What is an economy? Ultimately, it is a number of people, mostly strangers to each other, but connected through an intricate network that allows them to produce goods and services in quantities and varieties immensely larger than what they could obtain on their own.
People produce with the aid of capital, both human and non-human. Human capital provides labour services and earns a wage; non-human capital provides non-labour services such as risk bearing, lending, and housing, and earns dividends, interest and rents. The total value of production equals the total value of returns, or revenues.
People produce in order to consume. Consumption is their ultimate payoff. Therefore, just as the value of a company’s capital is the discounted sum of expected dividends, the value of an economy’s capital is the discounted sum of expected consumption. Consumption C corresponds to dividends D and production Y to earnings E. Hence retained earnings E-D are equivalent to savings Y-C=S and the retention ratio H=1-D/E is equivalent to the saving rate s=S/Y. As the sole factor of production, capital K, both human and non-human, corresponds to the company’s book value B. Just as the book value is increased by retained earnings, ∆B=E-D, capital is increased by savings: ∆K=Y-C. Therefore I=S, where I is new investment. In the long run, the return on capital R equals the production-capital ratio Y/K and the growth rate G=H∙R equals (S/Y)(Y/K) =S/K. Hence G=(S/Y)/(K/Y)=s/β, where β=K/Y is the capital-production ratio.
The latter equality is what Thomas Piketty calls the second fundamental law of capitalism: β=s/G (Le Capital au XXIe Siècle, p. 262). Since β=1/R, the law can also be written as s=G/R, which, since s<1, implies R>G. Much of the debate around Piketty’s book revolves around R-G. Our framework makes it clear that, just as a company that entirely retains its earnings is worth nothing to its shareholders, the capital of an economy where all production is saved and none is consumed is, unsurprisingly, worthless. Since consumption cannot be negative, R must be larger than G – for the same reason that the infinite sum of natural numbers must be positive.
Piketty’s first law (p. 92) also holds in our framework: α=R∙β, where α is the share of non-human capital revenues on total production. This is actually not a law but an identity. In our framework, total capital K is the sum of human capital and non-human capital: K=KH+KNH and total production is the sum of human capital revenues and non-human capital revenues: Y=YH+YNH. Hence α=YNH/Y. This can be decomposed as (YNH/KNH)(KNH/Y), where YNH/KNH=RNH is the return of non-human capital and KNH/Y=βNH is the non-human capital-production ratio. In Piketty’s example, βNH=6, RNH=5% and therefore α=30%: non-human capital earns 30% of total revenues.
A major difference between our framework and Piketty’s is that he defines capital just as non-human capital (p. 82), expressly excluding any consideration of human capital. This doesn’t mean, of course, that in Piketty’s economy there is no labour. In fact, in his example labour earns 70% of total revenues. In our notation, 1-α=YH/Y=(YH/KH)(KH/Y), where YH/KH=RH is the return of human capital and KH/Y=βH is the human capital-production ratio, and the shares of non-human and human capital revenues on total production add up to one:
RNHβNH + RHβH = 1
Although he does not consider it, in Piketty’s economy human capital has an implicit return RH and is implicitly worth a multiple of production βH. And, since the total capital-production ratio β is the sum of the non-human and the human capital-production ratios:
β = K/Y = βNH + βH
then βH=β-βNH, where remember β=1/R=s/G.
This shows an inconsistency in Piketty’s model. In his example, s=12% and G=2%, hence β=6, which is the same multiple he uses to calculate α. But this implies βH=0, which is impossible: if human capital earns 70% of revenues, it must have a value – however implicit – which must be worth some multiple of production. Hence β must be bigger than βNH, which requires either a higher saving rate or a lower growth rate. For example, with G=1% we have β=12 and therefore βH=6: human capital is worth as much as non-human capital and, since it earns 70% of revenues, it must return 70%/6=11.7%. Alternatively, βNH must be smaller than 6, which in turn means that either α must be smaller than 30% or the return of non-human capital must be larger than 5%. For example, with βNH=4 and RNH=7.5% (so as to preserve α=30%), we have βH=2 and RNH=35%.
One can play around with the numbers, but the important point is that A Country is Not a Company. While labour is a cost to a company and is not part of its capital, human capital is very much part of an economy. The saving rate s measures the savings of all revenues, from human as well as non-human capital, and G is not only the growth rate of non-human capital but incorporates the growth of human capital, including, very importantly, increases in the labour force.
The return on total capital R=G/s is the discount rate of the consumption stream that determines the value of the economy’s capital. As s<1, R is always larger than G. In the long run, R equals the production-capital ratio Y/K, and its inverse β can be seen as the economy’s “PE ratio”, with K=Y/R corresponding to the Tangible Value of the economy’s capital.
Our framework shows that there is no relationship between R-G and the distribution of revenues and wealth between workers, the owners of human capital, and so-called rentiers, the owners of non-human capital. Distribution depends on the relative size of non-human vs. human capital and on their relative returns. s, G and R have nothing to do with it.
