Abstract
We consider the problem of producing an impure public good in various jurisdictions formed through the strategic decisions of agents. Our environment inherits two well-known problems: (i) Under individual decisions, there is a tension between stability and efficiency; (ii) Under coalitional decisions, stable jurisdiction structures may fail to exist. The solution, we propose is the use of membership property rights: When a move among jurisdictions is subject to the approval of the agents whom it affects, coalitionally stable jurisdiction structures coincide with those which are efficient.
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Notes
As a famous example in this direction, we have the critique of Bewley (1981) who shows that equilibrium district structures may fail to exist or be inefficient.
Buchanan and Goetz (1972) and Flatters et al. (1974) are among the first to point to the possible inefficiency of free-mobility equilibria. The literature contains many explorations about this particular equilibrium concept, among which we have Richter (1982), Greenberg (1983), Konishi (1996) and Konishi et al. (1998).
We owe this concept to Sertel (1992) whose formulation in an abstract setting has been worked on by Eren (1993) and further treated in detail by Sertel (1998, 2003). The concept has been applied to more structured frameworks, such as the worker-partnership model proposed by Sertel (1982); the pure public good production problem analyzed by Asan and Sanver (2003) and the coalition formation analysis of Ozkal-Sanver (2005), Nizamogullari and Ozkal-Sanver (2011) in matching problems.
Bergstrom et al. (1986) give a detailed analysis of the voluntary contributions solution (VCS) which is well known to pave the way to inefficient allocations. Our choice of VCS as the prevailing allocation rule is rather arbitrary. It can be justified by assuming the non-existence of institutions which implement efficient allocation rules. After all, our interest is not toward the efficiency of the considered public good allocation rule, but toward the efficiency of institutions leading to jurisdiction formations under a given allocation rule. Moreover, we conjecture that our results prevail under any allocation rule which satisfies the population monotonicity condition used by Sertel and Yildiz (1998).
i.e., the entrance to a jurisdiction requires the consent of all members of that jurisdiction.
i.e., the exit from a jurisdiction requires the consent of all members of that jurisdiction.
Bergstrom et al. (1986) show that this will be unique in our assumed environment. From now on we let E be the set of economies (with normal private and public goods) where any \(K\subseteq N\) leads to a voluntary contributions allocation \(V(K)\) with \(y^{*}>0\), i.e., there is at least one agent who contributes to the public budget.
A partition of \(K\) is a finite family \(\{K_{t}\}\) of pairwise disjoint subsets of \(K\) such that \(\bigcup K_{t}=K\). While the usual definition requires each \(K_{t}\) to be non-empty, we relax the definition by treating the partitions \(\{K_{t}\}\) and \(\{K_{t}\}\cup \{\emptyset \}\) equivalently.
Individual stability of \(\pi \in \varPi \) under FE–FX is equivalent to the free mobility equilibrium as defined in Conley and Konishi (2002). Bogomolnaia and Jackson (2002) identify, in a general hedonic setting, conditions that ensure the existence of individually stable partitions. Note that individual stability under FE–FX (resp, AE–FX, AE–AX) is called Nash stability (resp., individual stability, contractual individual stability) by Bogomolnaia and Jackson (2002).
\( T\backslash S\) is empty when \(T\subseteq S\). This violates the usual definition of a partition. However, as noted in Footnote 11, we allow partitions to include the emptyset.
The fact that coalitional moves generalize individual moves does not imply that coalitional stability generalizes individual stability. For, conceiving the move of \(i\) from \(\pi (i)\) to \(L\in \pi \) as the \(\rho -\)move of \(L\cup \{i\}\) imposes an implicit approved entry requirement to the individual move of \(i\), as no member of \(L\cup \{i\}\) must be worsed-off by the \(\rho -\)move of \(L\cup \{i\}\). Note that coalitional stability of \(\pi \in \varPi \) under FE–FX is stronger than the Conley and Konishi (2002) definition of strong Tiebout equilibrium.
We say that \(\pi \in \varPi \) is efficient iff there exists no \(\pi ^{\prime }\in \varPi \) such that \(v_{i}(\pi ^{\prime })\ge v_{i}(\pi ) \forall i\in N\) and \(v_{i}(\pi ^{\prime })>v_{i}(\pi )\) for some \(i\in N\). Due to the finiteness of the society (hence the finiteness of the logically possible partitions), every \(e\in E\) admits an efficient jurisdiction structure.
The intuition behind this claim -which we will justify soon- is related to the seminal work of Buchanan (1965) which determines the optimal size of jurisdictions.
The first part of Theorem 4 below follows from Proposition 3 of Bogomolnaia and Jackson (2002) which shows, in a general hedonic setting, that all efficient coalition structures are “contractually individually stable”. (See Footnote 12).
As a matter of fact, the economies in the proof of Theorem 2 do not fit the definition of crowding-convexity. However, they can easily be rendered crowding-convex (by making \(M\) increase with the number of agents within a coalition) without harming the results they establish.
see Footnote 15.
which we non-exhaustively cite.
such as in Bogomolnaia and Jackson (2002). See Footnote 12.
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This paper has been presented at the Eighth International Meeting of the Association for Public Economic Theory, 6–8 July 2007, Nashville and the 29’th Bosphorus Workshop on Economic Design, 25 August–1 September 2007, Bodrum. We thank the participants. Remzi Sanver acknowledges the support of the Turkish Academy of Science Distinguished Young Scientist Award Program (TUBA-GEBİP).
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Asan, G., Sanver, M.R. The Tiebout hypothesis under membership property rights. Theory Decis 78, 457–469 (2015). https://doi.org/10.1007/s11238-014-9430-7
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DOI: https://doi.org/10.1007/s11238-014-9430-7