Cournot Competition had firms choose quantities. If instead they choose prices, the answer changes dramatically, at least when the products are identical.
Key idea
Two firms selling identical goods with the same constant marginal cost and no capacity limits, choosing prices at the same time, end up with $$p = MC$$ and zero profit (the Bertrand paradox). With different costs $c_1 < c_2$, the low-cost firm prices just below $c_2$ and serves the whole market. With differentiated products, prices stay above cost.
Why it works
Suppose both charge the same price above marginal cost and split the market. Either firm could cut its price by a cent and take the whole market, nearly doubling its profit. So no price above cost is stable. Below cost, a firm loses money on every sale and would rather raise its price. Only $p = MC$ survives. If products differ, a small price cut steals only some customers, and best responses settle above cost: with demand $q_i = A - p_i + e\,p_j$ (where $0 < e < 1$), firm $i$'s best response is $p_i = \dfrac{A + c + e\,p_j}{2}$.
Worked example: Identical goods
Two firms sell identical cement with $MC = 40$.
- At any common price above 40, undercutting by a cent takes the whole market.
- Equilibrium: $p = 40$, zero profit.
Answer: \$40.
Worked example: Different costs
$Q = 100 - P$, $c_1 = 20$, $c_2 = 30$.
- Firm 2 won't sell below 30. Firm 1 can price at \$29.99, serve the market and earn almost $(30 - 20)(70) = 700$.
- Firm 1's monopoly price would be $(100 + 20)/2 = 60$, above 30, so it doesn't want to go higher than 30 either: it would lose the market to firm 2.
Answer: Price about \$30; firm 1 sells about 70.
Worked example: Differentiated products
$q_i = 35 - p_i + \tfrac{1}{2}p_j$, $c = 10$.
- Firm $i$ maximizes $(p_i - 10)(35 - p_i + \tfrac{1}{2}p_j)$: $p_i = \dfrac{45 + \tfrac{1}{2}p_j}{2}$.
- Symmetric: $2p = 45 + \tfrac{1}{2}p$, so $p = 30$.
Answer: \$30.
Common mistake
Expecting two firms to keep prices well above cost, as in Cournot. With identical products, price competition, equal costs and no capacity limits, undercutting continues until $P = MC$. Check which assumption a real market breaks before predicting high prices: products that differ, capacity limits, or repeated play (see Repeated Games and Cooperation) can all keep prices up.
Capacity matters: if firms first choose how much capacity to build and then compete on price, the outcome can look like Cournot (a result due to David Kreps and José Scheinkman, 1983). Cournot and Bertrand are two ends of a range, and which fits depends on the industry.
See both models side by side in Competition: Cournot, Bertrand and Hotelling.