Proving that the utility is concave Consider a household which solves the following problem:
$$v(k,r,w)=\underset{c,l\in B{(k,r,ω)}}{\ {max}} \{u(c,l)\}$$
where $u : R_+^2 \rightarrow R$ is a strictly concave, twice continuously differentiable, strictly increasing function in its two arguments: consumption, $c$, and leisure, $l$. The constraints the household must obey in selecting $c, l$ are summarized by $B$:
$$B(k, r, w) = {\{c, l : 0 ≤ c ≤ rk + w(1 − l), 0 ≤ l ≤ 1\}}$$
Here, $k, r, w &gt; 0$ are numbers over which the household has no control.
Prove that $v$ is concave in $k$ and that the derivative of $v$ with respect
to $k$ exists for ‘interior $k$’. Display a formula for the derivative of $v$.
What I was thinking for solving this is by following Benveniste &amp; Scheinkman theorem on differentiability $ω : D → R$ defined on the neighborhood $D$ of $x_0$, i.e.
$D ⊂ X$ and $x_0 ∈ int(D)$ such that: $ω(x) ≤ v(x)$ and $ω(x_0) = v(x_0)$.
And $ω(.)$ is concave and differentiable, then $v(.)$ is differentiable at $x_0$ and $v'(x_0) =ω'(x_0)$.
I'm guessing we have to substitute c with $rk + w(1 − l)$ in the utility function, but I'm confused a bit with leisure. Because after that, I think we should just get the envelope theorem or not?
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 $k^*$.
$V_k^*=U_k(rk^* + w(1-l^*, l*)
      = U_c(rk^* + w(1-l^*, l*)$