Personnel Economics & Executive Compensation
Lazear-Rosen Tournament Prize Spread & Effort Formula
Design rank-order promotion tournaments via Edward Lazear and Sherwin Rosen’s (1981) optimal prize spread W_1 - W_2 = p / g(0), equilibrium effort, and participation-constrained pay levels.
Why CEO pay jumps 2x over VP runner-up pay even when the promotion margin is razor-thin!
Edward Lazear and Sherwin Rosen (1981) explained why executive pay reflects the incentives of the ENTIRE tournament below it rather than just the CEO’s current-year marginal product! When two symmetric risk-neutral contestants produce output q_i = p * e_i + eps_i (marginal effort product p > 0, convex effort disutility 0.5 * k * e_i^2, and luck-difference density at zero g_0 = g(0) > 0), contestant i chooses effort to maximize P(Win) * W_1 + (1 - P(Win)) * W_2 - 0.5 * k * e_i^2, yielding symmetric Equilibrium Effort e = (g_0 * (W_1 - W_2)) / k! Matching First-Best Efficient Effort e* = p / k requires Optimal Promotion Prize Spread Delta_W* = W_1* - W_2* = p / g_0 — which rises whenever output noise is large (small g_0) so luck doesn’t dull effort incentives!
Verify on exact integers: Marginal_Product p = 20 ($k/effort unit, B2), Zero_Noise_Density g_0 = 0.25 (B3), Effort_Cost_Slope k = 4 ($k/unit^2, B4), and Outside_Reservation_Utility U_bar = 70 ($k, B5): First-Best Effort is e* = 20 / 4 = 5.00 units (Disutility = 0.5 * 4 * 25 = 50.00 $k); Optimal Prize Spread is W_1* - W_2* = 20 / 0.25 = 80.00 ($k); Expected Pay needed for participation is U_bar + 50 = 120.00 ($k); giving Optimal Winner Prize W_1* = 120 + 40 = 160.00 ($k) and Runner-Up Wage W_2* = 120 - 40 = 80.00 ($k — exact 2.00x pay jump!)!
| Tournament Design Metric (p=20, g_0=0.25, k=4, U_bar=70) | Formula | Verified Output |
|---|---|---|
| First-Best Efficient Effort (e*) | p / k = 20 / 4 | 5.00 effort units |
| Optimal Promotion Prize Spread (W_1* - W_2*) | p / g_0 = 20 / 0.25 | 80.00 ($k Spread) |
| Winner Promotion Compensation (W_1*) | U_bar + 0.5*k*(e*)^2 + 0.5*Spread | 160.00 ($k) |
| Runner-Up Base Compensation (W_2*) | U_bar + 0.5*k*(e*)^2 - 0.5*Spread | 80.00 ($k) |
Lazear-Rosen Optimal Tournament Prize Spread Formula
Validate p > 0, zero-difference noise density g_0 > 0, and effort curvature k > 0.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Tournament Input", ROUND(B2 / B3, 2))Optimal Winner Prize W_1* & Runner-Up Wage W_2* Formula
Notice that noisier business environments (smaller g_0, say g_0 = 0.125) force a WIDER promotion spread (160 $k instead of 80 $k) to overcome luck dilution!
=LET(p_mp, B2, g_zr, B3, k_cs, B4, u_rs, B5, IF(OR(p_mp <= 0, g_zr <= 0, k_cs <= 0), "Invalid Tournament Input", LET(e_fb, p_mp / k_cs, w_mn, u_rs + 0.5 * k_cs * (e_fb ^ 2), sp_w, p_mp / g_zr, ROUND(w_mn + 0.5 * sp_w, 2))))Try the example
Use the original values or change them to test your understanding.