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Industrial Organization & Antitrust Economics

Cournot Oligopoly Nash Price and Cowling-Waterson HHI Lerner Formula

Solve N = 3 asymmetric Cournot-Nash equilibrium price P* = (a + SUM c_i) / (N + 1), firm outputs q_i = (P* - c_i) / b, and verify Cowling and Waterson’s HHI / |e_D| markup identity.

Connect Cournot quantity competition directly to antitrust HHI / |e_D| markups

Under linear inverse demand P(Q) = a - b * Q across N = 3 quantity-setting firms with marginal costs (c_1, c_2, c_3), interior Cournot-Nash equilibrium sets market price to P* = (a + c_1 + c_2 + c_3) / 4 and each firm’s output to q_i* = (P* - c_i) / b! Even more remarkably, Cowling and Waterson (1976) proved that the share-weighted industry Lerner markup SUM(s_i * (P* - c_i) / P*) equals exact decimal HHI divided by demand elasticity magnitude |e_D|!

Verify on choke price a = 100 (B2), slope b = 1 (B3), and asymmetric unit costs c_1 = 16 (B4), c_2 = 20 (B5), c_3 = 24 (B6): equilibrium price is P* = (100 + 60) / 4 = 40.0000, outputs are {24, 20, 16} totaling Q* = 60.0000, decimal HHI is (24^2 + 20^2 + 16^2) / 3600 = 0.3422, market elasticity at P* is |e_D| = 40 / 60 = 0.6667, and industry share-weighted Lerner markup equals HHI / |e_D| = 77 / 150 = 0.5133!

Cournot-Nash Quantity (a=100, b=1, c={16,20,24})FormulaVerified Output
Equilibrium Market Price (P*)(a + c_1 + c_2 + c_3) / 440.0000
Firm Outputs {q_1, q_2, q_3} & Total Q*(P* - c_i) / b{24, 20, 16} -> Q* = 60
Decimal Herfindahl Index (HHI_dec)SUM((q_i / Q*)^2)0.3422 (3,422 pts)
Cowling-Waterson Weighted LernerHHI_dec / (P* / (b * Q*))0.5133 (51.33% Margin)

Interior 3-Firm Cournot-Nash Equilibrium Price P* Formula (with Exit Guard)

Check MAX(c_1, c_2, c_3) < P*! If a high-cost rival has c_i >= interior P*, that firm shuts down (q_i = 0) and equilibrium must be re-solved over active firms only.

=LET(a_ch, B2, b_sl, B3, c_1, B4, c_2, B5, c_3, B6, IF(OR(a_ch <= 0, b_sl <= 0, c_1 < 0, c_2 < 0, c_3 < 0), "Invalid Cournot Input", LET(p_eq, (a_ch + c_1 + c_2 + c_3) / 4, IF(MAX(c_1, c_2, c_3) >= p_eq, "Corner Exit: Max c_i >= Interior Price", ROUND(p_eq, 4)))))

Cowling-Waterson Industry Share-Weighted Lerner Index Formula

Compute SUM(q_i * (P* - c_i)) / (Q* * P*) and verify that symmetric costs c = {20, 20, 20} collapse cleanly to (40 - 20) / 40 = 0.5000.

=LET(a_ch, B2, b_sl, B3, c_1, B4, c_2, B5, c_3, B6, p_eq, (a_ch + c_1 + c_2 + c_3) / 4, IF(OR(a_ch <= 0, b_sl <= 0, MAX(c_1, c_2, c_3) >= p_eq), "Corner Exit: Max c_i >= Interior Price", LET(q_1, (p_eq - c_1) / b_sl, q_2, (p_eq - c_2) / b_sl, q_3, (p_eq - c_3) / b_sl, q_tot, q_1 + q_2 + q_3, ROUND((q_1 * (p_eq - c_1) + q_2 * (p_eq - c_2) + q_3 * (p_eq - c_3)) / (q_tot * p_eq), 4))))

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