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VAT Compliance Incentives

In this work I clarify VAT evasion incentives through a game theoretical approach. Traditionally, evasion has been linked to the decreasing risk aversion in higher revenues (Allingham and Sandmo (1972), Cowell (1985) (1990)). I claim tax evasion to be a rational choice when compliance is stochastically more expensive than evading, even in absence of controls and sanctions. I create a framework able to measure the incentives for taxpayers to comply. The incentives here are deductions of specific VAT documented expenses from the income tax. The issue is very well known and deduction policies at work in many countries. The aim is to compute the right parameters for each precise class of taxpayers. VAT evasion is a collusive conduct between the two counterparts of the transaction. I therefore first explore the convenience for the two private counterparts to agree on the joint evasion and to form a coalition. Crucial is that compliance incentives break the agreement among the transaction participants' coalition about evading. The game solution leads to boundaries for marginal tax rates or deduction percentages, depending on parameters, able to create incentives to comply The stylized example presented here for VAT policies, already in use in many countries, is an attempt to establish a more general method for tax design, able to make compliance the"dominant strategy", satisfying the"outside option"constraint represented by evasion, even in absence of audit and sanctions. The theoretical results derived here can be easily applied to real data for precise tax design engineering.


Introduction
In this work I clarify VAT compliance incentives through a game theoretic approach. Traditionally, evasion has been linked to the decreasing degree of risk aversion as income increases (Allingham and Sandmo (1972), Cowell (1985), (1990) ). I claim tax evasion to be a rational choice when compliance is stochastically more expensive than evasion, also in absence of controls and sanctions. Infact, the phenomenon is not at all confined to high income agents only.
I create a framework able to measure the incentives for taxpayers to comply. The incentives here are detractions of specific VAT documented expenses from the income tax. The specific types of transaction are the ones with a short production chain for reasons that will be explained.
The issue is very well known and these policies at work in many countries. The aim here is to build a model useful to compute the right parameters (tax rates, sanction rates, audit frequency) for precise classes of taxpayers, defined by revenue, type and dimension of the transactions, cost structure and eventually more precise discriminating variables, using available population data.
To simplify the analysis and get simple first results, only the final segment of the transaction (the sale to the consumer), where the value added is easily defined, will be discussed.
VAT evasion is a collusive conduct between the two counterparts to a transaction. I therefore first explore the convenience for the two private parties to agree on the joint evasion. The setting I use is a game in which players are a buyer and a seller choosing whether to comply or to evade under different scenarios: normal VAT taxation and VAT offering some detractions to the final buyer. Being counterparts to a same transaction, buyer and seller need to agree on their strategy, whether to comply or to evade. If they agree, they actually form a coalition. The third player is the tax authority / government, choosing whether to audit. The probability or necessity to audit, in order to make compliance effective, is what is usually explored. Leaving aside the fact that auditing is costly, here I am interested in studying how its frequency/probability can be minimized as detractions can help compliance by themselves, reducing the necessary frequency, and therefore the cost, of compliance.
The game can be modelled as a Bayesian game in which we collapse normal form games, one for each tax-setting scenario, and all of them under a government that doesn't audit and one in which government audits and sanctions. Players payoffs of course vary across scenarios. "Almost equivalently" it's possible to simplify the games considering just two players: on one side the private coalition constituted by buyer and seller agreeing on their strategy, and on the other side the tax authority playing "audit" with a probability. The interesting part of the "almost" above, is that tax detractions alter agreement within the private coalition: the buyer prefers to comply, while the seller still prefers to evade (in the paper many more details arise). This means that among the coalition there is still a game in mixed strategies to be solved. Crucial is that compliance incentives break the agreement among the transaction participants coalition about evading, altering equilibria results. This is the study most original contribution.
The final Nash solution sets boundaries for marginal tax rates or detraction percentages, dependent on parameters, able to create incentives to compliance. I additionally explore how the solutions change when evasion is extended to the whole production chain. Essentially results prove numerically that the larger the evasion benefit, the larger detractions should be.
The stylized example presented here for VAT policies, already in use in many countries, is just a first attempt at establishing a more general method of tax design, able to make compliance the "dominant strategy", satisfying the constraint of the "outside option" represented by evasion.
In the first paragraph I define the transaction, the agents and their strategies, and the institutional scenarios. In the second paragraph the different games are presented, where we will observe that many events are redundant, and we reach a unique game for each tax policy scenario: absence or presence of detractions. The third paragraph solves for the mixed strategy bayesian game where the tax authorities audit with a probability. The final paragraph shows how the solution can be used to compute some of the variables of the problem as functions of others: (i) evasion probabilities as function of detraction amounts and viceversa, or (ii) which detraction percentage is necessary to make evasion unattractive; or (iii) which VAT rate is attainable given a maximum amount of auditing and / or detraction policy.

Setting
The analysis is at micro level and considers three agents: two private ones, the registered trader or "seller" (S) of the final good and the consumer or final "buyer" (B) on which the whole VAT accrues, and the government tax authority (G) that collects income tax and VAT revenue.
We focus on evasion just on the final transaction of a final good or service between the two private counterparts, the final seller and the final buyer. Each of the two agents has a mentioned and fully declared income, prior to the transaction, respectively   and    that determines their initial marginal tax revenue rate.

Definition 1
The transaction subject to VAT has a value (  −   ), where  0 is the value of the final good or service (output value), net of input costs   (input value).
Since we only want to deal on the opportunity to evade VAT of a single transaction, we assume that each agent is compliant over the rest of his income.
For notation, we add a superscript  to every declared value Assumption 1 "No Income Tax Evasion". Both the final seller and the final buyer are compliant on income that is not entering the considered transaction even while evading VAT on a single transaction, i.e.
Let's   and   be the marginal tax rates on respectively the seller and buyer income and let  be the   rate.
Depending on agents' income, their income marginal tax rates are different and may vary following the transaction. For simplicity the possible change in the tax rate due to the transaction will be not taken into account. (Note: Tax rate threshold are quite few, so that the average tax rate remains almost constant in the transaction). Therefore, for simplicity and no loss of generality, it will be assumed that: Assumption 2 Marginal income tax rates for the seller  and for the buyer  do not change because of the transaction.
All the computations will be done under the following alternative scenarios set by fiscal rules. Fiscal Rules Scenarios 1. No Taxes, meaningless, but used as a benchmark.

Games in absence of Government Audit and Sanctions ()
Let's consider these simple matrix form games, one for each institutional scenario.
In games with only buyer and one seller, being the two counterparts of a unique transaction, the strategies must be the same by both players. Therefore the off diagonal elements of the matrix are not defined. Moreover, the consumer, being only on the last transaction, by definition, he can just evade that single transaction, being counterparts of many ways of seller evasion
Let's define them The best strategy is the unique strategy.

Scenario 2. Taxes with no detractions ( − )
3.2.1 Agents strategies and relative payoffs ( − ) Net Incomes for each agent are the following: The agent claims to be a final buyer and can reduce his income.
The seller pays all costs without savings on imposable income.
The net total revenue (the surplus) due to the sum of the payoff in the economy must always be the sum of the private income before the transaction (  +   ), minus the value of the transaction inputs, excluded by assumption. The other items are just funds reallocation among the agents. This will be valid all along the model, so it will not be repeated.

Dominant Strategy for each agent
The buyer, in absence of government audits by accepting a no-VAT transaction, saves the VAT costs on the transaction.
Result 1 With no detractions, the buyer has no rational convenience to pay VAT The seller finds convenient to comply only when the income revenue saving, due to VAT on costs, is smaller than income tax on additional revenue.
Evasion is convenient if the income tax saving is greater than VAT reimbursement loss. The threshold   is the minimum tax rate over which conveniences guaranteed decreasing as the transaction value increases and the VAT refund decreases.() When VAT refunds are zero (  = 0)  because the transaction has only one segment (Professional labour services), the entire costs may disappear and gain from evasion magnify.
It's easy to see that for   = 0 the limit revenue tax rate threshold is and therefore evasion is always profitable. At the same time, the larger the VAT rate  the less convenient is to lose VAT refund. The threshold  under which evasion is convenient is proportional to the seller's income tax rate   and the transaction value The equation is more easily satisfied the larger the income tax on the transaction and the smaller the costs    In absence of input costs, VAT rate threshold becomes redundant, because the advantage of getting the VAT reimbursement vanishes.

In this section we have learned that
Proposition 3 In absence of government controls, the only incentive to comply is the VAT refund on costs. As soon as costs are negligible, there are no private incentives to comply.
Proof. By (3) and (4)  Let's see whether the consumer can harm this fraud. Which type of evasion is more easily attainable?
Result 2 Seller dominating strategy under no audit is "Evading just the last transaction" ( _ 1  ) Proof. By (3)  (4) and (6) the RHS is smaller in the first case. In other words, the seller has his best advantage in getting VAT traslation, cost refund and savings on income tax on the transaction revenue. In the second case he loses cost detractions from net disposable income and VAT traslation. In the third case he loses just cost detractions on net income, because he doesn't pay any VAT on the chain.

Scenario 3. VAT Detractions from Personal Income Tax (  − )
Assumption 3 Some consumer costs, with VAT payment invoice, can be deducted from the Personal Revenue Tax. The VAT rate can be reduced by a percentage

Dominant Strategy for each agent (  − )
Now the buyer strategies have to be compared with the new institutional setting of detractions.
The Buyer prefers to comply under allowed detractions of VAT expenses from his income tax The graph below shows the area of compliance.

Remark 5 Even without Audit!
Remark 6 For  respecting eq. (9) the agreement between Buyer and Seller breaks out, and there will be a bargaining about which joint strategy to adopt. (We leave this issue aside).

Games in presence of Tax Authorities Audit and Sanctions ()
We now repeat the procedure in presence of government successfully auditing and sanctioning evasion. with probability  = 1 i.e. evasion is certainly sanctioned. For each combination of strategies, we define payoffs. We remind Assumption 1 setting no evasion on income tax. The matrix form is the same Same as before.

Scenario 2. Taxation with no detractions ( − )
4.2.1 Agents strategies and relative payoffs ( − ) Net Incomes are same as before: Remark 7 In case of successful controls on E_LT1 & 2 and sanctions, VAT on inputs,   is paid twice: once by the consumer, included in the total added value, and a second time by the seller that couldn't transfer it, so paying it too.

Evasion on Whole Transaction
The seller cannot justify the costs and gets sanctions on whole  0 

Dominant Strategy for each agent ( − )
Buyer ( − ) Remark 9 Sanctions on VAT turn consumer preferences towards compliance.

Result 3 Again Compliance becomes dominant
The Seller looses the savings due on "reporting costs" and, of course, the additional cost of revenue tax sanctions.
The seller has lost the tax shield due to subtracting     from tax revenue and transferring   forward to the consumer.
Now the buyer strategies have to be compared with the new institutional setting of detractions.
Seller (  − ) Same as before, so Proposition 11 Under allowed detractions of VAT expenses from his income tax, in presence of sanctions, the Buyer prefers to comply It's interesting to compare the relative gain to comply in absence and in presence of sanctions.

Remark 12
Compliance with no sanctions is guaranteed by the magnitude of the detractions percentage  , function of VAT rate and its discounts , so it can be described as the area over the function  =  (1 + )  As soon as sanctions exist the function has an additional negative term ( (1 +   )) that makes its slope negative" determining a null probability of evasion.
In the next paragraph we check what happens when sanctions are applied with a probability  ∈ [0 1]  i.e. when the two games will be joined in a bayesian game.

Scenario 1. No Taxes ( − )
Compliance implies no sanctions and therefore payoffs are same as before.

Agents strategies and relative payoffs ( − )
For every strategy I take the expected value between the sanctions event with probability  and no sanctions event with probability (1 − )  1. Event "Compliance" {     }  Net Incomes for each agent are the following:

Event "Evasion in the Last Transaction only"
 where the seller claims to be a final buyer.
3. Event "Evasion in the Last Transaction only" {   _ 2  }  where the seller pays all costs without savings on imposable income

Dominant Strategy for each agent ( − )
Buyer ( − ) In the Bayesian Game, the expected sanctions decrease the necessary probability of controls.
Remark 13 For low , evasion is still convenient.
The seller finds convenient to comply only when the VAT savings on input (LHS) is larger than net income tax saving diminished by sanctions. Evasion convenience depends on the value of the transaction and has not a certain result.
The preference depends on parameters. For  = 0 we go back to eq. (5) and evasion wins.

Proposition 14
Evasion is convenient for high income tax rates. As  −→ 1 compliance dominates anyway.
Proof. By(14)  • Let's try a simulation for  = 022   = 03   = 100   = 50 (solid line),100 (dashed line)  Remark 15 Evasion is convenient in the area over the line. We see that as soon as we decrease input costs from 50 to 10, the area over the dashed function (evasion convenience) becomes dramatically higher.
The result depends on parameters.

Result 7 Seller compliance is dominant
and it decreases with the importance of the input costs.

Dominant Strategy for each agent (  − )
Seller (  − ) Same as before, so This time it's interesting to compare the relative gain to comply w.r.t. the absence of sanctions in the same scenario.
Proposition 16 Under allowed detractions of VAT expenses from his income tax, in stochastic presence of sanctions, the Buyer prefers to comply Proof. By(15)  Again, let's compare the relative gain to comply in absence and in stochastic presence of sanctions. The area of compliance (area over the curve) increases as the curve shifts lower. The parameter that regulates the descent of the curve is  in (16)  The lower  the probability that of successful audit and sanctions, the higher the curve and therefore the higher the detraction rate necessary to incentivize compliance.
In the figure: black ( = 0), blue( = 05), magenta( = 1)  Proposition 17 The necessary rate of detraction able to induce buyer and seller compliance is, ceteris paribus, a decreasing function of the probability of tax authorities auditing. 6 Are "VAT Detractions" Incentives for Compliance? () 1 Coalitions agree in all scenarios {( )  ( )}, apart the detraction one ( )  What have we learned? Trivially, in absence of audit and sanctions, evasion is more profitable both for buyer and for seller, as everyone knows, so it is possible to consider a unique coalition of private agents converging to a same strategy. Moral and general equilibrium considerations are excluded from this setting. If audit would happen with certainty, evasion is certainly not an option and again the private coalition definitely prefers to comply. Along the study we have explored larger and smaller advantages for each agent, depending on the entity of the transaction, agent income tax rates, VAT rate, but buyer and seller converge on the same strategy.
We can therefore consider the sum of the payoffs on the principal diagonal of the previous games between buyer and seller. Then putting them as column elements of a game in which we have the row player being the private coalition (B+S) and the column player being the government G, auditing with probability  In this way we are shrinking the many game in a single one, and we can compute the expected value of each row, being representing the expected value of each joint strategy, provided agents "agree" on the same strategy: complying or evading. The payoffs in the expected value (EV) column are those listed in the section Bayesian game, but summed.
The probability of auditing affects agent preferences towards compliance or evasion as shown while computing dominant strategy in each setting. The computed payoffs show that the only case of disagreement between buyer (B) and seller (S) happens in presence of detractions: buyers have convenience to comply even in absence of auditing, while sellers still prefer evasion.
The novelty is then to compute for which detraction rate the coalition B+S will agree on compliance. So how much surplus needs to be created by the buyer to convince the seller to accept to comply. To do this, we consider the following game where we just consider the sum of Scenario 3 payoffs for each ( + )strategy and compute the expected value of each joint action (values in the  column) to establish dominance between them.
This game, labelled Bayesian Coalition Game () should be built for each scenario: No Taxes (NT), Taxes with no detractions (T), Taxes with Detractions (TD).
In scenarios NT and T there is already agreement in the coalition, therefore computing the coalition payoffs does not lead to any new result.
Therefore only the TD scenario will be considered below. 1 Bayesian Game for the Coalition 6.1 Scenario 3. VAT Detractions from Personal Income Tax (  − ) 6.1.1 Private agents strategies and relative payoffs (  − ) 1.
We have to compare  (  +   ) with each of the Seller evasion strategies: Remark 18 Detraction rate decreases linearly in the probability of auditing.

Remark 19
Interesting that for  = 0 detractions able to enhance joint compliance must be around 18%, the probability of auditing to enhance compliance without detractions has to be around 70%. (Big data help) : Remark 20 Here, since costs are not deducted, the seller "suffers" from evasion and therefore joint compliance need less detractions.
3. Trade-offs between the need of detraction and/or sanctions, to have both buyer and seller agreeing on compliance, is a linear negative relationship. The slope is constant for all seller evasion strategies because the slope measures dependence on sanctions that same in all evasions. On the contrary the intercept measures the convenience to evade, different for the different strategies.

Proposition 21
The evasion on the all production chain is definitely the most convenient for the seller, therefore, in order to have compliance, consumer benefit through detractions must be more significant or, the production chain shorter.

Conclusions
This work clarifies the tax parameter relative magnitudes that may induce VAT transactions evasion or compliance. Literature explores incentives to compliance just as the outcome of the audit probability and/or the magnitude of sanctions. Here instead I try to make a model where the audit probability can be the endogenous variable needed to imply compliance as a function of existing tax rates, the transaction value with respect to previous costs, sanctions magnitude and probability to be audited.
In the scenarios where the final buyer is not allowed to deduct some VAT declared expenses, the only real threat against the seller evasion is the consequent missing VAT transfer and/or the impossibility to deduct his costs. In short production chains, like human capital services (professional services, artisans), cost detractions represent a small percentage of the revenue and therefor incentives to evade are strong. This fact has suggested partial detractions on consumers declaring these expenses. The model derives equations that can estimate evasion probability as functions of tax rates and entity of transactions and entity of production chain costs. The model is stylized to allow easy comprehension, but can be detailed and estimated with real data 2 .
The possibility for consumers to deduct some VAT certified costs on goods or services in short production chains (professionals services, artisans) decreases the nominal VAT revenue for the government, but allows the emergence of the large income tax evasion by sellers. This policy, successfully adopted in many countries, is able to break the coalition of the private agents ( + ) against the government , transforming it in a virtuous coalition ( + ) able to force sellers to comply.
The model here delivers the possibility to have the rate of detractions for consumer VAT expenses as an endogenous variable, function of tax parameters, transaction value, and, the type of seller evasion, whether on the specific transaction or on the full chain. The model determines how size detractions should be, as a function of seller benefit from evasion. If the buyer gets the sufficient, he gets sufficient compliance advantage, to convince the seller to join complying In other words, thresholds on parameters are able to invert the game equilibrium towards a compliance one.
Summarizing, the study measures which thresholds in fiscal rates, like VAT expenses detraction rate, are able to induce the sellers to comply, even in absence of audit and sanctions. The theoretical results derived here can be easily applied to real data for precise tax design engineering. A simple numerical example is presented in the Appendix.