Brad DeLong writes:
Department of “Huh?!”–I Don’t Understand More and More of Piketty’s Critics: Per Krusell and Tony Smith
As time passes, it seems to me that a larger and larger fraction of Piketty’s critics are making arguments that really make no sense at all– that I really do not understand how people can believe them, or why anybody would think that anybody else would believe them. Today we have Per Krusell and Tony Smith assuming that the economy-wide capital depreciation rate δ is not 0.03 or 0.05 but 0.1–and it does make a huge difference.
Let me do my best to try to educate Brad.
Here’s your first clue. Try reading Krusell and Smith’s summary of their core criticism, and tell me where the assumption of a specific numerical value for δ makes any appearance:
Piketty’s second law is not mathematically incorrect, but it relies on assumptions– as do all economic theories. The central assumption concerns how the economy saves. Piketty assumes that the ‘net’ saving rate is constant and positive, i.e. the economy increases its capital stock from year to year by an amount that is a constant fraction of (net) national income.
This assumption may sound standard but actually it is not– precisely because it is expressed in net terms. In particular:
- With zero growth in population or technology, the assumption that the capital stock is always growing (because net saving is positive) implies that more and more output must be diverted away from consumption towards investment.
- Eventually, because capital needs to keep rising, it is necessary to devote 100% of GDP to capital formation!
Here’s your second clue. I made exactly the same point as Krusell and Smith in an earlier Econbrowser post in which I made no claims whatever about how big the depreciation rate has to be. It’s true that I illustrated the implications of Piketty’s assumptions using a simple numerical example. My numerical example used GDP = $100 and δ = 0.10, but those numbers were chosen just to keep the arithmetic simple. I know that GDP isn’t really $100! If you run through the numerical example instead with δ = 0.05, or 0.02, or any positive number and any numerical value for GDP, you will arrive at exactly the same necessary implication of Piketty’s “second fundamental law of capitalism” as in my numerical example. His assumption of a constant net saving rate implies that capitalists always try to increase the capital stock if they have any level of positive net income whatever. That mathematically and necessarily implies that the economy’s total depreciation expense has to be higher every year. The necessary implication of Piketty’s assumed saving behavior is that as the growth rate becomes smaller and smaller, the capital stock would tend to an arbitrarily large multiple of net income (tending to infinity as the growth rate goes to zero) and the net income of capitalists after paying the depreciation bill would become arbitrarily small (tending to zero as the growth rate goes to zero).
To summarize: Piketty’s assumption that the ratio of net saving to net income remains constant as the economy’s growth rate falls is incompatible with any coherent model of saving behavior.
Brad further asks why do Krusell and Smith
imply that this is a point that Piketty has missed, rather than a point that Piketty explicitly discusses at Kindle location 10674?
One can also write the law β = s/g with s standing for the total [gross] rather than the net rate of saving. In that case the law becomes β =s/(g + δ) (where δ now stands for the rate of depreciation of capital expressed as a percentage of the capital stock).
Kindle location 10674, for those of you with a physical copy of Piketty’s book, will be found in footnote 12 on page 594. As far as I can determine, this footnote is the only point in the book at which the alternative formulation β =s/(g + δ) gets mentioned. The two versions of Piketty’s “law”, the one given in the text on page 168, and the one given in footnote 12 on page 594, cannot both be true. Either net saving is a constant fraction of net income (as Piketty assumes throughout his text) or gross saving is a constant fraction of gross income (as Piketty assumes in footnote 12). If one assumption is true, the other must be false. And if it’s not being asserted that either the gross or net saving rates are constant, then neither equation is a law at all, but instead could only be a definition of the saving rate (net or gross) that would be associated with a particular steady-state capital/income ratio β. In other words, if s is not a constant, the “second fundamental law” has no implications or predictions whatever for what will happen to the share of capital income in the economy or anything else, and claims that Piketty has uncovered some underlying principles of capitalism are completely without substance.
Having I hope clarified that the debate is not over whether the economy-wide depreciation rate is 10% or some other number, I nevertheless cannot resist entering the discussion of what is a reasonable number to assume for δ for purposes of characterizing steady-state growth paths. Brad writes as though the only sensible number to use for δ would be “0.02 or 0.03 or 0.05.”
The fact that Brad mentions such a wide range of possible values makes it obvious that defending a particular number for δ is anything but cut and dried. If you look at the assumed depreciation rates that underlie the national income accounts to which Brad appeals, you will find that rates of 10-20% are quite common for most forms of producers’ machinery and equipment (and perhaps it would be unfair at this point to mention the fine study by DeLong and Summers (1991) which concluded that this is the category of capital to which we should pay the most attention). Brad likewise casually insists that the U.S. capital/income ratio is somewhere between 4 and 6, another number that anyone who has looked at the details behind how this particular sausage gets made would treat with some caution. For example, the Census Department estimates the total net stock of fixed assets and durable goods in 2009 to have been $48.5 T, or 3.4 times 2009 GDP of $14.4 T. Steve Parente recommends using a value for K/Y of 2.75, while Paul Evans regards the relevant number for the U.S. to be around 2.
Update: Reader Salim points out that I was misinterpreting Piketty’s use of a 10% figure in his book’s calculations of depreciation. Piketty uses 10% for depreciation as a percent of GDP, not as a percent of capital as my original post suggested. In order not to mislead, I have deleted the inaccurate paragraphs that were included in the first version of this post.