Definition. A maintained productive knowledge stock [ftip-00O5]
Definition. A maintained productive knowledge stock [ftip-00O5]
Fix \(A,\eta ,\delta >0\) with \(g=\eta A-\delta >0\), and initial stock \(h_0\geq h_{\min }>0\). Productive output is \(Ah(t)\) per unit time, measured in a fixed consumption numeraire. Learning expenditure \(v(t)\) and maintenance \(m(t)\) exhaust output. Assume
\[v(t)+m(t)=Ah(t),\qquad \dot h(t)=\eta m(t)-\delta h(t)=gh(t)-\eta v(t).\]An allocation on \([0,T]\) is admissible when \(h\) is absolutely continuous, \(v\) is measurable, \(h(0)=h_0\), and almost everywhere \(0\leq v(t)\leq Ah(t)\), while \(h(t)\geq h_{\min }\) at every time. Randomized allocations must satisfy these conditions almost surely. The conversion \(\eta \) and depreciation \(\delta \) are model assumptions; no causal estimate of human deskilling is asserted.
The variable \(v\) measures expenditure, not compute. To connect it to \(u\) in the compute envelope, specify the service price and require \(p(t)u(t)\leq v(t)\). The model excludes other productive assets and outside output. Applications that admit them must enlarge its state and recompute the bound.