An Agent Based Model for Studying Optimal Tax Collection Policy Using Experimental Data: the case of Chile and Italy Nicol´as Garrido∗

Luigi Mittone†

November 24, 2008

Abstract This paper investigates optimal audit programs in an economy populated by artificial agents. The behaviors of the artificial agents are calibrated using data obtained from experiments on fiscal evasion made in the north of Chile (Antofagasta) and the north of Italy (Trento). The tax collection policy that we find is optimal, in the sense that its outperforms the tax payments made by the calibrated agents, using any other standard plans of collection used by governments. We found that the design of an optimal audit scheme depends on three components: the income distribution, the identification of patterns of behaviors and the number of periods in the history of an individual controlled with he suffers an auditory. ∗ Universidad Cat´ olica del Norte. Nuclei of the Millennium Science Initiative Program ”Regional Science and Public Policy” , [email protected] † Universit di Trento. Dipartimento di Economia, Computable and Experimental Economic Laboratory (CEEL)

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1

Introduction

According to [EER01] tax evasion in Chile during year 1999 was about 6% of GDP, while the tax collection was of 18% of GDP. In Italy, the hidden economy figure in 1998 was about 27.8% of GDP (see [Sch00]). The aim of an institution to enforce tax laws is to design a taxation, audit programs and punishment scheme to meet revenue objectives. The efficiency of the program, improves government revenues and reduces tax evasion. This paper is about the optimal design of an audit program for improving the net revenue of the government, given a tax and a punishment scheme. We explore the audit program in an economy populated by artificial agents. The behavior of the artificial agents is calibrated using data from experiments run in Chile and Italy. Modeling audit programs means to model the behavioral interaction between taxpayers and tax authority. There are two line of research for understanding tax audit selection decisions. On the one hand there are empirical work trying to discover what makes the audit more likely given characteristics of the taxpayers, and on the other hand there is a more intensive literature on theoretical model describing which are the optimal audit rule, considering specific behavioral assumption of the agents. The empirical literature has found that the likelihood of an audit is correlated with the values of certain tax return line items. According to the review done by [JAF98], reports of capital income or a large tax liability are associated with a greater chance of being audited. Moreover, tax agencies possess private information not recorded in the available tax return, which play an important role in audit selection. As indicated by the authors, this information is highly correlated with actual noncompliance detected during audits. On the theoretical side, the models of optimal taxation and audit programs differ on the assumptions about taxpayers and tax authority. As suggested by [San04], the decision of one taxpayer might be affected by the decision of other taxpayers. It is more costly to be honest in a country where corruption is common or similarly it may be less risky to evade taxes in country where evasion is widespread. As shown in [AM90] in order to model this effects a theory of social interaction is required. The design of an optimal plan in this context require a treatment of the social effects. Most of the model on optimal taxation consider an individual representative taxpayer, and the optimal taxation problem is posed as that of choosing a set of instruments to maximize a government measure of social welfare subject to a budget constraints. This problem is solved in the framework of principal-agent problem as in [RW85] when it is assumed that the government commit to an audit rule or in the framework of game theory, as in [RW86] where the government does not commit to a rule, and the agents make decision sequentially. In this paper we populate an artificial economy with a set of artificial agents. The behavior of 2

the taxpayers is calibrated according to the pattern of behavior observed in two experiments on fiscal evasion ran in the north of Chile (Antofagasta) and north of Italy (Trento). We define a proportional tax for all the agents in the economy, and the purpose of the paper is to identify an optimal control plan, i.e. the frequency of audits for every agent in the economy, so that the government obtain an optimal collection of taxes in the economy. Following [HD88] we assume that the tax authority is a revenue-maximizing agency. The agency can earn high net revenues by doing a good job on, first of collecting revenue and second of deterring underpayments which threaten to undermine self-enforcement. We use the theory of finite automata for capturing the experimental subject’s behavior and integrate it into the simulated model because it provides us with a dynamic explanation of agent behavior. Working with finite automata means that we assume that an individual decide whether to evade or not according to the state where she is. For instance evasion might be very likely when an individual is in the state ”audited in the previous period”, or in the state ”audited and fined three periods ago”. The standard expected utility model of the income reporting decision discussed in [AS72] or more behavioral alternatives as the one presented by [DaN07] using Prospect Theory, do not take into account the changes in individual state as consequence of the interaction with the tax authority. Even when it is possible to introduce modifications into these models for capturing the dynamics, we rather prefer to keep simple the simulated behavior of the artificial agents. We organize the paper presenting on the fist place the structure of our artificial economy. Next we present the experiments carried on in Chile and Italy. Through a short introduction to the theory of finite automata we show how do we capture the behavior of the subjects in the experiments. After this we run the simulated experiments and analyze the results. At the end we discuss the most relevant features of our experiment and finally we conclude.

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The Model

In our economy time evolves in discrete periods t = 1..T . In every period there is an exogenous given production of the economy Z. This production is is distributed among the population N of individuals populating in the economy. In every period of time, each individual i receives an income yi which is the same for all the periods T for the individual i. The production of the economy is distributed according to the rule of distribution Y (N, ·). Given the income distribution Y (N, ·) the income of the agent i is given by yi = Y (N, i). As a particular case, the income distribution Y0 (N, ·) represents the distribution where all the agents receive the same income. In another words, G(Y0 (N, ·) = 0, where the G(·) is the Gini Index of the income distribution. PN Notice that in every period Z = i=1 Y (N, i) At the beginning of each period, each agent receives the income yi and she face a tax rate τ . She 3

has to decide whether to pay or not the total amount τ yi 1 . We assume that every individual i use a choice function ai (·, t) to decide whether to evade or not. We represent the choice function by a finite automaton calibrated by the results obtained in the experiment made for the cases of Chile and Italy. This will be explained with more detail below. The government in every period of time selects from the population a subset of individuals to be audited. We denote by S(N, t) the rule used by the government to select individuals from the population N at period t. Because there exist decreasing economies in the number of audit made by the government, it is not possible to audit all the individuals of the population. The government wants to find out a selection rule S(·, ·) that optimize the collection of taxes across the T periods. Formally we can express the government problem as the maximization of the net revenue defined by,

R(x) =

T X t=1

 N X 1  ai (yi , t) − C(]S(N, t)) + (1 + r)t i=1

 X

fi (ai (yi , t))

(1)

i∈S(N,t)

where ai (·, ·) is the choice function to decide the tax to be paid used by the agent i. C(]S(N, t)) is the cost of auditing the number of agent suggested by the selection rule S(N, t) at time t2 and finally fi (ai (yi , t)) is the fine payed by the agent i if he evaded and he was audited during the period t. We represent the rule S(N, t) as a binary string of size N . If the string S(N, t) has a ’0’ in the position i < N this means that the agent i is not audited in the period t. Similarly, if the string S(N, t) has a ’1’ in the position j < N this means that the agent j is audited in the period t. Notice that the rule S(N, ·) represents the plan of the government for the T periods. The bit, (t − 1)N + i of the plan S(N, ·) indicates whether the agent i is audited or not during the period t. For the sake of exposition, suppose that our economy has N = 3 agents and the government is evaluation its plan of tax collection for a horizont of T = 3 years. Thus, the plan saying that in every period only the first agent has to be controlled will be represented by S(N, ·) = 100100100. The problem for the government is to find out the plan S ∗ (N, ·) so that,

S ∗ (N, ·) = arg max R(x) S(N,·)

(2)

The behavior of the N agents populating the economy is calibrated according to the data obtained from the experiments that will be explained below. Each agent has a memory, so every time that the 1 Notice that in this model individuals decide whether to pay or not. The literature on tax compliance ask the taxpayer to report their income and she decides to pay tax according to the reported income. We make our agents to decide how much to pay because we are following the design of the experiments run in Italy and Chile. 2 Notice that ] count the number of audited agents by the rule S(N,t).

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government control one of them its behavior is modified. This makes the problem of searching for the optimal plan nonlinear. In the following section we present the experiments, and the method that we use for capturing the behavior of the subjects. So we specify how do we simulate the choice function ai (·, t) to decide whether an agent evade or not during a period t.

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The Experiments

The experiments discussed in this work have been carried out using an identical experimental design - software, instructions, payoffs structure - in Italy at the CEEL laboratory of the Trento University and in Chile at Northern Catholic University. The experiments are better described and discussed on [Mit06] and [GM08]. The baseline experiment is based on a repeated choices setting. Two alternative treatments of the baseline experiment are here reported as well. Treatment one introduces a mechanism of redistribution of the tax yield among the participants while treatment two uses the tax yield collected to finance the production of a public good which is consumed outside the strict group of the participants. The hypothesis tested with treatment one and treatment two is that sentiments of other regarding can contrast tax evasion ceteris paribus. The baseline session as well as the treatments sessions have been run using a computer-aided game. Thirty undergraduate students participated in each session, 15 men and 15 women. All the experiments were of the same length (60 rounds). The parameters entered in the experiment are the followings: • income - 0,51 Euro cents from round 1 until round 48, then 0.36 Euro cents; • tax rate - 20 from round 1 until round 10, then 30 from round 11 until round 30, and finally 40 from round 31 until the end; • tax audit probability - 6 from round 1 until round 21, then 10 from round 22 until round 40, and finally 15 from round 41 until the end; • fees - the amount of the tax evaded plus a fee equal to the tax evaded multiplied by 4.5; the tax audit had effect over the current round and the previous three rounds. To approximate a real life situation more closely, the tax audit was extended over a period of four rounds in the base experiment. For this reason, and as the lottery structure changed during the experiment, computation of the expected value from evasion was rather complex. The lottery structure for the experiment was always unfair which means that a risk neutral tax payer should always pay the entire amount of tax due. (see [Mit06] for the details) The experimental subjects have recruited either in Italy and in Chile through announcements on bulletin boards. During the experiment the players were kept separated so that they cannot communicate in any way. The relevant pieces of information have been communicated only via the computer screen, which showed the following items: 5

1. the total net income earned by the player since the beginning of the game, 2. gross income in the active round, 3. the amount of taxes to pay in the active round, 4. the number of the active round. The subjects were divided into two groups, and they underwent a fiscal audit in the same rounds (specifically rounds 13, 31, 34, 48, 54, 58 for the first group, and rounds 3, 24, 27, 40, 46, 50 for the second group). The two sequences have been randomly extracted accordingly with the rules of the design (i.e. using the probability values corresponding to each of the three sub periods of the game). The changes in the audit probability have been communicated to the participants with a warning on the computer screen which informed them that the audit probability would change after three rounds. Each participant every round must write, using the computer keyboard, the amount of money that she has decided to pay and then she must wait for being informed on the result of the audit extraction. The main result is that the behavioral patterns into the two countries are very similar and in particular the so called ”bomb crater effect” (bce) appears as a robust phenomenon in all the samples. The bce as a central element of the decision making process in the tax payment setting will be the topic for a further research.

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Automaton and the Behavior of Subjects in the Experiments

In [GM08] we shown that there are two type of agents representing more than 70% of subjects behavior during the experiments in both countries. On the one hand there are honest subjects which will never evade and on the other hand there are subjects behaving according to the bce pattern. This last effect suggest that subjects are more likely to evade right after of been audited. This means, that the subjects decide whether to evade or not, according to their state; if they were audited in the last period, it is more likely that they will evade during the current period. For distinguishing these patterns we follow a four step procedure: first we propose a set of hypothesis about how subjects behave, second we translate each hypothesis in a Moore automaton with binary stochastic output function, third we estimate the probability of the stochastic output function using the data from the experiments and fourth we compute the success ratio of each hypothesis or automaton at anticipating the decision of evading of every agent. In order to make this paper self-contained, in this section we will briefly describe the procedure follow in [GM08].

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4.1

Hypothesis

Every individual j in an experiment has a history hj of evasions. Extensively the history might be represented by a string of sixties binary digits hj = {1, 0, 1..., 0}, where the first 1 means that the subject evaded in the first period and the zero in the second position means that the subject did not evade in the second period. We are interested on anticipating when a subject will evade. In order to predict the agent’s answer, we have to work with an hypothesis about what determines her behavior. We proposed ten hypothesis, H0 : The decision of evading depends on whether the subject was audited or not in the previous period. H1 : The decision of whether to evade or not depends on whether the agent was audited or not during the previous period and if when he was audited he was caught evading. H2 : The decision of whether to evade or not depends on the whether the agent was audited or not during the last two periods. H3 : The decision of whether to evade or not depends on whether the agent was audited or not during the last two period and if when he was audited he was caught evading. H4 : The decision of whether to evade or not depends on the whether the agent was audited or not during the last three periods. H5 : The decision of whether to evade or not depends on the whether the agent was audited or not during the last fourth periods. H6 : The decision of whether to evade or not depends on the whether the agent was audited or not during the last three periods and if when he was audited he was caught evading. H7 Subjects decision of evading depends on whether he was audited or not during the previous period and if he was audited during the first five periods of the experiment he will reduce his probability of evading. H8 Subjects decision of evading depends on whether he was audited or not during the previous period and if he was audited during the first five periods of the experiment he will reduce his probability of evading. Moreover the decision of whether to evade or not depends on whether the agent was caught or not the last time that he was audited. H9 Subjects decision of evading at time t depends on whether he was audited or not during the two previous period and if he was audited during the first five periods of the experiment he will reduce his probability of evading. 7

The highest success rate determines the best hypothesis. The success rate is the proportion of right predictions made by an hypothesis out of the total number of predictions. In order of computing the success rate, each one of these hypothesis was translated into a Moore automaton with binary stochastic output function, and we simulated the automaton to produce the predictions for every agent.

4.2

The Moore Automata with Binary Stochastic Output Function

In the theory of computation, a Moore machine is a finite state automaton where its outputs are determined only by its current state (see [Sip97]). The standard state diagram for a Moore machine include a deterministic output signal for each state. We introduce a variation in the output function of the Moore automaton: instead of producing a deterministic output signal, the output function can produce either of two values 0 or 1. Every state s has a probability ps of producing a 1 and a probability (1 − ps ) of producing a 0. Thus, the Moore machine with a binary stochastic output function can be defined as a 7-tuple Γ = {S, S0 , P, Σ, Λ, T, G} consisting of the following: • a finite set of states ( S ), • a start state (also called initial state) S0 which is an element of (S), • a set of probability values ( P ). Every state s ∈ S has a probability ps ∈ P . The initial state has a probability pS0 ∈ P . The numerosity of the set P is equal to the numerosity of S plus 1, i.e. the initial state, • a finite set called the input alphabet ( Σ ), • a finite set called the output alphabet ( Λ = {0, 1}), • a transition function (T : SΣ → S) mapping a state and an input to the next state • an output function (G : S → Λ) mapping each state and its probability in ( P ) to the output alphabet as follows

( G(ps ) =

1, 0,

if ² < ps ; otherwise.

(3)

where ² is a uniform random number between 0 and 1. We translate each one of the ten hypothesis in an automaton. Taking the number of states of an automaton as a measure of complexity, (see [Rub86]), a simple hypothesis requires an automaton with few states, whereas sophisticated hypothesis require more states in order of being represented. The hypothesis H0 is the simplest one. It requires only three state: an initial state, the state of audited in the previous period and the state of not audited in the previous period. 8

S0 0 1 0 1

State: 0 0 1

State: 1

Figure 1: The automaton representing H0 , denoted ΓH0

In figure 1 we represent the states together with the arcs representing the transition function. It turns to be that this automaton produces the highest success ratio. Given the automaton representation of an hypothesis we estimate from the experimental data the probabilities of evading of every state in the automaton.

4.3

Estimating the probabilities of evading in every state

For the sake of exposition we denote all the set of experiments as Ξ = {ec1 , ec2 ..., ec6 , ei1 , ei2 ..., ei6 }. Where the subindexes c and i stand for Chile and Italy respectively. For instance experiments ec1 y ei1 are two experiments with the same experimental design, but they differ because the former was run with chileans subjects whereas the second with italians one. Every experiment has 15 subjects. Therefore in Ξ there are 180 subjects. Every subject i has a history of evasion hi together with the periods where he was audited. Using this information we compute the empirical probability of evading in each state of an automaton. For instance, if we are using the automaton ΓH0 for explaining the behavior of an agent i ⊂ ec1 we will compute the probability of evading in each one of the three states for the agent i. Estimating the empirical probabilities of evading of all the subjects in Ξ using ΓH0 produces 180 automata differing only in the set of probability P of every automaton. We show that using ΓH0 about 70, 3% of the 180 agents can be clustered according to its set of probabilities in two groups. The first group, type 1, has probability of evading close to zero in each one of the three states, P1 = {0.01, 0.15, 0.13} ,whereas the second group, type 2, has a low probability of evading in the initial state and in the state of not audited. In the state of audited the subjects have high probabilities of evading. The probability set for the type 2 agent is P2 = {0.04, 0.36, 0.76}. The 29, 7% of the agents that do not fit with any of the two type of agents do not have a well recognized behavioral pattern. They evade in every state with probability close to 0.5. This makes 9

them random agents.

4.4

Computing the success ratio

The success rate of a given automaton is the percentage of times that the automaton makes the right prediction on the behavior of any subject i. Given an automaton ΓH with its the set or probabilities of evading P in every state s ∈ S it is ˆ j of the decision of evading or not. possible to simulate for a subject j a prediction h We define the success rate predicting the evasions of agent j P60 χj = 1 −

t=1

ˆ j (t) − hj (t)| |h 60

(4)

where | · | is the absolute value function. The performance of an hypothesis is given by the mean of the success rate obtained for the automaton that represent it, at predicting the extensive decision of every subject in Ξ. Using the automaton ΓH0 with the probability vector of type 1, i.e. honest, and type 2, i.e. bce strategy, the success rate is of χ = 0.6748. Thus, taking into account these results, the behavior of the agents ai (·, ·) in 1 will be captured by the automaton ΓH0 with the two probability vectors.

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Searching for optimal policy

The problem represented by R(x) in (2) is nonlinear and stochastic, making difficult to characterize the properties of S ? (N, .). The source of nonlinearity and stochasticity comes from the choice function ai representing the behavior of every agent i. In order find out a solution we will simulate the economy underlying the government problem R(x), and we will use genetic algorithm for searching the optimal plan S ? (N, .).

5.1

Base Line Parameters

We simulate an economy with N = 100 taxpayers, during T = 20 periods3 . Every taxpayer i is assigned a behavioral property ai (·, ·). There are 50 agents of type 1, and 50 agent of type 2. Given a set index I = 1..100, we associate an agent to each index i ∈ I. Taxpayers with odd indexes are type 1, and taxpayers with even index are type 2. In every period the economy distributes the income of Z = 5.000 monetary units among all the agents. We explore three different type of distributions. According to their Gini Index we use, G(Y1 (N )) = 0.796, G(Y2 (N )) = 0.379 and G(Y3 (N )) = 0. The distribution Y3 means that all the 3 This means that every plan has 2000 decisions to be made. We did not find qualitative differences in the analysis of the results as consequences of changing the number of agents or increasing the number of periods.

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agents receives the same income in every period and Y1 is the more unequal distribution of the simulations. We assume that the government applies a proportional tax rate to all the taxpayers of τ = 20%. The aggregate cost of auditing can be expressed as C(x) = cx2 , where x is the number of individuals controls. The cost of auditing a taxpayer increases with the parameter c = 3. Following the parameters of the experiments we set the fine λ = 4.5 of the evasion and the interest rate r = 0.

5.2

Results Analysis

Before of analyzing the simulated results, we explore some characteristics of the model which are relevant for future explanations. First notice that the total income during the 20 periods is Υ = 100.000 monetary units, which taking into account the marginal tax rate of τ = 20% means that the income for the economy if all the agents are honest and without any auditory is R? = 20.000. Second. Assuming that taxpayer probability of being honest is independent of his history and equal to θ, it is possible to approximate the net revenue as a function of the number of auditories q according to, ¡ ¢ R(q) = T Zτ θ − cq 2 + δqτ yλ

(5)

where Z is the income distributed within the population in every period, τ is the marginal tax rate, θ is the probability of being honest for every agent, λ is the fine and δ is a parameter measuring the efficiency of every auditory. If δ = 1 means that each one of the auditories q catch an evasor. Finally, y is the average income within the audited agents. The optimal number of auditories is approximated by the auditory that makes its marginal cost equal to its marginal revenue. In another words, we have to makes in every period q =

δτ yλ 2∗c

auditories.

Using the parameteres of our base line model, this means that the number optimal of auditories in every period can be approximated by q = 0.15δy. The efficiency parameter δ should increases with the quality of the program auditory, and y depends on the sub population which is targeted by the program auditory. In table 1 we present the best performance obtained by the optimal plan for each one of the three type of distributions. Notice that in all the three cases, the optimal plan has collected higher net revenues that the benchmark R? . Second, more unequal distributions produce higher net revenues and third in the three cases the mean number of auditories increases with the inequality of the distribution. This is because there is an increases in the income of audited subjects. 11

Table 1: Maximal revenue obtained for each distribution Distribution Y1 Y2 Y3

Gini 0.796 0.379 0

Net Revenue 42,042 27,810 23,718

Audit for period 9.4 9.05 8.8

Given the size of the space of problem, and the nonlinearity of the fitness function, the optimization landscape is complex. In this space it is difficult to qualify how good the solution is. In order to show how good the performance of the optimal plans are, we compare their results against a set of other plans that a government might use as its tax collection plan. The set of plans that we compare are, • S ∗ (N, ·), This is the optimal plan obtained using the optimization algorithm. • S0 (N, ·), The government does not control to any agent, therefore the government save all the auditory costs • S1 (N, ·), The government controls randomly in every period to a group agents • S2 (N, ·), The government controls the same number of agents in every period than the optimal plan. However, in this case the agents are selected randomly. • S3 (N, ·), The government controls using the cut-off rule proposed in [RW85]. We search for an optimal income threshold ν in every distribution. Any declared income, below ν is audited with probability

1 1+λ ,

where is the fine λ = 4.5

• S4 (N, ·), The government controls only the agent who happens to be more likely to evade. In this plan, the governments knows ex-ante who are agents of type 1 and type 2. The number of auditories is selected according to the one that produces the highest net revenue. For each income distribution, we simulate an economy using each one of the plans. In every case, we run 300 Monte Carlo simulations. In table 2 we report the mean of the net revenue obtained by the government, for every pair of plan and income distribution. For every income distribution the net revenue obtained by the optimal plan S ∗ (N, ·) outperform the net revenue obtained by the first four benchmark. The exception is the benchmark plan S4 . In this last case, the optimal plan only beat the benchmark when there is no inequality of the income distribution. For the plan S4 , beside the net revenue obtained by the plan, in parenthesis we Before explaining why is this effect, we have to show what are the main properties of the optimal plan. 12

Table 2: Revenue obtained from Optimal Plan, against Benchmarks Plan ∗ S (N, ·) S0 (N, ·) S1 (N, ·) S2 (N, ·) S3 (N, ·) S4 (N, ·)

Income Inequality Equal Medium Unequal 22,825 26,171 38,233 16,398 16,353 15,945 17,781 17,799 18,538 15,153 16,663 17,066 16,401 5,823 4,360 18,762 (6) 27,867 (9) 46,783 (9)

Table 3: Properties of the Optimal Plans Distribution Inequality Equal, Gini = 0 Medium, Gini = 0,396 Unequal, Gini = 0,769

5.3

Mean Audits for period 8.8 9.05 9.4

% of audits to 10th decil 12.5 % 25.5 % 35.91 %

% of audits to bce agents 98.37 % 89.89 % 69.61 %

% of bce strategy 4.1 % 19.42 % 24.7 %

Optimal Plan Attributes

In table 3 we analyze the properties of the optimal plan. There are three rows, one for every distribution. The second column, shows the mean number of auditories made in every period. As we noticed before, there is no differences between the number of auditories made for every distribution. The third column indicate the percentage of auditories made to the richest individuals out of all the auditories. Notice that higher the inequality, the auditories are more concentrated in the 10 % richest agent in the economy. The fourth column indicates the percentage of auditories made to the individuals that behave according to the bomb crater effect pattern. Notice that lower inequality, makes the optimal plan to be more focus on identifiying who are the agent of type 2, behaving strategically. In order of increasing the probability of applying a fine to an agent behaving according to the bce pattern, two audits in a row has to be applied to the same agent. The fifth columns indicates how many times this strategy was used. Notice that higher inequality increases the uses of this strategy for applying fines to the bce agents. It is interesting to summing up the properties of the optimal plan. As the inequality increases the optimal plan concentrates in the richest decil and uses more frequently strategy against bce. This concentration on the 10th decil is because the fine payed by this decil are high enough. Together with the volatility induced by the stochasticity of the automata, makes that auditing to poorest agents does not provide good information for maximizing the net revenue. In order to explain why the plan S4 beats the optimal plan when the distribution is unequal we will simplify our model. This simplification set aside effects which are not important for the explanation. 13

First, assume that the tax authority has to run a constant number of auditories M, so that we do not worry about the cost of controlling the individuals. Second, assume that the population of individual is composted by honest and bce agents. Moreover every individual has only two states which discriminates the probability of evading. In the state ”not audited” honest evade with probability zero and bce agents evade with probability pl . In the state of ”audited” honest evade with probability zero and bce agents evade with probability ph . Third, suppose that the revenue of the tax authority is defined by R = Ω + Λ, where Ω is the tax payed by the agents, and Λ is the expected fine payed by taxpayers who evade and they might be caught by the tax agency. We assume that Ω is constant, so that the difference on the audit program will comes only from the expected fine Λ payed by evaders. Λ is computed as the expected value because the tax agency audit a set of agent M which evade with some probability. If the optimal plan audit only the rich individuals behaving as bce agents the expected fine can be PM approximated by Λ1 = ph 1 f . Where ph is the probability that an agent type bce evade and f is the fine that the rich agent pays. Notice that in order of having ph = 0.76 in every period of time, the tax authority has to keep auditing the same agent in every period. This is the strategy against bce agents. On the other hand, in an unequal distributions the potential fine payed by an agent fall very fast if he does not belong to the highest deciles. Thus, strategy against bce together with fines falling fast, makes the M auditories to be concentrated in the highest deciles. The combinations of these effects explains the concentration of 35.91% of auditories in the 10th decil and the intensive use of strategy against bce shown for unequal distribution in table 3. When the distribution of income becomes more equal the optimal plan change. First, notice that every time that an agent suffers an audit the tax authority controls him during his last four periods and second, a fine can not be payed twice. Taking this in account, if the auditories are carefully spread across all the income distribution so that every agent is audited ones every four periods, without PM applying the strategy against bce, the expected fine can be expressed as Λ2 = pl 1 (mt + mt−1 + mt−2 + mt−3 ). Where ms for s = t..t − 3 is the fine that an agent has to pay for its potential evasion during his last four periods tax compliance. In our case, if the strategy of spreaded auditories are applied only to bce agents, pl = 0.36. So assuming the same number of auditories M in every case, the tax authority will audit only bce agents using strategy against them when Λ1 > Λ2 . However, as soon as the income distribution becomes more equal, eventually Λ1 < Λ2 , which makes the tax authority to change its audit strategy; it will keeps auditing the bce agents, but without using the strategy against them. Indeed, the agency will spread its auditories as suggested in Λ2 . This explains why in table 3 the percentage of strategy against bce agents is very low when there is an equal distribution and higher when the income distribution becomes unequal. Moreover notice 14

that when there is an equal distribution the tax authority target better the bce-agents. Indeed 98.37% of bce-agents are audited. On the other hand, when the income distribution is unequal, honest agents in high deciles are better target than bce-agents in the first decil. Thus when the distribution is more unequal, all the bce-agents are not targeted anymore. Only the ones with high income.

6

Discusion

The calibrated artificial agents used in the computational experiments does not guarantee neither, the realism of the model that we elaborated nor the realism of the agents that populates our artificial economy. Although the automaton capture the behavior of a sample of subjects during laboratory experiments, we can not say that people behave like them. The automata of this paper represent the intuition processing in the two system of reasoning described by Khaneman [Kha03]. In the dual process theory, the intuition (or system 1), is determined to be fast and automatic, usually with emotional participation included in the reasoning process. Khaneman says that this kind of processing is based on formed habits and they are difficult to change or manipulate. Reasoning, or system 2 is slower, and it is relatively easy to change, being subject to conscious judgments and attitudes. Our agents respond according to their state. In this sense we can say that they answer are mainly emotional. Because we tried to calibrate our artificial agents only with the observed and analyzed experimental behavior, we do not integrate a system 2 type of processing to our agents. An important consequence of this is that the artificial agents do not have learning capabilities. In our artificial economy, the tax authority might keep controlling a bce agent, and he will keep evading with 76% of probability. The standard taxpayer in the tax compliance literature is an agent with given risk preferences deciding the composition of a portfolio with a risky asset. The taxpayer is an optimizer who uses the system 2 type of reasoning. This is the taxpayers populating the models by [AS72], [RW85] and [RW86]. Although dual process models are diffused in social psychology, the taxpayer attitude is also result of moral rules and social interactions (see [San04]). Thus, more sophisticated models of taxpayer behavior should consider a tradeoff between parsimony and the interaction of system 1 and system 2 type of reasoning in a social dynamics framework. Tax authority in our model has powerful processing capacities. Even when we do not claim that the optimal plan is the best possible plan, we assume that government is capable of solving a problem exploring a big solution space. Moreover, in order to find out the, likely local, optimal solution the tax authority can experiment with the same economy many times. Capacity in the tax agency literature are powerful because they are able to solve Nash equilibria, with the reasoning process ad infinitum. For instance in the model of game theory used in [RW86] the tax agency does not know the true income 15

of any taxpayer prior to audit her and it computes the equilibrium of best response according to that. In all the cases, the optimal plan designed is a consequence of the behavioral or rational assumption made on taxpayers and tax authority. The accumulation of single contributions increases the understanding of the relationship between the tax authority and taxpayers. An important result of our experiment is that expected fine revenue, given by the probability of catching an evader and the fine that he has to pay, leads the design of the optimal plan. The fine is a function of the income and the evasion of the taxpayer; higher evasion means higher fine. In the experiment with subjects made in Chile an Italy we observed that the participants makes not only extensive decision but intensive decision of evading. In another words, it might be the case that a rich individual decides to evade an small amount, reducing the incentives by the government to audit him. However, we only deal with the extensive decision made by the taxpayers, i.e. evade or not evade. Despite the assumption on the taxpayer behavior and the tax authority the results that we present in this paper are important for understanding the characteristics of an optimal plan, and in the next section we move to the emphasizing the conclusions.

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Conclusions

In this paper we integrate techniques coming from behavioral economic and computational economic into an aggregate economic model to explore policy on optimal taxation. We assume that the taxation and the punishment scheme is given and tax authority, given imperfect ability to monitor taxpayers, has to design an audit schemes that optimize government revenues. The agents populating our economy behave according to two behavioral patterns that explain more than 70% of the behavior of subjects which has participated in experiments on taxation made in Chile and Italy. In those experiments we identify honest taxpayers and subjects that behave strategically according to the bomb-crater-effect pattern described in [Mit06] and [GM08]. These agents increase the probability of making an evasion right after being audited. In the framework of our model, we found that the design of an optimal audit scheme depends on three components; the income distribution, the identification of patterns of behavior and the number of periods in the history of an individual controlled when he suffers an auditory. When the distribution of income is unequal, it is worth for the tax authority to develop strategies for catching the richest taxpayers which are more likely to evade. Thus, in our case when there is income inequality the tax authority develops strategy targeting the agents behaving according to the bomb-crater-effect pattern in the last deciles. On the other hand, when the income is uniformly distributed the tax authority changes its strategy spreading its audits through the population so that every auditory takes the history of an individual increasing the potential fine that an evader would has to pay if he is audited. 16

The basic principle underlying this is that tax authority attempts to increases the expected revenue computed by the interaction between the probability of catching an evader and the fine that he would has to pay. According to the three components described before the tax authority will develop strategies for increasing the probability or increasing the amount of the potential fine. Finally, future integration of the type of strategy that we propose in this model requires attending the learning capacities of the agents. This eventually means that some experiments with subjects would have to be designed for learning how people learn in specific contexts, so that this can be also calibrated into the computational model.

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References [AM90] Jens Andvig and Karl Moene. How corruption may corrupt. Journal of Economic Behavior and Organization, 13:63–76, 1990. [AS72]

Michael Allingham and Agnar Sandmo. Income tax evasion: a theoretical analysis. Journal of Public Economics, 1:323–338, 1972.

[DaN07] Sanjit Dhami and Ali al Nowaihi. Why do people pay taxes? prospect theory versus expected utility theory. Journal of Economic Behavior and Organization, 64(1):171–192, September 2007. [EER01] Alexander Galetovic Eduardo Engel and Claudio Raddatz. A note on enforcement spending and vat revenues. The Review of Economic and Statistics, 83(2):348–387, 2001. [GM08] Nicols Garrido and Luigi Mittone. A description of experimental tax evasion behavior using finite automata: the case of chile and italy. Working Paper, IDEAR, 2008. [HD88]

Erekson Homer and Sullivan Denis. A cross-section analysis of irs auditing. National Tax Journal, 41(2):175–189, 1988.

[JAF98] Brian Erard James Andreno and Jonathan Feinstein. Tax compliance. Journal of Economic Literature, 36(2):818–860, June 1998. [Kha03] Daniel Khaneman. A perspective on judgement and choice. American Psychologist, 58:697– 720, 2003. [Mit06]

Luigi Mittone. Dynamic behaviour in tax evasion: An experimental approach. The Journal of Socio-Economics, 35:813–835, 2006.

[Rub86] Ariel Rubinstein. Finite automata play the repeated prioner’s dilemma. Journal of Economic Theory, 39:83–96, 1986. [RW85] Jennifer Reinganum and Louis Wilde. Income tax compliance in a principal-agent framework. Journal of Public Economics, 26:1–18, 1985. [RW86] Jennifer Reinganum and Louis Wilde. Equilibrum verification and reporting policies in a model of tax compliance. Internation Economic Review, 27:739–760, 1986. [San04] Agnar Sandmo. A theory of tax evasion: a retrospective view. Discussion Paper 31/04, December 2004. [Sch00]

F. Schneider. The increase of the size of the shadow economy of 18 oecd countries: some preliminary explanations. IFO Working Paper, (306), 2000. 18

[Sip97]

Michael Sipser. Introduction to the Theory of Computation. PWS Publishing Company, 1997.

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