# Hsieh-Moretti on Housing Regulation: A Gracious Admission of Error

##### By Bryan Caplan

Chang-Tai Hsieh and Enrico Moretti‘s “Housing Constraints and Spatial Misallocation” (*American Economic Journal: Macroeconomics*) is arguably the single most influential article ever published on housing regulation. It also contains a few large miscalculations.

I noticed them a couple weeks ago, and Hsieh and Moretti have graciously confirmed the mistakes via email. Since the gracious admission of error was always a rare bird, and practically went extinct circa 2016, I hope readers won’t judge Hsieh and Moretti (henceforth HM) too harshly. Though my public betting record is 22 for 22, I know I’ve made my share of mistakes, too.

Still, our priority should be to set the record straight. Where did HM go wrong? On pp.25-6 of their article, they write:

Starting with perfect mobility, the second row in Table 4 shows the effect of changing the housing supply regulation only in New York, San Jose, and San Francisco to that in the median US city. This would increase the growth rate of aggregate output from 0.795 percent to 1.49 percent per year—an 87 percent increase (column 1). The net effect is that US GDP in 2009 would be 8.9 percent higher under this counterfactual, which translates into an additional $8,775 in average wages for all workers.

On the next page, they re-estimate the results with imperfect mobility:

Table 5 shows that changing the housing supply regulation in New York, San Jose, and San Francisco to that in the median US city would increase the growth rate of aggregate output by 36.3 percent (second row). The net effect is that US GDP in 2009 would be 3.7 percent higher under this counterfactual, which translates into an additional $3,685 in average wages for all workers, or an increase of $0.53 trillion in the wage bill.

Both tables indicate that HM are covering the period from 1964-2009. How then can these enormous changes in the annual growth rate, compounded over 45 years, lead to relatively modest changes in total GDP? Answer: They can’t!

The correct estimate to derive from Table 4 is that GDP will be 1.0149^45/1.00795^45=*+36%* higher, not +8.9%.

Similarly, the correct estimate to derive from Table 5 is that growth will be 1.084% per year (.795%*1.363), so GDP will be 1.0108^45/1.00795=*+14%* higher, not +3.7%.

There is a completely distinct error in footnote 28 on p.26, which reads:

US GDP in 2009 was $14.5 trillion so a GDP increase of 8.9 percent implies an additional aggregate income of $1.95 trillion. Given a labor share of 0.65, this amounts to an increase of $1.27 trillion in the wage bill…

8.9% of 14.5T is actually $1.29T. Multiplied by .65, it comes to $.84T, much less than HM wrote. If they had used the correct +36% GDP estimate, however, they would have reported the much larger figure of +$3.39T ($14.5T*.36*.65).

There is a parallel error on p.27. They say that raising GDP by 3.7% raises the wage bill by $.53T. But the correct calculation given these numbers is $.35T ($14.5*.037*.65). And if you replace +3.7% with +14%, the change in the wage bill comes to +$1.32T.

Critics may rush to accuse HM of motivated reasoning, but the shoe does not fit. Their reported figures for the effect of housing deregulation on total GDP and the wage bill turn out to be gross *under*statements. The reasonable interpretation, rather, is that authors and referees alike focused so intently on the advanced mathematics that they glossed over some elementary yet crucial errors. And this is roughly what Hsieh told me: The referees requested some changes to the text (not the tables, which look fine), but these were inconsistently implemented.

Is there any meta-lesson? Perhaps not, but I would definitely like to see top journals audited to discover the frequency of errors of this sort…