A benchmark result in public economics is that it is possible to increase both tax revenue and welfare by making a monopoly subject to ad valorem taxes rather than unit taxes. We show that such revenue and welfare dominance does not hold in two-sided markets.Ad Valorem Taxes; Unit Taxes; Two-Sided Markets; Revenue-Dominance; Welfare-Dominance; Monopoly

Kind, Hans Jarle

Koethenbuerger, Marko

Schjelderup, Guttorm

English

Research Papers in Economics

Discussion paper
FOR 9  2009
ISSN:  1500-4066
SEPTEMBER 2009
INSTITUTT  FOR  FORETAKSØKONOMI
DEPARTMENT OF FINANCE AND MANAGEMENT SCIENCE
On Revenue and Welfare Dominance of
Ad Valorem Taxes in Two-Sided Markets
BY
HANS JARLE KIND, MARKO KOETHENBUERGER ANDGUTTORM SCHJELDERUPOn Revenue and Welfare Dominance of Ad
Valorem Taxes in Two-Sided Markets
Hans Jarle Kind￿ , Marko Koethenbuergeryand Guttorm Schjelderupz
This version; March 14, 2009
Abstract
A benchmark result in public economics is that it is possible to
increase both tax revenue and welfare by making a monopoly subject
to ad valorem taxes rather than unit taxes. We show that such revenue
and welfare dominance does not hold in two-sided markets.
Keywords: Ad Valorem Taxes, Unit Taxes, Two-Sided Markets,
Revenue-Dominance, Welfare-Dominance, Monopoly.
JEL Codes: H2; D4; L1.
￿Norwegian School of Economics and Business Administration, Bergen, Norway. E-
mail: hans.kind@nhh.no.
yUniversity of Copenhagen, Department of Economics, Studiestraede 6,
DK 1455 Copenhagen, Phone: +45 35324417, Fax: +45 35323000, Email:
marko.koethenbuerger@econ.ku.dk.
zNorwegian School of Economics and Business Administration, Bergen, Norway. E-
mail: guttorm.schjelderup@nhh.no.
1It is well known that ad valorem and unit taxes are equivalent in per-
fectly competitive markets, in the sense that appropriately chosen they yield
the same output and tax revenue. In the case of industries characterized
by imperfect competition, ad valorem and unit taxes have quite di⁄erent
e⁄ects. As shown by Suits and Musgrave (1953) ad valorem taxes revenue-
dominate unit taxes under monopoly, i.e. for any unit tax it is possible to ￿nd
an ad valorem tax which generates higher tax revenues while leaving quan-
tity choices una⁄ected. Subsequent literature shows that ad valorem taxes
welfare-dominate, and even Pareto-dominate, unit taxes under monopoly;
e.g. Delipalla and Keen (1992) and Skeath and Trandel (1994). The ￿ndings
are supportive for the widespread use of ad valorem taxes in practice. Almost
all ￿scally important commodity taxes are levied as ad valorem taxes.
In this paper we examine the e⁄ect of ad valorem taxes and unit taxes
in two-sided monopoly markets. The de￿ning characteristic of a two-sided
market is that there exists a platform ￿rm that caters to two distinct groups
of customers, whose demand is connected through positive quantity spillovers
from at least one of the groups to the other (positive externalities). Thus,
the ￿rm enables value-creating interactions between two di⁄erent groups of
end-users, and the pricing decision re￿ ects the demand externalities between
the groups; see e.g. Rochet and Tirole (2003, 2006). Two-sided platform
￿rms operate in many economically signi￿cant industries, such as the media
sector, the ￿nancial sector (payment card systems), real-estate brokerage,
and the computing industry (computer operating systems, software, game
consoles etc.).1
Contrary to what is the case in one-sided markets, we show that a shift
from ad valorem to unit taxes which holds monopoly output ￿xed in two-
sided markets may lead to higher tax revenue. This is true if the quantity
spillovers from one end-user group to the other are su¢ ciently strong. In the
same vein we ￿nd that unit taxes may yield higher welfare than ad valorem
taxes in two-sided markets.
We would like to emphasize that our analysis should not be confused with
the standard theory of two goods being complements. Complements are used
to describe a situation where an increase in the price of one good causes a
consumer to reduce consumption of both goods, as measured by the change
in his or her compensated demand (see e.g. Kreps 1990, p. 61). A two-
sided market, in contrast, consists of two distinct groups of customers, and
1See Evans 2003 for further examples.
2the groups may respond di⁄erently to changes in output on the other side
of the market (see Rochet and Tirole (2003) for a general discussion). An
example from the media industry may be clarifying; a newspapers serves two
distinct customer groups: readers and advertisers. The price of a newspaper
is irrelevant for advertisers per se, as are advertising prices for the readers.
However, to the extent that a higher newspaper price translates into reduced
sales of newspapers, demand for ads will typically fall.2 A lower advertising
volume (e.g. due to higher advertising prices), on the other hand, may either
increase or decrease demand for newspapers, depending on whether ads are
perceived as a good or a bad.
1 Analysis
We consider a two-sided market with two di⁄erent groups of customers, where
group i = A;B buys xi units of good i at price pi. Customer group i has
an inverse demand function pi (xi;xj) where the own-price e⁄ect as usual is
negative
￿
pi
xi < 0
￿
. A characteristic of a two-sided market is that there are
positive externalities from at least one side of the market to the other. For
example, higher sales of good j may increase the willingness to pay for good
i, that is, pi
xj > 0:
Let ti and ￿i denote the ad valorem and the unit tax rate, respectively,
on good i. The monopolist￿ s pro￿t under ad valorem taxation is
￿ =
xApA
1 + tA +
xBpB
1 + tB ￿ C(x
A;x
B) ,
whilst pro￿t under unit taxes equals
￿ = x
A ￿
p
A ￿ ￿
A￿
+ x
B ￿
p
B ￿ ￿
B￿
￿ C(x
A;x
B):
The cost function C (xi;xj) ￿ 0 satis￿es the standard conditions Cxi > 0
and Cxixi ￿ 0.
The ￿rst-order condition for pro￿t maximization under ad valorem taxa-
tion is
￿xi = 0 =>
pi + xipi
xi
1 + ti +
xjp
j
xi
1 + tj ￿ Cxi = 0: (1)
2Other things equal, the willingness to pay for an ad is increasing in the size of the
audience, whether we consider newspapers, TV stations or other media products.
3The ￿rst-order condition under unit taxes is likewise given by
￿xi = 0 =>
￿
p
i ￿ ￿
i + x
ip
i
xi
￿
+ x
jp
j
xi ￿ Cxi = 0. (2)
From equations (1) and (2), it follows that quantity choices are identical
under unit and ad valorem taxation i⁄
￿
A =
tB ￿ tA
1 + tB
￿
x
Bp
B
xA
￿
+ t
ACxA (3)
and
￿
B =
tA ￿ tB
1 + tA
￿
x
Ap
A
xB
￿
+ t
BCxB: (4)
In order to bring forward our main result in the simplest possible manner,
we shall make an assumption that reduces the complexity of the model with-
out a⁄ecting either standard results or the qualitative insight. We assume:
Assumption: The ad valorem tax rate is zero on good A and positive
on good B, that is, tA = 0 and tB > 0:
This assumption reduces (3) and (4) to
￿
A =
tB
1 + tB
￿
x
Bp
B
xA
￿
and ￿
B = ￿t
B ￿
x
Ap
A
xB
￿
+ t
BCxB: (5)
We de￿ne tax revenue under respectively ad valorem and unit taxation
as
R
AV T =
tB
1 + tBx
Bp
B and R
UT = ￿
Ax
A + ￿
Bx
B:
Using equation (5), we can express the change in tax revenue going from
ad valorem to unit taxes as
R
AV T ￿ R
UT = t
B
￿
pB
1 + tB + x
Ap
A
xB ￿ CxB
￿
x
B
| {z }
+
+
￿
￿
tB
1 + tBp
B
xAx
Bx
A
￿
: (6)
The fact that the ￿rst bracketed term on the right-hand side of (6) is
positive can readily be veri￿ed from ￿rst order condition (1) for good B.
Let us consider what would be the outcome if the markets in question
were one-sided, meaning that pB
xA = pA
xB = 0: Then the second term in (6)
4would be zero, and we clearly have RAV T ￿ RUT > 0.3 Thus, the standard
argument in the literature in favor of ad valorem taxation is reproduced: for
any given unit tax imposed on a monopoly, there exists an ad valorem tax
that revenue-dominates the unit tax.
In order to see that this result need not hold in a two-sided market,
suppose that pB
xA > 0: Then the last term on the right-hand side of (6) is
negative. Unit taxation may now generate higher tax revenue than does ad
valorem taxation. Ceteris paribus, this is more likely to happen the larger
the spillover e⁄ect pB
xA. In particular, inserting (1) (for good B) into (6), the
spillover e⁄ect must satisfy
p
B
xA > ￿p
B
xB
xB
xA , ￿pB;xA >
￿
￿￿pB;xB
￿
￿; (7)
where ￿pB;xA ￿ pB
xAxA=pB > 0 and ￿pB;xB ￿ pB
xBxB=pB < 0. Condition (7)
holds if, in absolute values, the willingness to pay for good B is more elastic
with respect to good A than with respect to good B. To see the intuition for
this result, it is useful to rewrite ￿rst-order condition (1) for good A under
ad valorem taxation as
p
A + x
Ap
A
xA = CxA ￿
xBpB
xA
1 + tB (8)
The left-hand side of (8) may be interpreted as the perceived marginal rev-
enue on good A, and the right-hand side as the perceived marginal cost.
The latter is decreasing in pB
xA; since larger quantity spillovers to good B
reduce the opportunity cost of producing good A. Note that the value of this
spillover is smaller the higher the ad-valorem tax on good B.
The ￿rst-order condition for unit taxation, equation (2), can likewise be
written as
p
A + x
Ap
A
xA = CxA ￿ ￿
A ￿ x
Bp
B
xA: (9)
Suppose that we start out in an equilibrium with tB > 0; and that pB
xA > 0 is
"large". By removing the ad valorem tax, we then see from equation (8) that
the perceived marginal costs of good A decrease signi￿cantly, thus giving the
monopoly incentives to produce more of that good. Thus, the government
will have to impose a high unit tax on good A to keep output at the initial
3By appealing to the continuity argument it is clear that this result also holds for
su¢ ciently small spillover e⁄ects, i.e. if the two-sidedness is not very pronounced.
5level.4 This explains why unit taxes yield higher tax revenue than ad valorem
taxes for su¢ ciently high values of pB
xA.
The following example demonstrates in a simple way that tax revenue
may rise if we switch from ad valorem to unit taxes. Let the inverse demand
functions be given by
p
A = 1 ￿ x
A and p
B = 1 ￿ x
B + ￿x
A; (10)
which means that pA
xB = 0 and pB
xA = ￿ > 0. We de￿ne the cost function
as C(xA;xB) = (0:1)
￿
xA + xB￿
; and set tA = 0 and tB = 0:2: Solving
d￿=dxA = d￿=dxB = 0, we obtain
x
A =
2(27 + 11￿)
5(24 ￿ 5￿2)
and x
B =
3(88 + 45￿)
25(24 ￿ 5￿2)
: (11)
These ￿rst-order conditions yield an optimum with non-negative prices if ￿ <
1:24 (for higher values of ￿ the non-negativity constraint on pA is binding).
Using (10) and (11) in (6) show that unit taxes yield higher tax revenue
than ad valorem taxes (RAV T ￿ RUT < 0) for ￿ > 1:05; as illustrated by
Figure 1. It should be stressed that the qualitative result does not hinge on
the assumption tA = 0; it is straightforward to construct examples where
RAV T ￿ RUT < 0 when all taxes are positive.
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.0 0.2 0.4 0.6 0.8 1.0 1.2
RAVT - RUT
Figure1: Tax Revenue Comparison.
Finally, can we draw any welfare implications from the analysis? Assume
that pB
xA is su¢ ciently large to make revenue from unit taxes higher than
4Note from equation (9) that ￿A has no direct e⁄ect on the value of the spillover pB
xA:
6from ad valorem taxes, other things equal (i.e. (7) holds). Instead of keeping
output constant when switching from ad valorem taxes to unit taxes, the
government might decide to keep tax revenue constant. It could then impose
lower unit taxes than those which are given by equations (3) and (4). This
in turn implies that output will increase subsequent to the switch from ad
valorem to unit taxes. Provided the monopoly undersupplies the two goods
from a social point of view, welfare will consequently increase.5
It is thus clear that neither revenue dominance nor welfare dominance of
ad-valorem taxes extends from one-sided to two-sided markets. In a wider
sense, the lesson is that in-depth knowledge of industry structure is more
important than what has previously been the conception.
References
[1] Evans, D. S. (2003), Some empirical aspects of multi-sided platform in-
dustries. Review of Network Economics, 2 , 191 - 209.
[2] Delipalla, S. and M. Keen (1992), The comparison between ad valorem
taxes and speci￿c taxation under imperfect competition. Journal of Public
Economics 49, 351-367.
[3] Kind, H.J., M. Koethenbuerger, and G. Schjelderup (2008), E¢ ciency-
enhancing taxation in two-sided markets. Journal of Public Economics
92, 1531￿ 1539.
[4] Kreps, D.M. (1990), A Course in Microeconomic Theory. Harvester
Wheatsheaf.
[5] Rochet, J.C., and J. Tirole (2003), Platform competition in two-sided
markets. Journal of the European Economic Association 1, 990-1029.
[6] Rochet, J.C. and J. Tirole (2006), Two-sided markets: A progress report.
Rand Journal of Economics 35, 645-667.
[7] Skeath, E. S. and G. A. Trandel. (1994), A Pareto comparison of ad val-
orem and unit taxes in non-competitive environments. Journal of Public
Economics 53-71.
5It can be shown that the goods are underprovided in our example. See Kind et al.
(2008) for a general discussion of when goods are under- or overprovided in a two-sided
monopoly market.
7[8] Suits D. and R. Musgrave (1953), Ad valorem and unit taxes compared.
Quarterly Journal of Economics 67, 598￿ 604.
8

On Revenue and Welfare Dominance of Ad Valorem Taxes in Two-Sided Markets

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On Revenue and Welfare Dominance of Ad Valorem Taxes in Two-Sided Markets

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