Exam Public Economics November 2011
This exam is part of the examination of the course Public Economics. In
order to pass the course you have to pass this exam as well as the written
group project.
The written exam and the group project are equally important for the overall
result, which together amounts to 100 points. This exam consists of 6 questions and 50 points in total. To pass the written exam 25 points are needed.
Hence, a maximum of 50 points will also be given for the group project. To
pass the course a total of 50 points is needed (with at least 25 points from
each of the exam and the written group project) and for High Pass 75 points.
Motivate your answers carefully and report your calculations.
You should answer in English (you may use a dictionary) and you may use
a calculator.
Good Luck!
Question 1 (5 points)
Explain the following shortly:
a) Progressive income tax
b) Libertarian paternalism
c) Revenue-matching grants
d) The top cycle
e) Pigouvian taxation
Question 2 (6 points)
a) To what extent would each of the following explain a government’s spending on the installation of community-wide high-speed wireless internet:
i) Baumol’s Law (2 points)
ii) Demographic changes (e.g. age) of the voting population (2 points)
b) Is it appropriate for the government to provide high-speed internet rather
than a private market? (3 points)
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Question 3 (6 points)
In a three-person economy, suppose we are deciding between two di¤erent
allocations and where utilities fU1 ; U2 ; U3 g are such that := f5; 8; 6g
and := f7; 7; 7g : In choosing between these two allocations the Paretocriterion is better than any voting system. Discuss.
Question 4 (10 points)
Consider consumers A and B in a pure exchange economy consuming goods
x and y. Their preferences are represented by the utility functions
U A = log xA + log y A and U B = log xB + log y B :
A
and consumer B the endowment
Consumer A has the endowment ! A
x ; !y
B
B
!x ; !y .
a) Show that the equation of the contract curve is
B
!A
y + !y A
y = A
x ;
!x + !B
x
A
and comment on how A’s relative consumption of the two goods on the
contract curve depends on the relative total endowment. (3 points)
Suppose we have the following social welfare function W = U A + U B
b) Comment on the ethical objectives or social planner’s distributive preferences embodied in this social welfare function. (1 point)
c) Suppose the total endowment of the economy is 20 units of good x and 10
units of good y. Show that the equitable distribution is chosen when we
use this social welfare function to select the socially optimal allocation
on the contract curve. (3 points)
d) Explain why we must assume some degree of utility function comparability to get a consistent social welfare function. (1 point)
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e) Suppose we assume Cardinal-unit comparability. Given U A and U B as
already de…ned, consider transformations of these utility functions supposedly representing the same preferences U^ A = 2 + 2U A and U^ B =
2 + U B . Prove by example that such transformations would give an
inconsistent utilitarian social welfare function, i.e. U and U^ are not
Cardinal-unit comparable. (2 points)
Question 5 (12 points)
Consider an individual with utility function U = x1=2 l, where x is consumption and l is number of hours worked a week.The individual receives an
hourly wage rate w, and faces a proportional tax rate t. Hence, the budget
constraint is x = w(1 t)l.
a) Find the labour supply function (2 points)
b) How many hours a week would the individual work and what would
his total income be if w = 100 and t = 0:3? (1 points)
c) How would labour supply change if the tax rate was reduced to t =
0:27? (1 point)
The government wants to prevent people from having a too low income
and therefore gives all individuals a tax free bene…t of 1300.
d) How would this bene…t alter the individual’s choice, i.e., …nd the new
general labour supply function and the number of hours worked if w =
100 and t = 0:3. (2 points)
e) How would labour supply change in this case if the tax rate was reduced
to t = 0:27? (1 point)
f) Sketch the solutions in b) and d) in a diagram and explain why the
answers in c) and e) di¤er. (3 points)
g) Suggest one way to increase the income of low-income households that
would be better than the tax free bene…t. You do not have to make
any calculations; a short discussion will do. (2 points)
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Question 6 (11 points)
A consumer with utility function U = ln C, where C is consumption, determines how much of his gross income Y to declare to the tax authority. Denote
the declared amount X. The tax rate t is paid for all declared income, and if
the consumer is caught evading, he has to pay a …ne, F , proportional to the
evaded tax amount. Hence, Y nc = Y tX and Y c = Y (1 t) F t(Y X),
where superscript c denotes caught and nc not caught. Moreover, the probability of getting caught is denoted p.
a) Derive the expression for the optimal value of X if the consumer maximises expected utility. Also derive the conditions for corner solutions.
(4 points)
b) What happens to the declared amount as t increases? Explain the
intuition (2 points)
c) In reality, tax evasion is less prevalent than predicted by this standard
model. Discuss two elements that that may change the utility function
and therefore may be crucial in determining the existence of tax evasion.
(4 points)
END
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