Saturday, March 01, 2008

Declining marginal utility, progressive taxes, and fairness

I recently had an interesting online exchange on the subject of taxation. It started with the question of whether and how progressive the U.S. tax system was--to what extent, if you took account of not only the federal income tax but also payroll taxes and state taxes, rich people paid a higher (or lower) fraction of their income than poor people. We mostly skirted the difficult but important issue of tax incidence--not who hands over how much money but who is how much poorer as a result. With that qualification, the conclusion to which the person I was arguing with agreed was that richer people probably paid a larger share of their income in taxes. His calculations suggested that the proportion varied over about a factor of two.

He argued, however, that fairness required greater progressivity. Asked to explain and defend the basis for that belief, he offered the usual argument for declining marginal utility of income, claimed that fairness required equal utility burdens on rich and poor, and concluded that the tax system ought to be highly graduated.

As some of you may realize, he was making a mathematical mistake. His argument, if true, implies that richer people should pay more dollars in taxes than poorer people. But it does not tell us whether they should pay a larger or smaller proportion of their income.

To see that, consider two taxpayers, one with an income of $40,000/year, one with an income of $80,000. Consistent with declining marginal utility of income, assume that the former has a marginal utility of income of two utiles/dollar, the latter of one utile/dollar. Assume a flat tax which collects $4,000 from the poorer taxpayer, $8,000 from the richer. The utility cost of the tax is then 8000 utiles for each--"fair" by the standard of equal utility burden. The utility cost to the richer person of each dollar he pays is half as much--but he is paying twice as many dollars.

Generalizing this example, we can see that if marginal utility of income declines with increasing income less than proportionally--if, say, MU(I)=AI^(-.9)--then equal utility shares imply a regressive tax, while if it declines more than proportionally to income, the same rule implies a progressive tax.

The next interesting question is whether the rule itself makes any sense. I do not see that it does. I can see a philosophical argument for the claim that everyone should end up with the same income, although I am not convinced by it. I can see a utilitarian argument for redistribution designed to transfer income from those with low MUI to those with high MUI, although I can also see utilitarian problems with such a policy.

But a rule of equal utility cost faces two obvious problems. The first is that taxpayers do not all get the same utility benefit from the state, the second that the state does not get the same benefit from all taxpayers.

Consider a program such as social security which collects money and pays out money. Dollars collected from the richer taxpayer probably cost him less utility than dollars collected from the poorer taxpayer cost him. But dollars paid to the richer taxpayers also provide less utility than dollars paid to the poorer. So a rule of equal utility burden means that the poor are getting, in utility terms, a much better deal than the rich.

Next note that if rich and poor are bearing equal utility burdens, the state is getting a much larger benefit from rich than from poor. If one is going to imagine taxation as some sort of exchange between taxpayer and state, shouldn't costs and benefits to both sides count?

Consider the same standard in a private transaction. I offer to mow your lawn. You are much richer than I am, so--as I kindly point out--in order for the payment to cost you as much utility as the mowing costs me, you will have to pay me a hundred dollars an hour.

Assume, I think reasonably, that an hour of work costs each of us the same amount of utility--say ten utiles. You are saving me ten utiles of lawn mowing at a cost to you of ten utiles of lawn mowing. I am paying you (say) ninety utiles of money--money having a high MUI to you--at a cost to me of nine utiles of money. On net I am giving up nine utiles to get ten, you are giving up ten utiles to get ninety. That does not look like a fair transaction.

I should probably add that it is not clear to me that there is such a thing as a fair tax, even if we are willing to separate questions of fairness from questions of justice. But since many people do believe that there is such a thing and that they know about what it would be I find it interesting to try to make sense of the idea. It's a harder project than they may suppose.


At 2:46 AM, March 02, 2008, Anonymous Anonymous said...

If you define "rich" as people who derive the bulk of their income from investments - they do pay a lower rate than low-income workers. The top tax rate on dividends and capital gains is 15% (and the capital gains tax can often be postponed and/or avoided). The tax rate on municipal bonds is 0%.

Workers face a 15.3% SocialSecurity/Medicare tax on the first dollar earned, plus federal and state income tax with a very low standard deduction (a few thousand). Plus sales tax, property tax, car tax, etc. The only escape is the Earned Income Tax Credit, but by design, most poor people don't qualify for it, or qualify for only a small amount of relief.

It isn't quite that simple. I believe it would be economically proper to attribute the corporate tax rate to shareholders (right?) And muni bonds have an implicit tax rate, although it's lower than the top tax rate on labor income.

I think the people who were arguing with you gave up too easily.

At 3:36 AM, March 02, 2008, Blogger John K. said...

I would list taxes from least unfair to most unfair in the following order: use taxes, property tax on the unimproved value of land, inheritance taxes, gift taxes, luxury taxes, taxes on income above the mean, tariffs, other sales taxes, taxes on income below the mean. (I think the distinction between income above and below the mean is better than the often-comparable distinction between unearned and earned income, in light of current CEO salaries on the one hand and a hypothetical retired person making a modest "unearned" income from previously earned capital on the other hand.)

Use taxes for obvious reasons, land value taxation for the reasons espoused by Henry George, inheritance taxes because it's impossible for a dead person to really own anything, gift and luxury taxes to shore up the inheritance tax as a source of revenue and because gifts and luxury purchases are relatively voluntary activities (though I would prefer we didn't have gift and luxury taxes and simply kept the inheritance tax modest enough that people wouldn't be driven to excessive gifting and splurging to avoid the tax).

If we're going to keep the evil income tax (and I'm afraid it looks like "we" are), then it should only apply to income above the U.S. mean income. Assuming this eminently fair exemption, I see no reason for taxes on income above the mean to be graduated, for reasons you've discussed. (I'd suggest a flat 10% "tithe," since that is all that the church has historically requested and in my view the church has historically done more good and less harm than the state.)

Although all income taxes are bad, taxes on income below the mean is doubly evil, because they prevent lower income people from accumulating the capital that is necessary for financial freedom and security. Every dime the state takes from a lower income person could be better spent -- or saved -- by the person from whom it's taken, on that person's own stability and happiness and therefore the stability and happiness of the wider society. On the other hand, it seems to me theoretically possible (bear with me here, in the realm of theory if not outright fantasy) that a well-run government serving the common good might make a better social use of 10% of an individual's income exceeding the mean than the individual himself might.

At 3:49 AM, March 02, 2008, Blogger John K. said...

BTW, David, I just realized from surfing the web that your old buddy J. Neil Schulman now apparently believes in God and supports the War on Terror. Whaddya think of that?

At 5:23 AM, March 02, 2008, Blogger John K. said...

I forgot to include taxes on property other than land in my list of least unfair to most unfair taxes. Hmmm, where should it go? As a "direct tax" (which the Constitution rightly presumes a problem with) without the natural law justifications of the land value tax, I'd characterize it as the most unfair tax except for taxes on income below the mean (which Adam Smith characterized, in its common "labor tax" manifestation, as the most "absurd" and "destructive" of taxes).

At 8:38 AM, March 02, 2008, Blogger Will McLean said...

David friedman writes:

"But a rule of equal utility cost faces two obvious problems. The first is that taxpayers do not all get the same utility benefit from the state, the second that the state does not get the same benefit from all taxpayers."

Yes. Consider Abe, who earns $15,000 a year and has $1,000 in assets, and Bob, who makes $150,000 a year and has $1,000,000 in assets. One of the most valuable things the government does is protect their property rights from theft, etc. This is worth 1,000 times as much to Bob as to Abe, measured in dollars. Paying for this benefit with a flat tax on income is a much better deal for Bob than for Abe.

At 9:34 AM, March 02, 2008, Anonymous Seth said...

Anonymous, if you make bogus definitions, you can show anything. You are claiming that a retired man getting $20,000 from investments is rich, while a working man earning $500,000 in W2 income (and with no income from investments, he spends it all on high living, art, etc.) isn't.

The bigger problem with the whole argument is that utility for different people cannot be compared; if I measure in utils and you measure in micro-utils, all decisions are the same.

At 11:48 AM, March 02, 2008, Anonymous Anonymous said...

"Anonymous, if you make bogus definitions, you can show anything. You are claiming that a retired man getting $20,000 from investments is rich, while a working man earning $500,000 in W2 income (and with no income from investments, he spends it all on high living, art, etc.) isn't."

What I meant was that if you consider the subset of rich people who derive the bulk of their income from investments (which is a high percentage of the very rich) to be "the rich", then...

Incidentally, the tax system is not uniformly friendly to investors. Think about what happens to money in a bank account earning 15% when the inflation rate is 10%. The real pretax return is 5%, but the 15% nominal income is what is taxed - so the effective tax rate can be greater than 100%.

At 3:17 PM, March 02, 2008, Blogger David Friedman said...

Anonymous points out that tax is on nominal interest, not real interest, which can result in a very high effective rate. The same point applies to capital gains, which means that the effective tax rate there may also be much higher than it appears.

He also mentions that "the tax rate on municipal bonds is 0%." That again is deceptive. Tax exempt bonds pay a lower interest rate than otherwise similar non-exempt bond, so the holder is, in effect, paying taxes in the form of a lower return.

I'm puzzled by the claim that "the top tax rate on dividends ... is 15%." I thought dividends were simply classified as income. Am I mistaken?

At 4:19 PM, March 02, 2008, Blogger Mike Huben said...

Here's a thought experiment for economists accustomed to marginalism.

Presume you're going to auction off residency in the US to the person expecting to earn X dollars who bids the highest percentage of that X dollars in taxes. That would give you the appropriate tax rate. Last person in sets the rate. People who don't like that rate can swap positions with foreigners who do. If Bill Gates doesn't like the bid rate at his income level, then he's welcome to swap positions with somebody (perhaps a Russian oil oligarch) willing to pay that rate.

We very much see something similar with illegal immigrants, who often invest large amounts of money to be able to work here: the equivalent of a tax.

If you think that auctions are fair, then this idea is also fair.

This idea is preliminary and kind of sloppy, but it could perhaps be cleaned up.

At 7:05 PM, March 02, 2008, Anonymous Anonymous said...

"He also mentions that "the tax rate on municipal bonds is 0%." That again is deceptive. Tax exempt bonds pay a lower interest rate than otherwise similar non-exempt bond, so the holder is, in effect, paying taxes in the form of a lower return."

Yes, I mentioned that in the same post.

"I'm puzzled by the claim that "the top tax rate on dividends ... is 15%." I thought dividends were simply classified as income. Am I mistaken?"

You're mistaken. It was reduced a few years ago (2003?)

The rationale was double taxation - companies pay income tax on the money that they pay out as dividends, unlike interest payments which are deductable. Note that they didn't choose the obvious solution of making dividend payments deductable from corporate tax.

At 10:37 PM, March 02, 2008, Blogger happyjuggler0 said...

Fairness is in the eye of the beholder. It is hardly an objective concept.

Fairness in the eyes of a Democrat means that however high the current tax rate is, "the rich" aren't paying enough. Yet. And once that rate is raised, in the next election it will come to light that they still aren't paying enough. Yet. It is a sliding scale over time.

At 5:21 AM, March 03, 2008, Anonymous Anonymous said...

The only fair taxes are those that are completely left to the taxpayer. Otherwise, if a taxpayer doesn't want to pay taxes, there aren't any to be collected. It could work in theory, but the massive needs supported by taxes are too cumbersome to fill if the monies are only supplied when a need arises, and taxpayers are willing to contribute.

At 9:24 AM, March 03, 2008, Anonymous nh said...

David, do you believe there is an objective way to measure someone's perception of value, and thus implement a "fair" tax?

The government doesn't need income taxes because they simply print more money and tax our savings through inflation. Why do you think they implement an income tax at all?

At 12:14 PM, March 03, 2008, Blogger David Friedman said...

nh asks about my beliefs wrt a fair tax and measuring values.

I don't see any good basis for defining a tax as fair. I think we do have some very limited ability to judge utility, even interpersonal utility--for instance, some reason to conclude that a loaf of bread provides more utility to a man who is starving than to one who is wealthy and well fed.

nh also writes:

The government doesn't need income taxes because they simply print more money and tax our savings through inflation. That's wrong in two respects. Inflation doesn't tax our savings, it taxes our money balances. And the tax doesn't bring in enough to come close to replacing income taxes.

Most savings take the form of ownership of assets (indirectly, via stocks) or of loans (bonds, bank deposits used to lend money to home buyers, and the like). Predictable inflation doesn't transfer any of that wealth to anyone, and greater than expected inflation transfers from creditors to debtors, not to the government, except when it is the debtor.

The total income the government can get by money creation in a society like ours is much less than current levels of expenditure, which is one reason income taxes exist.

At 3:06 PM, March 03, 2008, Blogger Charles said...

Quite apart from "fairness" (the meaningfulness of which - at least as the word is typically used in the tax context - I also question), I don't understand "marginal utility of income" per the $40K/80K example. My intuitive interpretation of MUI (and I think that of many other lay persons) is that in terms of satisfying "real" needs, the additional benefits attendant to additional income decrease as total income increases. Eg, the marginal utility (in this sense) of the n-th $40K increment of income might be 0.9^n. Then in the example, the tax system can < fairness concept> relatively painlessly </fairness concept> extract the difference, ie, $4K for the first $40K, $7.6K for the second, etc. (Perhaps more accurately called "utility of marginal income".)

In the SS example, I don't understand the reasoning leading to the conclusion that the richer person loses relative to the poorer person since both the utility cost (at least in the sense just described) of the marginal income loss due to deductions and the utility benefit due to subsequent payments are higher for the poorer person than for the richer person (assuming, of course, that their wealth statuses don't change). In the limit, the marginal utility of both the cost and the benefit for the truly rich person are essentially zero.

And I don't understand the lawn mowing example at all, partly because there appears to be an assumed but undefined mapping between the "utility" of dollars and of work hours.


- Charles

At 3:26 PM, March 03, 2008, Blogger Charles said...

"richer people probably paid a larger share of their income in taxes"

Unfortunately, I don't have the cite but FWIW there was a study a few years ago that IIRC suggested that this percentage rose as expected for incomes up to some high percentile and then actually declined slightly for the highest incomes. I think the percentage peaked in the low 20s.

- Charles

At 12:59 AM, March 04, 2008, Anonymous Nishant said...

actually you have not considered teh fact that the state provides things like security, which have an exceptionally high utility for rich in comparison to the poor.

At 1:41 AM, March 04, 2008, Blogger David Friedman said...

Charles has several questions which I will try to answer:

The marginal utility of income is a function of your income--it's the derivative of total utility wrt income. In my example, I assumed that increasing your income from $40,000 to $40,001 increased your happiness twice as much as increasing it from $80,000 to $80,001 would.Ignoring the fact that MUI will change a little between $36,000 and $40,000, and a little between $72,000 ad $80,000, that means that a $4000 tax on the poorer taxpayer reduces his utility as much as a $8000 tax on the richer.

My SS example wasn't a description of the present system but of what one would have if the definition of fairness I was critiquing were applied. The richer person would be paying enough more to make the utility cost to him the same as to the poorer person, but they would be getting back similar amounts of money, so the utility benefit to the richer would be much less than to the poorer.

In the lawnmowing example, I was assuming that an hour of work was about the same disutility for both people, but losing a dollar was a much smaller utility cost for the richer person.

Hope that clears things up a little.

At 9:17 AM, March 04, 2008, Blogger Charles said...

Prof F -

Thanks, that helped. I was not aware of the formal definition of MUI as a derivative, but it appears that the informal view that I described is effectively equivalent.

Given that deciding what might constitute a "fair" tax scheme seems so challenging (eg, Murphy & Nagel's "Myth of Ownership") and that whether fairness is even the preferred objective is debatable, I tend to be wary of discussions of tax policy that emphasize fairness. That skepticism is clearly appropriate in the political arena, and your interesting post seems to suggest that it may be appropriate even in more promising arenas.

- Charles

At 9:05 AM, March 05, 2008, Blogger Leonard said...

nh, what Dr. F wrote is right, but I think you are grappling with an aspect of the Austrian theory of inflation that confused me for years. Why is it that some prominent Austrian economists make such a big deal of the fact that inflation is a "hidden tax", when by itself inflation, as Dr. F. says, only "taxes" money balances? That is, cash you have in your pocket, or to some extent, savings accounts. (Note that even a savings account is only partially taxed by inflation, because the interest rate is a function of inflation).

In the modern era, most of our savings is not in money. Rather, as Dr. F. says, "most savings take the form of ownership of assets".

But what he did not spell out, is that inflation combined with income taxation does "tax our savings". We have income taxation on nominal capital gains. The values of all non-money assets should be expected to rise, in nominal terms, given sufficient inflation, which the state is perfectly capable of creating. Thus most of the increase in value in the stock market is an increase nominal terms; see this chart:
The DJIA is now about 10x as high as it would have been with 0% CPI-measured inflation since 1945. Much of that 10x gain has been taxed as a capital gain. Thus are savings converted into taxable capital gains by inflation.

At 10:49 PM, March 05, 2008, Anonymous Anonymous said...

A data point:

According to a WSJ article, the 400 top taxpayers (average income: $213.9 million) in 2005 paid an average tax rate on IRS "adjusted gross income" of 18.23%. (AGI doesn't include tax-exempt income).

I think Charles is correct - tax rates increase up to a point, then decline at very high levels of income. The decline is because the tax rate on investment income is lower than the tax rate on labor income.


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