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# Mankiw Defends his Tall Tale

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In an email that he gave me permission to quote, Greg writes,

Your instrumental variable analogy is good, but your assertion that it is a weak instrument is not. The calculations in the paper establish how good an instrument height is. In the conclusion, we state:

Our calculations show that a utilitarian social planner should levy a sizeable tax on height. A tall person making \$75,000 should pay about \$4,500 more in taxes than a short person making the same income.

To be clear, we are not advocating this policy, but rather raising the issue as a challenge to conventional utilitarianism.

Mankiw's idea of a good instrument and my idea are different.

[major update posted April 18. I am more convinced by Greg.]

An increase in US men’s heights from the 25th to the 75th percentile of the height distribution — an increase of four inches — is associated with an increase in earnings of 10 percent on average.

That is quantitatively significant, and they also show that it is statistically significant. If that is all you want for an instrument, fine.

I think that an instrument needs to be highly correlated with the variable for which it is going to serve as a stand-in. I want a lot of the variation in the target variable (income in this case) to be explained by the instrumental variable (height in this case). That clearly is not true here.

Suppose that the income distribution is binary, with 95 percent of people earning \$20K and 5 percent of people earning \$500K. Suppose that half the people are tall, and half the people are short, and that tall people have a 5.5 percent chance of being high-income and short people have a 4.5 percent chance of being high-income. Then average earnings for short people will be \$41,600 and the average earnings for tall people will be \$46,400, which is higher by 11.5 percent. The difference is quantitatively significant, but the correlation is low (my guess is that the real-world correlation might be even lower).

If you impose a tax surcharge on tall people and subsidize short people, then 94.5 percent of the people who pay the surcharge will be low-income. Looking at another conditional probability, 45 percent of high-income people will get a subsidy.

Suppose that the tax/subsidy is set at \$5000. Then after the tax/subsidy kicks in, the distribution of income will be as follows:

47.25 percent of people will be at \$15,000 (low-income tall people)
47.75 percent of people will be at \$25,000 (low-income short people)
2.75 percent of people will be at \$495,000 (high-income tall people)
2.25 percent of people will be at \$505,00 (high-income short people)

Most people would think of that as a less egalitarian income distribution than the pre-tax distribution.

So, on further reflection, I continue to think that using an instrument with a low correlation coefficient leads to silly results.

UPDATE: What is misleading about my example is that I only tax height, rather than a combination of height and income.

Greg refers me to table 3 in his paper, which is an actual distribution of incomes and heights. I compressed the table into 4 groups by combining medium and tall and then calling the bottom four rows high-income and the top 14 rows low-income. I also multiplied his income measure, the wage, by 2000. This leads to

29 percent of the men are short/poor, with average incomes of \$29,300.
66 percent of the men are med-tall/poor, with average incomes of \$29,300.
1 percent of the men are short/rich with average incomes of \$108,400
3 percent of the men are med-tall/rich with average incomes of \$108,400.

Suppose that non-short rich pay \$33,000 in taxes, the short-rich pay \$30,000 in taxes, the non-short poor get a \$1200 negative tax and the short poor get a \$1500 negative tax. This does not seem as silly as my example, because the differences in taxes across income classes are large relative to the differences within income classes.

One could play interesting games with this. Suppose that instead of height, we use demographic variables as instruments (we are assuming that all of our instruments are only correlated with ability, not effort). We would tax third-generation Jewish immigrants higher than first-generation Hispanic immigrants. We would tax people with high IQ's, and perhaps children of people with high IQ's, higher than others.

Does this make you queasy? I think that Greg and Matt could argue that this sort of modified progressive income tax should not make you any more queasy than a plain progressive income tax. In fact, it should make you feel better. If you're going to try to play God, might as well use all available information.

CATEGORIES: Economic Methods

Pearl writes:

And I think that using an instrument with a correlation coefficient is silly without an analysis of cause and effect. It seems pretty trivial as statement, but correlation and causality is too often considered as being the same.

writes:

I think part of the assumption here is that 1) income is to some extent susceptible to incentives, whereas 2) hight -- and the part of income that is explained by hight -- is not (or at least is much less). On some level, then, your poor people subsidizing your rich people are, to some extent, at fault for being poor. It can be combined with a progressive tax to make a system that is more efficient than one that doesn't include a transfer from tall to short but raises the same revenue. In fact, I bet this is what is demonstrating in the paper (which I haven't read).

conchis writes:

Arnold says:

"I want a lot of the variation in the target variable (income in this case) to be explained by the instrumental variable (height in this case)."

Well, actually what you want is for a lot of the residual variation (i.e. the variation not due to effort) to be explained by the instrument. That makes the case for height taxation somewhat stronger than your analysis would imply, though obviously still problematic for the reasons you point out.

Two other issues:

(1) Even with a weak instrument, (in the Arnold sense, rather than the econometric sense) the optimal tax on height is probably still positive. It's just smaller than it would be without the error.

(2) Height isn't purely exogenous. What does this do to incentives for e.g. childhood nutrition?

conchis writes:

P.S. Pearl, I think the paper claims that for optimal tax purposes, correlation with ability is all that is needed; causation is irrelevant, provided that height is orthogonal to effort (which I assume they must have dealt with, otherwise it wouldn't be a valid instrument). Am I wrong?

RWP writes:

How much tax incentive would it take to cut off your legs? Maybe even your head? I find it interesting that we would want to encourage shortness and discourage non-shortness. If IQ correlates with height, then we would want to tax shortness, right?

Matt writes:

The X axis is fundamental. It is ordered by poor to wealthy, lower to higher income, sort term long term out look.

So, the federal legislature price fixes for eight years, dropping the cost of government by 30%.
Ask, are all the players on the right well correlated with each other? Is this a set of economies with inertia?

If we have correlation on the right, (a sub class of all economies), then we get group phase and group intensity movements.

The group on the right over estimates assets, taking tomorrows assets today. Their little bulge meets resource constraints and "bounces" back (left side resitance).

Eight years later, we get a price fixing by minimum wage; and on and on.

Matt writes:

OK, give me a day to read the paper, it seems important.

Matt writes:

OK, I got the paper and read over the introduction, but I need more time. I can tell you my approach.

I will create the more general problem of social engineering by price fixing any particular good, not just federal goods. The approach Greg took specified that a particular good, government services was price fixed to effect income distributions.

I will try to effect incomes by variable price fixing on shoelaces and show that in the equilibrium, the shoelace industry disappears and incomes hardly effected.

Then I assume tall people are highly dependent on shoelaces so the use of shoelaces as an income modifier seems much more effective. Under this assumption we can look at the equilibrium state.

Then I deal with federal services. However I consider the additional condition that tall people and social engineers both have the power to fix government prices.

Then, we can ask, under what types of economic models will produce group price fixing by tall people. It is this effect that progressive income taxes are designed to compensate.

writes:

I think I'm against the Shaq tax. More at the link...

Matt writes:

I read the paper. My uneasiness is directed toward the standard method, not Greg, apologies.

My uneasiness is the unbounded description in Greg's model, which works and results are correct. But we know the budget for social redistribution is not 100%, as Greg's results would appear in a bounded assumption. The average guy knows things go on.

The real budget in the final answer, in Greg's paper, is output from labor, socialness, and height; a three eigenvector decomposition of the general set of economies consisting of random draws according the the presented probaility distributions.

Hence my comment on my favorite strawman, the shoelace industry; because in the common sense, shoelaces would disappear until one understands we are dealing with unbound functions, though bound relatively.

Greg proved two things, the space defined by labor alone and the space defined by labor and height are spectrally invariant.

Maybe we should have a website related to spectrally invariant decompositons of economies.

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