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

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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-

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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