<p>I'll try to translate the argument of <a href="http://www.dklevine.com/archive/refs4233.pdf" rel="nofollow noreferrer">Hansen's original (1985) paper</a></p>
<p>Let <span class="math-container">$h_t = \alpha_t h$</span> be aggregate labour supply at <span class="math-container">$t$</span>.</p>
<p>First define aggregate leisure <span class="math-container">$\ell_t = 1 - h_t = 1 - \alpha_t h$</span> which gives:
<span class="math-container">$$
\alpha_t = \frac{1 - \ell_t}{h}
$$</span>
Then the instantaneous utility function can be written as:
<span class="math-container">$$
u(c_t) + \alpha_t (v(1-h)-v(1)) + v(1) = u(c_t) + v(1) + \frac{v(1-h)}{h} - \left[\frac{v(1-h)}{h}\right]\ell_t
$$</span>
Here the choice variable <span class="math-container">$\alpha_t$</span> changed to <span class="math-container">$\ell_t$</span>. We see that this function is linear in <span class="math-container">$\ell_t$</span> which gives an infinite elasticity of substitution between leisure in different periods.</p>
<p>I'm not a macro person, so I might be totally wrong here. However, for another way to see the issue is to look at your Bellman equation:
<span class="math-container">$$
V(k_t) = \max_{c_t, \alpha_t, b_t} u(c_t) + \alpha_t v(1-h) + (1-\alpha_t) v(1) + \beta\left[\alpha_t V(k_{t+1}^g) + (1-\alpha_t) V(k_{t+1}^b)\right]
$$</span>
where:
<span class="math-container">$$
\begin{align*}
&amp;k_{t+1}^g = (1-\delta) k_t + w h - (1-\alpha)b_t - c_t\\
&amp;k_{t+1}^b = (1-\delta) k_t + b_t - (1-\alpha)b_t - c_t
\end{align*}
$$</span>
Here I'm assuming the premium to be based on the market risk (<span class="math-container">$\alpha$</span>) and not on the individual risk <span class="math-container">$\alpha_t$</span>, so <span class="math-container">$\alpha$</span> is fixed for the individual that is optimizing this bellman equation.</p>
<p>Let us show that 'in general' aggregate frish supply will be <span class="math-container">$h_t = h$</span> or <span class="math-container">$h_t = 0$</span> except for a particular value of <span class="math-container">$w$</span>. So let's go with the assumption that <span class="math-container">$\alpha \in (0,1)$</span>.
Assume that we already solved for the optimal values of consumption and insurance.</p>
<p>To find the optimal value of <span class="math-container">$\alpha_t$</span> take the first order condition of <span class="math-container">$V(k_t)$</span> with respect to <span class="math-container">$\alpha_t$</span>:
<span class="math-container">$$
v(1-h) - v(1) + \beta \left[V(k_{t+1}^g)- V(k_{t+1}^b)\right] = 0
$$</span>
As <span class="math-container">$k_{t+1}^g$</span> is increasing in <span class="math-container">$w$</span> and the rest is independent of <span class="math-container">$w$</span> there can be at most one value of <span class="math-container">$w$</span> for which this condition holds.</p>
<p>If <span class="math-container">$w$</span> is bigger than this value, it will be optimal to set <span class="math-container">$\alpha_t = 1$</span> so aggregate demand will jump to <span class="math-container">$h_t = \alpha_t h = h$</span>. (I'm hand-waving here as the optimal values of <span class="math-container">$c$</span> and <span class="math-container">$b$</span> may also of course depend on <span class="math-container">$w$</span>).</p>
<p>If <span class="math-container">$w$</span> is smaller than this value, it will be optimal to set <span class="math-container">$\alpha_t = 0$</span>, so aggregate demand will jump to <span class="math-container">$h_t = \alpha_t h = 0$</span>.</p>
<p>This means that aggregate elasticity is infinite.</p>

CC BY-SA 4.0
2022-01-13T04:11:40.850Z
I think your first solution is quite clear in macro point of view. Actually I was stuck in F.O.Cs of bellman equation. The F.O.C of $c_t$ gives $u'(c_t)=\lambda$ and F.O.C of $\alpha$ gives $v(1-h)=\lambda w h$. Since frisch elasticity assumes $u'(c_t)$ is constant, $v(1-h)$ is constant and this fact can be used in your solution? I mean, how to show that $(d \alpha /dw)*(w/\alpha)$ goes infinity using that fact?
guest
CC BY-SA 4.0
2022-01-12T14:18:07.913Z
2022-01-12T14:18:07.913Z
tdm
<p>It is from Hansen(1985)'s paper.<br />
Suppose that household has utility function <span class="math-container">$u(c)+v(1-h)$</span> and they can choose probability to work(<span class="math-container">$h$</span>) as <span class="math-container">$\alpha$</span>.<br />
There is an insurance <span class="math-container">$b_t$</span> giving <span class="math-container">$1$</span> unit of consumption goods and insurance market is complete.<br />
(It means the insurance price is <span class="math-container">$1-\alpha$</span>.)</p>
<p>I skipped detailed derivation process and result of this economy shows that every household has same consumption(<span class="math-container">$c_1=c_2$</span>) and capital(<span class="math-container">$k_1'=k_2'$</span>).<br />
It means budget constriants<br />
(i) <span class="math-container">$c_1+k_1'+(1-\delta)k_0 =wh+rk_0 -(1-\alpha)b$</span> w/ prob. <span class="math-container">$\alpha$</span> <br />
(ii) <span class="math-container">$c_2+k_2'+(1-\delta)k_0 =b+rk_0 -(1-\alpha)b$</span> w/ prob. <span class="math-container">$1-\alpha$</span> <br />
can be reduced like: <span class="math-container">$c+k'-(1-\delta)k_0 =\alpha wh+rk_0$</span></p>
<p>Maximization problem as bellman equation form:<br />
<span class="math-container">$V(k)=u(c)+\alpha v(1-h)+(1-\alpha)v(1)+\beta EV(k')$</span> <br />
s.t <span class="math-container">$c+k'-(1-\delta)k_0 =\alpha wh+rk_0$</span><br />
For simplicity, normalize <span class="math-container">$v(1)=0$</span>.</p>
<p>I want to show that the aggregate labor supply has infinite elasticity using frisch elasticity.<br />
How to derive frisch elasticity in this economy?</p>

CC BY-SA 4.0
2022-01-12T12:04:09.583Z
Your Bellman equation looks really strange. It has $V(1-h)$ and $V(1)$ where I would expect this to be instantaneous utility. It would be nice if you could give the derivation so we know where everything comes from. Also your budget constraint has no expenses on the insurance premium. Is the choice of taking an insurance a decision variable.
tdm
CC BY-SA 4.0
2022-01-12T12:31:58.803Z
@tdm You're right. That's typo and I corrected it. I'm sorry that I skipped lots of calculation parts. The insurance part is disappeared in budget constraint since the price of insurance is 1−α. I also added detailed budget constraints.
guest
<macroeconomics><elasticity><frisch-elasticity>
Infinite labor elasticity with frisch elasticity
guest
# Infinite labor elasticity with frisch elasticity

(Asked by: username_0 on 2022-01-12)

It is from Hansen(1985)'s paper.  
Suppose that household has utility function $u(c)+v(1-h)$ and they can choose probability to work($h$) as $\alpha$.  
There is an insurance $b\_t$ giving $1$ unit of consumption goods and insurance market is complete.  
(It means the insurance price is $1-\alpha$.)

I skipped detailed derivation process and result of this economy shows that every household has same consumption($c\_1=c\_2$) and capital($k\_1'=k\_2'$).  
It means budget constriants  
(i) $c\_1+k\_1'+(1-\delta)k\_0 =wh+rk\_0 -(1-\alpha)b$ w/ prob. $\alpha$   
(ii) $c\_2+k\_2'+(1-\delta)k\_0 =b+rk\_0 -(1-\alpha)b$ w/ prob. $1-\alpha$   
can be reduced like: $c+k'-(1-\delta)k\_0 =\alpha wh+rk\_0$

Maximization problem as bellman equation form:  
$V(k)=u(c)+\alpha v(1-h)+(1-\alpha)v(1)+\beta EV(k')$   
s.t $c+k'-(1-\delta)k\_0 =\alpha wh+rk\_0$  
For simplicity, normalize $v(1)=0$.

I want to show that the aggregate labor supply has infinite elasticity using frisch elasticity.  
How to derive frisch elasticity in this economy?

**Tags:** macroeconomics, elasticity, frisch-elasticity
> **Comments:**
> **username_1**: Your Bellman equation looks really strange. It has $V(1-h)$ and $V(1)$ where I would expect this to be instantaneous utility. It would be nice if you could give the derivation so we know where everything comes from. Also your budget constraint has no expenses on the insurance premium. Is the choice of taking an insurance a decision variable.  
> **username_0**: @tdm You're right. That's typo and I corrected it. I'm sorry that I skipped lots of calculation parts. The insurance part is disappeared in budget constraint since the price of insurance is 1−α. I also added detailed budget constraints.  

---

## 1 answers

### Answer by username_1 on 2022-01-12. Score: 3 | Accepted answer

I'll try to translate the argument of [Hansen's original (1985) paper](http://www.dklevine.com/archive/refs4233.pdf)

Let $h\_t = \alpha\_t h$ be aggregate labour supply at $t$.

First define aggregate leisure $\ell\_t = 1 - h\_t = 1 - \alpha\_t h$ which gives:
$$
\alpha\_t = \frac{1 - \ell\_t}{h}
$$
Then the instantaneous utility function can be written as:
$$
u(c\_t) + \alpha\_t (v(1-h)-v(1)) + v(1) = u(c\_t) + v(1) + \frac{v(1-h)}{h} - \left[\frac{v(1-h)}{h}\right]\ell\_t
$$
Here the choice variable $\alpha\_t$ changed to $\ell\_t$. We see that this function is linear in $\ell\_t$ which gives an infinite elasticity of substitution between leisure in different periods.

I'm not a macro person, so I might be totally wrong here. However, for another way to see the issue is to look at your Bellman equation:
$$
V(k\_t) = \max\_{c\_t, \alpha\_t, b\_t} u(c\_t) + \alpha\_t v(1-h) + (1-\alpha\_t) v(1) + \beta\left[\alpha\_t V(k\_{t+1}^g) + (1-\alpha\_t) V(k\_{t+1}^b)\right]
$$
where:
$$
\begin{align\*}
&k\_{t+1}^g = (1-\delta) k\_t + w h - (1-\alpha)b\_t - c\_t\\
&k\_{t+1}^b = (1-\delta) k\_t + b\_t - (1-\alpha)b\_t - c\_t
\end{align\*}
$$
Here I'm assuming the premium to be based on the market risk ($\alpha$) and not on the individual risk $\alpha\_t$, so $\alpha$ is fixed for the individual that is optimizing this bellman equation.

Let us show that 'in general' aggregate frish supply will be $h\_t = h$ or $h\_t = 0$ except for a particular value of $w$. So let's go with the assumption that $\alpha \in (0,1)$.
Assume that we already solved for the optimal values of consumption and insurance.

To find the optimal value of $\alpha\_t$ take the first order condition of $V(k\_t)$ with respect to $\alpha\_t$:
$$
v(1-h) - v(1) + \beta \left[V(k\_{t+1}^g)- V(k\_{t+1}^b)\right] = 0
$$
As $k\_{t+1}^g$ is increasing in $w$ and the rest is independent of $w$ there can be at most one value of $w$ for which this condition holds.

If $w$ is bigger than this value, it will be optimal to set $\alpha\_t = 1$ so aggregate demand will jump to $h\_t = \alpha\_t h = h$. (I'm hand-waving here as the optimal values of $c$ and $b$ may also of course depend on $w$).

If $w$ is smaller than this value, it will be optimal to set $\alpha\_t = 0$, so aggregate demand will jump to $h\_t = \alpha\_t h = 0$.

This means that aggregate elasticity is infinite.

> **Comments:**
> **username_0**: I think your first solution is quite clear in macro point of view. Actually I was stuck in F.O.Cs of bellman equation. The F.O.C of $c_t$ gives $u'(c_t)=\lambda$ and F.O.C of $\alpha$ gives $v(1-h)=\lambda w h$. Since frisch elasticity assumes $u'(c_t)$ is constant, $v(1-h)$ is constant and this fact can be used in your solution? I mean, how to show that $(d \alpha /dw)*(w/\alpha)$ goes infinity using that fact?  

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