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VEE Economics · Oligopoly Models and Outcomes · Tutorial 374 of 998

Comparing Oligopoly Outcomes with Competitive and Monopoly Benchmarks

Learn to solve and compare the four market outcomes using a consistent linear-demand model, with attention to assumptions, firm profits, and tie-breaking.

Advanced 10 min read

What You'll Learn

  • State the assumptions behind competitive, monopoly, Cournot, and homogeneous-good Bertrand benchmarks
  • Derive monopoly and symmetric Cournot outcomes from linear inverse demand
  • Calculate equilibrium prices, quantities, and profits under each market structure
  • Compare Cournot outcomes with competitive and monopoly benchmarks
  • Avoid confusing total market profit with profit per firm
  • Apply an equal-split rule correctly when Bertrand firms tie on price

Why Compare Market-Structure Benchmarks?

A market’s price and output depend not only on demand and costs, but also on how firms compete. A single seller chooses output with the market demand curve in mind; Cournot firms choose quantities while accounting for rivals’ quantities; homogeneous-good Bertrand firms choose prices; and price-taking firms treat the market price as given. Comparing these outcomes shows how the strategic variable and market structure affect price, total quantity, and profit.

The comparisons below use one consistent model. The resulting rankings are useful benchmarks, not universal predictions: changing product differentiation, capacity, costs, or other assumptions can change the result. The topic is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model: Market demand is represented by inverse demand \(P=a-bQ\), where \(P\) is the market price, \(Q\) is total market output, \(a>c\) is the price intercept, and \(b>0\) is the slope parameter. Each active firm has constant marginal cost \(c\), no fixed cost, and profit equal to revenue minus variable cost. Products are homogeneous. Unless noted otherwise, firms have no capacity constraints, and output is nonnegative. Prices and costs are per unit. These assumptions make the comparisons precise; the formulas should not be applied automatically when costs differ, products are differentiated, or capacity is binding.

Four Outcomes in the Linear-Demand Model

Under perfect competition, firms take the market price as given. With constant marginal cost \(c\), the benchmark market-clearing price is \(c\). Substituting that price into demand gives \(Q=(a-c)/b\). With no fixed costs, operating profits are zero. This benchmark describes the market price and total quantity; it does not, by itself, specify how output is allocated among individual price-taking firms.

A monopolist chooses total output to maximize profit. Its revenue is \(PQ=(a-bQ)Q\), so the marginal-revenue curve has twice the slope of inverse demand. A Cournot firm chooses its own quantity, taking rivals’ quantities as given. Under the symmetric assumptions here, the equilibrium quantities are equal. In homogeneous-good Bertrand competition, firms choose prices. With at least two firms, equal constant marginal costs, no capacity constraints, and the ability to meet demand, the equilibrium price is marginal cost. If firms tie at that price, a tie-breaking rule determines how sales are allocated, but does not change total market quantity or the price.

Benchmark formulas: For \(a>c\), \(b>0\), and the assumptions just stated, perfect competition and homogeneous-good Bertrand competition have \(P=c\) and \(Q=(a-c)/b\). Monopoly has \(Q=(a-c)/(2b)\), \(P=(a+c)/2\), and profit \((a-c)^2/(4b)\). With \(n\) identical Cournot firms, each produces \((a-c)/(b(n+1))\); total output is \(n(a-c)/(b(n+1))\), and price is \(c+(a-c)/(n+1)\). Each Cournot firm’s profit is \((a-c)^2/[b(n+1)^2]\).

Why the Monopoly and Cournot Formulas Hold

The monopoly formula follows from maximizing total profit. With output \(Q\), revenue is \(aQ-bQ^2\), and total cost is \(cQ\). Profit is therefore \((a-c)Q-bQ^2\). Its derivative with respect to output is \(a-c-2bQ\); setting this equal to zero gives the profit-maximizing quantity. Since the second derivative is \(-2b<0\), this quantity maximizes profit.

$$ \pi_M(Q)=(a-c)Q-bQ^2,\qquad \frac{d\pi_M}{dQ}=a-c-2bQ=0 $$

For Cournot competition, firm \(i\) chooses \(q_i\) while treating the other firms’ total output \(Q_{-i}\) as fixed. Its profit is \([a-b(q_i+Q_{-i})-c]q_i\). Differentiating with respect to its own quantity gives the condition for a best-response quantity. In a symmetric equilibrium, all \(n\) firms produce the same amount \(q\), so rivals together produce \((n-1)q\). Substitution gives the symmetric quantity in the benchmark formulas.

$$ \frac{\partial \pi_i}{\partial q_i} =a-c-bQ_{-i}-2bq_i=0 $$
$$ Q_{-i}=(n-1)q_i\ \text{in a symmetric equilibrium},\qquad a-c-b(n-1)q-2bq=0 $$

The resulting Cournot market price is above cost when \(a>c\), but below the monopoly price for \(n>1\). As the number of Cournot firms increases, the equilibrium price approaches marginal cost and total output approaches the competitive quantity. These comparisons follow from the formulas: \(P_C=c+(a-c)/(n+1)\) and \(Q_C=n(a-c)/(b(n+1))\).

Worked Example: Comparing All Four Outcomes

Worked Example: Comparing All Four Outcomes

Suppose inverse demand is \(P=100-Q\), marginal cost is \(c=\$20\) per unit for every firm, and there are no fixed costs. Find the competitive, monopoly, two-firm Cournot, and homogeneous-good Bertrand outcomes. For Bertrand, assume that if both firms charge the same price, demand is split equally.

Method and assumptions. Here \(a=100\), \(b=1\), and \(a-c=80\). Use the stated constant-cost model, and report prices and profits to the nearest cent.

Perfect competition. The price-taking benchmark sets price equal to marginal cost: \(P=\$20\). Demand at that price is \(Q=100-20=80\) units. With no fixed costs, total operating profit is \((20-20)\times80=\$0\).

Monopoly. The monopolist produces \(Q=(100-20)/(2\times1)=40\) units. The price is \(P=100-40=\$60\). Profit is \((60-20)\times40=\$1{,}600\). As a check, total revenue is \(60\times40=\$2{,}400\), total cost is \(20\times40=\$800\), and revenue minus cost is \(\$1{,}600\).

Two-firm Cournot. Each firm produces \(q=(100-20)/(1\times3)=80/3=26.666\ldots\) units, so total output is \(Q=160/3=53.333\ldots\) units. Price is \(100-160/3=140/3=\$46.666\ldots\). Each firm’s margin is \(140/3-20=80/3\), so its profit is \((80/3)(80/3)=6{,}400/9=\$711.11\) to the nearest cent. Total industry profit is \(2(6{,}400/9)=\$1{,}422.22\) to the nearest cent.

Bertrand. The equilibrium price is marginal cost, \(P=\$20\), and total demand is \(Q=100-20=80\) units. Under the stated equal-split rule, each firm sells 40 units and earns \((20-20)\times40=\$0\). The market price and total quantity are \(\$20\) and 80 units, respectively.

The check is the expected ordering: monopoly has the highest price and lowest output; competition and Bertrand have the lowest price and highest output; Cournot lies between them. The common mistake avoided here is treating total Cournot profit as one firm’s profit: each Cournot firm earns \(\$711.11\), while the two firms together earn \(\$1{,}422.22\).

Worked Example: A Different Demand Slope

Worked Example: A Different Demand Slope

Suppose inverse demand is \(P=76-2Q\) and each firm’s constant marginal cost is \(c=\$16\), with no fixed costs. Compare perfect competition, monopoly, duopoly Cournot, and homogeneous-good Bertrand competition. Assume equal demand sharing when Bertrand prices tie.

Method. Here \(a-c=60\) and \(b=2\). The changed slope matters: it appears in the denominator of the quantity formulas. Report prices and profits in dollars to the nearest cent.

Perfect competition. Price is \(P=\$16\). Solving \(16=76-2Q\) gives \(2Q=60\), so \(Q=30\). Profit is zero because price equals marginal cost.

Monopoly. Output is \(Q=60/(2\times2)=15\). Price is \(76-2(15)=\$46\). Profit is \((46-16)\times15=30\times15=\$450\). The implied revenue is \(\$690\), cost is \(\$240\), and the difference is \(\$450\).

Duopoly Cournot. Each firm produces \(q=60/[2(2+1)]=10\), so total output is 20. Price is \(76-2(20)=\$36\). Each firm earns \((36-16)\times10=\$200\), giving total industry profit of \(\$400\).

Bertrand. Price is \(P=\$16\), and demand is \(Q=(76-16)/2=30\). Under the equal-split rule, each firm sells 15 units. Each earns \((16-16)\times15=\$0\). Across the four outcomes, Cournot’s price of \(\$36\) is between the monopoly price of \(\$46\) and the competitive and Bertrand price of \(\$16\). A common mistake avoided is using \(Q=(a-c)/2\) for competition without dividing by the demand slope \(b\); here that would give the wrong competitive quantity.

Worked Example: How the Number of Cournot Firms Matters

Worked Example: How the Number of Cournot Firms Matters

Market demand is \(P=120-2Q\), marginal cost is \(c=\$40\), and there are four identical Cournot firms. Find the Cournot outcome and compare it with the competitive, monopoly, and Bertrand benchmarks. Assume equal demand sharing at a tied Bertrand price.

Cournot calculation. Here \(a-c=80\), \(b=2\), and \(n=4\). Each firm produces \(q=80/[2(4+1)]=8\) units, so total output is \(Q=4\times8=32\). Price is \(120-2(32)=\$56\). Each firm’s profit is \((56-40)\times8=16\times8=\$128\), and total industry profit is \(4\times128=\$512\).

Competitive benchmark. Price is \(\$40\). Demand at that price satisfies \(40=120-2Q\), giving \(Q=40\). With no fixed costs, profit is zero.

Monopoly benchmark. Output is \(80/(2\times2)=20\), and price is \(120-2(20)=\$80\). Profit is \((80-40)\times20=\$800\).

Bertrand benchmark. Price is \(\$40\), and total quantity is 40. If the firms tie, each sells 10 units under equal sharing and earns zero. Thus, the four-firm Cournot result has a price above cost and output below the competitive quantity, while its price and quantity remain between the monopoly and competitive benchmarks. The calculation also distinguishes per-firm Cournot profit, \(\$128\), from total industry profit, \(\$512\).

Common Traps and How to Avoid Them

  • Using the wrong strategic variable. Cournot firms choose quantities; Bertrand firms choose prices. Do not use the Cournot quantity formula to calculate a Bertrand price.
  • Forgetting the demand slope. For inverse demand \(P=a-bQ\), quantities depend on \(b\). A steeper inverse-demand curve changes equilibrium quantity even if \(a\) and \(c\) stay fixed.
  • Confusing firm profit with total profit. Under symmetric Cournot competition, multiply one firm’s profit by \(n\) to obtain total industry profit.
  • Misapplying the Bertrand tie rule. A tie-breaking rule can determine each firm’s sales. With equal prices and an equal-split rule, divide total demand equally; do not describe the allocation as indeterminate.
  • Applying benchmarks outside their assumptions. The Bertrand price-equals-cost result requires the homogeneous-good, equal-cost, unconstrained-capacity setting used here. Differentiated products, unequal costs, or capacity limits require a different analysis.
Key takeaway: Under the stated linear-demand and constant-marginal-cost assumptions, monopoly restricts output to raise price, Cournot lies between monopoly and the competitive benchmark, and homogeneous-good Bertrand competition reaches the marginal-cost price. Always calculate total quantity first, then use demand to verify price and use the firm’s own sales to calculate its profit.

Check Your Understanding

For each problem, use the stated linear-demand model and distinguish firm profit from total industry profit where relevant.

  1. Inverse demand is \(P=84-Q\), marginal cost is \(\$24\), and there are no fixed costs. Find the competitive price and quantity and the total operating profit. (Answer: \(P=\$24\), \(Q=60\), total profit \(\$0\).)
  2. Inverse demand is \(P=110-2Q\), and a monopolist’s marginal cost is \(\$30\). Find monopoly output, price, and profit. (Answer: \(Q=20\), \(P=\$70\), profit \(\$800\).)
  3. Inverse demand is \(P=90-Q\), marginal cost is \(\$18\), and two identical firms compete in Cournot quantities. Find each firm’s output, total output, market price, and each firm’s profit. (Answer: each produces 24, total output is 48, price is \(\$42\), and each earns \(\$576\).)
  4. Inverse demand is \(P=70-Q\), marginal cost is \(\$10\), and two firms compete in homogeneous-good Bertrand prices. At a tie, demand is split equally. Find the equilibrium price, total quantity, and each firm’s sales and profit. (Answer: price \(\$10\), total quantity 60, each sells 30 and earns \(\$0\).)
  5. Inverse demand is \(P=150-3Q\), marginal cost is \(\$30\), and four identical firms compete in Cournot quantities. Find each firm’s output, total output, market price, each firm’s profit, and total industry profit. (Answer: each produces 8, total output is 32, price is \(\$54\), each earns \(\$192\), and total industry profit is \(\$768\).)

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