Environmental Economics & Regulatory Design
Weitzman Prices vs. Quantities Instrument Choice Formula
Evaluate Martin Weitzman’s (1974) comparative advantage Delta_(P-Q) = sigma^2 * (C_slope - B_slope) / (2 * C_slope^2) between Pigouvian emission fees and tradable quantity caps.
Choose between an emission fee and a quantity cap using relative marginal slopes!
Under certainty, a Pigouvian emission fee p* and a tradable quantity cap q* achieve the exact same abatement! When regulators face a zero-mean compliance-cost shock theta (variance sigma_theta^2 > 0) in marginal abatement cost MC(q) = C_0 + theta + C_slope * q against marginal abatement benefit MB(q) = B_0 - B_slope * q, Martin Weitzman’s (1974) landmark "Prices vs. Quantities" theorem proves the exact expected welfare advantage of the Price Fee over the Quantity Cap is Delta_(P-Q) = (sigma_theta^2 * (C_slope - B_slope)) / (2 * C_slope^2)!
Verify the intuitive slope rule on B_0 = 120 (B2), C_0 = 20 (B3), marginal benefit slope B_slope = 0.50 (B4, flat stock-pollutant marginal damage), steep marginal cost slope C_slope = 2.00 (B5), and cost shock SD sigma_theta = 10 (B6 -> variance 100): expected efficient abatement is q* = (120 - 20) / (0.50 + 2.00) = 40.00 units at Pigouvian fee p* = 20 + 2 * 40 = 100.00, and Weitzman’s comparative advantage is 100 * (2.00 - 0.50) / (2 * 4.00) = +18.75 > 0 (Price Fee Preferred!). Flip B_slope to 5.00 (> C_slope = 2.00, steep toxic threshold damage!) and Delta_(P-Q) turns negative to -37.50 (Quantity Cap Preferred!).
| Pollutant Regime (B_0=120, C_0=20, C_slope=2, sig=10) | Weitzman Delta_(P-Q) | Optimal Policy Instrument |
|---|---|---|
| Stock Pollutant (B_slope = 0.50 < C_slope) | +18.75 | Price Fee Preferred |
| Equal Slopes (B_slope = 2.00 = C_slope) | 0.00 | Indifferent (Equal Welfare) |
| Threshold Pollutant (B_slope = 5.00 > C_slope) | -37.50 | Quantity Cap Preferred |
Weitzman Expected Welfare Advantage Delta_(P-Q) Formula
Validate interior abatement B_0 > C_0 >= 0, B_slope >= 0, C_slope > 0, and cost shock SD sigma_theta >= 0.
=LET(b_0, B2, c_0, B3, b_sl, B4, c_sl, B5, sd_c, B6, IF(OR(c_0 < 0, b_0 <= c_0, b_sl < 0, c_sl <= 0, sd_c < 0), "Invalid Weitzman Input", ROUND(((sd_c^2) * (c_sl - b_sl)) / (2 * (c_sl^2)), 4)))Instrument Recommendation Rule Based on Relative Slope C_slope vs B_slope Formula
Classify whether a Pigouvian price fee, tradable cap, or hybrid safety-valve collar maximizes expected social surplus.
=IF(ISTEXT(E4), E4, IF(ABS(E4) < 1E-9, "Indifferent (Equal Welfare)", IF(E4 > 0, "Price Fee Preferred", "Quantity Cap Preferred")))Try the example
Use the original values or change them to test your understanding.