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Capital in Netherlands 1921-2013 & Piketty’s formula

Summary of article in ESB Jaargang 100 (4704) 26 February 2015, by Dr. Marein van Schaaijk

Click here to download Kapitaal NLD.xls with the underlying data

Piketty (2014) presents in his splendid book for several countries for several periods a treasury of data concerning the private rate of return on capital, r, the rate of growth of income and output, g and the capital/income ratio, K/Y. Based on his empirical analysis using graphs, he finds his third formula: The Central Contradiction of Capitalism: r > g. His conclusion is:
“The principal destabilizing force has to do with the fact that the private rate of return on capital, r, can be significantly higher for long periods of time than the rate of growth of income and output, g. The inequality r > g implies that wealth accumulated in the past grows more rapidly than output and wages.”
However, ”significantly higher for long periods of time”  what does that exactly mean? So first we try to find out what is exactly the relation between, r, g and K/Y.

In sheet Formulas in Van Schaaijk, 2015, you can find that there exists a definition relation:
p*r –g = z
in which p = dK/W (the investment ratio / profit rate), and z the percentage growth rate of K/Y
Given this definition follows:  If  r > g/p then z > 0

However Piketty’s conclusion “The inequality r > g implies that wealth accumulated in the past grows more rapidly than output and wages”, is only true for certain values of p.

So if you change in the third formula of Piketty ”significantly higher for long periods of time”  into “1/p” then Piketty’s third formula becomes a definition equation, and that is true always and everywhere. So not only this century but also next and not only in the countries that Piketty studied but all countries.  So no empirical work needs to be done to come to this conclusion for example for the Netherlands during 1921 – 2013.

Actually, also concerning the second formula of Piketty a definition equation can be derived, see in sheet Afleiding formules in Kapitaal NLD.xls.

Capital in Netherlands 1921-2013
A lot of methodological problems are there when calculating longer time series for capital. See Coenen, ESB, 26-2-2015.  We made several time series, see KapitaalNLD.xls on www.micromacroconsultants.com

In the graph below we present two of them for the Netherlands for the period 1921-2013. One V1/Y NLD, based on wealth statistics, and the other one K1/Y NLD based on cumulated net real investments with starting point taken in 1947 from the figure of the wealth statistic.

Furthermore in the graph below we show the K/Y for the World 1870-2100  from Piketty’s graph 12.4  

K/Y World and Netherlands

Source: Piketty, 2014 Graph 12.4  and  van Schaaijk, KapitaalNLD.wks January 2015

The graph shows big differences between K1/K NLD and V1/Y NLD, but in both cases there was a big decrease between 1921 and the seventies (in Netherlands much more then in the World), then a slight increase, followed by stabilisation of the last thirty years in case of K1/Y, but some increase in V1/Y. In the World the last thirty years show an increase, and Piketty expects a further increase.

In this article is shown that Piketty’s third formula could be made more precisely by bringing in it a factor p (the ratio of investment/profits). Then a definition equation results. Such equations do not need to be proved by empirical analysis.  So instead of analyzing r and g one can immediately start analysing K/Y.  That is what has been done for Netherlands.
Is the decrease of K/Y between 1921 and the seventies the result of capital saving technical progress or changes in wage costs per product?  It might be interesting to analyse Dutch capital stock data that are available per sector of industry.
Answering those questions is beyond the scope of this brief article.


- Piketty, T., Capital in twenty-first century, The Belknap Press of Harvard University Press.
Cambridge, Massachusetts, London, England, 2014
- Van Schaaijk, M. van (2015) KapitaalNLD.xls (in Dutch) on
- Schaaijk,M. van (2015) Kapitaal in Nederland, ESB Jaargang 100 (4704) 26 February 2015

Addendum 2nd March 2015

We mentioned already that also concerning the second formula of Piketty a definition equation can be derived. What does this mean?

We know:  z = k – g   The percentage growth rate of K/Y z, is the percentage growth rate of capital k, minus the percentage growth rate of Y, g.  And savings rate s = S/Y*100.
k = dK/K *100  = (dK/S)*(S/K) *100 =  (dK/S)*(S/Y)*(Y/K) *100 
so k = (dK/S)*s/(K/Y)
so z = (dK/S)*s/(K/Y)   - g

so if z = 0 and dK=S then
K/Y = s/g   This is Piketty’s second formula

Please note that K/Y = s/g  is only true if z = 0 and dK = S     It is a little bit strange to use this formula that assumes K/Y to be stable, to forecast changes in K/Y.

Piketty assumes that world g will decrease from more then 3 now to below 1,5 and that s will stabilise around 10 percent. If in the formula K/Y = s/g  you divide g by 2, then K/Y will become twice as big. And that is what Piketty shows in his graph 5.8. His main conclusion and fear is that K/Y will increase a lot this century, and might even go up to the level of the beginning of last century.
However assume for example g becomes 0, then, following Piketty’s formula, K/Y becomes infinitive (!?) and should g change from 3 to – 3 then K/Y does not change in number, but becomes negative (!?)

So it looks like that Piketty’s second formula cannot be used to forecast K/Y.

Piketty wrote a beautiful book with a lot of interesting information, only he might reconsider to revise his second and third formula. That could also result in another conclusion concerning K/Y growth during this century.

He might just use z = k – g  if he wants to forecast the effect of X lower increase in g (because of lower population growth) on z. All other things equal (like k)  X lower g means X higher z.

But K and g are not independent of each other. If g goes down businesses need less growth of machinery and buildings, so sooner or later you might expect also k to go down. In case savings at the same time go up (because maybe expected ageing of population in the future one might save more now). Thanks to deficits on the capital account of the Balance of Payments, the national Wealth/ Y ratio might increase even when domestic K/Y is stable.