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Math/Physic/Economic/Statistic Problems

ECON2103 Business and Government

Problem Set 3

Pei Cheng Yu

Problem set 3 is worth 15% of your final grade. The total points for this problem set is
100.

This is due at 10pm (Sydney time) Friday of Week 10 (10pm of August 6th). 20%
will be deducted from the mark of late submissions, and submissions late by more
than 24 hours will not be marked. upload your answers to Moodle.

1. Optimal Income Taxation: 60 points

After Pawnee and Eagleton merged to become Pawgleton, Leslie Knope was tasked
with redesigning the income tax system for Pawgleton residents. Residents from Ea-
gleton are more productive than Pawnee residents. Specifically, Eagleton residents
have the following production function
yE = 2l,

while Pawnee residents have production function
yP = l.

In other words, Eagleton residents can produce twice as many output y as Pawnee
residents for a given labor input l. The labor market in Pawgleton is perfectly com-
petitive. Both Eagleton and Pawnee residents have the same utility function:
u (c, l) = ln (c) −1
2 l2.

Pawnee and Eagleton have the same population. Leslie Knope can identify where
each individual is from (Pawnee or Eagleton), and she can also force each agent to
produce a certain output. Furthermore, Leslie Knope is utilitarian.

a. What is the socially optimal allocation (cP∗, cE∗, lP∗, lE∗)? Hint: you will need
to maximize the sum of utilities subject to the resource constraint: yE −cE +
yP −cP = 0. (8 points)

b. What is the utility of Pawnee residents and Eagleton residents at the social
optimum? (5 points)

c. Suppose that the price of output is 1. Show why the competitive wages for
Pawnee residents is wP = 1 and the competitive wages for Eagleton residents
is wE = 2. (4 points)
1
d. Let T (y) denote the income tax function. Find an income tax function T (y)
that implements the socially optimal allocation. Hint: writing out the budget
constraints for both Pawnee and Eagleton residents would help. (6 points)

A law was passed by Councilman Jeremy Jamm that prohibits the discrimination of
Pawnee and Eagleton residents. This law prohibits Leslie from identifying where
each individual is from.

In essence, she cannot force productive individuals to pro-
duce more, because she is no longer even able to identify who is productive. Fur-
thermore, to prevent Leslie from monitoring the Pawgleton residents, Councilman
Jamm has also prohibited policies that are based on labor supply l.

e. Suppose Leslie tries to implement the same tax function as Part d. How much
labor supply does Eagleton residents and Pawnee residents provide? (4 points)
f. Explain why Leslie will no longer be able to implement the social optimal allo-
cation (cP∗, cE∗, lP∗, lE∗) in one sentence. (6 points)

Leslie is now stumped. Ben Wyatt explains to her that she needs to introduce a new
tax system that incentivizes the productive Eagleton residents to voluntarily pro-
duce more.

Ben suggests to Leslie that she should consider introducing a constant
income tax τ combined with a universal transfer of T. The Pawgleton government
has no external revenue needs, so all of the tax revenue from τ will be used to fund
the universal transfer.

g. Write out the individual’s budget constraint. (3 points)

h. Derive the labor supply of Pawnee and Eagleton residents as a function of τ
and T. Hint: you will need to solve the individual’s optimization problem. (8
points)

i. Without solving for the optimal policy, argue why Ben’s proposal is better than
the tax system in Part d. Hint: you will need to argue that, despite sacrificing
some equality, Ben’s proposal is more efficient. (6 points)

When Anne saw the problem that Leslie was facing, she immediately recalled her
suggestion to Ben for the health insurance problem in Problem Set 2. Anne suggests
to Leslie that she should introduce a menu of allocations
{(
cP, yP )
,
(
cE, yE )}

such that Eagleton residents voluntarily select the allocation (cE, yE ) while Pawnee
residents voluntarily select (cP, yP ). (Notice that Ben has suggested replacing labor
supply l with output y, because the policy can no longer be based on l.)

j. Rewrite the utilities for both Pawnee and Eagleton residents in terms of con-
sumption c and output y. Hint: output is linear in labor supply, so l = y
marginal productivity .
(2 points)
2

k. Write out the constraints such that Eagleton residents select (cE, yE )while Pawnee
residents select (cP, yP ). Hint: check out constraint (2) on page 20 of Week 7
Notes. (2 points)

l. Leslie found that the optimal tax system under Anne’s proposal is nonlinear.
Without solving for the optimum and comparing the social welfare under both
schemes, argue why Anne’s proposal is superior to Ben’s. (6 points)

2. Bequest Taxation: 40 points
Consider a two period model: t = 0, 1. The parent lives in t = 0 and has endowment
W >0 and needs to decide how much to bequeath the kid. Denote bequest as b ≥0.

The kid lives in period t = 1. There are no labor decisions taken in any period. The
parent loves the kid and puts an altruistic weight of γ on the kid. The parent has
utility
ln
(
cP0
)
+ γ ln
(
cK1
)
,
where δ = 1 is the discount factor both the parent and the kid use. Suppose the
interest rate is r = 0.

President Selina Meyer is deciding whether to tax bequest with a bequest tax of τ.

Let θ ∈[0, 1] denote weight Selina puts on the parent and 1 −θ the kid, i.e., Selina’s
objective is to maximize
θ
[
ln
(
cP0
)
+ γ ln
(
cK1
)]
+ (1 −θ) ln
(
cK1
)
.
a. Setup the parent’s optimization problem. (4 points)

b. Solve for the equilibrium bequest as a function of the bequest tax. (8 points)

c. Setup the social planner’s optimization problem. Hint: the sum of consump-
tions cannot be greater than W. (4 points)

d. Solve the social planner’s problem and find the allocations in terms of θ. (8
points)

e. Find the optimal bequest tax for θ = 1, 12 . (8 points)

f. Is it ever optimal for Selina to introduce a positive tax on bequests in this econ-
omy? Explain in one sentence. (8 points