Choosing between work and leisure with indifference curves

The trade-off. A day has only so many hours, and an hour used one way cannot be used another, so every use of time has a cost in the uses given up. For a worker the hours fall into two blocks: those sold to an employer and those kept for herself. Leisure is wanted for its own sake, while income is wanted for what it buys, and here the only route to income is to give up leisure. So deciding how much labour to supply is deciding how to divide time between leisure and income, and gaining more of one means accepting less of the other. Indifference curve analysis treats this as a choice between two goods, leisure (hours not worked, on the horizontal axis) and income (in rupees, spent on goods, on the vertical axis).

Indifference curves between leisure and income. Each curve joins the combinations of leisure and income that leave her equally satisfied. Since both are desirable, her preferences are monotonic: she prefers a combination with more of one and no less of the other. So:

• each curve slopes downwards, because an extra hour of leisure keeps her equally well off only if she has less income;
• a higher curve stands for greater satisfaction;
• the slope of a curve measures her MRS, the income she is willing to give up for one more hour of leisure, and as this falls with more leisure the curve is convex to the origin.

The budget line: what the wage allows. Suppose she has T hours a day available and the wage is w rupees an hour. If she takes L hours of leisure she works (T − L) hours and earns Y = w(T − L). Every hour of leisure costs her w rupees of income forgone, so the wage is the price of leisure and the opportunity cost of an hour off. Plotted, Y = w(T − L) is a straight line from T hours of leisure with no income to wT of income with no leisure. As with any budget line, the absolute value of its slope is the rate at which the market lets her exchange one good for the other, here w rupees per hour.

The best choice. A rational person picks the point on the budget line that reaches the highest indifference curve she can afford. That is where the line is tangent to an indifference curve, so the rate at which she is willing to give up income for leisure (her MRS) equals the rate at which the market lets her do so (the wage). If her MRS were above the wage, an extra hour of leisure would be worth more to her than the income it costs, and she would take more leisure; if it were below the wage, she would work more. Her hours of work are T minus her chosen leisure.

When the wage rises. A higher wage makes the budget line steeper, pivoting it about the point of T hours of leisure, and the new tangency may lie to the right or to the left of the old one, because the rise pulls her two ways:

• she is better off: every hour worked now earns more, so she can afford to take more time off;
• time off costs more: each hour of leisure now means giving up more income, which pushes her towards longer hours. Joining her chosen hours of work at successive wages gives her labour supply curve. Over the range where the dearer-leisure pull is the stronger, usually at low wages, it slopes upward; beyond the wage at which the better-off pull takes over, a further rise cuts her hours, so the individual labour supply curve bends backward.

Example. Our own figures. Kavya, a delivery rider in Pune, has 14 hours a day to divide between work and leisure, and earns ₹120 an hour. Her budget line runs from 14 hours of leisure with ₹0 to ₹1,680 with no leisure. She chooses 6 hours of leisure and 8 hours of work, earning ₹960; at that point an extra hour of leisure is worth exactly ₹120 to her. If the rate rises to ₹180, she may ride 9 hours (₹1,620) if the pull of the dearer leisure is stronger, or 7 hours (₹1,260) if the higher income makes her want more free time.

Assumptions. Only two goods, leisure and income; her hours are her own choice at the given wage; her tastes and any non-wage income stay the same; and leisure and income are both desirable to her.

Conclusion. The worker's choice is the tangency of her highest indifference curve between leisure and income with a budget line whose slope is the wage. The wage is the price of leisure, and a change in it alters her hours through two opposing effects.