What Is Oligopoly Models?

An oligopoly is a market structure with a small number of firms whose strategic choices (price or quantity) materially affect market outcomes. Oligopoly models formalize these strategic interactions and produce equilibrium predictions about prices, quantities, and profits. They lie at the intersection of microeconomics (profit maximization, marginal analysis) and game theory (Nash equilibrium, strategic reasoning).

Why care? Because many important industries are oligopolistic: airlines, telecommunications, energy generation, and some retail markets. Correctly predicting outcomes in these markets requires reasoning about how rival firms anticipate and react to each other's decisions.

Core intuition, stated upfront:

• •Cournot (quantity competition): Firms simultaneously choose outputs; the strategic variable is quantity. Each firm takes rivals' quantities as given and sets its own output. The equilibrium is a Nash equilibrium in quantities.
• •Bertrand (price competition): Firms simultaneously set prices for homogeneous goods. With constant marginal costs and no capacity constraints, the Bertrand outcome drives price to marginal cost (the Bertrand paradox).
• •Stackelberg (leader-follower): One firm commits to a quantity first (leader); the other firm(s) observe and then choose quantities (followers). The leader obtains a first-mover advantage.

These models are not just alternative formulations; they produce different equilibria and policy implications. To make their predictions precise we rely on two prerequisites:

• •In Nash Equilibrium, we solved for strategy profiles with no profitable unilateral deviation. Here, each firm's strategy is a price or quantity and the equilibrium is a profile of such choices.
• •In Profit Maximization, we use marginal revenue = marginal cost (MR = MC) first-order conditions to find best-responses (and check second-order conditions).

A standard common benchmark demand specification is linear inverse demand

where $Q$ is total market output, $P$ price, and $c$ constant marginal cost. A concrete numeric example: take $a=100$, $b=1$, $c=20$. Then $P=100-Q$ and marginal cost is $20$. If total output is $Q=50$, then $P=100-50=50$ and per-unit margin is $30$.

Every equilibrium concept we use will be constructed as a Nash equilibrium in a game whose strategy sets and timing differ. For Cournot and Bertrand these are simultaneous-move games; Stackelberg is sequential and solved by backward induction (subgame-perfect Nash equilibrium).

Comparative statics and ordering: A key take-away you will see proven below is a typical ordering of market outcomes (for standard linear costs/demand and homogeneous product):

• •Monopoly: highest price, smallest total output.
• •Cournot (duopoly): intermediate price and output.
• •Stackelberg (leader-follower): lower price and higher total output than Cournot; leader enjoys higher profit.
• •Bertrand (homogeneous goods, unconstrained capacity): price equals marginal cost $c$ (lowest price), output is largest.

All three models provide different lenses for regulator and manager decisions: antitrust enforcement, entry deterrence, capacity planning, and pricing strategy. In the next sections we'll derive closed-form formulas, compute numeric examples, and expose the mechanisms behind these orderings.

Core Mechanic 1: Cournot Quantity Competition

Cournot models assume firms choose quantities simultaneously and each firm's payoff depends on market price determined by aggregate quantity. Formally, for a duopoly (firms 1 and 2) with inverse demand $P=a-bQ$ and constant marginal cost $c$, firm $i$'s profit is

Reference to prerequisites: In Nash Equilibrium, we require that each $q_i$ is a best response to $q_j$. In Profit Maximization we apply MR = MC (first-order condition) to get that best response.

Derivation of best-response (duopoly). The firm maximizes $\pi_i(q_i,q_j)$ w.r.t. $q_i$. Compute the first-order condition (FOC):

Collect terms to isolate $q_i$:

This is firm $i$'s reaction (best-response) function:

Numeric example near the formula: with $a=100$, $b=1$, $c=20$, we get

Interpretation: the more the rival produces, the less you produce; slope $-1/2$.

Solve for symmetric Nash equilibrium (set $q_1=q_2=q^*$). Then

So

Numeric substitution: with $a-c=80$, $b=1$, $q^* = 80/3 \approx 26.667$ per firm. Total output $Q^C = 2q^* = 53.333$, price $P^C = a - b Q^C = 100 - 53.333 = 46.667$. Each firm's profit

Generalization to $n$ identical firms: each firm's symmetric Cournot output is

so total output is

Numeric check: with $n=2$, $q_i=(100-20)/(1\cdot3)=26.667$ recovers the duopoly.

Limit as $n\to\infty$: $Q^{(n)} \to \frac{a-c}{b}$ and price $P\to c$. That is, Cournot competition converges to the competitive outcome (price equals marginal cost) as the number of firms grows.

Graphical intuition: plot the two best-response lines in $(q_1,q_2)$ space. Each is downward sloping with slope $-1/2$; the intersection is the Nash equilibrium. As $n$ increases, each firm's reaction becomes flatter (less influence of a single firm), pushing equilibrium output higher.

Second-order condition check: ensure profit is concave in $q_i$. Compute second derivative

so the FOC gives a maximum (this uses the Profit Maximization prerequisite).

Comparative statics: if marginal cost $c$ increases, each firm's output drops linearly: $\partial q^*/\partial c = -1/(3b)$ for $n=2$; price increases. If firms differ in marginal cost, the lower-cost firm produces more; closed-form asymmetric solutions exist but require solving linear system of first-order conditions.

Takeaway: Cournot equilibrium is an application of Nash Equilibrium plus Profit Maximization MR=MC, with closed-form solutions under linear demand. It yields intermediate prices between monopoly and perfect competition.

Core Mechanic 2: Bertrand Price Competition and Stackelberg Leader-Follower

This section contrasts price competition (Bertrand) with sequential quantity competition (Stackelberg). Both highlight how the strategic variable (price vs quantity) and timing (simultaneous vs sequential) change equilibria.

Bertrand (homogeneous goods, unconstrained capacity):

Assume two firms set prices $p_1,p_2$ simultaneously for an identical product and face the market demand function $D(p)$ (a decreasing function). Each firm supplies the entire demand if it posts the lowest price; if prices are equal, they split demand. With constant marginal cost $c$ and unlimited capacity, the classical Bertrand argument shows the unique Nash equilibrium is $p_1=p_2=c$.

Reason (sketch): If a firm sets $p_j>c$, its rival can slightly undercut by setting $p_i=p_j-\varepsilon$ (for tiny $\varepsilon>0$) and capture nearly the entire demand at a price strictly above cost, securing positive profit — thus any $p_j>c$ is not a best response. If one firm sets $p=c$, the other cannot profitably set any $p'<c$ (loss) and any $p'>c$ is undercuttable. So $p_1=p_2=c$ is an equilibrium.

Concrete numeric example: demand $D(p)=100 - p$ ($A=100,B=1$) and $c=20$. If both set $p=20$, each gets half the demand $D(20)=80$, so quantity each $=40$ and profit per firm is $(20-20)\times 40=0$. Any $p>20$ invites undercutting and positive profit, any $p<20$ yields losses. Thus equilibrium price $p^B=20$.

This is the Bertrand paradox: with only two firms and no capacity constraints or product differentiation, price equals marginal cost — the competitive outcome. To get prices above $c$, one needs capacity constraints, product differentiation, dynamic considerations, or incomplete information.

Stackelberg (sequential quantities):

Stackelberg duopoly: firm 1 (leader) chooses output $q_1$ first; firm 2 (follower) observes $q_1$ and chooses $q_2$. The game is solved by backward induction. Use the same linear inverse demand $P=a-b(q_1+q_2)$ and marginal cost $c$.

Step 1 (follower's best response): As in Cournot, firm 2's best response to $q_1$ is

Numeric example: with $a=100$, $b=1$, $c=20$, we get $q_2(q_1)=40 - 0.5 q_1$.

Step 2 (leader anticipates this reaction): firm 1 chooses $q_1$ to maximize

Substitute $q_2(q_1)$:

Total output $Q = q_1 + q_2 = q_1 + \frac{a-c}{2b} - \frac{1}{2} q_1 = \frac{1}{2} q_1 + \frac{a-c}{2b}.$

So

Leader's profit:

FOC (maximize w.r.t. $q_1$):

Numeric substitution: with $a-c=80, b=1$, we get $q_1^* = 40$.

Follower's output at equilibrium:

Numeric: $q_2^* = 20$. Total output $Q^S = 60$, price $P^S = a - bQ^S = 100 - 60 = 40$. Profits: leader $\pi_1^S = (40-20)\times 40 = 800$; follower $\pi_2^S = (40-20)\times 20 = 400$.

Compare Cournot and Stackelberg (numbers):

• •Cournot: total $Q^C=53.333$, price $P^C=46.667$, each profit $\approx711.11$.
• •Stackelberg: total $Q^S=60$, price $P^S=40$, leader profit $800$, follower profit $400$.

Observations and intuition:

• •The leader produces more than a single Cournot firm ($(a-c)/(2b)$ vs $(a-c)/(3b)$) and captures a larger share; this is the first-mover advantage.
• •Total output under Stackelberg is higher and price lower than under Cournot—competition intensifies because the follower reacts to the leader's commitment
• •The leader's choice is equivalent to committing to a production level that incentivizes the follower to adopt a less aggressive response than if both moved simultaneously.

Ordering of outcomes (standard linear model):

Monopoly price: solving the monopoly problem for linear demand gives total monopoly output $Q^M = (a-c)/(2b)$ and price $P^M = (a+c)/2$. Numeric check: $Q^M=40$, $P^M=(100+20)/2=60$.

Bertrand with product differentiation or capacity constraints modifies the paradox: if capacity is limited, Bertrand need not drive price to marginal cost; with differentiated products, firms enjoy some price-setting power and equilibrium prices exceed $c$.

Technical note: Stackelberg equilibrium is a subgame-perfect Nash equilibrium: we used backward induction (solve follower's subgame then leader's problem). This leverages the sequential structure beyond static Nash.

Applications and Connections

Oligopoly models are central to applied industrial organization, antitrust analysis, and competitive strategy. They connect to regulatory policy, firm behavior, and dynamic strategic considerations.

1) Antitrust and merger analysis

Regulators use oligopoly models to assess how mergers affect prices and welfare. The Cournot model is often used as a benchmark: a merger that reduces the number of Cournot firms increases equilibrium price because each remaining firm's market power rises. For example, with linear demand $a=100,b=1,c=20$, reducing duopoly ($n=2$) to monopoly ($n=1$) moves total output from $53.333$ to $40$, raising price from $46.667$ to $60$ — a significant consumer harm. Practitioners estimate demand elasticities and marginal costs and compute counterfactual equilibrium outcomes.

2) Capacity planning and supply chains

If firms face capacity constraints, Bertrand competition may not deliver prices equal to marginal cost. Edgeworth cycles (no pure-strategy equilibrium) and mixed-strategy equilibria can arise. For example, if capacity is constrained so a firm cannot serve entire demand when undercutting, the incentive to undercut is limited and equilibrium prices can be above marginal cost.

3) Product differentiation and spatial competition

Differentiated-products models (Hotelling, Salop) convert price competition into outcomes where firms enjoy local market power; prices exceed marginal cost. These extensions bridge Bertrand and Cournot logic: as product differentiation increases, price outcomes move away from the Bertrand benchmark and resemble Cournot-like markup behavior.

4) Dynamic oligopoly, collusion, and repeated games

Repeated interaction allows firms to sustain collusive prices above the one-shot Nash outcomes. Using trigger strategies or Folk Theorem logic, oligopolists can sustain cooperative pricing if future punishment is credible. This requires understanding subgame-perfect equilibria in repeated games—a direct connection to the Nash Equilibrium prerequisite and to dynamic game theory.

5) Entry deterrence and limit pricing

A Stackelberg leader may choose a commitment (capacity or output) to deter entry. Limit pricing arguments model incumbent firms choosing prices or capacities to make entry unappealing. These incorporate strategic commitment and backward induction reasoning similar to Stackelberg analysis.

6) Empirical estimation and structural IO

Structural econometrics uses oligopoly models to estimate demand and cost parameters from observed prices and quantities, then simulates counterfactuals (e.g., the price after a merger). Cournot, Bertrand, and differentiated-product Bertrand models are common structural specifications.

Connections to prerequisites (looking back):

• •In Nash Equilibrium, we studied best-responses and fixed points; Cournot and Bertrand equilibria are explicit applications where each firm's strategy must be a best response.
• •From Profit Maximization, we used MR=MC and second-order conditions to derive best responses and ensure interior maxima.

What this enables (looking forward):

• •Dynamic oligopoly and repeated games: sustaining collusion, trigger strategies, and subgame-perfect equilibrium refinement.
• •Entry/exit models and real options: how firms invest and commit capacity over time.
• •Industrial organization empirical work: using structural models to quantify market power and policy effects.

Practical note for model selection: pick Cournot when firms compete on capacity/quantity (e.g., commodity producers), Bertrand for price-setting (retail), Stackelberg when leadership/commitment or sequential moves matter (a dominant firm sets capacity before smaller rivals). Always test sensitivity: small modeling choices (timing, capacity, product differentiation) materially affect policy-relevant predictions.

In the exercises and worked examples that follow we will compute numerical equilibria for these models, cementing the algebraic derivations above and making clear the economic intuition behind comparative static results.