Consider an unemployment insurance program, which is designed as follows.
If a person works, he or she should contribute 0.55% of the earnings as the UI premium. In addition, his or her employer contributes the same amount. Therefore, government takes 1.1% of the earnings of each worker as the UI premium. If the person becomes unemployed, he or she receives benefits, which amount to 50% of what he or she earned during the most recent working period. The UI benefits may last up to 8 months.
To analyze the impact of this program on welfare and work incentives in detail, let's assume that all individuals have the following utility function:
U= ln c+a ln (1-L),
Where c and L represent consumption and labor, respectively. Thus, 1-L is leisure. Also, parameter a indicates how much individuals value leisure relative to consumption. Individuals have a year of time endowment, which is allocated to work (L) and leisure (1-L). (So we measre leisure as a fraction of a year.)
Wage rate is w, which means if an individual devotes L to work, he or she earns y=Lw. The budget constraint differs depending on employment status.
Employed: c = (1-t)Lw,
Unemployed: c = B,
where t is the worker's contribution rate of the UI.
Question:
Solve the problem of the employed, who maximizes utility subject to budget constraint. You can solve this either by calculus or by indifference curve analysis. Define ce and Le as the optimal consumption and labor of the employed. Express them in terms of a, t, and w. Interpret the results.