Consider an unemployment insurance program, which is designed as follows. If a p
ID: 1218157 • Letter: C
Question
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.
Explanation / Answer
The employed contributes to UI when he is working and so enjoys a consumption that is left with him = (1 - t)Lw. When unemployed, he consumes whatever he receives as UI benefits which is given as B. Optimal consumption and Leisure are given as ce and Le .
Use Lagrangian method to solve this optimization problem
Set up the method:
Max U = Inc + aIn(1 - L) - (c - (1-t)Lw)
Find the partial derivatives and set them equal to zero
dU/dc = 0
1/c - = 0
= 1/c
And
dU/dL = 0
-a/1-L + w(1 - t) = 0
= a/w(1-t)(1/1-L)
Solve these two equations
1/c = a/w(1-t)(1/1-L)
c* = (w/a)(1-L)(1-t)
Substitute this value of c in the budget constraint
c = (1-t)Lw
(w/a)(1-L)(1-t)= (1-t)Lw
1/a(1-L) = L
Le = 1/1+a
optimum value of c can be found using this value of Le:
c = (1-t)Lw
ce = w(1-t)/(1+a)
Hence the optimum values are ce = w(1-t)/(1+a) and Le = 1/1+a.
Optimum labor when employed depends only on the value of parameter a that indicates how much individuals value leisure relative to consumption. The greater they value, the lesser will be the optimal amount of labor. This also suggest that when they value leisure more, they supply less labor which is true since by increasing the values of parameter a, we see that 1/1+a falls.
Similarly, optimum consumption is positively related to wage and negatively related to the parameter a and t.