Final Project 810

Model

• Households live for $ T $ periods, without retirement. Agents are heterogeneous in human capital $ h $ and assets $ k $. Agents are either employed or unemployed (but looking for a job). They spend $ s $ proportion of time in school each period. Moreover, $ S $ represents cumulative years of schooling.
• There are two kinds of firms: a high type and a low type.

Firms

• There are $2$ firm types $I \in {L,H}$, with $ \mu $ fraction of firms being the low type.
• Firm type $I=L$ hires all workers while firm type $I=H$ hires only workers with $S \geq \underline{S}$.

Workers

• Law of motion of human capital is: $h' = \exp(z) H(h,s)$ where $z \in \mathbb{R}_+$ is a random shock.

Unemployed Workers

• Search for a job with intensity $\gamma$, $(\gamma + s = 1)$.

• Receive a job offer with probability $\pi(\gamma, S)$, $(\pi(0,\cdot) = 0)$ -- $ \pi_t(\gamma, S) = \gamma \cdot \frac{S}{t} $.

• Dependent on $S$ they might receive an offer from just $L$ or both firms.

• Receive unemployment benefits $b$ while unemployed.

• There are $3$ state variables:

  • $h \in \mathbb{R}_+$ human capital. Law of motion $ h' = \exp{z'} H(h,s) $.
  • $k \in \mathbb{R}_+$ assets.
  • $S \in \mathbb{R}_+$ (accumulated) schooling. Law of motion $ S' = S + s $.

Value Function if $S < \underline{S}$

$$ U_t(h,k,S) = \max_{k',s} \left{ u(c) + \beta\mathbb{E}\left[\pi(\gamma, S)W^L_{t+1}(h',k',S') + (1 - \pi(\gamma, S))U_{t+1}(h',k',S') \right] \right} $$

Value Function if $S \geq \underline{S}$ $$ U_t(h,k,S) = \max_{k',s} \biggl{ u(c) + \beta\mathbb{E}\biggl[\pi(\gamma, S) \left[\mu W^L_{t+1}(h',k',S') + (1 - \mu) W^H_{t+1}(h',k',S') \right] + (1 - \pi(\gamma, S)) U_{t+1}(h',k',S') \biggr] \biggr} $$ with the budget constraint $$ c + k' \leq b + k(1+r).$$

Employed Workers

• Divide their time for $ s + l = 1 $.
• No on-the-job search allowed.

Value function $$ W^I_t(h,k,S) = \max_{k',s} \left{ u(c) + \beta\mathbb{E}\left[ (1 - \delta) W^I_{t+1}(h',k',S') + \delta U_{t+1}(h',k',S') \right] \right} $$ with the budget constraint $$ c + k' \leq R^I_t h l + k(1 +r) $$

Code Notes

• OG version:

  • Has A compounding over time.

• Version 2(V2)

  • Has A be a constant productivity difference over time.

• Version $\mu$ (_vmu files)

  • Has the alternative specification of $\mu = 0.95$.
  • Makes getting an offer from the high type firm even rarer.
  • npz files saved as files ending with ".._vmu.npz".

• Version $A$ (_vA files)

  • Has the alternative specification of $A = 1.05$.
  • Reduces the productivity difference between firms -- more directly reduces labor earnings difference.
  • npz files saved as files ending with ".._vA.npz".

• Version thr (_vthr files)

  • Has the alternative specification of $thr = 0.5$.
  • Increases the qualifications required for the high paying job.
  • npz files saved as files ending with ".._vthr.npz".