# Dynamic Programming and optimization problem

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.

• Either I am confused or there is something wrong with the wording... If $x_t$ is income and he is required to spend less than the income ($u_t \le x_t$) each year, then the capital doesn't matter since he cannot spend it, so the optimum is to just spend all income every year. OTOH if he can spend the capital, then (assuming $\theta > 0$) the optimum is clearly to spend nothing at all until the very last year when you spend the entire pot. Am I confused? – antkam Apr 10 '18 at 15:11
• @antkam Perhaps I'm mistaken but it seems to make sense because the capital becomes part of his income for next year, and thus consumable. – user3658307 Apr 11 '18 at 16:37
• @user3658307 - Sure it makes sense that he can consume capital that was previously invested. But if the interest rate > 0, and if the optimality critierion is maximize total consumption while it doesnt matter WHEN money is spent (i.e. no "net present value" style consideration), then the optimum is clearly: invest everything & spend nothing, until the very last year when you spend it all. Right? What I mean is this combo (can spend capital + positive interest rate + maximize total consumption) has a trivial solution. Or am I missing something? – antkam Apr 11 '18 at 17:22