A microeconomic model of corporate

A MICROECONOMIC MODEL OF CORPORATE
SOCIAL RESPONSIBILITYmeca_4087 69..95
Tommy Lundgren*
Centre for Environmental and Resource Economics (CERE),
Umeå School of Business
(December 2008; revised January 2010)
ABSTRACT
This paper explores the economic mechanisms behind corporate social responsibility (CSR) in a
microeconomic model of the firm. The study’s motivation is to shed light on the potential causes of the
observed phenomena of voluntary over-compliance among firms. We investigate how assumptions
about costs and benefits affect CSR behavior through a stock of goodwill capital. In optimum, the firm
must balance marginal costs and benefits of investing in CSR. We characterize the equilibrium and
examine comparative statics and dynamics from a parameterized model. Finally, we link some of the
model’s results to the empirical literature on CSR.
1. INTRODUCTION
According to conventional economic theory, firms maximize profits subject
to technological and other constraints. Without economic incentives like
taxes or quantitative regulations, the firm might, for example, pollute too
much, or engage in some other socially detrimental behavior. A cursory look
around business environments today suggests that this view might be a bit
old-fashioned. Indeed, firms spend resources to convince potential consum-
ers and other stakeholders that they are more socially responsible than what
the authorities or society demand. This paper seeks to explore the economic
* Financial support was provided by the MISTRA research program ‘Sustainable investments’.
Useful comments from Karl-Gustaf Löfgren, Bert Scholtens, Lammertjan Dam, Geoffrey Heal,
and participants at the workshop ‘Corporate social responsibility and socially responsible invest-
ments’, Umeå University, School of Business, Umeå-Sweden, 16 November 2007, ‘3rd Atlantic
Workshop on Energy and Environmental Economics’, Universidade de Vigo A Toxa, Galicia,
4–5 July 2008, and the Centre of Environmental and Resource Economics Workshop, 2 October
2008, improved the paper significantly. Helpful comments and suggestions from two anonymous
referees are also gratefully acknowledged.
Metroeconomica 62:1 (2011) 69–95
doi: 10.1111/j.1467-999X.2010.04087.x
© 2010 Blackwell Publishing Ltd

mechanisms behind corporate social responsibility based on a microeco-
nomic perspective of the firm. The ultimate objective is to shed some light on
the rationale behind the observed voluntary over-compliance among firms.
Since the publication of the report ‘Our Common Future’ (1987) by the
World Commission on Environment and Development, the terms sustainabil-
ity and sustainable development have become prominent in the public debate.
The sustainability debate has, in part, focused on what companies can do to
facilitate sustainable development, so-called corporate social responsibility
(CSR). One of the first attempts to bring CSR into the public debate was from
Milton Friedman who in his article in the New York Times Magazine argued
that ‘the corporate social responsibility of firms is to maximize its profits’
(Friedman, 1970). While this statement may appeal specifically to neoclassical
economists, it may also seem provocative and without nuance to others. But
as we shall see, profit maximization does not have to be in conflict with
social responsibility (see, for example, Husted and Salazar, 2006). Following
Friedman’s initial definition of CSR, a number of others have followed. For
example, Heal (2008) suggests that CSR is ‘. . . the interactions between
corporate behavior and civil and legal society, and how these interactions
structure the company’s incentives on social and environmental issues’.1
We
simply consider CSR to be actions that, to some degree, imply corporate
beyond-compliance behavior in the social and/or the environmental arena.
Most of the empirical studies of the effects of CSR on either a firm’s
economic or financial performance have been performed during the last two
or three decades (and most focus on financial performance). The plethora of
individual studies has led to at least 10–15 reviews, many of them assessed in
Margolis and Walsh (2001), which reviews nearly 100 separate studies. Also,
Reinhardt (2000), Orlitzky and Benjamin (2001), Orlitzky et al. (2003), Lyon
and Maxwell (2004) and Orlitzky and Swanson (2008) summarize to a large
extent the bulk of empirical CSR literature that is currently available. Hay
et al. (2005) offer a comprehensive review from the fields of economics, law
and business. The evidence from these review studies is not conclusive, but
empirical results seem to indicate that CSR leads to positive financial per-
formance for a firm.2
When it comes to CSR and economic performance, the research is less
extensive. Indeed, economic and financial performances are linked,3
but there
are some interesting differences worth noting. The Journal of Productivity
1
See also McWilliams and Siegel (2001), Hay et al. (2005) or Portney (2008) for similar defini-
tions of CSR.
2
Possibly because of publication bias, i.e. negative results tend not to be published.
3
If, for example, economic efficiency is improved by CSR this will most likely show in financial
performance such as stock price through higher profits.
70 Tommy Lundgren

Analysis recently published a theme issue on CSR and economic performance
(Paul and Siegel, 2006). They note that the vast number of studies of CSR
and the effect on financial measures is, from an economic perspective, unfor-
tunate. Instead, they suggest that a more salient issue is the relationship
between economic performance and CSR behavior, where economic perfor-
mance entails technological and economic interactions between production
of output and input demands, recognizing the opportunity costs of inputs
and capital formation. Their conclusion is that the costs of CSR must be
balanced by benefits to motivate firms to carry out such activities. Another
theme issue, Paton and Siegel (2005), can be found in Structural Change and
Economic Dynamics, where CSR is studied using empirical and theoretical
tools from both finance and economics.
Compared with the empirical literature on CSR, attempts to formally
model the microeconomic mechanisms behind voluntary over-compliance at
the firm level in a consistent way are less plentiful. However, a number of
studies are worth mentioning. Bergman (1995) provides several interesting
simple static micro-models, in terms of environmentally friendly firms, that
provide a rationale for CSR behavior. A game theoretical model of voluntary
over-compliance is proposed by Arora and Gangopadhyay (1995), who
assume that firms signal their ‘greenness’. If consumers prefer to buy prod-
ucts from a ‘greener’ firm, then the cost of being environmentally friendly
may be justified by higher revenues.
McWilliams and Siegel (2001) and Baron (2001) are probably the first two
papers that explicitly model profit-maximizing CSR in a way that accounts
for the wide range of costs and benefits to be considered when investing in
socially responsible projects.4
These papers identify the implications of CSR
when information is asymmetric between firms and consumers, discuss the
importance of reputation, and examine the strategic use of CSR (e.g. the
problem of ‘greenwashing’5
). The reputation and strategic dimension of CSR
is discussed further in a subsequent dialogue between McWilliams and Siegel
(2002), Piga (2002) and Siegel and Vitaliano (2007). Furthermore, McWill-
iams and Siegel (2001) explicitly argue that the demand for CSR will be
4
McWilliams and Siegel (2001) outline a model in which two firms sell identical goods, but one
company decides to add an additional social attribute to its product. This attribute is valued by
some consumers or, potentially, by other stakeholders. Firm managers conduct a cost–benefit
analysis to determine the level of resources to devote to CSR activities. That is, firms simulta-
neously assess the demand for CSR and the cost of satisfying that demand, and then determine
the optimal level of CSR to provide.
5
See, for example, Greer and Bruno (1996). ‘Greenwash’ refers to companies that disingenu-
ously spin their products and/or policies as environmentally friendly, e.g. presenting cost cuts as
reductions in resource use.
A Microeconomic Model of Corporate Social Responsibility 71

greater for experience and credence goods and services, which is empirically
confirmed by Siegel and Vitaliano (2007). Bagnoli and Watts (2003) extend
Baron (2001) and McWilliams and Siegel (2001) by analyzing how the struc-
ture of competition in the market for the private good affects CSR. Baron
(2007, 2008) provides further theoretical discussion of profit-maximizing
CSR, with a focus on managerial issues, reputation and stakeholder pressure/
incentives.
Closely related to the concept of reputation, Lundgren (2003) and Kris-
tröm and Lundgren (2003) formally introduce goodwill capital in a micro-
economic setting of the socially responsible firm. While Lundgren (2003)
concentrates on uncertainty in goodwill evolution and the timing of abate-
ment investment, Kriström and Lundgren (2003) develop a model where
voluntary abatement investments (one dimension of CSR) create a stock of
goodwill capital that enables the firm to differentiate their product. Other
product differentiation models connected to green consumerism include
Eriksson (2004) and Rodriguez-Ibeas (2007).
In our view, the most comprehensive and complete theoretical discussion
can be found in Heal (2005, 2008). Using a non-formal model he discusses
CSR from both economic and financial perspectives, and proposes how it is
reflected in financial markets. He discusses several cases in relation to CSR.
Heal defines CSR as actions to reduce externalized costs and/or to avoid
distributional conflicts. He suggests that there may be a resource allocation
role for CSR programs in cases of market failure through private–social cost
differentials. Furthermore, he argues that in sectors where social and private
costs are not in line, or where distributional conflicts are common, CSR can
play a valuable role in ensuring that ‘the invisible hand’ acts, as intended, to
produce the social good. It can also act to improve corporate profits and
guard against reputational risks.
In light of the condensed review of theoretical studies on CSR presented
here, we conclude that this type of analyses are on the rise. However, to our
knowledge, a formal dynamic microeconomic model of the firm that
accounts for several dimensions of CSR, in terms of both different types of
CSR and the various drivers and mechanisms, is non-existing. Paul R.
Portney, in Hay et al. (2005), points out that despite a vast amount of
empirical studies of CSR, very few, if any, have derived testable hypotheses
from an adequate theoretical model of the firm. Moreover, few studies clearly
identify the basic mechanisms of how socially responsible behavior leads to
a economic/financial advantage. Portney provides a general outline of how
such a model might work: by engaging in CSR, output price (price differen-
tiation), wages (higher worker productivity or lower wages through worker
satisfaction), and the cost of capital (risk reduction due to lower risk of
72 Tommy Lundgren

conflict with stakeholders) become to some degree endogenous to the firm.
Thus, profits would not depend solely on the cost of engaging in CSR, but
also on the benefits (Hay et al., 2005, p. 114). The purpose of this paper is to
build such a model within a dynamic framework and explore its properties.
To our knowledge, this is an original exercise and a contribution to the
theoretical literature on CSR.
The paper is organized as follows. In the next section, we present a dynamic
model of CSR. In section 2.1 we propose relevant benefits and costs of CSR,
introduce intertemporal features, and explicitly model goodwill capital as the
driving force for benefits. Sections 2.2 and 2.3 discuss the model properties and
establish the existence and characteristics of a stable equilibrium. In section 2.4
we offer insights from comparative statics and dynamic back-of-the-envelope
analysis. We then proceed to link some of the hypotheses that can be derived
from the model to the empirical literature on CSR. Finally, we offer some
concluding comments and suggestions for future research.
2. A FORMAL MODEL OF THE SOCIALLY RESPONSIBLE FIRM
This section outlines and discusses a firm-level model that sheds light on the
potential mechanism behind CSR. The model proposes that with stakeholders
(e.g. consumers, financial sector, government, employees, etc.) rewarding
CSR behavior, the cost of CSR may be balanced by benefits in terms of higher
profitability (see, for example, McWilliams and Siegel, 2001). Indirectly, we
are assuming that the firm is socially responsible for strategic reasons, i.e. it is
good business, or at least not bad business. The analysis relies on dynamics,
a pertinent feature of CSR since it potentially has effects on reputation or
goodwill, which has inherent intertemporal properties. Specifically, we assume
the notion of goodwill capital as a stock that acts upon a firm’s revenues and
costs in different ways. An intertemporal setting is reasonable when dealing
with CSR investments and goodwill capital. CSR projects build goodwill
capital over time, which can be seen as an intangible asset, a form of ‘reputa-
tion’. The reputational implications of CSR are highlighted and discussed in
detail in McWilliams and Siegel (2001, 2002), especially the potential effects
of asymmetric information about product attributes and firm activities. In
the model presented here, we abstract from the complications of asymmetric
information. That is, the activities of the firm and the product attributes are
perfectly transparent and known to the consumer.6
6
An asymmetric information case is investigated in Spremann (1985). He provides an interest-
ing advertising model set-up that accounts for asymmetry of information. Advertising can be

The notion of goodwill or reputation as an intangible asset is used exten-
sively in dynamic models of advertising.7
Assuming CSR efforts are signaled
(advertised) appropriately to the market, CSR investment projects can poten-
tially provide the firm with various benefits. We discuss these benefits below
using a model based on the green firm framework developed in Kriström and
Lundgren (2003), which, in turn, are inspired by advertising models such as
Dorfman and Steiner (1954), Nerlove and Arrow (1962), Gould (1970) and
Jacquemin (1973).
2.1 Benefits and costs of CSR
We begin by describing the main assumptions regarding the benefits and
costs of CSR.
Three main benefits of CSR are assumed:
B1. Consumers reward CSR efforts by a price premium (product differen-
tiation), or (equivalently) by buying more at the same price. Thus, all else
equal, this leads to increasing revenues and profits for the firm (see, for
example, Blend and Ravenswaay, 1999).
B2. Wage is to some degree endogenous to the firm; i.e. people are willing to
accept lower wages to work at a CSR firm, or work more productively at
the market wage rate (see, for example, Bolvig, 2005). This assumption
is based on the notion that employees may experience a ‘warm glow’
feeling when working at a socially responsible firm.
B3. Cost of capital is reduced because the financial sector, banks and port-
folio managers, etc., assign lower risk to a socially responsible firm (see,
for example, Heinkel et al., 2001; or Godfrey, 2005). The reason is a
lower probability of conflict with stakeholders and various interest
groups in the future (Heal, 2008, provides an extensive discussion of how
CSR can work as a risk management tool).
It is certainly possible to think of other potential benefits. For example,
utility or ‘warm glow’ from CSR experienced by firm owners (or investors)
and the potential impact on corporate strategies are not considered explicitly
used to signal product quality, augment reputation and increase sales, while consumer goodwill
is a consequence of actual experience of the product. This enables the firm to boost reputation
and sales with advertising in the short run, but over the long run ‘greenwashing’ becomes an
unsustainable strategy; reputation is damaged if the product does not live up to its expectations.
7
For excellent reviews of quantitative advertising models see Sethi (1977) and Feichtinger et al.
(1994).
74 Tommy Lundgren

here.8
Furthermore, societal benefits from CSR are not considered because
our model maintains a firm perspective. Based on a review of the CSR
literature, B1–B3 are the main benefits that emerge as the ‘usual suspects’
when trying to rationalize socially responsible behavior at the firm level.
The costs of CSR are sorted into three categories:
C1. Actual investment costs in CSR projects. Whether it is an environmental
project or a project related to human rights, there is always an invest-
ment cost of engaging in such projects. This cost is assumed to be linear
in the amount of CSR.
C2. Costs of promoting (advertising) CSR investments to stakeholders
(see, for example, Wang, 2008). Without stakeholders’ knowledge of a
firm’s socially responsible behavior, the benefits cannot be fully realized.
Together with the project investment cost (as suggested in C1), these
constitute the unit cost, or price, of CSR investments.
C3. Costs that stem from crowding-out effects of CSR. Productive invest-
ments and/or production are held back to give room for CSR (for a
discussion of the environmental investments case, see Gray and Shad-
begian, 1998). We assume that CSR may hamper ‘conventional’ firm
activities, and that this cost is increasing at an increasing rate in CSR.9
2.2 The model
Let us transform the model assumptions into a formal model in a fairly
general form. Define instantaneous profits of a firm, P, at time t as
Π Π= ( ) = ( ) − ( ) − ( )g G H R G H C G H A g, , , ,* * * (1)
The right-hand-side terms represent revenues and costs. The first two terms’
functions depend on the goodwill stock, G, and a set of parameters given by
H*, which are exogenous to the firm. We can think of H* as representing
inputs such as labor and capital that have been chosen optimally at a previ-
ous stage and are taken as given.10
This means we can abstract from H* in the
sequel, and thus focus solely on the intertemporal problem involving invest-
ment in CSR and the goodwill stock. The last term in the profit function,
8
For a discussion see, for example, Mackey et al. (2007).
9
Similar to the notion of adjustment costs in capital formation analysis.
10
The model could include these inputs as control variables, and potential links to CSR
investments and goodwill, but we keep our approach simple to capture and convey the essential
features of a socially responsible firm.

A(g), incorporates all costs associated with investments in CSR, g, a control
variable we here consider to be one-dimensional. More realistically, the
control variable could be defined as multidimensional, as CSR can take many
forms. However, to simplify, we treat CSR investment as a one-dimensional
control variable.11
This does not change the basic idea we want to convey
here, but simplifies our notation.
For the revenue function we assume that12
R R
R R
G GG> <
( ) = =
0 0
0
,
revenue with zero goodwill stock
(2)
This is the price premium or product differentiation effect. By increasing
goodwill the firm can increase revenue, but at a decreasing rate.
For the cost function we assume that
C C G C w G q G= ( ) = ( ) ( )[ ], (3)
where
w G q G( ) = ( ) =wage rate cost of capital,
w w q qG GG G GG< > < >0 0 0 0, , ,
w w0( ) = = market wage rate
q q0( ) = = cost of capital with zero goodwill
so that
C CG GG< >0 0,
C C0( ) = = production costs with zero goodwill stock
Costs are decreasing at a decreasing rate in G, due to the beneficial effects on
the wage rate and cost of capital. Both these effects are decreasing at a
decreasing rate, meaning that the firm cannot run the price of labor and
capital to zero by investing in goodwill.
For the CSR cost function we assume that
A A Ag gg> > ( ) =0 0 0 0, , (4)
11
It is straightforward to allow the control to be multidimensional.
12
Let subscripts denote partial derivatives from hereon.
76 Tommy Lundgren

A(g) is the total cost of investing in CSR, including promotional costs and
crowding-out costs. Crowding out means that CSR takes resources from
other productive activities at an increasing rate (since Agg > 0). This suggests
that small investments in CSR are relatively ‘cheaper’ than large investments
as a result of convexity in A(g).
Our assumptions about functional forms govern how revenues and costs
are affected by CSR investments, g, and goodwill, G, and ultimately the
behavior of the firm. Let us now introduce dynamics into this setting.
Given the above functional forms, the value function for the management
problem is written as
V e R G C G A g t
g
rt
= ( ) − ( ) − ( )[ ]−
∞
∫max d
0
(5)
where V is the value function at time t, and e-rt
is a discount factor where r is
the firm discount rate.13
Note that V is also the value of the firm since it is
defined as the perpetual discounted stream of profits. The management
problem is to chose g to build G as to maximize the future stream of dis-
counted profits, given an equation describing how goodwill evolves over
time. In general, it is assumed that goodwill develops over time according to
the following relationship:
G f g G
G t G
= ( )
=( ) =
,
0 0
(6)
where G˙ is the time derivative of G and f(g, G) maps CSR investments and
current goodwill capital level into changes in goodwill. G0 is a given starting
value for goodwill at time t = 0. We put no sign restriction on G in general or
the starting value G0. Negative goodwill can be considered ‘badwill’ and is
detrimental to profits (it implies a negative premium on price, and a positive
premium on the wage rate and the cost of capital). Assume the following
properties of the equation of motion specified in (6):
f f g G
f f f fg gg G GG
= ( )
> ≤ < ≥
,
, , ,0 0 0 0
(7)
13
More realistically, we could make the discount rate r also depend on goodwill; i.e. should the
perceived risk of the firm decrease through investment in CSR, then, as a consequence, the rate
of return, which is closely related to the firm discount rate, should also decrease. As a conse-
quence, the value of the firm also changes. However, the inclusion of this mechanism creates a
substantial increase in model complexity. Therefore, we opt to leave this exercise for future
analysis.

Investment in CSR has a positive effect on the change in goodwill. This effect
is either decreasing or linear with magnitude of g. Further, fG < 0 means that
the higher the level of goodwill, the smaller the relative change in goodwill,
ceteris paribus. No assumptions, at this point, are made concerning the
cross-effects between g and G. The following equation of motion for goodwill
is reasonable and easy to work with:
G g G= −α δ (8)
The parameter a is a CSR investment efficiency parameter, and d denotes the
rate of depreciation in the goodwill stock (exponential decay). The efficiency
parameter reflects the ability to transform CSR into goodwill stock. The
depreciation of goodwill over time, if not maintained, can be interpreted as if
stakeholders tend to forget about the firm’s historical social responsibility. It
could also be interpreted as an effect of other firms investing in CSR, and
therefore deflating the goodwill of this particular firm (as a particular firm is
becoming ‘more similar’ to other firms, the product differentiation effect gets
less pronounced).
Using equation (5) together with (8), we construct the current value
Hamiltonian,
H R G C G A g g Gc
= ( ) − ( ) − ( ) + −( )λ α δ (9)
where l is the adjoint variable or shadow price of goodwill. The shadow price
of goodwill represents the ‘theoretically correct’ price of goodwill should it be
traded in a competitive market. That is, l is closely related to the marginal
cost of investing in goodwill (as we shall see below).
The optimal conditions given by the maximum principle are
Hg
c
= 0 (10)
λ λ= −r HG
c (11)
lim
t
t
→∞
( ) =λ 0 (12)
The first two conditions must hold along the optimal path. The third condi-
tion is a transversality condition associated with infinite horizon autonomous
problems and is needed to provide a boundary condition at the limit.14
Equation (10) can be written
14
The boundary condition is typically replaced by the assumption that the optimal solution
approaches a steady state and ‘settles down’ (Kamien and Schwartz, 1991, section 9).
78 Tommy Lundgren

− + =
⇔
=
A
A
g
g
λα
λ
α
0
(13)
which simply states that the shadow price of goodwill is equal to the marginal
cost of investing in CSR normalized with the efficiency parameter. Expand-
ing the optimal condition (11) yields
λ λ λδ
δ λ
= − − −( )
= +( ) − −( )
r R C
r R C
G G
G G
(14)
which is the differential equation for the shadow price of goodwill. Now we
can use (13) and (14) to extract the differential equation for CSR investments,
g. First, take the time derivative of (13), i.e. λ = A ggg , and substitute the
result for λ in (14), then substitute l for Ag/a in (14). After isolating changes
in CSR—g˙ on the left-hand side—we have the following differential equation
for CSR investments:
g
r
A
R C
A
g
G G
gg
=
+( ) − −( )
+( )
+( )
+( )
δ
α (15)
The development over time for g is a function of both g and G. From (15) we
see that the difference between marginal costs and marginal benefits governs
changes in CSR investments over time. We see that if
A R C
r
g
g G G
α δ
≷
≷
−
+( )
⇒
0
(16)
This suggests that CSR investment/disinvestment will occur when the dis-
counted marginal benefits are smaller/larger than the marginal cost of invest-
ing in one extra unit of CSR. This means that when marginal costs are
larger/smaller than marginal benefits, the firm invests/disinvests to the point
where benefits equal costs.
The system is in steady state when g˙ = 0 and G˙ = 0 (and λ = 0). Assuming
that the efficiency parameter a = 1, then
A
R C
r
g
G G
=
−
+( )δ
(17)

g G= δ (18)
The marginal cost of investing in one extra unit of CSR is equal to the
benefits associated with the goodwill it creates, instantly and in the future,
discounted by the rate of return plus the rate of depreciation of goodwill.
According to (18), the level of goodwill is kept unchanged if the firm invests
an amount of CSR equal to the decay of goodwill.
2.3 Characterizing the equilibrium
When certain concavity conditions are satisfied for the Hamiltonian, the
conditions in (10), (11) and (12) are sufficient for maximization (see, for
example, Mangasarian, 1966; Kamien and Schwartz, 1971; or Arrow, 1998).
In short, these conditions require that the current value Hamiltonian is
concave in g and G, jointly, implying that the Hessian of the current value
Hamiltonian is negative semi-definite. The Hessian can be written
H H
H H
A
R C
gg gG
Gg GG
gg
GG GG
c c
c c
⎡
⎣
⎢
⎤
⎦
⎥ =
−
−
⎡
⎣⎢
⎤
⎦⎥
0
0
(19)
Since both terms in the diagonal are non-positive by assumption, the Hessian
of the current value Hamiltonian is negative semi-definite in the arguments g
and G, and thus there exists an interior solution, which is a maximum.
Let us portray the steady state in a phase diagram. It is now convenient to
use (14) together with (18). Note that using the differential equation for g will
generate the same qualitative results and conclusions since g is proportional
to l. Setting λ = 0 in (14) and isolating l on the left-hand side gives the locus
for the co-state in steady state,15
λ
δ
=
−
+
R C
r
G G
(20)
The right-hand side depends only on G, and since R is increasing and concave
in G, and C is decreasing and convex in G, the term (RG - CG) is decreasing
and convex in G. Around G = 0 the term (RG - CG) tends to infinity, and the
locus λ = 0 is not well defined at this point. The locus for the state variable,
G˙ = 0, is given by g - dG = 0 (with a = 1), which implies a linear curve with no
15
The locus for g˙ = 0 would look very similar, Ag = (RG - CG)/(r + d). Assuming a quadratic
function for A(g) implies that the locus for g˙ = 0 is proportional to the locus for λ = 0, so the
qualitative results would be the same whether we use either locus in our phase diagram.
80 Tommy Lundgren

constant and a coefficient equal to the depreciation rate, d. Since g is a linear
transformation of l, the locus g - dG = 0 can be written h(l) - dG = 0, where
g = h(l) and h-1
(g) = l.16
Plotting the loci in the [l, G]-space generates the
curves in figure 1. Studying the partials of λ and G˙ we can easily verify, for
the functional assumptions at hand, that ∂ ∂λ λ > 0, ∂ ∂λ G > 0, ∂G˙ /∂l > 0
and ∂G˙ /∂G < 0, which gives the directional arrows depicted in figure 1.
The equilibrium is a maximum and is given by l* and G* with a stable
branch leading into it from both left and right. It is easy to formally show that
this equilibrium is a saddlepoint maximum. See the Appendix for details.
2.4 Comparative statics and dynamic envelope results
In this section we parameterize the model and look at steady-state CSR
investment behavior (comparative statics) and some back-of-the-envelope
dynamic analysis. The steady-state analysis is standard. The derivation of
16
The slopes of the loci are given by the partial ∂l/∂G conditional on λ = 0 and G˙ = 0,
respectively.
Figure 1. Phase diagram in the [l, G]-space.

cost–benefit rules follows the back-of-the-envelope ‘recipe’ proposed by
Caputo (1990a, 1990b).17
We begin by parameterizing the model.
2.4.1 Model parameterization
Consider a firm that initially maximizes profits by choosing the levels of
capital and labor,
max ,
,K L
pY K L wL qKπ = ( ) − − (21)
where p, w and q are market prices for output, labor and capital, respec-
tively. Y(K, L) is a production function with capital and labor as arguments.
This gives us L* and K*, the optimal levels of labor and capital.18
Next, the
firm maximizes profits with respect to investment in CSR and building to a
stock of goodwill capital. At this stage L* and K* are taken as given.19
The
CSR management problem is written
V e t
e p G Y w G L q G K A g t
rt
rt
ω ω( ) = ( )
= ( ) − ( ) − ( ) − ( )[ ]
−
∞
−
∫max
max
Π d
* * * d
0
00
00
∞
∫
= −
( ) =
G g G
G G
α δ
(22)
where w is a vector of parameters.20
Assume the following parametric speci-
fication for the functions p(G), w(G), q(G) and A(g):
p G p G
w G w G
q G q G
A
( ) = + >
( ) = − >
( ) = − >
ε ε
θ θ
γ γ
ln ,
ln ,
ln ,
with
with
with
0
0
0
gg p p g g
p w q p p
( ) = +( ) + >
=
csr adv
csr adv
with
1
2
02
β β
ω ε θ γ β α
,
, , , , , , , , , ,, , ,δ r G0( )
(23)
These functions have the desired properties stated in the previous section (see
(2), (3) and (4)); i.e. there exists a steady-state saddlepoint equilibrium, which
is a maximum. The output price, wage and cost of capital functions are
17
See also, for example, Oniki (1973) for a similar methodology. Caputo (1990b) contains a
general discussion of comparative dynamics in optimal control problems.
18
Compare with the ‘H*’ in (1).
19
The problem would be considerably more complex if we include K and L as controls, without
really adding relevant richness to the analysis.
20
All parameters are non-negative.
82 Tommy Lundgren

specified so that they depend on the exogenously given market rate and an
endogenous premium determined by a parameter and goodwill capital. The
CSR cost function consists of a linear part that depends on the price of CSR
and the price of promoting CSR, and a non-linear part that represents costs
that occur due to crowding-out effects.
2.4.2 Steady-state properties (comparative statics)
First, let us look at the steady-state CSR investment behavior of the firm.
Recall that from the steady-state conditions in (17) and (18) we have
A g R G C G
r
g G
g G G* * *
* *
( )
=
( ) − ( )
+
=
α δ
δ
α
(24)
where g* and G* are steady-state levels of CSR investments and the goodwill
stock. Given the functional forms specified and parameterized in (23) we can
write
p p g G
Y L K
r
csr adv
* *
* * *
+ +
=
+ +( )
+
β
α
ε θ γ
δ
1
(25)
where G* = (a/d)g*. After some rearranging we get
a g bg c* *( ) + + =
2
0 (26)
where
a =1
b
p p
=
+csr adv
β
c
Y L K
r
= −
( )
+ +
+
α
β
α
δ
ε θ γ
δ
Solving for g* yields the following expression for the steady-state level of
CSR investments and the goodwill stock:
g
a
b ac b* = − ± − +( )1
2
4 2

G
a
b ac b* = − ± − +( )⎡
⎣⎢
⎤
⎦⎥
α
δ
1
2
4 2
Given a, b and c we can expand and cancel terms so that the steady-state
conditions become21
g
Y L K
r
p p p p
G
*
* * * csr adv csr adv
=
( )
+ +
+
+
+⎛
⎝⎜
⎞
⎠⎟ −
+1
2
4
1
2
2
α
β
α
δ
ε θ γ
δ β β
** *=
α
δ
g
(27)
As long as the efficiency parameter is positive, a > 0, marginal changes in its
value do not affect the level of steady-state rate of CSR investments, i.e.
∂g*/∂a = 0. However, if a = 0, the firm’s investment in CSR will be zero, since
it cannot convert CSR efforts into goodwill at any investment rate. The stock
of goodwill, G*, is proportional to the efficiency parameter, i.e. its steady-
state level is directly affected by changes in the parameter a.22
The signs of the
partial derivatives for g* and G* are obvious for the premium parameters, e,
q, g, and the discount rate, r, but it is less straightforward to assess in the case
of the rate of depreciation in goodwill, d, the price of CSR, pcsr
, the price of
promoting CSR, padv
, and the crowding-out parameter, b. It is easily verified
that ∂g*/∂d > 0 and ∂G*/∂d < 0. This means that the higher the rate of depre-
ciation in goodwill, the more the firm will have to invest in CSR to maintain
it. However, ceteris paribus, an increase in the depreciation rate decreases the
goodwill stock. Furthermore, we can derive the following partials for the
price of CSR and price of promotion/advertising:
∂
∂
∂
∂
∂
∂
∂
∂
g
p
g
p
p p
G
p
G
p
p
* *
* *
csr adv
csr adv
csr adv
s
= =
+( )−
= =
β
β
α
δ
Ψ
Ψ2 2
ccr adv
csr adv
* * *
+( )−⎡
⎣⎢
⎤
⎦⎥
=
+( ) +( ) + + +( )
p
r p p Y L K
β
β
δ βδ ε θ γ
Ψ
Ψ
Ψ
2
4
2
ββ δ2
0
r +( )
>
(28)
The signs of ∂g*/∂pcsr
, ∂g*/∂padv
, ∂G*/∂pcsr
and ∂G*/∂padv
are ultimately
determined by the term (pcsr
+ padv
) - bY, which is negative for reasonable
21
Note that only the negative root makes sense in this case.
22
An increase in a would simply scale the stock of goodwill upward since CSR now adds more
goodwill per unit invested.
84 Tommy Lundgren

parameter values. The partial derivatives of g* and G* with respect to the
crowding-out parameter are derived as
∂
∂
g r p p p p
r
Y L
*
* *
csr adv csr adv
β
δ β
β δ
βδ ε θ
=
+( ) +( ) − +( )[ ]
+( )
−
+
Ψ
Ψ2
2
3
++( )
+( )
=
γ
β δ
β
α
δ β
K
r
G g
*
* *
2 3
Ψ
∂
∂
∂
∂
(29)
which both have negative signs (for reasonable parameter values), as one
would expect.
2.4.3 Dynamic back-of-the-envelope analysis
Here we aim to investigate the dynamic effects on firm value of changes in
relevant parameters associated with CSR investments and the goodwill stock.
This is different from the previous exercise that analyzed CSR investment
behavior around the steady state. Instead, we now look at dynamic proper-
ties, i.e. we explore the partial effects on firm value over the whole planning
horizon using a back-of-the-envelope approach.
Consider the envelope properties of value function, V(w). The dynamic
envelope theorem postulates that the first partials of V(w) are found by (i)
differentiating the Hamiltonian for the optimal control problem directly with
respect to the parameters of interest, (ii) holding the state, co-state and
control fixed, then (iii) evaluating the partials along the optimal paths for
these variables, and (iv) finally integrating the result over the planning
horizon (Caputo, 1990b). This implies we can differentiate the Hamiltonian
directly with respect to parameters prior to substituting in the optimal
trajectories.
Write the present value Hamiltonian23
as
H e p G Y K L
e w G L q G K
e p
rt
rt
rt
= +( ) ( )[ ]
− −( ) − −( )[ ]
−
−
−
−
ε
θ γ
ln ,
ln ln
* *
* *
ccsr adv
+( ) +⎡
⎣⎢
⎤
⎦⎥
+ −( )p g g g Gβ ψ α δ
1
2
2
(30)
23
Here we follow Caputo (1990b) and work with the present value Hamiltonian instead of the
current value Hamiltonian. This is convenient and enables us also to explicitly investigate the
effect of changing the discount rate, r.

where y is the present value co-state variable. Following the Caputo-recipe24
for the parameter e generates
H e GY
H e G t Y
rt
rt
ε
ε ω
=
= ( )
−
−
ln
ln ;
*
* *optimal path
(31)
so that the dynamic envelope theorem yields
V e G t Y trt
ε ω ω( ) = ( ) >−
∞
∫ ln ;* *d
0
0 (32)
That is, if the output market’s sensitivity with respect to the firm’s goodwill
increases, then the value of the firm also increases. Since goodwill creates a
price premium whose size is determined solely by the parameter e, it comes
as no surprise that Ve(w) > 0. The same procedure for the other parameters
gives us
V e Y tp
rt
ω( ) = >−
∞
∫ *d
0
0 (33)
V e L tw
rt
ω( ) = − <−
∞
∫ *d
0
0 (34)
V e K tq
rt
ω( ) = − <−
∞
∫ *d
0
0 (35)
V e G t L trt
θ ω ω( ) = ( ) >−
∞
∫ ln ;* *d
0
0 (36)
V e G t K trt
γ ω ω( ) = ( ) >−
∞
∫ ln ;* *d
0
0 (37)
V e g t tp
rt
csr * dω ω( ) = − ( ) <−
∞
∫ ;
0
0 (38)
V e g t tp
rt
adv * dω ω( ) = − ( ) <−
∞
∫ ;
0
0 (39)
24
The derivation of this convenient result in Caputo (1990b) is rather lengthy and involved. A
much neater and less rigorous derivation is found in Aronsson et al. (2004, ch. 9). The trick is to
introduce an artificial state variable in terms of the parameter of interest (in their case it
represents a project). Assume we want to examine the project a. Then the co-state dynamics
is represented by α = 0, a(0) = a. The co-state variable or shadow price of a is la = ∂V/∂a. It
is now easy to show from the co-state optimal condition (co-state equation) that
λ αα α= =
∞
∫∂ ∂V H td
0
. The result is derived by integrating λα α= −H over (0, •) and setting
la(•) = 0 (transversality condition).
86 Tommy Lundgren

V e g t trt
β ω ω( ) = − ( )[ ] <−
∞
∫
1
2
0
2
0
* d; (40)
V e t g t trt
α ω ψ ω ω( ) = ( ) ( ) >−
∞
∫ * * d; ;
0
0 (41)
V e t G t trt
δ ω ψ ω ω( ) = − ( ) ( ) <−
∞
∫ * * d; ;
0
0 (42)
V te tr
rt
ω ω( ) = − ( ) <−
∞
∫ Π d
0
0 (43)
Let us briefly comment on the partials portrayed in (33)–(43). All partials are
unambiguously signed. None of the derived signs of the partials comes as a
surprise, since the Hamiltonian was carefully rigged to meet the conditions
necessary for a saddlepoint equilibrium. Note that these dynamic envelope
results recover cumulative discounted functions, and not instantaneous func-
tions as with static envelope analysis. Vq(w), Vg (w) > 0 implies that should the
parameters governing the size of the premiums on wage and cost of capital
increase—i.e. employees and capital markets become more sensitive to
CSR—then the value of the firm moves in the same direction. If price of CSR
and price of promoting CSR go up, Vpcsr ω( ), Vpadv ω( ) < 0, then, not surpris-
ingly, the firm value decreases. Crowding-out effects are measured by the
parameter b. Since Vb(w) < 0, one can conclude that more severe crowding-
out effects will lower the value function and thus firm value. Efficiency in
converting CSR efforts into actual goodwill is measured by the parameter a;
i.e. should a increase, the firm becomes more efficient in transforming CSR
policy into goodwill, and from Va(w) > 0 we see that the value of the firm also
increases.25
An increased depreciation rate of goodwill, d, or discount rate, r,
has a negative impact on firm value, Vd (w), Vr(w) < 0.
Another useful dynamic result that can be derived from optimal control
theory is that the change in the value of the firm is directly related to the
change in goodwill. This comes from a general result in optimal control (see,
for example, Brock, 1998), where the time derivative of the value function in
a dynamic optimal control problem is directly related to the net changes in all
stocks in the model in the following way:
V Si i
i
= ∑λ (44)
25
Note that the comparative statics showed that in equilibrium, the CSR investment rate is not
affected by changes in the efficiency parameter. However, the dynamic long-run effect on firm
value associated with a change in the efficiency to convert CSR investments into goodwill is
unambiguously positive.

where li is the shadow price of stock i, and Si is the ith stock. Since the present
control problem has only one stock, goodwill, we can write
V G= λ (45)
This result suggests that all changes in goodwill, positive or negative, as a result
of investing or disinvesting in CSR, will have direct effect on the value of the
firm given that there is a non-negative shadow price of goodwill capital.26
3. WHAT CAN EMPIRICAL EVIDENCE TELL US?
The model presented in this paper can help grasp how empirical hypotheses
concerning CSR can be derived in a consistent way, without having to
resort to speculation or poorly undermined reasoning. Here we identify
some possible hypotheses drawn from the model framework and then look
into whether the empirical literature on CSR supports or refutes them. The
purpose of this exercise is to point out the usefulness of the model pre-
sented to facilitate the understanding of the drivers behind CSR and, con-
sequently, how to build testable hypotheses consistently.27
It also reveals
that our model encompasses many of the known and tested features of
CSR.
H1: CSR behavior that increases goodwill may have a positive effect on output
price: ∂P(G)/∂G > 0.
Is there a price premium for CSR firms? Just a cursory look around
suggests this, e.g. actual forest product price lists suggest that certification
of forest products leads to higher price. Kriström and Lundgren (2003) find
evidence of a price premium for ‘green’ pulp in Swedish pulp industry.
Furthermore, empirical results in Blend and Ravenswaay (1999) suggest
that American consumers are willing to pay a premium for eco-labeled
apples, but not too much. Similar examples from the literature abound.
H2: CSR efforts that increase goodwill may have a beneficial effect (i.e. nega-
tive) on the wage rate: ∂w(G)/∂G < 0.
Empirical evidence of wage differentials for CSR firms are scarce. Bolvig
(2005) finds evidence of compensating wage differentials in CSR firms in a
26
This relationship is (implicitly) the most widely tested in the empirical literature on social
performance and financial performance. More on this below.
27
Bert Scholtens is thanked, without being implicated, for suggesting this section.
88 Tommy Lundgren

sample of 2000 US firms. Edmans (2007) explores the relationship between
employee satisfaction and long-run stock performance by reviewing a port-
folio of stocks selected by Fortune magazine as the ‘Best Companies to Work
For in America’ in 1998. The study finds the portfolio earned more than
double the market return, which would suggest that employee satisfaction
improves corporate performance.
H3: CSR efforts that increase goodwill may have a beneficial effect on the cost
of capital: ∂q(G)/∂G < 0.
By engaging in CSR, the firm signals a probability of less conflicts in the
future, and thus a more ambitious risk management strategy, which lowers
the specific risk that capital markets assign to the firm. The empirical evi-
dence supporting the connection between cost of capital and CSR is scarce.
Hart and Ahuja (1996) find empirical evidence in the USA that the cost of
capital decreases, and thus firm value increases, following good environmen-
tal performance. Derwall and Verwijmeren (2006) find in a US sample that
environmental performance and corporate governance have a negative effect
on the cost of capital, while human rights issues increase the cost of capital.
The net effect is ambiguous. Sharfman and Fernando (2008) study 267 US
firms and show that improved environmental risk management is associated
with a lower cost of capital.
H4: CSR investments cause crowding-out effects: ∂A(g)/∂g < 0.
Kriström and Lundgren (2003) try to measure crowding-out effects due to
abatement investments in the Swedish pulp industry during the period 1985–
90, but statistical analysis does not show such effects. Gray and Shadbegian
(1998) find evidence that environmental investments crowd out other produc-
tive investments as follows: a 1 per cent increase in environmental investment
causes a 1.88 per cent decrease in productive investments. Similar evidence is
found in an earlier study by Barbera and McConnel (1986), who find that
abatement investments retard capital and labor productivity and hence
impose an extra ‘adjustment’ cost, which, ceteris paribus, lowers profitability.
H5: Change in firm value is positively affected by changes in goodwill:
∂V/∂t = l∂G/∂t.
The bulk of the empirical work can be found within this category. Margolis
and Walsh (2001) or Hay et al. (2005) both provide excellent reviews. The
results are somewhat ambiguous. However, as mentioned in the introduction,
most studies show a positive relationship between firm value and different

measures of social responsibility. This could be a result of ‘publication bias’,
i.e. mainly positive results are submitted and subsequently published. But a
certain amount of heterogeneous results would be expected according to the
model presented here. The model predicts that some firms will be positively
affected by engaging in CSR, while others will not. The expectation of
heterogeneous results is attributed to whether such behavior is rewarded or
punished by a firm’s stakeholders. The existence of a goodwill capital stock is
solely dependent on whether the stakeholders care about firm social behavior.
In summary, it is not difficult to find empirical evidence for some of the
hypotheses that can be derived from the model framework presented here.
However, the current literature lacks a ‘joint test’ of several of the hypotheses
listed above.28
For example, consider a firm that experiences crowding out-
costs due to environmental investments. Does this firm also reap the benefits
in terms of an output price premium, and/or cost reductions via compensat-
ing wage differentials, and/or lower cost of capital due to effects via ambi-
tious environmental risk management?
4. CONCLUDING REMARKS
This paper provides theoretical underpinnings to help understand the mecha-
nisms and incentives behind the behavior of a socially responsible firm.
Profit-maximizing firms consider both the costs and benefits of CSR. The
implications of these findings are that firms will engage in CSR activities if
stakeholders, such as the government, the financial sector, consumers, non-
governmental organizations, etc., reward or pressure firms to engage in such
behavior. The link between profitability and different dimensions of CSR is
therefore likely to differ across countries, sectors and even firms. The model
in this paper provides a useful theoretical background for the understanding
of CSR incentives and for constructing relevant hypotheses in empirical
applications.
Future research should include the impact of uncertainty, e.g. what are the
effects of environmental incidents or ‘bad news’ that arrive over time in some
stochastic manner? A natural way to model this would be to include a
stochastic element to the evolution of goodwill capital, e.g. an Ito diffusion
process and/or a Poisson jump process.29
Another possible route of research
28
Although H5 can be considered a kind of ‘overall’ test of whether social performance is
associated with higher firm value.
29
The models in, for example, Tapiero (1977, 1978) could be adapted and modified to tell the
story of CSR under uncertainty. See also Lundgren (2003) for an Ito/Poisson process applied to
the evolution of goodwill in a highly simplified investment optimal stopping problem.
90 Tommy Lundgren

would be to allow for potential ‘Porter effects’ where some types of CSR
investments (e.g. investment in green technology) could have positive effects
on long-term efﬁciency of the capital stock and/or spur innovative processes
and investment in R&D. This could be modeled within the framework pre-
sented here, possibly combined with uncertainty components. It would also
be useful to explore the consequences of asymmetric information further;
that is, when reputation and sales can be boosted with advertising efforts
(see, for example, Spremann, 1985) without actually altering the product
attributes, so-called ‘greenwash’.
APPENDIX
In inﬁnite horizon autonomous problems with one control and one state
variable, and when the discount rate is small enough, the equilibrium is
usually a stable saddlepoint (Kamien and Schwartz, 1991, section 9).
However, this is more of a ‘rule of thumb’ than a mathematical certainty. In
general, the steady state may or may not exist, or there may be multiple
steady states. To characterize the equilibrium formally, we now proceed to
study the linear differential system that approximates (6) and (11).30
Again we
work with the differential equation for l instead of g. Since g is proportional
to l, compare (14) with (17), the qualitative results will be the same. Write the
Hamiltonian in general form as
H g G f g Gc
= ( ) + ( )Π , ,λ (A1)
Since g maximizes the value function (5), we have
H g G f g Gg g g
c
= ( ) + ( ) =Π , ,λ 0 (A2)
and
Hgg
c
< 0 (A3)
Recall that
λ λ λ= − ( )r H g GG
c
, , (A4)
Equations (A2) and (A3) implicitly gives g as a function of l and G,
g u G= ( )λ, (A5)
Now totally differentiate (A1) and (A5) to get the properties of u(l, G):
30
The procedure follows Kamien and Schwartz (1991, section 9).

d d d dc c c
H H g H G fgg gG g= + + λ (A6)
d d dg u G uG= + λ λ (A7)
Using (A6) we have
d d d
c
c c
g
H
H
G
f
H
gG
gg
g
gg
= −
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟ λ (A8)
From (A7) and (A8) we see that
u
H
H
u
f
H
G
gG
gg
g
gg
= − = −
c
c c
, λ (A9)
Substitute (A5) into (6) and (A4):
G f u G G G G
r H u G GG
= ( )( ) ( ) =
= − ( )( )
λ
λ λ λ λ
, , ,
, , ,
0 0
c
(A10)
The steady state, (l*; G*), exists if
f u G G
r H u G G
g u G
G
λ
λ λ λ
λ
* * *
* * * *
* * *
c
, ,
, , ,
,
( )( ) =
− ( )( ) =
− ( ) =
0
0
0
(A11)
The nature of the steady state is characterized by looking at a linear Taylor
expansion approximation of the system given by (A10) around l* and G*:
G f f u G G f u
H H u G G r f
G g G g
GG Gg G G
= +( ) −( ) + −( )
= − +( ) −( ) + − −
* *
*c c
λ λ λ
λ HH uGg
c
*λ λ λ( ) −( )
(A12)
Using (A9) we can eliminate uG and ul and write (A12) as
G a G G b
c G G r a
= −( ) + −( )
= −( ) + −( ) −( )
* *
* *
λ λ
λ λ λ
(A13)
where a f f H HG g gG gg= − c c
, b f g Hgg= −( )2 c
and c H H H HGg gg GG gg= ( ) −⎡⎣ ⎤⎦
c c c c2
.
The characteristic equation and roots for the system (A13) are
m rm a r a bc
m m
r r a bc
2
1 2
2
1
2
0
2
2 4
2
− + −( ) − =
= ±
−( ) +⎡⎣ ⎤⎦,
(A14)
92 Tommy Lundgren

If the roots are real, then the larger root is positive. The sign of the smaller
root is either positive or negative. It is negative if
bc a r a> −( )
or
−
⎛
⎝⎜
⎞
⎠⎟ ( ) −⎡⎣ ⎤⎦
> −
+( )
f g
H
H H H
H
f
gg
Gg gg GG
gg
G
2
2
c
c c c
c
ff H
H
r f
f H
H
g gG
gg
G
g gG
gg
c
c
c
c
⎛
⎝⎜
⎞
⎠⎟ − −
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣
⎢
⎤
⎦
⎥
−( )
If this holds, the roots are real and of opposite sign, and the steady state
satisﬁes the conditions to be a saddlepoint. It is easy to verify that for the
functional properties assumed in our model, this condition indeed holds.
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Tommy Lundgren
Centre for Environmental and Resource Economics
Umeå School of Busniness
S-901 87 Umeå
Sweden
E-mail: tommy.lundgren@usbe.umu.se

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