Time is the one thing everyone has the same amount of. Choosing how many hours to work is a consumer choice in disguise: the two "goods" are consumption and leisure, and the tools are the ones from Cobb-Douglas Demand and Substitution and Income Effects.
Key idea
With $T$ hours, wage $w$ and other income $N$, consumption is $C = w(T - \ell) + N$, or $$C + w\ell = wT + N.$$ Leisure is a good whose price is the wage, and $wT + N$ is full income. Hours worked are $h = T - \ell$. A higher wage has a substitution effect (leisure costs more, so work more) and an income effect (you are richer, so take more leisure). If the income effect wins, labor supply bends backward.
Why it works
Every hour of leisure costs the $w$ dollars you could have earned, so $w$ is leisure's price. Unlike a price you pay at a store, though, you are a seller of time: when the wage rises, the value of what you own (your $T$ hours) rises too. So a wage rise makes a worker richer, and if leisure is a normal good, that income effect pushes toward more leisure, against the substitution effect. Which effect is bigger is not settled by theory.
Worked example: Hours with Cobb-Douglas utility
$U = C\ell$, $T = 24$, $w = 10$, $N = 40$. How many hours does she work?
- Full income: $10(24) + 40 = 280$.
- With $U = C\ell$, half of full income goes to each "good": $w\ell = 140$, so $\ell = 14$.
- Hours: $24 - 14 = 10$; consumption $C = 10(10) + 40 = 140$.
Answer: 10 hours.
Worked example: A raise
Same person, but the wage rises to 20. Then do it again with $N = 0$.
- Full income: $20(24) + 40 = 520$, so $\ell = 260/20 = 13$ and $h = 11$. Hours rose.
- With $N = 0$: $\ell = \tfrac{1}{2}\cdot\dfrac{24w}{w} = 12$ at any wage, so $h = 12$ always.
- For Cobb-Douglas with no other income, the two effects of a raise cancel exactly. Other income makes the income effect relatively smaller, so hours rise with the wage.
Answer: 11 hours with $N = 40$; 12 hours at any wage when $N = 0$.
Worked example: A backward-bending case
$U = \min\{C, 10\ell\}$ (consumption and leisure in fixed proportion), $T = 16$, $N = 0$. Compare hours at $w = 10$ and $w = 30$.
- At the kink $C = 10\ell$, and the budget says $C = w(16 - \ell)$. So $\ell = \dfrac{16w}{w + 10}$.
- $w = 10$: $\ell = 8$, $h = 8$. $w = 30$: $\ell = 12$, $h = 4$.
- Fixed proportions leave no room for substitution, so only the income effect acts: a higher wage buys more leisure.
Answer: 8 hours, then 4 hours: labor supply bends backward.
Common mistake
Assuming a higher wage always means more hours. That is only the substitution effect. The income effect of a raise pulls toward more leisure, and for some people, at some wages, it wins. Check: decompose the change. If a raise leaves someone richer and leisure is normal, the two effects point in opposite directions, and you need the numbers to know the total.
What the data say: studies generally find that the hours of people who already work respond only modestly to their wage, while the decision whether to work at all responds more. How large these responses are, and how they differ across groups, is still studied and debated, and it matters for questions such as how taxes on earnings affect work.