Multiperiod effects of corporate social responsibility on supply chain

networks, transaction costs, emissions, and risk

Jose M. Cruz* and Tina Wakolbinger

Department of Operations and Information Management

School of Business, University of Connecticut, Storrs, CT 06269-2041

Department of Marketing and Supply Chain Management

Fogelman College of Business and Economics, University of Memphis

Memphis, TN 38152-3120

Revised July 2008

To appear in International Journal of Production Economics

Abstract

This paper develops a framework for the analysis of the optimal levels of corporate social

responsibility activities in a multiperiod supply chain network consisting of manufacturers,

retailers, and consumers. Manufacturers and retailers determine their production quantities,

transaction quantities, and the amount of social responsibility activities they want to pursue

that maximize net return, minimize emission, and minimize risk over the planning horizon.

We investigate the interplay of the heterogeneous decision-makers and compute the equi-

librium pattern of product outputs, transactions, prices, and levels of social responsibility

activities. The paper provides insights concerning the optimal allocation of resources to CSR

activities when considering a multi-period time frame.

Keywords: Supply chains; Environment; Corporate social responsibility; Risk management;

Network equilibrium; Pricing, Variational Inequalities, Multicriteria decision-making;

*Corresponding author: Tel.: +1 413 210 6241; fax: +1 860 486 4839.

E-mail address: jcruz@business.uconn.edu (J. Cruz).

1

1 Introduction

Corporate social responsibility encompasses the economic, legal, ethical, and philanthropic

expectations placed on organizations by society at a given point in time (Carroll and Buch-

holtz, 2002). Today, corporate social responsibility is not only a prominent research theme

but it can also be found in corporate missions and value statements (Svendsen et al., 2001).

Companies increasingly realize that their actions in purchasing and supply chain manage-

ment strongly affect their reputation and long-term success (Castka and Balzarova, 2008

and references therein). Corporations are held accountable for promoting and protecting the

environmental, health, and safety regulations of workers that make their products, regardless

if they are direct employees or work for their suppliers. For example, corporations like Nike,

Liz Claiborne, Disney, and Wal-Mart have faced damaging media reports, external pressure

from activists, and internal pressure from investors demanding that companies acknowledge

responsibility for labor rights abuses in factories that make their products (Arriaga, 2008).

McDonalds, Mitsubishi, Monsanto, Nestle, Nike, Shell, and Texaco have suffered damage to

their reputations and sales as a result of public awareness campaigns by advocacy groups

about their CSR practices (Svendsen et al. 2001). As a consequence, companies start

expanding their responsibility for their products beyond their sales and delivery locations

(Bloemhof-Ruwaard et al., 1995) and they start managing the CSR of their partners within

the supply chain (Kolk and Tudder, 2002; Emmelhainz and Adams, 1999).

Many researchers have tried to understand business motivation to adopt CSR programs

(Delmas and Terlaak, 2001; Marcus et al., 2002), legal and institutional factors shaping CSR,

the effects of attitudes of managers and consumers towards CSR (Williams and Aguilera,

2008), the effects of the dissemination of industry standards such as ISO 26000 (Castka

and Balzarova, 2008) and the relationship between the three concepts, CSR, risk, and profit

(Dowling, 2001; Fombrun, 2001; Clarkson, 1991; Kotter and Heskett, 1992; Collins and

Porras, 1995; Waddock and Graves, 1997; Berman et al., 1999; Roman et al., 1999).

Indeed, firms engage in CSR activities as a way to enhance their reputation (Fombrun,

2005), preempt legal sanction (Parker, 2002), respond to NGO action (Spar and La Mure,

2003), manage their risk (Fombrun et al., 2000; Husted, 2005), and to generate customer

loyalty (Bhattacharya and Sen, 2001, 2004). CSR can potentially decrease production in-

efficiencies, reduce cost and risk and at the same time allow companies to increase sales,

increase access to capital, new markets, and brand recognition.

While many companies see CSR as a means for damage control or PR, companies in-

creasingly realize that CSR activities offer opportunities to create value (Porter and Kramer,

2006). ”The practice of CSR is an investment in the company’s future; as such, it must be

planned specifically, supervised carefully, and evaluated regularly” (Falck and Heblich, 2007,

1 p.248). It is very important that organizations take the long-term benefits of CSR into

2

consideration when determining their optimal investment in CSR activities.

Merck & Co. Inc. is an example of a company that benefitted from reputational capital

created by CSR activities in the past (Fombrun, 1996). In 1995, Merck & Co. Inc.’s Flint

River plant in Albany, New York, leaked phosphorous trichloride. As a result of the leak,

forty-five people were taken to hospital and 400 workers were evacuated (Svendsen et al.

2001). However, the community response ranged from indifference to laudatory support of

Merck. Merck was given the benefit of the doubt because it had been a good CSR citizen

(Svendsen et al. 2001). While Merck & Co. Inc. benefitted from its reputational capital,

BP suffered negative financial and reputational consequences due to insufficient attention to

CSR activities in the past. In 2004, BP was fined a record $1.42 million for health and safety

offenses in Alaska even as the chief executive of BP, was establishing himself as a leading

advocate for CSR (Doane, 2005).

In reality, determining the “ideal level of CSR” activities (McWilliams and Siegel, 2001)

is difficult. Even more difficult, is it to set the right incentive structures into place to ensure

that this level is reached since pressures for short-term performance are often very strong

(Falck and Heblich, 2007). However, to plan and communicate the value of CSR activities,

its long-term effects need to be better understood (Porter and Kramer, 2006).

To contribute to this understanding, we build a multi-tiered multiperiod supply chain

model where decision-makers can not only decide about the product flows that they want

to transact with each other but where they can also strategically allocate resources to CSR

activities. The analysis of the model allows for insights on how CSR’ activities impact

companies performance in the long run and how ideal levels of CSR activities are influenced

by factors within as well as outside the firm.

Several of the assumptions in the model are similar to the assumptions of the conceptual

model by McWilliams and Siegel (2001). As in McWilliams and Siegel (2001) we assume

that firms try to maximize profits and that CSR can be viewed as an investment. However,

we do not model CSR as a differentiation strategy but consider its effect on transaction costs,

emissions and risk. As in McWilliams and Siegel (2001), we assume that firms must devote

resources for CSR activities. We, hence, consider the tradeoff between the costs to generate

CSR attributes and the benefits, which include lower risk, lower emissions and lower costs

in the long run.

We explicitly include the behavior of decision-makers within the supply chain as well

as the supply chain structure while we implicitly include institutional factors in the cost

and risk functions. The model is flexible enough to analyze how different objectives of firms

(McWilliams and Siegel, 2001), legal and institutional factors (Williams and Aguilera, 2008),

and country differences (Matten and Moon, 2008) impact optimal CSR levels.

Cruz (2008) considered corporate social responsibility activities and risk management

3

in a single period setting in addition to the concept of environmental decision-making. In

this paper, however, we turn to the critical issue of social responsibility activities and risk

management in a multiperiod supply chain network framework. As the previous section

highlighted, CSR activities lead to many long-term effects that are essential in the cost-

benefit analysis of CSR activities. These long-term effects were not considered in Cruz

(2008). The multiperiod framework allows us to explicitly capture these long-term effects

and, hence, provides a valuable extension of previous research. Furthermore, it allows us to

see how changes in the planning framework impact the decision-making, the resulting payoffs

and costs.

This paper is organized as follows. In Section 2, we develop the multitiered, multiperiod

supply chain network model. We describe decision-makers’ optimizing behavior and establish

the governing equilibrium conditions along with the corresponding variational inequality

formulation. In Section 3, we propose an algorithm and present computational studies.

In Section 4, we discuss the results. We conclude the paper with Section 5 in which we

summarize our results and suggest directions for future research.

2 The Multiperiod Supply Chain Network Model

In this section, we develop the multiperiod supply chain network model with risk man-

agement. We assume that all decision-makers consider a fixed planning horizon which is

discretized into periods: 1, ..., t, ..., T. The model consists of I manufacturers, J retailers,

and K demand markets as depicted in Figure 1. We denote a typical manufacturer by i, a

typical retailer by j, and a typical demand market by k. The links between the tiers repre-

sent transaction links. The variables for this model are given in Table 1. The equilibrium

solution is denoted by *. All vectors are assumed to be column vectors, except where noted.

The top-tiered nodes in Figure 1 represent the I manufacturers in the T time periods with

node (i, t) denoting manufacturer i in time period t. The manufacturers are the decision-

makers who produce a homogeneous product and sell it to the retailers in the second tier

of nodes in the supply chain network in Figure 1. A node (j, t) corresponds to retailer j in

time period t, where j = 1, ..., J and t = 1, ..., T. The consumers at the demand markets are

represented by the nodes in the bottom tier of the supply chain network. They acquire the

product from the retailers. Demand market k at time period t is denoted by node (k, t) with

k = 1, ..., K and t = 1, ..., T . The model developed in this section is based on the assumption

that manufacturers and retailers can perfectly predict the benefits of CSR activities during

their planning horizon T . All prices and costs are expressed in terms of their value in period

1.

We now turn to the description of the functions. We first discuss the production cost,

4

transaction cost, handling, and unit transaction cost functions given in Table 2. At each time

period t, each manufacturer is faced with a certain production cost function that depends on

his production output and the levels of social responsibility activities of current and previous

time periods. Furthermore, each manufacturer and each retailer are faced with transaction

costs. The transaction costs are affected by the amount of the product transacted and the

levels of social responsibility activities of current and previous time periods.

Each retailer is also faced with what we term a handling/conversion cost (cf. Table

2, Nagurney and Dong, 2002), which may include, for example, the cost of handling the

product. The handling cost of a retailer is a function of how much he has obtained of the

product from the various manufacturers in time period t.

The consumers at each demand market are faced with a unit transaction cost. As in the

case of the manufacturers and the retailers, higher levels of social responsibility activities

may potentially reduce transaction costs, which means that they can lead to quantifiable cost

reductions over the planning horizon. The unit transaction costs depend on the amounts of

the product that the retailers transact with the demand markets as well as on retailers’ social

responsibility activities of current and previous periods. We assume that the production

cost, the transaction cost, and the handling cost functions are convex and continuously

differentiable and that the unit transaction cost functions are continuous.

We now turn to the description of cost functions for social responsibility activities, the

emission functions, the risk functions and the demand functions. We assume that the cost

functions for social responsibility activities, the emission functions, and the risk functions are

convex and continuously differentiable. The demand functions are assumed to be continuous.

We start by describing the cost functions for social responsibility activities that are given

in Table 3. We assume that at each time period each manufacturer and each retailer may

spend money, for example, in the form of time/service, investment in new technology, train-

ing employees, and information sharing to promote social responsibility activities. Here,

social responsibility activities are activities that promote quality assurance, environmental

preservation, and compliance. The costs for social responsibility activities at time t depend

on levels of social responsibility activities of current and past periods.

These cost functions for social responsibility activities may be distinct for each manufac-

turer, each retailer, and each time period. We assume that these levels of social responsibility

activities (cf. Table 1) take on a value that lies in the range [0, 1]. No social responsibility

activity is indicated by a level of zero and the strongest possible level of social responsibility

activity is indicated by a level of one. This is consistent with the Corporate Responsibility

(CR) Report from Business in Community where businesses are graded on a scale of 0 to

100% for their CSR performance. The levels of social responsibility activities, along with

the product flows, are endogenously determined in the model.

5

We now describe the emission functions as presented in Table 4. Environmental issues

surrounding supply chains have only recently come to the fore, notably, in the context of

conceptual and survey studies (Hill, 1997 and the references therein) as well as applied studies

(Hitchens et al., 2000). In response to growing environmental concerns, researchers have

begun to deal with environmental risks (Qio et al., 2001). More significantly, the increased

focus on the environment is significantly influencing supply chains. Legal requirements and

changing consumer preferences increasingly make suppliers, manufacturers, and distributors

responsible for their products beyond their sales and delivery locations (Bloemhof-Ruwaard

et al., 1995). Nevertheless, in the supply chain context, models that explicitly include

minimization of emissions as an important goal are clearly needed.

We assume that the amount of emissions generated depends on the amount of product

produced and transacted as well as on decision-makers’ levels of social responsibility activi-

ties in current and previous periods (see, e.g., Lamming and Hampson, 1996; Florida, 1996;

Clift and Wright, 2000; Geffen and Rothenberg, 2000; Hall, 2000). We assume that each

manufacturer and each retailer seek to minimize the total emission generated in the produc-

tion process as well as in the process of product delivery to the next tier of decision-makers.

Hence, we truly capture the environmental decision making in the supply chain framework.

In terms of risk management, most of the research has focused on the study of the

relationship between corporate social responsibility (CSR) and financial performance (Graves

and Waddock, 1994; Griffin and Mahon, 1997; McGuire et al., 1988; McWilliams and Siegel,

2000; Preston and OBannon, 1997; Roman et al., 1999; Waddock and Graves, 1997). A

subset of these studies began to look at the relationship between CSR and risk. Spicer (1978)

looked directly at the CSR-risk relationship and found evidence for a negative correlation

between the two: as CSR increased, risk decreased. Orlitsky and Benjamin (2001) also

found support for negative relationship between CSR and risk. Bowman (1980) asserts that

firms with proactive CSR that engage in managerial practices like environmental assessment

and stakeholder management (Wood, 1991) tend to anticipate and reduce potential sources

of business risk, such as potential governmental regulation, labor unrest, or environmental

damage (Orlitzky and Benjamin, 2001). Feldman et al. (1996) suggest that adopting a

more environmentally proactive posture has, in addition to any direct environmental and

cost reduction benefits, a significant and favorable impact on the firm’s perceived riskiness

to investors and, accordingly, its cost of equity capital and value in the market place.

Table 5 describes the risk functions. Most research on CSR and risk relationship has

been empirical or conceptual research and did not focus on risk management as a function

of CSR activities on supply chain. Risk functions in our model are functions of both the

product transactions and the levels of social responsibility activities in current and previous

periods. Juttner et al. (2003) suggest that supply chain-relevant risk sources fall into three

categories: environmental risk sources (e.g., fire, social-political actions, or acts of God),

organizational risk sources (e.g., production uncertainties), and network-related risk sources.

6

Johnson (2001) and Norrman and Jansson (2004) argue that network-related risk arises

from the interaction between organizations within the supply chain, e.g., due to insufficient

interaction and cooperation. We use levels of social responsibility activities as a way of

possibly reducing risks.

The demand functions as given in Table 6 are associated with the bottom-tiered nodes of

the supply chain network. The demand of consumers for the product at a demand market at

time t depends, in general, not only on the price of the product at that demand market but

also on the prices of the product at the other demand markets. Consequently, consumers at

a demand market, in a sense, also compete with consumers at other demand markets.

We now turn to describing the behavior of the various economic decision-makers. The

model is presented, for ease of exposition, for the case of a single homogeneous product. It

can also handle multiple products through a replication of the links and added notation. We

first focus on the manufacturers. We then turn to the retailers, and, subsequently, to the

consumers at the demand markets.

2.1 Multicriteria Decision-Making Behavior of the Manufacturersand Their Optimality Conditions

Let ρi∗1jt denote the price charged for the product by manufacturer i in transacting with

retailer j in period t. The price ρi∗1jt is an endogenous variable and will be determined once

the entire multiperiod supply chain network equilibrium model is solved. The quantity of the

product produced by manufacturer i in time period t must satisfy the following conservation

of flow equation:

qit =J∑j=1

qijt, (1)

which states that the quantity of the product produced by manufacturer i in time period t

is equal to the sum of the quantities transacted between the manufacturer and all retailers.

The first objective of the manufacturers is to maximize the total profit over the plan-

ning horizon T. The decision variables for manufacturer i are the distribution quantities in

each period, qijt; j = 1, ..., J ; t = 1, ..., T, and the levels of social responsibility activities at

each period, ηit. Thus, manufacturer i is faced with an optimization problem which can be

expressed as follows:

MaximizeT∑t=1

[J∑j=1

ρi1jtqijt − f it (qit, η1

it)−J∑j=1

cijt(qijt, η

1it)− bit(η1

it)

](2)

subject to the non-negativity constraints: qijt ≥ 0, and 0 ≤ ηit ≤ 1, ∀i, j, t.

7

The first term in (2) represents the revenue and the subsequent three terms the production

costs, the transaction costs, and the costs for social responsibility activities for manufacturer

i. Note that we allow the specifications of all the cost functions to be time-dependent.

In addition to the criterion of profit maximization, we assume that each manufacturer

also seeks to minimize the total emissions (waste) generated in the production of the product

as well as its delivery to the next tier of decision-makers, the retailers, over the planning

horizon T.

Hence, the second criterion of each manufacturer can be expressed mathematically as:

MinimizeT∑t=1

eit(Q1it, η

1it) (3)

subject to the non-negativity constraints: qijt ≥ 0, and 0 ≤ ηit ≤ 1, ∀i, j, t.

The third criterion faced by manufacturer i, thus, corresponds to risk (cf. Table 5)

minimization and can be expressed mathematically as:

MinimizeT∑t=1

rit(Q1it, η

1it) (4)

subject to the non-negativity constraints: qijt ≥ 0, and 0 ≤ ηit ≤ 1, ∀i, j, t.

We can now construct the multicriteria decision-making problem facing a manufacturer

which allows him to weight the criteria of profit maximization (cf. (2)), total emission

minimization (cf. (3)), and total risk minimization (see (4)) in an individual manner.

Assume that manufacturer i assigns a nonnegative weight ωi2 to total emission gener-

ated by production and transaction processes. Furthermore, assume that he assigns the

nonnegative weight ωi3 to risk. The weight associated with profit maximization serves as

the numeraire and is set equal to 1. The nonnegative weights measure the importance of

emission and risk, and, in addition, transform these values into monetary units. We can now

construct a value function for each manufacturer (cf. Keeney and Raiffa, 1993) using a con-

stant additive weight value function. Therefore, the multicriteria decision-making problem

of manufacturer i can be expressed as:

MaximizeT∑t=1

[J∑j=1

ρi1jtqijt − f it (qit, η1

it)−J∑j=1

cijt(qijt, η

1it)− bit(η1

it)

−ωi2eit(Q1it, η

1it)− ωi3rit(Q1

it, η1it)]

(5)

subject to the non-negativity constraints: qijt ≥ 0, and 0 ≤ ηit ≤ 1, ∀i, j, t.

8

The first four terms in (5) represent the profit which is to be maximized, the next term

represents the weighted total emission, which is to be minimized, and the last term represents

the weighted total risk, which is to be minimized.

We assume that manufacturers compete in a noncooperative manner in the sense of Nash

(1950, 1951). The optimality conditions for all manufacturers i; i = 1, ..., I, simultaneously,

can then be expressed as the following variational inequality (cf. Cruz, 2008; Bazaraa et al.,

1993; Gabay and Moulin, 1980): determine (Q1∗, η1∗) ∈ K1 satisfying:

T∑t=1

I∑i=1

J∑j=1

[∂f it (q

i∗t , η

1∗it )

∂qijt+∂cijt(q

i∗jt , η

1∗it )

∂qijt+ ωi2

∂eit(Q1∗it , η

1∗it )

∂qijt+ ωi3

∂rit(Q1∗it , η

1∗it )

∂qijt− ρi∗1jt

]

×[qijt − qi∗jt

]+

T∑t=1

I∑i=1

J∑j=1

[∂∑Tt=1 f

it (q

i∗t , η

1∗it )

∂ηit+∂∑T

t=1 cijt(q

i∗jt , η

1∗it )

∂ηit+∂∑T

t=1 bijt(η

1∗it )

∂ηit

+ωi2∂∑T

t=1 eit(Q

1∗it , η

1∗it )

∂ηit+ ωi3

∂∑T

t=1 rit(Q

1∗it , η

1∗it )

∂ηit

]×[ηit − ηi∗t

]≥ 0, ∀(Q1, η1) ∈ K1, (6)

where K1 ≡[(Q1, η1)|qijt ≥ 0, 0 ≤ ηit ≤ 1,∀i, j, t

].

2.2 Multicriteria Decision-Making Behavior of the Retailers andTheir Optimality Conditions

The retailers, in turn, are involved in transactions both with the manufacturers since they

wish to obtain the product for their retail outlets, as well as with the consumers, who are

the ultimate purchasers of the product. Thus, as depicted in Figure 1, a retailer conducts

transactions both with the manufacturers and with the consumers. The retailers are also

assumed to be multicriteria decision-makers who seek to maximize profits, to minimize their

individual risk associated with their transactions and to minimize the emissions generated

by their transactions.

Let ρj2kt denote the price charged by retailer j for the product at time period t. This price

will be determined endogenously after the complete model is solved. We assume that the

objective of a retailer is to maximize his total profit over the planning horizon T . The decision

variables of retailer j include the transaction amounts in each period, qijt; i = 1, ..., I; t =

1, ..., T with total procurement qjt =∑I

i=1 qijt, the sales made with the demand markets at

each period, qjkt; k = 1, ..., K; t = 1, ..., T, and the levels of social responsibility activities at

each period, ηjt ; t = 1, ..., T . Hence, the profit maximization problem faced by retailer j is

9

given by:

MaximizeT∑t=1

[K∑k=1

ρj2ktqjkt − c

jt(q

jt )−

I∑i=1

cjit(qijt, η

2jt)−

K∑k=1

cjkt(qjkt, η

2jt)− b

jt(η

2jt)−

I∑i=1

ρi1jtqijt

](7)

subject to:K∑k=1

qjkt ≤I∑i=1

qijt ∀t, (8)

and the non-negativity constraints: qijt ≥ 0, qjkt ≥ 0, 0 ≤ ηjt ≤ 1, ∀i, k, t.

The first term in the objective function (7) represents the revenue of retailer j, whereas

the next four terms represent various costs (see Table 2), and the last term represents the

payout to the manufacturers. Constraints (8) state that the amount of product available

for distribution to the demand markets in a time period t is less or equal to the amount

obtained in that period from the manufacturers.

In addition, we assume that each retailer seeks to minimize the emissions and waste

associated with his transactions with manufacturers and demand markets over the entire

planning horizon (cf. Cruz, 2008). Hence, the second criterion of each retailer can be

expressed mathematically as:

MinimizeT∑t=1

ejt(Q2jt, η

2jt). (9)

Furthermore, we assume that each retailer is also concerned with risk minimization. For

the sake of generality, we assume, as given, a risk function rjt (Table 5), for retailer j in time

period t. The risk functions are assumed to be continuous and convex and a function of

both the product transactions and the levels of social responsibility activities in current and

previous periods. The third criterion of each retailer can be expressed mathematically as:

MinimizeT∑t=1

rjt (Q2jt, η

2jt). (10)

Retailer j assigns the nonnegative weight ωj2 to total emissions generated, and the non-

negative weight ωj3 to total risk. The weight associated with profit maximization is set equal

to 1 and serves as the numeraire (as in the case of the manufacturers). We are now ready

to construct the multicriteria decision-making problem faced by a retailer, which combines

with appropriate individual weights the criteria of profit maximization given by (7), emis-

sion minimization given by (9), and risk minimization given by (10). Let intermediary j’s

multicriteria decision-making problem be expressed as:

MaximizeT∑t=1

[K∑k=1

ρj2ktqjkt − c

jt(q

jt )−

I∑i=1

cjit(qijt, η

2jt)−

K∑k=1

cjkt(qjkt, η

2jt)− b

jt(η

2jt)

10

−I∑i=1

ρi1jtqijt − ω

j2ejt(Q

2jt, η

2jt)− ω

j3rjt (Q

2jt, η

2jt)

], (11)

subject to:K∑k=1

qjkt ≤I∑i=1

qijt ∀t, (12)

and the non-negativity constraints: qijt ≥ 0, qjkt ≥ 0, 0 ≤ ηjt ≤ 1, ∀i, k, t.

We assume that the retailers also compete in a noncooperative manner. The optimality

conditions for all retailers simultaneously can be expressed as the variational inequality:

determine (Q1∗, Q2∗, η2∗, λ∗) ∈ K2 satisfying:

T∑t=1

J∑j=1

K∑k=1

[∂cjkt(q

j∗kt , η

2∗jt )

∂qjkt+ ωj2

∂ejt(Q2∗jt , η

2∗jt )

∂qjkt+ ωj3

∂rjt (Q2∗jt , η

2∗jt )

∂qjkt+ λjt − ρj∗2kt

]×[qjkt − q

j∗kt

]

+T∑t=1

I∑i=1

J∑j=1

[∂cjit(q

i∗jt , η

2∗jt )

∂qijt+∂cjt(q

j∗t )

∂qijt+ ωj2

∂ejt(Q2∗jt , η

2∗jt )

∂qijt+ ωj3

∂rjt (Q2∗jt , η

2∗jt )

∂qijt+ ρi∗1jt − λjt

]×[qijt − qi∗jt

]+

T∑t=1

I∑i=1

J∑j=1

[∂∑Tt=1 c

jit(q

i∗jt , η

2∗jt )

∂ηjt+∂∑T

t=1 cjkt(q

j∗kt , η

2∗jt )

∂ηjt+∂∑T

t=1 bjt(η

2∗jt )

∂ηjt

+ωj2∂∑T

t=1 ejt(Q

2∗jt , η

2∗jt )

∂ηjt+ ωj3

∂∑T

t=1 rjt (Q

2∗jt , η

2∗jt )

∂ηjt

]×[ηjt − η

j∗t

]+

J∑j=1

T∑t=1

[I∑i=1

qi∗jt −K∑k=1

qj∗kt

]×[λjt − λ∗jt

]≥ 0, ∀((Q1, Q2, η2, λ) ∈ K2, (13)

where K2 ≡[(Q1, Q2, η2, λ)| qijt ≥ 0, qjkt ≥ 0, 0 ≤ ηjt ≤ 1, λjt ≥ 0, ∀i, j, k, t

].

Here λjt denotes the Lagrange multiplier associated with constraint (12) and λ is the

column vector of all the retailers’ Lagrange multipliers. These Lagrange multipliers can also

be interpreted as shadow prices. Indeed, according to the fifth term in (13), λjt serves as

the price to clear the market at retailer j at time t.

2.3 Equilibrium Conditions for the Demand Markets

We now describe the behavior of the consumers located at the demand markets. The con-

sumers take into account in making their consumption decisions not only the prices charged

for the product by the retailers, ρj∗2kt; j = 1, ..., J ; t = 1, ..., T, but also the unit transaction

costs to obtain the product. The equilibrium conditions for consumers at demand market

11

k, (cf. Samuelson, 1952) take the form: for all retailers j; j = 1, ..., J and time periods

t; t = 1, ..., T :

ρj∗2kt + ckjt(qj∗kt , η

2∗jt )

{= ρk∗3t , if qj∗kt > 0

≥ ρk∗3t , if qj∗kt = 0,(14)

and

dkt (ρ∗3t)

=

J∑j=1

qj∗kt , if ρk∗3t > 0

≤J∑j=1

qj∗kt , if ρk∗3t = 0.

(15)

Conditions (14) state that, in equilibrium, at each time period, if the consumers at demand

market k purchase the product from retailer j, then the price charged by the retailer for the

product at that time period plus the unit transaction cost is equal to the price that the

consumers are willing to pay for the product at that time period. If the price plus the unit

transaction cost is higher than the price the consumers are willing to pay at the demand

market then there will be no transaction between the retailer and demand market pair at

that time period. Conditions (15) state, in turn, that if the equilibrium price the consumers

are willing to pay for the product at the demand market at the time period is positive,

then the quantities purchased of the product from the retailers at that time period will be

precisely equal to the demand for that product at the demand market at that time period.

In equilibrium, conditions (14), and (15) will have to hold for all demand markets and

these can be expressed as an inequality analogous to those in (6) and (13) and given by:

determine (Q2∗, ρ∗3) ∈ K3, such that

J∑j=1

K∑k=1

T∑t=1

[ρj∗2kt + ckjt(q

j∗kt , η

2∗jt )− ρk∗3t

]×[qjkt − q

j∗kt

]+

K∑k=1

T∑t=1

[J∑j=1

qj∗kt − dkt (ρ∗3t)

]×[ρk3t − ρk∗3t

]≥ 0,

∀(Q2, ρ3) ∈ K3, where K3 ≡[(Q2, ρ3)|(Q2, ρ3) ∈ R(1+J)KT

+

]. (16)

2.4 The Equilibrium Conditions of the Multiperiod Supply ChainNetwork

In equilibrium, the optimality conditions for all manufacturers, the optimality conditions for

all retailers, and the equilibrium conditions for all demand markets must hold simultaneously

so that no decision-maker can be better off by altering his decisions. Also, the shipments

that the manufacturers ship to the retailers must be equal to the shipments that the retailers

accept from the manufacturers. Similarly, the quantities of the product obtained by the

consumers at the demand markets must coincide with the amounts sold by the retailers.

12

Definition 1: Multiperiod Supply Chain Network Equilibrium

The equilibrium state of the multiperiod supply chain network is one where the sum of (6),

(13), and (16) is satisfied, so that no decision-maker has any incentive to alter his decisions.

Theorem 1: Variational Inequality Formulation

The equilibrium conditions governing the multiperiod supply chain network model are equiv-

alent to the solution of the variational inequality problem given by:

determine (Q1∗, Q2∗, η1∗, η2∗, λ∗, ρ∗3) ∈ K4

T∑t=1

I∑i=1

J∑j=1

[∂f it (q

i∗t , η

1∗it )

∂qijt+∂cijt(q

i∗jt , η

1∗it )

∂qijt+ ωi2

∂eit(Q1∗it , η

1∗it )

∂qijt+ ωi3

∂rit(Q1∗it , η

1∗it )

∂qijt

+∂cjit(q

i∗jt , η

2∗jt )

∂qijt+∂cjt(q

j∗t )

∂qijt+ ωj2

∂ejt(Q2∗jt , η

2∗jt )

∂qijt+ ωj3

∂rjt (Q2∗jt , η

2∗jt )

∂qijt− λjt

]×[qijt − qi∗jt

]+

T∑t=1

J∑j=1

K∑k=1

[∂cjkt(q

j∗kt , η

2∗jt )

∂qjkt+ ωj2

∂ejt(Q2∗jt , η

2∗jt )

∂qjkt+ ωj3

∂rjt (Q2∗jt , η

2∗jt )

∂qjkt+ ckjt(q

j∗kt , η

2∗jt ) + λjt − ρk∗3t

]×[qjkt − q

j∗kt

]+

T∑t=1

I∑i=1

J∑j=1

[∂∑Tt=1 f

it (q

i∗t , η

1∗it )

∂ηit+∂∑T

t=1 cijt(q

i∗jt , η

1∗it )

∂ηit+∂∑T

t=1 bijt(η

1∗it )

∂ηit

+ωi2∂∑T

t=1 eit(Q

1∗it , η

1∗it )

∂ηit+ ωi3

∂∑T

t=1 rit(Q

1∗it , η

1∗it )

∂ηit

]×[ηit − ηi∗t

]+

T∑t=1

I∑i=1

J∑j=1

[∂∑Tt=1 c

jit(q

i∗jt , η

2∗jt )

∂ηjt+∂∑T

t=1 cjkt(q

j∗kt , η

2∗jt )

∂ηjt+∂∑T

t=1 bjt(η

2∗jt )

∂ηjt

+ωj2∂∑T

t=1 ejt(Q

2∗jt , η

2∗jt )

∂ηjt+ ωj3

∂∑T

t=1 rjt (Q

2∗jt , η

2∗jt )

∂ηjt

]×[ηjt − η

j∗t

]+

J∑j=1

T∑t=1

[I∑i=1

qi∗jt −K∑k=1

qj∗kt

]×[λjt − λ∗jt

]+

K∑k=1

T∑t=1

[J∑j=1

qj∗kt − dkt (ρ∗3t)

]×[ρk3t − ρk∗3t

]≥ 0,

∀(Q1, Q2, η1, η2, λ, ρ3) (17)

where K4 ≡[(Q1, Q2, η1, η2, λ, ρ3)| qijt ≥ 0, qjkt ≥ 0, 0 ≤ ηit ≤ 1, 0 ≤ ηjt ≤ 1, λjt ≥ 0,

ρk3t ≥ 0, ∀i, j, k, t].

Proof: Summation of inequalities (6), (13), and (16), yields, after algebraic simplification,

the variational inequality (17). We now establish the converse, that is, that a solution to

variational inequality (17) satisfies the sum of conditions (6), (13), and (16) and is, hence,

13

an equilibrium according to Definition 1. To inequality (17) add the term +ρi∗1jt - ρi∗1jt to the

fifth set of brackets preceding the multiplication sign. Similarly, add the term +ρj∗2kt−ρj∗2kt to

the term preceding the sixth multiplication sign in (17). The addition of such terms does not

alter (17) since the value of these terms is zero. The resulting inequality can be rewritten to

become equivalent to the price and material flow pattern satisfying the sum of the conditions

(6), (13), and (16). The proof is complete. �

We now put variational inequality (17) into standard form which will be utilized in the

subsequent sections. For additional background on variational inequalities and their appli-

cations, see the book by Nagurney (1999). In particular, we have that variational inequality

(17) can be expressed as:

〈F (X∗), X −X∗〉 ≥ 0, ∀X ∈ K4, (18)

where X ≡ (Q1, Q2, η1, η2, λ, ρ3) and F (X) ≡ (Fijt, Fjkt, Fit, Fjt, Fjt, Fkt) with indices: i =

1, . . . , I; j = 1, . . . , J ; k = 1, . . . , K; t = 1, . . . , T , and the specific components of F given by

the functional terms preceding the multiplication signs in (17), respectively. The term 〈·, ·〉denotes the inner product in N -dimensional Euclidean space.

We now describe how to recover the prices associated with the first two tiers of nodes in

the supply chain network. Clearly, the components of the vector ρ∗3 are obtained directly from

the solution of variational inequality (17). To recover the second tier prices associated with

the retailers one can (after solving variational inequality (17) for the particular numerical

problem) either (cf. (16)) set ρj∗2kt =[ρk∗3t − ckjt(q

j∗kt , η

2∗jt )], for any j, k, t such that qj∗kt > 0, or

(cf. (13)) for any qj∗kt > 0, set ρj∗2kt =

[∂cjkt(q

j∗kt ,η

2∗jt )

∂qjkt

+ ωj2∂ej

t (Q2∗jt ,η

2∗jt )

∂qjkt

+ ωj3∂rj

t (Q2∗jt ,η

2∗jt )

∂qjkt

+ λjt

].

Similarly, from (6) we can infer that the top tier prices comprising the vector ρ∗1 can be

recovered (once the variational inequality (17) is solved with particular data) thus: for any

i, j, t, such that qi∗jt > 0, set ρi∗1jt=[∂f i

t (qi∗t ,η

1∗it )

∂qijt

+∂cijt(q

i∗jt ,η

1∗it )

∂qijt

+ ωi2∂ei

t(Q1∗it ,η

1∗it )

∂qijt

+ ωi3∂ri

t(Q1∗it ,η

1∗it )

∂qijt

], or,

equivalently to[λjt −

∂cjit(qi∗jt ,η

2∗jt )

∂qijt

− ∂cjt (qj∗t )

∂qijt− ωj2

∂ejt (Q

2∗jt ,η

2∗jt )

∂qijt

− ωj3∂rj

t (Q2∗jt ,η

2∗jt )

∂qijt

](cf. (13)).

Under the above pricing mechanism, the optimality conditions (6) and (13) as well as the

equilibrium conditions (16) also hold separately (as well as for each individual decision-maker

at any time period).

3 Computational Procedure and Studies

In this section, we consider an algorithm for the computation of solutions to variational

inequality (17). The algorithm that is proposed is the Euler-type method, which is induced

14

by the general iterative scheme of Dupuis and Nagurney (1993).

3.1 The Discrete-Time Algorithm

The Euler Method

Step 0: Initialization

Set X0 = (Q10, Q20, η10, η20, λ0, ρ03) ∈ K4. Let T denote an iteration counter and set T = 1.

Set the sequence {aT } so that∑∞T =1 aT =∞, aT > 0, aT → 0, as T → ∞ (such a sequence

is required for convergence of the algorithm).

Step 1: Computation

Compute XT = (Q1T , Q2T , η1T , η2T , λT , ρT3 ) ∈ K4 by solving the variational inequality sub-

problem:

〈XT + aT F (XT −1)−XT −1, X −XT 〉 ≥ 0, ∀X ∈ K4. (19)

Step 2: Convergence Verification

If |XT −XT −1| ≤ ε, with ε > 0, a pre-specified tolerance, then stop; otherwise, set T := T +1,

and go to Step 1.

Note that this algorithm has been applied to-date to solve a plethora of network models

(see, e.g., Cruz, 2008; Nagurney et al., 2005; Nagurney and Dong, 2002).

3.2 Computational Studies

In this computational study, we analyze the impact of changes in parameters in the mul-

tiperiod supply chain network model on equilibrium product flows, prices, and CSR levels.

The supply chain model is represented in Figure 2. It consists of 2 manufacturers i, i = 1, 2;

2 retailers j, j = 1, 2; 2 demand markets k, k = 1, 2; and 2 time periods t, t = 1, 2. CSR

levels create the links between the first and the second time period.

The functions used in the numerical analysis are shown in Table 7. Manufacturers incur

transaction costs and risk for their transactions with retailers. Retailers incur transaction

costs and risk for their transactions with demand markets. Demand markets face unit

transaction costs when transacting with retailers. Transactions between manufacturers and

retailers and between retailers and demand markets cause emissions. We assume that all the

prices and costs at the second period are measured in terms of their dollar value at the first

period.

15

We assume that production costs, transaction costs, and emissions are linear increasing

in the amount of products produced/transacted. Fixed costs are assumed sunk costs and

are not considered. Demand at each demand market is linear decreasing with the price at

this demand market. Risk is a quadratic function of product transacted. In the field of

finance, the measurement of risk has included the use of variance-covariance matrices, yield-

ing quadratic expressions for risk (see also, e.g., Nagurney and Siokos, 1997). In addition,

in finance, the bicriterion optimization problem of net revenue maximization and risk mini-

mization is fairly standard (see also, e.g., Dong and Nagurney, 2001). We use quadratic risk

functions in a supply chain context in this sensitivity analysis as it has been suggested in,

for example, Nagurney et al. (2005).

Emissions, transaction costs, and risk in the first period are a function of CSR levels in the

first period. Emissions, transaction costs, and risk in the second period are a function of CSR

levels in the first and second period. These assumptions are consistent with many empirical

studies. The empirical work of Dyer and Chu (2003) indicate that, as the levels of social

responsibility activities increase the overall cost would decrease. Furthermore, as we mention

in the introduction, Spicer (1978) and Orlitsky and Benjamin (2001) looked directly at the

CSR-risk relationship and found evidence for a negative correlation between the two: as CSR

increased, risk decreased. This is because firms with proactive CSR that engage in managerial

practices like environmental assessment and stakeholder management (Wood, 1991; Bowman,

1980) tend to anticipate and reduce potential sources of business risk, such as potential

governmental regulation, labor unrest, or environmental damage (Orlitzky and Benjamin,

2001). Moreover, in addition to any direct environmental and cost reduction benefits, CRS

activities has a significant and favorable impact on the firm’s perceived riskiness to investors

and, accordingly, its cost of equity capital and value in the market place (Feldman et al.,

1996).

The costs for CSR activities are quadratic. This indicates that it gets more expensive to

increase CSR levels, the higher they already are.

F1it and A1ijt represent parameters ∀ i=1,2, j=1,2, and t=1,2. At the beginning of the

study we set A1ijt = 0.2 ∀ i, j, and t, and F1it = 1 ∀i, t. Our parameter settings reflect

that production costs are typically larger than the transaction costs. All the weights are set

equal to 1. The Euler method was implemented in Matlab to solve these numerical examples.

Alpha is set to 1/(2+0.4t). The parameter for convergence is set to 0.0001.

4 Discussion of Results

The initial settings lead to the following equilibrium product flows, prices, and CSR levels:

The product transaction amounts q∗ijt = q∗jkt= 14.33 for i=1,2, j=1,2, in the first time period.

16

q∗ijt = q∗jkt= 14.50 for i=1,2, j=1,2 in the second time period. The CSR levels are at their

upper limit of 1 for all time periods. The prices at the manufacturers ρi∗1jt = 40.27 for i=1,2,

j=1,2 at the first time period and ρi∗1jt=40.20 for i=1,2, j=1,2 at the second time period.

The prices that retailers charge ρj∗2kt= 70.53 for j = 1, 2, k = 1, 2 at the first time period and

ρj∗2kt = 70.40 for j = 1, 2, k = 1, 2 at the second time period.

Transaction costs, emissions, and risk in the first period are a function of CSR levels in

the first period. Transaction costs, emissions, and risk in the second period are a function of

CSR levels in the first and second period. In equilibrium, all decision-makers establish CSR

levels of 1 in both periods. Hence, decision-makers face a stronger reduction in transaction

costs, emissions, and risk in the second period than in the first period. This leads to higher

product flows and lower prices in the second period. This highlights that to understand the

development of product flows and prices, it is of utmost importance to consider CSR levels

and their impact in all time periods.

There are a lot of debates concerning the measurement of the costs for CSR activities

as well as the measurement of their impact. Hence, in this analysis we want to show how

changes in these parameters impact equilibrium results in the supply chain network. Due to

strong uncertainty in the measurement, we consider a wide range of values.

In the first two numerical examples we look at the impact of an increase in costs for CSR

activities on CSR levels, product flows, and prices. In the third numerical example, we look

at the effect of a change in the impact that CSR levels have on transaction costs on CSR

levels, product flows, and prices. In all the examples we use the following abbreviations: M1:

manufacturer 1, M2: manufacturer 2, R1: retailer 1, R2: retailer 2, DM: demand market,

T1: time period 1 and T2: time period 2.

Example 1

In Example 1 we vary the parameter in the cost function for CSR activities, F1it, for both

manufacturers in the range 1 to 25. The effects of these changes on CSR levels are shown in

Figure 3(a).

An increase in the parameters in manufacturers’ cost functions for CSR activities leads

to decreasing manufacturers’ CSR levels. CSR levels in the second period react to smaller

changes in the parameters than CSR levels in the first period. The reason is that transaction

costs, emissions, and risk in the first period are influenced by CSR levels in the first period;

however, transaction costs, emissions, and risk in the second period are influenced by CSR

levels in the first and second period. Hence, CSR levels in the first period have a stronger total

positive impact than CSR levels in the second period. This highlights that the equilibrium

CSR levels will be strongly influenced by the planning horizons of different companies. Figure

3(b) shows the impact of these changes on product flows.

17

We can see from Figure 3(b) that in the original scenario product flows in period 2 are

higher than product flows in period 1. The reason is that CSR levels in the first and second

period positively impact emissions, risk and costs in the second period. However, we see

that product flows in the second period are the first to decline since they are impacted by

CSR levels in the second period which decline earlier.

Example 2

In Example 2, we analyze the effects of changes in parameters in manufacturer 1’s cost

functions for CSR activities.

Figure 4 shows the effects of changes in F11t in the range 1 to 25. Increasing cost pa-

rameters lead to decreasing CSR levels for manufacturer 1. First CSR levels in the second

period decrease and then the CSR levels in the first period decrease. The CSR levels for

manufacturer 2 do not change.

Figure 4(b) highlights the impact of these changes on product flows. When manufacturer

1’s CSR levels decrease, product flows originating from manufacturer 1 decrease and product

flows originating from manufacturer 2 increase. Product flows reaching demand markets

decrease in both time periods. Since overall product flows decrease, prices at demand markets

increase. Hence, we can see that manufacturer 2 benefits from this change in the cost

structure of manufacturer 1.

Example 3

In Example 3 we want to highlight the effects of an increase of the impact of CSR levels

on transaction costs. Specifically, we want to highlight how the equilibrium solution changes

if the transaction costs of manufacturer 2 respond stronger to changes in CSR levels. We

chose the impact of CSR levels on transaction costs due to the high and rising importance

of transaction costs. Results from this section can also be used to understand the effects of

changes of the impact of CSR levels on emissions and risk.

We keep all the parameters as in the base case, except for the parameter for the cost for

CSR activities which we set to 6. We vary manufacturer 1’s parameter that measures the

impact of CSR activities on transaction costs (A11jt) between 0 and 0.9. Table 8 indicates

how these changes impact CSR levels.

We see that when A11jt = 0 CSR levels for manufacturer 1 for period 2 are below 1. Man-

ufacturer 1’s CSR levels increase as the impact of CSR levels on transaction costs increases

until they reach their upper limit of 1. Starting from A11jt= 0.3, CSR levels for manufacturer

2 for period 2 start decreasing.

The reason for this change can be found in Figure 5. We can see that with an increase

in A11jt, product flows for manufacturer 1 increase and product flows for manufacturer 2

18

decrease. This is true for both time periods. Since the benefit of CSR levels depends on

the product flows, reduced product flows lead to reduced CSR levels in the second period.

These changes do not impact CSR levels at period 1 which stay at their upper limit. Total

product flows increase and prices at demand markets decrease.

These examples, although stylized, have been presented to show both the model and the

computational procedure. Obviously, different input data and dimensions of the problems

solved will affect the equilibrium product transaction, levels of social responsibility activities,

and price patterns. They highlight that changes in benefits or costs associated with CSR

levels do not only impact the optimal CSR levels and product flows of the company that

faces these changes. These changes also have the potential to impact optimal CSR levels,

product flows and prices of other companies in the supply chain. Furthermore, these changes

can potentially impact competitors’ performance.

The numerical examples highlight that the best outcome for the supply chain as a whole

might not always be achieved if each member in the supply chain determines the optimal

levels of CSR based only on his/her own costs and benefits. Hence, the optimal investment

in CSR levels in a supply chain constitutes a social dilemma as described in McCarter and

Northcraft (2007) where decision-makers must choose between doing what is in their own

best interest or the overall supply chain best interest. We illustrate that if supply chain

decision-makers choose to do what is in their own best interest, this will lead to an outcome

that does not provide benefits for the supply chain (Dawes, 1980; McCarter and Northcraft,

2007).

In reality, we can see that many companies deal with this dilemma by expanding their re-

sponsibility for their products beyond their sales and delivery locations (Bloemhof-Ruwaard

et al., 1995) and by managing the CSR of their partners within the supply chain (Kolk and

Tudder, 2002; Emmelhainz and Adams, 1999). The model developed in this paper allows

managers to see how changes in supply chain partners’ CSR activities affect their own oper-

ations. The model, hence, provides managers with insights concerning potential increases in

supply chain performance that might be achieved if CSR activities are coordinated among

supply chain companies or centrally managed.

The framework developed in this paper represents a powerful decision-making tool with

which the potential stake holders, such as manufacturers, distributors, and/or retailers, will

be able to model a current market situation and explore the effects of various perturbations

to the data, different environmental strategies as well as the effects of changes in the number

of manufacturers, retailers, and demand markets. Moreover, each decision-maker can also

use this framework to determine what should be the optimal level of investment in CSR

that minimizes potential risk and environmental impact. In addition, our framework is not

only useful for these stake holders, but also for the policy makers. It will allow the policy

makers to model the market and explore the potential benefits of different policies related

19

to emission/waste regulations. The proposed computational procedure allows for massive

parallelization which, in turn, makes the computation of large models quick and efficient.

5 Conclusions

CSR can potentially decrease production inefficiencies, reduce cost and risk and at the same

time allow companies to increase sales. As a result of lower costs, lower risk and increase in

sales, companies become more profitable. However, we expect that as the investment in CSR

activities increases, the return on investment is decreasing. Therefore, it is very important

for managers to find the optimal level of investment in CSR activities so that he\she can

allocate the appropriate amount of resources to these activities over time. The optimal levels

of CSR activities are impacted by factors within the firm as well as its business environment.

In this paper, we develop a framework for the analysis of the optimal levels of corporate

social responsibility activities in a multiperiod supply chain network consisting of manufac-

turers, retailers, and consumers. The framework explicitly includes the behavior of decision-

makers within the supply chain as well as the supply chain structure while it implicitly

includes institutional factors in the cost and risk functions. Manufacturers and retailers are

multicriteria decision-makers who decide about their production and transaction quantities

as well as the amount of social responsibility activities they want to pursue to maximize

net return, minimize emissions, and minimize risk over the multiperiod planning horizon.

We construct the finite-dimensional variational inequality governing the equilibrium of the

multiperiod competitive supply chain network. The model allows us to investigate the in-

terplay of the heterogeneous decision-makers in the supply chain network and to compute

the resultant equilibrium pattern of product outputs, transactions, product prices, and lev-

els of social responsibility activities. A computational procedure that exploits the network

structure of the problem is proposed and then applied to several numerical examples.

We analyze the impact of the cost of CSR on the investment level in CSR activities.

We found that as the cost of CSR activities increases the firm will have less incentive to

invest in them. Here, as in McWilliams and Siegel (2001), the ideal level of CSR should be

determined by a long term cost benefit analysis. In the short run, the cost of CSR may seem

high, however, this cost would be less in the long run compared to the cost of liability for

pollution, compliance with regulation, dangerous operations, use of hazardous raw materials,

production of hazardous waste, and for health and safety issues. Moreover, these liabilities

may cost companies their reputation (Dowling, 2001; Frombrun, 2001), brand image, sales,

access to markets and financial investments (Feldman et al., 1997). In conclusion, man-

agers should treat their decision regarding CSR as they treat all their long term investment

decisions.

20

The numerical examples highlight that the best outcome for the supply chain as a whole

might not always be achieved if each member in the supply chain determines the optimal

levels of CSR only based on his/her own costs and benefits. It is important that CSR

activities are coordinated among different firms in the supply chain. Increased coordination

among firms in the supply chain leads to a multitude of additional positive effects. It

has the potential to reduce network related risk (Johnson, 2001; Norrman and Jansson,

2004). Furthermore, Simpson and Power (2005) indicate that strong relationships in the

network are capable of leading to programs of collaborative waste reduction, environmental

innovation at the interface, cost-effective environmental solutions, the rapid development and

uptake of innovation in environmental technologies, and allows firms to better understand

the environmental impact of their supply chains.

The model is flexible enough to analyze how different objectives of firms (McWilliams

and Siegel, 2001), legal and institutional factors (Williams and Aguilera, 2008), and country

differences (Matten and Moon, 2008) impact optimal CSR levels. The model developed in

this paper provides a foundation for future studies that attempt to test assumptions in the

conceptual literature. As a first step it is necessary to empirically validate the following rela-

tionships: 1) the relationship between levels of social responsibility activities and transaction

costs; 2) the relationship between levels of social responsibility activities and total emission

(waste) generated; and 3) the CSR\Risk\Profit relationships. Second, as operations of the

firms become more globalized it is important to analyze how the concept of CSR is applied

in different countries with different culture, as well as rules and regulation. Future research

will also include the extension of this framework to the international arena. Finally, we shall

develop a dynamic model that takes into consideration the rate of change in price, cost, risk

and profit as the investment in CSR increases or decreases over time.

Acknowledgements

The authors gratefully acknowledge the constructive and helpful comments of two anonymous

referees on the earlier version of the manuscript.

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26

Manufacturers

Demand Markets

CSR

CSR

Manufacturers

Demand Markets

Manufacturers

Demand Markets

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Figure 1: Time Evolution of the Supply Chain Network Model

Manufacturers

Retailers

Demand Markets

CSR Levels

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Retailers

Demand Markets

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Figure 2: Network Structure of the Numerical Examples

27

(a) Levels of CSR (b) Product Flows

Figure 3: Effects of Changes in Manufacturers’ Cost for CSR Activities on the Levels of CSRand Product Flows

(a) Levels of CSR (b) Product Flows

Figure 4: Effects of Changes in Manufacturer 1’s Cost for CSR Activities on the Levels ofCSR and Product Flows

Figure 5: Effects of Changes of CSR Levels on Transaction Costs

28

Table 1: Variables in the Supply Chain Network

Notation DefinitionQ1 IJT -dimensional vector of product flows transacted between each

manufacturer and each retailer at each time period with component ijtdenoted by qijt

Q1it J-dimensional vector of product flows transacted between manufacturer

i and each retailer at time tQ2 JKT -dimensional vector of product flows transacted between each re-

tailer and each demand market at each time period with componentjkt denoted by qjkt

Q2t JK-dimensional vector of product flows transacted between each re-

tailer and each demand market at time t with component jkt denotedby qjkt

Q2jt I + K-dimensional vector of product flows sent to and from retailer j

at time tη1 IT -dimensional vector of levels of social responsibility activities of each

manufacturer at each time period with component it denoted by ηitη1it t-dimensional vector of levels of social responsibility activities of man-

ufacturer i at time periods 1 to tη2 JT -dimensional vector of levels of social responsibility activities of each

retailer at each time period with component jt denoted by ηjtη2jt t-dimensional vector of the levels of social responsibility activities of

retailer j for time periods 1 to t with component t denoted by ηjtρ3 KT -dimensional vector of prices of the product at each demand market

with component kt denoted by ρk3tρ3t K-dimensional vector of prices of the product at each demand market

at time t with component k denoted by ρk3t

Table 2: Production, Handling, and Transaction Cost Functions

f it (qit, η

1it) production cost of manufacturer i at time period t with qit =

∑Jj=1 q

ijt

cijt(qijt, η

1it) transaction cost of manufacturer i with retailer j at period t

cjit(qijt, η

2jt) transaction cost of retailer j with manufacturer i at time period t

cjkt(qjkt, η

2jt) transaction cost of retailer j with demand market k at time period t

cjt(qjt ) handling cost of retailer j at time period t with qjt =

∑Ii=1 q

ijt

ckjt(qjkt, η

2jt) unit transaction cost of demand market k transacting with retailer j

at time period t

29

Table 3: Cost Functions for Social Responsibility Activities

Notation Definitionbit(η

1it) cost functions for social responsibility activities associated with

manufacturer i in time period t

bjt(η2jt) cost functions for social responsibility activities associated with

retailer j in time period t

Table 4: Emission Functions

Notation Definitioneit(Q

1it, η

1it) emission function associated with manufacturer i at period t

ejt(Q2jt, η

2jt) emission function associated with retailer j at period t

Table 5: Risk Functions

Notation Definitionrit(Q

1it, η

1it) risk incurred by manufacturer i at period t

rjt (Q2jt, η

2jt) risk incurred by retailer j at period t

Table 6: Demand Function

Notation Definitiondkt (ρ3t) demand function at demand market k at time period t

30

Table 7: Functions for Computational Study

Notation Definition

f it (qit) = 10× (

∑2j=1 q

ijt) Production costs faced by manufacturer i

at period tdkt(ρ3) = 100− ρ3kt Demand functions

cijt(qijt, η

1it) = qijt − A1ijt × (

∑tt=1 η

it)× qijt Transaction costs faced by manufacturer i

transacting with retailer j at period t

cjkt(qjkt, η

2jt) = qjkt − 0.2× (

∑tt=1 η

jt )× q

jkt Transaction costs faced by retailer j trans-

acting with demand market kat period t

ckjt(Q2t , η

2jt) = 1− 0.2× (

∑tt=1 η

jt ) Unit transaction costs faced by consumer

k transacting with retailer j at period t

rit(Q1it, η

1it)) =

∑2j=1(q

ijt

2 − 0.2× (∑t

t=1 ηit)) Risk faced by manufacturer i at period t

rjt (Q2jt, η

2jt) =

∑2i=1(q

jkt

2 − 0.2× (∑t

t=1 ηjt )) Risk faced by retailer j at period t

bit(η1it) = F1it × ηit

2Cost for establishing social responsibilityactivities associated with manufacturer iin time period t

bjt(η2jt) = ηjt

2Cost for establishing social responsibilityactivities associated with retailer jin time period t

eit(Q1it, η

1it) =

∑2j=1(q

ijt− 0.2× (

∑tt=1 η

it)× qijt) Emission associated with manufacturer i

at period t

ejt(Q2jt, η

2jt) =

∑2k=1(q

jkt−0.2×(

∑tt=1 η

jt )×q

jkt) Emission associated with retailer j at pe-

riod t

Table 8: The Changes in CSR Levels

A11jt 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9CSR for M1 at T2 0.511 0.754 1 1 1 1 1 1 1 1CSR for M2 at T2 1 1 1 0.998 0.996 0.993 0.991 0.989 0.987 0.984

31