Tax evasion: Is this a government fight,

or can anyone join?

Marcelo Arbex∗ Enlinson Mattos†

May 16, 2012

Abstract

In this paper, we study optimal tax and auditing policy in an economy where consumersplay a role in enforcing compliance with the tax code on the part of the firms. In particular,we model an environment where firms can evade retail sales tax, the government can engage instandard auditing and enforcement procedures as well as reward households for a role of indi-vidual tax enforcers. Upon the purchase of goods and services agents have to decide whetheror not they will request a receipt for their purchase. This request is costly in terms of time andconsumers might decide not to engage in such activity, which we denote individual auditing.We characterize optimal sales taxes, auditing probabilities (executed by tax authorities) andoptimal rebates for the consumers. We show that the optimal tax rebate rate is in generalpositive and higher than the audit probability. If the unit cost of audit is zero, the governmentwill set the retail sales tax and the audit probability such that the expected penalty for taxevasion is equal to the retail tax. If a consumer spends a fixed amount of his time regardlessthe proportion of his purchases he asks for a receipt, the optimal tax rebate rate is zero. Fi-nally, we show that the rate of substitution between audit probability and rebate rate shouldbe equal to marginal rate of substitution (utility constant) and that the distortion on optimalquantities should be lower the larger is the effect of the policy on tax revenues comparativelyto prices.

Keywords: Optimal Taxation, Indirect Tax Evasion, Tax Enforcement and Auditing.

JEL Classification: E62, H21, H26, K42.

∗Department of Economics, University of Windsor. Windsor, N9B 3P4, Canada. arbex@uwindsor.ca.†São Paulo School of Economics, Getulio Vargas Foundation. São Paulo, 01332-000, Brazil. enlin-son.mattos@fgv.br.

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1 Introduction

Tax evasion and informal activities are pervasive features of countries throughout the world and

recurrent themes in academic and business debates. There is a broad literature on different aspects

of this phenomenon and reasons why individuals would try to conceal their economic activities.

Factors influencing tax evasion and informal activity have been related to aspects of public policy

and public administration such as the burdens of taxation and social security contributions, the

complexity and arbitrariness of the tax system, the extent of bureaucracy and regulations, and the

incidence of corruption and rent-seeking (e.g., Friedman et al. 2000; Johnson et al. 1998a,b; Loayza

1996; Schneider and Enste 2000, 2002; Schneider and Neck 1993).1

As tax evasion has a significant impact on public finances, a great deal of effort is being invested

by tax authorities in developed and developing countries to design and test policy measures that

aim to tackle this problem. Two broad policy approaches can be identified: deterrence and enabling

compliance. The approach towards tax evasion in most countries remains focused on deterrence by

improving detection or increasing penalties. Evasion penalties are generally imposed as a percentage

of the additional tax payable, and most countries adopt a range for evasion penalties and the penalty

imposed varies according to the seriousness of the offence.2 Detection probabilities based on the

proportion of tax returns audited (frequency of audit) are reportedly very low.3

In recent decades, however, a more enabling approach to fighting informal activities have received

increasing attention in the literature (Renooy et al., 2004; SBC, 2004; Slemrod, 1992; Williams,

2006). And, in practice, there has also been a noticeable increase in the use of more measures that

encourage compliance by preventing people from engaging in tax evasion, enable the legalization

of previously informal work and change attitudes (Williams and Renooy, 2008). In this approach,

individuals are viewed as social actors who are usually inclined to comply with the law, partly

1For a comprehensive discussion of these (and other) factors, see Schneider and Enste (2000).2A common approach is to levy penalties for minor offences in the region of 10 to 30 percent of the tax evaded

while more serious offences involving deliberate evasion are in the region of 40-75 percent of the tax evaded (OECD,2004).

3In the United States, according to Andreoni et al. (1998), the audit rate for individual tax returns was 1.7 percentin 1995. Data from the Internal Revenue Service indicate that this rate decreased to 0. 93 percent in the period1996-2002. Similarly, Busato and Chiarini (2004) estimate a 3 percent probability of detection in Italy and Ratto,Thomas and Ulph (2005) found a 0.7 percent probability of being caught evading taxes in the United Kingdom. InCanada, according to Canada Revenue Agency, this audit rate fell from 1.01 percent in 2002 to 0.75 percent in 2006.

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because of their belief in the rule of law and partly as a matter of long-term self-interest (Kagan and

Scholz, 1984; Murphy, 2008). Thus, cooperation rather than coercion is pursued. In this context, the

role of the tax authorities is not so much to act as a strict law enforcer that punishes bad behavior,

but more as a service provider that secures future compliance (Murphy, 2005). Understanding the

interplay between the deterrence and enabling approaches and how alternative policy instruments

can be use to tackle tax evasion is an important avenue of research which we pursue in this paper.

There is a diverse range of enabling measures being used to tackle tax evasion. Of particular

interest for our discussion are policies of Member States of the European Union that have targeted

consumers with direct tax measures and wage cost subsidies. For instance, in the household repair,

maintenance and improvement sector, a sector notorious for the prevalence of informal (undeclared)

work, tax rebates on home maintenance expenses have been available in France under the Universal

Service Employment Cheque (Chèque Emploi Service Universel, CESU ) scheme. Individuals can

purchase cheques from their local bank or post offi ce to pay for domestic service work in the home.

For example, for a job valued at €20, the employer can contribute €10 and of the remaining €10

due to be paid by the individual household, 50% can be recovered in the form of a tax credit. Tax

reductions for labour costs on home repairs and household services have been introduced in Italy,

Luxembourg and Sweden. In Finland, it is possible to deduct thirty percent of the wage cost of

other domestic services (including household cleaning and gardening). In 2004, over ninety per-

cent of householders purchased the deductible service from a company. The Danish Home Service

Scheme (Hjemmeserviceordningen) is directed at lowering wage costs for certain activities. Busi-

nesses registered in the program provide services to households for which the government reimbursed

a proportion of the cost (Williams and Renooy, 2008).

In Brazil, the First Congress of Tax Authorities (Encontro Nacional de Administradores Trib-

utários) approved in 2004 a policy called Nota Fiscal Eletronica (Electronic Fiscal Receipt) which

substitutes the emission of fiscal paper by an electronic counterpart in order to improve auditing

in sales (Imposto sobre Circulação de Mercadorias e Serviços). These sales taxes are under the

authority of individual States. This means the States are in charge of collecting, auditing and

registering all transactions within their territory. This policy was successful to create an electronic

platform to further combat tax evasion. São Paulo, the most populated in Brazilian State, has im-

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plemented a program called São Paulo Fiscal Receipt (Nota Fiscal Paulista) in 2007. The main goal

of this scheme is to reach hard-to-tax establishments such as bars, restaurants and small bakeries

which usually do not provide any receipt after the transaction.4 This program entitles consumers a

rebate of up to thirty percent of the taxes collected in their retail purchases. To request the receipt

costumers have to use their social security number or a firm’s identification number for business

services.

We study optimal tax and auditing policies in an economy where consumers play a role in

enforcing compliance with the tax code by firms. We focus on retail sales tax (RST) evasion, a form

of evasion that has received rather less consideration in the literature. In an environment where

firms can evade retail sales tax the government can engage in standard auditing procedures as well

as reward households for a role of individual tax enforcers. Upon purchasing goods agents must

decide whether they request a receipt for their purchase. Receipts are not automatically issued as

firms try to evade sales taxes. For consumers, requesting receipts is time consuming because they

either spend time standing in line or filling up paperwork. If a consumer request a purchase receipt,

the firm has to remit sales taxes to the tax authority. The government then send a portion of the

tax collected back to the consumer as a way to reward her auditing effort. This kind of tax rebate

has elements of both the deterrence and the enabling approach. Consumers can elicit a change

in behavior by firms when they request a sales receipt and force them collect taxes (deterrence).

Moreover, when agents engage in individual auditing they change their own attitudes towards tax

evasion and promote a sense of tax morality in the economy (enabling).

The evasion of retail sales taxes constitutes a process involving sellers and consumers and their

roles depend critically on the tax remittance responsibility - that is, whether the buyer or seller

4More recently São Paulo’s government has adopted raffl e prizes (per receipt number) in addition to these taxrebates to attract more consumers in requesting such fiscal receipt. São Paulo is the richest state in Brazil andcollects on average almost US $ 3 billions per month on sales tax. This amount correspondents to the average forBrazil as a whole is much lower, only US$ 300 millions per month. In addition, São Paulo collects more than 65%of all taxes collected under ICMS. After São Paulo’s initiative, Alagoas State, situated on Northeast part of Braziland less significant in terms of tax collection (US$50 millions per month) also introduced a similar program in 2008.However such a program is not as transparent as Nota Fiscal Paulista. Similar programs started to be adopted inother states: Nota Legal for Brasilia (capital of Brazil) in 2008, Nota Vale Dinheiro for Ceara (Northeast region)in 2009, Cupom Mania for Rio de Janeiro (Southeast region) in 2009. These programs are really close in spirit toNota Fiscal Paulista because they either deliver tax rebates (first two) or raffl e prizes (last one). Finally, states ofGoias (Midwest region), Piaui and Bahia (both Northeast region) exchange fiscal receipts by tickets to shows, soccergames, books etc.

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of a commodity must remit any sales tax triggered by the sale. If all the information needed to

assess tax liability is observed costlessly by the government, remittance is not relevant. However,

in the presence of avoidance as well as evasion, the identity of the remitter (i.e., whether it is a firm

or a final consumer) matter as information about economic transactions are not readily available.

Collecting taxes from businesses makes use of the economy of scale of the tax authority dealing

with a smaller number of larger units. Firms are also central to information reporting, as they

provide the information that facilitates checking whether the correct amount of tax liability has

been collected from all the remitting parties (Slemrod, 2008).5 In the case of a Value Added Tax

(VAT) for instance, tax payment can be remitted by all businesses involved in the process.

The role played by firms in the tax remittance process is particularly relevant in the case of

retail sales taxes.6 In general, final consumers have no role in tax remittance to the government

and RST are remitted only by retail businesses, triggered by sales to final consumers. However,

taxes are only collected if consumers ask for the receipt. There are several reasons to request a

transaction receipt (or record of transaction), for instance, as proof of purchase for warranty and

exchange/return purposes, to obtain reimbursements of work-related expenses and, in some cases,

to prove a tax deductible expense. Due to the nature of tax evasion we study, agents would request

receipts of purchase to force firms to report transactions and remit sales tax to the tax authorities.

The involvement of consumers and firms in tax evasion is well-documented in the literature.

Boadway, Marceau and Mongrain (2002) study an economy where tax evasion requires a collusion

behavior for every pair of agents (firm and consumer). Chang and Lai (2004) introduce the possi-

bility of interaction between buyers and sellers and find that a collaborative tax evasion might be

an equilibrium result. Gordon and Li (2008) consider an environment where firms can avoid tax

payments by shifting entirely to cash transactions, a common feature in less developed economies.

When firms do not use the financial sector they avoid leaving any paper trail of their transactions.

This, in turn, reduces the effi cacy of traditional auditing and enforcement policies. Tanzi and Shome

5See also Marrelli (1984) and Virmani (1989) that study the effects of indirect tax evasion on the productiondecision on the part of the firms.

6Krauss and Bird (1971) characterize the difference between a consumption VAT and a retail sales tax as adifference in method (italics in original) rather than in essence. On the other hand, Slemrod (2008) points out thata VAT and an RST are equivalent if, but only if, one assumes away avoidance and evasion, as well as administrativeand compliance costs.

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(1993) argue that in order to improve tax collection a cross tax-control between buyer and seller

where the transaction receipt is the main proof of purchase and taxes should be remitted to the

government. Das-Gupta and Gang (1996) propose a matching mechanism to improve auditing for

value added tax evasion context. They address the role played by a transaction matching technology

and its role to enforce the tax code, which is essentially a comparison of buyer’s and seller’s record

of transactions. None of the papers explore the auditing role of the consumer to enforce the tax

code.

Traditional auditing procedures have been extensively studied in the literature and there is an

extensive literature on the optimal choice of enforcement tools by governments and tax authorities

for a variety of forms of tax evasion. Usher (1986) and Kaplow (1990) address the role of auditing

strategy in general and Cremer and Gahvari (1993) study the optimal commodity taxation in the

context of tax evasion and characterize the optimal purely random audit strategies. Such random

strategies are known to be suboptimal compared to more general policies, such as cut-off rules

(Reinganum and Wilde, 1985; Border and Sobel, 1987; Mookherjee and Pnj, 1989 and Cremer et

al., 1990).

Our work differs from the existing literature in the sense that we study individual auditing as

an additional tools available to the government to tackle tax evasion and characterize not only the

optimal commodity taxes when evasion is present (Usher, 1986) but also the optimal tax and audit

policies under uncertainty (different from Kaplow, 1990). We allow for different auditing policies

which are purely random probabilities and can be executed by the government (as in Cremer and

Gahvari, 1993) or by the consumer. More specifically, in our economy firms face two distinct risks of

being audited. First they are subject to individual auditing, i.e., consumers decide the proportion

of their purchase they will request receipts. Second, on the portion of sales that a receipt was not

issued, firms can decide whether to collect taxes or not. Each unit of output concealed from the

tax authorities by firms entails a resource cost which is increasing in the proportion of sales not

reported. If firms evade taxes they are subject to being detected by the government with a positive

probability, which we call government auditing. The production of goods is perfectly competitive

which excludes the possibility of under-reporting prices and collusion between consumers and firms.7

7Regarding the possibility of under-reporting prices see, for instance, Chang and Lai (2004).

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We focus on retail sales tax evasion and abstract from any other form of evasion.

The government chooses how much to audit (auditing probability), the correspondent fines in

case firms are caught cheating, commodity taxes and tax rebates. From the point of view of the

government both auditing procedures - individual and government - are costly. To reward agents

for their role of tax enforcers, the government pays a benefit in the form of tax rebate, while to

engage in auditing procedures itself, the government incurs a cost proportional to the fraction of

the firms it want to audit. The design of optimal tax enforcement policy follows the Ramsey primal

approach (Lucas and Stokey, 1983; Chari, Christiano and Kehoe, 1991). We show that the optimal

tax rebate rate is in general positive and higher than the audit probability. If the unit cost of audit

is zero, the government will set the retail sales tax and the audit probability such that the expected

penalty for tax evasion is equal to the retail tax. We can also show that if the consumer spends

a fixed amount of his time regardless the proportion of his purchases he asks for a receipt, the

optimal tax rebate rate is zero. Given that the time spent to request receipts is a sunk cost for the

consumer, the government has no need or reason to reward the agent’s behavior and the optimal

rebate is zero.

In this paper, we attribute a key role for the firms in remitting taxes to government and we are

concerned about an incentive mechanism that could induce agents to engage in auditing activities

when firms would prefer evade taxes. Our contribution is noteworthy in light of the fact that, with

some exceptions, firms are absent from the modern theory of optimal taxation (Slemrod, 2008;

Dharmapala, Slemrod, and Wilson (2008). Moreover, to the best of our knowledge, our paper is

the first to consider tax rebates as a mechanism to enforce the tax code and reward agents for

their role as tax enforcers. By requesting purchase receipts, consumers automatically disclosure the

transaction to the government and the amount of taxes due by the firms.

The paper proceeds as follows. Section 2 presents the model economy. In Section 3 we study

optimal tax and auditing policies as well as policy trade-off faced by the government in an economy

where consumers play a role in enforcing compliance with the tax code by firms. Section 4 presents

a numerical exercise and Section 5 offers concluding comments.

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2 The Economic Environment

2.1 Firms

Consider an economy, defined in discrete time, in which there are infinite-lived consumers, perfectly

competitive firms and a government. Each firm uses labor Lt to produce a single perishable good

yt according to a production technology given by yt = Lt. We assume that each firm solves a profit

maximization problem on a period-by-period basis, subject to a technological constraint and to

the possibility that it may be discovered evading taxes, convicted of tax evasion and subject to a

penalty surcharge.

The output of firms is sold at a consumer price pt and sales are subject to a retail sales tax (RST)

τ t. In this economy tax is remitted to the government only by retail businesses, triggered by sales

to final consumers (Slemrod, 2008). This remittance system creates opportunities for avoidance

and evasion, which is associated with firm’s decision of whether or not remit sales tax.

The enforcement of the tax rules is stochastic and occurs either due to consumer’s actions or due

to government’s auditing measures and procedures. We sketch the scenario of tax evasion by means

of the following three-stage game. During the first stage, firms sell the good to consumers at a price

pt and the seller must remit any sales tax triggered by this sale. However, the remittance of taxes

to the government will depend on whether consumers request or not a receipt of the transaction.

Let at be the proportion of a consumer’s purchases for which he asks for receipts in period t. We

denote this mechanism individual auditing, where the consumer acts to enforce the tax code. In this

case, firms are obliged to collect taxes in the amount of atτ tptLt. However, this activity is costly

(as we explain below) and individuals might decide not engage in such form of auditing.

In the second stage, out of the proportion of sales that is not subject to individual auditing, i.e.,

(1−at)ptLt, the firm will evaluate the benefits and costs of tax evasion in order to decide whether to

evade taxes or not. The firm may attempt to evade taxes by reporting only a fraction γt of its sales.

In order to avoid being detected by the tax authorities evading taxes the firm spends resources to

conceal it. Each unit of output concealed from the authorities by firms entails a resource cost of

H(ξt), where ξt = (1−at)(1−γt). The concealment cost is an increasing and convex function of the

proportion ξt of sales not reported, either because consumers did not asked for a receipt, (1− at),

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or firms decided to hide a fraction of its sales, (1− γt). Without such (concealment) strategy firms

would be automatically detected without the necessity of auditing by a simple process of verification.

Finally, firms may be discovered evading retail sales taxes with probability θt ∈ (0, 1). We

denote this kind of auditing government auditing. If caught evading taxes, a firm must pay a tax

rate ρ ≥ τ , which is the standard RST τ plus a surcharge factor. Hence, firms choose prices and

the proportion of sales it wants to collect taxes to maximize expected profits in each period t ≥ 0,

Πe = {1− atτ t − (1− at)[γtτ t + (1− γt)θtρt + (1− γt)H(ξt)]} ptLt − wtLt (1)

where wt is the nominal wage.

Firms’behavior is characterized by the following first order necessary and suffi cient conditions:

−(1− at)ptLt [τ t − θtρt −H(ξt) + (1− γt)Hγ(ξt)] = 0 (2)

[1− T et − ξtH(ξt)]pt − wt = 0 (3)

where T et = atτ t + (1 − at)[γtτ t + (1 − γt)θtρ] is the expected tax that a firm pays to the tax

administrator on each unit of output produced and sold. Notice that a firm will incur concealment

costs regardless whether it has to issue a sales receipt (and collect the appropriate amount of taxes)

or is detected by de government evading taxes (and has to pay a penalty).

Equation (2) reveals the optimum choice of firms regarding the percentage of their sales they

want to report to the tax authorities, after consumers decide the fraction of purchases with receipt

at. A firm will choose γt such that the cost of honestly report this amount and remit the appropriate

amount of sales taxes is equal to the hiding cost plus the penalty rate in the event of being detected

by the government. That is, τ t = H(ξt)− (1− γt)Hγ(ξt) + θtρt. The condition (3) rules out corner

solutions and allows us to focus on the interesting problem of tax evasion when a positive (and not

infinite) quantity of output is sold.

Combining equation (2) and (3) we obtain an equilibrium condition for the firm as follows:

pt = [1−H(ξt) + (at + (1− at)γt)(1− γt)Hγ(ξt)− θtρt]−1wt (4)

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which reflects how the sales price is affected by the amount of sales with receipts, the amount of

sales voluntarily reported and the concealment cost. We then define P (at) = [1−H(ξt) + (at + (1−

at)γt)(1− γt)Hγ(ξt)− θtρt] and write

p(at) = P (at)−1wt (5)

2.2 Households

The representative household supplies Lt units of labor to the market at the wage wt. He devotes

the remainder of his time to either leisure ht or auditing dt. Preferences are represented by the

following utility function∞∑t=0

βtu(ct, ht) (6)

where the instantaneous utility function u is increasing, concave and differentiable, β is the discount

rate which lies in (0, 1), ct is consumption and the time endowment is normalized to 1 so that leisure

is ht = 1− Lt − dt.

Upon the purchase of goods for consumption, a consumer has also to decide whether or not to

request a receipt of his purchase. The very act of asking for a receipt is what we call individual

auditing and this activity is time consuming (leisure cost). Every sale triggers remittances of sales

taxes to the government as long as receipts of purchases are generated. When a consumer does not

request the receipt, firms have the opportunity to evade taxes. When a receipt is issued, it leaves

a paper trail of the transaction and sales taxes must be remitted.

Auditing takes time. In developing economics, such request of fiscal receipt usually is associated

with a longer time standing in the line waiting for assistance or even to fill out paper work. The

amount of time employed in auditing is described by the technology

dt = d(at) = daδ (7)

where at is the proportion of the goods purchased and a receipt is requested, d ∈ (0, 1) an upper

bound on the leisure cost of auditing activities and δ ≥ 0 is the auditing technology parameter.8

8The idea that spending time on consumption activities might be substitute to labor is also addressed on Boadway

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For the range 0 < δ < 1, i.e. when the auditing technology exhibits decreasing returns to scale,

it is less costly, in terms of leisure time, for a consumer to request a receipt for an additional unit

of purchased good. On the other hand, if δ > 1 it becomes increasingly costly for consumers to

engage in auditing activities. Auditing time is increasing in the share of consumer’s purchases for

which he asks receipts for, i.e., d′(at) > 0. The auditing technology satisfies d(0) = 0, meaning

that if individuals do not request receipts in any of their purchases no time is spent in auditing and

d(1) = d. This last assumption is to avoid a degenerate behavior of time allocation and guarantee

an interior solution for labor and auditing time.9

Since these activities are costly, without further incentives consumers would not engage in any

auditing. To reward agents for their role of tax enforcers, the government promises a tax rebate

based on the amount of taxes collected due to consumers’auditing actions. The amount of tax

rebate an agent receives in time t depends on the amount of his purchases he requested receipts for

in the previous period, i.e., at−1pt−1ct−1.

The representative consumer faces the following budget constraint:

ptct = wtLt + pt−1φt−1τ t−1at−1ct−1 (8)

where pt is the consumer price (equal to the producer price since we abstract from consumption tax),

φ ∈ [0, 1] is the tax rebate rate. Notice that agent’s auditing effort in a previous period affects his

current choice of consumption and leisure as the tax rebate received from the government represents

additional income available for consumption.

At this point is relevant to notice that the consumer, when making his decision, has to take into

account the firm’s reaction to his receipt requests. That is, he has to consider that the sales price pt

is in fact p(at). Hence, using (5) into (8) we have a modified consumer budget constraint as follows:

p(at)ct = wtLt + p(at−1)φt−1τ t−1at−1ct−1 (9)

and Gahvari (2006), Gahvari (2007), Gahvari and Yang (1993) and Kleven (2004). Our interpretation of the timespent on calling for receipts has the same spirit of their idea about time spent on consumption activities.

9This function is usually related with the information transmission technology within the firm. Modern firms(better internet connections, computerized fiscal receipts emission) with better technologies make the transmissionless timely cost for consumers.

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The individual maximizes (6) subject to (9) by choosing values of ct, ht and at, taking as

given prices and government policy instruments. The individual’s optimization problem yields the

following first-order conditions

u1(t) + βµt+1p(at)φtτ tat = µtp(at) (10)

u2(t) = µtwt (11)

µt [pa(t)ct + d′(at)] = βµt+1φtτ tct [pa(t)at + p(at)] (12)

where µ is the Lagrange multiplier on the consumer budget constraint equation (8). From equations

(10)− (12), we obtain the following equilibrium conditions:

u1(t) + βu2(t+ 1)wtwt+1

P (at)−1φtτ tat = u2(t)P (at)

−1 (13)

u1(t)ct[P (at)

−1Pa(t)at − 1]

= u2(t)[atd′(at)− P (at)

−1ct]

(14)

Equation (13) shows the relative price of leisure with respect to consumption when the agent engages

in non-zero individual auditing. Equations (13) and (14) illustrate the trade-off agents face when

deciding to request sales receipts.

2.3 The government and aggregate resources

The government finances its expenditures Gt by collecting taxes from the firms. The government

budget constraint is given by

pt (Gt + θtNt) + pt−1φt−1τ t−1at−1ct−1 = (atτ t + (1− at)[γtτ t + (1− γt)θtρt]) ptLt (15)

where N is a fixed unit cost of audit. The terms on the left side are, respectively, government

expenditures, auditing spending and tax rebate payments. The right-hand side of this equation

contains government revenues generated by individual auditing, firm’s voluntary compliance and

penalties imposed on firms detected evading taxes.

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The resource constraint in this economy is as follows

ct +Gt = Lt − ξtH(ξt)Lt − θtNt (16)

3 Optimal Tax and Audit Policy

At the beginning of each period, the government announces its program of tax-enforcement instru-

ments and private agents behave competitively. The problem of the effi cient tax policy, known as

the “Ramsey problem,”prescribes that the utility of the representative consumer is maximized by

taking into account the competitive equilibrium with distortionary taxes and the constraint that a

given amount of revenue has to be raised (Chari and Kehoe, 1999).

In this problem we assume that evasion penalties (ρ) are at their statutory level and constant

over time. These policy instruments are determined by institutional arrangements and cannot be

easily changed. Hence, we focus on policy instruments that affect the auditing behavior of consumers

and the government monitoring strategy. That is, we allow the Ramsey planner to choose optimally

the retail sales tax τ , the rebate rate φ and the probability of detection θ.

In the primal approach to solve the Ramsey problem, i.e., the government’s choice of its tax

policy mix to maximize the household’s welfare by taking into account the equilibrium reaction

of private agents to the tax policy, we eliminate taxes and prices, so that the government can be

thought as directly using the quantities as controls. On the other hand, the dual approach does not

aim to describe optimal taxes, but seeks to characterize optimal quantities. That is, the government

uses the tax rates or prices as controls.

3.1 Policy Trade-off

In this section we use the dual approach to solve the Ramsey problem and to study the trade-off

the government faces when it chooses the auditing probability and the tax rebate tax. Government

expenditures are assume to be exogenously given and, for the purpose of what follows, can be

considered as a lump-sum transfer to consumer. This assumption would affect the consumer budget

constraint but not consumer allocation choices. The advantage of this particular assumption, to

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be modified later, is to make the problem simpler and the policy trade-off clearer since in this case

supply and demand of goods are equal.

We start by deriving ordinary demand functions c(p, w), L(p, w) and a(p, w) from the first-order

conditions (10) − (12). Assuming that the solution of the Ramsey problem converges to a steady

state, the consumer’s budget constraint, equation (8), can be rewritten as

p[1− φaτ ]c = wL (17)

Normalize the wage rate to 1, i.e. w = 1, and write p = p[1 − φaτ ], where p is endogenously

determined. Consumers take the price of the goods p as given even though in equilibrium their

decision regarding auditing (a) affects the price level. Therefore one can write p = F (p, φτ) and the

indirect utility function as v(p) ≡ v(p(p, φ, τ)). In this setup the Roy’s identity still applies, i.e.,

∂v/∂p = −µc where µ is the representative household’s marginal utility of income.

The government problem is then to choose a retail sales tax τ , an auditing probability θ and

a tax rebate rate φ in order to maximize the consumer’s indirect utility v(p) subject to its budget

constraints given by

pT ec− apφτc− pθN = G. (18)

where T e = aτ+(1−a)[γτ+(1−γ)θρ] is the expected tax that a firm pays to the tax administrator

on each unit of output produced and sold. The second term on the LHS of equation (18) is the cost

of the rebate or the fraction of the tax revenue collected via individual auditing that is returned

to consumers. Hence, we write the net expected tax as T (e,n) = pT e − apφτ . The third term on

the LHS, pθN , corresponds to the auditing costs technology, which is a linear function and N is

constant.

The structure of this problem now is analogous to the standard optimal commodity tax problem

(Sandmo, 1974, Atkinson and Stiglitz, 1980) with one relevant difference. As we normalize the

individual’s wage to one, equation (8) reveals that actual price is also dependent on φτ and θ.

Therefore the price p is used as an argument of the indirect utility of the consumer.

14

The government’s problem in the Lagrangian form is

L = v(p) + λ(T (e,n)c− pθN −G), (19)

where λ is the Lagrangian multiplier on the government budget constraint, equation (18). The first

order conditions for (θ), (φ) and (τ) are, respectively:

∂L

∂θ=

∂v

∂p

∂p

∂θ+ λ

[T (e,n)

∂C

∂θ+ C

∂T (e,n)

∂θ− pN

](20)

∂L

∂φ=

∂v

∂p

∂p

∂φ+ λ

[T (e,n)

∂C

∂φ+ C

∂T (e,n)

∂φ

](21)

∂L

∂τ=

∂v

∂p

∂p

∂τ+ λ

[T (e,n)

∂C

∂τ+ C

∂T (e,n)

∂τ

](22)

Proposition 1 Optimal auditing and tax-rebate must satisfy:

T (e,n)∂c

c∂p=

µ

λ−

∂T e,n

∂θ∂p∂θ

+pN/c∂p∂θ

(23)

T (e,n)∂c

c∂p=

µ

λ−

∂T e,n

∂φ

∂p∂φ

(24)

T (e,n)∂c

c∂p=

µ

λ−

∂T e,n

∂τ∂p∂τ

(25)

These expressions have an interesting interpretation. The LHS of these two equations shows

the proportional movement along the Marshalian demand caused by marginal changes in the policy

instruments, which in our case are θ, φ and τ , respectively. On the other hand, the RHS of equations

(23), (24) and (25) indicate that, in our environment, the optimum choice of policies θ, φ and τ

imply non-constant proportional movements the demand curve. This is different than the solution

of standard Ramsey problem where movements are constant and given by µ/λ.

Critical to this analysis are the terms [∂T e,n/∂θ]/[∂p/∂θ] and [∂T e,n/∂φ]/[∂p/∂φ]. Consider,

for instance, the trade-off faced by the government when deciding to change the audit probability

θ. For an increase in θ the government must consider its effect on tax revenues (∂T e,n/∂θ) and

on prices (∂p/∂θ). Increasing the rate at which the economy is audited will be optimal as long as

the impact on taxes collected is higher than the increase in prices, the later being a measure of

15

welfare loss for the consumers. In this case, the government wants to increase the distortion in the

economy via an increase of θ. This occurs when [∂T e,n/∂θ] > [∂p/∂θ], which implies that the RHS

of equation (23) is smaller. Consequently the LHS is smaller as well. More distortion occurs and

the substitution effect is greater. Lastly, the larger the term [pN/c]/[∂p/∂θ], the lower the net taxes

collected. In this case, it is optimal for the government to reduce θ in order to reduce distortions on

the demand side. The analysis for the tax rebate rate φ follows the same arguments and is omitted.

The following proposition summarizes the optimal trade-off of the government when choosing

between auditing the economy directly (θ) and promoting individual auditing via a tax rebate (φ).

Proposition 2 Trade-off between auditing and tax-rebate is represented by the following relation-

ship:∂p∂θ∂p∂φ

=[T (e,n) ∂C

C∂θ+ ∂T (e,n)

∂θ− pN

C]

[T (e,n) ∂CC∂φ

+ ∂T (e,n)

∂φ]

(26)

The welfare of the consumer in this dual-approach setup is measured by price p. The left hand

side of equation (26) shows the trade-off in terms of the costs of using these policies. The right

hand side shows the marginal rate of substitution of both policies keeping utility constant. The

numerator shows the revenues of government after a marginal increase in θ: the first two terms

denote the net increase in tax collection and the third term represents the increase in auditing

costs. The denominator corresponds to the counterpart effect of an marginal increase in φ on net

tax collection. Equation (26) states that these two rates of substitution must be equal, a necessary

condition in the Ramsey problem.10

To understand this result better, consider a vector of transformed consumer prices p. Any price

vector can be achieved by various combinations of taxes τ , audit probabilities θ, and tax rebates

φ. At the optimum, however, policies (τ , θ, φ) and consequently prices (p) must be such that

the government’s net tax revenue is maximal (conditional on p). Otherwise, more revenue could

be collected without changing consumer prices (keeping utility constant). That would contradict

the optimality condition since it would also imply that the same revenue could be raised at lower

consumer prices. One can also derive the trade-off between tax versus traditional auditing as well

as tax versus individual auditing. Our next proposition summarizes the result.10Cremer and Gahvari (1993) derive a similar expression when characterizing the relationship between an optimal

commodity tax and the audit probability for firms (equation 14, pp. 269)

16

3.2 Optimal Policy Analysis

We also study the Ramsey problem by using the primal approach to optimal taxation (Lucas

and Stokey, 1983). This approach to optimal taxation employs the implementability constraint

which is obtained from the consumers’ intertemporal budget constraint by expressing prices and

auditing instruments in terms of allocations through the marginal effi ciency conditions of consumers

- equations (10), (11), (12). That is,

∞∑t=0

βt [u1(t)ct − u2(t) (1− ht − d(at))] = W0 (27)

whereW0 = β−1u2(c−1, h−1)a−1d′(a−1). Since optimization starts at t = 0, we take c−1, h−1 and a−1

as given. The implementability constraint is similar to the objective function in the sense that both

are discounted infinite sum of terms. Thus, given the Lagrange multiplier λ1 (which is endogenous),

it is possible to write the pseudo-welfare function of the social planner as

W (ct, ht, at;λ1) = u(ct, ht) + λ1 [u1(t)ct − u2(t) (1− ht − d(at))] (28)

The social planner is also constraint to choose allocations that are feasible in this economy. This

implies that the resource constraint must be imposed on the planner’s problem. Notice that the

audit probability θ appears in the equation (16). Following the primal approach to the Ramsey

problem, we must express this constraint only in terms of allocations only. This implies that we have

to express θ in terms of allocations. To do so, we proceed as follows. First, combining equations

(3) and (13) we obtain:

[1− atτ t − (1− at)[γtτ t + (1− γt)θtρ]− ξtH(ξt)] = Z(ct, ht, at) (29)

Solving (2) for θtρ and substituting it into (29) we get an expression for τ t as follows:

τ t = 1 + ξ2tH′(ξt)− Z(ct, ht, at) (30)

17

which we use to substitute back into (2) to obtain

θt = ρ−1 [1− (1− ξt)ξtH ′(ξt)−H(ξt)− Z(ct, ht, at)] (31)

We then use this expression for the audit probability, equation (31), to express the economy’s

resource constraint in terms of allocations only:

(1−ξtH(ξt)) (1− ht − d(at))−Nρ−1 [1− (1− ξt)ξtH ′(ξt)−H(ξt)− Z(ct, ht, at)]−ct−Gt = 0 (32)

The planner’s maximization problem can thus be written as follows:

∞∑t=0

βtW (ct, ht, at, γt;λ1)−W0 (33)

subject to (32) and Gt = G, ∀t given. The first order conditions for this problem evaluated at the

steady state are

W ∗c − λ∗2(1−Nρ−1Z∗c ) = 0 (34)

W ∗h − λ∗2(1− ξ∗H(ξ∗)−Nρ−1Z∗h) = 0 (35)

W ∗a − λ∗2([1− ξ∗H(ξ∗)] d(a∗)−Nρ−1Z∗a) = 0 (36)

H(ξ∗) + ξ∗H ′(ξ∗)−Nρ−1(1− ξ∗)

[2H ′(ξ∗) + ξ∗H

′′(ξ∗)

](1− h∗ − d(a∗))

= 0 (37)

18

where

W ∗c ≡ u∗1 + λ1 [u∗11c

∗ + u∗1 − u∗21 (1− h∗ − d(a∗))] (38)

W ∗h ≡ u∗2 + λ1 [u∗12c

∗ + u∗2 − u∗22 (1− h∗ − d(a∗))] (39)

W ∗a ≡ +λ1 [u∗2d

′(a∗)] (40)

Z∗c ≡(u∗1u

∗21 − u∗2u∗11)(c∗)2 + (u∗2)

2a∗d′′(a∗)

(u∗1c∗ + u∗2a

∗d′(a∗))2(41)

Z∗h ≡(u∗1u

∗22 − u∗2u∗12)(c∗)2

(u∗1c∗ + u∗2a

∗d′(a∗))2(42)

Z∗a ≡ −(u∗2)

2[d′(a∗) + a∗d

′′(a∗)

]c∗

(u∗1c∗ + u∗2a

∗d′(a∗))2(43)

λ1 and λ2 represent the Lagrange multipliers on equations (27) and (32).

Proposition 3 If the solution to the Ramsey problem converges to a steady state, and the govern-

ment has access to a retail sales tax τ , an auditing probability θ and a tax rebate φ, then in steady

state, the tax and auditing instruments are as follows:

τ ∗ = 1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + ξ∗Ψ∗ (44)

θ∗ =1

ρ[1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + (ξ∗ − 1)Ψ∗] (45)

φ∗ =Z(c∗, h∗, a∗)d′(a∗)

βc∗ [1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + ξ∗Ψ∗](46)

where

Ψ∗ ≡ N

ρ

(1− ξ∗)[2H ′(ξ∗) + ξ∗H

′′(ξ∗)

](1− h∗ − d(a∗))

(47)

Z(ct, ht, at) ≡u2(t)ct

u1(t)ct + u2(t)atd′(at)(48)

Proof. In order to obtain the expressions for the optimal probability of auditing and optimal retail

sales tax, we proceed as follows. First notice that the steady state version of equation (2) is

τ ∗ − θ∗ρ = H(ξ∗) + ξ∗H ′(ξ∗) (49)

19

Comparing this expression with the first order condition of the Ramsey problem with respect to γ

we obtain

τ ∗ − θ∗ρ = Ψ∗ (50)

Solving this expression for θ∗ρ and combining it with equation (3) we obtain an expression for the

optimal retail sales tax:

τ ∗ = 1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + ξ∗Ψ∗

Plugging this expression for τ ∗ back into expression (49) and solving for θ∗ρ we get

θ∗ =1

ρ[1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + (ξ∗ − 1)Ψ∗]

In order to derive the optimal tax rebate tax we start by writing the first order condition of the

consumer problem, equations (10)− (12), in steady state:

u∗1 + βµ∗φ∗τ ∗a∗p∗ = µ∗p∗ (51)

u∗2 = µ∗w∗ (52)

µ∗w∗d′(a∗) = βµ∗φ∗τ ∗p∗c∗ (53)

From (52) and (53) we obtain

u∗2d′(a∗) = µ∗p∗ (βφ∗τ ∗c∗)

which combined with (51) yields

φ∗ =1

βτ ∗u∗2d

′(a∗)

u∗1c∗ + u∗2a

∗d′(a∗)

Using the definition of Z(ct, ht, at) and the expression for the optimal retail sales tax, equation (44)

we get the following expression for the optimal tax rebate rate

φ∗ =Z(c∗, h∗, a∗)d′(a∗)

βc∗ [1− ξ∗H(ξ∗)− Z(c∗, h∗, a∗) + ξ∗Ψ∗]

20

From expressions (44) and (45), notice that if the unit cost of audit N is zero, the government

will set the retail sales tax and the audit probability such that the following condition is satisfied:

τ ∗ = θ∗ρ. This means that the expected penalty for tax evasion is equal to the retail sales tax.

Moreover, the optimal tax rebate rate is negatively related the auditing probability and given

by φ∗ = Z(c∗, h∗, a∗)d′(a∗)/ [θ∗ρβc∗]−1. These two auditing policies are substitutes tools that the

government has to fight tax evasion and the higher the probability of detection the lower is the

incentive it has to reward consumers in their task of auditing firms.

It is clear from expressions (44)−(46) that the optimal policies depend on the auditing technology

d(a) and the concealment cost H(ξ). Consider the individual auditing technology described by

expression (7) and assume δ = 0. This means that consumers spend a fixed amount of their time on

auditing, i.e., d(a) = d and d′(a∗) = 0. Given that this is a sunk cost for consumers, the government

does not have to reward the agent’s behavior and the optimal rebate rate is zero (φ∗ = 0).

Corollary 1 If the unit cost of audit is zero (N = 0), then (1) the expected penalty for tax evasion

is equal to the retail sales tax, i.e., τ ∗ = θ∗ρ and the optimal tax rebate rate is negatively related to

the auditing probability and given by φ∗ = Z(c∗, h∗, a∗)d′(a∗)/ [θ∗ρβc∗]−1 and (2) the optimal rebate

rate is zero (φ∗ = 0) if δ = 0.

Based on our results presented in Proposition 1, we can show that there exist an optimal sales

tax τ ∗ such that the government audits the economy at the same rate as it rewards consumers for

their auditing efforts, i.e., θ∗ = φ∗. Moreover, this particular sales tax is increasing in the unitary

cost of auditing, meaning that the higher the cost for the government to audit firms the higher the

sales tax. The next proposition summarizes this result.

Proposition 4 There is an optimal retail sales tax such that the government audits the economy

at the same rate as it rewards consumers for their auditing efforts. That is, for θ∗ = φ∗ the optimal

sales tax is given by:

τ ∗ =Ψ∗ +

[Ψ∗

2+ 4ρZ(c∗, h∗, a∗)d′(a∗) (βc∗)−1

] 12

2(54)

21

Moreover, this sales tax is increasing in the unit cost of auditing N and when N = 0, it is equal to

τ ∗ =[ρZ(c∗, h∗, a∗)d′(a∗) (βc∗)−1

] 12 .

Proof. Combining equations (45) and (46) we obtain the following expression

τ ∗ −Ψ∗

ρ=Z(c∗, h∗, a∗)d′(a∗)

βc∗τ ∗(55)

which solving for τ ∗ results in (54). Since Ψ∗ = 0 when N = 0, we have that the optimal sales tax

in this case is τ ∗ =[ρZ(c∗, h∗, a∗)d′(a∗) (βc∗)−1

] 12 . Moreover since ∂Ψ∗/∂N ≥ 0, τ ∗ is increasing

the cost of auditing N (∂τ ∗/∂N > 0).

4 Numerical Results

4.1 Parameterization and Solution Method

We now use numerical methods to simulate calibrated versions of the model we have been examining.

We use these results to confirm the validity of our analytical expressions and to explore further the

relationship between retail sales tax, rebate tax and alternative auditing policies. The model is

calibrated such that the steady state is consistent with United States data. We assume the utility

function U(ct, ht) = ln ct + κ lnht and set the weight on leisure, κ, to match initial steady-state

labor supply. One quarter of an agent’s time is allocated to market activities (40 hours per week)

and in the steady state it implies h = 0.75. We set the discount factor β = 0.96 which implies a

rate of time preference of 4 percent on an annual basis.

We calculate the solution of the optimal tax problem for the calibrated version of the model

described above. The initial retail sales tax is set to 12 percent, i.e., τ 0 = 0.12. The probability of

detecting an agent working underground or evading taxes is related to auditing procedures and the

monitoring technology. In the United States, according to Andreoni et al. (1998), the audit rate

for individual tax returns was 1.70 percent in 1995. Data from the Internal Revenue Service (IRS)

indicate that this rate has dropped to 0.93 percent in the period 1996-2002. We set θ0 = 0.01.

Statutory penalties for understatements of tax liability are generally imposed as a percent of the

22

additional tax payable and vary according to the seriousness of the offence. In the United States,

evasion penalties are applied at rates of 20 and 75 percent of the portion of underpaid tax or fraud,

respectively (Andreoni, Erard and Feinstein, 1998). We assume the lower rate of 20 percent as our

baseline value. We set ρ = 0.14 and interpret it as follows. In the United States, a firm detected

working in the informal sector is expected to pay the tax rate it was trying to evade (τ = 0.12), plus

an additional penalty of 20 percent of the evaded amount, which implies a tax rate of 14 percent

(0.12 × (1 + 0.20) = 0.144). Government purchases, Gt, are such that the steady-state ratio of

government purchases to GDP generated by the model with initial policy is 20 percent of GDP.

The method to solve for the Ramsey equilibrium is standard in the literature and we briefly

describe it. We assume that the economy converges to a steady-state and the Ramsey equilibrium

is characterized by a system of non-linear equations. We solve for the Ramsey equilibrium by

adjusting the multiplier on the implementability constraint, equation (27), until this constraints are

satisfied.

[TO BE COMPLETED]

5 Conclusion

The paper investigates optimal tax and audit policy where both consumers and tax administrators

can engage in auditing. In particular, our model allows consumers to play a role in auditing business

to enforce compliance with the tax code and tax remittances to the government in accordance with

recent policy adopted in Brazil called Nota Fiscal Paulista, in which consumers requests fiscal

receipts in order to guarantee that the transaction is registered to the tax authorities. Those

consumers are entitled to receive rebates based on tax collected due to their actions. Different from

most papers in the field, we allow the firms to have the decision on how much to evade and also to

pay the sales taxes collected. This is assumed because we want to have consumers focused on the

possibility of individual auditing. These, consumers, in turn, decide consumption, time spent on

auditing and working. However they are paid by their auditing in the next period.

We characterize optimal sales taxes, auditing probabilities (executed by tax authorities) and

optimal rebates for the consumers that request sales receipt. Although we are able to find explicit

23

expressions for those policies using a very general model, only few conclusions can be made at this

stage. We show that tax rates should be equal to expected fines whenever the auditing cost is

zero, given that some evasion is pursued on the part of the firms. This result is reasonable since

both policies work as tax instruments but under two different states of nature. Taxes are paid

by firms when they decide to declare their sales (or are forced by the consumers), while fines are

those ’taxes’imposed by tax authorities when firms are caught cheating their revenues. Whenever

there is a cost of auditing, fines should be greater than the tax, otherwise, it is not necessary to

differentiate between the two policies. We also show that if the consumers choose a fixed proportion

of their time on requesting receipts the government should not provide any rebate since that policy

will not change their behavior in terms of more auditing. This is rather trivial since consumers are

not affected by tax incentives (fixed proportion of their time on auditing) then tax incentives are

not necessary.

Finally, we characterize the trade-off between the traditional and the consumer’s auditing. We

show that the rate of substitution between both policies, conditional on prices such that govern-

ment’s tax revenue net of auditing costs remains constant, should be equal to marginal rate of

substitution (utility constant) and that the distortion on optimal quantities should be lower the

larger is the effect of the policy on tax revenues comparatively to prices.

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