Public Finance & Welfare Economics
Harberger Deadweight Loss Triangle & Marginal Cost of Funds
Calculate Harberger excess burden DWL = 0.5 * eta * tau^2 * (P_0 * Q_0), quadratic distortion scaling, and Marginal Cost of Public Funds MCPF = (1 - eta*tau) / (1 - 2*eta*tau).
Audit the square law of price wedges: doubling an excise rate quadruples deadweight loss!
Arnold Harberger (1964) established that an ad-valorem commodity levy or regulatory price wedge tau_rate on baseline market expenditure P_0 * Q_0 with compensated equilibrium quantity elasticity eta_eq = (e_D * e_S) / (e_D + e_S) (reducing to e_D under horizontal supply) shrinks quantity by eta_eq * tau_rate * Q_0 and creates an excess-burden triangle DWL = 0.5 * eta_eq * (tau_rate^2) * (P_0 * Q_0)! Because tau_rate enters SQUARED, doubling the wedge rate quadruples deadweight loss — and Browning’s Marginal Cost of Public Funds MCPF = (1 - eta_eq * tau_rate) / (1 - 2 * eta_eq * tau_rate) measures total private surplus lost per incremental dollar of net public revenue raised!
Verify on baseline expenditure P_0 * Q_0 = 10,000,000 (B2), compensated elasticity eta_eq = 0.80 (B3), and ad-valorem excise rate tau_rate = 0.25 (B4 -> eta * tau = 0.20 < 0.50): Harberger DWL is 0.5 * 0.80 * 0.0625 * 10,000,000 = 250,000.00, net levy revenue is 0.25 * 10,000,000 * (1 - 0.20) = 2,000,000.00, and MCPF = (1 - 0.20) / (1 - 0.40) = 1.3333 (a public project funded at the margin needs a benefit-cost ratio >= 1.3333 to break even socially!). Double tau_rate to 0.50 and DWL quadruples to 1,000,000.00!
| Welfare Metric (P_0*Q_0 = 10M, eta = 0.80) | tau = 0.25 | tau = 0.50 (Doubled Rate) |
|---|---|---|
| Harberger Excess Burden (DWL) | 250,000.00 | 1,000,000.00 (4x Larger!) |
| Net Revenue Collected | 2,000,000.00 | 3,000,000.00 |
| Marginal Cost of Funds (MCPF) | 1.3333 | 3.0000 |
Harberger Deadweight Loss Triangle 0.5 * eta * tau^2 * Expenditure Formula
Validate positive baseline expenditure, elasticity eta_eq >= 0, and wedge rate 0 <= tau_rate < 1 / eta_eq.
=LET(exp_0, B2, eta_q, B3, tau_w, B4, IF(OR(exp_0 <= 0, eta_q < 0, tau_w < 0, eta_q * tau_w >= 1), "Invalid Wedge Input", ROUND(0.5 * eta_q * (tau_w^2) * exp_0, 2)))Marginal Cost of Public Funds MCPF = (1 - eta * tau) / (1 - 2 * eta * tau) Formula
Enforce the upward-sloping Laffer revenue bound 2 * eta_eq * tau_rate < 1 (when eta_eq * tau_rate >= 0.50, raising the rate further reduces net revenue while increasing DWL!).
=LET(exp_0, B2, eta_q, B3, tau_w, B4, IF(OR(exp_0 <= 0, eta_q < 0, tau_w < 0), "Invalid Wedge Input", IF(2 * eta_q * tau_w >= 1, "Past Revenue Peak (Require 2*eta*tau < 1)", ROUND((1 - eta_q * tau_w) / (1 - 2 * eta_q * tau_w), 4))))Try the example
Use the original values or change them to test your understanding.