Distributing the European structural and investment funds from a conflicting claims approach

Abstract

In order to support economic development across all European Union regions, €351.8 billion –almost a third of the total EU budget– has been set aside for the Cohesion Policy during the 2014–2020 period. The distribution of this budget is made through five main structural and investment funds, after long and difficult negotiations among the EU member states. This paper analyzes the problem of allocating the limited resources of the European Regional Development Fund as a conflicting claims problem. Specifically, we attempt to show how the conflicting claims approach fits this actual problem, and we propose alternative ways of distributing the budget via (i) claims solutions or (ii) the imposition of bounds (guarantees) to each of the regions. By applying this approach we also show that there is a claims solution that performs better than the others by reducing inequality and promoting convergence to a greater degree. It is clear that political bargaining will always be part of the allocation process. However, having an intuitive initial proposal may help politicians to find the best agreement. To that effect, we propose the use of a claims solution as a way to find an initial proposal for future policy changes concerning the allocations of the EU structural funds.

Zusammenfassung

Zur Unterstützung der wirtschaftlichen Entwicklung in allen Regionen der Europäischen Union wurden im Zeitraum 2014–2020 351,8 Mrd. EUR – fast ein Drittel des gesamten EU-Haushalts – für die Kohäsionspolitik bereitgestellt. Die Verteilung dieses Budgets erfolgt nach langen und schwierigen Verhandlungen zwischen den EU-Mitgliedstaaten über fünf große Struktur- und Investitionsfonds. In diesem Artikel wird das Problem der Allokation der begrenzten Mittel des Europäischen Fonds für regionale Entwicklung als widersprüchliches Anspruchsproblem analysiert. Konkret versuchen wir aufzuzeigen, wie der konfliktbehaftete Schadenansatz zu diesem realen Problem passt, und schlagen alternative Wege zur Verteilung des Budgets durch (i) Schadenslösungen oder (ii) die Auferlegung von Grenzen (Garantien) für jede der Regionen vor. Mit diesem Ansatz zeigen wir auch, dass es eine Lösung gibt, die andere in Bezug auf den Abbau von Ungleichheit und die stärkere Förderung der Konvergenz übertrifft. Natürlich werden politische Verhandlungen immer Teil des Zuteilungsprozesses sein. Ein intuitiver Erstvorschlag kann den politischen Entscheidungsträgern jedoch helfen, die beste Lösung zu finden. Zu diesem Zweck schlagen wir vor, mit Hilfe einer Schadenlösungslösung einen ersten Vorschlag für zukünftige politische Änderungen bei der Zuweisung von EU-Strukturfondsmitteln zu unterbreiten.

1 Introduction

The main objective of the European Union (EU) is to strengthen the social and economic cohesion of the EU regions, as well as to reduce the inequalities among them. In doing so, and in accordance with the objectives of the Europe 2020 strategy, the European Structural and Investment Funds (ESIF) are implemented through five main funds: the European Regional Development Fund (ERDF), the European Social Fund (ESF), the Cohesion Fund (CF), the European Agricultural Fund for Rural Development (EAFRD) and the European Maritime and Fisheries Fund (EMFF).¹

In order to support job creation, business competitiveness, economic growth, sustainable development, and improve citizens’ quality of life, the Regional Policy has allocated €351.8 billion -almost a third of the total EU budget- to the Cohesion Policy funds for the 2014–2020 period. According to the Panorama Inforegio magazine, support from the EU’s cohesion policy had led member states to experience a 5% growth in per capita gross domestic product. The bulk of Cohesion Policy funding, over 50%, is allocated to less developed European regions in order to help them to catch up and to reduce the economic, social and territorial disparities that still exist in the EU.²

Among all the aforementioned funds, the present paper focuses on the European Regional Development Fund (ERDF), which represents almost 44% of the total budget. These funds are allocated at the NUTS level 2, which is a regional classification providing a harmonized hierarchy of regions: the NUTS classification subdivides each member state into regions at three different levels, from larger to smaller areas. For practical reasons the NUTS classification generally mirrors the territorial administrative division of the member states, supporting availability of data and policy implementation capacity. Specifically, the NUTS regulation defines minimum and maximum population thresholds for the size of the NUTS regions: NUTS level 2 corresponds to regions with populations between 800,000 and 3,000,000 inhabitants. Taking into account this division, the regional eligibility for the ERDF is calculated on the basis of regional GDP per inhabitant (per capita), and NUTS level 2 regions are ranked and split into three different groups, according to their per capita GDP: $R1$ corresponding to the most developed regions, $R2$ which refers to transition regions, and $R3$ which includes the less developed regions.

Although the final decision on the way the budget is allocated is the result of a political bargaining process (between the European Commission and the Member States), an initial proposal is presented as a starting point. Nowadays, the European Commission proposes allocations using the so-called Berlin method. This is a methodology, devised in 1999, for allocating cohesion funds based on regional and national prosperity and unemployment. Our main objective is to propose a new initial proposal to distribute the ERDF budget. And we do it by using the claims problem approach.³

A claims problem involves a set of agents demanding a part of some (perfectly divisible) endowment. It is a conflicting claims problem if the endowment cannot honor all the claims in full. If we consider the ERDF budget as the endowment to be allocated, and the claims consist of the amounts required to develop some projects (mainly in infrastructures: airports, universities, hospitals, etc.) that regions could not afford individually, it is noteworthy that the available budget is not enough to satisfy all the claims that the regions have on it.

The most difficult and controversial part of our approach is to define the claim of each region. As the ERDF projects must be co-financed by the Member States (in a percentage that depends on the category of the region), it is up to these states to present the co-financed projects properly. An alternative way is to consider previous allocations and to observe the gap between the different region’s Gross Domestic Product (normalised in PPS euros). Once the projects have been selected, or each region’s claim has been fixed, the conflicting claims problem is well defined and the ERDF budget must be rationed by using well-known claims rules.

As far as we know, the recent paper by Fragnelli and Kiryluk-Dryjska (2019) is the only reference analyzing the ERDF distribution as a conflicting claims problem. As mentioned in that paper “this approach has the great advantage that solutions may be obtained with a fast computation.” In this context, we should also mention the papers by Kiryluk-Dryjska (2014, 2018) that propose a formal framework for rural development budget allocation by using fair division techniques. Conflicting claims problems have also been used to analyze other related economic and social problems: in the education sector Pulido et al. (2002) use this approach for obtaining an efficient allocation of the university funds; in the fishing sector, it is a useful tool for searching possible solutions to address fish shortages, by proposing fishing quotas among a number of agents within an established perimeter (Iñarra and Prellezo 2008; Iñarra and Skonhoft 2008; Kampas 2015); or, in the negotiations on CO2 emissions, a relevant issue nowadays, Giménez-Gómez et al. (2016) and Duro et al. (2020) propose an appealing distribution by analyzing this situation as a conflicting claims problem.

We consider the use of claims rules to propose an initial allocation for distributing the EU funds in order to achieve social cohesion and convergence among member states. In doing so, our first step is to formally introduce the distribution of the ERDF budget as a conflicting claims problem. Once this is implemented, we apply some of the usual claims rules and compare them from a convergence perspective (comparing changes in the inequality of regions once each of the proposals is applied). We define a convergence ratio to analyze this question. Our results show that the allocations proposed by all of the claims solutions reduce the divergence among regions. Moreover, we obtain that, among the analyzed claims solutions, the constrained equal losses performs better than the other ones and better than the current allocation, for the purpose of achieving the convergence objectives.

Even though the EU has made significant efforts to “regularize” and “rationalize” the formal process for policy-making and the procedures for the negotiation of regional development programmes, the empirical evidence suggests that the interactions remain very complex and uncertain (Conzelmann 1998). As mentioned in Dotti (2015) “first, the EU and the member states decide general policy goals, the total budget and regional eligibility criteria. Next, each member state designs its own regional development strategies, according to the general framework and with the support of the EC (European Commission). In the final step, national and regional authorities have to implement regional development programmes, as agreed during previous phases and under the supervision of the EC.” Our proposal (the use of claims rules to solve the distribution problem) is about eliminating discretionary decisions and making the process of allocating the EU funds transparent.

There are many papers analyzing the importance of ESIF funds in order to achieve greater social cohesion and economic growth among the European Union countries, most of them looking for the results obtained through the policies applied. For instance, Rodríguez-Pose and Fratesi (2004) apply cross-sectional and panel data analyses to observe the impact of European Structural Funds in Objective 1 regions; also Puigcerver-Peñalver (2007) studies the impact of the ESIF funds on the economic growth of the regions; Mohl and Hagen (2010) analyze the economic growth of the European Union countries, from a financial perspective, for the NUTS level 1 and NUTS level 2 regions; Bouayad-Agha et al. (2013) consider an econometric model to analyze the effect of the cohesion policies on the European economies; and Dall’Erba and Fang (2017) apply a meta-analysis with the objective of studying the impact generated by the ESIF funds on the development of the recipient regions.

Some recent papers deal with political issues of the governance of the funds and the political/economic challenges. Bouvet and Dall’erba (2010) advocate that the decision process involves interaction between the actors (European Commission and Member States). Bodenstein and Kemmerling (2012) point out that the process of the distribution of regional funds has been termed a two or three-level game and the bargaining occurs between the regional and national actors. Chalmers (2013) provides some evidence that constitutionally strong regions are better lobbying advocates for investment projects. In Charron (2016) it is argued that “the determination of Structural Funds is based on an interaction between a region’s formal institutions (the level of a regional autonomy) and informal institutions (its government quality level).” Papp (2019) analyzes, for the case of Hungary, the electoral connection between legislators and voters, and the European Union’s contribution to regime legitimization. Finally, Crescenzi et al. (2020) argue that “in a context of rising economic nationalism and Euroscepticism, the value added of a supranational Cohesion Policy of the European Union is constantly under scrutiny” and propose to explore new institutional and policy arrangements in order to offer more flexibility and that “EU policies need to buy-in ‘national’ policy agendas in a more timely and systematic manner, sharing responsibility for (and ownership of) key policy reforms.” In this Eurosceptic scenario, the possibility of offering a neutral and fair initial point, as offered by the claims rules, could help to reach more consensual budget distributions. Moreover, the proposed claims rules can be supported by equity and fairness criteria.

The remainder of the paper is organized as follows. Next, Sect. 2 formally presents the notion of conflicting claims problem and some of the main solutions in the literature. Sect. 3 presents the ERDF conflicting claims problems and applies the different claims solutions to the EU data. Sect. 4 analyzes and compares the proposed allocations from the point of view of convergence, and Sect. 5 studies the problem of ensuring some guarantees (in awards and in losses) for all regions. Some final comments in Sect. 6 conclude the paper.

2 Conflicting claims problems

A claims problem appears whenever several (economic and/or social) actors, the agents, demand a part of some (perfectly divisible) endowment. It is a conflicting claims problem if the endowment cannot honor all the claims in full. The typical example is that of bankruptcy: a firm does not have enough assets to pay all its debts and the endowment (the assets of that firm) must be distributed among its creditors. Another example would be the division of an estate amongst several heirs, particularly when the estate cannot meet all the deceased’s commitments.

Although some references to this situation appear in ancient literature (2000-year old Babylonian Talmud), modern literature begins with the seminal paper by O’Neill (1982), also originated in a Talmud rights arbitration problem. There are three simple methods for solving bankruptcy problems in practice: The proportional rule (divide the endowment proportionally to each agent’s claim), the constrained equal-awards solution (divide the endowment equally among the agents, ensuring that no agent gets more than their claim), and the constrained equal-losses solution (divide the losses equally, i.e., the difference between the total claim and the endowment, ensuring that no agent ends up with a negative transfer). Apart from these solutions, we will also introduce one additional method obtained by combining them: the $\alpha^{\min}$ rule (Giménez-Gómez and Peris 2014). Next, we formally define the problem and rules.

We study problems where an endowment $E> 0$ must be divided among a group of agents $N=\left\{1,2,\ldots,n\right\}$. Agents $i\in N$ are identified by their claim $c_{i}\geq 0$ on the endowment $E$. We will denote by $c=\left(c_{1},c_{2},\ldots,c_{n}\right)$ the vector of claims. The aggregate claim $C$ is given by $C=\underset{i=1}{\overset{n}{\sum}}c_{i}$ and a conflicting claims problem appears whenever the aggregate claim is greater than the available endowment: $C> E$. The pair $(E,c)$ represents the conflicting claims problem.

The question that arises is: how to divide the endowment among the agents? This question is answered by defining rules. A claims rule is a single valued function $\varphi$ such that for each conflicting claims problem $(E,c)$ it assigns an amount $\varphi_{i}(E,c)$ to each agent $i\in N$, fulfilling:

• a) $0\leq\varphi_{i}(E,c)\leq c_{i}$ (non-negativity and claim-boundedness); and

• b) $\underset{i=1}{\overset{n}{\sum}}\varphi_{i}(E,c)=E$ (efficiency).

That is, the endowment $E$ is completely distributed among the agents, and no agent receives neither a negative amount, nor an amount exceeding the corresponding claim. Some commonly used claims rules are:

• The proportional rule ($P$) is the most popular one, and it divides the endowment proportionally to the claim of the agents.

  For each $(E,c)$ and each agent $i\in N$, $P_{i}(E,c)\equiv\lambda c_{i}$, where $\displaystyle\lambda=\dfrac{E}{C}$.

• The constrained equal awards rule (CEA) (Maimoindes 2000) equalizes the amount each agent receives, such that no agent receives more than their demand.

  For each $(E,c)$ and each agent $i\in N$, $\textit{CEA}_{i}(E,c)\equiv\min\left\{c_{i},\lambda\right\}$, where $\lambda$ is chosen so that $\displaystyle\sum_{i=1}^{n}\min\left\{c_{i},\lambda\right\}=E$.

• The constrained equal losses rule (CEL) (Maimoindes 2000; Aumann and Maschler 1985) analyzes the problem from the point of view of losses (what the agents do not receive with respect to their claims) and proposes equalizing losses, such that no agent receives a negative amount.

  For each $(E,c)$ and each agent $i\in N$, $\textit{CEL}_{i}(E,c)\equiv\max\left\{0,c_{i}-\lambda\right\}$, where $\lambda$ is chosen so that $\displaystyle\sum_{i=1}^{n}\max\left\{0,c_{i}-\lambda\right\}=E$.

• The $\mathbf{\alpha}^{\mathbf{\min}}$ rule (Giménez-Gómez and Peris 2014) guarantees a minimum amount to each agent: if possible, all agents first receive an amount that coincides with the lowest claim and then, the remaining endowment is distributed proportionally to the reduced claims (the initial claims minus the amount already received). If the endowment does not allow each agent to receive at least the lowest claim, then all agents receive the same amount. That is:

  For each $(E,c)$,

  $$\alpha^{\min}(E,c)\equiv\left\{\begin{array}[]{lcl}\frac{1}{n}E&\text{ if }&E\leq nk,\\ \\ k+P(E-nk,c-k)&\text{ if }&E\geq nk,\\ \end{array}\right.$$

  where $k=\min\left\{c_{1},c_{2},\ldots,c_{n}\right\}$.

2.1 The socially accepted properties: axiomatic analysis

To analyze the behavior of the aforementioned claims rules, we propose two separate sets of properties that solutions of conflicting claims problems should fulfill: what we call minimal requirements and additional principles.⁴

The minimal requirements should contain the basic properties: equal treatment of equals, anonymity, order preservation and resource monotonicity. Note that these principles ensure that there is no discrimination among the agents (regions), in the sense that only the claim matters, and the regions with larger claims would not receive a smaller allocation than those regions with smaller needs. Note that, as Table 1 depicts, all these properties are satisfied by the claims rules we presented.

Appart from these basic requirements, there are some additional principles that differentiate one claims rule from another. In particular, we consider the properties of super-modularity, composition down and composition up. Super-modularity requires that regions with larger claims experience a greater increase in the ERDF budget. Composition down and up analyze the coherence of the rules whenever the endowment (the budget) decreases or increases. Table 1 depicts which of the aforementioned principles are fulfilled by the proposed claims rules.⁵

Table 1 Principles and claims rules. The table shows which principles are satisfied by the claims rules considered. These results can be found in Thomson (2019)

Note that the proposed claims rules satisfy all the axioms and further analysis is needed to select one of them in a particular scenario. In Sect. 4 we will introduce some criteria to select just one of these rules in the context of the distribution of the ERDF budget.

3 The distribution of the ERDF as a conflicting claims problem

Before presenting our model, it it noteworthy to observe that although ERDF resources are allocated between three categories of regions (NUTS level 2), the allocation of each region also depends on other variables. According to European Commission guidelines, this allocation depends on the category of the region (level of development), the gap between the region’s GDP and the average EU GDP, and the state in which the particular region is located. Additionally, some premiums are allocated to less developed and transition regions in order to promote employment, youth employment, increase of the education level, decrease of gas emissions, or for migration purposes.⁶

In what follows, we present a very simplified version of this scenario in which the actors are the three different categories of region in each country: less developed, transition and more developed regions. This defines 47 agents in the distribution problem. Our objective is to show how claims rules perform in this situation. A more complete analysis at regional level (without joining the regions of the same type in a country) could be carried out by using the conflicting claims approach at the cost of enlarging the number of agents involved in the claims problem to 256. As mentioned in Fragnelli and Kiryluk-Dryjska (2019), it is worthwhile to remark that the computational aspects may be easily dealt with. In any case, we propose the solution given by a claims rule as an initial distribution to be discussed by political actors. As mentioned in Bouvet and Dall’erba (2010) “political bargaining will always be part of the allocation process because there are too many potential recipient regions, and the decision process involves interaction between several levels of the political arena.”

First, some stylized descriptive facts about the situation (regarding the regions’ GDP) will be useful in our analysis. There are 256 NUTS2 regions in the EU. In 2018, regional GDP per capita, expressed in terms of purchasing power standards (PPS), ranged from 30% of the European Union (EU) average in Mayotte, an overseas region of France, to 263% in Luxembourg. As Fig. 1 depicts, there is considerable variation both between and within the EU Member States.

Fig. 1

EU GDP per capita in NUTS 2. Source: Eurostat, 2021

Although all EU countries belong to the so-called First World, there are notable differences in terms of regional GDP per capita. As Fig. 2 shows, 60% of the population which corresponds to the most developed regions (R1), generates 73% of the global GDP of the EU. While 27% of the population lives in the less developed regions (R3), in which 17% of the global GDP of the EU is generated.

Fig. 2

GDP and population in the EU. Source: Eurostat, 2021

Nowadays, in order to correct the above differences, the European Commission proposes allocations to the different regions using the so-called Berlin method. This is a methodology, devised in 1999, for allocating cohesion funds based on regional and national prosperity and unemployment. Although remaining consistent in focus, the criteria used in this method have evolved with each programming period to reflect new challenges and new policy objectives. The Berlin Formula key points are:⁷

• The eligibility of regions within the Cohesion Policy architecture (more, less developed or transition regions) is based on a reference period taking average economic data of three years.

• The methodology is largely based on regional statistics at NUTS level 2 regions.

• The different categories of regions (more, less developed and transition regions) are subject to different formulas for allocating funds.

• The methodology for allocating funds is publicly available. This, in theory, makes it the only EU policy based on shared management and pre-allocation to Member States that uses an objective formula.

Table 2 shows the data of the 47 regions that will define our problem: population, GDP per capita (expressed in terms of purchasing power standards [PPS]) and the allocation assigned using the Berlin method (both per capita and in total terms). Note that some countries do not contain all types of regions.⁸

Table 2 The agents: Nuts2 regions in each EU member state, population, GDP per capita, and current ERDF allocations per inhabitant and in total terms. Source: Eurostat, 2021

To define a conflicting claims problem associated with the distribution of ERDF funds, we need to specify:

• 1. The agents.

• 2. The endowment.

• 3. The claim of each agent.

As mentioned, in our applied analysis the agents are the different types of regions in each country. These regions are differentiated by the corresponding Gross Domestic Product per inhabitant $(\text{GDP}^{h})$. The official website of the European Commission states that “the ERDF aims to strengthen economic and social cohesion in the European Union by correcting imbalances between its regions.” Therefore, the agents in this problem should be the different types of regions as described in the Official Journal of the European Union: “Resources for the Investment (for growth and jobs goal) shall be allocated among the following three categories of NUTS level 2 regions:”

$R1:$ :

More developed regions: $\text{GDP}^{h}$ is above 90% of the average of the EU-27.

$R2:$ :

Transition regions: $\text{GDP}^{h}$ is between 75% and 90% of the average of the EU-27.

$R3:$ :

Less developed regions: $\text{GDP}^{h}$ is less than 75% of the average of the EU-27.

The endowment $E$ will consist of the ERDF budget to be allocated to all regions in the EU (in absolute terms). This budget is decided by the European Council and the European Parliament and covers a 7-year programming period. During the 2014–2020 programming period, the EU will spend over €350 billion on cohesion policy. That is equal to 32.5% of the overall EU budget. Around €199 billion is allocated to the European Regional Development Fund. This includes €10.2 billion for European Territorial Cooperation (ETC) and €1.5 billion of special allocations for outermost and sparsely populated regions.⁹

In our simulation, we use the ERDF budget as the endowment; that is, an amount of 188,008 million euros, which corresponds to the allocated budget without considering the European Territorial Cooperation and other special allocations. Note that the actual budget is 182,150 because we have removed United Kingdom from our analysis, since it does not belong to the EU anymore. When we analyze absolute budgets, we measurement the claims and allocations in millions of euros, M€. When we analyze the per capita distribution, the unity of measure is the euro, €.

Finally, the claim $c_{i}$ of each type of region in each country remains to be decided. This is the more difficult and controversial part in defining the conflicting claims problem. As mentioned, the ERDF projects must be co-financed by the Member States, so it is up to these states and the European Parliament through several negotiations to decide which projects deserve to be properly co-financed.

An alternative way that we use to present our simulation, is to make the claim depend on the difference between the GDP per inhabitant of the regions (expressed in terms of purchasing power standards [PPS]). More precisely, on the gap between the greatest GPD per capita and that of the specific region. Then, for each agent $i$ in our allocation problem, $i=1,2,\ldots,47$, we define the claim per capita as a linear function:

$$c_{i}=\delta+\epsilon\left(\text{GDP}^{h}_{*}-\text{GDP}^{h}_{i}\right)\quad\delta\geq 0,\> \epsilon\in\left[0,1\right]$$

where:

• $\text{GDP}^{h}_{*}$ is the greatest GDP per capita in the EU regions (Luxembourg);

• $\delta$ is a common amount per inhabitant that all regions receive (that can be interpreted as a minimal allocation); and

• $\epsilon$ is a coefficient that can be interpreted as a convergence speed fixed by the Member States.

For our computations, we fix $\delta$ as the allocation per inhabitant obtained by the region with the highest GDP per inhabitant (Luxembourg). That is, this region claims to receive the same amount as before, and other regions will claim this amount plus a part of their GDP gap. We set the $\epsilon$ coefficient at $2.5\%$; so, from Table 2 we obtain

$$c_{i}=32.40+0.025\left(79{,}300-\text{GDP}^{h}_{i}\right).$$

The above expression gives rise to a minimum claim per inhabitant (after Luxembourg which is 32.40 €) of 546.90 €, which corresponds to R1 region in the Czech Republic. The maximum claim is that of R3 region in Bulgaria, that rises up to $1629$ €. It should be noted that the way we have defined claims makes the regions with lower GDP per inhabitant have a higher claim; that is, the claim decreases with the GDP per capita of the region in question. In Table 3 we can find the claims of all regions.¹⁰

Once the problem of distributing the ERDF budget among the EU regions has been translated into a conflicting claims problem, as formulated in claims per inhabitant in each region, we need to adapt the claims rules introduced in Sect. 2 to the per capita analysis:¹¹

• The per capita proportional rule $P^{h}$ equalizes the portion of the claim that is satisfied,

  $\displaystyle P^{h}_{i}=\dfrac{c_{i}^{h}}{\sum_{j=1}^{n}c_{j}^{h}}\lambda$, $\lambda$ such that $\sum_{i=1}^{n}p_{i}P^{h}_{i}=E$.

• The per capita $\textit{CEA}^{h}$ rule equalizes the awards (constrained to no one receiving more than her claim),

  $\textit{CEA}_{i}^{h}=\min\left\{c_{i}^{h},\lambda\right\}$, $\lambda$ such that $\sum_{i=1}^{n}p_{i}\textit{CEA}^{h}_{i}=E$.

• The per capita $\textit{CEL}^{h}$ rule equalizes the losses (constrained to no one receiving a negative amount),

  $\textit{CEL}_{i}^{h}=\max\left\{0,c_{i}^{h}-\lambda\right\}$, $\lambda$ such that $\sum_{i=1}^{n}p_{i}\textit{CEL}^{h}_{i}=E$.

• The per capita $\left(\alpha^{\min}\right)^{h}$ rule ensures a minimum amount per capita to all regions and uses the per capita proportional rule to share the remaining estate (if any), with respect to the unsatisfied claim.

  $$\left(\alpha^{\min}\right)^{h}_{i}=\left\{\begin{array}[]{lll}\dfrac{1}{\sum_{i=1}^{n}p_{i}}E&\text{ if }&E\leq k\sum_{i=1}^{n}p_{i},\\ \\ k+P_{i}^{h}\left(E-k\sum_{i=1}^{n}p_{i},c^{h}-k\right)&\text{ otherwise, }\\ \end{array}\right.$$

  where $k=\min\left\{c^{h}_{1},c^{h}_{2},\ldots,c^{h}_{n}\right\}$ and $n$ is the number of agents.

Table 3 shows the distribution of the budget proposed by the different claims rules (per inhabitant in each region).

Table 3 Claims, current allocations, and proposals according to the different rules, in € per capita

Although the criteria for the allocation of ERDF funds are applied by region, the main negotiations in the different bodies of the European Parliament take place between representatives of the Member States. That is why it is interesting to observe the allocation of these funds at the country level. Table 4 contains the distribution of ERDF funds by country, depending on the solution chosen, and comparing these with the current distribution. It also shows the percentage of the funds allocated to each country.

Table 4 Absolute allocations of ERDF funds by country: current allocations and proposals according to the different rules (in M€). The figure given between brackets is the percentage of the funds allocated to each country

By observing the data in Tables 3 and 4, the way in which the ERDF budget is distributed varies from one proposal to another. On the one hand, we have the ‘most egalitarian’ proposal, given by CEA rule. This is a conservative approach, in the sense that the situation before and after the budget is allocated does not vary so much. At the other extreme, the CEL proposal is the most groundbreaking one, in the sense that it promotes serious changes in the previous status quo. The proportional $P$ and the $\alpha^{\min}$ proposals are located somewhere between both approaches. In order to choose one proposal from all the obtained allocations, the following section compares the different claims rules in terms of convergence and equity.

4 Convergence among regions

As already mentioned, one of the main objectives of the EU through the ERDF is to promote convergence between regions of different types. In this section we analyze how the introduced rules promote this convergence and compare the effectiveness of the allocation proposed by each of these rules. Note that by effectiveness we mean the faster path to achieve convergence among regions.

To do this, we define a divergence ratio that attempts to capture the differences among regions in terms of $\text{GDP}^{h}$. Let us consider two agents (two types of regions in some member states) $\alpha$ and $\beta$ such that:

• $p_{\alpha}<p_{\beta}$, where $p_{k}$ stands for the $\text{GDP}^{h}$ of region $k$.

• $c_{\alpha}> c_{\beta}$, where $c_{k}$ stands for the (per capita) claim of region $k$.

That is, region $\alpha$ is less developed than region $\beta$ and, consequently, the (per capita) claim of this region is greater than that corresponding to the most developed region.

We define the divergence ratio of (the less developed) region $\alpha$ versus (a more developed) region $\beta$ as the quotient:

$$d_{(\alpha,\beta)}=1-\dfrac{p_{\alpha}}{p_{\beta}}$$

Note that $d_{(\alpha,\beta)}$ is always greater than 0 and the ideal convergence will arrive whenever all divergence ratios are equal to 0.

From an initial divergence ratio, convergence initiatives will promote the reduction of such a ratio. To study the impact of the allocation on convergence across regions, trying to capture the effect of assigning an amount $x> 0$ to a region with a given $\text{GDP}^{h}$, $p$, we assume that this allocation originates a new $\text{GDP}^{h}$, $\hat{p}$, that can be defined as a function of the allocated amount and the previous purchasing power standard:

$$\hat{p}=p+F(p,x)\quad\text{where function }F\text{ fulfills}\quad F(p,x)\geq 0,\> \dfrac{\partial F}{\partial x}> 0,\> \dfrac{\partial F}{\partial p}\leq 0$$

We are stating that, when $p$ is fixed, the new $\text{GDP}^{h}$ strictly increases as $x$ increases. The negativity in the other partial entails that a fixed amount $x$ provides the greatest increase in $\text{GDP}^{h}$ for less developed regions.

Note that the present work only intends to show a new way of distributing the ERDF funds. Thus, we only analyze the effect that our proposal has on GDP in a very simplified way. For further analysis see, for instance, Becker et al. (2010). The easiest way of defining function $F(p,x)$ is to only consider an additive effect of the allocation; i.e., that the post-$\text{GDP}^{h}$ can be estimated simply as the initial $\text{GDP}^{h}$ added to the proposed allocation. This entails defining $F(p,x)=x$, so $\hat{p}=p+x$. But this is a rather simplistic approach, as it omits possible multiplier effects that would stem from investment spending initiated with ERDF funds. A more general case is to consider quasi-linear functions. A possible example appears in the following expression:¹²

$$F(p,x)=\delta p+v(x)\quad v^{\prime}(x)> 0\> \text{ (increasing) and }\> v^{\prime\prime}(x)\leq 0\> \text{ (concave)},\> \delta<0$$

Independently of the way in which we define function $F(p,x)$, we observe that the divergence ratio before the allocation is greater that the ratio after the allocation, when the rule for assigning the ERDF funds provides larger allocations to regions with larger claims. To show this fact, we denote by $d^{0}_{(\alpha,\beta)}$ the divergence ratio before the allocation and $d^{1}_{(\alpha,\beta)}$ the ratio after the allocation. Then, if $x_{\alpha}\geq x_{\beta}$

$$1-d^{1}_{(\alpha,\beta)}=\dfrac{p_{\alpha}+F(p_{\alpha},x_{\alpha})}{p_{\beta}+F(p_{\beta},x_{\beta})}\geq\dfrac{p_{\alpha}+F(p_{\alpha},x_{\beta})}{p_{\beta}+F(p_{\beta},x_{\beta})}> \dfrac{p_{\alpha}}{p_{\beta}}=1-d^{0}_{(\alpha,\beta)}$$

that implies that $d^{1}_{(\alpha,\beta)}<d^{0}_{(\alpha,\beta)}$.

It is noteworthy that each of the proposed claims rules satisfies the so-called order preservation property; that is, the larger the claim, the larger the resources allocated by the claims rule. Therefore, the proposed claims rules always reduce the divergence ratio.

On the other hand, it is easy to observe that $c_{\alpha}> c_{\beta}$ implies that the application of the CEL rule always provides an allocation to the less developed region that is greater or equal than the one provided by other rules:

$$\textit{CEL}_{\alpha}> \varphi_{\alpha}\qquad\text{for}\quad\varphi=P,\textit{CEA},\alpha^{\min}$$

for $\alpha$ such that $\text{GDP}^{h}_{\alpha}$ is low so,

$$d^{1}_{(\alpha,\beta)}(\textit{CEL})<d^{1}_{(\alpha,\beta)}(\varphi)\qquad\text{for}\quad\varphi=P,\textit{CEA},\alpha^{\min}$$

that is, the rule that best promotes convergence is CEL.

The above fact can also be deduced by using an additional equity criterion: Lorenz dominance, a useful tool to check whether a solution is more favorable to smaller claimants relative to larger claimants.¹³

Formally, let $\mathbb{R}^{n}_{\leq}$ be the set of positive $n$-dimensional vectors $x=\left(x_{1},x_{2},\ldots,x_{n}\right)$ such that the entries are ordered from small to large; i.e., $0<x_{1}\leq x_{2}\leq\ldots\leq x_{n}$. Let $x$ and $y$ be in $\mathbb{R}^{n}_{\leq}$. We say that $x$ Lorenz dominates $y$, denoted by $x\succ_{L}y$, if for each $k=1,2,\ldots,n-1$

$$x_{1}+x_{2}+\dots+x_{k}\geq y_{1}+y_{2}+\ldots+y_{k}\quad\text{and}\quad\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}y_{i}.$$

If $x\succ_{L}y$ and $x\neq y$, then at least one of these $n-1$ inequalities is a strict inequality. Given two claims rules, $\varphi$ and $\psi$, it is said that $\varphi$ Lorenz dominates $\psi$, $\varphi\succ_{L}\psi$, if $\varphi(E,c)\succ_{L}\psi(E,c)$, for each conflicting claims problem $(E,c)$. Bosmans and Lauwers (2011) obtain a Lorenz dominance comparison among several claims rules:

$$\textit{CEA}\succ_{L}\alpha^{\min}\succ_{L}P\succ_{L}\textit{CEL}$$

So, the CEA rule distributes the budget in the most egalitarian manner possible, maintaining the existing differences before the budget was allocated. On the contrary, the CEL rule provides the less egalitarian distribution of the funds. Then, if one of the objectives is to reduce the previous inequalities, our results indicate that the CEL rule may be most appropriate.

5 Establishing guarantees

An interesting focus in the conflicting claims problems literature addresses the possibility of ensuring a minimum amount for each agent (each region in our application), or to limit the maximum amount they can receive. These amounts will depend on the available budget and on the quantity that each region claims. The minimum amounts that agents (regions) should receive are known as lower bounds (or guarantees).

• The fair lower bound (F) (Moulin 2002) establishes that all regions should receive at least the amount assigned to each of them in an equal division, or their full claim. Formally,

  For each $(E,c)\in\mathcal{B}$ and each $i\in N$,

  $$F_{i}(E,c)=\min\left\{c_{i},\dfrac{E}{n}\right\}{.}$$

If we analyze the problem from the point of view of losses (the unsatisfied part of the claim), ensuring a lower bound in losses is equivalent to establishing an upper bound in awards.To define what we name the fair upper bound in awards, we denote by $L$ the aggregate losses, that is $L=\sum_{i}c_{i}-E$.

• The fair upper bound (U) establishes that all regions should incur the same loss, restricted to the fact that no region may end up with a negative allocation. Formally,

  For each $(E,c)\in\mathcal{B}$ and each $i\in N$,

  $$U_{i}(E,c)=\max\left\{0,c_{i}-\dfrac{L}{n}\right\}{.}$$

As the population is very different from one country to another, we need to compute these bounds in per capita terms and then we obtain the lower and upper bound by multiplying for each country population. Then,

$$\dfrac{E}{n}=408.22\text{ \EUR}\qquad\dfrac{L}{n}=853.97\text{ \EUR}$$

Table 5 shows the result (in percentages) that the bounds assign to each country of the ERDF budget.

Table 5 For each country, the percentage of the ERDF budget assigned by fair lower and upper bounds

Note that, for most of the countries, the current allocation does not remain within the lower and the upper limits. Nevertheless, the claims rules provide allocations within the ranges obtained for many countries (compare with Table 4). The fair lower and upper bounds must be understood as the desirable limits within which the proposals for the allocation of funds must be found. Among the cases in which current allocations do not meet these restrictions, it is interesting to look at the Netherlands, Denmark or Sweden (among others), receiving an amount below the recommended minimum. At the other extreme are Poland or Hungary, which receive an amount above the maximum proposed by the fair upper bound.

6 Conclusions

The European Union tries to promote the social and economic cohesion of the Member States, as well as to reduce the inequalities among them. To achieve this objective, the EU uses several financial instruments, one of which is the European Regional Development Fund (ERDF).

We suggest a way to obtain an initial proposal for the distribution of the ERDF budget, that is based on defining a conflicting claims problem. To define this problem, we easily identify the agents (the EU NUTS level 2 regions) and the endowment (the ERDF budget to be allocated). To complete the construction of the model, it is only necessary to define the claim of each region, a matter of political approach. This part is related to the political aspect of the distribution of cohesion funds in the EU and is beyond the scope of this paper, although we suggest a way to proceed: let the regions propose (co-financed) projects and select the credible/viable ones.

Once the conflicting claims model is completed, any of the rules defined to solve claims problems (claims rules) can be used to obtain a meaningful distribution of the ERDF budget. We use four claims rules to show the performance of our model: the proportional rule, the constrained equal awards rule, the constrained equal losses rule, and the $\alpha^{\min}$ rule. We show that all these rules promote convergence among regions and reduce inequalities. Among the analyzed rules, the one that performs best (promoting convergence) is the one that proposes the most unequal (per capita) distribution of the ERDF budget: the constrained equal losses rule.

We propose a simulation exercise in which the claims are defined in a linear way by adding a fixed (subsistence) amount with a factor that depends on the per capita gap between the GDP of different regions. Although this assumption may have some logic, the obtained (simulated) numbers are only used to illustrate how allocations are decided according to the claims rules and observe the behavior of such a distribution.

As mentioned, our empirical exercise only tries to compare current allocations with the proposals obtained through claims rules. Although it is a very simplified scenario, our empirical results have two remarkable features:

• 1. The results do not drastically differ from current allocations.

• 2. It promotes (theoretical) convergence better than the current allocation.

We propose the application of claims rules (the CEL rule, in particular) to obtain an initial proposal to be discussed by Member States.

As done in Fragnelli and Kiryluk-Dryjska (2019), a national-level analysis of the application of claims rules can be an interesting on-going research. This study may have two possible lines:

• To analyze the distribution of the ERDF allocated to a Member State among the different regions in this country (NUTS 3 analysis).

• To analyze the distribution of the ERDF allocated to a Member State among the different programs in this country (unemployment, youth unemployment, education, migration, etc.). In Fragnelli and Kiryluk-Dryjska (2019) this kind of study is carried out for Poland.