Game Analysis of Manufacturers and Gray Market Speculators
Under Different Government Policies
Fang Zhang
a
and Qianqian Zhang
*b
School of Marketing Management, Liaoning Technical University, Huludao, China
Keywords: Government Policies, Manufacturers, Gray Market Speculators, Stackelberg Game.
Abstract: This article aims to explore the impact of different government policies on the sales of licensed products by
manufacturers and the sales of parallel products by gray market speculators. A market structure with
manufacturers as dominant players and gray market speculators as followers is established. Based on the
Stackelberg game approach, three scenarios are considered in which the government does not implement a
policy, the government implements a subsidy policy for manufacturers, and the government implements a
regulatory policy for gray market speculators. In the three cases, the influence of government subsidy
amount and supervision intensity on the equilibrium solution of manufacturers and gray market speculators
is solved and compared. Research shows that when the government implements subsidy policies, both
manufacturers and gray market speculators will reduce the unit sales price of their products. When the
government implements regulatory policies, manufacturers will increase the unit sales price of licensed
products, and gray market speculators will reduce the sales price of parallel products. Regardless of whether
the government implements subsidies or regulatory policies, it will increase the after-sales service level,
sales volume and profits of manufacturers, and reduce the sales and profits of gray market speculators.
1 INTRODUCTION
1
Gray markets, also known as parallel imports, are
market channels that sell branded goods without the
authorization of the trademark holder (Liu et al.,
2020). Unauthorized sellers are called "gray market
speculators" and goods sold through gray market
speculators are called "gray market products" or
"parallel products". Factors such as volume
discounts implemented by manufacturers and large
fluctuations in exchange rates that cause price
differences in different markets for the same product
are the main reasons why gray markets occur
(Cavusgil et al., 1988). In recent years, with the
rapid development of e-commerce, Internet
technology and globalization of logistics, the gray
market phenomenon has become increasingly
prominent. For example, according to the Financial
Times, in the European Union, the gray market
accounts for billions of dollars in pharmaceutical
sales each year. Kanavos et al. (2004) find that from
1997 to 2002, the share of the gray market in the
a
https://orcid.org/0000-0002-2155-6759
b
https://orcid.org/0000-0003-2502-3260
overall pharmaceutical industry increased from less
than 2% to 10.1% in Sweden, from 1.7% to about 7%
in Germany, and from less than 1% to 21.6% in
Greece during the same period. A report released by
a company called iSuppli shows that 145 million
cell phones were shipped in the gray market in 2009,
with a nearly 13% market share of the global cell
phone market, a significant increase of 43.6%
compared to 2008. Gray markets are also found in
other sectors around the world, with the airline
industry, the automotive industry, watches and
jewelry, and beauty and health products all
involving gray market speculators (Wang, 2014).
The increasingly large size of the gray market can
cause a decline in profits from the perspective of
companies. It also reduces consumer satisfaction and
damages brand image in the long run as the quality
of after-sales service for consumers cannot be
guaranteed. From the stakeholder's point of view, it
will make gray market speculators and unscrupulous
black marketers gain higher profits, leading to
chaotic market channels and social harmony in
turmoil. Therefore, it is of some practical
importance to motivate companies to solve the gray
market problem from the government's perspective.
42
Zhang, F. and Zhang, Q.
Game Analysis of Manufacturers and Gray Market Speculators Under Different Government Policies.
DOI: 10.5220/0012069800003624
In Proceedings of the 2nd International Conference on Public Management and Big Data Analysis (PMBDA 2022), pages 42-52
ISBN: 978-989-758-658-3
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

The increasingly prominent issue of gray market
has attracted domestic and international scholars to
study it from different perspectives. For example, Su
et al. (2012) find that volume discount contracts and
revenue sharing contracts can reduce retailer
participation in gray market transactions. Hong et al.
(2018) establish compensation contracts to achieve
overall supply chain coordination and incentives in a
gray market environment. Since parallel imports are
a common form of gray market, scholars have also
studied parallel imports and manufacturer channel
choice. For example, Su et al. (2017) based on the
perspective of parallel imports find that
manufacturers' choice of channel depends on the
effects of parallel import costs, the elasticity of
market demand in two countries and production
costs. Hong et al. (2021) construct four channel
structures based on manufacturers selling directly or
distributing in two markets to investigate the
conditions for the existence of gray markets under
different channel structures and the optimal channel
choice for manufacturers. Yeung et al. (2013)
examine how official distribution channels of
multinational companies, asset specificity, limited
rationalization of franchised dealers, and parallel
traders lead to the sustainability of parallel imported
cars. Other scholars have studied gray markets in
terms of pricing and service strategies. For example,
Ahmadi et al. (2000) give manufacturers' pricing
strategies under different conditions by developing a
three-level supply chain model with manufacturers,
gray market speculators, and consumers. The
findings suggest that the presence of gray market
speculators may help manufacturers to expand the
global scope of their products and even increase
their global profits. Iravani et al. (2016) find that the
emergence of gray markets leads manufacturers to
increase service levels in both high- and low-price
markets and that service decisions can be used as a
non-price mechanism to manage gray markets. Rong
et al. (2020) consider a multinational manufacturer
and a local manufacturer selling products to two
independent markets and compare and analyze the
effects of different power structures on supply chain
members' pricing and profits with and without gray
markets. In addition, it has been found that RFID
technology (Ding et al., 2022), manufacturers'
after-sales service quality decisions (Hu et al., 2021),
and brands' remanufacturing decisions (Huang et al.,
2020) can be used to manage and control gray
markets.
Throughout the literature, manufacturers'
after-sales service quality decisions (Iravani et al.,
2016; Hu et al., 2021) can be an effective tool for
managing gray markets, but few scholars have
incentivized firms to improve after-sales service
quality from the government's perspective to address
the gray market problem. Most of the scholars are
combining government policies with green supply
chain and closed-loop supply chain. For example,
Shang et al. (2020) construct a green supply chain
model in which the government subsidizes R&D
costs and production costs, respectively, and find
that both government subsidy strategies positively
affect product greenness, sales volume, and supply
chain members' profits. Cao et al. (2020) establish
three scenarios based on secondary supply chains
with no government subsidies, government subsidies
for manufacturers, and government subsidies for
retailers. The results show that government subsidies
can stimulate the green effort behavior of
manufacturers and retailers, which is always
beneficial to the green development of the supply
chain. Xia et al. (2017) analyze the impact of
government adoption of subsidy policy, adoption of
regulatory policy, and no policy on the reverse
recycling of end-of-life vehicles in formal and
informal channels in three cases. Although Wu
(2017) studies the impact of government regulatory
policies on manufacturers in the presence of gray
markets, the relationship between government
subsidy policies and gray markets is not studied.
In view of this, this paper considers the existence
of a manufacturer and a gray market speculator in
the market, constructs three different scenarios in
which the government does not implement a policy,
the government implements a subsidy policy for the
manufacturer, and the government implements a
regulatory policy for the gray market speculator, and
uses a dominant-subordinate game approach to
study the decision problem of different government
policies for the manufacturer and the gray market
speculator, respectively.
2 MODEL INTRODUCTION
2.1 Model Description
In this paper, we consider a market structure with a
manufacturer (denoted as M) as the dominant player
and a gray market speculator (denoted as A) as the
follower. The manufacturer sells a licensed product
to consumers through an authorized channel at price
m
p
, and the gray market speculator sells a parallel
product to consumers through an unauthorized
channel at price
a
p
. In contrast to gray market
Game Analysis of Manufacturers and Gray Market Speculators Under Different Government Policies
43

speculators, manufacturers need to provide
after-sales services to consumers. The government,
in order to motivate manufacturers to provide
after-sales services and to reduce arbitrage by gray
market speculators, has considered three scenarios:
no government policy, a government policy of
subsidies to manufacturers, and a government policy
of regulation of gray market speculators.
2.2 Model Assumptions
(1) Assume that the manufacturer's cost of providing
after-sales service is
2
2
ks
, where k indicates the
manufacturer's after-sales service cost factor.
(2) Although both licensed and parallel products
are genuine in the market, parallel products do not
have access to after-sales service, quality assurance,
etc., and their perceived product quality is lower than
that of licensed products. Therefore, it is assumed
that in the market, the perceived quality of the
licensed product is 1 and the perceived quality of the
parallel product is
θ
( 10 <<
θ
).
(3) Assume that the consumer can only purchase a
maximum of one product.
(4) Denote consumers' marginal willingness to
pay for the perceived quality of a product by
x
. Let
x
obey a uniform distribution of
[
]
0,1 , which yields
the consumer's perceived value of the licensed
product as
x
and the perceived value of the parallel
product as x
θ
.
(5) Assume that transportation costs, production
costs, etc. are all zero.
2.3 Description of Symbols
Table 1: Symbol settings and their descriptions.
Symbol Symbol description
n
Government does not implement policies
v
Government imposes subsidy policy on
manufacturers, unit subsidy amount
τ
Government imposes regulatory policy on
g
ra
y
market s
p
eculators, unit re
g
ulation
o
m
s
The after-sales service level of a
manufacturer when the government
implements policy o, where
{
}
,,onv∈τ
o
m
p
The unit selling price of a manufacturer
selling a licensed product when the
government implements policy o, where
{
}
,,onv∈τ
S
y
mbol S
y
mbol descri
p
tion
o
a
p
The unit selling price of a gray market
speculator selling a parallel product when
the government implements policy o, where
{
}
,,onv∈τ
o
m
q
The number of sales of licensed products
sold by manufacturers when the
government implements policy o, where
{
}
,,onv∈τ
o
a
q
The number of sales of parallel products
sold by speculators in the gray market when
the government implements policy o, where
{
}
,,onv∈τ
o
m
π
Manufacturer's profit on the sale of licensed
products when the government implements
policy o, where
{
}
,,onv∈τ
o
a
π
The profits of gray market speculators from
the sale of parallel products when the
government implements policy o, where
{
}
,,onv∈τ
o
m
u
The utility function of a consumer
purchasing a licensed product when the
government implements policy o, where
{
}
,,onv∈τ
o
a
u
The utility function of a consumer's
purchase of a parallel product when the
government implements policy o, where
{
}
,,onv∈τ
2.4 Demand Function Creation
2.4.1 Government Non-Implementation
Policy and Government
Implementation of Subsidy Policy
According to the assumptions, the utility function of
a consumer purchasing a licensed product is
expressed as
=− +
ooo
mmm
uxps
and the utility
function of a consumer purchasing a parallel product
is expressed as
=θ −
oo
aa
uxp
, respectively, where
{
}
,∈onv . When the utility function
,0≥≥
ooo
mam
uuu
, the consumer chooses to buy the
licensed product; when
,0≥≥
ooo
ama
uuu
, the
consumer chooses to buy the parallel product. At this
point the demand function is
1
1
θ
−−
=−
−
ooo
o
mam
m
p
ps
q
(1)
PMBDA 2022 - International Conference on Public Management and Big Data Analysis
44

1 θθ
−−
=−
−
ooo o
o
mam a
a
p
ps p
q
(2)
2.4.2 Government Implementation of
Regulatory Policies
According to the assumptions, the utility function of
consumers purchasing licensed products is
τττ
=− +
mmm
uxps
, and the utility function of
consumers purchasing parallel products is
ττ
=θ − −τ
aa
uxp
. When the utility function
,0
τττ
≥≥
mam
uuu
, the consumer chooses to buy the
licensed product; when
,0
ττ
≥≥
o
ama
uuu
, the
consumer chooses to buy the parallel product. At this
point the demand function is
1
1
θ
τττ
τ
−−−τ
=−
−
mam
m
pps
q
(3)
1 θθ
τττ τ
τ
−−−τ +τ
=−
−
mam a
a
pps p
q
(4)
3 MODEL SOLVING
3.1 Government Non-Implementation
Policy
The government has neither a subsidy policy for
manufacturers nor a regulatory policy for gray
market speculators. At this point, the profit functions
for the manufacturer and the gray market speculator
are
2
()
(,)
2
π =−
n
nnn nn
m
mmm mm
ks
ps pq
(5)
() π =
nn nn
aa aa
ppq
(6)
Theorem 1: In the absence of government
intervention, the optimal solution for manufacturers
and gray market speculators is
2
4(1 )
(2 )(4k 4k 2)
θ
θθθ
∗
−
=
−+−−
n
m
k
p
2(1 )
442
θ
θθ
∗
−
=
+− −
n
m
s
kk
(1 )( 2 k 2 2 )
(2 )(4k 4k 2)
θθ θ θ
θθθ
∗
−+−−
=
−+−−
n
a
k
p
Substituting
∗
n
m
p ,
∗
n
m
s and
∗
n
a
p into equations
(1) (2) (5) (6), we get
2(1 )
442
θ
θθ
∗
−
=
+− −
n
m
k
q
kk
222
(2 )(4k 4k 2)
θθ
θθθ
∗
+− −
=
−+−−
n
a
kk
q
2
(2 2)
2(2 )(4 k 4 k 2)
θ
π
θθθ
∗
−
=
−+−−
n
m
k
2
22
(1 )( 2 2 2 )
(2)(4 4 2)
θθ θ θ
π
θθθ
∗
−+−−
=
−+−−
n
a
kk
kk
Proof 1: According to the reverse solution method,
n
a
p
is solved first. Since
2
2
2
0
() ( 1)
π
θθ
∂
=<
∂−
n
a
n
a
p
, it
is known that
π
n
a
has a great value about
n
a
p
. Let
0=
∂
∂
n
a
n
a
p
π
and solve for
()
2
θ −
=
nn
n
mm
a
ps
p
.
Substituting
n
a
p
into
π
n
m
, we find the Hessian
matrix of
π
n
m
with respect to
n
m
p
and
n
m
s
as
22
12(1)
2
2( 1)
θθ
θθ
θ
θ
−−




−−


=
−


−


−


H
k
.
When
0244 >−−+
θ
θ
kk
, we have
1
2
0
1
θ
θ
−
=<
−
H
and
2
2
(2)(4 4 2)
0
4( 1)
θθθ
θ
−− +− −
=>
−
kk
H
.
It is known that the matrix
H
is negative definite
and
π
n
m
has great values with respect to
n
m
p
and
n
m
s
. We combine
0=
∂
∂
n
m
n
m
p
π
and
0=
∂
∂
n
m
n
m
s
π
to
obtain the values of
n
m
p
and
n
m
s
. Substitute them
back into
()
2
θ −
=
nn
n
mm
a
p
s
p
to obtain the optimal
solution for
n
a
p
. The proof is complete.
3.2 Government Subsidy Policies for
Manufacturers
The profit functions for manufacturers and gray
market speculators when the government implements
a subsidy policy for manufacturers are
2
()
(p , ) ( )
2
π =+ −
v
vvv v v
m
mmm m m
ks
spvq
(7)
Game Analysis of Manufacturers and Gray Market Speculators Under Different Government Policies
45

() π =
vv vv
aa aa
ppq
(8)
Theorem 2: In the case of a government subsidy
policy for manufacturers, the optimal solution for
manufacturers and gray market speculators is
22
4(1 ) [(2 ) 2( 1)(2 )]
(2 )(4k 4k 2)
θθθθ
θθθ
∗
−+ −+ − −
=
−+−−
v
m
kv k
p
2(1 ) v(2 )
442
θθ
θθ
∗
−+ −
=
+− −
v
m
s
kk
(1 )( 2 k 2 2 2)
(2 )(4k 4k 2)
θθ θ θ θ
θθθ
∗
−+−−+−
=
−+−−
v
a
kkvkv
p
Substituting
∗
v
m
p ,
∗
v
m
s and
∗
v
a
p into equations
(1) (2) (7) (8), we obtain
(2 2 2 )
442
θθ
θθ
∗
−+−
=
+− −
v
m
kvv
q
kk
222 2
(2 )(4k 4k 2)
θθ θ
θθθ
∗
+− − + −
=
−+−−
v
a
kkkvkv
q
2
(2 2 2)
2(2 )(4 k 4k 2)
θθ
π
θθθ
∗
−+−
=
−+−−
v
m
kvv
2
22
(1 )(2 2 2 2)
(2)(4 4 2)
θθ θ θ θ
π
θθθ
∗
−+−−+−
=
−+−−
v
a
kkkvkv
kk
Proof 2: The proof procedure is the same as in
Proof 1 and is therefore omitted. It should be noted
that the condition for the existence of the optimal
solution is
22 2
(2 )
kk
v
k
θθ
θ
−+−
<
−
.
3.3 Government Regulatory Policies
for Gray Market Speculators
The profit function for manufacturers and gray
market speculators when the government imposes a
regulatory policy on gray market speculators is
2
()
(,)
2
τ
τττ ττ
π =−
m
mmm mm
ks
ps pq
(9)
()
ττ ττ
π =
aa aa
ppq
(10)
Theorem 3: The optimal solution for
manufacturers and gray market speculators in the
case of a government policy of regulation of gray
market speculators is
2( 1)(2 2 )
(2)(4 4 2)
τ
θτθ
θθθ
∗
−+−
=
−+−−
m
kk k
p
kk
22
442
τ
τθ
θθ
∗
−+
=
+− −
m
s
kk
(1)
(2)(4 4 2)
τ
θ
θθθ
∗
−
=
−+−−
a
m
p
kk
Where
τθθθ
τ
θ
θ
τ
θ
τ
kk
kkm
32
2422
22
++
−−+−−=
Substituting
∗
τ
m
p ,
∗
τ
m
s and
∗
τ
a
p into equations
(1) (2) (9) (10), we obtain
(22)
442
τ
τθ
θθ
∗
−+
=
+− −
m
k
q
kk
(2 )(4 4 2)
τ
θθ θ θ
∗
=
−+−−
a
m
q
kk
2
(22)
2( 2)(4 4 2)
τ
τθ
π
θθθ
∗
−−+
=
−+−−
m
k
kk
2
22
(1 )
(2)(4 4 2)
τ
θ
π
θθ θ θ
∗
−
=
−+−−
a
m
kk
Proof 3 The proof procedure is the same as in
Proof 1 and is therefore omitted. It should be noted
that the condition for the existence of the optimal
solution is
(2 2 2)
43 2
θθθ
τ
θθ
−+−
<
−+−
kk
kk
.
4 MODEL ANALYSIS
Proposition 1: 0
∗
∂
<
∂
v
m
p
v
, 0
∗
∂
>
∂
v
m
s
v
, 0
∗
∂
>
∂
v
m
q
v
,
0
π
∗
∂
>
∂
v
m
v
; 0
∗
∂
<
∂
v
a
p
v
, 0
∗
∂
<
∂
v
a
q
v
, 0
π
∗
∂
<
∂
v
a
v
.
Proposition 1 suggests that the government's
subsidy policy for manufacturers will result in a
decrease in the manufacturer's unit sales price for
selling licensed products and an increase in the level
of after-sales service, product sales and profits for
manufacturers selling licensed products. At the same
time, it will lead to a decrease in the unit selling price,
sales volume and profits of gray market speculators
selling parallel products. The main reason for this is
that manufacturers increase the number of licensed
products sold by lowering the unit sales price of
licensed products in order to receive more
government subsidies, at which point the
manufacturer's profit increases by selling more at a
lower price. As a result of this increase in profits,
manufacturers are more motivated to provide better
quality after-sales service to their customers. Gray
market speculators will reduce the selling price of
parallel products in order to maintain their original
PMBDA 2022 - International Conference on Public Management and Big Data Analysis
46

competitive advantage, but the sales volume will be
partially reduced due to the impact of government
subsidies, which ultimately leads to a reduction in
profits for gray market speculators.
Proposition 2:
0
τ
τ
∗
∂
>
∂
m
p
,
0
τ
τ
∗
∂
>
∂
m
s
,
0
τ
τ
∗
∂
>
∂
m
q
,
0
τ
π
τ
∗
∂
>
∂
m
,
/
1
/
τ
τ
τ
τ
∗
∗
∂∂
<
∂∂
m
m
s
p
;
0
τ
τ
∗
∂
<
∂
a
p
,
*
0
τ
τ
∂
<
∂
a
q
,
0
τ
π
τ
∗
∂
<
∂
a
.
Proposition 2 suggests that the government's
implementation of regulatory policies on gray
market speculators will increase the unit sales price,
after-sales services level, sales volume and profit of
licensed products sold by manufacturers, and the
increase in service level is smaller than the increase
in price. For gray market speculators, government
regulation will simultaneously reduce the unit sales
price, sales volume and profit of the parallel product.
The main reason for this is that when the government
imposes regulatory policies on gray market
speculators, it reduces the competition between
manufacturers and gray market speculators, allowing
manufacturers to restore the prices of licensed
products in the market and sell them in the market at
high prices. Although manufacturers will increase
costs by raising after-sales service levels as
regulation increases, manufacturers can increase their
profits with government protection as the increase in
after-sales service levels is less than the increase in
price. When the government implements regulatory
policies, it causes some consumers who buy parallel
products to switch to buying licensed products, and
gray market speculators sell fewer parallel products.
To prevent a reduction in demand for their products,
gray market speculators will continue to reduce the
unit selling price of their parallel-imported products,
ultimately leading to a reduction in their profits.
Proposition 3: When
v
v
≥
(
22 2
(2 )
θθ
θ
−+−
=
−
kk
v
k
), then
0
∗
≤
v
a
q
. When
τ
τ
≥ (
(2 2 2)
43 2
θθθ
τ
θθ
−+−
=
−+−
kk
kk
), then 0
τ
∗
≤
a
q
and
0 τ −<v
.
Proof 4: By
0
∗
∂
<
∂
v
a
q
v
, we know that the quantity
of parallel products sold by gray market speculators
will fall to a demand of 0 as the amount of
government subsidy increases, i.e. 0
∗
=
v
a
q , and we
get
22 2
(2 )
θθ
θ
−+−
=
−
kk
v
k
. Similarly, by
0
τ
τ
∗
∂
<
∂
a
q
,
we know that the number of gray market speculators
selling parallel products will fall to a demand of 0 as
government regulation increases. i.e.
0
τ
∗
=
a
q
, we
get
(2 2 2)
43 2
θθθ
τ
θθ
−+−
=
−+−
kk
kk
. In addition, we can
conclude that
2
(2 2 2)(4 5 2)
0
(2)(4 3 2)
θθ θθθ
τ
θθθ
+− − +− + −
−= <
−+−−
kk kkk
v
kkk
.
Proposition 3 suggests that whether the
government implements a regulatory or a subsidy
policy, gray market speculators will choose to exit the
market when either the level of regulation or the
amount of subsidy reaches a certain level. For the
government, it takes less effort to implement a
regulatory policy than a subsidy policy to fully
combat and regulate the gray market.
Proposition 4:
2
2
43 2
,
,
,
τ
ττ
ττ
ττ
τθθ
θθ
ππππππ
∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
−
<
−+−

>>


>> >>


>> >>


>> >>

vn
mmm
nvn v
mmma a a
vnnv
mmma a a
vnnv
mmma a a
kk
vkk
sss
pppppp
qqqqqq
;
2
2
2
43 2
,
,
,
τ
ττ
ττ
ττ
θθ τ
θ
θθ
ππππππ
∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
−
≤≤−
−+−

≥>


>> >≥


≥> >≥


≥> >≥

vn
mmm
nvnv
mmma a a
vnnv
mmma a a
vnnv
mmma a a
kk
kk v
sss
pppppp
qqqqqq
;
2
,
,
,
vn
mmm
nvnv
mmma a a
vnnv
mmma a a
vnnv
mmma a a
v
sss
pppppp
qqqqqq
τ
ττ
ττ
ττ
τ
θ
ππππππ
∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
∗∗∗∗∗∗
>−

>>


>> >>


>> >>


>> >>

.
Proposition 4 suggests that, for the manufacturer,
we obtain the following conclusion:
(1) From the manufacturer's after-sales service
level, when the ratio of government regulation to the
amount of government subsidy satisfies
/2 τθ <−v , the manufacturer provides the highest
Game Analysis of Manufacturers and Gray Market Speculators Under Different Government Policies
47

level of after-sales service when the government
implements a subsidy policy, followed by the
implementation of a regulatory policy and the lowest
when no policy is implemented. This is because
government-imposed subsidy policies can cover the
after-sales service costs of manufacturers, resulting
in the highest level of after-sales service. Compared
with the government does not implement the policy,
the manufacturer's after-sales services level is higher
when the government implements the regulatory
policy, in order to attract consumers to change from
unauthorized channels to authorized channels,
thereby supporting licensed products. Whichever
policy the government implements, it will help to
improve the level of after-sales service for
manufacturers.
(2) In terms of manufacturer prices for the sale of
licensed products, manufacturer unit sales prices are
highest when the government implements a
regulatory policy, followed by no policy and lowest
when a subsidy policy is implemented. The main
reason is that when the government implements a
subsidy policy, manufacturers will increase their
sales volume by reducing their prices in order to
obtain more government subsidies. When the
government imposes a regulatory policy, competition
between manufacturers and gray market speculators
is reduced. This will eliminate the need for
manufacturers to reduce their unit selling prices,
which are greater than they would be if the
government did not implement the policy.
(3) In terms of sales of licensed products by
manufacturers, when the ratio of government
regulation to the amount of government subsidy
satisfies /2 τθ <−v , the number of sales by
manufacturers is highest when the government
implements the subsidy policy, followed by the
implementation of the regulation policy and lowest
when the policy is not implemented. The reason is
that when the government implements the subsidy
policy, the manufacturer has the lowest unit sales
price and the highest level of after-sales service,
resulting in the highest number of sales. When the
government imposes a regulatory policy, the
manufacturer has the highest unit sales price and the
second highest level of after-sales service. Although
the manufacturer's unit sales price is higher than it
would have been in the absence of the government's
policy, the government's regulation of gray market
speculators has resulted in a shift of consumers to the
authorised channel, ultimately resulting in higher
sales than would have been the case in the absence of
the government's policy. Manufacturer profits are not
discussed here as they change in line with changes in
their sales volumes.
For gray market speculators, the conclusions we
can draw are as follows:
(1) From the perspective of the price of parallel
products sold by gray market speculators, when the
ratio of government supervision to government
subsidy amount meets
2
2
/
43 2
θθ
τ
θθ
−
<
−+−
kk
v
kk
,
the sales price of gray market speculators is the
highest when the government does not implement
policies, followed by the implementation of
regulatory policies, and the lowest is the
implementation of subsidy policies. The main reason
for this is that both government regulation and
subsidy policies are unfavourable to gray market
speculators, who will then lower their unit sales
prices in order to maintain their competitive
advantage. Compared with government supervision,
as the amount of subsidies increases, gray market
speculators will sell parallel products at lower prices.
(2) From the perspective of the sales volume of
parallel products sold by gray market speculators,
when the ratio of government supervision to
government subsidies meets
2
2
/
43 2
θθ
τν
θθ
−
<
−+−
kk
kk
, the sales volume of gray
market speculators is the highest when the
government does not implement policies, followed
by the implementation of regulatory policies, and the
least is the implementation of subsidy policies. The
main reason is that compared to government
regulation, when the subsidy is greater than a certain
value, the government subsidy is most beneficial to
the manufacturer, causing more consumers to turn to
authorized channels, and ultimately leading to the
lowest sales volume of gray market speculators.
Since the relationship between the quantity sold
and the profit of a gray market speculator satisfies
2
(1 )( ) πθ θ =−
ii
aa
q
,
{,,} τ ∈inv
, the change in the
profit of a gray market speculator is consistent with
the change in the quantity it sells.
PMBDA 2022 - International Conference on Public Management and Big Data Analysis
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5 NUMERICAL ANALYSIS
The above model mainly analyses the optimal
strategies of manufacturers selling licensed products
and gray market speculators selling parallel products
under different policies implemented by the
government. In the following, the relevant
parameters of the model are taken to further analyse
and verify the relevant findings of this paper.
According to Wu (2017), we assume that the
parameter 0.25 θ = and take 1.5=k in order
to ensure that the hypothesis
2
2(1 )
θ
θ
−
>
−
k
holds.
At this point, we can get
2
2
0.21
43 2
θθ
θθ
−
=
−+−
kk
kk
and 21.75 θ −= . Since the change in sales of
manufacturers and gray market speculators
coincides with the change in profits, the portrayal
and analysis of profits is omitted below.
5.1 The Impact of Government
Subsidies and Regulatory Policies
on the Unit Sales Prices of
Manufacturers and Gray Market
Speculators
As seen in Figure 1, as the amount of government
subsidies increases, the unit sales price of licensed
and parallel products will continue to decrease. As
government regulation increases, the unit sales price
of licensed products continues to increase and the
unit sales price of parallel products continues to
decrease. Neither regulation nor subsidy policies
imposed by the government are favourable to gray
market speculators and will result in gray market
speculators reducing their unit sales prices.
For the unit sales price of licensed products, the
price is from highest to lowest: government-imposed
regulatory policy, government-imposed no policy,
government-imposed subsidised policy. For the unit
sales price of a parallel product, when 𝜏/𝑣 >0.21,
the price is from highest to lowest:
government-imposed no policy, government-
imposed subsidised policy, government-imposed
regulatory policy. When 𝜏/𝑣  0.21, the price goes
from high to low: government-imposed no policy,
government-imposed regulatory policy,
government-imposed subsidised policy.
Figure 1: Impact of government subsidies and regulatory policies on unit sales prices.
5.2 Impact of Government Subsidies
and Regulatory Policies on Sales
Volumes of Manufacturers and
Gray Market Speculators
As seen in Figure 2, sales of licensed products have
been increasing and sales of parallel products have
been decreasing as the amount of government
subsidies or regulation has increased.
For the number of licensed products sold by
manufacturers, when 𝜏/𝑣 >1.75, the number of
sales goes from high to low: government-imposed
regulatory policy, government-imposed subsidy
policy, and government-imposed no policy; when
𝜏/𝑣 1.75, the number of sales goes from high to
low: government-imposed subsidy policy,
government-imposed regulatory policy, and
government-imposed no policy. For the number of
Game Analysis of Manufacturers and Gray Market Speculators Under Different Government Policies
49

gray market speculators selling parallel products,
when 𝜏/𝑣 >0.21, the number of sales goes from
high to low: no government policy, government
policy of subsidies, government policy of regulation;
when 𝜏/𝑣 0.21, the number of sales goes from
high to low: no government policy, government
policy of regulation, government policy of subsidies.
Figure 2: Impact of government subsidies and regulation on sales volumes.
5.3 The Impact of Government Subsidies
and Regulatory Policies on
Manufacturers' after-Sales Service
Levels
Figure 3: The impact of government subsidies and
regulatory policies on manufacturers' after-sales service
levels.
As seen in Figure 3, the level of manufacturer
after-sales service increases as the amount of
subsidy or regulation increases. When /1.75 τ >v ,
the level of service goes from high to low:
government-imposed regulatory policy, government-
imposed subsidy policy, and government-imposed
no policy; when /1.75 τ <v , the level of service
goes from high to low: government-imposed subsidy
policy, government-imposed regulatory policy, and
government-imposed no policy.
6 CONCLUSION
We consider a market in which manufacturers sell
licensed products and gray market speculators sell
parallel products. Based on the three scenarios of no
government policy, government subsidy policy for
manufacturers and government regulatory policy for
gray market speculators, a game model with
manufacturers as dominant players and gray market
speculators as followers is developed to solve and
analyse the impact of different government policies
on each equilibrium solution for manufacturers and
gray market speculators. The study shows the
following.
(1) As the amount of government subsidies
increases, the unit selling price of manufacturers
selling licensed products will decrease and their
sales volume and profits will increase. The unit sales
price, quantity and profit of parallel products sold by
gray market speculators will be reduced. With
increased government regulation, the unit sales price,
sales volume and profits of manufacturers selling
licensed products have increased and the unit sales
price, sales volume and profits of gray market
speculators selling parallel products have decreased.
Both government subsidies and regulatory policies
can improve the level of after-sales service for
manufacturers.
(2) For manufacturers, the unit sales price of
licensed products is highest under
government-imposed regulatory policies, followed
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50

by no government-imposed policies, and lowest by
government-imposed subsidy policies. When the
ratio of unit regulation to unit subsidy is greater than
a certain threshold value (i.e.
/2 τθ >−v
), the
highest level of sales, profits and after-sales service
of licensed products is achieved when the
government implements a regulation policy, the
second highest level of sales, profits and after-sales
service is achieved when the government
implements a subsidy policy, and the lowest level of
sales, profits and after-sales service is achieved
when the government does not implement a policy.
Conversely, sales, profits and service levels of
licensed products are highest when the government
implements a subsidy policy, second highest when
the government implements a regulatory policy and
lowest when the government does not implement a
policy.
(3) For gray market speculators, when the ratio
of government unit regulation to unit subsidy is
greater than a certain threshold value (i.e. 𝜏/𝑣 >



), the unit sales price, sales volume and
profit of the parallel product is highest when the
government does not implement the policy, the unit
sales price, sales volume and profit is second highest
when the government implements the subsidy policy,
and the unit sales price, sales volume and profit is
lowest when the government implements the
regulation policy. Conversely, the unit sales price,
sales volume and profit of a parallel product are
highest when the government does not implement a
policy, second highest when the government
implements a regulatory policy and lowest when the
government implements a subsidy policy.
The above research has some implications for
government and business control of gray market
transactions, but there are still many shortcomings:
for example, firstly, this paper only considers the
case where the demand function is deterministic,
and there are real-life cases where demand is
uncertain. Secondly, the paper does not break down
gray market speculators, and future research could
consider the entry of retailers into the gray market
separately.
ACKNOWLEDGEMENTS
This paper is one of the achievements of the
National Social Science Fund project "Research on
Online Behavior Mode of Rural Consumers Based
on Social Identity" (project number: 19BGL108)
and the Human and Social Sciences Research
Project of Liaoning Education Department
"Research on Recycling and Remanufacturing of
New Energy Vehicle Power Battery Based on Game
Theory" (project number: 21-A827).
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