The Marginal Value of Money

Hand a bum a $20 and he’ll party for a day. Hand a billionaire a $20 and he’ll be annoyed at your wasting his time. Money, like everything else, is subject to the Law of Diminishing Returns.

This, dear liberals, is your best justification for wealth redistribution. You need not resort to Keynesian quackery. Oligarchy can “work.” The Negro servants of the antebellum South had full employment. It was not a happy arrangement on average, however. (And it is important that you do drop your dependence on Keynes, as he rationalized subsidizing the rich and destroying the environment as much as he rationalized wealth transfer programs.)

And you, dear conservatives and libertarians, need to come to terms with the marginal value of money. Your dreams of a redistribution-free society are doomed to failure. Democracies attempt to be utilitarian. The promise of greatest good for the greatest number wins the most votes. It’s practically a tautology. Either dispense with democracy or deal with this reality.

This is not to say that liberals and other leftists are always right. The New Deal split the middle class into the prosperous and paupers. Dictatorships of the Proletariat are less egalitarian than advertised; it is better to be commissar than comrade. And general prosperity can offset inequality to a significant degree.

But there are tolerably effective redistribution schemes in place, schemes compatible with democracy and prosperity. The Scandinavian welfare states are doing rather well these days. So are some of the social democracies of Western Europe. Their fate is our future unless conservatives and/or libertarians come up with something better. That is, if we are lucky. If conservatives slavishly follow the talking points of Fox News, they doom themselves to eventual irrelevance and we might skip the Western European fate and jump straight to the Venezuelan option.

I’d rather this not happen. I prefer to have at least two viable political parties keeping each other semi-honest. I would also like the United States to remain a haven for the ambitious and the diligent. So, over the next few months I’ll present some egalitarian options compatible with family and economic liberty. (Liberals in the audience might find some of these options useful as well.)

That said, if ye on the Right can credibly propose a bigger overall economy, ye need not be as egalitarian as the left. But how much growth is needed to offset an uptick in the GINI Coefficient? Let’s run some rough numbers.

Money vs. Utility

We start our investigation of equality vs. prosperity with a passage from a skeptic of redistribution, and of government in general: the famous anarcho-capitalist David D. Friedman. In his excellent Price Theory textbook he wrote:

From both introspection and conversation, I have formulated a general law on this subject. Everyone feels that there is a level of income above which all consumption is frivolous. For everyone, that level is about twice his own. An Indian peasant living on $500/year believes that if only he had $1,000/year, he would have everything he could want with a little left over. An American physician living on $50,000/year (after taxes) doubts that anyone has any real use for more than $100,000/year.

Even as he pooh-poohs far left notions of “enough”, he implies a rule for the diminishing marginal value of money. Over a wide array of incomes, the value of “enough” is twice that of the person defining “enough.” The value of money is relative.

In this general law the value of money is inversely proportional to current income. If you are making $500/year, then an additional $1/year is 1/500 of “enough.” If you are currently making $100,000/year, then an additional $1/year is only 1/100,000 of “enough.” As Friedman writes in the next paragraph, this is a matter of limited horizon:

Both the peasant and the physician are wrong, but both opinions are the result of rational behavior by those who hold them. Whether you are living on $500/year or $50,000/year, the consumption decisions you make, the goods you consider buying, are those appropriate to such an income. Heaven would be a place where you had all the things you have considered buying and decided not to. There is little point wasting your time learning or thinking about consumption goods that cost ten times your yearly income, so the possession of such goods is not part of your picture of the good life.

Still, I think that this idea of money having a marginal value inversely proportional to current income is a good rule of thumb – if we apply it to small increments. That is:

Where du is our marginal change in utility, m is income and c is a constant of proportionality. Our choice for c is inherently subjective, but the choice has a limited effect on the results to follow. All we are claiming with this formula is that an additional dollar for a doctor making $100,000/year is worth a tenth that of an additional dollar for a burger-flipper making $10,000/year – on average. (For a really greedy doctor vs. an eco-hippie burger flipper, the difference will be much less.)

Given this rule of thumb for marginal utility vs. incremental changes in income, we can begin to ask how much better off the doctor is than the burger flipper. (To be precise, we are asking how much more utility the doctor is getting from his job. Work itself has a cost as well. Would you rather flip a burger or deal with sick people?) To answer this question, let’s apply this formula in dollar income increments:

This looks hard to so. However, if we make our increment even smaller, we can get a closed formula using undergraduate calculus:

We have a formula for the difference in utility gained by going from $10K to $100K per year, but we have no notion of absolute relation between utility and income. Without it, we have no way of saying whether the doctor is twice as well off as the burger flipper or a hundred times better off.

It’s time to calibrate further, and get rid of that pesky constant. For example, suppose we accept the idea of a Living Wage, and define the Living Wage for a single person living in Johnston County NC a utility of 1. Using this handy-dandy calculator from MIT, we multiply $10.15/hour by 2080 hours/year to get $21,000/year (rounded). Let us define a utility of 2 being the Living Wage to support two adults in Johnston County: $15.53/hour time 2080 hours/year = $32,000/year. We can plug into our prior formula and get rid of c and have a scale for total utility.

This gives us the formula:

u = 1 + 2.37 ln(m/$21000)

Or, if you prefer to work in base 10 logarithms, we can convert using lnx = (ln10)logx which gives:

u = 1 + 5.46 log (m/$21000)

Different normalizations of utility will give a different multiplier of the logarithm and a different reference income inside the logarithm. We are stuck with a degree of arbitrariness. But the general shape of the curve stays the same. It looks like this:

Utility is roughly the logarithm of money

No matter how you calibrate it, the curve goes to negative infinity as m goes to zero. Those who use per capita incomes to compare the wealth of nations are implicitly assuming a utility curve that looks like this:

The difference is nontrivial. A bit of welfare transfer can be worth some pretty big losses in total economic growth. That’s the qualitative takeaway. To actually compare states or nations, we have to handle some subtleties. I will do so in a future article. This one is getting too long as it is. So let’s wind up with a couple quick takeaways, while leaving it as an exercise to the reader to fill in the missing steps.

Hippytopia beats Galt’s Gulch. In Hippytopia no one is rich. Most of the time you walk or ride a bicycle. The beat up van is for special occasions. But everyone has enough for organic vegetables, bell bottoms, beads, Birkenstocks and weed. In Galt’s Gulch the billionaires float around in hovercars and take vacations on the moon. The average income is ten times Hippytopia. However, the unproductive live in shantytowns next to the toxic waste dump. Add up the utilities and Hippytopia beats Galts Gulch.

Overall income is an equalizer. Conversely, if two countries have identically shaped income distributions, the country with the higher average income is more equal. Growth is equalizing. The GINI Coefficient does not tell the full story. A semi-libertopia with a decent safety net can beat Hippytopia hands down.

I’ll put some numbers to these statements in a future article. For now, I leave it as a homework assignment.

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