General Problem
Let $k\in\mathbb{R}_+$ be the state variable, $k=k^*$ a fixed point (saddle) and $v(k)$ a value function. The problem is, that the value function has two distinct functional forms, where the switch is at the fixed point.
- Do you have any reference for theses kind of problems?
- Do you think a solution to the model below can be approximated?
- What about a kink in the control/policy function?
Model
Let $t\in[0,\infty)$ and $k\in[0,2]$ is the state. There are two players $i=1,2$ with controls $\tau_i\in[0,1]$. Production is given by $f:[0,2]\to\mathbb{R}_+$. The wages and interests are given by $w_1=f(k)-f'(k)k$, $w_2=f(2-k)-f'(2-k)(2-k)$ and $r_1=f'(k)-\tau_1$, $r_2=f'(2-k)-\tau_2$ respectively. The law of motion for the state is given by \begin{align} \dot{k}=r_1-r_2. \end{align} There is a saddle at \begin{align} k^*=1,~ \tau^*_1=\tau^*_2 \end{align} where $\tau_i^*$ depens on some parameters. Utility is given by the function \begin{align} u(g_i,c_i)=h(g_i)+c_i \end{align} where $u_g>0<u_c$. Public good is with $g_i=\tau_i k$ and consumption \begin{align} c_1=\begin{cases} w_1+r_1k+\int^1_k{[r_2-a(\theta)]d\theta},\quad & k\in[0,1)\\ w_1+r_1,& k\in[1,2] \end{cases} \end{align} and \begin{align} c_2=\begin{cases} w_2+r_2,\quad & k\in[0,1)\\ w_2+r_2(2-k)+\int^k_1{[r_1-b(\theta)]d\theta},& k\in[1,2] \end{cases} \end{align} with $a:[0,1]\to\mathbb{R}_+,~b:[1,2]\to\mathbb{R}_+$ representing some cost for $\theta\in[0,2]$. The payoff functionals are then given by \begin{align} j_i(\tau_i,\tau_j)=\int^\infty_0{e^{-\rho t}(h(g_i)+c_i)dt} \end{align} with $\rho>0$ time preference rate. And the value functions are \begin{align} v_i(k)=\sup_{\tau_i} j_i. \end{align} Setting up the Hamilton-Jacobi-Bellman (HJB) equations for each player yields \begin{align} \rho v_i(k)=\max_{\tau_i} h(g_i)+c_i + v'_i(k)\dot{k}. \end{align} Firstly, I will fix $\tau_2$ at the equilibrium level $\tau^*$. Then the HJB for player 1 reads
\begin{align} \rho v_1(k) &=\max_{\tau_1} h(g_1)+c_1 + v'_1(k)\dot{k}\\ &=\begin{cases}\max_{\tau_1} h(g_1) + w_1+r_1k+\int^1_k{[r_2-a(\theta)]d\theta} + v'_1(k)\dot{k},\quad & k\in[0,1)\\ \max_{\tau_1} h(g_1) + w_1 + r_1 + v'_1(k)\dot{k},& k\in[1,2] \end{cases} \end{align}
The FOCs are \begin{align} \tau_1= \begin{cases} (v_1'(k)+k)^{-1},\quad & k\in[0,1)\\ (v_1'(k)+1)^{-1},\quad & k\in[1,2] \end{cases} \end{align}
First pass implementation
I used the algorithm provided by Benjamin Moll. For reference: in his optimum growth model the value function converged within 8441 iteration and it took 0.825062 seconds.
I used a CB Production function $f(k)=k^\alpha$ with $Ak$-Technology, that is $f(k)=k$ (for $\alpha<1$ I have a problem) and log preferences a la $u(g,c)=\ln g +c$. A linear cost function with $a:\theta\mapsto (1-\theta)$.
I stopped after 200 iterations which already lasted for 21mins. In the first panel is the guess of the 1st and 200th value function. The second panel shows the difference for between the value function (convergence tendency). And the last panel ist the actual policy function, say, $\tau_1(k)$ (note the kink at $k=1=k^*$). It seems to be right from an intuitive perspective. That is, a country lowers its tax rate on capital if the capital stock is realitively small and vive versa. I'm not quite sure why the procedure is so time consuming.
Update
I was able to increase the speed. Then I increased the range of the grid to the respective endpoints of the state, as well as the number of grid points and ran the VFI (40000 Iterations) until I was out of memory. First panel contains first and final value function. The algorithms is converging with $|v^{40000}-v^{39999}|=0.0077$. The kink in the policy function still remains, but the shape slightly differs now. 
