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.

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