Pricing decision models for a two-echelon supply chain
with stochastic price-dependent demand
Tian Zhiyu
Supervisor: Xu Chen
Pan Jingming
Abstract: In this paper we study the impact of power structure, information and demand
fluctuation on the pricing decision of supplier and retailer in a two-echelon supply chain. We first
develop and analysis the case that supplier as the leader in the symmetric information scenario and
consider the way to improve the supply chain performance. Then we discuss the impact of
asymmetric information on pricing decision. We also explore the impact of power structure on the
pricing decision. A series of characters and principles are drawn.
Keywords: Supply chain, Stochastic price-dependent demand, Pricing decision, Game
1 Introduction
Typically in the literature on supply chain coordination, the decisions are set by a
central decision-maker to optimize total supply chain profit[1][2][3]. This centralized
approach requests all members’ interest to be consistent, or despite the conflict of
interest, the central decision-maker can control the system and other members submit
to the system interest optimization. Since most supply chain systems are decentralized,
the centralized approach ignored the competition of the supply chain members.
Competitive behavior between members may lower the supply chain efficiency, so the
centralized optimization approach is only an approximation of the real supply chain
system and the massiness and stability is irresponsible.
Most of the studies of the decentralized supply chain have been focused on the
design of all members’ sub-goals or performance measurement schemes to mitigate
the incentives problems of all members in the supply chain[4][5][6]. In these literatures
market parameters such as demand and selling price are exogenous. However, take
these factors into consideration can provide an excellent vehicle for examining how
operational problems interact with marketing issues to influence decision making at
The
anthor: Tian Zhiyu, an undergraduate of School of Management, University of Electronic Science and
Technology.
the firm level. We use the following two-echelon gaming structure: a supplier
wholesales a product to a retailer, who in turn retails it to the consumer. The retail
market demand is price-dependent and stochastic. How should the supplier and
retailer make their pricing and batch-size decisions? The purpose of this paper is to
analyze this problem.
2 Literature Review
There have substantial studies that range from marketing to operations research
and management on the pricing problems. The marketing literature often focuses on
the coordination of pricing decisions in a single period, without production and
inventory considerations. The operations literature, on the other hand, has traditionally
been focused on coordinating production and inventory decisions, assuming that price
and demand are given. Elmaghraby and Keskinocak[7]give an extensive literature
review of this literature. Now we will concentrate on those that are related to our
study.
The pricing model of the decentralized supply chains has been first studied under
Stackelberg game framework by Eliashberg and Steinberg[8]. The major concern in
their study is to characterize properties of the pricing and production policies of the
Stackelberg solution. Then many studies focused on this problem for a two-echelon
supply chain channel[9][10][11][12]. Their study is based on the assumption that the
price-dependent demand curve is deterministic, either liner or iso-elastic. Lau and
Lau[13]showed that the demand-curve’s shape can affect the system’s optimal solution
in some very counter-intuitive ways, thus demonstrating that results derived from
assuming one demand-curve shape cannot be safely generalized to other demand
curve shapes. Considering the stochastic and price-dependent demand and the
product’s savage and opportunity value, Whitin[14], Mills[15], Porteus[16] and Zabel[17]
formulated a newsboy model with stochastic price-dependent demand. This parallels
in many ways the observation that since the 1950s, the practice of operations has
emphasized functional efficiency at the expense of cross-functional effectiveness.
Comparing with the above studies, our work has three main differences. First, the
demand function is stochastic and price-dependent demand. The objective function is
a utility function. Second, we discuss the impact of asymmetric information on the
decision of the more powerful supplier. Third, the impact of power on the pricing
decision is studied.
3 Basic Definitions and Assumptions
Denotation definition: c－unit manufacturing cost; w－unit wholesale price; p－
unit retail price; q－quantity that the retailer orders; D－market demand. We assume
market demand for the product D is stochastic price-dependent as D(p,ε)=a-bp+ε
(a>0, b>0 and p∈(0,a/b))[15], where a-bp is a decreasing function of p that captures
the dependency between demand and price, and b is the demand elasticity in respect
to retail price, ε is a random variable with known distribution that shows randomness
in demand and is price-independent, we assume that ε follows Normal distribution
with mean μ=0 and standard variance σ, i.e., ε~N(0, σ2). In order to assure positive
demand in some rang of p, we require that a>σ. Obviously D(p,ε) is also Normal
distribution with mean μD= a-bp and standard variance σD=σ.
On the basis of above and with some common sense, we can educe p>w>c and
a>bw>bc. In addition, the supplier and retailer each has a reservation profit, and they
will not participate in the channel if their expected profit are less than that. We
assume it is zero for each, so they will choose to participate in the supply chain
channel on condition that their expected profits are non-negative.
4 Model Formulation and Analysis
4.1 Symmetric information supplier-Stg game
In this section, we assume that the supplier is a Stackelberg leader and the
retailer is a Stackelberg follower (hereafter the “supplier-Stg” game) and information
is symmetric. This relationship between the supplier and retailer is a sequential
noncooperative game. The process can be explained as follows. The supplier as the
leader first declares the wholesale price and the retailer as the follower then decides
on the retail price and quantity of products to order. The supplier maximizes its profit
by specifying the optimal wholesale price taking the behavior of the retailer into
account. The retailer’s decision is setting the optimal retail price to maximize its profit
at given wholesale price. The solution of this game is called Stackelberg equilibrium.
4.1.1. The retailer’s problem
We first study the retailer’s problem. As a Stackelberg follower, the retailer’s
problem is to find optimal retail price p, given the wholesale price w. His profit
function is πr=(p-w)D(p,ε), clearly it is a Normal distribution with mean
μπ=(p-w)(a-bp) and standard variance σπ=(p-w)σ. So the retailer should consider the
trade-off between the expected revenue and the revenue’s standard deviation in taking
a decision[18]. To explicitly effect this trade-off, we propose to define a utility function
Uπr(p) as the retailer’s objective (as Cheng, 1984) [19]: Uπr(p)=μπ-λσπ where λ is the
mean and standard deviation trade-off factor specified by retailer. The retailer’s
problem is to optimal the retail price at the given wholesale price to maximizing the
utility function, that is:
max U r  p    p  w a  bp    .
p
Hence, Uπr(p) is concave with respect to p where p∈(w,a/b) and there exists a
unique optimal retail price at given wholesale price w to make the retailer’s utility
function maximization. Setting ∂Uπr(p)/∂p=0, we get the optimal retail price, which is
a function of wholesale price: p(w)=(a+bw-λσ)/2b where w∈(c,(a-λσ)/b). Substituting
it into q(p)=a-bp-λσ, we can obtain the retailer’s optimal order quantity
q(w)=(a-bw-λσ)/2.
4.1.2 The supplier’s problem
The supplier has dominant bargaining power with symmetric-information in this
supplier-Stg game, and can correctly anticipate the retailer’s reacting for any
wholesale price and select optimal wholesale price which maximizes his profit.
The supplier faces demand curve q(w) and chooses w to maximize his expected
profit: Eπs=(w-c)q(w). We also introduce a utility function Uπs= Eπs(w) as the
supplier’s objective, i.e., Uπs= (w-c)(a-bw-λσ)/2. Since Uπs is concave with respect to
w where w∈(c,a/b), Hence, the supplier’s utility function is concave with respect to
wholesale price, and there exists a unique optimal wholesale price to make the
supplier’s utility function maximization. Setting ∂Uπs(p)/∂w=0, we get the optimal
wholesale price w* =(a+bc-λσ)/2b. Substituting it in to the above equation, we get the
optimal retail price p* =(3a+bc-3λσ)/4b and the optimal ordering quantity q*
=(a-bc-λσ)/4. (w*,p*) is the Stackelberg equilibrium. Then the retailer, the supplier and
the channel’s utilities are, respectively: Uπs*=(a-bc-λσ)2/8b, Uπr*=(a-bc-λσ)2/16b, Uπ*
=Uπs*+ Uπr*=3(a-bc-λσ)2/16b.
Proposition 1:
(1) The optimal wholesale price is decreasing in b, the optimal retail price is
decreasing in b.
(2) The optimal wholesale price is decreasing in σ, the optimal retail price is
decreasing in σ.
(3) The division of channel utility between the supplier and retailer, i.e., the ratio of
maximal utility, RU=Uπs*/ Uπr*=2.
4.1.3 The integrated benchmark
To establish a performance benchmark, we study the problem of an integrated
firm, this is the instance that the supplier and retailer choose the strategy of
cooperation and their objective is to maximize the channel’ profit. Consider first what
an integrated firm would do, it will choose optimal retail price to maximize the
channel’ profit. The integrated channel profit πi=(p-c)D(p,ε) is a Normal distribution,
i.e., πi ~N((p-c)(a-bp),(p-c)2σ2). Consider the trade-off between the expected profit
and the profit’s standard deviation in taking a decision and assume the trade-off factor
between the mean and standard deviation specified by the integrated firm is the same
as the retailer’s, we define a utility function Uπi(p) as the integrated firm’s objective:
max U i  p    p  c  a  bp     p  c   .
p
Thus, Uπi(p) is concave with respect to p where p∈(c,a/b). We get the integrated
firm’s optimal retail price pi*=(a+bc-λσ)/2b and maximal utility Uπi*(p)=
(a-bc-λσ)2/4b. We define channel efficiency as CE=Uπ*/Uπi* to describe the channel
performance. Thus, CE of supplier-Stg game is 0.75.
Proposition 2:
(1) The integrated firm’s maximal utility is decreasing in σ.
(2) The integrated firm’s optimal retail price is lower than the supplier-Stg channel’s.
Conclusion 1:
(1) The wholesale price and retail price both are decreasing in price elasticity of
demand, i.e., the lower the price elasticity of demand, the higher the wholesale price
and retail price. This is accord with the pricing principle in microeconomics.
(2) Taking the demand fluctuation into consideration, the supplier and retailer
both will set lower price to deal with the uncertainty. The retailer will order fewer
products with more fluctuating demand. The supplier’s and retailer’s utility at the
Stackelberg equilibrium strategy are decreasing in the standard deviation of demand.
(3) As the Stackelberg leader, the supplier’s utility is two times as the retailer’s.
So the leader position is an advantage to gain more utility than the follower position
for the supplier in the symmetric information scenario.
(4) The integrated firm’s optimal retail price is lower than the supplier-Stg
channel’s.
(5) The supplier-Stg channel’s efficiency: CE  0.75 .
4.1.4 How to improve the channel performance
We have learn that the supplier and retailer can improve channel performance by
cooperation, but the supplier and retailer will choose cooperation only when
cooperation will not decrease each other’s profit. Now we study how to negotiate the
wholesale price and the split of channel profit. Let p=(1+r)w, (1+r) is the retailer’s
markup ratio, so we get the integrated firm’s objective:
max U i  w, r    1  r  w  c   a  b 1  r  w   
w, r
i
Hence, Uπ (w,r) is concave with respect to (w,r), where w∈(c,a/b) and
r∈(0,(a-bc)/2bc). Hence there exists a unique optimal strategy set (wpe,rpe) to make
the utility function maximization. Solving it we get the optimal strategy
set:Z={((wpe,rpe))| rpe=(a+bc-λσ)/2wpe-1, wpe∈(c,a/b), rpe∈(0,(a-2b)/2bc)}.
Proposition 3: The integrated firm’s optimal strategy is Pareto efficient strategy and
there exits feasible Pareto efficient strategy:
{((wfpe,pfpe))|(a+3bc-λσ)/4b<wfpe<(3a+5bc-3λσ)/8b,pfpe=(a+bc-λσ)/2b}
that can improve the supplier and retailer’s condition synchronously.
Thus the retailer and supplier can choose to cooperation at the set (wfpe,pfpe) to
improve the channel performance, and the supplier’s profit is increasing in wholesale
price and the retailer’s profit is decreasing in wholesale price, so the wholesale price
is determined by the players’ bargain power, attitude toward risk, i.e. trade-off factor
and so on. Then we assume the channel utility split factor is α(0<α<1), then
Uπis*=αUπi*>Uπs* and Uπir*=(1-α)Uπi*>Uπr*. Thus, we obtain the optimal wholesale
price wfpe*=(2bc+α(a-bc-λσ))/2b at 0.5<α<0.75. And we can get ratio of utility
1<RUi<3. This is the condition that the supplier and retailer will choose cooperation
to improve the channel efficiency.
4.2 Asymmetric information Supplier-Stg Game
We assumed that the supplier is the powerful player, the market demand is
stochastic and price-dependent and information is symmetric, on this scenario the
supplier knows the retailer utility function, i.e., the price dependent demand a-bp, the
retailer’s trade-off factor between the mean and standard deviation λ and the market
demand’s standard deviation σ, so we get the exact deterministic Stackelberg
equilibrium (w*,p*). At this equilibrium, RU=2 and CE=0.75. The integrated firm
shows that 1<RUi<3 and CE=1. It seems that the supplier’s leading position ensures
him more power consequently much more profit than the retailer. Such result is based
on the symmetric information and stochastic scenario. This model has omitted the
factor that in the stochastic environment information can hardly be symmetric. The
retailer as the downstream partner facing the market demand directly and has more
precise information of the fluctuation of demand than the supplier as the upstream
firm.
Consider now information asymmetric and stochastic scenario. The factor σ are
known to be a constant, but the value is uncertain to the supplier at the time when he
has to set w, i.e., the supplier perceives D(p,ε) as D(p,ε)~N(a-bp, õ2), where õ is a
random variable. In contrast, we assume that the retailer can wait until the actual
σ-value is known before ordering, and that the supplier is aware of this. It appears
reasonable to assume that the supplier might consider the following arrangements:
(Ⅰ)Enforce the same actions that an integrated firm would take. This requires
pre-negotiating a percentage to split of channel profit, then the retailer and the
supplier cooperation to set the optimal retail price to maximize the channel profit after
the actual σ-value is known to the retailer.
(Ⅱ)Enforce a supplier-imposed retail price before knowing the actual σ-value.
(Ⅲ)Play the supplier-Stg game, i.e., the supplier sets only w, and leave the retail
pricing decision entirely to the retailer.
4.2.1 Alternative (Ⅰ) the integrated firm
Consider first what an integrated firm would do. The firm knows that after
observing the actual value of σ, it can choose optimal retail price to maximize the
channel’s utility. This scenario is the same as the integrated benchmark, so the
optimal retail price and the maximal utility are, respectively: pI*=pi*=(a+bc-λσ)/2b,
UπI*(p)=Uπi*(p)= (a-bc-λσ)2/4b.
Now we consider the actual situation in which the supplier and retailer are
separate entities. Assume that the players have agreed through solving the problem of
how the retail price is set that the supplier should receive α(0<α<1) of the channel’s
profit. Coordination can theoretically be done as follows: the supplier asks the retailer
to set p at whatever level necessary to maximize channel utility after σ is revealed to
the retailer. The supplier will then charge the retailer w that is pre-negotiated or will
lead to the pre-negotiated percentage split of channel profit.
In the integrated benchmark, the supplier and retailer will choose cooperation
when they will obtain more utility than they play the supplier-Stg game. But in this
arrangement, the supplier faces asymmetric-information and retailer has more market
demand information, i.e., the retailer has more negotiation power on the utility split
now. In the information asymmetric scenario, w and α is also determined by the
players’ bargain power, attitude toward risk, i.e. trade-off factor λ in our model and so
on. But the difference is at first the supplier have to charge initial shipments to the
retailer, it maybe higher or lower than the optimal w; then after σ is determined by the
retailer and the final total retail revenue is realized, the supplier refunds the retailer the
amount UπIs*-αUπI* (the higher w arrangement) or the retailer remits to the supplier a
supplementary payment UπIr*-(1-α)UπI* (the lower w arrangement). Both arrangements
require one player to trust that the other would reimburse an appropriate amount at the
end of the selling activity. Although there is no theoretical reason why this is
impossible, it does imply a much higher level of willingness to coordination and share
information than in the symmetric information situation, there is now a genuine
inconvenience in replicating the integrated firm.
4.2.2 Alternative (Ⅱ) supplier-imposed retail price
In this arrangement, we assume the supplier impose a retail price ps before
knowing the actual σ value. For any realized σ value, the quantity the retailer will
order is q=a-bps-λσ. So the supplier’s utility:UπIIs=(w-c)(a-bps-λõ). Hence the utility is
stochastic and the supplier’s expected utility: EUπIIs= (w-c)(a-bps-λEõ). So the
supplier will set the wholesale price as high as possible and the retail price as low as
possible as soon as the retailer’s expected utility is above retailer’s subsistence level θ.
Thus the supplier’s objective is:
max EUs   ps  c  a  bps   Eo    .
ps
Following the way above, we can obtain ps*=(a+bc-λEσ)/2b. After observation
the actual σ value, the retailer will order (a-bc+λ(Eσ-2σ))/2 unit products, so the
channel’s utility is: UπII*=(a-bc-λEσ) (a-bc+λ(Eσ-2σ))/4b.
4.2.3 Alternative (Ⅲ) play the supplier-Stg game
We now assume that before observing the actual σ value, the supplier sets
optimal w that maximizes his expected utility, knowing that after being informed of w,
the retailer will set p after observing both w and the actual σ. The supplier knows that
for any given w and observed σ, the retailer will set p(w,õ)=(a+bw-λõ)/2b and thus
q(w,õ)=(a-bw-λõ)/2. So the supplier’s objective is:
max EUs   w  c  a  bw   E  2 .
w
Hence, we obtain: pIII*=(3a+bc-λ(Eσ+2σ))/4b, wIII*=(a+bc-λEσ)/2b
UπIIIs*=(a-bc-λEσ)(a-bc+ λ(Eσ-2σ)) /8b,
UπIIIr*=(a-bc+λ(Eσ-2σ))2/16b,
UπIII*=(3a-3bc-λ(Eσ+2σ))(a-bc+λ(Eσ-2σ))/16b.
Conclusion 2:
(1) Alternative(Ⅰ) yields the highest expected total channel utility, but it is also
the most unrealistic.
(2) Alternative(Ⅱ)’s maximal utility is concave with respect to Eσ, and it reaches
maximal point where the Eσ=σ, i.e., the supplier estimate the σ-value precisely. At
this maximal point, the channel’s utility is the same as the Alternative(Ⅰ)’s.
Alternative(Ⅲ)’s maximal utility is increasing in Eσ. This can be explained as follows:
a higher Eσ implies a lower w and p but a higher retailer’s marginal profit and a
higher ordering quantity. Thus, a higher channel utility can be realized whereas the
division of channel utility between the supplier and retailer decrease.
(3) The difference between Alternative(Ⅱ) and Alternative(Ⅲ)’s maximal utility
is concave with respect to Eσ. Generally, Alternative(Ⅱ)’s utility is larger than
Alternative(Ⅲ)’s where Eσ<(a-bc+2λσ)/3λ and Alternative(Ⅱ)’s utility is smaller
than Alternative(Ⅲ)’s where Eσ>(a-bc+2λσ)/3λ. That is to say, the lower supplier’s
estimated
standard
variance
of
demand
implies
more
advantageous
of
Alternative(Ⅱ) than Alternative(Ⅲ) and the higher standard variance of demand
implies more advantageous of
Alternative(Ⅲ) than Alternative(Ⅱ). Exceptionally,
in the condition that σ>(a-bc)/2λ and -(a-bc-2λσ)/λ>Eσ>0, Alternative(Ⅱ)’s utility is
smaller than Alternative(Ⅲ)’s. This is the condition that the standard variance of
demand is sufficiently large, the lower supplier’s estimated standard variance of
demand will make Alternative(Ⅲ) performance better.
(4) At Alternative (Ⅲ), the supplier choose to play supplier-Stg game and the
supplier’s
utility is
smaller
than
the
retailer’
utility in
condition
that
Eσ>(a-bc+2λσ)/3λ in the general condition, and -(a-bc-2λσ)/λ>Eσ>0 in the
exceptional condition that σ>(a-bc)/2λ. In these conditions that the standard variance
of demand is small and the supplier’s estimated standard variance of demand is large
or the standard variance of demand is sufficiently large and the supplier’s estimated
standard variance of demand is small, the more power supplier’s utility is less than the
weaker retailer’s. So on the asymmetric information and stochastic scenario the
supplier-Stg game often require the more powerful player to either accept much lower
profits than the weaker counterpart or be able to compel the weaker party to
participate at other arrangements.
5 A More Powerful Retailer
In the literature of OR/MS supply chain coordination, the focus is on a
relationship in which supplier is the leader and retailer is the follower. This implies
the dominant power of supplier over retailer. Recent market structure reviews have
shown a shift of power from supplier to retailer. Based on this market phenomenon, in
this section we study two conditions that the retailer has more power: retailer as the
leader (hereafter the retailer-Stg game) and the retailer and supplier have equal power.
5.1 The retailer as Stackelberg leader
We first study the retailer-Stg game. The retailer takes an active role and declares
a nonnegative marginal profit v so the retail price is p=w+v. The supplier reacts by
choosing w that maximizes his utility given the declared value v. The retailer has
complete knowledge of the reaction w, so he can choose the optimal value of v to
maximize his profit. We assume the supplier has the same attitude toward risk as the
retailer, i.e. trade-off factor λs=λ. We also define the supplier and retailer’s objectives
are,
respectively:
Uπs(w)=(w-c)(a-b(w+v)-λσ)
and
Uπr(w)=v(a-b(w(v)+v)-λσ).
Following exactly the same approach we obtain: w** =(a+3bc-λσ)/4b ， p**
=(3a+bc-3λσ)/4b.
The channel efficiency CE=0.75 and the division of channel utility between the
supplier and retailer RU=1/2, i.e., this is similar to the supplier-Stg game. In the
retailer-Stg game, the leader retailer declares that he will have a unit profit margin (or
markup) of v, regardless of what the supplier does. It is not self-evident that the
retailer can be the leader, so imposing this game would require justification.
5.2 The equal power scenario
Now we study the case he retailer and the supplier have equal power and make
their
decision
of
retail
price
and
wholesale
price
simultaneously
and
non-cooperatively maximize their profit with respect to any possible strategies set by
other member in the system. This problem is a simultaneous move game and the
solution is called Nash equilibrium.
In this game, we assume the retailer make the decision of a markup ratio of price
(1+r) and at the same time the supplier make the decision of w. We also define the
supplier’s objective as Uπs(w)= (w-c)(a-b(1+r)w-λσ) and the retailer’s decision
function is p(w)=(a+bw-λσ)/2b as section 4.1.4. Following exactly the same approach
we get the Nash equilibrium (w***, p***):

 4b ,
8bc  a     b c  8b .
w***  bc  8bc  a     b 2 c 2

p***  4  a     bc 
2 2
Conclusion 3:
(1) From section 4.1, 5.1 and 5.2, we can educe w***<w*, w**<w*, Uπs***<Uπs*,
Uπs**<Uπs*. Hence, the supplier can set higher w and obtain more utility as the leader.
It is more complicated in the compare of the follower and equal power position.
w***>w* where a≥3bc, (a-3bc)/λ<σ<(a-bc)/λ or bc  a  3bc and at this condition
Uπs**<Uπs***. This is the general condition that the supplier set higher wholesale price
and obtain more utility with more power. Considering the standard variance of
demand, the exceptional condition is w***<w** where a>3bc, 0<σ<(a-3bc)/λ and at
this condition Uπs**>Uπs*** where a>(7+5×20.5)bc, 0<σ<(a-(7+5×20.5)bc)/λ. With
the growing market size and the small fluctuate of demand, the retailer’s leading
position will make the supply chain performance better and hence the supplier can
obtain more profit than the equal power situation could explain this exception.
(2) We can also educe p***<p*= p**, p*- w*< p***-w***< p**-w**, and
Uπr***>Uπr**> Uπr*. That is to say, the retailer’ can charge lower retail price but get
higher marginal profit hence higher utility with power grows but doesn’t exceed the
supplier’s power. As he becomes the Stackelberg leader, he can gain more marginal
profit hence more utility and at the same time keep the retail price unchanged.
(3) Uπ***>Uπ**= Uπ*, so CE of the Nash game is higher than of the Stackelberg
game.
6 Conclusion
The first part of this paper studied the optimal solutions of a supplier-Stackelberg
game and the integrated firm performance of the two-echelon supply chain system
with stochastic price-dependent demand and symmetric information. The way to
improve the performance of the supply chain was proposed. Then we take asymmetric
information into consideration. We present a simple scenario with stochastic demand
and asymmetric information and consider the actual situation in which the supplier
and retailer are separate entities, demonstrate how this leads to more plausible results.
This scenario is more practical because information asymmetric is ubiquitous in the
supply chain system, especially the stochastic market demand. After that, we studied
the scenario of a more power retailer, and analysis how the power effect the game
players’ decision.
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