Friday, September 23, 2011

Marginal utility

Frances Woolley asks a question:

In othe words, the reason we care about inequality is that it reduces the happiness achievable from a given amount of income. How much depends upon the happiness/income relationship. Does the marginal utility of income fall rapidly? Or is the happiness from the 100,000th dollar almost as great as the happiness from the 100th?

I think that this question has an obvious answer. Anybody who has lived in poverty can immediately answer that just a few dollars for food can make a huge difference. But how many wealthy executives would feel happy all day over a $3 bonus?

Of course this curve is non-linear. The arguments for inequality that are interesting seem to involve: 1) Whether a class of wealthy investors can stimulate the economy through free market investment and 2) Whether differences in economic outcomes is needed to incentive unpleasant but necessary labor. Now, the later is unlikely to explain Bill Gates or Warren Buffet (are their jobs really so vile?). SO the former really has to do a lot of heavy lifting . . . and is likely overstated.

But an assumption that income effects are perfectly linear seems so contrary to experience that I presume that this post is to stimulate thinking!


  1. My immediate thought is that income is linear on the log scale. I frequently use a log transform when analyzing money data, and a 3% bonus should mean something to anybody on the income scale. This would imply that a flat rate is fair.

    But that would imply that incomes have a lognormal distribution. But I seem to recall that real income distributions are actually more skewed than that (at least I was trying to make income distributions out of lognormals and they didn't look quite right). If this is the case, then maybe even a flat rate is not fair enough.

  2. I also used to model income on the log scale. I agree that it does not handle the skew but it is a much, much better approximation than linear. Defensible, even, and with some intuitive predictions.

    I think slightly greater than flat makes sense. I wonder how far we are from flat if you combine income taxes, FICA, state taxes and local taxes (the first of which is progresssive, the latter of which are regressive).

  3. Joseph - thanks for the link.

    Yes, you're right, clearly the 100,000th dollar isn't worth as much as the 100th. But how much less is it worth?

    Here's another way of phrasing the question: suppose you were redistributing income from a highly paid CEO to a poor person. But you have to redistribute income with a leaky bucket - that is, every time you take a dollar from the CEO, you lose some amount in 'deadweight loss'.

    How much waste are your prepared to accept before it's no longer worth trying to redistribute income. E.g., is it worth taking $1 from the CEO if only $0.20 makes it to the poor person? If only $0.50 makes it? If only $0.01 makes it?

  4. I think there are several income "ranges" and thus a very non-linear curve.

    For the very poor (people without basic food ans shelter), I think efficiency of transfer is a secondary issue. Unless we are in danger of running out of resources, there is no excuse for a rich society not to help the poor (even if the transfer is $0.01 on each dollar of tax).

    Remember Hayek: “I have always said that I am in favor of a minimum income for every person in the country.”

    For the very wealthy, those who make more than they can possibly consume (or their heirs consume) then I am not overly sympathetic to losses of income. Here I am talking about incomes of $100 million per year plus. People whose incomes is 1000's of times greater than a median worker. Nobody reaches this level of income without leveraging the structure of society as an individual's productivity simply cannot exceed that of a medium city.

    It is the middle range (notice the tremendous layering of non-linearity here), I seem to assume that log transforms of income seem to be a good reflection of marginal utility.

    BTW, Frances, I love Worthwhile Canadian Initiative and the great set of posts that show up there.