Contract Theory & Organizational Economics
Holmström-Milgrom Optimal Incentive Intensity (beta*) Formula
Design an optimal linear bonus contract w(x) = alpha + beta * x under CARA moral hazard using beta* = k^2 / (k^2 + r * c * sigma^2), base salary alpha*, and agency loss.
Quantify the classic Incentive-Insurance Trade-Off in sales and executive pay
In Holmström and Milgrom’s (1987) Linear-Exponential-Normal (LEN) principal-agent model, observable output is x = k_p * e + noise (noise variance sigma^2 >= 0), the agent has CARA risk aversion r >= 0, quadratic effort disutility 0.5 * c * e^2 (c > 0), and reservation certainty equivalent U_0. Facing linear compensation w(x) = alpha + beta * x, the agent chooses effort e(beta) = k_p * beta / c, and a risk-neutral principal maximizes net surplus by setting commission share beta* = k_p^2 / (k_p^2 + r * c * sigma^2) and fixed base salary alpha* = U_0 + 0.5 * (r * sigma^2 - k_p^2 / c) * (beta*)^2!
Verify on productivity k_p = 10 (B2 -> k_p^2 = 100), effort curvature c = 2 (B3), risk aversion r = 0.05 (B4), KPI variance sigma^2 = 1,500 (B5), and outside option U_0 = 40 (B6): risk-cost product r * c * sigma^2 = 150, so optimal commission share is beta* = 100 / (100 + 150) = 0.4000 (40% variable pay!), induced effort is e* = 2.0000 (vs First-Best 5.0000), fixed base pay is alpha* = 40 + 6 - 4 = 42.00, and principal net profit is 10.00 (agency risk loss = 15.00 vs First-Best 25.00!). Set r = 0 (risk-neutral agent) or sigma^2 = 0 (noiseless KPI) and beta* jumps to 1.0000 ("selling the firm to the agent")!
| LEN Contract Quantity (k=10, c=2, r=0.05, sig^2=1500) | Formula | Verified Output |
|---|---|---|
| Optimal Commission Share (beta*) | k_p^2 / (k_p^2 + r * c * sigma^2) | 0.4000 (40.00%) |
| Induced Agent Effort (e*) | k_p * beta* / c | 2.0000 (vs FB = 5.00) |
| Optimal Fixed Base Pay (alpha*) | U_0 + 0.5*(r*sigma^2 - k^2/c)*(beta*)^2 | 42.00 |
| Principal Net Expected Profit | (k_p^2 / (2 * c)) * beta* - U_0 (if U_0=0: +10) | -30.00 (Net Surplus +10.00) |
Holmström-Milgrom Optimal Piece-Rate Commission Intensity beta* Formula
Validate k_p > 0, effort curvature c > 0, CARA risk aversion r >= 0, and KPI noise variance sigma^2 >= 0.
=LET(k_pr, B2, c_ef, B3, r_av, B4, v_ns, B5, IF(OR(k_pr <= 0, c_ef <= 0, r_av < 0, v_ns < 0), "Invalid Contract Input", ROUND((k_pr^2) / ((k_pr^2) + r_av * c_ef * v_ns), 4)))Participation-Binding Base Salary alpha* and Joint Certainty-Equivalent Gain Formula
Compute exact base salary alpha* leaving the agent at reservation certainty equivalent U_0 after accounting for effort cost and income-risk premium 0.5 * r * beta^2 * sigma^2.
=LET(k_pr, B2, c_ef, B3, r_av, B4, v_ns, B5, u_res, B6, IF(OR(k_pr <= 0, c_ef <= 0, r_av < 0, v_ns < 0), "Invalid Contract Input", LET(bt_s, (k_pr^2) / ((k_pr^2) + r_av * c_ef * v_ns), ROUND(u_res + 0.5 * (r_av * v_ns - (k_pr^2) / c_ef) * (bt_s^2), 4))))Try the example
Use the original values or change them to test your understanding.