(i) At the beginning of year $n$, an investor makes decisions about his investment and consumption for the coming year. He first takes out an amount $c_{n}$ from his current wealth $w_{n}$, and sets this aside for consumption. He splits his remaining wealth between a bank account (unit wealth invested at the start of the year will have grown to a sure amount $r>1$ by the end of the year), and the stock market. Unit wealth invested in the stock market will have become the random amount $X_{n+1}>0$ by the end of the year.

The investor's objective is to invest and consume so as to maximise the expected value of $\sum_{n=1}^{N} U\left(c_{n}\right)$, where $U$ is strictly increasing and strictly convex. Consider the dynamic programming equation (Bellman equation) for his problem,

$$\begin{aligned} V_{n}(w) &=\sup _{c, \theta}\left\{U(c)+E_{n}\left[V_{n+1}\left(\theta(w-c) X_{n+1}+(1-\theta)(w-c) r\right)\right]\right\} \quad(0 \leqslant n<N), \\ V_{N}(w) &=U(w) . \end{aligned}$$

Explain all undefined notation, and explain briefly why the equation holds.

(ii) Supposing that the $X_{i}$ are independent and identically distributed, and that $U(x)=x^{1-R} /(1-R)$, where $R>0$ is different from 1 , find as explicitly as you can the form of the agent's optimal policy.

(iii) Return to the general problem of (i). Assuming that the sample space $\Omega$ is finite, and that all suprema are attained, show that

$$\begin{aligned} E_{n}\left[V_{n+1}^{\prime}\left(w_{n+1}^{*}\right)\left(X_{n+1}-r\right)\right] &=0 \\ r E_{n}\left[V_{n+1}^{\prime}\left(w_{n+1}^{*}\right)\right] &=U^{\prime}\left(c_{n}^{*}\right) \\ r E_{n}\left[V_{n+1}^{\prime}\left(w_{n+1}^{*}\right)\right] &=V_{n}^{\prime}\left(w_{n}^{*}\right) \end{aligned}$$

where $\left(c_{n}^{*}, w_{n}^{*}\right)_{0 \leqslant n \leqslant N}$ denotes the optimal consumption and wealth process for the problem. Explain the significance of each of these equalities.