MICROECONOMICS 2: WELFARE ECONOMICS
Topic: Fairness, Poverty and Inequality
Exercise 1: Fairness
Consider the situation in which a cake of size 1 has to be divided between two persons. We assume
that each person ! is selfish and has a utility function "! that is strictly increasing in the part of the cake
#! that he/she gets. Hence, more is better. Assume that you want to make a fair division, and that your
fairness criteria is that each person should not feel envious of what the other person has (let’s call this
envy-free criteria). In other words, after allocating the cake each person should weakly prefer his own
part to the part that the other got: "" (#" ) ≥ "" (## ) and "# (## ) ≥ "# (#" ).
a) Is the situation in which each person gets one-third of the cake and one-third is thrown away
envy-free? Is it also Pareto efficient?
b) Which divisions of the cake are both Pareto efficient and envy-free?
c) Now assume that the cake has a rectangular shape. Furthermore, the cake has a nice cherry
on the left hand side, while there is no cherry on the right hand side. Person 1 (from now
on (" ) does not care about the cherry (it leaves him indifferent), but Person 2, (from now on
(# ) is willing to give up one-fourth of the cake to get the cherry. Both individuals know this.
Which divisions of the cake are both Pareto efficient and envy- free?
d) Assume (" and (# are using the “divide and choose” method to divide the cake. (" divides the
cake into two pieces and then (# can choose which part he takes; the part that is not taken is
then left for (" . How will (" divide the cake? Is the outcome Pareto-efficient? Is it envy-free?
e) Now consider the case from part d), but assume that the roles of the individuals are reversed:
(# divides and (" chooses. What is the outcome in this case? Is the outcome Pareto efficient?
Is the outcome envy-free?
f) Think about the envy-free criteria: what assumptions are needed to make it implementable?
g) Assume that a society adopts envy-free as its fairness criteria: after all, a society where nobody
is envious could be a happy society. Discuss what could be the resulting income distribution
in such a society.
Exercise 2: The Gini Index before and after taxation
Consider a society in which there is a continuum of individuals. Specifically, the society is represented
by the interval [0,1] and we denote a typical individual from this society by *. The total size of the
society is normalized to 1 and the gross income of individual *+[0,1] is 10*.
a) Show that the total income in this society is 5.
1
,b) For #+[0,1], let 1(#) be the share in the total income that is earned by all *+[0, #] together.
Hence, 1(. ) Is the Lorenz curve. Calculate 1(#) for each value of #.
c) Calculate the Gini-index of the gross income distribution.
d) Now assume that proportional taxation is introduced. Hence, each individual * has to pay a
fraction 2 from his income, so that his net income is (1 − 2)10*. What is the Gini index of the
net income distribution?
2
,
Topic: Fairness, Poverty and Inequality
Exercise 1: Fairness
Consider the situation in which a cake of size 1 has to be divided between two persons. We assume
that each person ! is selfish and has a utility function "! that is strictly increasing in the part of the cake
#! that he/she gets. Hence, more is better. Assume that you want to make a fair division, and that your
fairness criteria is that each person should not feel envious of what the other person has (let’s call this
envy-free criteria). In other words, after allocating the cake each person should weakly prefer his own
part to the part that the other got: "" (#" ) ≥ "" (## ) and "# (## ) ≥ "# (#" ).
a) Is the situation in which each person gets one-third of the cake and one-third is thrown away
envy-free? Is it also Pareto efficient?
b) Which divisions of the cake are both Pareto efficient and envy-free?
c) Now assume that the cake has a rectangular shape. Furthermore, the cake has a nice cherry
on the left hand side, while there is no cherry on the right hand side. Person 1 (from now
on (" ) does not care about the cherry (it leaves him indifferent), but Person 2, (from now on
(# ) is willing to give up one-fourth of the cake to get the cherry. Both individuals know this.
Which divisions of the cake are both Pareto efficient and envy- free?
d) Assume (" and (# are using the “divide and choose” method to divide the cake. (" divides the
cake into two pieces and then (# can choose which part he takes; the part that is not taken is
then left for (" . How will (" divide the cake? Is the outcome Pareto-efficient? Is it envy-free?
e) Now consider the case from part d), but assume that the roles of the individuals are reversed:
(# divides and (" chooses. What is the outcome in this case? Is the outcome Pareto efficient?
Is the outcome envy-free?
f) Think about the envy-free criteria: what assumptions are needed to make it implementable?
g) Assume that a society adopts envy-free as its fairness criteria: after all, a society where nobody
is envious could be a happy society. Discuss what could be the resulting income distribution
in such a society.
Exercise 2: The Gini Index before and after taxation
Consider a society in which there is a continuum of individuals. Specifically, the society is represented
by the interval [0,1] and we denote a typical individual from this society by *. The total size of the
society is normalized to 1 and the gross income of individual *+[0,1] is 10*.
a) Show that the total income in this society is 5.
1
,b) For #+[0,1], let 1(#) be the share in the total income that is earned by all *+[0, #] together.
Hence, 1(. ) Is the Lorenz curve. Calculate 1(#) for each value of #.
c) Calculate the Gini-index of the gross income distribution.
d) Now assume that proportional taxation is introduced. Hence, each individual * has to pay a
fraction 2 from his income, so that his net income is (1 − 2)10*. What is the Gini index of the
net income distribution?
2
,
