Practice Problem 1
Riccardo M. Masolo
April 6, 2008
1 Question 3: Pareto Eﬃciency
a The set of feasible alternatives, maintaining the same assumptions as in
Question 2 for what concerns nonnegative consumption and possibility of
wasting goods, is:
X
=
{
((
x
1
,y
1
)
,
(
x
2
,y
2
)) :
x
1
,x
2
,y
1
,y
2
≥
0
and x
1
+
x
2
≤
10
.
5
, y
1
+
y
2
≤
6
}
b The set of Pareto Eﬃcient alternatives can be characterized noticing how
only agent one ”enjoys” good
y
. So any allocation such that all of good
y
is
consumed by agent 1 and good
x
is not wasted will be part of the PE set:
PE
=
{
((
x
1
,y
1
)
,
(
x
2
,y
2
)) :
x
1
,x
2
≥
0
y
1
= 6
, y
2
= 0
, and x
1
+
x
2
= 10
.
5
}
c Right from the deﬁnition of UPS in your notes one can describe it as:
UPS
=
{
(
u
1
,u
2
) :
u
1
=
1
3
x
1
+
y
2
1
, u
2
= 5
x
2
for some
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 Spring '08
 SARVER
 Microeconomics, KaldorHicks efficiency, Vilfredo Pareto, Riccardo M. Masolo

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