9.2.1.1 Purpose

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).

9.2.1.2 Entities, State Variables, and Scales

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.

	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.

Neoclassical A-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,

9.1

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

9.2

and

9.3

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

9.4

and the upper bound

9.5

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

9.6

Social Interacting B-Types

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

9.7

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.

Model Output: The Extent of Tax Evasion

At each tax relevant period the extent of tax evasion, , of the whole group of taxpayers is calculated as

9.8

while the group behavior of the different kinds of taxpayers is expressed as

9.9

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.

9.2.1.3 Process Overview and Scheduling

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.

Lapse of Time

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

9.10

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.

Public Goods Provision

If this feature is activated, the government provides a public good, , which can be nonrivalrously consumed by the whole population of taxpayers

9.11

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, .

12

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.

9.2.2.1 Theoretical and Empirical Background

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

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.

9.2.2.2 Individual Decision-Making

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.

9.2.2.3 Learning

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.

9.2.2.4 Individual Sensing

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.

9.2.2.5 Individual Prediction

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.

9.2.2.6 Interaction

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.

9.2.2.7 Collective

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.

9.2.2.8 Heterogeneity

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.

9.2.2.9 Stochastic

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.

9.2.2.10 Observation

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.

9.2.3.1 Implementation Details

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.

9.2.3.2 Initialization

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.

9.2.3.3 Input Data

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).

9.2.3.4 Submodels

There are no submodels.

9.3.3.1 Lapse of Time Effects

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.

9.3.3.2 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.