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.]Paxson and Case write,

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.