Illegal immigrants, crime, and sanctuary cities
Kaz Miyagiwa
Yunyun Wan
July, 2020
Discussion Paper No.2012
Revised August, 2020

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANIllegal immigrants, crime, and sanctuary cities
Kaz Miyagiwa∗and Yunyun Wan†
August 13, 2020

Abstract
In the United States today, more 300 jurisdictions (cities, counties andstates) refuse to deport illegal immigrants even with criminal records inspite federal immigration law. Do such sanctuary jurisdictions necessarily
attract more illegal aliens and as a consequence suffer from higher unemployment rates and more crime? In this paper we formally investigate
these questions and discuss the conditions leading to such results. Examined also are the effects on immigration from raising minimum wages,
enhancing tougher anti-crime policies and providing unemployment benefits to illegal immigrants.
Keywords: Illegal immigration, sanctuary city, frictional unemployment, crime, minimum wage

1

Introduction

In the U.S. there are more than 300 jurisdictions (cities, counties and states)
today which refuse to deport illegal immigrants even with criminal records in∗ Departmentof Economics, Florida International University, U.S.A. E-mail:kmiyagiw@fiu.edu† Graduate School of Economics, Kobe University, Japan. E-mail:jiangyun.wan@gmail.com

1spite of federal immigration law. In this paper we formally analyze the consequences of the decision for a jurisdiction to become a sanctuary - a policy we
dub sanctuarization. Our analysis focuses on local residents’ two main concerns.
Does sanctuarization worsen unemployment? Does it give rise to more crime?
To address these questions, we present a dynamic model in which both natives
and immigrants commit crimes when unemployed. Workers can be unemployed
for two reasons: minimum wage laws and search friction. More specifically, it isassumed that firms employing native workers are bound by the minimum wage
law but that those employing illegal immigrants can evade the law, instead negotiating the wages directly and continuously with their employees; see, e.g.,
Pissarides (2000) and Rogerson, Shimer and Wright (2005).
As for modeling crime, we assume that individuals commit crime only when
the benefit from crime exceeds its cost. Following Burdett et al. (2003), the costof crime consists of the disutility of being arrested, convicted and incarcerated
in prison. For illegal immigrants, there is an additional cost: the risk of being
deported to their home countries. A sanctuary city eliminates this risk, whichtends to increase the net benefit of crime and hence the immigrant’s welfare.
However, this risk elimination does not necessarily cause an influx of illegal
immigrants into a sanctuary city. The reason is that sanctuarization causes
the immigrant’s wage to rise and hence labor demand to fall. If labor demand
falls so much, immigrants move out of the jurisdiction instead of moving in
there despite its decision to become a sanctuary. Thus, sanctuary cities do notnecessarily attract more illegal immigrants and as a consequence suffer morecrime compared with non-sanctuary cities. However, sanctuary cities always
suffer from higher unemployment rates.
In addition to sanctuarization, we also examine the effects of other policies
sanctuary cities may undertake to protect their native residents as well as illegal2

immigrants. The following are our main findings. First, extending unemployment benefits to illegal immigrants enhances the effects of sanctuarization. On
the other hand, enhancing anti-crime policies countervails them. Raising minimum wages to protect native workers does not increase their unemployment
rates and may even increase aggregate employment - a result which may be ofinterest in the line of research investigating the positive relationship between
minimum wages and employment (e.g., Card and Krueger 1994).
We conclude this section with a brief review of relevant literature. Formal
analysis of illegal immigration dates back to Ethier (1986), who has studied immigration policy for a small host country which suffers from unemployment due
to wage rigidity. His work has stimulated numerous extensions; see, e.g., Djajić
1987, Bond and Chen 1987, and Woodland and Yoshida 2006. Recent attention
has turned to frictional unemployment.1 For example, Liu (2000) has studied
the model in which natives negotiate the wage with employers while immigrants
receive a given fraction of the native’s equilibrium wage. Chassamboulli and
Palivos (2014) and Battisti et al. (2017) have regarded immigrants as legal andheterogeneous from natives with respect to substitutability with capital. Thesestudies differ from ours in that they consider exogenous immigration inflows.2
The present paper is also related to the recent analysis of immigration policy
in a multi-country model by Miyagiwa and Sato (2018). Their focus however ison intergovernmental immigration policy conflict.3
The remainder of the paper is organized in four sections. The next section
presents the model. Section 3 presents the main results. Section 4 investigates
the effects of related policy measures: (a) extending unemployment benefit cov1 In addition, Carter (1998) has applied the Shapiro-Stiglitz efficiency-wage model to generate unemployment. In a two-country model Ortega (2000) has shown that the world is betterwith free labor mobility than labor autarky even in the presence of frictional unemployment.
2 All three works contain calibration results based on the data from the U.S. (Liu 2000,
Chassamboulli and Palivos 2014) and from multiple countries (Battisti et al. 2017).
3 Also see, e.g., Mayr et al. (2012), Giordani and Ruta (2013), and Bandyopadhyay and
Pinto (2017) in this literature3

erages to illegal immigrants, (b) enhancing anti-crime measures and (c) raising
the minimum wages. Section 5 concludes.

2

Model

In this section we present a continuous-time model of a jurisdiction (e. g., acity) in which firms produce the homogeneous good, which serves as numeraire.

2.1

Immigration decisions

A typical foreign-born worker has the lifetime welfare F if he stays in his country.
If he decides to immigrate (illegally) to the destination country, he incurs a real
travel cost τ to reach the border of that country. When crossing the border,
he is apprehended by border patrol agents with probability β ∈ (0, 1) and sent
home. If he evades apprehension, he enters the country, chooses a jurisdiction
j and looks for a job there. Thus, if we let Umj denote his lifetime welfare
during unemployment in jurisdiction j, an immigrant’s expected net welfare
equals βF + (1 − β)Umj − τ . For an interior equilibrium this welfare must
equal his stay-at-home welfare F , i. e., βF + (1 − β)Umj − τ = F . Further,
immigrants naturally choose to settle in the jurisdiction that gives them thehighest welfare, so in equilibrium all jurisdictions yield the common welfare
Umj = Um . Dropping the subscript j in the above equation, we can write:

Um = F + T(1)

where T ≡ τ /(1 − β) is the generalized cost of immigration. This equationshows that Um depends on the tightness of border control β but is independentof local government policy.

4

2.2

SearchWe now fix a jurisdiction and suppose that it is inhabited by Nn native workers
and Nm illegal immigrants. (The subscript n and m denote native workers
and immigrants, respectively.) We take Nn as exogenous but determine Nm
endogenously. If we let Ii denote the number of individuals serving time inprison, the actual labor force is given by Li = Ni − Ii .
There is no centralized labor markets in the model. Jobs are created when
job seekers and firms with vacancies are matched randomly according to somematching function. Suppose that a job seeker finds a firm at rate a(ν) and afirm finds a job seeker at rate q(ν) per instant, where ν denotes the numberof vacancies per job seeker. These two rates are not independent but related
negatively by the equation νq(ν) = a(ν) due to the homogeneity of the matching
function. We write this relation as q = q(a) and denote its derivative by q 0 (a) <0, assuming differentiability.

2.3

Crime and punishment

In modeling crime we follow Burdett et al. (2003) in that individuals commit
crime only if the benefit of crime exceeds its cost. However, for a complete
analysis the presence of two types of workers gives rise to eight separate cases
to consider, depending on the permutations over who commit crime and when.
To keep the analysis under control, we focus in this paper only on the case inwhich workers commit crime only when they are unemployed. The selection ofthis case is justified by only unemployed workers have sufficient time to look foropportunities to commit crime. This has corroborating evidence. For example,
Raphael and Winter-Ebmer (2001) have found that higher unemployment causes
more crime, especially on property crime.
An unemployed type-i worker has the welfare Ui . Assume that he stumbles

5

on an opportunity to commit crime at exogenous rate θ per instant. If he does
not commit crime, he secures welfare Ui . Should he commit a crime, however, he
immediately gains the gross benefit g > 0 but runs the risk of getting arrested,convicted and imprisoned, which occurs with probability α ∈ (0, 1) and reduces
his welfare to Pi . Of course he keeps welfare Ui if he is not arrested. Thus,
crime has the expected benefit g + αPi + (1 − α)Ui . An individual commit crime
if this net welfare exceeds the secure welfare Ui , org + α(Pi − Ui ) > 0.

We assume this condition holds within the relevant parameter values.
We next calculate the welfare Pi from incarceration. Assume that a convictedcriminal serves the average time 1/φ in prison, where the Poisson rate φ > 0reflects a priori uncertainty in sentencing. The Poisson rate also indicates thata convict is released from prison at rate φ per instant. When released, a native
worker moves to unemployment and looks for a job, causing the welfare change
from Pn to Un . If we assume that convicts receive no income, we can relatethese two welfare values by the Bellman equation

rPn = φ(Un − Pn ).

With r denoting the rate of time preference, the left-hand side of this equation
converts Ph into flow values whereas the right-hand side takes the standard formin financial economics for decomposing the asset value into the flow income (= 0here) and the expected capital gain φ(Un − Pn ). Thus

Pn = φUn /(r + φ).

(2)

Immigrants differ from natives in that they may face deportation risk when6

released from prison. Let δ ∈ [0, 1] denote this risk (δ = 0 in a sanctuary
jurisdiction). When an immigrant is deported to his home country, his welfarechanges from Pm to F ; otherwise, he moves to unemployment, causing thewelfare change Um − Pm . Thus Pm satisfies the Bellman equation:rPm = φ[δF + (1 − δ)Um − Pm ].

Recalling that Um is exogenous by (1), the above equation uniquely determines
the equilibrium Pm

Pm = [δF + (1 − δ)Um ]φ/(r + φ) = [F + (1 − δ)T ]φ/(r + φ)(3)

subject only to the jurisdiction’s policy parameters φ and δ.
Finally, victims of criminal activity suffers the exogenous welfare loss l. We
assume that everyone in the labor force is equally likely to fall victim to crime(thus, the unemployed are also victimized while convicts are not) and obtainthe victimization rate π endogenously below.

2.4

Firms

Every firm with a vacancy is identical and has the asset value V . A typicalfirm incurs search cost c and is matched with a worker at rate q(a) as alreadymentioned. Search is undirected; that is, a firm does not know beforehand
whether it will be matched with a native or an immigrant. It knows only that it
will be matched with an immigrant with probability µ ∈ (0, 1). After a match,however, a firm learns a worker’s type i and enjoys the asset value Ji . Theseasset values are related by the Bellman equationrV = −c + q[µ(Jm − V ) + (1 − µ)(Jn − V )],7where the bracketed term on the right denotes the “expected capital gain” froma match. Free entry drives V to zero, transforming the above equation into

µJm + (1 − µ)Jn − c/q(a) = 0.

(4)

The left-hand side is the normalized value of job creation. The equation will bereferred to as the job-creation equation.
When matched with a type-i worker, a firm produces yi units of the homogeneous good per instant. A firm matched with a native worker is bound by theminimum wage law and receives the flow profit yn − w > 0, where w denotes
the minimum wage. All active firms face idiosyncratic bankruptcy risk with the
Poisson rate λ. Thus, the asset value Jn of the firm in question satisfies the
Bellman equation

rJn = yn − w + λ(V − Jn ).

With V = 0, this equation gives us the asset value

Jn = (yn − w)/R(5)

where we writeR ≡ r + λ.
A firm employing an immigrant is assumed to be able to evade the minimum
wage law and negotiate the wage wm directly and continuously with a worker.
Since all firms bankrupt at rate λ, the asset value Jm satisfies the Bellman
equationrJm = ym − wm − λJm ,8yieldingJm = (ym − wm )/R.

2.5

(6)

Workers

When employed, a native worker receives the minimum wage w. When the firm
goes bankrupt (at rate λ), he moves to unemployment, suffering the welfare loss
Un − En . Thus, an employed native’s welfare En satisfies the Bellman equationrEn = w + λ(Un − En ) − πl,(7)where the last term on the right is the expected welfare loss due to crime per
instant. When unemployed, a native receives unemployment benefits bn andfinds a job at rate a, which boosts his welfare from Un to En . He also commits
crime at the rate θ per instant for the net benefit

g + α(Pn − Un ) > 0.

Thus, an unemployed native’s welfare Un must satisfy the Bellman equationrUn = bn + a(En − Un ) + θ[g + α(Pn − Un )] − πl,where the last term on the right-hand side implies that unemployed workers also
fall victim to crime. By virtue of (2) we can rewrite the net benefit as

g + α(Pn − Un ) = g − ηUn > 0,rαwhere η = r+φ. Substituting this and solving the two Bellman equations, we

obtain9En = [(r + a + θη)w + λ(bn + θg) − (R + a + θη)πl]/[R(r + θη) + ra](8)

Un = [aw + R(bn + θg) − (R + a)πl]/[R(r + θη) + ra].

(9)

Similarly, we have the Bellman equationrEm = wm + λ(Um − Em ) − πl(10)for an employed immigrant andrUm = bm + a(Em − Um ) + θ[g + α(Pm − Um )] − πl(11)

for an unemployed immigrant. We assume that bm , unemployment benefits forillegal immigrants, is initially zero until later we examine the implications ofextending unemployment benefits to illegals.

2.6

Determination of the immigrant’s wage

An immigrant’s wage is determined through Nash bargaining. Let γm ∈ (0, 1)
denote an immigrant’s relative bargaining power. Nash bargaining implies thatγm determines a worker’s share of the total surplus generated by a match; thatis, γm satisfies
Em − Um = γm (Em − Um + Jm − V ).
Recalling that V = 0, we can rewrite the above as(1 − γm )(Em − Um ) = γm Jm

10

(12)

We can use (6) to substitute for Jm . To evaluate the left-hand side expression,
let us rewrite the net benefit from crime as

S ≡ g + α(Pm − Um )(13)(note that S is exogenous). Substitute this into (11) and subtracting the resulting equation from (10) yields the desired expression

Em − Um =wm − bm − θS
.
R+a

(14)

Substituting this into 12), we getwm =γm (R + a)ym + R(1 − γm )(bm + θS).
R + γm a(15)

Substitution of the wage into (6) and (14) yields

Jm =(1 − γm )[ym − (bm + θS)]
R + γm a(16)γm [ym − (bm + θS)]
R + γm a(17)

Em − Um =2.7

Unemployment

We next determine the unemployment rates ui . Given the labor force Li , (1 −ui )Li of type-i workers are employed while ui Li are unemployed. Thus, λ(1 −ui )Li employed workers lose their jobs and join unemployment while aui Li of
the unemployed find jobs and move out of unemployment. In a steady statethese two numbers must be equal and henceui = λ/(λ + a) = u;

11

(18)

the unemployment rates are identical between natives and immigrants and depend on the job-acquisition rate a. Therefore the proportion of immigrants inthe unemployment pool equals(um Lm )/(um Lm + un Ln ) = Lm /(Lm + Ln ).

By the law of large numbers this ratio must equal the probability µ that a firm
is matched with an immigrant:µ = Lm /(Lm + Ln )(19)

Further, with uLi type-i workers unemployed, θuLi commit crime per instantand αθuLi are arrested and sent to prison. Meanwhile, there are Ii type-i
individuals in prison and φIi are released per instant. In a steady state thesenumbers must equal; that is, φIi = αθuLi . Hence

Ii = αθuLi /φ.

(20)

Substitution into the identity Ni = Li + Ii enables us to relate the size of thelabor force to the total population:

Li =

φ
Ni
φ + αθu(21)

Finally, we compute the victimization rate. Since θu(Lm + Ln ) crimes arecommitted on (Lm + Ln ) individuals per instant, the victimization rate equals

π = θu = θλ/(λ + a) = π(a).

12Figure 1: Gµ = Jm − Jn < 0This rate is inversely related to the job-acquisition rate due to the derivative

πa = −θu/(λ + a) < 0.

This completes the description of the model.

2.8

Equilibrium

To solve the model we first substitute from (17) into (11) to obtain

H(a, S) ≡aγm [ym − (bm + θS)]
+ (bm + θS) − lπ(a) = rUm
R + γm a(22)

The expression H(a, S) represents an unemployed immigrant’s flow welfare. Ina steady state, this welfare flow equals rUm defined by (1) due to free mobilityof immigrants across jurisdictions. Thus, the above equation enables us todetermine the equilibrium rate a, which we write a∗ . In figures 1 and 2 below,
equation (22) is represented by the schedule labeled H.

13

We next substitute for Jm and Jn from (16) and (5) respectively to rewrite
the job-creation equation (4) as

G(a, µ, S) ≡ µJm + (1 − µ)Jn − c/q(a) = 0.

(23)

The expression G(a, µ, S) represents the expected net profit from entry, whichmust equal zero in a steady state due to free entry of firms. In figures 1 and 2
we represent this equation by the schedule G. Differentiation yields

Ga = −

Gµ = Jm − Jn =µγm Jm+ cq 0 /q 2 < 0
R + γm a(1 − γm )[ym − (bm + θS)] yn − w−
R + γm aR(24)

so G schedule is downward-sloping if Gµ < 0 (or Jn > Jm ) as in figure 1 andupward-sloping if Gµ > 0 as in figure 2 below. Substituting a∗ into (23), we
can uniquely determine µ∗ , the equilibrium probability with which firms arematched with immigrants. In the figures, the equilibrium a∗ and µ∗ are givenby the intersection of the H schedule and the G schedule.
Once a∗ and µ∗ are determined, the other equilibrium values follow readily.
The unemployment rates depend only on a; thus u∗ = λ/(λ + a∗ ). So does theimmigrant’s wage via (15). Substituting u∗ into (21) gives us the equilibrium
native labor force
L∗n =

φ
Nn .
φ + αθu∗(25)

We can then get the number of natives in prison from In∗ = Nn −L∗n . Substitutionof L∗n and µ∗ into (19) gives us the immigrant labor forceL∗m = µ∗ Nn /(1 − µ∗ )

14

(26)

Figure 2: Gµ = Jm − Jn > 0
which shows that the immigrant labor force grows only if µ∗ increases. Substituting u∗ and L∗m into (20), we get the number of immigrants in prison:
∗

∗

αθu µ
∗
= φ(1−µIm∗ ) Nn . The total immigrant population is calculated from the iden-tity: Nm = Lm + Im ;namely,

∗
Nm
=

3

µ∗
1 − µ∗

1+αθλφ(λ + a∗ )

NnSanctuary City

We are now ready to study the effect of sanctuarization, i.e., a decision to
decrease deportation risk δ, given δ > 0. Since dS/dδ = (dS/dPm )(dPm /dδ) > 0
by (3), the effect of sanctuarization can be studied by observing the effect ofincreasing S. Differentiating (22) with respect to S yields

15da∗ /dS = −HS /Ha < 0

where
HS =

Ha =

θR
>0
R + γm a

R(Em − Um )− lπa > 0.
R + γm aThus, sanctuarization decreases the equilibrium a, shifting the H schedule downward, say to H 0 , as shown in figures 1 and 2. A falling a causes the unemployment rate and the immigrant’s wage to increase.
∗/dS > 0.
Proposition 1. (a) da∗ /dS < 0; (b) du∗ /dS > 0; (c) dwm

To see the effect on µ, we differentiate (23) to obtaindµ∗ /dS = −(Ga da∗ /dS + GS )/Gµ =HS Ga − GS Ha.
Ha G µThe derivative

GS = −(1 − γm )θµ< 0,
R + γm asays that sanctuarization decreases the expected profit from entry, other things
being equal. To sign dµ∗ /dS, we make appropriate substitutions to rewrite

HS Ga − GS Ha
=µγm Jm−+ cq 0 /q 2
R + γm a






Rθ
(1 − γm )θµ
R(Em − Um )
− −
− lπa
R + γm a
R + γm a
R + γm a

16

=



θ
Rcq 0
−
(1
−
γ
)µlπ
m
a .
R + γm a q 2(27)

where the second equality follows from 12). Both terms in the bracketed expression are negative because q 0 < 0 and πa < 0; hence (27) cannot be uniquely
signed. Recalling that Ha > 0, however, we obtain
Proposition 2. dµ∗ /dS > 0 if and only if



Rcq 0
− (1 − γm )µlπa Gµ > 0.
q2

(28)

We now show that condition (28) holds in figure 1 but is violated in figure
2. To this end, note that condition (28) is equivalent to(HS Ga − GS Ha )Gµ > 0.

Assume for now that HS Ga − GS Ha > 0. Then, condition (28) is satisfied
in figure 1, where Gµ < 0 but violated in figure 2, where Gµ > 0. We next
demonstrate that HS Ga − GS Ha > 0 if and only if

da/dS|H < da/dS|G < 0;

(29)

where
∂a/∂S|H = −

HS
θR
=−
< 0.
Ha
R(Em − Um ) − (R + γm a)πa∂a/∂S|G = −GSµθ(1 − γm )=−<0Ga
µγm Jm − (R + γm a)cq 0 /q 2

are obtained by differentiating (22) and (23), respectively. These derivatives
indicate how much each schedule shifts downward (a falls) as a result of sanctuarization. The demonstration of our claim is immediate because taking the17difference yields∂a/∂S|H − ∂a/∂S|G = −HS
GS
HS Ga − GS Ha
+
=−
.
Ha
Ga
Ha Ga

where Ga < 0 and Ha > 0. Thus under condition HS Ga − GS Ha > 0 the Hschedule shifts down more than the G schedule as in figures 1 and 2, where
the schedule H shifts to H 0 and the schedule G shifts to G0 . As a result, µ isincreased in figure 1 and decreased in figure 2, in consonance with proposition
2.
As we show in the appendix, a, µ and all the other variables (with the exception of the unemployment rates) jump immediately to the new stationary values
without transition dynamics - a common property of equilibrium unemployment
models (Pissarides 2000). However it is still useful to understand intuitively why
they change the way they do.
We have seen that elimination of deportation risk raises the unemployed
immigrant’s welfare above the steady-state value rUm . This cannot persist in
equilibrium. To reach a new steady state the job-acquisition rate a must fallsufficiently to counterbalance the initial welfare increase. At the same time, theinitial welfare increase implies that immigrants have a higher reservation utility
in wage bargaining, which emboldens them to demand higher wages (proposition
1) and turns the expected profit from entry negative in the job-creation equation.
This cannot persist in equilibrium, either, calling for changes in a and µ to bring
the profit up to zero in a new steady state.
Figure 1 shows that a must fall to a0 to counterbalance the initial welfare
increase. However, a needs to fall only to a∗∗ to bring the profit up to the steadystate value, holding µ constant. In fact, at a0 < a∗∗ , the profit overshoots its
target and becomes positive. This overshooting must be corrected by a change
in µ. Since Gµ = Jm − Jn < 0, the required correction calls for µ to rise to µ0
18to bring the profit down to zero. The situation depicted in figure 2 is similar
except that, since Gµ = Jm − Jn > 0 there, the profit can be brought down tozero by a fall in µ as stated in proposition 2.
The case with HS Ga − GS Ha < 0 is analyzed analogously. Since the schedule G shifts down more than the schedule H, when the welfare is restored toits equilibrium value uUm , the profit undershoots its target. This shortfall isredressed by a decrease in µ when Gµ < 0 and an increase in µ when Gµ > 0 in
consistence with proposition 2.
Since sanctuarization raises the unemployment rates (proposition 1), more
natives turn to crime and end up in prison. As a result, a new native labor
force is smaller by (25). In contrast, a new immigrant labor force may begreater; (26) shows that the immigrant labor force is larger (dLm /dS > 0)
if and only if dµ/dS > 0 . If there are more immigrants in the labor force, ahigher unemployment rate implies that more of them are unemployed and hence
committing more crime. Even if fewer of them are in the labor force, more
immigrants may be unemployed due to a higher unemployment rate. Thus, it
is still possible that there is more crime by immigrants. The next propositionsummarizes the above findings.
Proposition 3:(a) Sanctuarization increases crime by natives and diminishes the nativelabor force.
(b) If (28) holds, sanctuarization results in a greater immigrant labor force
and more crime by immigrants.
(c) If condition (28) is violated, sanctuarization diminishes the immigrantlabor force but there may still be more crime by immigrants.
We close this section by examining the welfare impact of sanctuarization.
The unemployed immigrant’s steady-state welfare is fixed at Um by (1) and19

hence insensitive to the local government’s policy changes. By virtue of (17)the employed immigrant’s welfare satisfies

Em = Um +γm [ym − (bm + θS)].
R + γm a

Differentiation yields

dEm /dS = −θγm /(R + γm a) − γm (Em − Um )/(R + γm a)(da∗ /dS)

(30)

Substituting for da∗ /dS and arranging the result, we get

dEm /dS =θγm lπa< 0;
R(Em − Um ) − (R + γm a)lπaemployed immigrants are always harmed, independently of the sign of Gµ . This
welfare loss is due to an increase in the victimization rate.
As for natives, differentiation of (9) and (8) with respect to a yields, respectively:

dUn /da =R[(r + θη)w − r(bh + θg) − θηπl](R + a)lπ̃a−>02[R(r + θη) + ra]R(r + θη) + ra

dEn /da =(R + a)lπ̃aλ[(r + θη)w − r(bh + θg) − θηπl]−> 0.
[R(r + θη) + ra]2R(r + θη) + ra

Since da/dS < 0, dUn /dS < 0 and dEn /dS < 0. Natives are harmed by sanctuarization not only because the victimization rate is higher but because thejob-acquisition rate has declined.
Proposition 4 : Sanctuarization always harms natives.

20

4

Extensions

4.1

Unemployment benefits for illegal immigrants

Many sanctuary cities not only shield illegal immigrants from deportation but
provide them with unemployment benefits. An inspection shows that bm and S
appear only in the form (bm + θS) and that neither term appears independentlyof each other anywhere in the model. It follows immediately that an increase inbm has qualitatively the same effect as an increase in S.
Proposition 5: Providing illegal immigrants with unemployment benefits
has qualitatively the same effects as sanctuarization.

4.2

Anti-crime policy

Our model is also useful in evaluating the effects of anti-crime policies. In ourmodel the jurisdiction can crack down on crime by catching more criminals (increasing α) and/or meting out longer sentences to convicted criminals (loweringφ). Both these measures generate the effects qualitatively opposite to thoseyielded by sanctuarization. The reason is that both policy parameters appear
only as components of S in (22) and (23) and nowhere else in the model. For
example, increasing α decreases S (dS/dα = Pm − Um < 0), decreasing theunemployment rates by proposition 1. However, the preceding analysis implies
that lower unemployment rates do not necessarily mean less crime. If (28) holds,
an increase in α induces a fall in µ, causing the immigrant labor force to shrink
by virtue of (26). Then, we can conclude with certainty that a low unemployment rate implies less crime by immigrants. However, if (28) is reversed, µincreases, expanding the immigrant labor force. In this case, there can be morecrime by immigrants in spite of a fall in unemployment rates. Turning to thenative labor force, it is evidently clear that a lower unemployment rate implies

21

natives commit fewer crimes. However, since the arrest rate is higher, more
native criminals are caught and imprisoned. As a result, the native labor force
can be smaller; cf. (25). The results from meting out tougher sentences (a fallin φ) have similar effects because dS/dφ = rαPm /(r + φ) > 0.
A community can also fight crime through more enhanced neighborhood
watch. Increased degrees of vigilance lower θ, making it more difficult for criminals to find opportunities to commit crime. Since θ appears only in the form
θS in (22) and (23), this policy also countervails sanctuarization like the othertwo anti-crime policies examined above. However, there is one difference. With
falling unemployment rates, we have dLn /dθ < 0 by (25); an increase in vigilance expands the native labor force unambiguously while reducing the numberof crimes committed by natives. We summarize the findings in
Proposition 6. (a) All three anti-crime policies countervail sanctuarization,
reducing unemployment rates and lowering the immigrant’s wage.
(b) If (28) is satisfied, all three anti-crime policies result in a smaller immigrant labor force and less crime by immigrants.
(c) Increased vigilance results in less crime by natives while enlarging thenative labor force.

4.3

Raising the minimum wage

In recent years, many jurisdictions have raised minimum wages. What are theimpacts of such a policy change in the presence of immigration? An increase in
w makes employment of native labor more expensive, decreasing the expected
profit through (5), but it has no impact on equation (22). In other words,
an increase in minimum wages shifts down the schedule G while keeping theschedule H intact. As a result, raising the minimum wage has no effect on the
job-acquisition rate. Hence, the unemployment rates are unaffected. It follows

22that it has effect neither on the size of the native labor force nor on the numberof crimes committed by natives.
A downward shift of the schedule G causes µ∗ to fall if Gµ < 0, shrinking
in the immigrant labor force and decreasing crime by immigrants. If Gµ > 0,
these results are reversed so that there are more immigrants in the labor force.
In this case, since the native labor force is unaffected, we observe an increase intotal employment as a result of higher minimum wages, the possibility that hascaught attention since the work by Card and Krueger (1994). We summarize
the effect of higher minimum wages in
Proposition 7.
(a) da∗ /dw = du∗ /dw = 0,(b) dL∗h /dw = 0; dIh∗ /dw = 0∗(c) dL∗m /dw < 0 and dIm/dw < 0 if and only if Gµ < 0.

4.4

Alternative wage determination schemes

We have so far assumed that natives are paid the minimum wage while illegal
immigrants negotiate wages directly with employers. While we believe this is amost relevant scenario in many jurisdictions, in this subsection we explore theimplications of alternative scenarios. First, suppose that both natives and immigrants negotiate wages with employers. Then, by an appeal to the analogous
procedures used above, we can determine the native wage and the asset valueof firms employing natives:Jn =(1 − γn )(yn − bn )R + γn a

where γh is the measure of a native’s bargaining power. Substituting this into
the job-creation equation (23) gives us the following restatement of proposition2:
23

Proposition 2’: dµ∗ /dS > 0 if and only if


Rcq 0− (1 − γm )µlπa −q2

γn (1 − µ)Jn
R + γn aRθ
R + γm aGµ > 0.

The addition of the third term makes the bracketed expression more likely to be
negative compared with our original model but does not affect our qualitative
results.
As a second alternative, suppose that both natives and illegal immigrants arepaid the minimum wage. Then, sanctuarization has no effect on the asset valueof firms employing immigrants and hence leaves the the G schedule intact. As
the H schedule shifts down, µ increases if and only if Gµ = Jm − Jn < 0. Thus
these alternative scenarios yield results similar to those stated in propositions 1
and 2.

5

Concluding remarks

We have presented a model of a jurisdiction (city, country and state) and studied
the effect of sanctuarization. Sanctuarization eliminates deportation risk for illegal immigrants, making the jurisdiction more attractive to illegal immigrants,
other things being equal. However, sanctuarization also diminishes labor demand, a point that has received little attention. If too many jobs are destroyed,
immigrants find the jurisdiction less attractive in spite of its sanctuary status.
Thus, sanctuary cities do not necessarily attract more illegal immigrants nor
suffer from more crime by immigrants compared with non-sanctuary cities.
We have also examined the effect of other related policies available to jurisdictions. Extending unemployment insurance benefits to illegal immigrants isqualitatively identical to sanctuarization whereas anti-crime policies have countervailing effects. Raising minimum wages has no effect on unemployment rates

24and may even increase total employment in the jurisdiction.
The foregoing results are derived from the model where both native andimmigrants commit crime when they are unemployed but not when they are
employed. Although we believe this is the most relevant scenario, it may be
useful to extend the analysis to other taxonomical cases. Alternative modeling strategies are also conceivable; for example, instead of having unemployedworkers commit crime, we could introduce heterogeneity among workers so thatcrime is committed only by those individuals having compunction towards crime
below a cutoff level. All these extensions are left for future research.

AppendixOut-of-steady-state dynamics
When the economy moves out of steady-state values, the equation for the asset
values of a firm employing an immigrant is modified torJm = ym − wm + λ(V − Jm ) + J˙m ,(31)where the dot indicates time derivatives (ẋ = dx/dt). The equations for thewelfare of an employed and unemployed immigrants undergo similar changes:

rEm = wm + λ(Um − Em ) − πl + Ėm(32)

rUm = a(Em − Um ) + bm + θS − πl + U̇m .

(33)

These three equations combine to yield(r +λ)(Em −Um +Jm −V ) = ym −bm −θS −a(Em −Um )+ J˙m + Ėm − U̇m (34)25

If we let Σm denote the total surplus from a match, thenΣm = Em − Um + Jm − Vand henceΣ̇m = Ėm − U̇m + J˙m − V̇ .
Since firms can enter freely and that immigrants can move freely across jurisdictions at any date, we can put V = V̇ = 0 and U̇m = 0 while Um is given by(1). Substituting these values, we can rewrite (34) as

RΣm = ym − bm − θS − a(Em − Um ) + Σ̇mwhere R = r + λ as in the main text. Nash bargaining implies that Em − Um =γm Σm . Substituting this, we can rewrite the above equation as(R + γm a)Σm = Σ̇m + ym − bm − θS.

(35)

Letting R + γm a(t) ≡ P (t) > 0 and ym − bm − θS ≡ z > 0, the above equation
is written more compactly:Σ̇m (t) − P (t)Σm (t) + z = 0

This differential equation has the general solution
´t

ˆ t

´x

´t

ze− 0 P (v)dv dx + ce 0 P (x)dx

Σm (t) =e 0 P (x)dx0where c is a positive constant. Regardless of a(t), P (x) > 0 and hence Σm (t)does not converge. Thus (35) holds only if Σ̇m = 0; that is,

26

Σm =

ym − bm − θS
R + γm ais stationary. This implies that when the economy is disturbed, a jumps immediately to its new steady-state value. This is in reflection of agents’ forwardlooking behavior (rational expectations) and is standard in the equilibrium unemployment models (Pissarides 2000). The absence of transition dynamics implies that the number of jobs and the immigrant’s wage also jump immediately
to their stationary values. Since V = V̇ = 0, the job-creation equationµJm + (1 − µ)Jn − c/q = 0.

holds at any date. The surplus-sharing rule gives us Jm = (1 − γm )Σm andhence the job-creation equation implies thatµ̇ = 0;thus, µ also jumps immediately to its stationary value, forcing Lm to do thesame. In contrast, the unemployment rate adjusts slowly according to the equationu̇ = −(λ + a)u + λconverging to the steady-state rate u∗ = λ/(λ + a). This indicates that the
number of immigrants in prison changes according to the differential equation
I˙m = θαu(t)Lm − φIm . (Although immigrants with criminal records in nonsanctuary cities are deported with probability δ when released prison, they
are immediately replaced by new immigrants.) Thus, if we do not care about
the identities of individual immigrants, the equilibrium outcome is as if all27

immigrants move to unemployment when released from prison.)

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