HAND-IN ASSIGNMENT 1
General instructions
In this assignment you should perform some experiments on the linear income tax model
that we have discussed in the lectures. The assignment should be uploaded to the Assign-
ments page on Studium latest on November 19th and should consist of your modified
MATLAB files as well as the answers to the questions in this document. Make sure the
code is well-documented by using ”%” at the beginning of a line to generate comments.
Background
The starting point of this lab is the linear income tax model. The government designs a
tax system consisting of a linear tax rate on income z equal to ⌧ and uses the resulting
tax revenue to finance a transfer given to everyone in the population equal to R. In
addition, each individual has some additional (non-labor) income, such as spousal income
or inheritance income, equal to yi.
Individuals are characterized by their potential income which is distributed according
to some probability distribution with density function f(zp) over the support zp > 0.
The discrete analogue of this distribution is a set of potential earnings levels {zip}Ni=1 with
associated probability mass {⇡i}Ni=1. To avoid having to compute integrals in MATLAB,
we will focus on this discrete formulation which implies that we can use sums (i.e.
P
)
instead of integrals. We interpret ⇡i as the fraction of individuals in the population that
have potential income equal to zip (
PN
i=1 ⇡
i = 1). Henceforth, we will refer to an individual
with potential income zip, simply as ”type i”. Thus, letting !
i denote the ”welfare weight”
attached by the government to agent i, the government wishes to choose ⌧ and R in order
to maximize the social welfare function:
⌦(V 1, V 2, . . . , V N) =
NX
i=1
!i⇡iV i
subject to the government budget constraint
⌧
NX
i=1
⇡iz(1 ⌧, R + yi, zip) =
NX
i=1
⇡iR
where
V i = max
c,z
u(c, z, zip) (1)
subject to
c = (1 ⌧)z + yi +R.
1
and z(1 ⌧, R + yi, zip) denotes the optimal choice of z for an individual with potential
income zip and non-labor income y
i, given the tax rate ⌧ and the transfer R. Notice that
this is a nested optimization problem! The government wants to find the optimal tax
system {⌧, R}, taking the optimal behavior of agents as given.
In most of the questions below, we assume that the utility function is:
u(c, z, zip) = c
zip
1 + 1
✓
z
zip
◆1+ 1
. (2)
In this hand-in assignment you will get a sample code (optimal linear tax.m
and auxiliary functions) available in Studium that you may use when answering the
questions.
Questions
1. Consider the individual optimization problem in (1) given the utility function (2).
(a) Show that the analytical solution to the optimal income choice of a type i
individual is z = z(1 ⌧, R + yi, zip) = zip(1 ⌧).
(b) How do the optimal choices of taxable income depend on R and yi?
(c) Calculate the compensated elasticity of taxable income.
2. How does the optimal linear tax system depend on the weights assigned by the
government to the utility levels of the di↵erent agents in the economy? Assume
= 0.5 and use the potential income distribution labeled ”A”.
(a) Find the optimal tax rates for the ”Max-min” (Rawlsian) case and the ”Utili-
tarian” case.
(b) Now let the weights take on the following geometric pattern:
!i =
1
ik
, i = 1, . . . , N,
where N is the total number of taxpayers. Find the optimal tax rate for the
case k = 2 and k = 4. Compare your results with the Max-min and Utilitarian
cases.
3. Now you will investigate how the optimal tax system depends on the degree of skill
inequality. Assume = 0.5 and the social welfare function in 2b) using k = 2.
(a) Consider changing the potential income distribution from ”A” to ”B”. What
do you conclude about the relationship between pre-tax inequality and the
optimal tax system?
(b) Assume everyone has the same skill. What is the optimal tax rate?
4. Experiment with di↵erent values of . How do the optimal tax rates depend on ?
Explain the intuition. What happens when becomes very large? What happens
when = 0?
Good Luck!
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Optional exercise (gives no points) Suppose now instead that the utility function
is given by
u(c, z, zip) =
c1+
1
⌘
1 + 1⌘
z
i
p
1 + 1
✓
z
zip
◆1+ 1
.
Set ⌘ = 3, = 0.5 and let yi = 0, i = 1, . . . , N , the skill distribution labeled A, and
the social welfare function given by 2b) using k = 2. What is the optimal tax rate in this
case? Notice in this exercise there does not exist an analytical solution to the individual’s
problem, so you will have to solve it numerically.
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