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\end{equation}  We can re-state the Planner's objective using stationary variables:  \[ E_0 \sum_{t=0}^{\infty} \beta^t  [ \nu \ln (A_{t-1} ) + \nu \ln C_t + (1-\nu) \ln (1- L_t)] \]  \[ E_0 \sum_{t=0}^{\infty} \beta^t  [ \nu \ln C_t + (1-\nu) \ln (1- L_t)] \, + \, \nu E_0 \sum_{t=0}^{\infty} \beta^t \ln (A_{t-1} ) \]  Exponential discounting together with log-utility produce a constant and bounded second term in the last equation, that we can safely ignore and proceed to find optimal values for leisure ($L_t$) and optimal ratios for consumption versus productivity ($C_t$) that maximize \eqref{eq:toy_preferences_u_stationary}  \begin{equation} \label{eq:toy_preferences_u_stationary}  E_0 \sum_{t=0}^{\infty} \beta^t u_t  \end{equation}