904
ASHER WOLINSKY
Ž .
Ž
.
Then, by the construction in the proof of Proposition A- ii , there exists an equilibrium ¨Љ, Љ
Ž .
Ž
.<
Ž
.<
Ž
.< .<
for all j - i, ¨ Љ , Љ i, i, i s ¨ Ј, Ј
i
Ž
such that ¨Љ i s i, ¨Љ , Љ i, j s ¨,
and
iy1, j
Ž
Ž
.<
Ž
.<
Ž
Ž
.
Ž
.
¨Љ, Љ i, i, iq1 s ¨Ј, Ј iq1. Since ¨Љ, Љ may differ from ¨Ј, Ј only after histories of the form
¨
Љ, Љ
¨Ј, Ј
.
.
Ž
Ž .
i, j such that j-i, where ⌸
i, j G⌸
i, j , it follows that, on the equilibrium path, the
Ž
.
.
Ž
.
Ž
.
only difference between ¨Љ, Љ and ¨Ј, Ј is that Љ i, i FЈ i, i . This together with qFi
¨
Љ, Љ
¨
Љ, Љ
¨
Ј, Ј
¨
Ј, Ј
¨Љ, Љ
Ž .
Ž . Ž .
i G⌸ i qqW
¨Љ, Љ
Ž .
Ž . Ž
i, j s⌸ i
implies that ⌸
y1 , condition 2.3 implies W
i qqW
i . Since max jF iy1
⌸
¨
Љ, Љ
.
Ž
.
Ž .
i s⌸
Ž .
Ž
.
i y⌸ iy1 qWU . It then follows that
¨
Љ , Љ
¨
Љ , Љ
¨Љ , Љ
Ž .
i qqW
Ž .
Ž
.
Ž .
Ž
.
⌸
i s qq1 ⌸
i yq⌸ iy1 qqWU
Ž
.
Ž .
Ž
.
Ž .
Ž .
F qq1 ⌸ i yq⌸ iy1 qqWU s⌸ i qqW i .
Therefore,
¨
Ј, Ј
¨Ј, Ј
Ž .
Ž .
Ž .
i qqW
Ž .
i
⌸ i qqW i G⌸
¨
,
¨ ,
w
Ž .
i qqW
Ž .x
i
w x
qg 0, i .
smaxŽ
⌸
,
for any
¨
, .
w
Ž .
w
Ž .
xx
Ž .
.
Ž .
Step 2: Let jgargmaxiF m ⌸ i qi W i yWU
subject to ⌸ i G⌸ m , and let kg
w
Ž . xw Ž . xx
w
Ž .
an equilibrium ¨, such that ¨ m sj, ¨ mq1 sk and thereafter it continues according to
.< .<
Ž
argmaxiF mq1 ⌸ i qmin i, m W i yWU subject to ⌸ i G⌸ mq1 . For mFN, there exists
Ž
.
Ž
.
Ž
.
Ž
Ž
¨, and ¨,
respectively.
j
k
Ž
.
To see this, let ¨, have the features just mentioned. In addition, after any history h starting
Ž
.<
Ž .
coincide with ¨, . Finally
with m, i such that i/j, or with mq1, i such that i/k, let ¨,
h
after all histories not covered above let it coincide with some arbitrary equilibrium. Since all
Ž .
Ž
.
Ž .
Ž
. Ž
.
continuations are chosen to be equilibria and since ⌸ j G⌸ m and ⌸ k G⌸ mq1 , ¨, is
clearly an equilibrium. Hence, there exists such an equilibrium.
Thus, Step 1 implies that the equality appearing in the statement of Claim B.1.2 holds as
inequality, with the left-hand side being smaller than or equal to the right-hand side. Step 2 then
implies that this is actually an equality, as required by the claim.
Q.E.D.
Claims B.1.1᎐2 imply Proposition B.1.
Q.E.D.
s
Ž .
i
Ž .
Ž .
PROPOSITION B.2:
NGN . ii For m)N, ¨ m sN.
Ž .
Ž
.
PROOF:
i
Suppose to the contrary that there exists m-Ns such that ¨ mq1 Fm. It follows
.<
Ž
from Proposition B.1 that all employment levels on the path ¨,
are smaller or equal to m. This
m
and the fact that is increasing for m-Ns imply ⌸ m F m r 1y␦ and hence mq1 ) 1
. Ž
Ž
.
Ž
.
Ž
.
Ž
Ž
.
Ž
Ž
.
.
Ž
.
.
Ž
.
Ž
.
Ž .
w
Ž
.
y␦ ⌸ m . Since, by definition, Smq 1 ¨, G⌸ mq1 G⌸ m , we have mq1 q mq
.
Ž
.x Ž
.
Ž
1 ⌸ m q␦Smq 1 ¨, r mq2 )⌸ m . From Proposition A- ii , there exists an equilibrium
¨
,
Ž
.
Ž
.
Ž .
Ž .
¨, such that ¨ mq1 smq1 and
⌸
mq1 )⌸ m , contrary to the supposition that
s
Ž
.
¨ mq1 Fm. Therefore, NGN .
Ž .
exists j)N such that ¨ j sj. Let m be the minimal such j. From the choice of N and m,
Ž .
ii Suppose to the contrary that there exists some j)N such that ¨ j )N. It follows that there
Ž .
Ž
.
Ž
w
Ž
.
Ž
.
Ž
.
m ) N q 1 and ¨ m y 1 - m y 1. This implies m y 1 q m y 1 ⌸ m y 2 q
Ž
.x Ž .
.
␦ maxŽ , . Smy1 ¨, rm-⌸ my2 , since otherwise Proposition A- ii would imply that there
¨
¨
,
Ž
.
Ž
.
Ž
. Ž .
my1 )⌸ my2 , in contradic-
exists an equilibrium ¨, such that ¨ my1 smy1 and ⌸
Ž
.
tion to ¨ my1 -my1. Rearrangement yields
Ž
.
Ž . Ž . Ž . Ž .
my1 q␦ maxŽ , . Smy1 ¨ , -⌸ my2 F⌸ my1 .
¨
B.4
Ž
.
Ž
.
Since maxŽ , . Smy1 ¨, G⌸ my1 , we get
¨
Ž
.
Ž . Ž . Ž . Ž . Ž .
my1 - 1y␦ ⌸ my1 F 1y␦ maxŽ , . Smy1 ¨ , .
¨
B.5