Accounting for Corruption: The Effect of Tax
Evasion and Inflation on Growth∗
Max Gillman
Cardiff University Business School
Michal Kejak
CERGE-EI
PRELIMINARY VERSION
Abstract
The paper studies the relation between taxes and growth when
there is evasion or avoidance of key taxes, on labor and capital income, on goods purchases, and on money holdings. The paper models
tax evasion using a decentralized corruption service sector, takes a
banking approach, and assumes a production function based on financial intermediary microfoundations for laundering undeclared income
and sales revenue. This loosens the linkage between tax rate levels and
the size of the shadow economy, as is consistent with correlation facts,
while still embodying the well-accepted notion of marginal substitution towards the shadow economy as tax rates increase. The results
are that taxes decrease growth, evasion decreases the negative effect
of taxes on growth, and the growth rate falls at a decreasing rate as
individual tax rates increase. This presents a fiscal principle of the
effect of flat taxes on growth with evasion, based on a rising demand
price sensitivity to higher tax rates.
JEL Classification: E13, E31, H26, O42
Keywords: Tax evasion, corruption, financial services, endogenous
growth, and inflation.
Preliminary Draft; Incomplete
∗
We thank Szilard Benk for research assistance, Dario Cziraky, Bye Jeong, Patrick
Minford, and Slava Vinogradov for comments, and the seminars at CERGE-EI, Prague,
WIIW, Vienna, Koc University, Cardiff Business School, and the 2nd CDMA Conference
at the University of St Andrews, and the Macro and Financial Economics/Econometrics
Conference at Brunel University. Research support of the World Bank GDN fund at WIIW
is kindly acknowledged.
0
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 nonlinear 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 increasingly 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 Cagan’s 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 Bailey’s (1956) inflation tax is legal, all taxes tend to be avoided
or evaded. This follows from Gary Becker’s (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 inflation tax avoidance, tax evasion of flat taxes on labor income, capital income
1
Less controversial than sometimes reported, there appears to be only a dispute over
what happens below a "threshold" inflation rate, which is found to be at low inflation
rates, 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 that
the negative nonlinear profile holds for all positive inflation rates for both developed and
less 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 paper’s approach is to follow the funds.2
We first show that the model accounts for certain stylized facts of corruption, 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, behavior.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 re2
This is part of the research agenda proposed by Lucas’s Nobel address (1996) to find
significant long run effects of monetary factors such as inflation, combined with the study
of flat taxes in Rebelo and Stokey (1996), and Easterly’s (2001) emphasis on the need to
include the non-market sector to explain the economy in a policy relevant way.
3
Corruption can be defined narrowly in terms of public officials taking kickback, as
in Schneider and - (2006); they then propose that corruption can act as a substitute to
expanding the size of the shadow economy. Our definition of corruption is more broad, of
illegal activity by anyone in the economy; but we then focus only on tax evasion within
this broad definition, and evasion and the size of the shadow economy end up moving
together, more as complements.
4
? reports that the shadow output equals 39% of the actual magnitude of reported
GDP 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 case
of Canadian tax rate changes, and ?, in which an increase in the tax rate
induces 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 positive correlation between personal or corporate tax rates, and the shadow
economy size, as the next section shows.
2.1
Correlation Evidence
Sweden’s 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 ?.
5
The 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 2. Full Sample: Personal Tax Rates and Shadow Economy Size
Figure 1. OECD: Shadow Size and Personal Tax Rates
60
35
y = -0.3104x + 28.085
R2 = 0.0947
y = -0.0019x + 19.175
R2 = 5E-06
50
Shadow economy size
Shadow economy size
30
25
20
15
40
30
20
10
10
5
0
0
0
5
10
15
20
25
30
35
0
40
10
20
30
40
50
60
Tax Rate, Effective Personal Income (% GDP per capita)(2001)
Tax Rate, Effective Personal Income (% GDP per capita)(2001)
Figure 3: OECD Corporate Tax Rates and Shadow Economy Size
35
y = -0.0742x + 21.513
R2 = 0.0054
Figure 4: Full Sample: Corporate Tax Rates and Shadow Economy Size
60
30
y = -0.0213x + 23.222
R2 = 0.0002
25
Shadow economy size
Shadow economy size
50
20
15
10
5
40
30
20
10
0
0
0
10
20
30
40
50
60
0
Tax Rate, Average Corporate (2002)
10
20
30
40
50
60
Tax Rate, Average Corporate (2002)
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
12
Figure 6: Full Sample Corruption and Shadow Economy Size
12
y = -0.161x + 9.6522
R2 = 0.4736
Transparency International Corruption
Perceptions Index (2001)
Transparency International Corruption
Perceptions Index (2001)
y = -0.2193x + 11.151
R2 = 0.5565
10
8
6
4
2
10
8
6
4
2
0
0
5
10
15
20
25
30
0
35
0
10
20
Shadow Economy size
30
40
50
60
Shadow Economy size
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 correlated 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 8: Full Sample Transparency and Revenues
Figure 7. Full Sample Shadow Economy Size and Revenues
12
60
Transparency International Corruption Perceptions
Index
y = -0.2781x + 31.39
R2 = 0.1006
Shadow Economy size
50
40
30
20
10
y = 0.1015x + 2.8447
R2 = 0.2396
10
8
6
4
2
0
0
0
10
20
30
40
50
0
60
10
20
30
40
50
Tax Revenues, Collected (% of GDP) (2001)
Tax Revenues, Collected (% of GDP) (2001)
5
60
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 economy 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 services sector. The consumer pays a competitive market price for the service,
and as representative agent, owns shares in the corruption sector and receives 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 normal funds through what we think of as a bank-related "laundering" service.
This interpretation guides the banking-related specification of this sector’s
production functions.
The model includes ? endogenous growth within the monetary setting,
6
an extension of ?. The financial intermediary sector, which supplies the
exchange 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 the
market 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 growth monetary economy
The economy is populated with infinitely-lived identical households with preferences over consumption, ct , and leisure, xt , streams given by
u (ct , xt ) = ln ct + α ln xt
(1)
where α > 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 household’s real assets, denoted by at , are physical capital kt and financial 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 .
6
(2)
In contrast, Koreshkova (2006) uses credit equally in both sectors; we add the complication of greater cash use in the shadow sector to make the model more realistic.
7
In 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 the
aggregate consumption good, denoted by ct , defined by ct = [νcεmt + (1 − ν) cεnt ]1/ε ,where
ν and ε are utility function parameters.
7
Along with the households there exist many identical firms owned by
households which produce goods output, yt , using a CRS technology in capital, sGt kt , and effective labor, lGt ht ,
yt = AG (lGt ht )β (sGt kt )1−β .
(3)
The economy considers a financial intermediation sector which is composed 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 revenues 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 ?
κit = Ai (lit ht )ωi (dit )1−ωi
(4)
where κit 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 ω 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
ḣt = AH (lHt ht )ε (sHt kt )1−ε − δ h ht .
(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 household, a representative firm, and four representative banks. To simplify the
economy’s 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
(7)
lt = lGt + lQt + lct + lkt + llt .
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 physical capital in the goods production
sGt = srt + sut
9
(11)
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,
(12)
sGt + sHt = 1.
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 how much money
and bonds to hold; it also decides on the deposits in the non-corrupt
financial intermediary, which will be used to buy the reported consumption goods via an ATM, Mrt , and a credit account, Pt qt . So in
real terms
(13)
dQt = mrt + qt ;
the rest of money, Mt − Mrt , will be kept in the household pocket to
buy 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 family’s 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 to
rent labor and capital services; workers travel to labor markets;
• the manager organizes the goods production, gives the goods to the
shop-owner and delegates him to sell the goods to the shopper;
• the shopper pays the reported goods purchases using the ATM and
credit cards, and the unreported goods purchases using the cash in his
pocket; the shop-owner sells the reported good at real price 1 + τ c in
8
In the continuous time framework there is no such thing as a period defined like in
discrete 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 continuous
time models with no uncertainty has no meaning we still consider it useful for a better
explanation of the flow of funds and goods in the economy where income can circulate
several times within a period.
10
the front part of the shop, and the unreported good at real price 1 + pct
in the back-yard of the shop. So there are two exchange constraints
imposed
mrt + qt ≥ (1 + τ 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 consumption to the consumption tax evading bank; the corrupt bank launders the receipts by setting up two accounts for the shop-owner, one
that is reported and one that is hidden; the banker sends the consumption 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 manager
receives 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 specialized
on the capital income laundering;
dkt = rt (sut + srt ) kt ;
(17)
• after being the part of capital income related to the unreported capital
laundered, 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;
9
For the sake of keeping the model simple taxes are paid by workers instead of by
the firms as it is common in actual economies according to the principle pay-as-you-earn
introduced in most developed economies after the World War II. Moreover, it is irrelevant
who 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 in
the goods sector and four banking sectors and sends it to the labor tax
evading bank
(18)
dlt = wt (lut + lrt ) ht ;
• after being the part of labor income related to the unreported labor
laundered, 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 services
the 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. Now
we 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
Corrupt
BANK K
Cash
Credit
Cash
HOUSEHOLDS
Reports
Fraction of
Income
Reports
Fraction of
Income
Non-Corrupt
BANK
Deposits
Goods Expenditures
Sales Deposits in
Producer's Account
Sales
Corrupt BANK C
Receipts
RETAIL STORES
Reports Fraction of Sales
GOVERNMENT TAX AUTHORITY
Figure 1: Flow Chart of the Economy
max
{ut }
subject to
Z
∞
[ln (crt + cut ) + α ln xt ] e−ρt dt
(19)
0
• the household budget constraint
ȧt = (1 − τ l ) wt lrt ht + (1 − plt ) wt lut ht + rlt dlt
+ (1 − τ k ) rt srt kt + (1 − pkt ) rt sut kt + rkt dkt
− (1 + τ c ) crt − (1 + pct )cut + rct dct + vt
−pQt qt + rQt dQt − δ K kt − π t mt + bt (Rt − π 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 corrupt
banks (16)-(18).
10
Both 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 ; χt ) , related to the household problem where χt is the vector of
shadow prices, χt = (λt , μt , ξ 1t , ξ 2t , ηt , ζ ct , ζ kt , ζ lt ) . So
H (ut ; st ; χt ) =
= (ln ct + α ln xt ) e−ρt
⎫
⎧
⎪
⎪
(1
−
τ
)
w
l
h
+
(1
−
p
)
w
l
h
+
r
d
⎪
⎪
l
t rt t
lt
t ut t
lt lt
⎪
⎪
⎪
⎬
⎨ + (1 − τ ) r s k + (1 − p ) r s k + r d ⎪
k
t rt t
kt
t ut t
kt
kt
+λt
⎪
⎪
− (1 + τ c ) crt − (1 + pct )cut + rct dct + vt
⎪
⎪
⎪
⎪
⎪
⎭
⎩ −p q + r d − δ k − π m + b (R − π ) ⎪
Qt t
Qt
K
t
t
t
t
t
t
Qt
©
ª
+μt AH [(1 − lrt − lut − xt ) ht ]ε [(1 − srt − sut ) kt ]1−ε − δ h ht
+ξ 1t {mrt + qt − (1 + τ c ) crt }
+ξ 2t {mt − mrt − (1 + pct ) cut }
+η t {mrt + qt − dQt }
+ζ ct {(crt + cut ) − dct }
+ζ kt {rt (sut + srt ) kt − dkt }
+ζ lt {wt (lut + lrt ) ht − dlt } .
(22)
Taking the first order conditions with respect to the state variables kt , ht , bt , mt
we get the conditions
λ̇ = ρλ − λ [(1 − τ k ) rsr + (1 − pk ) rsu − δ k ]
(23)
−μMP KH (1 − sr − su ) + ζ k r (sr + su )
(24)
μ̇ = ρμ − λ [(1 − τ l ) wlr + (1 − pl ) wlu ]
−μ [MP HH (1 − lr − lu − x) − δ h ] + ζ l w (lr + lu )
(25)
(26)
λ̇ = ρλ − λ (R − π)
(27)
λ̇ = ρλ − (ξ 2 − λπ)
(28)
14
and with respect to the decision variables mrt , crt , cut , xt , lrt , lut , srt , sut , qt , dQt , dlt , dkt , dct
we get the conditions
η + ξ1 − ξ2
1
e−ρt − λ (1 + τ c ) − ξ 1 (1 + τ c ) + ζ c
cr + cu
1
e−ρt − λ (1 + pc ) − ξ 2 (1 + pc ) + ζ c
cr + cu
1 −ρt
e − μMP HH h
x
λ (1 − τ l ) wh − μMP HH h + ζ l wh
= 0
(29)
= 0
(30)
= 0
(31)
= 0
(32)
= 0
(33)
λ (1 − pl ) wh − μMP HH h + ζ l wh = 0
(34)
λ (1 − τ k ) rk − μMP KH k + ζ k rk = 0
(35)
λ (1 − pk ) rk − μMP KH k + ζ k rk = 0
(36)
−λpQ + ξ 1 + η = 0
(37)
λrQ − η = 0
(38)
λrc − ζ c = 0
(39)
λrk − ζ k = 0
(40)
λrl − ζ l = 0
(41)
³ ´1−ε
³ ´ε
and MP HK = (1 − ε) AH slHHhk are the
where MP HH = εAH slHHhk
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 services
for capital and labor tax evasion and of credit services are equal to the re11
There exist other competitive equilibria in which e.g. some corruption sectors are
not used. However, as we limit our attention here only to a competitive equilibrium
representing interior solution, i.e. corruption equilibrium with respect to all the agents’
problems we are not interested in these other ones.
15
spective tax rates,
plt = τ l ,
(42)
pkt = τ k ,
(43)
pQt = R.
(44)
The price of corruption services for consumption tax evasion satisfies the
following condition
¶
µ
rQt
− 1.
(45)
pct = (1 + τ c ) 1 −
1 + pQt
Proof. The first expression follows directly from the household’s firstorder conditions given by (33) and (34) where we see that the interior equilibrium exists12 only when pl = τ l . Similarly, equations (35)-(36) imply that
pk = τ 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 market, (1 +³τ c ) λ, the cost
´ of using the exchange means, credit and ATM card,
(1 + τ c ) ξ 1 cqr + ξ 1 mcrr = (1 + τ c ) ξ 1 and the unit reward on putting the sales
revenues into the corrupt bank, ζ c . Similarly, according to (31) the marginal
(discounted) utility of unreported consumption is equal to the unit of consuming unreported consumption, which is composed of the purchasing cost,
(1 + pc ) λ, which already includes the fee for laundering, pc ,the cost of using
r
= (1 + pc ) ξ 2
the pocket cash as the only means of exchange, (1 + pc ) ξ 2 m−m
cu
and the unit reward on putting the sales revenues into the corrupt bank,
ζ 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 + τ c ) (λ + ξ 1 ) − ζ c = (1 + pc ) (λ + ξ 2 ) − ζ c .
12
(46)
If pl > τ l then nobody would be willing to work in the legal sectors and the equilibrium
will not exist. Similarly, when pl < τ l nobody would use the corruption services.
13
It is again a condition for the interior equilibrium where both reported and unreported
consumption are consumed.
16
Referring to equation (29), we get ξ 2 = ξ 1 + η, where the benefits of using
the pocket cash in exchange transactions are equal to the benefits of using deposited cash. In accordance with (38) and (37) expressed in units of
consumption, ξλ1 = pQ − rQ , the benefits of credit via exchange services, ξλ1 ,
(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, ξλ2 = 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 ξ 1 and ξ 2 into (28) we obtain the condition for the return on money
¸
∙
¸
∙
ξ1
ξ2
−π−ρ =λ
+ rQ − π − ρ = λ [pQ − π − ρ] .
(47)
−λ̇ = λ
λ
λ
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 − π = pQ − π.
(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 τ 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 τ 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 ≈ τ c − rQ . Since the
return, rQ , increases with the interest rate the services price decreases with R.
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, τ 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
MP HH
MP HK
μ
=
=
.
(49)
λ
(1 − τ l + rl ) w
(1 − τ k + rk ) r
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”, τ̃ l and τ̃ k , as τ̃ l = τ l −rl
and τ̃ k = τ k − rk , respectively.
Taking (40) for ζ k and substituting it together with (49) and (43) from
Proposition 1 into (23) we get
−λ̇ = λ [(1 − τ k + rk ) r − δ k − ρ] .
(50)
So the total net return on physical capital is equal to the after-tax return on
capital, (1 − τ̃ k ) r, where the relevant tax rate is the effective tax rate, τ̃ k ,
rather than the official rate, τ k , minus the rate of physical capital depreciation. 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
14
So for every nominal interest rate R there exists a threshold consumption tax rate,
τ c (R), below which there is no consumption tax evasion and ac = 1. The precise formula
for the threshold tax rate will be derived later - see Proposition 2.
18
cash:
(1 − τ k + rk ) r − δ k = R − π.
(51)
Using (41) for ζ l and substituting it together with (49) and (42) from
Proposition 1 into (25) we get
−μ̇ = μ [MP HH (1 − x) − δ h − ρ] .
(52)
Formulas (48) and (51)15 confirm the standard result of the general equilibrium 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 expression for the marginal rate of substitution between the reported consumption
and leisure
(1 + τ c ) (1 + R − rQ ) − rc
x
=
(53)
MRScr ,x =
αc
(1 − τ l + rl ) wh
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 hon the credit-account, both surcharged
by the consumption tax, i.e.
i
(1 + τ c ) 1 + cqr (pQ − rQ ) + mcrr (R − rQ ) minus 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 − τ l + rl ) wh.
If we define “the effective inflation and consumption tax rates”, R̃ and
τ̃ c , as R̃ = R − rQ and τ̃ c = τ c − rc , respectively, then formula (53) can be
expressed as
MRScr ,x =
1 + τ̃ c + (1 + τ c ) R̃
.
(1 − τ̃ l ) wh
15
(54)
Condition (51) can be written as the Fisher equation for interest rates Rt =
(1 − τ k + rkt ) rt − δ K + πt .
19
It means that the relevant tax rate for buying one unit of reported consumption is the effective consumption tax rate, τ̃ c , rather than the official rate,
τ c , - see the first term in the numerator; and the relevant inflation tax is the
effective inflation tax, R̃, 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 + τ 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 technology in capital and effective labor according to (3). The firm, taking the
prices of capital and labor services, rt , and wt , respectively, as given, maximizes its profit by choosing effective labor and capital inputs
max
{lGt ht ,sGt kt }
ΠGt = AG (lGt ht )β (sGt kt )1−β − wt lGt ht − rt sGt kt .
(55)
The firm producing the market good and the non-market good face no government taxes nor corruption service fees because these are assumed to fall
on the household. From the first-order conditions of the firms profit maximization problems, we obtain
3.4
wt = βAG (sGt k t )1−β (lGt ht )β−1 ,
(56)
rt = (1 − β) AG (sGt k t )−β (lGt ht )β .
(57)
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, κQt . Using the technology in (4) the
non-corrupt bank maximizes its profit ΠQt by choosing the effective labor
and the amount of deposits, i.e.
max ΠQt = pQt κQt − wt lQt ht − rQt dQt ,
{lQt ht ,dQt }
(58)
κQt = AQ (lQt ht )ωQ (dQt )1−ωQ .
(59)
subject to
The profit of the bank, ΠQt , is defined as the total revenue, pQt κQt , the credit
fee times the amount of demanded services, minus the labor cost, wt lQt ht ,
and the rental payment on the deposit, rQt dQt . The resulting equilibrium
demand for the credit bank labor and deposit are
µ
¶
dQt 1−ωQ
,
(60)
wt = pQt ωQ AQ
lQt ht
Ã
!ωQ
lQt ht
.
(61)
rQt = pQt (1 − ωQ ) AQ
dQt
Using the cash-in-advance constraint (14) and the condition for the deposit in the non-corrupt bank from (13) we find that
(62)
dQt = mrt + qt = (1 + τ c ) crt .
Assuming that the household acts in the sense of a Beckerian (1965) household that combines the credit service with the expenditures in order to get the
amount of credit, qt , equal to the supply of credit services, κQt , so qt = κQt .
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
≡
= AQ
(1 + τ c ) crt
µ
ωQ AQ Rt
wt
ωQ
¶ 1−ω
Q
where aQt is the share of cash transaction in the legal sales revenues.
21
(63)
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 Πit = pit κit − wt lit ht − rit dit
{lit ht ,dit }
(64)
subject to
κit = Ai (lit ht )ωi (dit )1−ωi .
(65)
Profit Πit 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, lit ht , 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
µ
¶1−ωi
dit
,
(66)
wt = pit ω i Ai
lit ht
µ
¶ω
lit ht i
rit = pit (1 − ω i ) Ai
.
(67)
dit
Similarly to the credit sector, we can derive the expression for the corruption output-deposit ratio as:
µ
¶ ωi
ωi Ai pit 1−ωi
κit
= Ai
(68)
dit
wt
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:
κlt = wt lut ht ,
(69)
κkt = rt sut kt ,
(70)
κct = 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
1 − akt
1 − act
¶ ωl
ω l Al τ l 1−ωl
wt
µ
¶ ωk
ωk Ak τ k 1−ωk
sut
≡
= Ak
srt + sut
wt
µ
¶ ωc
ω c Ac pct 1−ωc
cut
≡
= Ac
crt + cut
wt
lut
≡
= Al
lrt + lut
µ
(72)
(73)
(74)
where pct is given by (45) and alt , akt , and act are the relative sizes of noncorrupted 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, τ l , τ k , and τ 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
τ l wt Pt lrt ht + τ k rt Pt srt kt + τ c Pt crt + Ṁt + Ḃt − Bt Rt = Vt .
(75)
It is assumed that the money supply grows at a constant rate of σ,
Ṁt = σMt .
(76)
Consistently with the existence of the balanced growth path in equilibrium,
the nominal bonds supply has to grow at the same rate
Ḃt = σBt .
(77)
In real terms, dividing equation (76) by Pt implies that the government’s
real money is the supply growth rate net of the inflation-based depreciation
of Ṗt /Pt ≡ π t
23
ṁt = (σ − π t )mt .
(78)
Defining Bt /Pt ≡ bt , then (Ḃt − Bt Rt )/Pt = ḃt − bt (Rt − π t ), and the government constraint in real terms is
vt = τ l wt lrt ht + τ k rt srt kt + τ c crt + ṁt + πmt + ḃt − bt (Rt − π t ).
3.7
(79)
Social Resource Constraint
Substituting into the household’s 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 government’s fiscal {τ c , τ k , τ l , vt , σ B }
and monetary {σ M } policies, where σ B = Ḃt /Bt and σ M = Ṁt /Mt , with
σ B = σ M = σ, 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 ΠGt 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 ΠQt 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 Πct , Πkt , Πlt 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 capital
is equal to the return to physical capital,
µ ∗ ∗ ¶1−ε
s k
(1 − x∗ ) − δ H = (1 − τ̃ ∗k ) r∗ − δ K ,
(81)
εAH ∗H ∗t
lH ht
the real variables kt∗ , h∗t , c∗t , c∗rt , c∗ut , qt∗ , d∗it , m∗t , and b∗t , where i ∈ {Q, k, l, c}
grow at the rate g ∗
g∗ = (1 − τ̃ ∗k )r∗ − δK − ρ,
(82)
the inflation rate is equal to
π∗ = σ − g∗
(83)
and the nominal variables Mt∗ , and Bt∗ grow at rate σ, and τ̃ ∗k = τ k −rk∗ is the
constant effective capital tax rate. The shares of capital s∗m , s∗n , s∗H , of labor
∗
∗
, ln∗ , lQ
, lc∗ , lk∗ , ll∗ , x∗ , and the prices of effective labor, w∗ , and capital, r∗ ,
lm
25
∗
stay constant. The rental prices rQ
, rk∗ , rl∗ and the nominal interest rate, R∗ ,
are also constant and equal to
¡
¢
∗
= (1 − ωQ ) R∗ 1 − a∗Q
rQ
rl∗ = (1 − ωl ) τ l (1 − a∗l )
rk∗ = (1 − ωk ) τ k (1 − a∗k )
R∗ = σ + ρ
(84)
(85)
(86)
(87)
with the shares of real credit, 1 − a∗Q , of unreported labor, 1 − a∗l , and capital
income, 1−a∗k , given by (63), and (72)-(73), respectively. When τ c > τ c (R∗ ) ,
r∗
where τ c (R∗ ) = 1+R∗Q−r∗ with τ c (0) = 0 and τ c 0 (R∗ ) > 0, the rental price rc∗
Q
and the corruption fee p∗c are equal to
rc∗ = (1 − ωc ) p∗c (1 − a∗c )
∗
rQ
∗
pc = τ c − (1 + τ c )
;
1 + R∗
(88)
(89)
with unreported sales revenues, 1 − a∗c , given by (74); when τ c 6 τ c (R∗ ) ,
the consumption corruption services are not supplied, i.e. a∗c = 1 and thus
rc∗ = p∗c = 0; if R∗ = 0 then p∗c = τ c > 0 and
rc∗ = (1 − ω c ) τ c (1 − a∗c ) .
The leisure on BGP is given by
1 + (1 + τ c ) R̃∗ + τ̃ ∗c c∗t
x =α
(1 − τ̃ ∗l ) w∗
h∗t
∗
(90)
where we used the effective tax rates at BGP defined as τ̃ ∗l = τ l − rl∗ , τ̃ ∗c =
∗
.
τ c − rc∗ , and R̃∗ = R − rQ
Proof. It follows from equation (49) that the growth rates of shadow
prices of physical and human capitals are the same along BGP,
à ∗! µ ¶
μ̇∗t
λ̇t
=
.
λ∗t
μ∗t
26
Equations (50) and (52) directly imply that the returns on physical capital
and on human capital are the same along BGP
µ ∗ ∗ ¶1−ε
s k
(1 − x∗ ) − δ H = (1 − τ̃ ∗k ) r∗ − δ K .
εAH ∗H ∗t
lH ht
Taking logs of (30),
it and imposing BGP conditions gives
´
³ ∗ differentiating
∗
us − (ċ∗t /c∗t ) − ρ = λ̇t /λt . So using (50) we get
∗
g ≡
µ
ċ∗t
c∗t
¶
= (1 − τ̃ ∗k ) r∗ − δ K − ρ.
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
µ ∗¶ Ã ∗! Ã ∗!
Ṗt
Ṁt
ṁt
−
= g∗
=
∗
∗
mt
Mt
Pt∗
³ ∗ ´
and π ∗ = σ − g∗ . Using this and formula for (27) at BGP gives − λ̇t /λ∗t =
∗
, rc∗ ,
g∗ + ρ = R∗ − π ∗ = R∗ − (σ − g ∗ ) , so R∗ = σ + ρ. The formulas for rQ
rk∗ , rl∗ follow directly from the profit maximization conditions (61) and (67),
the production functions in (59) and (65), the bank services fees derived in
τc
=
Proposition 1, and (72)-(74). From (45) we see that p∗c = 0 if f (τ c ) ≡ 1+τ
c
´
³
∗
∗
∗
∗
rQ (R )
r
(R
)
Q
/∂R∗ > 0,
. Since f 0 (τ c ) > 0, f (0) = 0 and f (1) = 1/2 and ∂ 1+R
∗
1+R∗
³ r∗ (R∗ ) ´
Q
∗
rQ
(0) = 0, and 1+R
< 1/2, there is always unique 1 > τ c (R∗ ) > 0
∗
∗
R =1
r∗ (R∗ )
Q
∗
which satisfies f [τ c (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
¡
¢
∗
(91)
R̃∗ = a∗Q R∗ − ω Q R∗ 1 − a∗Q = a∗Q R∗ + w∗ ˜lQ
³ l∗ h∗ ´ ³ l∗ h∗ ´
Q t
∗
˜
as the labor in the credit sector
where we defined lQ ≡ Qd∗ t = (1+τ
∗
c )crt
Q
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,
∗
, since
a∗Q R∗ , plus the unit labor cost of producing the credit services, w∗˜lQ
∗
these costs diminish the return on deposits, rQ . Similarly, using the definitions
³ ∗ ∗ ´
³ ∗ ∗ ´
˜l∗ ≡ ll ∗ht∗ and ˜l∗ ≡ lk∗ht ∗ and formulas (85)-(86) and (66) for i = l, k
l
k
wl ht
rsG kt
we get
τ̃ ∗l = a∗l τ l + w∗ ˜ll∗
τ̃ ∗ = a∗ τ k + w∗ ˜l∗ .
k
k
k
(92)
(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, p∗c , depends
both on the cosumption tax rate, τ 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 p∗c is simply equal to the tax rate on consumption, p∗c = τ c , an according
to (88) and (66) for i = c we get the formula similar to the other effective
tax rates
³
∗
´
∗
τ̃ ∗c = a∗c τ c + w∗ ˜lc∗
(94)
and ˜lc∗ ≡ lcch∗t .
t
The situation is getting a little more complicated when we assume nonzero inflation tax, i.e. R∗ > 0. According to the results from Proposition 2
28
there is always a range of tax rates, τ c < τ 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
(
τ c , for R∗ > (τ c )−1 (τ c )
∗
(95)
τ̃ c =
τ c − (1 − a∗c ) p∗c + w∗ ˜lc∗ , for R∗ < (τ c )−1 (τ c )
where 0 < p∗c 6 τ c . Unless we explicitly mention otherwise we will assume
further 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∗ < (τ c )−1 (τ 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 (1−x), depending on the amount of leisure and the human capital sector’s
productivity, is the major determinant of the growth rate:
g = rh − δ H − ρ = AH (1 − x) − δ H − ρ.
16
(96)
The stationarity implies that all model extensive variables always grow at the same
constant growth rate. All other model variables, like the shares, keep always constant
values.
17
The 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 growth
when 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 production function is linear, and the real wage is the constant value of the
production function shift parameter, w = AG :
c = y = wlG h.
(97)
It says that the total labor income, wlG h, 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=
ρ
.
AH
(98)
Thus the rest of time is used either for leisure or invested in human capital,
i.e. x + lH = 1 − AρH . The total labor time, l, is allocated among the working
time in the goods production, lG , the credit production, lQ , and two corrupted
banks, ll and lc , i.e.
(99)
lG + lQ + lc + ll = l.
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
18
In comparison to this paper we use there an endogenous growth model with inflation
tax only.
19
Let us note the ratio of a variable z to human capital as ẑ ≡ z/h, so ĉ = ŷ = wlG .
30
labor time, ll = w˜ll l, 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 − w˜ll
1+
lQ
lG
+
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 revenues, respectively, we can express (100) the goods production time as
lG =
1 − w˜ll
l.
1 + (1 + τ c ) w˜lQ ac + w˜lc
(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 production 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)
1 + (1 + τ c ) R̃ + τ̃ c
lG .
(102)
1 − τ̃ l
Then putting together (102), (101), and (98) we get the closed-form solution
for leisure
x=α
x=
1 + (1 + τ c ) R̃ + τ̃ c
1 − w˜ll
αρ
.
1 − τ̃ l
1 + (1 + τ c ) w˜lQ ac + w˜lc AH
(103)
Via (96) there is a close negative link between leisure and growth. Further
results will be analysed in the following section.
20
21
Note that ˜ll = Ωl given by (141) in Appendix A.1.
Note that ˜lQ = ΩQ and ˜lc = Ωc where ΩQ and Ωc 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 output. 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 τ c , labor tax τ l , and inflation tax, R, each causes
increased leisure that decreases the ”capacity utilization rate” of human capital, which is 1 − x, and so decreases the marginal product of human capital
and 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 relative
size of the shadow economy from the respective corruption services activity. An increase in the inflation tax causes an increase in the inflation tax
22
As we already explained in the preceding sections the positive effect of corruption on
growth is the implication of a special setup of our economy. There are several caveats
we have to have in mind before we move to general conclusions. First, in our economy
we assume the corruption which allows to evade distortive taxation - all other types of
corruption which may have no benefits to the society are not considered. Second, despite
that there are definitely other roles of government which can be beneficial to welfare (which
may justify the government’s use of distortive taxes and perhaps outweight it) no positive
role for government is assumed in this paper. Third, the positive effect of corruption on
the growth rate of the economy is not necessarily welfare improving as there are resource
wasted by corruption. The polar setup used in our economy allows us to isolate the pure
interplay between taxation and its evasion and thus to distill a mechanism which would
be 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 ) /∂τ l >
0, and ∂ (1 − ac ) /∂τ c > 0, and ∂pc /∂τ c = 1 − (1 − ωQ ) (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 ceteris
paribus 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 higher
relative 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 proposition we use formula (63) we get the effect of inflation tax on the relative use of credit ∂ (1 − aQ ) /∂AQ > 0. And there is an additionally effect
on the size of consumption corruption services ∂ (1 − ac ) /∂AQ < 0 since
∂pc /∂AQ = − (1 + τ c ) (1 − ωQ ) ∂ (1 − al ) /∂AQ < 0.
33
Corollary 6 The size of the consumption shadow sector shrinks with higher
credit production productivity, other things equal, since all unreported transactions are performed only in cash and higher productivity makes higher relative costs of holding cash.
Corollary 7 In the light of the fact that in this paper we relate the higher
transparency index with the lower productivity of corruption services, the results 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, especially 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 country can have the same size of a shadow economy as a low tax, high corruptionproductivity country. And it shows how lower tax rates with more enforcement, 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 or
goods 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 more
government 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 growthtax relationships. For that purpose let us introduce the following notation:
Φ1 (R, τ c ) ≡ 1+(1 + τ c ) w˜lQ (R) ac (R, τ c )+w˜lc (R, τ c ) , Φ2 (τ l ) ≡ 1−w˜ll (τ l ) ,
Ψ1 (R, τ c ) ≡ 1 + (1 + τ c ) R̃ + τ̃ c and Ψ2 (τ l ) ≡ 1 − τ̃ l then leisure and the
growth rate are given by
x (R, τ c , τ l ) =
Φ1 (R, τ c ) Ψ2 (τ l ) αρ
,
Φ2 (τ l ) Ψ1 (R, τ c ) AH
(104)
and
g (R, τ c , τ l ) = AH [1 − x (R, τ c , τ l )] − δ H − ρ,
(105)
respectively.
Proposition 11 below says that the marginal rate of substitution - see
(102) - increases with taxes; and that consumption, ĉ, 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) τ̃ l = al τ l + w˜ll , and its derivative with
respect to the official tax rate is thus
∂al
∂ ˜ll
∂τ̃ l
=
τ l + al · 1 + w
∂τ l
∂τ l
∂τ l
˜
∂al
∂ ll
where ∂τ
< 0 and ∂τ
> 023 . There are three effects of the official tax rate on
l
l
the effective tax rate: one direct and two indirect ones. The first term in the
23
Remember that ˜ll = Ωl 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
2
l
> 0 and ∂∂ττ̃2l < 0. The key mechanism
to Proposition 11 we show that ∂τ̃
∂τ l
l
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 w˜ll . Clearly, as it is derived in the proof to Proposition
2˜
∂ l̃l
> 0 and w ∂∂τl2l > 0. The composition of the substitution and income
11 w ∂τ
l
l
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) R̃ = aQ R + w˜lQ , and its derivative with respect to
the official tax rate is
∂ R̃
∂ ˜lQ
∂aQ
=
R + aQ · 1 + w
∂R
∂R
∂R
∂a
∂ l̃
where ∂RQ < 0 and ∂RQ > 024 . There are again the three effects of the official
2
R̃
> 0 and ∂∂RR̃2 < 0.
tax rate on the effective tax rate. It was also proved that ∂∂R
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 by w˜lQ . Clearly, as it is derived in
∂˜
l
∂2˜
l
the proof to Proposition 11 w ∂RQ > 0 and w ∂RQ2 > 0. The actual unit labor
costs for tax-evasion and avoiding services influenced by R, when τ c > 0,
are a little more complicated (1 + τ c ) w˜lQ (R) ac (R) + w˜lc (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 τ c 6 τ c (R) (see Proposition
24
l̃Q = ΩQ given in (139).
36
2) there is no consumption tax evasion, i.e. τ̃ c = τ c . When the consumption
tax rate is sufficiently large with respect to R, i.e. τ c > τ c (R) , then there
is the consumption tax evasion in place and τ̃ c = τ c − (1 − ac ) pc + w˜lc . Its
derivative with respect to the official tax rate is thus
#
"µ
¶
∂τ̃ c
∂ ˜lc ∂pc
∂ (1 − ac )
=1−
pc + 1 − ac + w
∂τ c
∂pc
∂pc ∂τ c
∂ l̃c ∂pc
c
where ∂a
< 0 and ∂p
,
> 025 . This time there are four effects of the official
∂pc
c ∂τ c
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
2
∂pc
c
. It will be again proved that ∂τ̃
> 0 and ∂∂ττ̃2c < 0. The key
tax rate via ∂τ
∂τ c
c
c
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 w˜lc . Clearly, as it is derived in the proof
2˜
∂ l̃c
> 0 and w ∂∂τl2c > 0. The actual unit labor costs for taxto Proposition 11 w ∂τ
c
c
evasion and avoiding services influenced by τ c , when R > 0, are a little more
complicated (1 + τ c ) w˜lQ ac (τ c )+w˜lc (τ 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, τ c , τ 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
change
effect on the first fraction
µ
¶ a tax rate
¶
µ
αρ Ψ2 (τ l )
AH Ψ1 (R,τ c )
∂
Φ1 (R,τ c )
Φ2 (τ l )
∂
Φ1 (R,τ c )
Φ2 (τ l )
, with
> 0. The negative income
of (104),
∂∆
∂∆
efect is captured
by ¶a tax rate
change¶ effect on the second fraction of (104),
µ
µ
αρ Φ1 (R,τ c )
AH Φ2 (τ l )
25
∂
Ψ2 (τ l )
Ψ1 (R,τ c )
∂∆
∂
, with
Ψ2 (τ l )
Ψ1 (R,τ c )
∂∆
< 0. For each tax rate ∆ ∈ {R, τ c , τ l }
l̃c = Ωc given in (140).
37
©
ª
¯ ∈ R̄, τ̄ c , τ̄ l such that
there exists a tax rate ∆
´
´⎤
³
³
⎡
Φ1 (R,τ c )
Ψ2 (τ l )
∂
∂
Φ2 (τ l )
Ψ1 (R,τ c )
∂x (R, τ c , τ l )
αρ ⎣ Ψ2 (τ l )
Φ1 (R, τ c )
⎦ > 0,
=
+
∂∆
AH Ψ1 (R, τ c )
∂∆
Φ2 (τ l )
∂∆
¯ i.e. the substitution effect is the dominating factor in interfor ∆ ∈ [0, ∆),
¯ 26 . Formula (96) implies that the same forces with opposite signs
val [0, ∆)
influence the growth rate.
Proof. See proof in Appendix A.2.
Corollary 12 Since the income effect of any tax rate is exclusively caused by
the 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 economy’s 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;
26
Accordingly with the former literature, see e.g. Chari, Kehoe, and Manuelli (1997),
we consider as empirically relevant - and thus interesting for our analysis - the interval
where taxes have negative effect on growth, i.e. the interval in which the substitution
effect dominates the income effect.
27
The 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 = τ c = τ l = 0; leisure is
simply
αρ
;
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 + τ c ) R + τ c αρ
;
1 − τl
AH
(106)
3. an economy with banking and no tax evasion - noted as B,
xB =
1
αρ
1 + (1 + τ c ) R̃ + τ c
;
1 − τl
1 + (1 + τ c ) w˜lQ AH
(107)
4. an economy with banking and tax evasion in labor and capital income
when τ c 6 τ c (R) - noted it as E,
xE =
1 − w˜ll
αρ
1 + (1 + τ c ) R̃ + τ c
.
1 − τ̃ l
1 + (1 + τ c ) w˜lQ AH
(108)
5. an economy with banking and tax evasion in all sectors when τ c >
τ c (R) - noted it as F,
xF =
1 − w˜ll
αρ
1 + (1 + τ c ) R̃ + τ̃ c
.
1 − τ̃ l
1 + (1 + τ c ) w˜lQ ac + w˜lc AH
(109)
Proposition 14 The economies under different regimes but with the same
mix 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 the
presence of tax avoidance, are strictly lower than their nominal counterparts
- according to Proposition 2 e.g. τ̃ c = τ c − rc < τ c ; and (2) the labor cost
of evasion activities lowers the available amount of labor for production according to (101) e.g. the presence of w˜lc lowers lG . Having this in mind it
is trivial to show that for the given tax mix (τ c , τ l , τ 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, τ c , τ l } causes the growth rate to decrease at the rate
µ
∂g
∂∆
¶I
= −αρ < 0
when evaluated at the optimum where R = τ c = τ 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 operates
¶D
¶B
¶E
µ
µ
µ
∂g
∂g
∂g
<
<
< 0;
∂τ c τ c =0
∂τ c τ c =0
∂τ c τ c =0
¶D
¶B
¶E
µ
µ
µ
∂g
∂g
∂g
<
=
< 0;
∂τ l τ l =0
∂τ l τ l =0
∂τ l τ l =0
¶D
¶B
¶E
µ
µ
µ
∂g
∂g
∂g
<
<
<0
∂R R=0
∂R R=0
∂R R=0
when τ c 6 τ c (R) and
regimes; and
¶D
µ
∂g
∂τ c τ c =τ c (R)
¶D
µ
∂g
∂τ l τ l =0
µ
¶D
∂g
∂R R=0
there is no consumption corruption in any of the
¶F
∂g
<
<
< 0;
∂τ c τ c =τ c (R)
τ c =τ c (R)
¶B
¶F
µ
µ
∂g
∂g
<
<
< 0;
∂τ l τ l =0
∂τ l τ l =0
µ
µ
¶B
¶F
∂g
∂g
<
<
<0
∂R R=0
∂R R=0
µ
∂g
∂τ c
¶B
µ
when τ c > τ c (R) and there is consumption corruption in regime F ..
∂x
∂x
∂x
|R=τ c =τ l =0 = AαρH > 0; ∂τ
|R=τ c =τ l =0 = AαρH > 0; ∂R
|R=τ c =τ l =0 =
Proof. ∂τ
c
l
αρ
∂g
> 0. Since g = AH (1 − x) − δ H − ρ, ∂x < 0. It follows that, at the optiAH
mum of no taxes and no evasion/avoidance activity, the growth rate falls at
the same rate with an increase in either τ c , τ l , or R. The derivation of the
growth effects ordering under different regimes using the formulas (106)-(109)
³ ´B
³ ´D
∂x
1+R αρ
∂x
=
>
=
is straightforward. For example, ∂τ
1−τ l AH
∂τ c
c
τ c =τ c (R)
τ c =τ c (R)
³ ´F
1−w˜
ll
αρ
αρ
1+R̃
1
∂x
1+R̃
>
= 1−τ̃
. The reason
˜
1−τ l 1+(1+τ c (R))w˜
∂τ c
lQ AH
1+(1+τ
(R))w
lQ +wl̃c AH
l
c
τ c =τ c (R)
³ ´E
³ ´B
∂g
∂g
= ∂τ
is caused by the fact that unfor the equality between ∂τ
l
l
τ l =0
τ l =0
der the condition τ l = 0 both regimes have the same tax avoidance/evasion
structure - only banking to avoid the inflation tax as there is no consumption
corruption activity in regime E.
Consistently with the declining dominance of the substitution effect (see
Proposition 11) as the tax rate increases in the presence of tax avoiding
activities in the economy, the negative tax effect is strongest initially and
41
getting to be less strong as the tax rate increases. This and the relation to
tax elasticities will be demonstrated in the next subsection.
5.1.3
Tax Elasticities and Growth Effects
To get further insight into the behavior of the model we will analyze in this
section how the tax elasticity of the taxed quantity per unit of human capital
is related to the growth effect of tax. We mainly show that the negative
effect of taxes on growth is getting weaker as the tax rate increases due to
the increasing tax elasticity.
We start with the inflation tax. We show in Proposition 19 below that the
obtained results are the generalization of the results obtained in Gillman and
Kejak (2005). However, before getting there we derive the interest rate elasticity of money demand and its relation to the elasticity of the substitution
between money and credit services in the following proposition.
Proposition 16 The interest elasticity of money demand (per human capital), ηm̂
R , is the sum of the interest elasticity of consumption (per human
m̂/ĉ
capital), ηĉR , and the interest elasticity of the inverse velocity, ηR ,
m̂/ĉ
ĉ
η m̂
R = ηR + ηR
where
1
ηĉR = −ηΨ
R
m̂/ĉ
ηR
ã
= ηRQ =
aQ ac aQ (1 − aQ ) ac ac
η −
ηR
ãQ R
ãQ
a
ω
1−a
m
ω c 1−ac pc
with ãQ ≡ (1+τ
= 1−(1−aQ )ac , and ηRQ = − 1−ωQQ aQQ , η aRc = − 1−ω
ηR
c )c
c ac
being the interest elasticities of the share of cash-card transactions,
and of ´the
³ ˜
∂
l
1
c
+
= (1 + τ c ) w ∂RQ ac + w˜lQ ∂a
share of legal consumption, respectively; ∂Ψ
∂R
∂R
˜
∂ lc
w ∂R
> 0 and ηpRc < 0 is the interest elasticity of the consumption fee. The
money demand elasticity can also be decomposed as
ĉ
ηm̂
R = η R + (1 − ãQ ) ε
42
where ε is the elasticity of substitution between money and credit services,
and
¶
µ
1
aQ ac aQ
ac
ηR − ηR
.
ε=
1 − aQ
ãQ
As the interest rate R increases the elasticities of the share of cash transactions, ãQ , and of the consumption (per human capital) increase in absolute
value, ĉ, i.e.
¯
¯
¯ ĉ ¯
¯ ∂ηãQ ¯
¯ ∂η ¯
¯ R ¯
¯
¯ > 0 and ¯¯ R ¯¯ > 0
¯ ∂R ¯
∂R
so do the interest elasticity of money demand and of substitution between
money and credit services
¯ m̂ ¯
¯
¯
¯ ∂η R ¯
¯ ∂ε ¯
¯
¯
¯
¯
¯ ∂R ¯ > 0 and ¯ ∂R ¯ > 0.
Proof. See proof in Appendix A.2. Since m̂ ≡ m
, ĉ ≡ (1+τh c )c , and
h
a (1+τ )c
c )cu
u +mr
= m
= (1+p
+ Q(1+τ cc)c r = (1 − ac ) + aQ ac = 1 − (1 − aQ )ac ≡
(1+τ c )c
(1+τ c )c
m̂
ĉ
ã
Ψ2 (τ l )
Q
Ψ1
ĉ
ĉ
ãQ . Then ηm̂
R = η R + η R . Since ĉ = (1 + τ c ) Ψ1 (R,τ c ) l, η R = −η R . Since
(1−aQ )ac ac
ã
a
∂ãQ
∂a
a a a
c
c
= − ∂a
+ ∂a
a + ∂RQ ac , ηRQ = ãQQ c ηRQ −
ηR where ηRQ =
∂R
∂R
∂R Q
ãQ
ω
− 1−ωQQ
1−aQ
aQ
∂pc
∂R
ωc 1−ac pc
< 0, ηaRc = − 1−ω
ηR > 0 with ηpRc =
c ac
∂pc R
∂R pc
< 0 and28
1+ω R
Q
= − (1 + τ c ) (1 − aQ ) (1+R)
2 < 0. To derive the elasticity of substitution
, let us define first the moneybetween money and credit services, ε = ∂∂lnlnMC
R
ã
1−a
+a
a
ã
c
Q c
1
r
= 1−a
= 1−ãQQ . Then ε = 1−ã
η RQ
credit ratio, MC ≡ 1−amu +m
Q
(1+τ
)c
a
( Q)
c r
( Q) c
aQ
ã
ĉ
and thus³ ηm̂
=
η
+
(1
−
ã
)
ε.
As
η
ηaRc > 0, ηRQ <´0. Further
Q
R
R ´
R < 0 and
³
∂ ã
∂ãQ
∂aQ
∂a
ã
ca
c 1−a
a + ∂a
ãQ −aQ ac ∂RQ aQ ac aQ ∂ηaQ
− ∂RQ ac + ∂a
Q ) ãQ −(1−aQ )ac ∂R
∂ηRQ
∂R c
∂R Q
∂R (
R
=
η
+
−
ηaRc −
R
∂R
ã2Q
ã2Q ∂R
ã2Q
h ∂ac
a
pc i
c
c
ac (1−aQ ) ∂η a
∂η RQ
∂ηa
pc
ωc
1−ac ∂η R
∂R
R
R
,
<
0,
=
η
−
> 0 for low R
∂R
∂R
∂R
1−ωc
a2 R
ac ∂R
ã2
c
Q
∂η pRc
pc
→
0
and
→ −∞. Since the numerator in the
when
η
∂R ´
³ R
∂aQ
∂ã
∂ac
c
is − ∂R ac + ∂R (1 − aQ ) ãQ − (1 − aQ ) ac ∂RQ = (1 − aQ ) ∂a
∂R
0 and the numerator in the first term
c
aQ ∂a
+ ac (1 − ac )
∂R
ative and
28
ã
∂ηRQ
∂R
∂aQ
∂R
³
∂aQ
a
∂R c
+
∂ac
a
∂R Q
´
third term
∂a
− ac ∂RQ >
ãQ − aQ ac
∂ãQ
∂R
=
< 0 for low values
¯ ãof¯ R, all four terms are neg¯ ∂ε ¯
¯ ∂η Q ¯
R ¯
¯ ¯ > 0.
>
0
and
thus
< 0 which means that ¯¯ ∂R
∂R
¯
As we mentioned earlier we assume here the economy in regime F when there is present
consumption corruption activity, i.e. the situation when τ c > τ c (R).
43
³ 2
´
2
∂ l̃
∂˜
l
∂ 2 ac
c
˜
= (1 + τ c ) w ∂RQ2 ac + 2w ∂RQ ∂a
a
+
w
l
+ w ∂∂Rl̃2c . Since
Q ∂R2
∂R c
ωc
ωc
¡ ωc Ac ¢ 1−ω
∂ 2 ac
ωc
c p 1−ωc
=
A
c
c
2
∂R
1−ωc
w
According to Proposition above the share of money in total transactions,
ãQ , can be decomposed into the pocket cash uses in the illegal consumption
purchases, 1−ac , and the share of money (in the form of the cash card), aQ ac ,
where aQ is the share of cash transactions in the legal consumption purchases
only, so ãQ = 1 − ac + aQ ac . 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
£
¤
ã
a
that ac increases with R. Nevertheless, as η RQ = ãaQc aQ η RQ − (1 − aQ ) ηaRc
the interest elasticity ãQ is always negative.
Further,
∂ 2 Ψ1
∂R2
Corollary 17 In a distorted economy without any tax evasion activity aQ =
¡ ¢D ³ m̂/ĉ ´D
ac = 1, so ãQ = 1, and ηĉR = η R
= ε = 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 no
a
m̂/ĉ
tax evasion activity29 ac = 1 so ãQ = aQ < 1, and ηR = ηRQ , money demand
aQ
aQ
ω Q 1−aQ
ĉ
ĉ
elasticity is equal to ηm̂
R = η R +η R = η R +(1 − aQ ) ε where η R = − 1−ωQ aQ
a
1
and ε = ηRQ 1−a
.
Q
Proposition 19 There exists R̄ > 030 , such that any increase in the interest
rate for R ∈ (0, R̄) causes a decrease in the growth rate, g, according to
½
¾
¡
ηĉR
∂g
1 − w˜ll
aQ ¢
ĉ
(1 + τ c ) aQ 1 + ηR + η R + Γ1
= −αρ
+ Γ2 < 0 (111)
∂R
Ψ1 Ψ3
R
29
It is when the economy is in regime E when there is no consumption corruption
activity, i.e. the situation when τ c 6 τ c (R).
30
See Proposition 11.
44
where
i
h
Γ1 = τ c + (1 + τ c ) w˜lQ − pc (1 − ac ) > 0
"
#
∂a
∂ ˜lQ
∂ [(1 − ac )pc ]
c
Γ2 = (1 + τ c ) w (1 − ac )
− w˜lQ
−
>0
∂R
∂R
∂R
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 Ψ1 and Ψ3 are defined in Proposition 11. Further, the de, diminishes in absolute value with the increasing nominal intercrease, ∂g(R)
∂R
2 g(R)
> 0 for R ∈ (0, R̄).
est rate, i.e. ∂ ∂R
2
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
a
elasticities, ηĉR , and ηRQ , does not compose the money interest elasticity since
(1−aQ )ac ac
ãQ
ãQ
aQ ac aQ
ĉ
ηm̂
ηR .
R = η R + η R and η R = ãQ η R −
ãQ
c
in the bracketed part of formula
Note that the terms (1 + τ c ) aQ + ∂τ̃
∂R
∂g
for ∂R in (111) captures fully the substitution effect of the nominal interest
¡
a ¢
rate. The remaining terms of the formula, (1 + τ c ) aQ η ĉR + η RQ , compose
the income effect.
Corollary 20 In case of no tax evasion activities, i.e. the economy is in
regime B, there is a negative growth effect for R ∈ (0, R̄) where R̄ > 0 given
by
¶B
µ
n
¡
¢ τ c ĉ o
1
∂g
m̂
h
i
(1 + τ c ) aQ 1 + η R + ηR < 0.
= −αρ
∂R
R
1 + (1 + τ c ) w˜lQ (1 − τ l )
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 τ c = 0, obtained and documented by empirical evidence in Gillman and Kejak (2005). Additionally, according to Proposition
19 above 1 + η m̂
R is the interest rate elasticity of the inflation tax revenue.
Now we continue with the labor income tax.
45
Proposition 21 There exists τ̄ l > 031 , such that any increase in the interest
rate for τ l ∈ (0, τ̄ l ) causes a decrease in the growth rate, g, according to
(
)
³
´¡
˜ll
¢
Ψ2
∂
∂g
1 − w˜ll 1 + ηaτ ll + τ l w
<0
(112)
= −αρ
al
∂τ l
Ψ1 Ψ23
∂τ l
where ˜ll are the unit amounts of labor used in labor income services, η aτ ll =
ωl 1−al
< 0 is the interest elasticity of the share of reported labor income,
− 1−ω
l al
∂ l̃l
∂τ l
> 0, and Ψ1 , Ψ2 and Ψ3 are defined in Proposition 11. Further, the
l)
, diminishes in absolute value with the increasing labor income
decrease, ∂g(τ
∂τ l
∂ 2 g(τ l )
tax rate, i.e. ∂τ 2 > 0 for τ l ∈ (0, τ̄ l ).
l
Proof. See proof in Appendix A.2.
The next case is that of consumption tax.
Proposition 22 There exists τ̄ c > 032 , such that any increase in the interest
rate for τ c ∈ (0, τ̄ c ) causes a decrease in the growth rate, g, according to
⎧
h
i ⎫
¡
¢
∂ l̃c
1−ac ∂pc
⎨
˜
R̃
+
1
−
(1
−
a
)
1
+
η
+
w
Ψ
− ⎬
c
∂g
1 − wll
2
∂τ c
h ³
´pc
i∂τ c
<0
= −αρ 2
˜
⎭
∂τ c
Ψ1 Ψ3 ⎩
−Ψ1 ac 1 + ηaτ c 1+τ c w˜lQ + w ∂ lc
c
τc
∂τ c
(113)
where ˜ll , ˜lc are the unit amounts of labor used in labor income and consumption tax-evasion services, η aτ cc < 0 is the tax elasticity of the share of reported
c
> 0 is the fee elasticity of the share of unreported
consumption sales, η1−a
pc
∂ l̃c
consumption sales, ∂τ c > 0, and Ψ1 , Ψ2 and Ψ3 are defined in Proposition 11.
c)
, diminishes in absolute value with the increasing
Further, the decrease, ∂g(τ
∂τ c
∂ 2 g(τ c )
consumption tax rate, i.e. ∂τ 2 > 0 for τ c ∈ (0, τ̄ c ).
c
Proof. See proof in Appendix A.2.
Corollary 23 In case of no inflation there is a negative growth effect for
τ c ∈ (0, τ̄ c ) where τ̄ c > 0 given by
31
32
¢
∂g
1 − w˜ll ¡
= −αρ
ac 1 + ηaτ cc + ηĉτ c < 0.
∂τ c
Ψ1 Ψ3
See Proposition 11.
See Proposition 11.
46
Additionally, term 1 + η aτ cc + η ĉτ c is equal to the consumption tax rate
Tc
where Tc = τ c ac ĉ.
elasticity of the consumption tax revenue, η Tτ cc ≡ ∂∂ ln
ln τ c
Proposition 24 An increase in the corruption service productivity causes
an 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 the
tax 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 following Table. for the standard parameters were chosen: the share
47
of capital in the goods and human capital sectors, β = α = 0.36, physical
and human capital depreciation rates, δ K = δ H = 0.05, the discount rate,
ρ = 0.04 and the risk aversion parameter θ = 1. Given a growth rate of the
economy of g = 0.02, the weight of leisure in utility function φ = 2.5 and
productivity parameter in human capital sector AH = 0.236 yields a leisure
of x = 0.68 (in Parente et al. 2000 is x = 0.48 in Jones et al. 1993 = 0.7);
here the money growth rate is σ = 0.11, the inflation rate is π = 0.09, and
the net nominal interest rate is R = 0.15. For the credit sector technology,
the degree of diminishing returns is set at γ = 0.2 as based on the estimated
value of this parameter in the interval (0.2, 0.3) that is found for the US
and Australia in the money demand estimation of Gillman and Otto (2003).
The credit productivity parameter is put at AQ = 0.76 to yield a share of
cash in transaction equal to a = 0.7 (as in Gillman, Kejak, 2005 and as is
similar to Dotsey and Ireland, 1996). The labor shares in the corruption
sectors for capital, labor and consumption are set at ω k = ω l = ω c = 0.3
and the productivity parameters are assumed to be Ak = Al = Ac = 1. The
productivity parameters in the goods sector is AG = 1.5. The tax rates are
set to τ k = τ l = τ c = 0.15.
6.1
The Size of Shadow economy
Let us define the size of shadow economy as the ratio of the unreported
activity to the reported one
wlu h + su rk + cu
yu
=
y
y
where unreported activity is composed of unreported labor income, unreported capital income and unreported consumption. The reported activity
composes of the goods output, y.
6.1.1
The Effect of Tax Rate Changes
In Fig.4 we can see that the size of shadow economy defined above increases
with the increase of tax rates. The reason for positive effects of taxes on
48
a) shadow economy size
b) shadow economy size
0.25
0.4
τ only
K
0.2
all taxes
0.3
inflation only
all taxes
yU/y
yU/y
0.15
0.2
0.1
0.1
0.05
0
0
0.5
1
Money Growth Rate
0
1.5
0
c) shadow economy size
0.3
d) shadow economy size
0.4
τ only
τ only
C
L
all taxes
all taxes
0.3
yU/y
0.3
yU/y
0.2
τ
K
0.4
0.2
0.1
0
0.1
0.2
0.1
0
0.2
τ
0.4
0.6
L
0
0
0.2
τ
0.4
0.6
C
Figure 2: The effect of taxes on the size of shadow economy
the shadow economy size is straightforward. There is no unreported income
when inflation tax on is the only tax in the economy.
Otherwise in the comparison of the single tax case and the all taxes case
the unreported income shifts up as the existence of more taxes means higher
size of the shadow economy. The shape of the relationship between the money
growth rate and the size of shadow economy is U-shaped.
6.1.2
The Effect of Sectoral Productivity Changes
Fig.5 captures the effect of decreased productivity parameters on the size of
the shadow economy. The decrease of efficiency in the capital, labor and consumption corruption services leads to the lower size of the shadow economy.
The lower efficiency of banking sector leads to the higher shadow economy
size.
49
a) shadow economy size
b) shadow economy size
0.22
0.3
benchmark
decreased A
D
benchmark
decreased A
0.25
0.215
K
y /y
y /y
0.2
U
U
0.21
0.15
0.205
0.1
0.2
0
0.5
1
monetary growth
0.05
1.5
0
0.1
0.2
τ
0.3
K
c) shadow economy size
d) shadow economy size
0.3
0.35
benchmark
decreased A
0.28
benchmark
decreased A
L
0.3
C
0.26
0.25
U
y /y
U
y /y
0.24
0.22
0.2
0.2
0.18
0.15
0.16
0
0.2
τ
0.4
0.6
L
0.1
0
0.2
τ
0.4
0.6
C
Figure 3: The effect of the decreased efficiency on the size of shadow economy
50
a) tax revenue-output ratio
b) tax revenue-output ratio
0.2
all tax revenues/reported output
all tax revenues/reported output
0.8
inflation only
all taxes
0.6
0.4
0.2
0
0
0.5
1
Money Growth Rate
τ only
K
0.15
all taxes
0.1
0.05
0
1.5
0
c) tax revenue-output ratio
τ only
L
all taxes
0.4
0.3
0.2
0.1
0
0
0.2
τ
0.4
τ
0.2
0.3
K
d) tax revenue-output ratio
0.4
all tax revenues/reported output
all tax revenues/reported output
0.5
0.1
C
0.3
0.2
0.1
0
0.6
τ only
all taxes
0
L
0.2
τ
0.4
0.6
C
Figure 4: The effect of taxes on the tax revenues
6.2
6.2.1
Government Revenue
The Effect of Tax Rate Changes
Define government revenue as the sum of taxes and seignorage, and normalize
by total output
τ k sm rk + τ l (lm + lQ )wh + τ c cm + Rm
.
y
The tax revenue ratio is increasing in the tax rates until it reaches a point
behind which the revenues start to decline (as it can be seen in case of capital
taxes in Panel b).
6.2.2
The Effect of Sectoral Productivity Changes
The effect of sector productivities on the tax revenue is shown in Fig.7.
The decrease in the efficiency of the credit sector leads to higher relative
51
a) tax revenue-output ratio
b) tax revenue-output ratio
0.22
all tax revenues/reported output
all tax revenues/reported output
0.8
0.6
0.4
benchmark
decreased A
0.2
0
0
D
0.5
1
Money Growth Rate
0.19
0.18
0.17
0
c) tax revenue-output ratio
τ
0.2
0.3
K
0.5
all tax revenues/reported output
all tax revenues/reported output
0.1
d) tax revenue-output ratio
0.5
0.4
0.3
0.2
benchmark
decreased A
0.1
0
K
0.2
0.16
1.5
benchmark
decreased A
0.21
0
0.2
τ
0.4
L
benchmark
decreased A
0.3
0.2
0.1
0.6
L
C
0.4
0
0.2
τ
0.4
0.6
C
Figure 5: The effect of decreased efficiency on the tax revenues
tax revenues. The lower efficiency of corruption services leads to higher tax
revenues.
6.3
6.3.1
The Growth Effects
From Tax Rate Changes
First consider the case when there is tax evasion or avoidance, versus the
model in which there is zero tax evasion or avoidance. This latter case can
be derived by setting the productivity parameters in the corruption services
and credit production equal to zero. The simulations are presented in the
following figure. It shows the almost linear (negative) relation between the
growth rate and the tax rate for the case of no evasion/avoidance. While
with evasion/avoidance, the tax-growth profile is rather nonlinear, with the
growth rate decreasing at a significantly decreasing rate as the tax rises.
52
a) inflation tax only
b) capital tax only
0.08
0.065
no tax avoid
infl tax evas
no tax avoid
cap tax avoid
0.06
growth rate
growth rate
0.06
0.04
0.02
0
0.055
0.05
0.045
-0.02
0
0.2
0.4
0.6
0.8
monetary growth rate
0.04
1
0
c) labor tax only
0.2
0.3
d) consumption tax only
0.08
no tax avoid
labor tax avoid
0.06
no tax avoid
cons tax avoid
0.06
growth rate
growth rate
τ
K
0.08
0.04
0.02
0.04
0.02
0
-0.02
0.1
0
0.1
0.2
τ
0.3
0
0.4
L
0
0.2
τ
0.4
0.6
C
Figure 6: Comparison of Tax-Growth Relationship in cases when there is
only one tax and either no tax avoidance or only one avoidance channel
53
a) inflation tax only
b) capital tax only
0.06
0.06
0.058
growth rate
growth rate
0.05
0.04
0.03
0.056
0.054
0.02
0.01
0
0.5
1
monetary growth rate
0.052
1.5
0
0.1
0.2
τ
0.3
K
c) labor tax only
d) consumption tax only
0.06
0.06
0.055
growth rate
growth rate
0.04
0.02
0.05
0.045
0.04
0
0.035
-0.02
0
0.2
τ
0.4
0.03
0.6
L
0
0.2
τ
0.4
0.6
C
Figure 7: The inflation/tax-growth relationships
In the figures below we demonstrate the relationship between growth and
the tax rates on the balanced growth paths. Fig.1 shows the situation when
only one tax is in place: Panel a), b), c), and d) depict the growth effects
of the monetary growth rate, the capital tax rate, the labor tax rates, and
the consumption tax rates, respectively. All these relationships exhibit the
negative growth effects. The negative relationships are diminishing for all
taxes but the tax on labor, which is almost linear.
In Fig.2 we make a comparison of the case with one tax only with that
when all taxes are in place (e.g. in Panel a) other taxes are set at τ k =
τ l = τ c = 0.15). All the panels show that the presence of all taxes makes
the growth rate lower. There is almost no effect on the slope of the labor
tax-growth relationship when there exist other taxes (Panel c). In all other
cases (Panels a), b), and d) the slopes of the tax-growth relationship are
getting steeper when there are more taxes than a single one. [This result is
54
a) inflation-growth relationship
b) capital tax-growth relationship
0.08
0.08
infl only
all taxes
0.06
growth rate
growth rate
0.06
0.04
0.02
τK only
0.04
all taxes
0.02
0
-0.02
0
0.5
1
monetary growth rate
0
1.5
0
0.1
0.2
τ
c) labor tax-growth relationship
d) consumption tax-growth relationship
0.08
0.08
τ only
τ only
all taxes
all taxes
L
C
0.06
growth rate
growth rate
0.06
0.04
0.02
0.04
0.02
0
-0.02
0.3
K
0
0.2
τ
0.4
0
0.6
0
L
0.2
τ
0.4
0.6
C
Figure 8: Comparison of Tax-Growth Relationship in cases with only one
tax and with all taxes present
at odds with our prediction!]
6.3.2
From Sectoral Productivity Changes
In Fig.3 the effects of productivity parameters on the inflation-growth effect
are depicted. We can see that the decreased efficiency of producing credit,
capital, labor and consumption corruption services leads to a steeper taxgrowth relationship as the tax avoidance instrument is getting more costly.
7
Discussion
Without using a taste for corruption, the extent of corruption is explained as
based on relative prices: the demand price for services, being the government
tax rate, and the supply marginal cost of producing such services. Explaining
55
a) changes in A
b) changes in A
D
0.03
benchmark
decreased A
benchmark
decreased A
D
growth rate
0.02
growth rate
K
0.03
0.01
0
K
0.025
0.02
-0.01
-0.02
0
0.5
1
monetary growth
c) changes in A
0.015
1.5
τ
0.2
0.3
K
C
0.03
benchmark
decreased A
0.03
benchmark
decreased A
0.025
L
0.02
growth rate
growth rate
0.1
d) changes in A
L
0.04
0.01
0
-0.01
-0.02
0
C
0.02
0.015
0.01
0.005
0
0.2
τ
0.4
0
0.6
L
0
0.2
τ
0.4
0.6
C
Figure 9: The Tax-Growth Effect and the decrease in sectors’ efficiency
56
differences in corruption levels through the efficacity of corruption production, rather than through tastes for corruption, appears to be a plausible
approach. Consider countries in Africa, the Middle East, and Central Asia
for example, where tribal connections are very strong. Here corruption can
be easier to produce if a given tribe is able to keep the information about
the corruption activity within its tribe, which may also be represented in the
local and national government. This may allow economic activity that more
easily avoids government taxes because the many people involved act as an
extended family to some extent. This story, of keeping it within the family, is
also often told for the Western corruption providers, such as the mafia. The
existence of an extended family structure that guards information on illegal
activity, can make lower the cost of producing such activity.
In contrast, open, civil, societies in which family units give their allegiance to the government, interact readily in the public realm, and decentralize their education, leisure, and work efforts by demanding and supplying
labor and capital outside of the family or tribal group, makes for a more complex process in hiding illegal activity: the information on such activity is not
well-guarded and, by definition of an open society, easier to access by the
general public. This makes corruption activity more costly to produce. For
example, in the relatively open Scandinavian societies, such corruption is not
very feasible. In ex-communist countries, which thrived on closed societies
for more than half a century, corruption is more feasible because of longstanding institutional experience in such activity. The cost of corruption is
not so high, as information on such activity is easier to hide than in more
open societies. Russian and Eastern European countries after the government changes in 1989-1990 would be said to have had a certain efficiency
in producing corruption, that presumably is gradually decreasing as open
societies spread, and human capital accumulates.
Human capital productivity increases indicate an interesting policy implication. A greater efficacity of human capital production leads to less corruption activity, a very strong result in the model. It suggests that perhaps
instead of trying to suppress corruption activity per se, it may be more efficacious to spend on human capital investment so that corruption gradually
57
fades away.
Other policy implications left for future research include the optimal rate
of inflation is such an economy, as part of a Ramsey problem. The inflation
tax is the only tax that falls effectively on the non-market sector. Therefore
the ability to tax the otherwise non-taxed sector suggests that the optimal
inflation rate will not reside at the second-best Ramsey-Friedman optimum
of R = 0, which holds in related endogenous growth monetary economies
with labor, capital and goods taxes (?). Rather the optimum would likely
be at some positive level, although this may still be quite low.
8
Conclusion
Given the model’s ability explain certain correlation facts, it is put to work
in explaining the effect of tax evasion on economic growth. The principles
that guide the effect of avoiding inflation also determine the effect of tax
evasion. The equilibrium price of the corruption service equals the corresponding government tax rate: demand is perfectly elastic at the fixed tax
rate. A rise in the legal tax rate raises the price of the tax evasion. Following
from the Beckerian law-avoidance margin, there is movement up the upward
sloping marginal cost curve for corruption services; more evasion is supplied,
causing less labor or capital income, or goods revenues, to be reported. The
degree of the increase in shadow activity depends upon the productivity of
the corruption production function. Fact 1, concerning a lack of correlation, is explained by emboding the degree of transparency of the country’s
economy within productivity parameter for producing corruption: if everyone sees the corruption activity, then it is very difficult to engage in it and
the productivity parameter is small. Transparent but high tax Sweden, has
inefficient corruption production and so relatively little corruption; Russia is
efficient with high corruption productivity and so produces much tax evasion even at low tax rates. As a tax rate increases, corruption activity and
the shadow sector more together, as in Fact 2, but high tax rates and small
58
shadow sectors can coexist when the corruption is hard to produce.33
An important qualification is that corruption here means only tax evasion.
But note that all government regulations might be viewed as implicit taxes
on output, inputs, or finance. And typically, corruption avoids such myriad
regulations as well as direct taxation. While all tax revenues are returned
lump sum to the consumer in the model, with regulations the implicit tax
revenue is wasted. Therefore the paper can be thought of as capturing stylistically some of the broader elements of corruption beyond taxation, with the
change that a portion of government revenues would be a deadweight loss.
First the paper identifies some stylized facts as based on correlation evidence. Then it presents a model that can explain these facts. Having this
preliminary basis established, the paper then explores how taxes affect the
growth rate given the existence of evasive activity such as corruption and
exchange credit. A general principle of public finance within endogenous
growth emerges, an extension of the results in ?. The growth rate falls as
the tax rate increases, and the corruption (evasion) makes this fall occur at
a significantly decreasing rate. The reason is the same for each of the 4 taxes
in our model: the demand for the good (which is being taxed) is relatively
inelastic at a low tax rate and becomes more shadow-price elastic as the tax
rises and evasion occurs. This is for each tax when there are no other of the
4 taxes but the one. For when other taxes are positive, the principle is the
same but modified: The growth rate will decrease at a decreasing rate, but
the presence of the other taxes will make the demand already more elastic.
So as a particular tax rises, given the other taxes, the growth rate falls at a
decreasing rate, but it may be a smaller fall in the growth rate, as the tax
rises from zero, because other taxes already make the demand shadow-price
elastic. There is a turn-around point of the growth rate rising, only once
the elasticity becomes one, as occurs with the inflation tax in Gillman and
Kejak. And, it is never in the interest of a government, to be in the region
with an elasticity greater than one (in magnitude) since it means that tax
33
It is assumed that the corruption service provides the means to have unreported
income, with a type of certainty-equivalence that abstracts from the probabilities of getting
caught and the optimal penalty literature, for example as in ?.
59
revenues decrease as the tax rises!
The rate of growth, along the balanced-growth path, depends on the marginal products of human and physical capital, which are equal in equilibrium.
The marginal product of human capital is a product of the constant marginal
product of the effective labor in the human capital investment function and
of the variable capacity utilization rate, which is the employment rate of one
minus the leisure time. For the labor tax, goods value-added tax, and the
inflation tax, the channel of the growth effect is strictly through the human
capital capacity usage. The marginal product of physical capital is simple
the standard real interest rate; the capacity utilization rate is assumed to be
one. For the capital tax, the growth channel is different as a result, going
through the marginal product term and not the capacity utilization rate.
Instead the capital tax affects the marginal product of capital by lowering it
directly. Again, however, is the similar result that the growth rate falls, and
falls by less when there is corruption that allows equilibrium evasion.
The model developed to generate these results and the principle is based
on an explanation of corruption activity that does not rely on preferences.
With the representative agent getting the profits of corruption activity as a
dividend "kickback", proportional to its ownership in the corruption enterprice, the after-evasion tax rate is a simple function of the tax rate, the degree
of diminishing returns to the production factor, and the degree of evasion activity that occurs. This form occurs identically for each of the four taxes:
goods, labor, capital, and inflation. These results make the inflation tax effects exactly parallel to other tax effects, when there is avoidance/evasion of
the taxes through a market structure with a micro-based production function. Here the paper has applied the banking microeconomics of production
to the corruption sector. Corruption then is treated as a type of financial
enterprise that produces financial services, these being illegal ones.
Viewed from the model, Sweden, for example, is more transparent, has
less corruption, but suffers more in terms of its growth rate from given levels
of each of the four taxes studied. It may have a better level effect on its
consumption ratios along the balanced growth path, say as normalized by
human capital, because it wastes less resources in corruption activity, but its
60
growth rate is relatively lower in a ceteris parabus way. In terms of utility, its
hard to say analytically. These features fit casual observation of the Swedish
economy: rich, low growth, maybe better off, maybe not. And in doing so this
resolves the paradox the paper started with: no obvious correlation between
tax rates and the size of the shadow economy or the degree of corruption.
High tax rates may coincide with low productivity of corruption, and little
tax evasion, or with high productivity of corruption, and high tax evasion.
On this basis, of fitting this seemingly pradoxical stylized fact as well as the
other more straightforward ones, such as positive correlation of corruption
activity and shadow economy, lies the paper’s results on the growth effects
of corruption.
Lowering taxes has an increasingly bigger effect on growth. That is an
implication of the analysis. This suggests that gains in efficiency in the
provision of government services are worth it. The growth rate effect is
increasingly large as the tax rates go down. This can be important, for
example, if tax rates and government expenditure relative to GDP were to
decrease secularly; then the growth rate might trend upwards at a marginally
increasing rate because of the decreasing tax rates. While the limits to
growth are clear here for given levels of tax rates, the paper suggests that
trending over time towards a more efficient government that requires less
taxes as a percent of income allows for a bigger growth increase for each
knotch up towards such improved government efficiency. For the US, the
recent trend downwards in federal spending as a percent of GDP, and the
rising growth rate, are reasons why explanations such as in the paper may
be useful. Tax rates may matter for growth, and matter increasingly so if
they trend downwards. Other applications are why some cities grow quickly
and others less so, and why countries within the EU grow at different rates.
Tax harmonization would cause convergence if tax evasion were equal in all
countries, but is growth harmonization desired?
The concept gives a more generalized interpretation to the meaning of
the famous Baumol (1952) model: inflation is avoided up to the point where
the marginal cost of avoiding money use through banking is equal to the
61
nominal interest rate.34 This Baumol margin is sometimes considered an
additional margin that is not important in monetary economies. But adding
this tax avoidance aspect is non-trivial: it alone has been shown within a
general equilibrium to give rise simultaneously to the Baumol (1952) firstorder condition, the Cagan-type nature of the money demand function, and
to the nonlinear inflation-growth profile within endogenous growth [Gillman
and Kejak (2005)]. These are all part of the same effect of avoiding the
inflation tax and, further, a part of the Beckerian law-avoidance margin.
The resulting nonlinear tax-growth profile, while empirically supported
for the inflation tax and having some intuitive appeal, remains to be investigated empirically for non-inflation taxes. For inflation, the empirics
identifying the nonlinear profile were ahead of the theory; here we put forth
the theory ahead of any empirics, as a hypothesis for future testing. Another
qualification is that the analysis is a positive one about growth rate effects,
with normative questions left for future research on the optimal structure of
taxes in this environment; for example a second-best Friedman optimum will
not in general hold here, even though it does hold in a similar economy but
without tax evasion.35
A
A.1
Appendix
The Derivation of the Human-Capital-Only Model
We first build up the Hamiltonian
H (at , ht , bt , mt ; mrt , crt , cut , xt , lrt , lut , qt , dQt , dlt , dct ) =
34
The Baumol (1952) first-order equation is that the nominal interest rate R equals
the average cost of banking, or the number of withdrawals from the bank (velocity in
Baumol’s model) times the cost per withdrawal. The cost per withdrawal is constant and
so the average cost of banking equals the marginal cost of banking, which is set equal to
the average-marginal cost of using money, R. This condition is then used to solve for the
money demand, the well-known square-root formula.
35
See Gillman and Yerokhin (2005) for the Friedman optimum as second-best when
there is no tax evasion, and Koreshkova (2006) for the Friedman optimum as second-best
not holding when there is tax evasion.
62
= (ln ct + α ln xt ) e−ρt
⎧
⎪
(1 − τ l ) wt lrt ht − (1 + τ c ) crt + vt
⎨
+λt
+ (1 − plt ) wt lut ht + rlt dlt − (1 + pct )cut + rct dct
⎪
⎩
−pQt qt + rQt dQt − π t mt + bt (Rt − π t )
+μt {AH (1 − lrt − lut − xt ) ht − δh ht }
⎫
⎪
⎬
⎪
⎭
+ξ 1t {mrt + qt − (1 + τ c ) crt }
+ξ 2t {mt − mrt − (1 + pct ) cut }
+ηt {mrt + qt − dQt }
+ζ ct {(crt + cut ) − dct }
+ζ lt {wt (lut + lrt ) ht − dlt } .
The first order conditions with respect to ht , bt , mt ; mrt , crt , cut , xt , lrt , lut , qt , dQt , dlt , dkt , dct
are
μ̇ = ρμ − λ [(1 − τ l ) wlr + (1 − pl ) wlu ] − μ [AH (1 − lr − lu − x) − δ h ] + ζ l w (lr +
(114)
lu )
λ̇ = ρλ − λ (R − π)
(115)
λ̇ = ρλ + λπ − ξ 2 − η
(116)
η + ξ1 − ξ2 = 0
(117)
= 0
(118)
= 0
(119)
= 0
(120)
= 0
(121)
λ (1 − pl ) wh − μAH h + ζ l wh = 0
(122)
−λpQ + ξ 1 + η = 0
(123)
λrQ − η = 0
(124)
λrc − ζ c = 0
(125)
λrl − ζ l = 0
(126)
1
e−ρt − λ (1 + τ c ) − ξ 1 (1 + τ c ) + ζ c
cr + cu
1
e−ρt − λ (1 + pc ) − ξ 2 (1 + pc ) + ζ c
cr + cu
1 −ρt
e − μAH h
x
λ (1 − τ l ) wh − μAH h + ζ l wh
63
where w = AG is the wage rate. From (121) and (122) we see that the
equilibrium exists36 only when pl = τ l . Since from (117) ξ 2 = ξ 1 +η equations
(118)-(119) imply that
¶
µ
rQ
− 1.
pc = (1 + τ c ) 1 −
1 + pQ
Using (126) with (121) we get
μ
AH
.
=
λ
(1 − τ l ) AG + rl
(127)
Using (126) for ζ l and substituting it together with (127) into (114) we get
−μ̇ = μ [AH (1 − x) − δ h − ρ] .
(128)
According to (115) we get standard result
−λ̇ = λ [R − π − ρ]
(129)
According to (124) and (123), ξ 1 = λ (pQ − rQ ) , which means that the benefits of cash via exchange services, ξ 1 , are equal to the opportunity cost of
money net of the return on deposited money. Notice that the opportunity
cost of money is equal to the cost of credit, pQ , the fee paid for a unit credit.
Using (116) with the obtained result for ξ 1 we get
−λ̇ = λ [pQ − π − ρ] .
(130)
The formulas (129) and (130) imply that in equilibrium the total net returns
on bonds and money should be equal, i.e.
R − π = pQ − π.
So the cost of real credit is equal to the nominal interest rate on bonds,
pQ = R.
By using (30) and (32) we get the expression for the marginal rate of
substitution between the reported consumption and leisure
MRScr ,x =
x
(1 + τ c ) (1 + R − rQ ) − rc
=
αc
(1 − τ l + rl ) AG h
36
(131)
If pl > τ l then nobody would be willing to work in the legal sectors and the equilibrium
will not exist. Similarly, when pl < τ l nobody would use the corruption services.
64
is equal to the ratio of the price of one unit of consumption to the price of
one unit of leisure. Since there exists only BGP equilibrium in the model
with only human capital, according to (50) the total return to human capital
is equal to the return on real bonds and real cash:
AH (1 − x) − δ H = R − π
where again R = σ + ρ and further
g = AH (1 − x) − δ H − ρ.
(132)
c = y = AG (lm + ln )h.
(133)
Let us define first the shares of legal sectors: the cash transaction share, aQ ,
and the reported consumption and income labor shares, ac , and al :
m
aQ ≡
,
(1 + τ c ) cr
cr
ac ≡
,
cr + cu
wlr
al ≡
.
w (lr + lu )
Then it follows from (61)-(62) and (67)-(68) that the returns on deposits are
rQ = (1 − ωQ ) R (1 − aQ )
(134)
rc = (1 − ωc ) pc (1 − ac )
(135)
rl = (1 − ωl ) τ l (1 − al )
(136)
where the price of consumption corruption services is
pc = τ c − (1 + τ c ) (1 − ω Q ) (1 − aQ )
R
.
1+R
(137)
Let introduce further the following definitions
ΩG ≡ A−1
G
¶ 1
µ
ω Q AQ R 1−ωQ
ΩQ ≡
AG
¶ 1
µ
ω c Ac pc 1−ωc
Ωc ≡
AG
¶ 1
µ
ω l Al τ l 1−ωl
.
Ωl ≡
AG
65
(138)
(139)
(140)
(141)
From the equilibrium of goods market solution we get
³c´
.
lm + ln = ΩG
h
(142)
From (44) and (60) the equilibrium labor in the credit is
³c ´
r
,
(143)
h
and similarly from (42), (45), and (66) and and the labor used in the corruption sectors are
³c´
,
(144)
lc = Ωc
h
ll = AG Ωl l.
(145)
lQ = (1 + τ c ) ΩQ
Using (142) and (145) and the fact that lk = 0, and substituting for l into
(7) we can solve for first for l
l=
where we define
Ωc/h ≡
Ω−1
c/h
³c´
h
(146)
1 − AG Ωl
ΩG + (1 + τ c ) ΩQ ac + Ωc
and then we can solve for ll
ll =
³c´
AG Ωl
.
[ΩG + (1 + τ c ) ΩQ ac + Ωc ]
1 − AG Ωl
h
(147)
Formulas (131) and (134)-(136) imply
³c´
h
where we defined Ωxc as
Ωxc ≡
= Ω−1
xc x
α 1 + (1 + τ c ) R̃ + τ̃ c
.
AG
1 − τ̃ l
(148)
(149)
Using the equality between the growth rates given by equation (132) and
the human capital accumulation equation
g = AH (1 − l − x) − δH
66
we get the formula for total labor
l=
ρ
.
AH
Using (148) and (146) we can get the formula for leisure
x = Ωx
ρ
AH
(150)
where
Ωx ≡ Ωxc Ωc/h .
(151)
Substituting for rQ , rc , and rl in (134)-(136) into (151) we obtain
Ωx = α
A.2
1 − AG Ωl
1 + (1 + τ c ) R̃ + τ̃ c
.
1 − τ̃ l
1 + (1 + τ c ) AG ΩQ ac + AG Ωc
(152)
Proofs of Propositions
Proof to Proposition 9. The relative tax revenues are given as
t=
l
τ l lr wh + τ c cr + Rm
= τ l wal +τ c ac +R(1+τ c )aQ ac ≡ tl +tc +tR . (153)
y
ĉ
The effect of τ l
where
Further,
∂tc
∂τ l
∂tl
∂tc ∂tR
∂t
=
+
+
∂τ l
∂τ l ∂τ l
∂τ l
l
1 + (1 + τ c ) w˜lQ ac + w˜lc
tl = τ l wal = τ l al
.
ĉ
1 − w˜ll
= 0,
∂tR
∂τ l
= 0, and
¢
∂tl
wl ¡
= al
1 + ηaτ ll − η ĉτ l
∂τ l
ĉ
where η aτ ll < 0 and η ĉτ l < 0. There are three effects of τ l on the relative
government tax revenues: (1) positive direct effect; (2) negative effect due to
the increase in unreported labor income and thus the decrease in reported
labor income; and (3) positive effect due to the lower output, so ĉl increases.
67
∂tl
Total effect is positive for small τ l since ∂τ
(τ l → 0) = wl
(τ l → 0) = 1. We
ĉ
l
can compute
µ
¶
¢
∂ 2 tl
wl ∂al al ¡
al
ĉ
1
+
η
=
−
−
η
τl +
τl
∂τ 2l
ĉ ∂τ l
ĉ
!
Ã
µ
¶2
wl ∂al τ l ∂al
∂ 2 al
−
+ τl 2
+
ĉ ∂τ l al ∂τ l
∂τ l
!
Ã
µ
¶2
wl 1 ∂ĉ
τ l ∂ĉ
τ l ∂ 2 ĉ
.
−al
−
+
ĉ ĉ ∂τ l ĉ2 ∂τ l
ĉ ∂τ 2l
If
∂ 2 tl
∂τ 2l
< 0 then there clearly exists a τ̄ l such that
∂ 2 tl
2
τ l →0 ∂τ l
for τ l < τ̄ l . In line with this fact is that lim
we proved below:
∂tl
∂τ l
(τ̄ l ) = 0 and
∂tl
∂τ l
(τ l ) > 0
is initially equal to −∞ as
¸
∙
¢ ∂al ¡
¢
∂ 2 tl
wl ∂al ¡
∂ 2 al
al
ĉ
al
lim
= lim
1 + ητ l − ητ l +
1 + ητ l + τ l 2 =
τ l →0 ∂τ 2
τ l →0 ĉ
∂τ l
∂τ l
∂τ l
l
¶
µ
2
wl
∂ al
∂al
2
= lim
+ τl 2 =
τ l →0 ĉ
∂τ l
∂τ i
ωl
µ
¶ 1−ω
2ωl −1
l
wl
ωl
1
ω l Al
1−ω
τl l
= −∞.
= − lim Al
τ l →0 ĉ
w
1 − ωl
1 − ωl
The result holds under the general condition which we assume to hold in
our economy, ω l < 1/2, i.e. the labor share in the tax evasion/avoidance
activities is smaller that 50%.
The effect of τ c is given by
∂t
∂tl
∂tc
∂tR
=
+
+
∂τ c
∂τ c ∂τ c ∂τ c
∂tl
where ∂τ
= 0, and there are nonzero effects of the consumption tax rate
c
∂tc
and also on the relative
both on the relative consumption tax revenue ∂τ
c
∂tR
inflation tax revenue ∂τ c . First, the effect on the consumption tax revenues
is straightforwardly given by
¡
¢
∂tc
= ac 1 + η aτ cc > 0
∂τ c
where ηaτ cc < 0. The formula gives the standard public finance result two
effects of τ c on the relative government consumption tax tax revenues: (1) a
68
direct positive effect, and (2) an indirect negative effect due to the increase in
unreported sales revenues. The standard result of the Laffer curve says that
when the tax elasticity increases in absolute value above the point where the
elasticity equals minus one, i.e. ηaτ cc = −1, the tax revenue starts to decline
with increasing tax rate. Since all unreported consumption transactions are
performed only by the use of cash an increase in the size of consumption
shadow economy leads to higher inflation tax revenues:
µ
¶
1 + τ c ac
∂tR
>0
= RaQ ac 1 +
η
∂τ c
τ c τc
where ηaτ cc < 0. There are two effects of τ c on the relative inflation tax revenues: (1) positive direct effect; (2) negative effect due to the increase in
unreported consumption sales revenues. Initially for small τ c for small τ c
∂tc
total effect is positive since ∂τ
(τ c → 0) = 1. Putting the second order efc
fects due to the inflation tax revenues (at least for low inflation rates) we can
further analyze the effect on the consumption tax revenues. We can compute
∂ 2 ac
∂ 2 tc
∂ac
=
2
+
τ
c
∂τ 2c
∂τ c
∂τ 2c
2
∂tc
∂tc
If ∂∂τt2c < 0 then there clearly exists a τ̄ c such that ∂τ
(τ̄ c ) = 0 and ∂τ
(τ c ) > 0
c
c
c
2
for τ c < τ̄ c . In line with this fact is that lim ∂∂τt2c is initially equal to −∞ as
τ c →0 c
we proved below:
∂ 2 tc
=
τ c →0 ∂τ 2
c
lim
∂ 2 ac
∂ac
+ τc 2 =
τ c →0 ∂τ c
∂τ c
¶ ωc
µ
2ωc −1
1
ω c Ac 1−ωc ωc
= − lim Ac
τ c1−ωc
= −∞
τ c →0
w
1 − ωc
1 − ωc
lim 2
for ωc < 1/2.
Finally, we analyze the effect of the inflation tax rate, R, on total government tax revenues
∂tl
∂tc ∂tR
∂t
=
+
+
∂R
∂R ∂R ∂R
∂tl
= 0, and due to the interaction between the consumption tax
where ∂R
evasion and inflation tax avoidance activities there are nonzero effects of the
∂tc
and also
inflation tax rate both on the relative consumption tax revenue ∂τ
c
69
R
on the relative inflation tax revenue ∂t
. Firstly, since the inflation tax is also
∂τ c
the tax on the unreported consumption transactions there is a positive effect
on the consumption tax revenues:
∂ac
∂tc
= τc
>0
∂R
∂R
c
c
where ∂a
> 0 since ∂a
< 0 and
∂R
∂pc
tax revenues is given by
∂pc
∂R
< 0. Secondly, the effect on the inflation
∙
¸
¡
∂tR
∂ac
aQ ¢
= (1 + τ c ) aQ 1 + η R ac + R
∂R
∂R
a
c
> 0 and η RQ < 0. So there are three effects: two standard effects where ∂a
∂R
(1) a direct positive effect and (2) a indirect negative effect due to the decline
in the relative money demand; and one nonstandard positive effect due to
the decrease of the consumption shadow sector. The total effect is positive
R
(R → 0) = 1. We can compute
for small R since ∂t
∂R
¶¾
¶
µ 2
½ µ
∂ac
∂ 2 aQ
∂ 2 tR
∂aQ ∂ac
∂ aQ
∂aQ
.
ac + aQ
+R
+R
= (1 + τ c ) 2
ac + aQ
∂R2
∂R
∂R
∂R ∂R
∂R2
∂R2
¡ ¢
2
R
R
If ∂∂RtR2 < 0 then there clearly exists a R̄ such that ∂t
(R) > 0
R̄ = 0 and ∂t
∂R
∂R
2
∂ tR
for R < R̄. In line with this fact is that lim ∂R2 is initially equal to −∞ as
R→0
we proved below:
∂ 2 tR
∂ 2 aQ
∂aQ
+
R
=
lim
2
=
R→0 ∂R2
R→0 ∂R
∂R2
µ
¶ ωQ
2ω Q −1
1
ωQ AQ 1−ωQ ω Q
= − lim AQ
R 1−ωQ
= −∞
R→0
w
1 − ωQ
1 − ωQ
lim
for ωQ < 1/2.
Proof to Proposition 10. The relative tax revenue is according to
(153)
1 + (1 + τ c ) w˜lQ ac + w˜lc
tl = τ l al
1 − w˜ll
The effect of productivity Al on the relative tax revenue given in (153) above
is
∂tl
∂t
=
∂Al
∂Al
70
where
∂tl
τ l al wl
=
∂Al
Al ĉ
Ã
ηaAll
w˜ll
˜
ll
+
ηA
˜
1 − wll l
!
where ηaτ ll < 0 and ηl̃Al l > 0. There are two effects of Al on the relative
government tax revenues: negative effect due to the increase in unreported
labor income and thus the decrease in reported labor income; and positive
effect due to the higher labor input, ˜ll , into the production of evasion, and
thus output decline, wlG , and relative tax revenue increase. Since
h ³
´
i
∂tl
1
tl
=
Ωωl l − 1 − w˜ll + ωl τ l al
∂Al
al 1 − w˜ll 1 − ω l
∂tl
∂tl
Total effect is zero ∂τ
(τ l = 0) when τ l = 0 and negative ∂τ
< 0 for small
l
l
τ l.
The effect of productivity Ac on the relative tax revenue given in (153)
above is
∂t
∂tc
∂tR
=
+
.
∂Ac
∂Ac ∂Ac
There are nonzero effects of the productivity of consumer corruption services
∂tc
and also on the relative
both on the relative consumption tax revenue ∂A
c
∂tR
. First, the effect on the consumption tax revenues
inflation tax revenue ∂A
c
is straightforwardly given by
ac ac
∂tc
=
η <0
∂Ac
Ac Ac
where η aAcc < 0. The increased productivity leads to the increase in unreported
sales revenues. Second, the effect on the relative inflation tax revenue.
∂tR
ac
=
R (1 + τ c ) ηaAcc < 0
∂τ c
Ac
goes again through the increase in unreported sales revenues which decreases
the relative tax revenue.
Finally, we analyze the effect of the productivity, AQ , on total government
tax revenues
∂t
∂tc
∂tR
=
+
∂AQ
∂AQ ∂AQ
71
There are nonzero effects of the productivity of consumer corruption services
∂tc
and also on the relative
both on the relative consumption tax revenue ∂A
Q
∂tR
inflation tax revenue ∂AQ . First, the effect on the consumption tax revenues
is positive given by
∂ac ∂pc
∂tc
= τc
>0
∂AQ
∂pc ∂AQ
∂pc
∂pc
c
where ∂A
< 0 since ∂a
< 0 and ∂A
< 0. The productivity of the banking
∂pc
Q
Q
sector forces the corruption fee and the production of corruption to decrease,
so the tax revenue increases. Secondly, the effect on the inflation tax revenues
is ambiguous and given by
¸
∙
∂aQ
∂ac
∂tR
= R (1 + τ c ) ac
+ aQ
∂AQ
∂AQ
∂AQ
∂a
where ∂AQQ < 0. So there are two effects: a negative effect due to the decline
in the relative money demand; and a positive effect due to the decrease of
the consumption shadow sector. When τ c < τ c (R) there is no consumption
corruption and
∂t
∂tR
ac aQ aQ
=
= R (1 + τ c )
η < 0.
∂AQ
∂AQ
AQ AQ
For relatively small τ c > τ c (R) the effect of the decreased relative money
demand still dominates the increase in the reported sales and the total effect
is negative
∙
¸³
´
ac aQ τ c ac
∂t
a
=
ηAQ + R (1 + τ c ) ηAQQ + η aAcQ < 0.
∂AQ
AQ aQ
Proof to Proposition 11. In the text before Proposition 11 we used
the general results that the effective tax is increasing and concave with rel
= ∂τ∂ l (τ l − rl ) =
spect to the nominal rate. This is for the labor tax: ∂τ̃
∂τ l
∙
¸
ω
³
´ l
1
2
ω l Al τ l 1−ωl 1−ωl
ω l 1−al
∂
τ
= al > 0 and ∂∂ττ̃2l = − 1−ω
−
A
τ
< 0; for the
l
l
l
∂τ l
AG
l τl
l
inflation tax:
∂ R̃
∂R
=
∂
∂R
(R − rQ ) = aQ > 0 and
the consumption tax, when τ c > τ c (R),
72
∂τ̃ c
∂τ c
=
∂
∂τ c
∂ 2 R̃
∂R2
ω
= − 1−ωQQ
(τ c − rc ) =
1−aQ
R
∙
∂
∂τ c
< 0. For
¸
³
´ ωc
1
ωc Ac pc 1−ωc 1−ωc
τ c − Ac
=
pc
AG
1−(1 − ac )
∂pc
∂τ c
> 0 and 1 >
∂pc
∂τ c
=
³ ´2
rQ
∂pc
∂ 2 τ̃ c
ω c 1−ac
1− 1+R > 0; so ∂τ 2 = − 1−ωc τ c ∂τ
<
c
c
∂τ̃ c
∂ 2 τ̃ c
= 1, ∂τ 2 = 0. Another general results
∂τ c
c
0. When τ c 6 τ c (R), τ̃ c = τ c and
was the fact that the unit labor costs are increasing and convex in the nomi2˜
ll
l̃l
∂ l̃l
ωl
1 ˜
= 1−ω
> 0 and ∂∂τl2l = (1−ω
> 0;
nal rate. It says for the labor tax: ∂τ
)2 τ 2
l
l τl
for the inflation tax:
l
l
∂˜
lQ
∂R
=
1 l̃Q
1−ω Q R
> 0 and
∂ 2 l̃Q
∂R2
ωQ
l
l̃Q
R2
> 0. For
2
(1−ωQ )
2
1 l̃c ∂pc
= 1−ω
> 0 and ∂∂τl̃2c =
c pc ∂τ c
˜
=
∂ lc
the consumption tax, when τ c > τ c (R), ∂τ
c
c
³ ´2
˜
∂pc
ωc
lc
˜
> 0. When τ c 6 τ c (R), lc = 0.
∂τ c
(1−ωc )2 p2c
These results form the base for getting positive concave substitution effects and negative concave income
µ effects
¶ of the tax rates on leisure. In case of
Φ1 (R,τ c )
³
´
∂
Φ2 (τ l )
Φ1 (R,τ c ) ∂Φ2 (τ l )
Φ1 (R,τ c )
∂τ̃ l
−
>
=
−
=
−
labor tax: substitution effect ∂τ l
∂τ l
∂τ l
Φ22 (τ l )
Φ22 (τ l )
µ
¶
∙
¸
Φ (R,τ )
³ ´2
h
i
∂ 2 Φ1 τ c
Φ1 (R,τ c )
Φ1 (R,τ c )
2( l)
∂τ̃ l
∂ 2 τ̃ l
2al
ω l 1−al
2
=
−
0 and
=
−
+
−
Φ2 (τ l ) ∂τ l
Φ2 (τ l )
1−ω l τ l
∂τ 2
Φ2 (τ l )
∂τ 2
Φ2 (τ l )
2
l
2
l
when
τ l → 0, τ̃ l → 0 and
where Φ2 (τ l ) = µ 1 − τ̃ l ¶. We can see that
µ
¶
∂
al → 1, so lim
Φ1 (R,τ c )
Φ2 (τ l )
∂2
= 1 and lim
∂τ l
τ l →0
τ l →0
Φ1 (R,τ c )
Φ2 (τ l )
∂τ 2l
= −∞. Thus there al-
ways exists a non-zero interval for τ l on
µ which¶ the substitution effect is
∂
positive and concave. Income effect ∂2
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
∂τ 2l
¶
Ψ2 (τ l )
Ψ1 (R,τ c )
∂τ l
˜
∂ ll
w
= − Ψ1 (R,τ
< 0 and
c ) ∂τ l
∂ 2 l̃l
w
˜
= − Ψ1 (R,τ
2 < 0, where Ψ2 (τ l ) = 1 − w ll , is negative and
c ) ∂τ
l
concave.
In case of other taxes the derivations are slightly more complicated.
The
µ
¶
∂
Φ1 (R,τ c )
Φ2 (τ l )
=
case of inflation tax when R > (τ c ) (τ c ): substitution effect ∂R
∂Φ1 (R,τ c )
∂Φ1 (R,τ c )
1
∂ R̃
; Φ1 (R, τ c ) = 1+(1 + τ c ) R̃+τ c , ∂R
= (1 + τ c ) ∂R > 0 and
Φ2 (τ l )
∂R
−1
∂ 2 Φ1 (R,τ c )
∂R2
∂
2
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
¶
∂ l̃
2 (τ l )
= (1 + τ c ) ∂∂RR̃2 < 0; income effect = − ΨΨ2 (R,τ
(1 + τ c ) w ∂RQ <
∂R
)
c
1
µ
¶
¸
∙
Ψ (τ )
³ ´2
∂ 2 Ψ 2(R,τl )
c
1
lQ
∂ l̃Q
∂2˜
Ψ2 (τ l )
2
= − Ψ2 (R,τ c ) (1 + τ c ) w Ψ1 (R,τ c ) ∂R − ∂R2 , where Ψ1 (R, τ c ) =
0 and
∂R2
1
can see
that when τ c =
0, and¶ R → 0, then R̃ → 0 and
1 + (1 + τ c ) w˜lQ . We
¶
µ
µ
∂
Ψ2 (τ l )
Ψ1 (R,τ c )
∂2
Φ1 (R,τ c )
Φ2 (τ l )
∂τ 2l
= 0 and lim
= −∞. Thus there alaQ → 1, so lim
∂R
R→0
R→0
ways exists a non-zero interval for R on which the income effect is nonpositive
and concave. The case of inflation tax when R < (τ c )−1 (τ c ) is more com73
plicated due to the presence of the consumption corruption services. So we
only prove that the substitution and income effects are positive and negative,
respectively using the fact that as R further increases the effects will be concave
already been analysed in the former case. Substitution effect
µ as it has
¶
Φ1 (R,τ c )
Φ2 (τ l )
∂
-
(1 + τ c )
∂ R̃
∂R
∂Φ1 (R,τ c )
1
; Φ1 (R, τ c ) = 1 +
Φ2 (τ l )
∂R
∂τ̃ c
R̃
c
> 0 since ∂∂R
= aQ and ∂τ̃
∂R
∂R
=
∂R
+
(1 + τ c ) R̃ + τ̃ c ,
= − (1 − ac )
∂Φ1 (R,τ c )
∂R
=
∂pc
> 0 with
∂Rµ
¶
Ψ (τ )
∂ Ψ 2(R,τl )
¤
c
(1−aQ ) £
1
R
=
1 − (1 − ω Q ) 1+R
< 0; income effect 1+R
∂R
´
h
³ ˜
∂ lQ
∂Ψ1 (R,τ c )
∂ac
2 (τ l ) ∂Ψ1 (R,τ c )
˜
˜
˜
− ΨΨ2 (R,τ
;
Ψ
(R,
τ
)
=
1+(1
+
τ
)
w
l
a
+w
l
,
=
w
(1
+
τ
)
a
+
l
1
c
c
Q c
c
c
Q ∂R +
∂R
∂R
∂R c
c)
∂pc
∂R
= − (1 + τ c )
1
∂˜
lQ
∂R
ω c 1−ac ∂pc
l̃c
1 l̃c ∂pc
c
= − 1−ω
and ∂∂R
= 1−ω
where ∂p
= − (1 + τ c )
∂R
∂R
c pc
ic pc ∂R
h
(R,τ c )
c
0. At R = 0, aQ = 1 and ∂p
= 0, ∂Ψ1∂R
= 0; when R is close to
∂R
with
> 0,
∂ac
∂R
R=0
c
should
still be very close to 0 and the first effect will dominate, so
0µ∂p
∂R
¶
Ψ2 (τ l )
´
i
h
³
∂ Ψ (R,τ )
c
1
∂ l̃Q
Ψ2 (τ l )
∂ac
∂ l̃c
˜
+
< 0. Thus the in=
−
w
(1
+
τ
)
a
+
l
2
c
Q ∂R
∂R
∂R c
∂R
Ψ1 (R,τ c )
come effect should be negative.
The case
of consumption tax when τ c > τ c (R): substitution effect µ
¶
∂
Φ1 (R,τ c )
Φ2 (τ l )
∂τ c
c
R̃ + ∂τ̃
>
∂τ c ¶
µ
Ψ2 (τ l )
∂ Ψ (R,τ )
=
0
∂Φ1 (R,τ c )
1
;
Φ2 (τ l )
∂τ c
∂τ̃ c
since ∂τ c > 0;
Φ1 (R, τ c ) = 1 + (1 + τ c ) R̃ + τ̃ c ,
further
∂ 2 Φ1 (R,τ c )
∂τ 2c
=
∂ 2 τ̃ c
∂τ 2c
∂Φ1 (R,τ c )
∂τ c
=
< 0; income effect -
c
2 (τ l ) ∂Ψ1 (R,τ c )
= − ΨΨ2 (R,τ
, Ψ1 (R, τ c ) = 1 + (1 + τ c ) w˜lQ ac + w˜lc , and
∂τ c
∂τ c
c)
1
∂Ψ1 (R,τ c )
∂ac
∂ l̃c
∂ac
ωc 1−ac ∂pc
= w˜lQ ac + (1 + τ c ) w˜lQ ∂τ
+ w ∂τ
with ∂τ
= − 1−ω
< 0,
∂τ c
c
c
c
c pc ∂τ c
rQ
∂pc
∂ l̃c
1 l̃c ∂pc
= 1−ωc pc ∂τ c > 0, and ∂τ c = 1 − 1+R > 0; when τ c → τ c (R) then
∂τ c
∂ac
∂˜
lc
→ 0, lim ∂τ
= +∞ and positive efpc → 0 and ac → 1, so ∂τ
c
τ →τ (R) c
1
c
fects dominate
(R,τ c )
lim ∂Ψ1∂τ
c
τ c →τ c (R)
c
= +∞ and thus there are always τ c >
∂
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
¶
τ c (R) such that the income effect is negative
< 0. Further,
∂τ c
µ
¶
∙
¸
Ψ
τ
(
)
2
l
³
´2
∂ 2 Ψ (R,τ )
2
2
c
1
Ψ2 (τ l )
∂Ψ1 (R,τ c )
2
1 (R,τ c )
1 (R,τ c )
where ∂ Ψ∂τ
=
− ∂ Ψ∂τ
=
2
2
2
∂τ 2c
∂τ c
Ψ1 (R,τ c ) Ψ1 (R,τ c )
c
c
³ ´2
∂pc
∂ac
∂ 2 ac
∂2˜
∂ 2 ac
ωc 1−2ω c 1−ac
lc
˜
+
(1
+
τ
)
w
l
+
w
with
=
> 0 and
2w˜lQ ∂τ
c
Q ∂τ 2
∂τ 2c
∂τ 2c
1−ω c 1−ω c p2c
∂τ c
c
c
³
´
2
2
∂pc
∂ 2 l̃c
1
ω c l̃c
= 1−ω
> 0 using ∂∂τp2c = 0. We can see that if τ c → τ c (R)
2
∂τ 2
∂τ c
c 1−ω c p
c
c
c
74
1−aQ 1+ω Q R
1+R 1+R
<
∂ l̃c
∂R
i
then pc → 0, and since
lc
∂2˜
lim
2
τ c →τ c (R) ∂τ c
>
lim
τ c →τ c (R)
³
∂˜
lc
∂τ c
´2
∂2
,
lim
τ c →τ c (R)
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
∂τ 2c
¶
=
−∞. This implies that there are always τ c > τ c (R) such that the income
effect is concave. The last case is when τ c 6 τ c (R) ,however, then there are
no corruption services and the substitution
µ
¶ and income effects are linear in
τ c , i.e.
∂Ψ1 (R,τ c )
=
∂τ c
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
∂
∂Φ1 (R,τ c )
∂τ c
∂
= R̃ + 1 > 0 and
w˜lQ > 0,
Φ1 (R,τ c )
Φ2 (τ l )
∂τ c
¶
h
+
∂
µ
Ψ2 (τ l )
Ψ1 (R,τ c )
¶
Φ1 (R,τ c )
Φ2 (τ l )
∂τ c
∂2
< 0 and
∂τ c µ
¶
Ψ (τ )
∂ Ψ 2(R,τl )
c
1
Φ1 (R,τ c )
Φ2 (τ l )
∂τ c
R̃+1
1+R̃
wl̃Q
1+wl̃Q
i
µ
∂ 2 Φ1 (R,τ c )
∂τ 2c
¶
> 0 and
Ψ2 (τ l )
Ψ1 (R,τ c )
∂τ 2c
= 0, and also
= 0. The total effect is
h
i
wl̃Q
R̃+1
2 (τ l ) Φ1 (R,τ c )
,
= ΨΨ1 (R,τ
−
Φ1 (R,τ c )
Ψ1 (R,τ c )
c ) Φ2 (τ l )
dx
> 0 and total effect is dτ
−
> 0 and close to
if τ c = 0 then
c
linear one due to slightly changing weighting terms in the formula.
So we proved that with the exception of τ c 6 τ c (R) , there always exist
positive, concave substitution effect and negative concave income effect for
each tax. Since the substitution effect diminishes from its maximum value at
the zero rate of particular tax and the income effect increases from its zero
value at the zero rate of particular tax, there for each tax rate ∆ ∈ {R, τ c , τ l }
ª
©
¯ the
¯ ∈ R̄, τ̄ c , τ̄ l such that for ∆ ∈ [0, ∆)
there always exists a tax rate ∆
substitution effect is the dominating factor.
∂x
∂x
Proof to Proposition 24. [ ∂A
< 0; ∂A
< 0, for R = 0, small τ c
c
l
∂x
and τ l , and AG < 1; ∂AQ < 0 for τ c = τ l = 0. With these derivatives, the
growth rate g increases when the productivity parameters increase, under
the conditions given.
75