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The Role of Immigration in
Sustaining the Social Security
System: A Political Economy
Approach
By
Edith Sand and Assaf Razin
The Eitan Berglas School of Economics, Tel- Aviv University,
and Cornell University
Scope
In the political debate people express the idea that immigrants are good
because they can help pay for the old. The paper explores this idea in a
dynamic political-economy setup. We characterize sub-game perfect
Markov equilibria where immigration policy and pay-as-you-go (PAYG)
social security system are jointly determined through a majority voting
process. The main feature of the model is that immigrants are desirable for
the sustainability of the social security system, because the political
system is able to manipulate the ratio of old to young and thereby the
coalition which supports future high social security benefits. We
demonstrate that the older is the native born population the more likely is
that the immigration policy is liberalized; which in turn has a positive
effect on the sustainability of the social security system.
Outline
• Baseline Model: OLG model, repeated
voting, sub-game perfect equilibrium,
migrant-native differential population
growth rates.
• The Extended Model: private saving,
capital accumulation and endogenous
factor prices.
• The Effect of Aging.
Features of the Base-Line Model
• OLG two periods.
• PAYG Social Security with no private savings.
• Immigrants enter the economy when young, and
gain the right to vote only in the next period, when
old. They have the same preferences as those of
the native-born, except from having a higher
population growth rate.
• Offspring of immigrants are like native-born in all
respects (in particular, they have the same rate of
population growth).
Baseline model
• The utility of the young and old agents are logarithmic:
(1)
(2)

lt 1 
   ln bt 1 
U ( wt , t , bt 1 )  ln  (1   t )lt wt 
 1

U o(bt )  bt
y
where U i is the utility function,   [0,1] is the discount factor
and   0 is the labor supply elasticity with respect to the wage
rate.
Technology, social security system and
preferences
• The production function is a linear production function:
Yt  N t
(3)
where Yt is the output, and N t is the labor supply in period t .
• The tax/transfer is a “pay as you go”, where both
immigrants and native born contribute to and benefit from
the welfare state in the same way. The balanced government
budget constraint implies:
bt Nt 1   t lt wt Nt
(4)
• The labor-leisure decision of young individuals is given by:

lt  wt (1   t )
(5)
• A worker can be either native born or immigrant. The labor
supply in period t is:
N t  Lt lt (1   t )
(6)
where   [0,1] denotes the economy’s immigration quotas,
and Lt is the number of the native born workers in period t.
• The immigrant population, m  [1,1], has a higher population
growth rate than that of the native-born population, n  [1,1] :
(7)
nm
but their descendants’ population growth rate is the same.
• The number of native born individuals in period t is:
Lt  Lt 1 (1  n)   t 1Lt 1 (1  m)
(8)
Old voter’s preferences
• The indirect utility functions of old individual is:
(9)
V o ( Lt ,  t 1 , t ,  t ) 
 t wt lt [(1  n)   t 1 (1  m)](1   t )
(1   t 1 )
• The old favors the larger possible quotas,  t
"Laffer point" tax rate,  t*   .
1 
1
, and the
Young voter’s preferences
• The indirect utility functions of old individual is:
  t 1wt 1lt 1[(1  n)   t (1  m)](1   t 1 ) 



wt lt (1   t )    ln 
(1   t )
  1



(10) V y ( t ,  t , t 1,  t 1 )  ln 
• The young favors a minimal tax rate. Preferences regarding
the migration quota: 1. A higher migration quota increases
next period transfer payments. 2. Migration quota also
increases the number of next period young voters.
• The young voter favor the larger possible quotas, which on
the one hand increases social security benefits in the next
period, on the other hand changes next period decisive
voter's identity from young to old in the next period in
order to lead to a majority of old.
 t*  
n
 [0,1]
m
Markov Sub-game Perfect Political
Economy Equilibrium
The Markov subgame-perfect equilibrium means
that the vector of expected policy decision rules
(the tax rate, τ and the immigration quotas, γ),
which depends on the current state variable (i.e. the
migration quota at time t) is also the same vector
policy decision rules chosen by the current
decisive voter as a function of previous period state
variable (i.e. the migration quota at time t-1).
The Equilibrium
Proposition 1: The Markov equilibrium is:
  t  0 if ut  1
(14) T ( t 1 )   * 
t 
otherwise

 1

n
 *
 t  
if ut  1
(15) G ( t 1 )  
m

 t  1 otherwise
where ut is the ratio of old to young voters, and  t*  
 t-tax rate
 t -openness rate
1
1
*
*
0
ut  1
1
ut
0
ut  1
1
ut
n
 [0,1]
m
Strategic Voting
Because immigrants enter the country while young and gain
the right to vote only in the next period when they are old, the
equilibrium strategy adapted by current voters takes into
account the effect of the current level of immigration on
the composition of voters and their voting preferences in the
next period.
The equilibrium has a “switching” strategy: the
young decisive voter admits a limited number of immigrants,
in order to change the decisive voter's identity next period.
Equilibrium paths
There are three possible equilibrium paths:
1.
if n, m  0 , level of social security benefits is zero.
2.
if m  n  0 , migration quota is at the maximum, 
and the tax rate is at the "Laffer point”,    .
1
,
1 
3.
if n  0 and m  n  0 , there a “demographic switching”
equilibrium path : in a given period, the economy is fully
opened to immigration,   1 , and the tax rate is at the
"Laffer point”,    . In the next period the tax rate/social
1 
security benefits is set to zero and there is some
restrictions on immigration    n .
m
The First Equilibrium path
•
If n, m  0 , the level of social security benefits is zero.
The number of young voters exceeds the number of old
voters. Because the decisive voter is always young, and her
preferences are for zero labor tax, no social security benefits
will be paid to the old. The young is indifferent to
immigration because it does not influences her current
income, or the next period decisive voter's identity.
As a result, the equilibrium path is one where in every
period there is a majority of young voters, who therefore
destroys the social security system forever.
The Second Equilibrium path
•
If n  m  0 , migration quota is at the maximum,  t  1 ,
and the tax rate is at the "Laffer point”,  t   .
 1
The number of next period old voters exceeds the number
of next period young voters. Thus, along the equilibrium
path a majority of old will always prevail, which validates a
permanent existence for the social security system and a the
maximum flow of immigrants.
The Third Equilibrium path
• If n  0 , and n  m  0 , there is a “demographic switching”
equilibrium path which is characterized by an alternate
taxation/social security policy where some level of
immigration always prevails.
When there is a majority of old their preferable migration
quota is maximal and the tax rate is at the "Laffer point". But
since migration quota is maximal and n  m  0, the number of
young voters exceed the number of old voters in the next
period. Thus, the decisive voter in the next period is young
who votes for a minimal tax rate and votes strategically for a
certain level of migration quota which changes the identity of
the next period decisive voter to an old voter (there exist such
an migration quota since n  0 ).
Capital Accumulation and Endogenous
Factor Prices
The new features in the extended model are that young
individual is able to save. The aggregate savings of the young
which generates next period aggregate capital are being used as
a factor of production, in a constant return to scale production
function.
These new features create in addition to a very similar
“demographic switching” equilibrium as in the base line model,
another equilibrium: a "combined strategy" equilibrium.
The "combined strategy" equilibrium
The new equilibrium of the extended model, combines
strategies concerning both the old-young composition in
the population, and the level of capital: there is a range of
values of the capital per (native-born) worker, for which
the "demographic steady" strategy dominates; while for
values outside this range, the "demographic switching"
strategy dominates.
The "demographic steady" strategy
• The reason for the additional strategy results from the fact
that there is another channel of influence of the current
period policy variables on next period policy variables
through savings. Thus, the young decisive voter may adopt a
"demographic steady" strategy, where she admits the
maximum amount of immigrant. In so doing the young
decisive voter renders a majority for the young, every period.
•
The "demographic steady“ strategy is characterized by an
equilibrium tax rate which is a decreasing function of the
capital per (native-born) worker, and no restrictions on
immigration.
Aging: a decrease in n
I. Before: (1) the decisive voter switches between young and old
(m+n>0 and n<0), or (2) the young is the decisive voter every
period (n>0);
After (m+n<0): the old is in majority every period; the tax
rate is set at the "Laffer point", and there is no restriction on
immigration.
The aging of the native born liberalizes immigration policy,
and set the tax rate at the "Laffer point“.
II.
(1) Before (m+n>0; n<0): the decisive voter switches
between young and old.
After: (m+n>0; n<0): the decisive voter switches between
young and old .
The aging of the native born population enlarge
immigration quotas set by the young in the
"demographic switching" equilibrium path.
(2) Before: the decisive voter is young.
After: the decisive voter is young .
The aging of the native born population decreases the tax
rate in the "demographic steady" equilibrium path.
III.
(1)
Before: combined strategy equilibrium.
"demographic switching" equilibrium path.
After: combined strategy equilibrium. "demographic
steady" equilibrium path.
(2) Before: combined strategy equilibrium. "demographic
steady" equilibrium path.
After: combined strategy equilibrium. "demographic
switching" equilibrium path.
Aging affects the capital per (native-born) worker, and
thus can move the system from the "demographic
switching" equilibrium path to the "demographic
steady" equilibrium path or vice versa.
Result I:
Sharp aging trend of the native-born population,
can move the system to an equilibrium path where
the sum of the population growth rates are
negative, n + m <0. In this case, the old are in the
majority every period. The old liberalize
immigration policy as much as possible and
sustain the social security system by setting the tax
rate at the "Laffer point".
Result II:
The aging of the native-born population
enlarges immigration quotas set by the
young in the "demographic switching"
equilibrium path, while decreasing the tax
rate in the "demographic steady"
equilibrium path.
Aging and migration quota in the
"demographic switching" equilibrium path
Aging has the overall effect of raising the optimal immigration
quota of the young voter,  t  Min[ n ,  * ].
m
The effect of a decrease in n works itself out through next
period dependency ratio, ut 1 . This dependency ratio effects the
n
quota as follows: in the case where    m , the dependency
ratio effects the quota through the identity of next period
decisive voter; whereas in the case where  t   * , the
dependency ratio effect goes through next period capital per
(native born) worker, k t (due to the fact that a larger quota has
the overall effect of decreasing k t ). Since a larger quota
decreases the dependency ratio, it will decrease the ratio less
the lower is n.
t
Aging and the Tax Rate in the
"demographic steady" equilibrium path
Aging decreases the tax rate set by the young in the
"demographic steady" equilibrium path, .(kt )
This is due to the fact that aging increases total
savings which raises the amount of capital per
(native-born) worker. Since the tax rate is a
decreasing function of the capital per (native-born)
worker state variable, the aging of the native-born
population decreases the optimal tax rate in the
"demographic steady" equilibrium path.
Result III:
Aging affect the capital per (native-born) worker,
and thus can move the system from the
"demographic switching" equilibrium path to the
"demographic steady" equilibrium path or vise
versa, since the equilibrium paths are defined over
a closed range of the capital per (native-born)
worker state variable.
Thus, the older is the native born population,
the more likely is that the migration policy is
liberalized and that the social security system
survives.
The Razin-Sadka-Swagel Model
Innate ability parameter
e
G ( e)
g (e)  G ' ( e)
CDF of the innate ability parameter
Cutoff level of e
(1   ) w(1  e*)  (1   )qw  
e*  1  q 

(1   ) w
Y  wL  (1  r ) K
et*
Lt  {  (1  e)dG  q[1  G (et* )]}N 0 (1  n) t  l (et* ) N 0 (1  n) t
0
bt N 0 [(1  n) t 1  (1  n) t ]  wLt  wl (et* ) N 0 (1  n) t
1 n
bt 
wl (et* )
2n
N 0 (1  n)t G(eM )  N 0 (1  n)t (1  G(eM ))  N 0 (1  n)t 1
2n
e M ( n)  G [
]
2(1  n)
1
 0 (n)  arg max W ( , n, eM (n))

W ( , n, eM (n))
 B[ 0 (n), n]  0

Single Peak Conditions
 0 (n)  arg max W ( , n, eM (n))

W ( , n, eM (n))
 B[ 0 (n), n]  0

 2W ( 0 , n, eM (n))
 B [ 0 (n), n]  0
2

d 0 (n)
Bn [ 0 (n), n]

dn
B [ 0 (n), n]
Fiscal Leakage
Effect


Bn [ 0 (n), n]  
 1eM e*[0 ( n )
g{e *[ 0 (n)]}de * / d
1
wl{e *[ 0 (n)]}
 
2
( 2  n)
(1   )( 2  n) 2
deM
w
Median Voter Shift Effect
dn
deM
1
g { e*[ 0 ( n )]} de*/ d
 wl{ e*[ 0 ( n )]}

dn
( 2 n ) 2
(1 )( 2 n )2
eM e*[ 0 ( n )]
1
g { e*[ 0 ( n )]} de*/ d
wl{ e*[ 0 ( n )]}

( 2 n ) 2
(1 )( 2 n )2
w
1

r
1
B
A B
bt 1 
S{W (e, , bt , bt 1 ),
}dG

2n 0
1  (1   )r
B

r
b
1
B
A
b 
S{w(1  e)  b 
,
}dG 

2n 0
1  (1   )r 1  (1   )r
e*
B
r
b
1
A
S{wq  b 
,
}[1  G (e*)]
2n
1  (1   )r 1  (1   )r
b A  B A (t , n)
b B  B B (t , n)