Dr Seuss' wealth $x_{t}$ at time $t$ evolves as

$$\frac{d x}{d t}=r x_{t}+\ell_{t}-c_{t}$$

where $r>0$ is the rate of interest earned, $\ell_{t}$ is his intensity of working $(0 \leqslant \ell \leqslant 1)$, and $c_{t}$ is his rate of consumption. His initial wealth $x_{0}>0$ is given, and his objective is to maximize

$$\int_{0}^{T} U\left(c_{t}, \ell_{t}\right) d t$$

where $U(c, \ell)=c^{\alpha}(1-\ell)^{\beta}$, and $T$ is the (fixed) time his contract expires. The constants $\alpha$ and $\beta$ satisfy the inequalities $0<\alpha<1,0<\beta<1$, and $\alpha+\beta>1$. At all times, $c_{t}$ must be non-negative, and his final wealth $x_{T}$ must be non-negative. Establish the following properties of the optimal solution $\left(x^{*}, c^{*}, \ell^{*}\right)$ :

(i) $\beta c_{t}^{*}=\alpha\left(1-\ell_{t}^{*}\right)$;

(ii) $c_{t}^{*} \propto e^{-\gamma r t}$, where $\gamma \equiv(\beta-1+\alpha)^{-1}$;

(iii) $x_{t}^{*}=A e^{r t}+B e^{-\gamma r t}-r^{-1}$ for some constants $A$ and $B$.

Hence deduce that the optimal wealth is

$$x_{t}^{*}=\frac{\left(1-e^{-\gamma r T}\left(1+r x_{0}\right)\right) e^{r t}+\left(\left(1+r x_{0}\right) e^{r T}-1\right) e^{-\gamma r t}}{r\left(e^{r T}-e^{-\gamma r T}\right)}-\frac{1}{r}$$