Suppose that $\left(\varepsilon_{t}\right)_{t=0,1, \ldots, T}$ is a sequence of independent and identically distributed random variables such that $E \exp \left(z \varepsilon_{1}\right)<\infty$ for all $z \in \mathbb{R}$. Each day, an agent receives an income, the income on day $t$ being $\varepsilon_{t}$. After receiving this income, his wealth is $w_{t}$. From this wealth, he chooses to consume $c_{t}$, and invests the remainder $w_{t}-c_{t}$ in a bank account which pays a daily interest rate of $r>0$. Write down the equation for the evolution of $w_{t}$.

Suppose we are given constants $\beta \in(0,1), A_{T}, \gamma>0$, and define the functions

$$U(x)=-\exp (-\gamma x), \quad U_{T}(x)=-A_{T} \exp (-\nu x)$$

where $\nu:=\gamma r /(1+r)$. The agent's objective is to attain

$$V_{0}(w):=\sup E\left\{\sum_{t=0}^{T-1} \beta^{t} U\left(c_{t}\right)+\beta^{T} U_{T}\left(w_{T}\right) \mid w_{0}=w\right\},$$

where the supremum is taken over all adapted sequences $\left(c_{t}\right)$. If the value function is defined for $0 \leqslant n<T$ by

$$V_{n}(w)=\sup E\left\{\sum_{t=n}^{T-1} \beta^{t-n} U\left(c_{t}\right)+\beta^{T-n} U_{T}\left(w_{T}\right) \mid w_{n}=w\right\}$$

with $V_{T}=U_{T}$, explain briefly why you expect the $V_{n}$ to satisfy

$$V_{n}(w)=\sup _{c}\left[U(c)+\beta E\left\{V_{n+1}\left((1+r)(w-c)+\varepsilon_{n+1}\right)\right\}\right]$$

Show that the solution to $(*)$ has the form

$$V_{n}(w)=-A_{n} \exp (-\nu w),$$

for constants $A_{n}$ to be identified. What is the form of the consumption choices that achieve the supremum in $(*)$ ?