# Internal Migration and Public Policy

• Michele G. Giuranno and Rongili Biswas

## Abstract

This paper studies the relation between internal migration and public spending on public goods. We describe centralized public policy when a central government is comprised of elected representatives from local electoral districts. Internal migration determines the median voter in the districts. The median voters decide the equilibrium policy through bargaining. We find the conditions under which voters’ mobility results in larger or smaller public spending. Furthermore, the distance between the actual size and the efficient size of government spending depends on the way internal migration changes the distribution of income within and between districts.

JEL Classification: D30; D78; H0; H41; H50

# Appendix

A

Proof of Lemma 1. Denote by F the first order condition eq. (13),

(39)F=γ1m+Hgγ1mg+Hg+γ2m+Hgγ2mg+Hg=0.

We want to study dgdmFmFg. It is straightforward to verify that the second order condition is negative, Fg<0, while the numerator is

(40)Fm=γ1mϕ1+γ1mgϕ1gϕ12+γ2mϕ2+γ2mgϕ2gϕ22.

After rearranging we get

(41)Fm=gHgHgγ1mϕ12+γ2mϕ22.

Here, gHgHg is negative because the marginal benefit is smaller than the average benefit, i.e. Hg<Hg/g.[27] We conclude that Fm is positive when γ1mϕ12+γ2mϕ22 is negative. This proves the Lemma.

Proof of Proposition 4. In order to prove the proposition, we first show that the bargaining solution leads to the efficient solution when γ1=γ2=1. In this case, the bargaining first order condition eq. (13) becomes 21+Hgg+Hg=0. This is satisfied when Hg=1, as in eq. (5). Second, consider the case yy1y2. The efficiency condition eq. (5) does not change when the distribution of the electorate changes between regions. On the contrary, condition eq. (19) shows that the provision of g increases as the median mean income ratios decline for both median voters. Third, consider the case y1yy2. The impact on g of moving away from the situation γ1=γ2=1 is explained by Proposition 2. Therefore g may either increase or decrease.

Proof of Lemma 2. Denote by V the first order condition eq. (32),

(42)V=j=1Jγjm+Hgγjmg+Hg=0.

We want to study dgdmVmVg. It is straightforward to verify that the second order condition is negative, Vg<0, while the numerator is

(43)Vm=j=1Jγjmϕj+γjmgϕjgϕj2.

After rearranging we obtain

(44)Vm=gHgHgj=1Jγjmϕj2

where, we already know that gHgHg is negative. We conclude that Vm is positive when j=1Jγjmϕj2 is negative.

Proof of Proposition 6. In order to prove the proposition, we need to show that the bargaining solution with J > 2 jurisdictions leads to the efficient solution when γj=1 for all j = 1, ..., J. In this case, the bargaining first order condition eq. (32) becomes J1+Hgg+Hg=0, which is satisfied when Hg=1, as in eq. (5).

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Published Online: 2019-06-15

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