Antitrust Economics & M&A Strategy
Cournot Merger Paradox & 80% Threshold Formula
Prove Salant, Switzer & Reynolds’s (1983) Cournot Merger Paradox: without cost synergies, merging M of N symmetric Cournot rivals loses money unless M >= 80% of the industry!
Why horizontal Cournot mergers without cost synergies enrich non-merging rivals and lose money!
Stephen Salant, Sheldon Switzer, and Robert Reynolds (1983) stunned antitrust economists by proving the Cournot Merger Paradox: in an N-firm symmetric linear Cournot industry (P = a - b * Q, unit cost c), if M >= 2 firms merge without marginal-cost synergies, the post-merger industry simply becomes a symmetric (N - M + 1)-firm Cournot oligopoly! Pre-merger, the M insiders earned M * (a - c)^2 / (b * (N + 1)^2); post-merger, as a SINGLE combined Cournot player, they earn (a - c)^2 / (b * (N - M + 2)^2) because non-merging outsiders aggressively expand output! Solving 1 / (N - M + 2)^2 >= M / (N + 1)^2 reveals that a synergy-free Cournot merger is UNPROFITABLE unless M captures at least ~80% of the industry!
Check by hand with Choke_a = 140 (B2), Slope_b = 1 (B3), Cost_c = 20 (B4 -> a - c = 120), Initial_Firms_N = 5 (B5), and Merging_Firms_M = 2 (B6): pre-merger, each firm earns 14,400 / 36 = 400.00, so the 2 merging insiders earn 800.00 combined. Post-merger (4 firms remain), the merged firm earns ONLY 14,400 / 25 = 576.00 — LOSING -224.00 (-28.00%!) while each non-merging outsider jumps +44.00% from 400.00 to 576.00! Now test M = 4 of 5 firms (80% share): post-merger duopoly earns 14,400 / 9 = 1,600.00 = 4 * 400.00 (exact 0.00 breakeven!)!
| Merger Scenario (N = 5, a - c = 120, b = 1) | Combined Pre-Merger Profit | Post-Merger Profit & Net Change |
|---|---|---|
| 2-Firm Merger (M = 2, 40% Share) | 800.00 (2 x 400) | 576.00 (-224.00 Merger Paradox Loss!) |
| 3-Firm Merger (M = 3, 60% Share) | 1,200.00 (3 x 400) | 900.00 (-300.00 Loss!) |
| 4-Firm Merger (M = 4, 80% Threshold!) | 1,600.00 (4 x 400) | 1,600.00 (0.00 Exact Breakeven!) |
| Merger to Monopoly (M = 5, 100%) | 2,000.00 (5 x 400) | 3,600.00 (+1,600.00 Gain) |
Net Merging-Insiders Profit Change (Pi_post - M * Pi_pre) Formula
Validate a > c >= 0, b > 0, and integer firm counts 2 <= M <= N.
=LET(a_ch, B2, b_sl, B3, c_mc, B4, n_in, B5, m_mg, B6, IF(OR(a_ch <= c_mc, c_mc < 0, b_sl <= 0, n_in < 2, m_mg < 2, m_mg > n_in, INT(n_in) <> n_in, INT(m_mg) <> m_mg), "Require Integer 2 <= M <= N and a > c", LET(s_sq, ((a_ch - c_mc)^2) / b_sl, pi_pre, m_mg * s_sq / ((n_in + 1)^2), pi_pst, s_sq / ((n_in - m_mg + 2)^2), ROUND(pi_pst - pi_pre, 4))))Free-Riding Outsider Windfall Gain per Non-Merging Rival Formula
Every non-merging competitor enjoys the price hike without restricting its own plants — earning s_sq / (N - M + 2)^2 - s_sq / (N + 1)^2 (zero when M = N).
=LET(a_ch, B2, b_sl, B3, c_mc, B4, n_in, B5, m_mg, B6, IF(OR(a_ch <= c_mc, c_mc < 0, b_sl <= 0, n_in < 2, m_mg < 2, m_mg > n_in, INT(n_in) <> n_in, INT(m_mg) <> m_mg), "Require Integer 2 <= M <= N and a > c", IF(m_mg = n_in, 0, LET(s_sq, ((a_ch - c_mc)^2) / b_sl, ROUND((s_sq / ((n_in - m_mg + 2)^2)) - (s_sq / ((n_in + 1)^2)), 4)))))Try the example
Use the original values or change them to test your understanding.