Sascha Hokamp and Andrés M. Cuervo Díaz
“Essentially, all models are wrong, but some are useful!,” a well-known claim attributed to George E. P. Box (Box and Draper, 1987, p. 424). Of course, all models necessarily are a brief representation of a real problem and, thus, simplify the complexity of the real world. In addition, concerning tax evasion and the shadow economy, another issue arises, that is, how to measure what needs to be hidden in the shadow.
The early attempts to measure the shadow economy (Gutmann, 1977; Feige, 1980; Tanzi, 1980; Klovland, 1984) date back to the equations on currency demand by Cagan (1958) and the transaction approach by Feige (1979). Kirchgässner (2017, p. 99) identifies three approaches, which dominate the literature, (i) the “direct measurement” employing a survey method (Isachsen and Strøm, 1982; Feld and Larsen, 2005, 2012), (ii) the “indirect measurement” applying, a modified currency demand approach (Tanzi, 1980; Klovland, 1984), and, (iii) a model approach that originates from Weck (1983) and Frey and Weck-Hannemann (1984), well known as the (DYnamic) Multiple Indicators, MultIple Causes, (DY)MIMIC, approach. The latter is often used by Friedrich Schneider (with various co-authors, for example, Schneider, 2005, 2014; Schneider et al., 2011, 2015; Buehn and Schneider, 2012; Schneider and Enste, 2013) and their estimates truly dominate the field. Feige (2016a) and Kirchgässner (2017) find severe problems to measure tax noncompliance and the shadow economy by the (DY)MIMIC approach: (i) confusion with definitions and measurements of the shadow economy, it seems that Friedrich Schneider and co-authors have “simply defined the entity [they] have measured” (Feige, 2016a, p. 26), (ii) “estimations produce only relative weights” (Kirchgässner, 2017, p. 103), (iii) conclusions are influenced by “arbitrary choices of indicators and normalizing coefficients,” (Feige, 2016a, p. 26) and, (iv) impossibility “to draw statistically confirmed conclusions about causal relations in the real world […] from these estimates” (Kirchgässner, 2017, p. 103). Feige (2016b) and Schneider (2016) continue to deepen the discussion on flaws and strengths to measure the shadow economy. Hashimzade and Heady (2016) briefly summarize the dispute.
To overcome the severe problems in the measurement of tax compliance and the shadow economy, agent-based models (ABMs) might help via what–if studies. In this chapter, we focus on tax evasion at a micro level by agent-based computational simulations. We present novel insights into tax noncompliance driven by lapse of time, social norms, age heterogeneity, subjective audit probability, public goods provision, and Pareto-optimality within the framework presented by Hokamp and Pickhardt (2010) and Hokamp (2013, 2014). We reconsider the political cycle by Hokamp and Pickhardt (2010), the public goods provision cycle by Hokamp (2013, 2014) and, in addition, we provide new information on compliance at group level by the four behavioral types of taxpayers, i) neoclassical A-Types, ii) social interacting B-Types, iii) ethical C-Types, and iv) erratic D-Types. We find that neoclassical A-Type taxpayers play a key role within the model by pursuing the expected utility maximizing approach by Allingham and Sandmo (1972). Therefore, ceteris paribus we investigate the dynamics of tax evasion by neoclassical A-Types due to lapse of time, subjective audit probability, public goods provision, and age heterogeneity combined with social norms. The lapse of time effects strongly declines when raising the tax rate under a regime with a constant sanction rate and audit probability. Further, we build upon the calibration procedure of an econophysics ABM by Bazart et al. (2016) and calibrate our economic ABM with a tax declaration experiment by Alm et al. (2009). Finally, our sensitivity analysis provides numerical estimates that reveal the strong effect the penalty and tax rate have on the evasion by neoclassical A-Type and social interacting B-Type taxpayers. The chapter is structured as follows. Section 9.2 describes our ABM of tax evasion by a standard protocol. Section 9.3 discusses the scenarios and results while Section 9.4 provides the conclusion. Appendix 9A provides the political and public goods provision cycle by Hokamp and Pickhardt (2010) and Hokamp (2013, 2014).
This section describes the agent-based model of tax evasion by1 (Hokamp and Pickhardt, 2010; Hokamp, 2013, 2014) and our amendments according to the Overview, Design Concepts and Details (ODD) protocol by Grimm et al. (2006, 2010), including the extension to human decision-making by Müller et al. (2013), outlined in Chapter 1.
The agent-based tax evasion model by Hokamp and Pickhardt (2010) and Hokamp (2013, 2014) considers a heterogeneous set of taxpayers faced with a tax scheme explicitly modeling the environment consisting of a government and a tax authority. The model aims to analyze the tax compliance dynamics and the social interactions in a behaviorally heterogeneous population under lapse of time, social norm updating, age heterogeneity, adjustment of subjective audit probability, public goods provision, and Pareto-optimality. One out of the four behavioral archetypes of taxpayers rests on expected utility maximization by Allingham and Sandmo (1972); social norms evolve with the age of the taxpayers reflecting psychological findings by Kirchler (2007), Traxler (2010), and Traxler and Winter (2012), and social interactions are affected by the endorsement of Pareto-optimal allocations within closed groups or neighborhoods (Hokamp and Pickhardt, 2011; Pickhardt, 2012).
The purpose of our ABM is to investigate, at the macro- and the micro level, the dynamics of tax noncompliance via thought experiments, which allows users, for example, researchers, tax practitioners, and policy makers, a usage of our approach for their own purposes, for example, to estimate the effects of tax reforms on the extent of tax evasion. Hokamp (2014) and Bazart et al. (2016) show how simulations of tax noncompliance add to tax evasion theories and experiments. The computational simulation of the tax evasion problem with the opportunity to adjust parameters through an ABM gives users the possibility to overcome real life experimental barriers and to study how taxpayers at the micro level affect the aggregated outcome at the macro level, for example, the tax revenue, of the system. The ABM provides an expansion of neoclassical-theory prediction, as it is able to account for the complex interactions in heterogeneous environments. Furthermore, our model allows to investigate the characteristics of interest in what-if studies. In particular, examination of tax evasion dynamics under public goods provision and testing the relevance of Pareto-optimal allocations become possible (Hokamp, 2014).
In the model, time steps reflect tax relevant periods or tax years and space is not modeled but we consider the environment, in which the tax compliance decision takes place. Our model includes a behaviorally heterogeneous population of taxpayers2, a tax authority, and a government, described in Hokamp (2014) as three kinds of agents. The government provides a public good to the population of taxpayers; the provision level depends on total revenues, gathered by the government, consisting of all paid tax and penalties minus the payment for tax reimbursement3 and administrative expenses. The tax authority collects the tax voluntarily contributed by the taxpayers, and, in addition, it conducts at each time step a random audit process and charges the corresponding penalties and reimbursement from tax evasion or overpayment discovered, respectively. To run a tax scheme, the tax authority incurs administration expenses, which relate to the system efficiency. The tax authority is characterized by the effectiveness of the tax collection, , the effectiveness of the audit process, , and fixed costs, . The government decides on the tax rate, , the penalty rate, , the audit probability, , and the tax complexity, while providing a public good with the efficiency, . Hence, varying these parameters determines a political and public goods provision cycle (Hokamp and Pickhardt, 2010; Hokamp, 2013, 2014), which reflects the environment in which tax declarations, payments, and social interactions of taxpayers take place. A list of parameters and mathematical symbols is found in Table 9.1.
Table 9.1 Model parameters and mathematical symbols
Government | Tax authority | |||||||
Effectiveness of | ||||||||
Model parameter | Tax rate | Penalty rate | Audit probability | Public goods efficiency | Tax complexity | Audit process | Tax collection | Fixed costs |
Abbreviation |
We define behavioral archetypes of taxpayers based on Hokamp and Pickhardt (2010), which divide a heterogeneous population into four kinds of taxpayers showing different behaviors. In particular, the taxpayers are classified as, (i) neoclassical A-Types, (ii) social interacting B-Types, and (iii) ethical C-Types and (iv) erratic D-Types.
This kind of taxpayers shows behavior patterns described in the neoclassical theory proposed by Allingham and Sandmo (1972). At each time step the decision, made by neoclassical A-Type taxpayers, is the one that maximizes their utility, , so that the definition of this function governs their behavior. Reflecting the utility of money, an exponential utility function on the after tax and penalties net income, , is chosen4,
which is sensitive to a subjective audit probability, income, tax, and penalty rate, but also incorporates lapse of time effects and a dependency on the risk parameter, , which may change due to social norms. The incorporates the effects of public goods provision in the utility of the taxpayers via
and
where represents the income earned in a time step; is the decision variable of a taxpayer, how much income he voluntarily declares to the tax authority; describes the contribution to a public good by all other taxpayers, except the taxpayer under consideration; and denotes the penalties due to back audits for a lapse of time, ,5 described by Hokamp (2014). The is defined once for the case when the taxpayer is audited and once for no audit. In the version without an audit, the net income after tax and penalties is defined as the earned income minus the tax paid plus the benefits received from public goods. In the version when the taxpayer is audited, the charged penalty over evaded tax plus the change in public goods provision are taken into account.
Given Eqs (9.1)(9.2)(9.3) the neoclassical A-Type taxpayers apply the expected utility maximization procedure adopted by Allingham and Sandmo (1972). Hokamp (2014) presents the interval for the subjective audit probability, , which allows for an inner solution, given by the lower bound
and the upper bound
If the audit probability exceeds the upper limit in Eq. (9.5), the taxpayer becomes fully tax compliant, that is, , and when falls below the lower bound in Eq. (9.4), the taxpayer fully evades, that is, . For an audit probability in the range for an inner solution, the taxpayer voluntarily declares6
These taxpayers check the behavior of their vicinity, , which is of a parametrically specified size. For each taxpayer, the size of the social network, , is equal. If taxpayer is -close to taxpayer , the reverse may not be true. Further, -close is a relation that is not transitive. Hence, we assume a randomized list of acquaintances (Gilbert and Troitzsch, 2005; Hokamp, 2013, 2014) employed by the taxpayers to estimate an average benefit. If the average reductions of income were less than the tax rate in the previous period, then the condition to evade tax7 is fulfilled and they voluntarily announce
declaring the same as the mean in their neighborhood. If the condition to evade is not fulfilled then they honestly declare their total income. Full tax compliance also applies as a default option, when a B-Type is born as well as in the initialization period of the model. Finally, if the tax evasion of a B-Type is uncovered, this subject mutates to a fully compliant taxpayer for a preassigned number of periods, . To put this differently, an uncovered B-Type mutates into a C-Type, an agent type, which will be described in the next paragraph.
Such taxpayers are ethical or altruistically motivated. They stick to full declaration at all times; they are not affected by other taxpayers but affect the behavior of neoclassical A-Type and social interacting B-Type taxpayers, for example, via their contribution to the provision of public goods.
These taxpayers try to follow a behavior, for instance, full tax compliance, but they randomly fail to accomplish their goal since they are faced with the tax complexity, , of the system. Their voluntarily declared income follows a normal distribution centered in the desired behavior8. They are not influenced by other taxpayers but they indirectly interact with A-Type and B-Type taxpayers via a public good.
At each tax relevant period the extent of tax evasion, , of the whole group of taxpayers is calculated as
while the group behavior of the different kinds of taxpayers is expressed as
where is the subset of taxpayers of the same behavioral archetype. In Section 9.3 we present graphical visualizations of these indicators to show their evolution over time. Further, social interacting B-Types employ to estimate the extent of tax noncompliance in their social network.
At the beginning of each tax relevant period, the government and the tax authority inform the taxpayers about the tax environment, characterized by the parameters shown in Table 9.1. The taxpayers utilize this information to calculate (or optimize) their voluntarily declared income to the tax authority by applying the behavioral heuristics described in Section 9.2.1.2. After tax collection, the tax authority randomly conducts an audit, which might include back audits. The corresponding penalties and reimbursement are charged and the government may provide a public good. Finally, each agent grows one tax relevant period older and the taxpayers update their history of tax declarations and after 40 tax years the political and public goods cycle repeats. Further, the model allows to switch “on” and “off” the operation modes, (i) lapse of time, (ii) subjective audit probability, (iii) age heterogeneity, (iv) social norm updating, (v) public goods provision, and (vi) a check for Pareto-optimality.
Without considering lapse of time effects, , only the actual period might be subject to an audit by the tax authority. With lapse of time effects, , additional periods in the past might be audited. The whole population of taxpayers is affected by such a change in the audit procedure. However, lapse of time only affects the tax declaration by neoclassical A-Types and social interacting B-Types. A-Type taxpayers directly take into account that their penalties due to back audits, , may become greater and greater
which shifts the interval for an inner solution towards zero and, at the same time, the optimal declaration toward the true income. B-Types are indirectly influenced by these amendments via the average extent of tax noncompliance in their social network.
Typically, the tax declaration by neoclassical A-Types makes use of a (subjective) audit probability for the expected utility maximization proposed by Allingham and Sandmo (1972). However, after the experience of an audit, which uncovers a nonzero extent of tax evasion, the neoclassical A-Types endogenously adjust their subjective audit probability. Hence, they declare their true income in the following tax relevant period, since they set their subjective audit probability to unity, . Thereafter, is reduced in a stepwise fashion, by the updating parameter, , until the objective audit probability is reached. Hence, the subjective audit probability directly influences A-Types and indirectly affects B-Types via their social network.
The initial population of taxpayers is heterogeneous with respect to age. In particular, we assume that in the initial period the taxpayers are at an age between a minimum age, , and a maximum age, ; initial age is uniformly distributed on . Without “age heterogeneity” the taxpayers grow one year older after a tax relevant period and they live forever. In addition, if “age heterogeneity” is activated, the taxpayers die when they have reached the maximum age. In this case, to allow for ceteris paribus conditions, the dead are replaced by newborn taxpayers with identical initial properties, except for age, which is set to the minimum age. Thus, age heterogeneity affects social interacting B-Types, since they are fully tax compliant in their first period of life. Further, age heterogeneity influences neoclassical A-Types iff .
Social norms are updated via the risk parameter . If taxpayers are getting older, we assume that they become more and more risk averse, which results in a higher extent of tax compliance. Such behavioral patterns are reflected by raising, , for the elderly (Kirchler, 2007; Traxler, 2010; Nordblom and Z̆amac, 2012). Hence, social norm updating directly influences neoclassical A-Types via their utility function while social interacting B-Types are affected indirectly via their social network.9
If this feature is activated, the government provides a public good, , which can be nonrivalrously consumed by the whole population of taxpayers
and which is financed by tax, and penalties, , (consisting of fines due to tax cheating and reimbursement caused by tax overpayment) minus administrative costs, , described by Hokamp (2014). However, what really matters for expected utility maximization is the contribution to the public good by every other taxpayer, .
In addition to the tax authority efficiency parameters, , , the optimal solution, shown in Eq. (9.6), of the neoclassical A-Types is influenced by the effectiveness of the public sector, . Finally, public goods indirectly affect social interacting B-Types via their social network.
The operation mode “Pareto-optimality” only makes sense when a public good is provided. Social interacting B-Types check for Pareto-optimality and, in case of detecting a Pareto-optimal allocation, do not change their voluntarily declared income. Such a behavior reflects the efforts by B-Types to maintain a Pareto-optimal allocation. However, to identify Pareto-optimality of allocations, we make use of the procedure adopted by Hokamp and Pickhardt (2011) and Pickhardt (2012) which employs a maximum number of free-riders, , tolerated by the Pareto-optimality concept. If more than taxpayers do not pay their tax due, the allocation is not Pareto-optimal. Hence, the check for Pareto-optimality is a step function and directly influences social interacting B-Types as well as indirectly via their social network. Neoclassical A-Types are indirectly affected by the resulting change in the provision of public goods.
The model is able to capture a huge variety of scenarios because it allows for a wide selection of parameter values. For instance, in Sections 9.3.1–9.3.3 we examine an artificial political and public goods provision cycle by Hokamp and Pickhardt (2010) and Hokamp (2013, 2014). Targeting the tax declaration behavior of the taxpayers at the micro level, we employ theories as well as empirical findings. The neoclassical A-Types apply the theory of expected utility maximization proposed by Allingham and Sandmo (1972). Further, the investigation of lapse of time effects is motivated by changes to the German tax law. In tax year 2009, the relevant periods subject to an audit were changed from 5 to 10 years (Hokamp and Pickhardt, 2010), that is, and .10
To adjust the subjective audit probability, so that it reflects psychological impacts (Kirchler, 2007), we assume an update of per period in line with Hokamp and Pickhardt (2010). Furthermore, for the social interaction by the B-Types we follow Hokamp and Pickhardt (2010) and apply a mutation to full tax compliance, that is, , four tax years due to psychological reasons (Kirchler, 2007; Zaklan et al., 2008). Age heterogeneity and social norms date back to the findings by Kirchler (2007), Traxler (2010), and Nordblom and Z̆amac (2012), with the elderly showing a higher tax compliance than younger taxpayers. Following Hokamp (2014) we assume , and an update of social norms with respect to a taxpayer's is summarized in Table 9.2
Table 9.2 Social norm updating. Risk aversion changes with respect to age
Age | ||||
Risk aversion |
For the simulations in Sections 9.3.1–9.3.3 we employ the political and public goods provision cycle by Hokamp and Pickhardt (2010) and Hokamp (2014). For instance, the efficiency of tax collection, , is taken from Slemrod and Yitzhaki (2002). Thus, a public good is examined with a view to represent the standard theory and empirical findings on linear public goods experiments (e.g., by Ledyard, 1995; Zelmer, 2003; Croson, 2007) and we confirm the counter-intuitive result provided by Falkinger (1988, 1995) and Cowell (1992) that a higher provision level increases the extent of tax evasion. The check for Pareto-optimality originates from theoretical findings by Hokamp and Pickhardt (2011) and Pickhardt (2012).
Andreoni et al. (1998) estimate that about 7% of US taxpayers overpay their tax liabilities. In Sections 9.3.1 and 9.3.2 we use this empirical background for erratic D-Types to assume a share of about 15% in the population to account for unintended tax noncompliance (Hokamp and Pickhardt, 2010). In particular, we suppose that this group of taxpayers tries to behave like ethical C-Types, centering their declared income around perfect compliance, allowing for some overpayment, which results in a normal distribution with expectation and standard deviation . However, the tax declaration experiment conducted by Alm et al. (2009) does not allow for overpayment, so that we suppose as a best guess a normal distribution with expectation and standard deviation for our calibration and sensitivity analysis in Section 9.3.4.
Spatial aspects do not play a role in our agent-based framework but the environment is explicitly modeled via a government and tax authority. The government provides public goods and the tax authority collects tax and penalties; bureaucratic activities do not involve any kind of individual decision-making. In contrast, the taxpayers follow four behavioral archetypes or individual decision heuristics (i) expected utility maximization, (ii) social interaction, (iii) ethical motivation, and (iv) erratic perception described in Section 9.2.1.2. The taxpayers adapt their tax declaration behavior to exogenously changing parameters (e.g., tax rate , and penalty ), but also may use a subjective audit probability (A-Types) or an average extent of tax noncompliance within their social network (B-Types), which are derived endogenously. Social norms are implemented via an increase of risk aversion, , for senescent neoclassical A-Types. Hence, we model tax noncompliance as a social process, in which temporal aspects play an essential role in the declaration process via growing older and lapse of time effects. Erratic D-Types may err due to the complexity of the tax law. Further, uncertainty in the model is reflected by a random selection of the taxpayers subject to an audit.
The tax authority learns a taxpayer's true income and history via an audit. A neoclassical A-Type is able to learn the objective audit probability, iff “subjective audit probability” is activated, while a social interacting B-Type gathers information on the tax declaration behavior of his vicinity or social neighborhood. All other entities do not learn. While collective learning is not implemented in the model, the average extent of tax noncompliance learned by B-Types may be thought of as a type of “collective” learning.
The taxpayers correctly sense the information provided by the government and the tax authority. Nonetheless, erratic D-types may err in voluntarily declaring their income due to the complexity of the tax law, which reflects their misperception of the tax regime. If “subjective audit probability” is activated, the neoclassical A-Types misperceive the objective audit probability in some periods after an audit has discovered their cheating behavior on taxes. However, tax declarations are correctly perceived by the tax authority, but the government and the tax authority have no information on the true income besides the information that is provided by the taxpayers, voluntarily or via an audit. When considering the provision of a public good, the model also takes into account the costs for the tax collection and the audit process to uncover tax noncompliance.
The model is not forward looking and, therefore, does not include any individual prediction. However, the neoclassical A-Types use an expected utility maximization procedure and subjective audit probabilities that represent their perception of the future. The social interacting B-Types form beliefs of the tax compliance behavior in their neighborhoods. Further, iff “Pareto-optimality” is activated, B-Types do not change their contribution to the public good to maintain in the future the Pareto-optimal allocation observed. All decisions of the agents are based on present time and the tax relevant periods subject to back audits.
Between the government and the tax authority on the one side and the taxpayers on the other side, the interaction is direct via tax enforcement variables (, , and ), the audit process, tax payment, reimbursement and penalties, and the interplay takes place indirectly via public goods provision. The interaction among taxpayers is affected by the average rate of tax compliance in the social network for B-Types and public goods provision for A-Types. Hence, the communication channels between taxpayers are the level of public goods provision and the average extent of tax noncompliance.
Agents do not form a collective. However, the social network of B-Types may be considered as some kind of “collective.” The size, , of the social network, , is a parameter of the model, such that the chosen number of acquaintances is randomly selected. Neighborhoods are constructed in such a way that if taxpayer is in the social network of taxpayer , the reverse might not be true. In other words, the relationships between taxpayers within a social network are basically one-directional.
The parameters, announced by the government and the tax authority, may change with respect to time under the exception or condition to repeat every 40 tax years. The taxpayers might be heterogeneous concerning their initial properties, (i) age, (ii) income, (iii) risk aversion, and (iv) list of acquaintances. In addition, heterogeneity may be caused by the behavioral archetype of the taxpayers, (i) neoclassical A-Type, (ii) social interacting B-Types, (iii) ethical C-Types, and (iv) erratic D-Types. Further, when time evolves, the heterogeneity spreads regarding the individual experience of tax declarations, audits, and payments of penalties and reimbursement.
Age, income, risk aversion, and the list of acquaintances are randomly assigned in the initial period. However, the model allows for switching “off” the random assignment and, instead, to specify, for example, risk aversion, which we set to in Section 9.4. Further, the audits of taxpayers by the tax authority are random.
We collect data at each time step on the extent of tax compliance at the macro level and the behavioralgroup level described in Eqs (9.8) and (9.9), respectively. In particular, we are interested in the interaction among the taxpayers themselves within the tax declaration environment, and we study various scenarios as described in Section 9.3. We reaffirm the key findings by Hokamp (2014) that the dynamics by lapse of time and social norm updating have a particularly strong effect on the extent of tax evasion and we add an explanation at the behavioral group level. In addition, we visualize key aspects of the expected utility maximization approach adopted by Allingham and Sandmo (1972). Finally, in Section 9.3.4 we calibrate our model to the experimental tax compliance data collected by Alm et al. (2009) and provide novel insights into the influence of the tax rate, , the penalty, , the audit probability, , the tax complexity, , and the risk aversion, , on tax noncompliance.
The model is implemented in RePast making use of the Java software packages. Hokamp (2013) presents computational simulations with different random seeds and finds that the fluctuations due to random effects are rather small. Hence, we restrict our simulations in Section 9.3 to random seed 11223344 to allow for an identification of Pareto-optimal allocations. Following the work by Hokamp (2014), we updated the code with a view to reconsider the extent of tax compliance at a behavioral group level. Code is available upon request. Coincidently, such a procedure simplifies an independent reproduction of our results.
At the initial state of the artificial world no taxpayer evades tax and therefore his history of tax noncompliance does not reveal any kind of tax offence. The government and the tax authority have no costs and no revenues at the initial state. The initialization of the model may differ, for example, in the distribution of income and in the tax declaration by the taxpayers. For instance, a change in the composition of behavioral archetypes in the population of taxpayers might affect the audit process. To allow for ceteris paribus conditions it is possible to employ an identical random seed for generating the random numbers, which govern random effects such as the auditing process.
The model does not utilize input data from external sources. All input data are specified within the model, for instance, the political and public goods cycle, as referred in Hokamp and Pickhardt (2010) and Hokamp (2013, 2014).
There are no submodels.
This section provides the scenarios under consideration, the resulting outcome and a brief discussion of our key findings. Ceteris paribus and jointly, we examine the tax evasion dynamics by six operation modes, (i) lapse of time, (ii) social norms, (iii) age heterogeneity, (iv) subjective audit probability, (v) public goods provision and (vi) Pareto-optimality described in Section 9.2.1.3. First, we replicate (Hokamp, 2014) with a view to add novel insights into the behavioral group level in Sections 9.3.1 and 9.3.2. Second, we employ a behaviorally homogeneous population of neoclassical A-Types to elucidate the expected utility maximization approach by Allingham and Sandmo (1972) in Section 9.3.3. Finally, in Section 9.3.4 we calibrate our model with the experimental tax compliance data provided by Alm et al. (2009) and present our sensitivity analysis on tax and penalty rate, audit probability, tax complexity, and risk aversion. Table 9.3 summarizes the scenarios and related specifications.
Table 9.3 Overview: specification of agent-based simulations
Feature | Lapse | Tax | Age | Social | Public | Pareto- | Subjective | Figure |
of | relevant | heterogeneity | norm | goods | optimality | audit | ||
time | periods | updating | probability | |||||
Scenario inspired by Hokamp (2014) | ||||||||
Political and public goods cycle; Risk uniform distributed [0;1] | ||||||||
Distribution of behavioral types: A = 0.5; B = 0.35; C = 0; D = 0.15 | ||||||||
Reference 1; | LOT = 0 | 80 | Off | Off | Off | Off | On | 9.1–9.4 |
Reference 2; | LOT = 10 | 80 | Off | Off | Off | Off | On | 9.1–9.4 |
AH; | LOT = 0 | 80 | On | Off | Off | Off | On | 9.1 |
AH and SNU; | LOT = 0 | 80 | On | On | Off | Off | On | 9.1 |
SNU; | LOT = 0 | 80 | Off | On | Off | Off | On | 9.1 |
AH; | LOT = 10 | 80 | On | Off | Off | Off | On | 9.2 |
AH and SNU; | LOT = 10 | 80 | On | On | Off | Off | On | 9.2 |
SNU; | LOT = 10 | 80 | Off | On | Off | Off | On | 9.2 |
PGP; | LOT = 0 | 80 | Off | Off | On | Off | On | 9.3 |
… and AH and SNU; | LOT = 0 | 80 | On | On | On | Off | On | 9.3 |
… and PO (C = 0.15; D = 0); | LOT = 0 | 80 | On | On | On | On | On | 9.3 |
PGP; | LOT = 10 | 80 | Off | Off | On | Off | On | 9.4 |
… and AH and SNU | LOT = 10 | 80 | On | On | On | Off | On | 9.4 |
… and PO (C = 0.15; D = 0) | LOT = 10 | 80 | On | On | On | On | On | 9.4 |
Scenario inspired by Allingham and Sandmo (1972) | ||||||||
Audit probability ; Risk uniform distributed [0;1] | ||||||||
Distribution of behavioral types: A = 1; B = C = D = 0 | ||||||||
Allingham and Sandmo | LOT = 0 | 80 | Off | Off | Off | Off | Off | 9.5 |
Lapse of time effects | LOT = 5 | 80 | Off | Off | Off | Off | Off | 9.5 |
Lapse of time effects | LOT = 10 | 80 | Off | Off | Off | Off | Off | 9.5 |
Subjective audit probability | LOT = 0 | 80 | Off | Off | Off | Off | On | 9.6 |
Public goods provision | LOT = 0 | 80 | Off | Off | On | Off | Off | 9.6 |
Age heterogeneity and | LOT = 0 | 80 | On | On | Off | Off | Off | 9.6 |
social norm updating | ||||||||
Calibration to Alm et al. (2009) and Bazart et al. (2016) | ||||||||
Tax rate ; Penalty rate ; Audit probability ; Tax complexity | ||||||||
Distribution of behavioral types: A = 0.26; B = 0; C = 0.39; D = 0.35 | ||||||||
SAP On | LOT = 0 | 15 | Off | Off | Off | Off | On | 9.7 |
Risk uniform distributed [0;1] | ||||||||
Risk () | LOT = 0 | 15 | Off | Off | Off | Off | On | 9.7 |
Social interacting B-Types | LOT = 0 | 15 | Off | Off | Off | Off | On | 9.7 |
Distribution of behavioral types: A = 0.21; B = 0.15; C = 0.34; D = 0.30 | ||||||||
Sensitivity analysis | ||||||||
Distribution of behavioral types: A = 0.21; B = 0.15; C = 0.34; D = 0.30 | ||||||||
LOT = 0 | 15 | Off | Off | Off | Off | On | 9.7 | |
Scenarios: Tax rate ; Penalty rate ; Audit probability ; Tax complexity ; Risk |
Two scenarios are considered based on (i) Hokamp (2014) and (ii) Allingham and Sandmo (1972), and, in addition, a calibration to Alm et al. (2009) and Bazart et al. (2016) and a sensitivity analysis are presented. For each group of simulations the parameters of the model are specified, that is, tax rate , penalty rate , audit probability , tax complexity , risk , and the distributions of the four behavioral types, (i) neoclassical A-Types, (ii) social interacting B-Types, (iii) ethical C-Types, and (iv) erratic D-Types. The political and public goods cycle is taken from Hokamp and Pickhardt (2010) and Hokamp (2014) provided in the Appendix 9A. Further, the specifications of the 28 simulation runs are presented with respect to seven features, which are, (i) “lapse of time” (LOT), (ii) “tax relevant periods,” (iii) “age heterogeneity” (AH), (iv) “social norm updating” (SNU), (v) provision of pure “public goods” (PGP), (vi) a check for “Pareto-optimality” (PO), and (vii) an adjustment of the “subjective audit probability” (SAP). “on” marks activated features. “off” indicates options that are switched off. “…” extends the preceding simulation, for example, with “Pareto-optimality” (PO).
Figures 9.1 and 9.2 elaborate Hokamp (2014, p. 195, Figures 1.1 and 1.2) by showing the effects of age heterogeneity and social norm updating under lapse of time and an adjustment of the subjective audit probability as described in Sections 9.2.1.3 and 9.2.2.1. We adopt the size of the population, , and the behavioral type distribution: 50% A-Types, 35% B-Types, 0% C-Types, and 15% D-Types; and for income, , and risk aversion, , the uniform distribution on and , respectively (from Hokamp, 2014), and, in addition, we employ the political cycle from Hokamp and Pickhardt (2010) outlined in the Appendix 9A. The simulations are done with the active mode “subjective audit probability.” We present the extent of tax evasion for 80 tax relevant periods and the corresponding results at the behavioral group level for A- and B-Types. We do not show the graphs for C- and D-Types, since the former are fully tax compliant and the latter account for (unintended) tax noncompliance of less than 3%.
Figure 9.1(a) confirms that without lapse of time effects, , at the macro level no effect seems to be visible by age heterogeneity alone (Hokamp, 2014). However, the inspection of Figure 9.1(b) and (c) reveals that, contrary to A-Types, B-Types can be influenced by age heterogeneity alone (e.g., by % in tax relevant period 58). The B-Types become more tax compliant since newborn taxpayers declare their true income in their first tax relevant period. Social norms alone represent the lower bound since, after 30 tax relevant periods, each A-Type has reached the cohort with a risk aversion uniformly distributed on ; the effect on tax evasion is smaller for B-Types, strongly influenced by the size of the social network, , than it is for A-Types. Further, jointly considering social norms and age heterogeneity allows taxpayers to update their social norms, to die and to be born; creating a scenario in which we find a strong impact on the A-Types.
Figure 9.2(a) shows at the macro level the impact of age heterogeneity alone, (e.g., % in tax relevant period 8 and % in tax relevant period 12). Hence, there is no clear-cut conclusion regarding how age heterogeneity under a back auditing scheme influences the extent of tax noncompliance. Figure 9.2(b) and (c) provide novel insight that the impacts under lapse of time, with and without social norms, are mainly driven by A-Types. We also find that the peaks in Figure 9.2(b) are caused by lapse of time effects and the update of subjective audit probability, which forces the A-Types to become fully tax compliant after experiencing an audit and paying a penalty.
In Section 9.3.3 we investigate age heterogeneity and social norm updating in a behaviorally homogeneous population of 100% neoclassical A-Types and we answer related questions on lapse of time, subjective audit probability and public goods provision. However, in the next subsection we explore public goods provision and a check for Pareto-optimality in a behaviorally heterogeneous population, which is mainly driven by social interacting B-Types.
Figures 9.3 and 9.4 elaborate Hokamp (2014, p. 196, Figures 2.1 and 2.2) and show the effects by public goods provision and Pareto-optimality described in Sections 9.2.1.3 and 9.2.2.1. Following the methodology from Hokamp (2014), we take the size of the population, , and the behavioral type distribution: 50% A-Types, 35% B-Types, 0% C-Types, and 15% D-Types (and 15% C-Types, 0% D-Types, when the check for “Pareto-optimality” is activated, since the procedure cannot deal with tax overpayment). In addition, we suppose income, , and risk aversion, , to be uniformly distributed on and , respectively, and the political and public goods provision cycle outlined in the Appendix 9A. We ran our simulations with the active mode “subjective audit probability,” and, therefore, the neoclassical A-Types become more compliant for some endogenously determined periods. As output we present the extent of tax evasion for 80 tax relevant periods and the corresponding results at the behavioral group level for A- and B-Types and, again, we do not show the graphs for C- and D-Types for reasons discussed earlier.
Figure 9.3(a) shows at the macro level the tax compliance dynamics by public goods provision and a check for Pareto-optimality without lapse of time effects, ; for example, in the tax relevant periods 37–40 the extent of tax noncompliance is increased, (i) by % through the provision of a public good and (ii) through Pareto-optimality. Figure 9.3(b) visualizes that neoclassical A-Types account for an increase by % of tax evasion by the provision of a public good in period 37–40. Regarding Pareto-optimality, the effects on tax evasion are unclear, for instance, in period 19 we recognize a decrease while in period 38 there is an increase. Figure 9.3(c) presents the tax evasion dynamics by social interacting B-Types; for example, public goods provision and Pareto-optimality decrease tax evasion in periods 18–29 while they increase in periods 54–60.
Figure 9.4(a) shows at the macro level the extent of tax noncompliance by public goods provision and Pareto-optimality with lapse of time effects, . In particular, we confirm the existence of Pareto-optimal allocations in periods 10–17. Figure 9.4(c) shows two peaks in the extent of tax evasion by social interacting B-Types reaching 14.21 and 16.48% in period 13 and 17, respectively. Hence, under lapse of time effects, the check for Pareto-optimality leads to a higher extent of tax evasion than the reference case without public goods provision. The scenario with age heterogeneity and social norm updating provides a baseline of tax evasion under back auditing. Figure 9.4(b) represents the dynamics caused by neoclassical A-Types; among others, we recognize peaks in period 21 and 24, which are due to lapse of time effects and subjective audit probability updating. We investigate such effects in more detail in the next subsection.
We recognize that neoclassical A-Type taxpayers particularly strongly influence the extent of tax evasion at the macro level. Within the behaviorally homogeneous population of neoclassical A-Types, each taxpayer is endowed with a risk parameter, , uniformly distributed on . We assume a constant income, Tokens, a constant objective audit probability, and vary the tax rate between 0.05 and 0.65 (in steps of 0.05). We apply a constant sanction rate, %, to account for the critique by Yitzhaki (1974). Hence, the penalty rate ranges between 0.075 and 0.975. As output, we investigate for 80 tax relevant periods, the extent of tax evasion by A-Types with respect to, (i) lapse of time effects, (ii) subjective audit probability, (iii) public goods provision, and (iv) age heterogeneity combined with social norm updating.
Figure 9.5 visualizes the dynamic effects by lapse of time. The standard approach by Allingham and Sandmo (1972) without lapse of time effects, that is, , is presented as a reference case in Figure 9.5(a). We find that tax compliance takes place even for a comparatively low audit probability. A tax rate of allows for 95% tax evasion while a tax rate of 0.65 corresponds to 21%. The penalty rate and the uniformly distributed risk parameter are the key factors that explain this outcome since the interval described in Eqs (9.4) and (9.5) depends on both. Hence, a higher penalty rate shifts the lower bound, Eq. (9.4), toward zero, such that full evaders start to declare a part of their true income. The upper bound, Eq. (9.5), does not change, because of selecting a constant sanction rate. In addition, the uniform distribution of risk aversion allows that a neoclassical A-Type may fully evade, partially declare, or honestly announce his income. However, our results show that increasing the tax rate contributes to lowering the extent of tax evasion in an Allingham-and-Sandmo setting.
Figure 9.5(b) provides the extent of tax evasion with tax relevant periods subject to back audits. The results show peaks in tax evasion in periods that occur because (i) the first tax relevant period allows for a maximum extent of tax evasion, and (ii) periods drop out of the expected maximization procedure after six periods. Recall that in the model the actual period and, in addition, up to five periods can be subject to an audit by the tax authority. As a result, the upper level of tax evasion decreases with respect to time. A tax rate of 0.05 reaches 63% tax evasion in period 79 while a tax rate of 0.65 corresponds to 12%. The lower level is reached before the periods with a high extent of tax evasion drop out, that is, in periods . A tax rate of 0.05 corresponds to 25% tax evasion in period 78 while a tax rate of 0.65 allows for 4%. Thus, a back audit of 5 tax relevant periods significantly decreases tax evasion (e.g., 95 vs. 25 and 21 vs. 4%).
Figure 9.5(c) presents the tax evasion behavior when the actual period and 10 previous tax relevant periods, , are considered for the expected utility maximization procedure. Hence, periods are characterized by a peak in tax evasion. The upper level of tax evasion reaches 57% for a tax rate of 0.05 while the lower level is 17%. Considering a tax rate of 0.65 the upper level of tax evasion becomes 12% while the lower level is 2%. To conclude, the lapse of time of 10 tax relevant periods contributes to reducing the extent of tax evasion (e.g., 95 vs. 12 and 21 vs. 2%). The next subsection considers the dynamics by subjective audit probability, public goods provision, age heterogeneity, and social norm updating.
Figure 9.6 ceteris paribus shows the effects by an adjustment of the subjective audit probability, public goods provision (data by Hokamp, 2014, is provided in Tables 9A.1 and 9A.2 in the Appendix 9A), and age heterogeneity combined with social norm updating. Figure 9.6(a) elucidates the effects of adjusting the subjective audit probability described in Sections 9.2.1.3 and 9.2.2.1. After initializing our model, which needs five tax years, the extent of tax evasion fluctuates around a value significantly lower than without the adjustment of the subjective audit probability; for the reduction of tax evasion is % and the tax rate 0.65 corresponds to %.
Figure 9.6(b) investigates the dynamics of public goods provision explained in Section 9.2.1.3. We show that public goods provision increases the extent of tax evasion; for the maximum increase of tax evasion is in periods 5–8 while the tax rate 0.65 corresponds to . Note that this relationship is nonlinear. In behaviorally heterogeneous populations, the extent of tax evasion at the behavioral group level becomes even higher for neoclassical A-Types, for example, see Section 9.3.2.
Figure 9.6(c) jointly presents the extent of tax evasion by age heterogeneity and social norm updating outlined in Sections 9.2.1.3 and 9.2.2.1. After initialization of our model, which needs 11 tax relevant periods, the tax evasion fluctuates around a value significantly lower than without the inclusion of “age heterogeneity” and “social norm updating” effects; for the reduction of tax evasion is %, the tax rates 0.15 and 0.65 correspond to and %, respectively. Starting at time step 21 we identify a raise in the extent of tax evasion, for example, for , and this phenomenon repeats starting at period 61, showing a cohort effect of social norm updating. However, in the first period the tax evasion is higher than the benchmark, since social norms influence the distribution of risk aversion (tax rates 0.05, 0.15, and 0.65 cause an increase of tax evasion in the initialization period by , , and , respectively). The next section calibrates our ABM to experimental data assuming a behaviorally heterogeneous population.
We calibrate our economic agent-based tax noncompliance model with experimental data provided by Alm et al. (2009). In each round of the experiment the participants in the tax declaration game earn an income of 60, 70, 80, 90, or 100 Tokens. The individuals in the study are faced with a tax rate of , a sanction rate of 150%, and an audit probability of , which governs their decision on how much income to voluntarily declare to the tax authority. The population of 40 taxpayers is grouped into five social networks of eight group members. In addition, the taxpayers are informed on the average extent of tax noncompliance in their peer group, but they have no information on the individual outcome of the audit process; the environment, outlined ealier, does not change during the 15 rounds of the experiment. In particular, we make use of the experimental data from the session “official information (T2A)” (Alm et al., 2009) and we accordingly adjust the parameters of our ABM. We visualize the outcome of our calibration in Figure 9.7 and present our sensitivity analysis at the behavioral group level in Table 9.4.
Figure 9.7(a) shows the extent of tax noncompliance in the tax declaration game by Alm et al. (2009) as a benchmark. In addition, we present the corresponding results of the calibration by Bazart et al. (2016) to allow for a comparison. We recognize that the erratic D-Types are differently defined by Bazart et al. (2016), since Alm et al. (2009) do not consider tax overpayment. Hence, we recall our assumption that the tax declaration behavior by erratic D-Types is governed by a normal distribution with expectation and standard deviation (see Section 9.2.2.1). Further, let the tax complexity be . Following Bazart et al. (2016, p. 138, Figure 3a) we take their behavioral type distribution: 26% A-Types, 0% B-Types, 39% C-Types and 35% D-Types. The active mode “ subjective audit probability” described in Section 9.2.1.3 leads to a comparatively low level of tax noncompliance since neoclassical A-Types are highly tax compliant. Switching from a uniform distribution on to a constant risk aversion increases the extent of tax evasion by A-Types and, thus, raises tax noncompliance in the whole population. To include imitating behavior patterns (Bazart et al., 2016), we adjust the behavioral type distribution by increasing the share of social interacting B-Types at the expense of the three remaining archetypes to get a similar average extent of tax noncompliance as Alm et al. (2009), 33.6 versus 33.3%.
Hence, we conduct our sensitivity analysis under the behavioral type distribution: 21% A-Types, 15% B-Types, 34% C-Types, and 30% D-Types. Figure 9.7(b) visualizes the time evolution of tax compliance when investigating a 10% increase of the tax and penalty rate, the audit probability, the tax complexity, and risk aversion. Our results reveal that increasing the tax and penalty rate have the strongest impact on the extent of tax evasion, positively and negatively, respectively.11
Table 9.4 Calibration and sensitivity analysis: average extent of tax evasion
Average extent of tax evasion | Behaviorally heterogeneous society | Neoclassical A-types | Social interactig B-types | Ethical C-types | Erratic D-types |
Calibration | |||||
Scenario inspired by Alm et al. (2009) | 33.3 | ||||
SAP on | 23.5 | 3.8 | 0 | 0 | 50.5 |
Risk uniform distributed [0;1] | |||||
Risk () | 29.9 | 52.2 | 0 | 0 | 50.5 |
Social interacting B-types | 33.6 | 46.9 | 10.7 | 0 | 49.7 |
Sensitivity analysis | |||||
Tax rate | +2.59 | +6.96 | +1.17 | 0 | 0 |
Penalty rate | 0 | 0 | |||
Audit probability | 0 | 0 | |||
Tax complexity | +0.03 | 0 | +0.07 | 0 | +0.07 |
Risk | 0 | 0 |
“Calibration” shows the extent of tax evasion averaged over 15 tax relevant periods for a behaviorally heterogeneous society and the four groups of behavioral types according to the scenarios specified in Table 9.3. “Sensitivity Analysis” presents in absolute values the difference in the average extent of tax evasion compared to the benchmark “social interacting B-Types.”
Table 9.4 presents the extent of tax evasion averaged over 15 tax relevant periods. The tax declaration behavior by erratic D-Types differs when the behavioral type distribution is changed, although we made use of the same random seed 11223344. The A-Types are strongly affected by the tax () and penalty rate (), the effects by risk aversion () and audit probability () are orders of magnitudes smaller. The B-Types are influenced by all five parameters; again, the penalty rate () causes the strongest impact. Surprisingly, tax complexity identically changes the tax declaration by B- and D-Types (). To summarize, the higher the penalty rate the stronger the effect on hampering tax evasion () and the higher the tax rate the strongest the impact on the extent of tax evasion (). If we consider the aggregated influence of these two parameters, while keeping their ratio constant, then the net effect of increasing the tax rate will be a reduction of tax evasion, as the penalty rate dominates the effect.
In this chapter, we introduced an agent-based tax evasion model based on Hokamp and Pickhardt (2010) and Hokamp (2013, 2014), which consists of a government, a tax authority, and four behavioral groups of taxpayers governed by, (i) neoclassical expected utility maximization, (ii) social interaction, (iii) ethical motivation, and (iv) erratic perception. We addressed the impact of these behavioral groups on the extent of tax noncompliance both at the macro level and the group level. In particular, we replicated the scenarios provided by Hokamp (2014), confirmed his findings and elucidated the extent of tax evasion at the micro level. We showed a strong impact by neoclassical A-Types, which provides the incentive to reconsider the Allingham-and-Sandmo approach. The model incorporated psychological findings, for example, by Kirchler (2007) and Traxler (2010), and we measured the resulting effects on social norms and the subjective audit probability. In addition, we provided figures for lapse of time effects, age heterogeneity, and social norm updating, which reaffirm the notion that tax evasion models should incorporate back audits and the time evolution of tax declarations (Hokamp, 2014). Finally, we calibrated our economic ABM with the tax declaration game devised by Alm et al. (2009) and conducted a sensitivity analysis. Our findings revealed that the tax rate positively influences the extent of tax evasion (%) and that the penalty rate negatively impacts tax evasion (%), when the parameters are increased by 10%. Table 9.4 summarized our results.
With respect to data, an extension of our economic ABM might be to include as input the income distribution at a state level, for example, see Llacer et al. (2013). Additional extensions might be to allow the taxpayers to accumulate wealth and to engage in a labor market. Further, more experiments are needed to estimate lapse of time effects, to elucidate the risk aversion of neoclassical A-Types and to identify the social network of interacting B-Types. However, these issues delineate a rich research agenda, which we leave for the future.
The work originates from the Ph.D. Thesis by Sascha Hokamp entitled “Income Tax Evasion and Public Goods Provision – Theoretical Aspects and Agent-Based Simulations” (Hokamp, 2013) and was financially supported by the Westfälische Wilhelms-Universität Münster, Germany, the Brandenburg University of Technology Cottbus, Germany, the Deutsche Bundesbank, the German Academic Exchange Service (DAAD), and the European Social Simulation Association (ESSA). Investigating the micro perspective on tax noncompliance under social information was partly supported by the Cluster of Excellence “Integrated Climate System Analysis and Prediction” (DFG EXC 177 CliSAP), Universität Hamburg, Germany. We would like to thank two reviewers for valuable comments. However, all errors remain ours.
We present the political cycle and the public goods provision cycle by Hokamp and Pickhardt (2010), Hokamp (2014) used for our simulations in Sections 9.3.1–9.3.3.
In addition to the public goods provision cycle, Table 9A.1 presents resulting values for the voluntary marginal per capita return
the forced Marginal Per Capita Return
and the maximum number of freeriders, , tolerated by the Pareto-optimality concept (Hokamp and Pickhardt, 2011; Hokamp, 2014 and Pickhardt, 2012).
Table 9A.1 Political cycle
Parameter | Tax rate | Penalty rate | Tax complexity | Audit probability |
Abbreviation | ||||
Tax relevant period | ||||
1 | 0.20 | 0.30 | 0.10 | 0.01 |
5 | 0.20 | 0.30 | 0.10 | |
9 | 0.20 | 0.10 | 0.03 | |
13 | 0.45 | 0.10 | 0.03 | |
17 | 0.30 | 0.45 | 0.03 | |
21 | 0.45 | 0.20 | 0.03 | |
25 | 0.40 | 0.45 | 0.20 | |
29 | 0.40 | 0.45 | 0.20 | |
33 | 0.40 | 0.20 | 0.05 | |
37 | 0.50 | 0.20 | 0.05 | |
40 | 0.30 | 0.50 | 0.20 | 0.05 |
Source: Reproduced from Hokamp and Pickhardt (2010, p. 544, Table 1) and Hokamp (2014, p. 197, Table A1). Changing Parameters are denoted in bold.
Table 9A.2 Public goods provision cycle
Parameter | Taxcollection | Auditingprocess | Fixedcosts | Publicgoods | Voluntary | Forced | Maximumnumber offree-riders |
Abbreviation | |||||||
Tax relevant period | |||||||
1 | 0.006 | 0.50 | 10,000 | 0.2000 | 0.198800 | 0.100000 | 5 |
5 | 0.006 | 10,000 | 0.2000 | 0.198800 | 0.150000 | 5 | |
9 | 0.006 | 0.25 | 10,000 | 0.000994 | 0.000750 | 1006 | |
13 | 0.25 | 10,000 | 0.0010 | 0.000900 | 0.000750 | 1111 | |
17 | 0.100 | 0.25 | 10,000 | 0.004500 | 0.003750 | 222 | |
21 | 0.100 | 0.25 | 0.0050 | 0.004500 | 0.003750 | 222 | |
25 | 0.100 | 0.25 | 0 | 0.006750 | 0.005625 | 148 | |
29 | 0.100 | 0.25 | 0 | 0.009000 | 0.007500 | 111 | |
33 | 0.25 | 0 | 0.0100 | 0.009940 | 0.007500 | 100 | |
37 | 0.006 | 0.25 | 0 | 0.198800 | 0.150000 | 5 | |
40 | 0.006 | 0.25 | 0 | 0.2000 | 0.198800 | 0.150000 | 5 |
Source: Reproduced from Hokamp (2014, p. 195, Table 3).
Four parameters are assigned to values, that are, the effectiveness of “tax collection,” , for example, Slemrod and Yitzhaki, (2002), argue for on p. 1426 and on p. 1449) the effectiveness of “auditing process,” ; the “fixed costs” of tax authority, ; and the effectiveness of the “public goods” sector, . Given these four figures the “marginal per capita return” of voluntary () and forced () contributions to public goods as well as the “maximum number of free-riders” () are derived. “Abbreviation” lists related mathematical symbols. “Tax relevant period” denotes the point in time when changes take place, which is every fourth period. Changing parameters are denoted in bold. Fixed costs and the maximum number of free-riders are denoted in tokens and number of agents, respectively. Remaining figures are provided in percent.