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# System prepended metadata

title: Frisch elasticity

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# Frisch elasticity

## FOC from Lagrangian

$$  \frac{\partial L}{\partial l} = \alpha (1-l)^{-\zeta} - \lambda (1-m)* w   =0      $$

where $\lambda$ is a lagrange multipiler.
$$\lambda=\frac{\partial U}{\partial y} =\frac{\partial U}{\partial c} \frac{\partial c}{\partial y} =1/c*1= \frac{1}{c}$$
 
## Labor Supply Function

$$ l= 1- (\lambda (1-m) / \alpha *w)^{-1/\zeta}
$$

## labor supply w.r.t. wage

$$ \frac{\partial l}{\partial w} = \frac{1}{\zeta}  *(\lambda (1-m) / \alpha )^{-1/\zeta} *  w^{-1/\zeta-1} $$

## Frisch elasticity

$$ \frac{\partial l }{\partial w } \frac{w}{l} = \frac{1}{\zeta}  *  \frac{ (\lambda (1-m)*w / \alpha )^{-1/\zeta}}{1- (\lambda (1-m)*w / \alpha )^{-1/\zeta}} = \frac{1}{\zeta}  *  \frac{ ( (1-m)/ (\alpha*c) *w  )^{-1/\zeta}}{1- ( (1-m) / (\alpha * c)*w)^{-1/\zeta}}   $$ 

## 数値例
$\zeta=5, \alpha =0.2, \lambda=1 (c=1.0), m=0.1, w=1$の時、

$$\frac{\partial l }{\partial w } \frac{w}{l} =0.2*2.85=0.57$$

$\zeta=5, \alpha =0.2, \lambda=1.43 (c=0.7), m=0.2, w=0.9$の時、

$$\frac{\partial l }{\partial w } \frac{w}{l} =0.2*2.58=0.52$$

$\zeta=5, \alpha =0.2, c=0.5, m=0.3, w=1.4$の時、
$$\frac{\partial l }{\partial w } \frac{w}{l} =$$