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

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?

• Hi! Not sure what exactly you mean by "How do we proceed now? Do we compare leisure with $i$ or $w$?" but there are some nuances when you want to determine whether a good is normal and income is not independent of prices, because it is in the form of an endowment; if you are not familiar with the endowment effect you should research that a bit. Aug 5 at 8:55
• @Giskard By compare, I was referring to $\frac{\partial u}{\partial i}$ and $\frac{\partial u}{\partial w}$. As in, I am not sure whether income refers to the non-labour income or the labour income (wage) or a linear combination of the two. Aug 5 at 9:54
• I suppose you could have total income as a function of wage, leisure, and autonomous income. Then use implicit differentiation to find the derivative of leisure demand function w.r.t the income function. I don't think it will be monotonic, though. Aug 5 at 17:01