Holmstrom-Milgrom Multitask Incentive Formula

Understand why high-powered piece rates backfire when quality is unmeasurable!

Why do teachers, surgeons, airline mechanics, and enterprise engineers receive salaried pay or muted bonuses even when call volume or test scores are cheap to measure? Bengt Holmstrom and Paul Milgrom (1991) proved that when an agent allocates attention across Measured Task 1 (marginal value v_1, variance sigma_1^2) and Unmeasured Task 2 (marginal value v_2 > 0, no contractible metric beta_2 = 0) under quadratic joint effort cost C(e_1, e_2) = 0.5*c_1*e_1^2 + 0.5*c_2*e_2^2 + c_12*e_1*e_2 with effort substitution c_12 > 0 (and positive Hessian determinant det_C = c_1*c_2 - c_12^2 > 0), raising piece rate beta_1 diverts effort out of Task 2 at rate de_2 / dbeta_1 = -c_12 / det_C!

Verify on Task_1_Marginal_Value v_1 = 1.00 (B2), Unmeasured_Quality_Value v_2 = 0.80 (B3), Cost_Curvatures c_1 = 2.0 (B4), c_2 = 2.0 (B5), Cross_Substitution c_12 = 1.0 (B6 -> det_C = 4 - 1 = 3 > 0), CARA_Risk_Aversion r = 0.04 (B7), and Noise_Variance sigma_1^2 = 25 (B8): Naive Single-Task Holmstrom-Milgrom Commission is v_1 / (1 + r * c_1 * sigma_1^2) = 1.00 / (1 + 2.0) = 0.3333 (33.33%), whereas Exact Multitask Optimal Piece Rate is beta_1* = MAX(0, (v_1 - v_2 * (c_12 / c_2)) / (1 + r * sigma_1^2 * (det_C / c_2))) = MAX(0, (1.00 - 0.80 * 0.5) / (1 + 0.04 * 25 * 1.5)) = 0.60 / 2.50 = 0.2400 (24.00% — a 933 bps commission haircut)! Raise unmeasured safety stakes to v_2 = 2.20: numerator 1.00 - 2.20 * 0.5 = -0.10 <= 0, driving optimal commission to exact 0 ("Flat Fixed Salary (Zero Piece Rate)")!

Holmstrom-Milgrom (1991) Multitask Optimal Commission Formula

Always verify strict convexity of the two-task cost matrix: c_1 > 0, c_2 > 0, and c_1 * c_2 - c_12^2 > 0!

=LET(v_ms, B2, v_un, B3, c_m1, B4, c_u2, B5, c_x, B6, r_ra, B7, sg_v, B8, dt_c, c_m1 * c_u2 - (c_x ^ 2), IF(OR(v_ms <= 0, v_un < 0, c_m1 <= 0, c_u2 <= 0, dt_c <= 0, r_ra < 0, sg_v < 0), "Invalid Convex Cost Matrix", ROUND(MAX(0, (v_ms - v_un * (c_x / c_u2)) / (1 + r_ra * sg_v * (dt_c / c_u2))), 4)))

Single-Task Benchmark & Incentive Regime Formula

Notice that even when noise sigma_1^2 = 0 (perfectly noiseless volume tracking!), if v_2 * (c_12 / c_2) >= v_1, optimal variable pay is STILL zero!

=IF(ISTEXT(E3), E3, IF(E3 = 0, "Flat Fixed Salary (Zero Piece Rate)", "Muted Multitask Commission"))

Further reading

• Arte Sheets Lesson — Holmstrom-Milgrom Linear Moral Hazard Optimal Incentive Beta
• Arte Sheets Lesson — Lazear-Rosen Rank-Order Tournament Prize Spread & Effort