I have the following optimal control problem

$$\max_{c_t,l_t} \int_0^{\infty} [ln(c_t)+\theta ln(1-l_t)]e^{-pt}dt$$ st. $$\dot{k_t}=k_t^{1/2}l_t^{1/2}-c_t-\beta l_t$$ $$k_0>0$$

I do big part of its solution. But I'm stack at a part. Please share your ideas about that part with me.

Now, I typed my solution:

Firstly, $c_t, l_t$ are control variables, $q_t$ is co-state variable and $k_t$ is state variable (according to me, I hope it is correct)

current value Hamiltonian function is

$$H= ln(c_t)+\theta ln(1-l_t) + q_t[k_t^{1/2}l_t^{1/2}-c_t-\beta l_t]$$


(1) $${\partial H \over \partial c_t}= \frac{1}{c_t}-q_t \le 0$$, $$c_t [\frac{1}{c_t}-q_t]=0 ; c_t>0$$

(2) $${\partial H \over \partial l_t}= \frac{-\theta}{1-l_t}+q_t [k_t^{1/2}l_t^{-1/2}- \beta] \le 0 , $$

$$l_t[\frac{-\theta}{1-l_t}+q_t [k_t^{1/2}l_t^{-1/2}- \beta]=0; l_t>0$$

(3)$${\partial H \over \partial k_t}= q_t l_t^{1/2}k_t^{-1/2}=- \dot{q_t}+pq_t $$

(4) $${\partial H \over \partial q_t}=k_t^{1/2}l_t^{1/2}-c_t-\beta l_t=\dot{k_t}$$

(5) TVC: as $t$ goes to $\infty$ $$\lim q_t e^{-pt}k_t=0$$

I think that $c_t=0$ and $l_t=0$ are not optimal. so I have $c_t, l_t>0$

Since $c_t>0$, $q_t=\frac{1}{c_t}$ by (1). so, $\frac{\dot{q_t}}{q_t}=-\frac{\dot{c_t}}{c_t}$

By(3), $$\frac{\dot{q_t}}{q_t}=k_t^{-1/2}l_t^{1/2}-p$$

Form here, I have $$\frac{\dot{c_t}}{c_t}=k_t^{-1/2}l_t^{1/2}-p$$

Now, I have put the equation (1) into the equation(2) as follows:

Since $l_t>0$, I have $$\frac{\theta}{1-l_t}=q_t [k_t^{1/2}l_t^{-1/2}- \beta]$$

And I know that $q_t=\frac{1}{c_t}$ by (1).

So I have $$\frac{\theta}{1-l_t}=\frac{1}{c_t} [k_t^{1/2}l_t^{-1/2}- \beta]$$

$$c_t= {[k_t^{1/2}l_t^{-1/2}- \beta]\over \frac{\theta}{1-l_t} }$$

If I take its derivative,

$$\dot{c_t}= (-1/\theta)\dot{l_t}[k_t^{1/2}l_t^{-1/2}- \beta]+\frac{1-l_t}{2\theta} [\frac{l_t\dot{k_t}-k_t\dot{l_t}}{l_t\sqrt{k_tl_t}}]$$

And so what?

I am stack here.I try to find $\frac{\dot{l_t}}{l_t} $

I need to find all differential equations that the dynamic system requires in order to find optimal path.

This is a bit difficult question for me. What all I can do is that. I will be happy if you teach me and show me the true way. Thanks a lot

(The existence of two control variables makes solving it difficult according to me. Therefore, I am asking for.)


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