Question

For this problem, we will be working with an annual labor supply decision (rather than weekly). This means the person will have 5,000 hours available to spend on labor/leisure (50 weeks * 100 hours per week). We will explore the life and labor supply decisions of Sully. Sully is a single father of 2 that lives his life a quarter mile at a time. Suppose Sully can currently earn a wage of $10 per hour in the labor market. Because of his 2 beloved - though troubled - children, he is eligible to earn a tax credit (if he qualifies based on income level). The tax credit program for individuals with 2 kids has the following features: No credit is earned if no income is earned For annual earnings between 0 and $10,000, a 50% tax credit will be applied. This means, for example, an individual earning $1,000 in wage income would receive $500 in tax credit/refund (we can think of this as additional income). For annual earnings between $10,000 and $20,000, the individual keeps the tax credit earned (there is no payback in this range). For annual earnings over $20,000, the individual will begin to payback his or her tax credit at a rate of 20% (every $1 earned beyond $20,000 means paying back $0.20 worth of credit).

1. (2 points) Complete the following table by filling out earned income, tax credit earned, and total amount of spending money at each level of work hours. Determine the amount of hours/earned income where Sully will no longer qualify for any tax credit (he has paid it all back). Use this information to find his effective hourly wage (net wage) at different levels of work.

2. Hours Worked (annual)	Earned Income	Amount of Tax Credit earned	Total Spending Money	Effective hourly wage
   0	0	0	0	--
   500
   501
   1000
   1001
   2000
   2001
   		0
   5000

3. B (4 points) Create a sketch of Sully’s annual labor supply decision by showing the budget constraint with and without the tax credit program. Assume he has no non-labor income. Be sure to label consumption and leisure at any critical points on the budget constraint. Draw an indifference curve showing what you think is a reasonable choice, and label consumption and leisure at this point. Discuss how the income tax credit’s impact on net wages might impact Sully’s labor supply decision.

Answer 1

Consider utility maximization for a multiple job holder who is not constrained in
his choice of hours to work at the various jobs. To make things concrete we will
consider a Stone-Geary utility function for two different jobs or tasks:
U = (γ1 − h1)
α1 (γ2 − h2)
α2 (y − γ3)
1−α1−α2 (1)
where α1, α2, γ1, γ2, γ3 > 0, hj represents the time allocated to job j, and y is income.
The parameters γ1 and γ2 represent the upper bound on the time that can be ex-
pended on jobs 1 and 2, and still have the utility function defined. They satisfy the
following restriction:
X
2
j=1
γj = T
where T is the total time available for work and leisure. The parameter γ3 represents
the lower bound on the amount of income necessary in order to have the utility
function defined. The economic problem can be stated as
max
h1,h2,y
U = (γ1 − h1)
α1 (γ2 − h2)
α2 (y − γ3)
1−α1−α2
s.t. y = X
2
j=1
wjhj + I,
0 ≤ hj < γj , j = 1, 2 and
X
2
j=1
hj ≤ T,
where wj is the wage or pecuniary rewards to the jth job, and I is exogenous, noncan be shown that the labor supply functions to the two jobs are given by
h1 = (1 − α1) γ1 − α1γ2
µw2
w1
¶
+ α1γ3
µ 1
w1
¶
− α1
µ I
w1
¶
(2)
and
h2 = (1 − α2) γ2 − α2γ1
µw1
w2
¶
+ α2γ3
µ 1
w2
¶
− α2
µ I
w2
¶
. (3)
Accordingly, the earnings version of the labor supply functions are expressed as
w1h1 = α1γ3 + (1 − α1) γ1w1 − α1γ2w2 − α1I (4)
and
w2h2 = α2γ3 + (1 − α2) γ2w2 − α2γ1w2 − α2I. (5)
The uncompensated own wage effect for job i is given by
∂hi
∂wi
= αi
(wi)
2
¡
γjwj + I − γ3
¢
R 0, i, j = 1, 2 for i 6= j,
or in elasticity terms
ηii = wi
hi
∂hi
∂wi
= αi
wihi
¡
γjwj + I − γ3
¢
R 0.
Thus, the effect of an uncompensated increase in the own wage for job i can have
a positive, negative, or no effect on the labor supply to the ith job. An “inferior”
job might be defined as one in which an increase in its wage leads to a reduction in
labor supply to the given job and some combination of increases in leisure and labor
supplied to the other job, subsidized by the increased return to the given job.