A formal proof of the theorem that a decrease of the total gross wage sum is associated with a denser distribution of critical values in a population, like the dotted distribution in figure V would require the introduction of a utility function that does more than merely identify a minimum subsistence-threshold. This would require the introduction of a distribution of parameters of P utility functions. So far, we have not been able to give this a simple treatment. It should be observed that if the population's job preference is negligible relative to income preference, then everyone simply maximizes his income: s = 0, m = 1 and the optimal w/t = 1. The marginal utility of income can increase by taste or, if we assume decreasing marginal utility, by decrease of income.

There is a relationship of s to the more familiar concept of the "supply-elasticity of labour", but it is not straightforward: for the labour supply S as a function of w/t, labour supply elasticity e is defined by

Given some Nm _,s , e is not constant over all function values. A standard deviation s `> s implies a larger elasticity "in the center of the distribution", that is, where w/t is between m -s and m + s (these are the two function values of w/t, where

and a smaller elasticity for any w/t outside this this interval <m -s , m + s >.