Abstract
We study the consequences of the coexistence of different social protection systems on contribution rate levels and migration in a twocountry model. Before any migration takes place, the levels of contribution rates are chosen by a representative elected in each country. The migration of each agent depends on her anticipation of other agents’ migrations. We show that the richest agents are attracted to the Bismarckian country. The poorest agents tend to migrate toward one country or the other depending on the Beveridgean country contribution rate. The Beveridgean country can set a higher contribution rate to limit the departures of rich agents.
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Notes
The USA, Japan, and Australia, for example, have adopted a liberal regime.
See Besley and Coate (1997) for a complete analysis of the citizen–candidate notion.
If the relative risk aversion \(\lambda\) is lower than \(1,\) then the opposite effect occurs: the richer the agents are, the lower their preferred contribution rate.
We assume that there is a unique median income \(w_{m}\).
Note that the opposite is true for \(\lambda<1.\)
\(f\) is defined on whole \(R\), but \(f(x)\) is the expatriation cost probability density only for \(x>0\).
\(F\) is an increasing function from \(R\) to \([0;2]\), with \(F(0)=1\).
Boerner and Uebelmesser consider a median voter model. Migration leads to a reduction of the median income, but since the average income decreases as well, the effect on the tax rate is ambiguous.
\( w_{s} = t^{A} \overline{w} ^{p} {\left( {1  p} \right)}^{{\frac{\lambda } {{1  \lambda }}}} {\left[ {1  p^{\lambda } {\left( {1  t^{A} } \right)}^{{1  \lambda }} } \right]}^{{\frac{1} {{\lambda  1}}}} \) if \(t^{A}\leq 1p^{\lambda /(\lambda 1)}\) and \(w_{s}=0\) if \( t^{A}\geq 1p^{\lambda /(\lambda 1)}\).
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Acknowledgements
We would like to give special thanks to Hubert Kempf, Patrick Toche, Etienne Lehmann, and EUREQua’s “political economy” group for their comments. We also thank the Journal’s anonymous referees. We take full responsibility for any remaining errors.
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Appendices
Appendix
The firstorder condition in country \(B\) gives:
the secondorder condition is satisfied because we have, by concavity of \(U\!\!\):
and in country \(A\!\!\):
with a CRRA utility:
the secondorder condition is satisfied since we have:
Solving Eqs.12 and 13, we have for \(\lambda >1\!\!\):
and
Existence of a unique migration equilibrium for uniform costs probability density
We want to prove that there exists a unique equilibrium satisfying (i), (ii), and (iii) in Eq. 5, if the costs density function is uniform on \([0;a]\) (for \(a\) large enough).
We can rewrite Eq. 5 (iii) for any \(j\) with \(t^{B}=1p\) as:
where \(K\left( t^{A},t^{B}\right) =pU\left( 1t^{A}\right) U\left( p\right)\) and \(D=(1p)\left( t^{A}\frac{p}{1p}\right) ^{1\lambda }\!\!\). We set \(\phi (\overline{w}^{a})=\frac{\sum_{j}N_{j}F(x_{j})w_{j}}{\sum_{j}N_{j}F(x_{j})}\)
We assume that the cost density function is uniform in \([0;a]\!\!\). This means that \(f(x)=\frac{1}{a}\) for \(x\in [a;+a]\!\!\), and otherwise \(f(x)=0\!\!\). When \(a>0\) is sufficiently large, we obtain \(\left x_{j}\right<a\) for any \(j\!\!\), which means that no income group in any country can be totally emptied by the migration of its agents. We then have \(\phi (\overline{w}^{a})=\frac{ \sum_{j}N_{j}(1+\frac{x_{j}}{a})w_{j}}{\sum_{j}N_{j}(1+\frac{x_{j}}{a})}\!\!\).
Our aim is thus to prove that a unique migration equilibrium, i.e. a unique couple \((\overline{w}^{a},\overline{w}^{p})\) exists, such that \(\overline{w} ^{a}=\overline{w}^{p}=\phi (\overline{w}^{a})\!\!\). Clearly, we have \(\frac{dx_{j}}{d\overline{w}^{a}}=D(\overline{w} ^{a})^{\lambda }\!\!\!\), so we can write:
and \(\phi ^{\prime }(\overline{w}^{a})\geq 0\Leftrightarrow \phi (\overline{w }^{a})\leq \frac{\sum_{j}N_{j}w_{j}}{\sum_{j}N_{j}}=\bar{w},\) where \(\bar{w}\) is the average ex ante income.
Let us compute the second derivative:
We can see that this expression is negative for \(\phi (\overline{w}^{a})\leq \bar{w}\). We have thus shown that \(\phi \) is a decreasing function for \(\phi ( \overline{w}^{a})\geq \bar{w}\) and an increasing and concave function for \(\phi (\overline{w}^{a})\leq \bar{w}\!\!\). In the plane of coordinates \(( \overline{w}^{a},\overline{w}^{p})\!\!\!\), we denote the representative curve of \(\phi\) in \([w_{1},w_{M}]\) by \(C\phi ,\) and the line \(\overline{w}^{a}= \overline{w}^{p}\) by \(\Delta\!\).
Two cases must be considered:

If \(K(t^{A})<0\), \(x_{j}=w_{j}^{1\lambda }K(t^{A})+\frac{D}{1\lambda }( \overline{w}^{a})^{1\lambda }\) is an increasing function of \(w_{j}\!\!\). Setting \(N_{j}^{\prime }=N_{j}(1+\frac{x_{j}}{a})\!\!\), the ratio \(\frac{ N_{j}^{\prime }}{N_{j}}\) increases with \(w_{j}\!\), which implies, according to Lemma 2 (stated further), that \(\phi (\overline{w}^{a})=\frac{ \sum_{j}N_{j}^{\prime }w_{j}}{\sum_{j}N_{j}^{\prime }}\geq \bar{w}\!\!\). The curve \(C_{\phi }\) is thus decreasing and intersects \(\Delta\) at a unique point \(w^{\ast }\) (see Fig. 1).

If \(K(t^{A})>0\!\), then \(x_{j}\) is a decreasing function of \(w_{j}\!\). The ratio \(\frac{N_{j}^{\prime }}{N_{j}}\) decreases with \(w_{j}\!\), which implies, according to Lemma 2, that \(\phi (\overline{w}^{a})\leq \bar{w}\!\). The curve \(C_{\phi }\) is then increasing and concave: it thus intersects \(\Delta\) at a unique point \(w^{\ast }\) (see Fig. 2).
In both cases, we have proved the existence of a unique migration equilibrium.
Proof of Propositions 2 and 3
Propositions 2 and 3 can be rewritten using the \(x_{j}\). They assert that:

If \(t^{A}<t_{s}\!\), then:
\( x_{j} < 0 \Leftrightarrow w_{j} > w_{s} \)
\(x_{j}\) is an increasing function of \(w_{j}\!\!\).
\(\overline{w}\,^{p}<\overline{w}\)

If \(t^{A}>t_{s}\!\), then:
\(x_{j}<0\) for every \(j\)
\(x_{j}\) is a decreasing function of \(w_{j}\)
\(\overline{w}\,^{p}>\overline{w}\)
We first prove that \(x_{j}\) is a monotonous function of \(w_{j}\!\!\). Secondly, we study its sign and variations (according to \(t^{A}<t_{s}\) and \(t^{A}>t_{s}\) ) and finally compare \(\overline{w}^{p}\) and \(\overline{w}\!\).

1.
Monotonicity of \(x_{j}\) with respect to \(w_{j}\)
$$ x_{j} = {\left( {1  p} \right)}U{\left( {t^{A} \overline{w} ^{p} \frac{p} {{1  p}}} \right)} + pU{\left( {w_{j} {\left( {1  t^{A} } \right)}} \right)}  {\left( {1  p} \right)}U{\left( {t^{B} w_{j} \frac{p} {{1  p}}} \right)}  pU{\left( {w_{j} {\left( {1  t^{B} } \right)}} \right)}$$Taking \(t^{B}=1p\!\!\), we can write \(x_{j}\) in the following way:
$$ x_{j}=\left( 1p\right) U\left( t^{A}\overline{w}\,^{p}\frac{p}{1p}\right) +w_{j}^{1\lambda }K\left( t^{A}\right)$$where
$$ K\left( t^{A}\right) =pU\left( 1t^{A}\right) U\left( p\right)$$\(x_{j}\) is thus a monotonous function of \(w_{j}\!\!\).

2.
Sign and variations of \(x_{j}\!\!\) We can rewrite \(x_{j}\) as:
$$ x_{j}=w_{j}^{1\lambda }H\left( z\right) {\rm\text for }\; z={\frac{\overline{w}^{p} }{w_{j}}}$$and
$$ H\left( z\right) =\left( 1p\right) U\left( t^{A}z\frac{p}{1p}\right) +K\left( t^{A}\right)$$\(H\) is increasing on \(]0;+\infty [\), with \({\mathop {\lim H{\left( z \right)}}\limits_{z \to 0} } =  \infty \) and \( {\mathop {\lim }\limits_{z \to + \infty } }H{\left( z \right)} = K{\left( {t^{A} } \right)} \)
If this last limit is negative, then the equation \(H\left( z\right) =0\) has no solution. Observe that
$$ K{\left( {t^{A} } \right)} < 0 \Leftrightarrow t^{A} > t_{s} \;{\text{where}}\;t_{s} = 1  \frac{\lambda } {{p^{{\lambda  1}} }}$$(14)We have thus the two following cases:

For every \(t^{A}>t_{s}\!\), \(H\left( z\right)<0\) for every \(z\) i.e. \( x_{j}<0\) for any \(j\!\!\).
This is case (b) of Proposition 3: all agents are incited to migrate to country \(B.\) They migrate if their expatriation costs are lower than \(x_{j}\!\!\).
Since \(K\left( t^{A}\right)<0\!\!\), \(x_{j}\) is an increasing function of \( w_{j}, \) which means that the proportion of migrants per group is a decreasing function of income.

If \(t^{A}<t_{s},\) we have \(K\left( t^{A}\right) >0\!\!\). Thus, there exists \(z^{*},\) such that \(H\left( z^{*}\right) =0\!\!\). Solving this equation, we obtain:
$$ z^{*}=\frac{1}{t^{A}}\left( 1p\right) ^{\frac{\lambda }{\lambda 1}}\left[ 1p^{\lambda }\left( 1t^{A}\right) ^{1\lambda }\right] ^{\frac{1}{ 1\lambda }}$$We can set
$$ w_{s} = \frac{{\overline{w} ^{p} }} {{z*}} = t^{A} \overline{w} ^{p} {\left( {1  p} \right)}^{{\frac{\lambda } {{1  \lambda }}}} {\left[ {1  p^{\lambda } {\left( {1  t^{A} } \right)}^{{1  \lambda }} } \right]}^{{\frac{\lambda } {{\lambda  1}}}}$$(15)then, since \(H\left( z\right) \) is increasing, for \(w_{j}>w_{s}\) we have \( z = \frac{{\overline{w} P}} {{wj}} < z*, \) thus \(x_{j}<0\!\!\). Symmetrically, for \( w_{j}<w_{s,}\) we have \(x_{j}>0\!\!\).
This means that agents who are richer than \(w_{s}\) are prompted to migrate to \(B\!\!\), and those who are poorer than \(w_{s}\) are prompted to migrate to \(A\!\). Here, \( x_{j} \) is necessarily decreasing. Thus, the proportion of migrants to \(B\) is an increasing function of income and that of migrants to \(A\) a decreasing function of income.


3.
Variation of \(\bar{w}\!\!\). To compare \(\bar{w}\,^{p}\) and \(\bar{w}\!\!\), we first need the following lemma.
Lemma 2
Let \(N_{j}^{p}\)be the postmigration population of income group\(w_{j}\)and thus
Then:

(a)
If \(\left( \frac{N_{j}^{p}}{N_{j}}\right)\)is a decreasing function of \(j\!\!\), then \(\overline{w}\,^{p}\leq \bar{w}\!\!\).

(b)
If \(\left( \frac{N_{j}^{p}}{N_{j}}\right)\)is an increasing function of \(j\!\!\), then \(\overline{w}\,^{p}\geq \bar{w}\!\!\).
Proof

(a)
Let \(\overline{w}^{\prime }=\frac{\sum N_{j}^{\prime }w_{j}}{\sum N_{j}^{\prime }}\!\!\). It follows that if \(N_{j}^{\prime }\geq N_{j}\) for \( w_{j}\leq \overline{w}^{\prime }\) and \(N_{j}^{\prime }\leq N_{j}\) for \( w_{j}\geq \overline{w}^{\prime }\!\!\), then \(\overline{w}^{\prime }\leq \overline{w}\!\), since the \(w_{j}\) that are lower than \(\overline{w}^{\prime }\) get a higher weight in \(\overline{w}^{\prime }\) than in \(\overline{w}\!\!\). Now let \(d>0\!\!\), such that:
$$ \frac{d.N_{1}^{p}}{N_{1}}\geq ...\geq \frac{d.N_{l}^{p}}{N_{l}}\geq 1\geq \frac{d.N_{l+1}^{p}}{N_{l+1}}\geq ...\geq \frac{d.N_{M}^{p}}{N_{M}}$$where \(l\) satisfies \(w_{l}\leq \overline{w}^{p}<w_{l+1}\!\!\). Then, setting \(N_{j}^{\prime }=d.N_{j}^{p}\!\!\), we have \(\overline{w}^{\prime }=\overline{w} ^{p}\leq \overline{w}\!\!\).

(b)
Similarly, inverting \(\bar{w}\) and \(\overline{w}^{p}\!\!\).
We can now achieve the proof of Proposition 3. We have:
Adopting notations of Lemma 2, we set \(N_{j}^{\prime }=N_{j}F(x_{j})\!\!\), where we know that \(F\) is an increasing function.

For \(t^{A}<t_{s}\!\), \(x_{j}\) is decreasing; thus, \(\frac{N_{j}^{\prime }}{ N_{j}}=F(x_{j})\) is decreasing with \(j\!\!\). We then obtain \(\overline{w}^{p}\leq \bar{w}\) by applying Lemma 2.

For \(t^{A}>t_{s}\!\), \(x_{j}\) is increasing; thus \(\frac{N_{j}^{\prime }}{ N_{j}}=F(x_{j})\) is increasing with \(j\!\). We then obtain \(\overline{w}^{p}\geq \bar{w}\!\!\).
Proof of Lemma 1

Let us assume \(w_{k}\geq \overline{w}^{p}\)
Since \(\overline{w}^{p}>w_{s}\) agents with income \(w_{k}\) migrate from \(A\) to \(B\!\!\).

Let us assume \(w_{k}<\overline{w}^{p}\!\!\!\). We can then set:
$$ \begin{array}{*{20}c} {\Phi _{j} {\left( {t^{A} ,\overline{w} ^{p} } \right)} = x_{j} } \\ { = {\left( {1  p} \right)}U{\left( {t^{A} \overline{w} ^{p} \frac{p} {{1  p}}} \right)} + pU{\left( {w_{j} {\left( {1  t^{A} } \right)}} \right)}  {\left( {1  p} \right)}U{\left( {t^{B} w_{j} \frac{p} {{1  p}}} \right)}} \\ \end{array} $$for all given \(\overline{w}^{p}\) and \(t^{A}\!\!\).
If we take \(t^{A}=t^{B*}=1p\!\), we can then write:
When \(j=k\!\!\), it leads to:
i.e., \(x_{k}>0\) for \(t^{A}=t^{B*}\) and for any \(\overline{w}\,^{p}.\)
The level of taxation is chosen by the representative of income \(w_{k}.\) The program can be written as:
the maximum is obtained at \(t_{O}^{A\ast }.\) At this point, we obtain:
We then obtain that \(x_{k}>0\) at the optimum. Thus, \(t_{O}^{A\ast }<t_{s}\) according to Propositions 2 and 3.
Sign of \(\protect\eta\) terms
Indeed, for a CRRA utility function with \(\lambda >1\), the sign of \(\frac{ dx_{j}}{dt^{A}}\) at the optimum is given by (substituting Eq. 9 in Eq. 10):
thus, for \(\lambda >1\)
Moreover, recall that
\(\eta\) is positive if the sum of the absolute values of the terms for which \(w_{j}\) is between \(w_{k}\) and \(\overline{w}^{p}\) is smaller than the sum of the other terms. Moreover, Eq. 16 allows us to state that:
Sensitivity to migrate \(\frac{dx_{j}}{dt^{A}}\) is thus lower for agents with incomes that are close to the policymaker’s. Lastly,
which means that the reaction \(\frac{\partial \overline{w}^{p}}{\partial x_{j}}\) of the average income of country \(A\) to the net flow of migration of group \(j\) has an absolute value that is lower for agents with an income that is close to the ex post average income.
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Rossignol, S., Taugourdeau, E. Asymmetric social protection systems with migration. J Popul Econ 19, 481–505 (2006). https://doi.org/10.1007/s0014800600698
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DOI: https://doi.org/10.1007/s0014800600698
Keywords
 Social insurance
 Migration
JEL Classification
 D7
 H5
 F22