Session 15: Oligopoly — Strategic Interdependence and the Model Toolkit

Part of the Microeconomics Knowledge System. This post covers Session 15 — reaction functions, Cournot, Stackelberg, Bertrand, kinked demand, cartels, and the full numerical comparison across market structures.

Every market structure covered so far had a clean solution. Perfect competition: P = MC. Monopoly: MR = MC, read P from demand. Monopsony: MV = ME, read P from supply. The math is mechanical once you know the model.

Oligopoly does not work this way. Your optimal decision depends on what your rival does. Their optimal decision depends on what you do. This circular dependence — strategic interdependence — is what makes oligopoly the most complex structure to model, the most realistic to real industries, and the most useful for managerial thinking.

This session builds a toolkit of four models: Cournot, Stackelberg, Bertrand, and kinked demand. Each model makes different assumptions about what firms choose (quantity vs price) and what they believe about rivals (simultaneous vs sequential, matched vs unmatched). Each gives different predictions. The skill is knowing which model fits which industry and why.

The Central Problem: Why Oligopoly Is Hard

In monopoly, there is one decision-maker facing known demand. Optimise. Done.

In oligopoly with even just two firms, each firm's profit function contains the other firm's output or price as an argument. Formally:

π₁ = π₁(Q₁, Q₂) — Firm 1's profit depends on both its own output and Firm 2's output

Neither firm can choose Q₁ independently of Q₂, and neither knows for certain what Q₂ will be. Each firm must form a belief about what the rival will do, and that belief becomes part of the strategic problem.

With three or more firms, the problem escalates further — coalition formation becomes possible, and the combinatorial complexity of who coordinates with whom and on what terms makes prediction significantly harder than in the two-firm case.

The professor's key point: the hard part of oligopoly is not the calculus. It is deciding how to model rival behaviour. Every oligopoly model makes an explicit assumption about this. The model you choose determines the answer you get.

The Core Tool: Reaction Functions

Before any specific model, you need to understand reaction functions — they are the analytical engine behind every oligopoly equilibrium.

Definition

A reaction function (also called a best response function) gives one firm's profit-maximising action as a function of the rival's action:

Q₁ = R₁(Q₂) — "Given that Firm 2 produces Q₂, Firm 1's best response is Q₁"

This is not a demand curve. It is not an accounting identity. It is the output of a profit-maximisation problem, solved by Firm 1 treating Q₂ as a parameter.

How to derive a reaction function — general procedure

1. Write Firm 1's profit: π₁ = P(Q₁ + Q₂) · Q₁ − C₁(Q₁)
2. Treat Q₂ as given (a constant from Firm 1's perspective)
3. Differentiate with respect to Q₁ and set equal to zero: ∂π₁/∂Q₁ = 0
4. Solve for Q₁ as a function of Q₂ — this is the reaction function

Why the slope is typically negative in quantity competition

More output by Firm 2 → total market output rises → price falls → Firm 1's marginal revenue falls → Firm 1's optimal response is to produce less.

Reaction functions in quantity competition slope downward: when a rival expands, you contract. This is strategic substitutability — outputs are strategic substitutes.

In price competition with differentiated products, the opposite holds: when a rival raises price, your best response is to raise price too (you can charge more without losing customers to them). Prices are strategic complements — reaction functions slope upward.

Common mistake to avoid

Students often derive reaction functions by rearranging the demand equation. This is wrong. Reaction functions come from optimisation, not from algebra on demand. The two equations look superficially similar but mean completely different things.

Model 1: Cournot — Simultaneous Quantity Competition

The assumptions

• Two (or more) firms produce a homogeneous good
• Firms choose quantities simultaneously — neither knows what the other has chosen when deciding
• Each firm takes the rival's output as given when optimising
• Firms are experienced enough to correctly anticipate each other's equilibrium choices

The equilibrium concept: Nash

The Cournot equilibrium is a Nash equilibrium — a pair of outputs (Q₁, Q₂) such that:

• Q₁* is Firm 1's best response to Q₂*
• Q₂* is Firm 2's best response to Q₁*

Neither firm wants to deviate unilaterally. This is what makes it stable — once there, no one benefits from moving.

Graph: the intersection of reaction functions

Draw Q₁ on the vertical axis, Q₂ on the horizontal axis (or vice versa).

• Firm 1's reaction function: downward-sloping line in (Q₂, Q₁) space
• Firm 2's reaction function: downward-sloping line in (Q₂, Q₁) space (after re-expressing)
• Cournot equilibrium = the intersection point

Points off the intersection: at least one firm wants to move. Points on the intersection: neither firm wants to move. This is why the equilibrium is stable — the professor's phrase was "nobody wants to disturb the applecart."

Full worked derivation — P = 30 − Q, MC = 0

This is the example used in lecture. MC = 0 is a simplification for clean algebra — do not generalise MR = 0 to other problems.

Setup:

• Market demand: P = 30 − Q = 30 − Q₁ − Q₂
• Total revenue for Firm 1: R₁ = P · Q₁ = (30 − Q₁ − Q₂)Q₁ = 30Q₁ − Q₁² − Q₁Q₂

Firm 1's MR (partial derivative with respect to Q₁, treating Q₂ as fixed):

MR₁ = ∂R₁/∂Q₁ = 30 − 2Q₁ − Q₂

Set MR₁ = MC₁ = 0:

30 − 2Q₁ − Q₂ = 0

Reaction function for Firm 1:

Q₁ = 15 − (1/2)Q₂

By symmetry (identical firms), Firm 2's reaction function is:

Q₂ = 15 − (1/2)Q₁

Solving simultaneously:

Substitute Firm 2's reaction into Firm 1's:

Q₁ = 15 − (1/2)(15 − (1/2)Q₁) = 15 − 7.5 + (1/4)Q₁

(3/4)Q₁ = 7.5 → Q₁ = 10

By symmetry: Q₂ = 10

Market outcomes:

• Total Q = 20
• P = 30 − 20 = 10
• Revenue per firm = 10 × 10 = 100

Cournot outcome vs competition and monopoly

Cournot produces more than monopoly (Q=15) but less than perfect competition (Q=30, at P=MC=0). Price is below monopoly (P=15) but above competitive (P=0). It is the intermediate case — some market power, but competition limits its exercise.

Model 2: Stackelberg — Leader and Follower

The key difference from Cournot

In Cournot, both firms choose simultaneously. In Stackelberg, one firm moves first — the leader — and the other observes this and responds — the follower.

The leader's advantage: it knows the follower will best-respond to its output choice. So the leader can incorporate the follower's reaction function directly into its own optimisation.

The two-step procedure

Step 1: Solve the follower's reaction function (same as Cournot reaction function):

Follower (Firm 2): Q₂ = 15 − (1/2)Q₁

Step 2: Leader (Firm 1) substitutes this into its own revenue and maximises:

R₁ = (30 − Q₁ − Q₂)Q₁

Substitute Q₂ = 15 − (1/2)Q₁:

R₁ = (30 − Q₁ − (15 − 0.5Q₁))Q₁ = (15 − 0.5Q₁)Q₁ = 15Q₁ − 0.5Q₁²

MR₁ = 15 − Q₁

Set MR₁ = MC = 0: Q₁ = 15 (leader output)

Then: Q₂ = 15 − (1/2)(15) = 7.5 (follower output)

Market outcomes:

• Total Q = 22.5
• P = 30 − 22.5 = 7.5
• Leader revenue = 7.5 × 15 = 112.5
• Follower revenue = 7.5 × 7.5 = 56.25

The leadership premium

The leader produces more than in Cournot (15 vs 10) and earns more than the follower (112.5 vs 56.25). The follower produces less than in Cournot (7.5 vs 10) and earns less than either firm did under Cournot.

Why? The leader commits to a large output, which forces the follower to contract (due to the negative-slope reaction function). The leader exploits this — it knows that producing more will cause the follower to pull back, partially offsetting the price effect.

What first-mover advantage is actually about: not aggression or luck. It is about credible commitment. By moving first, the leader credibly locks in a quantity that the follower cannot ignore. The credibility is the advantage — if the leader could change its output after observing the follower, the commitment has no force.

Why total output is higher than Cournot

Stackelberg total Q (22.5) > Cournot total Q (20). The leader expands enough that even though the follower contracts, the net effect is higher total output — which means lower price. Consumers do better under Stackelberg than under Cournot, at the leader's initiative.

Model 3: Bertrand — Price Competition

The assumption change

Cournot and Stackelberg both use quantity as the strategic variable. Bertrand switches to price as the strategic variable.

For homogeneous goods with the same cost structure, this small change has a dramatic consequence.

The homogeneous goods case

If two firms sell identical products and compete on price:

• Any firm pricing above the other captures zero customers (buyers go to the cheaper firm)
• Any firm pricing above MC can be undercut by the rival and still profitably capture the entire market
• This logic continues until both firms price at MC

Bertrand equilibrium with homogeneous goods: P = MC

This is the "Bertrand paradox" — with just two firms, price competition produces the competitive outcome. Two firms are enough to eliminate all market power.

The cold drinks illustration

The professor used the example of two student groups bidding for the right to sell identical cold drinks at a festival. In a one-period game with perfect information, any price above MC can be undercut. The equilibrium bid drives price to MC.

In practice: real Bertrand markets look like this when products are close substitutes and capacity is unconstrained. Airline tickets on identical routes are a near-example. The lesson for firms: if you compete on price with a homogeneous product against a well-funded rival, you will likely drive margins to zero.

Why Bertrand with differentiated products is different

When products are differentiated, undercutting a rival's price does not immediately capture their entire market — some customers still prefer the rival's brand even at a higher price. This makes demand less price-elastic and allows a Nash equilibrium with both firms pricing above MC.

Model 4: Kinked Demand — Why Prices Are Sticky

The question it answers

In real oligopolies, prices often stay fixed for extended periods even when input costs change. Why? The kinked demand model offers an explanation grounded in strategic asymmetry.

The behavioural assumptions

At the current price P*:

• If Firm 1 raises price: rivals do not follow. Firm 1 loses customers to rivals who now look cheaper. Demand for Firm 1 is elastic above P*.
• If Firm 1 cuts price: rivals match the cut to defend their market share. No firm gains customers — all firms are now just earning less. Demand for Firm 1 is inelastic below P*.

The geometry

The demand curve facing Firm 1 has a **kink at P***:

• Relatively flat (elastic) segment above P* — rivals don't follow price increases
• Relatively steep (inelastic) segment below P* — rivals match price cuts

This kink in the demand curve produces a gap in the MR curve. MR drops discontinuously at Q* (the quantity corresponding to P*).

Why prices are sticky

If MC shifts — but stays within the vertical gap in MR — the profit-maximising output and price remain at Q* and P*. The firm has no incentive to change price as long as cost changes are moderate.

This is the key result: price rigidity is not necessarily the result of explicit collusion or managerial inertia. It can emerge naturally from the asymmetric incentives firms face in oligopoly.

Limitations — state these in exams

The kinked demand model is descriptively useful but analytically incomplete:

1. It does not explain how P* was established in the first place — the model takes the kink as given
2. Empirical evidence for the kink is mixed — prices in some oligopolies are not especially sticky
3. It cannot predict the price level, only explain rigidity around an assumed current price

The professor explicitly categorised this as a descriptive explanation for rigidity, not a predictive model of price levels. Use it to explain why prices stay stable — not to explain where they will settle.

Cartels and Price Leadership

Cartels — the coordination solution

If oligopolists can coordinate, they act jointly as a monopolist — restricting total output and raising price to the monopoly level. This is a cartel.

Two forms discussed:

Market-sharing cartel: firms divide geography, customers, or product lines. No one enters the other's territory. Classic form in older industrial cartels.

Production/price control cartel: firms agree on output quotas or minimum prices. OPEC is the most prominent example — member countries agree on production limits to influence global oil prices.

Why cartels are unstable — the cheating problem

Every cartel member has an incentive to cheat. If everyone else is restricting output and holding price high, any individual member can secretly expand output, sell at the cartel price, and earn more profit.

But if every member cheats, total output expands, price falls, and the cartel collapses back toward the competitive or Cournot outcome. This is a prisoners' dilemma — individual rationality produces a collectively worse outcome.

Cartel stability requires:

• Enforceable agreements (difficult without legal backing)
• Ability to detect and punish cheating quickly
• Repeated interaction (so the threat of future punishment deters present cheating)
• Few enough members that detection is feasible

The game theory session following this one addresses cartel stability through trigger strategies and repeated games.

Price leadership — coordination without explicit agreement

Price leadership is how oligopolies often coordinate in practice without running into antitrust law.

One firm — typically the largest, most efficient, or most respected — changes its price. Others observe the move and follow. No explicit communication is needed. The outcome approximates collusion without being collusion.

Examples from lecture:

• Indian banking: SBI as "Big Daddy" — when SBI moves lending rates after a policy rate change, others follow
• Indian telecom: price leadership has shifted over time (Tata Teleservices earlier; Reliance Jio disrupting the equilibrium through aggressive entry)
• Management education: IIM Ahmedabad setting fee benchmarks that others reference

Key managerial insight: price leadership explains why oligopoly prices sometimes move in coordinated steps even when firms appear to be competing. The coordination is implicit, achieved through a shared understanding of who leads and who follows.

The Full Numerical Comparison — Read This Table Cold

Using P = 30 − Q, MC = 0 throughout.

Perfect competition

P = MC = 0, so Q = 30. Maximum output, zero margin.

Cartel/monopoly

Joint output maximised: MR = 30 − 2Q = 0 → Q = 15, P = 15. Maximum price, restricted output.

Cournot (derived above)

Q₁ = Q₂ = 10 → Q = 20, P = 10.

Stackelberg

Q₁ = 15, Q₂ = 7.5 → Q = 22.5, P = 7.5.

The ranking to memorise:

• Output: PC > Stackelberg > Cournot > Cartel
• Price: Cartel > Cournot > Stackelberg > PC
• Total industry profit: Cartel > Stackelberg combined > Cournot combined > PC

Stackelberg leader vs Cournot: the leader earns more than either Cournot firm (112.5 vs 100). The follower earns less (56.25 vs 100). The leader extracts value from the follower by committing to a large output.

Cartel vs Cournot: the cartel earns more in total but requires that each firm restricts output and trusts the other not to cheat. The individual incentive to cheat is precisely the Cournot quantity — so the cartel is always one defection away from collapsing to Cournot.

Differentiated Products — Price Reaction Functions

When products are differentiated, firms compete on price. The analysis changes because:

1. The strategic variable is price, not quantity
2. Reaction functions are derived by differentiating profit with respect to price, not quantity
3. Reaction functions slope upward (strategic complements)

The setup from lecture

Demand functions:

• Q₁ = 12 − 2P₁ + P₂
• Q₂ = 12 − 2P₂ + P₁

Fixed cost = 20, variable cost = 0.

Note the cross-price term: +P₂ in Q₁'s demand. When Firm 2 raises price, demand for Firm 1 increases (the products are substitutes). This is what makes prices strategic complements.

Derivation of price reaction functions

Pitfall alert: Do NOT use MR = MC here. MR is defined with respect to quantity. Here we are optimising with respect to price.

Firm 1's profit:

π₁ = P₁ · Q₁ − FC = P₁(12 − 2P₁ + P₂) − 20

Differentiate with respect to P₁:

∂π₁/∂P₁ = 12 − 4P₁ + P₂ = 0

Price reaction function for Firm 1:

P₁ = 3 + (1/4)P₂

By symmetry: P₂ = 3 + (1/4)P₁

Solving simultaneously

Substitute P₂'s reaction into P₁'s:

P₁ = 3 + (1/4)(3 + (1/4)P₁) = 3 + 0.75 + (1/16)P₁

P₁(1 − 1/16) = 3.75 → P₁(15/16) = 3.75 → P₁ = 4

By symmetry: P₂ = 4

Equilibrium outcomes: Q₁ = 12 − 8 + 4 = 8, Q₂ = 8, π₁ = 4×8 − 20 = 12, π₂ = 12

The upward-sloping reaction function — why it matters

When Firm 2 raises its price, Firm 1's best response is also to raise price — because its demand has increased (substitutes), so it can extract more without losing customers. This positive feedback loop is what makes differentiated Bertrand less destructive than homogeneous Bertrand.

Strategic implication: in markets with differentiated products, price increases by one firm can pull others upward. This is the economic mechanism behind coordinated price increases in branded consumer goods markets — not explicit agreement, but strategic complementarity.

Concept Connections

• Reaction function → Nash equilibrium → Cournot: one logical chain
• Stackelberg leadership premium → first-mover commitment → credibility
• Bertrand paradox (two firms → P = MC) → explains why firms desperately seek product differentiation
• Kinked demand → price rigidity → why oligopoly prices seem "sticky"
• Cartel instability → prisoners' dilemma → repeated games (Session 16 preview)
• Price leadership → implicit coordination → avoids antitrust while achieving near-cartel outcomes

What the Professor Is Likely to Test

1. Derive Cournot reaction functions and solve Show the full derivation from profit function through partial derivative through reaction function through simultaneous solution. Do not skip steps.

Trap: writing MR = 0 as though it is generally true. It is only true because MC = 0 in the example. In general, set MR = MC.

2. Stackelberg — derive leader's output and compare to Cournot Show the substitution of the follower's reaction function into the leader's revenue. State why the leader produces more than in Cournot and earns more profit.

3. Kinked demand and price rigidity Asymmetric elasticity above and below P*. MR gap. MC shifts within the gap do not change optimal price. State the limitations explicitly.

4. Comparison table across structures The Q, P ranking: PC > Stackelberg > Cournot > Cartel for Q. Reversed for P. Know the numbers from the worked example.

5. Differentiated Bertrand — price reaction functions Derive by differentiating profit with respect to price (not quantity). State that reaction functions slope upward (strategic complements). Solve for equilibrium prices.

Trap: using MR = MC for price-setting problems. Wrong tool.

6. Why cartels are unstable Prisoners' dilemma logic. Individual incentive to cheat (produce more than quota) while everyone else holds back. Cartel collapses to Cournot outcome without enforcement.

Quick Reference

The benchmark numbers (P = 30 − Q, MC = 0):

Previous: Session 14 — Monopsony and Monopolistic Competition

Back to: Microeconomics Master Index

Source: Lecture slides by Prof. Kaushik Bhattacharya, IIM Lucknow (Session 15). Textbook: Pindyck & Rubinfeld, Microeconomics, Global Edition (2017), Ch. 12.