Post on 11-Mar-2018
Intergenerational Mobility and the Political Economy ofImmigration∗
Henning Bohn† Armando R. Lopez-Velasco‡
April 2017
Abstract
Flows of US immigrants are concentrated at the extremes of the skill distribution. We de-velop a dynamic political economy model consistent with these observations. Individuals careabout wages and the welfare of their children. Skill types are complementary in production.Voter support for immigration requires that the children of median-voter natives and of im-migrants have suffi ciently dissimilar skills. We estimate intergenerational transition matricesfor skills, as measured by education, and find support for immigration at high and low skills,but not in the middle. In a version with guest worker programs, voters prefer high-skilledimmigrants but low-skilled guest workers.
Keywords: immigration, political economy model, overlapping generations,intergenerational mobility, guest workers
JEL: F22, E24
∗We thank Finn Kydland, Peter Rupert, Giulio Zanella, Shawn Knabb, Stephen Trejo, Pia Orrenius, CarlosZarazaga, Erick French and Julian Neyra for valuable comments and discussions. We also thank the comments byNicola Pavoni as well as of two anonymous referees. All remaining errors are our own.†Corresponding Author. Department of Economics, University of California Santa Barbara; Santa Bar-
bara CA 93106; and CESifo network. Phone (805) 893-4532. E-mail: henning.bohn@ucsb.edu. Homepage:http://www.econ.ucsb.edu/~bohn‡Department of Economics, 253 Holden Hall. Texas Tech University. Lubbock, TX. Phone (806) 834- 8436.
Email: ar.lopez@ttu.edu. Webpage: https://sites.google.com/site/armandolopezvelascowebpage
1 Introduction
Stylized facts of international migration are that immigrants tend to be concentrated at the ex-
tremes of the skill distribution (high and low) and that high- and low-skilled immigrants are treated
very differently. Many countries allow or even encourage immigration of high-skilled workers, but
they prohibit low-skilled immigration or only accept low-skilled foreigners temporarily (e.g., as
guest workers or by allowing illegal immigrants subject to instant deportation to stay).
We examine the political economy of immigration in a dynamic model in which natives care
about their children and recognize that immigration influences the labor market for current and
future generations. Skill types are complementary and the majority of natives is medium-skilled.
Hence from a static perspective, the native majority benefits from foreign workers with skills far
from the middle, both high and low.
The challenge is to explain the differential treatment of high and low skilled foreigners. A
common argument is that natives worry about low-skilled migrants relying on welfare, whereas
the high-skilled pay more taxes. Our model includes a simple tax-transfer system to account
for this, but we find the welfare argument unconvincing, at least for countries like the US that
exclude migrants from most welfare benefits. Our main contribution is to provide an alternative
explanation: Using U.S. data on generational mobility, we show that children of unskilled workers
tend to compete in the labor market with the children of medium-skilled natives. In contrast,
children of high-skilled workers have a skill distribution more complementary to the children of
medium-skilled natives. Hence concern about children can provide a positive theory of why high-
skilled workers are allowed to enter permanently, i.e., with their children, whereas low-skilled
workers are not accepted permanently.1
Our data analysis focuses on the US. Since 1970, about 60% of immigrants are either "high-
skilled" (BA degree or more) or "low-skilled" (less than High School degree), while the rest is
"medium-skilled" (High School degree or some college). Because the share of medium-skilled
individuals in the US population is more than 50%, the amount of high-skilled and low-skilled
individuals as percentage of their native counterparts is larger than for the middle group. For
example, for the decade of the 90’s, we estimate that there were 5.65 low-skilled (net) immigrants
per 1000 unskilled natives in the US, about 2.02 medium-skilled per 1000 native medium skilled,
and 4.15 high-skilled immigrants per 1000 skilled US natives.2 In a 30 year period these numbers
would account for immigration quotas of roughly 18%, 6% and 13%, when compared to the
composition of the native population.
A diffi culty in interpreting these flow data is that control over immigration is highly imperfect.
1Undocumented workers are typically confined to low skilled work and cannot easily settle down as families,being under a constant threat of deportation. Hence it can be argued that de facto tolerance of illegal immigrantsis similar to a guest worker program for unskilled workers.
2See appendix for details on these numbers. We use the terms "unskilled" and "low-skilled" interchangeably;and we use "skilled" and "high-skilled" in the same manner.
2
Observed immigration is arguably a combination of legal and illegal flows, and of job-related and
other flows.3 To interpret the data, we set up a political-economy model to derive predictions
about optimal immigration under alternative asumptions, and we examine under what conditions
the model provides a positive theory.
The model has three types of labor inputs defined as low-, medium- and high-skilled; and two
types of migrants, permanent immigrants and temporary guest workers. Each worker supplies one
unit of their work-type to the production process, earns a wage, pays proportional taxes that are
then redistributed via lump-sum, has children according to a fertility profile that depends on skill
and place of birth. The model is calibrated to match the transition matrix of intergenerational
skill transmission for natives and immigrants in the US. The calibrated demographic process for
the US is such that with or without immigration, the medium-skilled type would be the absolute
majority in each generation.
Immigration policy is defined by a set of quotas indexed by skill level and type of immigration
permit (permanent vs guest-worker). Votes over immigration policy occur before the skill type of
children is revealed. Immigrants don’t have the right to vote, but the children of immigrants are
citizens identical in everything to natives, which in turn have the right to vote. We use Markov
Perfect as the equilibrium concept, as it is common in the literature of dynamic political economy.
Our analysis initially sets aside guest workers and focuses on the more challenging problem of
modelling permanent immigration. We show that the medium-skilled majority chooses a positive
level of unskilled immigration, zero medium-skilled migration, and substantial high-skilled immi-
gration. Thereafter, we add the possibility of guest workers, which is straightforward in our setting
because guest workers do not raise intertemporal issues. We show that in the presence of guest
workers and (full) immigration, the medium-skilled majority chooses a guest worker program but
no immigration for low-skilled foreigners, no immigration nor guest workers at the medium skill
level, and immigration but no guest workers for high-skilled foreigners.
U.S. immigration policy and recent reform proposals are broadly consistent with our model.
For the last several decades, the U.S. has in effect accepted significant low-skill immigration by
tolerating undocumented workers, minimized medium-skilled immigration by restricting undocu-
mented workers to menial jobs, and encouraged high-skilled immigration by providing work visas
(notably, the H1B program).
In our model, the difference between guest workers and immigrants is whether or not children
are excluded from the workforce —either by border controls that keep children out, by expelling
families that get in, or by some other means. We view the feasiblity of exclusions and the will-
ingness to separate families as beyond the scope of our model, as decisions that involves a mix of
diplomatic, altruistic and other non-economic motives and views about human rights. Changes in
3In the US, a significant share of immigration involves non-working family members of earlier immigrants whohave attained citizenship. We do not model family unification, but the phenomenon is consistent with our premisethat intergenerational issues are central to thinking about immigration.
3
immigration policy —reforms —are therefore interpreted as shifts between policies that are eco-
nomically optimal under different feasible sets as constrained by preferences about enforcement.
From this perspective, recent (pre-2016) US immigration policy is consistent with optimization un-
der the constraint that expulsions and family separations are unacceptable, i.e., over immigration
without guest workers. The pre-2016 reform discussion favored creating a guest worker program
for unskilled workers and increasing the immigration of the high skilled.4 Both changes would
increase the utility of medium-skilled natives (if feasible) and are thus consistent with our model.
We interpret the Trump election as a shift towards stricter enforcement of restrictions on low- and
medium-skilled immigration5 and possibly as shift towards low skilled guest workers. Although
President Trump initially expressed an intent to reduce all immigration drastically (which would
be inconsistent with our model), Congress will be needed to enact reforms and the President’s
position seems to be softening. Thus we maintain that, campaign rhetoric notwithstanding, fu-
ture U.S. policy will likely allow for substantial high skilled immigration and for some low skilled
immigrants or guest workers.
Two important objects in the model and of independent interest in this paper are the matrices
of intergenerational mobility for natives and for immigrants, which we estimate from the General
Social Survey. This survey collects information on education data on the respondents and their
parents, and it also identifies whether the parents are foreign born, among other variables. The
data required for our purposes is available since 1977. We find that on average the children of
unskilled and medium-skilled parents do better than their native counterparts, while there is no
statistical difference for children of high-skilled parents. Consistent with this, Card, DiNardo
and Estes (2000) find with regression analysis that children of immigrants (second generation
immigrants) have on average higher schooling and wages than children of native parents with
comparable education level. We now elaborate on findings of the paper.
Among other findings we show that the intergenerational mobility of children of low-skilled
workers can be a major determinant for the political support of low-skill immigration by the
medium-skilled majority. In this case 2nd generation immigrants from low-skilled parents mostly
4Major attempts at overhauling immigration include the Comprehensive Immigration Reform Act in 2006 whichpassed the Senate but not the House of Representatives (H1B visas would increase from 65,000 to 115,000, plusan annual increase each year). The Secure Borders, Economic Oportunity and Immigration Reform Act that wasintroduced in 2007 but never voted on proposed a merit based point system where skills was one of the mostimportant criterions for points, and the Border Securiy, Economic Opportunity and Immigration ModernizationAct in 2013 which in addition to moving to a point-based system, proposed to increase the H1B visas from 65,000to 205,000. All of these legislation projects also included some form of guest worker program or temporary visacategory for low skilled workers.
5Medium-skilled immigration is an issue because minimizing it requires monitoring of programs meant to allowhigh-skilled immigration, notably the H1B program. President Trump has suggested reducing the number ofHB1 visas and increasing the salary level required to apply. Currently, the number of H1B visas is capped (withexceptions for researchers/professors at universities), when applications exceed the cap, visas subject to the capare allocated via lottery. The lottery suggests that current policy is not maximizing applicant quality. Hencerestrictions that raise the skill requirement are not necessarily inconsistent with our model. Under the model,all medium skilled immigration is attributed to limited enforcement and non-economic (primarily family related)motives.
4
"complement" the children of medium-skilled natives. Also the higher the probability that children
of low-skilled immigrants become medium-skilled, the lower the low-skilled immigration quota
that is politically chosen because 2nd generation immigrants from low-skilled parents are mostly
"substitutes" of the children of medium-skilled natives. Since we estimate that on average children
of unskilled immigrants have a higher probability of upward mobility, and a lower probability of
becoming medium-skilled or staying unskilled than natives, these effects help explain why the US
allows significant quantities of unskilled immigrants. Another way to put this is the following.
If children of unskilled immigrants had on average the same probability of becoming skilled as
the unskilled natives, the number of unskilled immigrants allowed in equilibrium would decrease
considerably from current flows. In addition, the model suggests that intergenerational mobility
of 2nd generation immigrants from skilled parents is not as important for the admission of skilled
individuals, but does matter for the choice of immigration over guest worker permits. If the
children of the high-skilled immigrants were mostly concentrated in the medium-skill group and
therefore competing in the most likely state with the children of the majority, the medium skill
voters would choose guest worker permits for the high skilled as opposed to full immigration.
Finally, given that we establish this equilibrium result with a preference of high-skill immigration
over high-skill guest workers, our models show that the medium-skilled majority wouldn’t want to
restrict high-skill immigration from its current level (but increase it if possible). This is because
high-skilled immigrants and a large fraction of their children have skills that are complementary
to the native majority.
The paper is organized as follows. Section 1.1 reviews the literature, in section 2 we present the
model and the equilibrium concept. Section 3 presents the empirical evidence on intergenerational
mobility and fertility profiles of natives and immigrants. Section 4 presents the calibration of the
model as well as some important results from the calibration exercise. In section 5, we use the
model to ask applied questions. Section 6 does sensitivity analysis. Section 7 concludes.
1.1 Related Literature
This paper is closely related to Ortega’s politico-economic models on immigration with intergener-
ational mobility (2005, 2010). Ortega (2005) identifies political-economic trade-offs of immigration
in a model with two skill levels and skill upgrading while abstracting from redistribution issues.
Ortega (2010) extends his previous model by allowing agents to vote over immigration and redis-
tribution in a richer model with alternative immigration regimes. While we build on the insights
in Ortega’s work, we show that a generalization to three skill types implies different trade-offs,
produces new conceptual insights, and raises interesting quantitative questions that cannot be
addressed in a model with two skill types. We also show how our model can be used to explain
voter preferences for immigration versus guest workers, at each skill level.
Specifically, we use three instead of two skill levels for the following reasons: (1) the composi-
5
tion of unskilled workers (as traditionally defined as education level lower than a college degree)
is very different for natives and immigrants: a high percentage of immigrants has much lower
schooling than natives in this category; (2) the three-type classification helps to document that
most immigration to the US is in the extreme types of the skill distribution; (3) the demographic
profiles (mobility & fertility) of natives and immigrants are statistically different, particularly for
the unskilled group; (4) the model with three skill types has a natural middle. Moreover, the
middle type turns out to be the majority, which yield predictable voting outcomes. The result-
ing theory is centrally about the economic trade-offs and voting incentives of the medium-skilled
majority, and not about the political trade-off identified by Ortega (2005).
Other related literature on equilibrium political-economic models of immigration includes the
seminal work of Benhabib (1996) on immigration policy under heterogeneous agents. Dolmas and
Huffman (2004) study the interaction of immigration and redistribution. Sand and Razin (2007)
study the joint determination of immigration and social security with a Markov perfect equilibrium
concept, which we also use in this paper. Cohen, Razin and Sadka (2009), study the theory and
empirical relation between the size of the welfare state and the composition of the immigration
flows in a static model. Bohn and Lopez-Velasco (2015) study immigration games when immigrants
have a larger fertility rate than natives. On the dynamic fiscal effects of immigration there are
papers by Storesletten (2000) and Lee and Miller (2000). Ben Gad (2004) studies dynamic aspects
of immigration related to capital accumulation. The paper is also related to the macroeconomic
literature where current voters foresee the consequences of their choices on the future behavior of
voters. Some examples include Krusell and Rios-Rull (1999) and Hassler et.al (2003). Empirical
studies on the intergenerational mobility of immigrants include Borjas (1992) and Card, DiNardo
and Estes (2000). On the economics of guest worker programs Djajic (2014) has discussed welfare
implications of low skilled guest workers, the optimal design of temporary migration programs has
been discussed by Schiff (2007) and Djajic, Michael and Vinogradova (2013).
Regarding preferences over immigration, Dustmann and Preston (2005) find empirically that
competition in the labor market, the effects of immigration on the public burden, and effi ciency
considerations of immigration shape immigration attitudes. Mayda (2006) concludes that both
economic variables (mainly labor market outcomes in her analysis) and non-economic variables
shape the attitudes of natives toward immigration, and non-economic variables do not alter the
labor market results. The paper by Card, Dustmann and Preston (2009) concludes that immigra-
tion preferences are shaped not only by the economic effects of immigration, but by externalities
associated to the changes on the composition of the population induced by immigration policies.
This is one of the main arguments in our paper: that the current and future composition of the
population is affected by the composition of the immigration flows and their different-from-natives
demographic profiles (mobility and fertility), which in addition to the welfare state and labor mar-
ket effects would then shape views on immigration, and that ultimately can dictate policy.
6
2 The Model
2.1 Demographics
Adults live for one period, work full-time, and have children. Individuals are grouped by skill level
and immigration status. There are three skill types, which will earn different wages: low-skilled
workers (type 1), medium-skilled (type 2) and high-skilled (type 3).
Domestic-born individuals (henceforth: natives) can vote, as can their children. Immigrants
cannot vote, but their childen are considered natives, and as such have the right to vote. Guest
workers also cannot vote and they are assumed to leave their children abroad or return to their
countries of origin. They matter only for current-period labor supply and have no impact on
population dynamics.6
There is intergenerational mobility across skill types: a child’s skill level as adult has a dis-
tribution that depends on the skill type and immigration status of his/her parents. Children are
assumed not to take any economic decisions. Skill is realized at the start of adulthood.
We assume throughout that a majority of natives is medium-skilled. (The assumption will be
justified in the calibration of the model.) Thus policy is set by the medium-skilled. The other skill
levels are relevant not for voting, but because medium-skilled parents do not know their chidrens’
skill when voting over immigration. Hence they care about the impact of immigration on future
labor supply at all skill levels.7
To model population dynamics, let Ni,t denote the numbers of natives of skill-type i in period
t, let θIit ≥ 0 denote the quota of immigrants of skill type i as a percentage of the native group of
the same skill, and let θGit ≥ 0 denote the quotas of guest workers of skill type i (for i = 1, 2, 3),
again as percentage of natives. Let θit = θIit + θGit denote total migrants.
Let natives have ηi children, whereas immigrants have ηIi children. Let qij denote the proba-
bility that the child of a native parent of skill type i will have skill type j (i, j = 1, 2, 3). Similarly,
let qIij be the probability that the children of an immigrant parent of skill type i will have skill
type j. These fertility profiles and transition probabilities are estimated below by skill levels and
nativity.
With these definitions, the evolution of the native population by skill type is
Ni,t+1 =
3∑j=1
(ηjqji + ηIjq
Ijiθ
Ijt
)Nj,t, for i = 1, 2, 3, (1)
which includes the children of immigrants. Key state variables in the voting analysis will be
6The defining property for our purposes is the exclusion of their children. For example, if undocumentedimmigrants and their children were forced to return to their country of origin, these "immigrants" would be treatedin effect as guest workers.
7Due to the assumed medium-skill majority, our paper has a quite different focus than the literature on the po-litical economy of immigration, which examines under what conditions immigration might change voting majorities(e.g. Ortega (2010)).
7
the implied ratios of low- and high- relative to medium-skilled population, x1t = N1,t/N2,t and
x3t = N3,t/N2,t.
For reference below, we define a more compact vector notation. Let intergenerational transition
matrices for native and immigrants be
Q =
q11 q12 q13
q21 q22 q23
q31 q32 q33
and QI =
qI11 qI12 qI13
qI21 qI22 qI23
qI31 qI32 qI33
. (2)
The cells are probabilities. For example, q13 is the probability that a child of a low-skilled native
parent will be high-skilled. Hence the rows (denoted Q[i] and QI[i], respectively) add up one.
Define the population vector Nt = (N1t, N2t, N3t)′, where ′ denotes a transpose. Define the
vector of population ratios Xt = Nt/N2t = (x1t, 1, x3t)′ (with dummy x2t ≡ 1). Define policy
vectors θIt = (θI1t, θI2t, θ
I3t), θ
Gt = (θG1t, θ
G2t, θ
G3t), θt = θIt + θGt (dimension 1 × 3), and Θt = (θIt , θ
Gt )
(dimension 1× 6). Then population and population ratios have the dynamics
Nt+1 = St ·Nt, and (3)
Xt+1 = (St ·Xt)/(St[2] ·Xt), (4)
where St = S(θIt ) = Q′η +(QI)′ηI · diag
θIt,
St[2] is the second row of St, η = diag η1, η2, η3, and ηI = diagηI1, η
I2, η
I3
.
Policy overall is defined by the vector Θt. Constraints on policy are imposed by the available
supply of migrants and by the country’s ability or inability to prevent guest workers from settling
down as immigrants.
Limits on the supply of migrants are modeled as upper bounds θit ≤ θmaxi with θmax
i > 0
(i = 1, 2, 3). The bounds also ensure that policy spaces are compact. For a wealthy country
like the U.S., the economically most relevant constraint is the supply of high-skilled migrants,
θmax3 . We assume throughout that the supply of low- and medium-skilled migrants is effectively
unlimited; this is implemented by setting θmax1 and θmax
2 high enough to be non-binding.
If a country cannot prevent guest workers from settling down as immigrants, all migrants turn
into immigrants and the supply of guest workers effectively equals zero.
Combining these constraints, we consider three main policy settings:
Setting (I) assumes that the supply of high-skilled immigrants is effectively unlimited andthat all migrants can and will settle as immigrants. The former is arguably unrealistic and turns
out to have implausible implications, but it is instructive as starting point, because it treats type-3
like the others and is therefore least restrictive.
Setting (II) assumes a fixed "small" supply of high-skilled immigrants (θmax3 ), small enough
to be a binding constraint; as in (I), all migrants are immigrants. This setting yields the most
8
interesting results about equilibrium immigration and it will highlight the importance of intergen-
erational mobility.8
Setting (III) adds guest workers to the policy choices in (II), so policy can set separate quotasfor immigrants and guest workers at each skill level.
2.2 Production
Labor inputs are combined to produce output via a production function F,
Yt = F (L1t, L2t, L3t) , (5)
where F has constant returns to scale and Lit is the total labor supply of type i at time t.9 We
assume that F has partial derivatives Fi > 0 and Fii < 0, Fij > 0 for i, j = 1, 2, 3. Wages wit = Fi
equal the respective marginal products of labor.
Each agent supplies one unit of their labor-type inelastically. Natives, immigrants, and guest
workers are assumed to be perfect substitutes within each skill category,10 which defines the labor
supply of type i as
Lit = Nit
(1 + θIit + θGit
)= Nit (1 + θit) , for i = 1, 2, 3. (6)
Note that immigrants and guest workers of each type have the same impact on labor supply. Hence
policy Θt enters only through the sums θit. Constant returns to scale imply that wages depend
only on the labor supply ratios
z1t =L1t
L2t
= x1t1 + θ1t
1 + θ2t
and z3t =L3t
L2t
= x3t1 + θ3t
1 + θ2t
.
Hence wages depend on the elements of Xt and of θt, which we write as
wit = wi (Xt, θt) = wi (z1t, z3t) , for i = 1, 2, 3. (7)
For the numerical analysis, we assume F has the constant-elasticity-of-substitution form (CES),
8As extension, we examine a more general setting with wage-elastic supply of high-skilled immigrants, and weexamine the ramifications of alternative supply elasticities. The results are similar to case (ii) but more complicated.Hence the main analysis focus on the polar cases (i) and (ii).
9We abstract from capital inputs. An alternative interpretation is that production has constant scale returnsover capital and labor in an open economy with exogenous required (world) return to capital. Then capital wouldvary in a way that output net of capital costs is a function F with constant returns to labor.For example, suppose production is Q = AKαY 1−α, where A is technology, K is aggregate capital and Y is
composite labor given by (5), and suppose r is an exogenous world interest rate. Then K =(αAr
) 11−α Y varies with
Y , so Q− rK = A∗Y with A∗ = (1− α)A(αAr
) α1−α . Setting A
∗= 1 is without loss of generality, so we obtain (5).
10There is evidence by Ottaviano and Peri (2012) that immigrants and natives for a same level of school-ing/experience are not perfect substitutes. Borjas (2009) argues that for all practical purpose they can be consideredperfect substitutes. For the present purpose out of simplicity we assume that they are perfect substitutes.
9
Yt = [φ1 (L1t)ρ + φ2 (L2t)
ρ + φ3 (L3t)ρ]
1ρ , (8)
where the elasticity of substitution between labor types is ε = 11−ρ . and Cobb-Douglas, Leontief,
and perfect substitution are special cases of this function. The parameters φi are calibrated below
so that w1t < w2t < w3t holds under benchmark conditions.
There is one technical complication. Because the wage premiums w3t−w2t and w2t−w1t depend
on relative labor supplies, wage premiums could be negative for some immigration policies (e.g.,
if high θ3 raises L3t/L2t and reduces w3t relative to w2t). We rule out negative wage premiums
by assuming agents may work in any job with a lower skill requirement than their own; i.e., the
high-skilled can work in medium- or low-skilled jobs, the medium-skilled can work as low-skilled.11
This assumption ensures that wages satisfy w1t ≤ w2t ≤ w3t for all states and policies (Xt, θt).12
2.3 Redistributional Taxes
The model includes a simple tax-transfer system. Though the fiscal effects of immigration are not
our focus, the fiscal costs of low-skilled immigrants have been emphasized in the literature, and
hence their omission would raise questions.
To capture the fiscal impact of immigration in a simple way, we assume an exogenous tax rate
τ on wages that is redistributed lump-sum. Tax payments are τ · wit. The lump-sum transfer is
bt = τ · wt, wherewt =
Σ3i=1xit (1 + θit)witΣ3i=1xit (1 + θit)
is the average wage. High-skilled workers are net contributors to the system and low-skilled workers
are net beneficiaries, as they have above- and below-average wages, respectively. Medium-skilled
workers may have wages above or below the average, depending on relative productivities and
labor supplies. Hence their net contribution may be positive or negative.
Since wt depends on the elements of Xt and of θt, one may write wt = w (Xt, θt) .
2.4 Preferences
Utility of a skill-type i agent depends on its own consumption (cit) and on the expected utility of
their children (vjt+1). Consumption is derived from after-tax wages plus transfers:
cit = (1− τ)wit + bt, for i = 1, 2, 3. (9)
11This is consistent with the identification of skills by schooling level (as traditionally interpreted in the literature),because schooling is acquired sequentially. Ortega (2005) makes the same assumption in a setting with two skilllevels.12The job assignment is straightforward but tedious to formalize and therefore is relegated to the appendix. In
the calibration, wage premiums are strictly positive except in case of extremely large (unrealistic) immigration andfor Xt far from the steady state. Thus agents normally work at their own skill levels.
10
There are no bequests or other financial linkages across cohorts.
Overall utility for each type is obtained recursively from
vit = u (cit) + β
3∑j=1
qijvjt+1, for i = 1, 2, 3. (10)
where β > 0 is a scalar that governs the strength of the altruism motive; the sum can be interpreted
as expected value E [vt+1 (·) |i] =∑3
j=1 qijvjt+1 conditional on parental type i.
2.5 Equilibrium
The transition matrices and fertility rates in this paper are such that the medium-skilled remains
the majority each period, irrespective of the immigration quotas (justified with the empirical
demographic profiles in the calibration stage). Hence they dictate immigration policy every period.
The main equilibrium concept used is Markov perfect equilibrium (MPE), where the state
of the system is described by the composition of the native population (Xt). In the MPE the
equilibrium strategies are a function of the state but not of the history of the game. Under this
equilibrium concept -typically used in dynamic political economic games, agents fully incorporate
the expected induced effects on future policy due to changes in the current policy. In other words,
when making a voting decision the agents consider how future generations will respond to changes
in current policy.
For the medium skilled workers, who set policy, a strategy maps the state Xt into a policy
choice Θt ∈ ΩΘ, where ΩΘ is a compact policy space (e.g., one of the cases defined in Section
2.1). The medium-skilled have a majority if x1t + x3t < 1. To work with compact sets, we use
ΩX = (x1, 1, x3)|x1 ≥ 0, x3 ≥ 0, x1 + x3 ≤ 1 as domain of Xt.
An equilibrium is then defined as follows:
Definition. A politico-economic equilibrium is a policy rule P : ΩX −→ ΩΘ that defines
Θ = P (X) and a triplet of value functions (v∗1, v∗2, v∗3) such that for all X ∈ ΩX :
i) Given the policy rule P and implied rules θ = p (X) and θI = pI (X), continuation values
are given by
v∗i (X) = ui (X, p (X)) + β∑3
j=1qijv
∗j (Ψ (X,P (X))) (11)
for i = 1, 2, 3, where ui (X, θ) = u [(1− τ)wi (X, θ) + τ · w (X, θ)] and
Ψ(X, pI (X)
)= (S(pI (X))X)/(S[2](p
I (X))X) (12)
ii) The policy rule P solves:
P (X) = arg maxΘ∈ΩΘ
u2 (X, θ) + β
3∑j=1
q2jv∗j
(Ψ(X, θI
)). (13)
11
iii) Ψ(X, pI (X)
)∈ (x1, 1, x3)|x1 ≥ 0, x3 ≥ 0, x1 + x3 < 1.
The definition requires that the value functions v∗i , which are the expected lifetime utilities
of type-i agents, are consistent with the population process and with the state of the economy
induced by P . The policy P maximizes the expected lifetime utility of medium skilled natives
(type i = 2), which are the voting majority. Since types 1 and 3 do not control policy, their value
functions (v∗1 and v∗3) are computed under the policy set by type-2. The optimization takes into
account that type-2 offspring might be type-1 or 3 in the next generation, as well as the response
of the next generation to the induced state (which is Ψ(X, pI (X)
)). For completeness, condition
(iii) states that type-2 will retain its majority in the next period; this is not a binding constraint
in the analysis below.
Note that there may be multiple policies that solve (13) and yield the same utility. Notable,
wages are unchanged if migration is increased marginally at all skill levels in a way that relative
labor supplies (z1t, z3t) remain constant. Voter are generally not indifferent if this occurs through
immigration, because immigration impacts their children. Voters are indifferent, however, if labor
supplies were increased proportionally by guest workers. In our three policy settings, policies are
nonetheless generically unique: in (I) and (II) because there are no guest workers, and in (III)
because of a limited supply of high-skilled migrants.13
2.6 Static Analysis: β = 0 or Guest Workers Only
Dynamic effects would be absent if agents did not care about their offspring (if β = 0) or if all
migrants were guest workers (if θIit = 0 exogenously for i = 1, 2, 3). Under both assumptions,
agents would vote to maximize utility from current consumption. Since the medium-skilled are
the majority, policy in equilibrium would maximize c2.
We discuss these two special cases briefly, mainly to provide intuition; proofs and more details
(notably, tedious case distinctions) are relegated to the appendix. For the discussion here, assume
positive wage premiums and w2 ≤ w (which holds empirically).
Consumption depends on immigration policy through wages and transfers. On the margin,
∂ci∂θj
= (1− τ)∂wi∂θj
+ τ∂w
∂θj; i, j = 1, 2, 3. (14)
where time subscripts are omitted to reduce clutter. The wage effects ∂wi∂θjare negative for a guest
workers of the same type as the native and positive for a guest worker of a different type. Transfers
13To cover non-generic exceptions (e.g., if QI = Q and ηI = η), we adopt Ortega’s (2005) tie-breaker thatindifferent voters unanimously choose the policy with the minimal total number of migrants. This can be justifiedas selecting the unique policy that would be optimal with an infinitesimal cost of processing work permits orwith an infinitesimal element of decreasing returns to scale. The multiplicity issue motivates in part (apart fromplausibility) why we do not study guest workers combined with unlimited supply of high skilled migrants.
12
τw are increased if the guest worker has a skill that earns an above average wage ( ∂w∂θi
> 0 iff
wi > w), and decreased otherwise. Applied to the medium-skilled majority (c2), one finds:
(a) ∂c2∂θ3
> 0, because more high-skilled labor increases both w2 and w. Hence high skilled
migrants are admitted until θ∗3 = θmax3 .
(b) ∂c2∂θ2
< 0, because more medium-skilled labor reduces w2 and (under w2 ≤ w) reduces
transfers. Hence medium skilled migrants are not admitted, θ∗2 = 0.
(c) ∂c2∂θ1≶ 0 is ambiguous because more low-skilled labor increases w2 but reduces w. Hence θ1
may have corner solutions (0 or θmax1 ) or an interior solution (if ∂c2
∂θ1= 0 for some θ∗1 ∈ (0, θmax
1 )).
One can show that θ∗1 is decreasing in τ , as consistent with Razin et al.’s (2011) insight that welfare
discourages low skilled immigration.14
These findings determine unique immigration quotas when there are no guest workers (setting
(II) with β = 0). Then θI∗i = θ∗i (i = 1, 2, 3), which means high skilled immigrants are admitted,
medium skilled immigrants are not admitted, and low skilled immigrants may or may not be
admitted, depending on parameters.
Similarly, if all migrants were guest workers (θIi = 0 exogenously), one would obtain θG∗i = θ∗i ;
since guest workers leave, this applies for all β ≥ 0. High skilled guest workers would be admitted,
possibly low skilled guest workers, but not medium skilled guest workers.
Alternatively, suppose there are separate quotas for immigrants and guest workers (setting
(III)) and β = 0. Since wages depend on migration only through the sums θj = θIj + θGj , voters
are indifferent about immigrants versus guest workers.
Overall, the wage and tax effects documented in this section provide a motive for voters to
support high and (at not too high tax rates) low skilled migration. The indifference between
immigrants and guest workers for β = 0 shows that strict preferences for immigrants or for guest
workers in the general model must be driven entirely by voter concerns about their children.
3 Empirical Evidence
3.1 Intergenerational Mobility Matrices
Many studies about intergenerational mobility use transition matrices in order to analyze the
relative position of people in the income distribution compared to that of their parents. Such
studies typically suffer from possible life-cycle effects and a small number of observations (PSID,
NLS for example). Since we are interested in the intergenerational mobility of individuals based on
skills, we focus on the education levels of individuals and their parents in studying intergenerational
mobility.
14See appendix for more analysis. In principle, it is possible that θ∗3 < θmax3 (but only if w2 = w3) and/or thatθ∗2 > 0 (but only if w2 > w). Positive wage premiums require (in the static case, not in general) binding θmax3 ,which rules out setting (I).
13
We use the General Social Survey (GSS) for this matter, which is an annual survey since 1972.
Starting in 1977, this survey captures the schooling level of the respondent’s parents (in addition to
the respondent’s schooling). The survey also identifies whether the respondents and their parents
were born in the US. Hence, we can estimate transition matrices and perform some statistical tests
for natives and children of first generation immigrants.
We consider individuals who were born on or after 1945 and whose age at the time of the
interview was between 25 and 55 years old.15 The education variable used to classify individuals
is based on whether the individuals (either respondent or his/her parents) obtained any of the
following degrees: less than high school, high school, junior college, college and grad school. We
classify individuals as 2nd generation immigrants, whose transition matrix we are interested in, if
the respondent was born in the US but any of the parents were born outside the US.16 Natives in
turn are individuals whose parents were born in the US.
For the transition matrices we define an individual as "unskilled" if his/her education is less
than a high school degree, "medium-skilled" as either having a high-school or a junior college
degree, and "skilled" for individuals with college degrees and beyond. For individuals with in-
formation on both parents, we use the maximum degree obtained by any of them. Under these
filters, the total number of observations is 18, 999 for natives and 1, 447 for immigrants.
The transition matrices Q and QI are estimated from the GSS as follows. For each of their
elements qij and qIij , the estimate is the number of children with corresponding education level i
and parents with education level j divided by the total number of parents with educational level
j. The empirical frequencies differ somewhat across subsamples in the GSS (e.g. for men and
women; see appendix); but since they do not differ greatly -and we cannot reject that they are
different for native men and women, our main calibration uses results for all men and women,
given by
Q =
.256 .663 .081
.062 .707 .231
.010 .397 .593
, QI =
.211 .594 .195
.067 .633 .299
.022 .325 .653
. (15)
The first two rows of the transition matrices show that the children of unskilled and medium
skilled immigrants to the US appear to be more "successful" than natives. For unskilled parents,
children of immigrants have a lower probability of staying unskilled, and a higher probability of
15We cap it at 55 because of a well know relationship between mortality and education level, and also becauseusing older individuals from the early years of the GSS means using observations who were born early in the 1900’s.We use individuals born on or after 1945 as their average schooling years starting with that cohort has remainedapproximately constant (see the appendix for schooling statistics in the GSS).16We could classify second generation immigrants as those respondents born in the US whose both parents (rather
than only one) were born outside the US. We don’t do this because 1) the number of observations greatly decreasesfor second generation immigrants (from 1447 to 530) and 2) because the numbers don’t appear to be significantlydifferent according to the unskilled, medium-skilled, skilled classification used in this paper. The appendix showsthe estimated matrices under this definition.
14
upward mobility. Indeed, children of unskilled natives have an 8% probability of becoming skilled,
while for the children of unskilled immigrants it is almost 20%. Given medium-skilled parents, the
differences are not as marked as in the unskilled case but their odds seem to be slightly better.
We test formally if the probability distributions for natives and children of immigrants are
statistically the same, conditional on the skill of the parents. That is, we test if row i in matrix
Q is statistically the same as row i in QI . For all i, the null hypothesis is rejected at the 1%
level.17 We also test for the equality of both matrices, with a test given by summing over the
statistic for each row test.18 The null hypothesis that the transition matrices are the same is also
rejected at the 1% level. Further analysis with more disaggregated data (reported in detail in the
appendix) indicates that: (i) children of low-skilled immigrants are significantly more likely to
become high-skilled than the children of natives (qI13 > q13); (ii) children of natives and immigrant
with skilled parents have similar skill distributions for most partitions of the data; (iii) for children
of medium-skilled workers, differences between immigrants and natives are significant only for
women; and (iv) for all subgroups, intergenerational transition matrices of natives and immigrants
differ significantly.
3.2 Fertility Rates
In order to calibrate the number of children that agents have, we construct total fertility rates
by education level (TFR) for 3 different years: 1990, 2000 and 2005. This concept measures the
expected number of children that a woman would have in her lifetime if she was subject to the
current (cross-section) age-specific fertility profiles.19
Total fertility rates can be accurately calculated with birth data from the National Center
for Health Statistics (downloaded from their VitalStats system), and age-education groups from
Census data (1990, 2000) and the Current Population Survey (2005). However, the available data
on births doesn’t detail whether the mothers are US-born, or foreign-born. In order to arrive at
total fertility rates by place of births of mothers we also use information from several years of the
American Community Survey (ACS). Details on the construction of these estimates are in the
17Under the null, the statistic2∑
m=1nim
k∑j=1
(qij−qIij)2
qijis distributed chi-square with (k − 1) degrees of freedom
(k = 3), where nim is the number of counts of row i of sample m. The statistic for each row is to be comparedwith χ2(2) , which is 5.99 (9.21) at the 5% (1%) level of significance. The statistics are 58.74 for the first row, 18.63for the second and 10.16 for the third.18The null hypothesis that qij = qIij for all i = 1, 2, 3 and j = 1, 2, 3 (matrix equality) is rejected if the statistick∑i=1
2∑m=1
nimk∑j=1
(qij−qIij)2
qijis greater than χ2(k(k−1)) where the degrees of freedom are in this case k (k − 1) = 3 (2) = 6.
The test produces a statistic of 87.52, higher than χ2(6) = 16.81 which is the critical value at the 1% level. For moredetails on this test, see Amemiya Pp. 417 and Mood-Graybill-Boes Pp. 449.19Total fertility rates are constructed as follows. First divide the total number of births whose mothers are in
a specific age-education group (i.e. 20-24 year old mothers with 11 or less years of education) by the number ofwomen in that age-education group. The resulting number is multiplied by 5. Finally, those quantities are addedfor all age-groups for a given education level.
15
appendix.
Table 1. Total fertility rates by skill level. Years 1990, 2000 & 2005US-Born Foreign-Born All Women
Year Low Med High Low Med High Low Med High
2005 1.98 1.94 1.82 2.78 2.44 1.99 2.73 1.95 1.85
2000 2.24 2.00 1.59 3.04 2.50 1.76 2.26 2.00 1.84
1990 2.41 1.89 1.82 3.21 2.39 1.99 2.43 2.04 1.61
Average 2.21 1.94 1.75 3.01 2.44 1.91 2.47 2.00 1.77
The estimated fertility rates show the well-known negative relationship between education and
fertility, and also display that foreign-born women have higher fertility rates than US-born women
of the same skill level.
4 Calibration
4.1 Demographic Profiles
A population process is described in this paper by an intergenerational transition matrix (Q), a vec-
tor of fertility rates (η) and a vector of immigration quotas. In the absence of any immigration, we
denote the specific composition of the steady state native population by (xss1 , 1, xss3 ) = Xss (Q,η).
The model uses the matrices of intergenerational mobility Q and QI shown in equation (15).
For the fertility rates of the model, we divide by 2 the TFR’s shown in table 1 in order to obtain
implied model parameters. This yields η1 = 1.1, η2 = 0.97, η3 = 0.87 and ηI1 = 1.5, ηI2 = 1.22 and
ηI3 = 0.96.
One way to assess the quantitative significance of these differences between natives and immi-
grants is by examining the implied composition of the population. Table 2 displays steady state
ratios xss1 and xss3 implied by alternative assumptions about mobility and fertility.
Table 2. Steady State Composition of Population.Different
Mobility & Fertility
Mixed
Cases
(Q,η)(QI ,ηI
) (Q,ηI
) (QI ,η
)xss1 .0978 .1213 .1041 .1161
xss3 .5429 .7987 .5010 .8634
Using the data for natives (Q,η), assuming no immigration, one obtains xss1 = 9.78% (low
to medium skilled ratio) and xss3 = 54.29% (high to medium skill ratio). If a population had
(hypothetically) the mobility and fertility of first-generation immigrants(QI ,ηI
)forever, one
would obtain larger shares of the extremes in the skill distribution, with xss1 = 12.13% and xss3 =
16
79.87%.20 Considering populations that combine the mobility of natives with the fertility of
immigrants, or the fertility of natives with the mobility of immigrants (see columns 3-4), one finds
that differences in mobility are far more important than difference in fertility; that is, Xss for(Q,ηI
)is close to Xss for (Q,η), and Xss for
(QI ,η
)is close to Xss for
(QI ,ηI
).
4.2 Production, Preferences and Taxes
Production parameters are φ1, φ2, φ3 and ρ. The elasticity of substitution(ε = 1
1−ρ
)between
education groups has been carefully estimated in different studies that control for experience and
other observables for what is traditionally defined as high-skilled versus unskilled labor inputs,
as well as specifications with 4 skill types which correspond to the categories "less than high
school", "high school graduates", "some college" and "college graduates". When using the two-skill
specification, the empirical estimates in many studies range between 1.5 and 2.5 (see the references
in Ottaviano and Peri (2012) pp. 184), which would imply 0.4 ≤ ρ ≤ .66. In specifications with
more education types, the estimates range from 1.32 (Borjas (2003)) to the estimates in Borjas
and Katz of 2.42 (2007), with Ottaviano and Peri’s own estimates lying between those 2 extremes.
These other set of estimates imply 0.24 ≤ ρ ≤ .78. Even though the definition of the skill groups
is different, the intermediate value for ρ in these intervals is very close to ρ = 12, which we use as
baseline parameterization. We later perform sensitivity analysis.
The parameters φi are calibrated so that the model in steady state matches U.S. wage premiums
for educational attainment. We normalize φ1 + φ2 + φ3 = 1 and use the equations(φiφj
)=wi,twj,t
(Lj,tLi,t
)ρ−1
for i 6= j = 1, 2, 3, (16)
to relate wage premiums to calibrate the ratios φ2/φ1 amd φ3/φ2.
We use census data of the average hourly wage of workers that are between 25 and 65 years
old by educational attainment; that work at least 40 hours per week, and that worked at least 40
weeks in the previous year for census years 1990, 2000 and 2005. The skill categories are defined
in terms of schooling under the same definition as for the intergenerational mobility matrices. The
average wage ratios obtained are(
w2
w1
),(w3
w2
)= 1.315, 1.67.21
As noted above, the demographic profile of natives (Q,η) yields steady state ratios of xss1 =
0.0978 and xss3 = 0.5429. Taking into account immigration quotas of 18% for the unskilled
group, 6% for the medium skilled and 13% for the high skilled group yield labor ratios given
20We can also substitute the demographic profile of immigrants to those of natives, one skill-type at a time (i.e.substitute the demographic profile of the low-skilled immigrant for that of the low-skilled native, leaving the profileof the other natives unchanged). In all three hypothetical cases where we do this, the steady state shares of low-and high-skilled are greater than the shares implied by data on natives.21At a generational frequency the returns to skill have increased in the US over time. We do not attempt to
formally model these changes and so we calibrate the model by using the average ratios.
17
by(
L1
L2
),(L3
L2
)= .1089, .5788 . Using these data in (16), we calibrate φ1 = .0995, φ2 = .3966,
and φ3 = .5039.
For the period preferences, we assume log-utility (u(x) = ln x), which is a standard benchmark
in the literature. The sensitivity analysis will examine CRRA utility with alternative curvature
parameters.
In some initial parameterizations that explore the behavior of the model to alternative assump-
tions on the immigration choice space, we set the discount factor β as Ortega (2005), who uses
an annual value of β = .985 and that implies a model parameter of β = .98530 = .63546 when the
model-period represents about 30 years. In some versions of the model this parameter is calibrated
endogenously.
For the tax rate (and implied level of redistribution), we use an average tax rate of 30%,
approximately the current average from the series computed in McDaniel (2012) for labor and
consumption taxes for the US.22
4.3 Closing the Model: The Supply of Immigrants
As outlined in Section 2.1, we limit the policy space by making assumptions about the supply of
immigrants and guest workers. Having described the model, we can explain the assumptions in
more detail.
For the low-skilled, we model supply as effectively unlimited by setting θmax1 high enough so
that the maximum does not constrain immigration choices.
For the medium-skilled, arguments about θmax2 turn out to be moot, because in our calibrated
models the medium-skilled natives will not allow immigration at their own skill level. We set θmax2
to be a small non-zero number largely to verify that the optimization routine converges to zero as
optimal value.
For high-skilled immigration, we examine three policy settings, as follows:
(I) An "unrealistically large" pool of high-skilled immigrants. Our initial parameter-ization assumes that high-skilled immigration is essentially unrestricted, just like the other types.
Such large supply is arguably unrealistic, but instructive. Specifically, assume the supply of high
skilled immigrants θmax3 is large enough that it includes immigration rates that would lead to an
equalization of medium and high-skilled wages.23
(II) A "small" pool of high-skilled immigrants; no guest workers. Our preferredalternative is that the pool of high-skill immigrants is small enough to be a binding constraint,
22The particular series used are the average tax rate on labor income, average payroll taxes and the averageconsumption taxes. Using those series from McDaniel’s tax data from 1980 to 2010 results in a 28.7% average taxrate. We use a round number (30%). Sensitivity analysis on this parameter (not shown as the paper is alreadylong) show that small variations in the tax rate produce the same conclusions.23That is, a policy setting θ3 = θmax3 would imply w2t = w3t. For the parameterization of the initial model, a
high-skilled quota of about 200% would be needed in order to equalize wages between skilled and medium skilledagents.
18
and small enough to preclude wage equalization.
This case is interesting for several reasons. First, U.S. immigration laws have until recently
allowed high-skilled workers, especially those with advanced degrees, relatively easy access to
coming/staying in the country. Even though H1B visas are typically exhausted, people with
advanced degrees working in universities are exempt from the cap on H1B visas. Since this is
not a hard limit on skilled migration, this suggests that θmax3 can be calibrated from historically
observed levels of high-skilled immigration.
Second, a fixed value θmax3 is a polar case of a wage-elastic supply curve for high-skilled immi-
grants. In general, if countries compete with others for a pool of internationally mobile high-skilled
workers, each country will face an upward-sloping supply curve as function of the wage w3t. For
large, high-wage country like the U.S., supply is likely inelastic and well proxied by a constant.
(More general elastic labor supply is examined in Section 6.1; this is left as extension because
involves complications that would distract from the main analysis.)
Third, constrained high-skilled immigration is of interest for studying "piece-meal" immigra-
tion reforms, if one reinterprets the constraint as resulting from policy inertia. Notably, alternative
values of θmax3 (alternative reforms of high-skilled immigration) will have implications for subse-
quent policy choices over low-skilled immigration and guest-workers.
(III) A "small" pool of high-skilled immigrants with policy choice over guest work-ers. This case assumes the country has the ability to prevent (some) migrants from settling as
immigrants. Assumptions about the supply of high-skilled immigrants are as in case (II).
4.4 Results with a Large Pool of Skilled Immigrants
This section reports model results for policy setting (I). That is, we investigate which immigration
policies would be chosen by the majority in the absence of any meaningful constraint from the
supply side of immigration.
We interpret a model-period as 30 years, which is roughly the average age of mothers. Given
the 5.15 immigrants per 1000 natives annual estimate for unskilled workers, and 4.15 high-skilled
immigrants per 1000 high-skilled natives, the quotas for a 30 year equivalent are roughly θ1 = 18%
and θ3 = 13%.
We use a value function iteration algorithm in order to solve for the Markov perfect equilib-
rium (MPE) of the model, with discretized state and policy spaces and bilinear interpolation in
the evaluation of the value function.24 The state variables of the model are the ratios low-to-
medium-skill natives(x1 = N1
N2
)and high-to-medium-skill natives
(x3 = N3
N2
). Given the demo-
graphic profile of natives (Q, η), we obtain a steady state distribution of the skill types given byxSS1 , xSS3
= 0.09782, 0.5429. Then given the demographic profile of immigrants
(QI , ηI
)we
24Spline interpolation was also used. The results are identical to bilinear interpolation, but the computationalcost is much higher.
19
consider a grid in the state space that contains both the initial steady state and any possible future
steady state induced by the space of possible immigration policies.
We compute results for maximum immigration quotas of θmax1 = 450%, θmax
2 = 100% and θmax3
= 300%. These values are chosen high to ensure that the optimal choices will be in the interior of
the policy space [0, θmax1 ]× [0, θmax
2 ]× [0, θmax3 ] for all possible states.25
Table 3. Initial Parameterization
Demographic
Profiles
Production
and Taxes
Immigration
Pool and
Preferences
Q =
.256 .663 .081
.062 .707 .231
.010 .397 .593
[L1
L2
L3
L2
]=
[.1089
.5788
] θmax1 = 4.5
θmax2 = 1
θmax3 = 3.0
QI =
.211 .594 .195
.067 .633 .299
.022 .325 .653
[w2w1w3
w2
]=
[1.315
1.670
]β = .98530
η
ηI=
diag1.1, 0.97, 0.87diag1.5, 1.22, 0.96
τ = .30
The resulting optimal policy function implies extremely high immigration at high and low skill
levels. The optimal immigration quotas are θ∗1 = 281% and θ∗3 = 200% at the no-immigration
steady state, and the quotas are θ∗1 = 189% and θ∗3 = 101%, respectively, in the steady state with
optimal immigration. Medium-skilled immigration is always zero (θ∗2 = 0%).
These extremely high immigrations rates in this setting are clearly unrealistic. To put them
into perspective, note that 200% immigration would imply that immigrants are 2/3 of the labor
force at the respective skill level.26 The unrealistic results here mainly serve to motivate the cases
below that assume a more limited of supply of high-skilled immigrants.
Though the model overpredicts immigration, the optimal policy function has features that are
instructive and intuitive. Notably, the high-skilled quota θ∗3 is decreasing in x3 (the higher the share
of native skilled relative to the medium-skilled, the lower the demand for high-skilled immigration),
and is also increasing in x1, as more low-skill immigration makes the high-skill immigration more
25Greater or smaller maximum quotas would not change the results, provided the space considered doesn’t lead toa corner solution at a maximum value. At the steady state
xSS1 , xSS3
, high- and medium-skill wages are equalized
at high-skilled immigration of 198% or more, which explains why extremely high immigration quotas are neededto allow for wage-equalization.26We explored if alternative values of parameters (β, σ, ρ) might provide more plausible results, but we found
no combination of (β, σ, ρ), for which optimization with unlimited supply of high-skilled immigrant would yieldimmigration rates anywhere close to observed immigration rates.
20
valuable by increasing wages of the high skilled. The converse applies to unskilled immigration: θ∗1is increasing in x3 as more high-skilled immigration makes room for more unskill immigration by
not making unskilled immigration as expensive in fiscal terms, and θ∗1 is decreasing in x1, which
means that the higher the share of unskilled as percentage of medium skilled natives, the lower
the demand for unskilled immigrants.
We also studied a version of this model under a small constant marginal cost per immigrant that
is paid out of transfers. This could be justified in terms of externalities associated to the absortion
of very big immigration flows that are not captured in this parsimonious model (i.e. congestion
effects). Transfers in this case are b = b−cost∗[
Σ3i xiθi
Σ3i xi(1+θi)
], which reduces to the transfer equation
in the main text when cost = 0. For example, when the total cost of immigration is such that
it results in a loss of 1% of the initial transfer, the unskilled quota is reduced significantly, while
not reducing the skilled quota as much (to θ1 = 244% and θ3 = 198%, with an induced steady
state of θ1 = 171%, θ3 = 100%). Increasing this cost leads to lower overall immigration, with θ1
decreasing faster than θ3.
Taken literally, this section suggests that flows of high-skilled immigration in the U.S. are far
less than optimal. This is consistent with the strong political support for increased high-skilled
immigration in recent discussions about immigration reform.27 An alternative interpretation is
that this section’s implicit assumption of an elastic, effectively unlimited supply of high-skilled
immigrants is questionable. It leads to the implausible result that in equilibrium, high skilled
immigration reduces the wage premium for high-skilled work to zero, yet the supply of high-
skilled workers is assumed to be unaffected (given by an unchanged θmax3 ). A more plausible
assumption is that the supply of high-skilled immigrants is a limiting factor; this is examined in
the next section..
4.5 Results with a Small Pool of Skilled Immigrants
This section reports model results for policy setting (II). That is, we assume a perfectly inelastic
supply of high-skill immigration θmax3 that is not large enough as to equalize wages between medium
skilled and high-skilled workers; for this section, we assume no guest workers.
We perform experiments with two different levels of θmax3 . The first one is θmax
3 = 13% which is
the level that has been observed in the US for the analyzed period and that could represent a supply
side constraint (in light of the results of the above section), or more likely represents some other
exogenous constraint. The second value is θmax3 = 30%, which is large relative to historical levels
of US immigration and represents a scenario for studying policy reform. Under this interpretation,
13% would be suboptimal and studying the larger limit helps understand the effects of a policy
reform. These specific values matter only for the magnitudes of the chosen policies, but not for the
27As noted above, we view Trump administration expressions of intent to reduce high-skilled immigration asunlikely to succeed and/or intended to prevent medium skilled immigration under provisions meant for the highskilled, which would be consistent with the model.
21
qualitative effects since other levels of θmax3 also yield the same (qualitative) predictions provided
that high skill immigration is relatively scarce (as opposed to the assumption in the previous
section).
The parameter β can be calibrated to yield θ∗1 = 18% as it is found that θ∗1 and β are inversely
related (i.e. if β = 0, we obtain much higher low-skilled immigration than when β > 0). We
calibrate this parameter as β = .6312. With a higher β, more weight is given on the expected
utility of children, and agents pay attention to the bad state "low-skilled". The majority therefore
chooses less low-skilled immigration than otherwise. We label this case with θmax3 = 13% as the
"baseline" since this model reasonably and parsimoniously allows for the analysis of many issues
in the subsequent sections; while the qualitative predictions are robust to changes that are later
discussed.
The optimal policy function consists of (1) maximizing high skill immigration, (2) minimizing
medium skill immigration and (3) choosing a quota of low-skilled workers that is decreasing in x1
and increasing in x3. In words, the higher the number of low- relative to medium-skilled natives,
the lower demand there is for low-skilled immigration. There is less "room" for new low-skilled
workers whose positive effects on medium-skilled wages would be smaller but their net benefit
from transfers possibly larger. Similarly, the higher the number of high-skilled agents relative to
medium-skilled, the higher the equilibrium quota of low-skilled immigrants since the wages of low
and medium-skilled are higher and this in turn allows for more immigration from a redistributive
standpoint. In our calibrated model, medium-skilled workers are not accepted in equilibrium as the
"cost" of allowing some of them (lower current medium-skilled wages) outweighs future benefits.
Even though in reality we do observe some immigration of the medium-skilled type, in general it is
"small" when compared to the same category in the host economy and hence the model captures
the stylized fact of mostly observing immigration of the extremes in the skill distribution as an
equilibrium result.
These optimal policies induce a steady state with higher shares of the low-skilled and high-
skilled natives relative to the medium-skilled majority, with x1 = .10 and x3 = .589 (as opposed to
the initial steady state with xss1 = .0978 and xss3 = .5429). The slightly higher share of low-skilled
natives induces less unskilled immigration, but there’s an opposite effect due to the higher share
of skilled natives, for a total effect at the induced steady state of θ1 = 17.6% and θ3 = 13%.
Regarding the equilibrium effect of θmax3 on the immigration quotas of the other skill-types,
we find that the majority chooses the maximum high-skilled quota (θmax3 ), together with more
unskilled immigration. When θmax3 = 13% the low-skilled quota is θ∗1 = 18%, while the case with
θmax3 = 30% yields θ∗1 = 21.6% (and an induced steady state of θ1 = 29%, as opposed to 17.6%,
due to the higher share of high-skilled agents in the native population). Hence, a "piece-meal"
approach that relaxes first the amount of high-skilled immigrants would also lead to a politically
motivated higher demand for low-skilled immigrants.28
28We also analyze a case where we completely shut down high-skill immigration (θmax3 = 0). In this case the
22
The optimal policies are "state-contingent". If the country is in a state with a very small share
of low-skill natives, there would be (higher) demand for low-skilled immigration. The "bracero"
program that the US signed with Mexico in 1942 could be an example of this. Similarly, illegal
immigration (typically confined to low-skilled jobs) could also be interpreted as a demand side
factor (at least partially) as the native low-skilled population in the US as percentage of the labor
force has been shrinking over time.
4.6 Results when Guest-Workers are Available
In this section we enlarge the policy space to allow for the possibility of guest worker quotas, in
addition to immigration quotas, following policy setting (III). In the model, the only difference
between guest workers and immigrants is that immigrants affect the future composition of the
native population because they have children, while guest workers have a zero fertility rate (they
return to their home country). We perform this exercise under the exogenous limit on high-skilled
immigration, and in the sensitivity section we repeat the analysis with a wage-elastic supply for
high skill immigration.
Using the same (baseline) parameterization as in the previous section but allowing for 3 ad-
ditional choice-variables (guest worker quotas θG1 , θG2 , θ
G2 ), we find that the median voter chooses
full immigration to the available pool of high-skilled immigrants (no guest worker for them), no
immigration/guest worker for the medium-skilled, and for the low-skilled a positive quota of guest
workers (without immigration). We discuss the reasons below.
The median voter would offer full immigration to all available high-skilled agents (θ∗3 = θI∗3 =
13% with θG∗3 = 0%). As before, high-skilled workers —both immigrants and guest workers —are
desirable because their skills are complementary to medium-skilled voters. The voter preference
for immigration over guest workers is a notable result that relies on the estimated intergenerational
mobility matrices. Children of medium-skilled native have a high probability of being medium-
skilled like their parents (high q22 = .707) . High-skilled immigrants have a high probability of
having high-skilled children and a low probability of having medium-skilled children (qI33 = .653 vs
qI32 = .325). Hence medium-skilled voters can anticipate that the children of high-skilled migrants
would raise the expected utility of their own children, and hence lets high-skilled agents enter as
immigrants rather than guest workers.
Technically, the preference for high-skilled immigration over guest workers depends on all
elements of the intergenerational mobility matrices, as voters weight all possible combinations of
skill types for their children and for immigrants’children. The result is nonetheless quite robust
in the sense that large changes in intergenerational mobility would be needed to overturn it. For
example, suppose the children of high-skilled immigrants were less skilled in the sense that qI33 is
unskilled quota would be slightly lower in the initial steady state (θ∗1 = 17% as opposed to 18% when θmax3 = 13%),and because in the long run there wouldn’t be as many skilled individuals as in the baseline, the long run quota ofunskilled individuals is even lower (10%).
23
lower and qI32 is higher than in the estimated QI matrix, holding all other elements constant. We
find that immigrants are preferred provided qI33 ≥ .32 and qI32 ≤ .65, whereas guest workers would
be preferred (i.e. θ∗3 = θG∗3 = 13% and θI∗3 = 0) if qI33 < .32 and qI32 > .65. Thus the likelihood
ratio qI32/qI33 would have to more than quadruple (from .325/.653 to .65/.32) for voter preferences
to be reversed
In the case of the medium-skilled, allowing for guest workers doesn’t change the results since
there would only be "costs" of allowing guest workers of the medium-skill type (lower wages),
while there would be no benefits (no possibly advantageous change in the future composition of
native workers) for the medium-skilled native.
The main changes are observed in the low-skilled category as the majority prefers for them
guest worker permits as opposed to full immigration. When comparing across regimes, the quota
of low-skill guest workers is higher than the low-skill immigration quota when guest worker per-
mits are not available: one obtains θ∗1 = θG∗1 = 87% and θI∗1 = 0, whereas the model without
guest worker programs produces θ∗1 = θI∗1 = 18%. There are two reasons for this. First, low-skill
immigrants have a much higher fertility rate than natives. Hence, allowing low-skilled immigrants
can affect more easily the size and composition of the (future) native population than allowing the
same number of individuals of a different type; and second, low-skill immigrants have a majority
of children that become medium-skilled, which in turn would most likely compete with native
children of medium-skilled. When the dynamic effects of low-skill immigration are removed (via
allowing guest workers), the medium-skill majority allows low-skilled guest workers until the mar-
ginal benefit (higher wages for medium-skilled) equals the marginal cost (redistribution) to these
workers, everything else constant.29
For the interpretation, note that admitting low-skilled guest workers is economically equiva-
lent to a policy of tacitly tolerating low-skilled illegal immigrants/undocumented workers. One
way to implement such a policy is by neglecting border controls combined with measures that
exclude these individuals from medium-and high-skilled jobs, e.g., background checks of licensing
requirements. Thus the voting equilibrium in this section resembles US immigration policy, which
permits high-skilled immigration and seems to tolerate undocumented immigrants, provided they
do low-skilled work.
Also note that admitting a succession of low-skilled guest workers (without children) would
be equivalent to admitting low-skilled immigrants if all their descendents remained low-skilled;
for example, if their children were given low quality education. It may be an interesting issue
for future research to examine if the changing educational opportunities of immigrant children
(e.g., the end of bilingual education in California) have influenced voter support for immigration.
Our analysis implies, for example, that voters will be less tolerant of low-skilled immigration if a
29Another way to see this is the following: if a guest worker program is not feasible, then the immigration quotafor unskilled workers would be lower than the quota of unskilled guest workers due to the future competition thattheir children represent for the medium-skilled.
24
greater proportion of their children is medium-skilled rather than low- or high-skilled.
5 Using the Model
5.1 Is Intergenerational Mobility of Immigrants Important for Immi-
gration Policy?
In this section we describe the effects of changing entries in the transition matrix of immigrants
QI in the baseline case (θ3 ≤ 13% = θmax3 ). These effects are qualitatively robust when we
perform this exercise under a wage-elastic supply for high-skill immigration, which we discuss
later. Since we cannot change individual entries in Q without affecting other entries in that row
(i.e. ∆qI11+∆qI12+∆qI13 = 0), whenever we change a probability, we leave the ratio of the remaining
two probabilities unchanged. We use a 5 percentage point increase in each entry of the matrix QI
and document the results.
The most significant changes in equilibrium low-skilled immigration come from changes in the
probability distribution of low-skilled agents. In particular, an increase of 5 percentage points in
qI13 (which represents the probability that an low-skilled parent has a high skill child) increases θ∗1
to 42% in the initial steady state (from a baseline of 18%), with a long run quota (at the induced
steady state) of 34.5%. In turn, modifying qI11 changes the low-skilled quota marginally (decreases
to 15%), while an increase in qI12 would result in much less low-skilled immigration of 5% (8%
in the long run). We explain these results. Higher qI12 leads to lower low-skilled immigration
because those immigrant’s children would compete in the most likely scenario with the children
of the native medium-skilled majority (there is a 70% probability that children of medium-skilled
natives remain medium skilled). In turn, higher qI13 leads to more low-skilled immigration as the
possible complementarity of the children of immigrants with natives who are in states "low-skilled"
or "medium-skilled" outweighs the cost of the competition if the children of natives were in the
"high-skilled" state.
Changing the probability distribution of high-skill immigrants affects immigration quotas very
little at the initial steady state. However, at the induced steady state there’s a significantly lower
low-skill quota when qI31 is increased (decreases to θ∗1 = 12%), which happens because the high-
skill immigrants have also a higher probability of having low-skilled children than natives, thus
lowering the future demand for low-skill workers.
Changes to probabilities for medium-skill parents don’t affect the equilibrium quotas since
the medium-skilled quota is zero in the model and the "small" changes in probabilities that we
consider are not enough to produce a positive medium-skill quota.
25
5.2 What if Immigrants Had Identical Demographic Profiles to Na-
tives?
From the previous section, it is clear that our baseline results are driven in part by differences in
the transition matrices Q and QI . In this section we study the effects in equilibrium immigration
when we impose the demographic profiles of natives to immigrants, given the calibration in the
previous section. Holding fertility rates constant at their estimated levels (η,ηI) if we setQI = Q,
equilibrium low-skilled immigration is adversely affected because immigrants seem to have better
upward mobility odds than natives (though this might not be true for every immigration group
within a skill level, it is true in average30). Hence with lower mobility more children of low-
skilled immigrants would be in states that turn out to be undesirable for the medium-skilled
majority. Quantitatively, the low-skilled quota is shutdown (0%) at both the initial and induced
steady states. Repeating this experiment but with identical fertility doesn’t change the qualitative
results, but that model requires a different calibration for β since the calibration is contingent on
η and ηI . We don’t discuss this further as there’s not any extra insight gained from this. The
main point is that higher upward mobility of low-skilled immigrants leads to higher demand for
low-skilled immigration.
In order to better understand these results we also examine the effects of replacing one row at
a time in QI by the respective row in Q, thus imposing the mobility of natives to immigrants. For
low-skilled agents (QI[1] = Q[1], while QI
[j] 6= Q[j] for j = 2, 3), this exercise results in a shut-down
of low-skilled immigration (0% in both the short and long run). This result is robust to either
using identical fertility for natives and immigrants, or using the estimates for differential fertility.
The fact that natives have a higher share of low-skilled children than immigrants helps explain
this. The medium-skill majority demands less low-skill immigration since it would imply more
competition for their children in the low-skill state, and at the same time less complementarity of
the children as upward mobility of immigrants would be lower.
If children of high-skill immigrants have the same transition probabilities as natives, this pro-
duces more demand for low-skill immigration (θ∗1 = 20% as opposed to 18%). This is so because the
estimated probability of having low-skilled children is (slightly) higher for high-skilled immigrants.
This implies a lower number of low-skilled natives over time (as well as less high-skilled natives)
and the overall effect is to demand slightly more low-skilled immigration.31 Changing the medium-
skilled distribution doesn’t change the low-skilled immigration quota because in equilibrium there
is no medium-skilled immigration in our models.
From this exercise we conclude that more successful children of low-skilled immigrants lends
30The information available from the GSS is too small as to make inferences of different groups.31Holding constant the mobility matrices at their estimated levels, we can also study the effects of modifying
the fertility rates of each type of agent. The effect of fertility on low-skill immigration is significant. If low-skillimmigrants had children at the same rates as unskilled natives, equilibrium immigration of the low-skill types wouldbe higher, with a quota of 33% rather than 18%. The fertility of the medium skilled immigrant is not relevant,while the fertility of the high-skilled affects the low-skilled quota slightly (but not the high-skill immigration).
26
political support to a bigger low-skill quota, and the opposite is also true. And given current
flows of high-skilled immigration, mobility appears to be not as important as in the unskilled
case, altough as previously mentioned it is important for the composition of that flow: the choice
over high-skill immigration vs high-skill guest workers depends importantly on intergenerational
mobility of the high-skill immigrants.
6 Sensitivity Analysis
6.1 The Model Under a Wage-Elastic Supply of High-Skill Immigra-
tion
In this section we show that the qualitative predictions of the baseline model (setting (II) with
θ∗3 ≤ θmax3 = 13%) are robust to using a wage-dependent supply of high-skilled immigrants. The
particular functional form used for this exercise is
θmax3 (w3) = γ1
(w3
w3
)γ2
, (17)
where γ2 is the elasticity of θmax3 with respect to w3, the constant w3 is a reference point that
helps fixing the supply of high-skill immigrants to go through the point θmax3 (w3 = w3) = γ1 in
the space of w3 and θmax3 . This ensures that when considering alternative elasticity levels (γ2), the
supply curve goes through the reference point (w3, θmax3 (w3)), but the response of θmax
3 away of
that point depends on γ2.32
For the numerical choices of the parameters, it is not possible to calibrate them when it is
assumed that the observed immigration choices are suboptimal. Thus we proceed by studying
optimal choices to different elasticity scenarios. The elasticities considered range from 0 to 10,
where the 0 elasticity case was already analyzed in settings II (fixed supply).
The level of reference w3 used is the steady state wage of the high-skill agents in the absence
of immigration.33 Finally, the parameter γ1 has to be higher than the observed 13% (which is
observed inclusive of immigration) which we set at 30%. We tested other values for γ1 and obtain
qualitatively identical results. We don’t report other cases with the goal of keeping the paper as
32We also studied another specification of the supply for high skilled immigrants where θmax3 depends on theratio (w3/w2) (rather than on w3), where θ
max3 is positive provided that (w3/w2) is greater than or equal an
exogenous level given by (w3/w2). Since the ratio (w3/w2) depends negatively on skilled immigration (both raising
w2 or decreasing w3 would decrease θmax3 ) the agent would choose a level θ3 such that (w3/w2) = (w3/w2). Many
policies (θ1, θ2, θ3) yield the static condition (w3/w2) = (w3/w2), but the agent would rank policies according tothe dynamic effects. Under this specification the results were qualitatively similar; hence we don’t discuss it morethoroughly, and also because the immigrants would presumably respond to wages (w3) rather than to (w3/w2).33This number is w3 = .5424. It could be possible to back up w3 from some reference wage in the source countries
(i.e. average home-country wage of skilled workers migrating to the US). But doing this is a moot point since thelevel γ1 can’t be identified when θ3 is suboptimal and for any level w3 and given an elasticity γ2, the constant γ1can be normalized as to give the same supply with a different level w3.
27
short as possible. The results are very similar to the simpler versions of the model as we still
obtain immigration of the extremes, but quantitative results are slightly different. Under the
"high" elasticity scenario we obtain less skilled but more unskilled immigration than in the other
cases: the higher the elasticity, the bigger the quota response to the effect in wages. Thus in the
interest of keeping high-skill immigration at a higher level, the majority allows even more unskilled
immigrants than when the supply side is less responsive. Table 4 summarizes the numerical results.
Table 4. Immigration quotas under responsive supply (%)Elasticity of Supply: γ2 = 0 γ2 = .25 γ2 = 1 γ2 = 4 γ2 = 10
Initial steady state θ∗1(%) 26.2 27.8 29.0 35.7 42.7
θ∗3(%) 30 29.5 28.2 24.4 20
Induced steady state θ∗1(%) 30.2 31 31.9 33.5 34.8
θ∗3(%) 30 29.2 27.2 22.0 16.9
The effects on skilled migration are more pronounced at the induced steady state. Since skilled
immigrants have a high probability of having skilled children, increasing skilled immigration leads
to skilled wages that are also lower in the future (ceteris-paribus) by increasing the share of skilled
agents in the (future) native population. Since skilled wages are lower, less skilled agents are
willing to immigrate the more elastic the supply is. Low-skilled immigration responds in the
opposite direction: the higher γ2 is, the higher the equilibrium unskilled quota since this helps to
attract a higher number of skilled immigrants.
The optimal policy function for low-skill immigration is decreasing in x1 and increasing in x3
-as before, but now the optimal policy for high-skill immigration is decreasing in x3, and slightly
increasing in x1 (similar to the results of the model under a very big pool). In words, the higher
the native share of high-skilled agents, the higher the demand for low-skilled immigration and the
lower the high-skill quota. Similarly, the higher the share of low-skilled natives, the lower the
demand for low-skill immigration, and the higher the demand for high-skill immigration.
We also look at the effect of replacing the transition matrix of immigrants by the one of natives.
We find similar qualitative effects as before since replacing the probability distribution of low-skill
immigrants (holding fertility at its estimated levels) by those of natives also shuts down low-skill
immigration. Using the whole demographic profile of natives instead of the one estimated for
immigrants (fertility + mobility) affects mostly low-skill immigration, by reducing it (from 29%
to 15% in the unit elasticity case, and from 42% to 29% in the high elasticity case of 10).
Allowing for guest-workers in addition to immigration produces similar results as before: full-
immigration-only offered to high-skill immigrants, while low-skilled workers are only offered guest
worker permits. Quantitatively, in the case with a supply-elasticity of 1 (10) , the quotas chosen
are θG∗1 = 99% (93%) for the low-skilled-guest worker type, with a high-skill full-immigration
quota of θI∗3 = 28.4% (21%). The induced steady state has θG∗1 = 134% (109%) and θI∗3 = 27.2%
28
(16.7%). Similar to what was found before, the unskilled guest-worker quota is higher than the
unskilled immigration quota when guest worker programs are not available. Similar numbers are
found for the other elasticities considered.
6.2 Different degree of substitution in the Production Function ( 11−ρ)
We perform sensitivity analysis in the value of the elasticity of substitution between labor inputs,
given by ε = 11−ρ . We do it for ρ = 2
5and for ρ = 3
5, implying elasticities of ε
(ρ = 2
5
)= 1.66 and
ε(ρ = 3
5
)= 2.5 respectively. Using the same wage premia and labor ratios as before, we obtain
share parameters(φ1, φ2, φ3
)given by (.0836, .4160, .5004) when ρ = 2
5and (.1180, .3766, .5054)
when ρ = 35.
Just like before, the unconstrained model with a "huge" θmax3 yields quotas that are very high.
When ρ = 25, we obtain θ∗1
(ρ = 2
5
)= 352% and θ∗3
(ρ = 2
5
)= 227%. Alternatively, when ρ = 3
5we
obtain θ∗1(ρ = 3
5
)= 261% and θ∗3
(ρ = 3
5
)= 325%.34 Summarizing this information, the unskilled
quota is smaller when the elasticity of substitution is higher (for ε(ρ = 3/5) = 2.5), while the
opposite is true for high-skilled immigration. With higher substitution, both unskilled and skilled
migration become less attractive from the point of view of current and future complementarity,
while fiscal issues become more important. This shifts demand towards skilled migration but
overall similar to the baseline case.
For our main parameterization of the model (Setting (II) —inelastic supply θmax3 = 13%), we
calibrate β (ρ = 2/5) = .875 and β (ρ = 3/5) = .385. The comparative statics are as before, but
the specific magnitude of the effects differ. Replacing the mobility of immigrants by those of
natives shuts-down the demand for low-skill immigration. In both cases we obtain θ∗1 = 0. This is
driven by the mobility of the low-skilled immigrants (first row in QI) as those results are obtained
whenever we replace their mobility probabilities with the natives’probabilities. The comparative
statics for variations in the transition matrices yields identical qualitative results to the baseline
case, but with unskilled quotas that are more sensitive to these changes when ρ = 2/5.
As in previous sections, we find the result of a guest-worker program for low-skill immigrants,
and full immigration to the high-skilled ones (Setting (III)). Even though the numerical results
might differ somewhat, our findings are robust to the effects of this parameter.
6.3 Other levels of Relative Risk Aversion
Here we document the results of using CRRA preferences, given by u(x) = (x1−σ − 1) / (1− σ), as
opposed to the baseline case which corresponds to σ = 1 (log-utility). The behavior of the model
34Introducing a small cost of immigration in the same magnitude as previously done (that overall represents1% of transfer once all immigrants are taken into account), has the same effect as before since the magnitudesdecrease significantly to θ∗1
(ρ = 2
5
)= 256% and θ∗3
(ρ = 2
5
)= 153%, and in the case where ρ = 3
5 we obtainθ∗1(ρ = 3
5
)= 214% and θ∗3
(ρ = 3
5
)= 286%.
29
for other realistic levels of risk aversion (1 ≤ σ ≤ 4) leaves the qualitative results unchanged,
with just some small differences in magnitudes. Because of this, we briefly discuss the other level
typically used in the literature, which is σ = 2.
In Setting (I) (the case with unlimited supply of skilled workers), the chosen immigration quotas
are almost identical θ∗1 (σ = 2) = 277% and θ∗3 (σ = 2) = 200% as opposed to θ∗1 (σ = 1) = 281%
and θ∗3 (σ = 1) = 200%.
In Setting (II) (inelastic supply at θmax3 = 13%), and using σ = 2, we calibrate β = .5625 in
order to obtain θ∗1 (σ = 2) = 18%. We obtain identical comparative statics as before, with very
small differences in magnitudes. If the policy space is enlarged to include guest-worker programs
(Setting (III)), we again obtain full immigration for the high-skill and guest worker for the low-
high-skilled, with practically identical numbers as before(θG1 = 87%, θI1 = 0%
).
6.4 An Alternative Definition of the Medium-Skill Group
In this section we consider a different definition of the medium-skill group. We now put high-school
graduates with people who have a junior college or bachelor degrees in the middle category, with
low-skilled defined just like before, while high-skilled is defined as those with a master degree and
above.
The empirical transition matrices under the same filters as before yield
Q =
.256 .717 .027
.053 .875 .073
.007 .729 .264
& QI =
.211 .720 .069
.060 .827 .113
.013 .671 .315
,
where we still observe higher upward mobility for immigrants than from natives.
Under the new definitions we compute that during the 90’s there were 2.43 medium-skilled
immigrants per 1000 medium-skilled natives, 4.3 high-skilled per 1000 high-skilled natives, while
the low-skilled quota remains the same. These numbers yield θ3 = 13.75% for a model-period of
30 years, with θ1 = 18% as the definition of low-skilled remains constant, and θ2 = 7.5% .
Given the new skill definitions we obtain wage premia of w3
w2= 1.727 and w2
w1= 1.4845 for the
same samples and filters as previously discussed (except education). Using ρ = 1/2, the production
share parameters (φ1, φ2, φ3) are calibrated as (.1110, .5704, .3185). Total fertility rates using the
ACS for the years 2001-2008 yield 2.41, 2.04, 1.97 for native women and 3.23, 2.44, 2.08 forforeign-born women. This implies model parameters of η = diag 1.21, 1.02, .99 for natives andηI = diag 1.62, 1.22, 1.04 for immigrants. Given the demographic profiles of natives, the steadystate ratios in absence of immigration are xss1 , xss3 = 0.0760, 0.0988 . Naturally, in this casethere is a much bigger share of medium-skilled agents than in the case discussed in the main text.
We use the same tax rate used as before (τ = 30%) and evaluate the model for preference
parameters given by log-utility and β = .98530. In the case with a large supply of high-skilled
30
workers (Settings (I)), we obtain similar results as before with θ∗1 = 222% and θ∗3 = 231%, and
induced quotas of 143% and 132% respectively at the new steady state. A small immigration cost
as previously assumed yields somewhat smaller quotas given by θ∗1 = 207% and θ∗3 = 219%.
In Setting (II) (inelastic supply at θmax3 = 13.75%) and the rest of the parameters unchanged
produces qualitatively similar results as in the baseline, but with a larger share of low-skilled
immigration (θ∗1 = 141% at initial steady state and 86% in the induced one).
The comparative statics work in the same direction as previously (i.e. the unskilled quota
depends importantly on the transition matrix of low-skilled immigrants). When replacing proba-
bility distributions of immigrant types by those of natives we also obtain similar qualitative results
(i.e. using the probability distribution of low-skilled natives rather than the one for immigrants
yields less demand for low-skilled immigration). In Setting (III) we find again full-immigration for
high-skill individuals, and guest-worker programs for the low-skilled.
7 Conclusions
We study a dynamic macroeconomic model of intergenerational mobility and immigration with
three types of labor. In it, the demographic process is such that the medium-skill type is always
the majority. We parameterize several versions of the model and study the results.
We calibrate the model for the US and among other things find that children of low-skill
immigrants and medium-skill immigrants seem to be more "successful" than the children of natives
(there is a higher probability for their children to become high-skilled), using data from the General
Social Survey. We find the Markov Perfect equilibrium of the model and the optimal policy shows
that if the children of low-skilled immigrants are more successful than those of low-skilled natives,
there is more political support for low-skilled immigration by the majority (the medium-skilled).
The most preferred policy for the majority involves maximizing high-skill immigration, minimizing
medium-skill immigration and for low-skill immigration it is increasing in the share of high-skilled
natives and decreasing in the share of low-skilled natives.
In general, the current effect on wages and transfers from the high-skill agents are very impor-
tant for the welfare of the medium-skilled and thus their intergenerational mobility is relatively
unimportant for the political support of high-skilled immigration.
Under the interpretation that the observed flow of high-skill immigration is suboptimal, a
"piece-meal" approach to immigration reform allowing more high-skill immigrants would also lead
to an increase in the demand for low-skill immigrants.
If in addition to full immigration there are guest workers quotas available as policy tools, the
mobility matrices are such that the medium-skilled would choose a guest-worker quota for low-skill
immigrants, and full-immigration for the high-skill immigrants. Only if high-skill immigrants had
a very large probability of having medium-skilled children (as opposed to a high probability of
having high-skill children) would then the majority prefer guest worker permits for the high-skilled.
31
Finally, a version of the model with a positive-sloped supply of high-skill immigrants produces
similar results, and it is found that the higher the elasticity of supply, the higher the level of
low-skill immigration chosen optimally. Sensitivity analysis shows that the effects of the model
are robust.
References
[1] Anderson, T. W. and Leo Goodman (1957) “Statistical inference about Markov chains”.
Annals of Mathematical Statistics, Pp. 89-109.
[2] Amemiya, Takeshi (1985). Advanced Econometrics. Harvard University Press.
[3] Ben-Gad, Michael (2004) "The Economic effects of Immigration - a dynamic analysis". Jour-
nal of Economic Dynamics and Control 28 Pp. 1825-1845.
[4] – – (2008) "Capital-Skill Complementarity and the Immigration Surplus". Review of Eco-
nomic Dynamics 11. Pp. 335-365.
[5] Benhabib, Jess (1997) "On the Political Economy of Immigration". European Economic Re-
view 40. Pp. 1737-1743.
[6] Behrman, J., A. Gaviria and G. Szekely (2001) "Intergenerational mobility in Latin America".
Inter-American Development Bank working Paper #452, June.
[7] Benabou, Roland and Efe Ok, (2001). "Social mobility and the demand for redistribution:
the POUM hypothesis". Quarterly Journal of Economics.
[8] Bohn, Henning and Armando Lopez-Velasco (2015) "Immigration and demographics: can
high immigrant fertility explain voter support for immigration?". February 2015.
[9] Borjas, George (1992) "The intergenerational mobility of immigrants" NBER working paper
3972.
[10] Borjas, George (2003). “The Labor Demand Curve is Downward Sloping: Reexamining the
Impact of Immigration on the Labor Market”Quarterly Journal of Economics, 118, 1335—
1374.
[11] – – (2009). "The Analytics of the Wage Effect of Immigration". NBER working paper 14796.
[12] Borjas, George and Lawrence Katz (2007) "The evolution of the Mexican born workforce in
the United States". In Mexican Immigration to the United States, edited by George Borjas.
National Bureau of Economic Research Conference report, Cambridge, MA.
[13] Card, David (2005) "Is the New Immigration Really So Bad?" NBER working paper 11547.
32
[14] – – , John DiNardo and Eugena Estes (2000) "The more things change: Immigrants and the
Children of Immigrants in the 1940s, the 1970s, and the 1990s" in George Borjas, ed. Issues
in the Economics of Immigration. Chicago: University of Chicago Press.
[15] – – , Christian Dustmann and Ian Preston (2009) "Immigration, Wages, and Compositional
Ammenities". NBER working paper 15521.
[16] Cohen, A., A. Razin and E. Sadka (2009) "The Skill Composition of Migration and the
Generosity of the Welfare State". NBER Working paper 14738. February 2009.
Djajic, Slobodan (2014), "Temporary emigration and welfare: the case of low-skilled labor".
International Economic Review, Vol. 55, No. 2, May 2014.
– – , Michael S. Michael and Alexandra Vinogradova (2013) "Migration of high-skilled work-
ers: Policy interaction between host and source countries". Journal of Public Economics 96,
Pp 1015-1024.
[17] Dolmas, Jim and Gregory Huffman (2004). "On the Political Economy of Immigration and
Income Redistribution". International Economic Review Vol 45, Issue 4. Pp. 1129-1168.
[18] Dustmann, C. and Ian Preston (2005), "Is Immigration Good or Bad for the Economy?
Analysis of Attitudinal Responses". Research in Labor Economics, vol. 24, pp. 3-34, 2005.
[19] Gonzalez, L., and Francesc Ortega (2011). “How Do Very Open Economies Absorb Large
Immigration Flows? Evidence from Spanish Regions.”Labour Economics 18, pages 57—70.
[20] Hassler, J., J. V. Rodriguez-Mora, K. Storesletten and F. Zilibotti (2003) "The survival of
the welfare state". The American Economic Review, Vol. 93, No. 1 (Mar., 2003), pp. 87-112.
[21] Krusell, Per and J.V. Rios-Rull, (1999). "On the size of U.S. government: political economy
in the neoclassical growth model". American Economic Review.
[22] Mayda, Anna Maria (2006), "Who is against immigration? A crouss-country investigation of
individual attitudes toward immigrants". The Review of Economic Statistics, August 2006,
88(3):510-530.
[23] Mendoza, Enrique, Assaf Razin and Linda Tesar (1994) "Effective Tax rates in Macroeco-
nomics: cross country estimates of tax rates on factor incomes and consumption". Journal of
Monetary Economics 1994.
[24] McDaniel, Cara, (2012), “Average tax rates on consumption, investment, labor and capital
in the OECD 1950-2003”, mimeo, Arizona State University. Tax series updated April 2012.
[25] Mood, Graybill and Boes (1974) Introduction to the Theory of Statistics. McGraw-Hill, 3rd
edition.
33
[26] National Center for Health Statistics. Vital Stats system at
www.cdc.gov/nchs/VitalStats.htm
[27] Ortega, Francesc (2005) "Immigration Quotas and Skill Upgrading". Journal of Public Eco-
nomics. Vol 89 Pp. 1841-1863.
[28] – – (2010). “Immigration, Citizenship, and the Size of Government,”The B.E. Journal of
Economic Analysis & Policy: Vol. 10(1), Contributions, Art. 26.
[29] Ottaviano, Gianmarco and Giovanni Peri (2012) "Rethinking the effects of immigration on
wages". Journal of the European Economic Association. Volume 10, Issue 1, pages 152—197,
February.
[30] Ramos, Xavier (1999) "Earnings Inequality and Earnings Mobility in Great Britain: Evidence
from the BHPS, 1991-1994". Institute for Social and Economic Research, University of Essex,
Working paper 99-08.
[31] Razin, Assaf and Edith Sand (2007) "The Role of Immigration in Sustaining the Social
Security System: A Political Economy Approach". CESifo working paper 1979. April
[32] – – , Efraim Sadka and Benjarong Suwankiri (2011) Migration and the welfare state:
political-economy policy formation. The MIT press.
[33] Ruggles, S , J Trent Alexander, Katie Genadek, Ronald Goeken, Matthew B. Schroeder,
and Matthew Sobek (2010). Integrated Public Use Microdata Series: Version 5.0 [Machine
Readable Database]. Minneapolis: University of Minnesota.
[34] Storessletten, Kjetil (2000) "Sustaining Fiscal Policy through Immigration". The Journal of
Political Economy 108 Pp 300-323.
[35] US Census Bureau at http://www.census.gov/
34
8 Appendix to "Intergenerational Mobility andthe Political Economy of Immigration"
(Appendix not for publication but intended to be offered online)
by Henning Bohn and Armando R. Lopez-Velasco
8.1 Appendix to section 1. Foreign Born Population and implied Im-
migration Quotas
Table 5. Foreign born population by date of entry2000-04 1990-99 1980-89 1970-79
Number (1000’s) 3434 8679 6969 4494
Percentages of Corresponding Group
by Completed Schooling Levels
Less than High School 30.7 34.9 34.9 30.5
High School+Some College 35.0 37.5 40.7 40.6
Bachelor Degree & Beyond 34.3 27.6 24.4 28.9Source: US Census Bureau, Current Population Survey, Annual Social and Economic Supplement, 2004
Immigration Statistics Staff, Population Division. Table 2.5 Educational Attainment of the Foreign-Born
Population 25 Years and Over by Sex and Year of Entry: 2004
Table 5 shows the foreign born population by date of entry and schooling level. In order to
compute net immigration, we use the estimates in Ben-Gad (2004) of 3.2 immigrants per 1000
natives for the 1990-99 decade. Estimates of population by education level for individuals 25 years
and older are 19.1% for less than high-school, 58% for high school plus some college and 22.8%
for Bachelor degrees and beyond. Using these numbers we obtain estimates of 5.65 low-skilled
immigrants per 1000 low-skilled natives, 2.025 medium-skilled immigrants per 1000 natives and
4.14 high-skilled immigrants per 1000 high-skilled natives.
8.2 Appendix to section 2.2. Ruling out negative wage premiums
All agents works at their own (highest) skill level if the resulting "unconstrained" wages satisfy
w1 ≤ w2 ≤ w3. Otherwise, it is effi cient (both output maximizing and individually rational) for
some agents to take jobs at a lower skill level.
(Note to readers: The appendix is lengthy because we feel obliged to cover all theoretically
possible cases. Readers interested in the cases discussed in the text may focus on labor supply
ratios Z in the unconstrained set ΩuZ , both defined below.)
A-1
To model job assignment in general, we distinguish labor supplies Li = Ni(1 + θi) from actual
labor inputs Li, i = 1, 2, 3. Let Lij ≥ 0 denote labor supplied by workers with skill i at skill level
j < i. (Throughout, i, j = 1, 2, 3 unless noted.) Then L1 = L1 + L21 + L31, L2 = L2 − L21 + L32,
and L3 = L3 − L31 − L32. Note that L31 is redundant because any set of labor inputs (L1, L2, L3)
obtained with L31 > 0 can also be obtained with L31 = 0 if L32 and L21 are increased by the
original amount of L31. Hence we set L31 = 0 without loss of generality.
Assuming CES production, output is Y = [∑
i φi(Li)ρ]
1ρ . The resulting wages can be written
as
wi =∂Y
∂Li= Y 1−ρ ·
(Li/φ
11−ρi
)ρ−1
(A.1)
For given (L1, L2, L3), maximizing Y by choice of (L32, L21) ≥ 0 implies the Kuhn-Tucker condi-
tions L32 · (w3 − w2) = 0, L21 · (w2 − w1) = 0, and w3 ≥ w2 ≥ w1. With two sets of Kuhn-Tucker
conditions, the solutions divide into four cases: (1) L32 = 0 and L21 = 0, which is the case of
"unconstrained" wages; (2) L32 > 0 and L21 = 0, which implies w2 = w3; (3) L32 = 0 and L21 > 0,
which implies w2 = w1; (4) L32 > 0 and L21 > 0, which implies w1 = w2 = w3.
Wages depend naturally on labor supplies but can be written more compactly in terms of the
ratios z1 = L1/L2 and z3 = L3/L2, exploiting constant returns to scale. The latter is also useful
because
z1 =x1 (1 + θ1)
(1 + θ2)and z3 =
x3 (1 + θ3)
(1 + θ2)
are functions of the state (X, θ), whereas Li depends on native population Ni, which is not part
of the Markov state.
Define the vector of labor supply ratios
Z = (z1, z3) = Z(X, θ),
total labor L = L1 + L2 + L3, constants φ12 = (φ1
1−ρ1 + φ
11−ρ2 )1−ρ, φ23 = (φ
11−ρ2 + φ
11−ρ3 )1−ρ, and
φ123 = (φ1
1−ρ1 + φ
11−ρ2 + φ
11−ρ3 )1−ρ, the dummy variable z2 ≡ 1, and the sets
ΩuZ = Z : z3 ≤ z3, z1 ≥ z¯1, where z¯1 = (φ1/φ2)
11−ρ , z3 = (φ3/φ2)
11−ρ
Ω23Z = Z : z3 > z3, z1 ≥ z¯13(1 + z3), where z
¯13 = (φ1/φ23)1
1−ρ ,
Ω12Z = Z : z3 ≤ z31(1 + z1), z1 < z¯1, where z31 = (φ3/φ12)
11−ρ ,
Ω123Z = Z : z3 > z31(1 + z1), z1 < z¯13(1 + z3).
Note that z¯13(1 + z3) =z
¯1 and z31(1+z¯1) = z3. Hence the closures of the four sets above intersect
at (z¯1, z3), where Li/φ
11−ρi = L/φ
11−ρ123 for all i, so unconstrainted wages satisfy w3 = w2 = w1
without job reassignments. Wage differentials arise if high skills are relatively more scarce; job
reassigments must occur if high skills are in relatively greater supply. Specifically:
A-2
Lemma 2.2A (Four Cases): Exactly one of the following four cases applies for all Z:(1) If Z ∈ Ωu
Z , wages are unconstrained, Y = [∑
i φi (Li)ρ]
1ρ , and L21 = L32 = 0. Moreover,
z3 < z3 implies w2 < w3, and z1 >z¯1 implies w1 < w2.
(2) If Z ∈ Ω23Z , then w2 = w3, Y = [φ1 (L1)ρ + φ23(L2 + L3)ρ]
1ρ , L21 = 0, and L32 > 0.
Moreover, z1 >z¯13(1 + z3) implies w2 < w3.
(3) If Z ∈ Ω12Z , then w1 = w2, Y = [φ12(L1 + L2)ρ + φ3 (L3)ρ]
1ρ , L21 > 0 and L32 = 0. Moreover,
z3 < z31(1 + z1) implies w1 < w2.
(4) If Z ∈ Ω123Z , then w1 = w2 = w3, Y = (φ123)
1ρ L, L21 > 0, and L32 > 0.
Proof: Since the four sets partition the space Z : z1 ≥ 0, z3 ≥ 0, exactly one applies for anyZ. (1) From (A.1), wi = wj iff Li/φ
11−ρi = Lj/φ
11−ρj . Since ρ−1 < 0, wi ≥ wj iff Li/φ
11−ρi ≤ Lj/φ
11−ρj .
If z3 ≤ (φ3/φ2)1
1−ρ = z3 and z1 ≥ (φ1/φ2)1
1−ρ =z¯1, then L3/φ
11−ρ3 ≤ L2/φ
11−ρ2 ≤ L1/φ
11−ρ1 , so Li = Li
is consistent with w3 ≥ w2 ≥ w1. By analogous reasoning, strict inequalities z3 < z3 and/or z1 >z¯1
imply the corresponding strict inequalities for wages.
(2) If z3 > z3, then L3/φ1
1−ρ3 > L2/φ
11−ρ2 and L3/φ
11−ρ3 ≤ L2/φ
11−ρ2 imply L32 > 0, so w2 = w3. In
turn, w2 = w3 implies (L3 − L32)/φ1
1−ρ3 = (L2 + L32 − L21)/φ
11−ρ2 , so L32
L2=
φ1
1−ρ2 z3−φ
11−ρ3 +φ
11−ρ3
L21L2
φ1
1−ρ2 +φ
11−ρ3
≥
( φ2
φ23)
11−ρ · z3 − ( φ3
φ23)
11−ρ and L3
L2= z3 − L32
L2≤ (φ3/φ23)
11−ρ · (1 + z3).
Suppose for contradiction that L21 > 0. Then L1
L2= z1 + L21
L2> z1, and L32
L2≥ ( φ2
φ23)
11−ρ · z3 −
( φ3
φ23)
11−ρ would imply L3
L1= L3
L2/ L1
L2< (φ3/φ1)
11−ρ and hence w1 < w3 = w2, violating the Kuhn-
Tucker condition. By contradiction, L21 = 0, which implies L3
L2= (φ3/φ23)
11−ρ (1 + z3) and L2
L2=
(φ2/φ23)1
1−ρ (1+z3). With these labor inputs Li, Y = [∑
i φi (Li)ρ]
1ρ = [φ1 (L1)ρ + φ23 (L2 + L3)ρ]
1ρ .
(3) The proof for z1 <z¯1 and z3 ≤ z31(1+z1) is analogous to case (2), with analogous steps, first
showing that z1 <z¯1 implies L21 > 0, and then showing that L32 > 0 would lead to a contradition,
so L32 = 0.
(4) Conditions z3 > z31(1 + z1) and z1 <z¯13(1 + z3) imply z3/z1 = L3/L1 > φ1
1−ρ3 /φ
11−ρ1 . Since
L3/φ1
1−ρ3 ≤ L1/φ
11−ρ1 one cannot have L32 = L21 = 0. By similar reasoning z3
1+z1= L3
L1+L2> z31
rules out L32 = 0 and z11+z3
= L1
L3+L2<z¯13 rules out L12 = 0. Thus L32 > 0 and L21 > 0, hence
w1 = w2 = w3. Equality of Li/φ1
1−ρi for all i then implies Li/φ
11−ρi = L/φ
11−ρ123 , Li = φ
11−ρi /φ
11−ρ123 · L,
and Y = (φ123)1ρ L. QED.
Lemma 2.2B (Wages): (1) For Z ∈ Ωuz : w2 = φ2 φ1z
ρ1 + φ2 + φ3z
ρ3
1−ρρ , w1 = w2 · φ1
φ2·zρ−1
1 ≤w2, and w3 = w2 · φ3
φ2· zρ−1
3 ≥ w2.
(2) For Z ∈ cl(Ω23z ), where the closure covers cases with w2 = w3 in Ωu
z : Define z13 ≡ z11+z3
,
then w2 = w3 = φ23 φ1 (z13)ρ + φ231−ρρ and w1 = w2 · φ1
φ23· (z13)ρ−1 ≤ w2.
(3) For Z ∈ cl(Ω12z ), where the closure covers cases with w1 = w2 in Ωu
z : Define z31 ≡ z31+z1
,
then w2 = w1 = φ12 φ3 (z31)ρ + φ121−ρρ and w3 = w2 · φ3
φ12· (z31)ρ−1 ≥ w2.
(4) For Z ∈ cl(Ω123z ): w1 = w2 = w3 = (φ123)
1ρ .
Proof: Follows from differentiating the equations for Y in Lemma 2.2A and verifying that
A-3
wages are continous at boundaries between cases. QED.
8.3 Appendix to section 2.6. The model without dynamic effects.
Proofs and generalizations
8.3.1 Cases discussed in the text
We prove the claims in section 2.6 first for w1 < w2 < w3 and w2 ≤ w, and then generalize. The
wages w2 and w that determines voter consumption c2 have the following properties:
Lemma 2.6A (Derivatives of w2): (1) For Z ∈ int(Ωuz ):
∂w2
∂z1= (1 − ρ)w2w1
y> 0 and
∂w2
∂z3= (1 − ρ)w2w3
y> 0, where y = Y/L2 = φ1z
ρ1 + φ2 + φ3z
ρ3
1ρ . Moreover, ∂w2
∂θ1, ∂w2
∂θ3> 0 and
∂w2
∂θ2< 0.
(2) For Z ∈ int(Ω23z ): ∂w2
∂z13= (1− ρ)w2w1
y> 0, ∂w2
∂θ1> 0, and ∂w2
∂θ2, ∂w2
∂θ3< 0.
(3) For Z ∈ int(Ω12z ): ∂w2
∂z31= (1− ρ)w2w3
y> 0, ∂w2
∂θ3> 0, and ∂w2
∂θ1, ∂w2
∂θ2< 0.
(4) For Z ∈ int(Ω123z ): ∂w2
∂z1= ∂w2
∂z3= 0 and ∂w2
∂θi= 0 for all i.
Proof: Derivatives with respect to zi follow from differentiating the equations for w2 in Lemma
2.2B. Derivatives with respect to θi follow, using ∂zi∂θ2
= −LiL2
2
∂L2
∂θ2= −N2Li
L22< 0, ∂zi
∂θi= 1
L2
∂Li∂θi
= NiL2> 0
for i = 1, 3,∂z13
∂z1= 1
1+z3> 0, ∂z13
∂z3= − z13
1+z3< 0, ∂z31
∂z3= 1
1+z1> 0, ∂z31
∂z1= − z31
1+z1> 0, and applying
the chain rule. QED.
Remark: Since the derivatives differ across cases, w2 is generally not differentable at the
boundaries. For example, ∂z13
∂z1= 1
1+z3in Ω23
z implies ∂w2
∂z1= (1− ρ) w2w1
y(1+z3)which differs by a scale
factor from ∂w2
∂z1in Ωu
z .
Lemma 2.6B (Average wage w): The average wage w =∑
iLiLwi can be expressed as
w =∑i
zi1 + z1 + z3
wi =y
1 + z1 + z3
=1
1 + z1 + z3
φ1zρ1 + φ2 + φ3z
ρ3
1ρ ,
and its derivatives are
∂w
∂zi=
wi − w1 + z1 + z3
for i = 1, 3, and∂w
∂θi=Ni
L(wi − w) for all i
Proof: w = 1L
∑i Liwi = Y
Lby Euler’s law; w =
∑iLi/L2
L/L2wi =
∑i
zi1+z1+z3
wi and YL
= y1+z1+z3
follow. Differentiating w = 1L
∑j Ljwj with respect to wi yields ∂w
∂Li= wi
L+ 1
L
∑j Lj
∂wj∂Li−
1L2
∑j Ljwj = 1
L(wi− w) because
∑j Lj
∂wj∂Li
= 0 by constant returns to scale and 1L2
∑j Ljwj = w
L.
Differentiating y1+z1+z3
yields ∂w∂zi
= 11+z1+z3
∂y∂zi− y
(1+z1+z3)2 = wi−w1+z1+z3
since ∂y∂zi
= wi. Finally,∂w∂θi
= Ni∂w∂Li
since Li = Ni(1 + θi). QED.
Remark: Lemma 2.6B shows that ∂w∂θi
> 0 iff wi > w, as claimed in the text.
Lemma 2.6C: Suppose w1 < w2 < w3 and w2 ≤ w. Then (a) ∂c2∂θ3
> 0, (b) ∂c2∂θ2
< 0, and (c)∂c2∂θ1
> 0 for τ in a neighborhood of zero and ∂c2∂θ1
< 0 for τ in a neighborhood of one.
A-4
Proof: Differentiating c2(Z) = (1− τ)w2+ τw as function of Z = Z(X, θ), one obtains
∂c2
∂θi= (1− τ)
∂w2
∂θi+ τ
∂w
∂θi,
where Lemma 2.6A implies ∂w2
∂θ1, ∂w2
∂θ3> 0 and ∂w2
∂θ2< 0, since Z ∈ int(Ωu
z ) from w1 < w2 < w3. For
w2 ≤ w, Lemma 2.6B implies ∂w∂θ1
< 0, ∂w∂θ2≤ 0 and ∂w
∂θ3> 0. Combining terms, (a) follows from
∂w2
∂θ3> 0 and w3 > w; (b) follows from ∂w2
∂θ2< 0 and assumption w2 ≤ w; and ∂c2
∂θ1is a weighted
average of ∂w2
∂θ1> 0 and Ni
L(w1 − w) < 0 with weights (1 − τ , τ). Hence ∂c2
∂θ1> 0 as τ → 0 and
∂c2∂θ1
< 0 as τ → 1, proving (c). QED.
Corollary to 2.6C: Suppose w1 < w2 < w3 and w2 ≤ w for given X and for all θ ∈ Ωθ, and
β = 0. Then all optimal policies θ∗ satisfy θ∗2 = 0 and θ∗3 = θmax3 ; and Ortega’s (2001) tie-breaking
convention to minimize population selects a unique policy (i.e., unique θ∗1).
Proof: For β = 0, problem (13) reduces to finding θ∗ ∈ arg maxΘ∈ΩΘu2 (X, θ), which further
reduces to finding θ∗ ∈ arg maxΘ∈ΩΘc2. Since Lemma 2.6C implies ∂c2
∂θ3> 0 and ∂c2
∂θ2< 0,
arg maxΘ∈ΩΘc2 is attained at the corners θ∗2 = 0 and θ∗3 = θmax
3 . Since optimal policies vary only
by θ∗1, Ortega’s convention selects the unique minimum of θ∗1. QED.
Remark: Since θ∗3 = θmax3 and w1 < w2 < w3 requires z3 < z3, the corollary can only apply if
x3(1 + θmax3 ) < z3, i.e., if θ
max3 is suffi ciently small.
Lemma 2.6D: Suppose w1 < w2 < w3 and w2 ≤ w for all θ ∈ Ωθ, given X and assuming
β = 0. Consider a range of tax rates (τ−, τ+) ⊂ [0, 1] and the associated optimal policies θ∗(τ),
where in case of multiple solutions, θ∗(τ) is selected by Ortega’s (2001) tie-breaking convention
to minimize population. Then θ∗1 = θ∗1(τ) is decreasing in τ ; and strictly decreasing at interior
solutions θ∗1(τ) ∈ (0, θmax1 ).
Proof: The Corollary to 2.6C implies θ∗2 = 0 and θ∗3 = θmax3 for all τ ∈ (τ−, τ+). Let c2(θ1, τ)
denote consumption implied by immigration θ = (θ1, 0, θmax3 ). By the theorem of the maximum,
θ∗1(τ) ≡ arg maxθ1∈[0,θmax
1 ]c2(θ1, τ) is a non-empty u.h.c. correspondence.Consider any pair τa < τ b and any θ1a ∈ θ
∗1(τa), θ1b ∈ θ
∗1(τ b). Optimality implies c2(θ1b, τ b) ≥
c2(θ1a, τ b) and c2(θ1a, τa) ≥ c2(θ1b, τa). Since c2 is continuous and differentiable,
c2(θ1b, τ b)− c2(θ1a, τ b)− c2(θ1b, τa) + c2(θ1a, τa)
=
∫ τb
τa
∫ θ1b
θ1a
∂2c2(θ1, τ)
∂τ∂θ1
dθ1dτ =∂2c2(θ1m, τm)
∂τ∂θ1
· (θ1b − θ1a) · (τ b − τa) ≥ 0
where τm ∈ (τa, τ b) and θ1m is between θ1a and θ1b, using the mean value theorem. Differentiating∂c2∂θ1in Lemma 2.6C, ∂
2c2(θ1,τ)∂τ∂θ1
= −∂w2
∂θ1+ ∂w
∂θ1< 0 for all (θ1, τ). Since τ b − τa > 0, ∂
2c2(θ1m,τm)∂τ∂θ1
< 0
implies θ1b ≤ θ1a, proving that θ∗1(τ) is decreasing.
Note that ∂c2∂θ
(θ1b, τ b) = ∂c2∂θ
(θ1b, τa) +∫ τbτa
∂2c2(θ1b,τ)∂τ∂θ1
dτ < ∂c2∂θ
(θ1b, τa). Hence θ1b = θ1a would be
inconsistent with the first order condition ∂c2∂θ
= 0 for interior solutions. Hence θ1b < θ1a unless
θ1a = θ1b = 0 or θ1a = θ1b = θmax1 . Finally, uniqueness of θ∗2 = 0 and θ∗3 = θmax
3 implies that if
A-5
θ∗1(τ) has multiple values, population is minimized by setting θ∗1(τ) = min θ
∗1(τ). QED.
Lemmas 2.6A-D prove claims (a)-(c) in Section 2.6. Note that Lemma 2.6D would be much
easier to prove if ∂2c2/∂θ21 < 0, using the implicit function theorem; but ∂2c2/∂θ
21 < 0 may
not always hold. Note also that since the proof uses θ∗1(τ), the tie-breaking convention θ∗1(τ) =
min θ∗1(τ) is without loss of generality.
8.3.2 Generalizations
Now consider the static model without assumptions w1 < w2 < w3 and w2 ≤ w. (Note to readers:
We provide this analysis mainly to motivate our focus on small θmax3 in settings (II) and (III). That
is, this appendix section is meant to show that while it is possible to prove results for high θmax3 ,
the analysis is much more complicated and does not provide significant new insights. Readers may
skip this subsection without loss of continuity.)
(1) Regarding w2 ≶ w, note that w2 ≤ w was invoked only in the proof of Lemma 2.6C to sign∂c2∂θ2
< 0. If w2 > w for some (X, θ), the sign of ∂c2∂θ2
becomes ambiguous and hence θ∗2 > 0 cannot
be ruled out. However, θ∗1 and θ∗2 cannot both be positive, i.e., θ
∗1θ∗2 = 0. The proof is implied by
a more general insight:
Lemma 2.6E: Derivatives of ci = (1− τ)wi+ τw satisfy∑
j LjNj∂ci∂θj
= 0 whenever wages are
differentiable.
Proof: Consider∑
j∂ci∂Lj
Lj = (1− τ)∑
j∂wi∂Lj
Lj + τ∑
j∂w∂Lj
Lj. Constant returns to scale imply∑i∂w2
∂LiLi = 0. From Lemma 2.6B,
∑i∂w∂LiLi =
∑i
1L
(wi − w)Li =∑
iLiLwi − w = 0. Hence∑
j∂ci∂Lj
Lj = 0. Since ∂ci∂Lj
= Nj∂ci∂θj,∑
j LjNj∂ci∂θj
= 0. QED.
Corollary to 2.6E: Suppose w2 < w3 for given X and for all θ ∈ Ωθ, and β = 0. Then all
θ∗ = P (X) satisfty θ∗3 = θmax3 and θ∗1θ
∗2 = 0.
Proof: Since w2 < w3 implies w3 > w, ∂w2
∂θ3> 0, ∂w
∂θ3> 0 follow from Lemmas 2.6A-B. Hence
∂c2∂θ3
> 0 and θ∗3 = θmax3 . Then from Lemma 2.6E, either ∂c2
∂θ1< 0, which implies θ∗1 = 0, or ∂c2
∂θ2< 0
which implies θ∗2 = 0, or both, which combines to θ∗1θ∗2 = 0. QED.
Intuitively, θ∗2 > 0 is possible for w2 > w because medium skilled immigrants then generate
net taxes that reduce the tax burden on native medium skilled agents. However, medium-skilled
immigrants also reduce w2. Hence θ∗2 > 0 occurs only when the tax rate and initial value of x1
are so high that the fiscal benefits outweigh the reduction in wages; and under these conditions,
low-skilled immigrants would greatly reduce c2. This provides an intuition why θ∗1 = 0 when
θ∗2 > 0.
(2) Consider wage equalization. This is straightforward but requires multiple case distinctions
and new notation, because immigration has wage effects that differs across cases in Lemma 2.6A.
For given X, Z must be in the feasible set
ΩZ(X) ≡ Z : z1 =x1(1 + θ1)
1 + θ2
, z3 =x3(1 + θ3)
1 + θ2
for some Θ ∈ ΩΘ.
A-6
which may overlap with some or all of the sets in Lemma 2.6A. Note that ΩZ(X) is compact and
convex and that any Z ∈ ΩZ(X) can be implemented by the immigration policy
θo2 = maxx1
z1
− 1,x3
z3
− 1, 0, θo1 =z1(1 + θo2)
x1
− 1, θo3 =z3(1 + θo2)
x3
.
Moreover:
Lemma 2.6F: (a) The feasible set ΩZ(X) is equivalently to z1 ∈ [zmin1 , zmax
1 ] and z3 ∈[zmin
3 , zmax3 (z1)
], where zmin
1 = x1
1+θmax2, zmax
1 = x1 · (1 + θmax1 ), zmin
3 = x3
1+θmax2, and zmax
3 (z1) =x3(1+θmax
3 )
max(1,x1/z1).
(b) For any Z ∈ ΩZ(X), policy θo is the unique policy that satisfies Ortega’s (2001) tie-breaking
convention to minimize population. If z3 > x3, then θo1θo2 = 0.
Proof: (a) By construction, Z ∈ ΩZ(X) implies z1 ∈ Ωz1 and z3 ∈ Ωz3(z1). Conversely, for any
z1 ∈ Ωz1 and z3 ∈ Ωz3(z1), the policy defined in (b) satisfies Θ ∈ ΩΘ, hence Z ∈ ΩZ(X).
(b) Any policy θ′ that implements Z must satisfy (1 + θ′2)z1 = x1(1 + θ′1) and (1 + θ′2)z3 =
x3(1 + θ′3). Minimizing population L = N2
∑i xi(1 + θ′i) subject to these constraints implies
θ′2 = 0 if z1 ≥ x1(1 + θ′1) and z3 ≥ x3(1 + θ′3), implies θ′2 = x1
z1− 1 > 0 if z1 < x1(1 + θ′1) and
(1 + x1
z1)z3 ≥ x3(1 + θ′3), and θ′2 = x3
z3− 1 > 0 if z3 < x3(1 + θ′3) and (1 + θ′2)z1 ≥ x1(1 + θ′1). Hence
θ′2 = θo2. Then θo1, θ
o3 follow from the constraints. Moreover, z3 > x3 implies x3
z3− 1 < 0, hence
θo2 = maxx1
z1− 1, 0 and θo1 = z1(1+θo2)
x1− 1 = max z1
x1− 1, 0, which implies θo1θo2 = 0. QED.
Remark: Part (a) facilitates sequential choice of z3 followed by z1, which useful because ∂c2∂θ3
is easy to sign in all cases. Part (b) shows that Ortega’s (2001) tie-breaking convention yields a
unique policy in the static model whenever the set
Ω∗Z ≡ arg maxc2(Z) : Z ∈ ΩZ(X)
is single valued.
(3) To focus the analysis, suppose w1 < w2 < w3 applies at least at X, the starting point
without immigration; that is, Z(X, 0) = (x1, x3) ∈ int(ΩuZ). Also assume θmax
3 ≤ θmax2 , as implied
by the notion that high skilled immigrants are relatively scarce.35 One finds:
Lemma 2.6G: Suppose X ∈ int(ΩuZ). Then:
(a) If θmax3 ≤ z3/x3−1, all optimal policies satisfy θ∗3 = θmax
3 , θ∗1θ∗2 = 0, all imply unconstrained
wages, and Ortega’s (2001) tie-breaking convention generically selects a unique policy.
(b) If θmax3 > z3/x3− 1, there exists an optimal policy θ∗ with unconstrained wages (Z∗ ∈ Ωu
Z).
It is characterized by either:
(i) θ∗3 = z3/x3 − 1, θ∗1 ≥ 0, θ∗2 = 0, and w1 < w2 = w3, or
35Cases with X /∈ int(ΩuZ) would require numerous case distinctions that are utterly irrelevant in the context ofthe main model, where X is generated by transition matrices that tend to have x3/z3 << 1 and x1/z¯1
>> 1, whichmeans X is far from the boundaries of ΩuZ . If θ
max2 < θmax3 , optimal policy could be technically in Ω23Z though
economically equivalent to policies in ΩuZ , requiring needless elaboration.
A-7
(ii) θ∗3 = z3/x3 − 1, θ∗1 = 0, θ∗2 > 0, and w1 < w2 = w3, or
(iii) θ∗3 = θmax3 , θ∗1 = 0, θ∗2 >
x3
z3(1 + θmax
3 )− 1, and w1 < w2 < w3.
Moreover, Ortega’s (2001) tie-breaking convention generically selects a unique θ∗ in ΩuZ .
(c) If cases (b-i) or (b-ii) apply, Ω∗Z also includes a line segment in Ω23Z consisting of all Z ∈
ΩZ(X) such that z∗13 = z11+z3
is the same as for Z∗ ∈ ΩuZ ; all such policies imply the same wages.
In case (b-i), all policies in Ω23Z are eliminated by Ortega’s (2001) tie-breaking convention; in case
(b-ii), policies in Ω23Z are not eliminated, but all are economically equivalent to Z∗ ∈ Ωu
Z in the
sense that they imply the same wages and the same job assignments Li.
Proof: (a) If X ∈ int(ΩuZ) and θmax
3 ≤ z3/x3 − 1, then ΩZ(X) ⊂ ΩuZ ∪ Ω12
Z .
Suppose for contradiction that Z ∈ Ω12Z is optimal: Since x1 >z
¯1 whereas z1 <z¯1, θ2 >
x1/z¯1 − 1 > 0 for any policy θ that implement Z. Note that ∂c2∂θ2
< 0 on Z ∈ int(Ω12Z ) because
∂w2
∂θ2< 0 from Lemma 2.6A and ∂w
∂θ2≤ 0 from Lemma 2.6B with w1 = w2 < w3. Hence c2(Z) is
strictly less than c2 under the feasible policy θ′ that replaces θ2 by θ
′2 = x1/z¯1− 1 and implements
Z ′ ∈ ΩuZ (on the boundary to Ω12
Z ), contradicting optimality of Z ∈ Ω12Z and proving Ω∗Z ⊂ Ωu
Z .
Considering Z ∈ ΩZ(X) ∩ ΩuZ : If θ
max3 < z3/x3 − 1, then w2 < w3, so θ
∗3 = θmax
3 and θ∗1θ∗2 = 0
from Corollary 2.6E. If θmax3 = z3/x3−1, ∂c2
∂θ3> 0 for Z ∈ int(Ωu
Z) implies θ∗3 = θmax3 , and hence z∗3 =
zmax3 (z∗1). Hence the problem of maximizing c2(Z) on Ωu
Z reduces to maximizing c2(z1, zmax3 (z1))
by choice of z1. Since [maxzmin1 , z
¯1, zmax1 ] is compact, z∗1 = arg maxc2(z1, z
max3 (z1)) is non-
empty, showing existence of Z∗ ∈ ΩuZ . Moreover, z
∗1 = z∗1(τ) is strictly decreasing in τ for z∗1 ∈
(maxzmin1 ,z
¯1, zmax1 ) by arguments analogous to the proof of Lemma 2.6D (hence details omitted),
which implies uniqueness of (z∗1 , z∗3) except for a countable number of tax rates, i.e. generic
uniqueness. Then Lemma 2.6F implies generic uniqueness of θ∗ with tie-breaking convention, and
θ∗1θ∗2 = 0.
(b) For θmax3 > z3/x3 − 1, ΩZ(X) may overlap with Ω123
Z and Ω23Z .
Suppose for contradiction that Z ∈ cl(Ω123Z ) is optimal: Since x1
1+x3>z¯13 for X ∈ int(Ωu
Z),
whereas z13 ≤z¯13 on cl(Ω123Z ), implementing Z requires θ3 > 0 and/or θ2 > 0 and implies existence
of a feasible alternative θ′ with θ3 > θ′3 and/or θ2 > θ′2 that implements Z′ ∈ Ω23
Z ∩ cl(Ω123Z ). As
c2 is constant on cl(Ω123Z ), c2(Z) = c2(Z ′). Note that ∂w2
∂θ2, ∂w2
∂θ3< 0 in Ω23
Z and that ∂w∂θi
= 0 at Z ′.
Hence ∂c2∂θ2, ∂c2∂θ3
< 0 a neighborhood of Z ′ inside Ω23Z , which means there is a feasible Z” ∈ Ω23
Z for
which c2(Z”) > c2(Z ′) = c2(Z), contradicting optimality of Z ∈ cl(Ω123Z ). Analogous reasoning
rules out optimal Z ∈ Ω12Z , because there are superior alternatives either on the boundary between
ΩuZ and Ω12
Z or on the boundary between Ω23Z and Ω123
Z .
Thus Ω∗Z ⊂ ΩuZ ∪ Ω23
Z .
For Z ∈ cl(Ω23Z ), Lemma 2.2B implies that w2 and w are univariate functions of z13. Note
that Z ∈ ΩZ(X) ∩ cl(Ω23Z ) implies z3 ≥ z3 and z1 ≤ x1 · (1 + θmax
1 ), so z13 ≤ x1
1+z3· (1 + θmax
1 ) ≡zmax
13 ; also, , z13 ≥z¯13, and z3 ≥ z3 further impliesx3(1+θ3)
1+θ2≥ z3, 1 + θ2 ≤ x3
z3(1 + θmax
3 ), and
z13 ≥ x1
1+θ2+x3(1+θ3)≥ x1
(1+θmax3 )(x3/z3+x3)
. Thus, using zmin13 ≡ maxz
¯13, x1
(1+θmax3 )(x3/z3+x3)
, one findsΩZ(X) ∩ cl(Ω23
Z ) ⊂ Z : z3 ≥ z3, zmin13 ≤ z13 ≤ zmax
13 .
A-8
Conversely, note that Zu = (z13(1 + z3), z3) ∈ ΩZ(X) ∩ cl(Ω23Z ) for all z13 ∈
[zmin
13 , zmax13
],
and Zu ∈ ΩuZ . Define Ω∗13 ≡ arg maxc2(Zu), z13 ∈
[zmin
13 , zmax13
], which is non-empty because[
zmin13 , zmax
13
]is compact, and generically single-valued by arguments analogous to the proof of
Lemma 2.6D. Since w2 and w are univariate functions of z13 on cl(Ω23Z ), c2(Z) is constant for all
Z ∈ cl(Ω23Z ) with common ratio z1
1+z3= z13. Hence for any z∗13 ∈ Ω∗13, Z
u∗ = (z∗13(1 + z3), z3)
maximizes c2(Z) on Z : z3 ≥ z3, zmin13 ≤ z13 ≤ zmax
13 , and hence on the subset ΩZ(X) ∩ cl(Ω23Z ).
The set of policies that maximize c2(Z) on ΩZ(X) ∩ cl(Ω23Z ) therefore consist of Zu∗ and the
corresponding line segment(s) in Ω23Z with z13 = z∗13.
Now consider all Z ∈ ΩZ(X)∩ΩuZ . Since Z
u∗ ∈ ΩuZ , maxc2(Z) : Z ∈ ΩZ(X)∩Ωu
Z ≥ c2(Zu∗),
which leaves two possibilities:
If maxc2(Z) : Z ∈ ΩZ(X) ∩ ΩuZ = c2(Zu∗), Zu∗ ∈ Ωu
Z and the line segments in Ω23Z with
z13 = z∗13 are optimal. In addition, argmaxc2(Z) : Z ∈ ΩZ(X) ∩ ΩuZ may include policies
with z3 < z3. Since ∂c2∂θ3
> 0 on int(ΩuZ), z∗3 = minzmax
3 (z∗1), z3, which maxizes z3 within ΩuZ , and
θ∗1θ∗2 = 0 follows as in (a). This allows for three cases: (i) if z∗3 = z3 and θ
∗2 = 0, then θ∗3 = z3/x3−1
and θ∗1 ≥ 0; (ii) if z∗3 = z3 and θ∗2 > 0, then θ∗3 = z3/x3 − 1 and θ∗1 = 0; (iii) if z∗3 < z3, then
z∗3 = zmax3 (z∗1), so θ∗3 = θmax
3 and θ∗2 >x3
z3(1 + θmax
3 ) − 1 > 0; the latter implies θ∗1 = 0. Also,
z∗3 < z3 implies Z∗ ∈ int(ΩuZ), so w1 < w2 < w3. This proves the properties claimed in (i-iii).
Alternatively, if maxc2(Z) : Z ∈ ΩZ(X) ∩ ΩuZ > c2(Zu∗), then all optimal policies must satisfy
z3 < z3, and by the arguments above, they have properties (iii).
Note that all optimal policies inΩZ(X)∩ΩuZ must maximize c2 by choice of z1 ∈ [maxz
¯1, zmin1 , zmax
1 ]
with implied z3 = minz3, zmax3 (z1). For this univariate choice problem, arguments analogous to
the proof of Lemma 2.6D imply that z∗1(τ) is strictly decreasing in τ , hence generically unique,
which means Ortega’s (2001) tie-breaker selects a unique θ∗.
(c) The claimed line segments are defined in the proof of (b) above. In case (i), constant z∗13
on a line segment requires that θ∗1 and θ∗3 are increasing in z3. Hence Ortega’s tie breaker selects
(z∗13(1 + z3), z3), which is in ΩuZ . In case (ii), constant z
∗13 on a line segment requires that θ
∗3 is
increasing in z3 whereas θ∗3 is decreasing. Since θ
∗1 = 0, L1 = N1 is given, and constant z∗13 = N1
L2+L3
implies common values for L2 + L3, for wages, and hence the same job assignments Li. Ortega’s
tie breaker is unhelpful because N1 + L2 + L3 is common. In case (iii), z3 < z3 rules out policies
in Ω23Z . QED.
Remarks: Lemma 2.6G generalizes 2.6F and shows that there is always an optimal policy
with unconstrained wages. The multiplicity of optimal policies is largely resolved by Ortega’s tie-
breaking convention, except for non-generic cases with multiple solutions and in subcase (b)(ii),
when high-skilled immigrants are assigned to medium-skilled job. The latter is economically
irrelevant (not affecting the skill mix actually used (Li)), an artifact of labeling immigrants by
innate skills in a scenario that assignes them to the same job regardles of skill. A natural tie breaker
is to assume that immigrants are admitted at the skill level at which they are employed—then all
optimal policies have unconstrained wages.
A-9
8.4 Appendix to Section 3.1. Details on Mobility Matrices
We describe in more detail the analysis done in the text and present some robustness checks.
We use individuals whose interview was obtained during the period 1977-2012, who were aged
25-55 and born at least since 1945. Then we consider some subsamples. The total number of
observations used in the main text includes 18999 observations for children of natives, and 1477
observations for children of immigrants. Among natives, 8476 are men and 10523 are women.
Among immigrants, 636 are men and 841 are women. In what follows, we label "children of
immigrants" as "immigrants" for simplicity. We show in this section that the conclusions remain
even if use only sons or daughters, as well as controlling by white race.
Table 6 shows the estimated intergenerational (transition) matrices for immigrants. Matrix
#1′ is the transition matrix of men (sons of immigrants), matrix #2′ is the same concept for
women, while matrix #3′ is for both men and women (used in main text). Table 7 shows the
different matrices estimated for natives: matrix #1 is for men, matrix #2 is for white men, matrix
#3 is for women, matrix #4 is for white women and matrix #5 is for all men and women (used
in text).
Table 6. Estimated Transition Matrix for Children of ImmigrantsTransition Matrix (QI) Men/Women Race Age
1′.240 .544 .216 N1= 171
.068 .647 .285 N2= 323
.021 .338 .641 N3= 142
Men All 25-55
2′.192 .628 .180 N1= 250
.067 .622 .311 N2= 389
.023 .314 .663 N3= 172
Women All 25-55
3′.211 .594 .195 N1= 421
.067 .633 .299 N2= 712
.022 .325 .653 N3= 314
Both All 25-55
The results that low-skilled mobility of immigrants is higher for immigrants, as well as similar
mobility in the high-skilled category are very robust across subsamples, while for medium-skilled
men there’s no statistical difference in the mobility of immigrants and natives (native matrix #1
and immigrant matrix #1′, with a row statistic of 3.47, less than 5.99 which is the 5% significance
level), and there’s still significant difference for women (native matrix #3 and immigrant matrix
#2′, with a statistic of 17.45 > 5.99). For the test at the matrix level for these same subsamples
of men or women, we reject that they are statistically equal at the 1% significance level. Similar
results for white men vs immigrant men (native matrix #2 and immigrant matrix #1′) and white
women vs immigrant women (native matrix #4 and immigrant matrix #2′). The tests are reported
in table 8.
A-10
Table 7. Estimated Transition Matrices for Natives# Transition Matrix (Q) Men/Women Race Age
1
.281 .628 .091 N1= 1538
.068 .692 .240 N2= 4985
.008 .401 .590 N3= 1953
Men All 25-55
2
.282 .622 .096 N1= 1094
.064 .685 .251 N2= 4318
.007 .384 .609 N3= 1806
Men White 25-55
3
.239 .687 .075 N1= 2317
.057 .719 .224 N2= 6091
.011 .394 .595 N3= 2115
Women All 25-55
4
.232 .690 .078 N1= 1465
.047 .713 .231 N2= 4984
.009 .381 .610 N3= 1901
Women White 25-55
5
.256 .663 .081 N1= 3855
.062 .707 .231 N2= 11076
.010 .397 .593 N3= 4068
Both All 25-55
6
.253 .661 .086 N1= 2559
.055 .700 .245 N2= 9302
.008 .382 .609 N3= 3707
Both White 25-55
Table 8. Tests with 5% critical values (χ2)
H0 (Null
Hypothesis)
Row1 Test
CV=5.99
Row2 Test
CV=5.99
Row3 Test
CV=5.99
Matrix Test
CV=12.59
Q[#1] = QI [#1] 26.05∗∗ 3.47 5.80 33.87∗∗
Q[#2] = QI [#1] 21.53∗∗ 2.12 5.21 27.69∗∗
Q[#3] = QI [#2] 32.72∗∗ 17.45∗∗ 5.03 56.06∗∗
Q[#4] = QI [#2] 26.78∗∗ 14.68∗∗ 4.26 46.89∗∗
Q[#2] = Q[#4] 12.03∗∗ 19.83∗∗ 3.42 34.89∗∗
QI [#1] = QI [#2] 3.00 0.58 0.55 3.79
Q[#1] = Q[#3] 14.28∗∗ 11.27∗∗ 1.95 26.49∗∗
Q[#5] = QI [#3] 58.74∗∗ 18.63∗∗ 9.34∗∗ 87.52∗∗
Notes: The 1% critical value (CV) of the test for equality of matrices is 16.81 (6 degrees of freedom).
The 1% CV (∗∗) for tests of row equality is 9.21.
We also look at differences between men and women transition matrices, for natives-only and
immigrants-only. Comparing the transition matrices for native men and women (native matrix #1
vs native matrix #3), the first two rows suggest that men have slightly more extreme outcomes
A-11
than women. The row tests show that the probability distributions for sons of low-skilled and
medium-skilled parents are different to those of daughters, with statistics of 14.28 (low-skilled
parents) and 11.27 (medium-skilled), while we cannot reject that the probability distribution of
sons and daughters of high-skilled parents is the same (statistic=0.95 < 5.99). The matrix test
rejects that men and women have the same transition matrices, with an statistic of 26.49 > 12.59.
In the case of immigrants (immigrant matrix #1 and immigrant matrix #2), we cannot reject that
the rows and the whole matrices are statistically the same. The differences are similar to those of
natives, but in this case the lower number of observations is the cause that we can’t reject the null
of matrix equality. When we restrict the men-women comparison to native whites, the matrices
and tests results remain essentially unchanged.
8.4.1 Transition Matrices under an Alternative Definition of Second Generation Im-migrants
The following transition matrices are constructed when the definition of second generation immi-
grant is applied to individuals whose both parents were born outside the US under the same filters
as described in the main text. The sample size across all skills categories is only 530 observations.
Table 9. Transition Matrices of children of immigrants:both parents born outside US
Transition Matrix (QI) Men/Women Race Age
1′′.286 .464 .250 N1= 84
.085 .606 .309 N2= 94
.043 .391 .565 N3= 46
Men All 25-55
2′′.165 .617 .218 N1= 133
.058 .545 .397 N2= 121
.019 .288 .692 N3= 52
Women All 25-55
3′′.212 .558 .230 N1= 217
.070 .572 .358 N2= 215
.031 .337 .633 N3= 98
Both All 25-55
8.4.2 Average Schooling Years by Cohort in the GSS
The average number of schooling years for individuals born in the US and aged between 25 and
55 years old at the time of the interview ranges from 11.03 for those born in the period 1915-1924
to 13.86 for the cohort 1975-1984. We use individuals born on or after 1945 because the average
schooling years by cohort are roughly constant since then, while the average schooling years trend
upward for previous cohorts.
A-12
Table 10. Average schooling years by GSS cohort∗
Cohort born 1915-24 1925-34 1935-44 1945-54 1955-64 1965-74 1975-84
Mean 11.03 12.01 12.93 13.65 13.64 13.87 13.86
Std Dev 3.33 3.20 2.95 2.74 2.52 2.55 2.74
Sample (N) 115 1193 3342 7157 7277 3561 996∗Individuals born in the US, age 25-55 at the time of interview. Men and Women.
8.5 Appendix to Section 3.2. Estimation of TFR’s for Native and
Foreign Born Women in the US
We estimate total fertility rates (TFR) by education level and nativity, starting from the fertility
rates by education level that can be computed from birth and census data. Since birth data
available at the VitalStats website doesn’t distinguish whether the mother was US-born or foreign-
born, we construct estimates with the help of the American Community Survey (ACS), which
identifies the place where the mother was born and which can be used to estimate TFR’s. The
estimators are derived for each specific level of skill levels as defined in this paper. We estimate
TFR’s for years 1990, 2000 and ACS 2005. The results are in table 11.
Table 11.TFR’s for all women in US by education
Year less than HS HS + Some College BA+Beyond
2005 2.73 1.95 1.85
2000 2.26 2.00 1.84
1990 2.43 2.04 1.61
Average 2.47 2.00 1.77Census 1990, 2000. ACS-2005.
Now we show how we estimate TFR’s by nativity from the TFR’s in table 11 (for all women).
For the particular estimates, we keep notation simple by not including a particular skill level.
Define Xki = Number of children born during year to women type k in age-group i; Yk
i =total #
of women type k in age-group i; k = N,F , where N=US-born and F=foreign-born; n = # of
age-groups (5-year groups); TFRk =Total fertility rate of women type k ; TFR =Total fertility
rate of all women living in the US.
The goal is to obtain estimates for TFRN and TFRF as all data for their direct estima-
tion is not available. So we construct estimators of TFRN and TFRF based on the TFR for
all women. The TFR can be written as TFR = 5Σni
[(XNi +XF
i
)/(Y Ni + Y F
i
)]. It can be
rewritten as TFR = TFRF − 5Σni w
Ni mi, with weights wNi = Y N
i /(Y Ni + Y F
i
), and where mi =[(
XFi /Y
Fi
)−(XNi /Y
Ni
)]is the difference in births per foreign—born women and births per US-
born women in age group i (for a given level of education). Hence an estimate of TFRF is
obtained by computing TFRF = TFR + 5Σnimiw
Ni . Similar algebra yields native fertiliy as
TFRN = TFR− 5Σnimi
(1− wNi
).
A-13
Table 12.Estimated weights (wi)
Census 1990, 2000 and CPS-2005 by skill level (U, M, S)*
1990 2000 2005Age Group U M S U M S U M S
15-19 .082 .068 .250 .092 .083 .256 .078 .089 .209
20-24 .209 .079 .091 .296 .113 .135 .263 .109 .107
25-29 .229 .077 .106 .394 .127 .160 .383 .133 .157
30-34 .248 .075 .111 .400 .122 .161 .476 .148 .189
35-39 .257 .070 .096 .360 .107 .155 .452 .142 .177
40-44 .241 .075 .099 .339 .097 .135 .386 .117 .159
45-49 .208 .074 .101 .345 .087 .115 .348 .099 .141
50-54 .185 .077 .106 .303 .088 .106 .345 .090 .116
* U= Unskilled, M=Medium-Skilled, S=Skilled
The estimates for the weights wi come from the census of 1990, 2000 and the 2005 CPS.
However, data is not available for the differences mi for these exact years. Thus we estimate the
average differencemi for each age-education group for years 2001−2008 for from ACS data. Using
these numbers, we arrive at the fertility rates by skill level and nativity shown in the text.
Table 13. Estimates of differences in births per women(foreign - natives : mi =
[(XFi /Y
Fi
)−(XNi /Y
Ni
)])∗
Age Group 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54
Unskilled .014 .002 .042 .051 .036 .015 .000 .000
Med-skilled .002 .000 .0016 .037 .030 .012 .003 .000
Skilled -.003 .010 .005 -.004 .012 .009 .003 .001∗averages for 2001-2008 (ACS)
8.6 Appendix to section 4.1. Steady State Composition of the Native
Population in absence of Immigration
In the absence of immigration we have that St is time invariant, so we drop the time subscript.
Thus in this case we have that S = Q′η and the scalar given by(St[2]Xt
)=∑3
i xitηiqi2. Therefore
the evolution of the native population is given by
Xt+1 = Q′ηXt/(∑3
ixitηiqi2
).
Multiply both sides of the above expression by the scalar(∑3
i xitηiqi2), and evaluate the product
Q′ηXt in order to write
A-14
x1t+1
(∑3i xitηiqi2
)∑3i xitηiqi2
x3t+1
(∑3i xitηiqi2
) ≡ Xt+1
(∑3
ixitηiqi2
)= Q′ηXt ≡
∑3
i xitηiqi1∑3i xitηiqi2∑3i xitηiqi3
.Substracting Q′ηXt on both sides yields the [3x1] system of equations
Xt+1
(∑3
ixitηiqi2
)−Q′ηXt = 0.
At a steady state, this system is given byxss1
(∑3i x
ssi ηiqi2
)−∑3
i xssi ηiqi1
0
xss3(∑3
i xssi ηiqi2
)−∑3
i xssi ηiqi3
=
0
0
0
(A.2)
which is a quadratic system of 2 equations in the 2 unknowns (xss1 , xss3 ) , where one of the solutions
is the steady state composition of the population in the absence of immigration.
8.7 Appendix to sections 4.4 - 6.4: Numerical Analysis and Optimal
Policies
This section explains the algorithm used for the numerical analysis, including comments on the
choice of technical parameters.
We define the steady state pair ratios in absence of immigration asxSS1 , xSS3
, which is the so-
lution to (A.2) and turn out to be .09782, .5429. A grid ofNX1 points in[(1− a1)xSS1 , (1 + a2)xSS1
]and NX2 points in
[(1− b1)xSS3 , (1 + b2)xSS3
]for a total of NX1 ∗NX2 pairs in the resulting grid is
used for the value function iteration procedure, where a1, a2, b1 and b2 are positive constants that
represent the deviation from the no-immigration steady state. The specific nodes that are used
for interpolation are chosen as the roots of Chebyshev polynomials in the considered state spaces.
The specific constants that define the state grid depend on the policy space (i.e. big high-skilled
quotas might imply an induced state out of the grid if b2 is too small), which we now describe.
The policy space considers Nθ1 points for θ1 in the space [0, θmax1 ] , Nθ2 points for θ2 in [0, θmax
2 ]
and Nθ3 points for θ3 in [0, θmax3 ].
We first explain Setting (II), which has a perfectly inelastic supply of high skill immigrants
(θmax3 = 13%) in section 4.5 in detail as it provides our main results, then comment on the case
with wage-elastic supply of high-skilled immigrants (section 6.1), and finally on Settings (I) in
section 4.4. Comments on Settings (II) also apply to case (III), since allowing guest workers only
adds to choice variables but does not expand the range of parameters.
In the case of a "small pool" of high-skilled immigrants (θmax3 = 13%), the particular grid
used are points in [.07826, .176084] × [.434353, .977295] .The considered immigration policies are
A-15
elements (θ1, θ2, θ3) ∈ [0, θmax1 ]×[0, θmax
2 ]×[0, θmax3 ] ⊆ R3
+. For this version of the model, the number
of policy points in θ2 and the maximum immigration θmax2 quota turn out to be irrelevant since
the model always predicts (given parameterization of model) that the majority chooses θ∗2 = 0.
Similarly, given that the maximum level θmax3 = 0.13 is lower than the quota that would be
freely chosen under Settings (I), the majority chooses θ∗3 = θmax3 = 0.13. The only relevant policy
information in this version of the model are the number of policy points available for θ1, having a
large enough θmax1 in order to allow for an interior solution, and the specific level of θmax
3 (set at
13% in this version). We show the case where θmax1 = 1.25, and θmax
3 = 0.13, with 401 equidistant
points in the policy space for θ1 ∈ [0, 1.25] as in this case the number of points for high skill
immigration is irrelevant. Solving the model with a tolerance of .0000001 in the sup norm between
the value functions of the current and the previous iteration, we obtain an 18.09% quota of low-
skilled workers, and the high-skilled immigration quota is chosen at 13%. The induced steady state
is (.1006, .5886) . Cases where there are guest worker quotas available typically involve increasing
the size of the policy space.
Figure 1. Optimal Policy function for low-skilled immigration.
0.06 0.08 0.1 0.12 0.14 0.16 0.180.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
X3 (N3/N2)
X1 (N1/N2)
Uns
kille
dQ
uota
Case with θmax3 = 13%
As an example of the solution to the case with a wage-elastic supply as captured by (17) , we
use an elasticity of 10, while the rest of the parameters set as in the main text; with a state grid of
900 pairs of points in [.09478, .176084]× [.434353, .7492]. In this case the i− th immigration policypoint (θ1i, θ2i, θ3i) at the state node (x1, x3) is feasible if θ3i ≤ θmax
3 (w3 (x1, x3, (θ1i, θ2i, θ3i))) .
We use 90 points for θ1 ∈ [0, 2] and 150 points in θ3 ∈ [0, θmax3 (w3 (·))]. Again θmax
2 (and the
number of policy points for θ2) are irrelevant. At the no-immigration steady state the optimal
policies are (42.74%, 0%, 19.88%) . Below we show the immigration policies chosen in the state-grid
considered. The shape of the low-skilled immigration quota is just like before: increasing in x3
and decreasing in x1, while the shape of the high-skilled immigration policy is decreasing in x3
and slightly increasing in x1. Other numerical cases look qualitatively identical and are therefore
not discussed further.
A-16
Figure 2. Optimal policy functions with an elastic response of θmax3 (Elasticity=10)
Unskilled Immigration Skilled Immigration
0.08
0.1
0.12
0.14
0.16
0.18 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
0.1
0.2
0.3
0.4
0.5
0.080.1
0.120.14
0.160.18 0.4
0.5
0.6
0.7
0.80
0.2
0.4
0.6
0.8
In Setting (I), which is the case of a huge pool of high-skilled immigrants, we successively
expanded the maximum immigration quotas until we found a policy space such that the resulting
optimal functions were in the interior for all possible states. In the case of medium-skill immi-
gration, θmax2 = 100% was high enough, as the optimum policy is θ∗2 = 0. For the other types,
θmax1 = 4.5 and θmax
3 = 3 turned out to be suffi cient.
A-17