 Solution

The requested properties are not properties of the optimum under the printed assumption $\alpha +\beta >1$. We first derive their origin, and then solve the actual constrained problem.

Put $s=1-\ell$, so $0\le s\le 1$, and write the Hamiltonian of an optimal-control problemFor an interior stationary control, $H_c=H_s=0$ givesThis proves the printed identity only as a necessary stationarity condition for an interior candidate.

It fails the required maximization condition. For fixed positive $p$ and fixed $s$, maximization over $c$ givesSince $\beta /(1-\alpha )>1$, this is strictly convex in $s$. Its maximum on $[0,1]$ is at an endpoint, not at its interior stationary point. Thus the actual optimum uses $\ell =1,c=0$, or $\ell =0,c>0$. On any consumption interval the printed relation would require $\beta c=\alpha$, although the actual optimal consumption varies with time.

This proves a genuine failure of the printed premise, not just failure to check a boundary condition. If the parameter inequality were reversed and an interior solution stayed within the control bounds, these first-order equations would instead arise from a concave objective.