I find the labor-leisure model with utility functions interesting, but I find it lacks the factor of work experience, which is very important in the real life labor market.
This is a reason people why people might start working for a lower wage or even do unpaid internships.
I tried searching for a labor-leisure model including the work experience factor in the internet but had no luck.
I remember doing the following linear regression model in econometrics:
$\log(Wage_{it}) = \alpha + \beta \cdot \log(Experience_{it}) + \epsilon_{it}$
So maybe we could solve for the wage here and plug into the usual budget constraint?
By exponentiating, we get
$w_t = w_0 {E_t}^\beta$
where $w_0 := e^\alpha$ and $E_t$ is experience, which we could define as $E_t = \sum_{k=1}^{t} L_k$ (the total amount of time worked in the agent’s lifetime)?
With this we would get
$\max \sum_{t=0}^{\infty} \beta^t [U(c_t, l_t)]$
subject to
$w_0 (\sum_{k=1}^{t} L_k)^\beta + (1+r_{t+1}) s_t = c_t + s_{t+1}$
Here $l_t := T - L_t$ is leisure, where $T$ is the time endowment on each period.
I would appreciate any insight on an extension of this kind.
