AA MODEL FOR TAX EVASION WITH SOME REALISTIC PROPERTIES
RICHARD VALE
Abstract.
We present a discrete-time dynamic model of income tax evasion. The model is solved exactly in thecase of a single taxpayer and shown to have some realistic properties, including avoiding the Yitzhaki paradox. Theextension to an agent-based model with a network of taxpayers is also investigated. Introduction as this is equivalent to confiscation of all income. The problem of models predicting thathigher taxes should result in higher compliance is sometimes called the Yitzhaki Problem or Yitzhaki Paradox.1.5. The model presented in this paper predicts that higher taxes should result in lower compliance, providedthat the tax rate exceeds a certain critical value, and a 100% tax will result in total non-compliance. Again, theemotions of the taxpayers do not need to be taken into account. The relationship between tax rate andnon-compliance is derived from economic considerations and taxpayers seek only to maximise their profits.1.6. In the simplest terms, the reason why the new model predicts widespread compliance is because the taxauthority is assumed to collect all unpaid taxes rather than just taxes from the most recent period. Some authorshave previously considered models like this, although Seibold and Pickhardt (who call this feature back-auditing [5]) comment that it has been ‘largely neglected in the literature’. The reason why the new model predicts highernon-compliance at higher tax rates is that people will not have enough money to live on if the tax rate is too high.1.7. References.
The most famous model of tax evasion is the 1972 Allingham and Sandmo model [1]. TheYitzhaki Paradox originates in Yitzhaki’s 1974 paper [6]. The question of the effect of tax rates on tax compliancein the real world has been much studied. and a survey of models, empirical studies and experimental results is [2].The model introduced in the present paper is related to a model of Klepper, Nagin and Spurr [4] in whichtaxpayers make a decision whether to evade based on their savings rate. However, the model of [4] only has twoperiods ( t = 0 , Date : August 12, 2015. We consider only simple models in which a taxpayer’s entire income is taxed at a fixed rate. It is not clear that taxpayers with a marginal tax rate of 100% would be expected to evade. a r X i v : . [ q -f i n . E C ] A ug RICHARD VALE
Ising model of statistical mechanics. The idea of using the Ising model to study tax evasion is due to Zaklan,Westerhoff and Stauffer [7]. 2.
A Model of a Single Taxpayer τ , the savingsrate k , the probability of audit p and the penalty rate λ . We assume 0 < τ, k, p < λ > Verbal Description.
The taxpayer has an income of $1 and can choose whether to pay the tax of $ τ . If thetaxpayer pays the tax, the taxpayer is said to comply and receives $(1 − τ ). Otherwise, the taxpayer evades andreceives $1. The taxpayer saves a proportion k of this income and spends the rest. The taxpayer may then beaudited. If audited, the taxpayer must pay back all the tax evaded so far, together with a penalty which isproportional to the amount of tax evaded. Because the taxpayer spends some of his or her income, it may not bepossible to recover all of the evaded tax. In this case, the taxpayer pays as much as possible, and then continueswith $0 at the next time step.2.3. The decision whether to evade is based on the profit made from evasion so far. If the profit made fromevasion between time 0 and time t is positive then the taxpayer evades at time t , otherwise the taxpayer compliesat time t .2.4. The model is a very simplified version of how a real-world tax authority operates. The tax authority, uponuncovering evidence of evasion, attempts to recover all the evaded tax, not just the tax from the most recentperiod. But this is not always possible, because the taxpayer might have spent some of the evaded tax. Real-worldtax agencies might seize the taxpayer’s assets, but might still fall short if the evaded tax had been spent onperishable items such as meals, holidays or gambling. Interest is ignored in the model. In real life, the taxpayermight earn interest on savings, but this is balanced by the interest that must be paid on evaded tax. Of course,this is one of many assumptions that simplify the model. For example, we ignore utility and assume that tax ispaid on the whole income at the same fixed rate.2.5. Detailed Description.
We now give a more detailed mathematical description of the model. Time moves indiscrete steps t = 0 , , , . . . . At time t , the taxpayer may or may not evade (the condition for evasion will be givenbelow.) If the taxpayer evades, the income received is 1. If the taxpayer does not evade, the income received is1 − τ . A proportion k of the income is retained. Let f ( t ) denote the taxpayer’s fortune at time t . Then f ( t + 1) = (cid:40) f ( t ) + k if taxpayer evades at time tf ( t ) + k (1 − τ ) otherwise.Denote by pf( t ) the profit made from evasion up to and including time t . Thenpf( t + 1) = (cid:40) pf( t ) + τ if taxpayer evades at time t pf( t ) otherwise.Denote by n( t ) the number of times the taxpayer has evaded since the last audit. Thenn( t + 1) = (cid:40) n( t ) + 1 if taxpayer evades at time t n( t ) otherwise.Suppose the taxpayer is audited at time t with probability p . Audits are independent of one another. If thetaxpayer is audited then the taxpayer must pay λτ n( t + 1). If f ( t + 1) < λτ n( t + 1) then the taxpayer must paythe entire fortune f ( t + 1) instead. Thus, if audited, f , pf and n are updated according to the rules f ( t + 1) ← f ( t + 1) − min( f ( t + 1) , λτ n( t + 1))pf( t + 1) ← pf( t + 1) − min( f ( t + 1) , λτ n( t + 1))n( t + 1) ← t ) >
0. In otherwords, if the profit made from evasion so far is positive, the taxpayer chooses to evade because evasion looks like agood bet. We also assume f (0) = n(0) = 0 and pf(0) > MODEL FOR TAX EVASION WITH SOME REALISTIC PROPERTIES 3
Difference Equations.
The model can be described even more formally via the following system ofequations A t ∼ Bernoulli( p ) f ( t + 1) = f ( t ) + k (1 − τ ) + kτ δ pf( t ) > − min (cid:0) f ( t ) + k (1 − τ ) + kτ δ pf( t ) > , λτ (n( t ) + δ pf( t ) > ) (cid:1) A t pf( t + 1) = pf( t ) + τ − min (cid:0) f ( t ) + k (1 − τ ) + kτ δ pf( t ) > , λτ (n( t ) + δ pf( t ) > ) (cid:1) A t n( t + 1) = (n( t ) + δ pf( t ) > )(1 − A t )where δ pf( t ) > is 1 if pf( t ) > t ) ≤
0, and f (0) = n(0) = 0 and pf(0) > Solution of the One-Taxpayer Model t ) ≤ t , then thetaxpayer never evades again, so we restrict to the case pf( t ) >
0. In this case, pf( t ) cannot decrease until thetaxpayer is audited, so between audits the taxpayer always evades.3.2. Suppose the taxpayer is audited at time T and at time T > T and there are no audits in between. Weconsider the total profit made from evasion pf( T ) − pf( T )between T and T .3.3. In the first case, suppose λτ ≤ k . Then f ( t ) ≥ λτ n( t ) for all t . This follows by induction on t . It is true for t = 0. If f ( t ) ≥ λτ n( t ) and A t = 0 then f ( t + 1) = f ( t ) + k ≥ λτ n( t ) + λτ = λτ n( t + 1). If f ( t ) ≥ λτ n( t ) and A t = 1 then n( t + 1) = 0 and so f ( t + 1) ≥ λτ n( t + 1) in this case too. Therefore, in case λτ ≤ k , we have f ( t ) ≥ λτ n( t ) for all t andpf( T ) − pf( T ) = τ ( T − T ) − min( f ( T ) + k, λτ (n( T ) + 1))= τ ( T − T ) − λτ (n( T ) + 1)but n( T ) + 1 = T − T because the taxpayer evades at times T + 1 , T + 2 , . . . , T and sopf( T ) − pf( T ) = τ (1 − λ )( T − T )3.4. In the second case, suppose λτ > k . Then f ( t ) ≤ λτ n( t ) for all t . This follows by induction on t . It is truefor t = 0. If f ( t ) ≤ λτ n( t ) and A t = 0 then f ( t + 1) = f ( t ) + k ≤ λτ n( t ) + λτ = λτ n( t + 1). If f ( t ) ≤ λτ n( t ) and A t = 1 then f ( t + 1) = 0 and so 0 = f ( t + 1) ≤ λτ n( t + 1) in this case too. Therefore, in case λτ > k we have f ( t ) ≤ λτ n( t ) for all t andpf( T ) − pf( T ) = τ ( T − T ) − min( f ( T ) + k, λτ (n( T ) + 1))= τ ( T − T ) − ( f ( T ) + k )but f ( T ) + k = k ( T − T ) because f ( T ) = 0 in this case (the taxpayer is audited at time T by assumption andloses their entire fortune because f ( T −
1) + k ≤ λτ ( n ( T −
1) + 1)) and the taxpayer evades at times T + 1 , T + 2 , . . . , T , and so pf( T ) − pf( T ) = ( τ − k )( T − T )3.5. Although T and T are random variables, we see that in all cases the quantitypf( T ) − pf( T ) T − T = τ − min( k, λτ ) = (cid:40) τ − k τ > k/λτ (1 − λ ) τ ≤ k/λ is constant. This quantity represents the average change in pf between audits. Equivalently, it represents theaverage profit from evasion. We will use it extensively in this paper and denote it using the derivative-like symbol ∆ pf∆ T . RICHARD VALE
Figure 1.
Plots of pf( t ) versus t from simulations from the one-taxpayer model with k = 0 . p = 0 .
01, pf(0) = 5, λ = 1 . τ = 0 . , . , .
42. The dotted lines are y ( t ) = pf(0) + τ (1 − λ ) t (left) and y ( t ) = pf(0) + ( τ − k ) t (middle and right).3.6. We see that if τ > k then pf is an increasing function from one audit to the next, and therefore the taxpayeralways evades. If τ < k then pf will eventually become negative and the taxpayer becomes compliant. In this case,we can calculate the expected amount of time which elapses until the taxpayer becomes compliant. Let T comp bethe time at which pf first becomes negative. Each audit reduces pf by ∆ pf∆ T × ∆ T where ∆ T is the time elapsedsince the last audit. So pf will become negative as soon as there is an audit after time − pf(0) / ∆ pf∆ T has elapsed.The expected time until the next audit is a geometric random variable with a mean of 1 /p and therefore E [ T comp ] = − pf(0) (cid:18) ∆ pf∆ T (cid:19) − + 1 p = (cid:40) pf(0)( k − τ ) + p τ ≥ k/λ pf(0) τ ( λ − + p τ < k/λ (3.1)In particular, the expected time until the taxpayer becomes compliant is a linear function of − (cid:16) ∆ pf∆ T (cid:17) − .3.7. Examples.
The calculations of Section 3 can easily be tested by implementing the equations of Section 2.6on a computer. Plots of pf( t ) against t from three example runs are shown in Figure 1. Here k = 0 . λ = 1 . τ = 0 .
02 and τ = 0 .
39. These taxpayers eventually become compliant but atdifferent rates, as shown by the dotted lines with slope ∆ pf∆ T . The third taxpayer has τ = 0 .
42. This tax rate is toohigh for the taxpayer ever to become compliant.3.8.
Measuring Evasion.
When τ < k , the taxpayer always becomes compliant eventually. However, Equation3.1 shows that the expected time taken to reach compliance (which is the same as the number of times thetaxpayer is expected to evade until becoming compliant) is a linear function of − (∆ pf / ∆ T ) − with positivecoefficients. It is therefore reasonable to take − (∆ pf / ∆ T ) − as a measure of non-compliance. It is likewisereasonable to take ∆ pf∆ T as a measure of non-compliance because it measures how profitable evasion is for thetaxpayer on average.3.9. The one-taxpayer model is interesting because it exhibits phenomena which can be seen in the real worldbut not in many models of tax evasion. A graph of ∆ pf∆ T against τ for fixed values of k and λ is shown in Figure 2.Also sketched in Figure 2 is a graph of − (∆ pf / ∆ T ) − against τ . It will be useful to compare the U-shape of thisgraph with experimental results from Figures 3 and 4.3.10. Discussion. If τ > k/λ , increasing the tax rate causes compliance to decrease. This fact does not requireany arguments about the morality of taxpayers. It is a consequence of the cost to the taxpayer of non-compliancebeing bounded below, which means that evasion becomes more profitable to the taxpayer when the tax rate ishigher.3.11. When τ < k , the taxpayer becomes compliant. When τ < k/λ , increasing the tax rate causesnon-compliance to decrease. Non-compliance is minimised when τ = k/λ . This optimal tax rate increases with therate of saving and decreases when the penalties for evasion become harsher. MODEL FOR TAX EVASION WITH SOME REALISTIC PROPERTIES 5
Figure 2.
Left panel: graph of ∆ pf∆ T versus τ . Evasion becomes profitable when τ > k and isminimised when τ = k/λ . Right panel: graph of − (∆ pf / ∆ T ) − . The shape of this graph shouldbe compared with the experimental results in Figure 3 and Figure 4.3.12. Progressive taxation.
The result that a tax rate of τ = k/λ minimises non-compliance might help tojustify progressive taxation (which is a form of taxation in which income over a certain threshold is taxed at ahigher rate.) All other things being equal, it seems reasonable that a higher proportion of income will be saved ifit is above what is needed for survival. This corresponds to a higher k and therefore a higher k/λ . This argumentis similar to a common argument in favour of progressive taxation based on marginal utility but has a differentmotivation; the model suggests that taxes should be progressive if the tax authority wishes to minimisenon-compliance. 4. A Network of Taxpayers N taxpayers are connected in an undirected network. Let A xy = 1 if x and y are connectedand A xy = 0 otherwise. Assume that A xx = 0 for all x (there are no self-loops.) Suppose the tax rate τ , savingsrate k and probability of audit p are fixed quantities in (0 , λ > x , define the neighbourhood N ( x ) = { x } ∪ { y : A xy = 1 } to be the set of taxpayers consisting of x and the neighbours of x .4.4. Time proceeds in discrete steps t = 0 , , , . . . . Let f ( x, t ) be the fortune of taxpayer x at time t , pf( x, t ) bethe profit made by x from evasion up to and including time t , and n( x, t ) be the number of times x has evadedsince the last audit. Assume that taxpayers are audited independently of one another and that the probabilitythat a given taxpayer x is audited at a given time t is p . Assume f ( x,
0) = n( x,
0) = 0 for all x and that f , pf andn evolve for each taxpayer exactly as in the one-taxpayer model of Section 2, except for the criterion for choosingto evade. RICHARD VALE node 1 (centre) 2 3 4 5 6 7 8 9 10mean 420 410 394 434 435 408 415 412 408 425sd 153 239 240 248 242 237 236 254 231 237
Table 5.1.
Time until compliance for ten taxpayers in a star-shaped network for 50 simulationsof the model. See Section 5.4 for details.4.5. In the new model, the network structure enters through the rule which a taxpayer uses to decide whether toevade. A taxpayer is assumed to be aware of their own history and the histories of all of their neighbours and thedecision to evade is based on their best guess at the expected profit from evasion given this information. Letn total ( x, t ) be the total number of times that x evades between time 0 and time t . A taxpayer x chooses to evade if (cid:88) y ∈ N ( x ) pf( y, t ) (cid:46) (cid:88) y ∈ N ( x ) n total ( y, t ) > . (4.1)which is the same as (cid:88) y ∈ N ( x ) pf( y, t ) > . (4.2)5. Analysis of the network-of-taxpayers model x , pf( x, t ) evolves in much the same way as in Section 3. In particular, we need only considerthe case ∆ pf∆ T <
0. But it is no longer guaranteed that pf( x, t ) will become negative as t → ∞ ; the taxpayer maybecome compliant before this happens. Nevertheless, if taxpayer x is non-compliant, then pf( x, t ) will decreasewhen x is audited. Therefore, the taxpayers must eventually become compliant.5.3. We make only a crude estimate of the time until compliance. Let X be the set of taxpayers and A ⊂ X .(The case A = N ( x ) is of particular interest.) Definepf( A, t ) = (cid:88) x ∈ A pf( x, t ) . Ignoring the randomness in the audits and assuming that every taxpayer evades at every time step, we havepf( x, t ) ≈ pf( x,
0) + (cid:18) ∆ pf∆ T (cid:19) t and so pf( A, t ) ≈ (cid:88) x ∈ A pf( x,
0) + | A | (cid:18) ∆ pf∆ T (cid:19) t which becomes negative at time T comp = | A | (cid:80) x ∈ A pf( x, − (cid:16) ∆ pf∆ T (cid:17) . (5.1)Therefore, if all the pf( x,
0) are assumed to be equal, every subset of the taxpayers will become compliant at thesame time. In particular, all the taxpayers are expected to become compliant at the same time. The time taken tobecome compliant does not depend on the network topology.5.4.
Example.
The model was simulated for a star-shaped network with ten nodes labelled with the integers 1 to10. Node 1 was connected to all other nodes and there were no other connections. Taking pf( x,
0) = 1 for all x and k = 0 . , τ = 0 . , λ = 1 . , p = 0 .
01 the results of 50 simulations are shown in Table 5.1. For each node x , thetime in the table is the number of the last iteration on which x evaded. MODEL FOR TAX EVASION WITH SOME REALISTIC PROPERTIES 7
Figure 3.
Average number of evaders versus tax rate τ for a star-shaped network with 10 taxpayersrun for 1000 iterations. The line is at τ = k/λ , the tax rate which minimised non-compliance inthe one-taxpayer model. Compare Figure 2.5.5. There is no evidence of a difference between nodes in the times until the nodes become compliant. However,Equation 5.1 gives an estimate of 1 / ( k − τ ) = 10 for the time for a node to become compliant, which is clearlywrong. This is because Equation 5.1 ignores the audit probability p , which influences the time until compliance.Giving an accurate estimate of the time until compliance seems to be a more difficult problem than in Section 3,especially when the pf( x,
0) are allowed to take arbitrary values.5.6. Figure 3 shows the average number of evaders plotted against the tax rate for the star-shaped network with10 vertices and k = 0 . , p = 0 . , λ = 1 .
5. The vertical line is at the value τ = k/λ . Comparison with Figure 2shows that there is a wide range of parameters in the network case for which non-compliance is approximatelyminimised. All values of τ between about 0 . . Example.
Zaklan, Westerhoff and Stauffer [7] considered an Ising-like model in which taxpayers areconnected in a square lattice. Korobow, Johnson and Axtell [3] considered a different model with a similarnetwork topology but with diagonal connections. We simulated the model for 100 taxpayers arranged in a squarelattice. The lattice is wrapped around to form a torus, so for example the taxpayer at (1 ,
1) is adjacent to (1 , , ,
10) and (10 , τ = k/λ will minimise non-compliance.5.8. A plot of the average number of evaders against the tax rate τ is shown in Figure 4. It can be seen that theshape is very similar to Figure 3, which is further evidence to suggest that the topology of the network has noinfluence on compliance behaviour in this model.5.9. Varying the probability of evasion.
We can make the taxpayer model even more like the Ising model byspecifying a probability of evasion. The probability of taxpayer x evading at time t could be chosen to be anincreasing function of (cid:80) y ∈ N ( x ) pf( x, t ). By analogy with the Ising model, this could for example be taken to beexp β (cid:88) y ∈ N ( x ) pf( x, t ) (cid:44) (cid:88) y ∈ N ( x ) n total ( x, t ) RICHARD VALE
Figure 4.
Average number of evaders versus tax rate τ for a toroidal grid with 100 taxpayers runfor 1000 iterations. The line is at τ = k/λ , the tax rate which minimised non-compliance in theone-taxpayer model. Compare Figure 2 and Figure 3.where β > β → ∞ , this reduces to(4.2). For smaller values of β , there is more randomness in the decisions of the taxpayers and as t → ∞ , the modelapproaches Bernoulli trials with a probability exp( β ∆ pf∆ T ) of evasion. Otherwise, this choice of evasion probabilitydoes not affect the qualitative behaviour of the model when the parameters are held constant, so we choose not todiscuss it further.5.10. Effect of heterogeneous k . The model is still unrealistic in many ways. For example, real taxpayers donot inhabit a torus, and the savings rate k undoubtedly varies over time and from person to person. There aremany directions for further investigation. In this section we investigate what happens when each taxpayer isallowed to have an individual value for k . This follows the ideas of Korobow, Axtell and Johnson, who allowedvarious parameters to vary across taxpayers in their model.5.11. Under the conditions of Section 5.7, suppose that the savings rates of the 100 taxpayers are drawnindependently from a Beta(2 ,
3) distribution. This ensures that the mean savings rate is still about 0 .
4. The taxrate is chosen to be τ = 0 . p = 0 .
1. The model is run for 10000 iterations.Everything else is kept the same as in Section 5.7. We investigate the effect of varying k on compliance. Weexpect that individuals with higher k should be more compliant on average, but this might be tempered bynetwork effects.5.12. It is of interest to see how many times a given taxpayer evades in the 10000 iterations. Figure 5 shows aplot of a typical model run. The 100 taxpayers are plotted in a 10 ×
10 grid and the darker squares indicatetaxpayers who evade more often. The network is a torus, so the bottom row and top row are adjacent and the leftcolumn is adjacent to the right column. Note that the black and white regions in Figure 5 are almost contiguous,which is surprising because the savings rates of individual taxpayers were chosen at random. However,experiments with larger grids, for example 100 × k in the left panel of Figure 6.There seems to be no obvious pattern. In the right panel of Figure 6, the number of evasions is plotted against the MODEL FOR TAX EVASION WITH SOME REALISTIC PROPERTIES 9
Figure 5.
Total number of evasions for each taxpayer in a 10 ×
10 toroidal grid run for 10000iterations. The value of k varies from taxpayer to taxpayer. The other parameters have the values λ = 1 . , τ = 0 . , p = 0 . Figure 6.
Total number of evasions plotted against k (left panel) and average k in a five-taxpayerneighbourhood (right panel) for each taxpayer in a 10 ×
10 toroidal grid. The value of k varies fromtaxpayer to taxpayer. The other parameters have the values λ = 1 . , τ = 0 . , p = 0 .
1. The verticallines are at k = τ . average value of k , k avg = (cid:80) y ∈ N ( x ) k ( y ) for the 100 taxpayers x . It seems that taxpayers with k avg ( x ) < τ evadeevery time, but there is no obvious pattern for taxpayers with k avg ( x ) > τ . Also, the majority of taxpayers eitherbecome highly compliant or highly evasive, with few in between the two extremes.6. Acknowledgements
The author thanks D. Nagin and S. J. Spurr for providing a copy of their paper [4].7.
Conclusion
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