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Solutions for Problem Set 4
Question 1
(a) By definition, the utilitarian solution is the list of
h
1
,
h
2
,...,
h
n
that maximizes
the sum of all farmers’ valuations,
∑
i
1
n
qh
i
v
i
, where
q
10
−
∑
j
1
n
h
j
.I
fwede
f
ine
H
∑
j
1
n
h
j
,
which is the total number of cowhours of grazing, then
q
10
−
H
. Then
the sum of all farmers’ valuations becomes
10
−
H
∑
i
1
n
h
i
v
i
.
For a given
H
,
in order to maximize
∑
i
1
n
h
i
v
i
,
all the hours must be assigned to the
farmer with the highest value. That is, for every
i
,
h
i
Hi
f
v
i
is the highest value
;
0
otherwise
Hence the sum of all farmers’ valuations is
10
−
H
Hv
m
, where
v
m
is the highest
value of
v
1
,
v
2
,...,
v
n
.
The last step is to choose
H
to maximize
H
10
−
H
v
m
.
It is easy
to get that
H
5.
Therefore, the utilitarian solution of
h
1
,
h
2
,...,
h
n
is that, for each
i
,
h
i
5
if v
i
is the highest value
;
0
otherwise
,
which means only the farmer with the highest value will be able to use the land for
5 hours.
(b) (i) Suppose the announcement is
v
̂
v
̂
1
,
v
̂
2
,...,
v
̂
n
with
v
̂
1
6,
v
̂
2
4
and
v
̂
i
4
for
i
3,4,.
..,
n
,
then according to part (a), the utilitarian alternative
x
∗
v
̂
is that
only farmer 1 is allowed to use the land for 5 hours.
First consider farmer 1. The utilitarian alternative for the society consisting of
everyone except farmer 1,
x
∗
v
̂
−
1
, is that only farmer 2 is allowed to use the land for 5
hours. (This is because without farmer 1, farmer 2 is the one with the higest value.)
Hence, the VCG payment for farmer 1 is,
t
1
VCG
v
̂
∑
j
≠
1
v
̂
j
x
∗
v
̂
−
∑
j
≠
1
v
̂
j
x
∗
v
̂
−
1
0
−
10
−
5
∗
5
∗
v
̂
2
−
25
∗
4
−
100.
Now consider farmer
i
≠
1.
The utilitarian alternative for the society without
i
,
x
∗
v
̂
−
i
,
is that only farmer 1 is allowed to use the land for 5 hours. Hence,
x
∗
v
̂
−
i
x
∗
v
̂
.
Then the VCG payment for farmer
i
is,
t
i
VCG
v
̂
∑
j
≠
i
v
̂
j
x
∗
v
̂
−
∑
j
≠
i
v
̂
j
x
∗
v
̂
−
i
0
. (because
x
∗
v
̂
−
i
x
∗
v
̂
)
Hence, in this case, farmer 1 pays 100, and other farmers pay nothing.
(ii) Suppose farmer 1’s value is
v
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View Full Document If farmer 1 announce his value truthfully, then from the above part we know that
farmer 1 pays 100, and since he is allowed to use the land for 5 hours, his valuation is
10
−
5
∗
5
∗
6
150
, which means farmer 1’s payoff is
150
−
100
50.
If farmer 1 misreports and announces a value
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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.
 Spring '08
 Witte
 Macroeconomics

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