Strategic Pricing and Price Formation

Strategic pricing studies how firms set prices or quantities when each firm’s profit depends on the choices of its rivals; the standard models are Cournot (competition in quantities), Bertrand (competition in prices) and Stackelberg (a leader moves first), and in each the equilibrium is a Nash equilibrium of a game between the firms. In brief: with the demand price P = 100 − Q and a unit cost of 10, a monopolist sells 45 units at 55 and earns 2025. Two Cournot competitors each produce 30 units, the price falls to 40 and each earns 900. In Bertrand price competition the price falls to the cost, 10, and profits vanish.

Prices in a market are formed by the interaction of supply and demand, and when only a few firms compete, by their strategic reactions to one another. This guide starts with the market-clearing price, then solves the Cournot, Stackelberg and Bertrand models with numbers, shows when price differentiation softens competition and how repeated interaction can sustain high prices, with worked examples and a table.

How is a price formed in a competitive market?

The demand curve gives the quantity buyers want at each price, falling with the price, and the supply curve gives the quantity sellers offer, rising with the price. The equilibrium price clears the market: the quantity demanded equals the quantity supplied. Finding it means solving a pair of linear equations.

Example 1: a market-clearing price

Demand: Q_d = 100 − p. Supply: Q_s = p − 20. Setting 100 − p = p − 20 gives p = 60 and Q = 40. If the price is held at 50, demand is 50 but supply only 30: a shortage of 20, which pushes the price up. At 70, supply (50) exceeds demand (30): a surplus that pushes the price down.

What does a monopolist charge?

A single seller faces the inverse demand P = 100 − Q and has the unit cost c = 10. The profit is (100 − Q − 10)Q = (90 − Q)Q, maximal where the derivative 90 − 2Q = 0, that is Q = 45. The price is 55 and the profit is (55 − 10) · 45 = 2025. A perfectly competitive market would instead end up at P = c = 10 and Q = 90, with zero profit.

What is the Cournot model?

Two firms choose quantities q₁ and q₂ simultaneously; the price is P = 100 − q₁ − q₂ and each has the cost 10 per unit. Firm 1 maximizes (90 − q₁ − q₂)q₁: the derivative 90 − 2q₁ − q₂ = 0 gives the best response q₁ = (90 − q₂)/2, and symmetrically for firm 2. The Nash equilibrium is where both best responses hold: q = (90 − q)/2, so 3q = 90 and q = 30.

Example 2: the Cournot outcome

Each firm produces 30, the total is 60, the price is 100 − 60 = 40 and each profit is (40 − 10) · 30 = 900. The joint profit 1800 is lower than the monopoly profit 2025, but each firm earns more than zero. The monopoly output 45 would split as 22.5 each with the profit 1012.5 each, higher than 900, but this is not an equilibrium: given 22.5 from the rival, a firm’s best response is (90 − 22.5)/2 = 33.75, so it is tempted to cheat. This is a prisoner’s dilemma in quantities.

Example 3: more firms

With n firms the Cournot quantity per firm is 90/(n + 1), the total 90n/(n + 1), the price 10 + 90/(n + 1). For n = 1 this is the monopoly (45, price 55); n = 2 gives the price 40; n = 9 gives 19; as n grows the price approaches the cost 10, the competitive outcome.

What changes when one firm moves first?

In the Stackelberg model, a leader commits to a quantity first, and the follower observes it and responds. The leader anticipates the follower’s best response q₂ = (90 − q₁)/2 and maximizes q₁(90 − q₁ − (90 − q₁)/2) = q₁(45 − q₁/2). The derivative 45 − q₁ = 0 gives q₁ = 45.

Example 4: the first-mover advantage

The leader produces 45, the follower (90 − 45)/2 = 22.5, the total is 67.5, and the price is 32.5. The leader earns (32.5 − 10) · 45 = 1012.5, the follower 22.5 · 22.5 = 506.25. Moving first is valuable (1012.5 versus 900 in the Cournot game), and the price is lower than in Cournot, so consumers benefit.

What is the Bertrand model?

When firms with identical products and the same cost c choose prices, a firm charging less than the rival captures the whole market. If both set a price above c, each has an incentive to undercut slightly. The only equilibrium is p₁ = p₂ = c, with zero profit: two firms suffice to destroy all margin (the Bertrand paradox). The result contrasts with Cournot, which shows that the strategic variable matters.

Example 5: undercutting

With c = 10 and both charging 40, each sells half of the demand 60 and earns 30 · 30 = 900. A firm that lowers its price to 39.9 takes the whole demand (60.1) and earns 29.9 · 60.1 ≈ 1797: nearly twice as much. So 40 is not an equilibrium. At a price of 10, no firm gains by undercutting (it would sell at a loss) or by raising its price (it would lose all its customers), so (10, 10) is the equilibrium.

What if the products are differentiated?

When products differ in quality, brand or location, a small price cut does not capture the whole market, and prices stay above cost. Suppose the demand for firm i is q_i = 60 − 2p_i + p_j with a zero cost. The profit p_i(60 − 2p_i + p_j) is maximal where 60 − 4p_i + p_j = 0, so p_i = (60 + p_j)/4.

Example 6: differentiated Bertrand

In the symmetric equilibrium, p = (60 + p)/4, 3p = 60, p = 20. Then q = 60 − 40 + 20 = 40 and the profit of each firm is 20 · 40 = 800. The best response rises with the rival’s price (prices are strategic complements): if the rival raises its price, the firm raises its own.

Can firms sustain high prices?

In a one-shot game, firms cannot sustain the monopoly outcome. In a repeated game, they can if they are patient enough: a firm that cheats today is punished by the Cournot outcome forever (a grim trigger strategy).

Example 7: sustaining collusion in the Cournot market

Collusion pays 1012.5 per period to each firm. Deviating (producing 33.75 against the rival’s 22.5) pays (100 − 56.25 − 10) · 33.75 = 33.75² ≈ 1139.06 once, and then the punishment 900 forever. With the discount factor δ, collusion is stable if 1012.5/(1 − δ) ≥ 1139.06 + 900δ/(1 − δ), which simplifies to 1012.5 ≥ 1139.06 − 239.06δ, so δ ≥ 126.56/239.06 ≈ 0.53. Firms valuing the future enough (δ above about 0.53) can keep prices high without any written agreement, which is why price-fixing is combated by law and why the number of firms and the speed of detection matter.

Table of market outcomes

Where do these models come from?

Antoine Augustin Cournot published the quantity model in 1838. Joseph Bertrand criticized it in 1883 and proposed price competition, and Heinrich von Stackelberg studied the leader-follower structure in 1934. Harold Hotelling analyzed competition with differentiated locations in 1929. The formal connection with Nash equilibrium was made after 1950, and the study of collusion in repeated games was developed in the 1970s and 1980s, including by Jean Tirole, who received the 2014 Nobel Prize in economics for his work on market power and regulation.

Which mistakes are common?

Using the monopoly formula for competing firms

With rivals, each firm’s best response depends on the others’ quantities. Maximize against the rival’s choice, not against the whole market demand.

Forgetting the cost in the profit

Profit is (price − unit cost) × quantity. Using the price alone shifts every optimum.

Treating Cournot and Bertrand as interchangeable

Competition in quantities leaves positive profits, competition in prices with identical goods leaves none. The result depends on what the firms choose and on whether the goods are differentiated.

Assuming that collusion needs an agreement

Tacit collusion can arise from repeated interaction alone, if the firms are patient and deviations are detected quickly.