Illegal Immigration, Immigration Quotas, and Employer Sanctions
Akira Shimada
Faculty of Economics, Nagasaki University
Abstract
By assuming a small open economy with dual labor markets and efficiency wages, weinvestigate the effects of changes in the immigration quota on the welfare of native workers, legal
immigrants, and illegal immigrants under the different levels of employer sanctions, i.e., fines
imposed by the authorities on the firms that employ illegal immigrants. For this purpose, we
assume that the small open economy, which is faced with a large inflow of skilled and unskilled
immigrants, is endeavoring to accept only a limited number of skilled immigrants by enforcing theimmigration quota and employer sanctions. In this paper, we demonstrate that if fines are large,
increases in the quota are likely to increase the legal worker welfare. However, if fines are small,increases in the quota are likely to decrease it. We also show that the wages and expected
lifetime utilities of illegal immigrants do not depend on the quota and fines. These results
suggest that increasing the quota for skilled immigrants does not necessarily have positive impacts
on native workers and that the policy authority is required to manipulate the immigration quota
and employer sanctions simultaneously in order to increase the native worker welfare.

JEL Classification: F22, F41, J41, J42, J61

Keywords: International migration of labor; Efficiency wages; Dual labor markets; Small open
economy; Illegal immigration; Immigration quotas; Employer sanctions

01. Introduction
This paper investigates the effects of the quota for skilled immigrants on the welfare of native
workers, legal, and illegal immigrants under the different levels of employer sanctions, i.e., fines
imposed by the authorities on the firms that employ illegal immigrants, assuming a small open
economy with dual labor markets and efficiency wages.
Today, many countries are faced with a large inflow of foreign workers.1 Foreign workers
have various impacts on the host countries. It is often argued that the inflow of skilled and legal
immigrants should be encouraged because their inflow seems to have positive impacts on the hostcountries. On the other hand, we tend to argue that the acceptance of unskilled and/or illegal
immigrants should be prohibited since they are likely to have negative impacts on us. Moreover,there are strong arguments to support the implementation of concrete measures in order to control
foreign worker inflow, and many countries are, in fact, attempting to moderate this inflow byutilizing a variety of control methods.
In response to these developments and on the basis of studies on the effects of immigration on
the receiving countries, there has been a rapid proliferation of literatures suggesting effective
methods to control foreign worker inflow.2 Ethier (1986) provided the first formal analysis on
1

See Agiomirgianakis (2006) for recent developments of migration in Europe and related theories.

2

The effects of immigration have been analyzed by many authors by utilizing open economic

models with dual labor markets; this paper implements a similar approach. These analyses canbe divided into two groups.

One group of analyses that include Agiomirgianakis andZervoyianni (2001), Kemnitz (2003), and Shimada (2005) is based on the labor union models thatwere developed by Dunlop (1944), Oswald (1985), and McDonald and Solow (1981). Anothergroup assumes the efficiency wage hypothesis in the tradition of Shapiro and Stiglitz (1984) and
Bulow and Summers (1986). This group comprises researchers such as Carter (1998, 1999) andMüller (1999, 2003a, 2003b). Results regarding the effect of immigration on the receiving
countries are not uniform because these effects differ depending on the manner in which themodel of the receiving countries is formulated. For example, Agiomirgianakis and Zervoyianni(2001) focused on the problem arising from illegal unskilled foreign worker inflow anddemonstrated that its effects depend on the receiving country’s regime. In particular, whenskilled native workers and policymakers do not cooperate, illegal immigration is likely to have1the impact of illegal immigration by examining the effects of border and domestic enforcement
policies. Bond and Chen (1987) extended the analysis of Ethier (1986) by deriving a formula forthe optimal level of enforcement that can be implemented against firms that employ illegal
workers. Carter (2005) explored the effects of deportations, employer sanctions, and amnesties
by assuming an economy wherein undocumented workers are endogenously sorted into thesecondary labor market. Canto and Udwadia (1986) examined the effects of the immigration
quota on the average quality of migrating labor and income distribution.

Myers and

Papageorgiou (2002) made a comparison between the effectiveness of the immigration quota andthe proposed system of immigration tolls and emigration subsidies. Lundborg and Segerstrom(2002) analyzed the effects of the immigration quota on growth and welfare in a North–Southversion of the quality ladder growth model.
In actual economies, many governments usually manipulate multiple control methods at once tocontrol migration flow. However, most of the previous analyses on immigration controls did notadequately address the issues pertaining to the manipulation of the multiple control methods.3 In
particular, they did not pay sufficient attention to the probable relationship between control
methods. Accordingly, these analyses on immigration control are not necessarily acceptable.
The immigration quota and employer sanctions are the most frequently combined control
methods in many countries.4 Both of these considerations influence the decisions taken by thefirms regarding labor demand and wages. Moreover, the manipulations of these methods seemto be closely related through the labor market. This suggests that we cannot independently
determine the amount of the immigration quota from the levels of employer sanctions. However,many governments fail to take the probable relationship between them into account and they are

positive impacts on the entire native labor force. However, when they do cooperate, illegal
immigration may have negative impacts on native workers.
3

Ethier (1986) is one of a few exceptions. He examined the effects of combining borderenforcement policies and domestic enforcement policies.
4

See Robin and Barros (2000) for the actual measures that were implemented by OECD member

countries in order to prevent and combat the illegal employment of foreigners. Certainly, there
are doubts regarding the effectiveness of employer sanctions. See Chiswick (1988) and Boswell
and Straubhaar (2004) for the problems pertaining to employer sanctions.
2

often manipulated independently.5
Therefore, this paper introduces these most frequently combined control methods and aims todemonstrate that the immigration quota and employer sanctions are not independent for the
improvement of native worker and legal immigrant welfare by showing the effects of the changes
in the immigration quota for skilled immigrants on the welfare of legal and illegal workers at
different levels of employer sanctions.
For a country like Japan, which has been imposing fines on firms that employ illegal workers
and is now poised to accept foreign nurses as skilled immigrants by creating immigration quotas,
it is important to clarify the relationship between the abovementioned controls. People areconcerned about whether or not this new policy has positive impacts on the native worker welfare.
For this purpose, we assume a small open economy wherein the labor market possesses a dual
structure and the wages in the primary and secondary labor markets are determined according to
the non-shirk model. A large number of skilled and unskilled immigrants are flowing into thesmall open economy due to the higher wages in the primary and secondary labor markets thanthose in the rest of the world; however, this economy wants to accept only a limited number ofskilled immigrants. For this purpose, it sets up a quota for skilled immigrants and imposes fines
on firms that employ illegal immigrants. Skilled immigrants who meet the immigration quota
and thereby acquire legal status and all native workers can enter the primary labor market,
whereas the skilled immigrants who do not meet the immigration quota and all the unskilled
immigrants cannot acquire legal status and enter the secondary labor market as illegal immigrants.
The primary and secondary labor markets are distinct in the sense that jobs in the former arecharacterized by imperfect observations, whereas jobs in the latter have perfect observations.
They also differ in the sense that the firms run the risk of being detected and punished by theauthorities if they employ workers in the secondary labor market, whereas they have no risk in
employing workers in the primary labor market.
We demonstrate that if the authorities impose large fines on the firms that employ illegal
immigrants, increases in the quota for skilled immigrants are likely to benefit the legal workers,
5

The analysis by Gonzalez (1994) is also in accordance with actual policy-making.

He

introduced the border enforcement and employer sanctions into the same model; however, hemanipulated these control methods independently.
3

i.e., wages and the expected lifetime utilities of native workers and legal immigrants will begreater. This can be explained as follows: An increase in the immigration quota raises the legal
worker accession rate.

Since wages and expected lifetime utilities of legal workers are

increasing with respect to the legal worker accession rate, the larger quota implies higher wages
and higher expected lifetime utilities.
On the other hand, if the fines are small, increases in the quota for skilled immigrants are likely
to decrease wages and the expected lifetime utilities of legal workers. In other words, they mightworsen the legal worker welfare. This is because increases in the immigration quota lower thelegal worker accession rate, leading to lower wages and lower expected lifetime utilities of legal
workers.
We also demonstrate that wages and the expected lifetime utilities of illegal workers are
independent of the immigration quota and fines. This can be explained as follows: As with thelegal worker accession rate, the illegal worker accession rate also depends on migration flow andthe immigration quota. However, wages and the expected lifetime utilities of illegal workers
remain constant because the jobs in the secondary labor market—all of which are performed byillegal workers—are perfectly observable; this renders illegal worker wages and expected lifetime
utilities independent of the migration flow.
These results have the following implications: Increasing the immigration quota in order toaccept a larger number of skilled immigrants does not always impact native workers positively.
Accordingly, it is necessary for the government to manipulate the immigration quota taking into
account the magnitude of the fine that will be imposed on the firms that employ illegal workers.
In particular, if a country wants to accept a large number of skilled immigrants by expanding thequota for them, the government will have to impose larger fines accordingly.
The remainder of this paper is organized as follows:

Section 2 presents a small open

economic model with dual labor markets and the efficiency wages. In this section, we assume
the manner in which the foreign workers flow into the small open economy and how theauthorities penalize firms that employ illegal immigrants. Section 3 examines the effects of theimmigration quota on the welfare of legal and illegal workers, and tries to show that theimmigration quota and fines are not independent. The concluding comments are presented in
section 4.

4

2. The Model
We consider a small open economy with dual labor markets consisting of the primary andsecondary labor markets.6 Jobs in the primary labor market are attractive and require skill;
therefore, the marginal product of labor is high. In such a market, workers are required to beskilled and legal. In contrast, jobs in the secondary labor market are unattractive and do not
require skill; therefore, the marginal product of labor is low. Workers are required to be neither
skilled nor legal in order to enter this market.
A small open economy is connected to the rest of the world through migration. For economic
and non-economic reasons, the small open economy is making an attempt to accept only a limited
number of skilled immigrants by setting up an immigration quota M .
The number of native workers is denoted by N , which is assumed to be a constant. All
these workers are both skilled and legal by definition, and thereby, they participate in the primarylabor market.7 In the secondary labor market, jobs are unattractive to the extent that the native
workers do not want to be employed there. Accordingly, even if they are not employed in the
primary labor market, they do not enter the secondary market.
All immigrants can acquire legal status provided that they are skilled and their numbers do not
exceed the immigration quota. However, if the number of skilled immigrants exceeds the6

There are other models similar to our model. For their analyses on the effects of immigration,

both Carter (1999) and Müller (1999) built a small open economic model with efficiency wages
and dual labor markets. Carter (1999) assumed efficiency wages both in the primary andsecondary labor markets. On the other hand, Müller (1999) considered efficiency wages only inthe primary labor market, and the wages and employment in the secondary labor market were
determined competitively.
7

According to Massy and Taylor (2004), labor markets in developed nations become increasingly

segmented into a primary sector containing the “good” jobs that are attractive to the native
workers and a secondary sector of poorly paid, “bad” jobs that are shunned by native workers.
This explains why firms turn to legal and illegal immigrants. To model such an economy, it
would not be inappropriate to assume that unskilled native workers do not exist in the secondary
labor market. However, unskilled native workers who are considered as perfect substitutes forillegal immigrants are certainly assumed to exist in Agiomirgianaikis and Zervoyianni (2001).
5

immigration quota, only those skilled immigrants who meet the immigration quota can acquire
legal status. The remainder of the skilled immigrants is unable to acquire this status; therefore,
they become illegal immigrants.8
Due to wage differentials, skilled and unskilled workers migrate between a small open
economy and the rest of the world.9 In general, if the wages of skilled and unskilled workers arehigher (lower) in the small open economy as compared to the rest of the world, then skilled andunskilled workers will migrate from the rest of the world (the small open economy) to the small
open economy (the rest of the world). In this paper, we assume that the wages in the rest of theworld, i.e., the wages of skilled and unskilled foreign workers in their home countries—these are
given to the small open economy and remain constant throughout the analysis—are sufficiently
lower than those in the small open economy.

Consequently, the small open economy isconfronted by an inflow of skilled foreign workers that exceeds the immigration quota as well as
by a significant inflow of unskilled foreign workers.
Accordingly, the actual number of immigrants including the skilled and unskilled foreign
workers δ ( w2 − w* ) is always larger than the immigration quota, i.e., δ ( w2 − w* ) > M , andonly M of the skilled immigrants can acquire legal status, where w2 denotes the secondary
labor markets wages in the small open economy, w* denotes higher wages among those ofskilled and unskilled foreign workers in their home countries, and δ , which is a positive
constant, measures the sensitivity of migration flow to the changes in the differentials between

w2 and w* . 10 The remainder of the immigrants, i.e., the skilled immigrants who cannot acquire8

In order to simplify the analysis, it is assumed that legal and illegal immigrants do not tovoluntarily return to their home countries.
9

In this paper, the product price does not change and is assumed to be 1; therefore, we have noneed to make distinctions between the real and nominal wages.
10

Workers from the rest of the world are aware of the fact that, even though they are skilled, all ofthem cannot acquire legal status after migration; hence, some of them are left with no option thanto enter the secondary labor market. Moreover, no skilled foreign worker is likely to be certainof acquiring legal status before migration. Accordingly, skilled immigrants as well as unskilled
ones have to decide whether or not to migrate to the small open economy by comparing secondary
labor market wages in the small open economy with those in their home countries.
6

legal status and all unskilled immigrants, δ ( w2 − w* ) − M , are left with no other option than to
become illegal immigrants.
Therefore, there are N native workers and M legal skilled immigrants in the primary labormarket of a small open economy.11 There are also δ ( w2 − w* ) − M illegal skilled or unskilled
immigrants in the secondary labor market.
Assuming a Cobb-Douglas production technology and considering capital to be fixed, theoutput in a small open economy Y increases with an increase in employment in the primarylabor market, i.e., legal worker employment L1 and employment in the secondary labor market,
i.e., illegal worker employment L2 .
Y = L1a1 La22 , a1 > a2 > 0, a1 + a2 < 1. 12
Firms are perfectly competitive and they demand labor in the primary and secondary labormarkets in such a way as to maximize profits. The native workers and legal skilled immigrants,
who are regarded as perfect substitutes by firms, are employed in the primary labor market to an
extent wherein their marginal product is equal to the primary labor market wages w1.

In thesecondary labor market, similar to Agiomirgianakis and Zervoyianni (2001), firms are aware of
the fact that by employing illegal skilled or unskilled immigrants, they run the risk of being
detected by the authorities, in which case they will be forced to pay a fixed fine f , f > 0, foreach illegal immigrant employed by them; f is a constant and does not change throughout theanalysis.13 We assume that as the number of illegal immigrants increases, firms will be faced
11

We assume that all workers in the primary labor market have identical employment probabilities.

As with native workers, legal immigrants too do not enter the secondary labor market even if they
are not employed in the primary labor market.
The relation a 1 > a 2 reflects the assumptions that the marginal product of labor is high in the

12

primary labor market and low in the secondary labor market.
13

The authorities are fully aware that all the immigrants in the secondary labor market are illegal.

However, in order to penalize firms for employing illegal immigrants, the authorities are required
to prove that the firms are actually employing those immigrants in the secondary labor market.
Since many firms may circumvent rules by adopting strategies such as sub-contracting (Boswell
and Straubhaar 2004), it is not possible for the authorities to detect all the firms that employ illegal
immigrants.
7

with a higher probability of being detected. This assumption reflects the following facts: In
actual economies, we cannot perfectly control all the resources that contribute to tracing the firmsthat employ illegal immigrants.14 However, as the number of illegal immigrants increases, thegovernment tends to invest more resources into locating such firms; consequently, the authorities
are likely to detect them with more efficiency.15 We represent the probability that the authorities
will discover firms employing illegal immigrants by p(δ ( w2 − w* ) − M ), 0 < p < 1, p '> 0.
Accordingly, illegal immigrants are employed to the extent wherein their marginal product isequal to the sum of w2 and the expectation of a fine for employing an illegal immigrantp (δ ( w2 − w* ) − M ) f .

The demand functions for legal and illegal workers are derived asfollows:
1− a 2a2
1− a1 − a 2 1− a1 − a 2
1
2
a1
1− a1
1− a1 − a 2 1− a1 − a 2
12

L1 = aL2 = a

a

a

−

w1

1− a 21− a1 − a 2−{w2 + p(δ ( w2 − w ) − M ) f }
*

1− a1
−
1− a1 − a 2{w2 + p(δ ( w2 − w* ) − M ) f }a2
1− a1 − a 2

.

(1)

a1
−
1− a1 − a 2
1

(2)

w

.

Equations (1) and (2) indicate that legal and illegal workers are gross complements.
Since firms are unable to completely detect shirking by employed workers, they set wages in amanner that prevents shirking by assuming workers’ efforts as given (Shapiro and Stiglitz 1984).
As in Bulow and Summers (1984) and Carter (1999), this paper assumes that a shirker will be
detected and fired at different probabilities in the primary and secondary labor markets. This
comes from the assumption that jobs in the two labor markets are of different types.
The wages and expected lifetime utilities in the primary and secondary labor markets arederived similarly. If a representative employed worker in the primary labor market does not

14

For example, even if the government does not increase the direct expenditure on tracing them,

we will have a larger resource allocation and thereby a higher detection probability if thegovernment spends more on employing police officers. Therefore, this paper does not solve themaximization problem in order to determine the optimal level of the resource devoted to tracing
firms that employ illegal immigrants.
15

Since the number of firms does not change, under our assumption the authorities allocate moreresources for each firm as the number of illegal immigrants increases. This suggests that theauthorities become more efficient in tracing firms that employ illegal immigrants. Accordingly,
the detection probability becomes higher with increases in the number of illegal immigrants.
8

shirk, his instantaneous utility is measured by primary labor market wages minus effort. On theother hand, if he shirks, his instantaneous utility is measured by primary labor market wages.
However, in such a case, a representative employed worker in the primary labor market will be
detected and fired by firms at the probability ρ1.

Moreover, some of the employed workers in

the primary labor market separate from their jobs, even though they are not fired on the grounds ofshirking. The separation rate in the primary labor market, which is defined as the ratio ofseparations due to reasons other than shirking to the number of employed workers in the primarylabor market, is given by β .

The separation rate is assumed to be identical in the primary andsecondary labor markets.16
The expected lifetime utility of a representative employed shirker in the primary labor market

VES1 is,rVES1 = w1 + ( β + ρ1 )(VU1 − VES1 ),(3)

where r is the discount rate and VU 1 is the expected lifetime utility of a representative
unemployed worker in the primary labor market. Equation (3) can be rewritten as,
VES1 =

w1 + ( β + ρ1 )VU 1
r + β + ρ1

.

(3’)

On the other hand, the expected lifetime utility of a representative employed non-shirker in the
primary labor market VEN1 is,rVEN1 = w1 − e + β (VU1 − VEN1 ),(4)

where e is the effort exerted by a representative employed non-shirker in the primary labormarket. We assume that workers in the primary and secondary labor markets exert the samelevel of effort.17 As mentioned already, we also assume that the level of effort is exogenous andremains constant throughout the analysis.18 Equation (4) can be rewritten as,
16

The results of this paper remain unchanged even if the separation rates differ in the primary andsecondary labor markets.
17

Even if the workers in the primary and secondary markets exert different levels of effort, thepaper’s main results remain unchanged as long as the effort is exogenous.
18

The analysis will be more general if the employed workers determine the optimal level of effort

in a manner that will maximize their expected lifetime utility given that the wages are set by firms.
See Shimada (2007) for the determination of the optimal level of effort.
9

VEN1 =

w1 − e + βVU1
r+β

.

(4’)

The employed workers in the primary labor market may or may not shirk based on acomparison of VEN1 and VES1 .

In order to prevent the workers from shirking, firms have to set

primary labor market wages to ensure that VEN1 ≥ VES1 .

However, since there is no reason forfirms to pay more than what is essential to eliminate shirking by employed workers in the primarylabor market, they will set primary labor market wages such that VEN1 = VES1 (≡ VE1 ).

The

following is obtained by substituting Equations (3’) and (4’) into this condition:
w1 = rVU 1 +r + β + ρ1
e.
ρ1

(5)

Moreover, VU 1 is given by,
rVU 1 = w + α1 (VE1 − VU 1 ),(6)

where w is the unemployment benefit, which is a constant, and α1 is the accession rate in the
primary labor market, which is defined as the ratio of new hires in the primary labor market to thenumber of unemployed workers in the primary labor market. The unemployment benefits are
assumed to be identical for both primary and secondary labor markets. In steady state, α1 mustbe such that the flow out of unemployment in the primary labor market is equal to the flow into
unemployment in the primary labor market, i.e., α1 ( N + M − L1 ) = βL1.
In order to simplify the analysis, we assume that there are no unemployment benefits, i.e.,

w = 0 in both labor markets.

Substituting this assumption in Equation (6) and utilizing

Equations (3’), (4’), and (5), the primary labor market wages are derived as follows:
w1 = e +α1 + β + r
e.
ρ1

(7.1)

Equation (7.1) suggests that the primary labor market wages increase with an increase in theaccession rate in that market.
The expected lifetime utility of a representative employed worker in the primary labor market
under the non-shirk condition takes the form of,VE1 =1  α1 1 + e.
ρ1 
r 

(8.1)

According to Equation (8.1), the legal worker expected lifetime utility is also increasing withrespect to their accession rate.
10Based on a similar argument, the wages and the expected lifetime utility of a representative
employed worker in the secondary labor market under the non-shirk condition VE 2 are obtained
as follows;
w2 = e +VE 2 =

α2 + β + r
e,
ρ2

(7.2)

1  α2 1 + e,
r 
ρ2 

(8.2)

where α 2 is the accession rate in the secondary labor market—which is defined as the ratio of
new hires in the secondary labor market to the number of workers unemployed in the secondary
labor market.19 In steady state, α 2 must be such that the flow out of unemployment in thesecondary labor market is equal to the flow into unemployment in the secondary labor market, i.e.,
α 2{δ ( w2 − w* ) − M − L2 } = βL2 .
As assumed earlier, ρ1 and ρ 2 are different. In general, firms cannot easily monitor theefforts that are exerted by the workers on skilled jobs, whereas they can easily monitor efforts
exerted by the workers on unskilled jobs. Accordingly, we assume that jobs in the primary labormarkets have imperfect observations, i.e., ρ1 is finite, whereas jobs in the secondary labormarkets have perfect observations, i.e., ρ 2 is infinite.20
Taking the limit of Equations (7.2) and (8.2) as ρ 2 becomes arbitrarily large, the secondary
labor market wages and the expected lifetime utility of a representative employed worker in thesecondary labor market will be as follows:

w2 = e.
VE 2 = 0.

19(7.2’)(8.2’)

Illegal immigrants in actual economies might be deported to their home countries if theauthorities are able to locate the firms employing them. However, as long as the deportation
probability is a constant, the main results of this paper remain unchanged even if illegal
immigrants might be deported to their home countries.
As formulated by Carter (1999), we define ρ1 , ρ 2 as the probabilities per unit time that a20

shirker in the primary and secondary labor markets will be detected and fired. Therefore, ρ it ,i = 1,2, is the probability that a shirker will be detected and fired during a short interval,[T0 , T0 + t ].

With perfect observation, ρit is equal to 1—even with a very small t —i.e., ρiis infinite.
11

Equations (7.2’) and (8.2’) suggest that even in steady state, the wages and the expected lifetime
utility of a representative employed worker in the secondary labor market are independent of the
migratory flow.

This can be explained as follows:

In steady state, migration affects theaccession rate in the secondary labor market since α 2 = βL2 {δ ( w2 − w* ) − M − L2 }.

However,

the accession rate has no effect on the wages and the expected lifetime utility of a representative
employed worker in such a market since ρ 2 is infinite.
In contrast, the wages and the expected lifetime utility of a representative employed worker inthe primary labor market in steady state are affected by migration. On substituting α1 that
holds in steady state into Equations (7.1) and (8.1), we can solve them as follows:

w1 = e +VE1 ={( N + M ) ( N + M − L1 )}β + re.
ρ11  {L1 ( N + M − L1 )}β 
1+
 e.
ρ1 
r


(7.1’)(8.1’)

Equations (7.1’) and (8.1’) suggest that, in steady state, the wages and the expected lifetime utilityof a representative worker employed in the primary labor market are dependent on the migratory
flow. This can be explained as follows: As in Equation (1), the demand for skilled workers
changes with the number of immigrants. This suggests that the accession rate in the primarylabor market depends on migration indirectly as well as directly. Moreover, this rate affects thewages and the expected lifetime utility in such a market since ρ1 is finite.
3. Effects of the Immigration Quota on Legal and Illegal Workers under the Different
Levels of Employer Sanctions
In this section, we examine how changes in the immigration quota affect the welfare of theworkers in the primary and secondary labor markets when the levels of employer sanctions aredifferent.
For this purpose, we substitute Equation (7.2’) into Equation (1) and totally differentiate theresulting equation and Equation (7.1’). Eliminating dw1 from these equations, the effects of theimmigration quota on legal worker employment are derived as follows:
dL1
dM

β
1 − a2e
L1w1−1L1ρ1 ( N + M − L1 ) 2
1 − a1 − a2=β
1 − a2e1+
L1w1−1(N + M )1 − a1 − a2
ρ1 ( N + M − L1 ) 2
12a2
L1{e + p(δ (e − w* ) − M )}−1 p ' f
1 − a1 − a2+.
1 − a2
e
−1 β
L1w1
1+
(N + M )ρ1 ( N + M − L1 ) 2
1 − a1 − a2

(9.1)

Since both the terms on the right-hand side in Equation (9.1) are positive, increases in theimmigration quota will increase legal worker employment, which is the sum of native worker
employment and legal immigrant employment, i.e., dL1 dM > 0.

This result comes from thetwo effects; the effects on the legal workers’ wages in steady state that satisfy their non-shirk
condition (Equation 7.1’)—these effects are expressed by the first term on the right-hand side in
Equation (9.1)—and the effects on the demand for legal workers (Equation 1), which are
expressed by the second term on the right-hand side in Equation (9.1).
The effects of the former are as follows: For any demand for legal workers L1 , increases in
the immigration quota exacerbates legal worker unemployment N + M − L1 , whereas legal
worker separation due to reasons other than shirking βL1 remains constant. Accordingly, forany demand for legal workers, the accession rate in the primary labor market α1 decreases withincreases in the immigration quota. This causes the expected duration of being unemployed in
the primary labor market 1 α1 to be longer, suggesting that legal workers are more severely
punished for being unemployed. Therefore, for any demand for legal workers, the legal worker
wages that will prevent shirking by them will decrease with increases in the immigration quota.
If w1 is measured on the vertical axis and L1 is measured on the horizontal axis, the curve that
describes Equation (7.1’) is upward-sloping and shifts downwards to the right with increases in
the immigration quota.
The effects of the latter are as follows: Since the illegal worker wages in steady state satisfy
the non-shirk condition (Equation 7.2’), and thereby, the number of immigrants δ (e − w* ) do notdepend on the immigration quota, the number of illegal immigrants decreases with increases in theimmigration quota. This decreases the expected fine and thereby the expected cost that firms
have to pay for employing an illegal immigrant e + p (δ (e − w* ) − M ) f .

Since legal and illegal

workers are gross complements as in Equations (1) and (2), the demand for legal workers
increases with decreases in the expected cost for employing an illegal immigrant. Moreover, assuggested in Equation (1), the demand curve for legal workers is downward-sloping.
Accordingly, it shifts upward to the right with increases in the immigration quota.
These two effects always increase the legal worker employment when the immigration quota
13increases.
However, the effects on native worker employment and legal immigrant employment are notalways the same. For larger values of f , by increasing the immigration quota, the demand
curve for legal workers that describes Equation (1) shifts upward to the right to a larger extent.21
On the other hand, the curve that describes Equation (7.1’) shifts downward to the right to asmaller extent. 22

However, if f is sufficiently large, the former effect dominates, anddL1 dM becomes larger than the employment probability for legal workers L1 ( N + M ). 23 In
such a case, increases in the immigration quota increase the employment probability, i.e.,d{L1 ( N + M )} dM > 0. 24

Therefore, if

f

is sufficiently large, both native worker

employment and legal immigrant employment increase corresponding to an increase in theimmigration quota.
If f is small, a given increase in the immigration quota shifts the demand curve for legal
workers that describes Equation (1) upward to the right to a smaller extent, whereas a given
increase in the immigration quota shifts the curve that describes Equation (7.1’) downward to theright to a larger extent. If f is sufficiently small, the former effects dominate and dL1 dM is21∂ (∂L1 ∂M ) ∂f > 0 and lim ∂ (∂L1 ∂M ) ∂f = 0, where L1 and M satisfy Equation (1).
f →0

22

∂ (∂L1 ∂M ) ∂f < 0 and lim ∂ (∂L1 ∂M ) ∂f = 0, where L1 and M satisfy Equation (7.1’).
f →∞

23

The first term on the right-hand side in Equation (9.1) is smaller than 1 for any values of f .

On the other hand, the partial derivative of the second term on the right-hand side in Equation(9.1) with respect to f is equal to,
1−a2
1−a1 −a21

aa2
1−a1 −a2
2

a

1−a2
−
1−a1 −a21wa2p' (δ (w2 − w* ) − M )1 − a1 − a2
 p(δ ( w2 − w* ) − M ) fa2

×  −
− 1
+ 1.
*
 1 + p(δ ( w2 − w ) − M ) f
 1 − a1 − a2This becomes positive if the marginal product of labor in the secondary labor market is low anda2 is sufficiently small. Accordingly, in such a case, dL1 dM becomes larger for larger
values of f .
24d{L1 ( N + M )} dM = {1 ( N + M ) 2 }{(dL1 dM )( N + M ) − L1}.
14

smaller than L1 ( N + M ); hence, the employment probability for legal workers decreases as theimmigration quota increases.25 Therefore, for sufficiently small values f , the native worker
employment decreases; however, legal immigrant employment increases with increases in theimmigration quota.26
The immigration quota affects the legal worker accession rate as follows:
dα1
β
 dL1
 > 0, if dL1 dM > L1 ( N + M ).
=
(
N
+
M
)
−
L

=
1
dM ( N + M − L1 ) 2  dM
 < 0, if dL1 dM < L1 ( N + M ).
These results have the following interpretations: As shown by Equation (9.1), increases in theimmigration quota increase the legal worker employment.

Moreover, as legal worker

employment increases, legal worker separation due to reasons other than shirking increases andlegal worker unemployment decreases. These indirect effects lead to a larger accession rate.
However, for any demand for legal workers L1 , increases in the immigration quota directly
increase legal worker unemployment. This direct effect leads to a smaller accession rate. If f
is sufficiently large and ∂L1 ∂M is larger than L1 ( N + M ) , then the former indirect effects
dominate, and thereby, ∂α1 ∂M > 0.

On the other hand, if f is sufficiently small and∂L1 ∂M is smaller than L1 ( N + M ) , then the latter direct effect dominates, and thereby,∂α1 ∂M < 0.
These results suggest that in a scenario in which the authorities impose large fines, an increasein the immigration quota results in a higher likelihood for the unemployed legal workers to be
reemployed. However, in a scenario in which the authorities impose small fines, a decrease in
the immigration quota results in a higher likelihood for the unemployed legal workers to be
reemployed.
According to Equations (7.1) and (8.1), the wages and the expected lifetime utility of arepresentative employed worker in the primary labor market are increasing with respect to the

If f = 0, the second term on the right-hand side in Equation (9.1) is equal to 0 and the first25

term is lesser than L1 ( N + M ) .
26

Since the number of native workers does not change, decreases in the employment probability

reduce their employment. However, even if f is small, the sum of native worker employment
and legal immigrant employment increases with increases in the immigration quota as shown inEquation (9.1). Therefore, it is necessary for legal immigrant employment to increase.
15

legal worker accession rate. Therefore, the effects of changes in the immigration quota on thelegal worker wages and expected lifetime utility are derived as follows:

dw1 β
e
 dL1
 > 0, if dL1 dM > L1 ( N + M ).
=( N + M ) − L1  = 2 dM ρ1 ( N + M − L1 )  dM
 < 0, if dL1 dM < L1 ( N + M ).
dVE1
dM

=

(10.1)1 eβ
 dL1
 > 0, if dL1 dM > L1 ( N + M ).
( N + M ) − L1  = (11.1)2 
ρ1 r ( N + M − L1 )  dM
 < 0, if dL1 dM < L1 ( N + M ).

According to Equations (10.1) and (11.1), increases in the immigration quota are likely toincrease legal worker wages and expected lifetime utility if fines are large, whereas increases in
the immigration quota are likely to decrease these two if fines are small.27 Therefore, thechanges in the immigration quota have different effects on legal worker wages and expected
lifetime utilities depending on the values of fines, and the expansion of the immigration quota
does not necessarily lead to the improvement of the legal worker welfare. This suggests that theimmigration quota and employer sanctions are related for the improvement of the legal worker
welfare.
Our results have the following policy implications: The immigration quota and fines cannot
be manipulated independently. If large fines are imposed, the government can increase thenative worker welfare by increasing the immigration quota. On the other hand, if small fines are
imposed, the government will have to decrease the immigration quota in order to increase thenative worker welfare.
As shown in Equations (7.2’) and (8.2’), the immigration quota has no effects on the wages andthe expected lifetime utility of a representative employed worker in the secondary labor market.

dw2
= 0.
dM

(10.2)

27

If fines are large, legal worker employment increases with the immigration quota. Accordingly,

the legal worker welfare is increasing with respect to the immigration quota in terms of theiremployment as well as their wages and expected lifetime utilities. Moreover, it is almost certainthat if fines are small, the legal worker welfare is decreasing with respect to the immigration quota
in terms of their employment as well as their wages and expected lifetime utilities. This isbecause the ratio of native workers among legal workers is large and native worker employment
decreases with the immigration quota.
16

dVE2
dM

= 0.

(11.2)

Equations (10.2) and (11.2) suggest that as long as immigrants are illegal, their wages andexpected lifetime utilities remain unchanged even if the government manipulates the immigration
quota.
To summarize the results derived in this section, the effects of changes in the immigration quota
on the welfare of the native workers and the legal immigrants differ according to the values of thefine. However, the immigration quota and fines have no effects on the illegal immigrant welfare.

4. Conclusions
This paper investigated how changes in the quota for skilled immigrants affect legal and illegal
workers under the different levels of employer sanctions by utilizing a small open economic
model with dual labor markets and the efficiency wages hypothesis.
We demonstrated that the immigration quota cannot be manipulated independently of the levelsof employer sanctions if we try to improve the native worker welfare. Whether the immigration
quota should be increased or decreased depends on the levels of fines. We also showed that thewelfare of illegal workers is independent of the immigration quota and fines.
Our results suggest that expanding the immigration quota in order to accept a larger number ofskilled immigrants does not always improve the welfare of the native workers, which might partly
explain why the countries wishing to accept a large number of skill immigrants do not always
expand their quotas for these workers, and that if an attempt is made to increase the number ofskilled immigrants by raising the immigration quota, it is necessary for the government to impose
larger fines.

Acknowledgements
I am deeply indebted to George M. Agiomirgianakis. I am very thankful to him. This paper
was presented at the 6th annual meeting of the European Economics and Finance Society. I am
also grateful to the participants at the conference. Any shortcomings of this paper are, of course,
my responsibility.

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