Interregional transfers, group loyalty and the decentralization of redistribution

Abstract

We study the relative merits of centralized and decentralized redistribution in a political economy context assuming cross-regional heterogeneity in average income and identity. While centralizing redistribution allows to pool and redistribute resources at the country level, it may decrease the degree of solidarity in the society as a result of group loyalty. We show that total welfare maximization is closely linked to the minimization of income inequality within and between regions. Analyzing separately two particular cases under direct democracy—no interregional inequality and no group loyalty—we stress the existence of a scope effect and a pooling effect of centralized redistribution, respectively. In both cases, centralization welfare-dominates decentralization, from which it follows that the rationale for decentralizing redistribution only arises when the two sources of cross-regional heterogeneity interact.

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Fig. 1

Notes

  1. 1.

    “Redistribution policy” is meant here as assistance to the poor. That is, we focus on direct redistribution in the form of monetary transfers, such as pension or unemployment benefits.

  2. 2.

    In Belgium, for instance, the political landscape is clearly divided between the opponents and the partisans of a regionalization of social security competencies. In general, Flemish political parties push for organizing solidarity at the local level, while the French-speaking actors unanimously oppose it (Dandoy and Baudewyns 2005).

  3. 3.

    This finding—the fact that group loyalty tends to foster decentralization pressures in relatively richer regions—is consistent with recent empirical findings: Amat and Donnelly (2012) examine the link between Catalan identity and preferences for decentralization in Spain, and provide strong evidence for the proposition that strong Catalan identity causes support for (lower interregional redistribution and) more decentralization. Likewise, using survey evidence for the Basque country and Catalonia, Costa-i-Font (2010) finds that both fiscal imbalance and regional identity exert a robust effect on preferences for vertical state downscaling. For the case of Belgium, Dandoy and Baudewyns (2005) report survey evidence showing that around 55 per cent of the Flemish population feeling “only Flemish or more Flemish than Belgian” favor the regionalization of social security, against roughly 27 per cent for those feeling “only Belgian or more Belgian than Flemish”.

  4. 4.

    Alternative ways of introducing social preferences into the standard model of voting on redistribution (Meltzer and Richard 1981) have been considered in the literature, such as two-sided self-centered inequality aversion à laFehr and Schmidt (1999) (e.g., Dhami and al-Nowaihi 2010a, b; Tyran and Sausgruber 2006), Rawlsian altruism (e.g., Galasso 2003), or desert-sensitive altruism (e.g., Luttens and Valfort 2012).

  5. 5.

    Luttmer (2001) demonstrates that group loyalty is an important factor in explaining the support for redistribution programs. Using data from a social survey in the US in which the respondents report their attitudes on redistribution, the author shows that there is less support for redistribution programs in ethnically fragmented societies than in ethnically more homogenous ones. Using a wide cross-section of countries, Desmet et al. (2009) show that linguistic diversity has both a statistically and economically significant negative effect on redistribution. Likewise, Alesina et al. (2001) show that racial animosity in the US makes redistribution to the poor, who are disproportionately black, unappealing to many voters. Dahlberg et al. (2012) match Swedish data on refugee placement to panel survey data on inhabitants of the receiving municipalities to estimate the causal effects of increased immigrant shares on preferences for redistribution. The results show that a larger immigrant population leads to less support for redistribution in the form of preferred social benefit levels. Alesina et al. (1999) show that ethnic diversity tends to reduce both the supply of public goods and redistribution. Alesina and La Ferrara (2000) show that more homogeneous communities (in terms of income and race) have a higher level of social interactions leading to more social capital, which in turn influences economic outcomes and public policies. For a survey on the costs and benefits of ethnic fragmentation and the policy issues arising in diverse societies, see Alesina and La Ferrara (2005).

  6. 6.

    Shayo (2009) develops a general theoretical framework of social identification and applies it to a model of redistribution where individuals have a choice between identifying with their social class (rich or poor) and a common nationalist identity. Lindqvist and Östling (2013) study the interaction between individuals’ identity choices and redistribution, and show that redistribution is highest when society is ethnically homogeneous, while the effect of ethnic diversity on redistribution is not necessarily monotonic. More closely related to our analysis, Holm and Geys (2013) develop a theoretical framework linking jurisdictional identification and preferences for redistribution within a federation. They show that federal, rather than local, identification can lead individuals to shift their redistribution preferences against their self-interest.

  7. 7.

    The usual way to introduce an efficiency cost of taxation is to assume that the lump-sum transfer to individuals is given by \((t-t^2/2)\overline{y}\), and this typically allows to avoid corner solutions for the derivation of the preferred individual tax rates (e.g., Bolton and Roland 1997). Here, as we will assume that private utilities are strictly concave, together with altruism, we are guaranteed to have an interior solution for the equilibrium tax rates even without this assumption. For the sake of simplicity, and as our focus is entirely on distributional issues, we assume away the distortion that taxation involves for the individual choices of labor supply.

  8. 8.

    Although the uniformity assumption under centralization has been challenged on empirical and theoretical grounds (e.g., Besley and Coate 2003; Lockwood 2002), we believe that it remains appropriate in our setup. Indeed, as here the purpose of policy is pure redistribution, it is natural to assume a rule of horizontal equity within the geographical area in which the policy is implemented. That is, individuals should be treated equally under centralization regardless of the region they belong to.

  9. 9.

    Observe that the utility function in (5) is such that individuals value both the level and distribution of income in the two regions. This is consistent with recent experimental findings according to which individuals are more altruistic towards an ingroup match, and also more likely to choose welfare-maximizing actions when matched with an ingroup member (Chen and Li 2009).

  10. 10.

    Given that individuals exhibit altruistic preferences, we do not need the poor to be decisive to have positive redistribution in equilibrium. Notice, furthermore, that if the poor were decisive—and given that there is no efficiency loss from taxation—the equilibrium level of redistribution would be such that the poor end up consuming strictly more than the rich, which is unrealistic.

  11. 11.

    See also Romer (1975) and Roberts (1977).

  12. 12.

    That is, the final utility of a poor individual in region i is given by

    $$\begin{aligned} U_{i}^{P}=u\left( c_{i}^{P}\right) +\alpha \left\{ \beta \left[ \theta _i^Pu\left( c_{i}^{P}\right) +\theta _i^Ru\left( c_{i}^{R}\right) \right] +( 1-\beta ) \left[ \theta _j^Pu\left( c_{j}^{P}\right) +\theta _j^Ru\left( c_{j}^{R}\right) \right] \right\} \end{aligned}$$
  13. 13.

    Including an efficiency cost of taxation would clearly matter for aggregate welfare, as total consumption would be decreasing in t. Choosing the optimal tax rate would then imply solving a trade-off between efficiency and inequality. In particular, \(t^{*}\) would be such that \(c^{R}(t^*)>c^{P}(t^*)\) and thus \(\overline{c}_{A}\ne \overline{c}_{B}\). As mentioned earlier, our focus here is entirely on distributional issues.

  14. 14.

    Indeed, observe that if \(\theta ^P_A=\theta ^P_B\), it holds that \(t_A^*=t_B^*=t^*\) and \(W(t^*)=W(t_A^*,t_B^*)\).

  15. 15.

    In fact, one can show that if the social planner could implement interregional transfers under decentralization, he would choose \(t_A=t_B\) and a positive transfer from the rich to the poor region such that inequality within and between regions is fully eliminated. Thus, the social planner’s solutions under centralization and decentralization with transfers are equivalent in terms of welfare.

  16. 16.

    Indeed, if region i is richer than region j (i.e., \(\theta _i^P<\theta _j^P\)), any increase in taxation under centralization benefits relatively more region j, since it has more welfare recipients. In this case, therefore, an increase in group loyalty reduces the preferred tax rate of the rich individuals in the rich region (i.e., \(\partial t_i^C/\partial \beta <0\)), while it increases the preferred tax rate of the rich individuals in the poor region (i.e., \(\partial t_j^C/\partial \beta >0\)). In the absence of interregional transfers (i.e., \(\theta _i^P=\theta _j^P\)), in contrast, the preferred centralized tax rate of any rich individual does not depend on the strength of group loyalty (i.e., \(\partial t_i^C/\partial \beta =0\) for \(i=A,B\) and thus \(t_A^C=t_B^C\) for all \(\beta \in \left[ 1/2,1\right] \)).

  17. 17.

    In fact, if the rich individuals in region i could choose \(t_j\) under decentralization, they would de facto act as a social planner. Indeed, as the rich individuals in region i do not have to pay the tax in region j, they would choose the tax rate that fully eliminates inequality within region j.

  18. 18.

    Observe, however, that for \(\beta =1/2\) the average consumption of the poor is strictly higher under centralized redistribution (i.e., \(c^P(t)>\left[ c_A^P(t_A)+c_B^P(t_B)\right] /2\)). Therefore, although it is unclear whether a rich individual is willing to pay higher taxes under centralization, the level of redistribution is indeed higher.

  19. 19.

    Furthermore, when the strength of altruism and/or group loyalty differs across regions, the social planner solution under centralization typically involves some inequality among individuals (and thus between regions). In general, under the social planner approach, centralizing redistribution need not always be welfare-superior when allowing for those additional sources of cross-regional heterogeneity.

  20. 20.

    See, for instance, Musgrave (1959), Epple and Romer (1991), Wildasin (1994), and Dixit and Londregan (1998).

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Correspondence to Sabine Flamand.

Additional information

I wish to thank Caterina Calsamiglia and Hannes Mueller for helpful discussions, comments and suggestions. Financial support by the Spanish Ministry of Science and Innovation through Grant ECO2008-04756 (Grupo Consolidado-C) is acknowledged.

Appendix

Appendix

Proof of Proposition 1

\(\frac{\partial t_i^C}{\partial \beta }=\frac{\alpha }{\left( 1+\alpha \right) }\left( \theta _i^P-\theta _j^P\right) \left[ 1+\frac{\left( \theta _{i}^{P}+\theta _{j}^{P}\right) y^{P}}{\left( \theta _{i}^{R}+\theta _{j}^{R}\right) y^{R}}\right] >0\) if and only if \(\theta _i^P>\theta _j^P\) \(\square \)

Proof of Proposition 2

Assume that \(\theta _{A}^{P}\ne \theta _{B}^{P}\). It follows that \(W(t_{A}^{*},t_{B}^{*})>W(t^{*})\) if and only if

$$\begin{aligned} \theta _{A}^{R}\ln \frac{c_{A}^{R}(t_{A}^{*})}{c^{R}(t^{*})}+\theta _{A}^{P}\ln \frac{c_{A}^{P}(t_{A}^{*})}{c^{P}(t^{*})}+\theta _{B}^{R}\ln \frac{c_{B}^{R}(t_{B}^{*})}{c^{R}(t^{*})}+\theta _{B}^{P}\ln \frac{c_{B}^{P}(t_{B}^{*})}{c^{P}(t^{*})}>0 \end{aligned}$$

We know that \(t_{i}^{*}\) is such that \(c_{i}^{P}(t_{i}^{*})=c_{i}^{R}(t_{i}^{*})=\overline{c}_{i}\) for \(i=A,B\). Furthermore, \(t^{*}\) is such that \(c^{P}(t^{*})=c^{R}(t^{*})=\overline{c}\) in both regions. Therefore, \(W\left( t_{A}^{*},t_{B}^{*}\right) >W\left( t^{*}\right) \) if and only if \(\overline{c}_{B}\overline{c}_{A}>\overline{c}^{2}\), or, equivalently, \((1-t_{A}^{*})(1-t_{B}^{*})>(1-t^{*})^{2}\). Substituting, this condition becomes

$$\begin{aligned} \left[ \frac{(\theta _A^P-\theta _B^P)(y^R-y^P)}{4}\right] ^2<0 \end{aligned}$$

which never holds. Therefore, total welfare is strictly higher under the centralized solution. \(\square \)

Proof of Proposition 3

Assume that \(\theta _A^P=\theta _B^P=\theta ^P\). In that case, we know that \(U^{i}_A(t_{A},t_{B})=U^{i}_B(t_{A},t_{B})=U^{i}(t_{A},t_{B})\) and \(U^{i}_A(t)=U^{i}_B(t)=U^{i}(t)\) for \(i=R,P\) (i.e., final utility levels are the same across regions under both centralization and decentralization). An individual i is better off under decentralization if and only if \(U^{i}\left( t_{A},t_{B}\right) >U^{i}\left( t\right) \). When \(\theta _{A}^{P}=\theta _{B}^{P}\) and \(\beta =1\), we know that \(t_A=t_B=t\), and since there are no interregional transfers taking place under centralization, it follows that \(U^{i}\left( t_{A},t_{B}\right) =U^{i}\left( t\right) \), that is, all individuals in the economy are indifferent between centralized and decentralized redistribution. Then, as

$$\begin{aligned} \frac{\partial U^{i}\left( t\right) }{\partial \beta }= & {} 0 \quad \hbox {for} \quad i=R,P\\ \frac{\partial U^{P}( t_{A},t_{B})}{\partial \beta }= & {} \frac{\theta ^P(1+\alpha )+\theta ^R(1+\alpha \beta )}{\beta (1+\alpha \beta )\left[ \theta ^P+(1+\alpha \beta )\theta ^R\right] }>0 \\ \frac{\partial U^{R}( t_{A},t_{B})}{\partial \beta }= & {} \frac{\alpha (1-\beta )\theta ^P}{\beta (1+\alpha \beta )\left[ \theta ^P+(1+\alpha \beta )\theta ^R\right] }>0 \end{aligned}$$

it follows directly that all individuals are strictly better off under centralized redistribution for \(\beta <1\). \(\square \)

Proof of Proposition 4

If \(\beta =1/2\), the preferred centralized tax rate in region \(i=A,B\) is given by

$$\begin{aligned} t_i^C=\frac{1}{\left( 1+\alpha \right) }\left\{ \frac{\alpha }{2} \left( \theta _i^P+\theta _j^P \right) -\frac{y^{P}}{y^{R}}\frac{\left( \theta _i^P+\theta _j^P\right) }{\left( \theta _i^R+\theta _j^R\right) }\left[ 1+\frac{\alpha }{2} \left( \theta _i^R+\theta _j^R \right) \right] \right\} \end{aligned}$$

and thus the preferred tax rates of the rich individuals in region A and region B coincide. \(\square \)

Proof of Proposition 5

To prove Proposition 5, we will use the solution of an intermediary institutional arrangement between centralization and decentralization, namely decentralization with voluntary transfers.

Suppose that after the regional tax rates have been implemented, the rich region (region A) is allowed to transfer some proportion of its tax revenue to the poor region (region B). The poor individuals’ budget constraints in the two regions are now given by

$$\begin{aligned} c_{A}^{P}(t_A,\delta )= & {} y^{P}+\delta t_{A}y^{R}\frac{\theta _{A}^{R}}{\theta _{A}^{P}}\\ c_{B}^{P}(t_A,t_B,\delta )= & {} y^{P}+t_{B}y^{R}\frac{\theta _{B}^{R}}{\theta _{B}^{P}}+( 1-\delta )t_{A}y^{R}\frac{\theta _{A}^{R}}{\theta _{B}^{P}} \end{aligned}$$

where \(\delta \) is the proportion of the rich region’s tax revenue that remains in the region, and is to be determined endogenously. We consider a two-stage game: in the first stage, the regional tax rates are chosen, and in the second stage, the representative rich individual in the rich region chooses \(\delta \). We solve the game by backward induction.

In the second stage, the rich individual in region A chooses \(\delta \) given \((t_A,t_B)\) so as to maximize \(U_A^R(t_A,t_B,\delta )\) subject to the above constraints. The first-order condition for this problem is given by

$$\begin{aligned} \beta u^{\prime }\left( c_{A}^{P}\right) =(1-\beta )u^{\prime }\left( c_{B}^{P}\right) \end{aligned}$$

Assuming that \(u(c)=\ln c\), we get the transfer \((1-\delta )\) as a function of the regional tax rates. In the absence of group loyalty (i.e., \(\beta =1/2\)), the transfer reduces to

$$\begin{aligned} (1-\delta (t_A,t_B))=\frac{\theta _A^R\theta _B^Pt_A-\theta _B^R\theta _A^Pt_B}{\theta _A^R\left( \theta _A^P+\theta _B^P\right) t_A} \end{aligned}$$

Substituting for \(\delta (t_{A},t_{B})\) into \(U_{A}^{R}(t_{A},t_{B},\delta )\) and \(U_{B}^{R}(t_{A},t_{B},\delta )\), and solving the first stage of the game, we get the reaction functions \(t_A(t_B)\) and \(t_B(t_A)\). Finally, substituting for \(t_{A}\left( t_{B}\right) \) and \(t_{B}\left( t_{A}\right) \) into one another, we can finally solve for \(\left( t_{A},t_{B},\delta \right) \). For \(\beta =1/2\), this yields:

$$\begin{aligned} t_A= & {} \frac{\left[ \left( 2+\alpha \theta _A^R\right) \theta _B^P-\left( 2-\alpha \theta _A^R\right) \theta _A^P\right] y^R-\left( 2+\alpha \theta _A^R\right) \left( \theta _A^P+\theta _B^P\right) y^P}{2(2+\alpha )\theta _A^Ry^R}\\ t_B= & {} \frac{\left[ \left( 2+\alpha \theta _B^R\right) \theta _A^P-\left( 2-\alpha \theta _B^R\right) \theta _B^P\right] y^R-\left( 2+\alpha \theta _B^R\right) \left( \theta _A^P+\theta _B^P\right) y^P}{2(2+\alpha )\theta _B^Ry^R}\\ \delta= & {} \frac{\alpha \theta _A^P\left( \theta _A^R+\theta _B^R\right) (y^R-y^P)-4\theta _A^P}{\left[ \alpha \theta _A^P\left( \theta _B^R-\theta _A^P\right) -2\theta _A^P+(2+\alpha )\theta _B^P\right] y^R-\left( 2+\alpha \theta _A^R\right) \left( \theta _A^P+\theta _B^P\right) y^P} \end{aligned}$$

Let \(\theta _A^P=\gamma \theta _B^P\), where \(\gamma \in [0,1)\) (i.e., region A is the rich region). Comparing total welfare under centralization and decentralization with transfers, we have that \(W(t)-W(t_A,t_B,\delta )>0\) if and only if

$$\begin{aligned} A= & {} \gamma \theta _B^P\ln \left( \frac{c_A^P(t)}{c_A^P(t_A,t_B,\delta )}\right) +\left( 1-\gamma \theta _B^P\right) \ln \left( \frac{c_A^R(t)}{c_A^R(t_A,t_B,\delta )}\right) \\&+\,\theta _B^P\ln \left( \frac{c_B^P(t)}{c_B^P(t_A,t_B,\delta )}\right) +\left( 1-\theta _B^P\right) \ln \left( \frac{c_B^R(t)}{c_B^R(t_A,t_B,\delta )}\right) >0 \end{aligned}$$

Rearranging yields

$$\begin{aligned} A= & {} \theta _B^P\ln \left( \frac{c_B^P(t)/c_B^P(t_A,t_B,\delta )}{c_B^R(t)/c_B^R(t_A,t_B,\delta )}\right) +\gamma \theta _B^P\ln \left( \frac{c_A^P(t)/c_A^P(t_A,t_B,\delta )}{c_A^R(t)/c_A^R(t_A,t_B,\delta )}\right) \\&+\ln \left( \frac{c_B^R(t)}{c_B^R(t_A,t_B,\delta )}\frac{c_A^R(t)}{c_A^R(t_A,t_B,\delta )}\right) \end{aligned}$$

Substituting for equilibrium values and for \(\beta =1/2\), we get

$$\begin{aligned} A= & {} \theta _B^P\ln \left( \frac{\left[ 2+\alpha \left( 1-\theta _B^P\right) \right] \left[ 2-\theta _B^P(1+\gamma )\right] }{\left( 1-\theta _B^P\right) \left[ 2+\alpha \left( 2-\theta _B^P(1+\gamma )\right) \right] }\right) \\&+\gamma \theta _B^P\ln \left( \frac{\left[ 2+\alpha \left( 1-\gamma \theta _B^P\right) \right] \left[ 2-\theta _B^P(1+\gamma )\right] }{\left( 1-\gamma \theta _B^P\right) \left[ 2+\alpha \left( 2-\theta _B^P(1+\gamma )\right) \right] }\right) \\&+\ln \left( \frac{(2+\alpha )^2\left( 1-\theta _B^P\right) \left( 1-\gamma \theta _B^P\right) \left[ 2+\alpha \left( 2-\theta _B^P(1+\gamma )\right) \right] ^2}{(1+\alpha )^2\left[ 2+\alpha \left( 1-\theta _B^P\right) \right] \left[ 2+\alpha \left( 1-\gamma \theta _B^P\right) \right] \left[ 2-\theta _B^P(1+\gamma )\right] ^2}\right) \end{aligned}$$

Taking the derivative of this expression with respect to \(\alpha \) yields

$$\begin{aligned} \frac{\partial A}{\partial \alpha }= & {} \frac{2}{\alpha ^2}\left[ \frac{2+3\alpha }{2+\alpha (3+\alpha )}-\frac{2}{2+\alpha \left( 1-\theta _B^P\right) }-\frac{2}{2+\alpha \left( 1-\gamma \theta _B^P\right) }\right. \\&\qquad \left. +\frac{2}{2+\alpha \left[ 2-\theta _B^P(1+\gamma )\right] }\right] <0 \end{aligned}$$

Therefore, if \(A>0\) for \(\alpha =1\), it holds that \(A>0\) for all \(\alpha \in (0,1)\). Let \(\alpha =1\), the above expression becomes

$$\begin{aligned} A= & {} \theta _B^P\ln \left( \frac{\left( 3-\theta _B^P\right) \left[ 2-\theta _B^P(1+\gamma )\right] }{\left( 1-\theta _B^P\right) \left[ 4-\theta _B^P(1+\gamma )\right] }\right) +\gamma \theta _B^P\ln \left( \frac{\left( 3-\gamma \theta _B^P\right) \left[ 2-\theta _B^P(1+\gamma )\right] }{\left( 1-\gamma \theta _B^P\right) \left[ 4-\theta _B^P(1+\gamma )\right] }\right) \\&+\ln \left( \frac{9\left( 1-\theta _B^P\right) \left( 1-\gamma \theta _B^P\right) \left[ 4-\theta _B^P(1+\gamma )\right] ^2}{4\left( 3-\theta _B^P\right) \left( 3-\gamma \theta _B^P\right) \left[ 2-\theta _B^P(1+\gamma )\right] ^2}\right) \end{aligned}$$

Letting \(\gamma =0\) yields

$$\begin{aligned} A=\theta _B^P\ln \left( \frac{6-\theta _B^P\left( 5-\theta _B^P\right) }{4-\theta _B^P\left( 5-\theta _B^P\right) }\right) +\ln \left( \frac{3\left( 4-\theta _B^P\right) ^2\left( 1-\theta _B^P\right) }{4\left( 2-\theta _B^P\right) ^2\left( 3-\theta _B^P\right) }\right) >0 \end{aligned}$$

Given that for \(\alpha =1\), we have that

$$\begin{aligned} \frac{\partial A}{\partial \gamma }=\theta _B^P\left[ \frac{2\left( 1-\theta _B^P\right) }{\left( 3-\gamma \theta _B^P\right) \left[ \theta _B^P(1+\gamma )-4\right] }+\ln \left( \frac{\left( 3-\gamma \theta _B^P\right) \left[ 2-\theta _B^P(1+\gamma )\right] }{\left( 1-\gamma \theta _B^P\right) \left[ 4-\theta _B^P(1+\gamma )\right] }\right) \right] >0 \end{aligned}$$

It follows that for \(\alpha =1\), it holds that \(A>0\) for all \(\gamma \in (0,1)\). Therefore, \(A>0\) for all \(\alpha \in (0,1)\). Hence, it holds that \(W(t)-W(t_A,t_B,\delta )>0\) in the absence of group loyalty (i.e., \(\beta =1/2\)).

Comparing now total welfare under decentralization with and without transfers for \(\beta =1/2\), we have that

$$\begin{aligned} W(t_{A},t_{B},\delta ) -W(t_{A},t_{B})=(1+\alpha )n\ln \left[ \frac{\left[ \left( \theta _{A}^{P}+\theta _{B}^{P}\right) y^{P}+\left( \theta _{A}^{R}+\theta _{B}^{R}\right) y^{R}\right] ^2}{4\left( \theta _{A}^{P}y^{P}+\theta _{A}^{R}y^{R}\right) \left( \theta _{B}^{P}y^{P}+\theta _{B}^{R}y^{R}\right) }\right] >0 \end{aligned}$$

and so, by transitivity, \(W(t)>W(t_{A},t_{B})\) in the absence of group loyalty. \(\square \)

Proof of Proposition 6

As the decisive individual is partly self-interested, he always implements a tax rate such that the rich consume strictly more than the poor under both centralization and decentralization. Indeed, we have that

$$\begin{aligned} c^R(t)-c^P(t)= & {} \frac{\big \{\left[ 1+\alpha (1-2\beta )\right] \theta ^P_A+\left[ 1-\alpha (1-2\beta )\right] \theta ^P_B\big \}\left[ \left( \theta _A^P+\theta _B^P\right) y^P+\left( \theta ^R_A+\theta ^R_B\right) y^R\right] }{(1+\alpha )\left( \theta _A^P+\theta _B^P\right) \left( \theta ^R_A+\theta ^R_B\right) }{>}0\\ c_i^R(t_i)-c_i^P(t_i)= & {} \frac{\theta _i^Py^P+\theta _i^Ry^R}{(1+\alpha \beta )\theta _i^R}>0\quad \hbox {for} \quad i=A, B \end{aligned}$$

As total welfare under decentralization is maximized for \(t_i^*\) such that \(c_{i}^{R}=c_{i}^{P}\) for \(i=A, B\), and as

$$\begin{aligned} \frac{\partial W(t_i,t_j)}{\partial t_i}=(1+\alpha )ny^R\left[ \frac{\theta ^P_i\theta ^R_i}{\theta ^P_iy^P+\theta ^R_it_iy^R}-\frac{\theta ^R_i}{(1-t_i)y^R}\right] >0\quad \hbox {for} \quad t_i<t_i^* \end{aligned}$$

the result follows directly from the fact that \(\partial t_{i}/\partial \beta >0\) for \(i=A, B\). Similarly, as total welfare under centralization is maximized for \(t^*\) such that \(c^{R}=c^{P}\), and as

$$\begin{aligned} \frac{\partial W(t)}{\partial t}= & {} (1+\alpha )ny^R\left[ \frac{\theta ^R_A\theta ^P_A}{\theta ^P_Ay^P+\theta ^R_Aty^R}+\frac{\theta ^R_B\theta ^P_B}{\theta ^P_By^P+\theta ^R_Bty^R}-\frac{\theta ^R_A+\theta ^R_B}{(1-t)y^R}\right] \\> & {} 0 \quad \hbox {for}\quad t<t^* \end{aligned}$$

the result follows directly from the fact that \(\partial t/\partial \beta <0\). \(\square \)

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Flamand, S. Interregional transfers, group loyalty and the decentralization of redistribution. Econ Gov 16, 307–330 (2015). https://doi.org/10.1007/s10101-015-0169-6

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Keywords

  • Redistribution
  • Decentralization
  • Group loyalty
  • Inequality
  • Identity

JEL classification:

  • H77
  • D64
  • H23