# Infinite labor elasticity with frisch elasticity

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?

• 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
Jan 12, 2022 at 12:04
• @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. Jan 12, 2022 at 12:31

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.

• 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? Jan 13, 2022 at 4:11