Solution
The Unprofitable Merger
Show the problem again
Three identical firms with zero marginal cost compete in Cournot fashion with inverse demand P = 100 − Q. Two of them merge into a single firm that is otherwise identical (still zero cost, and it chooses one quantity), leaving a duopoly. By how much does the combined profit of the two merging firms fall as a result of the merger? Give the size of the fall as a reduced fraction.
Worked solution
With three firms, each produces 25, P = 25 and each earns 625, so the two future partners together earn 1250. After the merger there are two Cournot players, each producing 100/3 at price 100/3, and each earns (100/3)² = 10000/9. The merged firm earns 10000/9 ≈ 1111.1. The combined profit falls by 1250 − 10000/9 = 1250/9 ≈ 138.9. The merger lifts the price but the outsider expands its output and captures the gain, so the merging firms lose; mergers need a large enough share of the industry to pay.
Source: Salant, Switzer and Reynolds, 'Losses from Horizontal Merger' (1983). Statement written for AxiomIQ.