The work-leisure choice

In the neoclassical model of labour supply, a worker divides a fixed amount of time between paid work and leisure to maximize utility from consumption and leisure. At an interior optimum the highest attainable indifference curve is tangent to the budget line, so the marginal rate of substitution between leisure and consumption equals the wage. When no such tangency exists within the budget set, the worker chooses the corner with no work.

The intuition

An hour of leisure costs the wage: every hour not worked gives up $w$ dollars of consumption. A worker earning $20 an hour should take another hour off if that hour is worth more to them than $20 of goods, and work another hour if it is worth less. The optimum is where the two values are equal.

Preferences

The utility function $U = f(C, L)$ ranks bundles of consumption $C$, the dollar value of the goods and services bought in a week (the price of goods is set to $1), and leisure $L$, hours a week not spent in paid work. $U$ is an index: a higher number means a preferred bundle, but differences in $U$ have no meaning of their own. Both $C$ and $L$ are goods, so their marginal utilities $MU_C$ and $MU_L$ are positive.

An indifference curve is the set of bundles that give the same utility. Indifference curves have four properties:

1. They slope down. With more leisure, the worker must have less consumption to stay equally well off.
2. Higher curves are better. A bundle up and to the right has more of both goods.
3. They do not cross, as shown below.
4. They are convex to the origin. A worker with little leisure gives up a lot of consumption for one more hour of it; a worker with much leisure gives up little.

The slope of an indifference curve is $\Delta C / \Delta L = -MU_L / MU_C$. Its absolute value is the marginal rate of substitution, $MRS = MU_L / MU_C$: the consumption the worker would give up for one more hour of leisure. Convexity means a diminishing MRS along a curve. At a given bundle, a steeper curve means a higher MRS: that worker values leisure more.

Leisure at point A

60 hours

Weight on leisure b

0.60

Consumption at A

$1,262

MRS at A (dollars of consumption per hour of leisure)

$31.55

MRS at 72 hours of leisure on U₁

$20.00

Open in Graph Lab

Figure 1. Indifference curves for $U = C^{1-b}L^{b}$, each through a fixed bundle at 72 hours of leisure. With $b = 0.6$, the MRS along $U_1$ falls from $31.55 an hour at 60 hours of leisure to $7.79 at 105 hours. With $b = 0.8$, the curve through (72 hours, $960) is steeper, with an MRS of $53.33 instead of $20.00: a worker who values leisure more needs more consumption to give up an hour of it.

Why indifference curves cannot cross. Suppose two curves cross at bundle $Y$. Take $X$ on the first curve and $Z$ on the second, with $Z$ containing more leisure and more consumption than $X$. Then $Y$ is as good as $X$ (same curve) and $Y$ is as good as $Z$ (same curve), so by transitivity $X$ is as good as $Z$. But $Z$ has more of both goods, so it must be strictly preferred to $X$. The contradiction shows that curves of one person cannot cross.

Open in Graph Lab

Figure 2. Two supposed indifference curves crossing at $Y$. Transitivity makes $X$ and $Z$ equally good, but $Z$ has more leisure and more consumption than $X$, so it must be preferred: a contradiction.

Preferences differ across workers, and economists cannot observe them. The model’s predictions are therefore stated in terms of observable variables: the wage and non-labour income.

The budget constraint

A worker with $T$ hours a week works $h = T - L$ hours at wage $w$ and receives non-labour income $V$ (dividends, interest, transfers, gifts) regardless of work. With no saving, consumption equals income:

$$C = wh + V = w(T - L) + V = (wT + V) - wL .$$

In the $(L, C)$ diagram the budget line has slope $-w$ and vertical intercept $wT + V$, the consumption possible with no leisure. Its other end is the endowment point $E = (T, V)$: no work, all leisure, consumption $V$. Leisure cannot exceed $T$, so the budget set ends in a vertical line at $L = T$ below $E$. Hours of work are read from right to left, starting at $T$.

Wage w

$20.00

Non-labour income V

$200

Most consumption (no leisure)

$2,400

Consumption at E (no work)

$200

Slope of the budget line (−w)

−$20.00

Open in Graph Lab

Figure 3. With $T = 110$, a wage of $20 and non-labour income of $200, the budget line runs from the endowment point $E$ (110 hours of leisure, $200) to $2,400 at zero leisure, with slope $-20$. A change in the wage pivots the line around $E$; a change in non-labour income shifts it in parallel.

The hours-of-work decision

The worker picks the bundle on the budget line that reaches the highest indifference curve. At an interior optimum the indifference curve is tangent to the budget line, so their slopes are equal:

$$MRS = \frac{MU_L}{MU_C} = w \qquad \Longleftrightarrow \qquad \frac{MU_L}{w} = MU_C .$$

The second form compares the utility from the last dollar “spent” on leisure, by working one dollar’s worth less, with the utility from the last dollar spent on goods. If $MU_L / w > MU_C$, a dollar’s worth of leisure is worth more than a dollar of goods: work less. If $MU_L / w < MU_C$, work more. Any other point on the budget line lies on a lower indifference curve.

The tangency describes an interior solution, with $0 < L^* < T$ and $V < C^* < wT + V$. If the MRS at the endowment point exceeds the wage, the worker values even the first hour of leisure given up at more than the wage, and the best choice is the corner solution $L = T$: no work. The MRS at $E$ is the reservation wage.

Wage w

$20.00

Non-labour income V

$200

Leisure L

72.0 hours

Hours of work h = T − L

38.0 hours

Consumption C

$960

Reservation wage (MRS at E)

$2.73

Open in Graph Lab

Figure 4. With $U = C^{0.4}L^{0.6}$, a wage of $20, non-labour income of $200 and $T = 110$, the highest attainable indifference curve touches the budget line at $P$: 72 hours of leisure, 38 hours of work and $960 of consumption, where $MRS = w = 20$. With non-labour income of $1,600 the MRS at $E$ is $21.82, so at any wage below that the optimum is the corner $E$ and the person does not work.

Key idea

A worker who chooses to work sets the marginal rate of substitution equal to the wage. The wage is the price of an hour of leisure.

Worked example

A worker has utility $U = C^{0.4}L^{0.6}$, $T = 110$ hours, a wage of $20 and non-labour income of $200 a week.

1. Marginal rate of substitution. $MU_C = 0.4\,C^{-0.6}L^{0.6}$ and $MU_L = 0.6\,C^{0.4}L^{-0.4}$, so $MRS = \dfrac{MU_L}{MU_C} = \dfrac{0.6}{0.4}\cdot\dfrac{C}{L} = 1.5\,\dfrac{C}{L}$.

2. Tangency. $1.5\,C/L = 20$, so $C = \tfrac{40}{3}L$.

3. Budget line. $C = 20(110 - L) + 200 = 2{,}400 - 20L$.

4. Solve. $\tfrac{40}{3}L = 2{,}400 - 20L$ gives $\tfrac{100}{3}L = 2{,}400$, so $L^* = 72$ hours of leisure, $h^* = 110 - 72 = 38$ hours of work and $C^* = 20(38) + 200 = 960$ dollars.

5. Check the tangency and the corner. $MRS = 1.5 \times 960 / 72 = 20 = w$. At $E$, $MRS = 1.5 \times 200 / 110 = 2.73$, below the wage, so working beats not working: the solution is interior.

With Cobb-Douglas utility $U = C^{a}L^{1-a}$ the worker always spends the share $a$ of full income $wT + V$ on consumption and $1 - a$ on leisure: $C^* = a(wT + V)$ and $L^* = (1 - a)(wT + V)/w$. Here $C^* = 0.4 \times 2{,}400 = 960$ and $L^* = 0.6 \times 2{,}400 / 20 = 72$.

Evidence: do workers choose their hours?

The model lets each worker pick any hours at the going wage. Many jobs come with fixed schedules, but workers can still choose among full-time and part-time jobs, overtime and second jobs. Canadian data on why people work part time show both sides. In 2025, of 3.83 million part-time workers, 1.14 million worked part time because they were going to school and 1.02 million out of personal preference: hours chosen, as in the model. But 706,000, or 18 percent, worked part time because of business conditions or because they could not find full-time work.¹ At their wage they would supply more hours than they get, so they are off their labour supply curves.

The model’s sharpest predictions concern how hours respond to the wage and to non-labour income. Income and substitution effects derives them and the evidence on each.

Common mistakes

Traps on exams

• Inverting the MRS. The MRS is $MU_L / MU_C$, the slope of an indifference curve drawn with leisure on the horizontal axis.
• Drawing the budget line from the origin. It starts at the endowment point $(T, V)$, not at zero, and its slope is $-w$, not $-1/w$.
• Reading hours of work off the horizontal axis. The axis measures leisure; hours of work are $T - L$, measured leftward from $T$.
• Applying the tangency at a corner. If the person does not work, the MRS at $E$ exceeds the wage; the tangency condition does not hold.

On the exam

Put leisure on the horizontal axis with $T$ marked, consumption on the vertical axis, and the endowment point $E$ at height $V$ above $T$. Draw the budget line from $E$ with slope $-w$ to the intercept $wT + V$, draw an indifference curve tangent at $P$, and drop guides to read $L^*$ and $C^*$; label hours of work $h^* = T - L^*$.

Check yourself

1.

At the optimal consumption-leisure bundle of a worker who chooses to work, which condition holds?

1$MU_L / MU_C = w$2$MU_L = MU_C$3$MU_C / MU_L = w$4$MU_L / MU_C = 1/w$

Check answer

2.

True or false: at a given bundle of consumption and leisure, a worker whose indifference curve is steeper values leisure more than a worker whose indifference curve is flatter.

1True2False

Check answer

3.

A worker's utility is $U = C^{0.5}L^{0.5}$. She has 110 hours a week, earns $25 an hour and receives $250 a week of non-labour income. How many hours a week does she work at the optimum?

Your answer (hours per week)

Check answer

Practise more questions on this topic

Sources

1. Statistics Canada, Table 14-10-0029-01, Part-time employment by reason, annual. Canada, both genders, 15 years and over, 2025. Involuntary part time is the sum of the four reasons involving business conditions or not finding full-time work. ↩