I have to following functions describing an economic macro model with polution:
$ \begin{align} Y_t &=K_t^\alpha(A_tL_t)^{1-\alpha}z_t, \; \; L_t=1, \; \; z_t \in [0,1] \\ P_t & = Y_t z_t^\phi=K_t^\alpha(A_tL_t)^{1-\alpha}z_t^{1+\phi}, \; \; \phi>0 \\ A_t &= A_0e^{g_At}, \; g_A>0,\; A_0>0 \\ \underset{(c_t, z_t)_{t=0}^\infty}{\text{max}}W_0 &= \int_0^{\infty} e^{-\rho t}\left(\frac{c_t^{1-{\theta}}-1}{1-{\theta}}-B\frac{P_t^{1+\gamma}}{1+\gamma}\right)\mathrm{d}t\ \; \; \text{s.t. } \\ c &\geq 0, \\ \dot{{k}}_t &= k_t^\alpha z_t- c_t- (\delta+g){k}_t \\ k &\geq 0 \;\;\;\text{ for all t $\geq$ 0} \\ &\lim_{t\to\infty} \dot{k}_t\lambda_te^{-\rho t}=0 \end{align} $
Which when combined yields the following Hamiltonian function:
$ \begin{equation*} H(k, c_t, z_t,\lambda_t, t) = \frac{c_t^{1-{\theta}}-1}{1-{\theta}}-B\frac{(k_t^\alpha z_t^{1+\phi})^{1+\gamma}}{1+\gamma} +\lambda_t(k_t^\alpha z_t-{c_t}-(\delta+g){k_t}) \end{equation*}$
Where $c_t$ and $z_t$ are control variables and $k_t$ is a state variable. One of the challenges of solving this problem is that we have to ensure that $z_t$ is bounded between 0 and 1.
The Hamiltonian derivatives are as follows: $\begin{align} \frac{\partial H}{\partial c_t}&=0 \leftrightarrow c_t^{-\theta}=\lambda_t \\ \frac{\partial H}{\partial z_t} &= 0 \leftrightarrow -(1-\phi)B(k_t^\alpha z_t^{1+\phi})^{\gamma}k_t^\alpha z_t^\phi + \lambda_t k_t^\alpha = 0 \\ \frac{\partial H}{\partial k_t}&=-\dot{\lambda}_t+\rho\lambda_t \leftrightarrow -B\alpha(k_t^\alpha z_t^{1+\phi})^\gamma k_t^{\alpha-1}z_t^{1+\phi} + \lambda \alpha k_t^{\alpha-1}z_t - \lambda(\delta +g) = -\dot{\lambda}_t+\rho\lambda_t\\ \frac{\partial H}{\partial \lambda_t}&= k_t^\alpha z_t- c_t- (\delta+g){k}_t = 0 \end{align} $
A method of making sure that $z_t$ is bounded is by deriving the following equation:
$z_t=(B^{-1}(1+\phi)^{-1}k_t^{-\alpha\gamma}\lambda_t)^{\frac{1}{(1+\phi)\gamma+\phi}}$
If we define $B^{-1}(1+\phi)^{-1}=m$ and $\frac{1}{(1+\phi)\gamma+\phi}=\omega$, we can ensure that $z_t$ is bounded in the following two cases:
$z_t= \begin{cases} 1,& \text{if}\; \lambda_t\geq \frac{k_t^{\alpha\gamma}}{m} \\ (mk_t^{-\alpha\gamma}\lambda_t)^\omega & \text{if}\; \lambda_t < \frac{k_t^{\alpha\gamma}}{m} \end{cases}$
Having done this i seem at a loss of where to go next. I guess the idea is to simplify the expresson and isolate so that we get an expression for $c_t$ as function of time t and derivatives of $c_t$ with respect to time and the same for $z_t$. If anyone is able to point me in the right way or has seen any mistakes in what i've done so far it would be much appreciated.