Solution
How Many Firms Enter
Show the problem again
Identical firms may each pay a fixed entry cost of 50 to produce a homogeneous good with zero marginal cost. The inverse demand is P = 100 − Q. Once n firms have entered, they play the Cournot quantity game. Entry continues as long as one more firm would still earn a non-negative profit net of the fixed cost. How many firms enter?
Worked solution
With n firms and zero marginal cost, Cournot output is 100/(n + 1) per firm and the price is also 100/(n + 1), so gross profit per firm is (100/(n + 1))². A firm covers its fixed cost when (100/(n + 1))² ≥ 50, that is n + 1 ≤ 100/√50 ≈ 14.14. The largest integer n is 13, giving profit (100/14)² ≈ 51.02 ≥ 50, while 14 firms would each earn (100/15)² ≈ 44.4 < 50. So 13 firms enter.
Source: Original problem written for AxiomIQ, in the spirit of Cournot free-entry models.