Profitable gray market with asymmetric costs
Qing Hu
Tomomichi Mizuno
November 2023
Discussion Paper No. 2319

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANProﬁtable gray market with asymmetric costs∗
Qing Hu†ab , Tomomichi Mizuno‡baFaculty of Economics, Kushiro Public University of Economicsb

Graduate School of Economics, Kobe University

November 14, 2023
Abstract
When manufacturers sell their branded goods at diﬀerent prices in diﬀerent markets
or channels, gray marketers buy goods in the low-priced market and resell them in
the high-priced market to compete with the manufacturers’ authorized sellers. Conventional wisdom suggests the lost sales in the high-priced market resulting from thegray market’s diversion always make manufacturers worse oﬀ. However, by purely
considering the marginal production cost in the high-priced market is higher than the
low-priced market, we show that the manufacturer can gain from gray market, which
contradicts to the conventional result. It happens when the marginal production costin the high-priced market is suﬃciently large (or the transaction cost for the graymarketer is suﬃciently small in the linear demand case).

JEL codes: F12, F23, L13
Keywords: Gray market, parallel trade, cost asymmetry∗

The second author gratefully acknowledges the ﬁnancial support of the Japan Society for the Promotionof Science (JSPS), KAKENHI Grant Numbers JP22H00043 and JP23H00764. The usual disclaimer applies.
†
Corresponding author: Qing Hu, Faculty of Economics, Kushiro Public University of Economics, Ashino
4-1-1, Kushiro-shi, Hokkaido, 085-8585, Japan. Phone: (81)-154-37-3211. Fax: (81)-78-803-7289. E-mail:huqing549@gmail.com‡
Tomomichi Mizuno, Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo,657-8501, Japan. E-mail: mizuno@econ.kobe-u.ac.jp1

1

Introduction

Gray market (also known as parallel trade) goods are genuine branded products sold by
third parties (“gray marketers”) through manufacturer-unauthorized channels. When manufacturers sell their branded goods at diﬀerent prices in diﬀerent markets or channels, graymarketers buy goods in the low-priced market and resell them in the high-priced market tocompete with the manufacturers’ authorized sellers. Conventional wisdom suggests that graymarkets hurt manufacturers’ proﬁts by creating competition, which has also been shown inevidence in a wide variety of sectors. In the the survey of large high tech ﬁrms by KPMG
(2008), it estimates that gray markets result in $58 billion of lost sales annually. And according to a 2009 analysis by Deloitte LLP, gray markets account for $63 billion (4.5% of sales)
per year of lost U.S. sales in the consumer products sector (Wolf 2009). However, despiteof the large impact of gray markets, ﬁrms do not always implement systems to control ormonitor gray market distribution. For instance, in a survey conducted by KPMG (2008), it
was found that 42% of the participants did not possess any system to detect or track graymarket operations. Based on the above, one may have the following question: Why do someﬁrms allow the existence of gray markets and do nothing to curb gray market activity? This
study provides a possible reason for proﬁtable gray market by introducing cost asymmetryof the manufacturer in diﬀerent markets.
Speciﬁcally, we consider the marginal production cost in the high-priced market (e.g.
developed countries with higher consumers’ willingness to pay) is higher than the low-priced
market (e.g. developing countries with lower consumers’ willingness to pay). This cost
asymmetry may be because the input cost is higher in the high-priced market, which iswell-observed between developed countries and developing countries. Hence, it is natural tointroduce such cost asymmetry into the discussion of gray market.
We ﬁnd that a decrease in the transaction cost of the gray marketer, i.e., an increase of thetransaction volume of the gray marketer, can beneﬁt the manufacturer when the the marginal
cost of the production for the domestic market is suﬃciently large (or the transaction costfor the gray marketer is suﬃciently small in the linear demand case). The intuition for this2

result is as follows. In the presence of gray market, the manufacturer has two channels to
supply its product to a domestic market. One is the direct channel, where the manufacturer
supplies its product directly to the domestic market; the other is the indirect channel, where
the manufacturer supplies its product to the foreign market and the gray marketer resells it
to the domestic market. When the marginal cost of the production for the domestic market
is large (or the transaction cost for the gray marketer is small in the linear demand case),
the indirect channel becomes more eﬃcient. In this case, the manufacturer can make more
proﬁt by using the indirect channel. As a result, the existence of gray marketer makes the
manufacturer better oﬀ.
The conventional belief that gray markets hurt manufacturers’ proﬁts has been proved inmany previous literature (Richardson 2002; Maskus and Chen 2004; Hur and Riyanto 2006,
etc.). Our study is related to previous studies that tackle this well-known negative eﬀectof gray market for manufacturers in various contexts (Ahmadi and Yang 2000; Knox andRichardson 2002; Li 2006; Raﬀ and Schmitt 2007; Mukherjee and Zhao 2012; Autrey el al.
2015). These show that gray markets may beneﬁt the manufacturers and provide the reasons
why gray markets persist in the economy. However, no research among the above-mentioned
studies proves that gray markets may beneﬁt the manufacturers by purely considering theeﬀect of cost asymmetry of the manufacturer in diﬀerent markets.
The rest of this paper is organized as follows. Section 2 describes the model and Section
3 shows the conditions where a gray market beneﬁts the manufacturer in both general andlinear demand cases. Section 4 concludes.

2

Model

We consider a monopolist manufacturer M selling its products in a domestic market, market
1, and a foreign market, market 2 (where the market price is lower). The manufacturer has aplant in each of the country and oﬀers products to both countries from the respective plant.
We assume that the marginal production cost in the domestic market is c1 > 0 and that for
the foreign market is 0 (e.g. input cost in the domestic market).
3In our analysis, we assume that there is a ﬁrm G, which we denote by the gray marketer,
might divert the product of M from the foreign market to the domestic market.1 There is amarginal transaction cost cG > 0 for the gray marketer for trade between the two markets.
Note that when cG is suﬃciently large, the gray marketer might not be able to survive inthe market.
The production quantity in the domestic market and the foreign market are q1 and q2 ,respectively. We assume that the gray marketer G buys qG units of the products in theforeign market and sells to the domestic market. Therefore, for the manufacturer M , byconsidering the consumption of the gray marketer, it faces a demand which equals to thetotal supply in the foreign market as Q2 = q2 −qG and in the domestic market as Q1 = q1 +qG ,where q1 and q2 are the production quantities in the domestic market and foreign market,
respectively. We denote the inverse demand in the domestic market and the foreign market
as p1 (Q1 ) and p2 (Q2 ), respectively. We assume that the ﬁrst and second derivatives of
the inverse demands are p′i (Qi ) < 0 and p′′i (Qi ) ≤ 0 (i = 1, 2). Thus, the proﬁt of the
manufacturer M is πM ≡ [p1 (Q1 ) − c1 ]q1 + p2 (Q2 )q2 , while the proﬁt of the gray marketer G
is πG ≡ [p1 (Q1 ) − p2 (Q2 ) − cG ]qG .
The manufacturer M decides its production quantities q1 and q2 , and the gray marketer
G chooses the transaction volume qG , simultaneously.2 We employ Nash equilibrium as theequilibrium concept.

3

Analysis

In the constant-curvature demand model, for the simplicity of calculation, we assume thatcG is suﬃciently large so that the transaction volume of the gray marketer is close to zero. 3
1

Our set-up of the gray market diversion also applies to a situation where a branded manufacturer
attempts to price discriminate across diﬀerent channels and leakage across the channels takes place.
2
Even though some previous research assume that the gray marketer set the transaction volume as a
Stackelberg follower after the output decisions of the manufacturer, for the simplicity of calculation in thegeneral demand case, we assume all the players make the decisions simultaneously as Mukherjee and Zhao
(2012).
3
From the proﬁt function of the gray marketer, qG is negatively correlated to cG , thus an increase of cG
leads to a reduction of qG .

4

This assumption represents the situation where the size of the gray marketer is suﬃciently
smaller than that of the manufacturer. However, in the linear demand model, we do not set
this limitation of cG . We show similar results in both constant-curvature demand and linear
demand cases.

3.1

Constant-curvature demand case

The manufacturer M and the gray marketer G maximize the proﬁts by choosing qi and qG ,respectively. Hence, we can have the ﬁrst-order conditions as follows.
∂πM= F OCq1 = p1 (Q1 ) + q1 p′1 (Q1 ) − c1 = 0,
∂q1
∂πM
= F OCq2 = p2 (Q2 ) + q2 p′2 (Q2 ) = 0,
∂q2
∂πG
= F OCqG = p1 (Q1 ) − p2 (Q2 ) + qG [p′1 (Q1 ) + p′2 (Q2 )] − cG = 0.
∂qG

(1)
(2)
(3)

Then we derive the second-order conditions for the manufacturer: 4∂ 2 πM= SOCqi = 2p′i (Qi ) + qi p′′i (Qi ) < 0.
∂ 2 qi
Because of p′i (Qi ) < 0 and p′′i (Qi ) ≤ 0, we can easily get the second-order conditions arenegative. Moreover, from the above second-order conditions, we can also derive that SOCqi <p′i (Qi ).
Now, we discuss the eﬀects of the transaction cost of the gray marketer cG on the productquantities of the manufacturer qi and the transaction volume by the gray marketer qG . Substitute Q1 = Q1 (cG ) = q1 (cG )+qG (cG ) and Q2 = Q2 (cG ) = q2 (cG )−qG (cG ) into the ﬁrst-orderconditions F OCqi in (1) and (2) and F OCqG in (3). Since diﬀerentiating the ﬁrst-order conditions with respect to cG equals to zero, i.e., ∂F OCqi (cG )/∂cG = 0 and ∂F OCqG (cG )/∂cG = 0,
we use them solving for ∂qi (cG )/∂cG and ∂qG (cG )/∂cG . Recalling the second-order conditions
It is easy to ﬁnd that ∂ 2 πM /∂qi ∂qj = 0 and the Hessian condition in which ∂ 2 πM /∂qi2 · ∂ 2 πM /∂qj2 −[∂ πM /∂qi ∂qj ]2 = SOCqi SOCqj > 0.
4

2

5

SOCqi , we then obtain the following results of the comparative statics. Note that here we
consider the case where cG is suﬃciently large so that the transaction volume by the graymarketer is positive but suﬃciently close to zero, i.e., qG ≈ 0 and then Qi ≈ qi .
Lemma 1 When cG is suﬃciently large, a reduction in the transaction cost of the graymarketer decreases the domestic output level, but increases the foreign output level and thetransaction volume of the gray marketer. That is,∂q1 (cG )SOCq2 [p′1 (Q1 ) − SOCq1 ]=> 0,∂cG qG ≈0ξ∂q2 (cG )SOCq1 [SOCq2 − p′2 (Q2 )]=< 0,∂cG qG ≈0ξ∂qG (cG )SOCq1 SOCq2=< 0,∂cG qG ≈0ξwhere ξ = SOCq1 SOCq2 p′1 (Q1 ) + SOCq2 p′1 (Q1 )2 + SOCq1 p′2 (Q2 )[SOCq2 + p′2 (Q2 )] < 0.
An intuition behind Lemma 1 is simple. An increase in cG raises the marginal costof the gray marketer. Hence, qG decreases with cG , ∂qG (cG )/∂cG < 0. In the domestic
market, because of the strategic substitutability, the decrease of qG leads to an increase of
the output of the manufacturer q1 , i.e., ∂q1 (cG )/∂cG > 0. In the foreign market, the decreaseof qG causes a shrink of the demand, thus the output of the manufacturer q2 decreases, i.e.,∂q2 (cG )/∂cG < 0.
Then we analyze the eﬀect of cG on the proﬁt of the manufacturer M . The manufacturer’s
proﬁt is shown as πM (cG ) ≡ [p1 (Q1 (cG )) − c1 ]q1 (cG ) + p2 (Q2 (cG ))q2 (cG ), where Q1 (cG ) =q1 (cG ) + qG (cG ) and Q2 (cG ) = q2 (cG ) − qG (cG ). Substituting the results in Lemma 1 into the
manufacturer’s proﬁt, and then diﬀerentiating πM (cG ) with respect to cG , we have∂πM (cG )c1 SOCq2 [p′1 (Q1 ) − SOCq1 ] + ψ
=
∂cG
−ξ
qG ≈0

(4)

where ψ = SOCq2 p1 (Q1 )[SOCq1 −p′1 (Q1 )]+SOCq1 p2 (Q2 )[p′2 (Q2 )−SOCq2 ]−q1 SOCq2 p′1 (Q1 )2 +q2 SOCq1 p′2 (Q2 )2 . Note that ∂πM (cG )/∂cG in (4) is a linear function of c1 and the coeﬃcient6

of c1 is negative. Therefore, solving ∂πM (cG )/∂cG = 0 for c1 , we obtain the threshold valuec∗1 and then the following result.5
Proposition 1 When cG is suﬃciently large, a reduction in the transaction cost of the graymarketer increases the proﬁt of the manufacturer, which indicates that an increase of thetransaction volume by the gray marketer beneﬁts the manufacturer, ifc1 >ψ≡ c∗1 .
SOCq2 [SOCq1 − p′1 (Q1 )]We have a clean intuition behind this proposition. When the gray marketer is active, themanufacturer has two channels to supply its product to a domestic market. The ﬁrst is the
direct channel, where the manufacturer supplies its product directly to the domestic market;
the second is the indirect channel, where the manufacturer supplies its product to the foreign
market and the gray marketer resells it to the domestic market. When the marginal cost of
the production for the domestic market c1 is large, the direct channel becomes ineﬃcient.
In this case, the manufacturer can make more proﬁt by using the indirect channel. Thus, adecrease in the transaction cost (an increase of the transaction volume) of the gray marketer
brings more proﬁt to the manufacturer, i.e., πM (cG )/∂cG < 0.

3.2

Linear demand caseHere, we present the condition where the gray marketer beneﬁts the manufacturer undera linear demand model. Similar to the mechanism in the general demand model in theprevious subsection, the linear inverse demand functions in the domestic market and theforeign market are p1 (Q1 ) = 1 − q1 − qG and p2 (Q2 ) = F − q2 + qG , respectively. We assume
that the market size of the foreign market is smaller, hence F < 1.6 In addition, we assume
1 − F > cG > 0, which is a suﬃcient condition to enable the gray marketer to engage inthe transaction between the foreign market and the domestic market. Moreover, we assume
1/4 < F < 1 to guarantee all the outcomes in the equilibrium are positive.
Since the coeﬃcient of c1 is negative in (4), ∂πM (cG )/∂cG < 0 if c1 > c∗1 . Moreover, cG is inversely
related to qG .
6
The set-up of the linear inverse demand functions is similar to Mukherjee and Zhao (2012).
5

7

Substituting the linear inverse demand functions into the proﬁt functions of the manufacturer and the gray marketer, from the the ﬁrst-order conditions ∂πM /∂qi = 0 and∂πG /∂qG = 0, we have the following outcomes.
1 − 2cG + 5F + c1
5 + 2cG + F − 7c1
1 − 2cG − F + c1
L=
, q2L =
, qG
,
12
12
6
[13 + 4c2G + 13F 2 + 2F (5 − c1 ) − 34c1 + 25c21 + 8cG (1 − F − 2c1 )]
L
πM
=
,
72
(1 − 2cG − F + c1 )2
L
=
πG
,
18

q1L =

where the superscript L represents the case under linear inverse demand functions.
Now, we examine the eﬀect of cG on the proﬁt of the manufacturer M . Diﬀerentiating
the proﬁt of the manufacturer with respect to cG yields
1 + cG − F − 2c1
∂πM
=
.
∂cG
9
Note that when there is no marginal production cost for the manufacturer in the domestic
market, i.e., c1 = 0, we have ∂πM /∂cG = (1 − F + cG )/9 > 0, which means that an increaseof the gray marketer’s transaction cost always beneﬁts the manufacturer. In another word,
without cost asymmetry, gray market always hurts the manufacturer, which is a conventional
result. Here, with the marginal cost c1 for the manufacturer, solving ∂πM /∂cG < 0 for c1 ,we obtain the following result.
Proposition 2 With linear demand functions, a reduction in the transaction cost of thegray marketer increases the proﬁt of the manufacturer if c1 > (1 + cG − F )/2.
An intuition behind this proposition is similar to that behind Proposition 1. The manufacturer has two channels to supply its product to the domestic market: the direct channel
and the indirect channel. If the marginal cost of production in the domestic market, c1 ,is large or the marginal cost of the gray marketer, cG , is small, the direct channel is lessineﬃcient than the indirect channel. In this case, an increasing share of the gray marketer
by reducing cG makes more proﬁt for the manufacturer.
8

4Conclusions

When manufacturers sell their branded goods at diﬀerent prices in diﬀerent markets or channels, gray marketers buy goods in the low-priced market and resell them in the high-priced
market to compete with the manufacturers’ authorized sellers. Conventional wisdom andevidence show that gray market always hurt the manufacturers by creating competition.
However, by introducing the marginal production cost in the high-priced market, which ishigher than the low-priced market, we show that gray market may beneﬁt the manufacturer.
This result occurs when the marginal production cost in the high-priced market (cost asymmetry) is suﬃciently large (or the transaction cost for the gray marketer is suﬃciently small
in the linear demand case).

9References
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Wolf, C. (2009) Losing $63 billion to gray market is sleuth obsession (Update 2). April 9.
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