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

CC BY-SA 4.0
2020-12-07T18:52:51.840Z
Just few comments: 1). It is not (fully) clear to me how this is related to Benveniste and Scheinkman (who, in their 1979 paper, propose results that apply to dynamic problems).   2). Why do you write $k^*$? Is it optimally chosen? This fully change the statement of you question in which $k$ is exogenous.  3). You should avoid using the same notations for different functions $u(c,l)$ and then $u(k)$
Bertrand
CC BY-SA 4.0
2020-12-07T14:26:51.970Z
2020-12-07T14:26:51.970Z
Maybeline Lee
<p>Consider a household which solves the following problem:
<span class="math-container">$$v(k,r,w)=\underset{c,l\in B{(k,r,ω)}}{\ {max}} \{u(c,l)\}$$</span></p>
<p>where <span class="math-container">$u : R_+^2 \rightarrow R$</span> is a strictly concave, twice continuously differentiable, strictly increasing function in its two arguments: consumption, <span class="math-container">$c$</span>, and leisure, <span class="math-container">$l$</span>. The constraints the household must obey in selecting <span class="math-container">$c, l$</span> are summarized by <span class="math-container">$B$</span>:
<span class="math-container">$$B(k, r, w) = {\{c, l : 0 ≤ c ≤ rk + w(1 − l), 0 ≤ l ≤ 1\}}$$</span></p>
<p>Here, <span class="math-container">$k, r, w &gt; 0$</span> are numbers over which the household has no control.
Prove that <span class="math-container">$v$</span> is concave in <span class="math-container">$k$</span> and that the derivative of <span class="math-container">$v$</span> with respect
to <span class="math-container">$k$</span> exists for ‘interior <span class="math-container">$k$</span>’. Display a formula for the derivative of <span class="math-container">$v$</span>.</p>
<p>What I was thinking for solving this is by following Benveniste &amp; Scheinkman theorem on differentiability <span class="math-container">$ω : D → R$</span> defined on the neighborhood <span class="math-container">$D$</span> of <span class="math-container">$x_0$</span>, i.e.
<span class="math-container">$D ⊂ X$</span> and <span class="math-container">$x_0 ∈ int(D)$</span> such that: <span class="math-container">$ω(x) ≤ v(x)$</span> and <span class="math-container">$ω(x_0) = v(x_0)$</span>.</p>
<p>And <span class="math-container">$ω(.)$</span> is concave and differentiable, then <span class="math-container">$v(.)$</span> is differentiable at <span class="math-container">$x_0$</span> and <span class="math-container">$v'(x_0) =ω'(x_0)$</span>.</p>
<p>I'm guessing we have to substitute c with <span class="math-container">$rk + w(1 − l)$</span> 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?</p>

CC BY-SA 4.0
2020-12-07T11:38:20.537Z
Note that you could write $v(kr,w)$ instead of $v(k,r,w)$, this simplifies the expression of the derivatives of $v$ wrt $k$ and $r$.
Bertrand
CC BY-SA 4.0
2020-12-07T12:03:45.653Z
@Bertrand I'm sorry, I'm not following. Could you please elaborate more on that?
Maybeline Lee
CC BY-SA 4.0
2020-12-07T12:27:24.447Z
If you define $z = rk$ and replace it in your problem, you end up with $v(z,w)$. This establishes a relationship between the marginal utility of $k$ and of $r$.
Bertrand
CC BY-SA 4.0
2021-01-09T17:18:24.570Z
You should probably use the Inverse Function Theorem to establish that solution exists and is differentiable.
Jesper Hybel
<macroeconomics><utility><dynamic-programming>
Proving that the utility is concave
Maybeline Lee
# Proving that the utility is concave

(Asked by: username_0 on 2020-12-07)

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, $k, r, w > 0$ 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 & 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?

**Tags:** macroeconomics, utility, dynamic-programming

> **Comments:**
> **username_1**: Note that you could write $v(kr,w)$ instead of $v(k,r,w)$, this simplifies the expression of the derivatives of $v$ wrt $k$ and $r$.  
> **username_0**: @username_1 I'm sorry, I'm not following. Could you please elaborate more on that?  
> **username_1**: If you define $z = rk$ and replace it in your problem, you end up with $v(z,w)$. This establishes a relationship between the marginal utility of $k$ and of $r$.  
> **username_2**: You should probably use the Inverse Function Theorem to establish that solution exists and is differentiable.  

---

## 1 answers

### Answer by username_0 on 2020-12-07. Score: 1 | Accepted answer

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 & Scheinkman), $v$ is differentiable at $k^\*$.

$V\_k^\*=U\_k(rk^\* + w(1-l^\*, l\*)
= U\_c(rk^\* + w(1-l^\*, l\*)$

> **Comments:**
> **username_1**: Just few comments: 1). It is not (fully) clear to me how this is related to Benveniste and Scheinkman (who, in their 1979 paper, propose results that apply to dynamic problems).   2). Why do you write $k^*$? Is it optimally chosen? This fully change the statement of you question in which $k$ is exogenous.  3). You should avoid using the same notations for different functions $u(c,l)$ and then $u(k)$  

---

