The labor supply curve represents how the quantity of labor supplied by individuals varies with changes in wage rates while considering the trade-off between income and leisure. Its derivation is based on the fundamental concept of utility maximization, where individuals seek to balance their desire for income (from working) and their preference for leisure (time not spent working).

1. Utility Function: Start with an individual’s utility function, which quantifies the satisfaction or well-being derived from a combination of goods and leisure. Let’s denote this utility function as U(Y, L), where Y is income and L is leisure.
2. Budget Constraint: The individual faces a budget constraint, which relates income to the wage rate (W), hours worked (H), and leisure time. The budget constraint can be expressed as Y = W * H + R, where R represents non-labor income (e.g., from investments or other sources).
3. Optimization: The individual aims to maximize their utility subject to the budget constraint. This leads to the following maximization problem:Maximize U(Y, L) subject to Y = W * H + R.
4. First-Order Conditions: To find the optimal choice of labor (H), you can use the first-order conditions (partial derivatives):∂U/∂H = λ * ∂(W * H + R)/∂H,where λ is the Lagrange multiplier representing the marginal utility of income.
5. Solving for Labor Supply: This equation simplifies to:∂U/∂H = λ * W.This equation shows how the change in utility with respect to a change in hours worked is equated to the marginal utility of income.
6. Substitution Effect and Income Effect: Changes in the wage rate (W) can be separated into a substitution effect and an income effect. When wages increase, the substitution effect encourages individuals to work more (as leisure becomes relatively more expensive), while the income effect may lead them to work less (as they can afford more leisure with the increased income).
7. Labor Supply Curve: By varying the wage rate (W) and considering how it affects the individual’s choice of hours worked, you can derive the labor supply curve. This curve typically slopes upward because, at higher wages, the substitution effect tends to dominate the income effect, leading to an increase in labor supply.

In summary, the labor supply curve is derived from utility maximization, where individuals make choices based on the trade-off between income and leisure. It illustrates how individuals respond to changes in wage rates by working more or less, depending on the strength of the substitution and income effects.