Infinite labor elasticity with frisch elasticity

Question

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?

Answer 1

I'll try to translate the argument of Hansen's original (1985) paper

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.