Contract Theory & Corporate Governance
Grossman-Hart-Moore Hold-Up & Ownership Formula
Calculate non-contractible relationship-specific investment e_HU = (alpha + (1 - alpha)*lambda)*(k_S/c) and quantify how asset ownership shrinks hold-up deadweight loss.
Model the Fisher Body / GM hold-up problem and prove who should own critical assets!
In Sanford Grossman & Oliver Hart (1986) and Oliver Hart & John Moore (1990), a supplier makes an ex-ante non-contractible investment e at cost 0.5 * c * e^2 that generates relationship-specific value V(e) = k_S * e inside the partnership and outside disagreement value O(e) = lambda * k_S * e (0 <= lambda <= 1, determined by residual asset ownership!). Because contracts are incomplete, the parties split ex-post surplus V(e) - O(e) via Nash bargaining with supplier weight alpha in [0, 1], giving the supplier O(e) + alpha * (V(e) - O(e)) = w_eff * k_S * e where effective incentive weight is w_eff = alpha + (1 - alpha) * lambda!
Check by hand with Specific_Return_kS = 40 (B2), Investment_Slope_c = 4 (B3), Bargaining_Share_alpha = 0.50 (B4), and Buyer Ownership lambda = 0.00 (B5): First-Best investment is e_FB = 40 / 4 = 10.00 (net surplus S_FB = 200.00), whereas ex-post 50/50 hold-up cuts supplier investment to e_HU = 0.50 * 10 = 5.00, lowering realized surplus to 40*5 - 2*25 = 150.00 (Hold-Up Loss = 50.00!). Now transfer asset ownership to the Supplier so outside sensitivity rises to lambda = 0.60: effective weight jumps to w_eff = 0.50 + 0.50 * 0.60 = 0.80, lifting e_HU to 8.00 and slashing Hold-Up Loss by 84% from 50.00 down to 8.00!
| Property-Rights Regime (k_S=40, c=4, alpha=0.50) | Effective Weight w_eff & Investment e* | Realized Net Surplus & Hold-Up DWL |
|---|---|---|
| Complete Contract First-Best | w_eff = 1.00, e_FB = 10.00 | Surplus = 200.00 (DWL = 0.00) |
| Buyer Asset Ownership (lambda = 0.00) | w_eff = 0.50, e_HU = 5.00 | Surplus = 150.00 (DWL = 50.00) |
| Supplier Asset Ownership (lambda = 0.60) | w_eff = 0.80, e_HU = 8.00 | Surplus = 192.00 (DWL = 8.00!) |
Incomplete-Contract Hold-Up Investment e_HU Formula
Avoid the common mistake of adding alpha + lambda directly (which can exceed 100%!): the disagreement point enters with weight (1 - alpha) because alpha of O(e) is subtracted inside alpha * (V - O).
=LET(ks_v, B2, c_iv, B3, al_b, B4, lm_o, B5, IF(OR(ks_v <= 0, c_iv <= 0, al_b < 0, al_b > 1, lm_o < 0, lm_o > 1), "Invalid GHM Input", LET(w_ef, al_b + (1 - al_b) * lm_o, ROUND(w_ef * (ks_v / c_iv), 4))))Hold-Up Deadweight Surplus Loss (1 - w_eff)^2 * S_FB Formula
Quantify the exact value created by vertical integration or re-assigning IP/tooling ownership to the party with the more elastic non-contractible investment.
=LET(ks_v, B2, c_iv, B3, al_b, B4, lm_o, B5, IF(OR(ks_v <= 0, c_iv <= 0, al_b < 0, al_b > 1, lm_o < 0, lm_o > 1), "Invalid GHM Input", LET(w_ef, al_b + (1 - al_b) * lm_o, s_fb, (ks_v^2) / (2 * c_iv), ROUND(((1 - w_ef)^2) * s_fb, 4))))Try the example
Use the original values or change them to test your understanding.