ISSN 1791-3144
DEPARTMENT OF ECONOMICS
DISCUSSION PAPER SERIES
Illegal Immigration in a Heterogeneous Society
Theodore Palivos and Chong K. Yip
WP 2007 - 02
Department of Economics
University of Macedonia
156 Egnatia str
540 06 Thessaloniki
Greece
Fax: + 30 (0) 2310 891292
http://econlab.uom.gr/econdep/
Illegal Immigration in a Heterogeneous Society
Theodore Palivos∗
University of Macedonia, Salonica, Greece
Chong K. Yip†
Chinese University of Hong Kong, Hong Kong
June 29, 2007
Abstract
This paper examines the effects of illegal immigration in a neoclassical growth
model with two groups of workers, skilled and unskilled. We show that although
illegal immigration is a boon to a country as a whole, there are distributional effects,
whose sign is in general ambiguous. This is because all sources of income of both
groups are affected and some of these changes tend to move income in opposite
directions. Nevertheless, a calibration exercise shows that the wealth distribution
is likely to become more unequal as the number of illegal immigrants increases. We
confirm most of our calibration results analytically in a small open economy version
of the basic model.
Keywords: Illegal Immigration, Economic Growth, Income Distribution
JEL Classification: O41; F22
∗
†
[email protected]
Corresponding author: [email protected].
1. Introduction
This paper studies the effects of illegal immigration in a neoclassical growth model, in
which workers are heterogenous in terms of their production skills. The phenomenon of
illegal immigration is universal; it is present in almost every developed as well as in several
developing countries. Examples include, but are not limited to, Canada, the European
Union, Hong Kong, India, Japan, South Africa and the U.S.1 Illegal immigration is also
one of the most divisive issues in many of these countries (see, for example, the recent
discussion regarding immigration reform in the U.S.).
The majority of the existing literature studies illegal immigration in a static context.
The pioneering work is that of Ethier (1986), which considers the economic welfare of
a host country that confronts the inflow of unskilled illegal immigrants based on the
Beckerian crime-theoretic approach. This analysis has been extended to a two-country
setting by Bond and Chen (1987), Djajić (1987) and Woodland and Yoshida (2006). Also,
Kondoh (2004), adopting the efficiency wage hypothesis, argues that strengthening border
enforcement policy is welfare enhancing to native workers due to the outcomes of higher
wages, unemployment allowances and employment. The use of a dynamic framework for
analyzing illegal migration has begun less than a decade ago. For example, Djajić (1999)
examines the implications of border control policies and internal enforcement in a onesector setting, while Hazari and Sgro (2000) construct a two-sector model to analyze the
long-run growth effect of illegal immigration and border enforcement. Recently, following
the Ramsey tradition, both Hazari and Sgro (2003) and Moy and Yip (2006) develop a
one-sector optimizing dynamic model with two factors of production (labor and capital)
to investigate the welfare effects of illegal immigration.
All of the existing dynamic models that analyze the effects of illegal immigration adopt
a framework in which there is a homogeneous labor input. In reality, however, much of
the discussion has centered around the distributional effects of illegal immigration. That
is, illegal immigration may affect different groups of the society in a different way. Accordingly, the purpose of this paper is to examine in a systematic way the distributional
effects of illegal immigration on heterogenous workers within a dynamic context. Using a
standard neoclassical growth model, we allow for three factor inputs: capital, skilled and
1
According to the UN Population Division, the total number of undocumented immigrants is estimated
around 30-40 million, which corresponds to 15-20% of the world’s population of legal immigrants (see
United Nations 2004).
1
unskilled labor. We find that illegal immigration is a boon to a country as a whole, since
the average consumption, income and wealth increase both in the state and throughout
the transition. Nevertheless, it also has distributional effects, whose sign is in general
ambiguous. Under reasonable parameterization, calibration shows that the wealth distribution becomes more unequal as the number of illegal immigrants increases. In order to
shed further light on the analytics, we also study a small open economy version of the basic model, in which the real interest rate is given at the world level. In this case, the asset
holdings of the skilled workers must rise, following an increase in the number of illegal
immigrants, while those of the unskilled can go in either direction. Finally, we provide
a sufficient condition, regarding the fine imposed on firms employing illegal immigrants,
for the wealth distribution of the domestic economy to become more unequal both in the
steady state and during the transition after an increase in illegal immigration.
The organization of the paper is as follows. Section 2 develops a basic two-factor
neoclassical growth model with a homogeneous labor input, which serves as a benchmark
throughout the paper. Section 3, which constitutes the main part of the paper, extends
this basic model to the three-factor case. Section 4 analyzes the small open economy
version of the three-factor model. Finally, Section 5 concludes the paper.
2. A Model of Immigration with One Type of Labor
The primary purpose of our paper is to analyze the effects of illegal immigration on the
distribution of income and wealth in the host country. This presupposes the existence
of different groups, i.e., a heterogenous society. Nevertheless, to understand some of the
mechanisms at work and to establish notation, it is useful to start with an economy that is
inhabited by just one type of labor; more elaborate versions of this model are considered
in later sections.
2.1. The Model
Consider a Solow-type economy in which production takes place using two inputs, labor
and capital, according to a constant returns to scale (CRS) production function:
Y = F (K, L + M),
2
(2.1)
where Y denotes total output, K denotes aggregate capital, and L and M stand for domestic and foreign labor, respectively. Furthermore, F exhibits positive but diminishing
marginal products, i.e., Fi > 0 and Fii < 0, i = K, L, M, and satisfies standard Inada conditions. Obviously, with this formulation, L and M are perfect substitutes in production.
For simplicity, we abstract from population growth.
Using the CRS property of F , we can express output per domestic citizen as follows:
Y /L = F (K, L + M)/L = F (k, 1 + m)
(2.2)
where k ≡ K/L is capital per domestic citizen and m ≡ M/L is the immigration ratio
(number of immigrants per domestic citizen).
All immigration M ≥ 0 is assumed to be illegal (undocumented); that is, the model
abstracts from any legal immigration. In fact, the following two assumptions are intended
to characterize illegal immigration. First, there is a cost associated with the employment
of an undocumented immigrant. If caught employing an illegal immigrant, an employer
must pay a fine to the government. We denote the expected value of such a fine by γ (=
the probability of being caught employing an illegal immigrant times the fine). Second,
illegal immigrants do not save in terms of assets located in the host country. Instead,
they send all their savings abroad.
This is a private ownership economy, where all resources (besides foreign labor of
course) and firms are owned by households.2 Obviously, the CRS property of the production function allows us to assume that there is only one firm, whose shares are held by
consumers. We also assume that all markets are competitive.
The income (y) of each domestic citizen consists of his capital income (rk), where r is
the real interest rate, his labor income (w) and government transfers (τ ).3 Thus,
y = rk + w + τ .
(2.3)
Furthermore, each domestic citizen saves a constant fraction s ∈ (0, 1) of his income (y).
Thus, savings per domestic citizen are sy.
2
Later on, when we study the open economy version of the model, foreign households (but not illegal
immigrants) are allowed to own domestic firms and capital.
3
Note that in general y 6= Y /L; the former denotes the income of each domestic citizen, while the
latter is total output produced per domestic citizen. In other words, Y includes immigrant’s income. The
two variables are equal to each other only when there is no immigration (M = 0).
3
The representative firm maximizes its profit
Π = F (K, L + M) − rK − wL − wm M − γM − δK,
(2.4)
where wm is the wage paid to illegal immigrants and δ is the depreciation rate. The
first-order necessary conditions with respect to the three inputs are
F1 =
∂F
= r + δ,
∂K
F2 =
∂F
=w
∂L
F2 =
∂F
= wm + γ
∂M
(2.5)
Notice that illegal workers are paid a wage (wm ) that is lower than the wage rate w
(= F2 ) paid to domestic workers by the amount of the fine; that is, γ = w − wm > 0.
Put differently, since markets are competitive and each employer faces a perfectly elastic
supply of labor, immigrants bear the entire fine. This fine γ measures also the degree of
"exploitation" or the magnitude of the underpayment of immigrants, since their marginal
product is F2 = w.
The government raises revenue (R) by imposing fines on firms that employ undocu-
mented immigrants. As mentioned above, the expected value of the fine, which equals
the probability that the firm is caught employing an illegal immigrant times the value of
the fine, is γ. Hence, the government raises total revenue equal to
R = γM.
We assume that this revenue is distributed back to the households in a lump-sum manner
so that at the end of each period the government budget is balanced.4
2.2. Equilibrium Balanced Growth Path
A balanced government budget requires that revenue equals transfers, which implies that
the transfer received by each household is
τ=
R
M
=γ
= γm.
L
L
(2.6)
Furthermore, in equilibrium savings is equal to investment. Measuring both variables per
domestic citizen, we have
sy = k̇ + δk.
4
(2.7)
In reality, this transfer should be viewed more as a reduction in the fiscal burden of domestic citizens,
which a priori is taken as given.
4
Combining the CRS property of F , (2.5) and (2.6) yields F (k, 1+m) = F1 k +F2 (1+m) =
(r + δ)k + (1 + m)w = y + δk + m(w − γ). Thus, (2.7) becomes
k̇ = sy(k, m) − δk, where y(k, m) = F (k, 1 + m) − m[F2 (k, 1 + m) − γ] − δk
(2.8)
Equation (2.8), a differential equation in k, describes the dynamic behavior of the economy.
In steady-state equilibrium k̇ = 0 so that
sy(k∗ , m) = δk∗ .
(2.9)
To ensure the existence, uniqueness and stability of a positive steady-state capital stock
equilibrium, we make the following assumptions regarding the function y(k, m) (these
assumptions are analogous to those made in the standard Solow model). First, y is strictly
increasing and strictly concave in k, which in turn requires that ∂y/∂k = F1 −mF21 −δ > 0
and ∂ 2 y/∂k2 = F11 −mF211 < 0 ∀k ≥ 0.5 Second, y(0, m) ≥ 0. Third, limk→0 ∂y/∂k > δ/s
and limk→∞ ∂y/∂k < δ/s.6
2.3. Changes in Wealth, Income and Consumption
Totally differentiating (2.9) yields
dk∗
=
dm
∂y ∗
∂m
∂y∗
δ
−
s
∂k∗
>0
(2.10)
since ∂y ∗ /∂m = γ − mF22 > 0 and (δ/s) > ∂y ∗ /∂k∗ = F1 − mF21 − δ (from the conditions
for existence, uniqueness and stability imposed above).
Since all steady-state variables remain in fixed proportion, that is, y ∗ = (δ/s)k∗ ,
c∗ = [(1 − s)δ/s]k∗ , the effects of immigration on income and consumption, y ∗ and c∗ , are
proportional to that on capital:
dy ∗
δ dk∗
=
> 0,
dm
s dm
dc∗
(1 − s)δ dk∗
=
> 0.
dm
s
dm
Therefore, an increase in immigration raises domestic wealth, income and consumption.
This occurs for two reasons. First, an increase in immigration generates more revenue
5
Obviously, the aforementioned properties of the production function ensure that these conditions
hold for small values of m, that is, these conditions hold automatically in the neighborhood of m = 0.
6
As we show in Appendix A, all of these conditions hold at least for the family of production functions
that exhibit a constant elasticity of substitution (CES).
5
through the exploitation of new immigrants (this is captured by the term γ in ∂y ∗ /∂m;
see equation 2.10). Second, the increase in immigration lowers the domestic wage and,
thus, the wage paid to the existing immigrants (this is captured by the term −mF22 > 0
in ∂y ∗ /∂m). If γ = 0, then there is no exploitation and the first effect disappears. In
addition, if initially there is no immigration (that is, if we are examining an incremental
change in immigration starting from m = 0), then the second effect disappears as well.
These two special cases not withstanding, we can conclude that, in a framework where
there is only one type of labor, illegal immigration is in the long-run beneficial to domestic
citizens because it increases their steady-state level of consumption.
We can also analyze the response of capital to an increase in m during the transition
to the new steady-state equilibrium. The dynamic behavior of k is described by equation
(2.8). Given our assumptions on the function y(k, m), we have
∂ k̇
∂y
= s(F1 − mF22 − δ) − δ = s
− δ,
∂k
∂k
∂ 2 k̇
∂2y
=
s
< 0.
∂k2
∂k2
The curve k̇(k, 1 + m) is shown in Figure 1. Differentiating (2.8) yields
∂ k̇
= s [γ − mF22 ] > 0.
(2.11)
∂m
In other words, the locus k̇(k, 1 + m) curve shifts upwards in a manner similar to the one
shown in Figure 1. Moreover,
∂ k̇ dk
dk̇
=
=
k̈ =
dt
∂k dt
µ
¶
∂y
s
− δ k̇ < 0,
∂k
since s(∂y/∂k) − δ < 0 in the neighborhood of k∗ and k̇ > 0 for k < k∗ . Thus, we
can deduce the dynamic adjustment of k (see Figure 2). The paths of consumption and
income are similar. We conclude from this section that in a homogeneous society in which
domestic and foreign labor are perfect substitutes, illegal immigration is unambiguously
beneficial to domestic citizens because it raises their consumption level both in the steady
state and throughout the entire transition.
3. A Model of Immigration with Heterogeneous Workers
3.1. The Model
Next consider a competitive economy whose main difference from the one analyzed above
is that it is populated by two types of workers, L1 and L2 , say unskilled and skilled. Let
6
the relative size of L1 be φ; that is, φ ≡ L1 /L, where L = L1 + L2 . All agents have the
same saving rate (s). Also, all members of each group have the same income (yi ) as well
as the same wealth (ki = Ki /Li , i = 1, 2, where Ki is the total capital stock owned by
group i).
We assume that all foreign labor is unskilled. Moreover, it is a perfect substitute for
the domestic unskilled labor (L1 ). Hence, the production function is written as
(3.1)
Y = F (K, L1 + M, L2 ).
where K = K1 +K2 . The function F (.) is still assumed to exhibit constant returns to scale
and to satisfy standard Inada conditions with respect to all three inputs. Furthermore,
Fi > 0, Fii < 0, and Fij > 0 where i 6= j, i, j = K, L, M.7 To highlight the fact that
L1 is unskilled, we impose the following restriction for the relevant range of L1 and L2
throughout the paper:
F3 (K, L1 , L2 ) > F2 (K, L1 , L2 ) ∀K.
(3.2)
As before, using the CRS property, F is written as
Y /L = F (K, L1 + M, L2 )/L = F (k, φ + m, 1 − φ)
(3.3)
where k is the (weighted) average capital stock (k ≡ K/L = φk1 + (1 − φ)k2 ) and m
denotes again the immigration ratio (m ≡ M/L).
Let w1 and w2 represent the wage rate of unskilled and skilled labor, respectively.
Each domestic worker of type i saves syi , i = 1, 2, where
(3.4)
yi = rki + wi + τ i .
The representative firm maximizes its profit
Π = F (K, L1 + M, L2 ) − rK − w1 L1 − w2 L2 − wm M − γM − δK.
The first-order necessary conditions are
F1 =
∂F
= r + δ,
∂K
F2 =
∂F
= w1 ,
∂L1
F2 =
7
∂F
= wm + γ,
∂M
F3 =
∂F
= w2 .
∂L2
(3.5)
In the case of two inputs, the constant returns to scale property implies a positive cross-partial
derivative Fij > 0, i 6= j. On the contrary, if there are three inputs, then not all cross-partial derivatives
Fij need to be positive. Nevertheless, we assume so based on the empirical work of Gang and Rivera-Batiz
(1994), who find among others that education, unskilled labor and experience are complementary inputs;
that is, an increase in the supply of one input raises the price of the other.
7
Hence, our assumption (3.2) implies w2 > w1 .
Finally, the government behaves as before; it collects fines R = γM that are distributed
equally to the households in a lump-sum manner every period.
3.2. Steady-State Equilibrium and the Average Household
Once again, the balanced budget requirement implies,
τi =
R
= γm,
L
i = 1, 2.
(3.6)
Notice that, since the total revenue is distributed equally among all households, each one
of them receives the same transfer. Hence, from now on, we drop the index i from the
transfer τ .
The per capita asset accumulation by each group is described by
k̇i = syi − δki , where yi = rki + wi + γm
(3.7)
Next consider the "average household" with wealth equal to the average capital stock
of the economy k = φk1 + (1 − φ)k2 . Its income and consumption are defined similarly as
y = φy1 + (1 − φ)y2 and c = φc1 + (1 − φ)c2 . Moreover, the dynamics of k are described
.
by k = φk̇1 + (1 − φ) k̇2 . Using equation (3.7) and the fact that F (k, φ + m, 1 − φ) =
(r + δ) k + w1 (φ + m) + w2 (1 − φ) (CRS), we have
.
k = sy(k, m) − δk, where y(k, m) = F (k, φ + m, 1 − φ) − m[F2 (k, φ + m, 1 − φ) − γ] − δk.
(3.8)
.
In steady state k = 0, so that
∗
∗
sy(k , m) = δk ,
(3.9)
∗
where k is the steady-state average capital stock. One can notice at this point the
similarity between (3.8) and (2.8), as well as between (3.9) and (2.9). Hence, we can
expect all the conclusions reached in the previous section to hold also for the average
household in this heterogenous society. Let us show this result more formally by analyzing
∗
the determination of k .
First, to ensure existence, uniqueness and stability of a positive steady state level
∗
of capital stock in this economy as well, we impose conditions on the function y(k , m)
8
similar to those imposed in the one-type-of-labor economy. These conditions imply among
∗
∗
others that in steady state δ/s > F1 (k , φ + m, 1 − φ) − mF21 (k , φ + m, 1 − φ) − δ.
Second, equation (3.9) determines the equilibrium steady-state average capital stock,
∗
k . Notice that both the average and the aggregate capital accumulation are independent
of the distribution of wealth, that is, of the relative distribution of k among the two
groups. This is a result of the assumption that the saving rates of the two groups are the
same.8
Finally, we analyze the changes in the average income and wealth that are induced by
immigration. Differentiating (3.9) yields
∗
dk
=
dm
∂y ∗
∂m
∂y∗
δ
−
∗
s
∂k
(3.10)
> 0,
∗
∗
∗
where ∂y ∗ /∂m = γ − mF22 (k , φ + m, 1 − φ) > 0 and (δ/s) > ∂y ∗ /∂k = F1 (k , φ +
∗
m, 1 − φ) − δ − mF21 (k , φ + m, 1 − φ) (once again notice the similarity between (2.10)
and (3.10)). An increase in the immigration ratio leads to an increase in the steadystate average capital stock. Furthermore, there is an increase in average income and
∗
∗
consumption, y ∗ = (δ/s)k and c∗ = [(1 − s)δ/s]k . That is, as expected, the changes in
the steady-state levels of the average variables are similar to those in the economy with
one type of labor that was analyzed in the previous section. Furthermore, using equation
(3.7), we can also analyze the response of k to an increase in m during the transition.
In fact, once again, the analysis is identical to the one performed for k in the case of
only one type of labor [see equation (2.11) and Figures 1 and 2]. In other words, even in
the presence of heterogeneous agents, the economy gains on average from an increase in
immigration.
3.3. Movements in the Distribution of Wealth
∗
After having determined k , we can set k̇i = 0 in (3.7) and find the asset holdings of each
group:
ki∗ =
wi + γm
(δ/s) − r
i = 1, 2,
(3.11)
where all variables on the RHS of (3.11) are to be evaluated at their steady-state levels.
Note that δ is the rate of investment required to keep the per capita steady-state capital
8
Details on when this property is present in models where there exist heterogeneous agents can be
found in papers that focus on the distribution of wealth (see, among others, Stiglitz (1969) and Chatterjee
(1994)).
9
of each group ki∗ constant, while sr is the rate of savings out of capital income. Since
savings arises from all sources of income, it follows that for ki∗ to be positive it must be
the case that in steady state δ/s > r = F1 − δ (notice that this condition is stronger than
the stability condition δ/s > F1 − δ − mF21 in steady state, since F21 > 0).
Next we look at the changes in asset holdings of each group. In steady state we have
∗
syi∗ (k (m), ki∗ , m) = δki∗
where
yi∗
∗
(3.12)
i = 1, 2,
h
i
∗
∗
= F1 (k , φ + m, 1 − φ) − δ ki∗ + Fi+1 (k , φ + m, 1 − φ) + γm,
and k (m) is determined by (3.9). Differentiating (3.12) yields
dki∗
=
dm
∂yi∗ dk∗
∂y ∗
+ ∂mi
∗
∂k dm
∂y∗
δ
− ∂ki∗
s
i
(3.13)
,
where
∂yi∗
∗
∗ = F11 ki + F1i+1
∂k
∂yi∗
= F12 ki∗ + F2i+1 + γ
∂m
∂yi∗
= F1 − δ = r∗
i = 1, 2,
∂ki∗
∗
and the term dk /dm is given by (3.10). While the denominator in (3.13) is positive,
∗
the sign of the numerator is ambiguous because of the conflicting effects of k and m on
different sources of income. More specifically, an increase in m has a direct and an indirect
∗
(through k ) effect on the asset holdings of each group. None of these two effects can be
signed. An increase in m lowers the marginal product of unskilled labor (w1 ) but raises
the marginal products of capital and skilled labor (equal to r + δ and w2 respectively).
∗
On the other hand, an increase in k , following an increase in m, lowers the marginal
product of k (r + δ) but raises the marginal products of both types of labor (w1 and w2 ).
Thus, all sources of income (capital income, labor income and transfers) of both groups
are affected and some of these changes tend to move income in opposite directions. Hence,
the overall effect of an increase in immigration on the wealth of each group is ambiguous.
Note, however, that since the average wealth, income and consumption go up, it follows
that the corresponding variables for at least one of the two groups will also go up. We
just cannot identify the group whose welfare increases without being more specific about
10
some of the data of this economy (e.g., initial wealth of each group, relative size and the
degree of complementarity between each type of labor and capital).
To gain some insight, we run a calibration on the effects of illegal immigration on
wealth distribution. The benchmark parameterization is as follows: s = 25%, δ = 4%,
and φ = 0.5. The production technology is taken to be Cobb-Douglas with equal factor
shares, i.e., F (K, L1 + M, L2 ) = AK α (L1 + M)α L1−2α
where A = 1 and α = 1/3. For
2
the penalty, we assume it to be 30% of the initial equilibrium wage so that γ = 0.25.
Under this parameterization, in the absence of illegal immigration where m = 0, we have
∗
∗
k1∗ = k2∗ = k . We summarize the changes in (k1∗ , k2∗ , k ) due to a change in m in Figure 3.
First of all, the figure confirms the finding that the average capital holdings of the domestic
∗
workers (k ) increase with illegal immigration (equation 3.10). Moreover, as m increases,
the steady-state asset holdings of skilled workers (k2∗ ) rise while that of the unskilled (k1∗ )
∗
fall. As a result, the relative asset position of the unskilled worsens (k1∗ /k falls), while
∗
that of the skilled workers improves (k2∗ /k rises). Consequently, as we show below, the
steady-state wealth distribution becomes more unequal when m increases.
Before closing this section, we provide a discussion on the evolution of wealth distribution during the transition to the steady state. Consider first the following definition of
Lorenz ordering, which is borrowed from Chatterjee and Ravicumar (1999), p. 489. Let
the share of total capital held by household j at time t be ztj ≡ ktj /Lkt . The household
distribution of capital shares is the vector zt = (zt1 , zt2 , . . . , ztL ).
Definition 3.1. Arrange all L domestic households in order of increasing stock of capital.
Let zt and z0t be two different distributions of capital shares. Then, zt is said to be Lorenz
P
P
superior to (or to Lorenz dominate) z0t if lj=1 zt0j ≤ lj=1 ztj for all 1 ≤ l ≤ L and t, with
the inequality holding strictly for at least one l.
This definition is particularly useful because given two distributions any scalar measure
of inequality (such as the Gini coefficient, the coefficient of variation and the standard
deviation of the logarithms) will assign a lower inequality index to the distribution that is
Lorenz superior. For the case of the Gini coefficient and the particular problem analyzed
here, this result is shown in Appendix B.
According to assumption (3.2), the wage rate of unskilled labor is lower than that
of skilled, that is w1t < w2t . Assuming that the economy is initially in a steady state,
equations (3.11) and (3.7) imply that k1t < k2t ∀t. Thus, Definition 3.1 enables us
11
to analyze the impact of immigration on the distribution of wealth by looking at the
evolution of k1t /kt ; in particular, if an increase in immigration lowers k1t /kt , then it also
makes the distribution of wealth more unequal (see also Appendix B). From (3.7) and
(3.8), we can compute the following time derivative of the relative share kit /kt :
¡
¢
∙ µ
¶
µ
¶¸
d kit /kt
wit kit
kit
s
w̄t
+ γm 1 −
.
=
−
dt
wt
kt
kt
kt
In general, it is not possible to get definite results analytically and so we have to rely
on calibration to get further insight. We follow our baseline parameterization and plot
the time path of kit /kt in Figure 4. Two points are worth mentioning from the calibration. First, given the level of illegal immigration m, the wealth distribution becomes
more unequal along the transition k1t /kt (k2t /kt ) falls (rises). Second, when more illegal
immigrants enter the domestic economy, the time path of k1t /k t (k2t /kt ) shifts down (up)
so that the wealth distribution deteriorates over time.
4. The Small Open Economy Case: An Analytical Example
4.1. Steady-State Effects
To isolate some of the conflicting effects that we encountered above, we study next the
effects of immigration within a small open economy context. Thus, henceforth, the domestic interest rate is taken as given and equal to the world level re. As a result, we
have
F1 (e
k, φ1 + m, φ2 ) = re + δ.
(4.1)
e
e denotes the aggregate capital stock invested within the country.
where e
k = K/L
and K
Note that except in the steady state, in the case of an open economy, a country’s capital
stock does not necessarily coincide with the capital stock owned by its citizens, that
is, e
k 6= k.9 The steady-state equilibrium (e
k∗ , k1∗ , k2∗ ) is now given by (4.1) and the two
equations that determine individual asset holdings, either (3.11) or (3.12). Differentiating
(4.1) yields
de
k∗ /dm = −F12 /F11 > 0.
(4.2)
That is, since labor and capital are complementary inputs, an increase in immigration
raises the marginal product of capital, which in turn induces an inflow of capital up to
the point where the equality F1 = re + δ is restored.
9
e∗ = k∗ .
In Appendix C, we show that k
12
Next, from (3.12), we get
dki∗
=
dm
∂yi∗ dh
∂yi∗
k∗
+
∗
∂m
∂h
k dm
∂yi∗
δ
− ∂k∗
s
i
(4.3)
,
where
∂yi∗
= F1i+1 > 0
∂e
k∗
∂yi∗
= F2i+1 + γ
∂m
∂yi∗
= F1 − δ = re > 0
∂ki∗
i = 1, 2,
and de
k∗ /dm is given by (4.2). Notice that the impact of an increase in capital per
domestic citizen e
k on agents’s income, yi∗ , i = 1, 2, is now unambiguously positive. Thus,
an increase in immigration raises the wealth of skilled labor because their income from
all three sources (capital, labor and transfers) goes up. Indeed,
dk2∗
F11 F23 − F12 F13 + γF11
=
> 0.
dm
F11 ( δs − re)
(4.4)
As for the wealth of unskilled labor, we see that ∂y1∗ /∂m and hence dk1∗ /dm continue
to be ambiguous, since on the one hand the transfer that they receive goes up, but on
the other their labor income goes down (∂w1 /∂m < 0). Nevertheless, if the fine imposed
by the government and hence the degree of exploitation is small, then unskilled workers
experience a loss from immigration, since the second effect dominates on the first. Indeed,
using (4.3), we find
2
dk1∗
+ γF11
F11 F22 − F12
=
.
δ
dm
F11 ( s − re)
(4.5)
If γ = 0, then we have dk1∗ /dm < 0. In any case, we have as before that the country
on average gains from illegal immigration. To see this consider the average capital stock
k = φk1 +(1 − φ) k2 . Using (4.3) and the homogeneity property of factor demands derived
from a CRS production function, we have:
¸
∙
∗
2
dk
1
F11 F22 − F12
> 0.
=
γ−m
dm
(δ/s) − re
F11
Next, using (4.5) and (4.4), we get
2
F11 F23 − F12 F13 − (F11 F22 − F12
)
d(k2∗ − k1∗ )
=
> 0.
δ
dm
F11 ( s − re)
13
(4.6)
Thus, even if k1∗ increases, the gap between the steady-state asset holdings of the two
groups is amplified. This is also in accord with our numerical result depicted in Figure 3
of the previous section.
Finally, we study the impact of immigration on the steady-state distribution of wealth
and consequently on the steady-state distributions of income and consumption. From
(3.2), we have w1t < w2t . It follows then from equation (3.11) that k1∗ < k2∗ . Hence, from
the definition of Lorenz superiority we have that if an increase in immigration lowers
∗
k1∗ /k , then it also makes the distribution of wealth more unequal. Using (4.3) and (4.6),
we find that
¶
µ
¶¾
µ ∗¶
½ µ
2
d
1
1
F11 F22 − F12
k1∗
k1
k1∗
= ∗
1+m ∗
.
γ 1− ∗ +
dm k∗
F11
k̄
k (δ/s) − re
k
(4.7)
There are still two opposing effects on the distribution of wealth. A sufficient condition for
an increase in inequality, following an increase in immigration, is γ = 0, since in this case
∗
k1∗ /k decreases (see equation (4.7)). Thus, if there is no penalty then as we saw before
the wealth of unskilled labor goes down, while that of skilled labor goes up. Hence, the
relative wealth position of the unskilled workers deteriorates. Finally, since yi∗ = (δ/s)ki∗
and c∗i = (1 − s)yi∗ , the same conclusion holds for the steady-state distributions of income
and consumption.
4.2. Transitional Dynamics
We next study the effects of immigration on income distribution during the transition.
From (3.7) and (4.1), we have
h
i
k̇i = (se
r − δ)ki + s Fi+1 (e
k, φ + m, 1 − φ) + γm .
Solving the differential equation (4.8), we get
∙
¸
s (wi + γm) (shr−δ)t s (wi + γm)
+
e
.
kit = ki0 −
δ − se
r
δ − se
r
(4.8)
(4.9)
Thus, the effects of immigration on the individual asset holdings can be derived from
(4.9):
¤µ
£
¶
2
s 1 − e(shr−δ)t
F11 F22 − F12
dk1t
=
γ+
dm
δ − se
r
F11
¤
£
¶
µ
s 1 − e(shr−δ)t
dk2t
F11 F23 − F12 F13
> 0.
=
γ+
dm
δ − se
r
F11
14
(4.10)
(4.11)
When more immigrants enter the domestic economy, the capital holdings of the domestic
skilled workers go up in every period. However, the net effect on the capital holdings of
the domestic unskilled workers is ambiguous. The reason is the one that we saw before:
on the one hand, more immigration yields more transfers, but on the other it lowers the
real wage of the unskilled. Combining (4.10) and (4.11), we can derive the effect on the
average individual asset holdings:
¤∙
£
¸
2
s 1 − e(shr−δ)t
dkt
)
m (F11 F22 − F12
> 0.
=
γ−
dm
δ − se
r
F11
(4.12)
As more migrants enter the domestic economy, the average capital holdings of domestic
workers go up in every period.
Next, suppose k20 ≥ k10 so that the skilled workers have relatively higher initial asset
holdings than the unskilled. To compare the individual asset holdings, we first compute
the difference based upon (4.9):
¤
£
s 1 − e(shr−δ)t
k2t − k1t = k20 − k10 +
(4.13)
(w2 − w1 ) > 0
δ − se
r
¤
£
by (3.2). As 1 − e(shr−δ)t is increasing in t, the gap of the asset holdings between the
skilled and the unskilled widens over time. This prediction from (4.13) is consistent with
our calibration results in the previous section. Also, since
¤
£
2
s 1 − e(shr−δ)t F11 F23 − F12 F13 − (F11 F22 − F12
d (k2t − k1t )
)
=
> 0,
dm
δ − se
r
F11
(4.14)
we can also conclude that the gap of the asset holdings between the skilled and the
unskilled broadens over time following a one-time increase in m. This is also in accord
with our numerical findings in the previous section. What remains to be studied is the
effect on wealth distribution resulting from an exogenous change in m. As concluded in
the previous section, the distributional effects are in general ambiguous. However, in the
small open economy case, we are able to provide a sufficient condition in terms of an
upper bound on the penalty so that the wealth distribution deteriorates as the number of
migrants increases. We state our finding in the following proposition.
Proposition 4.1. Define a critical level of the fine γ c ≡ −
2
F11 F22 −F12
F11
> 0. Then for any
level of the fine γ ≤ γ c , the wealth distribution of the domestic economy worsens both in
transition and in the steady state, as the immigration ratio increases.
15
Proof: From (4.3) and (4.10), it follows that dk1∗ /dm < 0 and dk1t /dm < 0 for γ ≤
´
³
∗
γ c . These together with (4.6) and (4.12) yield the results that d k1∗ /k /dm < 0 and
¡
¢
d k1t /kt /dm < 0. Given that k1t /k t < k2t /kt , the result follows.
The intuition of the proposition is straightforward. If the compensation from the
transfer of the penalty revenue is not high enough to cover the loss due to the lower
real wage of the unskilled workers, then they will definitely be worse off. Given that
the skilled gain unambiguously and the unskilled is the relatively poor group, the wealth
distribution of the domestic economy must deteriorate over time. However, if the penalty
is high enough to compensate the domestic unskilled workers for their loss, then we cannot
obtain a conclusion analytically regarding the wealth distribution. For this reason, we
provide a numerical example which assesses quantitatively the magnitude of the critical
level of the fine.
Example 4.1.
Consider a parameterized example with the following Cobb-Douglas
production technology:
F (e
k, φ + m, 1 − φ) = Ae
k α (φ + m)β (1 − φ)1−α−β .
Notice that the restriction (3.2) that w1 < w2 is now equivalent to β (1 − φ) / (φ + m) <
1 − α − β, or that the relative size of immigration is sufficiently high:
w1 < w2 ⇔
β − φ (1 − α)
< m,
1−α−β
which under our baseline parameterization it becomes 0 < m.
Also, the critical level of the fine becomes
γc ≡ −
2
β−2(1−α)
1−α−β
α
α
1
F11 F22 − F12
(1 − α − β) β α−1
α A 1−α (φ + m) 1−α (1 − φ) 1−α (r̃ + δ) α−1 .
=
F11
1−α
(4.15)
Next, we impose the restriction that the fine is proportional to the unskilled wage, i.e.,
γ = θw1 . Combining this with (4.15), we have
γ ≤ γ c ⇔ θ ≤ θc ≡
1−α−β
.
(1 − α) φ + βm
Under our baseline parameterization, we have θc = 1. So the wealth distribution of the
domestic economy will deteriorate if the fine imposed on a firm employing an illegal immigrant is less than one period’s labor income of a home unskilled worker. To examine the
robustness of the distributional effects, we further perturb the baseline parameterization
16
(except m) within a ±20% range. The resulting θc then ranges from 0.80 to 1.25. In addition, with the baseline parameterization, as illegal immigration increases in the domestic
economy (from m = 0 to m = 0.5), θc falls from 1 to 0.67.10 Of course, in practice θ < 1,
since any value of θ higher than 1 will result in a negative wage for the immigrants [recall
that wm = w1 − γ = (1 − θ)w1 ]. Thus, our results suggest that for the baseline parameterization, an increase in immigration leads to a more unequal distribution of wealth. More
generally, we can say that unless the penalty is set above 2/3 of the unskilled wage, the
wealth distribution of the domestic economy will become more unequal as immigration
increases.11
5. Conclusions
In this paper, we have examined the effects of illegal immigration in a neoclassical growth
model with heterogenous workers. Illegal migrants enter domestic production as unskilled
workers which are Edgeworth complements to capital and skilled labor. From the perspective of resource allocation, the economy gains from illegal immigration both in the
steady state and during the transition, since the levels of per capita real income and consumption increase. Nevertheless, the distributional effects of immigration are in general
ambiguous. This is because all sources of income of both skilled and unskilled workers
are affected and some of these changes tend to move income in opposite directions. The
overall effect of an increase in immigration on the asset holdings of each group is also
ambiguous. However, a calibration exercise shows that the wealth distribution becomes
more unequal as the number of immigrants increases.
To gain additional insight on the analytics, we have also studied a small open economy
version of the basic model. With the real interest rate given at the world level, we can
show that the asset holdings of the skilled must rise while that of the unskilled can go on
either direction. Nevertheless, if the compensation from the transfer of the fine on illegal
immigration is not high enough to cover the loss of the domestic unskilled workers, due
10
In the U.S., employers who employ illegal immigrants are fined from $275 to $11,000 per employee,
depending on whether the violation is a first or a subsequent offense. However, the number of notices of
intent to fine to employers has declined from 417 in 1999 to 3 in 2004 (U.S. Government Accountability
Office August 2005). Also, the number of employers paying fines of at least $5,000 for hiring illegal
immigrants was 15 in 1990, 12 in 1994 and zero in 2004 (U.S. Government Accountability Office August
2005) (Recall that γ is the expected value of the fine).
11
The are two sets of winners in this model: skilled labor and illegal immigrants. A fine transfers
income from illegal immigrants to domestic citizens. A complementary way to narrow the resulting
wealth distribution is to tax skilled labor and compensate the unskilled.
17
to their lower real wage, then the wealth distribution of the domestic economy worsens
both in steady state and throughout the transition.
18
References
[1] Bond, E. W., Chen, T.-J., 1987, "The welfare effects of illegal immigration," Journal
of International Economics 14, 315-328.
[2] Chatterjee, Satyajit, 1994, "Transitional dynamics and the distribution of wealth in
a neoclassical growth model," Journal of Public Economics 54 (1), 97-119.
[3] Chatterjee, Satyajit and B. Ravicumar, 1999, "Minimum consumption requirements:
theoretical and quantitative implications for growth and distribution," Macroeconomic Dynamics 3, 482-505.
[4] Djajić, S., 1987, "Illegal aliens, unemployment and immigration policy," Journal of
Development Economics 25, 235-249.
[5] Djajić, S., 1999, "Dynamics of immigration control," Journal of Population Economics 12 (1), 45-61.
[6] Ethier, W. J., 1986, "Illegal immigration," American Economic Review 76, 258-262.
[7] Gang, Ira N. and Francisco L. Rivera-Batiz, 1994, "The labor market effects of
immigration in the United States and Europe: substitution vs. complementarity,"
Journal of Population Economics 7 (2), 157-175.
[8] Hazari, B. R. and P. M. Sgro, 2000, "Illegal migration, border enforcement and
growth," Review of Development Economics 4 (3), 258-267.
[9] Hazari, B. R. and P. M. Sgro, 2003, "The simple analytics of optimal growth with
illegal migrants," Journal of Economic Dynamics and Control 28 , 141-151.
[10] Kondoh, K., 2004. International immigration and economic welfare in an efficiency
wage model: the co-existence case of both legal and illegal foreign workers. Pacific
Economic Review 9 (1), 1-12.
[11] Moy, H. M. and C. K. Yip, 2006, "The simple analytics of optimal growth with
illegal migrants: A clarification," Journal of Economic Dynamics and Control 30,
2469-2475.
[12] Stiglitz, Joseph E., 1969, "Distribution of income and wealth among individuals,"
Econometrica 37 (3), 382-97.
[13] United Nations, 2004, World economic and social survey 2004, United Nations Department of Economic and Social Affairs, E/2004/75/Rev.1/Add.1.
[14] United States Accountability Office, 2005, "Immigration Enforcement: Preliminary
Observations on Employment Verification and Worksite Enforcement Efforts," June.
[15] United States Accountability Office, 2005, "Immigration Enforcement: Weaknesses
Hinder Employment Verification and Worksite Enforcement Efforts," August.
[16] Woodland, A. D. and Chisato Yoshida, 2006, "Risk preferences, immigration policy
and illegal immigration," Journal of Development Economics 81, 500-513.
19
Appendix A
Boundary Conditions on y(k,m)
Suppose the production technology takes the CES form:
¤1/ψ
£
,
F (k, 1 + m) = A ξkψ + (1 − ξ) (1 + m)ψ
ψ < 1.
Then we have
∙
¸
£ ψ
¤
∂y
mβ(1 − ξ)(1 + m)ψ−1
ψ 1/ψ−1
ψ−1
1 − (1 − ψ) ψ
> 0,
= A ξk + (1 − ξ) (1 + m)
ξk
∂k
ξk + (1 − ξ) (1 + m)ψ
∙
¸
βm[ψξkψ + (1 − ψ) (1 − ξ) (1 + m)ψ ]
(ψ − 1) Aξkψ−2 (1 − ξ)(1 + m)ψ
∂2y
1−
=
< 0,
∂k2
[ξkψ + (1 − ξ) (1 + m)ψ ](2ψ−1)/ψ
(1 + m)[ξkψ + (1 − ξ) (1 + m)ψ ]
lim y = 0 if ψ < 0
k→0
and
lim y = A(1 − ξ)1/ψ [1 + (1 − β)m] > 0 if ψ > 0,
k→0
∂y
∂y
and
lim
= Aξ 1/ψ if ψ < 0
= ∞ if ψ > 0,
k→0 ∂k
k→0 ∂k
∂y
∂y
lim
= 0 if ψ < 0
and
lim
= Aξ 1/ψ if ψ > 0.
k→∞ ∂k
k→∞ ∂k
Thus, under appropriate restrictions on the size of A, ξ, ψ, s, and δ all the boundary
lim
conditions imposed in the main text hold.
Appendix B
Lorenz Superiority and The Gini Coefficient
We first recall that the Gini coefficient applied to the distribution of wealth is
1 XX
G=
|ki − kj | .
2L2 k̄ i=1 j=1
L
L
This can also be written as
G=1+
2
1
− 2 (kL + 2kL−1 + ... + Lk1 ) ,
L L k̄
where kL is the capital of the wealthiest household, kL−1 of the next wealthiest, and so
on. Applying this to our case of just two groups with k1 < k2 , we have
2
1
− 2 [k2 + 2k2 + ... + L2 k2 + (L2 + 1)k1 + (L2 + 2)k1 + ... + Lk1 ]
L L k̄ ∙
¸
2 (1 + L2 ) L2
1
(1 + L1 + L) L1
k2 +
k1
= 1+ − 2
L L k̄
2
2
¶
µ
k1
.
= φ 1−
k̄
G = 1+
20
Hence, an increase in k1 /k lowers G. That is, if there is an increase in the relative asset
position of the poorest group then the Gini coefficient will decrease and hence the wealth
distribution will become more equal.
Appendix C
The Small Open Economy:
Using (3.11) we have
µ
∗
e
k∗ = k .
¶
δ
− r k̄∗ = φw1 + (1 − φ) w2 + γm.
s
(C1)
On the other hand, in steady state aggregate savings equal aggregate investment, or
i
h
∗
∗
s F (K̃ , L1 + M, L2 ) − wm M − δ K̃ = δ K̃ ∗ .
Dividing both sides by L and using the homogeneity property of F , we have
i
h
∗
∗
s F1 k̃ + (φ + m) F2 + (1 − φ) F3 − wm m − δk̃ = δ k̃∗
or
µ
¶
δ
− r k̃∗ = φw1 + (1 − φ) w2 + γm.
s
Comparing (C1) and (C2), the result follows.
21
(C2)
k&
k&(k ,1 + m), m > 0
k&(k ,1 + 0)
k*
Figure 1. The Effect of Immigration on Capital Accumulation
k *′
k
kt
k *′
k*
t
Figure 2. Capital Adjustment after an Increase in Immigration
Figure 3
9
k1*
k*
k2*
8.5
8
7.5
k
7
6.5
6
5.5
5
4.5
0
0.05
0.1
0.15
0.2
0.25
m
0.3
0.35
0.4
0.45
0.5
Figure 4
1.25
k1t/kt
k2t/kt
1.2
m=0.3
1.15
m=0.2
1.1
m=0.1
kit/kt
1.05
1
0.95
0.9
m=0.1
0.85
m=0.2
0.8
0.75
m=0.3
0
20
40
60
80
time
100
120
140