Two Firms Watching Each Other

In a town with only two cola firms, each firm sizes up the other and picks its output.

The price is set by the two firms' combined output. Once the rival picks a quantity, a reply rule gives your best answer. The market settles where the two replies cross. That reply rule is the formula below.

For example, only two booths sell 🍬 candy at the school fair. If the other booth puts out 10 📦 boxes, your share shrinks. If it puts out fewer boxes, you can add more without sinking the price.

P=a−b(q1+q2)P = a - b(q_1 + q_2)
q1=a−c−bq22bq_1 = \frac{a - c - b q_2}{2b}

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Examples & Exercises

Only two cola firms exist.

The market price is P=12−(q1+q2)P = 12 - (q_1 + q_2) and a bottle costs 33 to make.

If firm 2 produces 33, what is firm 1's best move — and why do both stop there?

Firm 2 makes 33, market size 1212 — right where the replies cross.
Show proof

  1. Firm 1's reply rule is q1=12−3−q22q_1 = \frac{12 - 3 - q_2}{2}. Plugging in q2=3q_2 = 3 gives q1=3q_1 = 3 — matching the rival is best.

  2. The total is 66 and the price is P=12−6=6P = 12 - 6 = 6.

    Checking firm 2's reply rule at this point also gives q2=3q_2 = 3 — each is answering the other perfectly, so nobody wants to move.

    That is the Cournot equilibrium.

In the same market, firm 2 suddenly floods production up to 66.

How should firm 1 respond, and what happens to the market?

Firm 2 makes 66 — firm 1 is pushed along its own reply line.
Show proof

  1. The reply rule with q2=6q_2 = 6 gives q1=12−3−62=1.5q_1 = \frac{12 - 3 - 6}{2} = 1.5. When the rival floods the market, pulling back is best — insisting on the old quantity would crash the price further.

  2. Even so, the total rises to 7.57.5 and the price falls to 4.54.5.

    Firm 1's profit is (4.5−3)×1.5=2.25(4.5 - 3) \times 1.5 = 2.25 — miserable next to the 99 it earned at the equilibrium.

    Losing the watching game is expensive.

1. Intro to Microeconomics