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I've been reading the book 'The Size of Nations' by Alberto Alesina and Enrico Spolaore (can be found on the net if you know where to look) and I'm having trouble following their "proof" of the first lemma for the overlapping jurisdictions model. First, I'll explain the model and the lemma, second I'll provide their proof and finally my (incomplete) attempt at the proof of the lemma with the questions I have. A similar model can be found in a paper of Spolaroe here: https://isites.harvard.edu/fs/docs/icb.topic107502.files/Alesina_and_Spolaore.On_the_Number_and_Size_of_Nations.pdf (not necessary)

The Model

In the model, the 'world' is a linear segment of length normalized to 1. There is also a mass of "identical" (except in location, so we can add up their utilities) individuals 1 uniformly distributed along the segment (one individual in each point of the segment). There are $M$ public goods, indexed by $j = 1,2,...,M_{j}$ Each good is available on a continuum of types. If a good $j$ is located at point $w ∈ (0,1)$ along the segment we call the type of good $j$ a type $w$ good (note: this is not a different good, it is a different type of the same good). Jurisdictions provide public goods. A jurisdiction for the $jth$ public good is defined by three points on the segment: for example, $A$, $B$, and $C$, with $A < B < C$. The middle point $B$ indicates where the public good is located; the other two are the borders of the jurisdiction. A jurisdiction that provides public good $j$ is defined as a $j$-level jurisdiction. Also, there are no externalities.

The utility of the $i$th individual is denoted by: $$ u_{i} = y - t_{i} + g - \sum_{j=1}^{M} a_{j}*l_{ij} $$ $y$ in the individual's gross income, $g$ the maximum value derived from the public goods for an individual who enjoys all of his most favored types. Let $t_{i}$ denote taxes payed by the $i$th individual. Alessina and Spolaore set $g$ = 0 for simplicity, without loss of generality (we can do the same for $y$ since it's an exogenous parameter). $l_{ij}$ denotes the distance from the $i$th individual to the $j$th public good. $a_{j}$ is a positive parameter that measures the marginal cost of distance. Each $j$th public good’s total cost $c_{j}$ in a certain jurisdiction is given by: $$ c_{j} = k_{j} + γ_{j}*s, $$ where $s$ is the size of the jurisdiction, $k_{j}$ is a fixed cost that is independent of the jurisdiction’s size, and $γ_{j}$ is a positive parameter (marginal cost of size). Total taxes in a $j$-level jurisdiction with borders at $A$ and $C$ must equal $c_{j}$ (balanced budget): $$ \int_{A}^{C} t_{i} di = c_{j}. $$ Therefore, the sum of everyone's utility is given by: $$ \int_{0}^{1} u_{i} di = y - \sum_{j=1}^{M} \left ( k_{j} * N_{j} + \sum_{x=1}^{N_{j}} \gamma_{j} * s_{jx} + a_{j} * \int_{0}^{1} l_{ij} di \right ) $$ where $N_{j}$ is the number of $j$-level jurisdictions and $s_{jx}$ is the size of a $j$-level jurisdiction $x$ for $x = 1,2,...,N_{j}$. Consider the problem of a utilitarian social planner that intends to maximize the sum of individual utilities defined as above.

Lemma

For every public good $j$, the social planner divides the world into $N_{j}$ jurisdictions of equal size $s_{j} = 1/N_{j}$ ,and locates at the middle of each jurisdiction a public good $j$. Nj is the integer close to: $$ \sqrt{a_{j}/4k_{j}} $$

Alesina and Spolaroe's "proof" of the Lemma (2.1 in the book)

The sum of utilities to be maximized is given by [the authors have dropped some notation]: $$ \int_{0}^{1} u_{i} di = y − γ −\sum_{j=1}^{M} \left [k_{j}*N_{j} + a_{j} *\int_{0}^{1} l_{ji} \right ] $$ For public good $j$ and for a given number of jurisdictions $N_{j}$, the sum of distances $\int_{0}^{1}l_{ji}$ is minimized if the public good is located at the midpoint of each jurisdiction. Hence the sum of distances is given as $\int_{0}^{1}l_{ji} = \sum_{x=1}^{N_{j}}s_{xj}^{2}/4$, where $\sum_{x=1}^{N_{j}}s_{xj} = 1$.The sum of squares is minimized by choosing jurisdictions of equal size, $s_{j} = 1/N_{j}j$. Therefore the solution for each $N_{j}$ is the positive integer that solves: $$ min_{N_{j}}: k_{j}*N_{j} + \frac{a_{j}}{4N_{j}} $$ The first-order condition for $N_{j}$ (ignoring the constraint that $N_{j}$ must be an integer) implies that: $$ N_{j}^{*} = \sqrt{\frac{a_{j}}{4k_{j}}} $$

My attempt

I start from: $$ \int_{0}^{1} u_{i} di = y - \sum_{j=1}^{M} \left ( k_{j} * N_{j} + \sum_{x=1}^{N_{j}} \gamma_{j} * s_{jx} + a_{j} * \int_{0}^{1} l_{ij} di \right ) $$ Let's now forget that there are $M$ public goods (since they do not interact with each other, they are independent, so a general solution for a $j$th public good well be able to apply it to $M$ public goods). We can also set $y$ to 0 without loss of generality. Due to this, the utilitarian social planner problem becomes: $$ min: k_{j} * N_{j} + \sum_{x=1}^{N_{j}} \gamma_{j} * s_{jx} + a_{j} * \int_{0}^{1} l_{ij} di $$ $j$-level jurisdictions do not overlap with each other (only with other levels), since an individual will always want to be to the $j$-level closes jurisdiction available, there are no benefits of being in more than one jurisdiction of the same level ($g$ fixed and set to 0) but there are costs of bigger jurisdictions, so the utilitarian social planner will never design $j$-level jurisdictions that overlap each other. $j$-level jurisdictions also occupy all of the segment since all individuals need to have access to each public good out there (so yeah, don't be a troll saying that the size and number of optimal jurisdictions is 0 since the authors, previously, set $g$ to 0). All of this imples that: $\sum_{x=1}^{N_{j}}s_{xj} = 1$, so the utilitarian social planner problem colapses to (I keep the $j$ subscripts for generality and set $\gamma_{j}$ to 0 since it's a given parameter that is not affected by any choice variables): $$ min: k_{j} * N_{j} + a_{j} * \int_{0}^{1} l_{ij} di $$ I'm stuck here, so I have 3 questions:

  • Is there a way to proof that, the sum of distances $\int_{0}^{1} l_{ji}$ is minimized if the public good is located at the midpoint of each jurisdiction? I understand the logic, but I don't know how to get the proof.

  • Why $\int_{0}^{1} l_{ji} = \sum_{x=1}^{N_{j}} s_{xj}^{2}/4$?

  • Why the sum of squares is minimized by choosing jurisdictions of equal size, $s_{j} = 1/N_{j}$? (proof would be helpful since I, somewhat, get the intuition of why it should be like this, but I'm not sure).

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Is there a way to proof that, the sum of distances $\int_{0}^{1} l_{ji}di$ is minimized if the public good is located at the midpoint of each jurisdiction?

We are told "there are no externalities" -so each individual enjoys only the public good offered in its own jurisdiction.

But this means that in order to determine the optimal position of the public good in each jurisdiction, we can initially consider a single jurisdiction. Then for this subset of individuals we have

$$i \in [A,C] : \int_{A}^{C} l_{ji}di=\int_{A}^{C} |i-w_j|di = \int_{A}^{w_j} (w_j-i)di + \int_{w_j}^{C} (i-w_j)di$$

Performing this integration we get

$$i \in [A,C] : \int_{A}^{C} l_{ji}di= w_j^2 - (A+C)w_j+\frac 12 (A^2+C^2)$$

Minimizing this gives

$$w^*_j = \frac {A+C}{2}$$

But this is the mid-point of the interval, and $A$ and $C$ where arbitrary except for $A < C$. So we conclude that the public good must be located at the midpoint of each jurisdisction.

Why $\int_{0}^{1} l_{ji}di = \sum_{x=1}^{N_{j}} s_{xj}^{2}/4$?

Using the previous result, we have that

$$i \in [A,C] : \min \int_{A}^{C} l_{ji}di= \left(\frac {A+C}{2}\right)^2 - \frac{(A+C)^2}{2}+\frac 12 (A^2+C^2)$$

$$...\implies i \in [A,C] : \min \int_{A}^{C} l_{ji}di = \frac {(A-C)^2}{4} = \frac {(C-A)^2}{4}$$

But the numerator is the squared length of this jurisdiction. We are also told that each public good must be offered in each jurisdiction, and also that all jurisdictions must cover the whole interval, without overlapping (for the $j$-level we are examining). These imply that for the whole unitary interval we get

$$\min \int_{0}^{1} l_{ji}di = \sum_{x=1}^{N_{j}} s_{xj}^{2}/4$$

So it is the total distance after optimizing for the location of the public good in each jurisdiction.

Finally,

Why the sum of squares is minimized by choosing jurisdictions of equal size, $s_{j} = 1/N_{j}$

In general notation you want to $\min \sum_{i=1}^n z^2_i \;\;\;\; s.t. \sum_i^nz_i=1$

The Lagrangian is $\Lambda = \sum_{i=1}^n z^2_i -\lambda \left(\sum_i^nz_i-1\right)$

and we have $n$ first order conditions $2z_i - \lambda = 0 \implies z_i = 2/\lambda,\;\;\;i=1,...,n$

But this means that $z_1=z_2=...=z_n=z^*$

The solution must respect the constraint so $ \implies \sum_i^nz_i = nz^*=1\implies z^*=1/n$

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