Macroeconomics & Consumption Theory
Kimball Prudence & Precautionary Saving Wedge Formula
Calculate Miles Kimball’s (1990) Coefficient of Relative Prudence P = gamma + 1, precautionary equivalent income haircut psi = 0.5*(gamma + 1)*sigma^2, and log-Euler consumption growth.
Distinguish risk aversion (-c*u''/u') from Kimball prudence (-c*u'''/u'')!
Why does quadratic utility (Certainty Equivalence, u'''(c) = 0) exhibit positive risk aversion yet ZERO precautionary saving against uninsurable income shocks? Miles Kimball (1990) proved that precautionary saving is governed not by concavity u''(c) < 0, but by the convexity of marginal utility u'''(c) > 0, measured by the Coefficient of Relative Prudence P(c) = -c * u'''(c) / u''(c)! Under Constant Relative Risk Aversion (CRRA) utility u(c) = c^(1 - gamma) / (1 - gamma), Relative Risk Aversion equals gamma while Relative Prudence equals P = gamma + 1!
Verify on CRRA_Risk_Aversion gamma = 3.0 (B2), Uninsurable_Log_Shock_StdDev sigma = 0.20 (20.0%, B3 -> sigma^2 = 0.0400), Safe_Real_Rate r = 0.040 (4.00%, B4), and Subjective_Discount_Rate rho = 0.025 (2.50%, B5): Relative Prudence is P = 3 + 1 = 4.0000; Arrow-Pratt Proportional Risk Premium is 0.5 * gamma * sigma^2 = 0.5 * 3 * 0.04 = 0.0600 (6.00%); Kimball Precautionary Equivalent Haircut (the certain drop in future resources inducing the exact same jump in expected marginal utility) is psi = 0.5 * (gamma + 1) * sigma^2 = 0.5 * 4 * 0.04 = 0.0800 (8.00%); and Second-Order Log-Euler Expected Consumption Growth E[Delta ln C] = (r - rho) / gamma + 0.5 * gamma * sigma^2 = (0.040 - 0.025) / 3 + 0.0600 = 0.0050 + 0.0600 = 0.0650 (6.50%, of which 6.00% is pure precautionary tilt!)!
| Kimball (1990) Precautionary Metric (gamma=3, sigma=20%, r=4%, rho=2.5%) | Formula | Verified Output |
|---|---|---|
| Coefficient of Relative Prudence (P) | gamma + 1 | 4.0000 |
| Arrow-Pratt Proportional Risk Premium (pi_risk) | 0.5 * gamma * sigma^2 | 0.0600 (6.00%) |
| Kimball Precautionary Equivalent (psi_prec) | 0.5 * (gamma + 1) * sigma^2 | 0.0800 (8.00% Buffer Wedge) |
| Log-Euler Consumption Growth (E[Delta ln C]) | (r - rho) / gamma + 0.5 * gamma * sigma^2 | 0.0650 (+6.50% / period) |
Kimball Precautionary Equivalent Premium psi_prec Formula
Compare sigma = 0 (complete insurance -> psi = 0, deterministic Euler growth = 0.0050) against sigma = 0.20 (incomplete markets -> psi = 0.0800) to see why Carroll (1997) buffer-stock consumers hold liquid wealth and track income closely!
=LET(gm_ra, B2, sg_sh, B3, IF(OR(gm_ra <= 0, sg_sh < 0), "Invalid Prudence Input", ROUND(0.5 * (gm_ra + 1) * (sg_sh ^ 2), 4)))Second-Order Log-Euler Consumption Growth Formula
Validate gamma > 0 and sigma >= 0.
=LET(gm_ra, B2, sg_sh, B3, r_rf, B4, rh_dc, B5, IF(OR(gm_ra <= 0, sg_sh < 0), "Invalid Prudence Input", ROUND((r_rf - rh_dc) / gm_ra + 0.5 * gm_ra * (sg_sh ^ 2), 4)))Try the example
Use the original values or change them to test your understanding.