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\begin{document}
\title{Migration and the Welfare State: Dynamic Political-Economy Theory}
\author{Assaf Razin, Cornell University and Tel-Aviv University \and Efraim
Sadka, Tel-Aviv University \and Benjarong Suwankiri, Cornell University}
\date{March, 2009}
\maketitle
\begin{abstract}
Milton Friedman, the Nobel-prize laureate economist, had it right: "It's
just obvious that you can't have free immigration and a welfare state." That
is, national welfare states can almost never coexist with the free movement
of labor. This fact underscores the relevance of the analysis in this paper,
which is a part of a forthcoming book on migration and the welfare state. It
focuses on the demographic, and economic, fundamentals behind
policy-restricted (political-economy based) migration, and the
policy-restricted (political-economy based) generosity of the welfare state.
\end{abstract}
\section{Introduction}
All over the world, the combination of declining population growth rates and
rising life expectancy presents a major fiscal challenge to social security
systems. From an economic perspective, a rise in the dependency ratio (i.e.,
the proportion of retirees per worker) increases the number of people
drawing from the system; while it decreases the number of contributors. From
a political perspective, the older is the decisive voter, the more relevant
is the pension spending in the political agenda. One of the policy tools
that are considered for mitigating these politico-economic forces which
result in higher demand for, and lower supply of, social security benefits
is migration policy.
The view that increased migration may come to the rescue of PAYG social
security systems reflects the fact that the flow of migrants can alleviate
the current demographic imbalance, by influencing the age structure of the
host economy. A few empirical studies address this point by calibrating the
equilibrium impact of a less restrictive policy towards migration according
to U.S. data. Storesletten (2000) finds in a general equilibrium model that
selective migration policies, involving increased inflow of working-age high
and medium-skilled migrants, can remove the need for a future fiscal reform.
By emphasizing the demographic side and abstracting from the migrants'
factor prices effects, Lee and Miller (2000) conclude in a similar analysis
that a higher number of migrants admitted into the economy can ease
temporarily the projected fiscal burden of retiring baby boomers.
This paper combines two fields of the existing political economy literature,
which have not been examined jointly, to our knowledge: the political
economy of the PAYG social security systems (Cooley and Soares (1999), Bohn
(2005), Boldrin and Rustichini (2000), Galasso (1999)) and the political
economy of migration (Benhabib (1997)). There are also a few studies which
deal with the effect of migrants on the PAYG social security system (Razin
and Sadka (1999) and Scholten and Thum (1996)). This paper addresses the
joint political economy decisions regarding both migration policy and social
security policy in a dynamic set-up.
The paper, a part of a forthcoming book, develops a dynamic
politico-economic model, in which both migration and taxes interact,
focusing on inter- and intra-generational aspect of social security. The
model is based on key demographic characteristics: that migrants are younger
and have higher birth rates than the native born population. To isolate the
inter-generational aspects, we abstract in this chapter from
intra-generational income transfers considerations. (These considerations
are brought up in subsequent chapters.) A standard dynamic equilibrium
concept is employed in which migration policy and pay-as-you-go (PAYG)
social security system are jointly determined through a majority voting
process.
\section{Background:Migration and Intergenerational Distribution Policy}
We briefly describe the model of inter-generational distribution policy and
migration is developed in Sand and Razin (2008). A perishable consumption
good is produced using only labor as input; transfers from young to old
(paid by flat tax rate on labor income) are an important supplement for
private savings guaranteeing old-age consumption. Each generational cohort
lives two periods, supplying labor elastically when young, and deriving
utility from consumption in both periods of life.
If there were not to be migration, it is a standard outcome in this
framework that if the population growth rate is positive, the young always
outnumber the old. Therefore, a pay-as-you-go social security system cannot
be sustained under majority voting. If, however, population growth is
negative, so that the old outnumber the young, then the pay-as-you-go system
can be sustained with a constant tax rate the maximizes the social security
benefits (the preferred point of old cohort at each period). Now, introduce
migration into the standard framework. Migrants arrive young but cannot vote
until they are old. Their children, who are identical to the young
native-born, can vote when young. Moreover, migrants (though not their
offspring) have a birth rate that is larger than the native-born rate.
Migration policy is described by an endogenously determined quota variable.
The central tension faced by today's young in thinking about migration
policy is that both the ratio of young to old in the next period, and the
ratio of taxpayers to old dependents in the next period increase when the
present period migration quota rises. A higher value of the latter this
period will raise the number of young taxpayers per old dependent next
period, but will also increase the voting power of the young next period,
perhaps putting them in the majority. If the native born and the migrants'
population growth rates are positive (while by assumption the latter rate
exceeds the former), then young voters always outnumber old voters and the
pay-as-you-go social security system will not be sustainable as a Markov
equilibrium. So migration is of no help in this case. On the other hand, if
the native-born population growth rate is negative, then the social security
system is sustainable in the absence of migration. In this case, the quest
is not whether migration helps sustain social security, but whether it
threatens its sustainability. Assuming that the population growth rate of
the native-born is negative, the sort of equilibrium that arises depends on
the sum of native-born and migrants' population growth rates. If this sum is
negative, admitting no migrants today guarantees an old majority tomorrow.
Even if the current young chooses the maximum allowable migration so as to
maximize next period's benefits, there will still be a majority of the old
in the next period. Both the current old and the current young agree on
letting in the maximal number of migrants, an except perhaps for the initial
period, the majority of voters will always be old. Therefore, the tax rate
is set at the "Laffer" rate. Migration does not yet add (nor subtract) much
to the survival of the social security system in this case.
But when the sum of the native-born and the migrants population growth rates
is positive and the native-born population growth rate is negative,
migration adds an interesting twist. In essence, it poses a threat to social
security that in the absence of migration will be assured. In this case, the
numbers of old and young next period are equal and by assumption, ties are
decided in favor of the old. Then current young's desire for higher
migration, to maximize their old-age benefits is constrained by their desire
to maintain an old majority next period. If the young are currently in the
majority, they set the current tax rate equal to zero (implying no benefits
for the current old), and set migration quota at an intermediate level that
barely preserves the old majority in the next period. In the next period,
the old median voter sets the tax rate at the "Laffer" rate and the
migration quota at the maximum level. The latter guarantees that the young
will be in majority in subsequent period; and the cycle repeats itself.
From this benchmark model, Sand and Razin (2008) develops a model which also
includes capital accumulation and endogenous factor prices. The extended
model has an additional demographic-steady equilibrium, where the young \ is
steadily the median voter. Most importantly, the young \ does set the\
social security tax to a positive level, and thus sustains the social
security system. As in Forni (2005), in the case of a positive native-born
population growth rate, when the young are always in the majority, a
pay-as-you-go social security system is sustained by a tax rate on labor
income which varies with the level of the capital stock (a second state
variable). Specifically, the tax rate on labor income is decreasing in the
capital stock. In the case in which the population growth rates of the
native-born and the migrants' are positive ($n,m>0$), the number of \ next
period young voters exceeds the number of next period old voters, which
means that the decisive voter is always young. Still, if the capital per the
native-born workforce is in some range, then the optimal strategy of the
young is always to vote for a positive tax rate, and maximum migration
quota, thus sustaining both migration and the social security system. The
size of the social security system depends on the capital per native-born
worker, and on the exogenously given ceiling on migration quota. Thus the
polico-economic sustainable migration boosts up the tax base for financing
the social security.
\section{Elements of Strategic Voting with Multiple Groups}
The initial motivation for our politico-economic setup is the class of
models with citizen-candidate structure. Before the introduction of the
citizen-candidate structure, earlier models in the fields of public choice
and political economics utilize heavily the Downsian candidate setup that
leads to the result of platform convergence of the candidates (Downs
(1957)). The model assumes purely office-motivated candidates competing for
a single office post. The competition to win the election will drive the
policy platforms of all the candidates to the bliss point of the median
voters, trying to attract as many votes as possible.\footnote{%
The politico-economic models we employed in the preceding chapters were in
this spirit too.} Thus the campaign among the candidates boils down to
pursuing what drives the preference of the median voter and what may shift
the distribution of voters. Moreover, the complete convergence in platforms
does not seem to be observed in practice in most elections. Furthermore,
candidates must arise from the citizen body and citizens are presumed to
have some preferences for the policy chosen, regardless of the number of
voters. Hence, assuming that candidates are only office-motivated misses out
key policy determinants of voting models. The citizen-candidate model stands
on the other end of the spectrum. First studied by Osborne and Slivinski
(1996) and Besley and Coate (1997), the citizen-candidate model seeks to
endogenize the candidates' selection from within the body of the citizens,
and how the policy is ultimately determined.
However, due to the richness of strategic choices in the model, the
citizen-candidate model is not easily applicable for applied research. In
particular, the model suffers from massive multiplicity of equilibria, even
in a static setting. For those seeking a dynamic politico-economic
framework, the citizen-candidate proves formidable. In a subsequent work,
Besley and Coate (1998) have extended the static model to a two-period
setting. Anything beyond two-period must face exponentiated complexity. All
in all, the citizen-candidate model is appropriate for an analysis focussing
on a small-scale election, and possibly static. Therefore, it remains just a
motivation for our exposition in this chapter, as we have adapted the model
into an easily applicable version.
\subsection{Many candidates}
Consider an economy with a continuum of citizens, normalizing the population
size to a unit. The citizens are divided into $N$ groups, indexed by $i\in
\left\{ 1,2,\ldots ,N\right\} $, and each has a mass of $\omega _{i}\geq 0$,
where $\sum_{i=1}^{N}\omega _{i}=1$. We imagine $N$ to be relatively small.
This means that, with a large population, people with similar interests
often get grouped together. This setup abstracts from the possibility that
one individual may belong to more than one group, sharing many interests.%
\footnote{%
This shortfall, nonetheless, is common even in literature concerning itself
primarily with interest groups' influence.}
To highlight the mechanics of the model, suppose that the voters must
collectively choose a one-dimensional policy (that is, $p\in P=\mathbb{R}$).%
\footnote{%
Besley and Coate (1997) studies a more general environment with possible
multi-dimensional policy space.} We assume that any two citizens belonging
to the same group will have identical preference over the policy. The
representative citizen from group $i$ has a preference defined over the
policy space, represented by the utility function $v^{i}(p)$. These
preferences are "singled-peaked" and we let $p_{i}^{\ast }$ denotes group $i$%
's preferred policy.
We assume that there are $N$ candidates running for office representing
directly the interest of the group they belong to. We denote with $j\in
\left\{ 1,\ldots ,N\right\} $ the identity of the candidates. This is fully
known to all voters. Only one candidate is present from each group. We
assume that, if the candidate representing group $j$ wins the election, the
implemented policy will be $p_{j}^{\ast }$. Under plurality rule, candidates
who receive the most votes win.
Each citizen has a single vote that can be cast for a candidate. In
particular, because voters from the same group have identical preference,
they will vote identically.\footnote{%
We allow no abstentions within the model. Abstention can be built directly
into voting choices. Depending on the context, however, it may appear
unrealistic because, if one voter from a group abstains, all members of the
same group must accordingly abstain.}
Let $e^{i}$ $\in \left\{ 1,\ldots ,N\right\} $ denote the vote casted by
voters of group $i$. How each chooses to vote depends on her preference and
what we allow them to consider while voting. We consider two canonical
voting behaviors:\ \textit{sincere} and \textit{strategic}.
\subsection{Sincere Voting}
Voting sincerely is the simpler of the two. Under sincere voting behavior,
voters will vote for candidates $j\in \left\{ 1,\ldots ,N\right\} $ whose
policy platform maximizes their utility, that is%
\begin{equation*}
\widetilde{e}^{i\ast }=\arg \max \left\{ v^{i}\left( p_{j}^{\ast }\right)
\mid e^{i}\in \left\{ 1,\ldots ,N\right\} \right\} .
\end{equation*}%
We can denote the voting vector as $\widetilde{\mathbf{e}}^{\ast }=\left(
\widetilde{e}^{1\ast },\ldots ,\widetilde{e}^{N\ast }\right) $. Under this
voting behavior, voters belonging to group $i$ will vote for candidate
representing their group. That is $\widetilde{e}^{i\ast }=$ $i$. The winner
of the election will be decided purely by the size of the groups. Under
plurality rule, the winning candidate will come from the group with the
largest size, as reflected by $\omega _{i}$. In the special case with two
groups ($N=2$), then the winning candidate will be represent the median
voter of the economy. However, as $N$ gets larger, it is no longer the case
that the winning candidate will represent the preference of the median
voter. When there are more fractions in the economy, and no collusion is
allowed (that is, assuming everyone votes sincerely), the preference of the
largest group in the economy will dictate the implemented policy.
\subsection{Strategic Voting}
Strategic voting relaxes the assumption of sincere voting. People are no
longer required to vote for the candidate they like most, but rather they
take into account the probability of that candidate winning the election. A
voter is said to be voting \textit{strategically} if she votes for the
candidate with a policy platform that maximizes her.expected utility, where
the expectation is taken over all the candidates and their probability of
winning the election. Moreover, the votes must be consistent with the
induced probability of winning of each candidate. Formally, voting decisions
$\mathbf{e}^{\ast }=(e^{1\ast },\ldots ,e^{N\ast })$ form a \textit{voting
equilibrium\footnote{%
The original definition of this voting equilibrium is due to Besley and
Coate (1997).}} if
\begin{equation*}
e^{i\ast }=\arg \max \left\{ \sum_{j=1}^{N}\mathcal{P}^{j}(e^{i},\mathbf{e}%
_{-i}^{\ast })v^{i}\left( p_{j}^{\ast }\right) \mid e^{i}\in \left\{
1,\ldots ,N\right\} \right\}
\end{equation*}%
for $i\in \left\{ 1,\ldots ,N\right\} $, where $\mathcal{P}^{j}(e^{i},%
\mathbf{e}_{-i}^{\ast })$ denotes the probability that candidate $j\in
\left\{ 1,\ldots ,N\right\} $ will win given the voting decisions, and $%
\mathbf{e}_{-i}^{\ast }$ is the optimal voting decisions of other groups
that is not $i$. Thus we also require that each vote cast by each group is a
best-response to the votes by the other groups. In addition, this also means
that the representative voter of each group must take into the account the
\textit{pivotal} power of her vote, because the entire group will also vote
accordingly. 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 her ideal set of
policies. The winning probability quantity, $\mathcal{P}^{j}(e^{i},\mathbf{e}%
_{-i}^{\ast })$, must be determined endogenously from the voting vector and
the groups' weight. Lastly, we define a \textit{political equilibrium} to
consists of two vectors, $\mathbf{e}^{\ast }$ and $\mathbf{p}^{\ast }$,
where the latter is the vector listing the policies preferred by every
candidate.
It is important to contrast the strategic voting scenario with the sincere
counterpart. We do this by a couple of examples, which will also demonstrate
how the probability a candidate would win is determined, $\mathcal{P}^{j}(%
\mathbf{e}^{\ast })$. Under sincere voting, voters assume that the policy of
their most-preferred candidate will be implemented with probability one,
while under strategic voting, the probability depends on how other groups
vote. A special case arises when a certain group form more than 50\% of the
population. In this case, the winning candidate, who will also represent the
preference of the median, will belong to this group, irrespective of the
voting profiles of the other groups. Therefore, the probability that its
candidate will win is 1. One can easily construct other examples with
different conclusions. For example, let $N=3$, and $\omega _{i}=\frac{1}{4},%
\frac{1}{3},\frac{5}{12}$ for $i=1,2,3$ respectively. No one group consists
of more than 50\% of the population; group $3$ is the largest. However, if
group 1 and 2 both dislike the policy preferred by group 3, they could
collude to surpass 50\% and win the election. The implemented policies will
be decided by the voting equilibrium. If collusion means voters from group 1
and group 2 both vote from group 2's candidate, the ideal policy of group 2
will be implemented in equilibrium. The probability of winning for
candidates representing group 1 and 2 are $\mathcal{P}^{1}(\mathbf{e}^{\ast
})=0$ and $\mathcal{P}^{2}(\mathbf{e}^{\ast })=1$. Likewise, group 1 and 2
could both vote for group 1's representative candidate, hence resulting in
policy preferred by group 1 in equilibrium. In this case, the probability of
winning for candidates representing group 1 and 2 are reversed $\mathcal{P}%
^{1}(\mathbf{e}^{\ast })=1$ and $\mathcal{P}^{2}(\mathbf{e}^{\ast })=0$. By
either collusions, the preferred policy of the largest group, group 3, will
be blocked in equilibrium. These two voting equilibrium will generate $%
\mathcal{P}^{3}(\mathbf{e}^{\ast })=0$.
Note that a rule for a tie breaker should be defined. That is, if two
candidates receive the same amount of votes, how will this be resolved.
Besley and Coate (1997) proposes equal probability across all leading
candidates. Alternatively, one can also assign some other arbitrary rules,
such as the candidate belonging the larger group always win or the candidate
with a smaller group index wins. Whichever rule one chooses, it should
complement the analysis underlying the usage of the model.
\section{Migration, \textit{Inter-} and \textit{Intra-}generational
Redistribution}
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 as in Part I of the book,
by a proportional tax on labor income to finance a demogrant in a
balanced-budget manner.\footnote{%
We draw heavily on Suwankiri (2009).} 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. As in
Chapter 5, we characterize subgame-perfect Markov politico-economic
equilibria consisting of the tax rate (which determines the demogrant),
skill composition and the the total number of migrants. We distinguish
between two voting behaviors: sincere and strategic voting (see Chapter 6).
As illustrated in that chapter, 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.
\subsection{Analytical Framework}
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 ly 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 intergenerational 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 ($\sigma _{t}>s_{t}$).
\subsection{Political Economy Equilibrium: Sincere Voting}
In this section, we study the politico-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
appendix 7A, 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}
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 $.
\subsection{Interpretation: Migration and Tax Policies}
The intuition for the aforementioned results is as follows. The skilled are
the net contributor to the welfare state, while the other two groups are net
beneficiaries. Preferences of the old retirees are simple. If the old cohort
is the largest, it wants maximal social security benefits, which means
taxing to the Laffer point ($\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{Conclusion}
In this paper, which is part of a forthcoming book, we built 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.
As in chapter 5, 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.
\section{Appendix 7A: 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 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 pairwise 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. For discussions on agenda-setting cycle, see Drazen
(2000, page 71-72), and Persson and Tabellini (2000, page 29-31).
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\end{document}