Utility functions visualised

Selected functions and their tangents
Selected functions and their tangentsEach selected function is drawn on shared axes with its tangent at the chosen input.-3-2-1012302468Consumption cUtility u(c)
0.57.5

Logarithmic

u(c)=ln⁡cu(c)=\ln c
u′(c)=1/cu'(c)=1/c
u′′(c)=−1/c2u''(c)=-1/c^2
Domain
c > 0
Minimum
None; u → −∞ as c → 0⁺
Maximum
None; u → ∞ as c → ∞

What to notice. Increasing and strictly concave. The curve is steep near zero.

Where it appears. A standard consumption payoff in growth and dynamic programming.

CRRA

u(c)=1−1/c(σ=2)u(c)=1-1/c\quad(\sigma=2)
u′(c)=c−2u'(c)=c^{-2}
u′′(c)=−2c−3u''(c)=-2c^{-3}
Domain
c > 0
Minimum
None; u → −∞ as c → 0⁺
Maximum
No attained maximum; ceiling 1.00 as c → ∞

What to notice. Increasing and strictly concave for σ > 0.

Where it appears. The curvature parameter σ also governs consumption smoothing in macro models.