  Because \(\{C^*(t)\}\) is by assumption the optimum, no feasible deviation from \(\{C^*(t)\}\) can increase \(V(0)\). That means the consumption rate of interest \((\varrho(t))\) must equal the social rate of return on investment \((F_{K(t)})\) at every \(t\). To see why, suppose in some vanishingly small interval of time \(F_{K(t)} \gt \varrho(t)\).  Then \(V(0)\) could be increased by consuming a unit less at \(t\) and enjoying the return of \((1+F_{K(t)})\) soon after. Alternatively, if \(F_{K(t)} \lt \varrho(t), V(0)\) could be increased by consuming a unit more at \(t\) and reducing consumption soon after by an amount equal to the return \((1+F_{K(t)})\).  But that means the consumption rate of interest \(\varrho(t)\) equals the social rate of return \(F_{K(t)}\) along \(\{C^*(t)\}\) at every date. Using equation (8) we have,