Tax evasion study in a society realized as a diluted Ising model with competing interactions
Introduction
Tax evasion is an issue that concerns the global economy, since tax collection is a first step in the economic growth of a country. However, the great benefits that an efficient tax system can bring to a nation can be overshadowed by the systematic abuse of tax collection or corruption [1], the feeling of getting rich, or the simple fact of not understanding the law, which inevitably leads to a sector of the population to become evader of its tax obligation, thus contributing to weaken the economy of a country [2]. Thus, tax evasion arises naturally as a field of study that must be tackled from a wide range of disciplines that allow a clearer knowledge of its most influential variables, so that in conjunction with the dynamics of government and society in general, this misconduct is minimized.
The first proposal can be traced to the work of Allingham & Sandmo [3], [4]. Such study dealt with the expected value of the profit obtained by an agent depending on the income declared, and the fraction of taxes to be paid according to what has been declared. This work also investigates the conditions dealing with the maximization of the profit, finding that, declaring less income than actually earned (which is nothing more than a form of evasion), is a sufficient condition to maximize such utility.
From the experimental point of view, Bosco & Mittone [5], and more recently Mittone [6] focused on how psychological factors influence tax evasion. In essence, it follows that a redistribution of the tax yield reduces tax evasion, and the way the agent perceives the detection and punishment risk becomes also determinant [6]. Such psychological factors tend to be correlated with the way the audit is done over the tax compliance population.
With the improve of computing features and algorithms, agent-based models [7], [8], [9], [10], [11] become an effective tool for describing the tax evasion of a society, including social parameters that take into account the possible scenarios in which a group of agents susceptible to declare taxes are involved.
Zaklan and Zaklan & Lima [12], [13], [14] used an Ising model [15] to describe a society composed of agents who can be compliant or non-compliant tax payers, applying social variables such as penalty periods and an audit probability within their study. In the context of an Ising model, it is assumed that agents that do not evade (the compliant ones) are identified with particles whose spin values are , while the evaders (the non-compliant ones) are identified with spin values equal to . Lima [16], [17], [18], [19] extend Zakland’s work [14] by investigating characteristics like phase transitions, and different networks in which agents lie in the framework of a pure Ising model.
The above opened a new range of possibilities: in the works of Seibold & Pickhardt [20], [21] they used physical variables in the framework of an Ising model to study the percentage of evasion of a system of agents. Such system was divided into 4 groups of agents according to their social conduct, namely: (i) totally honest or ethical agents, (ii) totally evading or selfish agents, (iii) agents influenced by their local environment, the so-called “copying” agents, and (iv) random agents who act by chance. Such classification was parameterized in terms of the interplay between temperature, magnetic field and exchange integral. In particular, both the magnetic field and temperature were considered as local variables whose values were taken different from site to site in the grid, dealing, in the case of temperature, with the personality of the agents and therefore with the degree of autonomy to make a decision. Their studies reproduced results of agent-based models that incorporate characteristics of the study made by Allingham & Sandmo [3]. This makes clear that, the models that are identified in the current literature as econophysical models, manage to extract the essentials of purely economic models, obtaining comparable results and making them more robust and accessible to a diverse scientific community [22]. For example, one of the latest works of Berger and collaborators [23], studied the “bomb crater effect” by means of a heterogeneous Ising model, following the same conditions as [20] or [21], about particularizing the magnetic field and the temperature as different for each agent.
The works mentioned above start by assuming an individual motivation to evade taxes (as in Allingham & Sandmo’s work [3]), and then towards the study of tax evasion for a full society (e.g Zakland & Lima’s work [13]). From an economical point of view, the economic behavior between agents and their interactions has been already addressed. Manski [24] discusses widely how economists gradually include social interactions as an influence in economic outcomes and strongly stress that concepts like preferences, expectations, constraints, and equilibrium, should be considered as core concepts to understand social science, of which, the economy is one of them. In Chetty’s work [25], a pragmatic perspective, a point of view where behavior factors strongly influence economical aspects making some conditions more flexible, is developed. He noticed that relaxing certain neoclassical assumptions can lead to new policy tools or better predictions about effects of existing policies. And as he concludes, the use of behavior factors go beyond philosophical questions, and becomes a versatile tool to understand a wide range of phenomena, helping to understand critical actual policy questions. More recently, Gallagher et al.[26] tackle the concept by considering the market as a cognitive institution, where the way the agents interact influence the market and the economy, and all of those interactions are influenced by social and cultural practices. Therefore, the market cannot be seen only as a mechanism regardless who interacts on it, but as a social institution. All the above show that not only the interactions between agents, but the way they interact between each other, matters, and of course, each society has its particular way to interact within.
As concerns to a case study, concretely for the Republic of Colombia, characterized historically by a high degree of tax evasion, a review of the state of the art has only been addressed by administrators or economists in recent years. The works of Parra Jimenez & Patiño Jacinto [27] or Rodríguez Rodriguez-Cuervo [28] seek to analyze the tax evasion rates for the periods 2001–2009 and 1997–2017, respectively. They studied the possible causes of tax evasion and some alternatives to reduce it, besides estimates of the percentage of evasion were also given. This has only been seen from a purely economic framework, wherewith the application of a general model to this particular case could give future perspectives to reduce, or at least control, tax evasion in Colombia. In such works, evasion rates based on official data obtained from the National Tax and Customs Directorate (DIAN), are presented, and therefore they are elements to be considered in this investigation.
The manuscript is organized as follows: In Section 2, we present the model, the physical insights involved in the simulations and the equivalence between physical and social variables. In particular, we show how the use of competitive exchange integrals and diluted bonds randomly distributed in the framework of a spin glass-type Ising model, allows to identify our system of agents like a disordered ternary alloy where stoichiometry plays a key role. In Section 3 methodology and computational details including the heat bath algorithm are presented. In Section 4 we analyze the effect of auditing and government policies for tax collection, the role that temperature plays in our approach as well as the exchange integrals, and a final analysis of application of our model for the Republic of Colombia. We finish with a discussion about the relevance of these results and we present the main conclusions.
Section snippets
Model
The two-state character of compliance of the agents is modeled by considering two possible values, namely: (i) a positive one corresponding to a compliant tax payer with spin and (ii) a negative one for a non-compliant tax payer with spin . On the other hand, we propose that instead of a predetermined agent classification in terms of personality, autonomy or even degree of selfishness, a range of subjective scenarios can arise in a natural way if the lattice of agents is realized
Methodology and computational details
Thermalization has been conducted by using the heat-bath algorithm [30], [31] where the probability of an agent to take on the values in a thermal reservoir at an absolute temperature, (in units of ), is given by: where is the energy change associated to a spin–flip at site and one time step per spin or per agent. A full sweep implies trying an inversion in every single agent of the system with the probability . In this work,
Auditing and government policies for tax collection
Due to the variety of parameters that have an effect on the system, first we start characterizing the degree of influence of the external field over the average magnetization per site, . Results are shown in Fig. 1, Fig. 2 for some fixed temperatures and for audit probabilities and respectively. These results allow to tune both the range of temperature values to be used and the range of field values where the government policies, or equivalently, the external applied field, can
Conclusions
The above results reveal that, from the perspective of the model stated here, the Eq. (1) under the conditions of Eq. (3) provides good results, which cover, not only works done before in the topic of tax evasion from an econophysics approach like the three-state kinetic agent-based model in [32], but also, allows to replicate results obtained by economical means as we have shown for the Colombian case. Even more, the model allows to identify the interaction between government policy and audit
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
J.R acknowledges University of Antioquia for the exclusive dedication program and the CODI-UdeA 2017-16253 project. J.G-B acknowledges Max Planck Tandem Group – Biophysics of Tropical Diseases for financial support and useful discussion. J.G-B and J.R acknowledge to Prof. Dr. Götz Seibold for providing us with the FORTRAN code used in [21], and for making the article referenced here as [23] available upon request. All authors contributed equally to the work
Additional information
Materials and data are promptly
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