Voting on Immigration Restriction¤
Francesco Magris
EPEE, University of Evry
Giuseppe Russo
DELTA, ENS-EHSS-CNRS; CNR-ISSM
Preliminary
Abstract
We study how immigration policies are determined in a model
where immigration raises capital income and lowers wages, migration
decisions are endogenous and there exists preference for home-country
consumption. We model migration policy as a pure entry rationing
rather than a complicated screening system, and the vote as a Stackelberg game. Our …ndings show that the most likely result is a polarization between voters who prefer total frontiers openness and voters
who prefer perfect frontier closure. The actual choice depends on
which group the median voter belongs to.
Keywords : immigration policy, voting.
Jel classi…cation : D72, F22, J18.
¤
We thank François Bourguignon for his help. Any error is our own.
1
1
Introduction
Immigration is one of the most compelling topics on the policy-makers agenda.
The collapse of the Soviet Union, the recent surge in regional con‡icts, as
well as long-term climate changes put enormous pressure on developed countries’ national borders. In the EU, high unemployment rates and uncertain
expectations contributed to spread immigration aversion. Anti-immigration
programs yielded immediate electoral consent, in some cases causing a sudden
and unexpected success of far-right parties. As a consequence, the legislative trend, as reported by OECD (2000, 2001), goes towards a generalized
increase in frontiers closure.
However, this has not always been the case. Goldin (1994) reports an
interesting reconstruction of the process leading to close the U.S. borders
in 1921, after more than 17 millions of entries over the previous 30 years.
Goldin …nds that this was correlated with the impact of migrants on domestic wages. Interestingly, she remarks that anti-immigration pressures were
stronger during economic downturns. On the other side, owners of capital 1
were consistently pro-immigration.
Goldin’s argument has an immediate economic intuition: as one would
expect, immigration lowers labour income and raises capital income, thus
the vote on frontier closure depends on the median voter’s share of capital
and labour income. Benhabib (1996) develops this point in a model where
electors vote on the immigrants’ capital requirements, taking into account the
e¤ect on their total income. Benhabib’s model, however, su¤ers from some
limitations: …rst, he assumes that a perfect immigrants’ screening and a
perfect enforcing of the rule are possible; moreover, the migrants’ behaviour
is passive: there exists in…nite individuals willing to enter the developed
country.
The combination of these assumptions boils down the immigration problem to identify the appropriate entry requirements. Once they are met, the
appropriate in‡ow is determined. This assumption is, in our opinion, unnecessarily restrictive: Chiswick and Hatton (2002) stress that migration is an
economic decision par excellence, and report that ‡ows are very sensitive to
market signals. Since immigration a¤ects wage di¤erentials, it is not correct
to assume that immigrants are available no matter what happens to the wage
1
For example, the National Association of Manufacturers, the National Board of Trade,
many Chambers of Commerce.
2
di¤erential.
In our model, we try to overcome these limitations by providing for endogenous migration decision and rational voters’ expectations in the form
of a Stackelberg-like equilibrium. In addition, we do not try to determine
any capital requirement to allow entry. Rather, we characterize the immigration policy as a probability to enter the destination country: since any
border closure implies some entry rationing, a restrictive policy is simply a
low probability of entering2 . So doing, we do not need to assume neither a
perfect screening, nor a perfect borders enforcement.
The paper is organized as follows: in the next section we develop our
model, and we characterize the migrants’ and voters’ decisions; then we analyze the voting behavior both in a referendum and in a pairwise alternatives
contest. Finally, we report our conclusions in section 5. The proofs are in
the Appendix.
2
The model
2.1
Destination country
We consider a static model including a given population of natives, i.e. individuals who live in the destination country, an aggregate CRS production
function in the destination country employing capital and labor to produce
a unique consumption good, a given size of potential immigrants and a CRS
aggregate production function in the origin country which produces an homogeneous consumption good out of labor force exclusively. Natives are
indexed by the unit of capital they own, denoted k. Following analogous
line as in Benhabib (1996), the density of native is given by the continuous
density function N (k) de…ned over [0; +1). Thus, the aggregate capital in
the destination country, K0, is given by
K0 =
Z 1
N (k) kdk:
(1)
L0 =
Z 1
N (k) dk:
(2)
0
and total population, L0; is
2 Note
0
that this …ts perfectly the quotas system by the U.S. until 1965.
3
For the sequel of our analysis, it is useful to introduce at this stage the
median type of natives, km, which corresponds to the individual whose capital
endowment solves
Z km
1
N (k) kdk = L0:
(3)
2
0
Each native is endowed with a unit of labor which she supplies inelastically in
a perfectly competitive labor market. In the destination country there is an
homogeneous consumption, c, good produced according to a constant returns
to scale aggregate production function F (K; L), where K and L denote,
respectively, aggregate capital and labor. Production can be expressed in
the intensive form f (a), with a ´ K=L, exhibiting the usual neoclassical
features.
Assumption 1. The intensive production function f : R+ ! R is
smooth, strictly increasing and strictly concave. Moreover f (0) = 0, limk!0
f 0 (k) = +1 and limk!+1 f 0 (k) = 0.
In absence of any immigration, the competitive interest rate, r, and real
wage, w, are, given by, respectively
and
where
w = f (R0) ¡ f 0 (R0 ) R0
(4)
r = f 0 (R0)
(5)
R
1
K0
N (k) kdk
R0 ´
= R0 1
(6)
L0
0 N (k) dk
stands for the pre-immigration capital-labor ratio. It follows that type ki
total income writes
w + rk i = f (R0) ¡ f 0 (R0) R0 + f 0 (R0) ki
(7)
For sake of simplicity and without any loss of generality, we assume that
natives share the same linear utility function3 de…ned over consumption4 ,
cN
ki ;
³ ´
N
ui c N
(8)
ki = cki for all ki 2 [0; +1)
3
Given the static nature of the model, agents do not save and consume their whole
revenue. Therefore, the choice of the utility function does not matter, the only requirement
being the monotonicity of agents’ preferences.
4 Therefore, capital endowment is the only source of heterogeneity across natives.
4
where N stands for ”native” It follows that pre-immigration natives’ welfare,
ui (cki) ; can be ranked according to their respective capital endowment
ui (cki ) = f (R0 ) ¡ f 0 (R0) R0 + f 0 (R0) ki :
2.2
Source country
Each potential immigrant supplies inelastically one unit of labor in a competitive labor market. Potential immigrants are indexed by their ”preference for
domestic consumption” µ and their density is given by the continuous density
function I (k) de…ned over [0; +1). The population, I0; of the source contry
is then
Z 1
I0 =
I (µ) dµ:
(9)
0
An homogeneous consumption good, cI ; where I stands for ”immigrant”, is
produced by mean of a CRS aggregate production function employing only
labor as input:
Z 1
c I = I0 =
I (µ) dµ
(10)
0
Thus immigrants di¤er across µ: such a parameter captures individual preference for domestic consumption, as a result of cultural and ethnic factors.
Assuming a linear utility function, pre-immigration utility of consumption of
type µ, cµ ; is
uIµ (cµ ) = µcIµ :
(11)
Obviously, when deciding whether to migrate or not, potential migrants will
compare the utility they get in their origin country with the level of utility they receive by migrating successfully. We assume that, once abroad,
migrated, each immigrant shares the same linear utility function
³
´
uµ c M
= cM
µ
µ
(12)
where M stands for ”migrated”. Therefore, origin country heterogeneity does
not translate into any post-migration heterogeneity.
3
Immigration policy
An immigration policy is a real number
¼ 2 [0; 1]
5
(13)
which captures the degree of ”frontier openness”. So doing, we don’t need
to use any screening mechanism, whose implementation can be di¢cult and,
moreover, uncertain because of illegal immigration5 . The choice of any policy
¼, from the point of view of any potential immigrant, represents the probability of a successful migration. When deciding whether to migrate or not,
they will compare the utilities they get within the two alternatives. Therefore we can describe the model by a Stackelberg game in which the timing is
the following:
(1) Natives choose an immigration policy ¼ 2 [0; 1]
(2) Potential migrants choose whether to try to migrate
(3) Nature chooses randomly a fraction ¼ of successful migrants among those
willing to migrate
Since immigrants are not endowed with capital, type µ will simply compare
destination competitive wage w with µ : if w > µ, she will try to migrate,
otherwise she will not. Actually, the potential migrants are those endowed
with a preference for domestic consumption such that µ · ^µ, where ^µ solves
Ã
K0
w
R
L0 + ¼ 0µ I (µ) dµ
!
= µ:
(14)
^µ denotes the type of potential migrant who is indi¤erent between sticking
to his native country and successfully migrating. Indeed, all type with µ
lower than ^µ; will strictly prefer to migrate, but only a fraction ¼ of them,
randomly chosen, will succeed. As a consequence, the equilibrium real wage
in the domestic country is given by the left-hand side of (14). Before going
through the sequel of the paper, let us prove that there exists an unique
solution ^µ of (14) for any ¼ included between zero and one, and that such a
solution is decreasing in ¼.
Lemma 1 For any ¼ 2 (0; 1], equation (14) admits an unique solution ^µ:
For a given ¼, denote such a solution
^µ (¼) :
5
(15)
An e¤ective screening mechanism can be very costly as well. In any case, from the
point of view of natives, all potential migrants are identical, so there is no reason to screen
them.
6
Then
R^
µ
I (^
µ)d^
µ
w0 (R (¼)) R (¼)2 0 k0
@ ^µ
=¡
<0
µ)
@¼
2 ¼I (^
0
w (R (¼)) R (¼) k0 + 1
where
R (¼) ´
:
2
R ^µ(¼)
L +¼
4 0
0
K0
3¡1
I (µ) dµ 5
(16)
(17)
Proof. To prove the …rst part of the Lemma, it is su¢cient to observe
that the right-hand side of (14) increases from zero to in…nity when µ goes
from zero to in…nity, and that its left-hand side is non-increasing and strictly
positive for µ = 0: The fact that @ ^µ=@¼ < 0 is easily proven once one observes
that left-hand side of (14) is decreasing in ¼: To get the expression (16), let
di¤erentiate totally
0
1
K
0
^
A=µ
w@
R ^µ
L0 + ¼ 0 I (µ) dµ
³
´R^
with respect to ¼ and ^µ, and observe that w 0 > 0 and @=@ ^µ 0µ I (µ) dµ =
³ ´
I ^µ :
Notice that expression (17) is continuous in ¼ = 0 since R (0) = R0 .
Indeed, for ¼ = 0 the number of potential migrants willing to migrate is
…nite and so the number of successful migrants is obviously zero.
Next Lemma is to show that the number of successful migrants increases
with ¼. Such a result is not as straightforward as it could seem at …rst sight.
Indeed, if on the one hand it is true that a larger ¼ increases the probability
of succeeding in migrating, on the other one it decreases the number of those
who chose to try to migrate, as expression (16) in Lemma 1 establishes.
Lemma 2 The number of successful migrants is increasing in ¼, i.e.
Ã
!
Z ^µ
@
¼
I (µ) dµ > 0
@¼
0
(18)
Proof. The derivative in the left-hand side of (16) is easily computed as
Z ^µ
0
³ ´ @^
µ
I (µ) dµ + ¼I ^µ
:
@¼
7
If we take into account expression (16) of
Z ^µ
0
³ ´
@^
µ
,
@¼
the expression becomes
2
0
w (R (¼)) R (¼)
I (µ) dµ ¡ ¼I ^µ
w 0 (R (¼)) R (¼)2
R ^µ
I (^
µ )d^
µ
k0
¼I( ^
µ)
k0 + 1
0
and, by appropriately rearranging terms
Z ^µ
0
2
³ ´
I (µ) dµ 6
µ
41 ¡ I ^
w (R (¼)) R (¼)2 k¼0
¼I (^
µ)
w0 (R (¼)) R (¼)2 k0 +
3
0
1
7
5
i.e.
Z ^µ
0
2
³ ´
¼I (^
µ)
k0
+ 1 ¡ I ^µ w 0 (R (¼)) R (¼)2
¼I (^
µ)
w 0 (R (¼)) R (¼)2 k0 + 1
2
0
6 w (R (¼)) R (¼)
I (µ) dµ 4
3
¼
k0 7
5
which …nally gives
Z ^µ
0
2
I (µ) dµ 6
4
3
1
¼I(^
µ)
w0 (R (¼)) R (¼)2 k0
+1
7
5
> 0:
Let now kI be the type indi¤erent to immigration policy ¼, so that his preimmigration and post-immigration incomes are identical. Then kI satis…es
f (R0) ¡ f 0 (R0) R0 + f 0 (R0) kI = f (R (¼)) ¡ f 0 (R (¼)) R (¼) + f 0 (R (¼)) kI
(19)
Notice that such a kI exists and is unique provided there exists at least
a native with capital endowment su¢ciently low and another one endowed
with a su¢ciently large amount of capital. Indeed, when ki tends to zero, the
left-hand side of (19) is greater that the right-end side, while for ki su¢ciently
high, the opposite is true.
Consider now the most preferred immigration policy for a native of type
ki: When deciding over ¼, she will take into account R (¼), in a Stackelberg
likewise game. More in details, she will try to maximize
f (R (¼)) ¡ f 0 (R (¼)) R (¼) + f 0 (R (¼)) ki
8
(20)
with respect to ¼. It is easy to prove that the objective function is convex with a minimum at ki = R (¼). Indeed, the …rst order condition for
(20) is ¡f 00 (R (¼)) R0 (¼) R (¼) + f 00 (R (¼)) R0 (¼) ki = 0 meanwhile the second order condition evaluated at the point where the derivative vanishes is
f 00 (R (¼)) R0 (¼) R (¼) > 0:
We can now immediately derive the following Proposition:
Proposition 3 (a) The average person of type ki = R0 obtains a higher
income than his pre-immigration income under any immigration policy ¼ > 0:
(b) For any given immigration policy ¼ > 0, R (¼) < kI < R0 and all
natives of type k > k I have a higher post-immigration income.
(c) If we assume that natives vote against an immigration policy that
reduces their income, then a policy ¼ will be defeated in a referendum if and
only if km · kI :
Proof. (a) follows since income of type ki = R0 reaches its minimum at
¼ = 0: (b) follows since, for any immigration policy, income is increasing in
type ki, since type solving ki = R (¼) obtains its minimum income, and by
the de…nition of kI . (c) is obvious and immediate.
4
Majority voting with pairwise alternatives
We want now to investigate the existence of an immigration policy able to
defeat any other policy in a pairwise contest under majority voting: Since
(20) reaches its minimum at ki = R (¼) and is continuous, the population
will then polarize between those who prefer ¼ = 0 and ¼ = 1. Let kL be the
type which is indi¤erent between ¼ = 0 and ¼ = 1, i.e. that solves for k i
f (R0) ¡ f 0 (R0) R0 + f 0 (R0) ki = f (R (1)) ¡ f 0 (R (1)) R (1) + f 0 (R (1)) ki
(21)
Such a type exists and is unique provided there is at least a native with
capital endowment su¢ciently low and another one endowed with an amount
of capital su¢ciently large. Indeed, for ki low enough, the left of (21) is
greater that the right, while for ki su¢ciently high, it is the right of (19)
to be larger. Then all types k < kL will prefer ¼ = 0 and types k > k L
will prefer ¼ = 1, since types with higher ki increase their income with ¼
relatively to low ki types. Therefore we have the following Proposition which
is immediately proven.
9
Proposition 4 If km < kL (km > kL ), the policy ¼ = 0 (¼ = 1) defeats all
other immigration policies under majority voting with pairwise alternatives.
5
Conclusions
In spite of its importance, voting on migrations is still a little developed …eld
in the theoretical literature. We have tried to extend our analysis to a more
general setting with respect to some recent works, allowing for less restrictive
assumptions. Our …ndings con…rm the Benhabib’s (1996) results in a more
realistic environment. A drawback of our analysis is that our migrants own
no capital at all. Nonetheless, in our opinion, immigration always dilutes the
capital/labor ratio: it is extremely di¢cult that immigrants, whose majority
is unskilled, on average bring more capital than natives.
The empirical evidence reported in Goldin (1994) …ts well our model.
Another limitation of the screening-based models is that they overlook the
importance of illegal in‡ows. Our …ndings do not depend on any screening
method, and suggest that modelling migration policies as a probability is a
better solution with respect to more complicated rules, whose implementation
is very di¢cult in practice.
10
References
[1] Benhabib J., 1996, On the political economy of immigration, European
Economic Review 40, 1737-1743.
[2] Chiswick B., Hatton T., 2002, International migration and the integration
of labor markets, IZA discussion paper n. 559.
[3] Goldin C., 1994, The political economy of immigration restriction in the
United States, 1890 to 1921, in C. Goldin and G. Libecap, eds., The regulated economy: an hystorical analysis of governement and the economy
(University of Chicago Press) and NBER working paper n. 4345.
[4] OECD, 1999. Trends in International Migration. SOPEMI, Paris
[5] OECD, 2001. Trends in International Migration. SOPEMI, Paris
11