Practice Pareto Optimality Questions
1)
There are three people (A, B, and C) and five possible states of their economy
summarized in the table below by the resultant utilities of each of the three people. Which
of the states are Pareto optimal?
State UA
UB
UC
1
10
10
10
2
12
9
7
3
12
10
7
4
11
13
10
5
9
14
14
2)
State
1
2
3
4
5
3)
State
1
2
3
4
5
Same as question 1)
UA
50
61
17
10
0
UB
40
31
18
10
1
UC
55
40
56
60
65
Same as question 1)
UA
15
16
16
16
13
UB
17
16
17
14
15
UC
13
14
14
12
14
4) There are two people (person B and person C) and two goods (good X and good Y) .
Person B has utility U B = X Ba YB1− a and person C has utility U C = X Ca YC1− a where X B is
defined as person B's consumption of good X and X C , YB , and YC are similarly defined.
Their economy is endowed with a total of X units of good X and Y units of good Y.
Characterize as fully as possible the set of Pareto optimal states of this economy.
Vaticrat
5) There are two people (person A and person B) and two goods (good X and good Y) .
Person A has utility U A = X Aa YA1− a and person B has utility U B = X Bb YB1− b where X A is
defined as person A's consumption of good X and X B , YA , and YB are similarly defined.
Their economy is endowed with a total of X units of good X and Y units of good Y.
Characterize as fully as possible the set of Pareto optimal states of this economy.
6) There are n people and 2 goods (X and Y). Person i's utility is given by Ui(Xi,Yi)
where i = 1,2,...,n; Xi is person i's consumption of good X, and Yi is person i's
consumption of good Y. The economy is endowed with X units of good X and Y units
of good Y. Show that in any Pareto optimal situation, for any two people (person i and
MU jX
∂ U i ( X i , Yi )
MU iX
X
person j),
=
where
MU
=
and MU iY , MU jX , and MU Yj
i
∂Xi
MU iY
MU Yj
are similarly defined.
7) There are 2 people (B and C) and m goods. Person B's utility is given by
U B ( X 1B , X 2B ,..., X mB ) where X iB is person B's consumption of good i. Person C's utility
is similarly given by U C ( X 1C , X 2C ,..., X mC ) . Their economy is endowed with X i units of
each good i ∈ {1,2,... , m} . Show that in any Pareto optimal outcome, for any two goods i
MU iB
MU iC
and j, it is the case that
=
.
MU jB
MU Cj
8) There are two people (B and C) who consume two goods (X and Y) but also enjoy
leisure (Z). Person B's utility is given by UB(XB,YB,ZB) and person C's by UC(XC,YC,ZC).
Goods X and Y must be produced, and the only input to their production is labour.
Output of good X is given by FX ( LBX + LCX ) where LBX is the amount of labour
contributed by person B to the production of good X and LCX is the amount of labour
contributed by person C. Output of good Y is similarly given by FY ( LYB + LYC ) . Each
person is endowed with a fixed amount of time L which is divided into leisure,
production of good X, and production of good Y. Show that in the Pareto optimal
MU BX
MU CX
MU BX
MU CX
outcome
=
and
=
(i.e. there is exchange efficiency) and
MU BY
MU CY
MU BZ
MU CZ
MPX
MU BY
MU BX
1
=
and
=
(i.e. there is match efficiency).
MPY
MPX
MU BX
MU :ZB
9) There are two consumers (B and C) and two goods (X and Y). Person B's utility is
given by UB(XB,YB) and person C's by UC(XC,YC). There are k inputs used in the
production of each good. Output of good X is given by FX ( Z1X , Z 2X ,..., Z kX ) where ZiX
is the amount of input i allocated to the production of good X. Output of good Y is
similarly given by FY ( Z1Y , Z 2Y ,..., Z kY ) . There is a fixed amount of each input available
that
for production, so Zi ≥ ZiX + ZiY for each input i ∈ {1,2, ..., k } . Derive the aspect of the
Pareto optimal outcome representing production efficiency.