Redistribution of tax resources: A cooperative game theory approach

Calvo, Emilio Article Redistribution of tax resources: A cooperative game theory approach SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Calvo, Emilio (2021) : Redistribution of tax resources: A cooperative game theory approach, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 12, Iss. 4, pp. 633-686, https://doi.org/10.1007/s13209-021-00253-5 This Version is available at: https://hdl.handle.net/10419/286546 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ SERIEs (2021) 12:633–686 https://doi.org/10.1007/s13209-021-00253-5 ORIGINAL ARTICLE Redistribution of tax resources: a cooperative game theory approach Emilio Calvo1 Received: 23 October 2020 / Accepted: 13 October 2021 / Published online: 19 November 2021 © The Author(s) 2021 Abstract We consider the problem of how to distribute public expenditure among the different regions of an economic entity after all taxes have been collected. Typical examples are: the regions that make up a country, the states of a federal country, or the countries of a confederation of countries. We model the problem as a cooperative game in coalitional form, called the tax game. This game estimates the fiscal resources collected in each region, or coalition of regions, by differentiating between what comes from economic activity within each region and what comes from trade with the other regions. This methodology provides a measure of the disagreement within a region, or coalitions of regions, with respect to the budget received. Similarly, the stability of a budget allocation can be inferred by its situation within the core of the corresponding tax game. We consider the Spanish case as an example and show that the current regional financial system has a moderate degree of instability. We introduce two budget allocation rules, both borrowed from the cooperative games literature: the balanced allocation, which coincides with the nucleolus and with the Shapley value of the tax game, and the weighted balanced allocation, which coincides with the weighted Shapley value. We compare both budget allocation rules with the current Spanish financial system. Keywords Fiscal balances ·Budget stability ·Coalitional games ·Shapley value JEL Classification H77 ·C71 1 Introduction This paper addresses the problem of how to distribute public sector spending among the regions of a country. Decisions about how much to spend in each region are increasingly up to local governments. Local institutions want to decide not only on BEmilio Calvo [email protected] 1Department of Economic Analysis and ERI-CES, Universitat de Valencia, Valencia, Spain 123 634 SERIEs (2021) 12:633–686 what and how to spend the public budget in their region, but also the total amount to be spent in their region. There are three basic principles1that appear recurrently in the search for a wellfunctioning regional financing system: 1. Non-discrimination Distributed funds must provide a uniform level of public services throughout the country. 2. Fairness in redistribution Allocation of funds should vary directly depending on fiscal needs and inversely according to the tax capacity of each jurisdiction. 3. Ordinality The results of the equalisation should be tolerable for donors and recipientsalike.They should narrowfinancingdisparitiesacrossregions without altering their per capita relative wealth ranking. They should not carry equalisation beyond a generally acceptable level. Unfortunately, how to make all these principles fully compatible with each other is not evident. Large disparities between regions in terms of their per capita wealth could make it difficult to apply the principle of non-discrimination, since it could imply high transfers of income from rich to poor regions, always viewed with suspicion by richer regions. When designing a regional financial system, the tax system should therefore minimise the complaints from a region, or group of regions, regarding the total budget obtained following that funding rule. Obviously, how to measure and compare such grievances becomes a key problem in determining the level of equalisation in per capita wealth that should be considered generally acceptable. This exercise can be transferred, point by point, to the problem of the distribution of the budget between the states of a federal country, or between the states of a confederation of countries, such as the European Union, simply by substituting the regions of a country for the federal states, or by the confederation countries. Until now, fiscal balances (FBs) were used for the analysis of such complaints. Fiscalbalancesdeterminethedifferencesbetweenpublicrevenuescollectedandpublic expenditures allocated in each region. These net per capita balances have been used to evaluate and compare the budget distribution between regions. WecanseeintheFBliteraturetwoopposite(moreorless explicit)fiscalsovereignty assumptions,whichwemightcallfull versus shared sovereignty. Inthefull sovereignty approach, all resources collected in a region are considered to belong exclusively to that region. In this case, fiscal balances are used as benchmarks by which the degree of satisfaction or disagreement with the total budgetary expenditure obtained by the region is measured. It should be noted that there is little room for negotiation on budget distribution with this approach. Significant differences between what is collected and obtained are viewed with suspicion. The objective of a stable budget distribution is therefore that what is spent in each region should be as close as possible to what is collected. This is so because the distribution of the budget is seen as a zero-sum game: What one region gains is at the expense of what another loses. The shared sovereignty approach is opposed to this point of view, in which all the regions that make up a country are considered part of the same economic entity. It is understood that all the resources collected come from the mutual cooperation 1We quote some of these principles (among others) from De la Fuente, Thöne and Kastrop (2016). 123 SERIEs (2021) 12:633–686 635 resulting from the exchange of goods and services between the economic agents of the country. From this perspective, regions are distributing all the economic gains from their cooperation between their citizens, regardless of where they are. In the first approach, any monetary transfer is only seen as a voluntary act of solidarity between regions. From the second perspective, it is nothing more than an obligation in order to obtain the common welfare of the country’s citizens. However, in our opinion, none of these perspectives are as irreconcilable as they appear at first glance. In this paper, we address this problem of budget distribution using tools from the theory of cooperative games. We believe this may help clarify the discussion on how much should be allocated to each region. Roughly speaking, the problem can be solved by determining more precisely how much of the fiscal resources belong exclusively to the region (or group of regions), and how much should be considered common property. We first enrich the FB model by explicitly incorporating interregional economic relationship in its construction. This is accomplished by constructing what we call the tax cooperative game, which determines for each region, or set of regions (coalitions), all tax revenues derived exclusively from the economic interaction within regions of the coalition, as well as any imports or exports from abroad. According to the total sovereigntyapproach,wecaninterpretthisamountastheminimum thatanycoalitionof regions should receive in any acceptable regional budget allocation. Here, we exclude all tax revenues that coalition members derive from their current interaction with the rest of the country, as according to the shared sovereignty approach they are common property, and consequently, their distribution can be subject to negotiation. We propose measuring the grievances of a region, or coalition of regions, against the budget distribution, by means of the cooperative game theory concept of the excess of a coalition. This is the difference between what the tax cooperative game gives the coalition minus what the budget allocation spends on it. The lower the allocation given, the greater the excess becomes, and therefore the greater the disagreement. The excess is therefore a measure of such disagreement. The goal is thus to find budget allocations that minimise such excesses. The familiar concept of the core of a cooperative game is associated with the excesses obtained by the coalitions, i.e. the set of all the budgetary distributions whose excesses are all negative. In this paper, we will continue with the usual custom in game theory of qualifying the stability of a budgetary allocation according to its situation with respect to the core of the tax game. In this way, we will evaluate the stability of the current budgetary distribution of a country, as well as any other alternative proposals we may consider. The first finding was that for any tax cooperative game there are always stable budget allocations. Thus, it is alwayspossibletopropose stable budget allocations. Secondly, we can undertake the theoretical exercise of applying solutions from the cooperative games literature to the tax game at hand. For example, the purpose of the egalitarian spending rule is that no citizen should be discriminated against in terms of public spending, matching the public spending per capita in all regions of a country. Unfortunately, wide disparities in per capita wealth between regions mean that this spending rule easily produces unstable allocations. 123 636 SERIEs (2021) 12:633–686 Twoother relevantrulesarethe nucleolus (Schmeidler1969),whichalwaysbelongs to the core when it is non-empty, and the Shapley value (Shapley 1953a,b). In general, both rules produce different allocations, and their computation is an arduous task. Nevertheless, given the matrix structure of the tax game, we demonstrate that the nucleolus and the Shapley value of every tax cooperative game coincide.Thevalue formula also takes a simple expression: it assigns each region its own fiscal resources plus half the common fiscal resources of its trade interaction with all other regions. We shall call this rule the balanced allocation ϕ, and this is placed at the centre of the core. However, ϕdoes not perform well in terms of wealth redistribution. To improve this poorredistributivebehaviour,weproposetoconsider theweighted balanced allocation ϕw, where the weights ware the per capita gross domestic product (GDP)ofregions. Now, the welfare contribution that each region makes to each other in the form of tax transfers is balanced inversely proportional to their GDP per capita. This allocation coincides through its construction with the weighted Shapley value (Shapley 1953a), and we show that it offers a greater degree of solidarity between regions than ϕ.Given the convexity of the tax game, this rule is also stable. We thus believe that ϕwis a reasonable trade-off between these two principles: stability on the one hand, and solidarity on the other one. Finally, the qualification of a budget distribution as stable or not, depending on whether it is inside or outside the core, is not entirely satisfactory. It is simply a binary assertion. To compare two alternative budget allocations, we would like to know the degree of stability they enjoy. For that purpose, we modify the tax game by means of a parameter α∈[0,1]. In the computation of the modified tax game vα twe add a proportion αof the initially excluded tax revenues (considered as a common sovereignty). This parameter allows a gradual transition from the concepts of stability of shared fiscal sovereignty to full fiscal sovereignty. As αincreases, therefore, the amount of those common fiscal resources that the region agrees to share with all the others decreases. Accordingly, the set of stable budget allocations reduces to the ideal full sovereignty target (α1), in which the budgetary expenditure obtained by the region is equal to the totality of taxes collected in it. In addition, αprovides anormalised measure of each region’s degree of dissatisfaction with respect to a given budget allocation. It is equal to the value αfor which the region’s excess over the modified tax cooperative game is equal to zero. This normalisation allows a better comparisonbetweentheexcesses of the regionsand, eventually, a comparisonbetween the stability observed in different countries. We illustrate how the theoretical concepts introduced here perform through a practical exercise: we consider the Spanish case by using fiscal data from 2011 to 2014. We start by illustrating how to build the tax game from FB and inter-regional commercial trade data. The current Spanish financial rule has a moderate redistributive effect of wealth between regions; the rule treats regions better the poorer they are. Surprisingly, we show that the current financial system belongs (unexpectedly) to the core of the tax game during those years. However, we should nuance the system stability. There are several autonomous communities (CAs) whose excesses are very close to being positive (some, such as Catalonia, with strong secessionist feelings), coexisting with other CAs whose situation in relative terms can be considered privileged, such as 123 SERIEs (2021) 12:633–686 637 the group of “foral communities” formed by the Basque Country and Navarre. This situation produces a relatively unstable political cocktail. Spaininvolvesdisparitiesin termsofpercapita wealthacrossregions,andtherefore, as expected, we show that computing the egalitarian rule yields an unstable allocation. The budget distribution of these four rules, the current system, the egalitarian, the balanced allocation ϕ, and the weighted balanced allocation ϕw, are compared. Weorganise thepaperasfollows.Following thisintroduction,Section2 isdedicated to reviewing the literature related to the problem in both public economics and game theory. Section 3 defines the tax cooperative game and summarises the main stability results. In Sect. 4, we introduce the budget allocation rules ϕand ϕw. In Sect. 5 we present the extended tax cooperative game associated with α. This parameter allows us to consider the degree of stability of any particular budget allocation. Section 6 is dedicatedtoapplying thesenewconceptstothe Spanishcase.Section7endswithsome final comments and remarks. We place the proofs of theorems, and some additional tables, in “Appendix”. 2 Related literature The connection between the game theory literature and public finance is not new. Aumann and Kurz (1977) made an early application of the Shapley value to political taxation, where the individual taxes are the Shapley value of an income distribution game, built as follows: each agent starts with an individual endowment. Redistribution decisions are made by majority voting, but each agent has the right to destroy part of their endowment. In some sense, we take a parallel approach to that work: in their case, players are citizens, and in our case, players are regions. In both, the Shapley value is applied. Another influential paper is O’Neill (1982). He has contributed to a body of work on fair division in the face of conflicting claims. Note that a taxation problem is formally identical to a claim problem. A claim rule specifies how to divide a fixed amount Ebetween a set of agents whose claims are d1,d2,..., and di>E. A taxation rule specifies how to divide the tax burden X(the amount of taxes to be collected) among the agents whose gross incomes are y1,y2,..., and yi>X. Starting with the work of Young (1988,1990) on equal sacrifice and distributive justice in taxation, there has been a trend of research applying tools from the claim problems literature to analysing tax rules and defining new ones. A good example is Moreno- Ternero and Villar (2006). For an extensive survey of this topic, see Thomson (2003, 2019). Differently to our problem, players in this literature are citizens and not regions; and secondly, and more importantly, our problem is purely re-distributive. However, as we have transformed the problem of distributing the public budget among regions in a cooperative tax game setting, many of the properties used to evaluate cooperative solutions could also be used to evaluate every tax spending rule, and that is a promising field of future research. Fiscal balances analysis has increased over time. For example, the FBs for Spain have been estimated by Castells et al. (2000), Uriel and Barberán (2007), López- Casanovas and Rosselló-Villalonga (2014), and De la Fuente et al. (2014). For Italy, they have been estimated by Ferraro and Zanardi (2011), and in Giannola et al. (2016); 123 638 SERIEs (2021) 12:633–686 for the UK by McLean and McMillan (2003), Oxford Economics (2008) and Office for National Statistics (2018a,b); for Ireland by Morgenroth (2010); for the USA by Dubay (2006) and the Tax Foundation (2007); and for Canada by Ruggeri (2010). De la Fuente (2014) compares Catalonia with similar regions in other countries. Monastell and Sánchez (2012) analyse the territorial redistribution of the public budget in Spain and make comparative studies of the FB involving several countries. Our contribution in this paper is to enrich the FB model by disaggregating the taxes collected in each region according to the origin of their commercial relationship with each other. We build a cooperative tax game from this, where the worth of a coalition of regions is the sum of the tax revenues derived exclusively from the economic interaction within them. As far as we know, this approach is novel within the FB literature. The cooperative tax game obtained has a special matrix structure, and in particular, it is the sum of an additive game plus a two-person game. This provides the convexity of the game and the simplicity of the computation of the balanced (and the weighted balanced) tax allocation. These two-person games were considered (and in general, k-person games) in van den Noweland et al. (1996), showing the coincidence of the Shapley value and the nucleolus. These games were applied to a telecommunication problem. Recent application of games with this matrix structure includes Bergantiños and Moreno-Ternero (2020) to broadcasting sports events and López-Navarerrete et al. (2019) to smart TV ecosystems. Convexity plays a crucial role in guaranteeing the existence of stable tax rules. Convex games have non-empty cores (Shapley 1971; Ichiishi 1981), and so tax games do too. The concept of ε-core was introduced for the analysis of games with empty cores (Shapley and Shubik 1996; Maschler et al. 1979), mainly to enlarge the set of quasi-stable allocations, recovering non-emptiness. The least core is at the centre of the ε-core. The ε-core converges towards the least core as εgrows. Motivated by the political consideration that the tax revenue vector Tis the target aspiration of an independent fiscal authority, we have replaced the least core by Tas the convergence point of the α-cores, CN,vα t. This modification of the least core concept is very specific to our tax game setting, and it has no clear counterpart in a general cooperative game. The balanced tax allocation introduced in Sect. 4 is just the Shapley value of the tax game (N,v0 t). Apart from Shapley’s original characterisation of his value (Shapley 1953a,b), several other characterisations have been outlined in the literature, for example, by Myerson (1980), Young (1988), Hart and Mas-Colell (1989), Feltkamp (1995), and van den Brink (2001), among others. We note that the property of equally sharing the profits of cooperation between every two regions, which we have used in the definition of the balanced tax rule, is just a forward translation of the property of balanced contributions introduced by Myerson (1980), for TU games. Myerson used efficiency and balanced contributions to give one of the simplest axiomatic characterisations of this value. The weighted balanced allocation is just the weighted Shapley value of the tax game. This weighted version of the value was introduced by Shapley (1953a). Its axiomatic characterisation by means of weighted balanced contributions was given in Hart and Mas-Colell (1989). For the role of weights in the value, see Kalai and Samet (1987), Monderer et al. (1992), and Calvo et al. (2000). Further 123 SERIEs (2021) 12:633–686 639 interpretation of the balanced contributions axiom can be found in Calvo and Santos (2000). 3 Tax cooperative model Denote by N{1,2,...,n}the set of regions in a country. The FBs determine the balance between the regional distribution of public expenditure and revenue flow. Broadly speaking, public sector expenditure is the total capital and current expenditure (mainly wages and salaries, goods and services, and expenditure on fixed capital, but also subsidies, social benefits, and other transfers) of central government and local government bodies, as well as public sector-controlled corporations. Public sector revenue isthetotalcurrent receipts(mainlytaxes,butalsosocialcontributions,interest, dividends, gross operating surplus, and transfers) received by central government and local government bodies, as well as public sector-controlled corporations. Net fiscal balance is the gap between total spending and revenue raised. A negative net fiscal balance represents a surplus, meaning that a region is receiving more in revenue than it is spending. A positive net fiscal balance represents a deficit, meaning a region is spending more than it is receiving in revenue.2 Our goal is to quantify how much of the fiscal resources collected in one region come from commercial interaction with the other regions in a country. We can build the cooperative tax game by means of this interregional disaggregation of the FBs. Thus, we use the commercial exchange matrix3Cof goods and services between regions of a country, and between them and abroad, and use it to obtain matrices Iand Rof indirect and direct tax revenues, respectively. We will obtain the tax cooperative game from them. In order to clarify the exposition, we will use a simple numerical example to illustrate the theoretical concepts as they are introduced. The trade exchange matrix Ccx mgathers the flow of goods and services between the regions of a country, and between them and abroad: 2Definitions taken from Office for National Statistics (2018a). A detailed explanation of the concepts used in the construction of the FBs can be found in Office for National Statistics (2018b), and De la Fuente et al. (2014). 3In the following, matrices will be in bold. 123 640 SERIEs (2021) 12:633–686 That is, for each region i,cij is the amount of goods and services from region i sold to region j,miare imports from abroad to i, and xiare exports from ito abroad. Rows thus indicate sales and columns purchases. For example, let N{1,2,3}be a country with three regions, and with the corresponding trade exchange matrix. The population of the three regions is given by the vector P(10,7,7), and the wealth of each region is given by the vector of gross domestic product GDP (16,24.5,22). The gross domestic product per capita of each region i; that is, GDP i/Pi, is given by the vector (1.6,3.5,3.14), and so we will consider regions 2 and 3 richer than region 1 in relative terms. Let us call Tithe total public revenue of the region i. The vector of regional public revenues is denoted by T(T1,...,Tn). We also denote the sum of all tax revenues by T(N)i∈NTi. In order to make a proper territorial allocation of the origin of these revenues, we must first separate indirect taxes from the rest of the income. This is because consumption bears indirect taxes: VAT and excise duties on alcoholic beverages, energy products and electricity, manufactured tobacco, etc. In fact, in the construction of the FBs, the total indirect tax collected in each region is calculated by distributing the total revenues of a country among the regions according to their consumption (purchases). According to this approach, we will follow the same procedure to assign the origin of the indirect tax collected in each region according to the origin of its consumption. Let Iibe the total indirect taxes collected on i. The sum of all the coefficients in column iof matrix Cyields the total consumption of region i. We therefore distribute these taxes in proportion of the column coefficients of C. Then, the coefficients of column iof indirect taxes are: Iji cji j∈Ncji +mi ·Ii,∀j∈N; andIm imi j∈Ncji +mi ·Ii(1) The Iji coefficient corresponds to the indirect taxes collected on ias a result of the purchase of goods and services cji from region j,and the Im icoefficient with the custom tariffs associated with the imports mi. Thus, we obtain matrix I, where each column gives the distribution of indirect taxes collected in each region by origin. 123 SERIEs (2021) 12:633–686 647 Fig. 1 Core representation which turns out to be the convex hull of the following vertices: C(N,v t)CH{(7,4,13),(4,4,16),(2,6,16),(2,12,10),(5,12,7)}. We can represent all these points graphically with the help of the equilateral triangle given in Fig. 1Core representation. This triangle represents all efficient and non-negative payoffs. For example, for region 3, the bottom side represents allocations where region 3 obtains zero. The vertex (0,0,24)is the opposite case, where region 3 obtains the total taxes to share. Each intermediate horizontal line represents a constant payoff for region 3 (between 0 and 24). The same happens for regions 1 and 2, with parallel lines of constant payoffs, between the maximum payoff of 24 in the vertex and the minimum 0 payoff on its opposite side. The core C(N,v t)of this game is the shadow area of the hexagon in Fig. 1Core representation, and we can see the vector of public revenues T(4,7,13)is inside the core. In our example, region 1 is the poorest region in relation to regions 2 and 3. The payoffs of the egalitarian rule are Eg (10,7,7). This allocation is not in the core, because coalition {2,3}has a positive excess, e({2,3})vt({2,3})−Eg2+Eg317 −(7+7 )3>0. In our example, the transfers of wealth imposed by egalitarian rule are incompatible with its stability, as it can see that Eg /∈C(N,v t). 123 648 SERIEs (2021) 12:633–686 4 Balanced allocations rules So far, we have identified those allocations that are in the core of the tax game as stable, and those that are outside of it as unstable. In the definition of the cooperative tax game vtwe have made the assumption that when we calculate the fiscal tax resources to which a coalition Shas access, those that are the consequence of the commercial exchange with the other regions N\Scannot be included as their own, since those resources should be considered as common property of Sand N\S. Under the stability approach followed here, we could ask which budget allocation is the most stable of all. As the goal is to minimise the excesses of coalitions (minimising their dissatisfaction), we have two possible answers, both drawn from cooperative game theory literature: the nucleolus and the Shapley value. The nucleolus was introduced in Schmeidler (1969). The idea behind this value is to select an allocation that minimises the dissatisfaction of the most dissatisfied coalition, where the dissatisfaction of an allocation xby coalition S⊆Nis measured by its excess e(S). For that purpose, it makes the largest dissatisfaction as small as possible. If there are several allocations to do this, then we make the second largest dissatisfaction as small as possible, and so on until we reach a unique allocation. The nucleolus is thus the unique allocation that minimises lexicographically the excesses of coalitions. In this sense, the nucleolus is similar in spirit to the maximin principle of distributive justice proposed by Rawls (1971). Moreover, the nucleolus is always in the core when the core is non-empty. There are several procedures to compute the nucleolus, but it can be quite hard.8 The Shapley value was introduced by Shapley (1953a,b). When a game is convex (as all tax games are9) the Shapley value turns out to be its core barycentre. Thus, it also becomes a good stable allocation proposal. To calculate it in a tax game, we need to obtain all the vertices that delimit the core, making its arithmetic mean (all being equally likely). Note that, in general, the Shapley value and the nucleolus select different points, even for convex games. This leads us to expect that the nucleolus and the Shapley value of a tax game could be different. Surprisingly, it turns out that both values coincide in any tax game. Moreover, its calculus is straightforward and has an intuitive interpretation in terms of fairness. For reasons that will be evident in the following, we will call it the balanced allocation ϕ. Following the common sovereignty assumption, the fiscal resources resulting from the economic exchange between regions iand jare the tax benefits of economic cooperation between both regions. Those amounts are Rij +Ijifor region iand Rji +Iijfor region j.Rij and Iij come from cij sales of goods and services from ito j, and Rji and Iji come from cji sales from jto i. How do we could distribute these benefits between both regions? A standard solution in two-person games is to equally share the benefits of cooperation. In accordance with this principle, we will assign each region its own fiscal resources, Rii +Iii +Rx i+Im i, plus half of the common fiscal resources of its commercial interaction with each of the other regions. 8See Maschler (1992) for a review of those procedures. The MATLAB toolbox package MatTuGames provided by Meinhardt (2012) is useful for practical computations. 9See Theorem 6 in “Appendix”. 123 SERIEs (2021) 12:633–686 649 To simplify the notation, we denote tii Rii +Iii +Rx i+Im i,and tij Rij +Iji,∀j i. Then, for any tax problem (N,R,I)∈TN, the balanced allocation ϕis defined by ϕi(N,R,I)tii +1 2 j∈N\itij +tji,∀i∈N(8) We may say that with the tax rule ϕthe budget contribution of region ito jand the budget contribution of jto iare the same, that is 1 2tij +tji. In the following theorem, the coincidence of the balanced allocation and the Shapley value of the tax game is established. Theorem 2 For every tax problem (N,R,I)∈TN, the balanced allocation ϕcoincides with the Shapley value and the nucleolus of the tax game (N,v t).10 Thus, ϕis a budget allocation located at the centre of the core. Going back to our numerical example, we show the calculation of ϕfor region 1. ϕ1t11 +1 2(t12 +t21)+1 2(t13 +t31) (0.7+0.7+0.2+0.4)+1 2[(0.4+0.6)+(0.6+0.4)] +1 2[(0.6+0.4)+(0.8+1.2)]4.5 Making the same calculation for regions 2 and 3, it is easy to check that the vector of balance allocations is equal to ϕ(4.5,8,11.5). We have made a close-up of the core in Fig. 2Close-up of the core below and it can be seen that ϕis located in the centre of it. It should be noted that regional redistribution of wealth was not an initial purpose in the design of the balanced allocations rule ϕ. Therefore, we cannot expect good redistributive behaviour accordingly when applied in countries with an uneven distribution of wealth. This is the case in our numerical example. We have assumed that the current spending rule is the egalitarian rule, Eg (10,7,7). Hence, the final gross domestic product associated with Eg is EgGDP (16,24.5,22). Given the population vector P(10,7,7), the per capita11 EgGDP is EgGDPh(1.6,3.5,3.14). The vector of total public revenues is T(4,7,13), and the vector of FBs associated with the egalitarian rule is FB(Eg)T−Eg (−6,0,6). The vector of initial gross domestic product is therefore IGDP GDP +FB(eg)(16,24.5,22)+(−6,0,6) (10,24.5,28), and the per capita IGDP is IGDPh(1,3.5,4). There is a wide 10 The τ-value (Tijs 1981) also coincide with the Shapley value and the nucleolus in the class of two-player games. It is easy to prove the same coincidence in the tax game problems. 11 In the following, when the superscript his added it indicates per capita. That is, EgGDPh iEgGDPi Pi, for each region i. 123 650 SERIEs (2021) 12:633–686 Fig. 2 Close-up of the core difference between the per capita wealth of region 1 and regions 2 and 3. The balanced allocation rule yields ϕ(4.5,8,11.5)as payoffs, and their FBs are FB(ϕ)(−0.5,−1,1.5). Its associated final GDP,ϕGDP IGDP −FB(ϕ), is ϕGDP (10.5,25.5,26.5)and the final GDP per capita associated with ϕ, ϕGDPh(1.05,3.64,3.79). Note that with the balanced allocation ϕwe can obtain a lower effect in the wealth redistribution comparing EgGDPhwith ϕGDPh. To overcome this weakness, we propose a budget allocation that incorporates a certain degree of wealth per capita redistribution in its formulation. To achieve this principle, we consider the differences in the IGDP per capita of the regions when distributing their common fiscal resources. That is, the profits of cooperation tij +tji are distributed inversely proportional to the IGDPhof regions iand j. Let be the vector of IGDP per capita of each region, i.e. IGDPh iIGDPi Pi.We define the weighted balanced allocation ϕwby ϕw i(N,R,I)tii + j∈N\i IGDPh j IGDPh i+IGDPh j ·tij +tji,∀i∈N.(9) For an alternative interpretation of ϕw, we consider the inverse of IGDPh i, that is (IGDPh i)−1Pi GDPi. We can interpret this inverse as a “normalised population per wealth”. Thus, given two regions iand jwith the same level of IGDP,ifihas a population greater than j, this means that iis poorer than j. Note that 123 SERIEs (2021) 12:633–686 651 IGDPh j IGDPh i+IGDPh j ·(IGDPh i)−1·(IGDPh j)−1 (IGDPh i)−1·(IGDPh j)−1(IGDPh i)−1 (IGDPh i)−1+(IGDPh j)−1 Therefore, we can define alternatively ϕwby ϕw i(N,R,I)tii + j∈N\i (IGDPh i)−1 (IGDPh i)−1+(IGDPh j)−1·tij +tji,∀i∈N. Thus, we distribute tij +tjibetween iand jin proportion to their normalised population. We have a parallel result to that of Theorem 2. Theorem 3 For every tax problem (N,R,I)∈TN, the weighted balanced allocation ϕwcoincides with the weighted Shapley value of the tax game (N,v t), i.e. ϕw(N,R,I)Shω(N,v t), where ωiIGDPh i−1, for all i ∈N. Moreover, ϕw(N,R,I)∈C(N,v t). The weighted Shapley value was introduced by Shapley (1953a). Although ϕwis also stable,12 it is now closer to the border of the core than ϕ. However, what it loses in stability, it gains in a greater degree of interregional solidarity. We also compute ϕwin our example. We only show the calculations for region 1. ϕw 1t11 +w2 w1+w2(t12 +t21)+w3 w1+w3(t13 +t31) (0.7+0.7+0.2+0.4)+3.5 1+3.5[(0.4+0.6)+(0.6+0.4)] +4 1+4[(0.6+0.4)+(0.8+1.2)]5.96 Making the same for regions 2 and 3, we find that the vector of weighted balance allocations is equal to ϕw(5.96,7.64,10.4).InFig.2we can see ϕwplaced close to the boundary of C(N,v t)in the direction of Eg. The final GDP associated with ϕw is equal to ϕwGDP (11.96,25.14,25.4), and the final GDP per capita associated with ϕwis ϕwGDPh(1.21,3.59,3.62). We obtain a greater wealth redistribution with ϕwthan ϕ, remaining in the core of the tax game. 5 Stability measure The binary assertion that a budget allocation is stable or not, simply because of the positive or negative sign of its associated excesses, is very limited for the political contextwe aredealingwith.The signoftheexcesses doesnotmakeitpossibletoassess the extent to which one spending distribution rule may attract a greater consensus than 12 This is always true because ϕwis a convex combination of the vertices of the core of the tax game, which is also a convex set. 123 652 SERIEs (2021) 12:633–686 another among the regions. It would be useful to have a measure of the intensity with which a region (or coalition of regions) accepts/rejects an allocation. One option might be to measure its distance to the centre of the core of the tax game (N,v t)(which is precisely ϕ). This could be done with the help of the ε-core concept, introduced by Shapley and Shubik (1996) and later named by Maschler et al. (1979) (see also Tijs and Driessen 1986). That is, the core CN,vε t, where vε t(N)vt(N), and vε t(S)vt(S)+ε, for all coalitions S N. The least core is the set CN,vε∗ t, where ε∗is the highest value (possibly negative) of εsuch that CN,vε t ∅.The case ε0 corresponds with the original game vt. A tax allocation is ε-stable if it belongs to the ε-core. As εgrows, the set CN,vε tdecreases to the least core. The degree of stability of an allocation will be the highest value of εcompatible with being ε-stable, where ε∗is the maximum degree of stability. The problem with this stability measure is that it is not consistent with the maximum budget allocation that a region aspires to manage if this were the only fiscal authority in charge of its management. This amount is given by the total revenue vector T. Clearly, Tiis the maximum amount of fiscal resources that region icould raise if it were an independent fiscal authority. Now, we have two opposite scenarios. On the one side, local authorities are not willing to share any fiscal resources with the rest of the regions. On the other side, local representatives are willing to negotiate regarding the total of their common fiscal resources. Between these two extremes, we can specify the αproportion,α∈ [0,1], of their common resources that they are not willing to share. With the help of this parameter αwe can obtain the degree of stability of a budget allocation ψ. This normalised index is a measure based on willingness to share their common tax resources. Moreover, this normalisation also allows us to make comparisons on the stability of spending rules between different countries, or to see the evolution that a rule has undergone over time. To do this, we define the extended tax game vα tas follows vα t(S) i∈S j∈S tij +α i∈S j∈N\SRij +Iji,∀S⊆N.(10) This amount is the total tax revenues that the coalition of regions Scollects within S, plus a proportion αof the tax revenue that comes from their commercial relationships withtheremaining regions N\S.Itistheminimum amount Sshouldgetto bewillingto accept an agreement on the distributionof the budget. Therefore, α1 corresponds to the full sovereignty approach where v1 t(S)i∈STi. Conversely, α0 corresponds to our original shared sovereignty approach of Sect. 2, v0 t(S)vt(S). Now, for each value of parameter α, we qualify as α-stable those allocations in the core of vα t, i.e. a budget allocation ψis α-stable if ψ∈CN,vα t. As, by construction, it holds that vα t(S)≤vα t(S)for all α≤α, it follows immediately that C(N,vα t)⊆ C(N,vα t). Therefore, to the extent that regions reduce the proportion of common tax resourcestheyarewillingtosharewitheachother,thesetofα-stablebudgetallocations is reduced. 123 SERIEs (2021) 12:633–686 653 Given a budget allocation ψ, the lower the value αfor the excess of coalition S to be zero, i.e. eα(S)i∈Sψi−vα t(S)0, the greater the amount of shared resources that coalition Shas transferred to N\Sto obtain ψ.Accordingly, for every coalition Sand budget allocation ψ, we define αS(ψ)asthevalueofαsuch that eα(S)0. Thus, αS(ψ)istheαthreshold beyond which a coalition Sobtains a positive excess in the game vα t. Therefore, αS(ψ)1 means that Shas not had to give to N\S any amount of fiscal resources to obtain ψ,that is i∈Sψii∈STiv1 t(S).A value αS(ψ)>1 implies that, even with all its fiscal resources at its disposal, Scould not achieve a budgetary expenditure equivalent to that obtained in ψ.Therefore, the bigger αS(ψ) is, the happier Swill be with what was obtained in ψ.In summary, from the point of view of the full sovereignty approach, low values of αS(ψ) will imply a greater reluctance of Sto accept ψ. Thus, we will define the degree of stability of a budget allocation ψas the highest value of αcompatible with being α-stable, and it will be denoted by α(ψ). The lower its value, the more unstable the budget allocation. What about the existence of α-stable allocations? We find again that its core is always non-empty. Theorem 4 For every tax problem (N,R,I)∈TN, the core of its associated tax game (N,vα t)is non-empty for every αin [0,1].In particular, T ∈CN,vα t. Note that when α1, the allocation Tis the unique stable allocation in the game (N,v1 t), that is CN,v1 t{T}. This happens because v1 tis an additive13 characteristic function. We have seen that the balanced allocation ϕis in the centre of C(N,v0 t). It is the most stable allocation for the α0 scenario. As far as αincreases, the set of α-stable allocations CN,vα treduces. The balanced allocation ϕwill therefore be unstable for some critical value of α. Moreover, this threshold value will depend on the data of each tax problem; however, it is possible to find a bound for αwhich guarantees that ϕremains α-stable in any tax problem, as the following theorem shows. Theorem 5 For every tax problem (N,R,I)∈TN, the balanced allocation rule is α-stable for all α≤1/2. We will again use the numerical example introduced in Sect. 2 to illustrate these new concepts. Now, the extended tax game N,vα tis given by vα t(1)2+2α,vα t(2)4+3α,vα t(3)7+5α,vα t({1,2})8+3α, vα t({1,3})12 + 5α,vα t({2,3})17 + 3α,vα t({1,2,3})24. To see the inequalities that satisfy any α-stable allocation x, note that, for region 1 it must hold that x1≥vα t(1)2+2α, and for regions 2 and 3 that x2+x3≥ vα t({2,3})17 + 3α, which jointly with i∈Nxivα t({1,2,3})24, implies that x1≤vt(N)−i∈Nxi24 −(17 + 3α)7−3α. Following the same reasoning for regions 2 and 3, we obtain that the core of the game N,vα tis given by the set CN,vα t(x1,x2,x3)∈R3:2+2α≤x1≤7−3α;4+3α≤x2≤12 −5α; 7+5α≤x3≤16 −3α 13 That is, v1 t(S)i∈STifor all S⊆N. 123 654 SERIEs (2021) 12:633–686 Fig. 3 Core representation for some values of α In Fig. 3Core representation for some values of we draw the core of the cooperative tax game N,vα tfor some values of α. We can see that the degree of stability of ϕ and ϕwis α(ϕw)0.384 and α(ϕ)0.75. In general, we can conjecture that the greater the differences in the per capita wealth among regions, the lower α(ϕw). In practice, calculating the excesses of all coalitions is not a feasible task.14 Nevertheless, we can calculate at least the individual excess of each region. In the following, we denote by ζα ithe individual regional value of iin the tax game vα t. That is, ζα i(N,R,I)vα t(i)tii +α j∈N\iDij +Iji,∀i∈N.(11) It is obvious that ζ1 i(N,R,I)Ti. For any budget allocation ψ, and region i,the threshold αi(ψ)is the value of αfor which ζα i(N,R,I)ψi(N,R,I). Equivalently, αi(ψ)isthevalueforwhichtheindividualexcessofregióniiszero.Forvaluesofαless than αi(ψ)the budget allocation ψgives a payoff to iwhich is not even individually rational, and therefore ψwill not definitely be α-stable.15 14 This is because the calculation time grows exponentially with the number of coalitions, which is 2n−1. 15 Of course, this definition can also be applied to any coalition. 123 SERIEs (2021) 12:633–686 655 5.1 Discussion of the interpretation of v˛ t We wish to stress that we should discard the use of the tax game vtas a tool to estimate the economic consequences under a secession process in terms of the amount of fiscal resources that a region, or a group of regions, would obtain at the end of secession. Under our approach, it simply gives us an estimate of the amount of fiscal resources over which a coalition of regions is sovereign, differentiating them from those of shared sovereignty with all regions, irrespective of what one may suspect will be the case in the event of secession. The excess in a tax game is therefore only a tool for the public budget spent in a region. If one wishes to consider what could happen after a secession process, the very pessimistic calculus of v0 t(S) gives a poor idea of the economic consequences of such rupture. This is because we have excluded all commercial relationships between regions Sand N\Sin its calculus. In a secessionist scenario we can expect that some trade activity would remain after the breakdown, to a greater or lesser extent. What would economic activity be like following a rupture within the country? There are too many factors to consider, each with a high degree of uncertainty. We can mention some of them here. Any hypothetical breakdown scenario that we can imagine will certainly restrict the commercial relationships between the secessionist and the rest of the regions that remain in the country. This is the well-known border effect.16 It establishes that domestic agents trade more with each other than with foreign agents of the same size and distance. A classic case study is that of US-Canadian trade. McCallum (1995) and Helliwell (1996) show that the interprovincial trade between Canadian provinces was more than 20 times larger than trade between Canadian provinces and American states in the period 1988–1990. This is a remarkable effect because both states have low custom tariffs, phased out by the 1988 Free Trade Agreement. In the European Union, Head and Mayer (2000) found that Europeans purchased 14 times more from domestic producers than from equally distant ones, for the average industry in 1985 (tariffs and quotas within the EU phased out by 1968). The range of assumptions made to determine the intensity in this border effect can range from a fully amicable process to a much more traumatic one. For example, Fidrmuc and Fidrmuc (2003) have studied some cases of disintegration in the former Eastern Bloc. They found evidence of a high level of economic integration before breakdown, with internal trade exceeding external trade intensity from 24-fold (for Slovenia and Croatia) to 43-fold (the former Soviet Union and Czechoslovakia). Disintegration is followed by a sharp fall in trade intensity. After breakdown, these levels decrease to twofold in the case of Slovenia and Croatia, sevenfold for the former Czechoslovakia, 13-fold for the Baltics, and 30-fold for Belarus, Russia, and Ukraine. However, Djankov and Freund (2002) reported that, between 1994 and 1996, Russian regions traded 60% more with each other than with former USSR Republics, while there was no significant difference before disintegration. This contrary result supports the hysteresis hypothesis: the tendency for established bilateral trade links to change 16 See Magerman et al. (2016) for a recent and comprehensive review of this phenomenon. 123 656 SERIEs (2021) 12:633–686 relatively slowly. De Sousa and Lamotte (2007) show that there is no empirical evidence to suggest that political disintegration favours either a severe fall in trade or a stable flow in commerce. Apart from the border effect, there are economic aspects in a secessionist process that are specific for each country in question. For example, Spain belongs to the European Union. This political fact implies that a region, which becomes independent, will be out of the EU, at least temporally. Banks could be tempted to change their headquarters to another region in the country to guarantee the financial support of the European Central Bank. Well-established companies in the Spanish and European market could also follow this offshoring process, fearful of losing their market share because of new (unknown) protective custom tariffs, or due to emotional (or rational) boycott campaigns. Such considerations are of an uncertain nature and difficult to assess in advance. Moreover, in each case, the productive structure and the institutional framework will condition the result. If region iproduces a product sold to region j, the entire VAT on this product is not necessarily lost in case of secession. The region could try to compensate for these losses by increasing the tax on the product during its production in the home region. This will increase their price decreasing their demand. The demand elasticity for the product will determine the ultimate tax effect. Once again, some legal rigidities (such as remaining the EU or not) could restrict freedom for such manoeuvres. In short, estimating a priori what the economic outcome of a region will be after a process of secession, although intellectually attractive, is not worthwhile for our practical purposes. For example, we could estimate what might happen if Catalonia separated from the rest of Spain, and then, once we have obtained a rough estimate of what could happen, carry out a rational cost–benefit analysis to find out if it is worth trying to achieve independence. For that purpose, we must compare alternative scenarios. First, we should establish whether the rupture is agreed or taken unilaterally. In the latter case, specify the kind of reaction that we can expect from Spain, with or without the use of force, and specify to what extent Catalonia would obtain international recognition as a state, particularly by the EU. Finally, specify the intensity of the border effect in each of these possible options. This means that we must estimate the corresponding matrix of interregional trade for all possible scenarios. In summary, there are too many parameters, all uncertain, and perhaps moving in opposite directions, to try to predict what might indeed come about.17 We believe that summarising all these parameters in a single optimistic/pessimistic parameter α,so that vα tis an acceptable linear approximation, does not seem like a good idea. 6 Spanish case We apply the theory developed in Sect. 2 to the Spanish case. We use fiscal data from 2011 to 2014, obtained from the System of Territorialised Public Accounts (SCPT) 17 Even if we insist on doing so, we will have to repeat this exercise for all possible coalitions, if we want to get vt. In Spain there are 18 ACs, so we will have to repeat that estimate 218 −1 times, in the EU, 228 −1 times, or in the USA, 250 −1 times, which in practical terms is an unfeasible task. 123 SERIEs (2021) 12:633–686 663 Fig. 4 Redistributive effects of the Spanish rule Sp We can also see this fact in Fig. 4. If the ranking is not altered, the points will be placed on the regression line, and dispersion will be null (R21). A greater dispersion (lower values of R2) corresponds to a greater number of alterations in the ranking. In summary, we can see that the results of the current Spanish rule in terms of equity, ordinality, and distribution of wealth seem, at the very least, quite arbitrary. We can explain this by the fact that, instead of being the result of the systematic application of a set of clear and transparent principles, it has been the product of successive negotiations over time between national political parties and small nationalist leaning parties. What can we say about the stability of Sp rule, from the point of view of the tax game (N,v t)? In the rest of this section, we will see that the Sp rule is located quite close to the area we have qualified as unstable. The first step is to build the Spanish tax game (N,v t). For that purpose, we use the matrix of the commerce inter-regional trades provided by the c-interg project institution. Unfortunately, the trade matrix only provides data for goods, and therefore, we also need to complete it for services. The National Institute of Statistics28 on the INEbase website provides this regional data on services. The statistics supply the data aggregated by CAs only. To make a territorial distribution of sales of services we will follow the same distribution as that provided for goods. At the national level only 13% of total services are destined for export, and therefore we will reduce the services distributedineachregionby87%.Inthis way,weobtainanestimatedapproximationof the matrix Cof interregional exchanges, exports and imports for the Spanish economy. We give these values in Table 9in the “Appendix”. We have added the values of the average types τR iand τI i. Once Cis obtained, we can calculate matrices Iand R.We give matrix Iin Table 8and matrix Rin Table 7in the “Appendix”. Recall that the excess of Sin the Sp tax rule is the difference between the total public resources collected in S(excluding all those derived from trade with the other 28 http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736176865&menu= resultados&dp=1254735576778. 123 664 SERIEs (2021) 12:633–686 regions) and the public expenditure incurred in S,that is e(S)vt(S)−i∈SSpi. Under our approach, vt(S)is the amount of fiscal resources that Sis not willing to share with the remaining regions (N\S) of the country. A positive excess, or very close to it, may therefore support a higher degree of disappointment against the distribution of resources obtained in Sp by S.29 In principle, we have to say that we have not found any region, or coalition of regions with a positive excess30 in the Spanish case. For example, Catalonia, which has high secessionist aspirations, has a negative excess of e({Cat},Sp,v t)e−9382 million If we consider the set of “Catalan Countries”, formed by Catalonia (Cat), Valencia (Va), and Balearics (Ba), its excess is also negative e({Cat,Va,Ba},Sp,v t)e−11,952 million Finally, the group of “foral communities”, formed by the Basque Country (PV) and Navarre (Na), which also have high secessionist aspirations, has a negative excess: e({PV,Na},Sp,v t)e−10,780 million This is undoubtedly a surprising result. Spain is a country with great differences in wealth across its regions. Looking at the IGDP per capita, the AC of Madrid (Ma) is 2.66 times richer than the AC of Ceuta and Melilla (CyMel). As we have just seen, in general, the Sp rule has an acceptable degree of inter-regional solidarity. It is noteworthy that the income allocation produced by the current Spanish financial system, Sp, has not caused an unstable distribution; mainly because it was not a goal pursued consciously by those who conceived it. To see how easily an ambitious redistributive target could generate unstable allocations in Spain, we will calculate what we would get if we wanted per capita public expenditure to be the same in all regions. We achieve it with the egalitarian spending rule Eg. We can compare the values obtained with Sp and Eg in Table 3. Income transfers between regions are very similar: e32,348 million for Sp and e32,288 million for Eg; however, the transfer of wealth has a different effect on both allocations. Consider the coalition formed by four of the latter in the ranking of IGDPh,S{CyMel,An,Ex,Cana},anN\Sis the set formed by the rest of Spanish regions. The excess for N\Swith the Spanish allocation was negative, but nevertheless, the excess with the egalitarian is now positive31: e(N\S,Sp,v t)e−74 million <0,and e(N\S,Eg,v t)e2757million >0 29 And those who think they would really improve their economic well-being if coalition of regions Swere an independent entity. 30 Note that the total number of possible coalitions is 218 −1, and we have not checked all of them. 31 The reader can compute these values easily in the downloadable Excel file TaxFederalism-X-M.xlsx. 123 SERIEs (2021) 12:633–686 665 Table 3 Comparison between the egalitarian rule and the Spanish rule ACs Sp Eg SphEghIGDPhSpGDPhEgGDPhAFB(Sp) AFB(Eg) An 67,924.6 73,992.4 8086 8808 15,600 16,516 17,238 −7689 −13,757 Ara 13,215.9 11,640.7 10,000 8808 24,152 24,791 23,599 −844 731 Ast 11,571.1 9,305.5 10,952 8808 17,491 19,477 17,333 −2098 167 Ba 8943.5 9723.6 8101 8808 25,182 23,809 24,515 1516 736 Cana 18,171.6 18,519.2 8643 8808 16,873 18,915 19,081 −4293 −4641 Cnt 5796.8 5169.5 9877 8808 19,476 20,354 19,286 −516 112 C-L 25,354.9 21,873.8 10,210 8808 19,186 20,910 19,508 −4280 −799 C-M 17,028.8 18,222.7 8231 8808 16,479 17,269 17,846 −1634 −2828 Cat 66,330.2 66,178.3 8828 8808 27,520 26,203 26,183 9892 10,044 Va 38,306.0 43,975.9 7672 8808 19,841 19,493 20,629 1735 −3935 Ex 10,347.0 9656.2 9438 8808 12,623 15,201 14,571 −2827 −2136 Ga 26,074.7 24,138.3 9515 8808 18,324 19,671 18,964 −3692 −1755 Ma 53,937.6 56,773.3 8368 8808 33,314 30,334 30,774 19,205 16,369 Mu 11,024.7 12,921.7 7515 8808 18,045 18,119 19,412 −108 −2005 Na 6345.0 5642.6 9904 8808 27,680 27,860 26,763 −115 587 PV 26,143.3 19,281.6 11,942 8808 27,670 29,217 26,083 −3387 3474 Ri 2910.4 2801.2 9151 8808 23,897 24,029 23,685 −42 67 CyMel 1884.8 1494.3 11,109 8808 12,543 17,393 15,091 −823 −432 123 666 SERIEs (2021) 12:633–686 If the Eg rule were applied, we can conjecture the emergence of a greater degree of rejection on the part of the rich regions of such budgetary regional distribution. In view of this, we might think that, despite everything, politicians have not ended up doing so badly in designing Sp, at least from the point of view of its stability. Unfortunately, this is not so. A closer look shows that Sp is certainly quite close to being unstable. After Madrid, the Balearic Islands is the second AC that loses more IGDPper capita whenapplying Sp(1.373epercapitaloss).Accordingly,itsregional excess is e({Ba},Sp,v t)e−461million <0 This is a relatively extreme situation, because it means that even without computing the tax revenues associated with all commercial relationships with the remaining Spanish ACs, this community can still manage a closer budget amount on its own than it could obtain with the current financial system. It is also worth noting that, without the poor regions formed by S, the other Spanish regions N\Sobtain an even greater excess equal to e−74 million <0 (although still negative). Let us consider again the Spanish allocation Sp, and compare it with what we obtain by applying the balanced allocation ϕand with the weighted balanced allocation ϕw. We show the data in the following Table 4(for a better visualisation of the redistributive effects of these rules, we have placed the CAs in decreasing order according to Table 4 Comparison of the Spanish rule with the balanced allocation and the weighted balanced allocation Acs Adjusted revenues Sp (ϕ)(ϕw) IGDPhSpGDPhjGDPhjwGDPh Ma 73,142.3 53,937.6 64,938 58,179 33,314 30,334 32,041 30,992 Na 6230.1 6345.0 6667 6478 27,680 27,860 28,363 28,068 PV 22,755.8 26,143.3 23,930 23,208 27,670 29,217 28,206 27,876 Cat 76,222.5 66,330.2 75,330 73,531 27,520 26,203 27,401 27,161 Ba 10,459.4 8,943.5 11,057 10,965 25,182 23,809 25,723 25,640 Ara 12,372.0 13,215.9 14,432 14,586 24,152 24,791 25,711 25,828 Ri 2868.6 2910.4 3024 3037 23,897 24,029 24,385 24,425 Va 40,040.6 38,306.0 41,681 42,713 19,841 19,493 20,169 20,376 Cnt 5281.2 5796.8 5442 5654 19,476 20,354 19,750 20,112 C-L 21,074.8 25,354.9 22,444 23,286 19,186 20,910 19,738 20,077 Ga 22,382.9 26,074.7 22,502 23,159 18,324 19,671 18,367 18,607 Mu 10,917.0 11,024.7 11,797 12,083 18,045 18,119 18,645 18,840 Ast 9472.6 11,571.1 9,579 10,027 17,491 19,477 17,592 18,015 Cana 13,878.1 18,171.6 14,595 15,345 16,873 18,915 17,214 17,571 C-M 15,395.1 17,028.8 16,662 18,255 16,479 17,269 17,091 17,861 An 60,235.3 67,924.6 58,247 60,903 15,600 16,516 15,364 15,680 Ex 7520.3 10,347.0 7804 8570 12,623 15,201 12,882 13,580 CyMel 1061.9 1884.8 1180 1330 12,543 17,393 13,238 14,125 123 SERIEs (2021) 12:633–686 667 their IGDPhand decreasing values with respect the Adjusted revenues have been highlighted in italics). We have italicised and shaded values that are worse than adjusted revenues AT,for each of the budget allocations Sp,ϕ, and ϕw. Now, in order to compare the stability of the three allocations, we compute the threshold regional values for each one. Next, we show them in Table 5. We add the additional column RGDPhto normalise the IGDPhvalues between 0 and 1. That is, RGDPh i IGDPh i−min j∈NIGDPh j max j∈NIGDPh j−min j∈NIGDPh j In this way, we show the results homogeneously. We can thus observe the budget allocation evolution over time, or compare their stability between different countries, if desired. Given the threshold values obtained, we can rank the regions in decreasing order. Those at the top have a lower value of αand thus are the ones that will have more reasons for complaint. Consideragainthe BasqueCountry(PV)andCatalonia(Cat),withhighsecessionist aspirations. The threshold value for the Basque Country is αPV(Sp)1.551 >1. It is then in a better position with the present system (Sp) than it would be if it were an independent fiscal entity (ζ1). Although the Balearic Islands (Ba)haveanegative excess, e({Ba},Sp,v t)e−461million <0, they have a very low threshold value Table 5 Spanish regions ordered by their threshold values with respect to Sp,ϕ,andϕw IGDPhRIGDPhα(Sp)α(ϕ)α(ϕw) Ma 33,314 1.00 Ba 0.233 Ma 0.761 Ma 0.565 Na 27,680 0.73 Ma 0.441 An 0.895 Cat 0.860 PV 27,670 0.73 Cat 0.487 Cat 0.954 An 1.035 Cat 27,520 0.72 Va 0.862 Ga 1.019 PV 1.074 Ba 25,182 0.61 Mu 1.027 Ast 1.029 Na 1.102 Ara 24,152 0.56 Ri 1.030 Cnt 1.072 Ri 1.120 Ri 23,897 0.55 Na 1.047 Ex 1.091 Ga 1.127 Va 19,841 0.35 Ara 1.160 Ri 1.111 Ast 1.149 Cnt 19,476 0.33 C-M 1.208 Va 1.131 Cnt 1.167 C-L 19,186 0.32 Cnt 1.230 C-M 1.162 Va 1.213 Ga 18,324 0.28 An 1.408 CyMel 1.165 Ba 1.256 Mu 18,045 0.26 C-L 1.550 Cana 1.167 C-L 1.284 Ast 17,491 0.24 PV 1.551 C-L 1.176 Mu 1.290 Cana 16,873 0.21 Ast 1.564 Na 1.180 Ex 1.337 C-M 16,479 0.19 Ga 1.603 PV 1.191 Cana 1.341 An 15,600 0.15 Ex 1.907 Mu 1.219 C-M 1.365 Ex 12,623 0.00 Cana 1.998 Ba 1.302 CyMel 1.376 CyMel 12,543 0.00 CyMel 2.153 Ara 1.390 Ara 1.419 123 668 SERIEs (2021) 12:633–686 Fig. 5 Hotspots for different αvalues αBa(Sp)0.223. Let Sthe coalition given by four of the latter in the ranking of GDP per capita, S{CyMel,An,Ex,Cana}, being N\Sthe set formed by the rest of Spanish regions. The excess of N\Sis e(N\S,Sp,v t)e−74 million <0. Their threshold value is αN\S(Sp)0.142 which is also very low. We can also show the regression line of these values in a graph. Typically, it should be as shown in Fig. 5. The negative slope of the linear regression clearly indicates a redistributive effect of wealth. The rule treats regions better the poorer they are. The lower shaded part of the graph shows the conflict zone from the point of view of stability (with αi(ψ)close to 0). It is also problematic that many regions are placed in unfair areas. Being in the northeast part of the graph means having a wealth per capita that is higher than the average and, at the same time, benefiting from a greater budget expenditure than if an amount equivalent to the total amount collected in the region were spent (αi(ψ)>1). The regions located in the southwestern area are clearly discriminated by ψ, since they have a lower than average per capita wealth and, at the same time, have a positive fiscal balance, which implies a net transfer of fiscal resources to the other regions.32 Figure 6provides an overview of the stability pattern of Spanish allocation Sp. Again, this shows the worrying cases of PV and Na with per capita wealth at the top of the ranking, and with threshold values above one, and with the case of Vawith a threshold value lower than one, and a per capita wealth lower than the average. We could consider the number of ACs with threshold values α(Sp)<1 excessive. Theyare{Ba,Ma,Cat,Va}.They include:Catalonia,with highsecessionistsupport; the Balearics, with a well-founded grievance against the present Spanish allocation; and Valencia, whose IGDPhis lower than average, has a positive adjusted fiscal balance,andends,after theapplicationofthe Spallocation,withafinal SpGDPheven worse than initially. Indeed, the coalition CC {Cat,Ba,Va}of so-called Catalan Countries has a threshold value of αCC(Sp)0.476 <1. If the present Spanish 32 αi(ψ)<1 is equivalent to ψi<Ti. 123 SERIEs (2021) 12:633–686 669 Fig. 6 Stability of Spanish rule Table 6 Threshold value evolution of the rest of Spain with respect to the foral communities Year 2011 2012 2013 2014 αN\F(Sp) 0.8684459 0.8373311 0.813369 0.6664546 financial system does not change, the nationalist concept of “Catalan Countries” could gain force in the future. The case of “Foral Communities” coalition, F{Na,PV}, formed by the Basque Country (PV) and Navarre (Na), is also interesting. Both communities have an IGDPhabove average and are currently among the richer regions in Spain. After the application of the Sp allocation, they end up with a final SpIGDPheven better than initially. Such a situation would be considered a privilege with respect to the remaining regions that follow the common financial system. There has been a tendency for such discriminatory positions to increase in recent years. Correspondingly, the threshold value for the rest of Spain without foral communities, i.e. N\F, decreases, as can be seen in next Table 6. It is desirable to find an agreement on a new regional financial system in Spain, improving its stability by minimising the reasonable grievances that ACs can hold. Otherwise, the current regional financial system will be a source of increasing political instability. Following Fig. 7shows the values for the balance allocation ϕand the weighted balanced allocation ϕw. From the point of view of stability, the behaviour of ϕimproves, as few regions are in the conflict zone, α(ϕ)<1, only {Ma,Cat,An}. Conversely, the slope of the regression line is almost zero, indicating a poor redistribution of wealth. 123 670 SERIEs (2021) 12:633–686 Fig. 7 Stability comparison of the rules ϕand ϕw For the case of ϕw, only two ACs are in the conflict zone, {Ma,Cat}, and the slope of the regression line is more negative, corresponding to a greater degree of wealth distribution. There is an additional issue related to the redistribution of wealth, which is the ordinality principle: the redistributive effect of a rule “should narrow financing disparities across regions without altering their needs-adjusted relative ranking.”33 In other words, it cannot be that a region iwith initial IGDPh igreater than of region j;itransfers wealth to j, and after that, iends with a final ψGDPh ilower than region j. We can therefore say that a budget allocation ψsatisfies ordinality if its application preserves the ranking of the regions by their wealth per capita: IGDPh i<IGDPh j⇔ψGDPh i<ψGDPh j As shown in Sect. 3, the Spanish allocation Sp breaks clearly such a principle: 12 pairs of ACs interchange their position ranking. The balanced allocation ϕworks much better, and only two pairs change their position: {(Ga,Mu),(CyMel,Ex)}. The weighted balanced allocation ϕwworks a little worse in this sense, as four pairs of regions change their position: {(Ara,Ba),(Mu,Ga),(C−M,Cana),(CyMel,Ex)}. We can observe an indirect relationship between the principles of solidarity (wealth redistribution) and ordinality. Given the unequal wealth distribution in Spain, the most solidary budget allocation considered here, the Spanish allocation Sp, turns out to be relatively unstable. Furthermore, it exhibits bad behaviour from the ordinality point of view. In the opposite direction, the balanced allocation ϕworks better from the stability and ordinality side, although at the cost of being supportive. The weighted balanced allocation ϕwis a trade-off between these two opposite sides: stability and ordinality on one hand, and solidarity/redistribution on the other. 33 De la Fuente et al. (2016). 123 SERIEs (2021) 12:633–686 671 7 Conclusions and final remarks The purpose of this exercise is twofold: theoretical and applied. From a theoretical perspective, we wish to show the stability analysis of the budget regional distribution that can be carried out with the help of the tax game. We believe it to be particularly relevant in the debate regarding regional fiscal balances. The Spanish case has been used to apply this analysis. We have presented two different concepts of stability. In one concept, the fiscal sovereignty over all resources collected in a region belongs exclusively to its regional authorities. In this case, we use the fiscal balances as benchmarks to measure the degree of satisfaction or disagreement with the total budgetary expenditure obtained by the region. There is little room for negotiation here. Significant differences between the collected and the obtained will always be viewed with suspicion. The consequence of this approach is that the only stable budget allocation involves spending in each region the equivalent of what is collected in it. In the other concept we share the fiscal sovereignty among all the regions that make upthecountry.Wecannotconsiderregionsaseconomicautarchies.Weshouldconsider a good part of the fiscal resources collected in each region as common property, given that they originate from the exchange of goods and services between economic agents residing in the different regions that form the country. According to this approach, at least that shared part of fiscal resources can be the subject of negotiation in its redistribution, widening the scope for negotiation. We have used the vα ttax game to estimate those resources that can be the object of negotiation. We approach the first concept of stability by v1 t, while in the second we use the core of the game v0 t. Apparently, the two approaches are incompatible; however, parameter αallows us to pass from one to the other smoothly. In this way, we can take a more nuanced view, and not see exclusively in black or white terms. In fact, the fiscal balances for Spain for the year 2014 present a certainly conflicting distribution. Although somewhat arbitrarily, the fiscal balances imply an acceptable degree of territorial solidarity. Unfortunately, Spain presents large differences in the territorial distribution of wealth, and the moderate degree of solidarity that the Spanish allocation Sp presents, causes its corresponding stability problems. We have followed the theoretical exercise of applying in this context two solutions brought from the literature of cooperative games: The balanced allocation ϕ, which coincides with the nucleolus and the Shapley value of the game v0 t, and the weighted balanced allocation ϕw, which coincides with the weighted Shapley value of v0 t.ϕis the most stable and best behaved in preserving regional rankings of wealth per capita, but, in comparison, it redistributes little wealth between rich and poor regions. ϕw is less stable and slightly worse in preserving rankings than ϕ, but it behaves better by redistributing wealth. We can therefore consider ϕwas an acceptable compromise between the principles of solidarity, stability and ordinality. The analysis of fiscal balances using the tax game tool relates only to the ratio between the total expenditures made by all public administrations in the territory and the total revenues obtained. We can use it to decide how much should be spent on each territory, irrespective of how to distribute this amount among the different public authorities. Once we decide how much to allocate to each region, how to distribute the 123 672 SERIEs (2021) 12:633–686 different expenditure items between the central government and the corresponding regional and local authorities becomes a different, though not trivial, problem. We should address both issues separately, and they ought not be intertwined. One aspect related to the process of agreeing on a budget allocation among the regions is the trust between them regarding the data provided on taxes collected. Depending on the tax rule selected, it opens the door to a possible strategic manipulation of the data. There are many papers devoted to designing rules that are immune to suchmanipulation.Thisnon-manipulabilityconditioniscalledstrategy-proofnees (see the classical papers of Gibbard (1973) and Satterwhite (1975)). An example of application to a claims/taxation problem is Ju et al. (2007), where agents can merge/split their claims; and an application to fiscal competition among jurisdictions is Wildasin (1988). We can make two comments here. First, a way to overcome this problem is to create a common and independent fiscal authority in charge of collecting all taxes, which provide the tax vectors Iand R; and a common and independent research centre in charge to collect and provide all data related to the matrix C. Obviously, the decision about how to distribute and how to spend the budget is a political decision for the political authorities. Second, there is an indirect way to deduce the plausibility of the data provided by a region. Suppose that each region iannounces as data the values of TiIi+Ri, and their corresponding values of matrix C, that is, (ci1,...,cin), (c1i,...,cni),xiand mi, from which announces the coefficients Iij and Rij. If a region iintends to underestimate its values to benefit from this misrepresentation, the data provided must be compatible with that provided by the others. For example, cij and cji announced by ishould be equal to that of coefficients announced by j. Otherwise, someone is cheating. We end by mentioning some issues that would be of interest for further research. From a theoretical point of view: 1. Converting the problem of redistributing the public budget between regions into the problem about deciding which solution to select in a cooperative tax game, opens a new line of exploration. There are many works that analyse the cooperative solution rules axiomatically. Many of the properties considered in this literature could be transferred to our context, allowing a better evaluation and comparison of different tax financing rules. 2. There is a computational problem in checking whether a budget allocation belongs to the core of the tax game N,v0 t. The size of the subcoalitions of Nis 2nand increases exponentially with n. It would be of interest to find efficient algorithms to check whether an allocation belongs to the core in polynomial time, helping with the convexity and simplicity of vα t. 3. It would be of interest to apply other cooperative solution concepts to the tax games setting, such as the Dutta–Ray egalitarian solution (see Dutta and Ray 1989). From a practical perspective: 4. Wecanextend this stability analysisofthefinancialregional system to anycountry, or confederation of countries, such as the EU, USA, whenever data for taxes and inter-regional commercial trade are available. 5. Totheextentthatapractical applicationissought,there isampleroomforimprovement in the data collected. There is a great deal of hidden work in constructing 123 SERIEs (2021) 12:633–686 679 Table 7 (continued) Matrix RVa Ex Ga Ma Mu Na PV Ri CyMel X exports Ara 490.8 28.7 46.5 291.0 28.9 299.0 376.9 52.5 6.8 2256.9 Ast 231.9 13.7 455.5 157.6 64.7 38.1 178.6 98.7 0.7 1377.5 Ba 139.9 0.0 56.8 16.5 43.1 0.0 6.3 0.0 5.7 445.2 Cana 94.4 1.1 95.4 197.8 245.8 6.2 670.9 2.5 2.8 850.0 Cnt 177.8 6.0 91.0 117.6 6.3 30.5 446.6 9.1 9.5 859.3 C-L 368.5 150.8 635.7 1171.6 116.7 161.5 905.8 111.4 17.2 3256.6 C-M 944.3 408.5 99.8 1798.4 235.4 61.5 77.4 20.8 6.2 1588.0 Cat 2555.4 26.2 424.5 1609.7 494.9 408.2 763.8 258.4 54.1 17,143.5 Va 10,090.7 69.7 200.5 1205.2 783.5 136.7 155.7 136.5 22.0 8053.7 Ex 91.8 2104.4 26.4 536.5 11.2 23.5 19.6 15.5 1.3 666.7 Ga 482.4 7.8 5574.3 421.8 34.6 32.0 368.1 19.8 20.0 4590.6 Ma 3856.4 819.4 1822.0 11,827.2 836.2 417.2 1922.5 313.9 93.8 10,779.1 Mu 1400.2 33.4 23.2 205.0 1650.7 15.5 40.2 7.0 9.5 2459.3 Na 143.6 9.2 22.0 108.8 25.2 927.4 352.7 101.5 0.6 1681.8 PV 385.7 30.9 290.1 316.5 91.5 612.1 5962.2 118.7 5.8 5722.9 Ri 66.0 11.4 6.2 99.4 1.0 153.5 160.4 559.0 0.2 365.2 CyMel 16.3 1.0 8.7 38.9 43.1 0.3 72.9 0.2 58.8 96.9 123 680 SERIEs (2021) 12:633–686 Table 8 Regional indirect taxes in Spain Matrix I An Ara Ast Ba Cana Cnt C-L C-M Cat Va Ex Ga Ma Mu Na PV Ri CyMel An 6059.7 51.5 132.5 112.3 399.8 10.1 160.1 355.8 438.5 492.6 493.5 65.4 664.9 317.6 39.8 73.0 1.6 168.4 Ara 157.3 661.9 20.4 3.5 17.1 10.2 88.9 70.9 490.0 212.2 14.9 21.3 166.0 10.4 100.2 124.4 17.8 2.6 Ast 31.8 14.8 726.4 7.5 6.2 75.8 165.6 13.3 35.0 67.2 4.8 139.7 60.3 15.6 8.6 39.5 22.4 0.2 Ba 16.8 3.5 7.8 1272.2 26.8 0.1 0.0 7.5 84.0 30.2 0.0 13.0 4.7 7.8 0.0 1.0 0.0 1.1 Cana 399.9 1.6 8.4 7.0 1624.7 5.7 1.7 4.4 88.1 27.6 0.4 29.5 76.4 60.0 1.4 149.9 0.6 0.7 Cnt 11.2 8.6 26.8 1.0 4.0 345.7 80.9 7.2 33.6 54.8 2.2 29.7 47.8 1.6 7.3 105.0 2.2 2.6 C-L 163.5 54.9 79.8 3.1 55.2 165.6 1730.9 138.0 120.0 150.1 73.7 273.9 629.7 39.6 51.0 281.5 35.5 6.1 C-M 395.5 29.0 14.1 6.0 27.8 11.0 97.8 930.7 138.2 333.9 173.2 37.3 839.2 69.5 16.9 20.9 5.8 1.9 Cat 648.3 787.0 72.3 451.7 295.7 89.1 237.1 370.9 6729.3 934.6 11.5 164.3 777.1 151.0 115.7 213.2 73.9 17.4 Va 454.1 121.2 16.8 214.8 86.9 26.0 83.0 264.4 447.1 3257.7 27.0 68.5 513.5 211.0 34.2 38.4 34.5 6.2 Ex 266.7 2.5 1.6 0.0 7.7 1.2 51.7 52.8 13.7 24.0 658.8 7.3 184.9 2.5 4.8 3.9 3.2 0.3 Ga 306.6 55.5 282.2 29.0 52.1 43.1 272.0 54.2 172.0 194.6 3.8 2379.7 224.6 11.6 10.0 113.3 6.2 7.1 Ma 1625.4 257.0 296.7 135.9 575.0 131.0 450.1 822.7 1046.4 1031.8 262.9 515.8 4176.5 186.7 86.5 392.6 65.7 22.0 Mu 351.1 10.8 16.0 26.2 18.1 3.7 11.9 95.3 66.0 618.2 17.7 10.8 119.5 608.1 5.3 13.5 2.4 3.7 Na 39.6 105.5 11.3 0.5 12.4 20.6 56.8 22.7 106.9 72.3 5.6 11.7 72.3 10.6 361.8 135.5 40.0 0.3 PV 101.4 75.5 99.4 8.6 30.5 94.4 313.7 50.1 135.3 157.7 15.2 125.5 170.8 31.2 193.9 1860.7 38.0 2.1 Ri 24.5 16.1 5.3 0.2 1.4 6.6 108.3 7.5 49.1 30.9 6.4 3.1 61.4 0.4 55.7 57.3 204.7 0.1 CyMel 15.0 0.0 0.0 0.0 0.5 0.0 0.4 0.1 0.4 0.7 0.0 0.4 2.1 1.5 0.0 2.3 0.0 2.1 M imports 4736.9 518.2 352.6 155.1 405.7 181.2 1008.1 572.3 6746.8 2222.7 122.3 1586.0 6981.5 1037.8 335.5 1359.8 93.9 58.6 123 SERIEs (2021) 12:633–686 681 Table 9 2014 trade exchange matrix in Spain Regional trades 2014 An Ara Ast Ba Cana Cnt C-L C-M Cat Va Ex Ga An 39,603.5 849.5 1255.6 1054.0 3503.9 103.3 1944.9 3713.6 4693.5 4736.9 3956.9 594.4 Ara 1028.2 10,923.6 193.4 33.3 150.1 105.2 1080.3 740.3 5245.1 2040.6 119.4 193.2 Ast 207.8 243.8 6884.6 70.5 54.2 778.8 2011.5 139.1 374.6 646.1 38.1 1269.2 Ba 109.6 57.4 73.7 11,937.5 235.0 1.3 0.0 78.7 898.9 290.3 0.0 117.9 Cana 2613.6 26.3 79.4 65.3 14,239.4 58.3 20.3 46.1 943.0 265.6 3.0 268.4 Cnt 73.0 141.2 254.4 9.3 34.8 3551.5 982.7 75.0 359.4 526.7 17.7 269.6 C-L 1068.6 906.3 755.9 28.8 484.0 1701.5 21,022.0 1440.8 1285.0 1442.8 590.6 2489.3 C-M 2584.6 479.0 133.4 56.2 244.0 113.1 1188.3 9715.3 1479.5 3210.5 1388.9 339.4 Cat 4236.8 12,988.8 685.3 4238.6 2591.8 914.9 2880.1 3871.6 72,032.9 8987.0 92.0 1492.8 Va 2967.5 2000.5 158.9 2015.0 761.5 266.7 1008.3 2760.2 4786.4 31,324.4 216.3 622.5 Ex 1743.3 41.7 15.3 0.0 67.8 11.9 627.5 550.8 146.3 230.5 5282.9 66.4 Ga 2004.0 916.6 2674.9 272.5 456.5 443.0 3304.0 566.0 1840.8 1871.5 30.1 21,626.3 Ma 10,622.7 4242.1 2812.4 1275.0 5039.6 1346.2 5466.4 8588.0 11,201.5 9921.3 2108.2 4687.4 Mu 2294.4 177.6 152.1 245.4 158.9 37.7 145.0 994.5 706.6 5944.3 141.7 98.5 Na 258.9 1741.0 107.0 4.5 108.3 212.1 690.4 236.5 1143.9 695.3 44.7 106.7 PV 663.0 1245.3 941.8 80.8 267.4 970.0 3809.6 523.1 1447.8 1516.4 121.6 1140.4 Ri 160.3 266.2 50.1 1.9 11.9 68.3 1315.5 78.0 525.7 297.2 51.2 27.9 CyMel 97.8 0.7 0.3 0.3 4.1 0.3 4.7 0.9 4.3 6.4 0.4 3.4 Internal purchases 32,734.3 26,323.9 10,344.0 9451.2 14,173.8 7132.6 26,479.4 24,403.0 37,082.4 42,629.5 8920.9 13,787.5 M Imports 30,957.9 8552.9 3342.0 1455.0 3556.0 1861.3 12,244.0 5974.2 72,220.5 21,372.9 980.4 14,413.3 123 682 SERIEs (2021) 12:633–686 Table 9 (continued) Regional trades 2014 An Ara Ast Ba Cana Cnt C-L C-M Cat Va Ex Ga TP total purchases 103,295.6 45,800.4 20,570.6 22,843.7 31,969.2 12,545.4 59,745.3 40,092.5 181,335.8 95,326.8 15,184.3 49,827.2 Indirect taxes 15,805.3 2775.1 2170.4 2434.6 3647.7 1221.1 4919.2 3840.8 16,940.3 9913.8 1893.6 5482.8 τI i15% 6% 11% 11% 11% 10% 8% 10% 9% 10% 12% 11% Regional trades 2014 Ma Mu Na PV Ri CyMel Internal sales X exports TS total sales DR direct revenues τR i An 4844.4 3660.3 493.5 919.1 19.9 1845.6 38,189.1 26,649.6 104,442.2 37,893 36% Ara 1209.6 120.2 1243.0 1566.7 218.4 28.4 15,315.3 9382.4 35,621.3 8569 24% Ast 439.0 180.3 106.2 497.5 274.9 1.9 7333.5 3838.4 18,056.5 6480 36% Ba 34.2 89.6 0.0 13.1 0.0 11.8 2011.5 924.0 14,873.0 7166 48% Cana 556.7 691.8 17.4 1888.3 7.1 7.9 7558.5 2392.5 24,190.4 8594 36% Cnt 348.5 18.6 90.5 1323.3 27.1 28.1 4580.0 2546.5 10,677.9 3603 34% C-L 4587.5 456.9 632.5 3546.7 436.0 67.4 21,920.7 12,751.9 55,694.6 14,223 26% C-M 6114.2 800.4 209.1 263.0 70.8 21.2 18,695.5 5398.9 33,809.7 9944 29% Cat 5661.2 1740.5 1435.5 2686.3 908.7 190.3 55,602.0 60,291.2 187,926.1 53,436 28% Va 3741.4 2432.1 424.5 483.3 423.7 68.2 25,137.2 25,001.1 81,462.6 26,242 32% Ex 1346.9 28.2 59.0 49.2 39.0 3.3 5027.3 1673.7 11,983.9 4774 40% Ga 1636.6 134.2 124.3 1428.0 76.8 77.4 17,857.2 17,809.7 57,293.3 14,768 26% Ma 30,427.8 2151.2 1073.4 4946.0 807.6 241.4 76,530.5 27,731.4 134,689.6 52,354 39% Mu 870.4 7007.9 65.7 170.5 29.7 40.2 12,273.3 10,440.7 29,721.9 7001 24% Na 526.8 122.1 4489.5 1707.4 491.5 3.1 8200.2 8141.1 20,830.8 4303 21% 123 SERIEs (2021) 12:633–686 683 Table 9 (continued) Regional trades 2014 Ma Mu Na PV Ri CyMel Internal sales X exports TS total sales DR direct revenues τR i PV 1244.3 359.9 2406.5 23,442.0 466.8 22.8 17,227.6 22,501.0 63,170.6 16,067 25% Ri 447.7 4.7 691.1 722.2 2516.7 1.0 4721.0 1644.4 8882.1 1973 22% CyMel 15.2 16.9 0.1 28.5 0.1 23.0 184.4 37.9 245.3 627 255% Internal purchases 33,624.7 13,007.8 9072.4 22,239.3 4298.2 2660.1 M Imports 50,863.6 11,959.8 4162.7 17,131.4 1154.4 642.1 TP total purchases 114,916.0 31,975.6 17,724.6 62,812.7 7969.4 3325.2 Indirect taxes 15,773.4 2774.7 1428.5 4985.8 648.3 303.4 τI i14% 9% 8% 8% 8% 9% 123 684 SERIEs (2021) 12:633–686 References Aumann R, Kurz M (1977) Power and taxes. 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