Industrial Organization & Supply Chain Pricing
Spengler Double Marginalization & Channel Fix Formula
Quantify Joseph Spengler’s (1950) vertical double-marginalization profit loss (25%) and consumer-surplus loss (75%) and solve the wholesale w* = c + franchise fee channel fix.
Why two successive monopolies hurt BOTH consumers AND their own combined profit!
Joseph Spengler (1950) uncovered one of the most counterintuitive welfare results in antitrust and supply-chain economics: when an upstream manufacturer (unit cost c) sells at linear wholesale price w to an independent downstream monopoly retailer facing P(Q) = a - b * Q, the retailer marks up ON TOP of the manufacturer’s markup! Specifically, upstream sets w_DM = (a + c) / 2, downstream sets retail price P_DM = (3 * a + c) / 4 and output Q_DM = (a - c) / (4 * b), and combined channel profit shrinks to (3 / 16) * (a - c)^2 / b — strictly 25% LOWER than integrated / two-part franchise profit (1 / 4) * (a - c)^2 / b at P_int = (a + c) / 2 and Q_int = (a - c) / (2 * b), while consumer surplus plunges 75%!
Verify on Choke_Price_a = 100 (B2), Demand_Slope_b = 1 (B3), and Unit_Cost_c = 20 (B4): Integrated / Coordinated Channel sets wholesale w* = c = 20 with fixed franchise fee F* = 1,600, selling Q_int = 40 at P_int = 60 for Total Channel Profit 1,600.00 and Consumer Surplus 800.00. Uncoordinated Double Marginalization sets w_DM = 60, P_DM = 80, Q_DM = 20, Upstream Profit 800 + Downstream Profit 400 = 1,200.00 (-400.00 channel profit lost!) and Consumer Surplus 200.00 (-600.00 consumer surplus lost -> 1,000.00 total welfare gained by coordination!)!
| Vertical Channel Regime (a = 100, b = 1, c = 20) | Retail Price P & Quantity Q | Combined Profit & Consumer Surplus |
|---|---|---|
| Uncoordinated Double Marginalization | w = 60, P = 80.00, Q = 20.00 | Profit = 1,200.00 | CS = 200.00 |
| Integrated / Two-Part Franchise (w* = c) | w* = 20, P = 60.00, Q = 40.00 | Profit = 1,600.00 | CS = 800.00 |
| Channel Coordination Gain | Price -20.00, Output +20.00 (+100%) | +400.00 Profit | +600.00 CS (+1,000 Total!) |
Uncoordinated Double-Marginalization Combined Channel Profit Formula
Validate choke demand a > c >= 0 and positive slope b > 0.
=LET(a_ch, B2, b_sl, B3, c_mc, B4, IF(OR(a_ch <= c_mc, c_mc < 0, b_sl <= 0), "Require a > c >= 0 and b > 0", ROUND((3 * ((a_ch - c_mc)^2)) / (16 * b_sl), 4)))Total Social Welfare Gain (Profit Gain + Consumer Surplus Gain) from Eliminating Double Marginalization
Eliminating double marginalization adds (1/16)*(a - c)^2/b in channel profit PLUS (3/32)*(a - c)^2/b in consumer surplus = (5/32)*(a - c)^2/b in total social surplus!
=LET(a_ch, B2, b_sl, B3, c_mc, B4, IF(OR(a_ch <= c_mc, c_mc < 0, b_sl <= 0), "Require a > c >= 0 and b > 0", ROUND((5 * ((a_ch - c_mc)^2)) / (32 * b_sl), 4)))Try the example
Use the original values or change them to test your understanding.