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\begin{document}
\title{The Welfare State and the Skill Mix of Migration: Dynamic Policy
Formation}
\author{Assaf Razin, \and Cornell University and Tel Aviv University \and %
Efraim Sadka, \and Tel Aviv University \and and \and Ben Suwankiri \and TMB
Bank Plc. Thailand \\
%EndAName
December 15 2009}
\maketitle
\begin{abstract}
Tthe native-born young, whether skilled or unskilled, benefit from letting
in migrants of all skill types, because their high birth rates can help
increase the tax base in the next period. In this respect, skilled migrants
help the welfare state more than unskilled migrants, to the extent that the
offspring resemble their parents with respect to skill. On the other hand,
more migrants in the present will strengthen the political power of the
young in the next period who, relatively to the old, are less keen on the
generosity of the welfare state. In this respect, unskilled migrants pose
less of a threat to the generosity of the welfare state then skilled
migrants.
\end{abstract}
\section{\protect\bigskip}
The comprehensive welfare state is characterized by both inter generational
redistribution (such as old-age social security) and intra-generational
redistribution (such as income maintenance programs)\footnote{%
Some features of the welfare state, such as national health insurance,
involve both inter- and intra- generational redistribution.}.This paper
delves into the theoretical analysis of the links between the generosity of
the welfare state and migration in a political-economy dynamic setting. The
framework brings to life \emph{inter}-generational aspects of redistribution
(that is, between the young and the old), in addition to the \emph{intra}%
-generational features redistribution. An overlapping generations model is
employed and voting about current migration and social security policy is
jointly conducted each period (where people live for two periods). We
plausibly assume that migrants have higher birth rates than the native-born.
As we aim to highlight this demographic difference, we assume that this is
the only feature by which migrants differ from the native-born. The latter
jointly determine in a political process the migration policy (that is, the
number of migrants allowed in) and the size of a pay-as-you-go (PAYG)
old-age social security. We employ a forward looking equilibrium concept
which means that each young voter takes into account the effect of her vote
on the evolution of the economy in the next period, which, in turn, affects
the voting outcome in the next period, especially with respect to the social
security benefit that she receives in the next period when she grows old;
voting in the next period is in turn influenced by the outcome of this
voting on the voting outcome in the following period, and so on.
We study how a more generous old-age social security system affects the
volume of migration ; how the volume of migration affects the generosity of
the old-age security system chosen by the native born; and how the
generosity of the old-age social security system and the volume of migration
are jointly determined by the native-born population .
The voting is conducted with respect to concurrent decisions on
redistribution between the old and the young, and between the rich (skilled)
and the poor (unskilled). In this setup there arise many more than two
voting groups. The skilled young does no longer share necessarily the same
interests as the unskilled young. Similarly, a distribution draws between
the skilled old and the unskilled old; and so on. We study the joint
determination of the generosity of the welfare state and the volume and
skill composition of migration. Of particular interest is the
characterization of the coalitions that are decisive in the
political-economic equilibria for different demographic and
skill-distribution parameters.
\bigskip
\section{Background}
Milton Friedman, reminded us that one cannot obviously have free immigration
and a welfare state. That is, a welfare state with open borders might turn
into a heaven for the poor and the needy from all over the world, thereby
draining its finances, and bringing it down.
Indeed, public opinion in the developed economies, with a fairly generous
welfare system, favors putting in some way or another restrictions on
migration. A skilled and young migrant may help the finances of the welfare
state; whereas an unskilled and old migrant may inflict a burden on the
welfare state. Of a particular interest is therefore the skill and age
composition of these restrictions. A welfare state with a heterogeneous (by
age, skill, etc) population does not evidently have a commonly accepted
attitude towards migration.
For instance, a skilled (rich) and young native-born who expects to bear
more than an average share of the cost of providing the benefits of the
welfare state is likely to oppose on this ground admitting unskilled
migrants. On the other hand, this same native-born may favor unskilled
migrants to the extent that it boosts up her wage. A native born old may
favor migration, even low-skilled, on the ground that it could help finance
her old-age benefits.
This variety of effects necessitates the use of a general equilibrium
framework in order to study how migration policies affect the native-born
voters. Furthermore, there are conflicting interests among the native-born
voters concerning these policies. This book develops a framework to study
how these many conflicts are resolved in a politico-economic setup.
\section{Fiscal Aspects of Migration: Evidence}
In 1997 the U.S. National Research Council sponsored a study on the overall
fiscal impact of immigration into the U.S.; see Smith and Edmonston (1997).
The study looks carefully at all layers of government (federal, state, and
local), all programs (benefits), and all types of taxes. For each cohort,
defined by age of arrival to the U.S., the benefits (cash or in kind)
received by migrants over their own lifetimes and the lifetimes of their
first-generation descendents were projected. These benefits include
Medicare, Medicaid, Supplementary Security Income (SSI), Aid for Families
with Dependent Children (AFDC), food stamps, Old Age, Survivors, and
Disability Insurance (OASDI), etc. Similarly, taxes paid directly by
migrants and the incidence on migrants of other taxes (such as corporate
taxes) were also projected for the lifetimes of the migrants and their
first-generation descendents. Accordingly, the net fiscal burden was
projected and discounted to the present. In this way, the net fiscal burden
for each age cohort of migrants was calculated in present value terms.
Within each age cohort, these calculations were disaggregated according to
three educational levels: Less than high school education, high school
education, and more than high school education. The findings suggest that
migrants with less than high school education are typically a net fiscal
burden that can reach as high as approximately US-\$100,000 in present
value, when the migrants' age on arrival is between 20--30 years.
Only three members of the EU-15 (the UK, Sweden and Ireland) allowed free
access for residents of the accession countries to their national labor
markets, in the year of the first enlargement, 2004. The other members of
the EU-15 took advantage of the clause that allows for restricted labor
markets for a transitional period of up to seven years. Focusing on the UK
and the A8 countries\footnote{%
The A8 countries are the first eight accession countries (Czech Republic,
Estonia, Hungary, Latvia, Lithuania, Slovenia and Poland.)}, Dustmann at al
(2009) bring evidence of no welfare migration. The average age of the A8
migrants during the period 2004\footnote{%
More accurately, the said period extends from the second quarter of 2004
through the first quarter of 2009.}-2008 is 25.8 years, considerably lower
than the native U.K. average age (38.7 years). The A8 migrants are also
better educated than the natives. For instance, the percentage of those that
left full-time education at the age of 21 years or later is 35.5 among the
A8 migrants, compared to only 17.1 among the U.K. natives. Another
indication that the migration is not predominantly driven by welfare motives
is the higher employment rate of the A8 migrants (83.1\%) relative to the
U.K. natives (78.9\%). Furthermore, for the same period, the contribution of
the A8 migrants to government revenues far exceeded the government
expenditures attributed to them\footnote{%
This finding does not yet indicate whether or not the A8 migrants impose a
net fiscal burden, because the latter takes in to account the present value
of all taxes paid by and revenues received by migrants through their life
time.}. A recent study by Barbone et al (2009), based on the 2006 European
Union Survey of Income and Living conditions, finds that migrants from the
accession countries constitute only 1-2 percent of the total population in
the pre-enlargement EU countries (excluding Germany and Luxemburg); by
comparison about 6 percent of the population in the latter EU countries were
born outside the enlarged EU. The small share of migrants from the accession
countries is, of course, not surprising in view of the restrictions imposed
on migration from the accession countries to the EU-15 before the
enlargement and during the transition period after the enlargement. The
study shows also that there is, as expected, a positive correlation between
the net current taxes (that is, taxes paid less benefits received) of
migrants from all source countries and their education level.
Hanson et al (2007) employing opinion surveys, find for the United States
that natives of states which provide generous benefits to migrants prefer to
reduce the number of migrants. This opposition is stronger among higher
income groups. Similarly, Hanson et al (2009), again employing opinion
surveys, find for the United States that native-born residents of states
with a high share of unskilled migrants among the migrants population prefer
to restrict in migration; whereas native-born residents of states with a
high share of skilled migrants among the migrants' population are less
likely to favor restricting migration. Indeed, developed economies do
attempt to sort out immigrants by skill (see, for instance, Bhagwati and
Hanson (2009)). Australia and Canada employ a point system based on selected
immigrants' characteristics. Recently the U.S. employs explicit preference
for professional, technical and kindred immigrants under the so-called
third-preference quota. Jasso and Rosenzweig (2009) find that both the
Australian and American selection mechanisms are effective in sorting out
the skilled migrants and produce essentially similar outcomes despite of
their different legal characteristics. A welfare state is typically engaged
in both \textit{inter-} and \textit{intra-}generational redistribution.
Therefore, in this chapter, we also introduce an elaborate and explicit
feature of intra-generational redistribution, and analyze the interactions
between inter and intragenerational conflicts. As was already pointed out,
not only the native-born contribute to, and benefit from, the welfare state,
migrants also contribute and benefit as well. Keeping this in mind, the
political process selects both the size of redistribution as well as he
migration policy. Therefore, the native-born voters must take into
consideration the costs and benefits of migrants when casting their votes.
Because of this interesting linkage between these two policy dimensions, we
study in this chapter the joint determination of redistribution and
migration policies.\footnote{%
Earlier studies include Dolmas and Huffman (2004) and Ortega (2005).} In
particular, the redistribution policy must have in mind both \textit{inter}
and \textit{intra}generational aspects, resembling a full-fledged
welfare-state system.
\section{Analytical Framework}
We employ a two-period, overlapping-generations model. The old cohort
retires, while the young cohort works. \ There are two skill levels: skilled
and unskilled. The welfare-state is modeled simply by a proportional tax on
labor income to finance a demogrant in a balanced-budget manner. Therefore,
some (the unskilled workers and old retirees) are net beneficiaries from the
welfare state and others (the skilled workers) are net contributors to it.
Migration policies are set to determine the total migration volume and its
skill composition. We characterize subgame-perfect Markov politico-economic
equilibria consisting of the tax rate (which determines the demogrant),
skill composition and the total number of migrants. We distinguish between
two voting behaviors: sincere and strategic voting . When participating in
political decisions, as we indeed have, sincere voting is too simplistic. We
therefore study also the case of strategic voting among the native-born in
order to enable the formation of strategic political coalitions.
Consider an economy consisting of overlapping generations. Each individual
lives for two periods, working in the first period when young, and retiring
in the second period when old. The population is divided into two groups
according to their exogenously given skills:\ skilled ($s$) and unskilled ($%
u)$.
\subsection{ Preferences and Technology}
The utility of each individual in period $t$, for young and old, is given,
respectively, by
\begin{align}
U^{y}(c_{t}^{y},l_{t}^{i},c_{t+1}^{o})& =c_{t}^{y}-\frac{\varepsilon
(l_{t}^{i})^{\frac{1+\varepsilon }{\varepsilon }}}{1+\varepsilon }+\beta
c_{t+1}^{o}\text{, }i=s,u \\
U^{o}(c_{t}^{o})& =c_{t}^{o}.
\end{align}%
where, as in Part I, $s$ and $u$ denote skilled and unskilled labor. Here, $%
y $ and $o$ denote to young and old, $l^{i}$ is labor, $\varepsilon $ is the
elasticity of the labor supply, and $\beta \in (0,1)$ is the discount factor.%
\footnote{%
This functional form of $U^{y}$ is similar to the one used in Part I.} Note
that $c_{t}^{o}$ is the consumption of an old individual at period $t$ (who
was born in period $t-1$). Agents in the economy maximize the above utility
functions subject to their respective budget constraints. Given the
linearity of $U$ in $c_{t}$ and $c_{t+1}$, a non-corner solution can be
attained on only when $1=\beta (1+r),$ where $r$ is the interest rate. We
indeed assume that the interest rate $r$ equal $\frac{1}{\beta }-1$ and
individuals have no incentive to either save or dissave. Fore simplicity, we
set saving at zero.\footnote{%
In fact, any saving level is an optimal choice. Assuming no saving is for
pure convenience. With saving, since old individuals do not work the last
period of their life, they will consume savings plus any transfer. Through
both these channels, the old individuals benefit from migration. To keep the
analysis short, we will just focus on the costs and benefits in terms of the
welfare state.} This essentially reduces the two groups of old retirees
(skilled and unskilled) to just one because they have identical preference
irrespective of their skill level. In addition to consumption, the young
also decide on how much labor to supply. Individual's labor supply is given
by
\begin{equation}
l_{t}^{i}=\left( A_{t}w^{i}(1-\tau )\right) ^{\varepsilon },\text{ }i=s,u
\label{eqLaborS}
\end{equation}%
where $w^{i}$ is the wage rate of a worker of skill level $i=s,u$.
There is just one good, which is produced by using the two types of labor as
perfect substitute.\footnote{%
This simplification, nonetheless, allows us to focus solely on the linkages
between the welfare state and migration, leaving aside any labor market
consideration. In Appendix 7A.1, we consider the case where the two inputs
are not perfect substitute.} The production function is given by
\begin{equation}
Y_{t}=w^{s}L_{t}^{s}+w^{u}L_{t}^{u}
\end{equation}%
where $L_{t}^{i}$ is the aggregate labor supply of skill $i=s,u.$ Labor
markets are competitive, ensuring the wages going to the skilled and
unskilled workers are indeed equal to their marginal products, $w^{s}$ and $%
w^{u}$, respectively. We naturally assume that $w^{s}>w^{u}$.
As before, we denote the demogrant by $b_{t}$ and the tax rate by $\tau _{t}$%
. The agents in the economy take these policy variables as given when
maximizing their utilities. Because the old generation has no income, its
only source of income comes from the demogrant. The model yields the
following indirect utility function (recall that saving is zero):%
\begin{align*}
V^{y,i}& =\frac{\left( (1-\tau _{t})w^{i}\right) ^{1+\varepsilon }}{%
1+\varepsilon }+b_{t}+\beta b_{t+1} \\
V^{o}& =b_{t},
\end{align*}%
for $i\in \{s,u\}$. For brevity, we will use $V^{i}$ to denote $V^{y,i}$
because only the young workers need to be distinguished by their skill level.
In addition to the parameters of the welfare state ($\tau _{t}$ and,
consequently, $b_{t}$), the political process also determines migration
policy. This policy consists of two parts: one determining the volume of
migration, and the other its skill composition. We denote by $\mu _{t}$ the
ratio of allowed migrants to the native-born young population and denote by $%
\sigma _{t}$ the fraction of skilled migrants in the the total number of
migrant entering the country in period $t$.
Migrants are assumed to have identical preference to the native-born. As
before, we assume all migrants come young and they are naturalized one
period after their entrance. Hence, they gain voting rights when they are
old, as in the inter-generational model of chapter 5.
As in chapters 2 and 3, let $s_{t}$ denote the fraction of native-born
skilled workers in the labor force in period $t$ (where $s_{0}>0$). The
aggregate labor supply in the economy of each type of labor is given by
\begin{equation}
L_{t}^{s}=\left[ s_{t}+\sigma _{t}\mu _{t}\right] N_{t}l_{t}^{s}
\end{equation}%
and%
\begin{equation}
L_{t}^{u}=\left[ 1-s_{t}+(1-\sigma _{t})\mu _{t}\right] N_{t}l_{t}^{u},
\end{equation}%
where $N_{t}$ is the number of native-born young individuals in period $t$.
\subsection{Dynamics}
The dynamics of the economy are given by two dynamic equations: one governs
the \textit{aggregate} population, while the other governs the \textit{skill}
composition dynamics. Because skills are not endogeneous within the model,
we assume for simplicity that the offspring replicate exactly the skill
level of their parents.\footnote{%
Razin, Sadka, and Swagel (2002a, 2002b) and Casarico and Devillanova (2003)
provide a synthesis with endogeneous skill analysis. The first work focuses
on the shift in skill distribution of current population, while the latter
studies skill-upgrading of future population.} That is,
\begin{align}
N_{t+1}& =\left[ 1+n+(1+m)\mu _{t}\right] N_{t} \label{eqPopDyn} \\
s_{t+1}N_{t+1}& =\left[ (1+n)s_{t}+(1+m)\sigma _{t}\mu _{t}\right] N_{t},
\notag
\end{align}%
where $n$ and $m$ are the population growth rates of the native-born
population and the migrants, respectively. As in chapter 5, we plausibly
assume that $ns_{t}$. Naturally,
when there is no migration the share of skilled workers out of (native-born)
young population does not change over time, by assumption. When migration is
allowed and its share of skilled labor is larger than that of the
native-born, the share of skilled labor in the population will grow over
time.
\subsection{The Welfare-State System}
As before, we model the welfare-state system as balanced period-by-period.
In essence, it operates like a pay-as-you-go system. The proceeds from the
labor tax of rate $\tau _{t}$ in period $t$ serve entirely to finance the
demogrant $b_{t}$ in the same period. Therefore, the equation for the
demogrant, $b_{t}$, is given by%
\begin{equation}
b_{t}=\frac{\tau _{t}\left( (s_{t}+\sigma _{t}\mu
_{t})w^{s}N_{t}l_{t}^{s}+\left( 1-s_{t}+(1-\sigma _{t})\mu _{t}\right)
w^{u}N_{t}l_{t}^{u}\right) }{\left( 1+\mu _{t}\right) N_{t}+\left( 1+\mu
_{t-1}\right) N_{t-1}},
\end{equation}%
which upon some manipulation reduces to%
\begin{equation}
b_{t}=\frac{\tau _{t}\left( (s_{t}+\sigma _{t}\mu _{t})w^{s}l_{t}^{s}+\left(
1-s_{t}+(1-\sigma _{t})\mu _{t}\right) w^{u}l_{t}^{u}\right) }{1+\mu _{t}+%
\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}}, \label{eqBudgetBalanced}
\end{equation}%
where the individual's labor supplies are given above in equation (\ref%
{eqLaborS}). It is straightforward to see that a larger $\sigma _{t}$
increases the demogrant (recall that $w^{s}l_{t}^{s}>w^{u}l_{t}^{u}$). That
is, a higher skill composition of migrants brings about higher tax revenues,
and, consequently, enables more generous welfare state, other things being
equal. Similarly, upon differentiation of $b_{t}$ with respect to $\mu _{t}$%
, we can conclude that a higher volume of migration enables a more generous
welfare system if the share of the skilled among the migrants exceeds the
share of the skilled among the native-born workers ($\sigma _{t}>s_{t}$).
\section{Political Economy Equilibrium: Sincere Voting}
In this section, we study the political-economic equilibrium in the model.
We imagine the economy with three candidates representing each group of
voters. In the text, we discuss only the equilibrium with sincere voting. In
the next section we consider the equilibrium with strategic voting.
We focus on "sincere voting," where individuals vote according to their
\textit{sincere} preference irrespective of what the final outcome of the
political process will be; see chapter 6. In this case, the outcome of the
voting is determined by the largest voting group.\footnote{%
Evidently, this assumption amounts to majority voting when there are only
two voting groups.} \ Therefore, it is important to see who forms the
largest voting group in the economy and under what conditions. Note that
there are only three voting groups: the skilled native-born young, the
unskilled native-born young, and the old (recall that there is no saving, so
that all the old care only about the size of the demogrant and thus have
identical interest.
\begin{enumerate}
\item The group of skilled native-born workers is the largest group ("the
skilled group") under two conditions. First, its size must dominates the
unskilled young, and, second, it must also dominate the old cohort.
Algebraically, these are
\begin{equation}
s_{t}>\frac{1}{2}
\end{equation}%
\qquad\ and
\begin{equation}
s_{t}>\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}
\end{equation}%
, respectively. It can be shown that, because $n\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}.
\end{equation}
\item The group of old retirees is the largest group ("the old group"), when
its size is larger than each one of the former groups, that is,
\begin{equation}
\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq \max \{s_{t},1-s_{t}\}.
\end{equation}
\end{enumerate}
\subsection{Equilibrium Characteristics}
We first describe what are the variables relevant for each of the three
types of voters when casting the vote in period $t$. First, $s_{t}$ is the
variable which describes the state of the economy. Also, each voter takes
into account how her choice of the policy variables in period $t$ will
affect the chosen policy variables in period $t+1$ which depends on $s_{t+1}$
(recall that the benefit she will get in period $t+1$, $b_{t+1}$, depends on
$\tau _{t+1},\sigma _{t+1}$, and $\mu _{t+1}$). Therefore each voter will
cast her vote on the set of policy variables $\tau _{t},\sigma _{t}$, and $%
\mu _{t}$ which maximizes \ her utility given the values of $s_{t}$, taking
also into account how this will affect $s_{t+1}$. Thus, there is a link
between the policy chosen in period $t$ to the one chosen in period $t+1$.
The outcome of the voting is the triplet of the policy variables most
preferred by the largest voting group.
The mechanism (policy rule or function) that characterizes the choice of the
policy variables ($\tau _{t}$, $\sigma _{t}$, and $\mu _{t}$) is invariant
over time. This mechanism relates the choice in any period to the choice of
the preceding period ($\tau _{t-1}$, $\sigma _{t-1}$, and $\mu _{t-1}$). \
This choice depend also on the current state of the economy, $s_{t}$. Thus,
we are looking for a triplet policy function $(\tau _{t},\sigma _{t},\mu
_{t})=\Phi (s_{t},\tau _{t-1},\sigma _{t-1},\mu _{t-1})$, which is a
solution to the following functional equation
\begin{align}
\Phi (s_{t},\tau _{t-1},\sigma _{t-1},\mu _{t-1})& =\underset{\tau
_{t},\sigma _{t},\mu _{t}}{\arg \max }V^{d}\left\{ s_{t},\tau _{t},\sigma
_{t},\mu _{t},\Phi (s_{t+1},\tau _{t},\sigma _{t},\mu _{t})\right\} \\
& \text{s.t. }s_{t+1}=\frac{(1+n)s_{t}+(1+m)\sigma _{t}\mu _{t}}{%
1+n+(1+m)\mu _{t}}, \notag
\end{align}%
where $V^{d}$ is defined in equations (7.5) and (7.11), and $d\in \{s,u,o\}$
is the identity of the largest voting group in the economy.
This equation states that the decisive (largest) group in period $t$
chooses, given the state of the economy $s_{t}$, the most preferred policy
variables $\tau _{t},\sigma _{t},$ and $\mu _{t}$. In doing so, this group
realizes that her utility is affected not only by these (current) variables,
but also the policy variables of the next period ($\tau _{t+1},\sigma
_{t+1},\mu _{t+1}$). This group further realizes that the future policy
variables are affected by the current variables according to the policy
function $\Phi (s_{t+1},\tau _{t},\sigma _{t},\mu _{t})$. Furthermore, this
intertemporal functional relationship between the policy variables in
periods $t+1$ and $t$ is \ the same as the one existed between period $t$
and $t-1$. Put differently, what the decisive group in period $t$ chooses is
related to $s_{t},\tau _{t-1},\sigma _{t-1},$ and $\mu _{t-1}$ in exactly
the same way (through $\Phi (\cdot )$) as what the decisive group in period $%
t+1$ is expected to be related to $s_{t+1},\tau _{t},\sigma _{t},$ and $\mu
_{t}$.
Denoting the policy function, $\Phi (s_{t},\tau _{t-1},\sigma _{t-1},\mu
_{t-1})$, by $\left( \tau _{t},\sigma _{t},\mu _{t}\right) $, we can show
that the outcomes of the policy rule are:%
\begin{align}
\tau _{t}& =\left\{
\begin{array}{cc}
0 & ,\text{ if the skilled group is the largest} \\
\frac{1-\frac{1}{J}}{1+\varepsilon -\frac{1}{J}} & \text{, if the unskilled
group is the largest} \\
\frac{1}{1+\varepsilon } & ,\text{ if the old group is the largest}%
\end{array}%
\right. \notag \\
\sigma _{t}& =\left\{
\begin{array}{cc}
1 &
\begin{array}{c}
\text{, if either the skilled or unskilled group} \\
\text{is the largest and }s_{t}<\frac{1}{1+n}%
\end{array}
\\
\widehat{\sigma }<\frac{1}{2} & \text{, if the skilled group is the largest
and }s_{t}\geq \frac{1}{1+n} \\
1 & \text{, if the old group is the largest.}%
\end{array}%
\right. \\
\mu _{t}& =\left\{
\begin{array}{cc}
\frac{1-(1+n)s_{t}}{m} &
\begin{array}{c}
\text{, if the unskilled group is the largest and }\Psi >0\text{ or} \\
\text{if the skilled group is the largest and }s_{t}<\frac{1}{1+n}%
\end{array}
\\
\widehat{\mu }<1 & \text{, if the skilled group is the largest and }%
s_{t}\geq \frac{1}{1+n} \\
1 &
\begin{array}{c}
\text{, if the unskilled group is the largest and }\Psi \leq 0 \\
\text{or if the old group is the largest.}%
\end{array}%
\end{array}%
\right. \notag
\end{align}%
where
\begin{eqnarray}
J &=&\frac{(s_{t}+\sigma _{t}\mu _{t})\left( \frac{w_{t}^{s}}{w_{t}^{u}}%
\right) ^{1+\varepsilon }+1-s_{t}+(1-\sigma _{t})\mu _{t}}{1+\mu _{t}+\frac{%
1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}} \label{taxwedge} \\
\Psi &=&b_{t}^{u}+\beta b_{t+1}^{o}-\widehat{b}_{t},
\end{eqnarray}%
where we denote by $\widehat{b}_{t}$ the demogrant period $t$ with $\mu
_{t}=1=\sigma _{t}$, and $b_{t}^{u}$ the demogrant in period $t$ with $%
\sigma _{t}=1$ and $\mu _{t}=\frac{1-(1+n)s_{t}}{m}$ (both demogrants are
associated with the tax rate preferred by the unskilled group). Similarly, $%
b_{t+1}^{o}$ is the demogrant in period $t+1$ associated with the set of
policy variables preferred by the old group.
Notice that the case $s_{t}>\frac{1}{1+n}$ cannot happen if the unskilled
group is the largest (because $n<1$). In this case, the special migration
policy variables preferred by the skilled group, $\widehat{\sigma }$, and $%
\widehat{\mu }$, are given implicitly from the maximization exercise%
\begin{align}
\left\langle \widehat{\sigma },\widehat{\mu }\right\rangle & =\underset{%
\sigma _{t},\mu _{t}}{\arg \max }V_{t}^{s}=\frac{\left(
A_{t}w_{t}^{s}\right) ^{1+\varepsilon }}{1+\varepsilon }+\beta b_{t+1}^{o}
\label{implicit} \\
& \text{s. t.\qquad }(1+n)s_{t}-1\leq \mu _{t}(1-(1+m)\sigma _{t}). \notag
\end{align}%
When the solution to the problem in (\ref{implicit}) is interior, we can
describe it by%
\begin{equation}
\frac{\frac{\partial V^{s}}{\partial \sigma _{t}}}{\frac{\partial V^{s}}{%
\partial \mu _{t}}}=\frac{\widehat{\mu }(1+m)}{(1+m)\widehat{\sigma }-1}.
\end{equation}%
There are also two possible corner solutions: $\left\langle \widehat{\sigma }%
,\widehat{\mu }\right\rangle =\left\langle 0,(1+n)s_{t}-1\right\rangle $ and
$\left\langle \widehat{\sigma },\widehat{\mu }\right\rangle =\left\langle
\frac{2-(1+n)s_{t}}{1+m},1\right\rangle $.
\subsection{ Migration and Tax Policies: Interpretation}
The intuition for the aforementioned results is as follows. The skilled are
the net contributor to the welfare state, while the other two groups are net
beneficiaries. Preferences of the old retirees are simple. If the old cohort
is the largest, it wants maximal social security benefits, which means
taxing to the Laffer point ($\frac{1}{1+\varepsilon }$). They also allow the
maximal number of skilled migrants in to the economy because of the tax
contribution this generates to the welfare system.
It is interesting to note that, although the unskilled young are net
beneficiaries in this welfare state, they are, nevertheless, still paying
taxes. Hence the preferred tax policy of the unskilled voters is smaller
than the Laffer point with a wedge $\frac{1}{J}$. (We will provide further
discussions on this deviation factor below.) Clearly, the unskilled workers
also prefer to let in more skilled immigrants due to their contribution to
the welfare state. How many will they let in depends on the function $\Psi $%
, which weighs the future benefits against the cost at the present.
Basically, if the unskilled workers are not forward-looking, it is in their
best interest to let in as many skilled migrants as possible. However, this
will lead to no redistribution in the next period because the skilled
workers will be the largest. Hence, the function $\Psi $ is the difference
between the benefits they get by being, as they are, forward-looking and
being myopic.
The skilled native-born prefer more skilled migrants for a different reason
than the earlier two groups. They prefer to let in skilled migrants in this
case because this will provide a higher number of skilled native workers in
the \textit{next} period. Thus, because the skilled are forward-looking,
they too will prefer to have more skilled workers in their retirement
period. However, they cannot let in too many of them because their high
birth rate may render the skilled young in the next period as the largest
group who will vote to abolish the welfare state altogether (similar to
chapter 5).
A common feature among models with subgame-perfect Markov equilibrium is the
idea that today's voters have the power to influence the identity of future
policymakers. Such feature is also prominent in our analysis here (as well
as in chapter 5). The migration policy of either young group reflects the
fact that they may want to put themselves as the largest group in the next
period. Thus, instead of letting in too many migrants, who will give birth
to a large new skilled generation, they will want to let in as much as
possible before the threshold is crossed. This threshold is $\frac{%
1-(1+n)s_{t}}{m}$. This strategic motive on migration quota is previously
fleshed out in chapter 5. Letting $s_{t}=1$ gets the result of the chapter.
There are two differences between this threshold and the one in chapter 5.
First, the equilibrium here has a bite even if the population growth rate is
\textit{positive}, which cannot be done when there are only young and old
cohort, as in chapter 5, unless there is a negative population growth rate.
Another fundamental is that, in order to have some transfer in the economy,
the young decisive largest group has a choice of placing the next period's
decisive power either in the hand of next period's unskilled or\ the old. So
we need to verify an additional condition that it is better for this
period's decisive young to choose the old generation next period, which is
the case.
When $s_{t}\geq \frac{1}{1+n}$, we have a unique situation (which is only
possible when $n>0$). In this range of values, the number of skilled is
growing too fast to be curbed by reducing migration volume alone. To ensure
that the decisive power lands in the right hand (that is, the old), the
skilled voters (who are the largest in this period) must make the unskilled
cohort grow to weigh down the growth rate of the skilled workers. This is
done by restricting both the skill composition as well as the size of total
migration.\footnote{%
Empirically, with the population growth rate of the major host countries for
migration like the U.S. and Europe going below 1\%, it is unlikely that this
case should ever be of much concern. Barro and Lee (2000) provides an
approximation of the size of the skilled. While Barro and Lee statistics
capture those 25 years and above, they also cite OECD statistics which
capture age group between 25 and 64. The percentage of this group who
received tertiary education or higher in developed countries falls in the
range of 15\% to 47\%.}
The tax choice of the unskilled young deserves an independent discussion. In
Razin, Sadka and Swagel (2002a, 2002b), it is maintained that the "fiscal
leakage" to the native-born and to the migrants who are net beneficiaries
may result in a lower tax rate chosen by the median voter. They assume that
all migrants possess lower skill than the native-born. Because this
increases the burden on the fiscal system, the median voter vote to reduce
the size of the welfare state, instead of increasing it. To see such a
resemblance the our result, we must first take the migration volume, $\mu
_{t}$, and the skill composition, $\sigma _{t}$, as given. Letting $\tau
_{t}^{u}$ denote the tax rate preferred by the unskilled group, one can
verify from equation (\ref{taxwedge}) that $\frac{\partial \tau _{t}^{u}}{%
\partial \sigma _{t}}>0$, and there exists $\overline{\sigma }$ such that,
for any $\sigma _{t}<\overline{\sigma }$, we have $\frac{\partial \tau
_{t}^{u}}{\partial \mu _{t}}<0$. Conversely, for any $\sigma _{t}>\overline{%
\sigma }$, we would get an expansion of the welfare state, because $\frac{%
\partial \tau _{t}^{u}}{\partial \mu _{t}}>0$.\footnote{%
Recall that the tax rate preferred by the unskilled young workers is less
than the level that is preferred by the old retirees. The tax rate preferred
by the old retirees, $\tau _{t}^{o}=\frac{1}{1+\varepsilon }$ is the Laffer
point that attains the maximum welfare size, given immigration policies.
Therefore the size of the welfare state is monotonic in the tax rate when $%
\tau \in \lbrack 0,\frac{1}{1+\varepsilon }]$. Thus, our use of "shrink" and
"expand" is justified.} The inequalities tell us that higher number of
skilled migrants will prompt a higher demand for intra-generational
redistribution. The fiscal leakage channel shows that unskilled migration
creates more fiscal burden, such that the decisive "unskilled" voters would
rather have the welfare state shrink. In addition, an increase in inequality
in the economy, reflected in the skill premium ratio $\frac{w_{t}^{s}}{%
w_{t}^{u}}$, leads to a larger welfare state demanded by the unskilled.
\section{Strategic-Voting Equilibrium}
Recall that we have only three groups: the skilled native-born, the
unskilled native-born, and the old. Let the set of three candidates be $%
\{s,u,o\},$ denoting their identity. Then, as in Chapter 6, the decision to
vote of any individual must be optimal under the correctly anticipated
probability of winning and policy stance of each candidate. Because
identical voters vote identically, we can focus on the decision of a
representative voter from each group. Let $e_{t}^{i}\in \{s,u,o\}$ be the
vote of individual of type $i\in \{s,u,o\}$ cast for a candidate. In the
same spirit as in Chapter 6, voting decisions $\mathbf{e}_{t}^{\ast
}=(e_{t}^{s\ast },e_{t}^{u\ast },e_{t}^{o\ast })$ form a \textit{voting
equilibrium} at time $t$ if
\begin{equation}
e_{t}^{i\ast }=\arg \max \left\{ \sum_{j\in \{s,u,o\}}\mathcal{P}%
^{j}(e_{t}^{i},\mathbf{e}_{-it}^{\ast })V^{i}\left( \Phi _{t}^{j},\Phi
_{t+1},\mathbf{e}_{t+1}\right) \mid e_{t}^{i}\in \{s,u,o\}\right\}
\label{eqVoteEqm}
\end{equation}%
for $i\in \{s,u,o\}$, where $\mathcal{P}^{j}(e_{t}^{i},\mathbf{e}%
_{-it}^{\ast })$ denotes the probability that candidate $j\in \{s,u,o\}$
will win given the voting decisions, and $\mathbf{e}_{-it}^{\ast }$ is the
optimal voting decision of other groups that is not $i$, and $\Phi
_{t}^{j}=\left( \tau _{t}^{j},\sigma _{t}^{j},\mu _{t}^{j}\right) $ is the
policy vector if candidate $j$ wins. Thus we require that each vote cast by
each group is a best-response to the votes by the other groups. In addition,
the representative voter of each group must take into the account the
\textit{pivotal} power of their vote, because the entire group will also
vote accordingly. The voting decision of the old voters is simple, because
they have no concern for the future,
\begin{equation*}
e_{t}^{o\ast }=\arg \max \left\{ \sum_{j\in \{s,u,o\}}\mathcal{P}%
^{j}(e_{t}^{o},\mathbf{e}_{-ot}^{\ast })V^{i}\left( \tau _{t}^{j},\sigma
_{t}^{j},\mu _{t}^{j}\right) \mid e_{ot}\in \{s,u,o\}\right\} .
\end{equation*}%
After the election, the votes are tallied by adding up the size of each
group that have chosen to vote for the candidate. The candidate with the
most votes wins the election and gets to implement his ideal set of policies.
Clearly, each individual prefers the ideal policies of their representative
candidate. Strategic voting opens up the possibility of voting for someone
else that is not the most preferred candidate in order to avoid the least
favorable candidate. For the skilled young, they prefer the least amount of
taxes and some migration for the future. Thus, they will prefer the policy
choice of the unskilled over the old candidate. As for the old retirees, the
higher the transfer benefits, the better. Clearly, the unskilled candidate
promises some benefits whereas the skilled promises none, so they would
choose the policies of the unskilled over the skilled.
As for the unskilled workers, both rankings are possible: either they prefer
the policy choice of the skilled over the old, or vice versa. The parameters
of the model will dictate the direction of their votes. The cut-off tax
policy, $\widetilde{\tau }$, is the break-even point for the unskilled
between getting taxed but receiving transfer (policies of the old candidate)
or pay no tax at all (policies of the skilled candidate).Formally, this tax
level, $\widetilde{\tau }$, is defined implicitly by the equation%
\begin{align}
& \frac{\left( w^{u}\right) ^{1+\varepsilon }}{1+\varepsilon }= \notag \\
& \frac{\left( (1-\widetilde{\tau })w^{u}\right) ^{1+\varepsilon }}{%
1+\varepsilon }+\frac{\widetilde{\tau }(1-\widetilde{\tau })^{\varepsilon
}\left( (s_{t}+\sigma _{t}\mu _{t})\left( w^{s}\right) ^{1+\varepsilon
}+\left( 1-s_{t}+(1-\sigma _{t})\mu _{t}\right) \left( w^{u}\right)
^{1+\varepsilon }\right) }{1+\mu _{t}+\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)%
}}. \label{EQtaxbreakeven}
\end{align}%
We know that such a tax policy exists, because, take next period's policy as
given, the payoff in this period to the unskilled is maximized at its
preferred policy and zero at $\tau =1$. Therefore, at some $\widetilde{\tau }
$, the equality will hold. This cut-off tax rate will play an important role
for the unskilled young' voting decision.
The main problem with ranking the utility streams of the voters is due to
the multiplicity of\textit{\ future} equilibria once we extend our work to
strategic voting. This makes it impossible for the voters to get a precise
prediction of what will happen as a result of their action today. Even if we
could pin down all the relative sizes of all possible payoffs in the next
period, multiple voting equilibria do not allow a prediction of which
equilibrium will be selected in the future. To deal with the problem, we
restrict the voting equilibrium to satisfy the stationary Markov-perfect
property, similarly to the policy choices in previous subsection. Now, we
are ready to define the subgame-perfect Markov political equilibrium under
strategic voting. We are looking for the a triplet policy function $(\tau
_{t},\sigma _{t},\mu _{t})=\Phi (s_{t},\tau _{t-1},\sigma _{t-1},\mu _{t-1},%
\mathbf{e}_{t}^{\ast })$ with the voting vector $\mathbf{e}_{t}^{\ast }$
that solve the following two problems:
\begin{align}
\Phi (s_{t},\tau _{t-1},\sigma _{t-1},\mu _{t-1},\mathbf{e}_{t}^{\ast })& =%
\underset{\tau _{t},\sigma _{t},\mu _{t}}{\arg \max }\quad V^{d}\left(
s_{t},,\tau _{t},\sigma _{t},\mu _{t},\Phi (s_{t+1},\tau _{t},\sigma
_{t},\mu _{t},\mathbf{e}_{t}^{\ast })\right) \\
& \text{s.t. }s_{t+1}=\frac{(1+n)s_{t}+(1+m)\sigma _{t}\mu _{t}}{1+n+\mu
_{t}(1+m)}, \notag
\end{align}%
where $d\in \{s,u,o\}$ is the identity of the the winning candidate, decided
by the voting equilibrium $\mathbf{e}_{t}^{\ast }$ that satisfies the
subgame-perfect Markov property and solves%
\begin{eqnarray}
e_{t}^{i\ast } &=&\mathbf{e}^{\ast }\left( s_{t},\tau _{t-1},\sigma
_{t-1},\mu _{t-1},\mathbf{e}_{t-1}^{\ast }\right) \label{eqVoteEqmMarkov} \\
&=&\underset{e_{t}^{i}\in \{s,u,o\}}{\arg \max }\sum_{j\in \{s,u,o\}}%
\mathcal{P}^{j}(e_{t}^{i},\mathbf{e}_{-it}^{\ast })V^{i}\left( \Phi
_{t}^{j},\Phi (s_{t+1},\tau _{t},\sigma _{t},\mu _{t},\mathbf{e}_{t}^{\ast
}),\mathbf{e}^{\ast }\left( s_{t+1},\tau _{t},\sigma _{t},\mu _{t},\mathbf{e}%
_{t}^{\ast }\right) \right) \notag
\end{eqnarray}%
where $\mathcal{P}^{j}(e_{t}^{i},\mathbf{e}_{-it}^{\ast })$ denotes the
winning probability of the representative candidate $j\in \{s,u,o\}$ given
the voting decisions, and $\mathbf{e}_{-it}^{\ast }$ is the optimal voting
decision of other groups that is not $i$, and $\Phi _{t}^{j}=\left\langle
\tau _{t}^{j},\sigma _{t}^{j},\mu _{t}^{j}\right\rangle $ is the vector of
preferred policy of candidate from group $j.$
The stationary Markov-perfect equilibrium defined above introduces another
functional equation exercise. The first exercise is to find a policy profile
that satisfies the usual Markov-perfect definition, as discussed in the case
of sincere voting in the text. The second exercise restricts the voting
decision to be cast on the belief that individuals in the same situation
next period will vote in exactly the same way. With this property, the
voters in this period know exactly how future generations will vote and can
evaluate the stream of payoffs accordingly.
Lastly, the keep the analysis simple, we focus on voting equilibria that are
consistent with policies derived in the text for the case of sincerely
voting. This will be the case if the policies are always coupled with a
voting equilibrium featuring the largest group always voting for its
representative candidate. In particular, if the group forms the absolute
majority, all votes cast from this group will go to its representative
candidate. The economy can go through different equilibrium paths depending
on $n$, $m$, and $s_{0}$, as follows:
\begin{enumerate}
\item If $n+m\leq 0$, the old group is always the absolute majority. Tax
rate is at the Laffer point and the economy is fully open to skilled
migration.
\item If $n+m>0$, then the dynamics depend on the initial state of the
economy, $s_{0}.$ If $s_{0}\geq \frac{1+\frac{n}{2}}{1+n}$, then the skilled
workers are the majority (controlling 50\% of the population), and zero tax
rate with limited skilled migration will be observed. If $\frac{n}{2(1+n)}%
\geq s_{0}$, the unskilled workers are the majority, then there will be a
positive tax rate (less than at the Laffer point) and some skilled
migration. If $n<0$, then \textit{initially} the old cohort is the majority;
the tax rate will be at the Laffer point and the skilled migration will be
maximal. Otherwise, the policies implemented are given in the equilibrium
below.
\end{enumerate}
The first equilibrium we look at is dubbed "Intermediate" because it
captures the essence that the preferred policies of the unskilled workers
are a compromise from the extremity of the other two groups. We can show
that the following strategy profile forms a subgame-perfect Markov
Equilibrium with strategic voting%
\begin{align}
e_{t}^{s\ast }& =\left\{
\begin{array}{cc}
s & \text{, if }s_{t}\geq \frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)} \\
u & ,\text{otherwise}%
\end{array}%
\right. \notag \\
e_{t}^{u\ast }& =u \\
e_{t}^{o\ast }& =\left\{
\begin{array}{cc}
o & \text{, if }\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq \max
\{s_{t},1-s_{t}\} \\
u & ,\text{otherwise}%
\end{array}%
\right. \notag
\end{align}%
and the policies implemented when no group is the absolute majority are
\begin{equation}
\Phi _{t}=\left( \tau _{t}=\frac{1-\frac{1}{J}}{1+\varepsilon -\frac{1}{J}}%
,\sigma _{t}=1,\mu _{t}=\frac{2+n-2(1+n)s_{t}}{m}\right)
\end{equation}%
where $J=J(\mu _{t},\sigma _{t},s_{t},\mu _{t-1})$ is as in equation (\ref%
{taxwedge}).
The equilibrium features the unskilled voters always voting for their
representative, whereas the other two groups vote for their respective
candidate only if they are the largest group, or for the unskilled candidate
otherwise. With these voting strategy, if no group captures 50\% of the
voting populations, the policy choice preferred by the unskilled candidate
will prevail. One notable difference is the policy related to the
immigration volume. In period $t+1$, as long as the skilled workers do not
form 50\% of the voting population, the policies preferred by the unskilled
workers will be implemented. To make sure that this is the case, skilled
migration is restricted to just the threshold that would have put the
skilled voters as the absolute majority in period $t+1$. The volume of
migration, $\mu _{t}^{\ast }=\frac{2+n-2(1+n)s_{t}}{m}$, reflects the fact
that the threshold value for this variable has been pushed slightly farther.
This level can be shown to be higher than the restricted volume in sincerely
voting equilibrium.
In the preceding equilibrium, we let the preference of the skilled workers
and the old retirees decide the fate of the policies. In the following
analysis, the unskilled workers consider who they want to vote for. This
will depend on how extractive the tax policy preferred by old is. We call
the next equilibrium "Left-wing", because it features a welfare state of the
size greater-than-or-equal to that of the intermediate policy equilibrium.
This may arise when the tax rate preferred by the old voters is not
excessively to redistributive. When $\frac{1}{1+\varepsilon }\leq \widetilde{%
\tau }$, we can show that we have an equilibrium of the following form%
\begin{align}
e_{t}^{s\ast }& =\left\{
\begin{array}{cc}
s & \text{, otherwise} \\
u & ,\text{if }\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq s_{t}\geq \frac{%
1+\frac{n-m}{2}}{1+n}%
\end{array}%
\right. \notag \\
e_{t}^{u\ast }& =\left\{
\begin{array}{cc}
u & \left\{
\begin{array}{c}
\text{, if }1-s_{t}\geq \frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)},\text{ or}
\\
\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq s_{t}\geq \frac{1+\frac{n-m}{2}%
}{1+n}%
\end{array}%
\right. \\
o & ,\text{otherwise}%
\end{array}%
\right. \\
e_{t}^{o\ast }& =o \notag
\end{align}%
and the policies implemented when no group is the absolute majority are
\begin{equation}
\Phi _{t}=\left\{
\begin{array}{cc}
\left( \tau _{t}=\frac{1-\frac{1}{J}}{1+\varepsilon -\frac{1}{J}},\sigma
_{t}=1,\mu _{t}=\frac{2+n-2(1+n)s_{t}}{m}\right) & ,\text{ if }\frac{1+\mu
_{t-1}}{1+n+\mu _{t-1}(1+m)}\geq s_{t}\geq \frac{1+\frac{n-m}{2}}{1+n} \\
\left( \tau _{t}^{\ast }=\frac{1}{1+\varepsilon },\sigma _{t}=1,\mu
_{t}=1\right) & ,\text{ otherwise }%
\end{array}%
\right.
\end{equation}%
where $J=J(\mu _{t},\sigma _{t},s_{t},\mu _{t-1})$ is as in equation (\ref%
{taxwedge}) and $\widetilde{\tau }$ is given implicitly in equation (\ref%
{EQtaxbreakeven}).
When the tax rate preferred by the old voters is not excessively
redistributive in the eyes of the unskilled, we could have an equilibrium
where the unskilled voters strategically vote for the old candidate to avoid
the policies preferred by the skilled voters. This will be an equilibrium
when the size of the skilled is not "too large." Recall that, voting to
implement the policies selected by the old candidate leads to opening the
economy fully to the skilled immigrants. If the size of the skilled group is
currently too large, there is a risk of making the skilled voters the
absolute majority in the next period and will result in no welfare state in
the retirement of this period's workers. The cutoff level before this
happens is given by $\frac{1+\frac{n-m}{2}}{1+n}$. Therefore, voting for the
old will only be compatible with the interest of the unskilled voters when
the tax rate is not excessively high and when the size of the skilled is not
too large.
We turn our attention to the next equilibrium. When $\frac{1}{1+\varepsilon }%
>\widetilde{\tau }$, we can show that there is an equilibrium with the
following functions:%
\begin{align}
e_{t}^{s\ast }& =\left\{
\begin{array}{cc}
s & \text{, otherwise} \\
u & ,\text{if }1-s_{t}\geq \frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}%
\end{array}%
\right. \notag \\
e_{t}^{u\ast }& =\left\{
\begin{array}{cc}
u & \text{, otherwise} \\
s & ,\text{ if }\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq \max
\{s_{t},1-s_{t}\}.%
\end{array}%
\right. \\
e_{t}^{o\ast }& =\left\{
\begin{array}{cc}
o & \text{, otherwise} \\
u & ,\text{if }s_{t}\geq \frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}%
\end{array}%
\right. \notag
\end{align}%
and the policies implemented when no group is the absolute majority are
\begin{equation}
\Phi _{t}=\left\{
\begin{array}{cc}
\left( \tau _{t}=0,\sigma _{t}=1,\mu _{t}=\frac{2+n-2(1+n)s_{t}}{m}\right) &
,\text{ if }\frac{1+\mu _{t-1}}{1+n+\mu _{t-1}(1+m)}\geq \max
\{s_{t},1-s_{t}\} \\
\left( \tau _{t}=\frac{1-\frac{1}{J}}{1+\varepsilon -\frac{1}{J}},\sigma
_{t}=1,\mu _{t}=\frac{2+n-2(1+n)s_{t}}{m}\right) & ,\text{ otherwise}%
\end{array}%
\right.
\end{equation}%
where $J=J(\mu _{t},\sigma _{t},s_{t},\mu _{t-1})$ is as in equation (\ref%
{taxwedge}) and $\widetilde{\tau }$ is given in equation (\ref%
{EQtaxbreakeven}).
When the Laffer point is higher than $\widetilde{\tau }$, the tax rate is
read as excessive. In this case, the unskilled voters will instead choose to
vote for the skilled over the old candidate. The resulting equilibrium as
the size of the welfare state less-than-or-equal to that in the intermediate
policy equilibrium, hence we refer to it as "Right-wing." When the tax
preferred by the old is excessive from the perspective of the unskilled, the
political process could implement the policies preferred by the skilled in
order to avoid the worst possible outcome. This happens when the old voters
constitute the largest group, and the unskilled voters vote strategically
for the skilled candidate. In other cases, however, the policies preferred
by the unskilled will be implemented, irrespective of the identity of the
largest group in the economy.
For our results with multidimensional policies, it is important to note here
that the ranking of candidates by individual voters allows us to escape the
well-known agenda-setting cycle (the "Condorcet paradox"). Such a cycle,
which arises when any candidate could be defeated in a pair-wise majority
voting competition, leads to massive indeterminacy and non-existence of a
political equilibrium. The agenda-setting cycle will have a bite if the
rankings of the candidates for all groups are unique: no group occupies the
same ranked position more than once. However, this does not arise here,
because, in all equilibria, some political groups have a \textit{common}
enemy. That is, because they will never vote for the least-preferred
candidate (the "common" enemy), the voting cycle breaks down to determinate
policies above, albeit their multiplicity. This occurs when voters agree on
who is the least-preferred candidate and act together to block her from
winning the election. The literature typically avoids the Condorcet paradox
by restricting political preferences with some ad hoc assumptions. For our
case, the preferences induced from economic assumption lead to the escape of
the Condorcet paradox.\footnote{%
For discussions on agenda-setting cycle, see Drazen (2000, page 71-72), and
Persson and Tabellini (2000, page 29-31).}
\section{Conclusion}
The paper develops a dynamic politico-economic model featuring three groups
of voters: skilled workers, unskilled workers, and retirees. The model
features both \textit{inter-} and \textit{intra-}generational
redistribution, resembling a welfare state. The skilled workers are net
contributors to the welfare state whereas the unskilled workers and old
retirees are net beneficiaries. When the skilled cohort grows rapidly, it
may be necessary to bring in unskilled migrants to counter balance the
expanding size of the skilled group.
The native-born young, whether skilled or unskilled, benefit from letting in
migrants of all types, because their high birth rates can help increase the
tax base in the next period. In this respect, skilled migrants help the
welfare state more than unskilled migrants, to the extent that the offspring
resemble their parents with respect to skill. On the other hand, more
migrants in the present will strengthen the political power of the young in
the next period who, relatively to the old, are less keen on the generosity
of the welfare state. In this respect, unskilled migrants pose less of a
threat to the generosity of the welfare state then skilled migrants.
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\end{document}