Accounting for Corruption: The Effect of TaxEvasion and Inflation on Growth

Max GillmanCardiff University Business School

Michal KejakCERGE-EI

PRELIMINARY VERSION

AbstractThe paper studies the relation between taxes and growth when

there is evasion or avoidance of key taxes, on labor and capital in-come, on goods purchases, and on money holdings. The paper modelstax evasion using a decentralized corruption service sector, takes abanking approach, and assumes a production function based on finan-cial intermediary microfoundations for laundering undeclared incomeand sales revenue. This loosens the linkage between tax rate levels andthe size of the shadow economy, as is consistent with correlation facts,while still embodying the well-accepted notion of marginal substitu-tion towards the shadow economy as tax rates increase. The resultsare that taxes decrease growth, evasion decreases the negative effectof taxes on growth, and the growth rate falls at a decreasing rate asindividual tax rates increase. This presents a fiscal principle of theeffect of flat taxes on growth with evasion, based on a rising demandprice sensitivity to higher tax rates.

JEL Classification: E13, E31, H26, O42

Keywords: Tax evasion, corruption, financial services, endogenousgrowth, and inflation.

Preliminary Draft; Incomplete

We thank Szilard Benk for research assistance, Dario Cziraky, Bye Jeong, PatrickMinford, and Slava Vinogradov for comments, and the seminars at CERGE-EI, Prague,WIIW, Vienna, Koc University, Cardiff Business School, and the 2nd CDMA Conferenceat the University of St Andrews, and the Macro and Financial Economics/EconometricsConference at Brunel University. Research support of the World Bank GDN fund at WIIWis kindly acknowledged.

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1 Introduction

Tax evasion through avoidance of the inflation tax can cause the growth rate

to decrease as the inflation rate rises, and to decrease at a decreasing rate

[Gillman and Kejak (2005a,b)]. Evidence tends to support such a nonlin-

ear profile, showing a more negative marginal growth effect the lower is the

inflation rate.1 As the tax rate rises and the shadow price of consumption

rises, the consumer becomes increasingly sensitive to money use, and so in-

creasingly uses credit as a means of tax avoidance to substitute away from

the taxed good. Set within endogenous growth, the tax avoidance via credit

causes the negative growth effect of the tax to be less. In monetary theoretic

terms, as the inflation rate rises, the money demand becomes increasingly

interest elasticity, as in Cagans model, and credit is increasingly used to

avoid inflation, instead of using leisure; this implies that velocity rises at a

faster rate, and the growth rate falls at a decreasing rate.

Tax avoidance is conceptually similar to tax evasion. While using credit

to avoid Baileys (1956) inflation tax is legal, all taxes tend to be avoided

or evaded. This follows from Gary Beckers (1968) study of how legislation

generally is complied with, and in particular, the notion that the marginal

cost of evading the law is set equal to the marginal benefit of the evasion. For

tax laws, the marginal benefit is the tax rate itself. In the case of inflation, the

marginal benefit of avoidance is the nominal interest rate, and the marginal

cost is that of using alternative means of exchange.

The paper applies the law avoidance approach more generally to fiscal

policy, while assuming zero enforcement of the tax laws. It shows that the

monetary concepts of tax avoidance in a growth context apply to major fiscal

taxes of our economic system. In particular we include, along with the infla-

tion tax avoidance, tax evasion of flat taxes on labor income, capital income

1Less controversial than sometimes reported, there appears to be only a dispute overwhat happens below a "threshold" inflation rate, which is found to be at low inflationrates, such as 1% for industrialized countries (Ghosh and Phillips). Insignificant, positive,effects of inflation are found below the threshold when not using instrumental variables.With instrumental variables, Gillman Harris and Matyas (2004) for example show thatthe negative nonlinear profile holds for all positive inflation rates for both developed andless developed samples.

1

and goods purchases. Tax evasion is produced in a competitive fashion in

order to lower the effective tax rates. The outcome is that tax avoidance

or evasion causes increasingly lessor negative growth effects as the tax rates

rise. More broadly it is an example of the consequences of increases in legal

restrictions causing increasingly elastic substitution away from the restricted

activity. More specifically, evasion is based in the financial intermediation

sector and the papers approach is to follow the funds.2

We first show that the model accounts for certain stylized facts of corrup-

tion, and then establish the nonlinear growth profile of the taxes. The next

section sets out the facts, followed by a model consistent with these. Here,

the production functions for avoidance and evasion are based on the Clark

(1984)-Hancock (1985) microeconomics of banking in which financial capital

is a third factor in the CRS production function of bank service output, and

which can be viewed as a general equilibrium formulation of the approach

of Benk and Green (2004). Propositions are set out and illustrated with

the simulations, in subsequent sections, followed by the conclusions on our

stylized banking story.

2 Taxes, Corruption, the Shadow Economy

Tax evasion is a part of the underground, shadow, economy; and evasion

takes place through non-law abiding, or what we call here corrupt, behav-

ior.3 Cash seems to be used more in the underground economy, and this

sector is estimated to be of significant size.4 There has also been found a re-

2This is part of the research agenda proposed by Lucass Nobel address (1996) to findsignificant long run effects of monetary factors such as inflation, combined with the studyof flat taxes in Rebelo and Stokey (1996), and Easterlys (2001) emphasis on the need toinclude the non-market sector to explain the economy in a policy relevant way.

3Corruption can be defined narrowly in terms of public officials taking kickback, asin Schneider and - (2006); they then propose that corruption can act as a substitute toexpanding the size of the shadow economy. Our definition of corruption is more broad, ofillegal activity by anyone in the economy; but we then focus only on tax evasion withinthis broad definition, and evasion and the size of the shadow economy end up movingtogether, more as complements.

4? reports that the shadow output equals 39% of the actual magnitude of reportedGDP in developing countries, 23% in transition countries and 14% in OECD countries;and the labor force, as a percent of the official labor force, is estimated to be about 50%

2

lation between tax levels and the size of the shadow sector, as in ? for a caseof Canadian tax rate changes, and ?, in which an increase in the tax rateinduces the agent to reallocate resources towards the untaxed non-market

sector and away from the market sector. However, standard international

correlation evidence, perhaps counterintuitively, is not consistent with a pos-

itive correlation between personal or corporate tax rates, and the shadow

economy size, as the next section shows.

2.1 Correlation Evidence

Swedens ratings from the Transparency International Corruption Perception

Index (TICPI) indicate it as one of the most transparent country, with a small

shadow economy, even though it has some of the highest tax rates. Russia

with its 13% income tax, one of the lowest personal tax rates, typically

appears in the ratings as one of the least transparent with a large shadow

economy. While possibly outliers, the figures below illustrate that

Fact 1: Tax rates are not positively correlated with the size of the shadow

economy.

Figures 1 and 2 show this for the effective personal income tax rate, in

the OECD, and in the larger sample that includes Latin America, Asian and

transition countries as well as the OECD, and in Figures 3 and 4 for the

corporate tax rate, in the OECD and in the broader sample.5

in developing and transition countries and 17% in OECD countries. See also ? ?, and ?.5The tax data are from The World Competitiveness Yearbook 2003, IMD, International

Institute for Management Development; the TICPI is from Transparency International,http://www.transparency.org/; and the shadow economy size data are from ?.

3

Figure 1. OECD: Shadow Size and Personal Tax Rates

y = -0.0019x + 19.175R2 = 5E-06

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40

Tax Rate, Effective Personal Income (% GDP per capita)(2001)

Shad

ow e

cono

my

size

Figure 2. Full Sample: Personal Tax Rates and Shadow Economy Size

y = -0.3104x + 28.085R2 = 0.0947

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Tax Rate, Effective Personal Income (% GDP per capita)(2001)

Shad

ow e

cono

my

size

Figure 3: OECD Corporate Tax Rates and Shadow Economy Size

y = -0.0742x + 21.513R2 = 0.0054

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60

Tax Rate, Average Corporate (2002)

Shad

ow e

cono

my

size

Figure 4: Full Sample: Corporate Tax Rates and Shadow Economy Size

y = -0.0213x + 23.222R2 = 0.0002

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Tax Rate, Average Corporate (2002)

Shad

ow e

cono

my

size

Rather than a positive correlation between the tax rate and the shadow

economy size, Figures 5 and 6 show that the correlation fact that does emerge

is that:

Fact 2: The corruption perception increases as the shadow economy size

increases.

The most widely used corruption index, the TICPI is typically interpreted

as being inversely related with the degree of corruption that is thought to

exist. Then Figures 5 and 6 show for the OECD and the broader sample that

as transparency falls, and corruption rises, the size of the shadow economy

increases.

4

Figure 5. OECD: Corruption and Shadow Size

y = -0.2193x + 11.151R2 = 0.5565

0

2

4

6

8

10

12

0 5 10 15 20 25 30 35

Shadow Economy size

Tran

spar

ency

Inte

rnat

iona

l Cor

rupt

ion

Perc

eptio

ns In

dex

(200

1)

Figure 6: Full Sample Corruption and Shadow Economy Size

y = -0.161x + 9.6522R2 = 0.4736

0

2

4

6

8

10

12

0 10 20 30 40 50 60

Shadow Economy size

Tran

spar

ency

Inte

rnat

iona

l Cor

rupt

ion

Perc

eptio

ns In

dex

(200

1)

Facts 1 and 2, the lack of a positive relation between tax rates and the

shadow economy size, in Figures 1-4, and the positive correlation betwen

corruption and shadow economy size, in Figures 5 and 6, together suggest

that:

Tax rates, corruption and shadow economy size do not all move together.

This warrants considering that corruption activity may be a separate

entity that is linked closely to the shadow economy, but not necessarily cor-

related to tax rates. Paradoxically, while tax rates are not correlated with

the shadow economy size, it emerges in Figures 7 and 8 that:

Fact 3: Tax revenues as a percent of GDP are negative correlated with

the shadow economy size.

Fact 4: Tax revenues rise as transparency increases.Figure 7. Full Sample Shadow Economy Size and Revenues

y = -0.2781x + 31.39R2 = 0.1006

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Tax Revenues, Collected (% of GDP) (2001)

Shad

ow E

cono

my

size

Figure 8: Full Sample Transparency and Revenues

y = 0.1015x + 2.8447R2 = 0.2396

0

2

4

6

8

10

12

0 10 20 30 40 50 60

Tax Revenues, Collected (% of GDP) (2001)

Tran

spar

ency

Inte

rnat

iona

l Cor

rupt

ion

Perc

eptio

ns

Inde

x

5

2.2 An Approach Consistent with Correlation Facts

Allowing for Fact 1, a lack of correlation between tax rates and the shadow

economy size, while maintaining the basic principle of using the shadow econ-

omy to evade taxes, the model here posits a competitive equilibrium supply

of corruption services that enable tax evasion. As corruption services are

supplied, the size of the shadow economy increases, consistent with Fact 2.

But an inefficient corruption sector can still produce little tax evasion even

in the face of high taxes, allowing for a possible lack of correlation between

tax rates and the size of the shadow sector, as in Fact 1. However, for any

given possible level of corruption efficiency, an increase in tax rates causes an

increase in the size of the shadow sector, resulting in lower tax revenues the

larger is the size of the shadow economy, as in Fact 3. Given the positive link

between corruption services and the shadow economy size, as in Fact 2, tax

revenue is negatively correlated with corruption service supply, consistent

with Fact 4.

Tax evasion is produced in a competitive decentralized corruption ser-

vices sector. The consumer pays a competitive market price for the service,

and as representative agent, owns shares in the corruption sector and re-

ceives its dividend profits (kickbacks). A preference for corruption does not

enter the model. Instead, sales of goods, and receipts of income, may or

may not be reported to the government tax authority, but there is just one

type of consumption good and one production sector for these goods. The

extensiveness of corruption depends solely on the efficacity in producing the

corruption service, which is tax evasion, as determined by the parameters of

the corruption service production functions. There are three such functions,

one for each type of tax evasion, that of evading the (VAT) sales tax, evading

(personal) income taxes, and evading (corporate) capital taxes. Tax evasion

allows the goods receipts or income to enter the market economy as nor-

mal funds through what we think of as a bank-related "laundering" service.

This interpretation guides the banking-related specification of this sectors

production functions.

The model includes ? endogenous growth within the monetary setting,

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an extension of ?. The financial intermediary sector, which supplies theexchange credit that enables avoidance of the inflation tax, is also made

explicit here, similar to ?. Since credit use typically leaves a "paper trail"that can be incriminating and is often avoided in the shadow economy [ ?,?], it is assumed that credit is available for avoiding the inflation tax in themarket sector by not in the shadow sector.6 Taxes decrease growth because

they lower the return to physical and human capital. But tax evasion, like

inflation tax avoidance, makes smaller the tax-induced decrease in the growth

rate.

3 The endogenous growthmonetary economy

The economy is populated with infinitely-lived identical households with pref-

erences over consumption, ct, and leisure, xt, streams given by

u (ct, xt) = ln ct + lnxt (1)

where > 0 is a relative weight of leisure with respect to consumption in the

households preferences. Here, there is just one consumption good produced

in the economy, although some goods sales are reported, denoted by crt, and

some are unreported, denoted by cut. And we assume that the reported and

unreported goods are perfect substitutes: the consumer does not feel bad in

any way about not reporting goods, so that7

ct = crt + cut.

The households real assets, denoted by at, are physical capital kt and finan-

cial capital, which consists of real money mt and bonds bt. Real money is

defined as the nominal money stock Mt divided by the nominal goods price

Pt; mt Mt/Pt; similarly, bt Bt/Pt :

at = kt +mt + bt. (2)6In contrast, Koreshkova (2006) uses credit equally in both sectors; we add the com-

plication of greater cash use in the shadow sector to make the model more realistic.7In a related economy such as ?, the market consumption good is denoted at time

t by cmt, and the non-market good by cnt, produced by different technologies, with theaggregate consumption good, denoted by ct, defined by ct = [cmt + (1 ) cnt]

1/ ,where and are utility function parameters.

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Along with the households there exist many identical firms owned by

households which produce goods output, yt, using a CRS technology in cap-

ital, sGtkt, and effective labor, lGtht,

yt = AG (lGtht) (sGtkt)

1 . (3)

The economy considers a financial intermediation sector which is com-

posed of a set of many identical banks which allow households to open an

account and receive credit for their transactions. Besides these non-corrupt

banks the households have an access to a tax-evading banking sector which is

corrupted and which allows the households to launder their incomes and rev-

enues and evade taxes. Particularly we assume that there are three sectors,

each for different money laundering, providing corruption services to evade

paying taxes on labor income, capital income, and sales revenue, respectively.

All the banks produce their services by using a CRS technology in effective

labor and deposits as in ? and ?

it = Ai (litht)i (dit)

1i (4)

where it is the amount of services produced in a bank of type i by the use of

labor, lit, and deposits, dit, when the technology is given by Ai and i. There

are four types of the banks: one non-corupt bank, Q, and corrupt banks for

labor, l, capital, c, and sales services, c, so i {Q, l, k, c}Each households is engaged in the production and accumulation of human

capital using the following technology

ht = AH (lHtht) (sHtkt)

1 hht. (5)

Further we will assume that each sector of the economy is represented by a

representative agent in a particular sector, so we have a representative house-

hold, a representative firm, and four representative banks. To simplify the

economys setup and make it structurally tractable we implement the Lucas

methodology (see e.g. Lucas (1990)) and assume one representative gigantic

household which consists of a shopper, a seller/shop-owner, five workers, one

manager and four bankers.

8

In the rest of the section we will proceed in the following way. We first

set up the problem of the representative household and derive the first-order

conditions. Then we set up the problems of the goods producer and the four

banks and derive the related first-order conditions. Finally, we define the

general equilibrium for the whole economy.

3.1 The Representative Household Problem

The households begins with cash, Mt, government bonds holding, Bt, and

the amount of physical capital, kt. It is endowed with one unit of time which

it allocates to working, lt, to study, lHt,and to leisure, xt, so

lt + lHt + xt = 1. (6)

The working time, lt, is allocated to the two legal sectors: goods production,

lGt, and credit production, lQt, and three corruption service sectors: labor

income tax evasion, llt, capital income tax evasion, lkt, and goods revenue

tax evasion, lct, so

lt = lGt + lQt + lct + lkt + llt. (7)

Workers evade labor taxes. They pay taxes only from the reported income

derived from the reported work in the goods production, lmt, the rest gets

unreported, lnt, and from the work in the non-corrupt bank, lQt, so

lrt = lmt + lQt, (8)

lGt = lmt + lnt (9)

where lrt stands for the total reported working time. The total unreported

working time, lut = lt lrt, is

lut = lnt + lct + lkt + llt. (10)

In order to avoid capital taxes, the household underreport its use of phys-

ical capital in the goods production

sGt = srt + sut (11)

9

where srt is the reported use of capital and sut is the unreported use of the

capital. Since the physical capital is used only in the goods and human

capital productions,

sGt + sHt = 1. (12)

Now we will trace the sequence of behavior of the household through a

period8. First, the family starts the period with the wealth, at, in the form

of portfolio, (kt,mt, bt) , and receives money transfers from the government.

the household trades on the asset market to determine howmuchmoneyand bonds to hold; it also decides on the deposits in the non-corrupt

financial intermediary, which will be used to buy the reported con-

sumption goods via an ATM, Mrt, and a credit account, Ptqt. So in

real terms

dQt = mrt + qt; (13)

the rest of money, Mt Mrt, will be kept in the household pocket tobuy the unreported consumption goods; then the family separates and

the individual members of the family travel to the respective markets;

the shoper will take with him the familys ATM and credit cards and

the pocket money in order to make purchases;

the manager and all the bankers travel to labor and capital markets torent labor and capital services; workers travel to labor markets;

the manager organizes the goods production, gives the goods to theshop-owner and delegates him to sell the goods to the shopper;

the shopper pays the reported goods purchases using the ATM andcredit cards, and the unreported goods purchases using the cash in his

pocket; the shop-owner sells the reported good at real price 1 + c in

8In the continuous time framework there is no such thing as a period defined like indiscrete time models. However, we can still consider an infinitesimal piece of time, dt,being decomposed into a sequence of the household activities. While timing in continuoustime models with no uncertainty has no meaning we still consider it useful for a betterexplanation of the flow of funds and goods in the economy where income can circulateseveral times within a period.

10

the front part of the shop, and the unreported good at real price 1+pctin the back-yard of the shop. So there are two exchange constraints

imposed

mrt + qt (1 + c) crt, (14)mt mrt (1 + pct) cut (15)

where mrt denotes the amount of cash withdrawals used in buying the

legal goods;

the shop-owner deposits sales revenues of unreported and reported con-sumption to the consumption tax evading bank; the corrupt bank laun-

ders the receipts by setting up two accounts for the shop-owner, one

that is reported and one that is hidden; the banker sends the consump-

tion tax to the authorities and takes the fee payment on laundering the

unreported goods sales demanded by the shop-owner demands,

dct = crt + cut; (16)

after being part of sales laundered and part of sales taxed the managerreceives the net payment for the sales from shop-owner and pays his

workers and the capital services in the goods sector;

the household sends its capital income9 to the corrupt bank specializedon the capital income laundering;

dkt = rt (sut + srt) kt; (17)

after being the part of capital income related to the unreported capitallaundered, the household withdraw its capital income from the corrupt

bank, pays tax payments to the government on the reported income

and the fee to the corrupt bank on the unreported income;

9For the sake of keeping the model simple taxes are paid by workers instead of bythe firms as it is common in actual economies according to the principle pay-as-you-earnintroduced in most developed economies after the World War II. Moreover, it is irrelevantwho actually pays the taxes, whether producer or worker, in a general equilibrium model.

11

the household will collect the labor incomes of its workers working inthe goods sector and four banking sectors and sends it to the labor tax

evading bank

dlt = wt (lut + lrt)ht; (18)

after being the part of labor income related to the unreported laborlaundered, the household withdraw its labor income from the corrupt

bank, pays tax payments to the government on the reported income

and the fee to the corrupt bank on the unreported income;

after receiving the fee payments on the credit, and corruption servicesthe bankers pay their profits/returns on the deposits to their owner,

household.

Because of the dispersion of the households and stores, the government

is unable to determine how much each household spends, how much income

it earns, and how many receipts each store takes in. It can only accurately

follow how much credit each household gets at the non-corrupt bank, and of

course what is reported to it by the households. Therefore, it is assumed that,

out of prudence, the amount that the household deposits into the non-corrupt

bank is equal to the amount of sales receipts reported by its store, or crt. This

way the household reports crt as its role as shop-owner, and this coincides

with the households transaction deposits from the non-corrupt bank; should

the government check, while also determining that all households are alike

with the same average reported consumption, there would be no obvious

inconsistency. The flow of funds, goods and reported information is depicted

in Fig. 1.

Let introduce the notation where the vector of state variables, st, and

the vector of decision variables, ut, are defined as st (at, kt, ht, bt,mt)and ut (mrt, crt, cut, xt, lrt, lut, srt, sut, qt, dQt, dlt, dkt, dct) , respectively. Nowwe are ready to set up the problem of the representative household which

maximizes its lifetime welfare

V (s0) =

12

GOODS PRODUCER Corrupt BANK L Wages

Rents Sales Receipts

Cash Non-Corrupt Corrupt Cash BANK BANK K Cash Credit

HOUSEHOLDS Deposits

Reports Goods Expenditures Sales Deposits inReports Fraction of Producer's AccountFraction of Income SalesIncome Corrupt BANK C

ReceiptsRETAIL STORES Reports Fraction of Sales

GOVERNMENT TAX AUTHORITY

Figure 1: Flow Chart of the Economy

max{ut}

Z 0

[ln (crt + cut) + lnxt] etdt (19)

subject to

the household budget constraint

at = (1 l)wtlrtht + (1 plt)wtlutht + rltdlt+(1 k) rtsrtkt + (1 pkt) rtsutkt + rktdkt (1 + c) crt (1 + pct)cut + rctdct + vtpQtqt + rQtdQt Kkt tmt + bt(Rt t) (20)

with the definition of financial wealth

at = kt +mt + bt, (21)

the human capital accumulation (5),

the two cash-in-advance constraints (exchange) constraints10 (14)-(15),

the bank deposits: in the non-corrupt bank (13) and in the corruptbanks (16)-(18).

10Both constraints will be binding in the equilibrium.

13

3.2 The First Order Conditions

In order to derive the first order conditions we first build up the Hamiltonian,

H (ut; st;t) , related to the household problem where t is the vector ofshadow prices, t = (t, t, 1t, 2t, t, ct, kt, lt) . So

H (ut; st;t) =

= (ln ct + lnxt) et

+t

(1 l)wtlrtht + (1 plt)wtlutht + rltdlt+(1 k) rtsrtkt + (1 pkt) rtsutkt + rktdkt (1 + c) crt (1 + pct)cut + rctdct + vt

pQtqt + rQtdQt Kkt tmt + bt(Rt t)

+t

AH [(1 lrt lut xt)ht] [(1 srt sut) kt]1 hht

+1t {mrt + qt (1 + c) crt}+2t {mt mrt (1 + pct) cut}+t {mrt + qt dQt}+ct {(crt + cut) dct}+kt {rt (sut + srt) kt dkt}+ lt {wt (lut + lrt)ht dlt} . (22)

Taking the first order conditions with respect to the state variables kt, ht, bt,mtwe get the conditions

= [(1 k) rsr + (1 pk) rsu k] (23)MPKH (1 sr su) + kr (sr + su) (24)

= [(1 l)wlr + (1 pl)wlu] (25) [MPHH (1 lr lu x) h] + lw (lr + lu) (26)

= (R ) (27) = (2 ) (28)

14

and with respect to the decision variablesmrt, crt, cut, xt, lrt, lut, srt, sut, qt, dQt, dlt, dkt, dctwe get the conditions

+ 1 2 = 0 (29)1

cr + cuet (1 + c) 1 (1 + c) + c = 0 (30)

1

cr + cuet (1 + pc) 2 (1 + pc) + c = 0 (31)

1

xet MPHHh = 0 (32)

(1 l)wh MPHHh+ lwh = 0 (33) (1 pl)wh MPHHh+ lwh = 0 (34) (1 k) rk MPKHk + krk = 0 (35) (1 pk) rk MPKHk + krk = 0 (36)

pQ + 1 + = 0 (37)rQ = 0 (38)rc c = 0 (39)rk k = 0 (40)rl l = 0 (41)

where MPHH = AHsHklHh

1and MPHK = (1 )AH

lHhsHk

are the

marginal product of human and physical capital in the production of human

capital, respectively.

First, the existence of an interior competitive equilibrium implies the

conditions for the equilibrium prices of tax evasion/avoidance services. These

are derived in the following Proposition.

Proposition 1 The competitive equilibrium11 prices of corruption servicesfor capital and labor tax evasion and of credit services are equal to the re-

11There exist other competitive equilibria in which e.g. some corruption sectors arenot used. However, as we limit our attention here only to a competitive equilibriumrepresenting interior solution, i.e. corruption equilibrium with respect to all the agentsproblems we are not interested in these other ones.

15

spective tax rates,

plt = l, (42)

pkt = k, (43)

pQt = R. (44)

The price of corruption services for consumption tax evasion satisfies the

following condition

pct = (1 + c)

1 rQt

1 + pQt

1. (45)

Proof. The first expression follows directly from the households first-order conditions given by (33) and (34) where we see that the interior equi-

librium exists12 only when pl = l. Similarly, equations (35)-(36) imply that

pk = k.

According to equation (30) the (discounted) marginal utility of reported

consumption is equal to the unit cost of consuming reported consumption. It

is composed of three terms: the purchasing cost on the reported goods mar-

ket, (1 + c), the cost of using the exchange means, credit and ATM card,

(1 + c)1

qcr+ 1

mrcr

= (1 + c) 1 and the unit reward on putting the sales

revenues into the corrupt bank, c. Similarly, according to (31) the marginal

(discounted) utility of unreported consumption is equal to the unit of con-

suming unreported consumption, which is composed of the purchasing cost,

(1 + pc), which already includes the fee for laundering, pc,the cost of using

the pocket cash as the only means of exchange, (1 + pc) 2mmrcu

= (1 + pc) 2

and the unit reward on putting the sales revenues into the corrupt bank,

c. Due to the perfect substitutability between the reported and unreported

consumption, the cost of consuming one unit of the reported and unreported

consumption must be equal13, i.e.

(1 + c) (+ 1) c = (1 + pc) (+ 2) c. (46)12If pl > l then nobody would be willing to work in the legal sectors and the equilibrium

will not exist. Similarly, when pl < l nobody would use the corruption services.13It is again a condition for the interior equilibrium where both reported and unreported

consumption are consumed.

16

Referring to equation (29), we get 2 = 1 + , where the benefits of using

the pocket cash in exchange transactions are equal to the benefits of us-

ing deposited cash. In accordance with (38) and (37) expressed in units of

consumption, 1= pQ rQ, the benefits of credit via exchange services, 1 ,

(which is equal to the benefits of ATM exchange services) are equal to its

cost, pQ (the fee paid for a unit of credit), net of the return on deposits, rQ.

On the other hand, the pocket cash provides benefits, 2= pQ. Using this

and the condition (46), under which both goods, reported and unreported,

will be consumed in equilibrium, we get the expression for the consumption

corruption fee, pc, given in (45).

If we further plug the results for 1 and 2 into (28) we obtain the condi-

tion for the return on money

= 2

=

1+ rQ

= [pQ ] . (47)

It confirms that in equilibrium all the means of exchange give the same

returns. Further, the formulas (27) and (47) imply that in equilibrium the

total net real return on bonds and money should be equal, i.e.

R = pQ . (48)

So the cost of real credit is equal to the nominal interest rate on bonds,

pQ = R, which can viewed as the inflation tax since the first-best Friedman

optimum claims R = 0.

Note that the price of the consumption laundering services, pc, is not

constant and equal to the evading tax rate like in the other evasion sectors

but never larger than the related tax rate c. In order to keep the total cost

of consuming illegal goods, which can be bought only by the use of pocket

cash, equal to that of the legal goods which is lower due to the rent on the

deposit (and thus implying that the household will consume both goods), the

price of laundering must always be lower than the cosumption tax rate. The

price pc is equal to c only if the nominal interest rate is zero, since in such a

situation there is no credit production and no rents paid on the bank account.

Neglecting the higher order effects the price of laundering services is lower

17

by the (nominal) return on the cash-card account, pc c rQ. Since thereturn, rQ, increases with the interest rate the services price decreases withR.

It also implies that for the price of consumption services to be nonnegative,

pc > 0, the wedge between the return on the pocket cash and that on the card

cash, rQ, must be approximately smaller than the consumption tax rate, c.

Otherwise, the consumption of illegal goods is too costly and it is better to

have all the money in the bank (either in the form of ATM or credit account)

and comply fully with consumption tax payments14.

Using (41) with (33) and (40) with (35) the relative price of human capital

in the units of physical capital can be expressed as either the ratio of the

marginal products of human capital in the human capital and physical capital

sectors or the ratio of the marginal products of physical capital in these two

sectors

=

MPHH(1 l + rl)w

=MPHK

(1 k + rk) r. (49)

Interestingly, there is an additional uncommon term in the returns to human

and physical capitals in the goods sector. It is the return on the labor and

capital income deposited in the corrupt banks, i.e. rl and rk, respectively,

which we can call the rates of tax evasion in the respective tax. We can

also define the effective labor and capital tax rates, l and k, as l = lrland k = k rk, respectively.Taking (40) for k and substituting it together with (49) and (43) from

Proposition 1 into (23) we get

= [(1 k + rk) r k ] . (50)

So the total net return on physical capital is equal to the after-tax return on

capital, (1 k) r, where the relevant tax rate is the effective tax rate, k,rather than the official rate, k, minus the rate of physical capital deprecia-

tion. Using (48)-(47) from the proof to Proposition 1 and (50) the net return

on physical capital should be equal to the real return on bonds and ATM

14So for every nominal interest rate R there exists a threshold consumption tax rate, c(R), below which there is no consumption tax evasion and ac = 1. The precise formulafor the threshold tax rate will be derived later - see Proposition 2.

18

cash:

(1 k + rk) r k = R . (51)

Using (41) for l and substituting it together with (49) and (42) from

Proposition 1 into (25) we get

= [MPHH (1 x) h ] . (52)

Formulas (48) and (51)15 confirm the standard result of the general equi-

librium that there are the same returns on all kinds of savings: on physical

capital investment, on bonds, on the cash used for reported and unreported

consumption purchases - so there is no arbitrage.

By using (30) and (32) and further (33), (39), and (42) we get an expres-

sion for the marginal rate of substitution between the reported consumption

and leisure

MRScr,x =x

c=(1 + c) (1 +R rQ) rc

(1 l + rl)wh(53)

which is equal to the ratio of the price of one unit of consumption to the

price of one unit of leisure. The unit price of the reported consumption

equals to the sum of the production price of consumption, of the share of

goods bought by cash at the price of cash minus the rent on the cash-account

and of the share of the goods bought by credit at the price of credit minus

the rent on the credit-account, both surcharged by the consumption tax, i.e.

(1 + c)h1 + q

cr(pQ rQ) + mrcr (R rQ)

iminus the return on the deposit in

the corrupt bank, kickbacks rate, rc, as it has been discussed above. The

unit price of leisure is equal to the opportunity cost of working time which

is the after-tax effective wage rate plus the rate of return on labor income

deposited in the corrupt bank, (1 l + rl)wh.If we define the effective inflation and consumption tax rates, R and

c, as R = R rQ and c = c rc, respectively, then formula (53) can beexpressed as

MRScr,x =1 + c + (1 + c) R

(1 l)wh. (54)

15Condition (51) can be written as the Fisher equation for interest rates Rt =(1 k + rkt) rt K + t.

19

It means that the relevant tax rate for buying one unit of reported consump-

tion is the effective consumption tax rate, c, rather than the official rate,

c, - see the first term in the numerator; and the relevant inflation tax is the

effective inflation tax, R, rather than the official inflation tax rate, R, - see

the second term in the numerator. Note that the base for the inflation tax

is 1 + c.

Since the unit price of the unreported consumption equals to the unit

price of the reported consumption, the marginal rates of substitution are

also the same

MRScu,x =MRScr,x.

Now we can proceed to the problem of the representative firm producing

goods.

3.3 Goods Producer Problem

The output of goods is produced by a representative firm using a CRS tech-

nology in capital and effective labor according to (3). The firm, taking the

prices of capital and labor services, rt, and wt, respectively, as given, maxi-

mizes its profit by choosing effective labor and capital inputs

max{lGtht,sGtkt}

Gt = AG (lGtht) (sGtkt)

1 wtlGtht rtsGtkt. (55)

The firm producing the market good and the non-market good face no gov-

ernment taxes nor corruption service fees because these are assumed to fall

on the household. From the first-order conditions of the firms profit maxi-

mization problems, we obtain

wt = AG (sGtk t)1 (lGtht)

1 , (56)

rt = (1 )AG (sGtk t) (lGtht) . (57)

3.4 Non-corrupt Intermediary Problem

The non-corrupt intermediary supplies exchange credit since it does not help

in tax evasion and is not concerned with records that can compromise its

20

clients. It supplies a credit card at price, pQt, to the household and makes

available a certain amount of credit, Qt. Using the technology in (4) the

non-corrupt bank maximizes its profit Qt by choosing the effective labor

and the amount of deposits, i.e.

max{lQtht,dQt}

Qt = pQtQt wtlQtht rQtdQt, (58)

subject to

Qt = AQ (lQtht)Q (dQt)

1Q . (59)

The profit of the bank, Qt, is defined as the total revenue, pQtQt, the credit

fee times the amount of demanded services, minus the labor cost, wtlQtht,

and the rental payment on the deposit, rQtdQt. The resulting equilibrium

demand for the credit bank labor and deposit are

wt = pQtQAQ

dQtlQtht

1Q, (60)

rQt = pQt (1 Q)AQ

lQthtdQt

!Q. (61)

Using the cash-in-advance constraint (14) and the condition for the de-

posit in the non-corrupt bank from (13) we find that

dQt = mrt + qt = (1 + c) crt. (62)

Assuming that the household acts in the sense of a Beckerian (1965) house-

hold that combines the credit service with the expenditures in order to get the

amount of credit, qt, equal to the supply of credit services, Qt, so qt = Qt.

Using it together with (44) in Proposition 1 obtained earlier, we get the

following formula for the share of credit transactions in the economy

1 aQt qt

(1 + c) crt= AQ

QAQRt

wt

Q1Q

(63)

where aQt is the share of cash transaction in the legal sales revenues.

21

3.5 Tax Evading Banks Production Problem

Using the technology in (4) and taking the prices of labor, deposits, and

corruption services as given, the tax evading bank in sector i {c, l, k}maximizes its profit

max{litht,dit}

it = pitit wtlitht ritdit (64)

subject to

it = Ai (litht)i (dit)

1i . (65)

Profitit is defined as the total revenue, the fee times the amount of produced

services, minus the labor cost and the rental payment on the deposit. As you

see from the expression above the homogenous effective labor input, litht, is

awarded by the effective wage rate, wt, identical across all sectors, the return

on the deposits, rit, differs among the sectors. The resulting equilibrium

demands for corruption labor and deposit are

wt = pitiAi

ditlitht

1i, (66)

rit = pit (1 i)Ailithtdit

i. (67)

Similarly to the credit sector, we can derive the expression for the cor-

ruption output-deposit ratio as:

itdit= Ai

iAipitwt

i1i

(68)

where fees, pit, for i {l, k, c} are given by (42), (43), and (45), respectively,in Proposition 1.

According to Becker (1965) we assume that the representative household

combines undeclared revenue and income in a one-to-one Leontieff-isoquant

fashion with the quantity of demanded corruption services that launders the

income or revenue:

lt = wtlutht, (69)

kt = rtsutkt, (70)

ct = cut. (71)

22

Putting (69)-(71) into (68) and using (42)-(43) from Proposition 1 we can

determine the shares of corruption activities in the respective sectors

1 alt lut

lrt + lut= Al

lAl lwt

l1l

(72)

1 akt sut

srt + sut= Ak

kAkk

wt

k1k

(73)

1 act cut

crt + cut= Ac

cAcpct

wt

c1c

(74)

where pct is given by (45) and alt, akt, and act are the relative sizes of non-

corrupted sectors: the share of reported labor and capital income, and the

shares of legal sales revenues, respectively.

3.6 Government

The agent faces proportional taxes on labor, capital and goods in the market

sector, l, k, and c, and receives from the government a nominal lump

sum transfer denoted by Vt. The government receives tax revenues only on

reported sales and incomes, prints money and issues nominal bonds, denoted

by Bt, and pays nominal interest on them of Rt. The government budget

constraint is given by

lwtPtlrtht + krtPtsrtkt + cPtcrt + Mt + Bt BtRt = Vt. (75)

It is assumed that the money supply grows at a constant rate of ,

Mt = Mt. (76)

Consistently with the existence of the balanced growth path in equilibrium,

the nominal bonds supply has to grow at the same rate

Bt = Bt. (77)

In real terms, dividing equation (76) by Pt implies that the governments

real money is the supply growth rate net of the inflation-based depreciation

of Pt/Pt t

23

mt = ( t)mt. (78)

Defining Bt/Pt bt, then (BtBtRt)/Pt = bt bt(Rtt), and the govern-ment constraint in real terms is

vt = lwtlrtht + krtsrtkt + ccrt + mt + mt + bt bt(Rt t). (79)

3.7 Social Resource Constraint

Substituting into the households income constraint in equation (20), for the

government lump sum transfer Vt, the prices of labor and capital services, wt,

and rt, the fees on services, pQt, plt, pkt, and pct, and the returns on deposits

in the financial intermediary and the corruption services, rQt, rlt, rkt, and rct,

the social resource constraint is

yt = crt + cut + it = ct + it. (80)

Based on the full specification of the behavior of all gents in the economy

we are now ready to summarize the whole in the following definition of general

equilibrium.

3.8 Definition of Equilibrium

A competitive equilibrium for this economy consists of a set of allocations

{at, kt, ht, bt,mt,mrt, lrt, lut, srt, sut, crt, cut, qt, dQt, dlt, dkt, dct}, a set of prices{Pt, wt, rt, Rt, pQt, pct, pkt, plt, rQt, rct, rkt, rlt}, the governments fiscal { c, k, l, vt, B}and monetary {M} policies, where B = Bt/Bt and M = Mt/Mt, withB = M = , and initial conditions {a0, k0, h0, b0,m0} such that

1. given the price level, Pt, prices of labor, wt, and capital services, rt, the

return on bond, Rt, the banking fees, pQt, pct, pkt, plt, and the returns

to deposits, rQt, rct, rkt, rlt, the household achieve the maximal lifetime

welfare V (a0, h0) in (19) subject to its budget constraint for the change

in real wealth (20), to the human capital investment constraint (5), to

the exchange technology constraints (14)-(15), and to conditions for the

deposits in the non-corrupt bank (13), and the corrupt banks (16)-(18);

24

2. given the prices of labor, wt, and capital, rt, the goods producing firm

maximizes its profit Gt in (58);

3. given the price of labor, wt, the return to deposit, rQt, and the fee for

credit services, pQt, the credit bank maximizes its profit Qt in (58);

4. given the price of labor, wt, the returns on deposits, rct, rkt, rlt, and

the fees for corruption services, pct, pkt, plt, the corrupt banks maximize

their profits ct,kt,lt in (64), respectively;

5. the government budget (79) is always satisfied;

6. and all markets clear at the given prices.

4 Balanced-Growth Path Equilibrium

In order to express the main properties of the competitive equilibrium along

the balanced growth path we set up the following Proposition.

Proposition 2 Along the balanced growth path the return to human capitalis equal to the return to physical capital,

AH

sHk

t

lHht

1(1 x) H = (1 k) r K , (81)

the real variables kt , ht , c

t , c

rt, c

ut, q

t , d

it, m

t , and b

t , where i {Q, k, l, c}

grow at the rate g

g = (1 k)r K , (82)

the inflation rate is equal to

= g (83)

and the nominal variablesMt , and Bt grow at rate , and

k = krk is the

constant effective capital tax rate. The shares of capital sm, sn, s

H, of labor

lm, ln, l

Q, l

c , l

k, l

l , x

, and the prices of effective labor, w, and capital, r,

25

stay constant. The rental prices rQ, rk, r

l and the nominal interest rate, R

,

are also constant and equal to

rQ = (1 Q)R1 aQ

(84)

rl = (1 l) l (1 al ) (85)rk = (1 k) k (1 ak) (86)R = + (87)

with the shares of real credit, 1 aQ, of unreported labor, 1 al , and capitalincome, 1ak, given by (63), and (72)-(73), respectively. When c > c (R) ,where c (R) =

rQ1+RrQ

with c (0) = 0 and c0 (R) > 0, the rental price rcand the corruption fee pc are equal to

rc = (1 c) pc (1 ac) (88)

pc = c (1 + c)rQ

1 +R; (89)

with unreported sales revenues, 1 ac , given by (74); when c 6 c (R) ,the consumption corruption services are not supplied, i.e. ac = 1 and thus

rc = pc = 0; if R

= 0 then pc = c > 0 and

rc = (1 c) c (1 ac) .

The leisure on BGP is given by

x = 1 + (1 + c) R

+ c(1 l )w

ctht

(90)

where we used the effective tax rates at BGP defined as l = l rl , c = c rc , and R = R rQ.

Proof. It follows from equation (49) that the growth rates of shadowprices of physical and human capitals are the same along BGP,

t

t

!=

tt

.

26

Equations (50) and (52) directly imply that the returns on physical capital

and on human capital are the same along BGP

AH

sHk

t

lHht

1(1 x) H = (1 k) r K .

Taking logs of (30), differentiating it and imposing BGP conditions gives

us (ct/ct ) =t/

t

. So using (50) we get

g ctct

= (1 k) r K .

The existence of BGP implies from the cash-in-advance constraints (14)-(15)

that the real money and credit should grow at the same rate as consumption,

so mtmt

=

MtMt

!P tP t

!= g

and = g. Using this and formula for (27) at BGP gives t/

t

=

g + = R = R ( g) , so R = + . The formulas for rQ, rc ,rk, r

l follow directly from the profit maximization conditions (61) and (67),

the production functions in (59) and (65), the bank services fees derived in

Proposition 1, and (72)-(74). From (45) we see that pc = 0 if f ( c) c1+c =rQ(R

)

1+R . Since f0 ( c) > 0, f (0) = 0 and f (1) = 1/2 and

rQ(R

)

1+R

/R > 0,

rQ (0) = 0, andrQ(R

)

1+R

R=1

< 1/2, there is always unique 1 > c (R) > 0

which satisfies f [ c (R)] =rQ(R

)

1+R for R (0, 1].

It is straightforward to get the expression for leisure (90) using the formula

for the marginal rate of substitution between consumption and leisure (53).

Expressions (84)-(86) and (88) in Proposition 2 show that the returns

on the bank deposits, or the tax evasion rates, increase when the relative

demand for them increases: i.e. the return on deposit in the credit bank

increases with the relative size of credit in the total of legal sales transactions;

similarly, the return on deposits in the corrupt bank raises with the relative

size of the respective shadow economy.

27

From the formulas for the effective tax rates we see that the shadow

economy acts as a way to evade taxes and this lessons the distortions of the

taxes on the margins. To understand better this mechanism let start with

the effective interest rate, effective inflation tax By the use of (84) and (60)

it can be expressed as

R = aQR QR

1 aQ

= aQR

+ wlQ (91)

where we defined lQ lQh

t

dQ

=

lQht

(1+c)crt

as the labor in the credit sector

per unit of the reported sales revenues. Formula (91) clearly states that the

effective inflation rate is equal to the relative tax base times the tax rate,

aQR, plus the unit labor cost of producing the credit services, wlQ, since

these costs diminish the return on deposits, rQ. Similarly, using the definitions

ll

ll ht

wlht

and lk

lkh

t

rsGkt

and formulas (85)-(86) and (66) for i = l, k

we get

l = al l + w

ll (92)

k = akk + w

lk. (93)

The formulas say that the effective tax rate is equal to the sum of the share

of the reported sector times the particular tax rate and of the unit labor cost

in the related tax-avoiding sector.

The situation for the consumption tax is more complicated due to the

fact that the price of the consumption-tax-avoiding services, pc , depends

both on the cosumption tax rate, c, and the nominal interest rate, R. Let

us consider first that there is no inflation tax, i.e. R = 0. In such case the

fee pc is simply equal to the tax rate on consumption, pc = c, an according

to (88) and (66) for i = c we get the formula similar to the other effective

tax rates

c = ac c + w

lc (94)

and lc lch

t

ct

.

The situation is getting a little more complicated when we assume non-

zero inflation tax, i.e. R > 0. According to the results from Proposition 2

28

there is always a range of tax rates, c < c (R) , at which there is no use

of consumption corruption services. The reason is that the consumption tax

evasion means that the transactions are performed in cash, however, the cash

is exposed to the inflation tax. So if the inflation tax rate is relatively high

with respect to consumption tax than it is better for agents to be exposed to

the lower tax only. Using this result we get a more general formula for the

effective consumption tax rate

c =

( c, for R > ( c)1 ( c) c (1 ac) pc + wlc , for R < ( c)

1 ( c)(95)

where 0 < pc 6 c. Unless we explicitly mention otherwise we will assumefurther in the text only the case when the consumption tax evasion is going

on, i.e. when the inflation tax rate is relatively small with respect to the

consumption tax rate, R < ( c)1 ( c) , so there is no substitution from

consumption to inflation tax.

5 Human Capital Only Case

To proceed further in our analysis we will set up a simplified economy with no

physical capital as the case which enables us to solve the model analytically

and to prove main propositions while keeping in live the major mechanisms

going on in the general economy. The derivation of the human-capital-only

model can be found in the appendix. Here we present only its solution.

The first result reveals that the model is always on its balanced-growth,

stationary16, path17, and that the gross return to human capital, rh =

AH(1x), depending on the amount of leisure and the human capital sectorsproductivity, is the major determinant of the growth rate:

g = rh H = AH(1 x) H . (96)16The stationarity implies that all model extensive variables always grow at the same

constant growth rate. All other model variables, like the shares, keep always constantvalues.17The model with only human capital can be seen as an AH model in the perspective

to the so called AK models with only physical capital and stationary dynamics.

29

This dependence on leisure is standard in the? model of economic growthwhen leisure is also included in the utility function. The monetary, public

finance, and shadow economy settings affect this basic relation only indirectly

through the effect of inflation, taxes, and corruption fees on the amount of

leisure that is used; in particular, inflation tends to increase leisure and reduce

growth, as focused on in Gillman and Kejak (2005)18.

A closed form solution results here by solving for leisure analytically, and

then the rest of the variables in the economy. Then comparative statics on

leisure, and hence growth, can be established. It is possible to see the effects

of taxes on the size of the shadow sector, and on the economic growth rate

[put more here Explain what we do in this chapter].With no physical capital, consumption equals output, the goods pro-

duction function is linear, and the real wage is the constant value of the

production function shift parameter, w = AG:

c = y = wlGh. (97)

It says that the total labor income, wlGh, is equal to the total output, y, and

the total consumption, c.19

The analytic solution for the equilibrium quantities derived in Appendix

A.1 provide us with the following results for total labor time, l,

l =

AH. (98)

Thus the rest of time is used either for leisure or invested in human capital,

i.e. x+ lH = 1 AH . The total labor time, l, is allocated among the workingtime in the goods production, lG, the credit production, lQ, and two corrupted

banks, ll and lc, i.e.

lG + lQ + lc + ll = l. (99)

According to (145) in Appendix A.1 the time used up in the production of

labor income corruption services, ll, can be expressed as a fraction of total

18In comparison to this paper we use there an endogenous growth model with inflationtax only.19Let us note the ratio of a variable z to human capital as z z/h, so c = y = wlG.

30

labor time, ll = wlll, with ll defined20 in the preceding section as the amount

of labor to produce labor income corruption services per unit of the total

labor income. It follows from (99) that the amount of the productive time

spent on production of consumption goods, lG, is

lG =1 wll

1 +lQlG+ lc

lG

l. (100)

Using lQ, lc, and ac defined in the preceding section as the unit credit and

consumption corruption services labor21 and the share of reported sales rev-

enues, respectively, we can express (100) the goods production time as

lG =1 wll

1 + (1 + c)wlQac + wlcl. (101)

Thus the formula (101) says that the productive time ratio, lG, and thus

the amount of consumption, c/h, decreases with more time used in the tax

evasion and the inflation avoidance sectors. Clearly, when there is no tax

evasion/avoidance the amount of production time is used only for the pro-

duction of consumption goods, so lG = l. The formula (101) captures the fact

that the presence of taxes decreases the productive time due to the increased

labor used in the process of avoidance/evasion.

To get the closed-form solution for the AH model we use first the formula

for leisure given in (90)

x = 1 + (1 + c) R+ c

1 llG. (102)

Then putting together (102), (101), and (98) we get the closed-form solution

for leisure

x =1 + (1 + c) R+ c

1 l1 wll

1 + (1 + c)wlQac + wlc

AH. (103)

Via (96) there is a close negative link between leisure and growth. Further

results will be analysed in the following section.

20Note that ll = l given by (141) in Appendix A.1.21Note that lQ = Q and lc = c where Q and c are given in (138)-(140), respectively.

31

5.1 Balanced Growth Tax Effects

A general result will be derived in this section: evasion leads to a higher

growth rate in a distorted economy22. However, it also leads to lower out-

put. First, it can be established that an increase in the government tax rate

causes a decrease in the growth rate. Second, evasion activity, either through

illegal corruption or through legal credit activity, enables indirectly, via the

decreased amount of leisure, the growth rate to decrease at a smaller rate

than without the evasion. This is not to say that evasion is overall good.

There are negative level effects of the evasion activity: real resources are

used up in evasion that cause less of both goods and leisure consumption.

However, the return to human capital and physical capital is increased by

the fact that the effective taxes are decreased by the tax evasion/avoidance

activity. The goods tax c, labor tax l, and inflation tax, R, each causes

increased leisure that decreases the capacity utilization rate of human cap-

ital, which is 1 x, and so decreases the marginal product of human capitaland the growth rate; but the tax evasion and credit activity decrease the

effective tax and increase the return to human capital.

First, a propositon and corollaries show the replication of the stylized

facts. Then the subsequent propositions establish the growth results.

5.1.1 Tax Effects and Stylized Facts

Proposition 3 An increase in the tax rate causes an increase in the relativesize of the shadow economy from the respective corruption services activ-

ity. An increase in the inflation tax causes an increase in the inflation tax

22As we already explained in the preceding sections the positive effect of corruption ongrowth is the implication of a special setup of our economy. There are several caveatswe have to have in mind before we move to general conclusions. First, in our economywe assume the corruption which allows to evade distortive taxation - all other types ofcorruption which may have no benefits to the society are not considered. Second, despitethat there are definitely other roles of government which can be beneficial to welfare (whichmay justify the governments use of distortive taxes and perhaps outweight it) no positiverole for government is assumed in this paper. Third, the positive effect of corruption onthe growth rate of the economy is not necessarily welfare improving as there are resourcewasted by corruption. The polar setup used in our economy allows us to isolate the pureinterplay between taxation and its evasion and thus to distill a mechanism which wouldbe present but blurred in much more realistic economy setups.

32

avoidance services, first, by using more credit in exchange transactions and,

secondly, by shrinking the consumption shadow economy.

Proof. The first part of Proposition 3 follows directly from equations(72), (74) and (89) [or (137) in Appendix A.1] which imply that (1 al) / l >0, and (1 ac) / c > 0, and pc/ c = 1 (1 Q) (1 aQ)R/(1+R) >0. For the proof of the second part of the proposition we use formula (63) we

get the effect of inflation tax on the relative use of credit (1 aQ) /R > 0.And there is additionally an effect on the size of consumption corruption

services (1 ac) /R < 0 as pc/R < 0.

Corollary 4 The size of the consumption corruption sector shrinks ceterisparibus with higher inflation rates. Since all unreported transactions are

performed only in cash and since higher inflation rates mean higher costs of

holding cash, which are reflected in higher price of corruption services pc, i.e.

ac/R = (1 ac) /pc and pc/R > 0.

The results of Proposition 3 supports common visdom that higher tax

rates lead to a larger shadow economy.

Proposition 5 An increase in corruption sector productivity induces a higherrelative quantity supplied of corruption services, and more unreported income

and sales receipts. An increase in credit sector productivity induces the higher

relative use of credit services and less of cash, first, by using more credit

in exchange transactions and, second, by shrinking the consumption shadow

economy.

Proof. The first part of Proposition 5 follows directly from equations(72), (74) and (89) [or (137) in Appendix A.1] which imply that (1 al) /Al >0, and (1 ac) /Ac > 0. For the proof of the second part of the propo-sition we use formula (63) we get the effect of inflation tax on the rela-

tive use of credit (1 aQ) /AQ > 0. And there is an additionally effecton the size of consumption corruption services (1 ac) /AQ < 0 sincepc/AQ = (1 + c) (1 Q) (1 al) /AQ < 0.

33

Corollary 6 The size of the consumption shadow sector shrinks with highercredit production productivity, other things equal, since all unreported trans-

actions are performed only in cash and higher productivity makes higher rel-

ative costs of holding cash.

Corollary 7 In the light of the fact that in this paper we relate the highertransparency index with the lower productivity of corruption services, the re-

sults of Proposition 5 conform with the stylized Fact 2, which states that the

transparency is negatively related to the size of shadow economy.

Conjecture 8 Taking into account the results of Proposition 3 and 5, es-pecially that an increase in the tax rate ceteris paribus increases the size of

shadow economy and that a decrease in the corruption productivity ceteris

paribus decreases the size of shadow economy, a simultaneous action of these

two changes can cause no change in the amount of unreported income or

sales receipts. Such a situation is consistent with the stylized Fact 1.

Conjecture 8 illustrates how a high tax, low corruption-productivity coun-

try can have the same size of a shadow economy as a low tax, high corruption-

productivity country. And it shows how lower tax rates with more enforce-

ment, in the sense that the corruption-productivity parameter falls, can still

yield higher tax revenues.

The effect of tax rates, and of corruption services productivity, on the tax

revenues relative to total output for the human capital case is the content of

the following proposition:

Proposition 9 An increase in the government tax rates on labor, capital orgoods causes an increase in government tax revenue, relative to output (Facts

2 and 3).

Proof. See proof in Appendix A.2.

Proposition 10 A decrease in corruption sector productivity induces moregovernment tax receipts (Fact 4).

Proof. See proof in Appendix A.2.

34

5.1.2 Growth Effects

In this section we focus on the growth effects of taxes and their non-linear

nature. We will be using formulas (96) and (103) for the growth rate, g, and

for leisure, x, respectively.

Let us first introduce a proposition which identifies the two main channels

via which taxes influence leisure and thus the growth rate in our human

capital economy. These lie in the heart of the nonlinearity of the growth-

tax relationships. For that purpose let us introduce the following notation:

1 (R, c) 1+(1 + c)wlQ (R) ac (R, c)+wlc (R, c) , 2 ( l) 1wll ( l) ,1 (R, c) 1 + (1 + c) R + c and 2 ( l) 1 l then leisure and thegrowth rate are given by

x (R, c, l) =1 (R, c)

2 ( l)

2 ( l)

1 (R, c)

AH, (104)

and

g (R, c, l) = AH [1 x (R, c, l)] H , (105)

respectively.

Proposition 11 below says that the marginal rate of substitution - see

(102) - increases with taxes; and that consumption, c, or working time, lG,

decreases with taxes - see (101).

The nonlinear nature of the tax-growth relationship is implied by the

nonlinearity of the substitution and income effects and the interplay between

them. The substitution effect is positive and concave and the income effect

is negative and concave. The key mechanism behind the weakening positive

substitution effect is a nonlinear relationship between the tax rate and its

effective tax rate. Let us start with the case of labor income tax. The

effective tax rate is according to (92) l = al l +wll, and its derivative with

respect to the official tax rate is thus

l l

=al l

l + al 1 + wll l

where al l

< 0 and ll l

> 023. There are three effects of the official tax rate on

the effective tax rate: one direct and two indirect ones. The first term in the23Remember that ll = l given in (141).

35

expression above captures the indirect effect of declining relative tax base, al.

The second term is the positive direct effect and the third one is the indirect

effect of increasing use of labor in the tax-avoiding activity. In the proof

to Proposition 11 we show that l l

> 0 and 2 l2l

< 0. The key mechanism

behind the strengthening negative income effect is the nonlinear relationship

between tax rate and the labor cost of producing tax-evading services. The

labor cost is given by wll. Clearly, as it is derived in the proof to Proposition

11 w ll l

> 0 and w 2 ll2l

> 0. The composition of the substitution and income

effects is responsible for the resulting non-linear tax-growth relationship as

it is in Proposition 11.

A similar situation appears for the inflation tax where the effective tax

rate is according to (91) R = aQR + wlQ, and its derivative with respect to

the official tax rate is

R

R=

aQR

R+ aQ 1 + wlQR

where aQR

< 0 and lQR

> 024. There are again the three effects of the official

tax rate on the effective tax rate. It was also proved that RR

> 0 and 2R

R2< 0.

The key mechanism behind the strengthening negative income effect is the

nonlinear relationship between the tax rate and the labor cost of producing

tax-evading services. The labor cost is given bywlQ.Clearly, as it is derived in

the proof to Proposition 11 w lQR

> 0 and w 2 lQR2

> 0. The actual unit labor

costs for tax-evasion and avoiding services influenced by R, when c > 0,

are a little more complicated (1 + c)wlQ (R) ac (R) +wlc (R). However, the

major tendency is not influenced this fact (see the proof). The composition of

the substitution and income effects is responsible for the resulting non-linear

tax-growth relationship as it is in Proposition 11.

In the case of consumption tax when R = 0 the effective tax rate is given

by the same formula (94) as for other taxes. However, the effective tax rate

is more complicated when R > 0 as it has to capture the effect of declining

the tax-evasion fee with increasing R and there are two regimes according

to equation (95): under the first regime when c 6 c (R) (see Proposition24 lQ = Q given in (139).

36

2) there is no consumption tax evasion, i.e. c = c. When the consumption

tax rate is sufficiently large with respect to R, i.e. c > c (R) , then there

is the consumption tax evasion in place and c = c (1 ac) pc + wlc. Itsderivative with respect to the official tax rate is thus

c c

= 1"

(1 ac)pc

pc + 1 ac+ w

lcpc

#pc c

where acpc

< 0 and lcpc

, pcc

> 025. This time there are four effects of the official

tax rate on the effective tax rate: there is a positive direct effect, the first

term, then there are three former effects of on pc which affect the effective

tax rate via pcc. It will be again proved that c

c> 0 and

2c2c

< 0. The key

mechanism behind the strengthening negative income effect is the nonlinear

relationship between tax rate and the labor cost of producing tax-evading

services. The labor cost is given by wlc. Clearly, as it is derived in the proof

to Proposition 11 w lcc

> 0 and w 2 lc2c

> 0. The actual unit labor costs for tax-

evasion and avoiding services influenced by c, when R > 0, are a little more

complicated (1 + c)wlQac ( c)+wlc ( c), however, the major tendency is not

influenced (see the proof). The composition of the substitution and income

effects is responsible for the resulting non-linear tax-growth relationship as

it is in Proposition 11.

Proposition 11 Any tax rate {R, c, l} influences leisure given by(104) via a positive substitution effect, where the households substitute away

from consumption to leisure, and via a negative income effect, where the level

of consumption is decreased via increased labor use in the tax avoiding sectors

and thus leading to a lower amount of available productive labor. The positive

substitution efect is captured by a tax rate change effect on the first fraction

of (104), AH

2( l)1(R,c)

1(R,c)

2(l)

, with

1(R,c)

2(l)

> 0. The negative income

efect is captured by a tax rate change effect on the second fraction of (104),

AH

1(R,c)2( l)

2(l)

1(R,c)

, with

2(l)

1(R,c)

< 0. For each tax rate {R, c, l}

25 lc = c given in (140).

37

there exists a tax rate R, c, l

such that

x (R, c, l)

=

AH

2 ( l)1 (R, c)

1(R,c)2( l)

+1 (R, c)

2 ( l)

2( l)1(R,c)

> 0,for [0, ), i.e. the substitution effect is the dominating factor in inter-val [0, )26. Formula (96) implies that the same forces with opposite signs

influence the growth rate.

Proof. See proof in Appendix A.2.

Corollary 12 Since the income effect of any tax rate is exclusively caused bythe avoiding activity, the income effect is non-existent in the economies where

tax avoidance sectors are not present. And thus the tax rates monotonnically,

and almost27 linearly, decrease the growth rate in such economies.

Corollary 13 Proposition 11 provides us with the key results of our paper.First, there is a negative relationship between growth and the tax rates. If

there is the tax avoidance present in the economy than the relationship is

getting nonlinear.

According to what has been said in Corollaries above it is the presence

of the tax evasion sectors in an economy that plays the key role in how

fiscal and monetary government policies affect the economys growth rate.

Further in the paper we will only consider the ranges of tax rates in which

the substitution effect dominates the income effect. This will be analysed in

the following Proposition. Let us first distinguish five different regimes under

which an economy can operate:

1. an economy without distortions noted as O;26Accordingly with the former literature, see e.g. Chari, Kehoe, and Manuelli (1997),

we consider as empirically relevant - and thus interesting for our analysis - the intervalwhere taxes have negative effect on growth, i.e. the interval in which the substitutioneffect dominates the income effect.27The slight deviation from the strict linearity is caused by the changing weights of

substitution and income effects which may depend on tax rates.

38

when there are no government policies, i.e. R = c = l = 0; leisure is

simply

xO =

AH;

and three more regimes when the government distortive fiscal and monetary

policies are present:

2. an economy without banking and tax evasion - noted as D,

xD =1 + (1 + c)R+ c

1 l

AH; (106)

3. an economy with banking and no tax evasion - noted as B,

xB =1 + (1 + c) R+ c

1 l1

1 + (1 + c)wlQ

AH; (107)

4. an economy with banking and tax evasion in labor and capital income

when c 6 c(R) - noted it as E,

xE =1 + (1 + c) R+ c

1 l1 wll

1 + (1 + c)wlQ

AH. (108)

5. an economy with banking and tax evasion in all sectors when c >

c(R) - noted it as F,

xF =1 + (1 + c) R+ c

1 l1 wll

1 + (1 + c)wlQac + wlc

AH. (109)

Proposition 14 The economies under different regimes but with the samemix of monetary and fiscal policies can be ordered according to their growth

rates in the following way

gO > gF > gE > gB > gD (110)

where gI is the growth rate of the economy which operates under the regime

I {D,B,E, F} , gO is the growth rate of the economy with no distortions.

39

Proof. The following two facts holds: (1) effective tax rates, in thepresence of tax avoidance, are strictly lower than their nominal counterparts

- according to Proposition 2 e.g. c = c rc < c; and (2) the labor costof evasion activities lowers the available amount of labor for production -

according to (101) e.g. the presence of wlc lowers lG. Having this in mind it

is trivial to show that for the given tax mix ( c, l, k, R) the adding more

evasion activities lowers leasure and increases the growth rate, so it proves

Proposition.

Proposition 14 says that the fastest, optimal, growth rate can be achieved

when there are no government distortive policies and so there is no reason

for tax avoidance - the economy in regime O. Interestingly, the worst growth

performance has the economy when there is no scope to avoid taxes and

inflation - the economy in regime D - and the second best outcome happens

when all the tax avoidance mechanisms are at work - the economy F . If the

consumption tax evasion is not at work the growth rate is lower - the economy

E. The situation when only inflation-avoiding banking sector is present in

economy - regime B - is the second worst.

Next Proposition demonstrates first that there is a complete symmetry

among the tax rates with respect to their effect when distorting the optimum.

Secondly, the severity of the initial tax effects on the growth rate can be

ordered according to the regime under which the economy operates.

Proposition 15 An increase in any of the tax rates or the inflation rate {R, c, l} causes the growth rate to decrease at the rate

g

I= < 0

when evaluated at the optimum where R = c = l = 0 where I {D,B,E} .The decreases of the growth rates implied by an increase in a tax rate can be

40

ordered according to the regimes under which the economy operatesg

c

Dc=0

0.

Proof. See proof in Appendix A.2. Since m mh, c (1+c)c

h, and

mc= mu+mr

(1+c)c= (1+pc)cu

(1+c)c+

aQ(1+c)cr(1+c)c

= (1 ac) + aQac = 1 (1 aQ)ac aQ. Then mR =

cR +

aQR . Since c = (1 + c)

2( l)1(R,c)

l, cR = 1R . SinceaQR

= acR+ ac

RaQ +

aQR

ac, aQR =

aQacaQ

aQR

(1aQ)acaQ

acR where aQR =

Q1Q

1aQaQ

< 0, acR = c1c1acac

pcR > 0 with pcR =

pcR

Rpc

< 0 and28

pcR= (1 + c) (1 aQ) 1+QR(1+R)2 < 0. To derive the elasticity of substitution

between money and credit services, = lnMC lnR

, let us define first the money-

credit ratio, MC mu+mr(1aQ)(1+c)cr

=1ac+aQac(1aQ)ac

=aQ1aQ . Then =

11aQ

aQR

and thus mR = cR + (1 aQ) . As

aQR < 0 and

acR > 0,

aQR < 0. Further

aQR

R=

aQR

ac+acR

aQ

aQaQac

aQR

a2QaQR +

acaQa2Q

aQR

RaQ

Rac+

acR (1aQ)

aQ(1aQ)ac

aQR

a2QacR

ac(1aQ)a2Q

acRR

,

aQR

R< 0,

acRR

= c1c

h acR

a2cpcR 1acac

pcRR

i> 0 for low R

when pcR 0 andpcRR . Since the numerator in the third term

isaQ

Rac +

acR(1 aQ)

aQ (1 aQ) ac aQR = (1 aQ)

acR ac aQR >

0 and the numerator in the first termaQR

ac +acR

aQaQ aQac aQR =

aQacR+ ac (1 ac) aQR < 0 for low values of R, all four terms are neg-

ative and aQR

R< 0 which means that

aQR

R

> 0 and thus

R

> 0.

28As we mentioned earlier we assume here the economy in regime F when there is presentconsumption corruption activity, i.e. the situation when c > c(R).

43

Further, 21R2

= (1 + c)w

2 lQR2

ac + 2wlQR

acR

ac + wlQ2acR2

+ w

2 lcR2

. Since

2acR2

= c1cAc

cAcw

c1c p

c1cc

According to Proposition above the share of money in total transactions,

aQ, can be decomposed into the pocket cash uses in the illegal consumption

purchases, 1ac, and the share of money (in the form of the cash card), aQac,where aQ is the share of cash transactions in the legal consumption purchases

only, so aQ = 1 ac + aQac. The share of legal consumption purchases, ac,increases with R via its negative effect on the consumption fee, pc. It means

that the use of pocket cash decreases with R. Since the use of the cash in

legal consumption purchases depends on the size of the legal purchases, there

are two effects of increasing R. The first, standard effect, is that the second

part of money demand decreases as there is growing demand for alternative

means of exchange, aQ decreases. The second effect, is driven by the fact

that ac increases with R. Nevertheless, as aQR =

acaQ

aQ

aQR (1 aQ) acR

the interest elasticity aQ is always negative.

Corollary 17 In a distorted economy without any tax evasion activity aQ =

ac = 1, so aQ = 1, andcRD=m/cR

D= = 0. Thus money demand (per

human capital) is inelastic with respect to the interest rate.

Corollary 18 In a distorted economy with the inflation avoidance and notax evasion activity29 ac = 1 so aQ = aQ < 1, and

m/cR =

aQR , money demand

elasticity is equal to mR = cR+

aQR =

cR+(1 aQ) where

aQR =

Q1Q

1aQaQ

and = aQR1

1aQ .

Proposition 19 There exists R > 030, such that any increase in the interestrate for R (0, R) causes a decrease in the growth rate, g, according to

g

R= 1 wll

13

(1 + c) aQ

1 + cR +

aQR

+ 1

cRR+ 2

< 0 (111)

29It is when the economy is in regime E when there is no consumption corruptionactivity, i.e. the situation when c 6 c(R).30See Proposition 11.

44

where

1 = c +h(1 + c)wlQ pc

i(1 ac) > 0

2 = (1 + c)

"w (1 ac)

lQR wlQ

acR

# [(1 ac)pc]

R> 0

where the consumption fee, pc, is given by (89), lQ, lc and ll are the unit

amounts of labor used in the credit, consumption and labor income services,

respectively, and 1 and 3 are defined in Proposition 11. Further, the de-

crease, g(R)R

, diminishes in absolute value with the increasing nominal inter-

est rate, i.e.2g(R)R2

> 0 for R (0, R).

Proof. See proof in Appendix A.2.Note that in the economy with tax evasions according to Proposition

16 the sum of the consumption and the cash-card-transactions-share interest

elasticities, cR, and aQR , does not compose the money interest elasticity since

mR = cR +

aQR and

aQR =

aQacaQ

aQR

(1aQ)acaQ

acR .

Note that the terms (1 + c) aQ + cR in the bracketed part of formula

for gRin (111) captures fully the substitution effect of the nominal interest

rate. The remaining terms of the formula, (1 + c) aQcR +

aQR

, compose

the income effect.

Corollary 20 In case of no tax evasion activities, i.e. the economy is inregime B, there is a negative growth effect for R (0, R) where R > 0 givenbyg

R

B= 1h

1 + (1 + c)wlQi(1 l)

n(1 + c) aQ

1 + mR

+

cRcR

o< 0.

The above Corollary says that as the nominal interest rate increases, the

money demand is getting more elastic, so the negative link between growth

and the nominal interest rate (inflation) becomes marginally weaker. This

is the exact result, when c = 0, obtained and documented by empirical ev-

idence in Gillman and Kejak (2005). Additionally, according to Proposition

19 above 1 + mR is the interest rate elasticity of the inflation tax revenue.

Now we continue with the labor income tax.

45

Proposition 21 There exists l > 031, such that any increase in the interestrate for l (0, l) causes a decrease in the growth rate, g, according to

g

l= 2

123al

(1 wll

1 + al l

+ lw

ll l

)< 0 (112)

where ll are the unit amounts of labor used in labor income services, al l =

l1l

1alal

< 0 is the interest elasticity of the share of reported labor income,ll l

> 0, and 1, 2 and 3 are defined in Proposition 11. Further, the

decrease, g( l) l

, diminishes in absolute value with the increasing labor income

tax rate, i.e.2g( l)2l

> 0 for l (0, l).

Proof. See proof in Appendix A.2.The next case is that of consumption tax.

Proposition 22 There exists c > 032, such that any increase in the interestrate for c (0, c) causes a decrease in the growth rate, g, according to

g

c= 1 wll

213

2hR+ 1 (1 ac)

1 + 1acpc

pcc+ w lc

c

i

1hac1 + acc

1+cc

wlQ + w

lcc

i < 0(113)

where ll, lc are the unit amounts of labor used in labor income and consump-

tion tax-evasion services, acc < 0 is the tax elasticity of the share of reported

consumption sales, 1acpc > 0 is the fee elasticity of the share of unreported

consumption sales, lcc

> 0, and 1, 2 and 3 are defined in Proposition 11.

Further, the decrease, g(c)c

, diminishes in absolute value with the increasing

consumption tax rate, i.e.2g(c)2c

> 0 for c (0, c).

Proof. See proof in Appendix A.2.

Corollary 23 In case of no inflation there is a negative growth effect for c (0, c) where c > 0 given by

g

c= 1 wll

13ac1 + acc +

cc

< 0.

31See Proposition 11.32See Proposition 11.

46

Additionally, term 1 + acc + cc is equal to the consumption tax rate

elasticity of the consumption tax revenue, Tcc lnTc ln c where Tc = cacc.

Proposition 24 An increase in the corruption service productivity causesan increase in the growth rate, near to the optimum.

Proof. See proof in Appendix A.2.The corollary means that the bigger is the size of the shadow economy,

or the credit sector, the smaller will be the use of leisure, and the higher will

be the growth rate.

Proposition 25 An increase in the inflation rate lowers the growth rate,and does so by more the higher is the corruption activity.

Proof. See proof in Appendix A.2.The propositions show that having the ability to avoid a tax on goods,

labor, or money, enables the consumer to feel the burden of the tax to a lessor

extent. Thus there is less substitution towards leisure, and the growth rate

does not fall as much, since these taxes work through the capacity utilization

rate on human capital, which equals the amount of time spent productively,

or 1 xt. So the growth rate falls at a substantially decreasing rate as thetax goes up. Now if there are already other taxes imposed, and one of these

taxes is increased, the most of the substitution is towards corruption activity

to avoid the tax, and not much towards more leisure. While if no other taxes

exist, and one of these taxes is increased, then the substitution towards leisure

is at first rather strong, and towards corruption weak. But the price-elasticity

rises as the tax increases, so that instead of moving from goods to leisure so

much, the move is towards corruption more, still consuming the goods, with

less increased leisure.

6 Full Economy Simulation

Parameter values for the simulation have been set at standard values, as in

the