Final answer:
An individual's labor supply function is derived from the utility function U(c,h)=c¹ᵃh¹ᵇ and constrained by income and total available time. The labor-leisure choices made in response to wage rate changes determine the labor supply function, which can vary depending on whether the wage increase leads to a preference for more work or more leisure.
Step-by-step explanation:
The task is to derive the labor supply function of an individual whose utility function is represented as U(c,h)=cᵃhᵇ, where α and β are positive and α+β=1. This individual's consumption, c, and labor, l, are constrained by c=wl+n (income is the wage rate times labor plus nonlabor income) and l+h=1 (total time available is divided between labor and leisure). In analyzing labor-leisure choices, as the wage rate changes, the individual will balance the marginal utility gained from income against the marginal utility gained from leisure.
When wages increase, the individual could respond by working more, the same, or fewer hours, leading to different parts of the labor supply curve: the upward-sloping part, the near-vertical portion, or the backward-bending supply curve. Through the utility maximization process, if an increase in wage leads to a higher utility from earning more than from additional leisure, the labor supply may increase, or vice versa. The individual will select the point along the labor-leisure budget constraint that maximizes utility, which in turn, determines the labor supply function.