Ramsey-Boiteux Regulated Utility Inverse-Elasticity Pricing Formula

Recover shared fixed infrastructure costs with minimum deadweight loss

When a regulated rail, grid, or water utility has large shared fixed network costs alongside low segment marginal costs MC_i, pure marginal-cost pricing P_i = MC_i runs an operating deficit! Ramsey (1927) and Boiteux (1956) proved that the second-best break-even tariff sets each segment’s Lerner markup inversely proportional to its own-price demand elasticity magnitude |e_i|: (P_i - MC_i) / P_i = R_ramsey / |e_i|, or P_i = MC_i / (1 - R_ramsey / |e_i|), where Ramsey number R_ramsey in [0, 1) reflects the break-even shadow multiplier.

Audit a two-tier transit authority at Ramsey number R_ramsey = 0.40 (B2): Peak Commuters (MC_1 = 12.00 in B3, |e_1| = 1.60 in B4) carry Lerner markup 0.40 / 1.60 = 0.2500 and pay P_1 = 12.00 / 0.75 = 16.0000; Off-Peak Leisure riders (MC_2 = 9.00 in B5, |e_2| = 4.00 in B6) carry markup 0.40 / 4.00 = 0.1000 and pay P_2 = 9.00 / 0.90 = 10.0000 — preserving the exact inverse-elasticity markup ratio 0.25 / 0.10 = 4.00 / 1.60 = 2.5000!

Ramsey-Boiteux Regulated Segment Tariff P_i = MC_i / (1 - R_ramsey / |e_i|) Formula

Enforce 0 <= R_ramsey < 1 and |e_i| > R_ramsey (so 1 - R_ramsey / |e_i| stays strictly positive and bounded below monopoly pricing).

=LET(r_rm, B2, mc_1, B3, el_1, B4, mc_2, B5, el_2, IF(OR(r_rm < 0, r_rm >= 1, mc_1 <= 0, mc_2 <= 0, el_1 <= r_rm, el_2 <= r_rm), "Invalid Ramsey Input", ROUND(mc_1 / (1 - r_rm / el_1), 4)))

Inverse-Elasticity Lerner Markup Ratio Verification Formula

Verify that Segment 1’s Lerner margin divided by Segment 2’s Lerner margin equals |e_2| / |e_1| whenever R_ramsey > 0.

=IF(OR(ISTEXT(E2), ISTEXT(E3), B2 = 0), 0, ROUND(((E2 - B3) / E2) / ((E3 - B5) / E3), 4))

Further reading

• Arte Sheets Lesson — Cournot Oligopoly Nash Price and Cowling-Waterson HHI Lerner Formula
• Microsoft Support — LET function