Inflation Tax and the Hidden Economy

In?ation Tax and the Hidden Economy.
Marco G. Ercolani?
Department of Economics, University of Essex,
Colchester, CO4 3SQ, United Kingdom.
marco@essex.ac.uk
May 2000
Abstract
Di?erential tax analysis is used to show how the optimal mix of in?ation tax
and direct taxation changes with the relative size of the hidden economy. The
larger the relative size of the hidden economy, the smaller the optimal ratio of
direct tax to in?ation tax. Anecdotal empirical evidence supports this result.
JEL Classi?cation: E62, H21, H26, O17.
Keywords: Hidden/Shadow economy, Optimal in?ation tax, Seignorage.
?I thank Alessandro Amelotti and Roy Bailey for their help. Any shortcomings are mine.

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Contents
1 Introduction.
3
1.1
What is the Hidden Economy? . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
In?ation Tax and Seignorage. . . . . . . . . . . . . . . . . . . . . . . .
3
2 The In?ation Tax: Concept and Measure.
4
2.1
The Government Sector. . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
The Private Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Measuring Seignorage. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Addressing Friedman’s Propositions.
. . . . . . . . . . . . . . . . . . .
7
3 Optimising the Revenue from In?ation.
7
3.1
The Private Sector’s Preferences. . . . . . . . . . . . . . . . . . . . . .
7
3.2
The Government Budget Constraint. . . . . . . . . . . . . . . . . . . .
8
3.3
The Private Sector Budget Constraint. . . . . . . . . . . . . . . . . . .
8
3.4
The Private Sector’s Optimising Behaviour.
. . . . . . . . . . . . . . .
8
3.5
The Government’s Optimising Behaviour.
. . . . . . . . . . . . . . . .
10
4 Implications for Government Policy.
12
4.1
The Limiting Cases, ? = 0 and ? = 1. . . . . . . . . . . . . . . . . . .
12
4.2
The Intermediate Cases, 0 < ? < 1. . . . . . . . . . . . . . . . . . . . .
12
4.3
Some Anecdotal Evidence. . . . . . . . . . . . . . . . . . . . . . . . . .
12
4.4
Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
5 Conclusion.
17
List of Figures
1
Tax to in?ation ratio and hidden economy size, OECD member states.
13
2
Tax to in?ation ratio and hidden economy size, Transition economies. .
14
3
Tax to in?ation ratio and hidden economy size, Developing economies. .
15
4
Tax to in?ation ratio and hidden economy size, all three groups. . . . .
16
2

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1
Introduction.
Raising tax revenue when the hidden economy forms a large proportion of the total
economy is a pressing problem in many countries. In?ation tax (seignorage) is seen as
one solution, however, this leads to price in?ation if the growth of the money supply
exceeds the growth of the total economy. This raises the question: How does the
relative size of the hidden economy a?ect the optimal in?ation tax?
The analysis is in the context of the optimal in?ation tax model proposed by Phelps
(1973). Phelps’s model is modi?ed to include a hidden economy as well as the visible
economy. Agents in the hidden economy are able to avoid direct taxation1 but unable
to avoid the in?ation tax because they require cash to carry out transactions. The
paper proceeds as follows. In this section the motivation for the paper is continued. In
section 2 the model is presented and in section 3 it is solved. In section 4 the results
are discussed along with some anecdotal empirical evidence and possible extensions.
A ?nal section concludes.
1.1
What is the Hidden Economy?
The hidden, black, informal, parallel, second, shadow or underground economy repre-
sents all economic activity which is not subject to direct taxation. In countries with
developed tax-gathering authorities the hidden economy is typically associated with
criminal activity. In developing countries, with less developed tax-gathering authori-
ties, the hidden economy is typically made up of a large agricultural economy which is
not subject to direct taxation because the costs of levying taxes outweigh the potential
revenues. In such a context the hidden economy need not be illegal.
1.2
In?ation Tax and Seignorage.
Seignorage is the di?erence between the face value of money and the cost of manu-
facturing it. Because of di?erent manufacturing costs, the seignorage value of coins is
quite low, the seignorage value of paper money is considerably higher and the seignor-
age value of the monetary base created through money on call nearly equals the full
value of the monetary base increase. In this paper, the simplifying assumption is made
that the seignorage value is exactly equal to the full value of the new money supply.
Subject to conditions (i) to (v) outlined in section 2, the “revenue from the in?ation
tax is simply its contribution to the government’s seignorage ... a tax on liquidity”.2
Seignorage can be a useful source of government revenue and all governments use
it to some extent. Fisher (1982) estimated the magnitude of seignorage for several
industrialised countries and found that this accounted for 6.1% of government revenue
during 1960-73 and 5.9% during 1973-78. The problem with raising large amounts of
revenue through seignorage is that it expands the monetary base, potentially leading to
high price in?ation. For example, Fisher (1982) found that during 1960-73 Italy raised
9.8% of government revenue by seignorage and experienced 4.66% average in?ation,
during 1973-79 seignorage rose to 16% and average in?ation rose to 16.38%.
1In this context direct taxation includes income and/or expenditure taxes.
2Phelps (1973), page 75.
3

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2
The In?ation Tax: Concept and Measure.
As in Phelps (1973), assume an economy where the government raises revenue by an
in?ation tax and by a direct tax. In contrast to Phelps (1973), the private sector
includes a hidden economy that is not subject to the direct tax. The private sector is
made up of the visible and hidden economies in the proportions 1?? and ? respectively.
The relative size of the hidden economy has implications for the government’s choice
of the optimal tax-mix between the in?ation and direct tax.
The simplifying assumptions underlying this analysis are that (i) agents forecast
in?ation perfectly, (ii) the natural rate of output is not a?ected by the level of in?ation,
(iii) the economy is closed so that in?ation is irrelevant as a stabilisation policy, (iv)
there are no costs to adjusting wage and prices, and (v) no interest is paid on money
balances.
2.1
The Government Sector.
The problem facing the government is one of having to maximise social welfare by
co-ordinating the actions of its three administrative branches; the expenditure branch,
the treasury and the central bank. The total tax yield is constant and di?erential
tax analysis is used to allocate taxes between in?ation and direct taxation in order to
maximise utility in the private sector. The notation and equation numbering follows
closely that of Phelps (1973) so as to facilitate any comparisons.
The expenditure branch determines government expenditure (G) and bene?ts (B)
as a function of time and independently of any other factors.
G = ?(t) ? 0,
B = ?(t) ? 0
(1)
The treasury, in ?nancing these expenditures, faces the budget constraint of match-
ing all costs and all revenues, including payments on the existing debt,
?
D
D?
T +
= G + B + iD
(2)
p
p
where T is direct taxation on the visible economy, D is the accumulated debt which
includes both the public debt (D?) and money (M), D? is the part of the accumulated
debt held by the private sector, p is the price level and iD is the nominal interest rate
on the debt.
The central bank sets the time path of the money supply (M) independently of the
treasury,
M = D ? D?
(3)
The money supply a?ects the price level, therefore, the central bank is able to determine
alternative price level programs in the form,
p(t) = ?(t; ?)
(4)
where ? is the target level of in?ation for the central bank.
4

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2.2
The Private Sector.
The behaviour of the private sector is determined by the following consumption de-
mand (C) and manhour supply (H) functions. Total consumption is determined by
consumption in the hidden economy (Ch) and consumption in the visible economy (Cv).
Total hours are determined by hours in the hidden economy (Hh) and hours in the
visible economy (Hv). The parameter ? determines the proportion of the population
N that is in each sector, so
C =
Cv( ˜
Y v, W ; ...; (1 ? ?)N; t) + Ch( ˜
Y h, W ; ...; ?N; t)
(5)
H =
Hv( ˜
Y v, W ; ...; (1 ? ?)N; t) + Hh( ˜
Y h, W ; ...; ?N; t)
(6)
where real net disposable income ˜
Y in each sector is determined by the population
proportions and by the di?erence between revenues and costs,
˜
D?
M + D?
Y v = (1 ? ?) ¯
Y + B + iD
? ?
? T
(7a)
p
p
˜
D?
M + D?
Y h =
?
¯
Y + B + iD
? ?
(7b)
p
p
where real disposable wealth is given by,
D
M + D?
W = K +
= K +
(8)
p
p
and potential pre-tax income is,
˜
Y = rKK + wH
(9)
where K is the real value of the capital stock, rK is the return on the capital stock
and w is the wage rate. Note that homotheticity is assumed throughout, that is,
the visible and hidden economies are scale values of one another. This implies that
Y i, Bi, Mi, Di, Hi, Ni and W i for i = v, h are purely functions of the population pro-
portions in the hidden and visible economies with the exception of the tax burden T
that only a?ects the visible economy.
2.3
Measuring Seignorage.
In this sub-section the seignorage equation (15) and the marginal rate of substitution
between direct taxes and in?ation (17) under the optimal policy trajectory are speci?ed.
Firstly, the tax burden is invariant to changes in the tax-mix, this is ensured by the
“forcing function” (10) which causes taxes T to change in response to changes in
in?ation ? such that the tax burden ? remains constant over time,
?M (iD ? ?)
?(t) = T +
(10)
p
p
where ? the “wondrous dynamic parameter”3 is a function only of time t. Secondly, the
level of wealth is also invariant to changes in the tax-mix. This ensures that the real
level of wealth is a function only of time and is not a?ected by the level of in?ation,
D
M + D?
?(t) =
=
(11)
p
p
3Phelps (1973), page 73.
5

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Thirdly, to avoid a wealth e?ect at time t = 0 from an increase in in?ation, we require
the price level to remain constant at this time,
p(0) = p0 > 0
(12)
of course, the rate of in?ation may change at time t = 0 following a change in policy
but the price level should not.
Substituting equations (3), (11) and (12) into (10) and re-arranging gives seignorage
tax at time zero,4
M
iD
= ?(0) ? T + (iD ? ?)?(0)
(13)
p0
where increased in?ation generates increased seignorage and reduces the tax burden.
There may be an additional increase in seignorage revenue insofar as the real rate of
interest (iD ? ?) may fall.
To further simplify the analysis, the arbitrage condition that the real rate of return
on the debt (D) equals the real rate of return on capital (K) and is a constant (?) is
imposed,
(iD ? ?) = rK = ?
(14)
Substituting in (14) for the arbitrage condition into (13) de?nes,
M
(? + ?)
= ?(0) ? T + ??(0)
(15)
p0
Equation (15) speci?es how much the private sector pays to hold a particular level of
liquidity. A behavioural equation describing money demand by the private sector is
required, thus liquidity preference is speci?ed by,
M = L(Y,rK + ?,K + D/p0)
(16)
p0
Thus, di?erentiating equation (13) with respect to in?ation gives equation (17), where
the implicit tax rate on liquidity is i = ? + ?, i.e. the foregone real return on capital
plus the in?ation tax revenue.
?dT
d
iM
M
dY
=
=
+ iLr+? + iLY
(17)
d?
d?
p0
p0
d?
Equation (17) describes the marginal change in tax to changes in in?ation.
4Start by substituting (3) into the second term on the right hand side of equation (10) to give, ?(t) =
T + ?(D?D?)
p
? (iD??)D?
p
. Then, cancel matching items in the second and third terms, ?(t) = T +
?D
p ? iDD?
p
. Substituting equation (11) into the second and third terms on the right hand side gives,
?(t) = T + ??(t) ? i
M
D ?(t) ? M
p
. Collect terms with ?(t), ?(t) = T + (? ? iD) ?(t) + iD p .
Substitute in using (12) for values at time zero, ?(0) = T + (? ? i
M
D) ?(0) + iD p . Re-arranging with
0
seignorage tax i M
D p on the left hand side gives equation (13).
6

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2.4
Addressing Friedman’s Propositions.
Phelps (1973) addresses two propositions by Friedman (1971). The ?rst proposition
is that seignorage revenue may actually fall following an increase in in?ation. Phelps
shows how his own model is consistent with this result subject to particular values for
the interest elasticity of liquidity preference. The second proposition is that there may
be no con?ict between full liquidity and in?ation tax revenue maximisation but Phelps
states that his own model is inconsistent with this: “If, with Friedman, we identify full
liquidity ... as occurring if and only if i ? 0, and if we assume, again with Friedman,
that
L(Y, 0, K + D0/p0) < ?
(18)
then the revenue from the in?ation tax, iDM/p0 must be non-positive at full liquidity.
At any in?ation rate too large for full liquidity, but not so large that M/p = 0, in?a-
tion tax revenue is positive. Hence there is a con?ict between acquiring revenue and
achieving full liquidity.”5
3
Optimising the Revenue from In?ation.
The government sets policy targets to maximise the private sector’s welfare and in
doing this takes into account the private sector’s responses. To solve the model, the
preferences of the private sector and the constraints facing both the private and gov-
ernment sectors are speci?ed.
3.1
The Private Sector’s Preferences.
The behaviour of the private sector is obviously relevant to the formulation of the
tax-mix. In this section the utility maximising behaviour of the visible and hidden
economies is obtained using Lagrangean functions. To begin an aggregate utility func-
tion is speci?ed for each economy, liquidity is included in the utility functions as it
provides a service to private agents,
Uv = U(Cv, Sv, Lv, Hv)
(19a)
Uh = U(Ch, Sh, Lh, Hh)
(19b)
where agents in both economies have the same preference structure over the level of
consumption (Uv,h
C
> 0), the level of saving (Uv,h
S
> 0), the level of liquidity (Uv,h
L
> 0),
and manhours worked (Uv,h
H < 0). So private agents are the same in all aspects except
that some happen to operate in the visible economy and others in the hidden economy.
For simplicity, Phelps assumes a short-run framework such that the capital stock is
taken as pre-determined. Just two taxes are assumed, the in?ation tax and a propor-
tional wage tax. To simplify factor pricing, capital and manhours are assumed to be
perfect substitutes with constant marginal returns. Government expenditure is ?xed,
liquidity is costless to produce and gross economic output is given by,
Y = ¯
wH + (¯
r + ¯
?)K = C + G + ?
K + ¯
?K
(20)
where ¯
w, ¯
r and ¯
? are all ?xed in time and where w is the pre-tax wage, r is the real
rate of interest and ? is the rate of capital depreciation.
5Phelps (1973), page 76.
7

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3.2
The Government Budget Constraint.
The government budget constraint is calculated with in?ation and direct taxes changing
such that the prescribed path of government debt ?(t) = ?
? = ( ?
D ? ?D)/p remains
unchanged. The proportional income tax rate is ? and pre-tax earnings are Z = ¯
wH.
Using equation (1) and the two equalities,6
D/p = ?
(21)
G + B + (¯
r + ?)(D/p ? L) ? ? Z ? ?D/p = ?
?
(22)
gives the budget constraint facing the government,
(1 ? ?)? Z + iL = ? + ? + ¯
r? ? ?
?.
(23)
The left hand side of equation (23) represents all revenues to the government and the
right hand side all expenditures. Marginal tax analysis requires that the magnitude
of equation (23) remains constant and that only the tax mix of the right hand side
change as policy changes.
3.3
The Private Sector Budget Constraint.
The budget constraint for the private sector, with all expenditures on the left hand
side and all revenues on the right hand side is given by,
C + S = (1 ? ? )(1 ? ?)Z + ?Z + (r + ?)K + ¯
B + i(? ? L) ? ?? ? ?K.
(24)
Assuming all aspects of the visible and hidden economies are a function of the popu-
lation proportions, with the exception of the income tax burden, equation (24) can be
re-arranged and split between the two sectors,
(1 ? ?)[C + S + iL] = (1 ? ?)[¯
rW + B + (1 ? ? )Z]
(25a)
? [C + S + iL] =
?
[¯
rW + B + Z]
(25b)
where wealth and savings are respectively given by,
W = K + ?,
S = ?
W .
(26)
3.4
The Private Sector’s Optimising Behaviour.
In order to derive the behaviour of the visible and hidden economies the ?rst order
conditions for the following two Lagrangeans must be satis?ed.
?v = Uv(Cv, Sv, Lv, Hv) ? ?v(1??)([C + S + iL] ? [B + ¯
rW + (1 ? ? )Z])(27a)
?h = Uh(Ch, Sh, Lh, Hh) ? ?h ? ([C + S + iL] ? [B + ¯
rW + Z])
(27b)
6At this point Phelps (1973) switches notation and uses t to denote the direct tax rate rather than
time. For the sake of continuity, here t continues to denote time and ? denotes the direct tax rate.
8

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Note that this optimisation, like that of Phelps, is in the context of a static analysis.
The resulting ?rst order conditions for the visible and hidden economies are,
UvC = UvS = ?v(1 ? ?)
UvL = ?v(1 ? ?)i = UvCi
(28a)
UvH = ??v(1 ? ?)(1 ? ?) ¯
w = ?UvC(1 ? ?) ¯
w
or, UvZ = ??v(1 ? ?)(1 ? ?) = ?UvC(1 ? ?)
UhC = UhS = ?h?
UhL = ?h?i = UhCi
(28b)
UhH = ??h? ¯
w = ?UhC ¯
w
or, UhZ = ??h? = ?UhC
Using these ?rst order conditions we can specify the maximised utility (U?) subject
to the tax rates from direct taxes ? and the in?ation tax rate i. Writing the value
functions for this optimisation in the visible and hidden economies,
V v(?, i) = Uv?[C(?, i), S(?, i), L(?, i), Z(?, i)]
(29a)
V h(?, i) = Uh?[C(?, i), S(?, i), L(?, i), Z(?, i)]
(29b)
The e?ect of the tax rates on the optimised level of utility is given by the derivatives
on equations (29a) and (29b),
V v
?C
?S
?L
?Z
? (?, i) = U v?
C ?? + U v?
S ?? + U v?
L ?? + U v?
Z ??
(30a)
V v
?C
?S
?L
?Z
i (?, i) = U v?
C ?i + U v?
S ?i + U v?
L ?i + U v?
Z ?i
V h
?C
?S
?L
?Z
? (?, i) = U h?
C ?? + U h?
S ?? + U h?
L ?? + U h?
Z ??
(30b)
V h
?C
?S
?L
?Z
i (?, i) = U h?
C ?i + U h?
S ?i + U h?
L ?i + U h?
Z ?i
Substituting the ?rst order conditions in (28a) and (28b) into (30a) and (30b) gives,
V v
?C
? (?, i) = U v?
C
?
?? + ?S
?? + i ?L
??
(1 ? ? )?Z
??
(31a)
V v
?C
i (?, i) = U v?
C
?
?i + ?S
?i + i ?L
?i
(1 ? ? )?Z
?i
V h
?C
? (?, i) = U h?
C
? ?Z
?? + ?S
?? + i ?L
??
??
(31b)
V h
?C
i (?, i) = U h?
C
? ?Z
?i + ?S
?i + i ?L
?i
?i
Di?erentiation of the budget constraint represented by equations (25a) and (25b) gives,
(1 ? ?)Z + (1 ? ?) ?C
?
?? + ?S
?? + i ?L
??
(1 ? ? )?Z
??
= 0
(32a)
(1 ? ?)L + (1 ? ?) ?C
?
?i + ?S
?i + i ?L
?i
(1 ? ? )?Z
?i
= 0
? ?C
? ?Z
?? + ?S
?? + i ?L
??
??
= 0
(32b)
?L + ? ?C
? ?Z
?i + ?S
?i + i ?L
?i
?i
= 0
Substituting equations (32a) and (32b) into (31a) and (31b) generates expressions for
changes in the optimal level of utility as the policy mix changes. These equations are
9

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of prime interest from the government’s standpoint in determining the optimal tax-mix
policy,
V v
? (?, i) = ?U v?
C Z
(33a)
V v
i (?, i) = ?U v?
C L
(33b)
V h
? (?, i) = ?U h?
C 0 = 0
(33c)
V h
i (?, i) = ?U h?
C L
(33d)
These equations are analogous to those numbered (33) in Phelps’s paper. Equation
(33c) is the only one that has no analogous equivalent, it suggests that at the margin,
the level of utility in the hidden economy does not diminish when the burden of direct
wage taxation increases.
3.5
The Government’s Optimising Behaviour.
A benevolent government would set tax policy to maximise utility as represented by
equations (29a) and (29b), subject to the constraint in equation (23). To establish this
optimal tax policy the following Lagrangean is speci?ed,
?(?, i) = (1 ? ?)V v(?, i) + ?V h(?, i) + µ[? (1 ? ?)Z + iL ? ¯
R]
(34)
Notice that the government weights the utility functions of the visible and hidden
economies in accordance to their population shares. Setting ¯
R = ? (1 ? ?)Z + iL,
the ?rst order derivatives for this Lagrangean are ??
?? = (1 ? ?)V v
? + ?V h
? + µ ?R
?? and
??
?i = (1 ? ?)V v
i + ?V h
i + µ ?R
?i . The corresponding ?rst order conditions are,
?R
(1 ? ?)V v
? + ?V h
? = ?µ ??
(35)
?R
(1 ? ?)V v
i + ?V h
i = ?µ ?i
The government sets the tax-mix policy that maximises utility (minimises tax distor-
tions) in both the visible and hidden economies, so the derivatives of utility with respect
to in?ation tax must be equal V v
i = V h
i . This implies through equations (33a,b,c,d)
that Uv?
C = U h?
C = U ?
C . This last condition, together with the conditions represented
by (33a,b,c,d) can be substituted into (35) to give,
?R
?R/??
(1 ? ?)U?CZ + ?U?C0 = µ
?
U?C =
??
µ
(1 ? ?)Z
?R
?R/?i
(1 ? ?)U?CL + ?U?CL = µ
?
U?C =
?i
µ
L
Equating the two expressions above gives the government policy target,
?R/??
?R/?i
U?
=
=
C ,
(36)
(1 ? ?)Z
L
µ
this de?nes the tax-mix that maximises social welfare subject to a constant total tax
revenue, ¯
R = ? (1??)Z+iL. The increases in overall income (I) required to compensate
for any change in the direct tax rate ? or the in?ation tax rate i are,
?I
?I
= ?Z,
= ?L,
(37)
?? ¯Vv,h
?i ¯Vv,h
10

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