An investor receives an annual income from a building society of $x_t$ dollars in year $t$. He consumes $u_t$ and adds $x_t − u_t$ to his capital, $0 ≤ u_t ≤ x_t$. The capital is invested at an interest rate of $θ×100%$. He wants to choose his consumption schedule $u_t, t= 1,...,h$ so as to maximize his total consumption over $h$ years,$C=\sum^{h−1}_{t=0}u_t$.
(a) Write down an equation for how his capital evolves.
(b) Write down the dynamic programming equation for this optimization problem.
(c) Write down the optimal consumption schedule for the particular case $h= 4$ and $θ=0.02$.
I think part (a) is $x_{t+1} = x_t + θ(x_t - u_t)$.
I know part (b) is supposed to take the form $V_n(x) = sup_{0≤u_n≤1}[f(u_nx)+V_{n+1}(R(1-u_n)x)]$ which is being derived from $\sum_{n=0}^{N-1}f(u_nX_n)+g(X_N)$, meaning that an agent receives a utility of $f(u_nX_n)$ and then for the amount of wealth left at the terminal time N the agent receives $g(X_N)$. I'm not sure how to replicate this with the constraints given in the problem.
I think that I just have to plug the information given in part c into the answer to part B in order to get the answer, which seems straightforward.