Diverse welfare provisions and migrant self-selection 1 by Łukasz Byra University of Warsaw Mailing Address: Łukasz Byra April 2019 Faculty of Economic Sciences, University of Warsaw Długa 44/50 00-241 Warsaw Poland E-mail Address: [email protected]ORCID ID: 0000-0003-1935-7264 I am indebted to Anna Janicka for thoughtful and inspiring comments. 1 This research was supported by the National Science Centre (Poland) under Grant ‘European Welfare Systems at Time of Mobility’ (number 2014/14/Z/HS4/00006).
Transcript
Diverse welfare provisions and migrant self-selection1
by
Łukasz Byra
University of Warsaw
Mailing Address: Łukasz Byra April 2019 Faculty of Economic Sciences, University of Warsaw Długa 44/50 00-241 Warsaw Poland
E-mail Address: [email protected] ORCID ID: 0000-0003-1935-7264 I am indebted to Anna Janicka for thoughtful and inspiring comments.
1 This research was supported by the National Science Centre (Poland) under Grant ‘European Welfare Systems at Time of Mobility’ (number 2014/14/Z/HS4/00006).
Abstract
We study the impact of welfare provisions in the destination country on migrants’ self-
selection under the assumption that migration is driven by an international difference in
returns to skills and by the incidence of unemployment in the source country. We consider two
types of provisions: a skill formation subsidy and an unemployment benefit. We show that
welfare provisions affect the returns from migration directly, via the balance between the
receipt of a provision and its cost in the form of taxation, and indirectly, by shaping the skill
composition of the destination country’s native workforce, which bears upon the workers’
wages and upon the tax rate. These indirect effects at least partly offset the direct effect,
reducing the intensity of welfare migration as compared to a setting in which the skill
composition of the destination country’s native workforce is unaffected by the generosity of
welfare provisions. A simulation illustrates the relative strength of the direct and indirect
effects, and reveals that the impact of welfare provisions on migrants’ self-selection can vastly
differ depending on the type of provision and on the characteristics of the sending country and
the destination country.
Keywords: Welfare provisions; Skill formation subsidy; Unemployment benefit; Size and
skill composition of migration; Selection into migration;
JEL classification: F22; H31; J24
1
1. Introduction
Two decades ago, Borjas (1999) introduced the “welfare magnet” hypothesis, which states
that the generosity of welfare provisions at destination is an important pull factor for the
migrants. The results of empirical research provide support for existence of the welfare
magnet effect, yet its strength is found to be relatively weak (Péridy 2006; De Giorgi and
Pellizari 2009; Razin and Wahba 2015). Notable efforts have been made to establish whether
immigrants are net fiscal contributors or net fiscal burden for welfare states (Boeri 2010;
Dustmann et al. 2010; Dustmann and Frattini 2014; Razin and Wahba 2015), and whether the
arrival of immigrants causes welfare states to increase or curtail redistributive spending
(Razin et al. 2002; Gaston and Rajaguru 2013), with inconclusive results. Another strain of
the literature focuses on the political process which leads to the formulation of the destination
country’s migration policy with regard to the preferred skill composition of migration (Cohen
and Razin 2008; Cohen et al. 2009, Razin and Wahba 2015; Suwankiri et al. 2016).
An important aspect of the interactions between the generosity of welfare provisions at
destination and migration is the impact of provisions on migrants’ self-selection. There is
evidence that welfare states attract relatively low-skill, low-earning migrants (Boeri 2010;
Razin and Wahba 2015). The theory behind this evidence is, however, ill-developed: simple
models yield the result that low-skill migrants are net beneficiaries of the tax and transfer
system, as opposed to high-skill, high-earning migrants, who are subjected to hefty taxation
(Razin and Sadka 2000; Razin et al. 2002; Cohen and Razin 2008; Cohen et al. 2009). We
argue that these simple models provide only a partial picture of the complex relationship
between the generosity of welfare provisions and the skill composition of migration, as they
abstract from three important considerations.
First, when welfare provisions are modelled as lump sum transfers of equal amount
regardless of a recipient’s characteristics, when these provisions are financed from a
proportional tax on workers’ wage earnings, and when wage earnings rise with skill level,
then the negative relationship between the generosity of provisions and the migrants’ skill
level is bound to obtain. However, it stands to reason that welfare states offer sets of
provisions of different types, each targeted to a specific group of individuals (Brown and
Kaufold 1988; Dellas 1997; Wigger 1999; Cremer et al. 2011). Some of these provisions,
such as skill formation subsidies, are directed only to (prospective) high-skill individuals.
With respect to such provisions, we should expect that they attract (prospective) high-skill
2
migrants rather than low-skill migrants, as the latter would participate in financing the
provision, yet would not benefit from it. In turn, the types of provisions that do not
discriminate between the recipients’ skill level, such as unemployment benefits, can indeed be
expected to attract low-skill migrants.
Second, the generosity of welfare provisions in a destination country affects not only
the skill composition of migration to that country, but, above all, the skill composition of its
native workforce. Other things equal, two countries differing by the generosity of a provision
will also differ by the relative supply of high-skill and low-skill work. If high-skill and low-
skill work are complementary factors of production, then we should expect the two countries
to differ by wages paid for each type of work. Moreover, if both countries finance welfare
provisions from a proportional tax on wage earnings, and if the earnings of high-skill workers
are higher than the earnings of low-skill workers, then a difference in the relative supply of
high-skill and low-skill work between the two countries means that, for a given tax rate, the
countries will collect different tax revenues. Therefore, to maintain balanced budget, the two
countries must differ by their tax rates not only to the extent required by different generosity
of (or expenditures on) the provision in these countries, but also to account for the fact that a
shift in relative supply of high-skill and low-skill work impinges on the collected tax revenue.
Because wages and the tax rate at destination are important inputs in the calculus of returns
from migration, the impact of welfare provisions on these variables must be accounted for
when modelling migration flows.2
Third, existing models studying the interactions between welfare provisions and the
skill composition of migration abstract from issues related to selection into migration; they
implicitly assume that all individuals in the sending country wish to migrate, albeit some more
than others. Consequently, the results yielded by these models are of qualitative nature,
namely that welfare provisions at destination have an adverse (a positive) effect on the
number of migrating high-skill (low-skill) workers. To study the quantitative relationship
between the generosity of welfare provisions and the skill composition of migration, explicit
modelling of selection into migration is required.
2 Razin et al. (2002) endogenize skill choices of the destination country’s native population, yet in their model high-skill workers and low-skill workers are perfect substitutes in output production. Thus, their model does not account for the impact of the generosity of welfare provisions on relative wages of the two types of workers.
3
We argue that a meaningful theoretical assessment of the impact of welfare provisions
on the skill composition of migration has to distinguish between different types of provisions;
it has to account for the response of the destination country’s native workforce to the
incentives emanating from welfare provisions; and it cannot be studied separately from
considerations related to selection into migration. In this paper, we provide such an
assessment. We construct a model of a developed country in which the consumption good is
produced by identical firms using a combination of high-skill and low-skill work. At the
beginning of their single-period lives, individuals choose whether to engage in costly skill
formation and become high-skill workers, or to (costlessly) become low-skill workers.
Individuals differ by their ability, which affects their productivity as high-skill workers, and
has no effect on their productivity as low-skill workers. Uncertainty hovers over individuals’
employment track: each individual faces an exogenous probability of temporary
unemployment, in the event of which his earnings are lower than if he was permanently
employed. The government of the developed country incentivizes skill formation by means of
the skill formation subsidy, and it provides the unemployment benefit for each temporarily
unemployed worker. These welfare provisions are financed from a proportional tax on wage
earnings. The equilibrium in the country’s labor market is determined by a cut-off level of
ability such that individuals with higher ability than the cut-off level choose to become high-
skill workers, whereas individuals with lower ability than the cut-off level choose to become
low-skill workers. We find that the welfare provisions shape the division of individuals
between the two types of workers such that the skill formation subsidy increases the supply of
high-skill work, whereas the unemployment benefit increases the supply of low-skill work.
Then, we introduce a foreign, sending country, which is technologically less advanced
than the developed, destination country. We characterize the equilibrium in the labor market
of the sending country. We allow for permanent migration between the two countries.
Prospective migrants in the sending country choose the timing of their migration. There are
two moments to choose from: just prior to skill formation, to which we will refer as the first
wave of migration, and just after entering the labor market and learning the employment
track, to which we will refer as the second wave of migration. Under specific conditions, the
first wave of migration is composed of the most able individuals in the sending country who
become high-skill workers in the destination country, and the second wave of migration is
composed of the least able individuals facing temporary unemployment in the sending
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country. We assume that skills are country-specific, indicating that the second-wave migrants
are employed exclusively as low-skill workers, regardless of their skill level.
In such a setting, we qualitatively assess the impact of an increase in the generosity of
the skill formation subsidy, and in the generosity of the unemployment benefit in the
destination country on the size of each wave of migration. We find that an increase in the
generosity of a provision impacts the size of each wave of migration directly and indirectly.
The direct impact is determined by the interplay between the positive effect of the increased
provision, and the negative effect of the increased taxation required to finance the provision.
The indirect impact is a resultant of a change in the division of the destination country’s
native workforce between high-skill workers and low-skill workers - a response to the shift in
incentives emanating from an increase in the generosity of a provision. This response
necessitates adjustments in the country’s tax rate (which is not the same as the increase in
taxation captured by the direct impact), and it affects the wages of the two types of workers.
To provide a quantitative assessment of the impact of an increase in the generosity of
welfare provisions on the size of each wave of migration, and on the skill composition of
migration, we apply a numerical simulation to our model. Contrary to our expectations, we
find that the skill formation subsidy discourages from migration not only the low-ability
second-wave migrants, but also the high-ability first-wave migrants: although the direct
impact of the skill formation subsidy on expected income in the destination country is positive
for the latter type of migrants, the indirect, adverse impact of the subsidy on the wage of high-
skill workers in that country more than offsets the positive, direct impact. The unemployment
benefit has the expected impact on the size of each wave of migration: it is a magnet for the
low-ability second-wave migrants, and a disincentive for the high-ability first-wave migrants.
Sensitivity analysis reveals that the impact of welfare provisions on the size and skill
composition of migration can significantly differ, depending on the characteristics of the
destination country and the source country. A 1.5-fold increase in the risk of unemployment
in the source country transforms the skill formation subsidy from being a disincentive to
migration for the ablest individuals in the source country to being a migration magnet for
these individuals.
The paper is organized as follows. In Section 2, we model the developed, closed to
migration country. We work out the equilibrium in the labor market of that country and we
establish how it is affected by the generosity of welfare provisions. In Section 3, we allow for
5
migration from a foreign country, and we characterize the skill composition of migration. We
study how the generosity of welfare provisions in the developed, destination country affects
the size and skill composition of migration both analytically and numerically. Section 4
concludes.
2. A closed to migration setting 2.1. Setup
Consider a developed country populated by a mass of individuals of measure one. At the
beginning of their single period lives, individuals choose the type of workers they wish to
become. The choice is between high-skill workers, denoted by H, and low-skill workers,
denoted by L. To become a high-skill worker, an individual must engage in costly skill
formation such that the cost is fixed at 0k > . In contrast, becoming a low-skill worker is
costless. Individuals choose whether to become high-skill workers or low-skill workers by
comparing expected incomes yielded by these choices. The expected income of individual j
whose type is ,l H L= is given by
, ,(1 )j j jl l g l bEI p I pI= − + , (1)
where I is income, with subscripts g and b indicating “good” and “bad” realizations,
respectively, and where p is the probability of a “bad” realization, 0 1p< < . The two
realizations label two distinct possibilities regarding an individual’s employment track. A
“good” realization obtains if an individual is employed throughout his entire life. Otherwise,
he is unemployed for a fraction 0 1β< < of his life, and he is employed for the remaining
fraction 1 β− of his life. The probability of temporary unemployment, p, is exogenously
given, and it is assumed to be the same for high-skill workers and for low-skill workers.
Initially, individuals differ only by their innate ability, θ , which is a random variable
defined over the interval (0, ]T according to a density function ( )f ⋅ and a cumulative
distribution function ( )F ⋅ , such that ( ) ( ) 0f z F z′= > for all (0, ]z T∈ . Ability determines an
individual’s performance as a high-skill worker. Specifically, over the course of his life, high-
skill worker j supplies inelastically jθ units of labor under a “good” realization (when he is
permanently employed) or (1 ) jβ θ− units of labor under a “bad” realization (when he faces a
spell of unemployment). Each low-skill worker supplies 1 unit of labor under a “good”
6
realization, and 1 β− units of labor under a “bad” realization. That is, labor supply of a low-
skill worker does not depend on his ability.
Individuals receive wage earnings for their work as high-skill workers or as low-skill
workers. Each high-skill worker earns Hw per unit of supplied labor, and each low-skill
worker earns Lw per unit of supplied labor. Therefore, gross lifetime earnings of high-skill
worker j are given by jHw θ under a “good” realization and (1 ) j
Hw β θ− under a “bad”
realization. In turn, gross lifetime earnings of a low-skill worker under a “good” realization
and under a “bad” realization are given by Lw and (1 )Lw β− , respectively.
The sequence of events is as follows. First, an individual chooses his skill level. Then,
he enters the labor market, learns his realization (“good” or “bad”), and earns income
according to the realization. Finally, the individual consumes his income.
A large number of competitive firms employ high-skill and low-skill workers to
produce the consumption good, which they sell at a unit price. The production of firm i is
given by
1( )i Hi Hi LiY A N Nα αθ −= , (2)
where 0A > is the country’s total factor productivity; α and 1 α− are the output elasticities
of high-skill and low-skill work, respectively; HiN is the size of high-skill workforce
employed by firm i; LiN is the size of low-skill workforce employed by firm i; and Hiθ is the
mean productivity of high-skill workers employed by firm i. For the purpose of subsequent
analysis, we define 11H Hi
iN N
pβ≡
− ∑ as the size of the country’s (employed and
unemployed) high-skill workforce, 11L Li
iN N
pβ≡
− ∑ as the size of the country’s (employed
and unemployed) low-skill workforce, 1(1 )H Hi Hi
iH
Np N
θ θβ
≡− ∑ as the mean productivity
of the country’s (employed and unemployed) high-skill workforce, and ii
Y Y≡ ∑
1(1 ) ( )H H Lp A N Nα αβ θ −= − as the country’s output and output per individual.
7
The government of the country under consideration taxes its working population,
collecting a fraction t of each worker’s wage earnings. The tax revenue is then used to finance
the provision of welfare benefits in two areas: skill formation, and unemployment insurance.
Specifically, the government subsidizes skill formation by decreasing the cost of becoming a
high-skill worker from k to (1 )s k− , where 0 1s< < is the share of the subsidy in the overall
cost of skill formation. Additionally, the government provides an unemployment benefit equal
Lu wβ to each worker facing a “bad” realization, where 0 1u t< < − is the replacement rate of
the unemployment benefit to the wage per unit of low-skill work.
The government is assumed to run a balanced budget, which relates the level of taxation
to the generosity of welfare provisions:
H LtY skN up wβ= + . (3)
2.2. Market equilibrium
An individual’s income is a sum of wage earnings net of taxation, and welfare benefits
provided by the government. Therefore, if individual j chooses to become a high-skill worker,
his income under a “good” realization, and his income under a “bad” realization, will be
given, respectively, by
, (1 ) (1 )j jH g HI t w s kθ= − − − , and , (1 ) (1 ) (1 )j j
H b H LI t w s k u wβ θ β= − − − − + . (4)
If instead j chooses to become a low-skill worker, his income under a “good” realization, and
his income under a “bad” realization, will be given, respectively, by
, (1 )jL g LI t w= − , and , (1 ) (1 )j
L b L LI t w u wβ β= − − + . (5)
Utilizing (4) in (1), on rearrangement, j’s expected income is rewritten as
(1 )(1 ) (1 )j jH H LEI t p w s k up wβ θ β= − − − − + (6)
if he is a high-skill worker or, utilizing (5) in (1), as
(1 )(1 )jL L LEI t p w up wβ β= − − + (7)
if he is a low-skill worker.
Individuals choose the type of work that returns higher expected income: j will
become a high-skill worker if j jH LEI EI> or, recalling (6) and (7), on rearrangement, if
8
(1 )(1 ) (1 )(1 )(1 )
j L
H
t p w s kt p w
βθβ
− − + −>
− −. If (1 )(1 ) (1 )
(1 )(1 )j L
H
t p w s kt p w
βθβ
− − + −≤
− −, j will become a low-skill
worker. We denote the individual who is indifferent between becoming a high-skill worker
and becoming a low-skill worker with an asterisk. For that individual we have H LEI EI=
which, recalling (6) and (7), on rearrangement, can be rewritten as
*(1 )( ) (1 )1H L
kt w w sp
θβ
− − = −−
. (8)
Profit-maximizing firm i employs high-skill workers and low-skill workers up to the
point where the wage per unit of high-skill work and the wage per unit of low-skill work
equal their respective marginal products, namely
1
LiH
Hi Hi
Nw AN
α
αθ
−
=
and (1 ) Hi HiL
Li
Nw AN
αθα
= −
. (9)
Because firms are identical and because they face the same wage costs, they employ high-skill
and low-skill workers in the same ratio, which is also the market ratio. This allows us to drop
subscripts i in (9), and write the firms’ aggregate profit maximization conditions,
1
LH
H H
Nw AN
α
αθ
−
=
and (1 ) H HL
L
Nw AN
αθα
= −
. (10)
Because individuals with ability *θ θ> will choose to become high-skill workers, whereas
individuals with ability *θ θ≤ will choose to become low-skill workers, we have the
following relationships: *
*( ) 1 ( )T
j jHN f d F
θ
θ θ θ= = −∫ ; *
*
0
( ) ( )j jLN f d F
θ
θ θ θ= =∫ ;
*
**
1 ( ) ( )1 ( )
Tj j j
H f dF θ
θ θ θ θ θ θθ
= ≡− ∫ ; and 1(1 ) ( )H H LY p A N Nα αβ θ −= −
* * * 1(1 ) [ ( )(1 ( ))] ( )p A F Fα αβ θ θ θ θ −= − − . Therefore, we rewrite (10) as
1*
** *
( )( )( )(1 ( ))H
Fw AF
αθθ α
θ θ θ
−
= − and
* **
*
( )(1 ( ))( ) (1 )( )L
Fw AF
αθ θ θθ α
θ −
= −
, (11)
where this time we stress that Hw and Lw are functions of *θ .
9
Utilizing *1 ( )HN F θ= − , * * * 1(1 ) [ ( )(1 ( ))] ( )Y p A F Fα αβ θ θ θ θ −= − − , and (11), we
where this time we stress that t is a function of *θ . Solving (12) for *( )t θ , we obtain the
formula for the tax rate as a function of *θ ,
*
** * * 1 *
(1 ( )) 1( ) .(1 ) [ ( )(1 ( ))] ( ) 1 ( )
k F pt s up A F F p Fα α
θ β αθβ θ θ θ θ β θ−
− −= +
− − − (13)
We now have all the building blocks needed to characterize the market equilibrium,
which is given by (8) upon substitution for Hw and Lw from (11), and for t from (13). That is,
the market equilibrium is determined by *θ which solves
* * * *(1 ( )) ( ) ( ) (1 )1H L
kt w w sp
θ θ θ θβ
− − = − −, (8’)
where *( )Hw θ and *( )Lw θ are given by (11), and where *( )t θ is given by (13). We have the
following claim.
Claim 1. *θ exists and is unique.
Proof. The proof is in the Appendix.
Having established the division of individuals between high-skill workers and low-
skill workers, we inquire if this division can be altered by means of the skill formation
subsidy and/or by means of the unemployment benefit. We have the following two claims.
Claim 2. Other things remaining the same, the higher the skill formation subsidy, the more
individuals choose to become high-skill workers, *
0s
θ∂<
∂.
Proof. The proof is in the Appendix.
Claim 3. Other things remaining the same, the higher the unemployment benefit, the fewer
individuals choose to become high-skill workers, *
0uθ∂
>∂
.
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Proof. The proof is in the Appendix.
Claim 2 and Claim 3 show that the skill formation subsidy and the unemployment benefit are
effective tools for inducing changes in the skill composition of the country’s workforce.
3. Introducing migration
3.1. A two country setup
Assume now that there is a second country, S (for “source”), which is less developed than the
country studied in Section 2, to which we now refer as D (for “destination”), and that
migration is possible from S to D. The cost of migration is the same for all individuals in S at
0m > . We assume that S is similar to D in the following respects. First, preferences of
individuals in S, just as preferences of individuals in D, are characterized by (1). Second,
ability is distributed in the population of S on the same interval and according to the same
cumulative distribution function as in the population of D. There are a few differences though.
The population of S is assumed to be of size 0SN > , where henceforth superscript S denotes
a parameter or a variable characterizing S. (To recall, the size of D’s population is one.) The
assumption that S is less developed than D is captured by the total factor productivity being
higher in D: the production technology in S is as per (2), yet with SA A< . We assume that S
also taxes its working population and uses the tax revenue to finance the skill formation
subsidy and the unemployment benefit, and that the generosity of these provisions can differ
between the two countries. Because the preferences of individuals, the distribution of ability,
and the production technology in S are akin to those in D, the equilibrium in the labor market
of S is determined in the same way as the equilibrium in the labor market of D, namely as per
Section 2. Specifically, the equilibrium is characterized by a unique ability level of an
individual who is indifferent between becoming a high-skill worker and becoming a low-skill
worker, *Sθ .
Migration from S to D is on a forever basis; there is no return migration. Skills are
country-specific: an individual who forms skills in S will work as a high-skill worker only if
he stays in S; if he migrates to D, he will work as a low-skill worker.3 Individuals who wish to
3 Our assumptions in the open-to-migration framework are an attempt to mimic the characteristics of migration between Poland and the UK following the 2004 EU enlargement. The assumption of country-specific skills is based on evidence that Polish migrants in the UK, although relatively skilled, usually work in low-skill, low-paid occupations and, by and large, are concentrated at the bottom of the pay distribution (Drinkwater et al. 2009).
11
migrate must decide on the timing of their migration. There are two moments in life when
migration is possible: just prior to choosing the type of work and just after learning the
realization in the labor market (“good” or “bad”). We will refer to migration prior to choosing
skill level as the first wave of migration, and to migration after learning the realization in the
labor market as the second wave of migration. We assume that the relationships between the
model’s parameters and the variables measuring the generosity of welfare provisions in the
two countries (namely, the relationships between k, p, β , A, s, u, and, consequently, t, and
their counterparts in S) are such that the following two sets of inequalities hold.
The first set of inequalities determines the skill composition of the first wave of
migration, and is given by SL LEI m EI− < , * *( ) ( )S S S
H HEI m EIθ θ− < , and ( )HEI T m−
( )SHEI T> . These three inequalities jointly imply that the first wave of migration is composed
of the most able individuals in S, who become high-skill workers in D.4 SL LEI m EI− <
indicates that those at the bottom of the ability distribution in S do not find it beneficial to
migrate to D to work there as low-skill workers. * *( ) ( )S S SH HEI m EIθ θ− < indicates that for
the individual with ability *Sθ it is more rewarding to become the least able high-skill worker
in S than to migrate to D to work there as a high-skill worker. (Because, by definition of *Sθ , *( )S S S
H LEI EIθ = , inequalities SL LEI m EI− < and * *( ) ( )S S S
H HEI m EIθ θ− < jointly imply that
none of the individuals in S with ability *Sθ θ< will migrate to D to work there as a high-skill
worker, and that the individual with ability *Sθ will not migrate to D to work there as a low-
skill worker.) ( ) ( )SH HEI T m EI T− > indicates that for the most able individual in S, migration
to D to work there as a high-skill worker is more rewarding than becoming a high-skill
worker in S.
Because in each country the expected income of a high-skill worker depends linearly
on a worker’s ability (cf. (6)), from a conjunction of * *( ) ( )S S SH HEI m EIθ θ− < and
( ) ( )SH HEI T m EI T− > it follows that there must exist a cut-off ability level, which we denote
4 That the first wave of migration is manned by individuals wishing to obtaining destination-country-specific skills is consistent with the intra-EU migration experience. For example, in 2016/17 in the UK, almost 135 thousand (almost 6 percent) of first-year higher education students came from EU member states other than the UK, which is a measure of the annual inflow of students to the UK from other EU member states. After graduation, these migrants should have little trouble finding a high-skill job in the UK labor market. (Unfortunately, because intra-EU migration is not regulated, there is no data on how many of these students remain in the UK after graduation.)
12
by 1θ , *1
S Tθ θ< < , such that individuals with ability 1θ θ> will choose to migrate and
become high-skill workers in D, whereas individuals with ability *1
Sθ θ θ< ≤ will choose to
stay and become high-skill workers in S. For the individual with ability 1θ , we must have that
his expected income when living and working as a high-skill worker in D equals his expected
income when living and working as a high-skill worker in S, namely 1 1( ) ( )SH HEI m EIθ θ− = .
On substitution for HEI and, likewise, for SHEI from (1), the condition 1 1( ) ( )S
H HEI m EIθ θ− =
becomes , 1 , 1 , 1 , 1(1 ) ( ) ( ) (1 ) ( ) ( )S S S SH g H b H g H bp I pI m p I p Iθ θ θ θ− + − = − + . Utilizing (4), and
solving for 1θ , we get that5
1(1 ) (1 )
(1 ) (1 ) (1 ) (1 )
S S S S S SL L
S S S SH H
s k s k u p w up w mt w p t w p
β βθβ β
− − − + − +=
− − − − −. (14)
The determination of 1θ is depicted in Figure 1. Dark blue lines correspond to
expected incomes in S, and light blue lines correspond to expected incomes in D. Straight
lines correspond to expected incomes yielded by optimal choices (of the country of residence
and of the chosen type of work), and dotted lines correspond to expected incomes yielded by
suboptimal choices. It can easily be verified that the three underlying inequalities S
L LEI m EI− < , * *( ) ( )S S SH HEI m EIθ θ− < , and ( ) ( )S
H HEI T m EI T− > hold.
Figure 1. The determination of 1θ .
5 From a conjunction of * *( ) ( )S S
H
SHEI m EIθ θ− < and ( ) ( )H
SHEI T m EI T− > , on substitution for ( )HEI ⋅ and
( )SHEI ⋅ from (6), it follows that the denominator of 1θ is positive.
13
The second set of inequalities determines the skill composition of the second wave of
migration, and is given by ,S
L L bEI m I− > , *, ( ) (1 )S S S S
L H bEI m I s kθ− > + − , and
, 1( ) (1 )S S SL H bEI m I s kθ− < + − . These three inequalities imply that the second wave of
migration is composed of the least able individuals in S, whose realization in the labor market
of S was “bad.”6 ,S
L L bEI m I− > indicates that all low-skill workers who face temporary
unemployment in S will choose to migrate to D. *, ( ) (1 )S S S S
L H bEI m I s kθ− > + − indicates that
for the high-skill worker with ability *Sθ who faces temporary unemployment in S, it is more
rewarding to migrate to D to work there as a low-skill worker than to stay and (temporarily
not) work in S. The term (1 )S Ss k− appears on the right-hand side of the latter inequality
because for the high-skill worker with ability *Sθ the cost of skill formation is sunk, and it
must not enter the balance of returns from migration and from staying put (adding (1 )S Ss k−
corrects *, ( )S S
H bI θ for the sunk cost of skill formation). , 1( ) (1 )S S SL H bEI m I s kθ− < + −
indicates that for the individual with ability *1
Sθ θ> it is more advantageous to face
temporary unemployment in S as a high-skill worker, than to migrate to D to work there as a
low-skill worker.
As in the case of the first wave of migration, in the case of the second wave of
migration there is a cut-off level of ability, which we denote as 2θ , such that workers with
ability 2θ θ< who face temporary unemployment in S will choose to live and work as low-
skill workers in D, whereas all individuals with ability 2 1θ θ θ≤ ≤ will choose to stay in S.
For the high-skill worker with ability 2θ , we must have that his expected income when living
and working as a low-skill worker in D equals his expected income when staying in S, namely
, 2( ) (1 )S S SL H bEI m I s kθ− = + − . On substitution for LEI from (1),
, 2( ) (1 )S S SL H bEI m I s kθ− = + − is rewritten as , , , 2(1 ) ( ) (1 )S S S
L g L b H bp I pI m I s kθ− + − = + − .
Substituting further for , 2( )SH bI θ from (4), and for ,L gI and ,L bI from (5), and solving for 2θ ,
the latter equality becomes 6 The unparalleled inflow of migrants to the UK from such countries as Poland after the 2004 EU enlargement can be at least partially explained by the high unemployment rate in Poland, which in 2004 was almost 20 percent.
14
2(1 ) (1 )
(1 ) (1 )
S S SL L L
S S SH
t w p pu w u w mt w
β β βθβ
− − + − −=
− −. (15)
The determination of 2θ is depicted in Figure 2. Light blue lines correspond to
expected incomes in D. Green and red lines correspond to actual incomes in S under a “good”
realization and under a “bad” realization, respectively. Straight lines correspond to expected
or actual incomes yielded by the optimal choice of the country of residence, and dotted lines
correspond to expected or actual incomes yielded by the suboptimal choice. It can easily be
verified that the three underlying inequalities ,S
L L bEI m I− > , *, ( ) (1 )S S S S
L H bEI m I s kθ− > + − ,
and , 1( ) (1 )S S SL H bEI m I s kθ− < + − hold.
Figure 2. The determination of 2θ .
Denoting the size of the first wave of migration and the size of the second wave of
migration as 1M and 2M , respectively, the size and the skill composition of the first wave of
migration is such that all 1 1(1 ( )) SM F Nθ= − migrants are high-ability individuals
(prospective high-skill workers), and the size and skill composition of the second wave of
migration is such that out of a total of 2 2( )S SM p F Nθ= migrants *( )S S Sp F Nθ are low-skill
workers and *2( ( ) ( ))S S Sp F F Nθ θ− are high-skill workers who take up low-skill jobs in D.
Therefore, the share of migrants who on migration to D will work as high-skill workers,
15
which is the most sensible measure of the skill composition of migration in our context, is
given by 1 1
1 2 1 2
1 ( )1 ( ) ( )S
M FM M F p F
θθ θ−
=+ − +
.
In the open-to-migration setting, the sequence of events is as follows. At the
beginning, the first wave of migration from S to D takes place. Then, individuals in S and
individuals in D choose the type of workers they wish to become. Next, individuals in S and
individuals (native and migrant) in D learn their realizations in the labor market. Then, the
second wave of migration from S to D takes place, and the migrants learn their realizations in
D’s labor market. Finally, all individuals consume.
Having established the size and the skill composition of migration from S to D, we
inquire how these variables are shaped by the generosity of welfare provisions in D. We
consider two settings. In Setting 1, D is initially closed to migration and is about to open up to
unrestricted migration from S. One could think of this setting as that of opening up of the UK
labor market for workers from such countries as Poland after the 2004 EU enlargement. In
Setting 2, D has a long history of migration from S, which means that D’s workforce is
already composed of native and migrant workers. This setting could correspond to
contemporary migration between Poland and the UK. In each of these two settings, we inquire
how a change in the generosity of a provision impacts the inclination of individuals in S to
partake in each wave of migration. By construction, in Setting 1, which abstracts from the
impact of migration on labor markets in S and in D, we will evaluate the potential of a
provision as a welfare magnet, whereas in Setting 2, which accounts for migration-induced
adjustments in the labor markets in S and in D, we will measure the actual impact of a
provision on the size and skill composition of migration.
Formally, the distinction between the two settings rests in the formulas for the size of
high-skill workforce and the size of the low-skill workforce in D and in S. In Setting 1, before
migration takes place, these formulas are given by *1 ( )HN F θ= − and *( )LN F θ= for D,
and by *(1 ( ))S SHN F Nθ= − and *( )S S
LN F Nθ= for S. In Setting 2, with migrants already
present in D’s labor market, these formulas are given by *1 11 ( ) (1 ( )) S
HN M F F Nθ θ+ = − + −
and *2 2( ) ( )S S
LN M F p F Nθ θ+ = + for D, and by *1 1( ( ) ( ))S S
HN M F F Nθ θ− = − and
2 2(1 ) ( )S S SLN M p F Nθ− = − for S.
16
Unfortunately, it is not possible to determine analytically the response of individuals
in S to a change in the generosity of welfare provisions in D in Setting 2. Therefore, in Sub-
section 3.2 we analyze the potential of a skill formation subsidy and of an unemployment
benefit as welfare magnets in Setting 1 only. Then, in Sub-section 3.3 we complement the
analysis from Sub-section 3.2 with a numerical simulation, and we conduct a separate
simulation for Setting 2.
3.2. The impact of welfare provisions on migration: Analytics
To calculate the impact of the generosity of welfare provisions in D on the inclination of
individuals in S to migrate, we would need to calculate first-order derivatives of 1M and 2M
with respect to s and u. The formulas for the first-order derivatives are, however, quite
complex, and they do not allow for an intuitive interpretation. As a response to this difficulty,
in this sub-section we provide a qualitative assessment of the impact of the generosity of
welfare provisions on the size of each wave of migration based on the impact of welfare
provisions in D on expected income of the migrants. Because the size of the first wave of
migration, and the size of the second wave of migration are determined by 1θ and 2θ which,
in turn, are solutions to 1 1( ) ( )SH HEI m EIθ θ− = and , 2( ) (1 )S S S
L H bEI m I s kθ− = + − ,
respectively, a study of the impact of the generosity of welfare provisions in D on 1( )HEI θ
and on LEI maps directly onto the size of each wave of migration.
A change of expected income in D of the first-wave migrant with ability 1θ in
response to a marginal increase in the generosity of the skill formation subsidy is given by
*1
1 1*
* *
1* *
*1
1*
* **
1* *
( ) (1 ) (1 )
(1 )(1 )
1 (1 )
( )(1 ( ))(1 )(1 ) ,( )
HH H
H L
HH
H H
dEI t tk p w p wds s s
w wt p ups s
tk p ws
w Ft p ups F
θ θβ θ β θθ
θ θβ θ βθ θ
θ θα β θθ θ
θ θ θθ β θ βθ θ
∂ ∂ ∂= − − − −
∂ ∂ ∂∂ ∂∂ ∂
+ − − +∂ ∂ ∂ ∂
∂ ∂= − − − ∂ ∂
∂ −∂+ − − − ∂ ∂
(16)
and the same change of expected income in response to a marginal increase in the generosity
of the unemployment benefit is given by
17
*1
1 1*
* *
1* *
*
1 1* *
* **
1* *
( ) (1 ) (1 )
(1 )(1 )
1 (1 )( )
( )(1 ( ))(1 )(1 ) ,( )
HL H H
H L
L H H
H H
dEI t tp w p w p wdu u u
w wt p upu u
tp w w p wF u
w Ft p upu F
θ θβ β θ β θθ
θ θβ θ βθ θ
α θβ θ β θθ θ
θ θ θθ β θ βθ θ
∂ ∂ ∂= − − − −
∂ ∂ ∂∂ ∂∂ ∂
+ − − +∂ ∂ ∂ ∂
− ∂ ∂= − − − ∂ ∂
∂ −∂+ − − − ∂ ∂
(17)
where * * *
*
* * 2
( ) ( ) (1 ( ))0
( )
Yf Y Ft sk upY
θ θ θθ β
θ θ
∂− − −∂ ∂= − <
∂ (that * 0t
θ∂
<∂
follows from Lemma
A1, which is embedded in the proof of Claim 1), *
* * * 1 * 1* * * *
( )1(1 ) ( )[ ( )(1 ( ))] ( ) 0( ) ( )(1 ( ))
H HH
H
w f F FF F
α α θ θα α θ θ θ θ θθ θ θ θ θ
− − ∂= − − + > ∂ −
, and
** * * *
* * * *
( )1(1 ) ( )[ ( )(1 ( ))] ( ) 0( ) ( )(1 ( ))
L HH
H
w f F FF F
α α θ θα α θ θ θ θ θθ θ θ θ θ
− ∂= − − − + < ∂ −
.
A change of expected income in D of the second-wave migrant with ability 2θ in
response to a marginal increase in the skill formation subsidy, and in response to a marginal
increase in the unemployment benefit, are given, respectively, by
* *
* *
* * *
* * *
(1 ) (1 ) [(1 )(1 ) ]
1 ( )(1 ) (1 ) [(1 )(1 ) ],( )
L LL L
LL
dEI wt tp w p w t p upds s s s
wF tk p w t p upF s s
θ θβ β β βθ θ
θ θ θα β β βθ θ θ
∂∂ ∂ ∂ ∂= − − − − + − − +
∂ ∂ ∂ ∂ ∂
∂− ∂ ∂ ∂= − − − − + − − +
∂ ∂ ∂ ∂ (18)
and
* *
* *
* *
* * *
(1 ) (1 ) [(1 )(1 ) ]
11 (1 ) [(1 )(1 ) ].( )
L LL L L
LL L
dEI wt tp w p w p w t p updu u u u
wtw p p w t p upF u u
θ θβ β β β βθ θ
α θ θβ β β βθ θ θ
∂∂ ∂ ∂ ∂= − − − − + − − +
∂ ∂ ∂ ∂ ∂
∂− ∂ ∂ ∂= − − − + − − + ∂ ∂ ∂ ∂
(19)
18
In (16) to (19), the overall change of expected income in D can be split into the direct
effect and the indirect effect of an increase in the generosity of a provision. The direct effect
is the sum of the (nonnegative) impact on expected income of a more generous provision, and
the (negative) impact on expected income of higher taxation required to finance the more
generous provision. It is captured by the first element of the sum on the most right-hand side
of (16) to (19). The indirect effect is a resultant of a more generous provision changing the
relative attractiveness of the two types of work among D’s native population, such that a more
generous skill formation subsidy induces more individuals to become high-skill workers (cf.
Claim 2), whereas a more generous unemployment benefit does the opposite (cf. Claim 3).
The resulting changes in the number of native high-skill workers relative to the number of
native low-skill workers in D’s workforce have an effect on the number of recipients of a
provision and on the revenue from taxation. The change in the government’s expenditures and
in the tax revenue mandates adjustment in the tax rate to keep the budget balanced. This effect
of an increase in the generosity of a welfare provision is captured by the middle element of
the sum on the most right-hand side of (16) to (19). Finally, changes in the number of native
high-skill workers relative to the number of native low-skill workers in D’s workforce have
an effect on the wage rates offered to the two types of workers, Hw and Lw . This effect is
captured by the last element of the sum on the most right-hand side of (16) to (19).
In Table 1, we present the impact of an increase in the generosity of welfare provisions
in D on the direction of a change in the size of each wave of migration separately for the
direct impact of a provision, the indirect impact via taxation, and the indirect impact via
wages. To ease the exposition, we denote * *
**
( )(1 ( ))( ) 0( ) 1
H F pF p
θ θ θ βψ θθ β−
= >−
.
19
Table 1. The impact of an increase in the generosity of welfare provisions in D on the size of
each wave of migration
Direct impact of a provision
Indirect impact via taxation
Indirect impact via wages
Firs
t wav
e
of m
igra
tion Skill formation
subsidy
Positive if *
11 ( )Hθ θ θα
< Negative Positive if
*1 ( )
1u
tθ ψ θ<
−
Unemployment benefit
Positive if *
*1
1 ( ) ( )HF θθ θ θα
−< Positive
Positive if *
1 ( )1
ut
θ ψ θ>−
Seco
nd w
ave
of
mig
ratio
n Skill formation subsidy
Negative Negative Positive
Unemployment benefit
Positive Positive Negative
From Table 1 it follows that more generous provisions directly increase the size of the
first wave of migration, if 1θ is not too high relative to the average ability among D’s native
high-skill workers. The rationale behind this result is the following. Whereas each welfare
provision is at an equal level for all (eligible) first-wave migrants, the cost of financing a
provision (taxation) is not; it is proportional to a migrant’s wage earnings, and, thus, it raises
with ability. Consequently, the ablest migrants may dislike generous welfare states, as their
contribution to financing welfare provisions is relatively high.
That the skill formation subsidy directly lowers the size of the second wave of
migration is straightforward: these migrants participate in financing the provision of the skill
formation subsidy, yet because they are post their skill formation choices, they are not eligible
to receive the benefit. In turn, that the unemployment benefit directly increases the size of the
second wave of migration follows because these migrants are at the bottom of the pay
distribution in D, and, consequently, their contribution to financing the benefit is relatively
low as compared to the benefit they may receive.
By increasing the generosity of the skill formation subsidy, D induces more of its
native inhabitants to become high-skill workers, as captured by Claim 2. This change in the
skill composition of D’s native workforce necessitates changes in the tax rate, such that the
20
higher the share of high-skill workers in D’s workforce, the higher the tax rate (cf. Lemma
A1). Consequently, a more generous skill formation subsidy mandates an increase in D’s tax
rate, which indirectly lowers the size of each wave of migration. The effects of an increase in
the generosity of the unemployment benefit for D’s tax rate are the opposite of the effects of
an increase in the generosity of the skill formation subsidy.
Finally, that a more generous skill formation subsidy increases the share of high-skill
workers in D’s native workforce has an effect on D’s wages such that the wage per unit of
high-skill work decreases, and the wage per unit of low-skill work increases. A reduction of
the wage per unit of high-skill work indirectly lowers the size of the first wave of migration,
while an increase of the wage per unit of low-skill work, which raises the level of the
unemployment benefit, increases the size of the first wave of migration. The net effect is
positive if *
*1 *
1 ( ) ( )1 ( ) 1 H
u F pt F p
θ βθ θ θθ β
−<
− − or negative if
**
1 *
1 ( ) ( )1 ( ) 1 H
u F pt F p
θ βθ θ θθ β
−>
− −,
with the latter being likely to hold in practice. This is because for *
*1 *
1 ( ) ( )1 ( ) 1 H
u F pt F p
θ βθ θ θθ β
−<
− −, pβ would need to be large; an unlikely characterization of
the developed, destination country. In turn, among second-wave migrants, the positive impact
on the wage per unit of low-skill work of an increase in the generosity of the skill formation
subsidy increases these migrants’ expected wage earnings and the level of the unemployment
benefit. Consequently, the indirect impact via wages of the skill formation subsidy on the size
of the second wave of migration is unambiguously positive. The effects of an increase in the
generosity of the unemployment benefit for D’s high-skill wage and low-skill wage are the
opposite of the effects of an increase in the generosity of the skill formation subsidy.
3.3. The impact of welfare provisions on migration: Simulation
As captured by (16) to (19), the overall impact of the generosity of a welfare provision on the
size of each wave of migration is the sum of the direct impact of a provision, the indirect
impact via taxation, and the indirect impact via wages. Because the signs of the three effects
do not align, we cannot determine whether the skill formation subsidy and the unemployment
benefit are a magnet for or a disincentive to migration in each of its waves for an arbitrary
distribution of ability and for arbitrary values of the model’s parameters. As a solution to this
difficulty, we provide a numerical simulation which illustrates the relative strength of the
three effects, and we calculate the overall impact of welfare provisions on the size of each
21
wave of migration, and on the skill composition of the overall migration. We proceed with a
simulation separately for Setting 1 and for Setting 2.
In our simulation, D is characterized by a uniform distribution of ability on a (0,1]
interval, and by the following parameter values: 1A = , 0.5α = , 0.1p = , 0.2β = , and
0.1k = . The generosity of welfare provisions in D is such that the share of the subsidy in the
overall cost of skill formation is 0.1s = , and that the replacement rate of the unemployment
benefit to the wage per unit of low-skill work is 0.5u = . Next, we assume that the population
of S is of size 0.6SN = , and that S is characterized by the same distribution of ability among
its native population as is D, namely by a uniform distribution on a (0,1] interval, and by the
following parameter values: 0.8SA = , 0.5Sα = , 0.1Sp = , 0.5Sβ = , and 0.1Sk = . That is,
aside from S being technologically less advanced than D, it is characterized by a higher spell
of unemployment than that in D. The generosity of welfare provisions in S is characterized by
0.3Ss = , and 0.2Su = . Finally, we assume that the cost of migration is 0.14m = .
We verify that the two sets of inequalities that govern the composition of migration in
each of its two waves hold for Setting 1: 0.201 0.257 SL LEI m EI− = < = ,
* *( ) 0.202 0.257 ( )S S SH HEI m EIθ θ− = < = , and ( ) 0.46 0.453 ( )S
H HEI T m EI T− = > = , which
ensure that the first wave of migration is by the most able individuals in S; and
,0.201 0.162 SL L bEI m I− = > = , *
,0.201 0.199 ( ) (1 )S S S SL H bEI m I s kθ− = > = + − , and
, 10.201 0.289 ( ) (1 )S S SL H bEI m I s kθ− = < = + − , which ensure that the second wave of
migration is by the least able individuals who face temporary unemployment in S. Likewise,
the respective inequalities hold for Setting 2: 0.202 0.256 SL LEI m EI− = < = ,
* *( ) 0.2 0.256 ( )S S SH HEI m EIθ θ− = < = , and ( ) 0.46 0.454 ( )S
H HEI T m EI T− = > = , which
ensure that the first wave of migration is by the most able individuals in S; and
,0.202 0.161 SL L bEI m I− = > = , *
,0.202 0.198 ( ) (1 )S S S SL H bEI m I s kθ− = > = + − , and
, 10.202 0.293 ( ) (1 )S S SL H bEI m I s kθ− = < = + − . Table 2 lists the model’s endogenous variables
yielded by the chosen distribution of ability and the parameter values under the two settings.
22
Table 2. Simulation of the model’s endogenous variables characterizing D and S, and of the
variables characterizing the skill composition of migration in the two waves of migration.
Country *θ Lw Hw t 1θ 2θ 1
1 2
MM M+
Setti
ng 1
D 0.623 0.351 0.713 0.017
S 0.624 0.280 0.572 0.042 0.957 0.633 0.415
Setti
ng 2
D 0.624 0.351 0.712 0.017
S 0.621 0.279 0.573 0.043 0.965 0.624 0.357
We study how the expected income in D of the first-wave migrants and of the second-
wave migrants changes in response to an increase in the generosity of the skill formation
subsidy in D from 0.1s = to 0.2s = , holding the generosity of the unemployment benefit
constant at 0.5u = , and, subsequently, to an increase in the generosity of the unemployment
benefit in D from 0.5u = to 0.6u = , holding the generosity of the skill formation subsidy
constant at 0.1s = . The results are reported for the least able first-wave migrant ( 1 0.957θ =
in Setting 1 and 1 0.965θ = in Setting 2), and for the most able second-wave migrant
( 2 0.633θ = in Setting 1 and 2 0.624θ = in Setting 2).7 The impact of an increase in the
generosity of the skill formation subsidy, and in the generosity of the unemployment benefit
on expected income in D, and on the size of each wave of migration are listed in Table 3
(Setting 1) and in Table 4 (Setting 2).
7 In fact, the results for the most able second-wave migrant apply to any second-wave migrant.
23
Table 3. The impact (in percent) of a change in the generosity of welfare provisions on
expected income of the migrants, and on the size of each wave of migration (Setting 1)
Table 6 reveals that the results of the simulation are sensitive to changes in the values
of the model’s parameters: an increase in the probability of temporary unemployment in S
from 0.1Sp = to 0.25Sp = transforms the skill formation subsidy into a magnet for the first-
wave migrants in Setting 1 and in Setting 2. Incidentally, in the case of the skill formation
subsidy, the indirect impact via wages is almost as strong so as to nullify the positive direct
impact. Thus, the results of the simulation in the revised setting continue to support our main
argument of the importance of accounting for the indirect channels via which welfare
provisions shape migration.
On an increase in the generosity of the skill formation subsidy in the revised setting, as
per Table 6 (Table 7), the overall size of migration increases by 1.1 percent (0.03 percent),
and the share of high-ability individuals in overall migration increases from 59 percent to 59.5
percent (from 44.9 percent to 45.1 percent). In turn, an increase in the generosity of the
unemployment benefit reduces the overall size of migration by 0.1 percent (0.08 percent), and
the share of high-ability individuals to 58.9 percent (to 44.87 percent).
4. Conclusions
We constructed a model of migration between two countries to study the impact of welfare
provisions in the destination country on the size and skill composition of migration to that
country both analytically and numerically. To the best of our knowledge, this model is a first
27
attempt to comprehensively address the complex relationship between welfare provisions at
destination, and the size and skill composition of migration. It delivers a more nuanced
assessment of the relationship than that provided in the received literature. The results of our
analysis indicate that the relationship between welfare provisions, and the size and skill
composition of migration can vastly differ depending on the type of provision, and on the
characteristics of the sending country and the destination country. They provide an
explanation for the relatively weak impact of welfare provisions on migration witnessed in the
received empirical literature: adjustments of the workers’ wages at least partly offset the
direct impact of provisions on migration.
The model can be extended in an interesting way by introducing a third country (a
second destination country), and adding a new dimension to individuals’ migration choices:
where to migrate? Such an extension would enrich the analysis of the impact of welfare
provisions on migration by allowing for a “switch” of the migration target following a change
in the generosity of a welfare provision in one of the destination countries. Relatedly, a three-
country framework would allow for a study of “welfare competition” between destination
countries aimed at skimming the ablest migrants.
28
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30
Appendix
Proof of Claim 1. We first show and prove the following Lemma.
Lemma A1. * 0tθ∂
<∂
.
Proof. Recalling (13) and * * * 1(1 ) [ ( )(1 ( ))] ( )Y p A F Fα αβ θ θ θ θ −= − − , we get that * *
**
* 2 *
( ) (1 ( )) (1 ) ( ) 01 ( )
Yf Y Ft p fsk uY p F
θ θ β α θθθ β θ
∂− − −∂ −∂= − <
∂ −. Because
*
*
(1 ) ( ) 01 ( )
p fup Fβ α θ
β θ−
>−
, a sufficient condition for * 0tθ∂
<∂
to obtain is that
* **( ) (1 ( )) 0Yf Y Fθ θ
θ∂
− − − <∂
. Because ** * * *
1( )( ) ( )(1 ( ))
Y f YF F
α αθθ θ θ θ θ
∂ −= − ∂ −
, we get
that * *
* * ** * *
1 ( )( ) (1 ( )) ( ) 1 (1 )( ) ( )
Y Ff Y F f YF
θ θθ θ θ α αθ θ θ θ
∂ −− − − = − + − − ∂
. In turn, a
sufficient condition for * *
** *
1 ( )( ) 1 (1 ) 0( ) ( )Ff Y
Fθ θθ α α
θ θ θ −
− + − − <
to obtain is * *( )θ θ θ< ,
which holds true by the definition of *
**
1( ) ( )1 ( )
T j j jf dF θ
θ θ θ θ θθ
≡− ∫ . Thus, we conclude
that * 0tθ∂
<∂
. Q.E.D.
We wish to show that *θ , which solves
[ ](1 ( )) ( ) ( ) (1 )1H L
kt w w sp
θ θ θ θβ
− − = −−
, (A1)
exists and is unique. We begin by studying the expression in square brackets on the left-hand
side of (A1). We have that (0) 0F = , and that 0 (0) Tθ< < , which, recalling (11), indicate
that (0) 0Hw = and (0)Lw = +∞ . Therefore, for 0θ → the expression in square brackets on
the left-hand side of (A1) approaches minus infinity. We also have that ( ) 1F T = , and that
( )T Tθ = , which, recalling (11) again, indicate that ( )Hw T = +∞ and ( ) 0Lw T = . Therefore,
for Tθ → the expression in square brackets on the left-hand side of (A1) approaches plus
