where the second equality follows because x(b, c) = 1 for c≤b and = 0 for c>b. A
similar analysis reveals
Realized surplus is b − c if
b≥c
and 0 otherwise. Hence, ex ante expected surplus is
If the mechanism is balanced, then
S
∗
=
E
b
{
u(b)
}
+
E
c
{
π(c)
}
. But, from (A.5),
where the last equality follows from (A.7). The mechanism can, thus, be balanced only
if u(0) = 0 and
E
c
{
π(
c
)
}
=0
; but, from (A.6),
E
c
{
π(
c
)
}
>
0
. Reductio ad absurdum,
no balanced, efficient, and interim IR mechanism can exist. ■
Proof of Lemma6.11: Consider b > b′. Then, for all
c,x(b,c)≥x(b
′
,c)
. Hence,
So ξ
B
(·) is nondecreasing ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month, and much more.