Compressing Reality

Utility Equals Log Consumption

Claim: Utility is equivalent to (a linear function of) the logarithm of consumption - at least withing typical ranges for income.

Evidence:
1. If the claim is correct, we can model the trade-off humans make between work and leisure like this $$U = A \ln{\left(Lw\right)} - f(L)$$ where $L$ is hours/week worked and $w$ is the wage they will be paid. If humans try to maximize utility, then we can mathematically prove this implies $L$ will be independent of $w$. The past 67 years have seen a 3.5-fold increase in the real wage in the US, while people still typically work 40 hours per week. While hours-worked-per-capita has increased by 12% during this time, hours worked per employee has decreased by 6%, so really we’ve seen very little change.
2. The most natural proxy for utility is asking people what their well-being is on a scale from 0-10. If we plot countries’ average log-consumption against their average self-reported happiness, we find a logarithmic relationship fits quite nicely
3. That’s all well and good between countries, but what about within countries? The chart below shows self-reported happiness within the US v. income as the darkest line "title": "High income improves evaluation of life but not emotional well-being". You might point out that there seems to be a tapering above \$100k, but I could equally point out that there is a tapering below \$20k. Perhaps it might be illustrative to plot an extrapolation from global happiness on to the same graph: While it appears Americans generally report less wellbeing than similar people around the world with the same incomes, the slopes are pretty similar between the two lines.

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