Is the Double Marginalization Unavoidable?

Is the Double Marginalization Unavoidable?
，
QU Yizhe YING Mingyou
School of Management, Donghua University, Shanghai, P.R.China, 200051
Abstract The double marginalization is an important issue in supply chain management study. It is
not only a well-known cause of supply chain inefficiency, but also shakes the rationality of supply
chains since it implies that supply chains are not economic. For a long time researchers have tried their
best to mitigate the effect of the double marginalization, but they cannot eliminate it. The double
marginalization seems to be unavoidable. This paper finds out that the double marginalization does not
exist when contracting in a proper quantity discount way, and that both players of the contract game can
be winners.
Keywords
Double Marginalization, Supply Chain Management, Contract Coordination,
Quantity-discount Contract
1
Introduction
Double marginalization was known in Spengler’s study on horizontal integration in industries in
1950 [1]. It is also an important issue in supply chain management study though it has not drawn
researchers’ enthusiasm. It was only found in many papers on supply chain coordination as an argument
in favor of manufacturer’s coordination in contracting. For examples, see Cachon [2] and Lariviere [3].
The double marginalization is often formulated through a system of a manufacturer and a retailer. The
retailer would order less quantity of product than that the manufacturer would offer if the manufacture
took over the retail business. Cheng [4] pointed out that the double marginalization is not only a
well-known cause of supply chain inefficiency as said by Tsay [5], but also shakes the rationality of
supply chains since it implies that supply chains are not economic. Cachon [2] thought this phenomenon
is unavoidable whenever the supply chain’s profits are divided among two or more players and at least
one of the players influences demand. For a long time researchers have searched ways to mitigate the
effect of the double marginalization. Pasternack [6] suggested that a buy-back contract be applied as a
way for manufacturer’s coordination. Lariviere [3] furthered discussion on buy-back contract approach
and added quantity flexible contract to the tool-kit of manufacturer’s coordination. Schuster et al [7]
showed that returns policies do not always coordinate the supply chain, and they are useful only on a
subset of the feasibility region. Recently, Liu [8] showed that under the game theoretical framework the
solution to a price-only contract problem does imply the double marginalization and that a buy-back
contract problem can be turned into one of the same structure as a price-only contract problem. Thus, a
manufacturer cannot eliminate the double marginalization with buy-back coordination.
So, is the double marginalization really unavoidable? This is the issue to be addressed in this paper.
Of course, it is absolutely possible to eliminate the double marginalization with the help of quantity
discount contract. The key idea of this paper is nested in Spengler’s viewpoint: the retailer does not care
the manufacturer’s profit margin. All the abovementioned efforts fail to eliminate the double
marginalization because the retailer arbitrates the order quantity. Hence, the manufacturer’s strategy has
to bind the wholesale price with order quantity during contract gaming, and such a kind of strategy is
just the term of a quantity discount contract.
2
Formulation of the problem
According to a general setup for addressing the manufacturer’s coordination problem as Cachon [2],
Lariviere [3] and others did, this paper assumes that the discussion is under the framework of a
newsvendor model, i.e. the life of the product is one period long and the demand has a known
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continuous distribution. Let y, r, w, cM, and F(x) be output of product (or order quantity), retail price,
wholesale price, manufacturing cost per unit, and the probability distribution function of demand, where
cM<w<r. Let v be the salvage of per unit of unsold product.
Then, let ∏I(y) and yI be the profit of the centralized case and the optimal output;
let y ( w) , Π R ( y ) , Π M ( w, y( w)) be the retailer’s optimal order quantity, the retailer’s profit and the
manufacture’s profit for the decentralized case. Adopting Cachon’s and Lariviere’s results [2] [3], we
have the following:
y
∏ I ( y ) = ( r − cM ) y − (r − v ) ∫ F ( x) dx .
(1)
0
y I = F −1 (
r − cM
)
r −v
.
(2)
∏ R ( y ) = ( r − w) y − ( r − v ) ∫ F ( x) dx ,
(3)
y
0
r−w
),
r −v
ΠM ( w, y( w)) = ( w − cM ) y( w) .
Apparently, y(w) is smaller than yI
y ( w) = F −1 (
(4)
(5)
and this phenomenon is referred to the “double marginalization”.
Cheng [4] questioned the above results since they do not accord with our life, because supply chains
should mean industrial specialization, and because specialization should mean absolute advantages and
competitiveness, whereas y(w)< yI implies that the “supply chain” produces less product and meets
demand at a lower service level. Cheng further referred the pitfall in Cachon or Lariviere’s theory to a
zero retail cost assumption implied in their models. Therefore, it is desirable to introduce retail cost into
our discussion and to assume the retailer is more advantageous than the manufacturer in retailing
business.
Let cRS be the retailer’s retail cost per unit. Now, for the decentralized case, the profits of the retailer
and the manufacturer are respectively given by equations (6) and (7):
y
Π R ( w, y ) = ( r − w − cRS ) y − ( r − v ) ∫ F ( x )dx ,
0
Π M ( w, y ) = (w − cM ) y .
(6)
(7)
Given w, the retailer’s optimal decision in order quantity is
 r − w − c RS
y ∗ = F −1 
 r−v

 .

(8)
If the centralized case describes merge of the manufacturer and the retailer, then the profit and the
optimal output of the case are revised by equation (9) and (10).
y
Π I ( y ) = (r − cM − cRS ) y − (r − v) ∫ F ( x)dx .
(9)
0
 r − c M − c RS
y I* = F −1 
r−v


 .

(10)
Obviously, the double marginalization still exists if we compare equation (8) with equation (10).
Because of Liu’s work [8] we are able to infer that neither price-only contract nor buy-back contract, nor
quantity flexible contract can eliminate the double marginalization.
Fortunately, Chen et al [9] threw lights on our study by proposing a new idea to explore the game
mechanism between the two parties though the purpose of that paper was to break through the
Stackelberg gaming via cooperative bargaining between the manufacturer and the retailer. They
introduced the rational player postulation and profit contour analysis. Both players could improve their
profits through bargaining as Chen et al suggested.
For any feasible price w and order quantity y, expressions (6) and (7) are payoffs of the retailer and
the suppler respectively. Let equations (6) and (7) equal arbitrary positive constant. Then they turn to be
profit contours of the retailer and the manufacturer accordingly as labeled with (11) and (12):
y
(r − w − cRS ) y − ( r − v) ∫ F ( x) dx = C2 ,
(11)
0
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( w − cM ) y = C1 .
(12)
Equation (12) can be considered as a set of the manufacturer’s quantity discount strategies with a
constant profit C 1 . In this circumstance, a quantity-discount contract problem can be expressed by the
following programming one:
max  Π R ( y, w) = ( r − w − c RS ) y − (r − v ) ∫ y F ( x )dx  ,
y
0


(13)
subject to (w − cM ) y = C1 .
If the maximization problem (13) yields its solution as same as expression (10), we do find the way
to avoid the double marginalization. Otherwise we have to try other forms of the manufacturer’s
quantity discount strategies.
3
Avoidance of the double marginalization
It is easy to show that the manufacturer’s profit contour (12) is strictly convex and the retailer’s
profit contour (11) is strictly concave. It is also easy to show that the manufacturer’s profit contour
moves up if C1 increases and the retailer’s profit contour moves down if C2 increases (see Figure 1).
Thus, the solution to the maximization problem (13) is the unique tangential point of the
manufacturer’s profit contour (12) with a retailer’s profit contour (11). Chen et al [9] found it as
expressed by (14). They called the solution a Pareto equilibrium.
 r − c M − c RS
y pe = F −1 
 r −v

 .

(14)
From the solution we get three observations. The first one is that the solution surprisingly eliminates
the double marginalization. The second one is that C1 is an arbitrary constant and y pe is free from C1 .
The third one is that the total profit of the manufacturer and the retailer keeps unchanged no matter how
C1 is valued. It can be derived from these observations that there exists a set of Pareto equilibriums
since any tangential pair of a manufacturer’s profit contour and a retailer’s profit contour bears a Pareto
Equilibrium. In fact, the set of Pareto equilibriums appears as a straight line as shown in Figure 1.
Figure 1. Pareto Equilibriums
Figure 2. Situations to the manufacturer’s profits
Although we have found that the double marginalization is avoidable under a quantity discount
contract, there remains a problem that whether the manufacturer should constant her expected profit to
C1 when addressing the strategy.
In fact, there are three situations to the manufacturer’s profit in her quantity discount strategy. The
first one is that it decreases when the retailer increases her order quantity y; the second one is that it
increases with the retailer’s order quantity; and the third one is that it is a constant. Since case 3 is the
proper way to avoid the double marginalization as we mentioned above and the manufacturer is
obviously undesirable to addresses such a strategy in case 1 as a rational player, we merely need to
discuss case 2 in detail.
As figure 2 shows, since the manufacturer’s profit contour moves upwards when the profit increases,
the manufacturer’s profit increases along curve BA in case 2, while curve BP represents her profit
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contour in case 3. Because curve BP is apparently steeper than curve BA, and all of the slopes on the
tangential points of Pareto Equilibriums are the same, it is easy to show that when the two parties’ profit
contour tangent to each other, the slope of the tangential point on curve BA must be larger than that on
curve BP, hence it is larger than that on Pareto Equilibrium point P1. Therefore the order quantity on
tangential point A is smaller than that on the Pareto Equilibrium point P1, since the slope of the retailer’s
profit contour decreases monotonously. However, this is just what the double marginalization performs.
Thus case 2 is improper to avoid the double marginalization.
So, it is the only way we have found to avoid the double marginalization that the manufacturer
addresses her quantity discount strategies under a constant profit C 1 . However, the total profit of the
manufacturer and the retailer keeps unchanged no matter how much the manufacturer’s expected profit
is, the higher the manufacturer’s profit is the lower the retailer’s profit will be. Therefore the retailer
would not be willing to do the business if the manufacturer’s profit is too high. Hereby it is important
for the manufacturer to set a reasonable expected profit when addressing the quantity discount strategy.
In fact, the manufacturer could address her expected profit according to the optimal profit derived from
the price-only contract that Liu [8] has found, since this optimal profit is acceptable to the manufacturer
and it will not cause any loss in the retailer’s profit in their bargaining process. Thus the quantity
discount strategy could be reasonable enough to be dealt with.
4
Conclusions
In this paper, we have provided enough evidence to show that the double marginalization could be
avoided when the manufacturer contract with the retailer by means of a quantity discount strategy under
a constant expected profit. Furthermore, the total profit of the two parties is maximized under this form
of contract, so that both players of the contract game can be winners.
References
[1]
[2]
Spengler, J. Vertical Integration and Antitrust Policy. Journal of Political Economy, 1950:347-352
Cachon, G.P. Competitive Supply Chain Inventory Management. S. Tayur, M. Magazine, R. Ganeshan,
Eds.Quantitative Models of Supply Chain Management, Kluwer Academic Publishers, Boston, MA.,
1999:111-145
[3] Lariviere, M.A. Supply Chain Contracting and Coordination with Stochastic Demand. S. Tayur, M. Magazine,
R. Ganeshan, Eds.Quantitative Models of Supply Chain Management, Kluwer Academic Publishers, Boston,
MA., 1999:232-265
[4] Cheng, Y.H. A Discussion on Contract Patterns Based on Cooperative Gaming (in Chinese). Master Degree
Thesis, Glorious Sun School of Business and Management, Dong Hua University, Shanghai, China,
2005:21-25
[5] Tsay, A.A., Nahmias, S., Agrawal N. Modeling Supply Chain Contracts: A Review. S. Tayur, M. Magazine, R.
Ganeshan, Eds.Quantitative Models of Supply Chain Management, Kluwer Academic Publishers, Boston,
MA., 1999:301-335
[6] Pasternack, B. Optimal Pricing and Return Policies for Perishable Commodities. Marketing Science, 1985,
4(2):166-176
[7] Barnes-Schuster, D., Bassok, Y., Anupindi, R. Coordination and Flexibility in Supply Contracts with Options.
Manufacturing & Service Operations Management, 2002, 4(3):174-207
[8] Liu, W.P. On the Best Solutions of Four Typical Contract Models (in Chinese). Master Degree Thesis,
Glorious Sun School of Business and Management, Dong Hua University, Shanghai, China, 2007:21-42
[9] Chen, Z.H., Liu, W.P. and Chen, J.X. Pareto Equilibriums of a Price-only Contract Game. International
Conference on Logistics and Supply Chain Management (LSCM) Hong Kong, China, Jan 5-7 2006, #223.
The author can be contacted from e-mail : [email protected]
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