Consider the model $\max U = \ln c + \ln l$ subject to $c=Lw$ and $1=l+L$, where $c$ denotes consumption, $l$ leisure, $L$ the labour and $w$ the wage rate.
The optimal choices of $L$ and $l$ can be easily computed to be 1/2. I don't understand the intuition why the labour supply decision is independent of the wage rate in this case. Is it possible to argue in terms of substitution effect and income effect of a wage rate change?