Wednesday, December 3, 2014

Thomas Piketty and the evolution of capital

Having finally read through Capital in the 21st Century (and spent considerable time with the supplied data) I find many critiques of Thomas Piketty very odd. Part of this is that Piketty (to my mind, anyway) tries very hard to take care with his words. Thus, I get annoyed when I see Branko Milanovic write
Just as a reminder: as we all know by now, $r>g$ implies that the stock of capital is increasing faster than net income.
We know this? Milanovic of course does not. Indeed, his post outlines a case where it may not be true. What irks me is that Milanovic seems to believe it to be generally understood. Frankly, I am not clear if Milanovic is critiquing Piketty or the public. With that in mind, let us begin to review the basic analytical framework presented in Capital.

Piketty defines “capital” or “wealth” as the market value of productive physical capital and net financial assets owned by households and government. Piketty’s capital includes residential real estate, but not consumer durables. It includes corporate-owned equipment through direct stock holdings of households and indirect holdings through pensions and other savings vehicles. Capital also includes holdings of foreign assets net of liabilities. Piketty divides the total value by national income (net of capital depreciation) and expresses the result, $\beta$, in terms of years.

Now define (net) savings as net investment and net acquisition of foreign financial assets so as to equal the change in real wealth, apart from miscellaneous volume adjustments and inflation-adjusted capital gains. If we presently assume these are zero, then the capital-income ratio must evolve as $$\beta_t=\frac{\beta_{t-1}+s_{t-1}}{1+g_t}$$ where $s_t$ is net savings as a share of net income in period $t$ and $g_t$ is the real (inflation-adjusted) growth rate of net income from period $t-1$ to $t$. Note that regardless of $\beta_{t-1}$ or $g_t$, if $s_{t-1} < g_t\beta_{t-1}$, then $$\beta_t<\frac{\beta_{t-1}+g_t\beta_{t-1}}{1+g_t}=\beta_{t-1}$$ Thus, Piketty’s “stock of capital is increasing faster than net income” if and only if there is sufficient net savings irrespective of the rate of return on capital.

What then is the significance of $r>g$? Stay tuned...