Inventory and Supply Chain Management with
Carbon Emissions
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Xi Chen
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Saif Benjaafar
July, 2014
c© Xi Chen 2014
ALL RIGHTS RESERVED
Acknowledgements
First and foremost I wish to express my sincere gratitude to my advisor Prof. Saif
Benjaafar for his continuous support throughout my PhD study, for his patient
guidance and motivation with which I have been able to overcome the difficulties
along this journey. His enthusiasm towards operations and supply chain manage-
ment and his never ending pursuit for new challenges have shown me the essence
of a true researcher.
Besides my advisor, I would like to thank the rest of my thesis committee:
Prof. William Cooper, Prof. John Carlsson, and Prof. Timothy Smith, for their
generous support for not only my dissertation but also my whole professional
development, their encouragement, insightful comments, and hard questions.
Last but not least, I would like to thank my parents, Xiaoqiang Chen and
Weiling Gong, who have been the most wonderful parents one could ask for with
their endless love and support. I am also blessed with a great boyfriend and
friends who were always there for me even during the most difficult times and to
whom I owe my most sincere gratitude.
i
Dedication
To my parents who are always understanding, always supportive,
and most of all always allowing me to pursue my dreams.
ii
Abstract
Although the literature on carbon emissions, from fields such as environmental
economics, public policy, and industrial ecology among others, is quite extensive,
the literature in supply chain management is relatively limited. Moreover, in
addressing concerns about carbon emissions, much of the focus has been on tech-
nological fixes (e.g., more carbon-efficient technologies and alternative sources of
energy). Much less attention has been paid to the potential of reducing carbon
emissions via adjustments in supply chain design and operation. This research,
utilizing optimization, game theory, deterministic and stochastic modeling, and
mechanism design, aims to bridge this gap. The first part of the study, using
the economic order quantity (EOQ) model framework, suggests that it is possible
to reduce emissions by modifying order quantities, and provides conditions under
which the relative reduction in emissions is greater than the relative increase in
cost. The second part examines the extent to which penalizing the emission of
harmful pollutants can successfully reduce overall emissions in decentralized sup-
ply chains, and shows that requiring each firm to pay for the emissions for which it
is directly responsible can paradoxically lead to higher overall supply chain emis-
sions and for this emission to increase in the price of emissions. The third part
includes preliminary results including the impact of price variability as well as
consumers’ preferences for low emission products on the ways firms manage their
inventory and the corresponding emissions.
iii
Contents
Acknowledgements i
Dedication ii
Abstract iii
List of Figures vii
1 Introduction 1
2 The Carbon-Constrained EOQ 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Problem Formulation and Results . . . . . . . . . . . . . . . . . . 7
2.3 Extensions to Systems with Carbon Prices . . . . . . . . . . . . . 21
2.3.1 Carbon Tax . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Cap-and-Offset . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 Cap-and-Price . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Extensions to Other Models . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 The Facility Location Model . . . . . . . . . . . . . . . . . 30
iv
2.4.2 The Newsvendor Model . . . . . . . . . . . . . . . . . . . 32
2.5 Social Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 On the Effectiveness of Emission Penalties in Decentralized Sup-
ply Chains 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 A Buyer-Supplier Model . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Potential Remedies . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.1 Supply Chain Coordination . . . . . . . . . . . . . . . . . 57
3.4.2 Shared Emission Responsibility . . . . . . . . . . . . . . . 60
3.5 Systems with Endogenous Demand . . . . . . . . . . . . . . . . . 66
3.6 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.1 A Make versus Buy Problem . . . . . . . . . . . . . . . . . 71
3.6.2 A Store Density Problem . . . . . . . . . . . . . . . . . . . 76
3.6.3 A General Model Formulation . . . . . . . . . . . . . . . . 81
3.7 Summary and Concluding Comments . . . . . . . . . . . . . . . . 84
4 Work-In-Progress and Future Research Directions 87
4.1 Systems with Stochastic Carbon Prices . . . . . . . . . . . . . . . 87
4.1.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . 88
4.1.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Inventory Management with Emission Sensitive Demand . . . . . 91
4.2.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . 92
v
4.2.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . 93
References 95
vi
List of Figures
2.1 The impact of the carbon cap on emission and cost . . . . . . . . 12
2.2 Cost versus Emissions . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 The impact of changes in order quantity on cost . . . . . . . . . . 15
2.4 Cost-emission tradeoff . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The impact of α on the maximum difference between δE and δZ . 20
2.6 The benefit of operational adjustment under carbon tax . . . . . . 23
2.7 The effect of the carbon cap under cap-and-offset . . . . . . . . . 26
2.8 The impact of carbon price on cost and carbon emissions . . . . . 29
2.9 The impact of changes in the number of facilities on cost . . . . . 31
3.1 The impact of the emission penalty on supply chain emission . . . 53
3.2 The impact of the emission penalty on supply chain emission when
m∗2 > 1 (A2 = 10, h2 = 1, Aˆ2 = 25, hˆ2 = 1, A1 = 5, h1 = 1.5, Aˆ1 =
3, hˆ1 = 2, cˆ1 = cˆ2 = 0, D = 100) . . . . . . . . . . . . . . . . . . . 55
3.3 Supply chain emission with and without coordination (A2 = 15, h2 =
1.4, Aˆ2 = 10, hˆ2 = 1.1, A1 = 21, h1 = 1, Aˆ1 = 5, hˆ1 = 1, cˆ1 = cˆ2 =
0, D = 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Supply chain profit and emission under different penalty schemes . 65
vii
3.5 The impact of the emission penalty on social welfare . . . . . . . 72
3.6 The impact of the emission penalty on the manufacturer’s emission
e1 and the supplier’ emissions e2 . . . . . . . . . . . . . . . . . . 75
3.7 The impact of the emission penalty on supply chain emission for
an example store density problem (p = 4, D0 = 1000, cc = 0.5, cr =
0.1, cˆc = 1, cˆr = 0.02, φc = 0.7652, φr = 1, γ = 0.1). . . . . . . . . . 80
viii
Chapter 1
Introduction
This thesis consists of two completed papers and some partially completed work.
The abridged version of Chapter 2 is published in the journal Operations Research
Letters. The completed paper in Chapter 3 is currently under review with a
refereed journal. The partially completed work, presnted in Chapter 4, forms the
basis of the proposed future work. All the work presented here is joint work with
my advisor Saif Benjaafar. Chapter 2 is also joint work with Adel Elomri.
A common thread that ties all the work described in this thesis is an attempt to
incorporate concern about carbon emissions in the management of supply chains.
Although the literature on carbon emissions, from fields such as environmental
economics, public policy, and industrial ecology among others, is quite extensive,
the literature in supply chain management is relatively limited (see the literature
review in Chapter 3). Moreover, in addressing concerns about carbon emissions,
much of the focus has been on technological fixes (e.g., more carbon-efficient
technologies and alternative sources of energy). Much less attention has been
paid to the potential for reducing carbon emissions via adjustments in supply
1
2chain design and operation.
In this thesis, we are motivated by two broad sets of questions. The first: “is
there an opportunity to make adjustments to the way supply chains are designed
and managed that could lead to significant reductions in emissions and without
resorting to investments in new technology?” The second: “what is the impact
of various environmental regulations, such as carbon taxes, emission caps, and
cap-and-trade, on the way supply chains are designed and managed and the cor-
responding cost/profitability?” In Chapter 2, we address the first question. In
Chapters 3 and 4, we address the second.
In particular, in Chapter 2, we provide analytical support for numerical re-
sults provided in a recent paper by Benjaafar et al. (2010) suggesting that it is
possible, by making only operational adjustments, to significantly reduce carbon
emissions without significantly increasing cost. We do so using the framework of
the economic order quantity (EOQ) model. We provide a condition under which
it is possible to reduce emissions by modifying order quantities. We also provide
conditions under which the relative reduction in emissions is greater than the rela-
tive increase in cost and discuss factors that affect the difference in the magnitude
of emission reduction and cost increase. We discuss the broader applicability of
these results to other operational models and to settings with different carbon
pricing schemes.
In Chapter 3, we examine the extent to which pricing emissions of harmful
pollutants can successfully reduce overall emissions in decentralized supply chains
(supply chains consisting of multiple independent firms or of independent busi-
ness units within a single firm). In particular, we show that requiring each firm
3to pay for the emissions for which it is directly responsible can paradoxically lead
to higher overall supply chain emissions and for this emission to increase in the
price of emissions. We show that this is because firms, in their efforts to reduce
their own emissions, can make decisions that adversely impact the emissions of
their suppliers or customers. We illustrate how this could arise in the context of
a buyer-vendor supply chain where the actions of each firm correspond to oper-
ational decisions regarding the frequency and size of replenishment orders, and
provide conditions under which higher emission prices can lead to higher total
emissions. We then discuss schemes under which this paradox can be eliminated,
including those involving coordinating the supply chain, sharing emission respon-
sibility across the supply chain, and penalizing consumption instead of production.
We also discuss examples of other settings where the paradox arises, including a
setting where a retailer decides on store locations and one where a manufacturer
decides on how much to outsource. Finally, we provide a general model for under-
standing how the actions of one firm could affect the costs and emissions of other
parties in the supply chain of which the previous examples can be considered as
special cases. We also provide a general necessary and sufficient condition for
when higher emission penalties lead to higher supply chain emissions.
In chapter 4, we briefly summarize our preliminary work on two problems. The
first is motivated by settings where carbon prices are stochastic, as in a cap-and-
trade system. Using the model described in Chapter 2, we examine the impact of
carbon price variability. Perhaps surprisingly, we show that higher variability can
be beneficial. The second problem, which is also built on the model described in
Chapter 2, considers a setting where consumer demand is sensitive to the firm’s
4emissions. We study how the sensitivity of consumers to emissions impact the
firm’s decisions and profitability.
Each chapter is self-contained, with related literature, notation, and ideas
for future research. In most cases, and unless explicitly stated, we maintain the
same notation, and the same definition of various concepts throughout the various
chapters.
Chapter 2
The Carbon-Constrained EOQ
2.1 Introduction
In pursuing carbon emission reduction efforts, firms have focused for the most
part on reducing emissions due to the physical processes involved (e.g., replacing
energy inefficient equipment and facilities, redesigning products and packaging,
deployment and use of less polluting sources of energy); see [16], the unabridged
version of the paper. These efforts are clearly valuable. However, they can over-
look a potentially significant source of emissions, one that is driven by business
practices and operational policies. In this paper, we examine the extent to which
operational adjustments alone could indeed be effective in reducing emissions. We
also examine the extent to which such adjustments could take place without sig-
nificantly increasing cost. This is important because resistance to environmental
regulation has often been based on concerns that such regulation would lead to
significantly higher costs.
Our analysis is in part motivated by a recent paper [5], in which the authors
5
6observe that it is possible to significantly reduce carbon emissions without signifi-
cantly increasing cost by making only operational adjustments. Their observations
are based on numerical results obtained for a lot sizing problem in which a firm
decides on production/procurement quantities over a finite planning horizon con-
sisting of discrete periods. In this paper, we use the framework of the economic
order quantity (EOQ) model to provide analytical support for similar observa-
tions. We provide a condition under which it is possible to reduce emissions by
modifying order quantities. We also provide conditions under which the relative
reduction in emissions is greater than the relative increase in cost and describe
when the difference between the two is maximized. We discuss the applicability
of these results to systems operating under a variety of regulatory policies and to
other operational models. We show that, the key requirements are that the cost
and emission functions yield different optimal solutions, implying that the cost
tradeoffs are different from the emission tradeoffs, and that the cost function is
flat around the optimal solution but can be steep elsewhere. We show that signif-
icant reductions in emissions can indeed be achieved without significant increases
in cost whenever the flat region of the cost function coincides with the steep region
of the emission function. In the unabridged version of the paper [16], we show
that these features are present in other operational models, including the facility
location and newsvendor models, among others.
The results in this paper indicate that the opportunity for reducing carbon
emissions via operational adjustments exists whenever the operational drivers of
emissions are different from the operational drivers of costs. In settings where this
is not the case (e.g., operational decisions that reduce cost tend to also reduce
7emissions), operational adjustments will obviously be ineffective. In that case, in-
vestments in efforts that modify the emission function (e.g., investments in efforts
or technologies that lead to reductions in the emission parameters of underlying
processes and activities) would be necessary.
Although there is growing literature that is concerned with issues of sustain-
ability in operations (see [5] for a review), papers that explicitly consider emissions
are relatively few. Hua et al. [38] consider a model similar to the cap-and-price
model we consider in Section 3. They compare the cost and order quantity ob-
tained under cap-and-price to those obtained without carbon considerations. Ca-
chon [9] studies the emission tradeooffs associated with the size and location of
retail facilities and shows that carbon pricing would have little impact on these
decisions; see also [30] for related analysis.
2.2 Problem Formulation and Results
Consider a firm that faces a constant demand with rate D per unit time. Each
time the firm places an order (either with its internal production facility or with
an external supplier), it incurs a fixed cost A per order. The firm also incurs a
holding cost h per unit kept in inventory per unit time, and a cost c per unit
purchased or produced. Without loss of generality, we assume that orders are
delivered with zero leadtime (a positive leadtime can be included and does not
affect the solution to the problem); we also assume that the firm must satisfy
all the demand (the analysis can be easily extended to settings with backorders).
8Total cost per unit time is then given by
AD
Q
+
hQ
2
+ cD.
Similar to cost, emissions are associated with ordering, inventory holding, and
production/purchasing, with Aˆ, hˆ and cˆ denoting the amount of carbon emissions
associated per order initiated, per unit held in inventory per unit time, and per
unit purchased or produced. Total emission per unit is therefore
AˆD
Q
+
hˆQ
2
+ cˆD.
The parameterization of emissions under the above formulation is flexible and can
be used to capture a variety of settings. For example, if emission from holding
inventory depends only on the maximum amount of inventory held (which would
correspond to Q), then total emission can be expressed as AˆD
Q
+ hˆQ+ cˆD, which is
equivalent to the original expression but with an inventory holding emission pa-
rameter equal to 2hˆ. Similarly, if emission from holding inventory is invariant to
the amount of inventory held, then total emission reduces to AˆD
Q
+ hˆ+ cˆD, which is
equivalent to the original model but with an inventory holding emission parameter
equal to 0. If the emission associated with initiating an order has both a fixed
and a variable component, say of the form Aˆ1+Aˆ2Q, then the corresponding total
emission is Aˆ1D
Q
+ hˆQ
2
+ (cˆ+ Aˆ2)D, which again has the same form as the original
model. Note that, depending on the setting, Aˆ can be higher or lower than hˆ. For
example, for some products transportation-related emissions are high but storage
emissions are low or even negligible (e.g., canned foods) while for others the re-
verse may be true (e.g., refrigerated foods). Walmart recently discovered that the
refrigerants used in grocery stores accounted for a larger percentage of Walmart’s
9greenhouse gas footprint than its truck fleet (http://www.walmartstores.com/
sites/responsibility-report/2012/sustainableFacilities.aspx). Tesco,
the largest retailer in the UK, found that 26 percent of its direct emissions
were due to refrigerant leakage while only 12 percent were due to transportation
(http://www.tesco.com/climatechange/carbonFootprint.asp). Our analysis
and results are applicable in all cases (see the end of this section for additional
discussion).
The objective of the firm is to choose an order quantity Q that minimizes its
cost per unit time subject to the constraint on the amount of carbon emitted (this
cap can reflect either government regulations imposed on the firm or a voluntary
effort by the firm to reduce its emissions by a specified amount). The amount of
carbon emitted is constrained to be less than a certain cap C. The problem can
then be formally stated as follows:
Minimize Z(Q) =
AD
Q
+
hQ
2
+ cD (2.1)
subject to
AˆD
Q
+
hˆQ
2
+ cˆD ≤ C (2.2)
Let Qˆmin denote the order quantity that minimizes carbon emission (the emission-
optimal solution), then it is easy to verify that Qˆmin =
√
2AˆD
hˆ
and the correspond-
ing emission level is Emin =
√
2AˆhˆD + cˆD. Consequently, the problem admits
a feasible solution if and only if C ≥ Emin. In the remainder, we assume that
this condition is always satisfied. Also, let Q∗ denote the order quantity that
minimizes the total cost while ignoring the carbon emission constraint (the cost-
optimal solution). Then, it is easy to see that Q∗ =
√
2AD
h
, which corresponds
to the standard EOQ solution. The following theorem characterizes the optimal
solution to (2.1)-(2.2).
10
Theorem 1 Let
Q1 =
Cˆ −
√
Cˆ2 − 2AˆhˆD
hˆ
and Q2 =
Cˆ +
√
Cˆ2 − 2AˆhˆD
hˆ
(2.3)
where Cˆ = C − cˆD. Then the optimal solution to problem (2.1)-(2.2) is
Qˆ∗ =

Q∗ if Q1 ≤ Q∗ ≤ Q2,
Q1 if Q
∗ ≤ Q1,
Q2 if Q
∗ ≥ Q2.
(2.4)
Furthermore, the emission level under the optimal order quantity is
E(Qˆ∗) =
 Emax if Q1 ≤ Q∗ ≤ Q2C otherwise, (2.5)
where
Emax = Aˆ
√
hD
2A
+ hˆ
√
AD
2h
+ cˆD (2.6)
and corresponds to the emission level in the absence of the carbon constraint (also
corresponds to the emission level when the optimal order quantity is Q∗).
Proof: From constraint (2.2), we can show that the optimal order quantity must
satisfy Q1 ≤ Qˆ∗ ≤ Q2. If Q1 ≤ Q∗ ≤ Q2, then obviously Qˆ∗ = Q∗ and the
corresponding emission is
E(Q∗) =
AˆD
Q∗
+
hˆQ∗
2
+ cˆD = Aˆ
√
hD
2A
+ hˆ
√
AD
2h
+ cˆD.
If Q∗ ≤ Q1 then Qˆ∗ = Q1 because Z(Q) is convex in Q and choosing a higher
value for Qˆ∗ will lead to a higher cost. Similarly, if Q∗ ≥ Q2 then Qˆ∗ = Q2
because choosing a lower value for Qˆ∗ will lead to higher cost. In both of these
cases, constraint (2.2) is binding and, therefore, E(Qˆ∗) = C. 
11
In the following proposition, we show that cost is indeed decreasing and convex
in the emission cap C while emission is linearly increasing in C, implying that
reducing the emission cap leads initially to a larger relative emission reduction
than the relative cost increase (e.g., in the example illustrated in Figure 2.1, an
emission reduction of 20% leads only to a 4% increase in cost).
Proposition 1 For C ≥ Emin, emission is linearly increasing in C while cost is
decreasing and convex in C.
Proof: For emission, the result follows immediately from Theorem 1. For cost,
consider first the case where Q∗ = Qˆ∗. In this case, cost and emissions are
unaffected by the cap C and the results hold. Next, consider the case where
Q1 ≥ Q∗. In this case, we have
Z(Qˆ∗) = Z(Q1) =
ADhˆ
Cˆ −
√
Cˆ2 − 2AˆhˆD
+
h(Cˆ −
√
Cˆ2 − 2AˆhˆD)
2hˆ
+ cD.
∂Z(Qˆ∗)
∂C
=
∂Z(Qˆ∗)
∂Cˆ
=
∂
∂Cˆ
(
ADhˆ
Cˆ −
√
Cˆ2 − 2AˆhˆD
+
h(Cˆ −
√
Cˆ2 − 2AˆhˆD)
2hˆ
)
=
Ahˆ+ Aˆh
2Aˆhˆ
+
Ahˆ− Aˆh
2Aˆhˆ
Cˆ√
Cˆ2 − 2AˆhˆD
.
∂2Z(Qˆ∗)
∂C2
=
∂2Z(Qˆ∗)
∂Cˆ2
=
Ahˆ− Aˆh
2Aˆhˆ
−2AˆhˆD
(
√
Cˆ2 − 2AˆhˆD)3
=
(Aˆh− Ahˆ)D
(
√
Cˆ2 − 2AˆhˆD)3
.
Since Q1 ≥ Q∗ implies
√
2AD
h
≤ Q1 ≤
√
2AˆD
hˆ
, we have Aˆh − Ahˆ ≥ 0. Therefore,
∂2Z(Qˆ∗)
∂C2
≥ 0. In the remaining case of Q2 ≤ Q∗ we can show using similar argu-
ments that ∂
2Z(Qˆ∗)
∂C2
≥ 0. Consequently, the optimal cost function is convex with
respect to the carbon cap C. 
From Proposition 1, we can see that imposing an emission cap leads to an emission
reduction only if the cap is sufficiently small, namely C ≤ Emax. Otherwise, the
12
Cost
Emissions
Emin 750 800 850 900 950 Emax 1050
3500
4000
4500
5000
5500
650.
700.
750.
800.
850.
900.
950.
1000.
Carbon Cap, C
C
o
st
Em
is
si
o
n
s
Figure 2.1: The impact of the carbon cap on emission and cost
(D = 600, A = 120, h = 2, c = 5, Aˆ = 2, hˆ = 3, cˆ = 1)
cost-optimal solution is feasible and the corresponding emission is Emax. If C ≤
Emax, emission is reduced by adjusting the order quantity (by either increasing
it or decreasing it from the cost-optimal order quantity). However, for this to
be possible, the cost-optimal order quantity must be different from the emission
optimal solution. This leads to the following important corollary.
Corollary 1 Reducing emissions by adjusting order quantities is possible if and
only if A
h
6= Aˆ
hˆ
.
This corollary follows from the fact that if A
h
= Aˆ
hˆ
then the cost-optimal solution
is also emission-optimal (i.e., Q∗ = Qˆmin). In that case, emissions are already at
their minimum and there is no operational adjustment that could further reduce
them. On other hand, if A
h
6= Aˆ
hˆ
, there is an opportunity to reduce emissions by
either increasing or decreasing the order quantity. In particular, if A
h
< Aˆ
hˆ
(A
h
> Aˆ
hˆ
)
13
Cost
Emissions
10 Q` min 150 Q* 250 300
3500
4500
5500
6500
7500
600.
700.
800.
900.
1000.
Order Quantity, Q
Co
st
Em
iss
io
n
s
(a) (D = 600, A = 120, h = 4, c = 5, Aˆ = 10, hˆ = 2, cˆ = 1)
Emissions
Cost
10 Q* 150 Q` min 250 300
600
700
800
900
1000
3000.
4000.
5000.
6000.
7000.
8000.
Order Quantity, Q
Co
st
Em
iss
io
n
s
(b) (D = 600, A = 10, h = 2, c = 1, Aˆ = 120, hˆ = 4, cˆ = 5)
Figure 2.2: Cost versus Emissions
14
then increasing (decreasing) the order quantity decreases emissions; see Figure 3.4
for a graphical illustration.
The broader implication of Corollary 1 is that operational adjustments can
lead to emission reductions only if the cost parameters are not strongly corre-
lated to the emission parameters, so that what drives cost more is not what also
drives emissions more. This is the case for example when fixed costs are higher
than inventory holding costs but fixed emissions are lower than inventory-related
emissions.
Although Corollary 1 identifies settings where it is possible to reduce emissions
by adjusting order quantities, it does not specify the extent to which this reduction
can be realized without significantly increasing cost. The examples shown in
Figures 2.1 and 3.4 do suggest that indeed a modest reduction in the order quantity
(away from the cost-optimal order quantity) leads to a modest increase in cost but
a significant reduction in emission (both cost and emission have similar functional
forms, with both being convex in Q and approaching ∞ as Q approaches either
0 or ∞). More importantly, the cost function for the EOQ is flat in the region
around the optimal solution. This means that in this region a relative change in
the order quantity leads to a lower relative increase in cost. In what follows, we
characterize this flatness and identify a condition under which the reduction in
emission is greater than the increase in cost (for further discussion and related
results see [24, 43, 58]).
Let Z ′(Q) and E ′(Q) refer to the components of cost and emission that are
affected by order quantity. That is, Z ′(Q) = AD/Q+hQ/2 and E ′(Q) = AˆD/Q+
15
hˆQ/2. Also, let
δQ =
Q−Q∗
Q∗
denote the relative change in the order quantity (with respect to the cost-optimal
order quantity), and
δZ =
Z ′(Q)− Z ′(Q∗)
Z ′(Q∗)
and δE =
E ′(Q∗)− E ′(Q)
E ′(Q∗)
denote respectively the corresponding relative change in cost and emission. Then,
we can show that
δZ =
δ2Q
2(1 + δQ)
and δE = −
(1− α)δQ + δ2Q
(1 + α)(1 + δQ)
, (2.7)
where α = Aˆ/hˆ
A/h
is the ratio of the emission parameters to the cost parameters.
-60 -40 -20 0 20 40 60 80
0
20
40
60
80
dQH%L
d
Z
H
%
L
Figure 2.3: The impact of changes in order quantity on cost
(Values of δZ below the 45 degree lines (shown in dashes)
correspond to cases where δZ is smaller than δQ)
First, it is easy to see that even relatively large values of |δQ| lead to relatively
small values of δZ . For example, increasing the order quantity by 30% (δQ = 0.3)
16
leads to an increase in cost of only 3.46% (δZ = 0.0346); see Figure 2.3 for a full
characterization of δZ as a function of δQ. Mathematically, we can show that
∂δZ
∂δQ
=
δQ(2 + δQ)
2(δQ + 1)2
,
which is equal to 0 for δQ = 0. Moreover, we can show that if δQ > 0, then
δZ ≤ δQ (in other words, the relative increase in cost is always lower than the
relative increase in the order quantity regardless of the size of the increase). If
δQ < 0, then δZ ≤ |δQ| as long as δQ ≥ −2/3 (i.e., we can reduce order quantity
by as much as 2/3 without increasing cost by as much). In contrast,
∂δE
∂δQ
=
α− (1 + δQ)2
(1 + α)(1 + δQ)2
and takes on a value equal to α−1
α+1
6= 0 for α 6= 1 when δQ = 0. That is, while
the cost function is always flat around Q∗, the emission function can be quite
steep (i.e., |α−1
α+1
| can be large). The fact that the flat region of the cost function
coincides with the steep region of the emission function means that there is an
opportunity to achieve more relative emission reductions than the corresponding
relative increase in cost.
Next, we describe, the range over which order quantity can be adjusted while
guaranteeing that the relative increase in cost is less than the relative reduction
in emission, that is δZ ≤ δE and δE ≥ 0.
Proposition 2 For α > 1, δZ ≤ δE if 0 ≤ δQ ≤ 2(α−1)3+α , and δZ ≥ δE otherwise.
For α < 1, δZ ≤ δE if 2(α−1)3+α ≤ δQ ≤ 0, and δZ ≥ δE otherwise.
The proof immediately follows from the expressions of δZ and δE in Equation
(2.7) and for brevity, we omit the details. As we can see, the interval over which
17
the order quantity can be varied depends solely on α with its width increasing in
the absolute value of the difference between Aˆ
hˆ
and A
h
. In the limit cases of either
α→ 0 or α→∞ the order quantity can be adjusted by as much as a factor of 3.
When δZ = δE (and δQ =
2(α−1)
3+α
) the resulting relative decrease in emission and
in cost is given by
δE=Z =
2(α− 1)2
(1 + 3α)(3 + α)
(2.8)
which is also increasing in the absolute value of the difference between the ratios
Aˆ
hˆ
and A
h
. In the limit, as either α → 0 or α → ∞, δE=Z → 2/3, implying that
emissions could be reduced by up to 2/3 without increasing cost by as much.
The following proposition further characterizes the tradeoff between cost and
emission reductions.
Proposition 3 Let δE(δZ , α) denote the relative emission reduction as a function
of the relative cost increase δZ and α. Then,
δE(δZ , α) =

− (δZ+
√
2δZ+δ
2
Z)(1−α+δZ+
√
2δZ+δ
2
Z)
(1+α)(1+δZ+
√
2δZ+δ
2
Z)
, if α > 1,
− (δZ−
√
2δZ+δ
2
Z)(1−α+δZ−
√
2δZ+δ
2
Z)
(1+α)(1+δZ−
√
2δZ+δ
2
Z)
, if α < 1.
(2.9)
Moreover,
• δE(δZ , α) is concave in δZ, and it achieves its maximum value (the emission
optimal solution) for δZ =
(1−√α)2
2
√
α
leading to an emission reduction of δE =
(1−√α)2
1+α
,
• for α > 1, δE(δZ , α) is increasing concave in α (the same percentage increase
in cost leads to a lower decrease on emission for a greater value of α), and
18
• for α < 1, δE(δZ , α) is decreasing convex in α (the same percentage increase
in cost leads to a lower decrease on emission for a lower value of α).
Proof: Expressing δQ as function of δZ leads to:
δQ =

δZ +
√
2δZ + δ2Z ≥ 0,
δZ −
√
2δZ + δ2Z ≤ 0.
Substituting into the expression of δE leads to (2.9). It is easy to verify that
∂δE(δZ ,α)
∂α
=

2
√
δZ(δZ+2)
(1+α)2
≥ 0, α > 1,
−2
√
δZ(δZ+2)
(1+α)2
≤ 0, α < 1.
and
∂2δE(δZ ,α)
∂α2
=

−4
√
δZ(δZ+2)
(1+α)3
≤ 0, α > 1,
4
√
δZ(δZ+2)
(1+α)3
≥ 0, α < 1.
Similarly,
∂2δE(δZ , α)
∂δ2Z
=

(1−α)
(1+α)(δZ(δZ+2))3/2
≤ 0, α > 1,
(α−1)
(1+α)(δZ(δZ+2))3/2
≤ 0, α < 1.
Furthermore, solving for ∂δE(δZ ,α)
∂δZ
= 0 leads to δZ =
(1−√α)2
2
√
α
and δE(
(1−√α)2
2
√
α
, α) =
(1−√α)2
1+α
, which completes the proof. 
The cost-emission tradeoff is illustrated in Figure 2.4, where values of δE above
the 45 degree lines (shown in dashes) correspond to cases where δE is larger than
δZ . Figure 2.4 suggests that there is a value of δZ (and corresponding order
19
a = 3
a = 4
a = 6
0 5 10 15 20 25 30
0
5
10
15
20
25
30
dZH%L
d
EH
%
L
Figure 2.4: Cost-emission tradeoff
quantity) which maximizes the difference δE − δZ . This value of δZ , to which we
refers as δmaxZ , is given by
δmaxZ =
2(1 + α)√
(3 + α)(3α + 1)
− 1, achieved for δmaxQ =
√
3α + 1
3 + α
− 1
The associated decrease in emission δmaxE and the maximum difference (δ
max
E −
δmaxZ ) are respectively given by
δmaxE =
1
1 + α
(1−
√
3 + α
3α + 1
)(
√
3α + 1
3 + α
− α), and
δmaxE − δmaxZ =
√
(3 + α)(3α + 1)
2(1 + α)
(1−
√
3 + α
3α + 1
)(
√
3α + 1
3 + α
− 1).
These values can be viewed as maximizing the benefit derived from operational
adjustments (the most environmental bang for the cost buck).
Proposition 4 δmaxE − δmaxZ is increasing in α > 1 and decreasing in α < 1 and
ranges from 0 to (2−√3) ≈ 26.8%.
20
Proof: It is easy to show that
∂(δmaxE − δmaxZ )
∂α
=
2(α− 1)
(1 + α)2
√
(3 + α)(3α + 1)

> 0 for α > 1
< 0 for α < 1
, and
limitα→0(δmaxE − δmaxZ ) = limitα→∞(δmaxE − δmaxZ ) = 2−
√
3. 
In the special cases of Aˆ = 0 and hˆ 6= 0, δmaxZ = 2−
√
3√
3
≈ 15.5% and δmaxE =
√
3−1√
3
≈ 42.3% (in other words, we achieve a 42.3% reduction emissions with only
a 15.5% increase in cost).
Figure 2.5: The impact of α on the maximum difference between δE and δZ
We conclude this section by noting that the key results and insights from this
section are applicable to settings where either Aˆ = 0 or hˆ = 0 – that is, settings
in which emissions are due only to initiating orders or only to keeping inventory.
In both cases, it is easy to see that the opportunity to reduce emissions without
significantly increasing cost is even greater; for brevity, we omit the details.
21
2.3 Extensions to Systems with Carbon Prices
In this section, we briefly describe how the analysis can be extended to settings
where emissions are regulated using carbon prices, instead of strict caps, or a
combination of caps and prices. In particular, we consider settings with a carbon
tax, cap-and-offset, and cap-and-price. In each case, we discuss the extent to
which the possibility of an operational adjustment can be leveraged to either
reduce emission costs or to generate additional revenue.
2.3.1 Carbon Tax
The pricing of carbon can take on a variety of forms. A simple mechanism is to
impose a financial penalty, a tax, per unit of carbon emitted. If we let t > 0 denote
the penalty per unit of carbon emitted (the tax rate), then the total cost incurred
by the firm, given a choice of order quantity Q, can be expressed as follows
Zt(Q) = Z(Q) + tE(Q) (2.10)
where, Z(Q) =
AD
Q
+ hQ
2
+ cD is the direct operational cost associated with Q,
E(Q) = AˆD
Q
+ hˆQ
2
+ cˆD is the corresponding emission.
To assess the extent to which operational adjustments can mitigate the cost of
emissions we compare two strategies available to a firm. The first is for the firm
to continue business as usual and to choose the order quantity that minimizes its
operational cost Z(Q) and then to pay the resulting tax. This means that the firm
chooses order quantity Q∗ =
√
2AD
h
. The second strategy is for the firm to adjust
22
its order quantity so that it minimizes the sum of its operational and emission
costs Zt(Q). In this case, the firm chooses
Q∗t =
√
2(A+ tAˆ)D
(h+ thˆ)
. (2.11)
Note that Q∗t 6= Q∗ only if Aˆhˆ 6= Ah . Otherwise, the “operational cost”-optimal
solution is also “total cost”-optimal.
Proposition 5 Let δZ′t =
Z′t(Q∗)−Z′t(Q∗t )
Z′t(Q∗)
and δE′ =
E′(Q∗)−E′(Q∗t )
E′(Q∗) denote respectively
the relative reduction in cost and emission due to the adjustment in the order
quantity, where Z ′t(Q) and E
′(Q) refer to the components of cost and emission
that are affected by order quantity, then when Aˆ
hˆ
6= A
h
• Both δZ′t and δE′ are positive and strictly increasing in t (δE′ is concave in
t), with δZ′t < δE′ and limt→∞
δZ′t = limt→∞
δE′ =
(1−√α)2
1+α
, and
• Both δZ′t and δE′ are strictly increasing (decreasing) in α > 1(α < 1).
Proof: First note that δZ′t and δE′ are respectively given by
δZ′t = 1−
√
(1 + ut)(1 + vt)
1 + t
2
(u+ v)
and δE′ = 1− ( u
u+ v
√
1 + vt
1 + ut
+
v
u+ v
√
1 + ut
1 + vt
),
where u = Aˆ
A
, v = hˆ
h
, so that α = u/v. Then, we can show that
∂δZ′t
∂t
=
(u− v)2t√
(1 + ut)(1 + vt)(2 + ut+ vt)2
> 0
and

∂δE′
∂t
= (u−v)
2
2(u+v)((1+ut)(1+vt))3/2
> 0
∂2δE′
∂t2
= − (u−v)2(u+v+2uvt)
4(u+v)((1+ut)(1+vt))5/2
< 0
.
23
Moreover, Z ′t(Q
∗)2−Z ′t(Q∗t )2 = AhD2 (u− v)2t2 > 0, E ′(Q∗)2−E ′(Q∗t )2 = AhD2 (u−
v)2 (u+v+uvt)t
(1+ut)(1+vt)
> 0, and δE′ − δZ′t = (a−b)
2t
(a+b)
√
(1+ut)(1+vt)(2+t(u+v)
> 0, therefore, 0 <
δZ′t < δE′ . It is easy to verify that limt→∞
δZ′t = limt→∞
δE′ =
(1−√α)2
1+α
. Finally, we can
show that
∂δE′
∂u
= (u−v)
2(u+v)2
t(1+vt)(u+3v+2uvt)
((1+ut)(1+vt))3/2
> 0(< 0) for u > v(u < v),
∂δZ′t
∂u
= (u−v)t
2
(2+t(u+v))2
√
1+vt
1+ut
> 0(< 0) for u > v(u < v),
and 
∂δE′
∂v
= (v−u)
2(u+v)2
t(1+ut)(v+3u+2uvt)
((1+ut)(1+vt))3/2
> 0(< 0) for u < v(u > v),
∂δZ′t
∂v
= (v−u)t
2
(2+t(u+v))2
√
1+vt
1+ut
> 0(< 0) for u < v(u > v).
Therefore, δZ′ and δE′ are strictly increasing (decreasing) in α > 1(α < 1). 
0 50 100 150
0
5
10
15
20
25
30
35
40
45
pe
rc
en
ta
ge
 (%
)
                t
δE ‘
δZ ‘t
Carbon tax rate,
Figure 2.6: The benefit of operational adjustment under carbon tax
The results of Proposition 5 show that operational adjustment can indeed lower
a firm’s emission cost (its tax burden) and for this reduction to be sufficient to
24
offset the associated increase in its operational cost. The relative benefit from this
adjustment (both in terms of cost and emission) increases with the tax rate, as a
higher tax rate justifies moving further away from the “operational cost”-optimal
order quantity. The fact that the emission reduction is concave in the tax rate
implies that there is a diminishing effect to increasing the tax rate; see Figure
2.6. It also implies that a modest tax rate can lead to a significant reduction
in emissions. This would be the case for example when α is large (small) for
α > 1(α < 1), or equivalently when the absolute difference between the ratios A
h
and Aˆ
hˆ
is large.
2.3.2 Cap-and-Offset
An alternative to taxing all emissions is to tax only emissions that exceed a certain
threshold. This can arise in settings where the regulatory agency sets an emission
cap on the firm and imposes penalties if the cap is exceeded. It can also arise in
settings where the regulatory agency sets an emission cap but allows firm to relax
its cap through the purchase of emission offsets through third parties (see [5] for
examples and references). Note that systems with a strict cap and a tax on all
emissions can be viewed as special cases of cap-and-offset, where in the first case
the penalty for exceeding the cap is infinitely large and in the second the cap is
set at zero.
Under a cap-and-offset system, the firm has again the option of either operating
business-as-usual by choosing an order quantity that minimizes the sum of its fixed
ordering and inventory holding costs (its direct operational costs) and then to pay
any resulting penalties if it exceeds its cap, or of adjusting its order quantity by
25
choosing one that minimizes the sum of its operational and emission costs. In the
latter, the problem the firm faces can be formulated as follows:
Minimize Zo(Q) =
AD
Q
+
hQ
2
+ cD + t(
AˆD
Q
+
hˆQ
2
− Cˆ)+, (2.12)
where Cˆ = C − cˆD, t is the penalty paid per unit emitted in excess of the cap,
and X+ = max(0, X).
Theorem 2 The optimal order quantity that minimizes (12) is given by
Q∗o =

Q∗t =
√
2(A+tAˆ)D
h+thˆ
, Cˆ < Cˆ1,
Q1 =
Cˆ−
√
Cˆ2−2AˆhˆD
hˆ
, Cˆ1 < Cˆ < Cˆ2 and
A
h
< Aˆ
hˆ
,
Q2 =
Cˆ+
√
Cˆ2−2AˆhˆD
hˆ
, Cˆ1 < Cˆ < Cˆ2 and
A
h
> Aˆ
hˆ
,
Q∗ =
√
2AD
h
, Cˆ > Cˆ2,
where Cˆ1 =
√
D
2
(Aˆ
√
h+thˆ
A+tAˆ
+ hˆ
√
A+tAˆ
h+thˆ
), and Cˆ2 =
√
D
2
(Aˆ
√
h
A
+ hˆ
√
A
h
).
Proof: Cˆ1 and Cˆ2 correspond to E(Qt) − cˆD, E(Q∗) − cˆD respectively. When
Cˆ ≥ Cˆ2, we know that for any Q, Zo(Q∗) ≤ Zo(Q), therefore Q∗o = Q∗. When
Cˆ1 ≤ Cˆ ≤ Cˆ2 and Ah < Aˆhˆ , we know that for any Q satisfying E(Q) ≥ C,
Q ≤ Q1 ≤ Qt. Therefore, Zo(Q1) ≤ Zo(Q). On the other hand, for anyQ satisfying
E(Q) < C, we know that Q∗ ≤ Q1 ≤ Q. Therefore, Zo(Q1) ≤ Zo(Q). Therefore,
when Cˆ1 ≤ C ≤ Cˆ2, we have Q∗o = Q1. The case of Cˆ1 ≤ C ≤ Cˆ2 and Ah > Aˆhˆ is
similar. When Cˆ ≤ Cˆ1 and Ah < Aˆhˆ , we know that for any Q satisfying E(Q) ≥ C,
Zo(Q
∗
t ) ≤ Zo(Q). On the other hand, for any Q satisfying E(Q) < C, we know that
26
Q > Q1 > Q
∗
t , because Cˆ < Cˆ1,
A
h
< Aˆ
hˆ
. Therefore, Zo(Q) > Zo(Q1) > Zo(Q
∗
t ).
Thus, when Cˆ < Cˆ1 and
A
h
< Aˆ
hˆ
, we have Q∗o = Q
∗
t . The case of Cˆ < Cˆ1 and
A
h
> Aˆ
hˆ
is similar. 
Figure 2.7: The effect of the carbon cap under cap-and-offset
(D = 100, A = 120, h = 2, c = 5, Aˆ = 1, hˆ = 0.5, cˆ = 0, t = 5)
Theorem 2 indicates that, depending on the cap, there are three regions of op-
eration. If the cap is sufficiently large (i.e., Cˆ ≥ Cˆ2), then the firm operates
business-as-usual and does not make any adjustments in its order quantity. If
Cˆ1 ≤ Cˆ ≤ Cˆ2, the firm makes adjustments in its order quantity so that its emis-
sions do not exceed the cap; the firm in this case does not incur any emission
penalties. Note that here the firm operates as if it were subject to a strict emis-
sion cap and therefore all the results of Section 2 apply, including the opportunity
to significantly reduce emissions without significantly increasing cost. Note also
that in this region the firm finds it more preferable to incur the increase in its
direct operational costs than to incur emission penalties. On the other hand, if
27
Cˆ ≤ Cˆ1, then the firm finds it more preferable to pay the emission penalty than to
reduce its emissions below C1 = Cˆ1+ cˆD. In this region, the firm emits exactly C1
and pays the penalty t(C −C1) associated with the difference. The three regions
are illustrated for an example system in Figure 2.12. Note that for the strategy
of not adjusting order quantity, cost is linear in the cap for Cˆ ≤ Cˆ2. Hence, the
difference in cost between adjusting and not adjusting order quantity can be sig-
nificant in the region where Cˆ1 ≤ Cˆ ≤ Cˆ2. It can also be significant when Cˆ ≤ Cˆ1,
where the difference is constant and equal to t(C2 − C1).
We conclude this section by highlighting the effect of the emission penalty t.
In particular, an increase in t has the effect of decreasing the value of Cˆ1 and,
therefore, increasing the width of the interval [Cˆ1, Cˆ2]. This means that a larger
t increases the region in which incurring higher operational costs is preferable to
incurring emission penalties.
2.3.3 Cap-and-Price
Instead of only penalizing emissions that exceed the specified cap, as in cap-and-
offset, the regulating agency may choose to also encourage emissions that are
lower than the cap by rewarding firms that emit less than their cap. Rewarding
lower emissions can take on a variety of forms. At its simplest, it consists of a
reward (penalty) t′ (t) per unit of emission below (above) the cap C. We refer
to this scheme as cap-and-price since there is now monetary value for emissions
regardless of whether or not the cap is exceeded (this is similar to the so-called
cap-and-trade system, except that in that case prices are established through a
market mechanism for carbon trading). All the previous schemes can be viewed
28
as special cases of cap-and-price (for example cap-and-offset corresponds to the
case where t′ = 0).
It is easy to show that when t′ = t the problem is similar to one with a carbon
tax, with a similar expression for total cost except for an additional revenue term
−tC (this extra term is due to the fact that cap-and-price is equivalent to a
scheme where firms receive a reward equal to tC and are then taxed for each
unit of emission with rate t). Therefore, many of the results obtained for the
case of carbon tax continue to apply. However, in contrast to a system with a
carbon tax, the effect of t on cost is not monotonic. In particular, if the cap
C is sufficiently large (larger than the emission that would be incurred if the
firm maintained business as usual), then cost would be strictly decreasing in t,
eventually becoming negative with the firm realizing a net profit. On the other
hand, if the cap is sufficiently small (smaller than the emission that would be
incurred if the firm chose the emission-optimal order quantity), then cost would
be strictly increasing in t since the amount of emissions would always exceed the
cap. If the cap falls between these two thresholds, then cost is not monotonic
in t; it initially increases then decreases. In particular, when t is small, the
firm chooses to exceed the emission cap and, therefore, incurs the corresponding
emission penalties. However, when t is large, the firm chooses to emit less than
the cap and realizes the corresponding revenue. This effect is illustrated in Figure
2.8. Note that the benefit from adjusting the order quantity to take acount of
carbon price can be significant, in terms of both cost and emission, when carbon
price is high and the firm can emit less than its cap.
29
170
180
190
200
210
220
230
240
250
t
Cost
Amount of carbon purchased
C
os
t
0 5 10 15 20 25 30
−6
−4
−2
0
2
4
6
8
10
A
m
ou
nt
 o
f c
ar
bo
n 
pu
rc
ha
se
d
Carbon price,
Figure 2.8: The impact of carbon price on cost and carbon emissions
purchased when Emin ≤ C ≤ Emax
(D = 100, A = 120, h = 2, Aˆ = 1, hˆ = 0.5, c = 0, cˆ = 0)
2.4 Extensions to Other Models
In the Section 2, we showed how it may be possible to significantly reduce emis-
sions without significantly increasing cost via only an operational adjustment. In
particular, we showed that the key requirements are that the cost and the emission
functions yield different optimal solutions and that the cost function exhibits less
sensitivity around the cost-optimal solution than the emission function. These
features can be shown to be present in several other operational settings and
models. Therefore, the insights obtained have applicability to those cases as well.
In what follows, we briefly describe two other widely used operational models for
which this is the case.
30
2.4.1 The Facility Location Model
Consider the problem of determining the number of facilities N to serve demands
that are uniformly distributed over a region with an area a and demand density ρ
(amount of demand per unit of area). There is a fixed cost f incurred with each
facility and a variable cost c incurred with each unit of demand transported one
unit of distance. There is a tradeoff between the fixed costs and the transportation
costs, with a higher number of facilities leading to higher fixed costs but, because
the service area of each facility is a/N , lower transportation costs. Under the
assumptions that a is a square area and travel occurs along paths oriented at 45
degrees to the sides of the square, the total expected cost Z(N) can be shown to
be given by (see for example [19] and [20])1
Z(N) = fN + cρa
2
3
√
a
2N
. (2.13)
Similarly to cost, there is a fixed emission fˆ associated with each facility and
variable emission cˆ associated with each unit of demand transported one unit of
distance, leading to an expected total emission E(N) given by
E(N) = fˆN + cˆρa
2
3
√
a
2N
. (2.14)
We can then show that the cost-optimal and emission-optimal number of facilities
are given by N∗ = K( c
f
)2/3 and Nˆ∗ = K( cˆ
fˆ
)2/3, respectively, where K = a( ρ
2
√
3
)2/3.
As in the EOQ case, there is an opportunity to reduce emissions via an adjust-
ment in the number of facilities whenever cˆ
fˆ
6= c
f
. Moreover, we can show that the
cost function is flat around the optimal solution and that whenever the flat region
1 In general, Z(N) = fN + c√
N
g(a), where g(a) is a function dependent on the shape of
area a.
31
of the cost function coincides with the steep region of the emission function, a
significant reduction in emissions can be achieved without a significant increase in
cost. To see this let α = cˆ
fˆ
/ c
f
, δN =
N−N∗
N∗ , δZ =
Z(N)−Z(N∗)
Z(N∗) , and δE =
E(N∗)−E(N)
E(N∗) .
Then, it is easy to verify that,
δZ =
1
3
(δN − 2 + 2√
1 + δN
) and δE =
1
1 + 2α
(1 + δN +
2α√
1 + δN
).
Figure 2.9: The impact of changes in the number of facilities on cost
The flatness of the cost function is apparent from Figure 2.9 (e.g., a 50%
increase in N leads to less than 5% increase in cost). Mathematically, the cost
ratio δZ is strictly convex in δN with
∂δZ
∂δN
(0) = 0. In contrast, the emission ratio
δE is strictly concave in δN with
∂δE
∂δN
(0) = α−1
1+2α
, which is increasing (decreasing)
in α > 1 (α < 1) (recall that α is increasing in the absolute value of the difference
cˆ
fˆ
− c
f
). Moreover,
• δZ ≤ δE, for |δN | ≤ |12( 9α2+α −
√
3(2+7α)
2+α
)|, and
• δE − δZ is maximal for δmaxN = (1+5α4+2α)2/3 − 1.
32
In the limit case of α → 0, δE(δmaxN ) − δZ(δmaxN ) → 2 − 22/3 ≈ 0.412 (we achieve
approximately 60% reduction in emissions with only about 19% increase in cost)
and in the limit case of α → ∞, δE − δZ → 2 − (5/2)2/3 ≈ 0.15 (we achieve
approximately 26% reduction in emissions with only about 10% increase in cost).
2.4.2 The Newsvendor Model
Consider the problem of determining the order quantity Q for a single selling
period given that demand for the period is stochastic and can be described by a
continuous random variable D. Ordering too little could lead to shortages while
ordering too much could lead to leftover inventory, with a shortage cost cs incurred
per unit of demand that exceeds the quantity ordered Q and an overage cost co
incurred per unit ordered that exceeds demand. Thus, the total expected cost for
the period is given by
Z(Q) = csE[(D −Q)+] + coE[(Q−D)+], (2.15)
where E[.] refers to the expected value operator. Similarly, there are emissions
associated with being short and with being over. In particular, cˆs is the amount
of emissions per unit of shortage (e.g., emissions caused by resorting to fulfilling
demand through more emission-intensive means) and cˆo is the amount of emissions
per unit of overage (e.g., emissions associated with disposing of leftover inventory
or with storing inventory until the next selling season). This leads to a total
expected emission given by
E(Q) = cˆsE[(D −Q)+] + cˆoE[(Q−D)+]. (2.16)
Then, it is easy to show that the cost-optimal order quantity Q∗ is the solution
33
to the critical fractile equation (see for example [58]): F (Q) = cs
cs+co
, and the
emission-optimal order quantity Qˆ∗ is given by the solution to F (Q) = cˆs
cˆs+cˆo
,
where F is the probability distribution of demand D.
As with the EOQ and the facility location models, there is an opportunity to
reduce emissions by modifying the order quantity whenever cs/co 6= cˆs/cˆo. For a
variety of demand distributions, it is possible to show that the cost function is
flat around the cost-optimal solution while the emission function is steep in the
same region, leading to the possibility of significantly reducing emissions without
significantly increasing cost.
Consider for example the case where demand is uniformly distributed over an
interval [a, b], with b > a ≥ 0. We can easily verify that the cost-optimal order
quantity is given by Q∗ = aco+bcs
co+cs
, the corresponding expected cost Z(Q∗) and
emission level E(Q∗) are respectively given by
Z(Q∗) =
csco
2(cs + co)
(b− a) and E(Q∗) = (b− a)c
2
s cˆo + c
2
ocˆs
2(cs + co)2
. (2.17)
We can show that δZ is strictly convex in δQ while δE is strictly concave in δQ,
with ∂δZ
∂δQ
(0) = 0, and ∂δE
∂δQ
(0) 6= 0.
Moreover, we can show that the maximum value δ0Q by which the order quantity
can be adjusted (increased (decreased) when α > 1(α < 1)) while guaranteeing
that 0 ≤ δZ ≤ δE is given by δ0Q = | 4(a−b)(1−α)co
cs
+1(a+b cs
co
)(1+α)
|, where α = cˆs/cˆo
cs/co
. We can also
show that δmaxQ , the relative change in the order quantity for which the difference
(δE−δZ) is maximum, is given by δmaxQ = 2(a−b)(1−α)co
cs
+1(a+b cs
co
)(1+α)
.When the order quantity
is adjusted by δmaxQ , we can easily verify that, δE(δ
max
Q ) = (2 +
α
1+ co
cu
α
)δZ(δ
max
Q ).
This means that, independently of the parameters of the demand distribution,
adjusting the order quantity by δmaxQ leads to a reduction in emissions that is at
34
least twice the increase in cost.
2.5 Social Welfare Analysis
In this paper, we have assumed that the parameters of the emission regulation are
exogenous. It is however possible to take the perspective of a social planner who
decides on the parameters of the regulation to maximize social welfare. We define
social welfare (SW) as the difference between the sum of producer surplus (PS),
consumer surplus (CS) and tax revenue (TR) on one hand and environmental
damage (ED) on the other, so that SW = PS + CS + TR − ED. For the sake
of brevity we consider only systems operating under a strict emission cap or an
emission tax. A similar analysis can be carried out for other policies.
In this paper, we assume that both demand and selling prices are fixed. This
assumption is consistent with a firm that is a price taker (e.g., a firm that does not
have the scale to affect the prevailing price). Consequently, consumer surplus is
fixed and remains the same regardless of the emission regulation policy. Similarly,
the revenue component of the producer (selling price multiplied by demand) is
fixed. Therefore, the surplus of the producer is determined only by its operational
cost in the case of a strict emission cap and by the sum of its operational cost and
tax payments in the case of an emission tax. Therefore, social welfare reduce to
the difference between the producer’s cost and environmental damage.
Consider first the case of a carbon tax. In this case, the social planner’s
objective is to choose a tax rate t that maximizes social welfare. Given a tax
rate t, the firm’s optimal order quantity is Q˜ =
√
2A˜D
h˜
. Letting w denote the
estimated environmental cost per unit of emission, maximizing the social welfare
35
is equivalent to
Minimize
t
AD
Q˜
+
hQ˜
2
+ w(
AˆD
Q˜
+
hˆQ˜
2
), (2.18)
Let ˜˜A = A + wAˆ, ˜˜h = h + whˆ, and ˜˜Q =
√
2 ˜˜AD
˜˜
h
. Then, it is easy to see
that t = w maximizes social welfare, with a corresponding emission level given by
E( ˜˜Q) and a producer cost given by AD˜˜Q
+ h
˜˜Q
2
+ tE( ˜˜Q).
Consider now the case of a strict emission cap. In this case, the social planner’s
objective is to choose an emission cap C that maximizes social welfare. The
producer’s optimal order quantity, denoted by Qˆ∗, is either Q∗ or Q1,2 depending
on the carbon cap C. Therefore, maximizing the social welfare is equivalent to
Minimize
C
AD
Qˆ∗
+
hQˆ∗
2
+ w(
AˆD
Qˆ∗
+
hˆQˆ∗
2
). (2.19)
It is easy to see that social welfare is maximized when Qˆ∗ = ˜˜Q, which can be
achieved by setting the optimal carbon cap such that C∗ = E( ˜˜Q). Therefore,
under the optimal strict emission cap policy, the emission level is E( ˜˜Q) and the
corresponding optimal social welfare is the same as the one achieved under the
optimal tax policy. The producer surplus is however higher since the producer
incurs the same operational cost but makes no tax payments.
2.6 Conclusion
In this paper, we provide analytical support for the notion that it may be possible,
via operational adjustments alone, to significantly reduce emissions without signif-
icantly increasing cost. Using the EOQ model, we provide a condition under which
it is possible to reduce emissions by modifying order quantities. We also provide
36
conditions under which the relative reduction in emissions is greater than the rela-
tive increase in cost and discuss factors that affect the difference in the magnitude
of emission reduction and cost increase. We then show that the treatment can be
easily extended to other operational settings. We illustrate the broader applicabil-
ity of our key results to two other important classes of operational models, namely
facility location modelsmodels, where costs and emissions are driven by a fixed
component and a distance-dependent component; and newsvendor models, where
costs and emissions are driven by inventory overages and inventory shortages. In
both cases, we show that the cost function is remarkably flat around the optimal,
providing an opportunity under some conditions to significantly reduce emission
without significantly increasing cost.
We also take the perspective of a social planner who decides on the parame-
ters of the regulation to maximize social welfare. We show that for the setting
described in this paper, a strict emission cap and an emission tax, when designed
optimally, both achieve the same level of social welfare, including the same level
of emissions. However, a strict emission cap is preferred by the producer since it
does not involve any tax payments.
Chapter 3
On the Effectiveness of Emission
Penalties in Decentralized Supply
Chains
3.1 Introduction
There is growing pressure on governments and firms worldwide to reduce the
emission of harmful pollutants, such as carbon dioxide, or the consumption of
scarce natural resources, such as water, to levels that are deemed environmentally
sustainable. Imposing penalties on emissions1 is one of the mechanisms adopted
by several countries to achieve these goals. These penalties can be direct in the
form of taxes imposed on the emissions of individual firms or facilities, or indirect
1 For the remainder of the paper, we will use the term emission exclusively. However,
our analysis extends to the production of negative externalities, broadly defined, including the
consumption of scarce resources.
37
38
because of limits (caps) imposed on the amount a firm or a facility can emit. In
the case of the latter, the penalties correspond to the marginal cost of meeting
emission caps, the price of purchasing emission offsets (offsets are reductions in
emissions in one place that can be used to compensate for emissions elsewhere),
or the prevailing market price of tradable emission permits in cap and trade-like
systems.
Imposing penalties on emissions is also a mechanism that an increasing number
of firms have adopted voluntarily in their efforts to mitigate their emissions. For
example, several firms have committed to becoming carbon-neutral through the
purchase of carbon offsets (a carbon-neutral firm is one that is credited with
reducing global emissions by an amount equal to the one it produces). In order
to pay for these offsets, some firms, such as Microsoft and Disney, are imposing
internal taxes on their business units for the carbon emissions for which these
business units are directly responsible; see DiCaprio (2013) and Davenport (2013).
Other firms are putting a price on emissions in evaluating the attractiveness of new
investments or to justify investments in conservation projects or renewable energy.
A recent report by CDP found at least 30 companies, ranging from utilities, such as
Xcel Energy, to energy companies, such as Exxon, and retailers, such as Walmart,
setting an internal price ranging from $6 to $60 per metric ton on their carbon
pollution (see CDP (2013a)). Firms that are water-intensive, such as Coca Cola,
are putting a price on water that is substantially higher than the prevailing price
(see Fellow (2013)).
Clearly, everything else remaining the same, imposing a penalty on emissions
makes business-as-usual more expensive and provides an incentive for firms and
39
business units to reduce their emissions. Thus, it is often argued that such penal-
ties should lead (through the emission reduction efforts of firms or business units)
to an economy-wide (or a firm-wide) reduction in total emissions. In this paper,
we examine the validity of this premise in the context of a decentralized supply
chain (a supply chain consisting of independent firms or of independent business
units within a single firm). Perhaps surprisingly, we find that penalizing each
firm for the emissions for which it is directly responsible can paradoxically lead
to higher overall supply chain emissions and for these emissions to increase in the
emission penalty. We show that this paradox arises because firms, in their efforts
to reduce their own emissions, can make decisions that adversely impact the costs
and emissions of their suppliers or customers.
We base our analysis on a buyer-supplier model that has been extensively
studied in the supply chain coordination literature, where the buyer and the sup-
plier make decisions regarding the size of order replenishments. We show how
the buyer can choose order sizes that reduce its own emissions but increase the
emission of the supplier, with the net effect being an increase in total supply chain
emissions and we provide necessary and sufficient conditions under which this oc-
curs. We examine how this effect could arise in other settings and discuss two
such examples. The first involves a make versus buy problem, where a manufac-
turer decides on how much to produce in-house and how much to outsource. The
second involves a retail store density problem, where a retailer decides on how
many stores to operate in a region. Then, we describe a general model of which
the various models we discuss can be viewed as special cases. We also provide a
general necessary and sufficient condition for when higher emission penalties lead
40
to higher supply chain emissions.
Remedies that can mitigate the potential negative impact of emission penalties
require that one firm internalizes either partially or fully the cost of the externality
it imposes on the other firm. We study three such remedies. The first involves
the firms coordinating their ordering decisions, either explicitly or implicitly, so
that total supply chain profit is maximized. The second is one where both the
buyer and the supplier are penalized for a fraction of total supply chain emissions
instead of only the emissions for which they are directly responsible. The third
involves penalizing consumers instead of the firms. We show that while all three
remedies guarantee that higher penalties lead to lower supply chain emissions, the
resulting social welfare can be different. More significantly, none of the schemes
guarantees that, for a given penalty, emissions would be lower than those obtained
when firms are penalized only for their own emissions.
To our knowledge, our paper is among the first to highlight the possibility
for emission penalties (and, more generally, emission reduction efforts) to backfire
because of supply chain interactions. Much of the existing literature in both oper-
ations management and environmental economics treats cases where the costs and
emissions of one firm are unaffected by the emission mitigation efforts of another
firm, The recognition that this may not be the case could have implications in how
either policy makers or firms go about designing schemes to reduce total supply
chain emissions and in how emission responsibility should be allocated across the
supply chain. Our results also highlight the importance of understanding supply
chain interactions in assessing the effectiveness of different approaches to emission
mitigation.
41
Our work is of course not the first to point out that well-intentioned actions
can lead to undesirable consequences. In the area of energy conservation, there
are several well-known paradoxes. Jevons’ paradox (or the rebound effect) refers
to improvements in energy efficiency leading to more energy consumption not less
(see for example Alcott (2005) and Jevons (1865)). For example, as air condition-
ing becomes more energy-efficient, more residential and office buildings install air
conditioning, leading to an increase in air conditioning-related energy consumption
and not less. The indirect rebound effect refers to savings from greater efficiency in
one area leading to increased consumption in other more emission-intensive areas
(see for example Greening et al. (2000) and Sorell (2009)). For example, improved
home insulation leads to less household spending on heating, which in turn leads
to more spending on other goods and services that may be more energy intensive
(e.g., air travel). The global rebound effect refers to reduction in energy consump-
tion in one country leading to lower global prices triggering more consumption in
other countries (see for example Sorrell and Dimitropoulos (2008)). The leakage
effect refers to pollution regulation in one country leading to production moving
to another country where production is more emission-intensive and regulation is
more lax (see for example Babiker (2005)). The effect we describe in this paper
appears to be distinct from those described above and caused by the fact that
costs and emissions of one firm can be affected by the actions of another.
The rest of the paper is organized as follows. In Section 3.2, we provide a brief
review of related literature. In Section 3.3, we describe the buyer-supplier model
and specify conditions under which higher emission penalties lead to higher supply
chain emissions. In Section 3.4, we discuss potential remedies. In Section 3.5, we
42
extend our analysis to the case where demand is endogenous. In Section 3.6, we
discuss other examples and a general model. In Section 3.7, we offer concluding
comments.
3.2 Related Literature
The literature on supply chain management is extensive, including literature that
deals explicitly with the misalignment of incentives among firms within the same
supply chain; see for example Cachon and Lariviere (2005) and Cachon (1998)
for reviews. This literature is primarily focused on identifying contracting mech-
anisms that coordinate the supply chain and ensure that supply chain profit is
maximized. (In this paper, we show that coordination does not necessarily re-
sult in less emission than no-coordination; with coordination, firms may choose
to emit more if it leads to higher profits.) There is also growing literature that
is concerned with issues of sustainability in supply chains; see for example Drake
and Spinler (2013), Dekker and Bloemhof (2012), and Tang and Zhou (2012) for
recent reviews. Within this literature, papers that explicitly consider emissions
are relatively few. We review some of the most relevant ones below. More ex-
tensive reviews can be found in Plambeck (2012), Benjaafar et al. (2013), and
Cachon (2014).
Caro et al. (2013) consider a setting where multiple firms contribute to total
supply chain emissions. They address the question of how to allocate emission
responsibility among the firms and show that over-allocating emissions, or double-
counting, is necessary for firms to exert their first-best emission reduction efforts.
An important difference between our work and that of Caro et al. is that in their
43
case the emission reduction, or abatement, costs of one firm are not affected by
the emission reduction effort of another. In our case, this interaction is central to
the results we describe (see also the discussion in Section 3.6.3).
Granot et al. (2014) study emission allocation in a setting where firms are
accountable for their own emissions as well as the emissions from all upstream
suppliers. Using a cooperative game theory approach, they show that an emission
responsibility sharing scheme where the direct emission of one firm is allocated
equally among all its downstream firms is in the core and corresponds to the Shap-
ley value. Their analysis relies on the assumption that, absent emission sharing,
firms are held accountable for the emissions of all their upstream suppliers. They
also assume that emissions are exogenously specified and not affected by how
emission responsibility is allocated and by the corresponding actions of individ-
ual firms. In this paper, we consider settings where firms modify their actions in
response to allocations and these actions can affect the costs and emissions of all
the firms in the supply chain.
Cachon (2014) considers the problem of a retailer deciding on the number and
size of stores taking into account the costs and emissions incurred by both the
retailer and the consumers. He shows that an emission tax would have to be
relatively high to affect the design of the retail network. In contrast to our setting
of a decentralized supply chain, Cachon considers a setting where the retailer acts
as a central planner and fully compensates consumers for their travel and emission
costs. In Section 3.6.2, we revisit the model in Cachon (2014) and show that, if the
retailer and the consumers act independently, then the retailer’s effort to reduce
its own emissions can lead to higher consumer emissions, which in turn can lead
44
to higher overall emissions.
Sunar and Plambeck (2014) consider the impact of a cross border adjustment
tax (a tax on the emissions caused by a product’s creation and transportation
to the border) incurred by an importer. They show that, in the presence of co-
products and for a specific type of allocation of emission among the products, it
is possible for a higher tax to lead to higher emissions. This is due to the fact
that higher taxes can lead the supplier of the imported good to lower its selling
price resulting in more demand and therefore more emissions.
Benjaafar et al. (2013) show how carbon emission parameters could be incor-
porated in lot sizing models. They investigate numerically the impact of imposing
an emission cap on supply chain cost and observe that there are cases where emis-
sions can be significantly reduced without significantly increasing cost. Chen et
al. (2013) validate these numerical observations for a setting with a single firm
using the EOQ model. Subramanian et al. (2007) study abatement investment in
a deterministic oligopoly where the price of emissions is endogenous. Vela´zquez-
Mart´ınez et al. (2014) compare different approaches to carbon footprint calcu-
lations in supply chains and show that carbon emission aggregation can lead to
poor estimation of actual footprint. Hoen et al. (2014) consider carbon emissions
in transportation and find that accounting for emission cost would have only a
limited effect on the relative desirability of different transportation modes.
Drake (2012) studies the effect of carbon trading on carbon leakage and do-
mestic and foreign firms’ comparative advantage. Drake et al. (2012) consider
capacity investment and production decisions under both a cap-and-trade system
(where carbon price is stochastic) and a carbon tax (where carbon price is fixed)
45
and find that expected profits are greater, and expected emissions are lower under
cap-and-trade. Krass et al. (2013) find that the choice of green technology in
response to an increase in emission taxes may be non-monotone. Gong and Zhou
(2013) analyze the joint production control and emission trading problem in the
presence of alternative production technologies.
There is also significant research that draws on the methodology of life cycle
analysis (LCA) to document carbon emissions that can be attributed to a prod-
uct at various stages of its production, distribution, consumption, and end-of-life
disposal; see for example Reap et al. (2008) and Matthews et al. (2008). An
important stream in this literature uses economic input-output (EIO) modeling
to account for carbon emissions at a sector level in the economy and to study
the impact of one sector on the carbon emissions of other sectors; see Wiedmann
(2009).
Finally, there is extensive literature from environmental economics that ex-
amines the optimal design of environmental policies from the perspective of a
social planner (see for example Baumol (1972), Barnett (1980), and Katsoulacos
and Xepapadeas (1995)). Results from this literature show that taxing emis-
sions always increases abatement effort and reduces emissions. They also show
that a Pigouvian tax (taxing emissions using the true social cost of emissions) is
socially-optimal under perfect competition but may not be otherwise (e.g., im-
posing a Pigouvian tax on a monopolist is not socially optimal and could hurt
consumption by increasing price and shrinking demand). This literature does not
account for the supply chain dynamics we model in this paper. In particular, it
46
assumes that the abatement effort of one firm does not affect the cost of abate-
ment of other firms. As we discuss in Section 5, when supply chain interactions
are accounted for, the socially-optimal tax can be either lower or higher than a
Pigouvian tax.
3.3 A Buyer-Supplier Model
We consider a buyer-supplier model that is perhaps the most widely studied model
of supply chain interactions, with literature that spans more than three decades;
see for example Jeuland and Shugan (1983), Lal and Staelin (1984), Weng (1995),
Corbett and De Groote (2000), Cachon (2003), Cachon and Kok (2010) and the
references therein. The model is popular because of its ability to capture impor-
tant supply chain dynamics, the robustness of the results it yields despite of its
succinctness, and because of its mathematical tractability. Specifically, we con-
sider a serial supply chain consisting of a buyer (Firm 1) and a supplier (Firm
2). The buyer faces external demand, constant with a fixed rate that may depend
on the selling price, and decides on the frequency with which to order from the
supplier (or equivalently on the order quantities). In doing so, the buyer trades
off fixed ordering costs with variable inventory holding costs. The supplier, in
response to the orders from the buyer, decides similarly on the frequency of or-
dering from its own supplier (or on the frequency of producing internally) trading
off fixed ordering (or production setup) costs with inventory holding costs.
Let D denote demand per unit time and p denote the unit selling price (we
treat demand as being exogenous for the time being and relax it later in Section
5 by making price a decision variable). The buyer incurs a fixed ordering cost
47
A1 per order, a holding cost h1 per unit kept in inventory per unit time, and
a variable cost c1 per unit purchased from the supplier in addition to the unit
price w the supplier charges. The supplier incurs similar costs with corresponding
parameters A2, h2, and c2, respectively. Without loss of generality, we assume that
orders are delivered from the supplier to the buyer with zero leadtime (a positive
leadtime can be included and does not affect the solution to the problem); we
also assume that each firm must always keep enough inventory to satisfy all the
demand from its customers (the analysis can be easily extended to settings with
backorders). It is possible to consider more general cost functions, including when
cost parameters are non-linear or exhibit discontinuities. However, the qualitative
insights we derive remain the same and we keep, consistent with much of the
existing literature, the simpler form for the sake of tractability.
In addition to costs, there may be emissions associated with ordering, process-
ing, and inventory holding. We let Aˆi denote the amount of emissions associated
with each order placed by firm i (e.g., due to transportation or process setup), hˆi
the amount of emissions due to a unit held in inventory per unit time (e.g., due to
heating or refrigeration of stored inventory), and cˆi the amount of emissions per
unit purchased or produced (e.g., due to energy consumed during handling and
processing). We only require that these parameters be non-negative so that it is
possible for some to be zero (for example, a firm may have negligible emissions
associated with initiating orders, holding inventory, or with order processing).
Similar to cost, it is possible to consider more general emission functions and
the results we describe continue to hold when the emission parameters exhibit
non-linearities (see for example, Section 3.6.1).
48
Similar to costs, characterizing emission parameters in practice requires a data
collection and documentation effort. Fortunately, emission data is increasingly
available on standard processes, including transportation and storage; see for ex-
ample Cachon (2014) and Chen et al. (2013). There are also increasingly ac-
cepted standards for measuring and reporting different categories of emissions, or
so-called emission scopes2 .
We consider a setting where each firm incurs a penalty at rate t per unit of
emissions it emits (scope 1 and 2 emissions). As discussed in the introduction,
the penalty can be self imposed or externally mandated. In the case where the
penalty is self imposed, the rate t may correspond to the prevailing market price
of offsets. If it is externally mandated, it may correspond to a tax imposed by
a regulating entity. It is possible to consider settings where the penalty rate is
non-linear and varies with the amount of emissions. It is also possible to consider
settings where only one of the firms incurs an emission penalty (e.g., the case
where one firm chooses to offset its emissions but not the other) or where the
emission penalties are different for different firms. In what follows, we treat the
case where both firms are subject to the same penalty (e.g., both firms seek to be
emission-neutral by purchasing offsets from the same offset market). However, the
effects we describe are also present in the more general settings mentioned above
(see Section 6.2 for an example where only one supply chain party is subject to
an emission penalty).
2 A prominent example of such a standard is the Green House Gas (GHG) Protocol developed
by the World Resource Institute and the World Business Council for Sustainable Development.
According to the GHG protocol, emissions are classified into scope 1, scope 2, and scope 3
emissions, with scope 1 referring to direct emissions by the firm at its installations, scope 2
to indirect emissions by the firm via electricity usage, and scope 3 to emissions by the firms’
upstream and downstream supply chains (GHG Protocol (2011)).
49
The objective of each firm is to maximize its own after penalty profit by
choosing an ordering quantity, qi for firm i = 1, 2. Since demand and price are
fixed, revenue is unaffected by the ordering decisions (we consider the case where
price is a decision variable that affects demand in Section 3.5). First, the buyer
chooses its order quantity; then does the supplier, taking into account the ordering
decision of the buyer. It is not difficult to show that in such a setting the optimal
ordering policy is a nested policy with the supplier ordering an integer multiple
m2 of the buyer’s order quantity q1 (see for example Chapter 2 of Zipkin (2000)).
Note that the buyer is unaffected by the decisions of the supplier but the reverse
is not true.
The problem faced by the buyer can be formulated as follows:
max
q1
z1(q1) = (p− w − c1)D − A1D
q1
− h1q1
2
− t(Aˆ1D
q1
+
hˆ1q1
2
+ cˆ1D), (3.1)
while the problem faced by the supplier, given that the buyer orders quantity q1,
can be formulated as
max
m2
z2(m2, q1) = (w − c2)D − A2D
m2q1
− h2(m2 − 1)q1
2
(3.2)
−t( Aˆ2D
m2q1
+
hˆ2(m2 − 1)q1
2
+ cˆ2D),
where the second argument of the function z2 is used to highlight the dependency
of the profit function of the supplier on the decision of the buyer.
We let q∗1 denote the buyer’s optimal order quantity and m
∗
2 the supplier’s
optimal order quantity multiplier. Also, let A˜i = Ai + tAˆi and h˜i = hi + thˆi, for
i = 1, 2. Then, we can show that
q∗1 =
√
2A˜1D
h˜1
, (3.3)
50
and m∗2 is an integer that satisfies the inequalities
m∗2(m
∗
2 − 1) ≤
A˜2h˜1
A˜1h˜2
≤ m∗2(m∗2 + 1). (3.4)
Given order quantity q∗1 and order multiplier m
∗
2, the buyer’s and the supplier’s
emissions are respectively given by
e1(q
∗
1) =
Aˆ1D
q∗1
+
hˆ1q
∗
1
2
+ cˆ1D, (3.5)
and
e2(m
∗
2, q
∗
1) =
Aˆ2D
m∗2q
∗
1
+
hˆ2(m
∗
2 − 1)q∗1
2
+ cˆ2D. (3.6)
Total supply chain profit and total supply chain emission can then be obtained
respectively as
zSC(m
∗
2, q
∗
1) = z1(q
∗
1) + z2(m
∗
2, q
∗
1),
and
eSC(m
∗
2, q
∗
1) = e1(q
∗
1) + e2(m
∗
2, q
∗
1).
Let us first consider the case where the supplier uses a lot-for-lot policy (i.e.,
m2 = 1). By virtue of inequalities (3.4), the lot-for-lot policy is optimal when
0 ≤ A˜2h˜1
A˜1h˜2
≤ 2. The lot-for-lot policy is commonly used even when it is not optimal
because of its simplicity and because it is consistent with the principle of just-in-
time adopted by many firms; see for example Corbett and De Groote (2000). The
results below apply regardless of whether the lot-for-lot policy is optimal or not.
Under a lot-for-lot policy, the expression of supplier’s emissions simplifies to
e2(q
∗
1) = Aˆ2
√
h˜1D
2A˜1
+ cˆ2D, (3.7)
51
and that of total supply chain emissions to
eSC(q
∗
1) = (Aˆ1 + Aˆ2)
√
h˜1D
2A˜1
+ hˆ1
√
A˜1D
2h˜1
+ (cˆ1 + cˆ2)D, (3.8)
where, for ease of notation, we have dropped the argument m2(= 1) from the
notation of the emission functions.
In the following theorem, we characterize the effect of the emission penalty on
total supply chain emission.
Theorem 3 Total supply chain emission eSC(q
∗
1) is increasing in the emission
penalty t if and only if Aˆ1
hˆ1
≤ A1
h1
and Aˆ2 ≥ hˆ1A1−h1Aˆ1h˜1 .
The result follows immediately from noting that
deSC(q
∗
1)
dt
≥ 0 if and only if the
condition specified in the theorem holds. The following are immediate corollaries.
Corollary 2 A sufficient condition, independent of t, for total supply chain emis-
sion to increase in the emission penalty is given by Aˆ1
hˆ1
≤ A1
h1
and Aˆ2 ≥ hˆ1A1−h1Aˆ1h1 .
Corollary 3 If Aˆ1
hˆ1
≤ A1
h1
but Aˆ2 <
hˆ1A1−h1Aˆ1
h˜1
, then there exists a threshold t′
on the emission penalty such that if t ≤ t′, then total supply chain emission is
decreasing in t and if t > t′, then supply chain emission is increasing in t, with
t′ = hˆ1A1−h1Aˆ2−h1Aˆ1
hˆ1Aˆ2
.
The fact that higher emission penalties can lead to higher supply chain emis-
sion can be explained as follows. When Aˆ1
hˆ1
≤ A1
h1
, the buyer’s optimal order
quantity q∗1 decreases with the emission penalty (the introduction of a penalty
moves the buyer from the cost-optimal order quantity, given by
√
2A1D/h1, in
the direction of the emission-optimal order quantity, given by
√
2Aˆ1D/hˆ1). In
52
doing so, the buyer of course reduces its own emissions. However, this comes
at the expense of the supplier whose emissions increase. This is because, un-
der lot-for-lot, the supplier uses the same order quantity as the buyer. Doing so
leads the supplier to incur higher fixed emissions (associated with more frequent
ordering) without benefiting from lower inventory-related emissions, as does the
buyer (recall that, under lot-for-lot, the supplier does not keep any inventory). If
the emission increase by the the supplier is higher than the emission reduction
experienced by the buyer, then total supply chain emissions see a net increase. A
condition for this to occur is for the supplier’s emission parameter Aˆ2 to be larger
than a threshold as specified by the condition in the theorem. Note that because
this threshold is decreasing in t, it is possible for total emissions to decrease as t
increases initially and then to increase as t increases further.
In figures 3.1(a) and 3.1(b), we illustrate the above results. Figure 3.1(a)
shows that the increase in emissions can be significant (as large as 10% in the
example shown). Figure 3.1(b), illustrates the non-monotonicity of emissions in
the emission penalty t.
Higher emission penalties can also lead to higher total supply chain emission
when the supplier does not use a lot-for-lot policy and instead uses a multiplier,
other than one, of the buyer’s order quantity (regardless of whether or not this
multiplier is optimized). First, consider the case where the supplier uses an ex-
ogenously specified multiplier m2 (m2 ≥ 1). Let A′2 = A2m2 , h′2 = (m2 − 1)h2,
Aˆ′2 = Aˆ2m2 and hˆ
′
2 = (m2 − 1)hˆ2. Then the total supply chain cost and emission
can be rewritten, respectively, as follows:
zSC(q
∗
1,m2) =
(A1 + A
′
2)D
q∗1
+
(h1 + h
′
2)q
∗
1
2
+ (c1 + c2)D, (3.9)
53
(a) (A2 = 10, h2 = 1, Aˆ2 = 5, hˆ2 = 1, A1 = 6, h1 = 1.1, Aˆ1 = 3, hˆ1 = 1.1, cˆ1 =
cˆ2 = 0, D = 100)
(b) (A2 = 15, h2 = 1.4, Aˆ2 = 10, hˆ2 = 1.1, A1 = 21, h1 = 1, Aˆ1 = 5, hˆ1 =
1, cˆ1 = cˆ2 = 0, D = 100)
Figure 3.1: The impact of the emission penalty on supply chain emission
54
and
eSC(q
∗
1,m2) =
(Aˆ1 + Aˆ
′
2)D
q∗1
+
(hˆ1 + hˆ
′
2)q
∗
1
2
+ (cˆ1 + cˆ2)D. (3.10)
Note that when m2 > 1, the supplier holds inventory and therefore incurs inven-
tory holding costs and emissions.
The following theorem provides necessary and sufficient conditions for the total
emission to increase with the emission penalty.
Theorem 4 Given multiplier m2, total supply chain emission increases with the
emission penalty t if and only if one of the following conditions holds:
• Aˆ1
hˆ1
≤ A1
h1
and Aˆ′2 ≥ A˜1(hˆ1+hˆ
′
2)
h˜1
− Aˆ1, or
• Aˆ1
hˆ1
≥ A1
h1
and Aˆ′2 ≤ A˜1(hˆ1+hˆ
′
2)
h˜1
− Aˆ1.
The result again follows from verifying that
deSC(q
∗
1 ,m2)
dt
≥ 0 if and only if the condi-
tions specified in the theorem hold. The result generalizes that of Theorem 1 and
can be explained as follows. When Aˆ1
hˆ1
≤ A1
h1
, the buyer’s optimal order quantity
q∗1 decreases with the emission penalty, per the explanation given for the result
in Theorem 3. This has again the effect of reducing the buyer’s emission but in-
creasing the supplier’s fixed emission. When Aˆ′2 ≥ A˜1(hˆ1+hˆ
′
2)
h˜1
− Aˆ1, the increase in
the supplier’s fixed emissions is sufficiently large to lead to a net increase in total
supply chain emission. When Aˆ1
hˆ1
≥ A1
h1
, the buyer reduces its emission by increas-
ing its optimal order quantity q∗1. However, this has the effect of increasing the
inventory-related emission of the supplier. if Aˆ′2 ≤ A˜1(hˆ1+hˆ
′
2)
h˜1
− Aˆ1, the increase in
the supplier’s inventory-related emission is sufficiently large to lead to an increase
in total supply chain emission. Note that when m2 = 1 (the lot-for-lot case), only
55
the first condition of Theorem 4 can hold. In that case, substituting for m2 = 1
leads to the result in Theorem 3.
When m2 is not fixed and is instead chosen optimally by the supplier, the
optimal value m∗2 varies with the penalty rate t. However, for any value of m
∗
2,
there is always a non-empty interval for t, say [t1(m
∗
2), t2(m
∗
2)], over which m
∗
2 is
constant (recall that m2 is an integer). For each such interval whose internal is not
empty, and corresponding fixed m∗2, it is possible for total supply chain emission
to increase if one of the conditions in Theorem 4 are satisfied. In Figure 3.2, we
illustrate this effect for an example system. Note that the discontinuities in total
supply chain emission are due to the integrality of m2.
Figure 3.2: The impact of the emission penalty on supply chain emission when
m∗2 > 1 (A2 = 10, h2 = 1, Aˆ2 = 25, hˆ2 = 1, A1 = 5, h1 = 1.5, Aˆ1 = 3, hˆ1 = 2, cˆ1 =
cˆ2 = 0, D = 100)
We conclude this section by making the following two remarks. First, our
56
analysis can be readily extended to serial supply chains with multiple firms, where
each firm, other than the end buyer, is a supplier to the one directly downstream
from it. We can show that, if the the conditions in Theorem 4 hold, the potential
negative impact of higher emission penalties continues to be present and in fact
can be amplified. For example, in the case where all suppliers are symmetric in
their parameters and all employ a lot-for-lot policy, total supply chain emission
increases linearly in the number of firms if the conditions of Theorem 3 hold.
Second, although we have considered settings where firms are subject to the
same emission penalty, the potential negative impact of higher emission penalties
is also present when the penalties faced by the firms are different or when only the
buyer faces a penalty (e.g., this is easy to verify in the case of lot-for-lot policy).
More significantly, this effect is present when neither firm is subject to an emission
penalty but instead the buyer decides to voluntarily reduce its own emission down
to a specified target (i.e., the buyer maximizes its profit subject to a constraint
on emissions).
In Section 3.6, we discuss other examples of applications where a penalty
on the emissions (or the emission reduction effort) of one or more of the firms
can backfire and lead to higher overall emissions. We also a describe a general
model for supply chain interactions, of which the various applications we discuss,
including the model in this section, can be viewed as special cases.
3.4 Potential Remedies
In this section, we consider remedies that can mitigate the possibility of emission
penalties backfiring and resulting in higher supply chain emission and not less.
57
A remedy would naturally require that the buyer internalizes either partially or
fully the cost of the externality it imposes on the supplier. We consider three
such remedies. The first involves the firms coordinating their ordering decisions,
either explicitly or implicitly, so that total (after-penalty) supply chain profit
is maximized (i.e., the buyer and the supplier adopt the centralized solution).
The second is one where both the buyer and the supplier are penalized for a
fraction of total supply chain emission instead of only the emissions for which
they are directly responsible. The third involves penalizing consumption instead
of production. This remedy applies to settings where demand is sensitive to
pricing and we postpone its discussion until Section 3.5 where we consider the
case of endogenous demand.
3.4.1 Supply Chain Coordination
It may be possible for the buyer and the supplier to coordinate their ordering
decisions so that total supply chain profit is maximized. This could take place
voluntarily, with the buyer and the supplier agreeing to collaborate and share
the resulting additional profit. It could also be imposed by headquarters if the
buyer and the supplier are business units within the same firm. Coordination
could also occur if either the buyer or the supplier has enough market power or if
either is in the position to enforce a coordinating contract (see for example Weng
(1995), Corbett and De Groote (2000), Sarmah et al. (2006)) for a discussion of
coordinating contracts. When the buyer and the supplier coordinate their ordering
decisions so as to maximize total supply chain profit, the problem for the central
58
planner can be formulated as follows:
max
m2,q1
zSC(m2, q1).
The resulting optimal order quantity for the buyer is then given by
qC1 =
√√√√ 2( A˜2mC2 + A˜1)D
h˜1 + (mC2 − 1)h˜2
, (3.11)
while the optimal order quantity multiplier for the supplier is an integer that
satisfies the inequalities
mC2 (m
C
2 − 1) ≤
A˜2(h˜1 − h˜2)
A˜1h˜2
≤ mC2 (mC2 + 1). (3.12)
In the case when a lot-for-lot policy is used (optimal when 0 ≤ A˜2(h˜1−h˜2)
A˜1h˜2
≤ 2), the
order quantity simplifies to
qC1 =
√
2(A˜1 + A˜2)D
h˜1
, (3.13)
and total supply chain emission is given by
eSC(q
C
1 ) = (Aˆ1 + Aˆ2)
√
h˜1D
2(A˜1 + A˜2)
+ hˆ1
√
(A˜1 + A˜2)D
2h˜1
+ (cˆ1 + cˆ2)D. (3.14)
It is easy to see that the order quantity for the buyer is always larger than the order
quantity in the decentralized system (the optimal order quantity now accounts for
the fixed ordering cost at both the buyer and the supplier). It is obvious that with
coordination a higher emission penalty always leads to lower total supply chain
emission and it can be directly verified that
deSC(q
C
1 )
dt
< 0. Hence, if feasible, supply
chain coordination would always ensure that imposing a penalty on emissions (or
increasing the penalty) always has the intended consequence of reducing emissions.
59
However, we must caution that coordination may not always be feasible or
practical. For example, decentralized supply chains are prevalent in practice and
so are contracts that do not guarantee coordination such as those involving whole-
sale prices. More importantly, we must caution that coordination may not nec-
essarily lead to emissions that are lower than those experienced in the absence of
coordination. In fact, as we show below, given the same emission penalty, it is
possible for a coordinated supply chain to have higher emissions than a decentral-
ized one. To see this, consider the case of a lot-for-lot policy (the result readily
extends to the more general case). Then the following holds.
Proposition 6 Given a penalty t, a coordinated supply chain has strictly higher
emissions than a non-coordinated supply chain if and only if hˆ1
h˜1
> Aˆ1+Aˆ2√
(A˜2+A˜2)A˜2
.
The proof follows immediately from comparing total supply chain emission when
there is coordination (eSC(q
C
1 ))with supply chain emission in the absence of coor-
dination (eSC(q
∗
1)).
The result in Proposition 6 can be explained as follows. With coordination,
the order quantity increases (relative to the order quantity without coordination).
This leads to lower fixed emissions for both the buyer and the supplier but higher
inventory-related emissions for the buyer (the supplier does not carry any in-
ventory). If the decrease in the fixed emissions is less than the increase in the
inventory-related emissions, total supply chain emission would see a net increase.
This can be most readily seen in the special case of t = 0, where the condition in
Proposition 6 can be rewritten as hˆ1 >
h1(Aˆ1+Aˆ2)√
(A1+A2)A2
. In other words, if the buyer’s
inventory emission intensity is sufficiently high, then total supply chain emission
in the coordinated supply chain would be higher. In all cases, coordination of
60
course leads to higher supply chain profit. However, with coordination, firms may
choose to reduce operational cost at the expense of increasing emission cost. Fig-
ure 3.3 illustrates for an example system with and without coordination and, as
we can see, a coordinated supply chain could have a significantly higher emission.
Figure 3.3: Supply chain emission with and without coordination (A2 = 15, h2 =
1.4, Aˆ2 = 10, hˆ2 = 1.1, A1 = 21, h1 = 1, Aˆ1 = 5, hˆ1 = 1, cˆ1 = cˆ2 = 0, D = 100)
3.4.2 Shared Emission Responsibility
An alternative to penalizing each firm for its own emission is to penalize each
firm for a fraction of total supply chain emission. We refer to such a scheme as
shared emission responsibility. Shared responsibility clearly ensures that emission
externalities inflicted by one firm on another are internalized. Shared emission
responsibility could be imposed if the firms correspond to business units within
61
the same firm or if the firms voluntarily agree to share the emission burden. The
latter might arise if firms are held accountable for the emissions of their entire
supply chain. For example, a growing number of publicly-traded firms are being
pressured by their shareholders to disclose not only their own emissions but the
emissions of their suppliers and customers; see CDP (2013b). It might also arise
if firms see an opportunity in positioning their products as having an overall low
emission footprint.
Under shared emission responsibility, firm i, for i = 1, 2, is penalized for a
fraction αi ≥ 0 of total supply chain emission, with
∑2
i=1 αi = 1. Then, given
penalty t and emission allocation (α1 = α, α2 = 1 − α), the buyer’s problem is
given by (for simplicity, we assume the supplier follows a lot-for-lot policy, but
the analysis can again be extended to the general case)
max
q1
z1(q1) = (p− w)D − (A1D
q1
+
h1q1
2
+ c1D) (3.15)
−αt((Aˆ1 + Aˆ2)D
q1
+
hˆ1q1
2
+ (cˆ1 + cˆ2)D),
with a corresponding optimal order quantity given by
q∗1(α) =
√
2[A1 + αt(Aˆ1 + Aˆ2)]D
h1 + αthˆ1
. (3.16)
Given q∗1(α), the supplier’s profit is given by
z2(q
∗
1(α)) = wD − (
A2D
q∗1(α)
+ c2D)− (1− α)t((Aˆ1 + Aˆ2)D
q∗1(α)
(3.17)
+
hˆ1q
∗
1(α)
2
+ (cˆ1 + cˆ2)D).
From the above, it is easy to verify that supply chain emission is decreasing in t.
62
Although any allocation that satisfies
∑2
i=1 αi = 1 and αi ≥ 0 accounts for all
the emission and guarantees that penalties lead to lower total supply chain emis-
sion, different allocations lead to different costs and to different levels of emissions
for both the individual firms and the supply chain. In the remainder of this sec-
tion, we examine the impact of three plausible allocation mechanisms. The first
allocates emission responsibility so that total supply chain profit is maximized;
the second so that total emission is minimized; and the third so that emission re-
sponsibility is proportional to the base-line emissions in the absence of penalties.
An allocation that maximizes supply chain profit is obtained by choosing α
such that zSC(q
∗
1(α)) is maximized. The implementation of this allocation could
be the outcome of a negotiation between the buyer and the supplier, which may
include additional financial transfers between the two parties, or imposed by a
third a party, such as headquarters in the case where the buyer and the supplier
are business units within the same firm. The optimal emission allocation problem
can be stated as follows:
max
α
zSC(q
∗
1(α)) = (p− w)D − (
(A1 + A2)D
q∗1(α)
+
h1q
∗
1(α)
2
(3.18)
+(c1 + c2)D)− t((Aˆ1 + Aˆ2)D
q∗1(α)
+
hˆ1q
∗
1(α)
2
+ (cˆ1 + cˆ2)D),
subject to
2∑
i=1
αi = 1 and αi ≥ 0.
If we ignore the constraints on α, we can show that the optimal allocation is
given by
α∗ =
A2
hˆ1
+ t( Aˆ1+Aˆ2
hˆ1
− A1
h1
)
t( Aˆ1+Aˆ2
hˆ1
− A1+A2
h1
)
,
63
and the corresponding optimal order quantity is
q∗1(α
∗) =
√
2[A1 + A2 + t(Aˆ1 + Aˆ2)]D
h1 + thˆ1
.
This order quantity is the same as the optimal order quantity in the coordinated
system since optimizing with respect to α is equivalent to optimizing with respect
to q1. This means that, whenever it is feasible, shared emission responsibility
with an unconstrained allocation can coordinate the supply chain. However, the
unconstrained optimal allocation may not be feasible if either α∗ > 1 or α∗ < 0.
The former occurs if Aˆ1+Aˆ2
hˆ1
≥ A1+A2
h1
and the latter occurs if A1
h1
− A2
thˆ1
< Aˆ1+Aˆ2
hˆ1
<
A1+A2
h1
. Given the unimodularity of the function zSC with respect to α, it is optimal
to set α∗ to 1 in the case of the former and 0 in the case of the latter. The solution
to the constrained problem in (3.18) is summarized below.
Proposition 7
α∗ =

0 if A1
h1
− A2
thˆ1
< Aˆ1+Aˆ2
hˆ1
< A1+A2
h1
,
1 if Aˆ1+Aˆ2
hˆ1
> A1+A2
h1
, and
A2
hˆ1
+t(
Aˆ1+Aˆ2
hˆ1
−A1
h1
)
t(
Aˆ1+Aˆ2
hˆ1
−A1+A2
h1
)
otherwise,
(3.19)
and
q∗1(α
∗) =

√
2A1D
h1
if A1
h1
− A2
thˆ1
< Aˆ1+Aˆ2
hˆ1
< A1+A2
h1
,√
2[A1+t(Aˆ1+Aˆ2)]D
h1+thˆ1
if Aˆ1+Aˆ2
hˆ1
> A1+A2
h1
, and√
2[A1+A2+t(Aˆ1+Aˆ2)]D
h1+thˆ1
otherwise.
The second allocation we consider is one that, instead of maximizing supply
chain profit, minimizes total supply chain emission. In this case, it is straight-
forward to show that it is optimal to set α = 1 and the optimal order quantity
64
to
√
2[A1+t(Aˆ1+Aˆ2)]D
h1+thˆ1
. That is, in order to ensure that total emission in the sup-
ply chain is minimized, we must allocate the entire emission responsibility to the
buyer.
Both the profit-optimal and emission-optimal allocations require that the firms
share their cost and emission parameters with each other or with a neutral third
party verifier. Also, both may be perceived as unfair, given that one firm could be
assigned most or all of the emission responsibility. To address these concerns, the
third allocation we consider is one that assigns emission responsibility proportion-
ally to the business-as-usual emissions of each firm. That is, a firm is responsible
for a fraction of total supply chain emission equal to the fraction of total emission
for which it would have been responsible absent the emission penalty. We let
αF denote the emission fraction assigned to the buyer. Then, αF is specified as
follows:
αF =
Aˆ1D
q∗1,BAU
+
hˆ1q∗1,BAU
2
+ cˆ1D
(Aˆ1+Aˆ2)D
q∗1,BAU
+
hˆ1q∗1,BAU
2
+ (cˆ1 + cˆ2)D
,
where q∗1,BAU =
√
2A1D
h1
and corresponds to the order quantity chosen by the
buyer under business-as-usual (BAU). This allocation is arguably fairer than the
other two and requires no information sharing (beyond disclosing business-as-usual
emissions).
In Figures 3.4(a) and 3.4(b), we illustrate for an example system the perfor-
mance of the three different allocations, along with the performance of the system
where no sharing takes place and each firm takes full responsibility for its own
emission (i.e, the system without coordination). As we can see, the different
65
(a) D = 100, p = 4.5, A1 = 1, h1 = 1, A2 = 6, c1 = 0, c2 = 0, Aˆ1 = 0.1, hˆ1 =
3, Aˆ2 = 0.2, cˆ1 = 0, cˆ2 = 0.1
(b) D = 100, A1 = 1, h1 = 1, A2 = 4, c1 = 0, c2 = 0, Aˆ1 = 0.01, hˆ1 = 1, Aˆ2 =
0.3, cˆ1 = 0, cˆ2 = 0.1
Figure 3.4: Supply chain profit and emission under different penalty schemes
66
approaches can lead to different profit and emission profiles. Although the profit-
optimal allocation achieves the highest profit, it can lead to significantly higher
emissions (up to 6.1% higher than the fair allocation and 7.3% higher than the
emission-optimal allocation in the example shown). Similarly, the fair allocation
and the emission-optimal allocation can lead to significantly lower profit than the
profit-optimal allocation (up to 20.5% for the fair allocation and up to 26.3%
for the emission-optimal allocation). More significantly, all three shared emission
responsibility schemes can lead to higher emissions than those incurred in the
absence of emission responsibility sharing.
3.5 Systems with Endogenous Demand
In this section, we extend our analysis to the case where the price of the end
product is a decision made by the buyer, with demand being sensitive to price (a
higher price implies lower demand) and show that it is still possible for emission
penalties to backfire. To allow for closed form results, we consider the case where
demand follows the widely studied constant elasticity demand function D(p) =
dp−2, where d > 0, but the analysis can be carried out for more general demand
functions. Also, for ease of exposition, we assume that the supplier follows a
lot-for-lot policy.
Assuming that, for a given price p, the buyer chooses order quantity q1(p) =√
2A˜1D(p)
h˜1
, the buyer’s problem can be stated as follows (see Weng (1995) for a
similar model and analysis):
max
p
z1(p) = (p− w − c˜1)D(p)−
√
2A˜1h˜1D(p). (3.20)
67
Given the buyer’s optimal price p∗, the supplier’s profit is given by
z2(p
∗) = (w − c˜2)D(p∗)− A˜2
√
h˜1D(p∗)
2A˜1
.
Substituting dp−2 for D(p), it is easy to verify that the profit function of the buyer
is unimodal in price, with the optimal price given by
p∗ =
2(w + c˜1)
1−
√
2A˜1h˜1/d
, (3.21)
and corresponding optimal demand D(p∗) = dp∗−2.
From (3.21) we can verify that p∗ is increasing in t, resulting in demand that is
decreasing in t. Consequently, emissions that scale up with demand also decrease
with t. We refer to this effect as the demand curtailment effect. However, the
emission reduction due to demand curtailment may not be sufficient to result in a
net reduction in total supply chain emission. As we show in the theorem below, it
may still be possible for higher emission penalties to result in higher supply chain
emissions.
Theorem 5 Total supply chain emission increases with the emission penalty t if
and only the following condition holds:
A1
h1
≥ Aˆ1
hˆ1
, Aˆ2 ≥ AˆL2 , and −
dD(p∗)
dt
D(p∗)
≤ A1hˆ1 − Aˆ1h1
A˜1h˜1
.
where AˆL2 =
A1hˆ1−Aˆ1h1
h˜1
− (
hˆ1
2D(p∗) q1(p
∗)+cˆ1+cˆ2)dD(p
∗)
dt
d
dt
(
D(p∗)
q1(p
∗) )
.
To understand the conditions in Theorem 5, note that the first two are similar
to those of Theorem 3 for the exogenous demand case. In particular, the second
condition is a requirement that the fixed emission parameter Aˆ2 is sufficiently
68
large (in fact, the requirement on Aˆ2 in Theorem 5 implies the requirement on
Aˆ2 in Theorem 3, Aˆ2 ≥ A1hˆ1−Aˆ1h1h˜1 ). The third inequality guarantees that the
demand curtailment (and the associated emission reduction) is sufficiently small.
That is, despite the emission penalty, demand continues to be sufficiently large.
This requirement is specified in terms of a threshold on the relative decrease in
demand as t increases. Note that this third condition is explicit and can be written
in closed form.
With regard to remedies, it is easy to show that those considered for the case
of exogenous demand continue to be effective. Given that price is now a decision
variable (and demand is sensitive to price), we can in addition consider remedies
that involve penalizing the consumer, either fully or partially, for supply chain
emission. In particular, let us consider the case where the consumer, instead of
the firms, is penalized for total supply chain emission. Such a scheme would of
course require that emissions over the entire supply chain are documented and
shared with the consumer.
Under a consumption-based penalty scheme, the consumer faces an effective
price, pe, that is the sum of the price the buyer charges and the emission penalty.
This effective price, in turn, determines demand. Hence, the buyer’s problem can
be restated as one involving choosing effective price and order quantity. In other
words, the buyer’s problem is given by:
max
pe
z1(pe) = (pe − t[Aˆ1 + Aˆ2
q1(pe)
+
hˆ1q1(pe)
2D(pe)
(3.22)
+(cˆ1 + cˆ2)]− w − c1)D(pe)− A1D(pe)
q1(pe)
− h1q1(pe)
2
,
It is easy to see that the above problem is the same as the one the buyer faces
when it is allocated all the emission cost (i.e., when α = 1 under a shared emission
69
responsibility scheme). In particular, both schemes lead the buyer to choose the
same demand level and order quantity, leading to the same buyer and supplier
profits, emission levels, and consumer surplus.
In Figure 3.5, we illustrate the differences in performance of five different
penalty schemes involving different ways of allocating emission responsibility. To
allow for a fair comparison between the different schemes, we evaluate the re-
sulting social welfare for each, where social welfare is defined as: consumer sur-
plus + producer surplus (supply chain profit) + revenue collected through the
penalty scheme - environmental damage. Consumer surplus is defined in the usual
way as
∫ +∞
p∗ D(p)dp, while environmental damage is given by ωeSC(q1(p
∗), p∗) =
ω( (Aˆ1+Aˆ2)D(p
∗)
q1(p∗)
+ hˆ1q1(p
∗)
2
+ (cˆ1 + cˆ2)D(p
∗), where ω corresponds to the social cost
of one unit of emission (for the consumption-based penalty scheme, p∗ is replaced
by p∗e).
The first of the penalty schemes depicted in Figure 3.5 corresponds to the one
that allocates emission responsibility among the two firms so that total supply
chain profit is maximized (profit-optimal emission allocation); the second corre-
sponds to allocating responsibility proportionally to business-as-usual emissions
(fair emission allocation); the third corresponds to allocating responsibility so
that the social welfare is maximized (socially-optimal allocation); the fourth cor-
responds to the baseline scheme of penalizing each firm for its own emissions
(no emission responsibility sharing); and the fifth corresponds to penalizing the
consumer and not the firms (consumption-based emission penalty).
From the figure, we can make the following observations:
• As expected, the socially-optimal allocation dominates the other emission
70
responsibility sharing schemes. More significantly, there are ranges of t for
which these other allocations (the fair allocation, the profit-optimal alloca-
tion, and the consumption-based penalty) perform poorly with regard to
social welfare.
• There are ranges of values of t for which the no emission responsibility
sharing scheme outperforms all the other penalty schemes, implying that a
policy that is socially-optimal may not necessarily involve fractional sharing
of total supply chain emission. In fact, a scheme, where firm i is penalized
at rate tij for emission due to firm j, with tij possibly different from tji can
be shown to outperform all the schemes considered so far (and of which all
can be viewed as special cases). Such a differentiated penalty scheme may,
however, be difficult to implement when the price of emission is determined
by a common market (e.g., a market for offsets or a market for tradable
emission credits).
• There is a value of t that maximizes social welfare for each of the penalty
schemes considered. Perhaps surprisingly, this value can be either lower or
higher than the social cost of emission (ω). In some cases, a value lower than
the social cost may be necessary so that consumer surplus is not degraded
too much because of demand curtailment (this is consistent with results from
environmental economics; see the discussion in Section 2). In other cases, a
value higher than the social cost (i.e., overcompensating for emissions) may
be preferable if a higher penalty reduces supply chain costs. For example, a
higher penalty can lead the buyer to increase its order quantity which then
reduces the supplier’s fixed ordering cost. In turn, this can lead to an overall
71
reduction in total supply chain cost.
3.6 Other Examples
In this section, we show that higher penalties leading to higher supply chain emis-
sion is not unique to the buyer-supplier problem we have studied so far and that it
arises in other settings. In particular, we discuss two other example applications.
The first involves make versus buy decisions, where a manufacturer decides on how
much to produce in-house and how much to outsource. The second involves retail
store density decisions, where a retailer decides on how many stores to operate in
a region. We also describe a general model formulation, of which the applications
we discuss can be viewed as special cases.
3.6.1 A Make versus Buy Problem
Consider two firms, a manufacturer and a supplier. The manufacturer decides on
how much to produce in-house and how much to outsource to an external supplier.
The supplier, in turn, decides on how much to charge the manufacturer. The
manufacturer must fulfill a constant demand which, for simplicity, we normalize to
one. The selling price, which we denote by p, is exogenously specified (the analysis
can be extended to the case where the manufacturer also decides on the price).
The manufacturer decides on the fraction of demand to produce in-house, which
we denote by x1, where 0 ≤ x1 ≤ 1, and the corresponding fraction x2 = 1−x1 to
outsource to the supplier (x1 can be alternatively viewed as the amount of work
content associated with each unit of the end product the manufacturer carries out
72
(a) (d = 400, A1 = 0.2, h1 = 8, A2 = 15, h2 =
1.5, c1 = 1, c2 = 0.5, w = 1, Aˆ1 = 1, hˆ1 = 0.5, Aˆ2 =
2, hˆ2 = 2, cˆ1 = 0, cˆ2 = 1, ω = 0.1)
(b) (d = 400, A1 = 0.2, h1 = 8, A2 = 15, h2 =
1.5, c1 = 1, c2 = 0.5, w = 1, Aˆ1 = 1, hˆ1 = 0.5, Aˆ2 =
2, hˆ2 = 2, cˆ1 = 0, cˆ2 = 1, ω = 1)
(c) (d = 600, A1 = 0.2, h1 = 8, A2 = 0, h2 =
1.5, c1 = 1, c2 = 1, w = 2, Aˆ1 = 0.5, hˆ1 = 0, Aˆ2 =
0, hˆ2 = 0, cˆ1 = 0.5, cˆ2 = 1, ω = 3)
Figure 3.5: The impact of the emission penalty on social welfare
73
in-house, and 1 − x1 the work content that is outsourced). The supplier decides
on the wholesale price w to charge the manufacturer.
The manufacturer and the supplier both face production cost and emission
functions that are increasing and convex in the fraction of demand they are allo-
cated. The convexity reflects settings where producing more becomes progressively
more expensive and more emission-intensive. This would occur, for example, if
limitations on capacity lead to more congestion (longer production leadtimes and
higher levels of work-in-process inventories) or poorer quality and lower yields, all
associated with higher cost and higher emissions and all typically convex in the
amount of work in the system.
We model the interaction between the supplier and the manufacturer as a
Stackelberg game. The manufacturer acts as the Stackelberg follower. It observes
w and chooses the amount of demand (or work content) to fulfill in-house. The
supplier, acts as the leader and chooses the price it charges the manufacturer,
anticipating the decision of the manufacturer. For simplicity, we consider the case
where the cost and emission functions are quadratic in the amount of demand (or
work content) allocated to each firm. In particular, for specified x1, the manu-
facturer incurs production cost a1x
2
1 and emission aˆ1x
2
1 while the supplier incurs
production cost a2(1− x1)2 and emission aˆ2(1− x1)2, where ai, aˆi ≥ 0 for i = 1, 2.
Given price w, the manufacturer’s problem can be stated as
max
x1∈[0,1]
z1(x1, w) = p− a1x21 − w(1− x1)− taˆ1x21. (3.23)
We can show that the manufacturer’s best response function is given by
x∗1(w) =
 1, if w ≥ 2a˜1w
2a˜1
, otherwise
,
74
where a˜1 = a1 + taˆ1.
The problem of the supplier, who anticipates the manufacturer’s choice x∗1(w),
is given by
max
w≥0
z2(w, x
∗
1(w)) = w(1− x∗1(w))− a2(1− x∗1(w))2 − taˆ2(1− x∗1(w))2.(3.24)
Using the fact that the relationship w = 2a˜1x
∗
1(w) always holds, the supplier’s
problem can be restated as follows:
max
x1∈[0,1]
2a˜1x1(1− x1)− a˜2(1− x1)2,
where a˜2 = a2 + taˆ2. The supplier’s profit function is concave in x1 and we can
show that the optimal solution is given by
x∗1 =
a˜1 + a˜2
2a˜1 + a˜2
, (3.25)
with a corresponding optimal wholesale price
w∗ =
2a˜1(a˜1 + a˜2)
2a˜1 + a˜2
. (3.26)
The following proposition follows directly from the expressions in (3.25) and
(3.26).
Proposition 8 The optimal wholesale price of the supplier w∗ is always increas-
ing in t. However, the optimal amount produced in-house x∗1 is increasing in t if
and only if aˆ1
a1
≤ aˆ2
a2
.
It is interesting to note that, while higher emission penalties always lead to higher
supplier prices (this is perhaps expected since the cost of both supplier and man-
ufacturer are both higher), higher emission penalties may or may not lead to less
75
outsourcing. The manufacturer outsources less if it has a lower emission to pro-
duction cost ratio than that of the supplier and outsources more otherwise. This
is because, in the case of the former higher emission penalties increase the rela-
tive cost advantage of producing in-house while in the latter, it improves that of
outsourcing.
Theorem 6 Total supply chain emission increases in the emission penalty if and
only if one of the following conditions holds:
• aˆ2
a2
≥ aˆ1
a1
and t ≥ aˆ2a1−aˆ1a1−aˆ1a2
aˆ21
, or
• aˆ2
a2
< aˆ1
a1
and t ≤ aˆ2a1−aˆ1a1−aˆ1a2
aˆ21
.
Figure 3.6: The impact of the emission penalty on the manufacturer’s emission e1
and the supplier’ emissions e2
The increase in supply chain emission can be explained as follows. Consider first
the case when aˆ2
a2
≥ aˆ1
a1
. In this case, the manufacturer outsources more as the
76
emission penalty increases (Proposition 8). The manufacturer’s emission is convex
increasing in the amount produced in-house (x1). On the other hand the supplier’s
emission is convex decreasing in x1. Hence, for sufficiently large t, or equivalently
sufficiently large x1, the rate of increase in the manufacturer’s emission could ex-
ceed the rate of decrease in the supplier’s emission, with the net effect being an
increase in supply chain emission with an increase in t. This effect is illustrated
graphically in Figure 3.6. The case of aˆ2
a2
< aˆ1
a1
can be explained similarly, except
that now an increase in the emission penalty leads to more outsourcing (Proposi-
tion 8). If t is sufficiently small, the increase in the supplier’s emission rate exceeds
the reduction in that of the manufacturer with the net effect being an increase in
supply chain emission with an increase in t.
Potential remedies along the lines discussed in Section 3.4 can be constructed.
Additionally, the analysis can be extended to the case where the manufacturer also
decides on the selling price and demand is decreasing in this price (for brevity, we
omit the details).
3.6.2 A Store Density Problem
Consider the problem posed recently by Cachon (2014) of a retailer who decides
on the number and size of stores to serve a specified geographic region over which
consumer are uniformly and continuously distributed. Consumers travel to the
nearest store using the shortest path and along a straight line. Deliveries to the
store are made from a warehouse that is co-located with one of the stores. Deliv-
eries to the stores are carried out such that total delivery distances are minimized.
In other words, the retailer solves a traveling salesman problem (TSP). The stores
77
are located such that the region is segmented into equally sized sub-regions, with
a store at the center of each subregion.
Consumers are more likely to shop with the retailer the shorter the distance
they have to travel to a store. Hence more stores generate more demand for the
retailer3 . However, more stores imply longer delivery routes and, therefore, more
transportation cost and possibly other fixed and variable costs for the retailer.
Hence, the retailer makes a decision regarding the optimal number of stores by
trading off the higher revenue generated from more stores with the corresponding
higher cost.
We assume that sub-regions are similarly shaped polygons and, for ease of
analysis, we restrict our treatment, to equilateral hexagons, squares, or triangles.
We let a denote the area of the entire region, so that each sub-region has area
a/n, where n denotes the number of stores. Given these assumptions, we can
show that the average travel distance for the consumer is given by dc = φcn
− 1
2
while the transportation distance for the retailer is given by dr = φrn
1
2 , where φc
and φr are constants that depend only on the shape of the sub-regions (here the
subscripts c and r refer respectively to the consumer and the retailer); see Cachon
(2014) for details and further discussion.
Both the retailer and consumers incur transportation costs and emissions per
unit of demand per unit of distance traveled, which we denote by ci and cˆi, for
i = r, c respectively. In addition the retailer incurs an emission penalty t per unit
of emissions. These unit costs (emissions) may reflect fixed and variable costs
(emissions) and account for the quantity transported by either the consumers
3 Cachon (2014) treats the case where consumers are insensitive to travel distances and
would always shop at the nearest store.
78
or the retailers4 . It is possible to consider other costs, including fixed and
variable costs associated with operating each store. We omit these for the sake
of brevity, but the analysis can be easily extended to those cases and our main
insight discussed below continues to hold (see also discussion at the end of this
section).
We assume the overall demand function for the retailer, arising from the de-
cisions of individual consumers, is decreasing in the average traveling distance
(or equivalently, average traveling cost) of the consumers and given by D =
D0(ccdc)
−γ = D0c−γc φ
−γ
c n
γ
2 , where c−γc φ
−γ
c n
γ
2 can be interpreted as the proba-
bility that a consumer decides to shop. The problem faced by the retailer can
then be stated as follows:
max
n
zr(n) = (p− c˜rφrn 12 )D0c−γc φ−γc n
γ
2 , (3.27)
where p denotes the unit selling price, and c˜r = cr+tcˆr. Treating n as a continuous
variable, we can show that the optimal solution to (3.27) is given by:
n∗ = [
γp
(γ + 1)c˜rφr
]2.
We can see that n∗ is decreasing in the emission penalty t. That is, the higher is
the penalty, the fewer is the number of stores, as doing so reduces the retailer’s
transportation distances. However, by doing so, the retailer affects consumers
in two ways. It reduces the fraction of consumers who shop and increases the
traveling distances for those who do. Therefore, if the reduction in demand is
4 For example, the cost and emission parameters could be determined as follows: ci =
vi+fipi
qi
and cˆi =
fiei
qi
, where vi is the non-fuel cost incurred by a transport vehicle per unit of distance,
fi is the amount of fuel consumed per unit distance traveled, pi is the price per unit of fuel, ei
is the emissions per unit of fuel consumed, and qi is the load carried per vehicle (see further
discussion in Cachon (2014)).
79
sufficiently small (or the emission-intensity of consumer travel is sufficiently high),
it may be possible for consumer-related emissions to increase. It may also be
possible for this increase to be higher than the decrease in the retailer’s emissions.
In what follows, we provide a necessary and sufficient condition for this to occur.
Let eSC denote total supply chain emission (the sum of retailer and consumer
emissions). Then,
eSC = (cˆrφrn
∗ 1
2 + cˆcφcn
∗− 1
2 )D0c
−γ
c φ
−γ
c n
∗ γ
2 .
The following proposition directly follows from differentiating eSC with respect to
t.
Theorem 7 Total supply chain emission eSC increases in the emission penalty t
if and only if
γ ≤
√
c˜2r cˆcφrφc
p2cˆr + c˜2r cˆcφrφc
.
The above result indicates that total supply chain emission would increase if
consumers are sufficiently insensitive to traveling distance as measured by the
parameter γ. The threshold than which gamma must be smaller is increasing in
the emission-intensity of consumers as specified by cˆc.
Results similar to Proposition 7 can still be obtained when other features are
added to the model, including a penalty on emissions borne by the consumers or
fixed costs associated with operating each store for the retailer. A similar result
can also be obtained for systems where the retailer decides on the selling price and
consumers are sensitive to both cost of travel and price. Finally, results can be
obtained, although not analytically, for the case where n is treated as a discrete
variable. Details for these cases are again omitted for the sake of brevity.
80
Figure 3.7: The impact of the emission penalty on supply chain emission for an
example store density problem (p = 4, D0 = 1000, cc = 0.5, cr = 0.1, cˆc = 1, cˆr =
0.02, φc = 0.7652, φr = 1, γ = 0.1).
The potential negative impact of emission penalties on supply chain emission
could be mitigated if the retailer were to bear the cost of emissions for the entire
supply chain (its own and those of the consumers). It would also be mitigated if
consumers were to be penalized for their own emissions and those of the retailer
(e.g., consumers pay an emission tax on both the fuel they purchase to travel and
the emission associated with the products they buy from the retailer). The effect
would of course also be mitigated if the retailer took on the full cost of the supply
chain (its own and that of the consumers) in deciding on the number of stores.
This is the case treated by Cachon (2014) who models a system where the retailer
takes on the role of a central planner and compensates consumers for their full
travel cost, including their emission cost if any. As we discussed in Section 4.2,
81
a centralized supply chain may not however guarantee that total supply chain
emission would be lower than that of a decentralized supply chain. Figure 3.7
illustrates this for an example system.
3.6.3 A General Model Formulation
In this section, we briefly describe a general model formulation of which the various
models we have considered so far can be viewed as special cases. We do so to
illustrate the broader applicability of the results we have obtained and to further
explain how the phenomenon of higher emission penalties leading to higher supply
chain emissions may arise.
Consider a supply chain consisting of two firms, which we denote by firm 1 and
firm 2 respectively. (As mentioned earlier, we use the term firm loosely to refer
to economic agents which may include independent firms, business units within
a single firm, or consumers.) Firm i takes an action, an operational decision in
our context, which we denote by xi, within a feasible set Si for i = 1, 2. The
action taken by a firm, in addition to affecting its profit and emissions, affects the
profit and emissions of the other firm and we let fi(x1, x2) and ei(x1, x2) denote
respectively the profit and emissions of firm i for i = 1, 2.
This formulation of course allows for the possibility of the action of only one
firm affecting the profit and emissions of the other. The buyer-supplier problem
we studied falls in this category with only the buyer’s action (the order quantity)
affecting the supplier. On the other hand, the make versus buy and the store
density problems fall in the general category. In the make versus buy problem,
the amount to produce in-house decided by the manufacturer affects the profit
82
and emissions of the supplier and, similarly, the wholesale price decided by the
supplier affects the profit and emissions of the manufacturer. Likewise, in the store
density problem, the number of stores chosen by the retailer affects the travel costs
and emissions of the consumers, while the collective action by the consumers on
whether or not to shop, modeled implicitly via the demand function, affects the
profit and emissions of the retailer.
Each firm i (i = 1, 2) is subject to an emission penalty ti per unit of emissions
(note that this allows for t1 = t2 = t or either t1 or t2 being zero, the cases
treated in the paper). The firms engage in a Stackelberg game (the analysis can
be extended to other forms of strategic interaction, such as when the firms choose
their actions simultaneously). Firm 2 observes Firm 1’s action x1 and chooses x2
to maximize its after penalty profit. Firm 1, anticipating the action of firm 2,
chooses x1 to maximize its own after penalty profit. If we consider settings where
Firm 2’s best response function and Firm 1’s optimal solution, which we denote
as x∗2(x1) and x
∗
1, are unique, as they are in the models we have considered, then
the problem faced by firm 2, given the choice x1 of firm 1, can be specified as
follows:
max
x2∈S2
z2(x1, x2) = f2(x1, x2)− t2e2(x1, x2), (3.28)
while the problem faced by firm 1 can be specified as:
max
x1∈S1
z1(x1, x
∗
2(x1)) = f1(x1, x
∗
2(x1))− t1e1(x1, x∗2(x1)). (3.29)
In equilibrium, this leads to a total supply chain emission, which we denote by
eSC , given by
eSC(x
∗
1, x
∗
2) = e1(x
∗
1, x
∗
2) + e2(x
∗
1, x
∗
2), (3.30)
83
where (x∗1, x
∗
2) is the equilibrium solution.
From the above equation, we can see that whether higher emission penalties
lead to higher or lower supply chain emission depends on how the individual
actions of the firms in equilibrium affect the emissions of both firms. In the case
where the supply chain emission function eSC is continuous and differentiable in
ti, which is again the case in the models we studied, the necessary and sufficient
condition for supply chain emission to increase with ti is given by
deSC(x
∗
1, x
∗
2)
dti
=
de1(x
∗
1, x
∗
2)
dti
+
de2(x
∗
1, x
∗
2)
dti
≥ 0. (3.31)
The above condition holds in two cases. The first is when the emissions of both
firms increase as the emission penalty increases. The second is when the emissions
of one firm decrease but those of the other firm increase, with the rate of increase
for one firm higher than the rate of decrease for the other firm (this would be the
case when the action of only one of the firms affects the emissions of the other
firm, and not vice-versa).
It is easy to verify that the above condition is what leads to the conditions
described in Theorems 3-7 for the models we studied. We can easily recast much of
the discussion of the remedies considered in terms of this general formulation. We
can also extend the model formulation to consider settings where each firms takes
a vector of actions instead of just one (to accommodate, for example, situations
where firms make both operational and pricing decisions, as in the model of Section
5 with endogenous demand). Given the generality of the model formulation, we
expect that there would be various other applications that would conform to the
specification of the model and exhibit similar supply chain effects.
84
3.7 Summary and Concluding Comments
In this paper, we examined the impact of emission penalties in decentralized sup-
ply chains. We showed that, perhaps surprisingly, emission penalties (if they are
imposed on the emission of each firm) can lead to higher overall supply chain
emission. We showed that this is because firms, in their efforts to reduce their
own emissions, can make decisions that end up increasing the emissions of other
firms. Although we based our analysis primarily on a buyer-supplier model where
the decisions the firms make involve choosing order quantities, we showed that
the same effects arise in other settings. We also described a general model for-
mulation of which the various models we studied can be viewed as special cases.
We described potential remedies that involve internalizing, to varying degrees, the
emission externality a firm may impose on another firm, and analyzed the per-
formance of these remedies in terms of emission, supply chain profit, and social
welfare. Although all the remedies guarantee that higher penalties lead to lower
emissions, we showed that they do not guarantee that the resulting emissions
would be lower than those obtained when firms are penalized only for their own
emissions.
A contribution of the paper is in shedding light on an effect that, to our knowl-
edge, has not been widely studied. Accounting for this effect appears important in
assessing whether or not well-meaning initiatives undertaken by individual firms or
policies enacted by governments lead to their intended outcomes. Our results pro-
vide support for efforts to hold firms accountable not only for their own emissions
but for the emissions of their suppliers and customers and for the development of
standards for documenting such emissions.
85
Although in much of our analysis we have treated settings where the firms
are motivated by an explicit penalty on emissions (or consumption), the effect
we document is also present when firms or business units decide to reduce their
emissions for other reasons (e.g., for reputational reasons). In this case again, a
firm in its pursuit of emission reduction can create externalities that lead other
firms to emit more and result in a net increase in total emission. Also, in much
of our analysis, we have assumed that all firms in the supply chain are subject to
a common price on emissions. The results we describe however hold even if only
one firm in the supply chain is subject to a penalty or if the penalties are different
for different firms.
In this paper, we considered settings where firms affect emissions by mak-
ing either operational or pricing adjustments. Firms of course may also invest
in technology that affects their underlying emission parameters for production,
transportation, and storage, among others. Our analysis can be extended to those
settings and to settings where firms first make technology investments and then
operational adjustments. In fact, it is not difficult to find instances where such
investments can amplify the negative impact of emission penalties (e.g., having
affected a change in technology, a firm finds it less costly to make operational
decisions that adversely affect the emissions of other firms).
Avenues for future research are many. It would be useful to investigate other
applications where the effect of higher penalties leading to higher emissions is
particularly prominent. It would also be useful to identify other remedies that
are easy to implement and manage. In settings in practice where firms have
86
undertaken emission reduction efforts, it would be of interest to empirically inves-
tigate whether their suppliers and customers saw increases or decreases in their
emissions.
Chapter 4
Work-In-Progress and Future
Research Directions
In this chapter, we describe work-in-progress and future research directions, con-
sisting of two parts. In the first part, we consider a model where the price of
carbon is stochastic. The second part deals with two models where demand is no
longer assumed to be exogenous. For the sake of brevity, we omit all the technical
developments and simply highlight some of the main results obtained so far.
4.1 Systems with Stochastic Carbon Prices
In this section, we describe preliminary analysis we carried out to understand
better the impact of variability in carbon prices. In Chapter 3, we considered a
setting where there is a tax on emission and this tax, which corresponds to the
price of carbon, is fixed. However, there are settings where the price of carbon
may vary over time. For example, this is the case under a cap-and-trade system
87
88
where the emission of firms are capped but firms are allowed to buy and sell
emission credits, based on whether they exceed or fall below their emission caps.
The analysis we carry out is also more generally relevant to settings where the cost
parameters (purchasing, fixed ordering, and holding costs) are stochastic. There
is a rich literature dealing with purchase cost variability (see for example Ho et
al. (1998) and Berling and Mart´ınez-de-Albe´niz (2011)). However, very little of
this literature addresses settings where variability is also associated with fixed
costs and inventory carrying costs. In this section, we use the framework of the
economic order quantity (EOQ) model as discussed in Chapter 2 to investigate
the impact of cost variability.
4.1.1 Preliminary Results
Consider the setting of Chapter 2. A price t is associated with emissions, except
that t is now a random variable. Prices are assumed to be taking on discrete
values t1, t2, ..., tS and are independently and identically distributed (i.i.d.) with
pii corresponding to the probability associated with price ti, for i = 1, 2, ..., N .
In particular, we assume that the carbon price observed at any time is i.i.d.
and can take on values from {t1, t2, ..., tS} with probabilities {pi1, pi2, ..., piS}. We
assume that at the start of each replenishment cycle, the current price is observed
and then a decision on the order quantity is made. The objective of the firm is
to choose an order quantity each time it observes price ti such that its long run
average cost per unit time is minimized.
Noting that the replenishment times form a renewal process, it can be shown
that, for a given vector of order quantities Q(t1), Q(t2), ..., Q(tS), the long run
89
average cost per unit time is given by
Z(Q(t1), Q(t2), ..., Q(tS)) =
1
S∑
s=1
Q(ts)
D
pis
(A+
S∑
s=1
[(c+ tscˆ)Q(ts) (4.1)
+tsAˆ+
(h+ tshˆ)Q
2(ts)
2D
]pis),
where Q(ts) is the order quantity used if price ts is used. The following theorem
characterizes the optimal vector of order quantities.
Theorem 8 The optimal ordering quantities {Q∗(t1), Q∗(t2), ..., Q∗(tS)} are given
by
Q∗(ts) =
1
(h+ tshˆ)E[
1
h+ thˆ
]
(−cˆDE[ ts − t
h+ thˆ
] (4.2)
+
√
cˆ2D2(E2[
ts − t
h+ thˆ
]− E[ (ts − t)
2
h+ thˆ
]E[
1
h+ thˆ
]) + 2D(A+ Aˆµt)E[
1
h+ thˆ
]),
and the optimal long run average cost per unit time is given
Z(Q∗(t1), Q∗(t2), ..., Q∗(tS)) =
Dcˆ
hˆY
+
chˆ− cˆh
hˆ
(4.3)
+
D
√
cˆ2
hˆ2
− cˆ
2
hˆ2
(h+ hˆµt)Y +
2(A+ Aˆµt)Y
D
Y
where µt = E[t] is the expected carbon price, and Y = E[
1
h+ thˆ
].
Using Theorem 8, a number of results can be obtained, including the following:
• A system where the price of carbon is a random variable always leads to a
lower expected cost than one where the price of carbon is fixed and equal to
90
the expected value of the random price (a phenomenon due in part to the
fact that price variability allows the firm to order less when the prices are
low and more when prices are high). This results suggests that, everything
else remaining the same, a system where the price of carbon is subject to
variability (perhaps as in a cap-and-trade system) is more preferable to firms.
• Although a variable price is preferable to a fixed price, high variability (as
measured by variance) does not always lead to lower cost. This is in contrast
to the result that would be obtained if variability was only associated with
the purchase cost. This effect is due to the fact that savings on purchasing
less (more) when prices are higher (lower) must be traded-off against the
corresponding savings in fixed ordering and holding costs.
• In general, higher variability is more preferable for settings where the ex-
pected cost function is concave in the carbon price. We can show this is
true for several special cases, including the case when hˆ = 0 and when t is
uniformly distributed. We can also show that higher variability is always
preferable if it is measured using the stochastic convex ordering; more if
t ≤cv t˜, then Z(t) ≥ Z(t˜).
4.1.2 Future Directions
We plan to extend the above analysis in several ways. First, we plan to consider
systems where the firm does not necessarily incur the cost of carbon at the be-
ginning of the replenishment period or as emissions are incurred. The firm may
be accountable for its emissions at fixed points in time. We also plan to consider
91
systems where firms are subject to a specified cap and can actively buy and sell
emission credits, either continuously or at fixed points in time. Moreover, we
note that variability in the price of emissions essentially reduces to variability in
the cost parameters. In the analysis we carried out so far, we assume that the
costs are perfectly correlated, we plan to study systems where this may not be
the case, with different costs having varying degrees of correlation, either positive
or negative.
4.2 Inventory Management with Emission Sen-
sitive Demand
With their increasing awareness of environmental concerns and their negative
impacts, consumers are beginning to exhibit preferences towards greener products,
including products with less carbon footprints. Harris (2007) finds that “consumer
demand for products that are clearly identified as genuinely sustainable, even
though they may be perceived to be more expensive than traditional products”.
Walmart’s success with its sustainability program is contributed partially to the
additional revenue generated through a premium for its “green” products and
the increased consumer demand (see for example Plambeck (2012)). Companies
have implemented measures such as obtaining sustainability certifications (e.g.,
USDA Organic in the US), and labelling the carbon content of their products
(e.g., Tesco). Such practices have allowed consumers to make more informed
decisions which encourage them to purchase “green” products. Vanclay et al.
(2011) for example confirm the positive impact of carbon labelling on the sales of
92
grocery items.
As pointed out by Tang and Zhou (2012), studies on emissions and pollution
reductions have largely ignored the market forces. As a result, there is a need
for research addressing consumer response and preference towards greener prod-
ucts in the context of supply chain and operations management. Gosh and Shah
(2012) study a two firm apparel supply chain facing consumers that prefer green
products, and compare the green innovation and other decisions between various
supply chain structures. There are a number of literature in the field of environ-
mental economics incorporating consumers’ prefernces towards green products.
For example, Moraga-Gonza´lez and Padro´n-Fumero (2002) study a duopoly facing
consumers with varying preferences for green products. Conrad (2005) studies the
effectiveness of environmental policies when considering consumer environmental
awareness using a duopoly model.
In this section, we consider consumers’ preference towards low emission prod-
ucts in the context of supply chain and inventory management.
4.2.1 Preliminary Results
We base the following analysis on the setup from Chapter 2 and consider the
setting that customer demand is not only sensitive to the price of of the product
but to the environmental image of the firm as well. Specifically, we assume that
customer demand D is linearly decreasing in the price of the product p and the
annual emission of the firm e: D = d − ap − be, with a and b being customers’
level of sensitivity to price and emission respectively. The objective of the firm is
93
to maximize the annual profit by choosing the order quantity Q:
max
Q
pi(Q) = (p− c)D − (AD
Q
+
hQ
2
) (4.4)
s.t. D = d− ap− be
The optimal solution to (4.4) is given by:
Q∗ =
√√√√2( hAˆ1+bcˆ + Ap−c)(d− ap+ b2Aˆhˆ2(1+bcˆ))
bhˆ+ h(1+bcˆ)
p−c
− bAˆ
1 + bcˆ
.
Using the optimal solution, we obtained some preliminary results and summa-
rize below:
• Moderate carbon emission sensitivity can have significant impact on regu-
lating firms and reducing emissions, while high carbon sensitivity does not
add much value in terms of emission reduction while it can significantly hurt
firms profitability.
• In the case of Aˆ = 0, we show that the optimal order quantity and corre-
sponding emission decrease with the product price, while profit first increases
and then decreases with the product price.
• In the case of hˆ = 0, we show that emission decreases with the product price,
while the optimal order quantity first increases and then decreases with the
product price.
4.2.2 Future Directions
There are several directions for future research. One is to study the effect of
customer sensitivity on the emissions of the firm and extend the current results
94
to more general settings. Secondly, it would be interesting to extend the anal-
ysis to incorporate interactions among firms within a supply chain and study
how consumers’ emission sensitivity affects supply chain decisions, performance
as well as the social welfare. It would also be interesting to examine the effect of
supply chain coordination on firms’ profitability as well as emission abatement.
Additionally, another useful avenue of research is to use advanced simulation tool
to incorporate more fully the preferences of various parties as well as their in-
teractions, through which more precise estimates of the implications of different
environmental policies and supply chain inputs can be obtained.
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