Industrial Organization & Pricing Strategy
Adams-Yellen Pure vs Mixed Bundling Profit Formula
Compare Separate Component Pricing, Pure Bundling, and Mixed Bundling profits across negatively correlated customer segments and verify Adams & Yellen’s Sub-Cost Exclusion Theorem.
Understand why negatively correlated tastes make software suites profitable—and when Mixed beats Pure Bundling!
William James Adams and Janet Yellen (1976, Quarterly Journal of Economics) showed why multiproduct firms bundle software suites, cable channels, and value meals! Suppose three equal-sized segments of N_seg customers each evaluate Symmetric Products 1 and 2 (each with unit marginal cost c_unit): Segment A Specialists value (r_high, r_low), Segment B Specialists value (r_low, r_high), and Segment C Generalists value (r_mid, r_mid) with r_low < r_mid < r_high and r_high + r_low < 2 * r_mid! Under Separate Pricing, the monopolist chooses MAX(N_seg * (r_high - c), 2 * N_seg * (r_mid - c), 3 * N_seg * (r_low - c)) per good. Under Pure Bundling at P_bundle = r_high + r_low, variance across negatively correlated specialists collapses so ALL 3 * N_seg buyers purchase at profit 3 * N_seg * (r_high + r_low - 2 * c)! Yet under Mixed Bundling (selling single goods at P_single = r_high to Specialists A & B while selling the Bundle at P_B = 2 * r_mid to Generalists C), whenever c_unit > r_low, Mixed Bundling avoids forcing Specialists to consume a second good whose marginal cost c_unit exceeds their valuation r_low!
Verify on Segment_Size N_seg = 100 buyers per segment (300 total, B2), Unit_Marginal_Cost c_unit = $10 (B3), Specialist_High_Val r_high = $90 (B4), Specialist_Low_Val r_low = $15 (B5 -> Specialist Bundle Sum = $105), and Generalist_Mid_Val r_mid = $60 (B6 -> Generalist Bundle Sum = $120): (1) Optimal Separate Pricing ($60 each to 200 buyers/good) earns 2 * 200 * ($60 - $10) = $20,000 (E2); (2) Pure Bundling ($105 bundle to all 300 buyers) jumps to 300 * ($105 - $20) = $25,500 (E3); and (3) Mixed Bundling ($90 single to 200 specialists + $120 bundle to 100 generalists) earns 200 * ($90 - $10) + 100 * ($120 - $20) = $26,000 (E4, +$500 Mixed Advantage in E5)! Now raise Unit_Marginal_Cost c_unit = $25 (> $15 r_low!) in cell B3: Pure Bundling drops to $16,500 while Mixed Bundling earns $20,000 (+$3,500 Sub-Cost Exclusion Advantage)!
| Adams-Yellen Pricing Regime (N=100/seg, r=(90,15) & (60,60), c=$10 vs c=$25) | Formula | Profit @ c=$10 (vs c=$25) |
|---|---|---|
| Optimal Separate Component Profit | 2 * MAX(N*(rH-c), 2N*(rM-c), 3N*(rL-c)) | $20,000 ($14,000 @ c=$25) |
| Pure Bundling Profit (@ P_B = rH+rL) | MAX(3N*(rH+rL - 2c), N*(2*rM - 2c)) | $25,500 ($16,500 @ c=$25) |
| Mixed Bundling Profit (@ rH & 2*rM) | 2N*(rH - c) + N*(2*rM - 2c) | $26,000 ($20,000 @ c=$25) |
| Mixed vs Pure Bundling Gain ($) | Profit_Mixed - Profit_Pure | +$500 (+$3,500 @ c=$25) |
Mixed Bundling Incentive-Compatible Profit Formula
Check incentive compatibility r_high - P_single >= (r_high + r_low) - P_bundle: at P_single = r_high (zero specialist surplus) and P_bundle = 2 * r_mid > r_high + r_low, Specialists strictly prefer buying their single valued component over the $120 bundle!
=LET(n_sg, B2, c_un, B3, r_hi, B4, r_lo, B5, r_md, B6, IF(OR(n_sg <= 0, c_un < 0, r_lo < 0, r_md <= r_lo, r_hi <= r_md, 2 * r_md <= r_hi + r_lo), "Invalid Bundling Matrix", ROUND(2 * n_sg * MAX(0, r_hi - c_un) + n_sg * MAX(0, 2 * r_md - 2 * c_un), 2)))Mixed-Over-Pure Sub-Cost Exclusion Gain Formula
Notice that for zero-marginal-cost digital SaaS (set c_unit = $0 in B3), Pure Bundling ($31,500) ACTUALLY BEATS Mixed Bundling ($30,000) by -$1,500 because giving Specialists the second module costs $0 and extracts their extra $15!
=IF(ISTEXT(E4), E4, ROUND(E4 - E3, 2))Try the example
Use the original values or change them to test your understanding.