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Two producers supply the same market. They never meet, never agree, and each wants to earn as much as possible. Left alone they could split the monopoly profit. They will not manage it, and the reason is worth watching.

Cournot and Bertrand

Give two firms the same product and one market, and how they compete depends on what they choose. Set quantities, as in Cournot, and they land between monopoly and perfect competition. Set prices, as in Bertrand, and a single undercut drives profit to zero.

From earlier lessons

Two quick ones before we start.

A best response is the move that pays the most...

A firm changes its output while its rival holds output fixed. Which kind of change is this?

Predict, then play

Demand is $P = 120 - Q$ , where $Q$ is total output, and production is free. Set your quantity; the rival best-responds. The two reaction lines cross at the equilibrium. Then flip to price competition and watch what happens.

Why did that happen?

Take a guess, then reveal the reason

Each firm treats the other’s output as fixed and asks: given that, how much should I make? The answer is always a bit more than the cooperative share, because expanding steals revenue from the rival while only slightly lowering the price you get. Both reason this way, so both expand, and the extra output drags the price down for everyone. Under price competition the same logic is sharper: undercutting the rival by a penny wins the entire market, so both keep cutting until price equals cost.

The math

Firm 1 chooses $q_1$ to maximize profit $(120 - q_1 - q_2)\,q_1$ , taking $q_2$ as given. The first-order condition is

120 - 2q_1 - q_2 = 0 \quad\Rightarrow\quad q_1 = \frac{120 - q_2}{2}.

That is firm 1’s reaction curve. By symmetry firm 2’s is $q_2 = (120 - q_1)/2$ . Solving both, or imposing $q_1 = q_2$ , gives

q_1 = q_2 = 40, \quad Q = 80, \quad P = 40, \quad \pi_1 = \pi_2 = 1600.

A monopolist would set $Q = 60$ , $P = 60$ , and earn $3600$ . The duopoly produces more and earns less in total, and as you add firms the price keeps falling toward marginal cost.

Definition, the Bertrand paradox

If the same two firms compete on price with identical products and equal marginal cost $c$ , the unique Nash equilibrium is $p_1 = p_2 = c$ , with zero profit. Any price above cost invites the rival to undercut and take the whole market, so competition collapses margins entirely, even with just two firms. Differentiated products or capacity limits restore positive margins; the paradox is the homogeneous-goods benchmark.

Work it out

Solve Cournot, then change the cost

Firm 1 maximizes $(120 - q_1 - q_2) q_1$ . Differentiate in $q_1$ and set to zero: $120 - 2q_1 - q_2 = 0$ . Impose symmetry $q_1 = q_2 = q$ : $120 - 3q = 0$ , so $q = 40$ and total output is $80$ , price $40$ , profit $40 \times 40 = 1600$ each.

Faded. Now let marginal cost be $c = 30$ . The condition becomes $120 - 2q_1 - q_2 - 30 = 0$ , and with symmetry $90 - 3q = 0$ , so $q = 30$ each. Higher cost, lower output.

Solo. In the commitment lesson, three ahead, one firm gets to move first. Guess now: does committing to a large quantity help or hurt the leader?

Key idea

Competition on quantity lands between monopoly and cost; competition on price, with identical goods, drives all the way to cost. What firms choose to compete over shapes the outcome as much as how many of them there are.

Check yourself

Check yourselfapply

At the Cournot equilibrium each firm makes 40, for a total of 80. A monopolist would make 60. What does the comparison show?

Check yourselfanalyse

Two firms sell an identical product and compete on price, with the same marginal cost. What is the Nash equilibrium?

Check yourselfbrings back: best response

A firm's reaction curve slopes downward: the more the rival produces, the less it makes. Why?

Sources

Cournot, A. A. (1838). Recherches sur les principes mathematiques de la theorie des richesses. Hachette.
Bertrand, J. (1883). Review of "Theorie mathematique de la richesse sociale". Journal des Savants, 67, 499-508.
Gibbons, R. (1992). Game Theory for Applied Economists. Princeton University Press.