CRRA

CRRA stands for Constant Relative Risk Aversion, a utility specification widely used in economics to model preferences over consumption under uncertainty and over time. It assumes that the relative, not absolute, risk aversion remains constant as wealth changes, making it convenient for dynamic optimization and asset pricing.

In a one-period setting, the CRRA utility is given by U(C) = (C^(1-σ))/(1-σ) if σ ≠ 1, and U(C) = ln(C) if σ = 1. The parameter σ is the coefficient of relative risk aversion, and the relative risk aversion is constant and equal to σ for all consumption levels C. The first and second derivatives are U'(C) = C^(-σ) and U''(C) = -σ C^(-σ-1), implying the standard relation that the coefficient of relative risk aversion is σ.

In intertemporal models, lifetime utility is typically written as E[ sum β^t U(C_t) ], with discount factor β in (0,1). The elasticity of intertemporal substitution (EIS) is 1/σ, so a higher σ reduces the willingness to substitute consumption across different periods. CRRA utilities are common in macroeconomic models, finance, and structural estimation because they deliver tractable Euler equations and align with a range of observed risk behaviors.

Limitations include potential misfit at extreme consumption levels and state-dependent risk preferences. CRRA is part of the broader HARA family, and alternatives such as Epstein-Zin preferences separate risk aversion from intertemporal substitution to capture more nuanced risk–time trade-offs.