Industrial Organization & Game Theory
Milgrom-Roberts Limit-Pricing Separating Formula
Calculate Paul Milgrom and John Roberts’s (1982) least-cost separating limit quantity Q_sep = MAX(Q_m(L), Q_m(H) + SQRT(Delta_H / b)), limit price P_sep, and signaling profit distortion.
Understand why a low-cost incumbent cuts Period-1 price below monopoly to signal efficiency!
Under complete information, cutting Period-1 price before entry cannot deter a rational entrant because pre-entry prices do not alter post-entry marginal costs! Paul Milgrom and John Roberts (1982) solved the limit-pricing puzzle via asymmetric information: when an entrant does not observe whether the incumbent has Low Marginal Cost c_L or High Marginal Cost c_H > c_L under linear inverse demand P(Q) = A - b * Q, a High-Cost incumbent earns Deterrence Prize Delta_H (present value of Monopoly minus Duopoly Period-2 profit) if mistaken for Low-Cost!
Verify on Demand_Intercept A = 120 (B2), Low_Cost c_L = 20 (B3 -> Q_m(L) = (120 - 20)/2 = 50, P_m(L) = 70), High_Cost c_H = 40 (B4 -> Q_m(H) = (120 - 40)/2 = 40, P_m(H) = 80), Slope b = 1 (B5), and High_Type_Deterrence_Gain Delta_H = 324 (B6): Because High-Cost’s Period-1 profit loss from expanding output from Q_m(H) = 40 to Q equals exact quadratic b * (Q - Q_m(H))^2, producing Q_m(L) = 50 costs High-Cost only 1 * (50 - 40)^2 = 100 < 324 (so High-Cost would bluff)! Under Cho-Kreps Intuitive Criterion, the Least-Cost Separating Quantity expands to Q_sep = MAX(Q_m(L), Q_m(H) + SQRT(Delta_H / b)) = MAX(50, 40 + SQRT(324)) = 58 units, driving Separating Limit Price down to P_sep = 120 - 58 = 62 (8 below monopoly price 70) at Period-1 Signaling Cost b * (58 - 50)^2 = 64 for the Low-Cost incumbent!
| Limit-Pricing Signaling Metric (A=120, c_L=20, c_H=40, b=1, Delta_H=324) | Formula | Verified Output |
|---|---|---|
| High-Cost Monopoly Output Q_m(H) | (A - c_H) / (2 * b) | 40.00 (P_m(H) = 80.00) |
| Low-Cost Monopoly Output Q_m(L) | (A - c_L) / (2 * b) | 50.00 (P_m(L) = 70.00) |
| Least-Cost Separating Quantity (Q_sep) | MAX(Q_m(L), Q_m(H) + SQRT(Delta_H / b)) | 58.00 units |
| Separating Limit Price & Signal Cost | P_sep = A - b*Q_sep; Loss = b*(Q_sep - Q_m(L))^2 | P_sep = 62.00; Cost = 64.00 |
Least-Cost Separating Limit Quantity Q_sep Formula
Test Delta_H = 64 in cell B6: Q_m(H) + SQRT(64 / 1) = 48 <= Q_m(L) = 50, so Low-Cost separates costlessly at its unconstrained monopoly output Q_sep = 50 (Signaling Distortion = 0)!
=LET(a_dm, B2, cl_mc, B3, ch_mc, B4, b_sl, B5, dl_h, B6, IF(OR(b_sl <= 0, cl_mc < 0, ch_mc <= cl_mc, a_dm <= ch_mc, dl_h < 0), "Invalid Limit-Price Input", LET(qm_l, (a_dm - cl_mc) / (2 * b_sl), qm_h, (a_dm - ch_mc) / (2 * b_sl), ROUND(MAX(qm_l, qm_h + SQRT(dl_h / b_sl)), 4))))Separating Limit Price & Low-Cost Period-1 Signaling Distortion Formula
Validate A > c_H > c_L >= 0, b > 0, and Delta_H >= 0.
=IF(ISTEXT(E4), E4, ROUND(B5 * ((E4 - E3) ^ 2), 4))Try the example
Use the original values or change them to test your understanding.