Strategic Competition & Industrial Organization
Dixit-Spence Capacity Commitment & Entry Deterrence
Calculate Avinash Dixit’s (1980) entry-deterring limit capacity K_det = (a - w - r - 2*SQRT(b*F)) / b and audit subgame-perfect credibility against K_max = (a - w + r) / (3*b).
Distinguish credible sunk-capacity deterrence from empty non-credible threats!
Sylos-Labini and Bain assumed an incumbent could deter entry simply by threatening to hold output high after entry — a bluff Reinhard Selten’s subgame perfection exposed unless the incumbent alters its OWN post-entry marginal cost! Avinash Dixit (1980), building on Michael Spence (1977), showed how irreversible pre-committed plant capacity K_1 (costing r per unit upfront, leaving only operating marginal cost w post-entry under demand P = a - b*(q_1 + q_2)) makes aggressive post-entry output credible! Specifically, Entrant 2 (who still faces full unit cost w + r plus fixed entry cost F > 0) is deterred at Limit Capacity K_det = ((a - w - r) - 2 * SQRT(b * F)) / b — PROVIDED K_det does not exceed Dixit’s Maximum Credible Threat Ceiling K_max = (a - w + r) / (3 * b) (beyond K_max, even with sunk capacity, the incumbent would voluntarilyleave excess capacity idle if entered!)!
Verify on capital-intensive Demand_Intercept a = 120 (B2), Slope b = 1 (B3), Operating_Cost w = 15 (B4), Sunk_Capacity_Cost r = 45 (B5 -> full marginal cost w + r = 60, Pure Monopoly Q_m = 30.00), and Entrant_Fixed_Cost F = 100 (B6 -> 2 * SQRT(100) = 20): Required Entry-Deterring Capacity is K_det = (120 - 60 - 20) / 1 = 40.00 units (> Monopoly 30.00); Maximum Credible Capacity Ceiling is K_max = (120 - 15 + 45) / 3 = 50.00 units; since 30.00 < 40.00 <= 50.00, strategic over-investment is 100% Subgame-Perfect ("Credible Deterrence"), yielding Limit Price 120 - 40 = 80.00 and Incumbent Profit (80 - 60) * 40 = 800.00! Now test smaller F = 16: K_det jumps to 52.00 > K_max = 50.00 -> "Non-Credible Deterrence (Accommodate)"!
| Dixit Capacity Metric (a=120, b=1, w=15, r=45, F=100) | Formula | Verified Output |
|---|---|---|
| Required Entry-Deterring Capacity (K_det) | ((a - w - r) - 2*SQRT(b*F)) / b | 40.00 units (vs Monopoly 30) |
| Dixit Max Credible Threat Ceiling (K_max) | (a - w + r) / (3 * b) | 50.00 units (>= 40.00!) |
| Credible Deterrence Incumbent Profit | (a - b*K_det - w - r) * K_det | 800.00 (at P = 80.00) |
| Regime Audit if F Drops to 16 (K_det = 52) | IF(K_det <= K_max, ...) | Non-Credible Deterrence (Accommodate) |
Dixit Entry-Deterring Limit Capacity K_det Formula
Validate a > w + r, b > 0, w >= 0, r > 0, and F > 0 (when 2*SQRT(b*F) >= a - w - r, entry is already Blockaded even at pure monopoly output!).
=LET(a_dm, B2, b_sl, B3, w_op, B4, r_cp, B5, f_en, B6, IF(OR(b_sl <= 0, w_op < 0, r_cp <= 0, a_dm <= w_op + r_cp, f_en <= 0), "Invalid Dixit Input", ROUND(MAX(0, ((a_dm - w_op - r_cp) - 2 * SQRT(b_sl * f_en)) / b_sl), 2)))Subgame-Perfect Credibility Regime Audit Formula
Distinguish Bain’s three classic regimes rigorously: Blockaded Entry (K_det <= Monopoly Q_m = (a - w - r)/(2*b)), Credible Deterrence (Q_m < K_det <= K_max), and Non-Credible Deterrence / Accommodation (K_det > K_max)!
=LET(a_dm, B2, b_sl, B3, w_op, B4, r_cp, B5, f_en, B6, IF(OR(b_sl <= 0, w_op < 0, r_cp <= 0, a_dm <= w_op + r_cp, f_en <= 0), "Invalid Dixit Input", LET(k_dt, ((a_dm - w_op - r_cp) - 2 * SQRT(b_sl * f_en)) / b_sl, q_mn, (a_dm - w_op - r_cp) / (2 * b_sl), k_mx, (a_dm - w_op + r_cp) / (3 * b_sl), IFS(k_dt <= q_mn, "Blockaded Entry", k_dt <= k_mx, "Credible Deterrence", TRUE, "Non-Credible Deterrence (Accommodate)"))))Try the example
Use the original values or change them to test your understanding.