Technical Appendix to
The Survival of the Welfare State1
John Hassler, José Vicente Rodríguez Mora, Kjetil Storesletten, and Fabrizio Zilibotti
December 2001
1
Not for publication.
1.1
E¢ cient redistribution
The following Proposition characterizes the set of e¢ cient sequences of bene…ts.
Proposition 1 A social planner whose objective function is given by
max
fbt g1
t=1
(
0s (1
u1 ) V^ os (b1 ; b2 ; u1 ) +
^ ou (b1 ; b2 ; u1 ) +
0u u1 V
1
X
)
^ y (bt ; bt+1 ; bt+2 ; ut ) ;
tV
t=1
subject to bt 2 [0; 1] 8t
will choose
bt = 0 8t
b1 = min max 2
1
+ 1
provided
1
t
> 0 8t
0u
0s (1
0s (1
u1 ) +
u1 ) +
(1
u1
0u u1
1.
Not for publication
1
0u u1
)
2
2
1 u1
;0 ;1
Proof. Substituting for the individuals’objective function (4), the planner’s objective
function can be rewritten as
max
fbt g1
t=1
=
(
0s (1
0s (1
fbt gt=1
1
2
((
(
t
u1 ) +
(1
u1
0s (1
1
X
ou
0u u1 V
(b1 ; b2 ; u1 ) +
tV
y
)
(bt ; bt+1 ; bt+2 ; ut )
t=1
max
1
1
u1 ) V os (b1 ; b2 ; u1 ) +
1
X
)
2
u1 ) +
+
t 1)
t=2
0u u1 b1
b1
0u u1 )
4
b2t +
(
0s (1
t=1
0s (1
2
t
u1 ) +
u1 ) +
1
2 ) b1 b2
4
+
1
X
(
(1 + )
4
)
0u u1 )
1
X
(1
0u u1 )
) + 2u1
b1
4
1 2
b
4 1
t+1
+
t 1
1
bt bt+1
4
2
t=2
As the objective function is strictly decreasing in bt 8t
2, regardless of b1 and the
planner weights f t g1
t=1 , it follows directly that e¢ cient bene…ts are given by bt = 0 8t
Now reconsider the planner’s problem, substituting in the optimal solution for t
2.
2:
max V (b1 ; u1 j fbt g1
t=2 )
b0 2[0;1]
= max f
b0 2[0;1]
+(
0s (1
u1 ) +
0u u1
1)
0s (1
1
2
u1
u1 ) +
(1
0u u1 b1
)
2
b1
(
(
0s (1
0s (1
u1 ) +
u1 ) +
0u u1 ) u1 b1
0u u1 )
1 2
b :
4 1
Taking the …rst-order condition and implementing the constraint b1 2 [0; 1], the optimality
of b1 follows.
1.2
Complete characterization of the anti-welfare equilibrium
Proposition 2 Suppose that
M
M
and suppose beliefs are given by
0:555 is the real solution to the equation
d(
)=
and
d(
2[
d(
) ; ], where
) is de…ned below. Then
an AWE, hB aw ; U aw i, exists and has the following characteristics:
Case 1 Suppose that
p
5 1
2
' 0:618 and that beliefs are given by
2
2[
a(
) ; ], where
a(
)
1+
period”; namely:
p
(1 + ) . Then there exists an AWE implying “switch in one
B aw (ut ; ) =
U aw (bt ; ) =
8
>
>
<
if ut > 1=2
>
>
: 0 if ut 2 0; 1
2
8
>
>
< 21 (1
+
+ bt ) if bt >
>
>
: U pl (bt )
else
where U pl (bt ) is de…ned in Proposition 1.
Case 2 Suppose
b(
)
and suppose beliefs are such that 2 [ b ( ) ; min f ; a ( )g), where
q
n
o
p
1
2
1
max 1 +
; 2(2+
' :570 is
3
+
4
1
+
4
+
8
, and
)
the real solution to the equation
b(
) = , (which occurs at 1 +
p
1
= ). Then
there exists an AWE involving “switch in two periods”; namely:
8
>
>
>
if ut ua ( ; )
>
>
>
<
B aw (ut ; ) =
1 if ut 2 (0:5; ua ( ; ))
>
>
>
>
>
>
: 0 if ut 2 0; 1
2
8
>
>
< 12 (1
+
+ bt ) if bt >
aw
U (bt ; ) =
>
>
: U pl (bt )
else
where ua ( ; )
1
2
(2 +
(2 + 3
2=3
Case 3 Suppose
2
M; 2
h
n
p
(
)
;
min
; 1+
d
) ) (1
)
1
.
[0:555; 0:630] and suppose beliefs are such that
1
d(
o
)
, where
2 (1 + ) = 3 + 2 + 2
3
p
:
2
Then there exists an AWE involving “switch in two periods”; namely:
8
>
>
>
if ut 2 ud ( ; ) ; 1
>
>
>
>
>
>
>
< 23 + (1
) =2 ut if ut 2 uc ( ; ) ; ud ( ; )
aw
B (ut ; ) =
>
>
>
1
if ut 2 (0:5; uc ( ; ))
>
>
>
>
>
>
>
: 0
if ut 2 0; 21
8
>
>
< 21 (1
+
+ bt ) if bt >
aw
U (bt ; ) =
>
>
: U pl (bt )
else
1
2
where uc ( ; )
de…ned above, and 2
+
(1 )
,
2
2=3
3+
2
ud ( ; )
2+
2
+
p
, and
is the real solution to the equation 1 +
p
and
M
1
=
d(
are
).
Proof omitted
Figure 1. Region of
and
where AWE are sustained. The area between the 450 line and
dashed line represents the range of
and
considered in Proposition 3 (i.e. cases 1 and 2
only).
COMMENT:
1. Intuition for the function
The …rst part, i.e.
b(
1+
)
p
max
1
n
1+
p
1
;
1
2(2+ )
3+4
, implies that the constraint bt
ing for all u 2 (0:5; ua ( ; )). In particular, for
= 1+
p
1
q
1+4 +8
1 is strictly bind-
and u = ua ( ; ), the
old unsuccessful are indi¤erent between setting b = and b = 1. The second part,
q
1
i.e.
3
+
4
1 + 4 + 8 2 guarantee that U aw (b; ) ua ( ; ) is a
2(2+ )
rational expectations equilibrium outcome for any b > . This restriction is necessary
to ensure that U aw (b; ) satisfy rational expectations for all b 2 [0; 1].
4
2
o
.
2. Intuition for the function
d(
). The function
d(
) = 2 (1 + ) = 3 + 2 + 2
p
, is
de…ned as the solution to the equation
lim fU aw (b; )g = ud ( ; ) ;
b!
i.e. the minimum beliefs
+
which would guarantee U aw (b; )
expectations equilibrium outcome for any b >
ud ( ; ) as a rational
. This restriction is necessary to
ensure that U aw (b; ) satisfy rational expectations for all b 2 [0; 1].
1.3
Pro-welfare equilibrium with myopic voting
In order to characterize the political equilibrium under myopic majority voting, we show,
as in the rational voting case, …rst the equilibrium under dictatorship. The following
proposition characterizes the myopic dictatorship equilibria.
Proposition 3 The myopic PL equilibrium is identical to B pl ; U pl .
The myopic DP equilibrium, B mdp ; U mdp , is characterized as follows;
8
>
1 bmdp
>
< 2 1 + mdp ut umdp + bmdp if ut > umdp
2(1+ mdp )
mdp
B
(ut ) =
>
mdp
>
: 1
if ut 2 0; u 2 11+b mdp
(
)
ut+1 =
mdp
ut
1
umdp + umdp ;
where,
2+2
< bdp ;
3+2
5+3
umdp =
< udp ;
6+4
q
1
2+
4+ 2
2
bmdp =
mdp
=
1
2
The economy converges asymptotically to bmdp and umdp . Since ut+1 > umdp
for all ut , the constraint bt
1 never binds except possibly in the …rst period.
5
1 bmdp
2(1+ mdp )
Proof. The policy function under DP maximizes utility of old unsuccessful, given bt ,
solving
(1
B mdp (ut ) = arg max bt
) + (bt + bt+1 ) + 2ut
bt ;
4
bt
with a …rst-order condition
bt =
3+
(1 bt+1 )
2
ut :
Rearranging terms yields,
2
bt+1 =
2
bt
ut +
3+
:
(1)
Optimal investment behavior implies
ut+1 =
1
2
+ bt +
2
bt
ut +
3+
bt
2
=
=2
(2)
ut + 2:
Equations (1) and (2) form a system of linear di¤erence equations, with a steady state given
by bmdp and umdp as expressed in the proposition. Both roots of the system are negative,
mdp ,
one stable and one unstable. The stable root, denoted
mdp
=
1
2
The associated eigenvector is
q
2+
2 1+
4+
2
1
;
2
2
increases in
and is given by
3 1p
5 :
+
2 2
mdp
; 1 and the solution to the system is therefore
u0
umdp
given by
bt =
2 1+
mdp
ut = u0
umdp
mdp
Using this, we can derive the policy function B mdp (ut ) =
and the law-of-motion for ut =
mdp
ut
1
t
mdp
t
(3)
+ umdp
2 1+
umdp + umdp .
6
+ bmdp ;
mdp
ut
umdp +bmdp
Finally, the constraint b 2 [0; 1] is to be taken into account. While b
0 never binds, b
1
binds if
ut < umdp
1 bmdp
:
2 (1 + mdp )
mdp
From the law-of-motion for ut , we …nd that for all t > 0, ut
umdp
1 bmdp
.
2(1+ mdp )
Thus, the constraint bt
1
umdp + umdp >
1 can only bind in the initial period.
Now, consider majority voting. Clearly, if u0
1=2, bt = 0 for all t
0.2 If, instead,
u0 > 1=2, the welfare state will never break down under myopic voting. Under myopic
voting, if 1 > u0 > 1=2, then ut > 1=2 and bt > 0 8t, i.e., the welfare state survives.
Assume the opposite, then there is a period t
1=2; bt+1
0 such that ut > 1=2; bt > 0 and ut+1
n
o
= 0: Then, the choice of bene…ts satis…es bt = min 1; 3+2
ut ,
ut+1 =
1
1
3+
2
3+
+ min 1;
2
+ min 1;
ut
=2
1
=2
1=2;
which contradicts the initial assumption.
Using the previous proposition, the next proposition immediately follows;
Proposition 4 Under myopic voting. The equilibrium is given by
8
>
>
< B mdp (ut ) if ut > 1=2
m
B (ut ) =
>
>
:
0
if ut 12
8
>
>
< B mdp (ut ) if bt > 0
m
U (bt ) =
>
>
1
:
if bt = 0
2
2
This is due to rational investments, since ut+1 = (1
7
+ bt+1 ) =2
1=28bt+1:
1.4
A stochastic pro-welfare equilibrium
Proposition 5 3 For all
and beliefs, (p; ) 2 [0; 1]
2 (0; 1) ; there exists a dense compact subset of parameters
[0; ], including p = 1 and
= 0; that sustains the following
“stochastic pro-welfare equilibrium”, hB spw ; U spw i;
8
>
1 3
>
>
ut
if ut > u ( ; p)
>
p 2
>
>
>
h
i
>
>
>
2p
< bs 2p1 2 (ut us ) if ut 2 32 2+
; u ( ; p)
B spw (ut ) =
;
i
>
>
2p
1 3
>
1
if ut 2 2 ; 2 2+
>
>
>
>
>
>
>
:
0
if ut 1=2
U spw (bt ) =
where
us
bs
u ( ; p) =
8
>
>
>
us +
>
>
>
<
>
>
>
>
>
>
:
2p
4p
(bt
bs )
1+bt
2
1
if bt 2
if bt 2
+bt
2
h
;2
+4(1 p)
2+
;
if bt
4 + (5 + )
3 (2 + ) (1 + 2p)
2 (4 + )
bdp + (1 p)
3 (2 + ) (1 + 2p)
q
2p (2p
6+
2 (1 p) (2p
)
udp + (1
i
2 +4(1 p)
;1
2+
p)
) ( + 2 (1
2 (2 + )
p))2
:
The equilibrium law of motion is
8
>
>
< U spw (B spw (ut ))
with probability p
ut+1 =
>
>
:
with probability (1 p) ;
t
where
t
is i.i.d. with a p.d.f. f ( t ) and E ( t ) = 1=2:
Proposition 3 nests the equilibrium of the deterministic economies of Proposition 2. For
p = 1, the stochastic pro-welfare equilibrium is identical to the PWE of Proposition 2. For
8
p < 1, an existing pro-welfare or anti-welfare majority regenerates itself with probability
(1 + p) =2 each period; whereas a change of majority occurs with probability (1
p) =2:
Proof.
We …rst prove existence of the equilibrium under DP.
Optimal e¤ort depends on bt and bt+1 , where the latter is conditional on no shocks, and
satis…es
et (bt ; bt+1 ) = (1 +
Thus,
ut+1 =
(bt + bt+1 )) =2:
8
>
>
< et (bt ; bt+1 ) with probability p
>
>
:
:
with probability density (1
p) f ( )
To balance the budget, 2 t = (ut + ut+1 ) bt =p, yielding
8
>
>
< (1 +bt +4 bt+1 )+2ut bpt with probability p
t =
>
>
: +ut bt
with probability (1
2 p
and the expected tax rate being
e
t
=
(1
+bt + bt+1 )+2ut
bt + (1
4
;
p)
p) E(
)+ut bt
2
p,
with E ( ) =
1=2. The equilibrium conditions can thus be written
1) B sdp (ut ) = arg maxbt V ou bt ; ut ; B sdp U sdp (bt )
2) U sdp (bt ) = 1
+ bt + B sdp U sdp (bt )
, subject to bt 2 [0; 1]; and
=2,
where
V sdp bt ; ut ; B sdp U sdp (bt )
bt
=
p
p 1
+ bt + B sdp U spd (bt )
4
9
V^ ou (bt ; ut )
+ 2ut + (1
p) bt
p
!
:
Using the expressions for B sdp , and U sdp ,
8
>
>
< p1 + 2+
sdp
sdp
U (bt ) =
B
>
>
: 1
1
2p bt
if bt 2
u
U
p p
1 4 + 2 2 (2
(1) =
4
2+
i
2 +4(1 p)
;1
2+
h
if bt 2 0;
where we used the fact that for p = 1,
sdp
h
2 +4(1 p)
2+
(4)
2
)
> 0;
implying that there is a p < 1 such that any p 2 (p; 1];
u > U sdp (1):
Now, we can evaluate
8
>
>
1 6+ (1+2p
<
4p
2+
ou
^
V (bt ; ut ) =
>
>
: bt 3 pbt 2ut
4p
)
1
2p ut
bt
2
b2t
8
p
if bt 2
else,
h
i
2 +4(1 p)
;1
2+
;
which is maximized by bt = B sdp (ut ) : To see this in more detail, de…ne
V2 = V^ a (ut )
=
8
>
>
<
1
16p2
(3
>
>
: ( + 2 (1
h max
2 +4(1
bt 2 0;
2+
V^ a;int (ut ) if ut
p)) 6+
2ut (2+ )+(1 p)2(
2(2+ )2 p
=
>
>
:
bt 2
1
8
(
2
i bt
2ut )2
V1 = V^ b (ut )
8
>
>
<
p)
h
max
2 +4(1 p)
;1
2+
2
(1+2p)+2ut (2+ ) 6)
p(2p
1 8+ (6
8
pbt 2ut
4p
16
2
2p)
4(1 p)(2p
2+
)
1
2p ut
)
V^ a;cor (ut ) else,
^ ou (bt ; ut )
iV
V^ b;int (ut ) , if ut
)(2+ )2
)+(1 p)2(2
p(2+ )
3
3
2
2p
2+
V^ b;cor (ut ) , else,
which we see are generalizations of the corresponding expressions in the proof of proposition 1. The functions V^ a;cor (ut ) and V^ b;cor (ut ) are calculated when the constraints b
10
2 +4(1 p)
and
2+
1, respectively, are binding implying that for any ut ; V^ a;int (ut )
b
V^ a;cor (ut ) and V^ b;int (ut )
does not bind for p
2 +4(1 p)
2+
2 +4(1 p)
,
2+
V^ b;cor (ut ). The constraint bt
for V^ b;int (ut )
1=2; as is easily veri…ed by standard algebra. We also note that
< 1 if p > 21 + 14 , which precludes the degenerate solution B sdp U sdp (bt ) = 18bt .
Now, by de…nition, V^ a;int (u) = V^ b;int (u) and for ut > (<) u; V^ a;int (u) > (<) V^ b;int (u)
@ V^ a;int
@u
since
16
2
+4(1 p)
2+
>
@ V^ b;int
@u 8u
8p(1 p)
=
2 [0; 1];
p
2
1
2
> 0 and p > 1=2: Furthermore, for p = 1; u
(4 2 )
2+
V^ a (u) = V^ a;int (u) > V^ b;int (u)
> 1;implying, again by continuity, that for ut
u;
V^ b (u) for any p in a neighborhood below unity. Thus,
for ut > u B sdp (ut ) is the interior optimum
1
p
3
2
ut .
Now, consider u < u, where we need to show that V^ b (u) > V^ a (u) : We already know
that V^ b;int (u) > V^ a;int (u)
ut <
2p
2+
3
2
; V^ b (u) = V^ b;cor (u)
p > 1=2 + =2, then
all u <
3
2
V^ a (u) when u < u: Thus, it remains to be shown that for
2p
2+
16
2
+4(1 p)
2+
V^ a (u) : To see that this is the case, note that if
8p(1 p)
>
3
2
2p
2+
, implying V^ a (u) = V^ a;cor (u) for
: Now,
V^ b;cor (u)
V^ a;cor (u) jut
(2p
3
2
2p
2+
) (2 +
4p)2
> 0;
8 (2 + )2 p
implying V^ b (u) > V^ a (u) as required.
Finally, to show that U sdp (bt ) satis…es the equilibrium condition, we use (4) to substitute
for bt+1 yielding
1
et bt ; B sdp U sdp (bt )
=
8
>
>
< 1
>
>
:
1+
bt +
1
p
+
2+
1
2p bt
=2 if bt 2
h
h
2 +4(1 p)
;1
2+
if bt 2 0; 2
1+bt
2
= U sdp (bt ) :
Comment on proof of Proposition 3:
11
+4(1 p)
2+
i
By plotting the di¤erent conditions for p, we can see that the strongest condition is
implied by
U sdp (1)
u ( ; p)
0:
This condition is a fourth-order equation in p; given by
p4 + 1 +
2
2
p3
3+
2
+
3
1
5
+
+
2 2
8
p2 +
2
p=
1
(2 + )2
16
where we note that the relevant root satis…es 0:5 + 0:45 < p < 0:6 + 0:4 :
Proof. Now, consider the case of majority voting and the two equilibrium conditions
1) B spw (ut ) = arg maxbt fV pw (bt ; ut ; B spw (U spw (bt )))g, subject to bt 2 [0; 1]; and
2) U spw (bt ) = (1
+ bt + B spw (U spw (bt ))) =2;
where
V spw (bt ; ut ; B spw (U spw (bt )))
=
8
>
>
>
<
p(1
bt
p
>
>
>
:
1
p
+bt + B sdp (U spd (bt )))+2ut +(1 p) bt
4
p
p(1
if ut > 1=2
:
+bt + B sdp (U spd (bt )))+2ut +(1 p) bt
4
p
else.
Starting with condition 2), we substitute B spw (U spw (bt )) for bt+1 in the optimal investment
expression, giving,
(1
+ bt + B spw (U spw (bt ))) =2
8
>
>
1
>
1
1+
bt +
>
p + 2+
>
>
<
=
(1 + bt ) =2;
>
>
>
>
>
>
: 1 +bt
1
2p bt
=2 if bt 2
bt 2
h
2 +4(1 p)
;1
2+
2
; 2+
i
;
bt
2
= U spw (bt ) ;
provided that, as in the non-stochastic case,
:
As for condition 1), consider …rst the range where ut
12
1=2; then Vtspw (bt ; ut ) is maximized
by setting bt = 0.
Now, consider ut > 1=2: Noting that U spw (bt ) > 1=2 for all bt > , we …nd that for any
bt > , any ut > 1=2 and any p su¢ ciently close to unity, V spw (bt ; ut ; B spw (U spw (bt ))) =
V sdp bt ; ut ; B sdp U sdp (bt )
: We thus only need to verify that for these values of bt ; ut and
p;
max V sdp bt ; ut ; B sdp U sdp (bt )
max V spw (bt ; ut ; B spw (U spw (bt )))
bt >
bt
1 3+p
= max bt
bt
4
Since, maxbt
1
16p2
1 3+p
4 bt
> 0; provided that
pbt 2ut
p
<
1
2p
pbt
p
2ut
(5)
:
tends to zero as approaches zero, while maxbt > V sdp bt ; ut ; B sdp U sdp (bt )
there exists a
> 0 such that for any
< ; (5) is satis…ed
and an existing pro-welfare majority chooses not to induce a breakdown.
13