The optimal safety net depends on the heterogeneity of the population?

Comparisons between heterogeneous countries like the US versus Sweden or Finland have always bothered me.  It seems intuitive that all kinds of public policy should optimally depend on the degree of heterogeneity in the underlying population.  A more homogeneous population should be able to provide a better safety net, for example, because there will be less of an incentive problem in providing a minimum level of welfare.   Here is an attempt to formalize that intuition.  I suppose this has been done before.  I don't have the math worked out entirely but it seems right...

Let individuals in the population be defined by a parameter a, which we will think of the individual's ability to create wealth in the market economy. (I do not presume a negative connotation here. This is a very narrow definition of ability -- the ability to create wealth in the market economy.  Great artists might not have much ability by this definition!)  More precisely, each individual has a function

     W = W(E | a)

where W is wealth, a is an individual-specific parameter, and E is effort.

We will assume that dW/dE = a

and let a be normally distributed.

Every individual will have an increasing disutility of effort, but this is the same for everyone.

Each individual will in the market economy choose her optimal level of effort and therefore her optimal level of wealth.  The optimum occurs where the marginal disutility of effort equals the marginal effect of effort on wealth, which is given by a.  With marginal disutility of effort increasing, it will be the case that individuals with higher a choose higher levels of wealth.  This makes sense.  If an individual has the capability to create more wealth, they will choose to do so.  This is the essential heterogeneity I am dealing with.

Now bring in public policy in the sense of a minimum safety net level of wealth, S, or a subsidy that would be available to anyone who has less wealth than S.

If there is no disutility associated with receiving the safety net subsidy, then a rational individual should compare the wealth less disutility they would achieve in the market economy to S and choose whatever is greater.  Since individuals with higher a choose higher wealth, there will be some critical level of the ability parameter, let's denote it a*, below which it will be optimal to elect the subsidy and above which it will be optimal to engage in the market economy.

The final step is to think about how public policy sets S.  There must be some value associated with equity, letting the less able achieve a minimum level of net wealth.  The cost of achieving equity, however, is the loss of effort from individuals who choose the subsidy.  As the base subsidy S increases, several marginal effects occur:  One, the marginal value of increased equity falls, because individuals getting the subsidy are increasingly well off.  Two, the marginal wealth loss increases as more-able individuals drop out of the market economy.  Wealth loss has to be a negative.  And third, as S increases, we move along the normal curve and experience an increasing slope of that density.  This last point is critical, as it addresses the issue of heterogeneity.  I assume that S is further than one standard deviation less than the mean of a.  Since the inflection point of the normal is at one SD away from the mean, that means we are in the range of increasing slope.  So as S increases in that range, more individuals are caught up, and more wealth is lost even if marginal ability were not increasing.

The optimal S, S*, has to balance the marginal benefits of greater equity against the marginal cost of wealth loss.  Remember that with S* there is also an implied a*.  Individuals with ability below a* take the subsidy; individuals above  a* participate in the market economy.

Here is a picture of two normals.  The tighter one represents a more homogeneous society to begin with (Finland, Sweden).  The less tight one we can think of the US.

Now remember we are far to the left on these densities, where the slope is increasing in a (which is on the horizontal axis).

Here is where it starts getting a little tricky, and I will have to check the math by getting the derivative of the normal density as a function of the standard deviation.

But suppose that S* for the tighter distribution, the more homogeneous population, implies an  a* right about where the density curve in the picture above ends.  Then go up to the density for the less homogeneous population, the wider normal curve.  Note that the slope of that curve will be greater than for the other curve -- essentially we are closer to the inflection point, where the slope reaches a maximum.

If the slope of the density for the more heterogeneous population is greater, that means that the marginal cost of increasing S is also greater, for we are picking up more of the population.

All the other marginal effects are the same, for we are at the same level of a.

A higher marginal cost of S means that the more heterogeneous population would optimally choose an S*, and an  a*, less than that of the more homogeneous population.

The optimal safety net depends on the heterogeneity of the population?