A MICROECONOMIC MODEL OF CORPORATESOCIAL RESPONSIBILITYmeca_4087 69..95Tommy Lundgren*Centre for Environmental and Resourc...
mechanisms behind corporate social responsibility based on a microeco-nomic perspective of the firm. The ultimate objective...
Analysis recently published a theme issue on CSR and economic performance(Paul and Siegel, 2006). They note that the vast ...
greater for experience and credence goods and services, which is empiricallyconfirmed by Siegel and Vitaliano (2007). Bagno...
conflict with stakeholders) become to some degree endogenous to the firm.Thus, profits would not depend solely on the cost of...
The notion of goodwill or reputation as an intangible asset is used exten-sively in dynamic models of advertising.7Assumin...
here.8Furthermore, societal benefits from CSR are not considered becauseour model maintains a firm perspective. Based on a r...
A(g), incorporates all costs associated with investments in CSR, g, a controlvariable we here consider to be one-dimension...
A(g) is the total cost of investing in CSR, including promotional costs andcrowding-out costs. Crowding out means that CSR...
Investment in CSR has a positive effect on the change in goodwill. This effectis either decreasing or linear with magnitud...
− + =⇔=AAggλαλα0(13)which simply states that the shadow price of goodwill is equal to the marginalcost of investing in CSR...
g G= δ (18)The marginal cost of investing in one extra unit of CSR is equal to thebenefits associated with the goodwill it ...
constant and a coefficient equal to the depreciation rate, d. Since g is a lineartransformation of l, the locus g - dG = 0 ...
cost–benefit rules follows the back-of-the-envelope ‘recipe’ proposed byCaputo (1990a, 1990b).17We begin by parameterizing ...
specified so that they depend on the exogenously given market rate and anendogenous premium determined by a parameter and g...
Gab ac b* = − ± − +( )⎡⎣⎢⎤⎦⎥αδ124 2Given a, b and c we can expand and cancel terms so that the steady-stateconditions beco...
parameter values. The partial derivatives of g* and G* with respect to thecrowding-out parameter are derived as∂∂g r p p p...
where y is the present value co-state variable. Following the Caputo-recipe24for the parameter e generatesH e GYH e G t Yr...
V e g t trtβ ω ω( ) = − ( )[ ] <−∞∫12020* d; (40)V e t g t trtα ω ψ ω ω( ) = ( ) ( ) >−∞∫ * * d; ;00 (41)V e t G t trtδ ω ...
where li is the shadow price of stock i, and Si is the ith stock. Since the presentcontrol problem has only one stock, goo...
sample of 2000 US firms. Edmans (2007) explores the relationship betweenemployee satisfaction and long-run stock performanc...
measures of social responsibility. This could be a result of ‘publication bias’,i.e. mainly positive results are submitted...
would be to allow for potential ‘Porter effects’ where some types of CSRinvestments (e.g. investment in green technology) ...
d d d dc c cH H g H G fgg gG g= + + λ (A6)d d dg u G uG= + λ λ (A7)Using (A6) we haved d dcc cgHHGfHgGggggg= −⎛⎝⎜⎞⎠⎟ −⎛⎝⎜⎞...
If the roots are real, then the larger root is positive. The sign of the smallerroot is either positive or negative. It is...
Derwall, J., Verwijmeren, P. (2006): ‘Corporate social responsibility and the cost of equity’,Unpublished mimeo, Departmen...
McWilliams, A., Siegel, D. (2001): ‘Corporate social responsibility: a theory of the firm perspec-tive’, Academy of Managem...
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A microeconomic model of corporate

  1. 1. A MICROECONOMIC MODEL OF CORPORATESOCIAL RESPONSIBILITYmeca_4087 69..95Tommy Lundgren*Centre for Environmental and Resource Economics (CERE),Umeå School of Business(December 2008; revised January 2010)ABSTRACTThis paper explores the economic mechanisms behind corporate social responsibility (CSR) in amicroeconomic model of the firm. The study’s motivation is to shed light on the potential causes of theobserved phenomena of voluntary over-compliance among firms. We investigate how assumptionsabout costs and benefits affect CSR behavior through a stock of goodwill capital. In optimum, the firmmust balance marginal costs and benefits of investing in CSR. We characterize the equilibrium andexamine comparative statics and dynamics from a parameterized model. Finally, we link some of themodel’s results to the empirical literature on CSR.1. INTRODUCTIONAccording to conventional economic theory, firms maximize profits subjectto technological and other constraints. Without economic incentives liketaxes or quantitative regulations, the firm might, for example, pollute toomuch, or engage in some other socially detrimental behavior. A cursory lookaround business environments today suggests that this view might be a bitold-fashioned. Indeed, firms spend resources to convince potential consum-ers and other stakeholders that they are more socially responsible than whatthe 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 AtlanticWorkshop 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 October2008, improved the paper significantly. Helpful comments and suggestions from two anonymousreferees are also gratefully acknowledged.Metroeconomica 62:1 (2011) 69–95doi: 10.1111/j.1467-999X.2010.04087.x© 2010 Blackwell Publishing Ltd
  2. 2. mechanisms behind corporate social responsibility based on a microeco-nomic perspective of the firm. The ultimate objective is to shed some light onthe rationale behind the observed voluntary over-compliance among firms.Since the publication of the report ‘Our Common Future’ (1987) by theWorld 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 tofacilitate sustainable development, so-called corporate social responsibility(CSR). One of the first attempts to bring CSR into the public debate was fromMilton Friedman who in his article in the New York Times Magazine arguedthat ‘the corporate social responsibility of firms is to maximize its profits’(Friedman, 1970). While this statement may appeal specifically to neoclassicaleconomists, it may also seem provocative and without nuance to others. Butas we shall see, profit maximization does not have to be in conflict withsocial responsibility (see, for example, Husted and Salazar, 2006). FollowingFriedman’s initial definition of CSR, a number of others have followed. Forexample, Heal (2008) suggests that CSR is ‘. . . the interactions betweencorporate behavior and civil and legal society, and how these interactionsstructure the company’s incentives on social and environmental issues’.1Wesimply consider CSR to be actions that, to some degree, imply corporatebeyond-compliance behavior in the social and/or the environmental arena.Most of the empirical studies of the effects of CSR on either a firm’seconomic or financial performance have been performed during the last twoor three decades (and most focus on financial performance). The plethora ofindividual studies has led to at least 10–15 reviews, many of them assessed inMargolis and Walsh (2001), which reviews nearly 100 separate studies. Also,Reinhardt (2000), Orlitzky and Benjamin (2001), Orlitzky et al. (2003), Lyonand Maxwell (2004) and Orlitzky and Swanson (2008) summarize to a largeextent the bulk of empirical CSR literature that is currently available. Hayet al. (2005) offer a comprehensive review from the fields of economics, lawand business. The evidence from these review studies is not conclusive, butempirical results seem to indicate that CSR leads to positive financial per-formance for a firm.2When it comes to CSR and economic performance, the research is lessextensive. Indeed, economic and financial performances are linked,3but thereare some interesting differences worth noting. The Journal of Productivity1See also McWilliams and Siegel (2001), Hay et al. (2005) or Portney (2008) for similar defini-tions of CSR.2Possibly because of publication bias, i.e. negative results tend not to be published.3If, for example, economic efficiency is improved by CSR this will most likely show in financialperformance such as stock price through higher profits.70 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  3. 3. Analysis recently published a theme issue on CSR and economic performance(Paul and Siegel, 2006). They note that the vast number of studies of CSRand the effect on financial measures is, from an economic perspective, unfor-tunate. Instead, they suggest that a more salient issue is the relationshipbetween economic performance and CSR behavior, where economic perfor-mance entails technological and economic interactions between productionof output and input demands, recognizing the opportunity costs of inputsand capital formation. Their conclusion is that the costs of CSR must bebalanced by benefits to motivate firms to carry out such activities. Anothertheme issue, Paton and Siegel (2005), can be found in Structural Change andEconomic Dynamics, where CSR is studied using empirical and theoreticaltools from both finance and economics.Compared with the empirical literature on CSR, attempts to formallymodel the microeconomic mechanisms behind voluntary over-compliance atthe firm level in a consistent way are less plentiful. However, a number ofstudies are worth mentioning. Bergman (1995) provides several interestingsimple static micro-models, in terms of environmentally friendly firms, thatprovide a rationale for CSR behavior. A game theoretical model of voluntaryover-compliance is proposed by Arora and Gangopadhyay (1995), whoassume that firms signal their ‘greenness’. If consumers prefer to buy prod-ucts from a ‘greener’ firm, then the cost of being environmentally friendlymay be justified by higher revenues.McWilliams and Siegel (2001) and Baron (2001) are probably the first twopapers that explicitly model profit-maximizing CSR in a way that accountsfor the wide range of costs and benefits to be considered when investing insocially responsible projects.4These papers identify the implications of CSRwhen information is asymmetric between firms and consumers, discuss theimportance of reputation, and examine the strategic use of CSR (e.g. theproblem of ‘greenwashing’5). The reputation and strategic dimension of CSRis 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 be4McWilliams and Siegel (2001) outline a model in which two firms sell identical goods, but onecompany decides to add an additional social attribute to its product. This attribute is valued bysome consumers or, potentially, by other stakeholders. Firm managers conduct a cost–benefitanalysis 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 determinethe optimal level of CSR to provide.5See, 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 asreductions in resource use.A Microeconomic Model of Corporate Social Responsibility 71© 2010 Blackwell Publishing Ltd
  4. 4. greater for experience and credence goods and services, which is empiricallyconfirmed by Siegel and Vitaliano (2007). Bagnoli and Watts (2003) extendBaron (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-maximizingCSR, 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 wherevoluntary abatement investments (one dimension of CSR) create a stock ofgoodwill capital that enables the firm to differentiate their product. Otherproduct differentiation models connected to green consumerism includeEriksson (2004) and Rodriguez-Ibeas (2007).In our view, the most comprehensive and complete theoretical discussioncan be found in Heal (2005, 2008). Using a non-formal model he discussesCSR from both economic and financial perspectives, and proposes how it isreflected in financial markets. He discusses several cases in relation to CSR.Heal defines CSR as actions to reduce externalized costs and/or to avoiddistributional conflicts. He suggests that there may be a resource allocationrole for CSR programs in cases of market failure through private–social costdifferentials. Furthermore, he argues that in sectors where social and privatecosts are not in line, or where distributional conflicts are common, CSR canplay a valuable role in ensuring that ‘the invisible hand’ acts, as intended, toproduce the social good. It can also act to improve corporate profits andguard against reputational risks.In light of the condensed review of theoretical studies on CSR presentedhere, we conclude that this type of analyses are on the rise. However, to ourknowledge, a formal dynamic microeconomic model of the firm thataccounts for several dimensions of CSR, in terms of both different types ofCSR and the various drivers and mechanisms, is non-existing. Paul R.Portney, in Hay et al. (2005), points out that despite a vast amount ofempirical studies of CSR, very few, if any, have derived testable hypothesesfrom an adequate theoretical model of the firm. Moreover, few studies clearlyidentify the basic mechanisms of how socially responsible behavior leads toa economic/financial advantage. Portney provides a general outline of howsuch a model might work: by engaging in CSR, output price (price differen-tiation), wages (higher worker productivity or lower wages through workersatisfaction), and the cost of capital (risk reduction due to lower risk of72 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  5. 5. conflict with stakeholders) become to some degree endogenous to the firm.Thus, profits would not depend solely on the cost of engaging in CSR, butalso on the benefits (Hay et al., 2005, p. 114). The purpose of this paper is tobuild such a model within a dynamic framework and explore its properties.To our knowledge, this is an original exercise and a contribution to thetheoretical literature on CSR.The paper is organized as follows. In the next section, we present a dynamicmodel of CSR. In section 2.1 we propose relevant benefits and costs of CSR,introduce intertemporal features, and explicitly model goodwill capital as thedriving force for benefits. Sections 2.2 and 2.3 discuss the model properties andestablish the existence and characteristics of a stable equilibrium. In section 2.4we offer insights from comparative statics and dynamic back-of-the-envelopeanalysis. We then proceed to link some of the hypotheses that can be derivedfrom the model to the empirical literature on CSR. Finally, we offer someconcluding comments and suggestions for future research.2. A FORMAL MODEL OF THE SOCIALLY RESPONSIBLE FIRMThis section outlines and discusses a firm-level model that sheds light on thepotential mechanism behind CSR. The model proposes that with stakeholders(e.g. consumers, financial sector, government, employees, etc.) rewardingCSR behavior, the cost of CSR may be balanced by benefits in terms of higherprofitability (see, for example, McWilliams and Siegel, 2001). Indirectly, weare assuming that the firm is socially responsible for strategic reasons, i.e. it isgood business, or at least not bad business. The analysis relies on dynamics,a pertinent feature of CSR since it potentially has effects on reputation orgoodwill, which has inherent intertemporal properties. Specifically, we assumethe notion of goodwill capital as a stock that acts upon a firm’s revenues andcosts in different ways. An intertemporal setting is reasonable when dealingwith CSR investments and goodwill capital. CSR projects build goodwillcapital over time, which can be seen as an intangible asset, a form of ‘reputa-tion’. The reputational implications of CSR are highlighted and discussed indetail in McWilliams and Siegel (2001, 2002), especially the potential effectsof asymmetric information about product attributes and firm activities. Inthe model presented here, we abstract from the complications of asymmetricinformation. That is, the activities of the firm and the product attributes areperfectly transparent and known to the consumer.66An 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 beA Microeconomic Model of Corporate Social Responsibility 73© 2010 Blackwell Publishing Ltd
  6. 6. The notion of goodwill or reputation as an intangible asset is used exten-sively in dynamic models of advertising.7Assuming 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 belowusing a model based on the green firm framework developed in Kriström andLundgren (2003), which, in turn, are inspired by advertising models such asDorfman and Steiner (1954), Nerlove and Arrow (1962), Gould (1970) andJacquemin (1973).2.1 Benefits and costs of CSRWe begin by describing the main assumptions regarding the benefits andcosts 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 elseequal, this leads to increasing revenues and profits for the firm (see, forexample, Blend and Ravenswaay, 1999).B2. Wage is to some degree endogenous to the firm; i.e. people are willing toaccept lower wages to work at a CSR firm, or work more productively atthe market wage rate (see, for example, Bolvig, 2005). This assumptionis 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 alower probability of conflict with stakeholders and various interestgroups in the future (Heal, 2008, provides an extensive discussion of howCSR 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 explicitlyused to signal product quality, augment reputation and increase sales, while consumer goodwillis a consequence of actual experience of the product. This enables the firm to boost reputationand sales with advertising in the short run, but over the long run ‘greenwashing’ becomes anunsustainable strategy; reputation is damaged if the product does not live up to its expectations.7For excellent reviews of quantitative advertising models see Sethi (1977) and Feichtinger et al.(1994).74 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  7. 7. here.8Furthermore, societal benefits from CSR are not considered becauseour model maintains a firm perspective. Based on a review of the CSRliterature, 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 environmentalproject 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 linearin the amount of CSR.C2. Costs of promoting (advertising) CSR investments to stakeholders(see, for example, Wang, 2008). Without stakeholders’ knowledge of afirm’s socially responsible behavior, the benefits cannot be fully realized.Together with the project investment cost (as suggested in C1), theseconstitute 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 adiscussion of the environmental investments case, see Gray and Shad-begian, 1998). We assume that CSR may hamper ‘conventional’ firmactivities, and that this cost is increasing at an increasing rate in CSR.92.2 The modelLet us transform the model assumptions into a formal model in a fairlygeneral form. Define instantaneous profits of a firm, P, at time t asΠ 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 byH*, which are exogenous to the firm. We can think of H* as representinginputs such as labor and capital that have been chosen optimally at a previ-ous stage and are taken as given.10This means we can abstract from H* in thesequel, and thus focus solely on the intertemporal problem involving invest-ment in CSR and the goodwill stock. The last term in the profit function,8For a discussion see, for example, Mackey et al. (2007).9Similar to the notion of adjustment costs in capital formation analysis.10The model could include these inputs as control variables, and potential links to CSRinvestments and goodwill, but we keep our approach simple to capture and convey the essentialfeatures of a socially responsible firm.A Microeconomic Model of Corporate Social Responsibility 75© 2010 Blackwell Publishing Ltd
  8. 8. A(g), incorporates all costs associated with investments in CSR, g, a controlvariable we here consider to be one-dimensional. More realistically, thecontrol variable could be defined as multidimensional, as CSR can take manyforms. However, to simplify, we treat CSR investment as a one-dimensionalcontrol variable.11This does not change the basic idea we want to conveyhere, but simplifies our notation.For the revenue function we assume that12R RR RG GG> <( ) = =0 00,revenue with zero goodwill stock(2)This is the price premium or product differentiation effect. By increasinggoodwill the firm can increase revenue, but at a decreasing rate.For the cost function we assume thatC C G C w G q G= ( ) = ( ) ( )[ ], (3)wherew G q G( ) = ( ) =wage rate cost of capital,w w q qG GG G GG< > < >0 0 0 0, , ,w w0( ) = = market wage rateq q0( ) = = cost of capital with zero goodwillso thatC CG GG< >0 0,C C0( ) = = production costs with zero goodwill stockCosts are decreasing at a decreasing rate in G, due to the beneficial effects onthe wage rate and cost of capital. Both these effects are decreasing at adecreasing rate, meaning that the firm cannot run the price of labor andcapital to zero by investing in goodwill.For the CSR cost function we assume thatA A Ag gg> > ( ) =0 0 0 0, , (4)11It is straightforward to allow the control to be multidimensional.12Let subscripts denote partial derivatives from hereon.76 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  9. 9. A(g) is the total cost of investing in CSR, including promotional costs andcrowding-out costs. Crowding out means that CSR takes resources fromother productive activities at an increasing rate (since Agg > 0). This suggeststhat small investments in CSR are relatively ‘cheaper’ than large investmentsas a result of convexity in A(g).Our assumptions about functional forms govern how revenues and costsare affected by CSR investments, g, and goodwill, G, and ultimately thebehavior of the firm. Let us now introduce dynamics into this setting.Given the above functional forms, the value function for the managementproblem is written asV e R G C G A g tgrt= ( ) − ( ) − ( )[ ]−∞∫max d0(5)where V is the value function at time t, and e-rtis a discount factor where r isthe firm discount rate.13Note that V is also the value of the firm since it isdefined as the perpetual discounted stream of profits. The managementproblem is to chose g to build G as to maximize the future stream of dis-counted profits, given an equation describing how goodwill evolves overtime. In general, it is assumed that goodwill develops over time according tothe following relationship:G f g GG t G= ( )=( ) =,0 0(6)where G˙ is the time derivative of G and f(g, G) maps CSR investments andcurrent goodwill capital level into changes in goodwill. G0 is a given startingvalue for goodwill at time t = 0. We put no sign restriction on G in general orthe starting value G0. Negative goodwill can be considered ‘badwill’ and isdetrimental to profits (it implies a negative premium on price, and a positivepremium on the wage rate and the cost of capital). Assume the followingproperties of the equation of motion specified in (6):f f g Gf f f fg gg G GG= ( )> ≤ < ≥,, , ,0 0 0 0(7)13More realistically, we could make the discount rate r also depend on goodwill; i.e. should theperceived risk of the firm decrease through investment in CSR, then, as a consequence, the rateof 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 asubstantial increase in model complexity. Therefore, we opt to leave this exercise for futureanalysis.A Microeconomic Model of Corporate Social Responsibility 77© 2010 Blackwell Publishing Ltd
  10. 10. Investment in CSR has a positive effect on the change in goodwill. This effectis either decreasing or linear with magnitude of g. Further, fG < 0 means thatthe higher the level of goodwill, the smaller the relative change in goodwill,ceteris paribus. No assumptions, at this point, are made concerning thecross-effects between g and G. The following equation of motion for goodwillis reasonable and easy to work with:G g G= −α δ (8)The parameter a is a CSR investment efficiency parameter, and d denotes therate of depreciation in the goodwill stock (exponential decay). The efficiencyparameter reflects the ability to transform CSR into goodwill stock. Thedepreciation of goodwill over time, if not maintained, can be interpreted as ifstakeholders tend to forget about the firm’s historical social responsibility. Itcould also be interpreted as an effect of other firms investing in CSR, andtherefore deflating the goodwill of this particular firm (as a particular firm isbecoming ‘more similar’ to other firms, the product differentiation effect getsless pronounced).Using equation (5) together with (8), we construct the current valueHamiltonian,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 priceof goodwill represents the ‘theoretically correct’ price of goodwill should it betraded in a competitive market. That is, l is closely related to the marginalcost of investing in goodwill (as we shall see below).The optimal conditions given by the maximum principle areHgc= 0 (10)λ λ= −r HGc (11)limtt→∞( ) =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 autonomousproblems and is needed to provide a boundary condition at the limit.14Equation (10) can be written14The boundary condition is typically replaced by the assumption that the optimal solutionapproaches a steady state and ‘settles down’ (Kamien and Schwartz, 1991, section 9).78 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  11. 11. − + =⇔=AAggλαλα0(13)which simply states that the shadow price of goodwill is equal to the marginalcost of investing in CSR normalized with the efficiency parameter. Expand-ing the optimal condition (11) yieldsλ λ λδδ λ= − − −( )= +( ) − −( )r R Cr R CG GG G(14)which is the differential equation for the shadow price of goodwill. Now wecan 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 theresult for λ in (14), then substitute l for Ag/a in (14). After isolating changesin CSR—g˙ on the left-hand side—we have the following differential equationfor CSR investments:grAR CAgG Ggg=+( ) − −( )+( )+( )+( )δα (15)The development over time for g is a function of both g and G. From (15) wesee that the difference between marginal costs and marginal benefits governschanges in CSR investments over time. We see that ifA R Crgg G Gα 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 arelarger/smaller than marginal benefits, the firm invests/disinvests to the pointwhere benefits equal costs.The system is in steady state when g˙ = 0 and G˙ = 0 (and λ = 0). Assumingthat the efficiency parameter a = 1, thenAR CrgG G=−+( )δ(17)A Microeconomic Model of Corporate Social Responsibility 79© 2010 Blackwell Publishing Ltd
  12. 12. g G= δ (18)The marginal cost of investing in one extra unit of CSR is equal to thebenefits 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 investsan amount of CSR equal to the decay of goodwill.2.3 Characterizing the equilibriumWhen certain concavity conditions are satisfied for the Hamiltonian, theconditions in (10), (11) and (12) are sufficient for maximization (see, forexample, Mangasarian, 1966; Kamien and Schwartz, 1971; or Arrow, 1998).In short, these conditions require that the current value Hamiltonian isconcave in g and G, jointly, implying that the Hessian of the current valueHamiltonian is negative semi-definite. The Hessian can be writtenH HH HAR Cgg gGGg GGggGG GGc cc c⎡⎣⎢⎤⎦⎥ =−−⎡⎣⎢⎤⎦⎥00(19)Since both terms in the diagonal are non-positive by assumption, the Hessianof the current value Hamiltonian is negative semi-definite in the arguments gand 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 touse (14) together with (18). Note that using the differential equation for g willgenerate the same qualitative results and conclusions since g is proportionalto l. Setting λ = 0 in (14) and isolating l on the left-hand side gives the locusfor the co-state in steady state,15λδ=−+R CrG G(20)The right-hand side depends only on G, and since R is increasing and concavein G, and C is decreasing and convex in G, the term (RG - CG) is decreasingand convex in G. Around G = 0 the term (RG - CG) tends to infinity, and thelocus 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 no15The locus for g˙ = 0 would look very similar, Ag = (RG - CG)/(r + d). Assuming a quadraticfunction for A(g) implies that the locus for g˙ = 0 is proportional to the locus for λ = 0, so thequalitative results would be the same whether we use either locus in our phase diagram.80 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  13. 13. constant and a coefficient equal to the depreciation rate, d. Since g is a lineartransformation of l, the locus g - dG = 0 can be written h(l) - dG = 0, whereg = h(l) and h-1(g) = l.16Plotting the loci in the [l, G]-space generates thecurves in figure 1. Studying the partials of λ and G˙ we can easily verify, forthe functional assumptions at hand, that ∂ ∂λ λ > 0, ∂ ∂λ G > 0, ∂G˙ /∂l > 0and ∂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 stablebranch leading into it from both left and right. It is easy to formally show thatthis equilibrium is a saddlepoint maximum. See the Appendix for details.2.4 Comparative statics and dynamic envelope resultsIn this section we parameterize the model and look at steady-state CSRinvestment behavior (comparative statics) and some back-of-the-envelopedynamic analysis. The steady-state analysis is standard. The derivation of16The 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.A Microeconomic Model of Corporate Social Responsibility 81© 2010 Blackwell Publishing Ltd
  14. 14. cost–benefit rules follows the back-of-the-envelope ‘recipe’ proposed byCaputo (1990a, 1990b).17We begin by parameterizing the model.2.4.1 Model parameterizationConsider a firm that initially maximizes profits by choosing the levels ofcapital and labor,max ,,K LpY 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.18Next, thefirm maximizes profits with respect to investment in CSR and building to astock of goodwill capital. At this stage L* and K* are taken as given.19TheCSR management problem is writtenV e te p G Y w G L q G K A g trtrtω ω( ) = ( )= ( ) − ( ) − ( ) − ( )[ ]−∞−∫maxmaxΠ d* * * d00000∞∫= −( ) =G g GG Gα δ(22)where w is a vector of parameters.20Assume the following parametric speci-fication for the functions p(G), w(G), q(G) and A(g):p G p Gw G w Gq G q GA( ) = + >( ) = − >( ) = − >ε εθ θγ γln ,ln ,ln ,withwithwith000gg p p g gp w q p p( ) = +( ) + >=csr advcsr advwith1202β βω 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, whichis a maximum. The output price, wage and cost of capital functions are17See also, for example, Oniki (1973) for a similar methodology. Caputo (1990b) contains ageneral discussion of comparative dynamics in optimal control problems.18Compare with the ‘H*’ in (1).19The problem would be considerably more complex if we include K and L as controls, withoutreally adding relevant richness to the analysis.20All parameters are non-negative.82 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  15. 15. specified so that they depend on the exogenously given market rate and anendogenous premium determined by a parameter and goodwill capital. TheCSR cost function consists of a linear part that depends on the price of CSRand the price of promoting CSR, and a non-linear part that represents coststhat 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 haveA g R G C Grg Gg G G* * ** *( )=( ) − ( )+=α δδα(24)where g* and G* are steady-state levels of CSR investments and the goodwillstock. Given the functional forms specified and parameterized in (23) we canwritep p g GY L Krcsr adv* ** * *+ +=+ +( )+βαε θ γδ1(25)where G* = (a/d)g*. After some rearranging we geta g bg c* *( ) + + =20 (26)wherea =1bp p=+csr advβcY L Kr= −( )+ ++αβαδε θ γδSolving for g* yields the following expression for the steady-state level ofCSR investments and the goodwill stock:gab ac b* = − ± − +( )124 2A Microeconomic Model of Corporate Social Responsibility 83© 2010 Blackwell Publishing Ltd
  16. 16. Gab ac b* = − ± − +( )⎡⎣⎢⎤⎦⎥αδ124 2Given a, b and c we can expand and cancel terms so that the steady-stateconditions become21gY L Krp p p pG** * * csr adv csr adv=( )+ ++++⎛⎝⎜⎞⎠⎟ −+124122αβαδε θ γδ β β** *=αδg(27)As long as the efficiency parameter is positive, a > 0, marginal changes in itsvalue 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, sinceit cannot convert CSR efforts into goodwill at any investment rate. The stockof goodwill, G*, is proportional to the efficiency parameter, i.e. its steady-state level is directly affected by changes in the parameter a.22The signs of thepartial 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 caseof the rate of depreciation in goodwill, d, the price of CSR, pcsr, the price ofpromoting CSR, padv, and the crowding-out parameter, b. It is easily verifiedthat ∂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 maintainit. However, ceteris paribus, an increase in the depreciation rate decreases thegoodwill stock. Furthermore, we can derive the following partials for theprice of CSR and price of promotion/advertising:∂∂∂∂∂∂∂∂gpgpp pGpGpp* ** *csr advcsr advcsr advs= =+( )−= =ββαδΨΨ2 2ccr advcsr adv* * *+( )−⎡⎣⎢⎤⎦⎥=+( ) +( ) + + +( )pr p p Y L Kββδ βδ ε θ γΨΨΨ242ββ δ20r +( )>(28)The signs of ∂g*/∂pcsr, ∂g*/∂padv, ∂G*/∂pcsrand ∂G*/∂padvare ultimatelydetermined by the term (pcsr+ padv) - bY, which is negative for reasonable21Note that only the negative root makes sense in this case.22An increase in a would simply scale the stock of goodwill upward since CSR now adds moregoodwill per unit invested.84 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  17. 17. parameter values. The partial derivatives of g* and G* with respect to thecrowding-out parameter are derived as∂∂g r p p p prY L** *csr adv csr advβδ ββ δβδ ε θ=+( ) +( ) − +( )[ ]+( )−+ΨΨ223++( )+( )=γβ δβαδ βKrG g** *2 3Ψ∂∂∂∂(29)which both have negative signs (for reasonable parameter values), as onewould expect.2.4.3 Dynamic back-of-the-envelope analysisHere we aim to investigate the dynamic effects on firm value of changes inrelevant parameters associated with CSR investments and the goodwill stock.This is different from the previous exercise that analyzed CSR investmentbehavior 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 planninghorizon using a back-of-the-envelope approach.Consider the envelope properties of value function, V(w). The dynamicenvelope theorem postulates that the first partials of V(w) are found by (i)differentiating the Hamiltonian for the optimal control problem directly withrespect to the parameters of interest, (ii) holding the state, co-state andcontrol fixed, then (iii) evaluating the partials along the optimal paths forthese variables, and (iv) finally integrating the result over the planninghorizon (Caputo, 1990b). This implies we can differentiate the Hamiltoniandirectly with respect to parameters prior to substituting in the optimaltrajectories.Write the present value Hamiltonian23asH e p G Y K Le w G L q G Ke prtrtrt= +( ) ( )[ ]− −( ) − −( )[ ]−−−−εθ γln ,ln ln* ** *ccsr adv+( ) +⎡⎣⎢⎤⎦⎥+ −( )p g g g Gβ ψ α δ122(30)23Here we follow Caputo (1990b) and work with the present value Hamiltonian instead of thecurrent value Hamiltonian. This is convenient and enables us also to explicitly investigate theeffect of changing the discount rate, r.A Microeconomic Model of Corporate Social Responsibility 85© 2010 Blackwell Publishing Ltd
  18. 18. where y is the present value co-state variable. Following the Caputo-recipe24for the parameter e generatesH e GYH e G t Yrtrtεε ω== ( )−−lnln ;** *optimal path(31)so that the dynamic envelope theorem yieldsV e G t Y trtε ω ω( ) = ( ) >−∞∫ ln ;* *d00 (32)That is, if the output market’s sensitivity with respect to the firm’s goodwillincreases, then the value of the firm also increases. Since goodwill creates aprice premium whose size is determined solely by the parameter e, it comesas no surprise that Ve(w) > 0. The same procedure for the other parametersgives usV e Y tprtω( ) = >−∞∫ *d00 (33)V e L twrtω( ) = − <−∞∫ *d00 (34)V e K tqrtω( ) = − <−∞∫ *d00 (35)V e G t L trtθ ω ω( ) = ( ) >−∞∫ ln ;* *d00 (36)V e G t K trtγ ω ω( ) = ( ) >−∞∫ ln ;* *d00 (37)V e g t tprtcsr * dω ω( ) = − ( ) <−∞∫ ;00 (38)V e g t tprtadv * dω ω( ) = − ( ) <−∞∫ ;00 (39)24The derivation of this convenient result in Caputo (1990b) is rather lengthy and involved. Amuch neater and less rigorous derivation is found in Aronsson et al. (2004, ch. 9). The trick is tointroduce an artificial state variable in terms of the parameter of interest (in their case itrepresents a project). Assume we want to examine the project a. Then the co-state dynamicsis represented by α = 0, a(0) = a. The co-state variable or shadow price of a is la = ∂V/∂a. Itis now easy to show from the co-state optimal condition (co-state equation) thatλ αα α= =∞∫∂ ∂V H td0. The result is derived by integrating λα α= −H over (0, •) and settingla(•) = 0 (transversality condition).86 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  19. 19. V e g t trtβ ω ω( ) = − ( )[ ] <−∞∫12020* d; (40)V e t g t trtα ω ψ ω ω( ) = ( ) ( ) >−∞∫ * * d; ;00 (41)V e t G t trtδ ω ψ ω ω( ) = − ( ) ( ) <−∞∫ * * d; ;00 (42)V te trrtω ω( ) = − ( ) <−∞∫ Π d00 (43)Let us briefly comment on the partials portrayed in (33)–(43). All partials areunambiguously signed. None of the derived signs of the partials comes as asurprise, since the Hamiltonian was carefully rigged to meet the conditionsnecessary for a saddlepoint equilibrium. Note that these dynamic enveloperesults recover cumulative discounted functions, and not instantaneous func-tions as with static envelope analysis. Vq(w), Vg (w) > 0 implies that should theparameters governing the size of the premiums on wage and cost of capitalincrease—i.e. employees and capital markets become more sensitive toCSR—then the value of the firm moves in the same direction. If price of CSRand 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 theparameter b. Since Vb(w) < 0, one can conclude that more severe crowding-out effects will lower the value function and thus firm value. Efficiency inconverting CSR efforts into actual goodwill is measured by the parameter a;i.e. should a increase, the firm becomes more efficient in transforming CSRpolicy into goodwill, and from Va(w) > 0 we see that the value of the firm alsoincreases.25An 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 controltheory is that the change in the value of the firm is directly related to thechange in goodwill. This comes from a general result in optimal control (see,for example, Brock, 1998), where the time derivative of the value function ina dynamic optimal control problem is directly related to the net changes in allstocks in the model in the following way:V Si ii= ∑λ (44)25Note that the comparative statics showed that in equilibrium, the CSR investment rate is notaffected by changes in the efficiency parameter. However, the dynamic long-run effect on firmvalue associated with a change in the efficiency to convert CSR investments into goodwill isunambiguously positive.A Microeconomic Model of Corporate Social Responsibility 87© 2010 Blackwell Publishing Ltd
  20. 20. where li is the shadow price of stock i, and Si is the ith stock. Since the presentcontrol problem has only one stock, goodwill, we can writeV G= λ (45)This result suggests that all changes in goodwill, positive or negative, as a resultof investing or disinvesting in CSR, will have direct effect on the value of thefirm given that there is a non-negative shadow price of goodwill capital.263. WHAT CAN EMPIRICAL EVIDENCE TELL US?The model presented in this paper can help grasp how empirical hypothesesconcerning CSR can be derived in a consistent way, without having toresort to speculation or poorly undermined reasoning. Here we identifysome possible hypotheses drawn from the model framework and then lookinto whether the empirical literature on CSR supports or refutes them. Thepurpose 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.27It also revealsthat our model encompasses many of the known and tested features ofCSR.H1: CSR behavior that increases goodwill may have a positive effect on outputprice: ∂P(G)/∂G > 0.Is there a price premium for CSR firms? Just a cursory look aroundsuggests this, e.g. actual forest product price lists suggest that certificationof forest products leads to higher price. Kriström and Lundgren (2003) findevidence of a price premium for ‘green’ pulp in Swedish pulp industry.Furthermore, empirical results in Blend and Ravenswaay (1999) suggestthat American consumers are willing to pay a premium for eco-labeledapples, 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 a26This relationship is (implicitly) the most widely tested in the empirical literature on socialperformance and financial performance. More on this below.27Bert Scholtens is thanked, without being implicated, for suggesting this section.88 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  21. 21. sample of 2000 US firms. Edmans (2007) explores the relationship betweenemployee satisfaction and long-run stock performance by reviewing a port-folio of stocks selected by Fortune magazine as the ‘Best Companies to WorkFor in America’ in 1998. The study finds the portfolio earned more thandouble the market return, which would suggest that employee satisfactionimproves corporate performance.H3: CSR efforts that increase goodwill may have a beneficial effect on the costof capital: ∂q(G)/∂G < 0.By engaging in CSR, the firm signals a probability of less conflicts in thefuture, and thus a more ambitious risk management strategy, which lowersthe 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 ofcapital decreases, and thus firm value increases, following good environmen-tal performance. Derwall and Verwijmeren (2006) find in a US sample thatenvironmental performance and corporate governance have a negative effecton the cost of capital, while human rights issues increase the cost of capital.The net effect is ambiguous. Sharfman and Fernando (2008) study 267 USfirms and show that improved environmental risk management is associatedwith 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 toabatement 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 investmentcauses a 1.88 per cent decrease in productive investments. Similar evidence isfound in an earlier study by Barbera and McConnel (1986), who find thatabatement investments retard capital and labor productivity and henceimpose 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. Margolisand Walsh (2001) or Hay et al. (2005) both provide excellent reviews. Theresults are somewhat ambiguous. However, as mentioned in the introduction,most studies show a positive relationship between firm value and differentA Microeconomic Model of Corporate Social Responsibility 89© 2010 Blackwell Publishing Ltd
  22. 22. measures of social responsibility. This could be a result of ‘publication bias’,i.e. mainly positive results are submitted and subsequently published. But acertain amount of heterogeneous results would be expected according to themodel presented here. The model predicts that some firms will be positivelyaffected by engaging in CSR, while others will not. The expectation ofheterogeneous results is attributed to whether such behavior is rewarded orpunished by a firm’s stakeholders. The existence of a goodwill capital stock issolely dependent on whether the stakeholders care about firm social behavior.In summary, it is not difficult to find empirical evidence for some of thehypotheses that can be derived from the model framework presented here.However, the current literature lacks a ‘joint test’ of several of the hypotheseslisted above.28For example, consider a firm that experiences crowding out-costs due to environmental investments. Does this firm also reap the benefitsin 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 REMARKSThis 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. Theimplications of these findings are that firms will engage in CSR activities ifstakeholders, such as the government, the financial sector, consumers, non-governmental organizations, etc., reward or pressure firms to engage in suchbehavior. The link between profitability and different dimensions of CSR istherefore likely to differ across countries, sectors and even firms. The modelin this paper provides a useful theoretical background for the understandingof CSR incentives and for constructing relevant hypotheses in empiricalapplications.Future research should include the impact of uncertainty, e.g. what are theeffects of environmental incidents or ‘bad news’ that arrive over time in somestochastic manner? A natural way to model this would be to include astochastic element to the evolution of goodwill capital, e.g. an Ito diffusionprocess and/or a Poisson jump process.29Another possible route of research28Although H5 can be considered a kind of ‘overall’ test of whether social performance isassociated with higher firm value.29The models in, for example, Tapiero (1977, 1978) could be adapted and modified to tell thestory of CSR under uncertainty. See also Lundgren (2003) for an Ito/Poisson process applied tothe evolution of goodwill in a highly simplified investment optimal stopping problem.90 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  23. 23. would be to allow for potential ‘Porter effects’ where some types of CSRinvestments (e.g. investment in green technology) could have positive effectson long-term efficiency of the capital stock and/or spur innovative processesand investment in R&D. This could be modeled within the framework pre-sented here, possibly combined with uncertainty components. It would alsobe 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 productattributes, so-called ‘greenwash’.APPENDIXIn infinite horizon autonomous problems with one control and one statevariable, and when the discount rate is small enough, the equilibrium isusually a stable saddlepoint (Kamien and Schwartz, 1991, section 9).However, this is more of a ‘rule of thumb’ than a mathematical certainty. Ingeneral, the steady state may or may not exist, or there may be multiplesteady states. To characterize the equilibrium formally, we now proceed tostudy the linear differential system that approximates (6) and (11).30Again wework with the differential equation for l instead of g. Since g is proportionalto l, compare (14) with (17), the qualitative results will be the same. Write theHamiltonian in general form asH g G f g Gc= ( ) + ( )Π , ,λ (A1)Since g maximizes the value function (5), we haveH g G f g Gg g gc= ( ) + ( ) =Π , ,λ 0 (A2)andHggc< 0 (A3)Recall thatλ λ λ= − ( )r H g GGc, , (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):30The procedure follows Kamien and Schwartz (1991, section 9).A Microeconomic Model of Corporate Social Responsibility 91© 2010 Blackwell Publishing Ltd
  24. 24. d d d dc c cH H g H G fgg gG g= + + λ (A6)d d dg u G uG= + λ λ (A7)Using (A6) we haved d dcc cgHHGfHgGggggg= −⎛⎝⎜⎞⎠⎟ −⎛⎝⎜⎞⎠⎟ λ (A8)From (A7) and (A8) we see thatuHHufHGgGggggg= − = −cc c, λ (A9)Substitute (A5) into (6) and (A4):G f u G G G Gr H u G GG= ( )( ) ( ) == − ( )( )λλ λ λ λ, , ,, , ,0 0c(A10)The steady state, (l*; G*), exists iff u G Gr H u G Gg u GGλλ λ λλ* * ** * * ** * *c, ,, , ,,( )( ) =− ( )( ) =− ( ) =000(A11)The nature of the steady state is characterized by looking at a linear Taylorexpansion approximation of the system given by (A10) around l* and G*:G f f u G G f uH H u G G r fG g G gGG Gg G G= +( ) −( ) + −( )= − +( ) −( ) + − −* **c cλ λ λλ HH uGgc*λ λ λ( ) −( )(A12)Using (A9) we can eliminate uG and ul and write (A12) asG a G G bc G G r a= −( ) + −( )= −( ) + −( ) −( )* ** *λ λλ λ λ(A13)where a f f H HG g gG gg= − c c, b f g Hgg= −( )2 cand c H H H HGg gg GG gg= ( ) −⎡⎣ ⎤⎦c c c c2.The characteristic equation and roots for the system (A13) arem rm a r a bcm mr r a bc21 2212022 42− + −( ) − == ±−( ) +⎡⎣ ⎤⎦,(A14)92 Tommy Lundgren© 2010 Blackwell Publishing Ltd
  25. 25. If the roots are real, then the larger root is positive. The sign of the smallerroot is either positive or negative. It is negative ifbc a r a> −( )or−⎛⎝⎜⎞⎠⎟ ( ) −⎡⎣ ⎤⎦> −+( )f gHH H HHfggGg gg GGggG22cc c ccff HHr ff HHg gGggGg gGggcccc⎛⎝⎜⎞⎠⎟ − −⎛⎝⎜⎞⎠⎟⎡⎣⎢⎤⎦⎥−( )If this holds, the roots are real and of opposite sign, and the steady statesatisfies the conditions to be a saddlepoint. It is easy to verify that for thefunctional properties assumed in our model, this condition indeed holds.REFERENCESArora, S., Gangopadhyay, S. (1995): ‘Toward a theoretical model of voluntary over-compliance’, Journal of Economic Behavior and Organization, 28, pp. 289–309.Aronsson, T., Löfgren, K.-G., Backlund, K. (2004): Welfare Measurement in ImperfectMarcets—A Growth Theoretical Approach. Edward Elgar, Northhampton, MA.Arrow, K. J. (1998): ‘Applications of control theory to economic growth’, in Dantzig, G. B.,Veionott, A. F. Jr. (eds): Mathematics of Decision Sciences, Part 2, American MathematicalSociety, Providence, RI.Bagnoli, M., Watts, S. (2003): ‘Selling to socially responsible consumers: competition and theprivate provision of public goods’, Journal of Economics and Management Strategy, 12, pp.419–45.Barbera, A., McConnel, V. (1986): ‘Effects of pollution control on industry productivity: afactor demand approach’, Journal of Industrial Economics, 35, pp. 161–75.Baron, D. P. (2001): ‘Private politics, corporate social responsibility, and integrated strategy’,Journal of Economics and Management Strategy, 10, pp. 7–45.Baron, D. P. (2007): ‘Corporate social responsibility and social entrepreneurship’, Journal ofEconomics and Management Strategy, 16, pp. 683–717.Baron, D. P. (2008): ‘Managerial contracting and corporate social responsibility’, Journal ofPublic Economics, 92, pp. 268–88.Bergman, L. (1995): ‘Simple observations on the “Green Firm” ’, in Wolff, R., Ytterhus, B. E.(eds): Environmental Management—Where Do We Stand?, Cappelen Akademisk Forlag,Oslo.Blend, J. R., Ravenswaay, E. O. (1999): ‘Measuring consumer demand for ecolabeled apples’,American Journal of Agricultural Economics, 81 (5), pp. 1072–7.Bolvig, I. (2005): ‘Firm-provided social concerns? Just another compensating wage differentialsstory?’, International Journal of Manpower, 26 (7–8), pp. 673–704.Brock, W. (1998): ‘Optimal control and economic dynamics’, in Eatwell, J., Milgate, M.,Newman, P. (eds): The New Palgrave Dictionary of Economics, Palgrave Publishers,Houndmills.Caputo, M. (1990a): ‘Comparative dynamics via envelope methods in variational calculus’,Review of Economic Studies, 57 (4), pp. 689–97.Caputo, M. (1990b): ‘How to do comparative dynamics on the back of an envelope in optimalcontrol theory’, Journal of Economic Dynamics and Control, 14, pp. 655–83.A Microeconomic Model of Corporate Social Responsibility 93© 2010 Blackwell Publishing Ltd
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