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 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.
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.
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 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.