3. The Solow-Swan model with only capital input (6 pts)
As human workers are increasingly replaced with new automation technologies, economists
have started investigating the so-called von Newmann singularity, a hypothetical scenario
1
in which capital share of income is equal to 100%.1 We now consider the implications of
singularity in the context of the Solow-Swan model. Production technology is now given by
$Y_t = AK_t$,
where there is no labor input because production is fully automated. The law of motion of
capital still follows
$K_{t+1} = K_t + I_t - \delta K_t$,
with the investment $I_t = sY_t$.
(a) (2 pts) Suppose $\bar{A} = 0.5$, $s = 0.2$, and $\delta = 0.2$. Draw the Solow diagram and find the
steady state capital stock. (Hint: the investment curve is now linear!)
(b) (2 pts) Now suppose $\bar{A} = 2$, $s = 0.2$, and $\delta = 0.2$. Would this fully automated
economy achieve sustained long-run growth? If so, what will be the growth rate for aggregate
output?
(c) (2 pts) What is the steady state capital stock if $\bar{A} = 1$, $s = 0.2$, and $\delta = 0.2$? Is it
unique?
