An Interactive Introduction to the Labour Supply Model

§ 1The Worker's Problem

A worker has a fixed amount of time, $T$ hours, to allocate between leisure $\ell$ and work $h = T - \ell$. Each hour worked earns the wage $w$, and the worker also receives non-labour income $V$. All earnings are spent on consumption $c$:

$c = w(T - \ell) + V = (wT + V) - w\,\ell$

This is the budget constraint: its slope is $-w$ (the price of leisure is the wage forgone), and the vertical intercept $wT+V$ is what the worker would consume by working all $T$ hours.

Preferences are summarised by a utility function $u(c,\ell)$. The worker maximises utility subject to the budget. At an interior optimum, the slope of the indifference curve equals the slope of the budget line:

$\text{MRS}_{\ell, c} = \dfrac{u_\ell}{u_c} = w$

The marginal rate of substitution of leisure for consumption equals the real wage. The optimum $P$ in the figure below is where the highest reachable indifference curve $U^*$ is tangent to the budget line.

§ 2Interactive Model

Move the sliders. The left panel reproduces the consumption–leisure choice; the right panel traces out the labour supply curve by recording the optimal hours of work at each wage you visit. Hit Sweep wage to draw the full curve in one go.

Parameters

$w$ wage rate$10

$V$ non-labour income$100

$T$ time endowment110 h

$\alpha$ taste for leisure0.55

leisure $\ell^*$—
hours worked $h^*$—
consumption $c^*$—
utility $U^*$—

Consumption–Leisure Choice

indifference curves & budget line

Labour Supply Curve

optimal hours worked at each wage

§ 3Reading the Two Panels

Left panel

The optimum $P$

The black line is the budget constraint, slope $-w$. The faint red curves are indifference curves; the bold one ($U^*$) is the highest reachable level. Their tangency at $P$ gives the optimal bundle $(\ell^*, c^*)$.

Right panel

Tracing the supply curve

Each time you change $w$, the worker's optimal $h^* = T - \ell^*$ is plotted against the new wage. With CES and a low $\sigma$, you recover the backward-bending shape: at high enough wages, hours fall.

Two effects

Income vs. substitution

A higher $w$ makes leisure more expensive (substitution effect: work more) but also makes the worker richer (income effect: work less). Which dominates depends on preferences — controlled here by $\sigma$.

§ 4Closed-Form Solutions

For Cobb–Douglas preferences $u(c,\ell)=c^{1-\alpha}\,\ell^{\alpha}$, optimal leisure is a constant share of full income $M = wT + V$:

$\ell^* = \alpha\,\dfrac{M}{w}, \qquad c^* = (1-\alpha)\,M, \qquad h^* = T - \alpha\,\dfrac{M}{w}.$

Note that with no non-labour income ($V=0$) Cobb–Douglas labour supply is perfectly inelastic at $h^* = (1-\alpha)\,T$ — income and substitution effects exactly cancel. To generate a backward-bending curve, switch to CES:

$u(c,\ell) = \big[(1-\alpha)\,c^{\rho} + \alpha\,\ell^{\rho}\big]^{1/\rho}, \qquad \rho = \dfrac{\sigma - 1}{\sigma}$

The first-order condition $\alpha\ell^{\rho-1}/[(1-\alpha)c^{\rho-1}] = w$ together with the budget gives:

$\ell^* = \dfrac{M}{\,w + (1-\alpha)^{\sigma}\alpha^{-\sigma}\,w^{\sigma}\,}, \qquad c^* = M - w(T - \ell^*) \cdot 0 + \big(M - w\ell^*\big)$

(The model below solves CES numerically to keep the code transparent.) When $\sigma < 1$, consumption and leisure are complements: a wage rise raises desired consumption, which requires more leisure to enjoy, so the income effect can overwhelm substitution at high wages — the curve bends backward.