Supplier and Buyer Driven Channels in a Two-Stage Supply Chain

Ertek, G., and Griffin, P. (2002). “Supplier and buyer driven channels in a two-stage supply chain.”
IIE Transactions, 34, 691-700.

Note: This is the final draft version of this paper. Please cite this paper (or this final draft) as
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Supplier and Buyer Driven Channels in a
Two-Stage Supply Chain

Gürdal Ertek and Paul M. Griffin

School of Industrial and Systems Engineering

Georgia Institute of Technology

765Ferst Dr., Atlanta, GA, 30332

Supplier and Buyer Driven Channels in a Two-Stage
Supply Chain
G¨ rdal Ertek & Paul M. Griffin∗
u
School of Industrial and Systems Engineering
Georgia Institute of Technology
765 Ferst Dr., Atlanta, GA, 30332
e-mail:pgriffin@isye.gatech.edu
Phone: (404)894-2431, Fax: (404)894-2300

November 3, 2001

ABSTRACT
We explore the impact of power structure on price, sensitivity of market price, and profits in a two-stage
supply chain with single product, supplier and buyer, and a price sensitive market. We develop and
analyze the case where the supplier has dominant bargaining power and the case where the buyer has
dominant bargaining power. We consider a pricing scheme for the buyer that involves both a multiplier
and a markup. We show that it is optimal for the buyer to set the markup to zero and use only a
multiplier. We also show that the market price and its sensitivity are higher when operational costs
(namely distribution and inventory) exist. We observe that the sensitivity of the market price increases
non-linearly as the wholesale price increases, and derive a lower bound for it. Through experimental
analysis, we show that marginal impact of increasing shipment cost and carrying charge (interest rate)
on prices and profits are decreasing in both cases. Finally, we show that there exist problem instances
where the buyer may prefer supplier-driven case to markup-only buyer-driven and similarly problem
instances where the supplier may prefer markup-only buyer-driven case to supplier-driven.

∗
corresponding author

1 Introduction
An interesting issue that has arisen in recent years in the area of supply chain man-
agement is how decisions are effected by the bargaining power along the channel. For
example, with the advent of the internet in recent years, customers have access to much
more information including price, quality and service features of several potential sup-
pliers. This information has in many cases increased their position to acquire goods
and services. On the other hand, if the number of suppliers is limited, then clearly a
supplier’s position is increased.
In this paper, we explore the impact of bargaining power structure on price, sensitiv-
ity of market price, and profits in a two-stage supply chain. We examine the case when
a single product is shipped from a supplier to a buyer at a wholesale price and then sold
to a price-sensitive market. Operational costs (namely distribution and inventory) are
included in the analysis.
Throughout this paper we will refer to the supplier as he and the buyer as she. Two
models arise when either the supplier or buyer has dominant bargaining power in the
supply channel, similar to the economics literature where a dominant (or leader) firm
moves first and a subordinate (or follower) firm moves second (Gibbons [?], Stackelberg
[?]).
The supplier-driven channel occurs when the supplier has dominant bargaining power
and the buyer-driven channel occurs when the buyer has dominant bargaining power.
Even though buyer-driven models are encountered less frequently in literature (compared
to supplier-driven models), there is practical motivation: Messinger & Narasimhan [?]
provide an interesting discussion on how the bargaining power has shifted to the retailer
(buyer) in the grocery channel.
The organization of the paper is as follows: In Section 2 a brief literature review is
provided. In Section 3 we describe the assumptions of our analysis. The supplier-driven
and buyer-driven models are then developed in Section 4. An example is presented
in Section 5, including a comparison to the coordinated case where total net profit
throughout the supply chain is maximized. An analytical comparison of supplier-driven
and markup-only buyer-driven cases is given in Section 6. We also show that there are
cases under which a supplier will prefer the buyer-driven channel and where a buyer
will prefer the supplier-driven channel. Conclusions and future research are described in
Section 7.

2

2 Literature Review
A significant amount of research has been done in the area of supply chain coordination.
Much of it has focused on minimizing the inventory holding and setup costs at different
nodes of the supply chain network. The classic Clark & Scarf [?] and Federgruen [?] are
examples. Traditionally, this type of research has assumed that the task of designing
and planning the operations is carried out by a central planner. However, the increased
structural complexity and the difficulty of obtaining and communicating all the infor-
mation scattered throughout the different units of the supply chain is a major block
in applying central planning. Research along decentralized control has been done by
a number of people and good reviews are found in Whang [?], Sarmiento & Nagi [?],
Erenguc et. al. [?], and Stock et. al. [?].
Although a significant portion of the coordination literature assumes vertical integra-
tion, some recent research focuses on contractual agreements that enable coordination
between independently operated units. Tsay et. al. [?] present a comprehensive review
on contract-based supply chain research. Jeuland & Shugan [?] present an early treat-
ment of coordination issues in a distribution channel. Porteus [?] establishes a framework
for studying tradeoffs between the investment costs needed to reduce the setup cost and
the operating costs identified in the EOQ. He also addresses the joint selection of the
setup cost and market price, comparable to Section 4.1 of our research. Abad [?] formu-
lates the coordination problem as a fixed-threat bargaining game, characterizes Pareto
efficient solutions and the Nash bargaining solution and proposes pricing schedules for
the supplier. Weng [?] also focuses on role and limitation of quantity discounts in chan-
nel coordination and shows that quantity discounts alone are not sufficient to guarantee
joint profit maximization under a model where both the market demand is decreasing
in prices and the operating cost depends on order quantities.
Ingene & Parry [?] investigate a model with fixed and variable costs at the two
stages and establish the existence of a menu of two-part tariffs that mimic all results of
a vertically integrated system. Wang & Wu [?] consider a similar model and propose
a policy that is superior for the supplier when there are many different buyers. Other
examples of related research include McGuire & Staelin [?] and Moorthy [?].
An extensive body of literature focuses on optimizing two-stage supply chains with
stochastic demands. Cachon [?] and Cachon & Zipkin [?] extend this literature by
developing game-theoretic models for the competitive cases of continuous review and
periodic review models. Moses & Seshadri [?] consider a periodic review model with
lost sales. Netessine & Rudi [?] develop and analyze models of interaction between a
supplier (wholesaler) and a single buyer (retailer) for “drop-shipping” supply chains.

3

In this paper we try to extend the current literature by considering operating costs
explicitly along with different power structures of the market channel.

3 Model Assumptions
Figure 1 illustrates the flow of products, information and funds in our models. The
product is shipped from the supplier to the buyer at wholesale price t [$/unit], and is sold
to the market by the buyer at market price p [$/unit]. The deterministic market demand
µ[p] is linear in p: µ[p] = m(a − p), m = d , a − b ≤ p ≤ a, 0 < b ≤ a, d > 0.
b
The buyer operates under a simple deterministic EOQ model. She places an order to
the supplier for a shipment of size q [units/order], τ [years] before she runs out of her
stock. The carrying charge (interest rate) that the buyer uses for calculating cost of
in-site and in-transit inventory is r [%/year]. The buyer has a reservation net profit RB ,
and she will not participate in the channel if her net profit is less than RB . The supplier
does not have any operational costs, but incurs a unit variable cost c independent of any
decision variables. This would occur when the supplier is functionally organized and the
sales department acts independently of operations (which is typical for large firms).
The profit function of the supplier is given by:

πS [t] = (t − c)m(a − p) (1)

The supplier’s wholesale price t is freight on board (f.o.b.) origin; that is, the buyer
makes the payment and then takes responsibility for the product at the origin by bearing
the shipment and material handling costs for the shipments to her facility. The buyer
pays the carrier at the time the shipment reaches her facility. The shipment takes a
deterministic time τ [years] to arrive at the buyer’s facility and costs k [$/order]. The
transit time and the cost of an order are independent of the order quantity q [units/order]
and the carrier always has sufficient capacity. The cost of an order is incurred at the
moment the shipment arrives at the buyer’s facility.
The value of the product on-site is accounted as t [$/unit]. Assuming the value
and the holding cost of product dependent on t (rather than a fixed value) makes the
profit function nonconcave at certain ranges. Thus, our analysis will yield the type of
results related with the regions of concavity and nonconcavity, similar to Porteus [?] and
Rosenblatt & Lee [?].
The buyer’s decision variables are the market price p and the order quantity q. The
profit function of the buyer is given by:

4

πB [p, q] = (p − t)µ[p] − kµ[p]/q − qrt/2 − µ[p]τ rt (2)

The approach of considering fixed unit costs at a certain stage and costs as a function
of operational costs at another stage of the supply chain is similar to the analysis of Ford
Customer Service Division done by Goentzel [?]. He focused on the customer allocation
problem where the cost per unit product at a warehouse that faces the customers is fixed,
but the cost of routing to a cluster of customers is depends on the choice of customers.
The assumption regarding the channel structure in this paper bears discussion. Mon-
ahan [?] notes that single supplier single buyer channels are often invisible to the public,
and lists examples where such a relationship may exist:

• “Small, closely-held or privately owned companies, producing exclusively for other
larger manufacturers or distributors;

• Common job-shops, supplying customized products for an individual buyer;

• Manufacturing arms or divisions of independently run parent companies.”

These examples define a very narrow scope of the economy, and it is true that most
of the economic markets are characterized by oligopoly or perfect competition. On the
other hand, Stuckey & White [?] explain how site specificity, technical specificity and
human capital specificity may create bilateral monopoly. They describe many indus-
tries, including mining, ready-mix concrete and auto assembly, that operate as bilateral
monopolies.
Single-supplier and single-buyer relationships are common also due to benefits of
long-term partnership. Some of these benefits are as follows (Tsay et. al [?]):

• Reduced ordering costs (i.e. reduced ordering overhead, due to established rela-
tionship)

• Additional efforts towards compatibility of information systems

• Additional information sharing

• Collaborative product design/redesign

• Process improvement, quality benefits

• Agreement on standards on lead-times and quality measures

5

Therefore a significant portion of firms in the economy can benefit by such a relationship.
We assume that the parties are interested in a long-term relationship and are using
a contract to enforce commitment to the relationship. Even though the dominant party
may deviate, we assume that s/he does not do so, since this might have associated costs.

4 Two-Stage Supply Chain Models
Both the supplier-driven and buyer-driven two-stage supply chain models are developed
in this Section. Before developing these models, we first discuss the buyer’s decision
problem given the wholesale price t.

4.1 Buyer’s Decision Problem When t is Given
In this section we investigate the decision problem of the buyer, whose objective is to
find the optimal market price p∗ , given the wholesale price t. As mentioned previously,
0
this problem was studied in detail in[?], and we summarize and extend the results here.
dπB [p,q] dπB [p,q]
By setting dq
= 0 and dp
= 0 and solving for q and p, we obtain:

2km(a − p)
q = (3)
rt
p = w/2 + k/2q (4)

where w = a + (1 + rτ )t.
The extreme points of πB [p, q] solve these equations. When these equations are solved
simultaneously, we obtain a cubic polynomial equation:

Φ(p) = p3 + Φ2 p2 + Φ1 p + Φ0 = 0 (5)

where

Φ2 = −(a + w),
Φ1 = ((4a + w)w)/4,
Φ0 = −aw 2 /4 + krt/(8m)

We use πB [p] to denote the net profit as a function of p alone: πB [p] = πB [p, q ∗ [p]].
Substituting the expression for q from (3) into the net profit function πB [p] yields:

6

m(a − p)p − (1 + rτ )m(a − p)t − ψ (a − p)t if a − b ≤ p < a
πB [p] = (6)
0 if p = a
√
where ψ = 2krm.
Notice that the profit function is defined as 0 when p = a since a discontinuity would
otherwise arise. We now define (p∗ , q0 ) as the optimal (p, q) pair when the constraint
0
∗

a−b ≤ p in the definition of the linear demand function is taken into account. πB [p∗ , q0 ] =
∗
0
∗

πB [p∗ ] is the optimal solution value:
∗
0

Definition 1 Let p∗ = argmax{a−b≤p≤a} πB [p], q0 = q[p∗ ] and πB [p∗ , q0 ] = max{a−b≤p≤a} πB [p].
0
∗
0
∗
0
∗

The regions where πB [p] is concave and convex can be found by investigating the
second order derivate of πB [p]:

d2 πB [p] ψt2 ((a − p)t)−3/2
= −2m +
dp2 4
which gives us the following:

Theorem 1 πB [p] is concave when p ≤ p and convex when p > p, where p = a −
˜ ˜ ˜
√ 2/3
ψ t
8m
.

Now let us characterize the cubic equation Φ(p) = 0, when it has three roots:

Theorem 2 If t ≤ t then Φ(p) = 0 has three real roots p0,1 , p0,2 and p0,3 . Let p0 =
argmax{p=p0,1 ,p0,2 ,p0,3 } πB [p]. p0 < p.
˜ p0,min = min {p0,1 , p0,2 , p0,3 } < w .
2
p0 =
min{ {p0,1, p0,2 , p0,3 }{p0,min } }.

Proof: Given in A.1. ✷

Theorem 2 tells us that among the three roots of (5), the middle one is the one that
maximizes πB [p].

4.2 Supplier-driven Channel
Let us assume that the supplier has dominant bargaining power and has the freedom
to decide on any t value that maximizes his net profit with no consideration for the
buyer. The buyer reacts to the wholesale price t declared by the supplier by selecting
her optimal price p∗ [t] (and the corresponding q0 [t]) that maximizes her net profit given
0
∗

t.

7

Since the supplier knows the cost structure and the decision model of the buyer, he
also knows the reaction p0 [t] of the buyer to t. Since p0 [t] determines p∗ [t] as described
0
earlier, we will present some of our results in terms of p0 [t].
We deﬁne the sensitivity of the buyer’s optimal price ξ[t, p0 [t]] as follows:

Deﬁnition 2 Let ξ[t, p0 [t]] denote the sensitivity of the buyer’s optimal price p0 [t] with
respect to the supplier’s wholesale price t. ξ[t, p0 [t]] is the ratio of marginal change in
p0 [t] to marginal change in t at the point (t, p0 [t]).

An expression for ξ[t, p0 [t]] is provided in the following Theorem:
ξ0 [t,p0 [t]]
Theorem 3 ξ[t, p0 [t]] = ξ1 [t,p0 [t]]
, where

ξ0 [t, p] = (1 + rτ )(p2 − (2a + w)p/2 + aw/2) − kr/(8m) (7)

where w = a + (1 + rτ )t, and

ξ1 [t, p] = 3p2 + 2Φ2 p + Φ1 . (8)

Proof: Given in Appendix A.2. ✷
The following Theorem gives a lower bound for ξ[t, p0 [t]] that is independent of any
parameters and decision variables:

Theorem 4 ξ[t, p0 [t]] > 1/2.

Proof: Given in Appendix A.3. ✷
An instance that achieves this lower bound is one that has all the logistics related
costs equal to zero (k = 0, r = 0, τ = 0). This instance is equivalent to the classic
bilateral monopoly model in the Economics literature (Spengler [?], Tirole [?]), where
the sensitivity of market price to wholesale price is always 1/2.
The optimal wholesale price for Model 3 can be determined by taking the partial
derivative of πS [t] with respect to t and setting it equal to zero:

a−p
= ξ[t, p0 [t]] (9)
t−c
We will refer to solution of the above equation as t2 . Since (9) does not yield a
closed form expression for t2 , one has to resort to numerical methods for solving the
equation. The following Theorem, with proof given in Appendix A.4, guarantees that
the numerical solution would indeed be the global optimum:

Theorem 5 πS [t] is concave in t.

8

4.3 Buyer-driven Channel
In this case the buyer takes an active role and declares a nonnegative price multiplier
α and a nonnegative markup β and states that she will set p = αt + β. The supplier
reacts by choosing the t that maximizes his net profit given the α and β declared by
the buyer. The buyer has complete knowledge of the reaction wholesale price t3 [α, β]
that the supplier will respond with to her declared α and β. She chooses her (α, β) so
as to maximize her net profit PB [α, β], given t3 [α, β]. We assume that the supplier will
participate in the supply chain channel as long as his profit is nonnegative.
One interesting result is that the buyer would use only a multiplier under these
conditions:

Theorem 6 β ∗ = 0.

Proof: The supplier’s profit function is πS = (t − c)m(a − p). The optimal t value
is found by taking the partial derivative of πS [t] with respect to t and setting equal to
zero.

∂πS [t] ∂p
= m (a − p) − (t − c) = 0
∂t ∂t
a−p
t = +c (10)
α
Assuming that p is fixed, α and β can be calculated so as to obtain p once t is
determined. Therefore, the selection of t determines α and β. Equation (10) shows that
the supplier would select a smaller t as α increases, all other things being fixed. The
largest α can be obtained by setting β = 0. Since this fact holds for any fixed p value,
β ∗ = 0. ✷
Since the optimal markup is equal to zero (β ∗ = 0) then P (α, β) can be optimized
with respect to α alone to find α∗ . By differentiating the supplier’s profit function
with respect to t and setting it to 0, we get the supplier’s optimal wholesale price
t3 [α, β] = a+αc−β . The corresponding market price p3 [α, β] = a+αc+β is then found by
2α 2
substituting into the market demand function.
The terms t3 [α, β] and t3 [α, β] can be substituted into (6) to obtain the buyer’s net
profit function PB [α, β] in the buyer-driven channel.
PB [α, β] cannot be solved analytically to find (α∗ , β ∗). On the other hand, Theorem
6 tells us that the buyer will always set β ∗ = 0 at the optimal, and will decide only on α.
The first order condition ∂PB [α,0] = 0 is not analytically solvable. Yet, in the following
∂α
Theorem, we show concavity of PB [α, β] in α, suggesting that a simple search algorithm
would enable us to find α∗ which is globally optimal:

9

Theorem 7 PB [α, 0] is concave in α.

Proof:
The expression in (6) is composed of three components, the first is a function in p
multiplied by the positive constant m, the latter components are functions of p and t
multiplied by negative constants. We will prove that f1 [α] = (a − p3 [α, 0])p3[α, 0] is
concave, and f2 [α] = (a − p3 [α, 0])t and f3 [α] = f2 [α] are convex, which will prove the
concavity of PB [α, 0].
a2 −α2 c2
First we focus on f1 [α] = (a − a+αc )( a+αc ) =
2 2 4
. Since df1 /dα2 = −c2 /2 < 0, f1
2

is concave in α.
a2 −c2 α2 a2
Next we focus on f2 [α] = ( a−αc )( a+αc ) =
2 2α 4α
. Since d2 f2 /dα2 = 2α3
> 0, f2 is
convex in α.
a2 −c2 α2 3a4 −6a2 c2 α2 −c4 α4
Finally we focus on f3 = 4α
. The second derivative is d2 f3 /dα2 = !
a2
.
8α3 (a2 −c2 α2 ) α
−c2 α
Since the denominator is the product of 8α2 , (a2 −c2 α2 ), and a square-root expression, it
is always positive. The numerator can be expressed as (2a4 − 2a2 c2 α2 ) + (a4 − 2a2 c2 α2 −
c4 α4 ), which can further be expressed as 2a2 (a2 − c2 α2 ) + (a2 − c2 α2 )2 . Since both of
2
the terms in this expression are positive, we have d f23 > 0, and thus f3 is convex.
dα
Having shown the concavity of its three components, we conclude that PB [α, 0] is
concave. ✷

5 An Example Channel Setting
Consider a supply chain channel where c = 5.600, a = 12.800, k = 480, τ = 0.010, r =
0.100, m = 2180, b = 12.800, RB = 0. Figure 2 shows the profits/costs related with the
buyer when the supplier sets the wholesale price as t = 11.000.
In Figure 2, w/2, p0 and p = 12.577 are indicated with dashed vertical lines. If there
˜
were no logistics related costs, the buyer would set her price to w/2 = 11.906. The
optimal price when the logistics costs are considered is greater: p∗ = p0 = 12.116. The
0
optimal profit of the buyer is πB [p∗ ] = πB [12.112] = 392.848. As long as a − b is less
∗
0
than 12.116, the buyer would clearly keep pricing at p∗ = p0 = 12.116. If a − b were
0
greater than 12.116, the buyer would see if setting p = a − b would bring any profits or
not, and if πB [a − b] ≥ RB she would set p∗ = a − b.
0
In order to compare the buyer-driven and supplier-driven models to “optimal”, we
use the coordinated net profit function πC [p, q] can be defined by:

πC [p, q] = (p − c)µ[p] − kµ[p]/q − qrc/2 − µ[p]τ rc (11)

10

This function is similar to πB [p, q] in (2), but there are some differences. Here, the
profit per unit is (p − c), as opposed to profit per unit (t − c) of πB [p, q]. The value of
the product on-site and in transit is accounted as c [$/unit] in πC [p, q], as opposed to t
[$/unit] of πB [p, q].
Cost c reflects the true value of the product, rather than the artificially created
wholesale price t. A central planner would use the true value of the product in calculating
the logistics costs. Notice that t does not appear in this cost function at all.
Analysis of the coordinated channel, where t = c = 5.600, tells us that the optimal
market price is p∗ = p0 = 9.269. The optimal profit of the buyer is πB [p∗ ] = πB [9.269] =
0
∗
0
26165.053.
Next we consider the supplier-driven model, and investigate the best pricing policies
of the supplier. We are implicitly assuming that the buyer would prefer to operate at
exactly zero profit, rather than destroy the channel. We focus on the supplier’s profit
maximization problem in this case. The supplier solves a−p0 [t] = ξ[t, p0 [t]] as given in
t−c
(9). We perform a bisection search to find (p2 , t2 ). The solution is found to be (t2 , p2 ) =
(t2 , p0 [t2 ]) = (8.974, 11.009), with buyer’s net profit πB [p2 ] = πB [11.009] = 6075.747 as
opposed to her best possible profit of πB [c] = 26165.053. Meanwhile the supplier attains
πS [8.974] = 13176.616.
A plot of the change of p0 with respect to t seems to suggest a linear relationship
(constant sensitivity). However, a plot of the change of the sensitivity ξ of market price
with respect to t in Figure 3 shows that this is not the case. Between values of t = c
and t = t there is a change of ∼10%. This shows that the buyer’s reaction price p0
¯
becomes increasingly sensitive to the supplier’s wholesale price t. This can be explained
as follows: As the wholesale price becomes greater, the buyer has to account not only for
this increase in prices, but also with the fact that the operational costs become a greater
percentage of total costs. The market becomes more and more of a niche market, and
market price increases nonlinearly.
Next we consider the buyer-driven case where the buyer is interested in maximizing
his net profit function PB [α, β] shown in the contour plot of Figure 4. It is easy to see
that α ≈ 1.6 and β = 0 at the highest point of the surface. The precise values are
(α∗ , β ∗ ) = (1.600, 0). The reaction wholesale price of the supplier is t∗ = 6.799 and the
3
market price is p3 = p3 = 10.880. Even though it is optimal to have β = 0 at α∗ , that’s
∗ ∗

not the case for other α values. So if value were fixed to α = αF due to certain conditions
in the channel, one would set β to the unique positive value that corresponds to the β
value of the point where the surface α = αF is tangent to the contour line of PB [α, β].
On the other hand, for a fixed RB (an isoprofit curve in Figure 4) and a fixed α, there
exist two possible choices of β, and the buyer should select the one that yields greater

11

profits for the supplier. Finally, we compute optimal β for α = 1 as β ∗ = 3.765. This is
where the buyer is restricted to declaring only a markup (thus “β-only buyer-driven”).
Tables 1, 2 and 3 show the profits, prices and sensitivities respectively in different
channel structures. The total profit in the coordinated channel structure is the max-
imum, followed by the buyer-driven structure. This result is due to the fact that our
model assumes logistics costs only at the buyer, not at the supplier, and in the buyer-
driven case the buyer declares her multiplier (and thus sensitivity) a high value to force
the supplier choose a lower wholesale price. The increase in total profit when one goes
from supplier-driven to buyer-driven is ∼ 4.3%.

Channel Structure Supplier’s Profit Buyer’s Profit Total Profit
Coordinated 0 26165.053 26165.053
Supplier-driven 13176.616 6075.747 19252.363
Buyer-driven 5020.9396 15053.371 20074.3106
β-only Buyer-driven 6429.454 12111.658 18541.112

Table 1: Profits in different channel structures

Channel Structure Wholesale Price Market Price
Coordinated t1 = 5.600 p1 = 9.269
Supplier-driven t2 = 8.974 p2 = 11.009
Buyer-driven t3 = 6.800 p3 = 10.880
β-only Buyer-driven tβ = 7.317 pβ = 11.083

Table 2: Prices in different channel structures

Channel Structure Sensitivity at the Optimal
Coordinated 0.5112
Supplier-driven 0.531
Buyer-driven 1.600
β-only Buyer-driven 1.000

Table 3: Sensitivity ξ[t, p0 [t]] of market price to wholesale price in different channel
structures

We also performed a numerical analysis to observe the impact of changing (m, k) and
(m, r) on prices and sensitivity of market price. For the supplier-driven model we used
Mathematica and searched for t2 by a bisection routine. For the buyer-driven model we
used online SNOPT nonlinear optimization solver at NEOS server [?], and expressed the
model in AMPL language.

12

Figure 5 shows the change of p2 with changing values of the market demand multiplier
m and the ordering cost k. One observation is that as m increases, p2 decreases, which
suggests economies of scale in the supply chain due to reduced ordering cost per unit.
This tells that as the market size increases, decreasing the price of the product is more
beneficial for the buyer (since smaller market price takes place only when the wholesale
price is smaller, we infer that the wholesale price also decreases with larger m). Similar
economies of scale results helping all players. This pattern is due to the fact that the
ordering cost does not increase proportionally as demand increases. Also, the impact of
increased demand is much more pronounced in smaller values of m (up to a point). For
a fixed m value, p2 (and thus t2 ) increase with increasing values of the ordering cost k.
However, the increase in p2 becomes smaller at higher values of k and m. Meanwhile,
it’s interesting to note that this economies of scale take place only after a certain market
size, and higher k requires greater demands for economies of scale to come into effect.
A similar analysis of change with respect to (m, r) shows that for a fixed m value,
p2 (and thus t2 ) increase with increasing values of the ordering cost r. However, the
increase in p2 becomes smaller at higher values of r and m.
The same patterns for the price (p3 ) is observed in the buyer-driven case and the
sensitivities (ξ[t2 , p2 ] and α∗ ) in supplier and buyer-driven cases.

6 Comparison of Supplier-driven and β-only Buyer-
driven Channels
In this section we compare the supplier-driven channel to β-only buyer-driven channel.
We show that there exist problem instances where the buyer may prefer the supplier-
driven to β-only buyer-driven channel and similarly problem instances where the supplier
may prefer the β-only buyer-driven channel to supplier-driven. For notational simplic-
ity, we show these results only for the classic bilateral monopoly (BM) model, where
operational costs are zero (k = 0, r = 0).
In BM, by setting ∂πB = 0 we find that the buyer selects p(0) = (a + t)/2 as the
∂p
market price. When p(0) is substituted, the supplier’s profit function becomes πS =
(t − c)m(a − t)/2. By setting ∂πS = 0 we find that the supplier selects t(0) = (a + c)/2
∂p
as the wholesale price, and the buyer sets p(0) = (a + t)/2 = (3a + c)/4 as the market
(0)
price. The optimal profits can be easily derived by substitution as πS = m(a − c)2 /8
(0)
and πB = m(a − c)2 /16.
(α,β)
∂πB
For a fixed α, the optimal β can be derived by setting ∂β
= 0 and solving for β.
The two roots are:

13

β = {a + cα2 − α c(2a − c + 2cα + cα2 ), (12)
a + cα2 + α c(2a − c + 2cα + cα2 )} (13)

β should be less than a − c, since a larger β would cause zero market demand. Since the
second root is always greater than a (and thus a − c), we use only the first root.
In realistic settings, we can normalize to a fixed α = 1 (i.e. the multiplier-markup pair
(1, β)). The supplier selects t(β) = (a−β)/2, and the buyer sets the market price to p(β) =
(β) √ √
(a + β)/2. When these are substituted into πB , one obtains β = (a + c) − 2c a + c.
Substituting this back into the price and profit functions, we obtain:

t(β) = c(a + c)/2, (14)
p(β) = (a + c) − c(a + c)/2, (15)
(β)
πS = m −2c + 2c(a + c) /4, (16)
(β)
πB = −m −2c + 2c(a + c) −(a + c) + 2c(a + c) /2 (17)

The pricing scheme that involves only β is not the most advantageous scheme for the
buyer. The buyer would always prefer a pricing scheme that involves a multiplier when
possible. Similarly, the supplier would always prefer supplier-driven to the buyer-driven
(where multiplier is allowed).
An interesting question is whether the buyer would ever be willing to give up the
β-only pricing scheme and prefer a supplier-driven. The next theorem answers this
question, and is obtained by comparing πB under β-only and supplier-driven models:

Theorem 8 In BM, the buyer would prefer supplier-driven setting to β-only buyer-
driven when

(a − c)2 − 8 −2c + 2c(a + c) −(a + c) + 2c(a + c) > 0 (18)

Proof:
(0) (β)
πB > πB (19)
m(a − c)2 /16 − m −2c + 2c(a + c) −(a + c) + 2c(a + c) /2 > 0 (20)

(a − c)2 − 8 −2c + 2c(a + c) −(a + c) + 2c(a + c) > 0 (21)

✷
The instances where (18) holds are characterized by very high a compared to c. If we
similarly consider the supplier’s preferences:

14

Theorem 9 In BM, the supplier would prefer supplier-driven setting to β-only buyer-
driven when

(a − c)2 + 4c − 2 2c(a + c) > 0 (22)

Proof:
(0) (β)
πS > πS (23)
2
m(a − c) /8 − m −2c + 2c(a + c) /4 > 0 (24)
(a − c)2 + 4c − 2 2c(a + c) > 0 (25)

✷

The instances where (22) holds are characterized by a being greater than c. Theorems
8 and 9 tell us that there does not exist a region where both parties prefer the same
case. Thus what determines which case will occur is the distribution of the bargaining
power in the channel.

7 Conclusions
In this paper we have explored the impact of power structure on price, sensitivity of
market price and proﬁts in a two-stage supply chain. Following analysis of the buyer’s
decision problem for a given wholesale price, we analyzed both the supplier-driven and
buyer-driven cases.
We showed that if the buyer uses a linear form of a price increase that it is optimal
for the buyer to set the markup to zero and use only a multiplier. We also observed that
the market price and its sensitivity with respect to the wholesale price becomes greater
in the supplier-driven case compared to a channel where operational costs at the buyer
are ignored. We found that the sensitivity of the market price increases non-linearly
as the wholesale price increases. In addition, we observed that marginal impacts of
increasing shipment cost and carrying charge (interest rate) on prices and proﬁts are
decreasing in both cases. Finally, we showed that there are cases when a buyer will
actually prefer a supplier-drvien channel to a buyer-driven one and where a supplier will
prefer a buyer-driven channel to a supplier-driven one.
Although, as we discussed earlier, there are many cases where the assumption of full
information is applicable, there are many cases where information assymetries arise. In
this case additional strategic considerations must be incorporated into the analyses.

15

Appendix

A Proofs

A.1 Theorem 2
Proof: We are interested in characterizing the three roots of Φ(p) = 0 given that they
exist. The case where a real cubic equation has all real and distinct roots is known as
the “irreducible case”, and holds when its discriminant

∆ = 18Φ0 Φ1 Φ2 − 4Φ0 Φ3 + Φ2 Φ2 − 4Φ3 − 27Φ2
2 1 2 1 0 (26)

is positive. (Dickson (1939) [?], p48-50) describe a “trigonometric solution” method
for this case.
If we let p0 = argmax{p=p0,1 ,p0,2 ,p0,3 } πB [p] then p0 < p. This follows from Theorem 1:
˜
πB [p] is concave when p < p and has a maximum in the interval (t, p). πB [p] is convex
˜ ˜
when p > p and has a minimum in the interval (˜, a). Thus the root with the maximum
˜ p
net profit value satisfies p0 < p.
˜
We derived the polynomial equation Φ(p) = 0 starting with (p − w/2) = k/(2q).
Note that taking the square of both sides, we introduce a new solution to the equation,
namely the p value such that |p − w/2| = p − w/2, even though p − w/2 is restricted to
positive values. So we have a new p value, which is neither an extreme nor an inflection
point, that satisfies Φ(p) = 0. This p value also satisfies p − w/2 < 0 and thus p < w/2.
Since the other roots satisfy p − w/2 > 0 and thus p > w/2, the mentioned p value is
the minimum p0,min of the three roots {p0,1 , p0,2 , p0,3 } of Φ(p) = 0. We can also deduce
that the maximum of πB [p] is achieved at p0 ∈ (w/2, p), and is the smaller of the two
˜
roots that satisfy p > w/2, that is, p0 = min{p0,1 , p0,2 , p0,3 }{p0,min }. ✷

A.2 Theorem 3
Proof: Equation (5) gives the relation between a given t and the supplier’s optimal
price p0 . Taking the partial derivative of both sides with respect to t we obtain:

16

∂p ∂Φ2 2 ∂p ∂Φ1 ∂p ∂Φ0
3p2 + p + 2p Φ2 + p + Φ1 + = 0
∂t ∂t ∂t ∂t ∂t ∂t
∂p
3p2 + 2Φ2 p + Φ1 −
∂t
(1 + rτ )p2 +
((1 + rτ )(4a + w) + (1 + rτ )w) p/4 −
a(1 + rτ )(2w)/4 + kr/(8m) = 0

∂p
−
ξ1 [t, p]
∂t
(1 + rτ )p2 − (1 + rτ )(2a + w)p/2 + a(1 + rτ )w/2 − kr/(8m) = 0
∂p
ξ1 [t, p] −
∂t
(1 + rτ ) p2 − (2a + w)p/2 + aw/2 − kr/(8m) = 0
∂p
ξ1 [t, p] − ξ0 [t, p] = 0
∂t
∂p ξ0 [t,p] ∂p0 [t]
From the last line we have ∂t
= ξ1 [t,p]
. When we substitute p = p0 [t] we have ∂t
=
ξ0 [t,p0 [t]]
ξ1 [t,p0 [t]]
. ✷

A.3 Theorem 4
Proof: Our proof has two steps: First we show that F [w, p] = ξ0 [t, p] − ξ1 [t, p]/2 < 0.
Next we will show that ξ1 [t, p]/2 < 0, which enables us to state that ξ[t, p] = ξ0 [t,p] .
ξ1 [t,p]
¿From (4), we can obtain w = 2p − ∆, where ∆ = k/2q > 0. We substitute this into
F [w, p] at the very ﬁrst step to simplify our analysis:

F [w, p] = F [2p − ∆, p]
1
= ξ0 [t, p] − ξ1 [t, p]
2
= (−kr/m − ∆2 − 4ar∆τ + 4pr∆τ )/8
= (−kr/m − ∆2 − 4r∆τ (a − p))/8

The last line is composed of a numerator with three negative components, thus
F [w, p] < 0.
Since F [w, p] = ξ0 [t, p] − ξ1 [t, p]/2 < 0, we can just take the second term to the right
hand side and divide both sides by ξ1 [t, p]. If ξ1 [t, p] < 0 then ξ[t, p] > 1/2. The last and

17

necessary step is showing that ξ1 [t, p] < 0, so that the inequality changes signs when
division by ξ1 [t, p] takes place.
A careful inspection of (5), (6) and (8) reveals that ξ1 [t, p] = dΦ(p) . From Theorem
dt
2b, for a given t value, p0 is the second largest root of Φ(p) = 0. Since Φ(w/2) > 0,
Φ(p) decreases from the positive value of Φ(w/2) at p = w/2 to the value of Φ(p0 ) = 0
at p = p0 . More formally,

dΦ(p0 )
ξ1 [t, p0 ] =
<0
dt
Thus when we divide both sides of the inequality ξ0 [t, p] < ξ1 [t, p]/2 by ξ1 [t, p], the sign
changes direction. ✷

A.4 Theorem 5
Proof:
a−p2
(9) tells us that t2 −c
= ξ[t, p0 [t]]
We are interested in showing that this indeed gives us the global optimum, by showing
the concavity of πS [t]. Thus, we are interested in showing that

∂ 2 πS [t] ∂p ∂ 2 p
= −2 − 2 (t − c) < 0
∂t2 ∂t ∂t
∂p ∂2p
We can use (9) in the above expression when substituting ∂t
and evaluating ∂t2
:

∂p ∂ 2 p
− 2 (t − c)
−2 =
∂t ∂t
a−p
∂p
(t − c) − (a − p)
−2 − ∂t (t − c) =
t−c (t − c)2
−2(a − p) − a−p (t − c) + (a − p)
t−c
=
(t − c)
(a − p)
−2 < 0
(t − c)

where the last inequality follows from (a − p) and (t − c) being positive. ✷

18

Figure 1: The supply chain channel

Figure 2: Proﬁts/Costs for the buyer when t = 11

19

Figure 3: Change of the sensitivity (ξ) of p with respect to t in the supplier-driven case

Figure 4: PB [α, β], contour plot

20

Figure 5: Change of p2 with respect to m and k in the supplier-driven case

21

Supplier and Buyer Driven Channels in a Two-Stage Supply Chain

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