Public Procurement & Mechanism Design
Laffont-Tirole Procurement Incentive Power Formula
Calculate Jean-Jacques Laffont & Jean Tirole’s (1986) optimal cost-sharing contract slope b* = 1 - (lambda_f/(1 + lambda_f))*gamma_e*(beta - beta_L) between Fixed-Price and Cost-Plus.
Balance cost-cutting incentives against contractor information rents via Laffont & Tirole (1986)!
Should defense, transit, and infrastructure agencies award Fixed-Price contracts (contractor bears 100% of cost overruns, power b = 1) or Cost-Plus contracts (government reimburses 100% of realized cost C = beta - e, power b = 0)? Jean-Jacques Laffont and Jean Tirole (1986) solved the exact optimal menu when contractor baseline cost beta ~ Uniform[beta_L, beta_H] is private information, cost-reduction effort e has disutility 0.5 * gamma_e * e^2, and public funds carry Harberger deadweight shadow cost lambda_f > 0! Optimal incentive power for type beta is b*(beta) = MAX(0, 1 - (lambda_f / (1 + lambda_f)) * gamma_e * (beta - beta_L)), inducing effort e*(beta) = b*(beta) / gamma_e!
Verify on Shadow_Cost_lambda = 0.25 (B2 -> lambda / (1 + lambda) = 0.20), Effort_Convexity_gamma = 0.20 (B3 -> First-Best effort e_FB = 1 / 0.20 = 5.00), Most_Efficient_Cost_betaL = 50 (B4), and Evaluated_Contractor_beta = 60 (B5 -> gap beta - beta_L = 10): Optimal Contract Power is b* = 1 - 0.20 * 0.20 * 10 = 0.6000 (firm bears 60% of cost overruns, agency reimburses 40%!), Optimal Effort is e* = 0.60 / 0.20 = 3.0000, and Realized Cost is 60 - 3 = 57.00! Test beta = 50 (most efficient contractor) to verify "No Distortion at the Top": b*(50) = 1.0000 (100% Fixed-Price!) and e*(50) = 5.0000!
| Laffont-Tirole Contract Metric (lambda_f=0.25, gamma_e=0.20, beta_L=50) | Formula | Verified Output |
|---|---|---|
| Top-Efficiency Type (beta = 50) | b* = 1.0000 (Pure Fixed-Price) | Effort e* = 5.0000 (= e_FB) |
| High-Cost Type (beta = 60) | 1 - (lambda_f/(1+lambda_f))*gamma*(60 - 50) | b* = 0.6000 (40% Reimbursed) |
| Induced Cost-Saving Effort e*(60) | b*(60) / gamma_e | 3.0000 (Realized Cost = 57.00) |
| Zero Public Funds Distortion (lambda_f = 0) | b* = 1.0000 for ALL types | Effort e* = 5.0000 |
Laffont-Tirole Optimal Contract Incentive Power b*(beta) Formula
Clamp b*(beta) at 0 (pure Cost-Plus floor) if extreme inefficiency gaps push the virtual-cost adjustment above 1.
=LET(lm_f, B2, gm_e, B3, bt_l, B4, bt_c, B5, IF(OR(lm_f < 0, gm_e <= 0, bt_l <= 0, bt_c < bt_l), "Invalid Procurement Input", LET(b_raw, 1 - (lm_f / (1 + lm_f)) * gm_e * (bt_c - bt_l), ROUND(MAX(0, b_raw), 4))))Second-Best Cost-Reduction Effort e*(beta) = b*(beta) / gamma_e Formula
Compare e*(beta) directly against First-Best effort 1 / gamma_e to quantify the exact effort sacrifice made to curb information rents on efficient contractors.
=IF(ISTEXT(E2), E2, ROUND(E2 / B3, 4))Try the example
Use the original values or change them to test your understanding.