Question

Consider a tax system as follows. The tax rate for income between 0 and $1,000 is 25%. Any additional income
above $1,000 is taxed at 70%.
Now think of an individual who makes $100 an hour and has a utility function:
u(c, 100-l) = ln(c) + 2ln(100-l)

(a) Compute how many hours a week would this individual work. What is the fiscal revenue?
(b) Now consider a fiscal reform that lowers the highest marginal tax to 60%. What is the impact on hours worked?
What is the impact on fiscal revenues?

Answer 1

a)

u(c,100-l)=ln(c)+2ln(100-l)

Marginal utility of leisure, (100-l)=2/(100-l)

Marginal utility of consumption=1/c

Case 1

Let the agent works only for maximum of 10 hours

w=100*(1-25%)=$75

Consumption is given by

c=w*l=75l

Now set MUl/MUc =w for utility maximization

(2/(100-))/(1/c)=75

2c/(100-l)=75

2c=7500-75l

c=3750-37.50l

Put c=3750-37.50l in consumption function

3750-37.50l=75l

l=33.33 hours

It is not the case as we have assumed that agent works for a maximum of 10 hours.

We ignore this case.

Case 2

Let the agent works for more than 10 hours

w=100*(1-70%)=$30

Consumption is given by

c=100*10*(1-25%)+(l-10)*30

c=750+30l-300=450+30l

Now set MUl/MUc =w for utility maximization

(2/(100-))/(1/c)=30

c/(100-l)=15

c=1500-15l

Put c=1500-15l in consumption function

1500-15l=450+30l

1050=45l

l=70/3=23.33 hours

Number of working hours=23.33 hours

Fiscal Revenue=10*100*25%+((70/3)-10)*70=$1183.33

b)

Now assume highest marginal tax is 60%

Let the agent works for more than 10 hours

w=100*(1-60%)=$40

Consumption is given by

c=100*10*(1-25%)+(l-10)*40

c=750+40l-400=350+40l

Now set MUl/MUc =w for utility maximization

(2/(100-))/(1/c)=40

c/(100-l)=20

c=2000-20l

Put c=2000-20l in consumption function

2000-20l=350+40l

1650=60l

l=1650/60=27.50 hours

Number of working hours=27.50 hours

Fiscal Revenue=10*100*25%+(27.50-10)*60=$1300