# Solve the Ben-Porath Model (Optimal Control Problem)

Suppose we have a Ben-Porath style human capital investment model, in which the representative agent maximize her lifetime earnings: $$V(h, a)=\max \int_{a}^{R} e^{-r(t-a)}\left[ w h(t)(1-n(t))-px(t)\right] d t$$ s.t. $$\dot{h}(t)=z_{h}(n(t) h(t))^{\gamma_{1}} x(t)^{\gamma_{2}}-\delta_{h} h(t)$$, where the constraint is the human capital production function.

I try the current Hamilton and obtain the FOCs: $$w h(t) n(t) = \mu(t) \gamma_{1} z_{h}(n(t) h(t))^{\gamma_{1}} x(t)^{\gamma_{2}}$$ $$px(t)=\mu(t) \gamma_{2} z_{h}(n(t) h(t))^{\gamma_{1}} x(t)^{\gamma_{2}}$$ $$\dot{\mu}(t) =r \mu(t)-\mu(t) \left[\gamma_{1} z_{h}(n(t) h(t))^{\gamma_{1}} x(t)^{\gamma_{2}} h(t)^{-1}-\delta_{h}\right]-w(1-n(t))$$ $$\dot{h}(t)=z_{h}(n(t) h(t))^{\gamma_{1}} x(t)^{\gamma_{2}}-\delta_{h} h(t)$$ $$\mu(R)=0$$.

I have no idea how to solve this analytically and get the the optimal solution and the value function, which should be $$V(h, a)=w\left\{\frac{m(a)}{r+\delta_{h}} h+\frac{1-\gamma}{\gamma_{1}}\left[\frac{z_{h} \gamma_{1}}{r+\delta_{h}}\left(\frac{\gamma_{2}}{\gamma_{1}} \frac{w}{p_{w}}\right)^{\gamma_{2}}\right]^{1 /(1-\gamma)}\times \int_{a}^{R} e^{-r(t-a)} m(t)^{1 /(1-\gamma)} d t\right\}$$ and $$n(t) h(t)=\left[\frac{z_{h} \gamma_{1}^{1-\gamma_{2}} \gamma_{2}^{\gamma_{2}}}{r+\delta_{h}}\left(\frac{w}{p_{w}}\right)^{\gamma_{2}}\right]^{1 /(1-\gamma)} m(t)^{1 /(1-\gamma)}$$, where $$m(t)=1-e^{-\left(r+\delta_{h}\right)(R-t)}$$ and $$\gamma =\gamma_{1}+\gamma_{2}$$.

Update: I find that Christopher Taber's slide solves this problem. I follow his approach and get the $$n(t)h(t)$$ exactly the same as above. I then try to use the $$n(t)h(t)$$ to solve the differential equation of $$h(t)$$ given some certain $$h(a)$$, and get \begin{aligned} h\left(t\right)= e^{-\delta_{h}\left(t-a\right)} h(a)+\frac{r+\delta_{h}}{\gamma_{1}}\left[\frac{z_{h} \gamma_{1}}{r+\delta_{h}}\left(\frac{\gamma_{2}}{\gamma_{1}} \frac{ w}{p_{w}}\right)^{\gamma_{2}}\right]^{1 /(1-\gamma)} \times \int_{a}^{t} e^{-\delta_{h}\left(t-s\right)} m(s)^{\frac{\gamma}{1-\gamma}} d s \end{aligned}. I then try to use $$n(t)h(t)$$ and $$h(t)$$ to get $$V(h,a)$$. However the result is quite messy and I fail to get the $$V(h,a)$$ as above.