Looking at the math of dividing a fixed pool of resources among a non-fixed number of people, a feature of log(r) that matters a lot is that log(0)<0. The first unit of resources that you give to a person is essentially wasted, because it just gets them up to 0 utility (which is no better than just having 1 fewer person around).
That favors having fewer people, so that you don’t have to keep wasting that first unit of resource on each person. If the utility function for a person in terms of their resources was f(r)=r-1 you would similarly find that it is best not to have too many people (in that case having exactly 1 person would work best).
Whereas if it was f(r)=sqrt(r) then it would be best to have as many people as possible, because you’re starting from 0 utility at 0 resources and sqrt is steepest right near 0. Doing the calculation… if you have R units of resources divided equally among N people, the total utility is sqrt(RN). log(1+r) is similar to sqrt—it increases as N increases—but it is bounded if R is fixed and just approaches that bound (if we use natural log, that bound is just R).
To sum up: diminishing marginal utility favors having more people each with fewer resources (in addition to favoring equal distribution of resources), f(0)<0 favors having fewer people each with more resources (to avoid “wasting” the bit of resources that get a person up to 0 utility), and functions with both features like log(r) favor some intermediate solution with a moderate population size.
Looking at the math of dividing a fixed pool of resources among a non-fixed number of people, a feature of log(r) that matters a lot is that log(0)<0. The first unit of resources that you give to a person is essentially wasted, because it just gets them up to 0 utility (which is no better than just having 1 fewer person around).
That favors having fewer people, so that you don’t have to keep wasting that first unit of resource on each person. If the utility function for a person in terms of their resources was f(r)=r-1 you would similarly find that it is best not to have too many people (in that case having exactly 1 person would work best).
Whereas if it was f(r)=sqrt(r) then it would be best to have as many people as possible, because you’re starting from 0 utility at 0 resources and sqrt is steepest right near 0. Doing the calculation… if you have R units of resources divided equally among N people, the total utility is sqrt(RN). log(1+r) is similar to sqrt—it increases as N increases—but it is bounded if R is fixed and just approaches that bound (if we use natural log, that bound is just R).
To sum up: diminishing marginal utility favors having more people each with fewer resources (in addition to favoring equal distribution of resources), f(0)<0 favors having fewer people each with more resources (to avoid “wasting” the bit of resources that get a person up to 0 utility), and functions with both features like log(r) favor some intermediate solution with a moderate population size.