I am once again flummoxed by the number of economists of note and reputation who have been commenting on Piketty’s Capital in the 21st Century without, apparently, bothering to do the work to understand the basic arithmetic scaffolding of the book.

I think that the most fruitful way to understand the basic arithmetic is via the road I took in my “Mr. Piketty and the ‘Neoclassicists'”, focusing on the equilibrium rate of accumulation n+g, the wedge ω between the rate of accumulation and the warranted rate of net profit, the warranted rate of net profit rnw, and the resulting equilibrium wealth-to-annual-net-income ratio K/Yn.

But there is another road–one that goes not through prices but through the quantities of the Solow growth model: gross savings, depreciation, and population and technology growth. I think this road is less illuminating and more likely to cause confusion. But perhaps it will help some people who do not currently do so to understand the arithmetic scaffolding of the book.

James s Kindle for Mac 3 Capital in the Twenty First Century

Piketty and his coauthors estimate that back before 1914 the wealth-to-annual-net-income ratio for northwest European economies was on the order of 700%: the total holdings of property by the economy’s wealthholders amounted to seven times annual income. They also calculate a northwest European rate of growth of income of 1.2%/year–about half from population and labor force, and about half from labor productivity. In the framework of the Solow growth model, you can calculate the rate of savings out of gross income required to maintain that wealth intensity from the formula:

MathType Untitled 1

if you know the annual rate at which wealth is subject to depreciation δ.

Three depreciation rates have been proposed. I tend to favor a depreciation rate of 3.33%/year–a lot of the wealth held in the economy back before World War I was in the form of land, which does not depreciate, or various property rights guaranteed by successful rent-seeking of one sort or another. With a depreciation rate of 3.33%, we calculate that 25.7% of gross output was saved and reinvested in maintaining and building the wealth stock; of this 8.4% of gross output keeps the capital stock growing at the 1.2%/year rate of income, and 17.3% of gross output covers the depreciation.

Alex Tabarrok favors a depreciation rate of 5.0%/year–what one would get out of a standard long-run growth model. This seems to me a little high given what kinds of property make up the claims to income included in Piketty’s definition of “capital”. But it is certainly in the admissible range. For this depreciation rate, we calculate that 32.1% of gross output in northwest Europe back before World War I was saved and reinvested in maintaining and building the wealth stock; of this, once again 8.4% keeps the capital stock growing at the 1.2%/year rate of income, and 23.7% to cover depreciation.

Krusell and Smith favor a deprecation rate of 10%/year–and I genuinely do not understand why they think it is appropriate. We are not, after all, dealing with short-run business-cycle fluctuations in which the pieces of the capital stock that vary are made up mostly of inventories and machines here. We are talking about land, very durable buildings, powerful property rights and the ability to summon the police to protect them–claims over future output that do not, I think, erode away at anything like 10%/year. And, indeed, if you try to understand the pre-WWI northwest European economies with such an assumed depreciation rate, you get results that strike me as completely nonsensical: 46.1% of gross output in gross investment; I think we would have noticed if, in pre-WWI northwest Europe’s economies, two out of every five workers spent their days simply keeping society’s collective capital stock from depreciating by repairing rust and wear-and-tear. That is simply not what the pre-WWI northwest European economies looked like.

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But the Belle Époque comes to an end. World War I, the Bolshevik Revolution, the Great Depression, and World War II wreak their effects on the North Atlantic. The wealth-to-annual-net-income ratio falls to 300% or so as the growth acceleration of the twentieth century raises the rate of growth of income to 3.0%/year. And then during the Thirty Glorious Years of post-WWII social-democracy and growth the wealth-to-annual-net-income ratio shows no signs of rising swiftly back to its previous equilibrium of 700%. Viewed from the perspective of the Solow growth model, the economy of the generation after 1950 is and remains much less capital-intensive than the economy before World War I, and to support this lower capital-intensity equilibrium requires savings rates out of gross output reduced by a third from their pre-WWI shares:

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Starting around 1980, Piketty argues, the North Atlantic shifted out of its Social-Democratic Era and is now moving into a new configuration, with increasingly-concentrated wealth, savings no longer reduced by highly progressive capital taxation and fear of expropriation, and slower rates of population and labor productivity growth. Piketty expects the consequence to be a rise in the savings rate back to Belle Époque levels and a return to the capital intensity and inherited-wealth dominance of those days. In my view, the next questions are two:

  1. Would this be a good thing? More savings and wealth accumulation by the rich that increase the capital intensity of the economy increase real wages for the working class and the poor, no? Here I think the answer is perhaps–and I think this is what the debate over Piketty should be about. Unfortunately, that is not the debate we are having. Instead, we have:

  2. Can this happen? And Krusell and Smith and company are saying: no, it cannot. As the capital-output ratio rises, the desire to consume wealth pushes the gross savings rate goes down and the fact that capital depreciates at 10%/year pushes the net savings rate down much further, and so there are no macroeconomic forces in play that could push the wealth-to-annual-net-income ratio far up above its current value of 300%.

But if the savings rate necessarily falls as the wealth-to-annual-net-income ratio rises, why was the (gross) savings rate half again as high back before World War I when the economy was wealth-dominated as it is today? And from where comes the 10%/year depreciation rate assumption?

What we clearly have here is a failure to communicate. And I really, really do not think that it is the result of a failure to try on Thomas Piketty’s part.

Calculations here: https://www.icloud.com/iw/#numbers/BAKWqmBPWn9zULDVGZGBDMJheVCgDvXgeXaF/20140605_Piketty_Solow.numbers

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