Actuarial Economics & Asymmetric Information
Rothschild-Stiglitz Insurance Screening Deductible Formula
Solve the incentive-compatible Low-risk partial coverage deductible D_L* and test Rothschild-Stiglitz separating equilibrium existence lambda_H >= lambda_H*.
Why Low-risk insureds face mandatory deductibles — and when equilibrium breaks down!
In Rothschild and Stiglitz’s (1976) competitive insurance market with loss size L > 0, High-risk loss probability p_H, and Low-risk probability p_L < p_H, competition forces zero expected profit per contract: High-risk buyers receive full coverage (deductible D_H* = 0 at fair premium p_H * L), while Low-risk buyers must accept an incentive-compatible deductible D_L* > 0 so High-risk buyers do not mimic the cheaper p_L rate! Under mean-variance / local CARA risk aversion gamma > 0, High-risk indifference (p_H - p_L)*(L - D_L) = 0.5 * gamma * p_H * (1 - p_H) * D_L^2 yields exact closed-form separating deductible D_L* = (-(p_H - p_L) + SQRT((p_H - p_L)^2 + 2 * gamma * p_H * (1 - p_H) * (p_H - p_L) * L)) / (gamma * p_H * (1 - p_H))!
Verify with loss L = 1,200 (B2), p_H = 0.50 (B3), p_L = 0.20 (B4), risk aversion gamma = 0.004 (B5), and High-risk share lambda_H = 0.40 (B6): discriminant 0.09 + 0.72 = 0.81 = 0.90^2 yields exact separating deductible D_L* = (0.90 - 0.30) / 0.001 = 600.00 (Low-risk buyers can insure only q_L* = 600 of the 1,200 loss!). Moreover, Low-risk risk-bearing cost 0.5 * 0.004 * 0.16 * 600^2 = 115.20 is smaller than pooling cross-subsidy lambda_H * (p_H - p_L) * L = 360 * lambda_H iff lambda_H >= lambda_H* = 115.20 / 360 = 0.3200 — below lambda_H = 32%, NO pure-strategy Nash equilibrium exists!
| Rothschild-Stiglitz Metric (L=1200, p_H=0.5, p_L=0.2, gamma=0.004) | Formula | Verified Output |
|---|---|---|
| Separating Low-Risk Deductible (D_L*) | Exact IC Quadratic Root | 600.00 (50% Coverage) |
| Low-Risk Partial Coverage (q_L*) | L - D_L* | 600.00 (Prem = 120.00) |
| Existence Threshold Share (lambda_H*) | (0.5*gamma*p_L*(1-p_L)*D_L*^2) / ((p_H-p_L)*L) | 0.3200 (32.00%) |
| Market Regime at lambda_H = 0.40 | IF(lambda_H >= lambda_H*, Separating, No Nash) | Separating Equilibrium |
Exact Incentive-Compatible Separating Deductible D_L* Formula
Validate L > 0, 0 < p_L < p_H < 1, and risk aversion gamma > 0 before evaluating the positive root of High-risk self-selection.
=LET(loss_l, B2, p_hi, B3, p_lo, B4, gam_r, B5, IF(OR(loss_l <= 0, p_lo <= 0, p_hi <= p_lo, p_hi >= 1, gam_r <= 0), "Invalid Screening Input", LET(dp_hl, p_hi - p_lo, a_qd, gam_r * p_hi * (1 - p_hi), disc_v, (dp_hl^2) + 2 * a_qd * dp_hl * loss_l, ROUND((-dp_hl + SQRT(disc_v)) / a_qd, 4))))Rothschild-Stiglitz Critical High-Risk Share lambda_H* for Equilibrium Existence Formula
When High-risk share lambda_H < lambda_H*, a cream-skimming full-coverage pooling policy attracts both types and breaks separation.
=LET(loss_l, B2, p_hi, B3, p_lo, B4, gam_r, B5, IF(OR(loss_l <= 0, p_lo <= 0, p_hi <= p_lo, p_hi >= 1, gam_r <= 0), "Invalid Screening Input", LET(dp_hl, p_hi - p_lo, a_qd, gam_r * p_hi * (1 - p_hi), d_low, (-dp_hl + SQRT((dp_hl^2) + 2 * a_qd * dp_hl * loss_l)) / a_qd, cost_sep, 0.5 * gam_r * p_lo * (1 - p_lo) * (d_low^2), ROUND(cost_sep / (dp_hl * loss_l), 4))))Try the example
Use the original values or change them to test your understanding.