Chapter 33: Oligopoly: Cournot, Bertrand, Capacity, and Leadership. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Cournot duopoly with a cost shock

When one firm's cost rises, how much of its lost output does the rival replace?

Quantities are strategic substitutes: when Beacon produces less, Atlas's best response rises, but only by half of Beacon's cut, so total output falls and price rises.

$Equation, written in LaTeX: q_A=\frac{90-q_B}{2}, q_B=\frac{90-q_A}{2}.$

$Equation, written in LaTeX: q_A=\frac{90-q_B}{2}, q_B=\frac{66-q_A}{2}.$

$Equation, written in LaTeX: \pi_A=(48-10)38=\$1{,}444, \pi_B=(48-34)14=\$196.$

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Inverse demand is P = 100 - Q with Q = q_A + q_B. Atlas has marginal cost 10; Beacon's marginal cost is the control. Each line is a firm's best output given the other's.

Predict first. When Beacon's cost rises from 10 to 34, does Atlas replace all of Beacon's lost output?

Your prediction Yes, fully No, only partly

Beacon marginal cost

Figure: Cournot duopoly with a cost shock. Reaction lines for Atlas and Beacon cross at q_A = 30.00, q_B = 30.00 when Beacon's marginal cost is 10. — Beacon marginal cost: 10
Constructed example: the chapter's hypothetical cement duopoly (costs 10 and 10, then 10 and 34); Beacon costs of 22 and 46 are added for comparison.

Atlas output q_A: 30.00
Beacon output q_B: 30.00
Price P: $40.00
Atlas profit: $900.00
Beacon profit: $900.00
Change in total output from 60: 0.00

Solving the two reaction lines gives P = (100 + 10 + 10) / 3 = 40.00, q_A = 30.00 and q_B = 30.00. Profits are 900.00 and 900.00. Costs are equal, so the firms split output evenly.

Worked steps

P = (100 + 10 + 10) / 3 = 40.00
q_A = 40.00 - 10 = 30.00
q_B = 40.00 - 10 = 30.00
pi_A = (40.00 - 10) x 30.00 = 900.00
pi_B = (40.00 - 10) x 30.00 = 900.00

Use the idea

When a rival suffers a cost shock, expect your best output to rise by less than its cut and the market price to rise.

Where the conclusion applies

Simultaneous quantity choice, homogeneous output, linear demand and constant marginal costs.

Check your understanding: At c_B = 22, what are quantities and price?

q_A = (100 - 20 + 22) / 3 = 34, q_B = (100 - 44 + 10) / 3 = 22, P = 100 - 56 = 44.

Chapter 33 source: section "Cournot quantity competition".

Demonstration 2 of 4

When does a cartel hold?

How patient must members be, and how likely must detection be, to deter secret discounts?

Cheating pays a one-time gain; punishment costs the cartel margin forever after. Patience and detection both scale the future loss, so a cartel holds only when their product is large enough.

$Equation, written in LaTeX: \delta^*=\frac{220-120}{220-70}=\frac{2}{3}\approx0.667.$

$Equation, written in LaTeX: \frac{q\delta(\pi^C-\pi^P)}{1-\delta}.$

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Payoffs are thousands of dollars per quarter: pi^C = 120 under the quota, pi^D = 220 from a secret discount, pi^P = 70 after punishment. delta is the discount factor and q the probability a discount is detected.

Predict first. Does raising detection from 0.40 to 0.60 restore deterrence at delta = 0.80?

Your prediction Yes No

Discount factor

Detection probability

Figure: When does a cartel hold? Bars comparing the 100 thousand temptation to cheat with the expected punishment loss of 80 thousand at discount factor 0.80 and detection probability 0.40. — Discount factor: 0.8, Detection probability: 0.4
Constructed example: the chapter's hypothetical three-firm cartel (120, 220, 70; delta 0.80; detection 0.40 and 0.60); discount factors 0.60, 0.70 and 0.90 and full detection are added.

Critical discount factor: 0.667
Compliance value: 600
Deviation value (sure detection): 500
Temptation: 100
Expected punishment loss: 80
Cheating: Not deterred

With sure detection, compliance is worth 600 against 500 for cheating; delta = 0.80 is above delta* = 0.667. With detection probability 0.40, the expected loss is 0.40 x 0.80 x 50 / 0.20 = 80 against a temptation of 100, so cheating is not deterred.

Worked steps

delta* = (220 - 120) / (220 - 70) = 0.667
Comply: 120 / (1 - 0.80) = 600
Deviate: 220 + 0.80 x 70 / (1 - 0.80) = 500
Expected loss = 0.40 x 0.80 x 50 / 0.20 = 80
Compare with the temptation 220 - 120 = 100

Use the idea

Judge a coordination risk by the gain from cheating, the speed and probability of detection and how much members value the future.

Where the conclusion applies

Grim punishment forever after detection, constant payoffs and a simplified expected-loss comparison rather than a full stochastic repeated game.

Check your understanding: At delta = 0.90 and q = 0.40, does the cartel deter cheating?

0.40 x 0.90 x 50 / 0.10 = 180 > 100, so yes.

Chapter 33 source: section "Cartel instability".

Demonstration 3 of 4

The Cournot merger paradox

Why can two Cournot rivals lose profit by merging, and how large a cost saving reverses it?

A merger removes one quantity setter, so the merged firm cuts output while outsiders expand. Price rises, but the insiders give up market share to the outsiders and lose profit unless the merger also cuts cost.

$Equation, written in LaTeX: q(6)=\frac{60}{7}\approx8.57.$

$Equation, written in LaTeX: (\frac{60}{7})^2=\frac{3600}{49}\approx\$73.47,$

$Equation, written in LaTeX: P=\frac{115}{6}, q_M=\frac{85}{6}, q_O=\frac{55}{6}.$

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Inverse demand is P = 70 - Q, every firm has marginal cost 10 and there are six firms before the merger. The merged firm's marginal cost is the control; the four outsiders keep cost 10.

Predict first. Without a cost saving, do the merging firms gain profit?

Your prediction Yes No

Merged firm marginal cost

Figure: The Cournot merger paradox. Bars comparing the two insiders' combined profit before the merger, $146.94, with the merged firm's profit $100.00 at merged marginal cost 10.0. — Merged firm marginal cost: 10
Constructed example: the chapter's hypothetical six-firm market (merged cost 10 and 5); merged costs 7.5 and 2.5 are added for comparison.

Insiders' profit before: $146.94
Merged profit after: $100.00
Gain from merging: -46.94
Price before: $18.57
Price after: $20.00
Merged output: 10.00

Before the merger each of six firms makes 60/7 = 8.57 and earns 73.47, so the two insiders earn 146.94. After merging with marginal cost 10.0, price is (70 + 10.0 + 40) / 6 = 20.00 and the merged firm earns 10.00^2 = 100.00. The insiders lose 46.94.

Worked steps

Before: q = 60 / 7 = 8.5714, profit 73.4694 each, 146.94 for two
After: P = (70 + 10.0 + 4 x 10) / 6 = 20.0000
q_M = 20.0000 - 10.0 = 10.0000
Merged profit = 10.0000^2 = 100.00
Gain = 100.00 - 146.94 = -46.94

Use the idea

Before assuming a merger is privately profitable, model how rivals expand in response and how large an efficiency the deal needs.

Where the conclusion applies

Homogeneous output, Cournot conduct, linear demand, constant costs and no entry.

Check your understanding: At merged cost 7.5, what is merged profit?

P = (70 + 7.5 + 40) / 6 = 19.583; q_M = 12.083; profit = 12.083^2 = 146.01, still just below 146.94.

Chapter 33 source: section "Cournot merger paradox".

Demonstration 4 of 4

Stackelberg commitment

How much is moving first worth, and is more output always better for the leader?

The leader takes the follower's reaction into account: more leader output pushes the follower back, but also lowers price. Profit peaks at 40, above the Cournot 26.67.

$Equation, written in LaTeX: R_F(q_L)=\max\{0,\frac{80-q_L}{2}\}.$

$Equation, written in LaTeX: \pi_L=q_L[80-q_L-\frac{80-q_L}{2}]=\frac{q_L(80-q_L)}{2}.$

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Inverse demand is P = 100 - Q and both firms have marginal cost 20. The leader commits to q_L; the follower sees it and picks q_F on its reaction R_F.

Predict first. Does the leader's profit rise if it commits to 60 instead of 40?

Your prediction Yes No

Leader committed output

Figure: Stackelberg commitment. Leader profit q_L(80 - q_L)/2 peaks at 40. At the committed output 40.00 the leader earns 800.00 and the follower 400.00. — Leader committed output: 40
Constructed example: the chapter's hypothetical leader and follower (q_L = 40 and the Cournot 80/3); leader outputs 20 and 60 are added for comparison.

Follower output q_F: 20.00
Price P: $40.00
Leader profit: $800.00
Follower profit: $400.00
Consumer surplus: $1,800.00

Committing to 40.00 leads the follower to (80 - 40.00) / 2 = 20.00. Price is 40.00, the leader earns 800.00 and the follower 400.00; leader profit is the leader's best commitment. Consumer surplus is 1,800.00.

Worked steps

q_F = (80 - 40.00) / 2 = 20.00
Q = 40.00 + 20.00 = 60.00, P = 100 - 60.00 = 40.00
pi_L = (40.00 - 20) x 40.00 = 800.00
pi_F = (40.00 - 20) x 20.00 = 400.00
CS = 0.5 x 60.0000^2 = 1,800.00

Use the idea

A commitment to capacity is worth having only when it is observable and irreversible, and only up to the output that maximizes profit along the rival's reaction.

Where the conclusion applies

Known demand and costs, homogeneous output, observable and credible commitment.

Check your understanding: At q_L = 60, what are q_F and pi_L?

q_F = (80 - 60) / 2 = 10; pi_L = 60 x (80 - 60) / 2 = 600 < 800.

Chapter 33 source: section "Stackelberg leadership".

Cournot duopoly with a cost shock

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

When does a cartel hold?

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

The Cournot merger paradox

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

Stackelberg commitment

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Calculated values

Worked steps

Use the idea

Where the conclusion applies