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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?

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  • $\begingroup$ 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. $\endgroup$
    – Giskard
    Aug 5 at 8:55
  • $\begingroup$ @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. $\endgroup$
    – Isa Pérez
    Aug 5 at 9:54
  • $\begingroup$ 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. $\endgroup$ Aug 5 at 17:01

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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).

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    $\begingroup$ Hi! "An optimal new point under the relaxed constraint (assuming monotonicity, convexity) would have both more leisure and consumption" This does, in fact, depend on the utility function. There are convex and monotonic utility functions where one of the goods is an inferior good. For an example see Sørensen 2007. $\endgroup$
    – Giskard
    Aug 5 at 20:05

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