Proving that the utility is concave
Sharing my answer out there, correct me if I'm wrong.
$u(c,l)=u(rk+w(1-l), l)$
$U$ is strictly concave and differentiable. Let $max$ u attained at $(c^*, l^*)$ i.e $(k^*, l^*)$.
Then $v(k^*)=u(k^*)$ and $u(k) ≤ v(k)$
Then by Theorem 4.10 (Benveniste &amp; Scheinkman), $v$ is differentiable at $k^*$.
$V_k^*=U_k(rk^* + w(1-l^*, l*)
      = U_c(rk^* + w(1-l^*, l*)$