Bertrand Competition between Manufacture and Remanufacture in a
Two-period Model
JIA Xiaokui, ZHANG Hanjiang
School of Economics and Trade,Hunan Univ,Changsha,Hunan,410079,China
[email protected]
Abstract: Take-back laws require that firms take responsibility for the collection/disposal costs of their
products. This paper develop a general two-period model, especially use Bertrand Game for the price
decision between manufacture and remanufacture in the second period to investigate questions of
interest to managers in industry. This paper is different from other papers in the demand analysis. We
thought the analysis by Bertrand, thus narrowing the scope, making the result simple. The results
suggest that enactment of collective WEEE take-back will result in higher manufacturer and
remanufacturer profits while simultaneously spurring remanufacturing activity. A negative effect is
higher consumer prices in the market.
Keywords: Remanufacturing; Environmental laws; Pricing; Competitive strategy; Reverse logistics
1 Introduction
Xerox corporation recycled their own waste products in the sale of their new products, using these
waste products to remanufacture new products saved the company 40%-50% of the manufacture costs[1].
Kodak cororation recover their disposable camera,an average of 76% of the components are reused in
the remanufacturing[2]. The motivation for remanufacturing is driven by economic considerations
(Toffel, 2003)[3]. In recent years, legislators have started to mandate it. In December 2002, New Jersey
enacted the Universal Waste Regulations Act, which governs the disposal of all electronic devices
including computers. The European Union (EU) has adopted a Directive on Waste Electrical and
Electronic Equipment (WEEE) such that, effective August 2005, EU member states must establish
collection systems for electrical and electronics waste. In January 2005, California enacted the
Electronic Waste Recycling Fee (SB20), which requires retailers to collect from consumers an electronic
waste recycling fee on electronic devices (presently limited to video displays) to cover the net cost of a
state authorized collector who collects, consolidates and transports electronic wastes.
Majumder and Groenevelt (2001)[4] study a two-period horizontal competition model of
remanufacturing in which an OEM (who manufactures new products and also remanufactures) competes
with a local remanufacturer under different allocation mechanisms for returns. The authors solve for
Nash equilibrium, and show that while the OEM would like to increase the local remanufacturer’s
remanufacturing cost, the local remanufacturer would prefer to reduce the OEM’s remanufacturing cost.
Ferguson and Toktay (2005)[5] develop a two-period model with a monopolist manufacturer in the first
period and duopoly of manufacturer and remanufacturer in the second period. They show that an
independent entrant could make profits even though a monopolist OEM may not find it more profitable
to remanufacture, in part because the OEM incurs an opportunity cost when selling remanufactured
products while also selling a new product. Ferrer and Swaminathan (2006)[6] study a firm that makes
new products in the first period and competes with a remanufacturer in the second period by selling both
new and remanufactured product.
In this paper, We assume a two-period game-theoretic mode, especially use Bertrand Game for the
price decision between manufacture and remanufacture in the second period. This model explicitly
addresses the way production quantity decisions in a period affect end-ofperiod costs for collection and
disposal.
2 Model and Pricing decisions
In this paper, we define and analyze a model of an industry comprised of a manufacturer producing
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new products and a remanufacturer. The manufactureroperates without competition from a
remanufacturer for a period of time after a new product is introduced. Eventually, as products are
returned, a remanufacturer may enter the market and compete with the manufacturer. We use our model
to predict industry behavior and resulting performance measures over alternative implementations of
take-back laws, including the alternative of no take-back law.
2.1Manufacturer and remanufacturer profit functions
We consider a market for a single product that is sold as new and as remanufactured. The
remanufactured product is labeled as such, and this labeling affects consumer perceptions of value. In a
given period, each consumer will buy, at most, one product and only if his valuation is not less than the
price.
The perceived value v of new product is uniformly distributed on [0, 1] within the consumer
population.Without loss of generality, we assume a density of 1 on the interval [0, 1. For each consumer,
the valuation ratio of remanufactured product to new product value is denoted θ ∈ (0,1) , θ is an
exogenous measure of acceptance of remanufactured product that holds for all consumers.
The following cost parameters and model accommodate the cost structures associated with the two
preceding implementations of take-back laws, as well as the case of no take-back law.
c m(1)
cost per new unit produced in period 1;
——
c —— cost per usable unit returned but not remanufactured at the end of period 1;
c —— cost per new unit produced in period 2;
c —— cost per remanufactured unit produced in period
p ——the manufacturer unit sales price in period 1
p ——the manufacturer unit sales price in period 2
p ——the remanufacturer unit sales price in period 2
γ ——the fraction of first period demand that is returned for remanufacturing. γ ∈ (0,1)
The period 1 demand function is： D ( p ) = 1 − p
and thus the manufacturer profit in period 1 is： Π
= (1 − p )( p − c )
d
m(2)
rm
m (1)
m (2)
rm
m (1)
m (1)
m (1)
m (1)
m (1)
m (1)
m (1)
A consumer with valuation v would consider buying a new product if v − pm (2) > 0 , and
would
consider
buying
a remanufactured
product
if
θ v − prm > 0
.
In
cases,
where
v − pm (2) > 0 and θ v − prm > 0 , the consumer selects the product that offers the most surplus. The
consumer whose valuation v = pm (2) is indifferent to buying from the manufacturer or not at all.
let v =
v
m
. The consumer whose valuation θ v = prm is indifferentto buying from the remanufacturer
or not at all, let v =
prm
= v Also, if v − pm (2) > θ v − prm , then the consumer prefers the
r
θ
manufacturer to the remanufacturer. Solving for v gives us the consumer whose valuation
v − pm(2) = θ v − prm → v =
let v =
v
rm
pm( 2) − prm
1−θ
is indifferent between the manufacturer and remanufacturer.
.
Consider the case of p m ( 2 ) ≥
p rm
θ
. We can have
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p rm
θ ≤ pm (2 ) ≤
p m ( 2 ) − p rm
1−θ
,
r
m
rm
v ≤v ≤v
demand is min{v
i.e.
Consider
i.e..
v
rm
the
.For this case, the manufacturer demand is1 − min{
v
rm
m
,1} ; and the remanufacturer
r
,1} − v .
case
≤v ≤v
rm
r
of
pm (2) ≤
prm
θ
,
We
can
have
p m ( 2 ) − p rm
For this case, the manufacturer demand is 1 −
1−θ
v
m
≤ p m ( 2) ≤
p rm
θ
,
; and the remanufacturer
demand is 0.
So Manufacturer and remanufacturer profit functions can be expressed in table 1:
Table 1: Manufacturer and remanufacturer profit functions
profit functions
Drm
Dm (2)
interval
pm (2) ≤
prm
prm
1 − pm (2)
θ
θ ≤ pm(2) ≤ prm + 1 − θ
1−
pm (2) ≥ prm + 1 − θ
pm (2) − prm
1−θ
0
0
all demand goes to the
manufacturer
θ pm (2) − prm
θ (1 − θ )
p
1 − rm
θ
both players are in the
market
all demand goes to the
remanufacturer
As long as the manufacture put the price over the limit pm (2) ≥ prm + 1 − θ ,his demand will be
0.so, when pm (2) ≥ prm + 1 − θ we let the price be
Similarly, when pm ( 2) ≤
prm
θ
pm (2) = prm + 1 − θ , pm (2) = prm + 1 − θ .
, we let the price be prm = θ pm (2) .so that the second phase of the
demand function can be written:
Dm 2 = 1 −
pm (2) − prm
1−θ
prm
θ ≤ pm ( 2) ≤ prm + 1 − θ
θ pm (2) − prm
Drm =
θ (1 − θ )
pm (2) − (1 − θ ) ≤ prm ≤ θ pm (2)
(3)
(4)
Thus the manufacturer’s total profit is :
Π m = Π m (1) + (1 −
prm
pm(2) − prm
1− θ
)( pm(2) − cm(2) ) − cd [γ (1 − pm(1) ) −
θ pm(2) − prm
]
θ (1 − θ )
θ ≤ pm (2) ≤ prm + 1 − θ
(5)
the manufacturer’s total profit is :
θ pm (2) − prm
( prm − crm )
θ (1 − θ )
max{ pm(2) − (1 − θ ), θ [ pm (2) − γ (1 − θ )(1 − pm (1) )]} ≤ prm ≤ θ pm (2)
Π rm =
2.2Price decision
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(6)
Note that
2 p − prm − cm (2) − cd ∂ 2Π m
∂Π m
∂Π m
,
= 1 − m ( 2)
= 0 and
2 < 0 . Setting
∂pm ( 2)
∂p
1−θ
∂pm (2)
m(2)
solving for p m(2) , we set
Note that
pm(2)
*
=
prm + cm (2) + cd + 1 − θ
(7)
2
∂Π rm θ pm (2) + crm − 2 prm ∂ 2Π r
∂Π rm
,
= 0 and solving for
=
2 < 0 . Setting
∂prm
∂p
∂prm
θ (1 − θ )
r
p rm , we set
prm
*
=
θ pm (2) + crm
(8)
2
By (7)and (8) , we can solve for p m(2) and p rm , we set
*
2(cm (2) + cd + 1 − θ ) + crm

p
 m(2) =
4 −θ

*
θ (cm (2) + cd + 1 − θ ) + 2crm

=
p
rm

4 −θ
Note that
（ 9）
2
∂Π m
∂Π m
= 0 and solving for p m(1) ,
= 1 − 2 pm(1) + cm (1) + γ cd , ∂ Π m2 < 0 . Setting
∂pm (1)
∂pm (1)
∂pm (1)
we set pm (1)* =
1 + γ cd + cm (1)
2
.
3 Numerical study
This paper consider two alternative implementations of take-back laws that are distinguished by the
degree of control that the manufacturer has on returns sold to the remanufacturer. In one implementation,
known as collective WEEE take-back, the manufacturer has no control over returns sold to the
remanufacturer. The other implementation, known as individual WEEE take-back, gives complete
control to the manufacturer. This paper develop a general two-period model to investigate questions of
interest to policy-makers in government and managers in industry.
3.1Select datas
In our numerical study, we assume that the unit cost to manufacture and distribute new product
( k m ) is the same in both periods. Similarly, we assume that the collection/disposal cost ( k d ) and the
fraction of useable units returned ( γ ) are the same in both periods. The unit cost to remanufacturer and
distribute product excluding the cost to purchase a return is k r .we can have cost parameters in terms
of fundamental cost elements for each scenario in the table 2:
table 2: Model cost parameters in terms of fundamental cost elements for each scenario
cm (1)
cd
cm(2)
crm
cm (1) = km
cd = 0
cm(2) = km
crm = k r + kd
S1 Individual take-babck law
cm (1) = km − γ ke
cd = kd + ke
cm (2) = km + γ k d
crm = kr + kd + ke + γ kd
S2 Collective take-babck law
cm (1) = km
cd = kd
cm (2) = km + γ k d
crm = kr + kd + γ kd
S0
No take-babck law
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In this paper we take manufacturer profit, remanufacturer profit and remanufacturer volume as the
indicators to measure the impact of take-back laws.
We let the return rate γ range from 0.1 to 0.9 in increments of 0.1.consider three values of
km ,i.e. k m ∈ (0.2, 0.4) .We consider a range of values for k d between the extremes of zero and the
max
value that equalizes the production cost of new and remanufactured product, i.e. kd
k d ∈ {0,
k
max
d
8
,
2k
max
d
8
,..., kdmax } ,in this paper,we select k d =
k
max
d
8
= k m − kr ,
.we use as a Am = k r
θ km
measure of the manufacturer’s relative competitive advantage. We consider A m = 0.8 , i.e.the
manufacturer be comes more competitive. identify a set of alternative values of θ for our numerical
study, we analyzed laser printer cartridge pricing data in the Tonertopup database. we let θ be 0.65.
3.2 Result and analysis
When
Am = 0.8 , km = 0.4
， θ = 0.65 , k
d
=
kdmax
8
,with the change of
γ
,how the
manufacturer profit, remanufacturer profit and remanufacturer volume change is expressed in figure 1:
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
制造商利润
S1
S2
S0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
再制造商利润
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
35.00%
30.00%
25.00%
S1 20.00%
S2 15.00%
S0 10.00%
5.00%
0.00%
再制造占总产量的百分比
S1
S2
S0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure1: the manufacturer profit, remanufacturer profit and remanufacturer volume with the change of
When Am = 0.8 , km = 0.2 , θ = 0.65 , kd = k
max
d
8
, with the change of
γ
γ
,how the manufacturer
profit, remanufacturer profit and remanufacturer volume change is expressed in figure 2(see page 6):
Fig.1-2 shows manufacturer profit, remanufacturer profit and remanufacturer percent of total
volume by scenario, return rate and four combinations of collection/disposal cost and manufacturer
relative competitive advantage.
Under S1,we can see that the manufacturer generally finds that it is more profitable to compete
with the remanufacturer than to operate as a monopolist. When the remanufacturer cost structure is high
relative to the manufacturer (e.g., Am = 0.8), potential remanufacturer volume is low, and the
manufacturer has less incentive to sell to the remanufacturer. Nevertheless, at low return rates, the
manufacturer prices returns just low enough to cause the remanufacturer to purchase all returns. While
the manufacturer reduces her margin on returns as the return rate increases, this is not enough to offset
the increase in the remanufacturer cost structure through the increasing cost of collection/disposal at the
end of period 2.
Fig.1-2 show that all parties are sometimes better (or no worse) off under collective WEEE
take-back (S2) relative to the alternative to no take-back law (S0).
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再制造商利润
制造商利润
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
S1
S2
S0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
再制造占总产量的百分比
S1
S2
S0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
S1
S2
S0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure2: the manufacturer profit, remanufacturer profit and remanufacturer volume with the change of
with the change of
γ
γ
4 Conclusions
This paper develop a general two-period model, especially use Bertrand Game for the price
decision between manufacture and remanufacture in the second period to investigate questions of
interest to managers in industry. The results of our study lead to two main conclusions.
First, even though the manufacturer has the power to keep a remanufacturer from entering the
market under individual WEEE take-back, this power is not exercised when return rates and unit
collection/disposal cost are moderate. The manufacturer profits more from the revenue associated with
returns sold to the remanufacturer than by operating as a monopolist.
Second, the enactment of a collective WEEE takeback law can lead to increases in manufacturer
and remanufacturer profitability while simultaneously spurring remanufacturing activity. The
responsibility for collection/disposal cost under collective WEEE take-back reduces the intensity of
competition between the manufacturer and remanufacturer, and both firms become more profitable. A
negative effect is higher consumer prices in the market.
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