Microeconomic Consumer Theory & Hedonic Pricing
Lancaster Characteristics Shadow Price & Frontier Formula
Solve Kelvin Lancaster’s (1966) 2x2 Hedonic Attribute Shadow Prices (pi_1, pi_2) from two frontier goods and test whether a third competing SKU expands or falls inside the Characteristics Efficiency Frontier.
Price product attributes—not SKU labels—along Kelvin Lancaster’s Efficiency Frontier!
In A New Approach to Consumer Theory (1966, Journal of Political Economy), Kelvin Lancaster revolutionized demand modeling: consumers derive utility not from goods themselves, but from the objective Characteristics Vector z = (z_1, z_2) embodied per unit! Given two linearly independent frontier products A at price P_A with attributes (z_1A, z_2A) and B at price P_B with attributes (z_1B, z_2B) (determinant D = z_1A * z_2B - z_2A * z_1B != 0), Cramer’s Rule solves the exact Implicit Shadow Prices of Characteristic 1 and Characteristic 2: pi_1 = (P_A * z_2B - P_B * z_2A) / D and pi_2 = (z_1A * P_B - z_1B * P_A) / D! Any new Product C offering attributes (z_1C, z_2C) has Frontier Maximum Competitive Price P_C* = pi_1 * z_1C + pi_2 * z_2C: if P_C > P_C*, buyers synthesize the exact same attributes cheaper by mixing A and B, driving Good C’s rational market share to ZERO!
Verify on Cloud/Hardware SKU A priced at P_A = $12.00 (B2) delivering z_1A = 6 vCPU (B3) and z_2A = 2 GB-RAM-Score (B4), and SKU B priced at P_B = $15.00 (B5) delivering z_1B = 3 vCPU (B6) and z_2B = 6 GB-RAM-Score (B7), testing Candidate SKU C delivering z_1C = 4.5 vCPU (B8) and z_2C = 4.0 GB-RAM-Score (B9) at Launch_Price P_C = $15.00 (B10): Determinant D = 6*6 - 2*3 = 30; Attribute 1 Shadow Price is pi_1 = (12*6 - 15*2) / 30 = $1.40/unit (E2); Attribute 2 Shadow Price is pi_2 = (6*15 - 3*12) / 30 = $1.80/unit (E3); SKU C’s Frontier Parity Price is 4.5 * $1.40 + 4.0 * $1.80 = $13.50 (E4); so launching at $15.00 leaves SKU C "Dominated Inside Frontier (Overpriced)" by $1.50 (E5)! Reprice P_C to $12.00 in cell B10: SKU C flips to "Expands Frontier Outward (Competitive)"!
| Lancaster Characteristics Metric (A:$12->(6,2), B:$15->(3,6), C:(4.5,4) @ $15) | Formula | Verified Output |
|---|---|---|
| Characteristic 1 Shadow Price pi_1 | (P_A * z_2B - P_B * z_2A) / (z_1A*z_2B - z_2A*z_1B) | $1.40 / unit |
| Characteristic 2 Shadow Price pi_2 | (z_1A * P_B - z_1B * P_A) / (z_1A*z_2B - z_2A*z_1B) | $1.80 / unit |
| SKU C Frontier Parity Price P_C* | pi_1 * z_1C + pi_2 * z_2C | $13.50 (50/50 Blend of A & B) |
| Competitive Frontier Verdict (@ $15) | P_C vs P_C* | Dominated Inside Frontier (Overpriced) |
Cramer’s-Rule Attribute Shadow Price & Parity Price Formula
Check collinearity D = z_1A * z_2B - z_2A * z_1B != 0 and positive shadow prices pi_1 > 0, pi_2 > 0 (if either shadow price is <= 0, one of the two baseline goods A or B already dominates the other)!
=LET(p_a, B2, z_1a, B3, z_2a, B4, p_b, B5, z_1b, B6, z_2b, B7, z_1c, B8, z_2c, B9, dt_z, z_1a * z_2b - z_2a * z_1b, IF(OR(p_a <= 0, p_b <= 0, B10 <= 0, dt_z = 0), "Collinear or Invalid Goods", LET(pi_1, (p_a * z_2b - p_b * z_2a) / dt_z, pi_2, (z_1a * p_b - z_1b * p_a) / dt_z, IF(OR(pi_1 <= 0, pi_2 <= 0), "Baseline Good Dominated", ROUND(pi_1 * z_1c + pi_2 * z_2c, 2)))))Lancaster Efficiency-Frontier Dominance Verdict Formula
Use a 0.001 tolerance around P_C = P_C* to distinguish exact linear-combination neutrality from strict frontier expansion or interior dominance!
=IF(ISTEXT(E4), E4, IF(ABS(B10 - E4) < 0.001, "On Frontier (Exact Blend Parity)", IF(B10 < E4, "Expands Frontier Outward (Competitive)", "Dominated Inside Frontier (Overpriced)")))Try the example
Use the original values or change them to test your understanding.