Financial & Accounting
Ramsey Rule & Keynes-Ramsey Euler Equation: Social Discount Rate & Consumption Growth
Calculate the social discount rate r = rho + eta * g for climate/infrastructure cost-benefit analysis and optimal per-capita consumption growth g_c = (r - rho) / eta using Frank P. Ramsey’s 1928 intertemporal optimization rule.
Why Stern, Nordhaus, and HM Treasury Green Book hinge on Frank Ramsey’s 1928 equation r = rho + eta * g!
Frank Plumpton Ramsey (1928, Economic Journal — extended by David Cass and Tjalling Koopmans in 1965) derived the foundational intertemporal optimality condition of modern macroeconomics and public cost-benefit discounting! Under isoelastic CRRA utility u(c) = c^(1 - eta)/(1 - eta) with pure rate of time preference rho and elasticity of marginal utility of consumption eta (= 1 / EIS), optimal per-capita consumption growth satisfies the Keynes-Ramsey Euler equation: g_c = (r - rho) / eta!
Rearranging for public infrastructure and climate-damage discounting gives the famous Ramsey Social Discount Rate: SDR = rho + eta * g! At real net return r = 6.5% (0.065), pure time impatience rho = 2.5% (0.025), and CRRA curvature eta = 2.0 (EIS = 0.50), optimal consumption grows at g_c = (0.065 - 0.025) / 2.0 = 0.0200 (2.00%/yr)—and conversely, under HM Treasury Green Book parameters (rho = 1.5%, eta = 1.0, g = 2.0%), SDR = 3.50%!
| Intertemporal Preference Profile | Keynes-Ramsey Growth g_c = (r - rho)/eta | Implied Social Discount Rate rho + eta*g_c |
|---|---|---|
| r = 6.5%, rho = 2.5%, eta = 2.0 | 0.0200 (2.00% / yr) | 6.50% |
| Lower Risk Aversion / Higher EIS eta = 1.0 | 0.0400 (4.00% / yr) | 6.50% |
| Invalid Curvature eta <= 0 | Non-Concave Utility! | "Requires Positive Curvature eta > 0" |
Keynes-Ramsey Optimal Per-Capita Consumption Growth Rate g_c = (r - rho) / eta
Evaluates optimal consumption growth from real net return r (B2), pure rate of time preference rho (B3 >= 0), and elasticity of marginal utility eta (B4 > 0).
=IF(OR(B3 < 0, B4 <= 0), "Requires Positive Curvature eta > 0", ROUND((B2 - B3) / B4, 4))Ramsey Social Discount Rate (%) = (rho + eta * g_growth) * 100
Computes the public social discount rate percentage when interpreting B2 as expected per-capita growth g (or checking Euler identity at g = 0.02).
=IF(OR(B3 < 0, B4 <= 0), "Requires Positive Curvature eta > 0", ROUND((B3 + B4 * 0.02) * 100, 2))Try the example
Use the original values or change them to test your understanding.