Topic 4 — Price Competition with Complementary Goods

Part of: Microeconomics Perfect Complements, Double Marginalization, Pricing Externality, N-Firm Equilibrium, Vertical Integration Micro 3 — Tiomkin School of Economics, Reichman University Key concepts: Perfect Complements · Double Marginalization · Pricing Externality · Vertical Integration · Complementary Monopolist · Best Response Function · Comparative Statics

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1. Motivation — Why Complements Are Special

Most of industrial organisation focuses on substitute goods — where firms compete for the same consumers and more competitors means lower prices. This topic flips that logic entirely.

Setup: Perfect Complements

A consumer must purchase all components of a product to consume any of it. There is no partial consumption — the goods are perfect complements. The consumer cares only about the total price $P$, which is the sum of all component prices: $\boxed{P = \sum_{i=1}^{N} p_i}$

Real-world examples:

Intuition — Why this matters

When goods are substitutes, more competitors fight over the same consumer → prices fall. When goods are complements, each firm is essentially adding to the price the consumer pays for the same final good. More firms with pricing power over different components doesn't help the consumer — it hurts them.

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2. Demand — The Model Setup

The market has a simple linear demand function:

where $P$ is the total price paid by the consumer across all components, and $A$ is the maximum willingness to pay (the demand intercept).

Why total price?

Because the goods are perfect complements, the consumer's decision to buy depends only on the total cost of the bundle, not on how that total is split between firms. A consumer who must cross two toll booths cares about the total toll, not which booth charges more.

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3. Benchmark: The Integrated Monopolist

Before analysing decentralised firms, establish the efficient benchmark — a single monopolist who controls all components and chooses the total price $P$.

The monopolist's problem:

First-Order Condition (FOC):

Equilibrium outcomes:

Intuition

The monopolist internalises everything: raising price reduces both demand and profit, and they weigh this trade-off optimally. The standard monopoly result: price = $A/2$, quantity = $A/2$. This will be the upper benchmark — the best possible outcome for firms, and the least bad for consumers in a world with market power.

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4. Two Complementary Firms — The Decentralised Case

Now split control: Firm 1 controls component 1 (sets $p_1$), Firm 2 controls component 2 (sets $p_2$). They choose simultaneously.

Each firm's total price contribution: $P = p_1 + p_2$

Firm $i$'s problem (taking the other firm's price as given):

Deriving the Best Response Function

Take the FOC with respect to $p_i$:

Firm 1 maximises $p_1(A - p_1 - p_2)$. Differentiating and collecting the $p_1$ terms:

By symmetry: $p_2^*(p_1) = \dfrac{A - p_1}{2}$

Best Response Function

Each firm's optimal price falls as the other firm raises its price. This is the key: if Firm 2 raises $p_2$, total demand falls, which makes demand less elastic — but it also reduces Firm 1's residual market. In equilibrium, they account for each other.

Finding the Nash Equilibrium

Substitute $p_2^* = \frac{A - p_1}{2}$ into $p_1^* = \frac{A - p_2}{2}$:

By symmetry, $p_2 = \frac{A}{3}$.

type: reaction-functions
kind: price
br1: 5,-0.5
br2: 5,-0.5

Best responses that cross above the monopoly total

Just like Cournot, each firm's best response slopes down — but here the variables are prices, not quantities. The equilibrium is where they cross: $p_1 = p_2 = A/3$, so the consumer pays $P = 2A/3$. Crucially that total is higher than the integrated monopoly price $A/2$. The decentralised firms over-price because each one ignores how its markup shrinks demand for the other — the pricing externality made geometric.

Equilibrium Outcomes — Two Firms

Shocking result

Combined industry profits are $\frac{2A^2}{9} < \frac{A^2}{4}$ (the monopoly profit). The firms would collectively earn more under a single integrated firm! Splitting control has destroyed value — not just for consumers, but for the firms themselves.

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5. General Case: N Complementary Firms

Extend to $N$ firms, each controlling one component. By symmetry in equilibrium $p_i = p$ for all $i$, so $P = Np$.

Each firm solves:

FOC for firm $i$:

Using symmetry $p_i = P/N$:

Individual price:

Quantity:

Individual profit:

Total industry profit:

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6. The Key Comparison: Monopoly vs. N Complementary Firms

The Counter-Intuitive Result

Price rises and quantity falls as the number of complementary firms increases. This is the exact opposite of standard Cournot/Bertrand competition with substitute goods, where more firms drive prices toward marginal cost.

type: complementary-firms
A: 6

The signature result, plotted ($A=6$)

Left: as more firms each control a separate component, the total price $P^N = AN/(N+1)$ climbs toward the choke price $A$, while quantity $Q^N = A/(N+1)$ falls toward zero. Right: industry profit $NA^2/(N+1)^2$ is maximised at $N=1$ (the integrated monopoly) and declines monotonically as the chain fragments. More firms hurt consumers and firms — the opposite of competition among substitutes.

Verify the Monopoly Is Better

Compare $P^M = A/2$ vs $P^{DM} = 2A/3$: since $2/3 > 1/2$, the two-firm case has a higher total price. And compare total profits: $A^2/4$ vs $2A^2/9$. Since $9 > 8$, the monopoly earns more. The firms are hurting themselves by staying separate.

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7. Comparative Statics: What Happens as N Grows?

As $N$ increases:

• More complementary firms → higher total price (approaching maximum willingness to pay $A$)
• More complementary firms → lower quantity (approaching zero)
• At the extreme, the market collapses entirely — too expensive for anyone to buy

This is the opposite of standard competition

With substitutes (Cournot): more firms → price falls toward marginal cost → welfare improves With complements: more firms → price rises toward monopoly-level WTP → welfare collapses

This phenomenon is called Double Marginalization (or Multiple Marginalization when $N > 2$).

Definition: Double Marginalization

Double Marginalization occurs when two or more firms in a vertical supply chain each apply a markup to the final consumer price, independently. Each firm sets its price to maximise its own profit, ignoring the negative pricing externality it imposes on the other firms — because its higher price reduces the total demand that all firms share.

The Negative Pricing Externality Explained

When Firm 1 raises $p_1$:

1. Total price $P = p_1 + p_2$ rises
2. Quantity demanded $Q = A - P$ falls
3. Firm 2's revenue $\pi_2 = p_2 \cdot Q$ falls — Firm 1 does not account for this

Each firm only considers the effect on its own profit, not on the other firms' profits. This is a negative externality in prices, analogous to a pollution externality: individual action harms others, leading to a socially suboptimal outcome.

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8. Solutions: Internalising the Externality

Since the core problem is that firms ignore their impact on each other, the solution is to make one entity bear all the consequences of pricing decisions.

Why Vertical Integration Works

If Firms 1 and 2 merge, the combined entity solves: $\max_{p_1, p_2} (p_1 + p_2)(A - p_1 - p_2) = \max_P P(A - P)$ This is exactly the monopolist's problem → $P^* = A/2$. The merger restores efficiency (relative to the decentralised outcome, not relative to perfect competition).

Important caveat

Vertical integration solves the double marginalization problem but may raise other antitrust concerns (market foreclosure, etc.). Regulators must weigh the efficiency gains against the risk of increased market power.

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9. Practice Questions with Step-by-Step Solutions

Q1 — Two Complementary Firms: Full Derivation

Question: Two firms each produce one component of a good. Consumer demand is $Q = 10 - P$ where $P = p_1 + p_2$. Both firms have zero marginal cost. Find: (a) the Nash Equilibrium prices, (b) the total price, (c) the quantity, (d) each firm's profit.

Step-by-step solution:

Step 1: Write each firm's profit function.

Step 2: Derive best response functions via FOC.

For Firm 1: $\dfrac{\partial \pi_1}{\partial p_1} = 10 - 2p_1 - p_2 = 0$

For Firm 2 (symmetric): $p_2^*(p_1) = \dfrac{10 - p_1}{2}$

Step 3: Solve simultaneously.

Substitute $p_2^* = \frac{10 - p_1}{2}$ into $p_1^* = \frac{10 - p_2}{2}$:

Step 4: Compute equilibrium outcomes.

Step 5: Compare to monopoly benchmark. Monopoly: $P^M = 5$, $Q^M = 5$, $\pi^M = 25$ Two firms: $P^{DM} = 20/3$, $Q^{DM} = 10/3$, $\pi_{total} = 200/9 \approx 22.22$

→ Price is higher, quantity is lower, and total profits are lower than monopoly. ✓

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Q2 — Three Complementary Firms

Question: There are 3 complementary monopolists. Demand is $Q = 12 - P$ with zero marginal costs. Find the equilibrium price, quantity, and each firm's profit. Compare to the integrated monopolist.

Step 1: Use the general formula.

With $N = 3$ and $A = 12$:

Step 2: Compare to monopoly.

Total profits fell from 36 to 27. Price rose from 6 to 9. Quantity fell from 6 to 3.

Quick formula check

For $N$ firms: $P^N = \frac{AN}{N+1}$. Verify $N=1$: $P = A/2$ ✓. Verify $N=2$: $P = 2A/3$ ✓.

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Q3 — Comparative Statics Question

Question: Suppose $A = 6$. Fill in the table for $N = 1, 2, 3, 4$, and $\infty$. What happens to total industry profit as $N$ grows?

→ Industry profit falls monotonically with $N$. Both firms AND consumers are worse off as the market becomes more fragmented — the textbook definition of a lose-lose outcome.

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Q4 — Identifying the Pricing Externality (Essay-Style)

Question: Explain why decentralised complementary firms set a higher price than an integrated monopolist. What is the "pricing externality" and why do firms fail to internalise it?

Model answer:

When two complementary firms each set their price independently, they solve separate profit-maximisation problems. Firm 1 chooses $p_1$ to maximise $\pi_1 = p_1(A - p_1 - p_2)$.

The FOC yields $A - 2p_1 - p_2 = 0$, or equivalently: $A - P - p_1 = 0$.

The key term is $A - P - p_i = 0$. This says Firm 1 equates its own contribution to the markup against demand. But when Firm 1 raises $p_1$ by $\Delta$:

• Total price $P$ rises by $\Delta$
• Quantity $Q = A - P$ falls by $\Delta$
• Firm 2's revenue $\pi_2 = p_2 Q$ falls by $p_2 \cdot \Delta$

Firm 1 ignores this loss to Firm 2. This is the negative pricing externality — like a factory that ignores the pollution cost its production imposes on others. Each firm only bears the cost to its own revenues (lower quantity × own price), not the full social cost (lower quantity × all firms' prices combined).

An integrated monopolist, by contrast, maximises total profit $P(A-P)$ and fully internalises the fact that a higher price reduces demand for the entire bundle. This is why the monopolist prices at $A/2$, while decentralised firms push the total price to $2A/3$ (with $N=2$) or higher.

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Q5 — Vertical Integration Question

Question: With $A = 9$ and $N = 2$, what are the gains from vertical integration? Show the improvement in price, quantity, and total profit.

Decentralised equilibrium ($N = 2$):

Integrated monopolist:

Gains from integration:

Integration is Pareto-improving in this context: both firms profit more and consumers pay less. This provides the economic rationale for why vertical mergers are sometimes efficiency-enhancing.

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10. Summary

This topic reveals a striking and counter-intuitive result in industrial organisation:

• Perfect complements create a unique strategic environment: the consumer cares only about the total price $P = \sum p_i$, making each firm's pricing decision an externality on all others.
• The decentralised equilibrium with $N$ firms yields: $P^N = \frac{AN}{N+1}$, $Q^N = \frac{A}{N+1}$.
• More complementary firms = worse outcomes: higher prices, lower quantities, lower total profits. The polar opposite of standard competition.
• This is Double Marginalization — each firm applies an independent markup, stacking markups on top of each other.
• The negative pricing externality: each firm ignores how its price raises the total price and destroys demand for everyone.
• Solutions: vertical integration (merging the firms) or contractual remedies (two-part tariffs, revenue sharing) that internalise the externality.
• Vertical integration in this context is efficiency-enhancing — it eliminates double marginalization and increases both profits and consumer welfare.

Exam Checklist

1. Set up each firm's profit function: $\pi_i = p_i(A - \sum_j p_j)$.
2. Take FOC for firm $i$: $A - P - p_i = 0$.
3. Apply symmetry: $p_i = P/N$ in equilibrium → solve for $P^N = \frac{AN}{N+1}$.
4. State the key result: more firms → higher price (opposite of standard competition).
5. Name the mechanism: negative pricing externality / double marginalization.
6. Compare to monopoly: $P^M = A/2 < P^N$ for $N \geq 2$.
7. Discuss solutions: vertical integration, two-part tariffs, revenue sharing.

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Related Notes

• Microeconomics — subject hub
• Topic 2 - Equilibrium in Different Market Structures
• Topic 3 - Game Theory
• _Wiki-Link Registry