## Monday, May 29, 2017

### The Evolution of Capital, Part III

In the previous post, we saw how, under restrictive assumptions, $r < g$ means that capital cannot self-perpetuate. Holders of wealth— in the aggregate— must save more than capital income provides or the wealth-income ratio $\beta$ will fall.

Unfortunately, the assumptions behind this conclusion are surely overly restrictive. In particular, we should at the very least investigate the dynamics when there are long-run capital gains. When there are no miscellaneous volume adjustments, $$\beta_t=\frac{1+q_t}{1+g_t}\left(1+g^{ws}_t\right)\beta_{t-1}$$ where $q$ is the rate of inflation-adjusted capital gains, and $g^{ws}$ is the pure rate of growth of wealth due to saving (that is, $g^{ws}={S}/{W}$– the savings-wealth ratio. We may rewrite the evolution of $\beta$ as $$\beta_t=\frac{1+q_t}{1+g_t}\left(\beta_{t-1}+s_{t-1}\right)$$ and therefore $$\left(\beta_t-\bar{\beta}_t\right)=\frac{1+q_t}{1+g_t}\left(\beta_{t-1}-\bar{\beta}_t\right)$$ where $$\bar{\beta}_t=\frac{1+q_t}{g_t-q_t}s_{t-1}$$ Thus, so long as $g>q$— the rate of capital gains is less than the growth rate of net income— then $\beta$ tends toward a finite ratio. However, if $q>g$, then $\beta$ grows without bound. The rate at which wealth appreciates may become more critical to the dynamics than the interest and dividends it may provide.

### The Evolution of Capital, Part II.

A long while back, I promised to get into the significance of $r>g$ to Piketty’s framework. To review where I left off,
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.
This result depended upon the assumptions that there are zero miscellaneous volume adjustments to the capital stock and zero inflation-adjusted capital gains. Under these assumptions, the evolution of the wealth-income ratio $\beta$ follows $$\beta_t=\frac{1}{1+g_t}\left(\beta_{t-1}+s_{t-1}\right)$$ Equivalently, we may write $$\left(\beta_t-\bar{\beta}_t\right)=\frac{1}{1+g_t}\left(\beta_{t-1}-\bar{\beta}_t\right)$$ where $\bar{\beta}_t={s_{t-1}}/{g_t}$. That is, $\beta$ is always tending toward ${s}/{g}$ so long as there is real growth in net income ($g>0$). This is Piketty’s Second Law in its simplest form.

Now, $s$ is defined as net savings as a share of net income. If we put savings instead in terms of net capital income, $$\zeta\equiv\frac{S}{Y^k}$$ then starting with Piketty’s First Law (the identity $\alpha=r\beta$) we find that the economy is tending toward $$\frac{\alpha}{r}=\beta=\frac{s}{g}=\frac{\alpha\zeta}{g}$$ If we then assume that all net savings come out of net capital income, we find $$\frac{r}{g}=\frac{1}{\zeta}\geq 1$$ or $r>g$.

Put another way: if, in the long run, Piketty’s Second Law holds and $r < g$, then $\zeta>1$. That is, under these very restrictive conditions, capital owners must save more than their capital income— in the aggregate, capital cannot self-perpetuate.

Unfortunately, real capital gains are something we do observe in the real world, so the story is surely more complex. We’ll look at that in the next (very mathy) post.