Labour supply and intertemporal choice

Every hour has to go somewhere. She has T hours; an hour of leisure is an hour not worked, and it costs her the wage she gives up. So leisure is a good, and its price is w, the hourly wage. Her other good is consumption c, which costs Re 1 per rupee’s worth. If she also has income m₀ that does not come from work (rent, a pension), her budget is

Plot leisure on the horizontal axis and consumption on the vertical. The budget line runs from the point (T, m₀) (all leisure, no work, only outside income) up and to the left with slope −w: each hour of leisure given up buys w rupees of consumption. Fred has 112 − 80 = 32 hours of work at $75, so $2,400 (JNU 2019).

It is Lesson 5 again. She picks the bundle (ℓ, c) on the line that reaches the highest indifference curve: her MRS between leisure and consumption equals the wage. The hours she works are H = T − ℓ.

Take Cobb–Douglas tastes between consumption and leisure. GATE 2024 gives U = (24 − H)^(1−a)(wH)^a with a = ½. The first factor is leisure and the second is earnings. Maximise: the first-order condition is a/H = (1−a)/(24−H), so H = 24a = 12 hours whatever the wage, because the wage multiplies utility without changing the trade-off. With outside income m₀ the same tastes give H = 24a − (1−a)·m₀/w, which does rise with the wage.

Move the wage and watch her choose. The right-hand plot traces the number of hours she works at every wage up to Rs 40; the dot marks the wage you have set. Lower ρ makes leisure and consumption harder to swap, which strengthens the income effect. Add outside income and turn on overtime to see the kink in the budget line.

A higher wage does two things at once, and they pull in opposite directions.

• Substitution effect. Leisure now costs more in forgone earnings, so she takes less of it and works more.
• Income effect. A higher wage makes her richer. If leisure is a normal good, she buys more of it and works less.

At low wages the substitution effect usually wins and the labour supply curve slopes up. At high wages the income effect can overtake it, and hours fall: the backward-bending labour supply curve. The lab shows it with outside income of Rs 20 and ρ = −1: hours rise to a peak near a wage of Rs 4.5 and then decline.

Overtime adds a kink. If hours beyond 8 pay 1.5 times as much, the budget line steepens at that point (a higher price for leisure beyond it), and a worker who would otherwise stop at 8 hours often works more.

Let c₁ be consumption now and c₂ consumption next period, with incomes y₁ and y₂. She can save or borrow at an interest rate r. A rupee saved now becomes 1 + r next period, so a rupee of consumption next period costs only 1/(1+r) of a rupee now. The budget is

Plot c₁ on the horizontal axis and c₂ on the vertical. The slope is −(1 + r): giving up a rupee of consumption now buys 1 + r tomorrow. If she consumes less than y₁ now she is a saver; if more, she borrows against y₂.

Give her utility u = ln c₁ + ln c₂/(1 + ρ), where ρ measures impatience. The tangency (the MRS equals 1 + r) gives the growth of consumption:

So when r = ρ she smooths consumption completely: c₁ = c₂. When r > ρ consumption grows, and when r < ρ it falls. Whether she saves or borrows depends on where her income sits: with y₁ = 90, y₂ = 0 and r = ρ = 25%, she consumes 50 in each period and saves 40; with y₁ = 0, y₂ = 90 she consumes 40 in each period and borrows 40. This lesson stops here. The Euler equation in continuous time, and the consumption theories built on it, belong to the macroeconomics modules.

Move the incomes, the interest rate and her impatience. The hollow dot is where she would be if she simply consumed her income each period. The gold dot is her choice. Make r equal to ρ and the dot sits on the 45° ray where c₁ = c₂.

The Delhi School entrance asks seven labour-supply and two-period questions above Easy; they are gathered, with their solutions, in Lesson 17.