Definition
Log utility is the utility function in which each additional dollar of wealth contributes less to a decision-maker's satisfaction than the dollar before, following the specific pattern of the logarithm — historically distinctive because maximizing expected log utility under multiplicative wealth dynamics turns out to coincide with maximizing the long-run growth rate of wealth.
Why it matters
Log utility occupies a unique place in the history of economics because Daniel Bernoulli proposed it in 1738 to resolve the St. Petersburg paradox, and the same function reappears throughout the literature on decisions under uncertainty. Ergodicity economics offers an interpretation of why it keeps reappearing: maximizing expected log utility is mathematically equivalent to maximizing the long-run growth rate of wealth under multiplicative dynamics — the log utility result is not about subjective preferences but about the correct objective for an individual subject to multiplicative dynamics over time.
How it works
For an individual facing a multiplicative wealth process — one where wealth grows by multiplication of period growth factors — the long-run growth rate of wealth equals the expected value of the log of the period growth factor, because the logarithm converts a multiplicative process into an additive one. Maximizing this quantity is exactly what an expected-log-utility maximizer does — the log function converts the multiplicative dynamics into additive form, and the expected-value operator then takes the right average for the additive form. The equivalence is mathematical, not preferential: log utility is the function that, when used in the expected-utility framework, happens to produce the same recommendations as direct maximization of the long-run growth rate. Other utility functions produce different recommendations that, under multiplicative dynamics, systematically diverge from what would maximize the individual's long-run growth.
In practice
For an individual evaluating investment or wealth-management decisions under long-horizon multiplicative dynamics, log utility is the framework that produces recommendations aligned with maximizing the long-run growth rate of wealth. The Kelly criterion — which sizes bets and positions to maximize long-run growth — is the operational form of log-utility maximization under multiplicative dynamics. The practical move is to recognize that log utility, when used in long-horizon wealth management, is not a subjective preference choice but a structural alignment with the dynamics of the underlying process. Different utility functions remain valid for different problems, but the special status of log utility under multiplicative dynamics is a mathematical feature, not a stylistic one.
In the Longevity Standard Framework
Log utility is the fundamental formulation in ergodicity economics of the utility function whose expected-utility maximization coincides with long-run growth-rate maximization under multiplicative wealth dynamics, and is the historical bridge between the expected-utility framework and the time-average orientation of the Longevity Standard framework. The Kelly criterion is the operational form of the same underlying mathematics.
Related terms
- Kelly criterion
- Expected utility theory
- Multiplicative dynamics
- Time-average return
- St. Petersburg paradox
- Wealth trajectory
- Ergodicity
- Geometric mean