two questions regarding a bellman optimality proof

Question

Hi: I'm reading a proof of "The Bellman Optimality Principle"

First, if someone knows a clearer proof using any reasonably rigorous methodology, I'm open to reading that one instead. I've only read the first two pages of this one and I already have two questions. Two pages may not sound like a lot but it's pretty involved so it's much appreciated if anyone can be bothered taking the time required. There are a lot of details.

My two questions are below. Both questions have to do with equation (7) on the second page which the author refers to as the Bellman-Euler equation.

1. What is the justification for using the replacement function

$$ g(t, k_{t}, c_{t}) = k_t + h(t, k_{t}, c_{t}) $$

2. Immediately below what is done in question 1., the author then defines, $\Delta_t \phi$ and claims that (8) then reduces to (7). I cannot figure out how (8) reduces to (7). It's just algebra as far as I can tell but my algebra is not leading to (7).

I emailed the author and asked him about both questions but he didn't respond. Again, thanks for any help. Pointing to a different and possibly clearer proof would also be helpful. I'd like to think that I looked pretty hard for a good proof but there could be a better one that I've missed. The author is a finance professor with a Ph.D in mathematics so he could have missed one that lives in the economics world.

Answer 1

The idea behind the function $h(t,k_t,c_t)$ is that it gives the increments of capital as it evolves. By definition of $g$, $g(t,k_t,c_t)=k_{t+1}$, so $h(t,k_t,c_t)=k_{t+1}-k_t$. This $h$ function allows us to make a connection between the discrete and continuous time settings. This connection is highlighted by the author in equations (13) and (14) later on in the text. Without $h$, the connection between discrete and continuous time is unclear since there is no analogous $g$ function in continuous time as there is no "next" time step from the current one. Expressing the discrete time case in terms $h$ rather than $g$ allows us to view the continuous time problem as the "limiting" case of the discrete time problem as time increments become infinitely small.

For point (2), it is just algebra. The equation in (2) is evaluated at the optimum. Given $g(t,k_t,c_t)=k_t+h(t,k_t,c_t)$, we have by an envelope argument that $g_k(t)=1+h_k(t)$ and $g_c(t)=h_c(t)$. Therefore,

$$\begin{align} f_k(t+1)-\frac{f_c(t+1)}{g_c(t+1)}g_k(t+1)&=-\frac{f_c(t)}{g_c(t)} \\ f_k(t+1)-\frac{f_c(t+1)}{h_c(t+1)}[1+h_k(t+1)]&=-\frac{f_c(t)}{h_c(t)} \\ f_k(t+1)-\frac{f_c(t+1)}{h_c(t+1)}h_k(t+1) &=\frac{f_c(t+1)}{h_c(t+1)}-\frac{f_c(t)}{h_c(t)} \\ f_k(t+1)-\frac{f_c(t+1)}{h_c(t+1)}h_k(t+1) &= \Delta_t\left(\frac{f_c(t)}{h_c(t)}\right)\end{align}$$

For alternative references, I think one of the best presentations of recursive methods is in Stokey-Lucas-Prescott "Recursive methods in economic dynamics" (1989). However, the presentation there is, if anything, too maths heavy. But still, it is a great reference text.