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Tax Evasion and Auditing

in a Federal Economy*

Sven Stöwhase Christian Traxler

Universitiy of Munich† Universitiy of Munich‡

First Draft – This Version: April 2004

Please do not quote or cite without permission of the Authors.

Abstract

This paper analyzes the relation between tax auditing and fiscal equalization in the

context of fiscal competition. We incorporate a model of tax evasion by firms into a

standard tax competition framework where regional governments use their audit rates

as a strategic instrument to engage in fiscal competition. It is well known that in such a

situation fiscal equalization can mitigate the inefficiencies from tax competition. We

compare the regions choice of audit policies for three different cases: A scenario of

unconfined competition without interregional transfers, a scenario with a gross revenue

equalization scheme and finally, a scenario with net revenue equalization, where not

only the revenues from taxation but also the regions auditing costs are shared. We

show that fiscal competition leads to audit rates which are inefficiently low from the

perspective of total revenue maximization. While in general gross revenue equalization

aggravates the inefficiency, net revenue equalization makes the decentralized choice of

auditing policies more efficient.

JEL-classification code: H26; H71; H77

Keywords: Tax Evasion; Fiscal Competition; Fiscal Equalization; Auditing

* We would like to thank Andreas Haufler for helpful comments and discussions. † Department of Economics, Akademiestr. 1/II, D-80799 Munich, Germany. Email: Sven.Stoewhase@lrz.uni-muenchen.de ‡ Munich Graduate School of Economics, Kaulbachstrasse 45, D-80539 Munich, Germany.

Email: Christian.Traxler@lrz.uni-muenchen.de, Phone/Fax: +49 89 2180 -3303/-6296; Corresponding Author.

1

1. Introduction

Much of the work in the theory and empirics of taxation takes tax collection as given

or costlessly executed, effectively assuming that tax authorities have full information

about individuals or firms’ tax liabilities. Of course, this is not very realistic. In fact,

we observe significant levels of tax evasion in almost all developed countries.1 In

order to fight evasion, tax authorities have to spend some resources on auditing. If

revenue maximization would be the primary objective, a tax planner would choose

expenditures on auditing such that the marginal gains (less tax evasion, higher

revenues from taxation and detected evasion) equal the marginal resource costs from

an increase in tax collection efforts.2

However, there is some evidence that tax authorities spend less on tax

enforcement than what would be optimal. Lenk et al. (1998) as well as a report by

the Arbeitnehmerkammer Bremen (2001) argue that some regions in Germany put

too little effort into tax investigation. In Belgium, there repeatedly were public

discussions which claimed that the Flemish region are too lax in their tax

enforcement policy. Anecdotal evidence comes from income tax evasion in Germany:

In 1998, German tax authorities inspected hundreds of banks which were under

suspicion to support tax evasion by transferring non declared income of their

customers to bank accounts abroad. Although these inspections were highly successful

in detecting cases of tax evasion, they ended in a disaster. In fact, due to the low

number of tax investigators, authorities could only examine a very small fraction of

all cases in detail. As a result, most evaders escaped punishment and a considerable

amount of revenue was lost for the government. Clearly, one could have prevented

such a situation by spending more on tax collection, i.e. by increasing the number of

tax investigators. If we take this evidence as indicating an inefficiently low level of

spending on auditing, we have to ask for the reasons that lead to this inefficiency.

1 According to estimations of Schneider and Ernste (2000) the informal sector accounts for

approximately 10 to 30 percent of total GDP in OECD countries and for more than 50 percent in

some less developed countries. If we take this as a proxy for the degree of tax evasion, these figures

emphasize how important it is to allow for an imperfect tax collection system. 2 This is the point made in the early literature on tax evasion, see for example Kolm (1973). For the

case of a welfare maximizing government, Slemrod and Yitzhaki (1987) argue that the optimal level of

auditing lies below this point.

2

For the German case, there are at least two alternative explanations which both

originate from the federal structure of the country.3

One is tax competition. At first glance, tax competition between German regions

(Länder) seems to be negligible. The statutory tax rate is determined on the national

level by the federal government and, unlike as in other federal countries such as

Switzerland, regions have no direct instrument for taxation (e.g. region specific

surtaxes). On the other hand, while taxing power lies in the hands of the federal

government, tax collection and thus the choice of the auditing policy is in commission

of the regional governments. In the presence of tax evasion, however, tax revenues

are not determined exclusively by the statutory tax rate set at the federal level, but

also by the enforcement policy set at the regional level. By choosing the auditing

probability, each region can determine its effective tax rate. Accordingly, the audit

policy becomes an alternative strategic tool for tax competition and regions might

compete via their effective rather than their statutory tax rate.4 As has been shown

by Cremer and Gahvari (2000), countries will then end up with less than optimal

audit rates, even if (statutory) tax rates are harmonized.

An alternative explanation for the rather low audit rates in Germany can be

found in the fiscal equalization scheme. In the current German interstate transfer

system (Länderfinanzausgleich), redistribution is based on per-capita gross revenues

rather than on per-capita net revenue (tax revenues net of tax enforcement costs).

This implies an asymmetric treatment of auditing costs on the one hand and tax

revenues on the other. While additional tax revenue that emerges from a higher audit

rate in one region has to be shared with all other regions via the fiscal equalization

scheme, the costs for an increase in the audit rate have to be fully born by the region

itself and cannot be deducted from the tax base. Considering this fact, it is

straightforward to argue that the current German fiscal equalization scheme distorts

the regions incentives, causing inefficiently low expenditures on tax collection and

hence an audit frequency that is too low. In the presence of tax competition,

however, a fiscal equalization scheme can mitigate harmful competition. As has been

shown by Köthenbürger (2002), fiscal equalization may induce the regions to

internalize their fiscal externalities. If this is the case, fiscal equalization could lead to

3 To some extent, our arguments can also be applied on other federal countries such as Canada, Spain,

Belgium and some Latin American Countries. 4 Keen and Marchand (1997) and Sinn (2003, Chapter 2) describe a similar problem where

expenditures on public infrastructure can be used as a strategic substitute for taxes.

3

higher tax rates which are an equivalent to higher audit frequencies in our case. As a

result, it is not clear whether the current German fiscal equalization scheme tends to

decrease or increase audit rates as compared to the case without any fiscal

equalization.

As has been outlined above, the choice of the audit rate may be affected by tax

competition and by the design of the fiscal equalization scheme. While these aspects

have been discussed separately in the literature, the aim of the present paper is to

combine both aspects in a single framework and to analyze how their interaction

affects the choice of the audit policy. We use a modified version of the standard

model of tax evasion by the firm (e.g., Cremer and Gahvari, 1993) and incorporate it

into a typical tax competition setting. In each region of a federal economy a

representative firm uses a fixed factor and mobile capital to produce a consumption

good. Firms have to pay a tax on capital and decide on how much of the taxes to

evade. In making these decisions, each firm takes the evasion costs as well as the tax

rate and the auditing probability into account. Local tax authorities choose their

audit rates (considering its auditing costs and the impact on the evasion level) in

order to maximize net revenues. To keep the analysis tractable, we take the central

government policy, especially the statutory tax rate, as exogenously given.

For this setting, we compare the choice of region specific audit rates for three

different cases. The first scenario describes a situation without any fiscal equalization

scheme where the choice of the audit rate is only affected by fiscal competition. We

show that this will result in inefficient low levels of auditing, which resembles the

classical tax competition result (Zodrow and Mieszkowski, 1986) as well as the

findings of Cremer and Gahvari (2000). In scenario two we introduce a system of

gross revenue equalization, which captures the central features of the current German

transfer mechanism (the Länderfinanzausgleich). Finally, scenario three describes an

alternative fiscal equalization scheme under which redistribution is based on a net

revenue concept, where not only the revenues from taxation but also the regions

auditing costs get shared. We show that, in general, a system of gross revenue

equalization leads to even lower spending on tax enforcement than the case of

unconfined competition. However, we can define some conditions under which the

positive effect from the gross revenue equalization – the internalization of fiscal

externalities – dominate the distortion due to the asymmetric treatment of revenues

and auditing costs. Under these conditions, gross revenue equalization will lead to

higher audit rates than under scenario one. We further show, that – maintaining the

4

degree of interregional redistribution – a system of net revenue equalization would

unambiguously increase audit rates in comparison to the case of gross revenue

equalization.

The paper is organised as follows: Section 2 presents the basic model. Section 3

provides us with the benchmark scenarios where tax authorities maximize their

revenue in the absence of a fiscal equalization scheme. Next, we analyze the results

under the two different designs of a fiscal equalization scheme and compare the

choice of the audit rate for several cases. In Section 5 we sum up and discuss our

results. Section 6 concludes with some policy implications.

2. Basic Model

In this section we characterize the private sector of the model. After the central and

regional government set the tax rate and the local enforcement policy (as discussed in

the following sections), firms and consumers make their decisions, taking the policy

variables as given.5

2.1 Firms

Consider an economy with n regions, each inhabited by a single representative

household. In a perfectly competitive industry firms produce one homogenous private

good (numeraire). The production process in each region i uses perfectly mobile

capital ik and a fixed, immobile factor. To simplify the analysis, we assume that all

regions are endowed with the same amount of the fixed factor, normalizing it to

unity. This can also be interpreted as only considering regions of the same size. The

technology is allowed to differ between regions and is represented by a standard

neoclassical production function ( )i if k , where the fixed factor is suppressed.

Firms have to pay a unit tax on capital at a rate t . This statutory tax rate is

equal for the whole economy. However, each firm can try to evade taxes by

concealing a share ie of the capital employed. To conceal inputs requires the use of

resources by the firm. Following the literature, we assume that the costs of evasion

are convex in ie and linear in the tax base:6 ( )i ig e k with 0g ′ > and 0g ′′ > . With a

5 Hence we omit commitment problems. For a game theoretical analysis with no commitment see

Reinganum and Wilde (1986). 6 Assuming that tax evasion costs increase linearly in the tax base simplifies the analysis a lot. With

this specification the firms’ evasion decision will be independent of the amount of capital employed

5

probability ip the evasion gets detected, and the firm has to pay the statutory taxes

plus a fine that is proportional to the taxes evaded (compare Yitzhaki, 1974). The

penalty rate, denoted by 1s − , is equal for all regions. With probability 1 ip− the

firm gets away with the evasion and pays only taxes on the declared amount of

capital. Expected profits eiπ are defined as

( )( ) ( )( )( ) ( ) 1 1 1ei i i i i i i i i i i if k rk g e k p tk s e tk p e tkπ = − − − + − − − − ,

where r is the factor price for capital. We can simplify this expression to

( )( ) ( )e ei i i i i if k r g e t kπ = − + + with ( )1e

i i i it t e e p s≡ − + , (1)

where eit is the expected or effective tax rate7 in region i.

A risk neutral firm now chooses ik and ie to maximize its expected profit. The

first order conditions are given by

( ) ( ) ei i i if k r g e t′ = + + , (2)

( )( ) 1i ig e p s t′ = − . (3)

Equation (3) defines a firm’s optimal evasion level. For the rest of the paper, we will

assume that there is an interior solution with evasion in equilibrium (i.e., 1ip s < has

to hold). From these equations one can easily derive two basic results. First, firms

will conceal more if the statutory tax rate increases or the detection probability

decreases – a standard result for models of firm tax evasion. Second, an increase in the

audit rate will raise (per unit) capital costs and hence decrease capital demand. This is

analogous to the effect of a tax increase on mobile capital in tax competition models.8

2.2 Capital Market

In each region i a representative household is endowed with capital ik (and the

immobile factor). The total capital supply to the economy k is fixed and market

clearing requires

1 1

n n

i ii i

k k k= =

= =∑ ∑ . (4)

(compare equation 3). Our qualitative results derived in the following hold also under less restrictive

assumptions. 7 Note that revenues from detected evasion (including penalties) are also included in the definition of

the effective tax rate. 8 These results are derived in Lemma 1 in the Appendix.

6

Individuals invest their capital in a large number of firms distributed over the whole

economy. By holding a fully diversified portfolio, they avoid the potential risks

associated with tax evasion by firms. In the capital market equilibrium there will be a

capital allocation which equates the net return, so the arbitrage condition

( ) ( ) ei i i if k g e t r′ − − = (5)

has to be fulfilled for all regions i.

3. Regional Governments’ Policy without Fiscal Equalization

In this and the subsequent sections we analyse the public sector’s decision making.

As a benchmark scenario we consider a federal economy without any interregional

redistribution. Our analysis concentrates on the decision problem of a typical regional

government which takes the policy variables set by the central government, i.e. the

tax and penalty rate as well as the redistribution parameter (introduced in the next

section), as exogenously given. We assume revenue maximizing regional governments

(or tax authorities), which is standard in the literature on tax evasion.9 This can be

justified either by a ‘Leviathan’ government or by a welfare maximizing government

in the case of consumers which receive significantly higher marginal utilities from

public than from private consumption (see eg. Kanbur and Keen, 1993).

The only policy variable controlled by the regional government is the audit rate,

which determines the capital allocation and hence the regions’ revenues from taxes

and penalties. To provide a certain audit frequency the region has to incur some

costs. These costs are convex in the detection probability and do also depend on the

size of the tax base, i.e. the level of firms’ capital inputs.10 Analogous to the evasion

costs of the firm we assume the government’s auditing costs to be linear in the

capital base11. The total detection costs of a region are then defined as ( )i ic p k , where

9 In Section 5 we discuss the case of a welfare maximizing governments. 10 In the presence of a planner who perfectly knows the size of the tax base, it is not clear at first sight,

why tax evasion should be possible. However, if there are many producers and the tax authority does

not know the exact distribution of the capital among the firms, tax evasion is a possible outcome. 11 One could easily think of a detection technology characterized by marginal auditing costs which are

increasing (decreasing) in the tax base. This would tend to lower (raise) the detection probability in

regions with a high level of capital employed. However, non-linearity would not affect our results in a

qualitative way.

7

c(pi) denotes the auditing costs per unit of capital (for a given pi), with 0c ′ > ,

0c′′ > and (0) 0c = . In the absence of an interregional distribution mechanism, the

net revenue of region i is given by

( )( )ei i ik t c p− , (7)

with the effective tax rate eit as defined above.

3.1 Closed Regions

Let us briefly consider the case of immobile capital. The revenue would be given by

(7) with i ik k= and a revenue maximizing planner would set a detection probability

according to the condition

( )ei

ii

t c pp∂ ′=∂ , (8)

where ei it p∂ ∂ is derived in Lemma 2 in the Appendix.12 The LHS of (8) depicts the

marginal benefits (per unit of capital) of a higher audit rate: Spending more effort on

auditing will increase effective taxation and revenues since a smaller share of the tax

base will be concealed. We can state that, in the absence of capital mobility, each

region sets its audit rate to equate marginal benefits per unit of capital and marginal

auditing costs per unit of capital.13

3.2 Open Regions

Let us now allow for capital mobility between regions. While the statutory tax rate in

the economy is ‘harmonized’ and hence there is no scope for standard tax competition

between regions, the detection policy acts as a strategic substitute for the regional

policymaker:14 By reducing its auditing probability, a region can lower the effective

tax rate. This will reduce capital costs and attract mobile capital from other regions.

Taking the capital market responses into account and considering the policy of the

other regions as fixed, the first order condition then becomes

12 The second order condition is fulfilled since 0c ′′ > and e

it is concave in pi (shown in Lemma 2 in

the Appendix). For the rest of the paper, second order conditions are discussed in the Appendix A.1. 13 Condition (8) is independent of the amount of capital, since capital supply is fixed in the closed

economy. Clearly this independence is also driven by the assumption that firms evasion costs as well

as total auditing costs are linear in ki . 14 The argument is comparable to the finding that competition via public expenditures (e.g. on

infrastructure) can serve as a substitute for competition in tax rates. See Keen and Marchand (1997),

and Sinn (2003, Chapter 2).

8

( )( ) ( )ei ie

i i i i ii i

t kk t c p c p kp p

∂ ∂ ′+ − =∂ ∂

. (9)

In order to ease notation and ensure comparability with the results derived in the

next sections, we rearrange condition (9) and get

ei ie

i i ii i

t kk t MCp p

∂ ∂+ =∂ ∂

with ( ) ( ) ii i i i

i

kMC c p k c pp

∂′≡ +∂

, (10)

where iMC denotes the ‘extended’ marginal costs.15 Note that ik is a function of the

detection policies of all regions, ( , )i i ik p p− , where the vector ip− is defined as

( )1 1 1,..., , ,...,i i i np p p p p− − += . The first order conditions derived for the

uncoordinated choice then determine a system of reaction functions ( )i i ip R p−= and

the equilibrium of the Cournot-Nash game is given by the intersection of these

reaction functions.16

Compared to the case of a closed economy, we now find the total marginal costs

on the RHS of condition (9). The marginal benefit on the LHS consists of two effects:

The marginal increase in the effective tax rate (weighted with the amount of capital

employed) and the marginal capital outflow which follows from an increase in pi

(weighted with the net revenue per unit of capital). While these capital outflows

clearly lower the marginal benefit of the region, they enlarge the foreign tax base and,

ceteris paribus, raise net revenues in the rest of the economy. Since regional decision

makers ignore these well known fiscal externalities, they set detection probabilities at

an ‘inefficiently low’ level from the perspective of the whole economy. This will, in

turn, lead to ‘inefficiently high’ levels of tax evasion by the firms.

Before we summarize these results, we want to clarify the concept of inefficiency

in our framework with revenue maximizing governments:

Definition 1. (i) Detection probabilities are set inefficiently if a central planner could

increase total net revenues in the economy through a change in the audit rate(s) (in

one or more regions). (ii) Otherwise, detection probabilities are set efficiently. (iii)

The firms’ evasion rates are called (in)efficient if they are associated with an

(in)efficient detection probability.

15 Note that we incorporate a partial effect from the capital outflow – the reduction in total auditing

costs due to a decrease in the tax base – into the definition of iMC . 16 In section A.2 in the Appendix we show that for all scenarios considered in the paper, the regions’

audit rates are strategic complements.

9

We neglect standard welfare aspects in this definition. Hence, a ‘more efficient’ audit

rate just means that this policy will induce higher total net revenues. Moreover, we

call a firm’s evasion level inefficient if it is associated with an inefficient detection

rate (albeit firms always choose an evasion rate which is optimal for them, i.e. profit

maximizing; Compare condition (3)). Using Definition 1 we can state

Proposition 1. In an economy with open regions and without interregional

redistribution, the uncoordinated choice of detection probabilities by revenue

maximizing governments will lead to an inefficiently low level of detection efforts and,

therefore, to an inefficiently high level of tax evasion.

Proposition 1 is similar to the classical ‘underprovision result’ in tax competition (see

Zodrow and Mieszkowsky, 1986) where the uncoordinated choice of tax rates leads to

inefficiently low levels of taxation. Since in our model the statutory tax rate is

‘harmonized’, it is the detection rate which serves the regions as strategic substitute.

In fact, we observe tax competition, but with an alternative instrument. This result

resembles the findings by Cremer and Gahvari (2000) who study tax competition

with audit probabilities as an alternative instrument. They show that competition in

tax and audit rates leads to an underprovision of public goods. In addition, they

demonstrate that harmonizing tax rates alone will not solve the underprovision

problem. Barred from competing in the statutory tax rate, regions lower their

effective tax rates through cutting audit rates. As a result, tax evasion increases.

Proposition 1 describes our benchmark result which we will compare to the

regions’ uncoordinated choice in the presence of two different fiscal equalization

mechanisms. Before we come to these cases in section 4, let us take a look at the

centralized solution to the problem.

3.3 Centralized Auditing Policy

Suppose that a central planner would choose a detection policy for each region in

order to maximize the sum of all revenues,

( )1,..., 1max ( )

n

ne

i i ip p ik t c p

=−∑ .

The n first order conditions are given by

( ) ( )( ) ( ) ( ) 0e

ji i e ei i i i j j

i i ij i

kt kk c p t c p t c pp p p≠

∂∂ ∂ ′ − + − + − = ∂ ∂ ∂∑ . (11)

10

These conditions then characterize a set of efficient audit rates for the economy.

While the first and second term also show up in equation (9), the last term represents

the revenue spillovers created by a region’s detection policy.17 For the case of

identical regions, the second and third term in condition (11) cancel out and we get

( )ei

ii

t c pp∂ ′=∂ , (12)

which is equivalent to the first order condition in the case of a closed economy.

Moreover, one can show that even with regional differences in production

technologies, this is the only solution which satisfies (11).18 The intuition for this

result is clear: Since the problem of maximizing the net revenue can be solved in ‘per

unit of capital’-terms, heterogeneity with respect to production technologies – which

basically leads to an asymmetric capital allocation – does not alter the result derived

under perfect symmetry.

In addition, this shows that there is a unique efficient audit rate for the whole

economy, even in the case of different production technologies. However, one cannot

interpret this as a strong point in favour of a central planner implementing a unique

efficient auditing policy, since the result is driven by two simplifying assumptions:

First, we assume the firms’ evasion costs as well as the auditing costs of the regions

to be linear in the tax base.19 Second, we also make the assumption that auditing

costs per unit of capital are identical across regions. (One can easily see from (12)

that heterogeneous costs would result in different audit rates.)

4. Regional Governments’ Policy with Fiscal Equalization

In this section, we introduce interregional redistribution systems. We analyse two

different fiscal equalization schemes: Gross revenue equalization and net revenue

equalization.20

17 Note that condition (11) corresponds to the case where one would provide each region with a

corrective subsidy in order to internalize its fiscal externalities. This replicates the result of Wildasin

(1989) for our framework of fiscal competition in audit rates. 18 A prove is given in section A.3 in the Appendix. 19 Therefore, neither the evasion decision by firms nor the regions’ choice of a detection policy is

affected by the amount of capital employed. Compare Lemma 1 in the Appendix and condition (12). 20 In a comparable framework, Köthenbürger (2002) studies tax competition under tax revenue as well

as tax base equalization.

11

4.1 Gross Revenue Equalization

Consider the following simple redistribution mechanism: Each of the n regions

contributes a share α ( 0 1α< ≤ ) of its gross revenues ei it k to the redistribution

system and receives a share 1n of the total revenues distributed, ej jt kα∑ .21 This

fiscal equalization scheme captures a central feature of the current German interstate

transfer system (Länderfinanzausgleich): While revenues obtained under a given

detection policy have to be shared with all the other regions, the costs for

maintaining a certain audit rate have to be fully borne by the region itself. The net

revenue of a region is then given by

( )1

1 ( )n

e ei i i i j j

jt k c p k k t

nαα=

− − + ∑ . (13)

The first order condition for the uncoordinated choice of a revenue maximizing

regional government becomes

( )1

1ne e

ji i ie ei i j i i

i i i ij

kt k tk t t k MCp p n p p

αα=

∂∂ ∂ ∂ − + + + = ∂ ∂ ∂ ∂ ∑ (14)

with iMC as defined above. The first term on the LHS depicts the distortion

introduced by the fiscal equalization scheme, whereas the second term shows that a

part of the fiscal spillovers will be internalized via the redistribution mechanism. For

a proper analysis, we have to distinguish between an economy with many small or

few large open regions.

A) Small open regions

In the case of many small regions (n → ∞ ), the impact of a single region’s detection

policy on the tax base in the rest of the economy becomes negligible. Hence, the

second term on the LHS of (14) vanishes and the first order condition becomes

( )1ei ie

i i ii i

t kk t MCp p

α ∂ ∂ − + = ∂ ∂. (14a)

If we compare (14a) with condition (10), the result in the scenario without any

interregional redistribution, one can easily see that the gross revenue equalization

mechanism distorts the region’s choice of detection efforts. While the marginal costs

iMC are unaffected, the redistribution system clearly reduces the marginal gains

from an increase in the audit rate. This is the case, since fiscal equalization leads to

21 Apparently the mechanism is budget balancing for any α .

12

an implicit taxation of a region’s gross revenue. Hence, regions will unambiguously

choose a lower audit frequency than in the absence of fiscal equalization.

Furthermore, it is straightforward to show that the audit rate will decrease as α

increases.

Proposition 2. (i) Under a system of gross revenue equalization, a small open region

maximizing its net revenue will choose a detection probability which is even lower than

in the case without fiscal equalization. (ii) The detection probability is decreasing in α .

This result is equivalent to Proposition 3 in Köthenbürger (2002). As in his analysis,

the inefficiency due to the distorted incentives adds to the inefficiency due to fiscal

competition: Detection efforts will be reduced further below the efficient level and the

amount of evasion will be higher than in our reference case without any interregional

redistribution system.

B) Large open regions

Since the decisionmaker of a large region takes into account the impact of the local

detection policy on the total revenues distributed, we can rearrange condition (14)

and get

( )( )11 1e

ji ie ei i j i

i i ij i

kt kk t t MCn p p n p

αα≠

∂∂ ∂ − − + + = ∂ ∂ ∂∑ . (14b)

As in the case of small open regions, gross revenue equalization introduces a

distortion. However, since large regions consider the strategic effect of their policies –

the impact of their audit rate on total revenues redistributed – the distortion becomes

smaller: ( )1 1 1 1nα α− − > − . Stated differently, for large regions the implicit

taxation of gross revenues drops below the rate α observed for small regions.22

Furthermore, as reflected in the second term on the LHS of condition (14b), a part of

the fiscal externalities23 gets internalized via the redistribution mechanism. Again,

this is due to the fact that large regions – in contrast to small ones – take into

account the strategic effect of their policies.

If we compare this case with the benchmark scenario, the introduction of the

gross revenue equalization scheme shows an ambiguous effect on the choice of audit

22 This result, which prevails in a similar way in Köthenbürger (2002), is also confirmed in an

empirical study for Germany by Baretti et al. (2002).

23 Note that the total spillover would be ( )1

( )j ej j

ij

kt c p

p≠

∂−

∂∑ .

13

rates. While the implicit taxation of gross revenues tends to lower auditing efforts,

the internalization of the spillovers will act in the other direction. Comparing

condition (14b) and (10), we can find a threshold level such that, for

ˆ 1

jej

ij ii e

i iei i

i i

kt

pn n t kk t

p p

≠

∂∂

< ≡ + ∂ ∂+∂ ∂

∑, (15)

the second effect dominates. We sum up these results in

Proposition 3. Suppose a revenue maximizing regional government under gross

revenue equalization. (i) For every large open region there exists a threshold in such

that for given auditing policies ip− in the other regions and in n< ( in n> ), the

detection probability chosen under a system of gross revenue equalization will be

higher (lower) than in the case without fiscal equalization. (ii) For in n< ( in n> )

the detection probability is increasing (decreasing) in α . (iii) The detection

probability is decreasing in n. (iv) If minˆn n≤ , with { }min 1minˆ ˆ ˆ,..., nn n n≡ , the

equilibrium detection probabilities under a system of gross revenue equalization will be

unambiguously higher than in the case without fiscal equalization.

While gross revenue equalization has an unambiguously negative effect in the case of

small regions (see Proposition 2), it could increase audit rates for large open regions.24

This result is to some extent surprising. At first glance, a gross revenue equalization

scheme seems to lead to a clear inefficiency: The asymmetric treatment of auditing

costs on the one hand, and tax revenues on the other, unambiguously distorts a

region’s choice of the auditing effort. However, in the presence of fiscal competition,

the redistribution system has a further effect: It works as a ‘corrective subsidy’ and

induces (large) regions to internalize part of the fiscal externalities. While the distortion

from the implicit taxation tends to lower audit rates, the corrective subsidy works in

the other direction. If there are in n< jurisdictions, the second effect dominates the

first one and gross revenue equalization makes the decentralized choice more efficient,

compared to the benchmark scenario. The intuition for this result is clear-cut: If the

number of regions diminishes, the implicit taxation of gross revenues becomes smaller

and the degree of internalization gets higher - irrespective of α (see condition (14b)).

24 Another issue we do not take up here, is the fact that large regions face a lower capital elasticity

than small regions because of a different impact of their detection policies on the interest rate.

Compare Wildasin (1988), Bucovetsky (1991).

14

Bearing this in mind, also the second part of Proposition 3 becomes obvious: If the

gross equalization system induces a region to raise (lower) its auditing effort for the

case of in n< ( in n> ), an increase in α further amplifies the efficiency enhancing

(reducing) effect of the mechanism. The third part of the proposition holds, since

more regions will always increase the distortion of the redistribution and decrease the

degree of internalization. In part (iv) of the proposition we just use the result from (i)

to make a statement about the equilibrium audit rates. It is straightforward that

audit rates must increase if all reaction functions shift ‘upward’. However, since

minˆn n≤ is only a sufficient but not a necessary condition, there is scope for an

increase in audit rates, even if the condition is not fulfilled.

4.2 Net Revenue Equalization

Let us now introduce an alternative system of interregional redistribution. Instead of

gross revenue equalization, we consider a mechanism which is based on net revenue

equalization. Each region contributes a share 0 1α< ≤ of its net revenues – tax

revenues net of auditing costs – and receives a share 1n of the total revenues

distributed, ( )( )ej j jk t c pα −∑ . With this mechanism, the revenue of a region

becomes

( ) ( ) ( )1

1 ( ) ( )n

e ei i i j j j

jk t c p k t c p

nαα=

− − + −∑ . (16)

The first order condition is given by

( ) ( )1

1 ( ) ( ) 0ne e

ji i ie ei i i j j i i

i i i ij

kt k tk t MC t c p k c pp p n p p

αα=

∂∂ ∂ ∂ ′ − + − + − + − = ∂ ∂ ∂ ∂ ∑ . (17)

As before, we discuss this condition separately for the cases of small and large open

regions.

A) Small open regions

As under gross revenue equalization the second term on the LHS of condition (17)

vanishes for the case of many small regions. We can rewrite (17) as

ei ie

i i ii i

t kk t MCp p

∂ ∂+ =∂ ∂

, (17a)

which is identical to condition (10) in the benchmark scenario. While gross revenue

equalization distorts the auditing policy of small regions (Proposition 2), the

distortion disappears under net revenue equalization. We can state

15

Proposition 4. Under a system of net revenue equalization with 1α < , a small open

region maximizing its net revenue will choose the same audit rate as in the benchmark

scenario without fiscal equalization.

The intuition for this result is clear: In contrast to the scenario of gross revenue

equalization, there is no asymmetric treatment of revenues and auditing costs under

net revenue sharing. Therefore, revenue maximizing regions would unambiguously

choose a higher audit rate after the change from gross to net revenue equalization.

The only inefficiency remaining arises from the competition for the mobile tax base.

As in the benchmark scenario, auditing efforts will be inefficiently low and the

evasion level of the firms will be inefficiently high.

An interesting point to note is that Proposition 4 holds true for any 1α < . Since

interregional redistribution does not introduce any distortion, there is no equity-

efficiency trade off in the choice of α .

B) Large open Regions

As before, the government of a large open region considers the impact of its auditing

policy on the tax base in the rest of the economy. We can rearrange condition (17)

and get

( )( )e

ji ie ei i j i i

i i ij i

kt kk t t c p MCp p p

β≠

∂∂ ∂+ + − =∂ ∂ ∂∑ with

( )1nαβ α α≡

− +. (17b)

As in the case of small open regions discussed above, redistribution via net revenue

equalization scheme does not distort the decentralized choice of detection

probabilities. An inefficiency due to the fiscal competition between regions still

remains. However, since the decisionmaker of a large open region incorporates the

strategic effect of its auditing policy, a fraction β of the fiscal externality gets

internalized. If we contrast the impact of gross revenue equalization with that of net

revenue equalization for large regions, the latter system introduces no distortion but

a higher corrective subsidy.25 Hence, we can state

25 Although nαβ > the comparison with the case of gross revenue equalization (equation (14b)) is

not that clear, since the redistribution volume is smaller in the case of net revenue equalization.

Nevertheless we will argue, that (for a given α ) the degree of the externality which gets internalized is

higher with net than with gross revenue sharing. This will hold if auditing costs c(pi) per unit are

rather small relative to the per unit effective tax rate.

16

Proposition 5. (i) Under a system of net revenue equalization, a large open region

maximizing its revenue will set an audit rate which is higher than the rate chosen

under a gross revenue equalization mechanism, and (ii) also higher than the rate

chosen in the benchmark scenario without any fiscal equalization. (iii) Furthermore,

the detection probability is increasing in α , and (iv) for 1α = , the auditing policy is

set efficiently. (v) The detection probability is decreasing in n.

Proposition 5 is the main result of our analysis: For large open regions, a net revenue

equalization scheme (partly) internalizes fiscal externalities without introducing any

distortion. Hence, under this mechanism, the decentralized choice of detection policies

will unambiguously result in more efficient (i.e. higher) auditing efforts as compared

to the choice under both, the case of a gross revenue equalization scheme and the

case without any interregional redistribution. Incorporating auditing costs into the

redistribution mechanism will, if at all, only slightly reduce the amount

redistributed26 (for a given α ) but increase detection probabilities. Firms would

evade less and tax revenues would increase. Since a higher degree of redistribution

does not induce any distortion but increases the degree of internalization, auditing

efforts will increase as α increases. Hence, as in Proposition 3 (ii), there is no equity-

efficiency trade-off. For the limit case of 1α = , the net revenue sharing rule provides

a mechanism to implement an efficient solution (according to our notion of efficiency

from Definition 1). Part (v) of the proposition is straightforward and follows the

intuition of Proposition 3 (iii), discussed above.

5. Discussion

Let us briefly sum up the results presented so far. In our benchmark scenario, we

showed that regions choose inefficiently low auditing efforts, in order to attract

mobile capital. While in such a setting fiscal equalization can mitigate this

inefficiency, this is not quite true in the case of a gross revenue equalization scheme.

Here, the asymmetric treatment of tax revenues and costs creates an additional

inefficiency. However, if we switch to net revenue equalization, i.e., if we incorporate

the auditing costs into the sharing rule, this will remove the distortion inherent to

the gross sharing system. Since net revenue equalization also provides for a

26 We assume, realistically, that ( )e

i it c p .

17

mechanism to incorporate (part of) the fiscal externality, it dominates both, the case

without fiscal equalization as well as the case of gross revenue sharing.

By now, there are two important questions we have not addressed. First, we want

to ask if our results extends for the case of welfare maximizing governments. This is

equivalent to the question whether or not our definition of ‘efficiency’ (Definition 1)

is misleading. Second, one might ask if the central government can use its policy

tools, especially the statutory tax rate, to counterbalance the regions’ inefficiently low

detection probabilities (i.e., the inefficiently low effective tax rates).

5.1 Welfare Maximization27

One can easily extend the model presented in this paper to a framework for a

standard welfare analysis. We add consumers, who receive income from their

endowments of capital and the fixed factor and who derive utility from private

consumption as well as from a regional public good. A welfare maximizing regional

planner would then choose an audit rate (or equivalently, a level of the public good)

that maximizes the utility of the representative consumer.

The most important difference to the case of revenue maximization is the

appearance of a further externality. A welfare maximizing planner will now also

consider the pecuniary effects of its detection policy: a higher audit rate will reduce

the capital income as well as the fixed factor income of the regions consumer. The

latter effect arises, since a capital outflow will lower the marginal productivity of the

fixed factor. However, the loss of capital in one region will lead to an inflow of capital

and therefore to higher fixed factor incomes in the rest of the economy. On the other

hand, all capital owners, domestic as well as foreign, will receive lower income from

capital since an increase in audit rates will decrease the interest rate. Since a regional

planner does not take into account the impact on factor incomes in the rest of the

economy, there is a pecuniary externality (which can be either positive or negative).

Hence, in the case of unconfined fiscal competition, there are now two externalities –

a fiscal and a pecuniary spillover – which render the decentralized choice of audit

rates inefficient (in terms of welfare maximization).

How does the introduction of gross or net revenue equalization affect efficiency in

this case? As discussed above, revenue equalization provides a mechanism which

makes the decentralized planner internalize (a part of) the fiscal externalities. This

27 In an extension to this paper (available upon request) we provide for a formal model with regional

governments maximizing welfare instead of revenues.

18

also holds for the welfare maximizing planner. The pecuniary externality, however,

will not get internalized via an interregional redistribution mechanism. It is not

possible to reach the first best solution. Nevertheless, our main result of section 4

holds in the welfare maximization framework: the decentralized choice under the

system of net revenue equalization will lead to more efficient auditing policies than

under gross revenue equalization.28 Again, this is due to the asymmetric treatment of

tax revenues and auditing costs inherent in the gross revenue sharing scheme.

5.2 Central Government Policy Tools

Since we do not model the decision of a central government, we can not quantify

whether or not an inefficiently low effective tax rate will prevail. The central

government might counterbalance inefficiently low detection probabilities by raising

the statutory tax rate. However, a comparative static analysis shows that this policy

is not necessarily feasible: applying the implicit function theorem on the first order

conditions of the different scenarios, we get

0ipt

∂∂

for all the cases considered.29 Hence, an increase in the statutory tax rate might

further reduce auditing efforts.

The other policy instrument available to the central government is the penalty

rate. Since punishment is costless and enforces a lower level of evasion, a central

planner could set s → ∞ and there would be no evasion at all (see Kolm, 1973).

However, strong penalties are probably not feasible: If a firm would go bankrupt

because of a very severe punishment, the penalty may not be credible. Keeping this

restriction in mind and assuming that s is fixed at some credible finite rate (where

1ps < holds) appears plausible.

6. Conclusion

In this paper, we have analyzed the decentralized choice of audit rates, assuming that

regional planners maximize their tax revenues. While the statutory tax rate, modeled

as a source tax on capital, is exogenously fixed by the central government, regions

28 In the extension we show under which conditions the net revenue equalization will lead to more

efficient audit rates compared to the case without any interregional redistribution. 29 The comparative static results are derived in section A.4 in the Appendix.

19

decide on how much to spend on tax auditing. By setting a certain detection

probability, regions determine the effective tax rate firms have to pay. Thus, regional

auditing policies determine the allocation of capital as well as the extent of tax

evasion by firms. Net revenues are given by the revenues from taxation plus penalties

from detected evasion minus the costs to provide the chosen audit rate.

As a benchmark result we show that fiscal competition will lead to detection

probabilities which are inefficiently low from the perspective of total revenue

maximization. This inefficiency is created by the fact that regional auditing policies

generate (positive) fiscal externalities. In such a framework, a fiscal equalization

scheme has an efficiency enhancing potential, since interregional redistribution

provides a mechanism to internalize the fiscal externality (see Köthenbürger, 2002).

However, a system of gross revenue equalization, which makes regions bear the full

auditing costs while tax revenues will be shared, also introduces a distortion. The

asymmetric treatment of enforcement costs and gross revenues drives a wedge

between marginal benefits and marginal costs of an increase in the detection rates.

This tends to lower audit rates. For the case of many small regions, only the

distortionary effect remains. Hence, the gross revenue equalization mechanism will

lead to even lower levels of tax enforcement than in the benchmark scenario without

fiscal equalization. In the case of large regions, the effect from the internalization and

the distortion work in opposite directions. If the number of regions in the economy is

below a certain threshold, the efficiency enhancing effect will dominate, and regions

will choose a higher audit rate than in the benchmark case.

Next, we introduce an alternative interregional redistribution system: net revenue

equalization. Compared to the gross revenue sharing, there is no distortion since

under this new mechanism tax revenues as well as auditing costs get shared. For

small regions this leads to the same result as in the reference case without any fiscal

equalization. However, due to the (partial) internalization of the fiscal externality,

net revenue sharing would unambiguously induce higher detection probabilities for

large regions. Therefore, a switch from a gross to a net revenue equalization system

would make the decentralized choice of audit rates more efficient.

The policy implications of these results are straightforward. A federal government

which, on the one hand, equalizes tax revenues between its regions but, on the other,

imposes the costs of tax collection upon these regions, will face a higher degree of tax

evasion. This may well be the case for Germany, since the current German interstate

transfer system (Länderfinanzausgleich) corresponds to gross revenue equalization.

20

Switching to a net revenue equalization scheme would hardly affect redistribution,

the primary objective of fiscal equalization. At the same time, audit rates would

increase, tax evasion decrease and total net revenues would be higher. Clearly, the

first best solution could be reached through a centralized choice of tax enforcement

policies, as practised in most other federal countries and as recently proposed by the

German Ministry of Finance. However, building up a new institutional framework

will be costly and will take some time. In the light of current budgetary problems, a

switch to a net revenue equalization system could be a more effective policy in the

short run.

Appendix [Preliminary]

Lemma 1

Using the implicit function theorem on equations (2) and (3) and simplify, we get

0i

i i

i

rstek pp g

∂+∂ ∂= <′′∂. (A.1)

0i

i

e stp g

∂ = − <′′∂. (A.2)

1 0i ie p st g

∂ −= >′′∂. (A.3)

Note that 0i

rp

∂ =∂ for small regions.

Lemma 2

The effective tax rate is given by ( )1ei i i it t e e p s≡ − + . We can easily derive

( )1ei i

i ii i

t et p s e sp p

∂ ∂ = − − + ∂ ∂. (A.4)

Using (A.2) from Lemma 1 and 1ip s < we get

( )1 0ei

i ii

t stt p s e sp g

∂ = − + > ′′ ∂. (A.5)

From this we can derive

( )

( )( )

2 2

2 2 1ei i

iii

t st e stgt p s spgp g

′′′∂ ∂ = − + − − + ′′ ∂ ∂ ′′ , (A.6)

21

which is negative if the first order effect dominates.

A.1 Second Order Conditions

(i) SOC for the case of an open economy without any redistribution; FOC is given by

(9).

( )2 2

2 2( ) 2 ( ) ( ) 0e e

i i i iei i i i i

i ii iIII III

k k t tt c p c p k c pp pp p

∂ ∂ ∂ ∂ ′ ′′− + − + − < ∂ ∂∂ ∂ , (A.7)

- From the FOC we see that the term in brackets in II must be positive. Since

0i

i

kp

∂ <∂

(Lemma 1), II is negative.

- From Lemma 2 and 0c ′′ > we know that term III is negative.

- Since the sign of ( )

2

2 2

2 2

i i

i i ii i

i

e r r kst f ste fp p pk p

p f

∂ ∂ ∂ ∂ ′′ ′′′+ − + ∂ ∂ ∂∂ ∂ =∂ ′′

is not clear

(negative, if 0f ′′′ = ), term I could be either positive or negative.

Since I is an effect of second order, SOC < 0 should always hold.

For small regions the partial effects on the interest rate would vanish, but the

analysis would not change.

(ii) …

A.2 Strategic Complements

Using the implicit function theorem on FOC (9), we obtain

( )

2( ) ( )

SOC

A Be

i i iei i i

i j j ii

j

k k tt c p c pp p p pp

p

∂ ∂ ∂ ′− + − ∂ ∂ ∂ ∂∂ = −∂

, (A.8)

where SOC stands for the the LHS of condition (A.7).

Since i

j

kp

∂∂

and the term in the brackets in B must be positive, the whole term is

positive. The sign of ( )

2

2

2

i

i j i ji

i j

r r kf ste fp p p pk

p p f

∂ ∂ ∂ ′′ ′′′− + ∂ ∂ ∂ ∂∂ =∂ ∂ ′′

is ambiguous

(negative, if 0f ′′′ = ), hence term A could be either positive or negative. Since SOC

22

is negative, the RHS in (A.8) is positive, if A is positive or if A is negative and gets

dominated by B. Since A is an effect of second order, 0i

j

pp

∂ >∂

should always hold.

…

A.4 – Comparative Statics

(i) Impact of the statutory tax rate t on pi

Note first, that in an asymmetric equilibrium eeji tt

t t∂∂ ≠

∂ ∂. Therefore any change in

the tax rate statutory will have an impact on the capital allocation.

* The Case without interregional redistribution. Using the implicit function theorem

on condition (9) we can derive

( )2 21 ( ) ( )

SOC

I IVIIIIIe e e

i i i i i i iei i i i

i i i i

p k k t k t tt c p c p kt p t p t t p p t

∂ ∂ ∂ ∂ ∂ ∂ ∂ ′= − − + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

. (A.9)

The sign of term I is ambiguous. (However the term becomes unambiguously negative

if the first order effect in 2

i

i

kp t∂∂ ∂

, i.e. the first term on the RHS of

( )

2 2

21i i i

i i i

k se e r r kst f ste fp t t p t p tf f

∂ ∂ ∂ ∂ ∂ ′′ ′′′ = + + − + ′′ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ′′ dominates.)

The first order effect, depicted in II, is negative. The sign of the term III is

determined by the ambiguous sign of ikt

∂∂ . From Lemma 2 one can easily show

that term IV is positive.

(ii) Impact of the the redistribution parameter α on pi

* The Case with gross revenue sharing. Using the implicit function theorem on

condition (16), we get

1 1SOC

e eji i i ie e

i i j ii i i ij

kp t k tk t t kp p n p pα

+− ∂∂ ∂ ∂ ∂ = − − + + + ∂ ∂ ∂ ∂ ∂

∑ . (A.10)

- For small regions, n → ∞ and the positive term in the brackets vanishes: 0ipα

∂ <∂

.

- For large regions it is straightforward to show, that the term in the brackets

becomes positive (negative), if in n< ( in n> ). Hence ˆ 0ii

pn nα

∂⇒∂

.

23

(iii) Impact of the the number of regions n on pi

* The Case with gross revenue sharing. Using the implicit function theorem on

condition (16), we get for 0α >

21 0

SOC

eji ie

j ii ij

kp tt kn p pn

α− ∂ ∂ ∂ = − − + < ∂ ∂ ∂

∑ . (A.11)

* The Case with net revenue sharing. Using the implicit function theorem on

condition (19), we get

( )1 ( ) ( ) 0SOC

eji ie

j j i ii ij

kp tt c p k c pn n p p

β− ∂ ∂ ∂ ∂ = − − + − < ∂ ∂ ∂ ∂

∑ , (A.12)

since 0nβ∂ <

∂.

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