I am playing with a simple, continuous time optimal growth problem to learn optimal control. The social planner chooses $c_t$ to maximize: $$\int_0^\infty e^{-\rho t}u(c_t)dt$$ where $$\dot k_t=f(k_t)-c_t,$$ $u$ is CRRA, $f(k_t)=k_t^\alpha$ and $k_0$ is given.

This has straightforward solution with optimal trajectory $(k_t,c_t)$ converging to stationary point $(k^*,c^*)$.

Numeric computations using a Python implementation of Ben Moll's MATLAB algorithm yield appropriate looking consumption-capital diagrams:

Optimal Policy Function

where the dashed vertical line represents the stationary point.

The optimization algorithm uses an approximation where, roughly speaking, there is a discrete guess for the value function, its derivative is approximated using its first difference and then approximate consumptions and costates are computed to update the value function.

The problem I am running across is when I try to evaluate the property that the current value Hamiltonian evaluated at the optimal $k$, $c$ and $\lambda$ should be constant. Following is a plot where values of $\mathcal H(k,c(k),V'(k))=u(c(k))+V'(k)[f(k)-c(k)]$ on the vertical axis depend on $c(k)$, the approximated optimal consumption and $V'(k)$, the first difference in the approximated value function.

Optimized Current Value Hamiltonian

As I understand from Chiang (1992), $\mathcal H$ should be constant but this is clearly not reflected in my plot -- all of my other plots are spot on. Am I making a mistake in my plot of the optimized current value Hamiltonian or is there some approximation error that is being magnified in my plot?


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