Is labour considered normal by someone whose utility function is $u = cl$?

Question

If $l$ is hours of leisure and $c$ is the consumption units, can we show that leisure is a normal good for someone whose utility function is $u = lc$?

If $w$ denotes wage per hour, $i$ denotes the non-labour income and $p = 1$ denotes the price of consumption, then the expenditure will be $pc \leq w \times (\text{labour hours}) + i = w(24-l) + i$. The constraint is binding as consumption-leisure indifference curves are convex.

We can rewrite the utility function as $u(l) = \left[w(24-l) + i\right] \cdot l = 24wl - wl^2 + il$. Maxima exists as $U(l)$ is a downward parabola. $u'(l) = 0 \implies l^{*} = \frac{T}{2} + \frac{N}{2w}$.

How do we proceed now? Do we compare leisure with $i$ or $w$? Moreover, since $i$ and $w$ are probably related (which I am not sure of and would like a confirmation), we can't directly say $\frac{\partial u(l)}{\partial w} = -N/2w^2 < 0$ or $\frac{\partial u}{\partial N} = 1/2w > 0$. Is that true?

Answer 1

I think answering this question requires less math than you are giving to it.

First, a normal good is a good that experiences an increase in its demand due to a rise in consumers' income. So suppose that your individual was given more money. This is equivalent to their budget constraint being expanded. An optimal new point under the relaxed constraint (assuming monotonicity, convexity) would have both more leisure and consumption than the previous optimal point. So leisure is a normal good.

Intuitively, you can think of this as: I have more money, so I'm going to buy a little bit more and work a little bit less (such that I can still buy slightly more than before).