each firm's output is $b(n+1)=2(4)=8$.

Initially, each firm produces $q_i^*=(120-20)/8=100/8=12.5$ units. Total output is $Q^*=3(12.5)=37.5$ units. The market price is $P^*=120-2(37.5)=120-75=45$ dollars. Each firm's profit is $(45-20)(12.5)=25(12.5)=312.50$ dollars.

After the intercept rises, $a-c=136-20=116$, so each firm produces $q_i^*=116/8=14.5$ units and total output is $Q^*=3(14.5)=43.5$ units. Price is $136-2(43.5)=136-87=49$ dollars. Profit per firm is $(49-20)(14.5)=29(14.5)=420.50$ dollars.

The change in the intercept raises each firm's output by $14.5-12.5=2$ units, total output by $43.5-37.5=6$ units, price by $49-45=4$ dollars, and profit by $420.50-312.50=108.00$ dollars per firm. As a formula check, new profit is $(136-20)^2/[2(4)^2]=116^2/32=13{,}456/32=420.50$ dollars. The result is consistent: the stronger demand raises output, price, and profit.

Common mistake avoided: A higher intercept does not leave price unchanged just because firms produce more. The correct price must be recalculated from the new inverse demand curve and new total output.

Worked Example: Higher Marginal Cost

Four identical firms face $P=150-3Q$. Their marginal cost increases from $30$ dollars to $42$ dollars per unit. Calculate the initial and new equilibrium quantities, price, and profit per firm. Round monetary amounts to the nearest cent.

Method. Use $n=4$, $a=150$, and $b=3$. The output denominator is $b(n+1)=3(5)=15$.

At $c=30$, each firm produces $q_i^*=(150-30)/15=120/15=8$ units. Total output is $Q^*=4(8)=32$ units, and price is $P^*=150-3(32)=54$ dollars. The margin is $54-30=24$ dollars per unit, so profit is $24(8)=192.00$ dollars per firm.

At $c=42$, each firm produces $q_i^*=(150-42)/15=108/15=7.2$ units. Total output is $Q^*=4(7.2)=28.8$ units, and price is $P^*=150-3(28.8)=150-86.4=63.6$ dollars. The new margin is $63.6-42=21.6$ dollars per unit, so profit is $21.6(7.2)=155.52$ dollars per firm.

Each firm reduces output by $8-7.2=0.8$ units, and total output falls by $32-28.8=3.2$ units. Price rises by $63.6-54=9.6$ dollars, but the higher cost still reduces each firm's margin and profit. The formula check gives new profit $108^2/[3(5)^2]=11{,}664/75=155.52$ dollars.

Common mistake avoided: The increase in price does not imply that firms earn more. Profit depends on the margin $P-c$ as well as quantity; here the margin falls from $24$ to $21.6$ dollars and output also falls.

Worked Example: Adding Firms

A market has inverse demand $P=130-2Q$ and identical firms with marginal cost $c=10$. Compare equilibrium outcomes when the market has three firms and when it has five. Find each firm's output, total output, price, and profit.

Method. The intercept-cost difference is $a-c=130-10=120$. Calculate both integer firm counts using the symmetric equilibrium formulas.

With $n=3$, each firm produces $q_i^*=120/[2(4)]=120/8=15$ units. Total output is $Q^*=3(15)=45$ units, and price is $P^*=130-2(45)=40$ dollars. Profit per firm is $(40-10)(15)=30(15)=450$ dollars.

With $n=5$, each firm produces $q_i^*=120/[2(6)]=120/12=10$ units. Total output is $Q^*=5(10)=50$ units, and price is $P^*=130-2(50)=30$ dollars. Profit per firm is $(30-10)(10)=200$ dollars.

Adding two firms reduces each firm's output from 15 to 10 units and profit from 450 to 200 dollars, while total output increases from 45 to 50 units and price falls from 40 to 30 dollars. This matches the model's comparative statics. The profit formula also checks: at $n=5$, $\pi_i^*=120^2/[2(6)^2]=14{,}400/72=200$ dollars.

Common mistake avoided: It is incorrect to infer that total output falls because output per firm falls. Total output must account for both the number of firms and the output of each firm.

Interpreting the Results and Avoiding Traps

Comparative statics isolate the consequence of changing one input while holding the rest of the model fixed. If the demand intercept and marginal cost both change, for example, the resulting price and quantity changes reflect both effects; the one-parameter conclusions cannot be applied without recalculating the equilibrium. Likewise, a change in the slope $b$ is not the same as a change in the intercept $a$: the formulas include $b$ in the quantity and profit denominators, so a slope change requires its own calculation.

A common exam trap is to focus on individual output alone. From $q_i^*=(a-c)/[b(n+1)]$, more firms means less output per firm. But aggregate output is $Q^*=n(a-c)/[b(n+1)]$, which rises with $n$. For example, in the preceding example, per-firm output drops by one-third, from 15 to 10, but total output rises from 45 to 50 units. The two statements are not contradictory: there are more firms sharing the market.

Another useful check is to verify price in two independent ways: substitute total output into $P=a-bQ$, and calculate $P=(a+nc)/(n+1)$. If these disagree, the likely causes are an incorrect total quantity or a misplaced firm count. Also check that equilibrium price exceeds marginal cost when $a>c$; the equilibrium margin is $(a-c)/(n+1)>0$.

Key takeaway: In the symmetric linear-demand Cournot model, the demand intercept and marginal cost determine the market's surplus over cost, while the slope and firm count determine how that surplus is translated into quantities, price, and per-firm profit. Always distinguish a firm's outcome from the market total.

Check Your Understanding

For each problem, use the active symmetric Cournot equilibrium and show your intermediate calculations before checking the final answer.

Two firms face $P=90-2Q$, with marginal cost $c=10$. Find each firm's output, total output, price, and profit per firm. (Answer: $q_i^*=13.3333$, $Q^*=26.6667$, $P^*=\$36.6667$, $\pi_i^*=\$355.56$.)
Three firms face $P=140-2Q$. Compare their outcomes when marginal cost rises from $20$ to $32$: find total output, price, and profit per firm in each case. (Answer: at $c=20$, $Q^*=45$, $P^*=\$50$, $\pi_i^*=\$450$; at $c=32$, $Q^*=40.5$, $P^*=\$59$, $\pi_i^*=\$364.50$.)
For $P=110-Q$ and $c=30$, compare the equilibrium with two firms and four firms. Find each firm's output, total output, price, and profit per firm. (Answer: with $n=2$, $q_i^*=26.6667$, $Q^*=53.3333$, $P^*=\$56.6667$, $\pi_i^*=\$711.11$; with $n=4$, $q_i^*=16$, $Q^*=64$, $P^*=\$46$, $\pi_i^*=\$256$.)
A market has $P=150-2Q$ and $c=30$. What is the smallest integer number of firms that makes equilibrium price no greater than $\$50$? At that count, find total output, each firm's output, and profit per firm. (Answer: $n=5$; $Q^*=50$, $q_i^*=10$, $\pi_i^*=\$200$.)
Three firms face $P=a-2Q$, with $c=20$. The intercept increases from $100$ to $120$. Calculate the old and new output per firm, total output, price, and profit per firm. (Answer: at $a=100$, $q_i^*=10$, $Q^*=30$, $P^*=\$40$, $\pi_i^*=\$200$; at $a=120$, $q_i^*=12.5$, $Q^*=37.5$, $P^*=\$45$, $\pi_i^*=\$312.50$.)

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