Topic 2 — Equilibrium in Different Market Structures

Part of: Microeconomics Micro 3 — Semester A | Tyomkin School of Economics, Reichman University Previous topic: Topic 1 - Asymmetric Information Key concepts: Monopoly, Price Discrimination, Bundling, Bertrand Competition, Cournot Competition, Stackelberg Model, Perfect Competition, Vertical Relations

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Overview

This topic builds a ladder of market structures — from the firm's cost structure all the way up to welfare analysis — and shows how equilibrium price, quantity and surplus depend on who has market power and how they compete.

The ladder

1. Costs — the firm's side of every market
2. Monopoly (single output, then multi-output with bundling)
3. Market power & elasticity
4. Price discrimination (1st, 2nd, 3rd degree)
5. Oligopoly — Bertrand, Cournot, Stackelberg
6. Perfect competition
7. Vertical relations — double marginalisation

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Costs of Production

Economic vs accounting costs

• Explicit costs — require an outlay of money (wages, rent, materials).
• Implicit costs — opportunity costs with no money changing hands (owner's forgone salary).
• Accountants ignore implicit costs → economic profit < accounting profit.

Key distinctions

Sunk cost fallacy

If you've already paid a non-refundable deposit on Building A, that deposit must not enter the decision to switch to Building B. Compare only the avoidable costs going forward.

Marginal cost shapes

• Type 1 — Fixed MC: $TC(Q) = F + cQ$, so $MC = c$ (horizontal line).
• Type 2 — Increasing MC: $TC$ curve gets steeper as $Q$ rises (diminishing marginal product of inputs).

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Monopoly — Single Output

A monopoly is the sole seller of a product with no close substitutes — a price maker subject only to the consumer's demand curve.

Why monopolies arise

1. Monopoly resources — sole ownership of a key input.
2. Government regulation — patents, copyrights, exclusive licenses.
3. Natural monopoly — economies of scale over the relevant output range (one firm supplies the whole market more cheaply than two could).

Profit maximisation

FOC: set $\partial \Pi / \partial Q = 0$:

Then charge the price the demand curve supports at that $Q$: $P^* = P(Q^*)$.

Linear demand worked example

Demand $P = A - bQ$ with $MC = c$:

Setting $MR = MC$:

The MR shortcut for linear demand

If $P = A - bQ$, then $MR = A - 2bQ$ — same intercept, twice the slope. Worth memorising.

type: monopoly-cs-dwl
a: 10
b: 1
mc: 2

The classic monopoly diagram

The monopolist produces where $MR = MC$ (purple dashes meet green) and then reads the price off the demand curve above that quantity — never off the MR curve. Compared with the competitive outcome ($P = MC$, where the green line meets demand at $Q = A-c$), the monopolist restricts output and charges more. The red triangle is the deadweight loss: trades that are worth more to buyers than they cost to produce but don't happen.

Non-flexible (perfectly elastic) price

When the price is fixed (e.g. the firm faces a world price), $MR = P$ at every $Q$. Profit max condition collapses to $MC = P$.

Example B — a tax on profit does not change the monopoly price

The OPEC oil-tax puzzle (1970s)

Oil producers paid OPEC countries a fixed tax of $9 per barrel (a per-unit tax). A proposal was floated to switch to a tax of 90% of gross profit instead. Would producers raise their price in response?

No. A proportional tax on profit scales the objective $\,(1-t)\,\Pi(Q)$ but leaves its maximiser unchanged: if $\Pi(Q)$ is largest at $Q^*$, so is $(1-t)\Pi(Q)$. The firm prices "as if there were no tax." A per-unit tax, by contrast, raises effective $MC$ and does push price up. This is the key distinction: profit taxes are non-distortionary for the monopolist's price; per-unit taxes are not.

Example C — unitary-elastic demand

If demand has constant unit elasticity ($E = 1$ everywhere, e.g. $P = k/Q$), then total revenue $PQ = k$ is constant and $MR = 0$ at every quantity. With positive $MC$ the firm wants to sell the smallest possible quantity at the highest price — there is no interior profit-maximising quantity, so the standard $MR = MC$ tangency breaks down. A useful edge case to recognise in exam questions.

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Monopoly — Multi-Output & Bundling

The bundling intuition

Gone With the Wind (1939)

LOEW's told theatres that wanted Gone With the Wind (GWTW) they must also rent Getting Gertie's Garter (GGG). Why?

Theatre	RP(GWTW)	RP(GGG)
A	$12,000	$1,000
B	$10,000	$4,000

Selling separately: best price for GWTW is $10k (both theatres buy) → $20k; best for GGG is $4k (only B buys) → $4k, which beats $1k to both ($2k). Total $24k.

Bundle price = $13k (A's total RP) or $14k (B's total RP). At $13k, both buy → $26k. Bundling wins by $2k.

Bundling rule

Bundling is profitable when the two goods' reservation prices are negatively correlated across consumers. The bundle smooths out heterogeneity and extracts more surplus.

type: bundling
consumers: C1:90,10,1;C2:80,40,1;C3:40,80,1;C4:10,90,1

Why negative correlation is the whole game

Each dot is a consumer placed by how much they value good X (horizontal) versus good Y (vertical). When valuations are negatively correlated — high-X consumers are low-Y and vice versa — every consumer's total willingness to pay clusters near the same value, here ≈ $100. A single bundle price (green line) then captures almost everyone with little surplus left on the table. Selling separately forces the monopolist to pick one price per good and lose either the low-value or the high-value buyers.

Bundling strategies

1. Pure bundling — only sell the bundle.
2. Mixed bundling — offer bundle and individual goods, usually at a premium.
3. Separate pricing — optimal when RPs are positively correlated.

Mixed bundling beats both pure strategies (Example 3)

Four customers value Excel and Word as follows:

Customer	RP(Excel)	RP(Word)	Total RP
1	90	10	100
2	80	40	120
3	40	80	120
4	10	90	100

• Separate pricing: best single price for Excel is $80 (customers 1–2 buy → $160); same for Word ($160). Total ≈ $320.
• Pure bundling at $100: all four buy → $400.
• Mixed bundling: offer the bundle at $120 and each good alone at $90. Customers 2 and 3 take the bundle ($240); customers 1 and 4 buy just their high-value good at $90 ($180) → $420.

Mixed bundling dominates because it sells the bundle to the "balanced" consumers while still skimming the high-value single-good buyers at a premium.

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Market Power and the Lerner Index

Elasticity of demand

• $E > 1$ — elastic (close substitutes exist) → $MR > 0$.
• $E = 1$ — unit elastic → $MR = 0$.
• $E < 1$ — inelastic (few substitutes) → $MR < 0$.

The MR–elasticity formula

At every point on the demand curve:

type: elasticity-mr

Why a monopolist never operates on the inelastic part of demand

Along a linear demand curve, MR is positive on the elastic upper half ($E>1$), zero at the midpoint ($E=1$), and negative on the inelastic lower half ($E<1$). Since $MC \geq 0$, setting $MR = MC$ always lands the monopolist on the elastic portion — selling into the inelastic region would mean cutting price while losing revenue. The formula $MR = P(1-1/E)$ makes this visible: as $E \to 1$, MR collapses to zero.

Pricing "rule of thumb"

Setting $MR = MC$ and rearranging:

This is the Lerner Index — a direct measure of monopoly power:

• $L \to 0$: price competition (perfect competition).
• $L \to 1$: maximum monopoly power.

Cake-mix problem

$MC = \$0.75$, $E = 3$: $P^* = \dfrac{MC}{1 - 1/E} = \dfrac{0.75}{2/3} = \$1.125$

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Price Discrimination

Definition: charging different prices for similar goods that don't reflect cost differences. Goal — capture more consumer surplus.

Essential conditions

1. Segmented markets with different elasticities.
2. Control over price.
3. Ability to infer willingness to pay (age, location, timing, etc.).
4. Prevent arbitrage — or the cheap segment resells to the expensive one.

Arbitrage-prevention toolkit

• Warranties void on resale (value drops).
• Services (dental, haircut) — inherently non-resellable.
• Adulteration — make the cheap version unfit for premium uses.
• Transport costs / remote outlets — stores "out of the way".
• Contractual — resale bans.
• Coupons — require time/effort, filtering by willingness to pay.

Third-degree (segmentation)

Segment the market; in each segment set $MR_i = MC$. More inelastic segment → higher price.

Two-segment monopoly

$MC = \$50$, Segment 1: $P_1 = 100 - Q_1$, Segment 2: $P_2 = 80 - 2Q_2$. In each: $MR_i = MC \Rightarrow$ pick $Q_i^*$ and $P_i^*$ separately.

type: price-discrimination-3rd
a1: 100
b1: 1
a2: 80
b2: 2
mc: 50
name1: Segment 1 (less elastic)
name2: Segment 2 (more elastic)

One firm, two prices

The monopolist treats each segment as its own little market and sets $MR_i = MC$ in each. Here that gives $P_1 = \$75$ in the less-elastic segment and $P_2 = \$65$ in the more-elastic one. The rule is general: the segment with less elastic demand pays the higher price, because those consumers are less willing to walk away from a price rise.

First-degree (perfect price discrimination)

Charge each consumer her exact reservation price. In practice approximated by:

• Two-part tariff (TPT): entry fee $F$ + per-unit price $P$. Optimal: $P = MC$, $F = CS$ at that price → captures all surplus from the segment.
• Examples: gyms, amusement parks, Gillette razors, telephone plans.

Second-degree (package / non-linear pricing)

Monopolist knows the types but can't identify which consumer is which. Offer a menu of bundles so each type self-selects.

The logic (IC-constraint style):

1. If you tried perfect-PD with menu, the "rich" consumer would mimic the "poor" bundle and pocket the surplus difference.
2. Reduce the poor consumer's bundle quantity by $\Delta q$:
   • Loss ($b$): revenue on the removed units from the poor consumer.
   • Gain ($a$): shrinks the rich consumer's mimicking surplus → allows charging them more.
3. Keep reducing while $a > b$. Optimum: $a = b$.

When to distort

If $a \geq b$ everywhere — distort until only the rich type buys. If $a \leq b$ everywhere — keep the poor's bundle at its first-best quantity; the rich end up buying the same bundle (pooling). Works for any demand curves, not just parallel ones.

type: second-degree-pd

The a-vs-b trade-off, drawn

Shrinking the poor type's bundle by $\Delta q$ has two effects. The loss $b$ (red, under the poor type's demand) is the revenue the monopolist gives up on those units. The gain $a$ (green, between the two demand curves) is how much more it can now charge the rich type, because the poor bundle is less tempting to mimic. The monopolist keeps cutting while $a > b$ and stops at $a = b$. This is why the low type's bundle is distorted downward but the high type's never is ("no distortion at the top").

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Oligopoly

Oligopoly — a small number of firms, each aware that its price/quantity choices move the market. Each firm must predict its rivals' strategies. The equilibrium concept is Nash Equilibrium.

Bertrand (1883) — Price competition

Assumptions: homogeneous product, firms set prices simultaneously, consumers buy from the cheapest.

Symmetric case: two firms with the same $MC = c$. The unique Nash equilibrium is:

"Bertrand paradox": just two firms suffice for perfect-competition outcomes.

Asymmetric case: firm 1 has $MC_1 = c_1 < c_2$. Firm 1 undercuts firm 2 just enough — equilibrium price = $\min(c_2, P_1^M)$, where $P_1^M$ is firm 1's unconstrained monopoly price.

Bertrand's knife-edge

The result collapses if any assumption breaks: non-constant MC, differentiated products, multiple periods, or imperfect information. These extensions motivate everything that follows.

Monopolistic (price) competition

Differentiated products with inverse demands:

where $g \in [0,1]$ measures how similar the products are. Each firm has some market power — derive best-response functions from $\partial \Pi_i / \partial P_i = 0$, intersect them for the Nash equilibrium.

Cournot (1838) — Quantity competition

Firms simultaneously choose quantities; market price = $P(q_1 + q_2)$.

Symmetric duopoly: $P = A - Q$, $C_i(q_i) = c q_i$.

Firm 1's FOC: $\dfrac{\partial \Pi_1}{\partial q_1} = A - 2q_1 - q_2 - c = 0$

Best response: $q_1 = \dfrac{A - c - q_2}{2}$ (symmetric for firm 2).

Solving the system:

type: reaction-functions
br1: 4.5,-0.5
br2: 4.5,-0.5

Where the best responses cross

Each firm's best-response line shows its profit-maximising output given the rival's output — both slope downward because quantities are strategic substitutes (if your rival floods the market, you cut back). The Nash equilibrium is the single point where both are simultaneously best-responding: $q_1 = q_2 = (A-c)/3$. Note each firm produces less than the monopoly output $(A-c)/2$ but the two together produce more, which is why price falls below the monopoly level.

Comparison table — linear demand, $MC = c$

Intuition

More competitors → more quantity, lower price, less industry profit. Cournot sits between monopoly and perfect competition.

type: structure-comparison
structures: monopoly,cournot,bertrand
A: 10
c: 2

The three structures side by side ($A=10$, $c=2$)

Reading left to right — monopoly to Cournot to Bertrand — quantity climbs (blue: 4 → 5.3 → 8), price falls (orange: 6 → 4.7 → 2 = MC), and industry profit collapses (green: 16 → 14.2 → 0). Cournot is genuinely intermediate; Bertrand with just two firms already reaches the competitive outcome (the "Bertrand paradox").

Stackelberg (1934) — Sequential quantity

Firm 1 (leader) sets $q_1$ first; firm 2 (follower) observes and responds with its Cournot best-response $q_2(q_1)$. Firm 1 maximises knowing this.

Plugging firm 2's BR into firm 1's profit and optimising:

type: reaction-functions
br1: 4,-0.5
br2: 4,-0.5
points: Stackelberg|The leader moves first and picks its best point on the follower's BR: q₁ = (A − c)/2.|4,2

The leader exploits the follower's best response

Because the follower will always react along its best-response line, the leader treats that line as a constraint and picks the point on it that maximises its own profit. That point is further down the line than the Cournot equilibrium — the leader commits to a larger quantity $(A-c)/2$, forcing the follower to scale back to $(A-c)/4$. Commitment is the whole advantage: it only works because the leader moves first and can't be undone.

First-mover advantage: the leader produces more and earns more than in Cournot; the follower earns less.

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Perfect Competition

Assumptions: many buyers and sellers, identical product, free entry/exit, everyone is a price taker.

Equilibrium mechanics

• Excess demand (shortage): $Q_d > Q_s$ at current $P$ → $P$ rises.
• Excess supply (surplus): $Q_s > Q_d$ at current $P$ → $P$ falls.
• Equilibrium: $Q_s = Q_d$, no tendency for $P$ to change.

Welfare

First Welfare Theorem (informal): competitive equilibrium maximises total surplus — Adam Smith's invisible hand.

Benchmark role

Perfect competition is the efficiency benchmark. Deadweight loss (DWL) in every other market structure is measured relative to this outcome.

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Vertical Relations — Double Marginalisation

A producer sells via a retailer — each stage marks up price above marginal cost. The stacked markups can hurt both firms and consumers.

Setup

• Market demand: $P = A - Q$
• Producer's $MC = k$; charges wholesale $w$ per unit to retailer.
• Retailer faces $w$ as its own MC; sets $P$ for consumers.

Backward induction

Stage 2 (retailer): Retailer's problem: $\max_Q (P - w)Q$ with $P = A - Q$. $MR_r(Q) = A - 2Q = w \Rightarrow Q = \dfrac{A - w}{2}$.

This implicitly defines the retailer's derived demand for the wholesale good:

Stage 1 (producer): Producer faces derived demand $w(Q) = A - 2Q$, so $MR_p(Q) = A - 4Q$. Set $MR_p = k$:

Compare to vertical merger (integrated monopoly)

A single firm with $MC = k$ would set:

Double marginalisation

The two-stage chain produces half the quantity and higher final price than the integrated monopoly — even though both are monopolies. Worse for consumers and worse for total industry profit.

type: double-marginalisation
A: 10
k: 2

Three outcomes on one demand curve ($A=10$, $k=2$)

Perfect competition prices at $MC$ ($P=2$, $Q=8$). A single integrated monopolist restricts to $Q=4$, $P=6$. The vertical chain stacks two markups and ends up even worse — $Q=2$, $P=8$ — because the producer marks up over its cost, then the retailer marks up over the wholesale price. Each link ignores the demand it destroys for the other, so the final price overshoots even the monopoly level. Merging the two (or a two-part tariff) collapses the chain back to the integrated point.

Remedy: vertical integration, two-part tariff wholesale contracts, or resale price maintenance — all recover the integrated outcome.

Welfare ordering (linear demand)

(in terms of total surplus; DWL grows along the chain).

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Summary — What Topic 2 Teaches

1. Market structure determines the wedge between price and marginal cost. The Lerner index $L = (P-MC)/P = 1/E$ captures this in one number.
2. Monopoly pricing balances the marginal-revenue trade-off; bundling extracts more surplus when RPs are negatively correlated.
3. Price discrimination needs segmentation, price control, willingness-to-pay inference, and arbitrage prevention. Second-degree PD uses a self-selecting menu — distort the low type's bundle until marginal loss $b$ = mimicking gain $a$.
4. Oligopoly ranges from Bertrand (price → $MC$) to Cournot (interior) to Stackelberg (first-mover advantage). More firms in Cournot ≈ perfect competition in the limit.
5. Perfect competition is the welfare benchmark; every departure creates DWL.
6. Vertical relations cause double marginalisation — stacked monopolies are worse than one integrated monopoly.

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Related Notes

• Previous topic: Topic 1 - Asymmetric Information
• Foundational: Nash Equilibrium, Consumer Surplus, Producer Surplus, Deadweight Loss
• Related: Lerner Index, Price Discrimination, Bundling, Cournot Competition, Bertrand Competition, Stackelberg Model, Vertical Integration