This preview shows pages 1–3. Sign up to view the full content.
Problem Set 2 — Solutions
TA: Guannan Luo
Econ 3102, Winter 2011
1 Transformation of Utility and SWFU
(a)
Formally, the problem we have to solve is Max
SWFU
=
u
1
+
u
2
subject to (
u
1
,u
2
) being on
the
UPS.
There are (at least) 2 methods to solve this problem.
Method 1:
Substituting the utility functions into the SWFU and using the feasibility constraint
we get
W
(
u
1
(
x
1
)
,u
2
(
x
2
)) =
u
1
(
x
1
) +
u
2
(
x
2
)
=
x
1
+ 2
√
x
2
= 8

x
2
+ 2
√
x
2
Setting the ﬁrst order condition equal to zero gives
0 = 0

1 +
1
√
x
2
x
2
= 1
x
1
= 8

1 = 7
Hence the allocation that maximizes the utilitarian SWFU is given by
x
1
= 7 and
x
2
= 1, which
corresponds to
u
1
= 7
,u
2
= 2. See ﬁgure 1.
Method 2
: We know that when the solution is interior (which is true in this exercise), welfare
maximization occurs where the UPF and the social indiﬀerence curves are tangent (or, in other
words, where the slope of the SWFU equals the slope of the UPF). Therefore
slope of SWFU = slope of UPF

1 =

1
√
8

u
1
u
1
= 7
Using the equation for the UPF we get
u
2
= 2
√
8

7
u
2
= 2
Notice that this is the same answer (
u
1
= 7 and
u
2
= 2
,
which corresponds to
x
1
= 7 and
x
2
= 1) as before.
(b)
Formally, the problem we have to solve is Max
SWFU
= min
{
u
1
,u
2
}
subject to (
u
1
,u
2
)
being on the
UPS.
You should remember from previous courses that we can’t solve this problem
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentusing calculus (why?). Instead, we get the solution by equating the utilities of the individuals
(why? If they are not equal, we can increase the Rawlsian SWFU by transferring some goods from
the individual with higher utility to the individual with lower utility).
Method 1
: Using the utility functions and the feasibility constraint we get
u
1
(
x
1
) =
u
2
(
x
2
)
x
1
= 2
√
x
2
x
1
= 2
√
8

x
1
x
2
1
= 4(8

x
1
)
x
2
1
+ 4
x
2

32 = 0
(
x
1

4) (
x
1
+ 8) = 0
x
1
= 4 (solution)
, x
1
=

8 (not feasible)
x
2
= 8

4 = 4
The allocation that maximizes the Rawlsian SWFU is given by
x
1
= 4 and
x
2
= 4, which
corresponds to
u
1
= 4
,u
2
= 4
.
See ﬁgure 2.
Method 2
: Using the equation for the UPF we get
u
1
=
u
2
u
1
= 2
√
8

u
1
u
2
1
+ 4
u
1

32 = 0
(
u
1

4) (
u
1
+ 8) = 0
u
1
= 4 (solution)
, u
1
=

8 (not feasible)
u
2
= 2
√
8

4
u
2
= 4
Notice that this is the same answer as before (
u
1
= 4
,u
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 SARVER
 Microeconomics, Utility

Click to edit the document details