# 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 $0 < r < 1$. Also, there is a fixed transfer of $R > 0$ 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!

• economics.stackexchange.com/q/5933/11590 – Bayesian Mar 3 '17 at 8:17
• I don't understand why this question has been flagged for closing. The OP solved fully the exercise, and has difficulty understanding and interpreting the result -which in my opinion is one of the most valid reasons why a forum like ours should exist, to help people interpret and understand. – Alecos Papadopoulos Mar 3 '17 at 18:31

$$U(c,l) = c^{1/2} - \frac{1}{2}l^2$$
Does now $R$ affect the labor-supplied decision?