Law & Economics & Regulatory Compliance
Becker Optimal Enforcement Audit & Sanction Formula
Compute Becker / Polinsky-Shavell maximal-fine audit probability p* = H/W_max - (c_p*G_max)/W_max^2 and explain optimal under-deterrence when auditing is costly.
Why Becker sets fines at the solvency cap W_max and calibrates p* below H / W_max!
Gary Becker (1968) and A. Mitchell Polinsky & Steven Shavell (1979) revolutionized regulatory compliance and environmental enforcement with two sharp insights. First, because raising a monetary penalty F is socially costless whereas raising inspection probability p consumes real enforcement budget c_p * p, a risk-neutral regulator sets F* = W_max (the offender’s maximum collectible wealth/solvency cap) before spending on extra audits! Second, when private offense gains g are uniform on [0, G_max] against external harm H <= W_max, minimizing social harm + audit expenditure yields p* = MAX(0, MIN(1, (H / W_max) - (c_p * G_max) / (W_max^2))) — intentionally setting expected sanction p* * W_max slightly BELOW harm H whenever auditing is costly (c_p > 0)!
Verify on External_Harm_H = 5,000 (B2), Max_Fine_Cap_Wmax = 20,000 (B3), Max_Private_Gain_Gmax = 10,000 (B4), and Audit_Cost_Slope_cp = 2,000 (B5): zero-cost Pigouvian audit rate would be H / W_max = 5,000 / 20,000 = 0.2500 (25.00%). Factoring in inspection cost c_p * G_max / W_max^2 = 20,000,000 / 400,000,000 = 0.0500 lowers optimal audit probability to p* = 0.2000 (20.00%), setting Optimal Expected Sanction at 0.20 * 20,000 = 4,000.00 and deterring 4,000 / 10,000 = 40.00% of potential violations!
| Becker-Polinsky-Shavell Metric (H=5k, W_max=20k, G_max=10k, c_p=2k) | Formula | Verified Output |
|---|---|---|
| Zero-Cost Pigouvian Audit Benchmark | H / W_max | 0.2500 (25.00% Audit Rate) |
| Cost-Adjusted Optimal Audit Rate (p*) | H / W_max - (c_p * G_max) / W_max^2 | 0.2000 (20.00% Audit Rate) |
| Optimal Expected Sanction (p* * W_max) | p* * W_max | 4,000.00 (< Harm 5,000.00) |
| Deterred Violation Share | MIN(1, (p* * W_max) / G_max) | 0.4000 (40.00% Deterred) |
Cost-Adjusted Optimal Inspection Probability p* Formula
Validate positive harm, wealth cap W_max >= H, positive gain support G_max, and non-negative audit cost c_p >= 0.
=LET(h_ex, B2, w_mx, B3, g_mx, B4, cp_a, B5, IF(OR(h_ex <= 0, w_mx < h_ex, g_mx <= 0, cp_a < 0), "Invalid Enforcement Input", LET(p_raw, (h_ex / w_mx) - (cp_a * g_mx) / (w_mx^2), ROUND(MAX(0, MIN(1, p_raw)), 4))))Optimal Expected Sanction p* * W_max Formula
Test c_p = 0 in cell B5 to verify expected sanction snaps back to exact Pigouvian harm H = 5,000.00!
=IF(ISTEXT(E3), E3, ROUND(E3 * B3, 2))Try the example
Use the original values or change them to test your understanding.