Example. Productive investment expands the envelope [ftip-00OL]
Example. Productive investment expands the envelope [ftip-00OL]
Consider a single productive capital stock \(K_0>0\), output \(AK\), depreciation \(\delta _K>0\) and allocation fractions \(s,\theta >0\) with \(s+\theta \leq 1\). Invest \(sAK\), spend \(\theta AK\) on training, and allocate the remainder to other uses. If \(A>0\) and \(\gamma =sA-\delta _K>0\), then
\[\dot K=\gamma K,\qquad K(t)=K_0e^{\gamma t},\qquad \int _0^T\theta AK(t)\,dt =\frac {\theta AK_0}{\gamma }(e^{\gamma T}-1).\]These identities follow by solving the linear stock equation and integrating output. With service price \(p>0\) and sufficient installed service capacity, division by \(p\) gives affordable compute. For \(T>0\), this expenditure exceeds the frozen-capital estimate \(\theta AK_0T\), since \(e^{\gamma T}-1>\gamma T\). That estimate cannot bound this policy. The model assumes investment converts into usable capital without delay; construction lags and essential complements require additional states.
Aghion, Jones and Jones study AI and economic growth with production and idea-generation mechanisms. Caballero analyzes a richer financing and capital-installation mechanism with alternative long-run outcomes. Neither citation makes the exponential path here an empirical forecast.