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shokwave comments on Rationality Quotes: November 2010 - Less Wrong
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If there is some mathematical reason why, I would love to know. I was going off the observation that the natural logarithm approximates the kind of diminishing returns that economists generally agree applies to the utility of wealth. This means that, very roughly, the logarithm of dollars is the 'revealed preference' utility.
It was actually more of a joke about that assumption, because it suggests that a 50 dollar meal is preferred four times as much to a 3 dollar candy bar - a bit odd, but perfectly natural if you like candy bars.
Well, log does that. But so does square root also. Lots of functions have diminishing marginal returns.
I can think of two good reasons to model diminishing returns with the natural log.
Logs produce nice units in the regression coefficients. A log-lin function (that is -- log'd dependent, linear independent) says that a percent increase in X results in a <coefficient> unit increase in Y. Similar statements are true for lin-log and log-log, the latter of which produces elasticities.
y=ln(x) and y=sqrt(x) will both fit data in a similar manner, so it makes sense to go with the one that makes for easy interpretation.
Additionally, the natural log frequently shows up in financial economics, most prominently in continuous interest but also notably in returns, which seem to follow the log-normal distribution.
Of course, there's the problem with pathological behavior near 0.
Or the utility of money could quite reasonably be bounded.
Hmm. If we grab some study data on wealth's mathematical relationship with utility, we might be able to decide what function best approximates it. As it is, yeah, there is no reason to prefer log to square root to anything other function.
Oooh, okay. Diminishing returns, certainly. Just not obvious that it would be "log" or near that.
:)