Immigration and Conflict in Democracies

Immigration and Conflict in Democracies Santiago Sánchez-Pagés Ángel Solano García June 2008 Abstract Relationships between citizens and immigrants may not be as good as expected in some western democracies. As social needs are different from both groups, immigrants may declare their dissent with the distribution of public expenditures. We introduce this possibility into a two party political competition model in which policies are the provision and the share of a public expenditure among natives and immigrants. Our results suggest that in countries where natives care strongly for public expenditures devoted to cover their needs, there is an efective provision of public services that cover the needs of both natives and immigrants in a peaceful environment. The threat of conflict is the main channel through immigrants get a part of total revenues. We also discuss how the enfranchisement of immigrants shapes policies. 1 The Model A country is populated by an exogenously determined mass of n natives and m immigrants with n > m > 1. 1.1 Natives Natives care about private consumption c n and about a certain groupspecific public service G n. More precisely, native i s preferences is represented by the following utility function: U i n(c n, G n ) = c n + β i ng n, where β i n R + stands for the native i s relative importance towards the public service. In particular we assume that β i n is distributed according Solano acnowledges financial support from the Junta de Andalucia. Economics, University of Edinburgh, 50 George Square, EH8 9JY Edinburgh, United Kingdom. E-mail: ssanchez@staffmail.ed.ac.uk. Departamento de Teoria e Historia Economica, Universidad de Granada, Campus de La Cartuja, 18011 Granada, Spain. E-mail: asolano@ugr.es. 1

to a certain pdf F(β i n) in the support (0, β max ]. Let β and β be the mean and the median of that distribution respectively. All natives are endowed with the same income level y n. Income after taxes is fully spent in consumption i.e. c n = y n (1 τ) 1.2 Immigrants Immigrants also care about private consumption c m and about their groupspecific public service G m according to the following utility function: U m (c, G m ) = c m + β m G m, where, β m R +. We are assuming that all immigrants have the same preferences, and for the sake of simplicity, they care about private consumption in the same way as natives do. All immigrants are endowed by the same income level y m such that y m < y n. Income after taxes is fully spent in consumption i.e. c m = y m (1 τ) 1.3 Government Government uses taxes to fund total resources spent in both public services. The total tax revenue is given by : R = τy, where Y = ny n + my m. We assume that government budget constraint is balanced, that is: G n + G m = R Moreover, government decides how to distribute the total tax revenue among natives and immigrants. In particular, government has to decide a certain α [0, 1] such that G m = αr and G n = (1 α)r. We choose policy instruments of the government to be τ and α. Notice that given these two instruments the values of G n, G m and R are determined. Government is formed by the winner of a two-party election in which each party announces a policy proposal that is a pair {τ, α} and voters vote for their most preferred policy. We assume full commitment on policy proposals, that is the winner by majority voting has to implement her policy proposal. 2

1.4 Timing 1. Political parties make policy proposal over policy instruments {τ, α}. 2. Elections take place and the winner implements the policy previously announced. 3. Given such a policy immigrants decide whether they go to conflict. 4. If immigrants decide to start a conflict, each group decides the amount of resources spent in the conflict. If immigrants do not start a conflict then they obtain payoffs in peace, otherwise they obtain the payoffs in conflict. We have then two main stages: the conflict stage and the political stage. We solve the game backwards. First, we solve the conflict stage for certain policies already implemented and then solve the political stage in order to characterize the SPE of the game. Let first study what will be the equilibria in the conflict stage. 2 Conflict Stage 2.1 A World in Peace. In a peaceful situation for a given pair {τ, α} the utilities that a native i and any immigrant obtain are given by: U i n(τ, α) = (1 τ)y n + (1 α)β i nr, U m (τ, α) = (1 τ)y m + αβ m R. In a world in peace, natives and immigrant have opposite preferences for the share of the public transfer α. In this scenario the optimal α for any native is α = 0. However this policy may not be compatible with peace as we see in the next subsection. 2.2 A World in Conflict Given the policy instruments implemented by the government, both groups natives and immigrants may declare their dissent regarding the public transfer they receive. Later on, we will describe in more details the process through which conflict erupts. The group who wins this confrontation can seize all the tax revenues. We will assume that agents can contribute to the 3

victory of the group they belong to by expending resources. Denote by a j the expenditure in conflict by a member of group j.then, given that taxes are levied before the conflict starts, it must hold that c j + a j (1 τ)y j. The probability of victory depends on the total amount of resources each group devotes to conflict. Denote as p m the probability that immigrants win. We assume that p m has the following functional form: p m = ma m A, where a m is the per capita amount of money each immigrant devotes to the conflict, and A = n i=1 ai n +ma m is the total amount of money that natives and immigrants dedicate to conflict activities. We assume that all natives contribute the same to the conflict activity, i.e. n i=1 ai n = A n = n a n. Given certain pair of policy instruments {τ, α}, the utility functions for both groups in case of conflict are the following: v i n(τ) = c n + (1 p m )β i nr = (1 τ)y n a n + (1 p m )β i nr, v m (τ) = c m + p m β m R = (1 τ)y m a m + p m β m R. Notice that α does not affect agents utilities under conflict. Then, we can calculate both groups decisions about the expenditure devoted to conflict activities (a n, a m ) maximizing their utility functions. The first order conditions of these problems uniquely characterize agents best reply functions, given implicitly by the following expressions: a n = p m (1 p m )βr, a m = p m (1 p m )β m R. We will assume throughout the paper that y n and y m are sufficiently high so an interior solution to the above system of equations always exists. From here, the following Proposition is straightforward. Proposition 1 There is a unique interior Nash Equilibrium of the conflict stage in which natives and immigrant spend A n and A m respectively in conflict activities such that: A n = (1 p m) 2 mβ m R, (1) where p m = mβm B+mβ m and B = n i=1 βi n. A m = (p m) 2 B R, (2) 4

Let us now perform some comparative statics on the equilibrium conflict expenditures. First, we study the relation between conflict expenditures and the intensity of preferences over the public service. As it turns out, in the case of natives, such preferences are summarized by the average β. Lemma 1 Under conflict, expenditures are increasing in the group average valuation of the public service. Moreover, a native expends more money in conflict than an immigrant if and only if β > β m. Proof. Let us first compute the derivative of each group equilibrium expenditure with respect to their average preference for the public service, i.e. β and β m. A n β = 2n(1 p m)p 2 mr > 0, A m β m = 2mp m(1 p m) 2 R > 0. Observe now that from the Nash Equilibrium levels of expenditure in conflict activities for both natives and immigrants we have that a n > a m if and only if: mβ m n ( ) B 2 ( mβm R > B + mβ m B + mβ m B n > β m, ) 2 B m R which is equivalent to β > β m. This Lemma shows also that whether at the individual level one group expends more resources in conflict than the other crucially depends on their relative average valuation of their respective public service. This implies that if β > β m natives spend in the overall more resources than immigrants in the conflict. Depending on the size of the immigrant group, this may not be the case when β < β m. Thus, we analyze now the effect of an increase in the number of immigrants, m, on the equilibrium total conflict expenditures. Lemma 2 The overall level of expenditures in conflict is increasing in the β size of the immigrant group if β m m n. Proof. Denote A = A n + A m. We know that A we have that: m = A n m + A m ( ) 2 A mβm m = B τ(ny n + my m ). B + mβ m m. From 2 5

It is straightforward that A m is a increasing function on m. Nevertheless, from 1 we have that: ( ) A B 2 n = mβ m τ(ny n + my m ). B + mβ m We calculate the derivative A n m and we obtain the following: ( B mβm B + mβ m ) A n m + y mτ (1 p m) 2 mβ m. Then a sufficient condition for A n m > 0 is B mβ m > 0, that is equivalent to β β m m n. This Lemma identifies a sufficient but not necessary condition for more immigrants to induce more natives expenditure in conflict. This is because the effect of m has an ambiguous effect: From 2 we can see that an increase in the size of the immigrant group produces a rise on the total expenditure devoted to conflict by that group since both the winning probability of that group and total revenues increase. But on the other hand, it is clear from 1 that natives expenditures may not increase with the number of immigrants. That is because as the group of immigrants becomes bigger natives winning probability gets smaller. Let us compute now the equilibrium utilities under conflict for both natives and immigrants. Given both groups equilibrium levels of effort in conflict and a tax τ, the utility derived from conflict for both natives and immigrants are respectively: V i n(τ) = (1 τ)y n + (1 p m)(β i n p mβ)r, V m (τ) = (1 τ)y m + p 2 mβ m R. Natives who have stronger preferences for the provision of the public service (relative to the average group preferences) are better off under conflict than those with weaker preferences. Furthermore, notice that natives with β i n below a p m proportion of the average valuation β strictly prefer the public service not to be provided. 2.3 When Does Conflict Start? We assume that immigrants may start a conflict if they obtain a larger utility from it than from the peaceful situation. That is, for any pair {τ, α} immigrants start a conflict if and only if: V m (τ) > U m (τ, α) 6

( ) 2 mβm (1 τ)y m + β m R > (1 τ)y m + αβ m R. mβ m + B Simplifying we rewrite the previous expression as: α < α ( ) 2 mβm. mβ m + B For any share of the tax revenues smaller than α immigrants are willing to start a conflict. Notice that the share of total revenues that natives need to give to immigrants in order to elude conflict is increasing both in the size of the immigrant group m and on the intensity of their preferences β m. On the other hand, we also assume that natives start a conflict if the majority of them obtain a larger utility from the conflict situation than under peace. That is, for any pair {τ, α} natives start a conflict if and only if: V i n(τ) > U i n(τ, α) (1 τ)y n + (1 p m)( β p mβ) βr > (1 τ)y n + (1 α) βr Simplifying we rewrite the previous expression as: α > α p m(1 + (1 p m) β β) It is easy to check that it is always the case that α < α. Notice that, that contrary to the immigrants case, for some parameters natives will not start a conflict even if they do not get any tax revenue. In fact, α > 1 if and only if β < p mβ. However, we should expect that in most societies β β since preferences for the public services are likely to be inversely related to income levels. We will assume throughout the paper that this is the case and thus the upper bound on α has a bite. Therefore, a peaceful situation requires a share of total revenues such that α [α, α]. 3 Natives Optimal policies and Preferences for Conflict In this section, we characterize voters preferences for policies and conflict before analyzing the political competition stage. We first assume that only natives have the right to vote. Therefore, we are interested on natives preferences over policies {τ, α}. Then, given these, we can identify those natives who prefer peace over conflict. In case of peace, the natives optimal level of α is straightforward and it is the minimum level that guarantee the peace, that is α = α. On the 7

other hand, preferences over the income tax τ are less trivial. We have to maximize natives utility function in case of peace with respect to τ, that is: Max τ (1 τ)y n + (1 α)β i nτy s.t. α = α ( mβm mβ m+b ) 2 Because of the linearity of the utility, we obtain corner solutions for this maximization program. In particular we have that: 1 if β τ p n i > β p = [0, 1] if βn i = β p, 0 if βn i < β p where β p = In case of conflict, that is, when α / [α, α], the share of tax revenue that goes to immigrants becomes irrelevant for natives because the whole revenue is seized by the winner of the conflict. In order to calculate the optimal income tax for a native i τ c under conflict, we have to maximize natives utility function Vn(τ), i that is: yn (1 α)y. Max τ (1 τ)y n + (1 p m)(β i n p mβ)r Because of the linearity of the utility, from the FOC we obtain also corner solutions for the maximization program. In particular we have that: 1 if β τ c n i > β c = [0, 1] if βn i = β c, 0 if βn i < β c where β c y = n Y (1 p m ) + p mβ. Tedious but straightforward computations show that β c > β p. Due to the linearity of utility functions, preferences are such that natives either prefer their specific public service to be provided (by using the whole tax payers income) or to not be provided at all. Hence, income redistribution occurs only if natives specific public service is provided because immigrants must receive a positive share of the tax revenues in order for peace to prevail. It is then conflict what creates the opportunity for income redistribution. Comparing the threshold values β p and β c we can identify those natives who prefer the provision of the public transfer in both, one or none of our two possible scenarios. In particular. This result indicates that there are more natives who prefer peaceful income redistribution than natives who prefer immigrants to start conflict. Lemma 3 All natives weakly prefer peace to conflict. 8

Proof. We have three types of natives depending on whether i) β i n > β c ii) β p < β i n < β c and iii) β i n < β p. Natives of type i) weakly prefer peace to conflict if and only if: (1 (p m) 2 )β i ny (1 p m)(β i n p mβ)y (1 (p m) 2 )β i n (1 p m)(β i n p mβ) β i n(p m p 2 m) (1 p m)p mβ Because (p m p 2 m) 0 we have that natives of type i) weakly prefer peace to conflict and their optimal policies are τ p = 1 and α = α. Natives of type ii) will weakly prefer peace to conflict if and only if: y n Y (1 (p m) 2 )β i ny y n (mβ m + B) 2 B(2mβ m + B) β p β i n β i n Therefore natives of type ii) weakly prefer peace to conflict and their optimal policies are τ p = 1 and α = α. Natives of type iii) are indifferent between conflict and peace since they always prefer no provision of the public service, i.e. τ p = τ c = 0. This Proposition states that if natives have sufficiently strong preferences for the public service devoted to satisfy their needs they will support redistribution of income towards immigrants in a peaceful environment. Otherwise, they prefer no provision of any public service, so they are indifferent about conflict and peace. One interesting question is how the size of the immigrant population may change natives preferences for redistribution. We find that if average natives preferences for their specific public service are strong enough relative to immigrants preferences for their own public service, bigger m increases the number of natives who prefer redistribution. Otherwise, if natives do not care strongly enough about their specific public service, the number of natives who prefer redistribution towards immigrants follows an inverted U- shaped pattern with respect of the number of immigrants. In this case, we find a level of immigration, m, such that maximizes the number of natives who prefer redistribution. β y n Proposition 2 If β m 2 3 y m an increase in the number of immigrants always increases the number of natives who prefer redistribution of tax revenues. Otherwise, there exists a level of immigration m > 0 that maximizes the number of natives who prefer redistribution. 9

Proof. An increase of the number of immigrants increases the number of natives who prefer no redistribution of tax revenues if and only if βp m > 0. That is, if and only if: y n y m 1 Y 2 1 p 2 m y 2 m Y + β p + 2p p m m m y n (1 p 2 m) 2 mβ m B+mβ m mβ mb (B+mβ m) 2 1 ( mβm B+mβ m ) 2 Y > 0 > 0 2mβ 2 m (B + mβ m )(B + 2mβ m ) > y m Y mβ m (2β m ny n 3y m B) > y m B 2, which boils down to the condition in the text of the Proposition. Otherwise, β i.e. β m < 2 y n 3 y m, then there exists a value m y = mb 2 β > 0 such m(2β mny n 3y mb) that for all m m we have βp m 0 and for all m m we have βp m 0. Thus m = arg min{β p (m)}. That implies that m maximizes the number of natives who prefer redistribution of tax revenues. The intuition behind Proposition 3 is the following. An increase in the number of immigrants produces two opposite effects on natives preferences for redistribution. First, an 0increase in the number of immigrants yields an increase in immigrants winning probability in conflict. This effect may drive natives to prefer no provision of any public service because in order to avoid conflict they must yield a large share of tax revenues to immigrants. Second, an increase in the number of immigrants rises total tax revenues via a higher tax base. This gives an incentive to natives to redistribute total tax revenues. To what extent one effect dominates the other depends on two factors: the relative intensity of preferences for their specific public services and the relative income of the two groups. Regarding the first factor, if natives care very strongly about their specific public service relative to immigrants the second effect will dominate the first one. On the other hand, more immigration will increase the support for redistribution when immigrants income is closer to natives income. 4 Political Competition Stage In this section, we analyze the political competition stage that takes place before the government is elected. We consider a model of two-party competition. As we explained above, parties choices are represented by the share of total tax revenues that will be devoted to finance immigrants public service and the income tax {τ j, α j }. Thus, the policy space is X = [0, 1] [0, 1]. 10

We assume that every voter votes for a party (there is no abstention). Immigrants have no right to vote. Thus, only natives can vote. The game takes place in two stages. In the first stage, parties propose a certain policy in X. In the second stage each voter votes for the party whose proposal would give her a higher utility. In case of indifference, a voter is assumed to vote for each party with equal probability. We assume that parties are fully committed to their policy proposals. This means that the party that wins the election will implement the policy chosen in the first stage. The winner of the election is decided according to majority rule. In case of a tie, both parties win with the same probability (equal to 1 2 ). We assume that parties maximize the probability of winning. Thus, the payoff function of a party can be defined as: 1 if # { i : U i (α l,τ l ) > U i (α k,τ k ) } + I 2 > n 2 1 Π l (α l, τ l ) = 2 if # { i : U i (α l,τ l ) > U i (α k,τ k ) } + I 2 = n 2 0 if # { i : U i (α l,τ l ) > U i (α k,τ k ) } + I 2 < n 2 (3) where # { i : U i (α l, τ l ) > U i (α k, τ k ) } is the number of voters who prefer to vote for party l and I = # { i : U i (α l, τ l ) = U i (α k, τ k ) } is the number of voters that are indifferent between the two parties. So, if the number of voters that prefer to vote for party l plus half of voters which are indifferent between the two parties is larger than the half of total voters, party l will win the election. From here the next Proposition is straightforward. Proposition 3 In equilibrium, both parties propose maximum redistribution and the minimum immigrants share of tax revenues compatible with peace, i.e. τ l = τ k = 1 and α l = α k = α, if and only if β β p. Otherwise, there is no redistribution, i.e. τ l = τ k = 0. As expected, office-motivated parties will fight for the support of the median voter. In this case, even though we have two dimensions, all natives have homogenous preferences over α. Then, the citizen with the median β i becomes the median voter. Parties will then propose redistribution depending on whether she supports redistribution or not, that is, depending on whether β is bigger or smaller than β p. As one might expect, in societies with more intense preferences for public services it will be more likely that public services will be provided. But more interestingly, our model predicts that this higher intensity will also reduce the amount of tax revenues that will be allocated to immigrants and therefore generated lower levels of income redistribution. We are also particularly interested on how equilibrium policies change as the number of immigrants increases. We know from Proposition 2 that 11

there exist some parameter values for which an increase in the number of immigrants has a stronger effect on immigrants winning probability than in total tax revenues. If this is the case, there will exist a non-monotonic relationship between the stock of immigrants and the support of redistribution. Hence, in populations with a large number of immigrants, an increase of immigration may drive natives to prefer no redistribution. 5 Enfranchisement Let us assume now that naturalization policies are such that the whole immigrant population can vote. We will keep our assumption of office motivated parties so in this environment there is no room for a potential incumbent to manipulate the enfranchisement process to gain votes (in that case, both parties would end up in a tie anyway) 1. Hence, we will compare the equilibrium policies of the competition stage with and without the full enfranchisement of immigrants. The analysis of partial enfranchisement would yield analogous results. In order to do this, we first have to study immigrants preferences on policies {τ, α}. Then, given these, we can identify whether immigrants prefer peace to conflict. In case of peace, the immigrants optimal level of α is straightforward and it is equal to the maximum level that is compatible with peace, that is, α = α. Preferences over the income tax τ are less trivial. We have to maximize immigrants utility function in case of peace with respect to τ, that is: Max τ (1 τ)y m + αβ m τy s.t. α = α p m(1 + (1 p m) β β). Because of the linearity of the utility function, we again obtain corner solutions for this maximization program. In particular we have that: 1 if β m > β τm p m p = [0, 1] if β m = βm p (4) 0 if β m < βm p where β p m = ym αy. In case of conflict, that is, when α / [α, α], the share of tax revenue that goes to natives becomes irrelevant for immigrants because the whole revenue is seized by the winner of the conflict. In order to calculate immigrants optimal income tax, τ c m under conflict, we have to maximize natives utility function V m (τ), that is: 1 However, if we consider that the incumbent has preferences over the policy finally implemented, then she may take advantage of such enfranchisement. 12

Max τ (1 τ)y m + p 2 mβ m τy. Because of the linearity of the utility, from the FOC we obtain also corner solutions for the maximization program. In particular we have that: 1 if β m > β τm c m c = [0, 1] if β m = βm c, 0 if β m < βm c where βm c = ym. As in the natives case, it is easy to show that β Y p 2 m c > βm. p m Once we have characterized immigrants preferences for policies, the next question is what scenario, peace or conflict, immigrants prefer. Lemma 4 Immigrants weakly prefer peace to conflict. Proof. The proof is identical to the proof of Proposition 2. We have now established that voters optimal policies (both natives and immigrants ) are one of the following three: x {1, α}, y {0, α} for any α [0, 1], and z {1, α}. Notice that natives optimal policies are either x or y, while immigrants optimal policies are either y or z. Let us define N x as the set of natives who prefer redistribution, that is the number of natives who prefer x to y. Denote by n x the number of natives in the subset N x. They are actually natives who care strongly enough about their specific public service, i.e. β i n β p. Finally, let us denote by N y the set of natives who prefer no redistribution, and denote its cardinality by n y = n n x. From here, the characterization of the equilibrium of the electoral competition stage will involve conditions on both the distribution of voters and the intensity their preferences over these three policies. First, since m < n, the policy z cannot be the optimal policy for a majority of voters. On the other hand, if no group of natives with the same optimal policy constitutes a majority of voters immigrants may join with any of two and obtain the majority of votes. This enables us to state the following Lemma. Proposition 4 If the proportion of natives with optimal policy x constitutes a majority of the population then both parties will propose that policy in equilibrium. Otherwise, if immigrants do not care strong enough about their specific public service, i.e. β m < β p m, both parties will propose y. Proof. The first part of the Lemma is trivial. On the other hand, if no group of natives with the same optimal policy constitutes a majority of n voters, that is h n+m < 1/2 for h = x, y, then immigrants may join with any of two and obtain the majority of votes. Therefore, in equilibrium both parties will propose either x n if x n+m 1/2 or y n if y n+m 1/2. When 13

β m < β p m, we know from (4) that immigrants optimal policy is y. In that case, they align with natives who support no redistribution and altogether constitute a majority. When no native group holds a majority and immigrants care strongly enough about their specific public service, i.e. β m β p m, parties proposals in equilibrium become far less straightforward. In this case, we have that more than a half of voters prefer public services to be provided but the conflict of interests between immigrants and natives persists. Immigrants prefer α and natives prefer α. Moreover, within the native group, some individuals have more intense preferences for the public service than others. So they are willing to accept a smaller share of tax revenues in order to ensure the provision of the public service. Given the intensity of preferences, there is a critical share of tax revenues for both natives and immigrants that makes them indifferent between the provision of the public service and no provision at all. Let β be the highest intensity of preferences for the public service such that natives with at least as intense preferences plus immigrants constitute a majority of the population. More formally, β is the value of β i n such that #{i N x with β i n β} = n m+1 2.Define α as the maximum share of tax revenue that leaves the native with β indifferent between the provision and the no provision of the public service, that is α = 1 y n. Y β Observe that α > α since α = 1 yn Y β and β p < β. p For the case of immigrants, define α as the minimum share of tax revenue such that immigrants are indifferent between the provision and the no provision of the public service. That is U m (1, α) = U m (y ) α = y m Y β m. Again, since α = ym Y β p m and β m > β p m then α < α. In the electoral competition stage, parties will take into account the support that the different policy combinations obtain. Redistribution with a level of α higher than α will be supported by all immigrants against no redistribution. On the other hand, natives will support redistribution only if they do not need to give away too much to immigrants. Their heterogeneous preferences imply that a new median voter can be defined, a native who is able to swing the election in favor of redistribution if offered a sufficiently 14

large share of the tax revenue, that is 1 α 1 α. By combining all these condition we are able to state the following Proposition. Proposition 5 When no native group holds a majority and immigrants care strongly enough about their specific public service, i.e. β m β p m, then: (i) If α α then both parties propose no redistribution in equilibrium. (ii) If α (α, α) there exists a unique mixed strategy equilibrium in which both parties propose no redistribution, redistribution with α = α and redistribution with α = α with probability 1 3. (iii) If α α then both parties propose redistribution with α = α in equilibrium. Proof. Let us consider first the case in which α α. In this case there is no combination of {1, α} that can beat y : Suppose that a party proposes y. Then a party proposing α > α will obtain votes from immigrants only. If instead, it proposes α α it will receive n x votes, that do not constitute a majority. Finally, if it proposes α (α, α) then it will receive no votes. Hence, both parties will propose y to ensure a tie. Let us now consider the case in which α < α < α. In this case, there is no Nash equilibrium in pure strategies. We have indeed a cycle between policies y, x, and {1, α}. One can show however that there exists a Nash equilibrium in mixed strategies in which both parties propose these three policies with probability equal to 1 3. We now show that parties are indifferent all these three strategies. Suppose that party l proposes y. Then, with probability 1 3 party k proposes y to and they both tie, with probability 1 3 party k proposes x and then party l gets n y + m votes and wins. Finally, with probability 1 3 party k proposes {1, α} and gets n x + m votes, so l loses. So l s expected payoff is then 1 2. Similar calculations show that l obtains the same expected payoff if proposes x or {1, α}. Secondly, we show that given this randomization any other pure strategy gives a lower payoff. Suppose on the one hand that party l selects a policy with τ = 1 and α (α, α). Then party l loses when party k proposes y or x and wins otherwise, so its expected payoff is 1 3. If party l selects a policy with τ = 1 and α (α, α] then it loses if party k proposes x and {1, α} because all natives will join against it. Its expected payoff is thus 1 3. Finally, in the special subcase in which α < α, if l proposes a policy with τ = 1 and α (α, α] it always loses. This proves that the strategy profile in which bot parties propose policies y, x and {1, α} with probability 1 3 constitutes a Nash equilibrium of this case. Let us finally consider the case in which α α. In this case, both parties will propose x. On the contrary, suppose that a party proposes y instead. Then, it will receive only n y votes that do not constitute a majority. Suppose 15

that it proposes {1, α} with α > α. Then all immigrants will vote for it, since they will prefer that proposal to x but all natives, n x and n y, will support the latter. Hence, the party will lose. Two factors contribute to these different cases: The relative intensity of immigrants preferences for the public service β m and their income level y m. For intermediate levels of such that we are in case (i) there is a stark conflict of interest between immigrants and natives who support redistributions. None of the two groups are willing to give away a sufficiently big share of the tax revenues to make their demands compatible with a supporting majority. Hence, no redistribution prevails. This is also the case if immigrants income is relatively high. When β m is high, we are in case (ii) and a compatible policy proposal exists, i.e. {1, α}. However, the fact that this proposal will be blocked by all natives and that immigrants can swing the election in favor of no redistribution against x generates a cycle. Parties will not choose just one policy since they must content somehow all social groups, all of them being decisive. Finally, if immigrants have very strong preferences for the public service or they are sufficiently poor, the most preferred policy of natives who support redistribution will prevail. This is not necessarily bad for immigrants since in this case it means that natives are giving them a relatively high share of tax revenues in order to avoid conflict. One last word regarding the comparison between the enfranchisement and the benchmark case. Enfranchisement can make a difference only when the number of natives who support either no redistribution or redistribution with α do not constitute a majority. As expected, if in the benchmark no redistribution took place, enfranchisement will introduce redistribution if immigrants preferences for the public service are high or very high (cases (ii) and (iii)). When redistribution already took place without enfranchisement, the intensity of immigrants preferences creates different effects: If they are low or intermediate (so we are either in Proposition (4) or in case (i) of Proposition (5)) immigrants will swing the election in favor of no redistribution. Their conflict of interests with natives who support redistribution is too acute for τ = 1 to prevail. This is reminiscent of the anti-solidarity effect described by Lee et al. (2006). As mentioned, enfranchisement with a high β m makes the three groups of voters decisive. Hence, parties implement higher levels of redistribution and no redistribution at all with positive probability. Finally, if immigrants have very intense preferences they will always align with natives who support redistribution so enfranchisement does not change anything. 16

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