Intermediate Microeconomics (Besanko & Braeutigam, 5th ed.) | Chapter 13: Market Structure and Competition
Difficulty: Intermediate | Prerequisites: Profit maximisation (MR = MC), linear demand curves, basic calculus (derivatives)
This set of notes walks through the quantitative side of Chapter 13: how to solve for equilibrium price, quantity, and profit under perfect competition, monopoly, and Cournot duopoly, given a linear demand curve and constant marginal cost. These are the bread-and-butter calculations for any exam on oligopoly. The first problem covers the symmetric case (identical firms); the second introduces asymmetric costs, which adds one layer of complexity to the reaction functions. If you can do both cleanly, you are well prepared for anything this chapter will throw at you.
Perfect competition sets P = MC and earns zero profit. A monopolist sets MR = MC, restricts output, and earns positive profit. Cournot duopolists each write a reaction function (their best quantity given the rival's), solve the system, and land on a price and output between the two extremes. When firms have different costs, the lower-cost firm produces more and earns higher profit.
Inverse demand function
The demand curve written as P = f(Q), giving price as a function of total quantity. This is the form you need for setting up the profit function. If the demand curve is given as Q = f(P), invert it first.
Reaction function (best response function)
The equation that gives one firm's profit-maximising quantity as a function of the rival's quantity. Derived by taking the first-order condition of that firm's profit function with respect to its own output. In simple terms, it answers: "Given what the other firm is doing, how much should I produce?"
Residual demand
The demand left over for one firm after subtracting the rival's output. If total demand is P = a - b(q₁ + q₂), then Firm 1's residual demand is P = (a - bq₂) - bq₁. The firm treats (a - bq₂) as its effective intercept.
Marginal revenue (for a Cournot firm)
Derived from the residual demand. For a linear residual demand, MR has the same intercept but twice the slope coefficient on the firm's own quantity.
Setup: P = 70 - 2Q, constant marginal cost c = 10, zero fixed costs.
In perfect competition, P = MC.
70 - 2Q = 10
2Q = 60
Q = 30, P = 10
Profit = (P - c) x Q = (10 - 10) x 30 = 0. As expected, perfectly competitive firms earn zero economic profit.
Total revenue = P x Q = (70 - 2Q) x Q = 70Q - 2Q².
Marginal revenue = 70 - 4Q.
Set MR = MC:
70 - 4Q = 10
4Q = 60
Q = 15, P = 70 - 2(15) = 40
Profit = (40 - 10) x 15 = 450.
The monopolist produces half the competitive quantity and charges a much higher price.
Total quantity Q = q₁ + q₂. Price: P = 70 - 2(q₁ + q₂).
Firm 1's profit:
π₁ = [70 - 2(q₁ + q₂)] x q₁ - 10q₁
π₁ = 70q₁ - 2q₁² - 2q₁q₂ - 10q₁
π₁ = 60q₁ - 2q₁² - 2q₁q₂
First-order condition (differentiate with respect to q₁):
∂π₁/∂q₁ = 60 - 4q₁ - 2q₂ = 0
Firm 1's reaction function:
q₁ = 15 - 0.5q₂
By symmetry (both firms have the same cost), Firm 2's reaction function is:
q₂ = 15 - 0.5q₁
Solve the system. Substitute Firm 2's reaction function into Firm 1's:
q₁ = 15 - 0.5(15 - 0.5q₁)
q₁ = 15 - 7.5 + 0.25q₁
0.75q₁ = 7.5
q₁ = 10
By symmetry, q₂ = 10.
Equilibrium values:
Total output: Q = 10 + 10 = 20
Price: P = 70 - 2(20) = 30
Each firm's profit: π = (30 - 10) x 10 = 200
Q (total) | P | Profit (total) | |
|---|---|---|---|
Perfect competition | 30 | 10 | 0 |
Cournot duopoly | 20 | 30 | 400 |
Monopoly | 15 | 40 | 450 |
The Cournot outcome sits neatly between the two extremes, exactly as the theory predicts.
Setup: Q = 4000 - 40P (demand), MC₁ = 20, MC₂ = 40.
Step 0: Invert the demand curve.
Q = 4000 - 40P → 40P = 4000 - Q → P = 100 - Q/40.
Write Q = q₁ + q₂, so P = 100 - (q₁ + q₂)/40.
Firm 1's profit:
π₁ = P x q₁ - MC₁ x q₁ = [100 - (q₁ + q₂)/40] x q₁ - 20q₁
π₁ = 100q₁ - q₁²/40 - q₁q₂/40 - 20q₁
π₁ = 80q₁ - q₁²/40 - q₁q₂/40
First-order condition:
∂π₁/∂q₁ = 80 - 2q₁/40 - q₂/40 = 0
∂π₁/∂q₁ = 80 - q₁/20 - q₂/40 = 0
Multiply through by 40:
3200 - 2q₁ - q₂ = 0
Firm 1's reaction function:
q₁ = 1600 - q₂/2
Firm 2's profit:
π₂ = [100 - (q₁ + q₂)/40] x q₂ - 40q₂
π₂ = 60q₂ - q₂²/40 - q₁q₂/40
First-order condition:
∂π₂/∂q₂ = 60 - q₂/20 - q₁/40 = 0
Multiply through by 40:
2400 - 2q₂ - q₁ = 0
Firm 2's reaction function:
q₂ = 1200 - q₁/2
Notice that Firm 2's reaction function has a lower intercept (1200 versus 1600) because of its higher marginal cost. The higher-cost firm produces less at every level of rival output.
Substitute Firm 2's reaction function into Firm 1's:
q₁ = 1600 - (1200 - q₁/2)/2
q₁ = 1600 - 600 + q₁/4
0.75q₁ = 1000
q₁ = 1333.33 (or 4000/3)
Plug back into Firm 2's reaction function:
q₂ = 1200 - 1333.33/2 = 1200 - 666.67 = 533.33 (or 1600/3)
Total output:
Q = 1333.33 + 533.33 = 1866.67 (or 5600/3)
Equilibrium price:
P = 100 - 1866.67/40 = 100 - 46.67 = 53.33 (or 160/3)
Profits:
π₁ = (53.33 - 20) x 1333.33 = 33.33 x 1333.33 = 44,444.44
π₂ = (53.33 - 40) x 533.33 = 13.33 x 533.33 = 7,111.11
The low-cost firm produces roughly 2.5 times the output of the high-cost firm and earns more than six times the profit. Cost advantage translates directly into market share and profitability in Cournot competition.
Write the inverse demand function in the form P = a - b(q₁ + q₂). If the demand curve is given as Q = f(P), invert it first.
Write each firm's profit function: πᵢ = P x qᵢ - MCᵢ x qᵢ. Substitute the inverse demand so profit is expressed entirely in terms of q₁ and q₂.
Take the first-order condition: differentiate πᵢ with respect to qᵢ, treating qⱼ as a constant. Set equal to zero.
Solve for qᵢ as a function of qⱼ. This is the reaction function.
Repeat for the other firm. With symmetric costs, write it by symmetry. With different costs, redo steps 2 through 4.
Solve the two reaction functions simultaneously: substitute one into the other, solve for one quantity, then plug back to get the second.
Compute the equilibrium price from the demand curve, using Q = q₁ + q₂.
Compute each firm's profit: πᵢ = (P - MCᵢ) x qᵢ.
Sanity check: the Cournot price should be above MC and below the monopoly price. Total output should be above monopoly output and below the competitive output.
Students sometimes forget to invert the demand curve. If the problem gives Q = 4000 - 40P, you must rewrite it as P = 100 - Q/40 before setting up the profit function. Working with the original form leads to errors in the first-order conditions.
A frequent algebra mistake: when differentiating the profit function, students forget the cross-term (the q₁q₂ term). This term is what creates the strategic interdependence and generates the reaction function. If your reaction function does not contain the rival's quantity, something has gone wrong.
Students sometimes try to use the general Cournot formula (q = (a - c)/3b) when costs are asymmetric. That formula only works when both firms have the same marginal cost. With different costs, you must derive each reaction function separately.
When checking the answer, students occasionally compare the Cournot price to marginal cost and declare it "too high" or "too low" without reference to the monopoly price. The correct benchmark is: MC < P_Cournot < P_monopoly.
⚠️ The most common exam format: you are given a linear demand curve and marginal costs, and asked to find equilibrium price, quantity, and profit under two or three market structures (competition, monopoly, Cournot). Practise until you can do this in under ten minutes.
⚠️ Asymmetric-cost Cournot problems are a favourite for exams that want to test whether you can derive reaction functions from scratch rather than plugging into a memorised formula.
⚠️ Always show the reaction functions explicitly. Partial credit depends on demonstrating the method, not just the final numbers.
⚠️ If the problem asks you to "compare" the outcomes, present a summary table. Examiners like seeing the three market structures side by side.
Q: Suppose P = 50 - Q and both firms have MC = 10 with zero fixed costs. Find the Cournot equilibrium price, each firm's output, and each firm's profit.
A: Reaction functions: q₁ = 20 - 0.5q₂, q₂ = 20 - 0.5q₁. Solving: q₁ = q₂ = 40/3 ≈ 13.33. Total Q = 80/3 ≈ 26.67. Price = 50 - 26.67 = 23.33. Each firm's profit = (23.33 - 10) x 13.33 = 177.78.
Q: Using the same demand curve P = 50 - Q, what is the monopoly price and output?
A: MR = 50 - 2Q. Set MR = MC: 50 - 2Q = 10, Q = 20, P = 30. Profit = (30 - 10) x 20 = 400.
Q: In the asymmetric Cournot problem (P = 100 - Q/40, MC₁ = 20, MC₂ = 40), which firm has the steeper reaction function, and why?
A: Both reaction functions have the same slope (-1/2 on the rival's quantity), because the demand curve is linear and both firms face the same demand parameters. The difference is in the intercept: Firm 1's is higher (1600 vs 1200) because its lower cost means it would produce more if the rival produced zero.
Q: If a third firm with MC₃ = 30 entered the market in Problem 2, how would you set up its reaction function?
A: Write Firm 3's profit as π₃ = [100 - (q₁ + q₂ + q₃)/40] x q₃ - 30q₃. Differentiate with respect to q₃, set equal to zero, and solve for q₃ as a function of q₁ + q₂. Then solve the three reaction functions simultaneously. The method is the same; there are simply three equations and three unknowns.
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