« Lesbian Albatrosses | Main | Is a technical background necessary for patent lawyers? »

Tuesday, December 14, 2010

Taxes and Marginal Utility

This post by Stephen Bainbridge quotes approvingly from a WSJ editorial entitled "'Billionaires on the Warpath'?":

In fact, the desire for higher taxes often seems to justify itself solely by the motive to level down. . . . . For all the talk about "fairness," Mr. Obama, Mr. Sanders and their fellow Democrats never really tell us what the magic number for fairness is. Is it 35% of income? 50%? 75%? Though they never commit themselves to an actual number, in each and every case we get the same answer: Taxes should be higher than they are now, for their own sake.

... [T]he politics of higher taxes now rests almost purely on stoking resentment. If Republicans hope to regain the moral high ground, they need to remind citizens that the argument for lower taxes and government that lives within its means is not an argument about numbers or federal revenues. It's an argument about the ability of all our citizens to realize their dreams and opportunities.

Why is anyone who suggests higher taxes on income over $250,000 "stoking class warfare"?  Isn't it a matter of simple economics?

One of the basic principles of economics is diminishing marginal utility.  Marginal utility represents the change in utility from the increase in consumption of a particular good or service.  As you get more and more of a good, your marginal utility with each increment generally decreases.  Eventually, you can have so much that an additional increment adds nothing to your overall utility.  Money is not a good or service, but it represents the ability to obtain goods and services.  And thus one would expect money to have diminishing marginal utility as well.  The more money you have, the less utility you get from each addition dollar.  

So if you're constructing a tax code, it makes sense to keep this in mind.  The lower the income, the higher the utility each dollar represents to that individual.  Since the government is indifferent as to which dollar it takes, it makes sense to take more money that has a lower utility to the taxpayer.  Doesn't $100 mean something different to someone who makes $30,000, as opposed to someone who make $30,000,000?  Isn't that what the parable of the widow's mite is all about?

There are reasons not to have higher taxes on higher income -- they may hurt productivity or investment.  But to claim that their only justification is class warfare ignores some basic economics.

UPDATE: Sarah Lawsky has a paper that challenges the assumption of diminishing marginal utility.  Here's an abstract, which claims: "while some evidence does support declining marginal utility, other evidence suggests that a significant number of people actually experience increasing marginal utility, at least over some range of wealth."  I'm curious about this evidence.  The paper is apparently forthcoming in Minn. L. Rev., but I could not find a copy on the Interwebs.  If anyone has any evidence contra diminishing marginal utility, please chime in.

Posted by Matt Bodie on December 14, 2010 at 05:04 PM in Current Affairs, Tax | Permalink


TrackBack URL for this entry:

Listed below are links to weblogs that reference Taxes and Marginal Utility:


I guess I'm making a variant of the point that Edgeworth made in 1897: The Pure Theory of Taxation, 7 Economic Journal 46–70, 226–238, & 550–571. Greg Mankiw discusses it here: http://www.economics.harvard.edu/faculty/mankiw/files/Spreading%20the%20Wealth%20Around.pdf
Again, I'm leaving out effects on productivity, etc., which makes this a much more theoretical exercise. But I still think the exercise is not about taking the same amount of utility from everyone, any more than it is about taking the same amount of money from everyone. It's about taking money based (in part) on the utility that the person has left. That's why we tax everyone's initial income at a low amount -- that initial income has the most value when it comes to utility. When folks move beyond that, their income get taxed at a higher rate.

Posted by: Matt Bodie | Dec 17, 2010 12:09:48 PM

Matt, in your example, assume that utility=log(income), as Paul Edelman suggested, which does give diminishing marginal utility. In log base 10, the first person you describe starts with $10,000 and ends up with $6700, so he loses log(10,000)-log(6700)= .17 utils. The second person loses log(100,000)-log(67,000)= .17 utils. This is not a crazy coincidence--the loss in utility will always be log(1/(1-rate)), because log(x)-log(y)=log(x/y), so log(income)-log(income*(1-rate))=log(1/(1-rate)). (Actually, many people use ln(income) in models in which they want to posit diminishing marginal utility, because the natural log is so easy to work with.)

In their famous article (later a book) on progressive taxation, Walter Blum and Harry Kalven, Jr., made the same point as Paul Edelman: even under the assumption of declining marginal utility, a principle of "equal sacrifice," or taking the same amount of utility from each taxpayer, does not lead to progressive taxation, whereas "proportionate sacrifice," or taking the same percentage of utility from each taxpayer, almost always does.

However, you may be getting at a larger point, which is that utilitarianism combined with diminishing marginal utility and identical utility functions would require continuing to redistribute until everyone is perfectly equal, because until then, we can always increase overall utility by taking from someone who has more and giving to someone who has less.

Posted by: Sarah L. | Dec 16, 2010 4:39:37 PM

And here's an interesting post from Matt Yglesias on a more streamlined "continuous function" approach:


Posted by: Matt Bodie | Dec 16, 2010 12:33:34 PM

Paul: Maybe I'm being obtuse, but I don't understand what you're saying. Are you saying that a 33% flat tax would result in everyone's utility dropping by the same amount? If so, you do *not* seem to be agreeing with the diminishing marginal utility argument. Sure, the person who makes $10,000 will lose $3300, and the person who makes $100,000 will lose $33,000 -- ten times what the first person lost. But the cut into utility will be much deeper for the person who has to get by on $6700 rather than $67,000. Leaving the first person with $9000 and the second person with $60,000 is better from a utility perspective if we assume diminishing marginal utility, no?

Posted by: Matt Bodie | Dec 16, 2010 12:24:38 PM

I think the marginal utility analysis may show the limits of a subjective utility approach once people have reached a certain level of economic security. I'm sure the Koch brothers approached infinite ecstasy after the Obama-GOP tax deal; that really shouldn't matter. Thoughtful libertarians appear to agree:


The "marginal utility" point also comes up in Charles Karelis's book on the poor:


He brings up the "following scenarios:

"In the first, a poor worker with no car or bus fare must walk six miles to work. And let’s say this long walk results in six blisters, and six unwashed dishes in the sink at home, and workplace mistakes that bring six reprimands from the boss. Suppose too that getting a bus ride for part of the way would reduce the worker’s troubles proportionately, so that each mile she didn’t have to walk would mean one fewer blister, unwashed dish, and reprimand. What will the poor worker give up to get a one-mile ride, given that she still has five miles to walk? Probably not much. After all, the sixth blister, unwashed dish, and reprimand tends to be drowned out, like a shout in a riot, by the other five anyway.

"But now imagine she has just been given a five-mile bus ride, free. She has only one mile left to walk. What will she give up to get a one-mile ride now? Probably much more than in the first scenario because the difference between the discomfort of one blister, unwashed dish, and reprimand and the discomfort of none is far greater than the difference in discomfort between six and five. If the effect of getting a one-mile bus ride in the first scenario is like that of quieting a shout in a riot, in this scenario the effect of the one-mile bus ride is like that of quieting a shout in an otherwise quiet street."

I do get that as an account of increasing utility.

Finally, as for "Leviathan's" insatiability, I highly recommend this writer:

who compares the US economy to other OECD countries. Here's one point from an article of his:

"Using data from the Organization for Economic Cooperation and Development -- the “rich countries’ club” whose members represent most of the world’s leading economies -- Sabina Dewan and Michael Ettlinger showed that between 2004 and 2007, the U.S. ranked 24th out of 26 countries in overall government spending as a share of our economic output. Only Ireland and South Korea, both relative newcomers to the club, had a more “limited government” than we did during that span."

That low tax thing is working out great for Ireland, isn't it?

Posted by: Frank | Dec 15, 2010 5:07:05 PM

If the goal is to tax in such a way that everyone loses the same amount of utility, then one has to assume more than just decreasing marginal utility to get to progressive taxation. For example, if the utility function looks like log(money) (which has decreasing marginal utility) then a flat percentage will result in everyone's utility dropping by the same amount. To get to progressive taxation you either have to assume some more specific utility function or decide that you want to tax at a level that, for example, drops everyones utility by the same percentage. But then what would the economic reason be to do that?

Posted by: Paul Edelman | Dec 15, 2010 3:56:37 PM

Hi Matt, I think your response to the Friedman-Savage curve makes a lot of sense. Friedman and Savage were trying to come up with a reason that people buy insurance and play lotteries that was also consistent with assuming people are rational because, as you note, once you give up the assumption that people are rational, there goes the traditional rational actor economic approach, which Friedman (in particular) really wanted to keep alive.

Anyway, the economic approach is, I think, best understood as a model--roughly speaking, a simplification of the world that, we hope, helps us understand the actual world better. A lot of assumptions and simplifications and plain old wrongness are required to make a useful model (all models are wrong, of course, on purpose). My main point is just that it's important to know when we're making assumptions, and why, and in particular when we're making claims (assumptions) that are normative instead of descriptive.

I suspect that there's no irrefutable argument that answers the question of whether the tax system should be progressive. Maybe the best we can do with hard questions like this is identify our assumptions, which will in turn help us figure out the fundamental normative commitments about which we disagree, which are probably based on things we all learned from our respective parents.

Posted by: Sarah L. | Dec 14, 2010 11:37:30 PM

Interesting, Sarah. I love these sorts of papers (although I haven't read yours yet).

Quick thought: I don't dispute that declining marginal utility is an assumption. But it seems to me the evidence in its favor is a bit stronger than mixed. If I buy two identical eggs, cars, or shirts, I'm not willing to pay more per unit for the second (identical) item. And, looking around, most people aren't. If it were normally the case that the nth unit of a thing provided increased marginal utility to the user, then we'd expect to commonly see people offering to pay more, per unit, for each additional unit. We don't. Right?

Posted by: Brendan Maher | Dec 14, 2010 11:21:02 PM

I'm glad my post making a rather mundane point has generated this terrific commentary. In particular, I would encourage readers to check out Sarah's article and blog posts, which are great fun while being very informative.

But here's my problem with Sarah's argument: I don't buy that the Friedman-Savage work "problematizes" diminishing marginal utility. Friedman-Savage is based almost entirely on the "irrationality" of lotteries, in that you pay more than the expected value of the ticket (based on your chance of winning), and you may accept a lower rate of return for higher awards. I don't think it blows out the rationality model, though, to say that people may be willing to spend small amounts of money to get a very small chance of winning a lot of money. That huge number associated with, say, PowerBall is part of the attraction; it catches the attention. It does not mean that utility rises per dollar between $45 million and $65 million when those dollars are actually doled out. Given the difficulty of processing the differences between remote levels of risk (say, 0.001% versus 0.0001%), we may "irrationally" go for the bigger number even if the expected return is lower. It's all about how we process risk, not how we value money. (E.g., 2008 financial crisis, as to the flip/loss side.) The strange wiggly slalom line that Friedman & Savage concoct is, in my view, no more than a thought experiment that doesn't pan out. And the fact that Friedman & Savage base their curve on *class-based* notions of utility must have Marx smiling somewhere.

I'm also comfortable assuming that everyone has the same baseline of utility, which for me solves the Rich-Penny problem. If you get too empirical here, then you blow utility theory out of the water, because utility can never really be measured. If that's your intent, fine. But then we have to throw out economics in its entirety, unless we make the even more problematic assumption that utility = money.

Steve seems to be making a broader point about the overall level of taxation. I don't think my post really touched on this. I don't believe people should be taxed just to bring down income levels. My only point is that, assuming some level of money needs to be collected to provide for government services, there's an economic argument for raising that money through progressive taxation. And that argument seems fairly noncontroversial from an economic perspective.

Finally, as to q, no, my argument does not depend on the money being reallocated to poorer citizens. If the question is simply how to tax -- how to get money from your citizens to pay for government services -- then declining marginal utility would say that you tax incomes at a higher rate as you proceed up the income level. What the government does with the money is a separate question.

Posted by: Matt Bodie | Dec 14, 2010 10:59:21 PM

If you take an objective view of a life well-lived, rather than assuming all preferences should be taken as given and fully credited, it becomes a good bit more reasonable (though not without needing further argument) to suggest that declining marginal utility is not an assumption but is in fact closer to a normative fact. Such talk, of course, gives crude Benthamites the heeby jeebies...

Posted by: Anon | Dec 14, 2010 10:10:52 PM

"Eventually, you can have so much that an additional increment adds nothing to your overall utility." When? Who decides? So what? Why does that mean the government gets to come along and take it by threat of force? And when the f*ck does the government's marginal utility start declining? At what point has Leviathan ever been satiated?

Posted by: Steve Bainbridge | Dec 14, 2010 7:53:35 PM

"But to claim that their only justification is class warfare ignores some basic economics."

But in this context, he is correct. Your analysis requires that the taxed revenue be allocated to the poor. Yet here we have "progressives" who have completely ignored the unemployment and tax benefits for the poor and middle class that Obama was able to extract from Republicans, and instead criticize him for not raising taxes on the rich.

Raising taxes on the rich should be used as a means for increasing the utility of the non-rich, not for its own sake which for some reason progressives desire.

Posted by: q | Dec 14, 2010 6:43:46 PM

Hi Matt--

(1) You can get a preliminary copy of my article here--I will post a more recent draft on SSRN when I get page proofs from Minnesota. The main point of the article is not that I have knock-down evidence against the assumption of declining marginal utility, but rather that declining marginal utility is indeed an assumption. That is, whether marginal utility declines is an empirical question, and a difficult, open empirical question at that. I discussed this a little before on this blog (here, here, and here), but essentially whether individuals experience declining marginal utility is, I think, a question about the world, not something that can be determined a priori through "basic economics."

(2) Even if you assume declining marginal utility, you have to assume that everyone has the same utility function to get from declining marginal utility to progressive taxation. Otherwise, while it's true that the next dollar for Rich, who's very rich, might have less utility to him than his last dollar did, it might also be true that his next dollar has more utility to him than the next dollar for Penny, who's very poor, has for her. My article doesn't discuss this issue, but in some ways it's even more of a problem for the "declining marginal utility implies progressive taxation" claim than whether people have declining marginal utility.

(3) That said, you can certainly assume declining marginal utility and identical utility functions if you like--if, for example, you have a prior commitment to increasing equality--but absent convincing evidence from the world, making these assumptions is a normative choice that probably starts arguments instead of ending them.

(4) All that said, I suspect (though don't know) that you and I agree on the bigger questions here. For example, I am more than happy to make the declining marginal utility assumption, if we're going to be analyzing tax policy strictly through welfare economics.

Posted by: Sarah L. | Dec 14, 2010 5:55:48 PM

Exactly. Those in the "politic[al]" arena "almost always" go for class warfare. That's a very far cry from saying that "anyone" who wants to tax the rich always is going for class warfare. Try again.

Posted by: confused | Dec 14, 2010 5:49:50 PM

"[T]he politics of higher taxes now rests almost purely on stoking resentment."

Posted by: Matt Bodie | Dec 14, 2010 5:32:13 PM

The article doesn't say that anyone who wants higher taxes necessarily wants to engage in class warfare. The article simply says that the debate is frequently couched in those terms and the article then provides examples. You've misread the article.

Posted by: confused | Dec 14, 2010 5:21:11 PM

The comments to this entry are closed.