There has been much discussion of Thomas Piketty’s new book, Capital in the 21st Century. Some of the criticisms I agree with, and some I do not.

The latest concern was raised by Chris Giles, who found “a series of problems and errors” in Piketty’s publicly posted data and spreadsheets. Here is Paul Krugman’s reaction:

Giles finds a few clear errors, although they don’t seem to matter much…. But is it possible that Piketty’s whole thesis of rising wealth inequality is wrong? … That can’t be right– and the fact that Giles reaches that conclusion is a strong indicator that he himself is doing something wrong.

I suspect Krugman is correct in this evaluation. An error in one spreadsheet does not refute the conclusion that inequality in income and wealth have been increasing over the last generation. We have abundant evidence for that conclusion from a good many sources besides Piketty.

Of course, Krugman would also have been correct if he replaced “Giles” in the first sentence above with “Herndon, Ash, and Pollin”. The contrast between Krugman’s defense of Piketty and the zeal with which he jumped on top of the Reinhart-Rogoff dog pile is amusing. But kudos to Justin Wolfers ([1], [2]) for even-handed summaries of both disputes.

A more telling criticism of Piketty’s thesis relates to his treatment of depreciation, an issue raised by Larry Summers and Matt Rognlie. The need to keep up with what would otherwise be a growing cost of depreciation over time is a key factor that pins down the stable growth path in most economic models and is one reason I am puzzled by Piketty’s claims of an inherent instability in capitalist economies.

On page 168 of Piketty’s book the reader is introduced to “the second fundamental law of capitalism” according to which β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate. Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.

To arrive at such a conclusion, Piketty is defining (page 174) the saving rate s to be the ratio of net investment to net income, where “net” here refers to net of depreciation. To see why under these definitions his “second fundamental law” would require an infinite capital/income ratio in the absence of growth, consider the following modification of the numerical example presented in Robert Solow’s sympathetic review of Piketty’s book. Consider an economy with GDP = $100, a depreciation rate of 10%, a net saving rate of 10%, and a capital stock of $100. The annual depreciation would be $10, which is subtracted from the $100 GDP to arrive at $90 in net income. The economy is assumed to save 10% (or $9) of this net income, meaning gross investment is $19, or 19% of GDP. Because gross investment exceeds depreciation, the capital stock would have to grow even if there is no other source of growth in productivity or population. Thus a capital stock of $100 is too low to be consistent with a steady state for this economy.

Let’s take those same initial conditions ($100 GDP, 10% saving and depreciation rates) and now suppose that the capital stock is $500. Then annual depreciation would be $50, leaving $50 in net income, of which the economy is again supposed to save 10%, or $5, making gross investment a total of $55, or 55% of GDP. But because we have added $5 net of depreciation to the capital stock, the capital stock would still have to grow if these were the initial conditions. So $500 is still too low a number for the capital stock for this economy.

For an economy with GDP of $100 and 10% saving and depreciation rates, the steady-state capital stock turns out to be $1,000. At this level, depreciation is $100, leaving zero net income with which anybody could do any further net investing. 100% of GDP is going to gross investment, but this doesn’t increase the capital stock any further above $1,000 because it is just sufficient to cover depreciation. Unfortunately, nothing is left over from the $100 GDP for capitalists or anybody else to consume, and everybody would starve. The ratio of the capital stock to net income is infinity because we have driven net income all the way down to zero.

This is the mechanism whereby Piketty’s law would predict a value of infinity for the steady-state ratio of the capital stock to net income. When the growth rate g = 0, Piketty’s capitalists save themselves into exhaustion until, with zero net income, they still dutifully try to salt away 10% of what is now nothing.

Thinking these numbers through leads you to understand that the assumption of a constant net saving rate, regardless of the levels of income and the capital stock, could not possibly be a sensible characterization of the decisions that would be made by real people in the real world.

A more plausible and standard assumption (though one that can still easily be improved on further if one thinks of the saving decision as the outcome of some kind of rational calculation) would be that the gross saving rate s* is constant. Under this alternative specification, the relevant characterization of the steady state would be β = s*/(g + d) where d is the depreciation rate. For example, when g = 0, with GDP of $100 and s* and d both 10%, the economy would be in a steady state with a capital stock of $100. Depreciation is $10 annually, and gross investment is just sufficient to cover depreciation each year with no growth in the economy and no reason for any variables to change over time. But the good news is that $90 would be left over for people to consume on a sustained basis every year. That’s a rather boring but much more plausible notion of what an economy with no growth could look like.

But a sensible characterization of the implications of depreciation for the dynamics of wealth accumulation would not have caught the attention of as many commentators as Piketty’s book seems to have done.