David R. Henderson  

Barro on Marginal Utility and Wealth Transfers

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In an otherwise good article defending Tyler Cowen, Josh Barro states the following:

There is declining marginal utility of money: A person who makes $10,000 gets more value out of an extra dollar than a person who makes $100,000 does. So, transferring wealth from rich people to poor people makes the public better off in the aggregate.

Tyler writes that Josh "is completely correct." Actually, Tyler and Josh are wrong. I went on Tyler's site to say why but for some reason it didn't accept my comment and when I tried again, it suggested that I was commenting too frequently.

Then it occurred to me that this would be a good "teachable moment," as Barack Obama likes to say. So find the flaw in the above quoted statement from Josh. HINT: It has nothing to do with ideology; it's straight mainstream economics.


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COMMENTS (58 to date)
Anonymous writes:

Utility is subjective

Taras Smereka writes:

The wealth transfer affects incentives, decreasing the higher income individual's willingness to live a high marginal product lifestyle and supply resources to the low marginal product individual.

The wealth transfer also decreases the willingness of the low marginal product individual to increase his income.

Additionally, if the donation of resources from the high income individual to the lower income individual was utility maximizing, the high income individual would have a high propensity to do it willingly without government intervention.

David R. Henderson writes:

@Taras Smereka,
Your first two paragraphs are correct and your last one is not. But none of these is the answer.

cedric writes:

Taras, I think that's right. One of my philosophy professors made Barro's point, and then I replied with your point. Then everyone in the class called me a fascist or something.

Eric writes:

Utilities aren't additive across individuals (or at least there's no way to be sure that they are). Since behavior is invariant in utilities under U*=a+b*U, it's possible that decreasing marginal utility for each individual doesn't imply that two different individuals would desire this transfer.

tom writes:

Hmmm- I'd go with S&D. Transferring wealth between the two increases demand for products likely to be bought by the person making 10,000 which drives up prices on those products which decreases the MU for all others making 10,000.

Jason Brennan writes:

In addition to some of the points above:

If I give a beggar a dollar, he buys food and eats it. If you give Steve Jobs another dollar, he'll invest it. The rich are less likely to consume their extra dollars and are more likely to invest them. They are thus more likely to transform those extra dollars into even more dollars and thus make the public better off in the aggregate.

Brian Albrecht writes:

Interpersonal comparisons are not possible, even with something as universal as money. There is no way to know whether person A will enjoy a good more the person B, no matter what their income.

Even Bentham recognized this point and stopped his utilitarian analysis at the individual.

Foobarista writes:

I suspect the magic bit is "gets more value". The value of money (at least at a given time) is a constant and isn't a function of one's net worth.

Luke writes:

Foobarista: I think that "value" is ambiguous. It could be stated that a head of cabbage has a value of one dollar no matter who is willing to pay the dollar. But certainly someone with no food will value the head of cabbage more than someone with a cupboard full of food.
Perhaps the answer lies in the marginal propensity to save/consume, and long run v short run.
The term "mainstream" is somewhat ambiguous, as well, so we cannot really be certain of the size of the pool from which we gather our guesses.

Jared writes:

1. Utility, at least to economists, is ordinal and not cardinal. It's a function that produces a ranking of bundles of goods, states of the world, or what-have-you. To say that U(x)>U(x') is just to say that you prefer x to x', but there is no sense of "how much" more you prefer it (due to the point made by Eric).

Without cardinal utility, there's no way to compare two people's utility, and so there's no way to make the claim Barro is making.

2. Even if we ignore (1) and let utility be cardinal, there's no guarantee that declining marginal utility of money for each person means that transferring from rich to poor maximizes welfare. It is entirely possible that the rich, even though they have high total utility (as U(money) is probably everywhere increasing), have a very high marginal utility as well. And though I personally have difficulty believing it, it is certainly possible that there are poor people whose marginal utility an additional dollar is quite low, even lower than that of a much wealthier person. In such a situation, taxing the poor person and giving it to the rich increases total utility.

Those, I suspect, are what Prof Henderson is looking for. Yes? In the ballpark?

Foobarista writes:

You're right, at least in non-economic English. But as I recall, "value" in economics is fairly precisely defined, as is "utility", and they aren't the same thing. The market price or value of the cabbage is the same whether a starving person buys it or Bill Gates buys it after a big meal, but the utility to the starving person is obviously greater.

Wallace Forman writes:

Even economists have a hard time getting this though their heads. Here is another person who makes the mistake:

http://m.theatlantic.com/magazine/archive/2011/12/i-was-wrong-and-so-are-you/8713/

And in an article talking about how people don't get economics, no less.

August writes:

Cowen and Barro are making claims that they cannot possibly know.

Yes it might be possible that the person making $10,000 might value an extra dollar more than a person making $100,000. But it is also equally as plausible that the person making $100,000 might value an extra dollar more than the person making $10,000. There is a declining marginal utility of money, but that doesn't mean one can compare marginal utilities between people. Trying to compare the marginal utility of money between person A and B is meaningless because it doesn't tell you anything relevant.

What Cowen and Barro are saying is that they have super human knowledge in being able to identify the marginal utility of a dollar for a rich and a poor man, compare the two values, and then make relevant economic decisions that improve society. So my question is - how do you have the ability to identify these two values?

Trey writes:

The last sentence ignores the cost of the transfer. Even if you're charitable and assume that you can compare utilities across people and that, all else equal, the public is better off in aggregate under more equal distributions, the last sentence might not be true. The gains from giving the money to people with higher marginal utility of money must outweigh the dwl of the transfer.

MikeP writes:

Jason Brennan has it best.

On average in a static world, transferring a dollar from someone earning $100,000 to someone earning $10,000 "makes the public better off in the aggregate".

But the world is not static, and the person who has $100,000 is probably more likely to know how to invest the marginal dollar to "make the public better off in the aggregate" than the person who has $10,000 is.

Mentha Trecenta writes:

[Comment removed for using crude language. Email the webmaster@econlib.org to request restoring your comment privileges. A valid email address is required to post comments on EconLog and EconTalk.--Econlib Ed.]

Terrier writes:

The quote from Barro confuses income with wealth.

While it may be true that the $10,000 person's marginal utility of income is greater than the $100,000 person's marginal utility of income, this statement does not therefore mean that transfers of wealth from the $100,000 person to the $10,000 person will make society at large any better off.


Patrick writes:

Along the lines of what a couple others have said, while the individual with an income of $10,000 may personally value that marginal dollar more, there is no reason to believe his use of that dollar will be more valuable TO SOCIETY.

Wallace Forman writes:

Brian and August have the correct answer.

Cody writes:

There is diminishing marginal utility of wealth, not income. The numbers Barro uses are for yearly incomes but he doesn't say anything about how rich or poor the folks otherwise are.

Fmb writes:

Can't be Brian and August. Best their point can do is say Barro might be wrong, with some p that most would say was strictly less than 1/2.

Other guesses in this thread seem to have approx the same flaw.

Torben writes:

It might seem plausible to assume that due to declining marginal utility a sufficiently richer person values an additional dollar less than a poorer person, but it is not necessarily true. Everybody should be aware of this after taking only one (micro)economics course...

Jody writes:

1) Considering utility functions:
a) It's obviously not Pareto efficient.
b) Assuming a simple sum utilitarian metric, it may not increase that either as studies have shown that those surveyed with the greatest desire to make more money tend to be the ones making more money when they're older.

2) Considering GDP as the metric, the S=I identity, and that we're transfering savings:
a) You're robbing I for C, so there's no benefit.
b) Real growth comes from I, not C, so decreasing I is deleterious for longer term social welfare.

3) Let income = C + S (no wealth drawdowns or debt). So if we're not taking savings, then we're just trading one type of consumption for another.

4) If we're literally consuming wealth in the transfer (i.e., taking stock instead of diverting flow), then we're making society less wealthy with the transfer from wealth to consumption.

Then as an aside that is less textbook (i.e., non-definitional), I subscribe to Say's law so I find the underlying theory for wealth transfers increasing production bass ackwards.

Eric Hosemann writes:

When taking from B to give to A, from the perspective of the economy as a whole, A's gains are offset by B's losses.

Charles R. Williams writes:

Utility to whom? The notion of utility has to do with what choices some rational agent makes. To talk about utility as Barro does is nonsensical since he doesn't specify a rational agent as chooser.

What he is doing is implicitly treating some abstract reader of his comment as the rational agent and to make unfounded assumptions about this hypothetical agent's preferences.

MikeP writes:
So, transferring wealth from rich people to poor people makes the public better off in the aggregate.

Seriously, read that sentence all by itself. It is quite obviously empirically wrong, and societies constructing themselves around this precept have produced inconceivably hellish results for "the public".

But that's not mainstream economics: that's just history. Refining my earlier answer more strictly toward mainstream economics might yield...

Because the marginal dollar has less value to the person making $100,000 than to the person making $10,000, the former is more likely to invest it than consume it. And such marginal investment is what actually "makes the public better off in the aggregate" as time progresses.

David R. Henderson writes:

I’m on the West Coast and I just got up.

Brian Albrecht and Jared got it. You can’t make interpersonal utility comparisons. Utility is ordinal, not cardinal.

Philo writes:

First, Barro should have said, “declining marginal utility of *wealth*.” His point has nothing to do with *money*, as opposed to other forms of wealth, and it has nothing to do with *income* as opposed to *wealth*. Furthermore, he should have specified that the two people who differed in wealth were *otherwise exactly alike* as regards their circumstances and their propensity for happiness. Really, the point is counterfactual, comparing two different scenarios, involving *a single person* (qualitatively the same in the two scenarios), which differ only as regards the individual’s wealth. Finally, to get his redistributive conclusion Barro assumes that the redistribution takes place accurately and costlessly, which would not be the case in practice.

Bill Nichols writes:

Treating ordinal as rational data is a common mistake. I see two other lessor problems even if we assume an underlying rational scale.

First, individual utility need not sum to society utility. There was a leap from individual to societal with an assumption that the distribution has no global effects on society. This just restates some other arguments.

Second and more seriously, prospect theory is pretty mainstream. If we reframe the transfer of the additional dollar to a loss and a gain, it is no longer obvious that the (assumed-rational) utility sum would increase.

Seth writes:

Love the straw grasping here. Wouldn't it be easier to, for example take a survey or conduct an experiment to see who wants the dollar more. Another approach would be to compare incomes in countries with highly redistributive programs. You can mark your theories to market. Also, for social policy you don't need some magic method to aggregate a non-continuous utility function, It can be measured in the aggregate where it is likely to be much smoother.

Roger writes:

In general I am willing to accept that on average, an additional dollar has more utility to those with $10,000 than those with $100,000. Let's not haggle over whether it is wealth or income or whether it is true for any given individual.

The reason forced transfers don't necessarily increase utility are as follows:
1). We are now comparing earned dollars to gifted dollars the utilities are different
2). People place a higher value (roughly double) on losses vs gains. Thus we are comparing lost utility to gained utility and relationship works the other way.
3). It reduces the incentives to earn on both parties. The greater the redistribution, the greater the effects.
4). Since wealth creation is a positive sum game in free markets, the reduced incentives will result in lower utility for those that would have interacted with the previously dis incentivized party.
5). Transfering income is not a cost less process. Further utility is destroyed in the process.
6). Most importantly of all though, the redistribution process creates winners and losers. The natural reaction is for the winners to invest time and money in getting more money, and for the losers to spend time and money to resist it. You will create arms races of offense and defense with middle men. It is entirely possible for the process to self amplify where more is spent in defense and offense than is actually transferred. (they could both approach $.99 spent in offense or defense to avoid or gain $1) Arms races even encourage proactive unrelated assaults or attacks on the other party to be used to weaken their position or to be used as a bargaining chip. Zero sum interactions spin out of control and tend to destroy value and utility.

To ensure utility is increased, it is essential that we create institutions which make the transfer voluntary, at least at the meta level (meaning that the parties agree at least to the rules of the game. Not necessarily every transfer).

Julien Couvreur writes:

Ok, the problem is that subjective value does not permit interpersonal utility comparisons.

I understand that from a rigorous and scientific point of view.

Yet, on an intuitive level this is hard to shake. Obviously people at large make this kind of comparison, which leads to all sorts of fallacious cost-benefit analyses.

I wonder whether this bad reasoning results from education (or lack of economic education) or results somehow from our evolutionary wiring.

Taylor Harris writes:

Simple. Dead weight loss. (In addition to incentive distortions)

Robert writes:

What if neither A|$100k nor B|$10k want to live in a society in which C expropriates from A to give to B?

Steve Z writes:
What if neither A|$100k nor B|$10k want to live in a society in which C expropriates from A to give to B?

That doesn't seem to be the world we live in.

Robert writes:
That doesn't seem to be the world we live in

Yes, my bad.

Floccina writes:

It is true for most individuals that the more money you have the less you value an additional dollar but people are different and a rich guy may value an addition dollar more.

BTW: Presumably if poor people valued money more they would work more for it and have more of it. Interestingly Democrats will say that poor people have more demand for money but they will also say that poverty makes people work less at school.

MingoV writes:
A person who makes $10,000 gets more value out of an extra dollar than a person who makes $100,000 does. So, transferring wealth from rich people to poor people makes the public better off in the aggregate.

The first sentence of the quote is correct, but the second sentence does not follow from the first.

Scenario A: Both individuals earn an extra dollar. Both spend that dollar quickly. In this scenario, the dollar had utility for both individuals and probably more utility for the $10,000 earner than the $100,000 earner.

Scenario B: A dollar was taken from the $100,000 earner and given to the $10,000 earner. For the $100,000 earner, the negative utility of having a dollar taken away and given to someone else exceeds the positive utility in Scenario A. That individual also has less incentive to earn more dollars. For the $10,000 earner, the gifted dollar has the same spending utility as the earned dollar in Scenario A, but the gift reduces the drive to earn more money. The negative effects (both have reduced drives to earn money, and the $100,000 earner loses spending power) outweight the one dollar benefit to the $10,000 earner.

What about incentive effects and the bureaucratic vig? The $100k earner has an incentive to spend his own money more carefully than the $10k earner that gets handed an unearned dollar. Plus the bureaucrats take their cut in the transfer.

Also, the utility of money seems strange to me. It's just a store of value and has the same utility for either person. It stores a dollar's worth of value in either case. It's the goods and services that can be exchanged for that might have different utilities, no?


As a final note, I'm not a fan of discussing 'aggregate utility' for the 'public'.

Les Cargill writes:

SFAIK, the only use of utility in this case is that there exists a nonzero probability P that a dollar transferred from a richer person to a poorer person will increase money velocity. If we agree to think in aggregates, in the manner of Scott Sumner, then increased velocity is a good and it's largely public ( without being a public good ).

If we otherwise hold that preferences are the thing and are inscrutable, then there should only be private money and it's all likewise private. The only use of macro is then score-keeping.

That seems to be the thesis and the antithesis and I don't see a synthesis on the horizon. One side seems to bash evolutionarily tribal "equality" by saying such evolutionarily based tribalism is non-information, and the other seems forced to claim utility has properties it lacks.

JohnE writes:

David,

Ordinality has nothing to do with it. Even when preferences are cardinal (as with expected utility), interpersonal comparisons of utility are meaningless. This is because cardinal preferences are unique up to positive affine transformation. Thus you can rescale any cardinal utility to get any marginal utility you want (for some fixed wealth level).

Eric writes:

What he said (meaning JohnE)

Tom West writes:

You can’t make interpersonal utility comparisons.

Nonsense! Anyone involved in selling anything does this all the time. It's how they decide at which price to sell their product.

It's just that you can't be right 100% of the time. You can, however, be right 99% of the time, which is good enough, at least when talking about reality rather than philosophy.

That said, it's a bit surprising that economics professors would make this mistake.

Philo writes:

“You can’t make interpersonal utility comparisons. Utility is ordinal, not cardinal.” I would like to see you try to support this claim. My prediction: your attempt would be unconvincing.

R. Jones writes:

Some people like to earn dollars, some people simply like to have dollars, and some people like to spend dollars. These are distinct.

You generally have to earn dollars in order to have them, and have them in order to spend them.

It may be true that the rich value earning dollars more and that is why they earn more. I assume that this was Mr. Henderson's point.

But it's still possible that the poor get more utility from having and spending dollars, but not earning them.

David Friedman writes:

(I haven't read through the comments, so don't know if someone else has already made these ones)

1. Obviously, the transfer might impose a deadweight cost larger than the utility gain.

2. Less obviously, it isn't clear that we should expect rich people to have lower marginal utility of income than poor people--because whether you are rich or poor is not a random selection.

Consider two simple models

A. Everyone has the same utility function for income and for leisure; people differ in their ability to convert leisure into income. That's the model implicit in the standard argument for redistribution, since it leads to marginal utility declining with income across individuals.

B. Everyone has the same ability to convert leisure into income; people differ in their utility function for income but not in their utility function for leisure. Now the conclusion reverses--the rich people are rich because they value income more than the poor people, and it's straightforward to show that, in equilibrium, marginal utility of income increases with income across people.

In both models, both marginal utility diminishes in the usual sense for both income and leisure.

Maurizio writes:

"Marginal utility of money is decreasing" means that the Nth unit of money allows me to accomplish ends that are more valuable to me than the ends I accomplish with the N+1-th unit. And this in turn means that I prefer the Nth unit to the N+1-th unit. And this is tautologically true: if I buy X with the Nth unit of money and Y with the N+1th unit of money, it is obvious that I prefer X to Y, otherwise I would have bought Y before X.

Philo writes:

The following comment is obviously quite intelligible:

“I slightly prefer spinach to kale, and I greatly prefer either to asparagus.”

But this requires that preference, and thus utility, be cardinal, not merely ordinal.

John T. Kennedy writes:

David,

You write: "You can’t make interpersonal utility comparisons. Utility is ordinal, not cardinal."

Doesn't that rule out most of what passes for economic efficiency (Everything beyond Pareto efficiency)?

Ken B writes:

@Philo: David is using 'cardinal' a bit loosely to avoid getting into boffinish detail. I think th precise term in this instance is "interval data". Utility is not interval data so it cannot be added and subtracted in the way Barro does it. And even if *your* utility were interval data for the purposes of your own decisions that would not mean yours and mine are commensurable.
That's an extra, rather dubious, assumption.

Andy writes:
"Marginal utility of money is decreasing" means that the Nth unit of money allows me to accomplish ends that are more valuable to me than the ends I accomplish with the N+1-th unit. And this in turn means that I prefer the Nth unit to the N+1-th unit. And this is tautologically true: if I buy X with the Nth unit of money and Y with the N+1th unit of money, it is obvious that I prefer X to Y, otherwise I would have bought Y before X.

This is not always true:

a) Minimum amount needed for living. For example, say you will die unless you get a medical procedure that costs $100,000. The utility of the first $99,999 of money to you far less than that of the $100,000-th dollar. For a less extreme example, if you can't afford enough food to survive, any money you have is useless.

b) Complementary goods. For example, say your car costs $10,000 and gas costs $100. Buying each of them without the other is useless. So just because you buy one "first" doesn't mean you prefer it to the other.

c) Relative comparisons. For example, you might be really happy if you make more money that your neighbor, and otherwise not very happy. In this case your marginal utility of money increases at the level of your neighbor's income.

There are many other situations as well.

Glen writes:

I've seen a lot of answers here, but I'm still waiting to hear what Henderson thinks is the error.

FWIW, if you read the original Barro article, you'll see that Barro himself said that higher marginal tax rates can discourage effort and production. So that couldn't be the error.

With respect to the impossibility of interpersonal utility comparisons, I'd say that's a philosophical/metaphysical position rather than a matter of settled economics. Lots of economists would dispute it. So even if it's true, I wouldn't call it an error according to "straight mainstream economics." (And I say that as someone who is sympathetic to that position.)

So my bet is that Henderson is focusing on the confusion between income and wealth.

David R. Henderson writes:

@John T. Kennedy,
Doesn't that rule out most of what passes for economic efficiency (Everything beyond Pareto efficiency)?
Good question. Not quite. If we use a compensation test--a change is efficient if the winner could use some of the gain to compensate the loser--then no. But we sometimes get sloppy in class--I know I do--and fail to tell students that there are definite losers. That they “could be” compensated by the gainers is small comfort.
@Glen,
I've seen a lot of answers here, but I'm still waiting to hear what Henderson thinks is the error.
I did answer.
You wrote:
With respect to the impossibility of interpersonal utility comparisons, I'd say that's a philosophical/metaphysical position rather than a matter of settled economics. Lots of economists would dispute it.
I don’t think that’s true.

Philo writes:

@ Ken B:
The assumption—interpersonal comparability of utilities—may be rather dubious, but it is not *obviously false*, as David implies.

Maurizio writes:

Andy, thanks for the insights.

a) Minimum amount needed for living. For example, say you will die unless you get a medical procedure that costs $100,000. The utility of the first $99,999 of money to you far less than that of the $100,000-th dollar. For a less extreme example, if you can't afford enough food to survive, any money you have is useless.

This only shows that 1 dollar cannot be chosen as a proper unit of money. You should have chosen 100.000$ as the unit. The second 100.000 are less valuable to you than the first 100.000, and so on.

in general, when I say "the Nth unit of money is less valuable to you than the N-1-th unit", this does not mean that you are free to choose 1 dollar as the unit of money.

to constitute a "supply", units of a good must be interchangeable and homogeneous. If the N-th unit produces your life and the N-1-th does not, this means they are not the same good. They are different goods, just like "strawberries in August" and "strawberries in December".

You can find a more detailed discussion of this in "Man, Economy and State", here and here.

b) Complementary goods. For example, say your car costs $10,000 and gas costs $100. Buying each of them without the other is useless. So just because you buy one "first" doesn't mean you prefer it to the other.

Again, the law only holds for interchangeable units of a supply of a good. a car and gas are different goods.

c) Relative comparisons. For example, you might be really happy if you make more money that your neighbor, and otherwise not very happy. In this case your marginal utility of money increases at the level of your neighbor's income.

This looks like the first objection. an additional dollar which enables you to reach the income of your neighbor cannot be considered as interchangeable with a dollar which does not. They are different goods, just like "strawberries in August" and "strawberries in December".

JH writes:

Preferences are heterogeneous.

Greg Jaxon writes:

The abstract idea of money is that it has constant marginal utility. This is why economists' charts can have axes labeled with "price", "cost" and various money-based measures: this sets subjectivity aside for the purpose of analysis; units of money are fully substitutable throughout the algebra of economics.

Of course, any physical (or electronic) form of money in a real economy has a diminishing marginal utility to its various users. But it has been chosen as the money because it is the basic good with the lowest possible rate of diminishing marginal utility (to its users) - the one nearest to an economists' ideal. Over the range of amounts from poverty level budgeting to commercial banking reserves, most modern money has effectively constant marginal utility because of its extraordinarily high salability (liquidity).

The initial observation about the different "value" of the marginal dollar to a rich vs a poor man is the necessary first step to explaining why they will enter into some voluntary trade - i.e. the poor guy will get a job that lets him convert his illiquid labor value into liquid cash value.

To transfer the dollar involuntarily is to negate the concept of any organic economy at all.

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