1.29 An infinitely lived agent owns 1 unit of a commodity that he consumes over his lifetime. The commodity is perfectly storable and he will receive no more than he has now. Consumption of the commodity in period \( t \) is denoted \( x_{t} \), and his lifetime utility function is given by
\[
u\left(x_{0}, x_{1}, x_{2}, \ldots\right)=\sum_{t=0}^{\infty} \beta^{t} \ln \left(x_{t}\right), \quad \text { where } \quad 0<\beta<1
\]
Calculate his optimal level of consumption in each period.