Effect of changing Non Labor Income on Consumption-Leisure problem Given a Utility function $U(c,l) = c - \frac{1}{2}l^2$ , where $c$ is the consumption and $l$ is the number of hours of labour. Let $L$ be the maximum amount of labor, so the amount of leisure is $L - l$. The wage rate is $w$.

Assume a linear tax rate $r$, where $0 &lt; r &lt; 1$. Also, there is a fixed transfer of $R &gt; 0$ to every individual that is irrespective of labor supply choice. Solve for labor supply l that maximises utility.

My solution: 

$c = (1-r)wl + R$ therefore we maximise $U(l) = (1-r)wl + R - \frac{1}{2}l^2$.

Differentiating $U$ w.r.t $l$ we get $(1-r)w - l = 0$; therefore $l = (1-r)w$. So this shows that $R$ has no relation to $l$. However, it should make a difference due to the income effect, so I'm not sure what I'm doing wrong :/

Thanks!
This has to do with the form of the utility function. Assume instead that,say, we had

$$U(c,l) = c^{1/2} - \frac{1}{2}l^2$$

Does now $R$ affect the labor-supplied decision?

Solve it and explore. Check also the link offered in a comment to your question.