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Question 1
Part (a)
To maximize social welfare, the public good should be produced if and only if
i.e.
v
1
+
v
2
c
x
= 1
the decision that the public good is produced; and
x
= 0
the decision that the public good is not produced The utilitarian decision
rule is given by
x
(
v
1
; v
2
; c
) =
1
if
v
1
+
v
2
c
0
if
v
1
+
v
2
< c
Part (b)
No. There is no mechanism that satis±es the following at the same time:
1. implements the utilitartian decision rule
x
2. provides dominant strategy to tell the truth
3. balances the budget.
We have seen in class that even when the production cost
c
is commonly
known, a mechanism that satis±es all of
(1)
,
(2)
, and
(3)
simultaneously does
help.
To be more precise, we know from class that the only mechanism that sat
is±es
(1)
and
(2)
above is a VCG mechanism. The decision rule is given by
x
above and the VCG transfer rule is given by
t
V CG
1
(
v
1
; v
2
; c
) =
min
f
v
2
±
c;
0
g
if
v
1
+
v
2
c
0
otherwise
for individual 1, and
t
V CG
2
(
v
1
; v
2
; c
) =
min
f
v
1
±
c;
0
g
if
v
1
+
v
2
c
0
otherwise
for individual 2,and
t
V CG
s
(
v
1
; v
2
; c
) =
v
1
+
v
2
if
v
1
+
v
2
c
0
otherwise
If the good is produced, i.e.
v
1
+
v
2
c
, the total transfer is given by
t
V CG
1
+
t
V CG
2
+
t
V CG
s
= min
f
v
2
±
c;
0
g
+ min
f
v
1
±
c;
0
g
+
v
1
+
v
2
&
0
1
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View Full Documentwhich is strictly positive if
v
1
+
v
2
> c
. Hence a VCG mechanism runs a
(1)
(3)
above.
Part (c)
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 Spring '08
 Witte
 Macroeconomics, Public Good

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