and T is prone to secession.
Since it is trivially satisfied for K = 1, we may consider only partitions into K > 1 regions.
where supremum is taken over the set of all partitions P of the nation I into more than one region and |P| stands for the number of regions in P. Note that ge ≤ D(1)/2 is bounded. Thus, the cooperation is efficient if and only if g ≥ ge.
the allocation (½, b) is secession-proof and gs is, indeed, a finite number.
that is, the cooperation is efficient.
Proof of Lemma 3.3: Let x ∈ X be a cost allocation such that a region S is prone to secession. Assume, without loss of generality, that m(S) ≤ ½. We shall carry out the proof of the lemma in four steps:
and S1 is prone to secession.
Thus, S2 is prone to secession.
and S3 is prone to secession.
4. Suppose that there is q > ½ such that [½, q] ⊂ S and S ∩ [1 − q ½] = Ø. Then S4 = S \ [½, q] ∪ [a − q, ½] is prone to secession: Since D(S4) ≤ D(S), the symmetry property (b) implies that the region S4 is prone to secession.
It is easy to verify that the proof of the lemma follows from steps 1 through 4.
Proof of the Main Result: It suffices to demonstrate that if g ≥ ge, the allocation xg is secession-proof. It is useful to recall that g ≥ ge, implies that (1) holds.
To proceed, we need the following lemma:
We shall show that inequality (3) would not be violated by the function xg(t) = a(t) + λ, i.e.,
which follows from (1). This completes the proof of the main result.
Since the maximum of the right-hand side is ⅛, it follows that gs = ⅛, i.e., the cooperation is stable and efficient if g ≥ ⅛.18
that is, for the range of government costs g, satisfying .125 < g < .134, the inequality (4) is violated. Thus, the allocation NC is not secession-proof, since as for this range of values of g, there are regions Lt, in particular for t = .423, that are prone to secession.
Two cases should be considered:
Since g ≥ ⅛, it is easy to verify that Ψ1(g) ≤ 1 − 4g ≤ Ψ2(g).
Therefore, in Case 1, the range of secession-proof values of α is the interval [max(0, Ψ1(g)), max(0, 1 − 4g)].
which is always negative. Therefore, the only upper bounds for α are 1 and 4g. Thus, in Case 2, the range of secession-proof values of α is the interval [max(0, 1 − g), min(4g, 1)].
the range of secession-proof values of a is the union of the two intervals, [Ψ1(g), 1 − 4g] (generated by Case 1) and [1 − 4g, 4g] (generated by Case 2). Thus, we obtain the interval [Ψ1 − (g), 4g].
Finally, by setting interval Ψ(g) = max[Ψ1(g), 0], we complete the proof of the proposition.
Ahmad, Ehtisham, and Jon Craig, 1997, “Intergovernmental Transfers,” in Fiscal Federalism in Theory and Practice, ed. by Teresa Ter-Minassian (Washington: International Monetary Fund).
Ahmad, Ehtisham, and Jon Craig, Keping Li, Thomas Richardson, and Raju Singh, 2002, “Recentralization in China?,” IMF Working Paper 02/168 (Washington: International Monetary Fund).
Alesina, Alberto, and Enrico Spolaore, 1996, “International Conflict, Defense Spending and the Size of the Countries,” NBER Working Paper No. 5694 (Cambridge, Massachusetts: National Bureau of Economic Research).
Alesina, Alberto, and Enrico Spolaore, 1997, “On the Number and Size of Nations,” Quarterly Journal of Economics, Vol. 112 (November), pp. 1027 –56.
Alesina, Alberto, and Enrico Spolaore, and Romain Wacziarg, 2000, “Economic Integration and Political Disintegration,” American Economic Review, Vol. 90 (December), pp. 1276 –96.
Alesina, Alberto, and Romain Wacziarg, 1998, “Openness, Country Size and the Government,” Journal of Public Economics, Vol. 69 (September), pp. 305 –22.
Boadway, Robin W., and Paul A.R. Hobson, 1993, “Intergovernmental Fiscal Relations in Canada,” Report of Canadian Tax Foundation (Toronto: Canadian Tax Foundation).
Bolton, Patrick, and Gerard Roland, 1997, “The Break-Up of Nations: A Political Economy Analysis,” Quarterly Journal of Economics, Vol. 112 (November), pp. 1057 –90.
Bolton, Patrick, and Gerard Roland, and Enrico Spolaore, 1996, “Economic Theories of the Break-Up and Integration of Nations,” European Economic Review, Vol. 40 (April), pp. 697 –706.
Buchanan, J.M., and R.I. Faith, 1987, “Secessions and the Limits of Taxation: Toward a Theory of Internal Exit,” American Economic Review, Vol. 77, pp. 1023 –31.
Caplin, Andrew, and Barry Nalebuff, 1991, “Aggregation and Social Choice: A Mean Voter Theorem,” Econometrica, Vol. 59 (February), pp. 1 –24.
Casella, Allesandra, 1992, “On Markets and Clubs: Economic and Political Integration of Regions with Unequal Productivity,” American Economic Review, Papers and Proceedings, Vol. 82 (May), pp. 115 –21.
Casella, Allesandra, and Jonathan Feinstein, 2002, “Public Goods in Trade: On the Formation of Markets and Political Jurisdictions,” International Economic Review, Vol. 43 (May), pp. 437 –62.
Clark, Douglas H., 1997, “The Fiscal Transfer System in Canada,” in Financing Decentralized Expenditures: An International Comparison of Grants, ed. by Ehtisham Ahmad (Cheltenham, England: Edward Elgar Publishing).
Craig, Jon, 1997, “Australia,” in Fiscal Federalism in Theory and Practice, ed. by Teresa Ter-Minassian (Washington: International Monetary Fund).
Cremer, H., A.M. De Kerchove, and J. Thisse, 1985, “An Economic Theory of Public Facilities in Space,” Mathematical Social Sciences, Vol. 9, pp. 249 –62.
Dabla-Norris, E., J. Martinez-Vasquez, and J. Norregaard, 2000, Fiscal Decentralization and Macroeconomic Performance: The Case of Russia, Ukraine, and Kazakhstan (Washington: International Monetary Fund).
Dabla-Norris, E., and Shlomo Weber, 2001, “Regional Disparities and Transfer Policies in Russia: Theory and Evidence,” in Institutional Change in Transition Economies, ed. by Michale Cuddy and Ruvin Gekker (United Kingdom: Ashgate Publishing Ltd.).
Easterly, William, and Sergio Rebello, 1993, “Fiscal Policy and Economic Growth: An Empirical Investigation,” Journal of Monetary Economics, Vol. 32 (December), pp. 417 –58.
Feinstein, Jonathan, 1992, “Public Good Provision and Political Stability in Europe,” American Economic Review, Papers and Proceedings, Vol. 82 (December), pp. 323 –29.
Fidrmuc, Jan, 1999, “Stochastic Shocks and Incentives for (Dis) Integration,” CEPR Working Paper No. 2104 (London: Centre for Economic Policy Research).
Friedman, David, 1977, “A Theory of the Size and Shape of Nations,” Journal of Political Economy, Vol. 85 (February), pp. 59 –77.
Greenberg, Joseph, and Shlomo Weber, 1986, “Strong Tiebout Equilibrium Under Restricted Preferences Domain,” Journal of Economic Theory, Vol. 38 (February), pp. 101 –117.
Guesnerie, Roger, and Claude Oddou, 1987, “Increasing Returns to Size and Their Limits,” Scandinavian Journal of Economics, Vol. 90, No. 3, pp. 259 –73.
Haimanko, Ori, Michel Le Breton, and Shlomo Weber, 2003, “Transfers in a Polarized Country: Bridging the Gap between Efficiency and Stability,” forthcoming in Journal of Public Economics.
Hu, C. and C. Tan, 1996, “Empirical Study on the Regional Economic Disparities in China,” DRC Report No. 42 (Beijing: State Council Research Development Center).
Hu, Dapeng, and Masahisa Fujita, 2001, “Regional Disparity in China 1985–1994: The Effects of Globalization and Economic Liberation,” Annals of Regional Science, Vol. 35, No. 1, pp. 3 –37.
Jehiel, Philippe, and Suzanne Scotchmer, 2001, “Constitutional Rules of Exclusion in Jurisdiction Formation,” Review of Economic Studies, Vol. 68, pp. 393 –411.
Krelove, Russell, Janet G. Stotsky, and Charles L. Vehorn, 1997, “Canada,” in Fiscal Federalism in Theory and Practice, ed. by Teresa Ter-Minassian (Washington: International Monetary Fund).
Moreno, L., 2001, “Divided Societies, Electoral Polarization and the Basque Country,” CSIC Working Paper No. 01–07 (Unidad de Politicas Comparadas).
Olofsgard, A, 1999, “Secessions and Nationalism in a Model with Size Externalities and Imperfect Mobility” (unpublished; Stockholm: Institute for International Economic Studies).
Perroni, Carlo, and Kimberly Scharf, 2001, “Tiebout with Politics: Capital Tax Competition and Constitutional Choices,” Review of Economic Studies, Vol. 68, pp. 133 –54.
Persson, Torsten, and Guido Tabellini, 1996a, “Federal Fiscal Constitutions: Risk Sharing and Moral Hazard,” Econometrica, Vol. 64 (May), pp. 623 –46.
Persson, Torsten, and Guido Tabellini, 1996b, “Federal Fiscal Constitutions: Risk Sharing and Redistribution,” Journal of Political Economy, Vol. 104 (October), pp. 979 –1009.
Spahn, Paul B., and Wolfgang Föttinger, 1997, “Germany,” in Fiscal Federalism in Theory and Practice, ed. by Teresa Ter-Minassian (Washington: International Monetary Fund).
Ter-Minassian, Teresa, 1997, “Intergovernmental Fiscal Relations in a Macroeconomic Perspective: An Overview,” in Fiscal Federalism in Theory and Practice, ed. by Teresa Ter-Minassian (Washington: International Monetary Fund).
Triesman, Daniel, 1996, “The Politics of Intergovernmental Transfers in Post-Soviet Russia,” British Journal of Political Science, Vol. 26 (July), pp. 299 –335.
Triesman, Daniel, 1998, “Deciphering Russia’s Federal Finance: Fiscal Appeasement in 1995 and 1996,” Europe-Asia Studies, Vol. 50 (July), pp. 893 –906.
Weber, Shlomo, and Shmuel Zamir, 1985, “Proportional Taxation: Nonexistence of Stable Structures in an Economy with Public Good,” Journal of Economic Theory, Vol. 35, pp. 178 –85.
Wei, Shang-Jin, 1991, “To Divide or to Unite: A Theory of Secessions” (unpublished; Berkeley: University of California at Berkeley).
Wittman, Donald, 1991, “Nations and States: Mergers and Acquisitions, Dissolution and Divorce,” American Economic Review, Papers and Proceedings, Vol. 81 (May), pp. 126 –29.
Wooders, Myrna H., 1978, “Equilibria, the Core, and Jurisdiction Structures in Economies with a Local Public Good,” Journal of Economic Theory, Vol. 18, pp. 328 –48.
Young, R., 1998, “Secession Games,” in Palgrave Dictionary of Economics and the Law, ed. by P. Newman (New York: Stockton Press).
Michel Le Breton is Professor of Economics at the Université de Toulouse I, GREMAQ and IDEI, Toulouse, France. Shlomo Weber is Robert H. and Nancy Dedman Trustee Professor of Economics in the Department of Economics, Southern Methodist University, Dallas, Texas, USA; and Fellow, the Center for Operations Research and Economics (CORE), the Catholic University of Louvain-la-Neuve, Belgium. A previous version of this paper was written while the second author was visiting the Department of Fiscal Affairs at the IMF. We wish to thank Ehtisham Ahmad, Francis Bloch, Youngsub Chun, Jacques Drèze, Michel Goemans, Philippe Jéhiel, Michael Keen, Ozgür Kibris, Luc Leruth, Lionel McKenzie, Jean-Charles Rochet, Karl Shell, Partho Shomme, Martin Skutella, Enrico Spolaore, Yuval Weber, an anonymous referee, and an editor of this journal for their valuable comments and suggestions. We are grateful to seminar participants at CORE, IMF, London School of Economics, and the Universities of Brussels, Copenhagen, Rochester, Stockholm, Toulouse, Brown, Georgetown, Kyoto, New York, Peking, and Seoul for their remarks. We would also like to thank Nicholas Gaspard and Monica Sarratt for their help in preparing the manuscript.
A large population of taxpayers can share the cost of public goods such as roads, a telephone network, defense, civil servants, and education. Alesina and Wacziarg (1998) show that small countries tend to have bigger governments, and bigger government consumption, as a share of GDP. Smaller countries also face substantial costs of maintaining their distinctive language and culture. For example, the economic cost of Iceland’s language is about 3 percent of the country GNP (Economist, 1998).
See Alesina and Spolaore (1996). In many countries a majority of citizens do not particularly value their country’s political and military might, but in some other countries, particularly China, France, Russia, India, and Pakistan, the citizens do care about their country’s standing and influence in the world. As evidence of this phenomenon, Easterly and Rebello (1993) confirm that large countries spend relatively more on their defense.
In our formal analysis, we opt for a cooperative approach to address this issue. The choice of a cooperative versus noncooperative model is a rather delicate task that does not obey very stringent rules. If an interaction among agents is governed by precise rules and protocols, it is appropriate to model it as a strategic form game where all potential moves are described very accurately without room for mistake. Even in this case one incurs the risk of deriving predictions based on a fragile structure of a specific construction. In the absence of a priori protocol for negotiations among parties involved, one may abandon a noncooperative mode in favor of an alternative cooperative approach based only on a surplus available to each coalition of players. We believe that in the context of secessions and monetary compensations, it is worthwhile to adopt the protocol-free cooperative approach of this paper. However, one has to recognize that constitutional constraints on secessions, as in Canada and France, could be modeled as a normal form game. Thus, a mixture of two approaches could be used for an analysis of the issues discussed in this paper.
Called C-stability in AS.
They assume that each jurisdiction decides not only upon the location of its government but also on its size.
Since S consists of a finite number of connected regions, there always exists an optimal location of the government and, therefore, the cost function is well defined. It is useful to note that for every set S the total transportation cost is minimized when the government chooses its location at the ideal point of its “median citizen,” m(S). that satisfies
With an obvious change of the variables the analysis remains unchanged, with g/α instead of g.
Log-concavity is a special case of a more general concept of p-concaviiy studied in Hardy, Little wood, and Poly a (1934). The applications of log-concavity are relatively novel to economic and political science theory (see Caplin and Nalebuff, 1991, and Weber, 1992). The difference between our setup and the models discussed in Caplin and Nalebuff (1991) is that they impose log-concavity on density functions whereas we consider log-concavity of the distribution function.
Jacques Drew: pointed out to us that, it’ the citizens were distributed (uniformly) over the entire real line, rather than the bounded interval [0, 1], the full equalization would be the unique secession-proof compensation scheme. Indeed, in this case, for any level of government costs the gains from cooperation are maximized for a partition of the real line into intervals of equal length.
It is easy to verify that a further partitioning of the interval [0, t] into smaller intervals would impact the efficiency bound, and it suffices to check the partitions with two sets only.