Money, Taxes, Audits and the Underground
Stephen Eli Ahiabu†
University of Toronto
JOB MARKET PAPER
Economic literature on the underground economy is restricted to (i) the
ratio of aggregate underground output to formal sector output. What in-
sights can one draw about micro-level transactions, given what we know
about published macro estimates? This paper introduces a two-sector mon-
etary search environment to study the additional ratios: (ii) the quantity-
per-trade ratio, (iii) aggregate private quantity ratio and (iv) price ratio
between sectors, among others. A search framework is essential for sep-
arating (ii) from (iii), while bargaining is ideal for generating (iv). I then
assess how monetary policy aﬀects all of these ratios. Monetizing part of the
government budget helps lower the tax burden on formal sector traders and
hence increase this sector. Apart from this “seigniorage eﬀect”, I identify
a residual “Tanzi eﬀect”, which acts in the opposite direction and partially
reverses gains in the formal sector.
Keywords: Seigniorage Eﬀect, Tanzi Eﬀect, Underground Economy
JEL classiﬁcation: E26, H26, L51, O23
∗This paper is a chapter in my doctoral dissertation at the University of Toronto. My gratitude
goes to Professors Shouyong Shi (supervisor), Andres Erosa and Miquel Faig for helpful comments
and suggestions. All remaining shortcomings are mine.
†Department of Economics, Rm 4072, 100 St. George Street, Toronto, Ontario, M5S 3G7,
Canada. Email: firstname.lastname@example.org, Tel/Fax: +1 (416) 963 4457
Economic knowledge about the underground economy at the micro level is limited
owing to non-reporting. Empirical studies are therefore conﬁned to the ratio of
aggregate underground output to aggregate formal sector output using latent vari-
able analysis. Can we draw any insights about transactions by the representative
ﬁrm, given what we know about published macro estimates? In this paper, I pro-
vide a theoretical contribution that achieves this. I extract the output-per-trade
ratio and the price ratio in an underground sector trade relative to a formal sector
trade, among others. I also show how these individual transactions collect into the
aggregate ratios. These results present an opportunity to revisit many questions
on the underground economy and to provide new answers on the micro level eﬀects
of economic policy. Considering monetary policy, I show that the eﬀect of inﬂation
on the aggregate underground economy ratios may be stronger than it’s impact on
the micro level ratios.
Despite the measurement handicap, overwhelming conclusions point to a signif-
icant and growing underground sector in most countries [see Schneider and Enste
(2000)]. Rogoﬀ (1998) estimates that notwithstanding the advent of the cashless
society, currency circulating outside the banking system in the US in 1996 was just
shy of $1500 per American, young and old inclusive.1 It is particularly puzzling
that each US resident is estimated to hold on average 24 $1 bills and $70 in coins.
At least one way to give meaning to the evidence is to acknowledge the existence
of a ﬂourishing underground sector. These observations require that increased at-
tention be paid to the underground economy and government policies that aﬀect
On the subject of monetary policy, the literature is dominated by the subopti-
mality of Friedman rule and the quest for the optimal mix of taxes and seigniorage
1 The corresponding ﬁgure for Canada is the equivalent of $611 US. Admittedly, a large portion
of this money, especially high denomination US dollar notes, are held abroad. In the case of small
denomination notes, a fair quantity may have been accidentally destroyed or lost.
that jointly maximize social welfare. Where there exists an underground sector,
oﬃcial sector taxation is distortionary since it shifts productive resources toward
the unoﬃcial sector. Alternative sources of government ﬁnancing are thus useful.
Seigniorage is one such alternative. Traditionally, inﬂation is considered to dimin-
ish the real value of government debt and this reduces the required tax hikes to
service old bonds. By lowering taxes, seigniorage can indeed contribute to raising
the relative size of the formal sector. Using an environment with the Walrasian
auctioneer, Nicolini (1998) provides quantitative estimates of the optimal seignior-
age rate for Peru, a country with underground output being about 40% of reported
GDP, to be about 10% per annum. This outcome leaves a substantial gap between
optimal and observed rates of inﬂation in many developing countries. Koreshkova
(2003) reconciles the pervasiveness of high inﬂation in poor countries to the socially
optimal choice by a benevolent government.
In the current paper, I reexamine the eﬀect of money growth on sectoral output.
Rather than add to the discussion concerning the optimal rate of seigniorage, I
focus solely on the eﬀect of money growth on the two sectors. The aim is to
separate two important eﬀects, namely the seigniorage eﬀect - exactly as described
above - and the Tanzi eﬀect. Tanzi (1978) proposed a caveat to the traditional
view on seigniorage by demonstrating that the longer the time lag between sending
out the tax bill and receiving payments, the lower the value of the collection. This
introduces a trade-oﬀ between the seigniorage motive for inﬂation and the erosion
of outstanding government revenue. The mechanism of the Tanzi eﬀect described
in this paper acts similarly. Rather than using bonds, government and households
are joint recipients of monetary transfers. We require both the government and
households to hold reserves in order to transact in the next period.2
the time lag constant, the higher the inﬂation rate, the lower the value of these
reserves. Depending on the rate of transfers to government vis-a-vis households,
the tax rate can adjust in either direction, which in turn aﬀects the relative size of
2 One can reinterprete government reserves as outstanding government revenues.
the underground sector. To summarize our ﬁndings, seigniorage income beneﬁts
the formal sector, but these gains are dampened by the erosion of government
reserves. While the total eﬀect has long been recognized, its components have not
been explicitly separated as done in this paper.
I employ a monetary search environment, since anonymity and imperfect infor-
mation readily motivate a tax-evading sector.3 The search framework also permits
an endogenous role for money as the sole medium of exchange. A third advantage
of a monetary search framework emanates from our ability to get price diﬀer-
entials between formal sector and underground sector goods, as observed in real
economies. In Walrasian environments, even though the formal and underground
markets may be physically separated, market clearing conditions ensure that prices
are equal between sectors. If formal and underground goods are substitutes and
equally weighted in the utility function, any sectoral diﬀerences in the unit price
will cause excess demand or excess supply, requiring a reallocation of resources.
If sales taxes are implemented as in Koreshkova (2003), the Walrasian auctioneer
sets the same price for both sectors and ﬁrms internalize the tax in their supply
decisions by relatively under-producing in the formal sector. When consumption
taxes are used instead as in Nicolini (1998), prices remain equal, while consumers
internalize the tax eﬀect by under-consuming formal sector goods. Just as taxes
in the formal sector, the detection and revenue-conﬁscation rate acts as a “duty”
on underground sellers. In the same way, the detection and conﬁscation rate is
internalized by underground sector producers and consumers. By using one-on-one
matching and bargaining in this paper, we permit higher “duty-augmented prices”
to arise in the formal market, echoing the observed reality.4
In the environment considered, households send a fraction of member buyers to
3 Perfect information has its advantages, since it is easier to integrate credit, where it is
4 With the Walrasian auctioneer, the inclusion of exogenous assumptions such as a cost for
visiting the one of the markets can deliver output price diﬀerentials between sectors. As will
diﬀerences in the quality of products in the two markets. Other non-Walrasian propositions such
as monopolistic competition and pricing-to-market can also deliver price diﬀerences.
the formal market and others to the underground market. Formal and underground
buyers may be allocated diﬀerent sums of money per capita. The intensive margin,
which describes what occurs within a representative transaction, can be fragmented
into two parts. First, the quantity purchased by a representative buyer may diﬀer
between sectors because money brought into a match is diﬀerent between sectors.
Secondly, each unit of money acquires relatively higher quantities underground,
since prices here are lower. The intensive margin is fully summarized by the re-
sulting quantity-per-trade ratio. The extensive margin on the other hand considers
the number of successful matches in each sector. In the Walrasian environments,
only a single transaction is necessary to clear each sector market. As a result, the
aggregate informal-to-formal output ratio and the quantity-per-trade ratio are one
and the same. With search however, the aggregate ratio is the ratio of a sum of
representative trades in each sector, which depends on the number of matches in
each sector. If seigniorage policy helps reduce taxes, the duty-augmented price
diﬀerence narrows between sectors. A unit of money buys more in the formal mar-
ket compared to previously, and households substitute by sending more buyers to
the formal market. Markets congestion worsens for buyers in the formal market
and to compensate, each formal buyer is handed more money. Thus, the intensive
margin unequivocally leads to relatively higher quantity-per-trade in the oﬃcial
sector. With more buyers in the formal market, the number of formal matches
increase. Thus, the extensive margin reinforces the intensive margin and hence
the aggregate ratio adjusts faster to inﬂation than the quantity-per-trade ratio.
The contributions of this paper are as follows. To the best of our knowledge, this
is the ﬁrst paper to break down the relative size of the underground economy into
the relative output-per-trade, relative aggregate private output and endogenous
relative price ratios. Also, the ﬁrst to explicitly separate the seigniorage eﬀect of
money growth from the Tanzi eﬀect on these ratios. Finally, a reasonable prognosis
is that the relative price, which is a nominal ratio, will respond to changes in the
inﬂation rate at least as fast as the response of the aggregate relative output ratios.
I show that in the a two-sector monetary search economy, the reverse may be the
This paper adds to the existing literature on the informal sector, which also
include Chaudhuri (1989) and Fugazza and Jacques (2003). The next section
presents a monetary search framework with households, a monetary authority,
ﬁscal authority and a regulatory body. In section 3, I characterize the model and
describe some properties of the equilibrium. Section 4 derives the price and output
ratios and section 5 considers the eﬀect of inﬂation. In section 6, I calibrate the
model to data from the US and Nigeria and present quantitative estimates of the
impact of money growth. Section 7 considers robustness and compares the results
to some forerunners. I conclude in section 8.
I extend the tractable framework introduced by Shi (1999) to allow for two sectors,
formal and underground/informal. These are denoted by the subscripts f and i
respectively. Goods are perishable between periods, irrespective of the sector in
which they are produced. By this, we preclude the emergence of commodity money.
Self-produced goods yield no utility and hence trade is essential for worthwhile
consumption. These restrictions are standard in monetary search models, as they
permit trade and an endogenous role for ﬁat money.
The economy is inhabited by a large number of anonymous and inﬁnitely-lived
agents who are either sellers or buyers. Agents visit one of two markets called
the formal and underground markets. For now, the two markets are assumed to
be on separate islands. Formal sector sellers pay taxes on income at the rate
τ . Underground sellers risk detection at the rate a, upon which sales income is
conﬁscated, a ≤ τ. A seller visiting island j enters the market with a production
capability. This allows him to produce output, qj, j = f, i, using the technology
qf = Alf or qi = li respectively. A ≥ 1 is a constant, while lj is labour input. A
buyer entering market j carries emj unitsofmoney.
Once in the market, agents match randomly and one-on-one. Anonymity for-
bids credit transactions and trade is quid pro quo. Whenever a buyer is matched
with a seller, trade can occur if the oﬀer is acceptable to both sides. Suppose an
oﬀer made by a buyer is the pair (qj, xj), where qj is the quantity requested and
xj is the amount of monetary compensation. Such monetary payments cannot
exceed the buyer’s money holding on entering the match: xj ≤ emj,j=f,i. This
feasibility constraint is intrinsic to the environment, given that trade is quid pro
Let ω be the value of money and Φ (lj) = lφ be the disutility of labour, φ > 1.
Then, for an oﬀer to be accepted, it must satisfy the seller’s individual rationality
constraint. These are (1 − τ)xfω ≥ Φ(lf) for the prospective formal seller and
(1 − a)xiω ≥ Φ(li) for the prospective underground seller. On both islands, we
allow buyers to hold all the bargaining power and to make take-it-or-leave-it of-
fers. Optimal oﬀers ensure that these individual rationality constraints hold with
equality. Combined with the feasibility constraints, we have:
(1 − τ)
− a) em
I name (1) and (2) the cash-and-carry constraints.
Sellers act as “oﬀer takers”, and take the quantity requested as given. Tem-
porarily assume that the cash-and-carry constraints hold with equality in both
sectors.5 Then labour employment in each match is known and one can rewrite
5 I later show that this is indeed the case in equilibrium.
the level of output per trade in each sector as:
qf = A h(1−τ) emfωi1φ and
qi = h(1−a) emiωi1φ .
With these quantities determined, we can proceed to ﬁnd the quantity-per-trade
ratio. I return to this later.
Here, I collect agents into decision-making families or households.6 From this stage,
the focus is on the representative household, who’s state and choice variables are
in lower-case letters. Capital-case variables represent those of other households
and the aggregate economy, which the representative household takes as given.
Time is discrete, denoted t. A household is constituted by a unit measure of
“formal sector sellers”, a unit measure of “underground sellers” and the measure
b of “private buyers”; b ∈ (0,1]. Buyers are incapable of making sales and sellers
do not buy goods. With this assumption, we disallow barter trades. The house-
hold chooses bft and bit, being the number of member buyers to visit the formal
and underground markets respectively. The allocation of buyers across sectors is
bft + bit ≤ b .
There is no population growth; the number of households, sellers and b being
There is a supply of perfectly divisible money, Mt + Mgt per capita household,
of which the representative household has mt. The household allocates its stock
of money into two folds, mft for formal sector buyers and mit for underground
6 A related environment that proceeds with agents - rather than households - is provided in
Lagos and Wright (2005).
buyers. The allocation of money is constrained by:
mft + mit ≤ mt .
After these allocations, a buyer bound to visit market j holds money to the sum
of emjt= mjt,j=f,i. Onlytheseresourcesarecarriedalongtothemarket. There
exists a government that sends a ﬁxed and unmodelled number of “public buyers”,
bg ∈ (0,1], to the formal market, each with money, Mgt. I return to discuss the
I specify the timing of events next. Starting a period with money holdings
mt, the representative household makes decisions on the allocation of buyers and
money. The household also instructs member buyers on (qjt, xjt), trade oﬀers to
make in the market and sellers on (Qjt, Xjt), the oﬀers to accept. The price of
output in sector j is hence xjt , j = f, i.
t + 1
Depart for market
−−−−−−−−−−−−−−→ −−−−−−−−−−−−−−−−−−−−−−−→ −−−−−−−−−−−−−→ −−−−−−−−−−−−→
Private Buyers → mjt, j = f,i Match, Bargain
Public Buyers → Mgt
Next, the markets open and matching occurs. Public and formal buyers visit only
the formal market while informal buyers go to the underground market. After
a bargain is reached, a successfully matched seller produces the desired output.
Trade is then completed and taxes paid. Audits are conducted and a fraction of
underground sellers have their incomes conﬁscated.
Markets close and agents go to their respective households. The monetary
authority then delivers lump sum transfers of Tt to each household. Purchased
goods and sales receipts are gathered. To take stock of the volume of pooled
resources, we must account for the number of successful matches in each market.
The eﬀective matching rates depend on the density of buyers and sellers, as well
as how these agents are distributed between sectors. First, the density of sellers
in the formal market is
and we denote this by
Sft. It also represents
the matching probability for a formal sector buyer. Likewise, the probability of
meeting an underground seller, formal buyer, underground buyer or public buyer
are Sit = 1 ,
Bft = Bft
Bit = Bit
suppose a households sends the measure bft of buyers and a unit measure of sellers
to the formal market. The expected number of successful matches are respectively,
Z bftSftdz=bftSft, and
Z 1 Bftdz+Z 1 Bgtdz=Bft+Bgt.
The two terms in the second equation are expected successful matches of formal
sector sellers with private formal buyers and public buyers. Similarly, the expected
number of successful matches are bitSit for underground buyers, bgSft for public
buyers and Bit for underground sellers.7
The household’s pooled resources therefore include goods of volume bjtSjtqjt
from sector j, j = f, i and these are instantly consumed. Money, of volume
(1 − τ)BftXft and (1 − a)BitXit, which were received from private buyers and
(1 − τ)BgtXgt from public buyers are gathered. Here, Xgt is money paid per public
buyer during trades.8
Let U (ct) be utility from consuming ct units of the consumption good, but
zero for the household’s domestically produced goods. Household members view
7 It is easy to see that Bft ≡ bftSft, Bit ≡ bitSit and Bg ≡ bgSft. Since it takes two to trade,
one successfully matched seller implies a successfully matched buyer.
8 The results do not depend on taxes being paid uniformly on all formal transactions, including
those involving public buyers.