Organizational Economics & Incentive Design
Holmström Team Moral Hazard & 1/N Free-Rider Formula
Quantify Bengt Holmström’s (1982) 1/N team partnership free-rider effort e_Nash = k_p/(N*c) against first-best e_FB = k_p/c and prove the exact ((N - 1)/N)^2 surplus loss.
Why equal-sharing partnerships underinvest by 1/N — and why Holmström’s budget-breaker restores full effort!
Bengt Holmström (1982) proved a foundational impossibility theorem in organizational economics: when N team members jointly produce unattributed team output Y = SUM(k_p * e_i) with private quadratic effort cost 0.5 * c * e_i^2, NO budget-balanced sharing rule SUM(s_i(Y)) = Y can achieve First-Best effort e_FB = k_p / c! Under an equal 1/N revenue-sharing partnership, each partner bears 100% of personal effort cost c * e_i while capturing only (1 / N) * k_p of marginal output — shrinking Nash effort to e_Nash = k_p / (N * c) and destroying an exact fraction ((N - 1) / N)^2 of First-Best net surplus unless an external principal ("budget-breaker") enforces a target-output contract!
Build a hand-checkable 5-partner practice table FIRST: Team_Size_N = 5 (B2), Marginal_Product_kp = 20 (B3), Effort_Slope_c = 2 (B4). First-Best effort is e_FB = 20 / 2 = 10.00 per partner for team net surplus S_FB = 5 * (200 - 100) = 500.00. Under 1/5 equal sharing, Nash effort plunges to e_Nash = 20 / (5 * 2) = 2.00, realized team net surplus drops to S_Nash = 5 * (40 - 4) = 180.00, and Deadweight Free-Rider Loss reaches 500 - 180 = 320.00 (exact (4/5)^2 = 64.00% loss!). A classic analyst mistake is thinking surplus falls linearly as 1/N; before revealing a team-bonus formula, ask a hint-first AI prompt: "Why does partnership net surplus equal S_FB * (2*N - 1) / N^2 instead of S_FB / N when e_Nash = e_FB / N?"
| Holmström Team Metric (N=5, k_p=20, c=2) | Formula | Verified Output |
|---|---|---|
| First-Best / Budget-Breaker Effort (e_FB) | k_p / c | 10.0000 |
| Equal-Share 1/N Nash Effort (e_Nash) | k_p / (N * c) | 2.0000 (1/5 of First-Best) |
| First-Best Team Net Surplus (S_FB) | N * k_p^2 / (2 * c) | 500.00 |
| 1/N Partnership Net Surplus (S_Nash) | S_FB * (2*N - 1) / N^2 | 180.00 (Loss = 320.00, -64%!) |
Equal-Sharing 1/N Partnership Nash Effort & Net Surplus Formula
Validate integer/positive team size N >= 1, marginal productivity k_p > 0, and effort convexity c > 0.
=LET(n_tm, B2, kp_v, B3, c_ef, B4, IF(OR(n_tm < 1, kp_v <= 0, c_ef <= 0), "Require N >= 1 and Positive Parameters", LET(s_fb, n_tm * (kp_v^2) / (2 * c_ef), ROUND(s_fb * (2 * n_tm - 1) / (n_tm^2), 4))))Exact Free-Rider Deadweight Surplus Loss S_FB * ((N - 1) / N)^2 Formula
In a shared Arte Sheets compensation model, have a suggester test N = 1 (solo proprietor -> 0.00 free-rider loss!) before an editor accepts the budget-breaker bonus threshold.
=LET(n_tm, B2, kp_v, B3, c_ef, B4, IF(OR(n_tm < 1, kp_v <= 0, c_ef <= 0), "Require N >= 1 and Positive Parameters", LET(s_fb, n_tm * (kp_v^2) / (2 * c_ef), ROUND(s_fb * (((n_tm - 1) / n_tm)^2), 4))))Try the example
Use the original values or change them to test your understanding.