Econ 131

Spring 2015

Emmanuel Saez

Problem Set 2

DUE DATE: March 11

1. True False Statements/Questions

Explain your answer fully based on what was discussed in class (no more than 10 lines per

question), since all the credit is based on the explanation.

a) The corporate income tax is highly progressive.

b) If incomes are fixed, then it is optimal for the government to fully redistribute incomes.

c) Suppose the elasticity of average earnings with respect to one minus the tax rate is equal

to 1. Can it be desirable to set the tax rate at 60% if the government cares a lot about

redistribution?

d) Suppose the elasticity of reported incomes of high income earners with respect to one

minus the top marginal tax rate is large because high income earners can exploit tax loopholes

to avoid taxes when top tax rates are high. The government should not impose a high tax rate

on high income earners.

e) The EITC has a positive effect on labor force participation but reduces hours of work

conditional on working.

f) If labor supply responses to taxes and transfers are concentrated along the extensive

margin (whether or not to work), it is desirable to provide benefits to those with no earnings

but then phase-out these benefits quickly as earnings increase.

g) Evidence from the Israeli Kibbutz shows that redistribution reduces incentives to acquire

an education.

2. Optimal Income Taxation

Suppose that individuals have the following utility function on consumption c and labor l:

U(c, l) = c− l2/2

Let us assume that the maximum amount L of labor an individual can supply is large enough

that the constraint l ≤ L never binds. Leisure is defined as L− l. All individuals have the same

wage rate w. If an individual works l and has wage w, he earns wl.

a) Assume that there is no tax and that individuals have no other income but their labor

income. Solve for the labor supply l as a function of w that maximizes the utility of the

individual.

b) The government imposes a linear tax on labor income at rate 0 < t < 1.

Show graphically in a consumption-leisure diagram how introducing the linear tax on labor

income modifies the budget constraint of the individual.

Solve for the labor supply l as a function of w and t that maximizes the utility of the

individual. What is the effect of increasing t on l?

c) Suppose that in addition to the linear tax, the government is transferring a fixed amount

R > 0 to every individual. That amount R is independent of labor supply choices.

Show graphically how the budget constraint is affected by the introduction of R.

Solve for the labor supply l as a function of w, t, and R that maximizes the utility of the

individual. What is the effect of increasing R on l?

d) Assume that R = 0 and t > 0 as in b). Compute income taxes collected by the government

per individual as a function of t and w. Draw a graph of taxes collected as a function of t. What

is the tax rate t∗ maximizing tax revenue?

e) Suppose that the government has set the tax rate t larger than t∗ (defined in d)). Can we

conclude, no matter what the redistributive tastes of the government are, that it is desirable to

decrease t? Make sure to explain your answer.

f) Suppose now that, wage rate w differs across individuals, but that all individuals have

the same utility function as defined above. What is in this case the income tax rate maximizing

tax revenue? Is it larger or smaller than the t∗ that you obtained in d)?

g) Suppose that, the utility function is now given by:

U(c, l) = c− lk+1/(k + 1)

where k > 0 is a fixed parameter. Solve for the labor supply l as a function of w, t, and k

that maximizes the utility of the individual. How is k related to the elasticity of labor supply

with respect to the net-of-tax wage rate w(1 − t)?

Solve for the tax rate maximizing tax revenue in that situation as a function of k.

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3. Labor supply and benefits

Pam is a single mother with two kids eligible for welfare: she receives $4,000 a year in

benefits and also receives health insurance for her family through Medicaid that she values at

$2,000 a year. If she works, she earns $20 per hour and can work up to 2,000 hours per year.

She gets taxed at 50% for every hour worked. In addition, she loses her Medicaid coverage if

she earns more than $20,000.

a) Draw Pam‘s opportunity set in the earnings-consumption space (carefully labeling the

slope of the budget constraint, intercept points, and any discontinuities) and write down the

opportunity set equation. How much earnings does Pam need to pay as much in taxes as she

receives in transfers?

b) Are there any hours of work that she definitely wont choose? Why or why not?

c) Is loosing Medicaid past $20,000 likely to produce a substitution effect, an income effect,

or both? Justify your answer.

Congress is concerned that there is too high a tax on work for welfare recipients and passes

a new law. Under the new law benefits are only $3,000 but the tax rate is reduced to 25%

Medicaid rules stay exactly the same.

d) Draw and write the equation of Pam‘s new opportunity set. What it the point where

Pam pays as much in taxes as she receives in transfers?

e) Discuss how the new tax and transfer system affects her labor choice compared to the

previous one. Discuss separately what would happen for a low earning Pam and a high earning

Pam.

f) How might this law affect the total number of people on welfare? Given your knowledge

from the class, is the extensive margin of work an important one?

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