A Real Options Approach to Abatement Investments ... - Springer Link

Environmental and Resource Economics 25: 17–31, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

17

A Real Options Approach to Abatement Investments and Green Goodwill TOMMY LUNDGREN Department of Forest Economics, S-901 83 Umeå, Sweden (E-mail: [email protected]) Accepted 16 October 2002 Abstract. In this paper we adopt the green goodwill argument as to why firms voluntarily invest in abatement capital. We investigate the effects on the abatement investment decision of changes in uncertainty about future green goodwill, competitor abatement investments, regulations, etc., using a real options framework. Our results indicate that increased uncertainty about consumers’ willingness to pay for “green” products in the future discourage voluntary abatement investments. The model also suggests that voluntary abatement investments are promoted by an increased threat of regulation and competitor abatement investments. Furthermore, the benefit-cost ratio of the abatement investment project, at the point where it is optimal to invest, is independent of what regulatory regime (stringent or lenient) the firm operates in. We also conclude that despite the fact that voluntary abatement investment exists, there may still be room for environmental policy. Key words: abatement capital, environmental policy, irreversibility, uncertainty JEL classification: C61, D81, E22, Q20

1. Introduction Voluntary investments in abatement capital by firms are becoming increasingly common.1 A cursory look around business environments today suggests that there is a growing tendency for firms to adopt an image of being green, and the recent overcompliance with environmental standards in some industries reveals a behavioral change among firms. For example, Brännlund and Kriström (1997) and Brännlund et al. (1996) show that during the last two decades pulp and paper mills in Sweden have made efforts far beyond regulatory requirements, to lower emissions of toxic substances. This has resulted in a rapid reduction of emissions while, during the same period, pulp and paper production has grown steadily. There are several reasons why firms’ emissions are below regulatory limits: (1) Regulations may be suboptimal and some firms pollute less, and others more, than the allowed level. In principle, the regulator can correct this by introducing a market for trading emission permits.2 (2) If emissions are stochastic and difficult to control, firms may try to “be on the safe side”, to avoid penalties. (3) Improved environmental performance may have a positive impact on a firm’s stock price and hence increases the wealth of its stockholders. This creates an incentive for

18

TOMMY LUNDGREN

the firm to invest voluntarily in abatement equipment. (4) Consumer preferences play an important role in promoting abatement investments. That is, investment in abatement equipment may create a competitive edge for the firm, through a boost in green goodwill, which has positive effects on profits. If consumers prefer to buy products from a “greener” firm, the additional cost of being green may easily be justified by higher revenues. Kriström and Lundgren (2001) show that green goodwill generated by abatement investments may have a positive effect on Swedish pulp firms’ profits. Non-voluntary abatement investments are analyzed in, for example, Beavis and Dobbs (1986), Forster (1988), Hartl (1992), Hartl and Kort (1996), and Farzin and Kort (1998). These papers consider a firm which has to acquire a certain amount of abatement equipment in order to comply with regulatory standards. A model of overcompliance can be found in Arora and Gangopadhyay (1995), where overcompliance is attributed to consumer preference for environmental quality.3 Brännlund and Löfgren (1995) suggest that overcompliance among pulp and paper firms in Sweden exists because the cost of exceeding the regulatory limits may be high and emissions stochastic. The firm therefore chooses to be on the safe side by emitting significantly less than what is allowed. In the environmental finance literature, Feldman, Soyka and Ameer (1997), and Hamilton (1995), have all suggested that improved environmental performance has an impact on stock prices. By lowering the environmental risk via investments in abatement capital, the company lowers its systematic risk (market risk), and as a consequence its total risk. This tends to, ceteris paribus, increase the current stock price. Hart and Ahuja (1996) and Klassen and McLaughlin (1996) study the relationship between environmental performance and overall firm performance. They find evidence of a positive correlation between these two variables. As in Kriström and Lundgren (2001) this paper adopts the green goodwill argument as to why some firms overcomply. Our model extends the Kriström and Lundgren model by also taking into account irreversibility, uncertainty, and investment timing. These are features which certainly characterize an abatement investment project. Using a real options framework we are able to capture these important aspects of the abatement investment decision. We investigate the effects on the decision to voluntarily invest in abatement capital of different regulatory regimes, regulation intensity, competing firms’ abatement investments, etc. The effects of regulatory activity and different regulatory regimes are especially interesting to investigate, since a potential problem for regulators is how to design and implement environmental policy in industries where a fraction of firms voluntarily make efforts to reduce emissions. One of our conclusions is that even though voluntary abatement investment exists, there may still be room for environmental policy. The analytical framework in this paper is based on the continuous-time stochastic optimal stopping model of irreversible investment developed by McDonald and Siegel (1986).4 In our model the value of the abatement investment project is assumed to be driven by green goodwill and the threat of regulation. We solve

ABATEMENT INVESTMENTS AND GREEN GOODWILL

19

for the optimal investment rule for the project and investigate the properties of the solution with some numerical examples. The remainder of this paper is structured as follows: In Section 2 the model framework is developed. In Section 3 we investigate the properties of the solution with some numerical illustrations. Concluding remarks are offered in Section 4. 2. Model Formulation The decision to voluntarily invest in abatement equipment is assumed to be uncertain and irreversible. Uncertainty arise from the fact that the future evolution of green goodwill is not known with certainty, due to the lack of knowledge about the consumers’ willingness to pay for environmentally friendly products. Irreversibility arises because abatement equipment is so specialized that it cannot be used in any other firm activity, and also because there is no (or a limited) functioning second hand market for this kind of capital. These features make the real options framework suitable when analyzing the investment problem. Consider a firm which can invest in abatement equipment that possibly boosts green goodwill and profits in the future. The firm has some market power and can influence the demand for its product by being more or less “green”. Competing firms may invest in similar abatement projects, and a regulator may implement an environmental policy requiring all firms to install new abatement technology. If a regulator implements a policy forcing the firm and its competitors to install green technology, the value of the abatement project drops, perhaps to zero, for the firm and its competitors, since the consumers cannot rank the firms if they are all equally “green”. We start by deriving the value process of the abatement investment project without regulations. The project value, here denoted V, is the expected discounted future flow of benefits generated by this particular project. If the firm knows the irreversible cost of the project, the problem reduces to deciding at which point it is optimal to pay C in return for an investment project, V, whose value is dependent upon green goodwill G (time index suppressed below), V = f (G)

(1)

It is assumed that there are no operating costs associated with the abatement investment project.5 For simplicity, green goodwill is assumed to follow a geometric Brownian motion,6 dG = µGdt + σ GdW,

(2)

where µ and σ are the drift and uncertainty parameters respectively, and dW is an √ increment of a Wiener process (= ε dt, where ε is N[0,1]). Other firms engaging in abatement investment projects is modeled by setting µ < 0. That is, for a given start value of G, there is a downward trend in green goodwill due to competitor abatement investments. At the time when the firm is faced with the opportunity to

20

TOMMY LUNDGREN

invest, it is endowed with the initial green goodwill stock, G0 . This initial stock is assumed to reflect historical abatement investments, and the particular project’s potential to generate green goodwill. Uncertainty about consumers’ willingness to pay for environmentally friendly products is represented by σ . Define the abatement project value V in (1) as f (G) = δGθ (3),

(3)

where θ is the elasticity of the abatement project value with respect to green goodwill. Given (1) and (3) we want to find the stochastic differential for the abatement project value V. For simplicity, from this point on we set δ = 1. Applying Ito’s lemma7 (see Appendix for details) to V = f(G) yields 1 2 dV = θµ + θ(θ − 1)σ V dt + θσ V dW, 2 (4) = µ V dt + σ V dW. The stochastic process governing the value of the abatement investment project inherits the geometric Brownian motion structure of the green goodwill process in equation (2). To account for the threat of regulation, we append a Poisson jump process to the investment project value process,8 dV = µ V dt + σ V dW − V dP .

(5)

The term dP is a Poisson increment with mean arrival rate λ. The increment dP is 0 with probability (1 − λ)dt, and γ with probability λdt. By assumption E[dWdP] = 0. If the regulator implements an environmental policy, P falls by a fixed percentage given by 0 < γ < 1 with probability 1. In other words, most of the time V fluctuates as a geometric Brownian motion with the trend µ and uncertainty parameter σ as in (4), but in each time interval dt there is a probability λdt that V will drop to (1 − γ ) times its value before the “event” (regulation) as suggested in (5). That is, at the date of compliance the value of this particular abatement project drops, possibly to zero, since all firms are forced to be equally “green”. In other words, the regulation takes away the potential competitive edge the firm may experience of being a “first mover”. The parameter λ is the mean arrival rate of the Poisson process. From this point on we interpret λ as regulatory intensity.9 Note that in this model the firm’s opportunity to invest is equivalent to an American (perpetual) call option, i.e., it has the right, but not the obligation, to invest in the abatement project at a pre-specified price (sunk cost). The problem is then to decide when to exercise this option. Luckily, well developed option valuation techniques can easily be applied to this setting, and the solution is in the form of a free boundary or threshold value for the abatement investment project, V. The payoff from investing at time t is Vt − C. We want an investment rule that maximizes its expected present value. This rule is given by O(V ) = max E[(VT − C)e−ρT ] T

(6)


21

where ρ denotes the firm discount rate, O(V) is the option value or value of waiting, T is the unknown future time of investment, and the maximization is subject to (5). We solve the maximization problem by using dynamic programming techniques. The main idea of dynamic programming is to split the sequence of decisions into two parts; the immediate choice and the remaining decisions, all of whose effects are summarized in the so called continuation value. Assuming risk neutrality,10 the value of the option to invest in the continuation region, i.e., the interval where the firm does not exercise its option to invest, is given by O(V ) =

1 E[O(V + dV )], 1 + ρdt

(7)

which, when taking expectations, can be written O(V ) =

1 (O(V ) + E[dO(V )]). 1 + ρdt

(8)

By multiplying both sides by 1 + ρdt we obtain the optimal investment rule as given by the solution to the Bellman equation,11 ρO(V )dt = E[dO(V )].

(9)

This value function makes explicit the idea that the entitlement to the flow of profits generated by the abatement investment, described in (5), is by itself an asset, and O(V) is its value. The left-hand side of the equation represents normal return per unit time that a decision maker would require to hold this asset. The right-hand side is the expected rate of capital gain. Equation (9) becomes a no-arbitrage or equilibrium condition, expressing the investor’s willingness to hold the asset. We apply Ito’s lemma12 (see appendix for details) to dO(V), take expectations, and rearrange (9) to obtain 1 2 2 σ V OV V + µ V OV − (ρ + λ)O + λO[(1 − γ )V ] = 0. 2

(10)

of the option value with respect to V. OV and OV V are first and second derivatives Recall that σ 2 = (θσ )2 , and µ = θµ+ 12 θ(θ − 1)σ 2 . The solution to (10) must satisfy the following boundary conditions, O(0) = 0 O(V ∗ ) = V ∗ − C, O (V ∗ ) = 1

(11) (12) (13)

where V∗ is the threshold value or free boundary at which the firm is indifferent to making the investment or not. Condition (11) means that if V goes to zero it will stay there, and as a consequence the option value is zero. Equation (12) is the value matching condition, and (13) is the so-called smooth pasting condition.13

22

TOMMY LUNDGREN

The solution to (10) is assumed to be of the form14 O(V ) = AV b ,

(14)

where b is the positive solution to the fundamental quadratic associated with equation (10), 1 2 σ b(b − 1) + µ b − (ρ + λ) + λ(1 − γ )b = 0. 2

(15)

For given values of the parameters of the model the positive root b can be obtained. Note that if γ = 1 we can easily obtain an analytical expression for b. A complete solution to the investment problem involves solving for the threshold value V∗ , i.e., the value at which it is optimal to exercise the abatement investment option. Substituting (14) into (12) and (13) and rearranging yields b ∗ C, (16) V = b−1 and A=

V∗ −C . (V ∗ )b

(17)

The investment rule at time t is then if Vt ≥ V ∗ → invest, if Vt < V ∗ → wait.

(18)

Irreversibility and uncertainty drive a wedge between the threshold V∗ and the cost of investing, C. The size of the wedge is determined by the underlying parameters of the model. The next section is devoted to investigating the effect on this wedge when varying the parameters of the model. First, we proceed to describe two alternative regulatory regimes in which the firm may operate. We consider two alternative regulatory regimes. Regime 1 involves an ex-post subsidy, whereby the regulator compensates firms that have not yet installed less pollution intensive technology at date of compliance. Regime 2 considers a regime where non-compliant firms are forced to install new abatement capital and carry all costs at date of compliance. Both regimes have counterparts in reality. Regime 1 is similar to the subsidies used early in the 1970’s in the OECD area.15 Regime 2 is similar to BACT, Best-Available-Control-Technology, an approach frequently advocated by environmental regulatory agencies worldwide. The regulator decides on a level of emissions for a pollutant and leaves it up to the firms to use BACT to lower emissions to the new standard. There is a difference between end-of-pipe type regulation and abatement investments undertaken voluntarily by the firm. However, the firm can use any end-ofpipe or other abatement investment as a signal of its commitment to environmental


23

improvements. Indeed, the model allows for a general interpretation of how costs inflicted voluntarily or involuntarily are mapped into signals about the status of the firm’s product in the eyes of the consumer. In both scenarios, the firm must balance the net benefits of acting now against the net benefits of waiting. Acting early on provides a possible market edge, while waiting risks competitor abatement investments and/or the regulatory measure. Regime 1 and regime 2 can be considered polar cases. We could allow for any regime in between these two extreme cases. However, including intermediate regimes would not enrich or change the general interpretation of the results. In regime 2 the sunk cost C must be modified to accurately take into account that firms have to carry all costs of compliance. Firms have to pay C sometime in the future with probability 1 if λ > 0. The density function of first intervention by the regulator is λexp(−λTR ), where TR is the date of regulation. Hence, in regime 2 the expected and unavoidable cost due to regulation in the future, when the firm does not choose to invest prior to intervention, is given by ∞ λ C. (19) (λe−λTR Ce−ρTR )dTR = λ+ρ 0 Therefore, in regime 2 the right hand side of equation (19) must be subtracted from the sunk cost C, to accurately describe the true cost of investing voluntarily in abatement capital. The modified cost is ρ λ C= C = C2 . (20) C− λ+ρ λ+ρ The regime 2 sunk cost depends on the parameters ρ and λ. C2 is not well defined for ρ ≤ 0. From this point on, the threshold values and sunk costs in regime 1 and regime 2 are denoted V∗1 , V∗2 , C1 and C2 respectively. Note that, for given values of ρ and λ, V∗2 is simply a multiple V∗1 : V∗2 = [ρ/(λ + ρ)]V∗1 , with V∗1 > V∗2 , if λ > 0 and ρ > 0. If λ = 0 then V∗1 = V∗2 . Also, it is obvious that V∗1 /C1 = V∗2 /C2 , i.e., the regime 1 and regime 2 threshold-cost ratios are always equal. 3. Numerical Illustrations In this section we investigate how the abatement investment threshold values, V∗1 and V∗2 , are affected by changes in the underlying parameters of the model. If V∗1 and/or V∗2 increase (decrease) as a result of a change in one of the model parameters, it implicates that the option value or value of waiting becomes larger (smaller) and this would discourage (promote) actual investment in real abatement capital. Recall that V∗2 is always proportional to V∗1 . Hence, unless we change ρ and λ it is not necessary to investigate the effects on V∗1 and V∗2 separately. In Table I we display the effects on V∗1 /C1 and V∗2 /C2 , i.e., the value of benefits relative to investment cost at which abatement investment is optimal, when changing the parameters µ, θ, and σ .

24

TOMMY LUNDGREN

Table I. Threshold-cost ratios without regulation (λ = 0 → V∗1 = V∗2 and C1 = C2 ) µ θ σ 0.10 0.15 0.20 0.25

0.50

−0.01 1.00

1.50

0.50

−0.05 1.00

1.50

0.50

−0.10 1.00

1.50

1.11 1.18 1.25 1.32

1.25 1.44 1.67 1.92

1.43 1.85 2.50 3.56

1.04 1.09 1.14 1.19

1.09 1.19 1.32 1.47

1.15 1.32 1.58 1.93

–a 1.05 1.08 1.12

1.05 1.11 1.19 1.28

1.08 1.17 1.31 1.49

Constant parameters: ρ = 0.05, λ = 0, γ = 1, C1 = 1, C2 = C1 [ρ/(λ + ρ)] = C1 . a V∗ and V∗ not well defined. 1 2

Table I entries refer to both regime 1 and regime 2 threshold-cost ratios. The investment threshold-cost ratio is increasing in the parameter representing competitor abatement investment activity, µ. Remember that by assumption µ < 0, and as competitors invest more in similar abatement projects, the value of µ drops even further below zero. This implies that as competitor abatement investment activity increases (i.e. µ decreases), the firm is more likely to invest in the project. A drop in the value of µ increases the downward trend in green goodwill which indicates that environmental awareness among competitors has increased. As a consequence the expected rate of change in the project value, µ = θµ+ 12 θ(θ − 1)σ 2 , decreases, and hence the value of waiting decreases, which promotes actual investment sooner.16 An increase in the elasticity of the investment project value with respect to green goodwill, θ, tends to augment the threshold value, and hence discourage investment in the abatement project. This result may seem counter-intuitive at first. As the project value becomes more sensitive to green goodwill changes, abatement investments are more likely to be postponed. The effect of increasing θ is twofold. First, it increases the expected rate of change in the project value in the future by increasing µ . This tends to increase the value of waiting. Second, it increases the variance of dV, the expected change of the project value, which also tends to raise the value of waiting.17 Higher volatility in future green goodwill, σ , increases the threshold value, i.e., the abatement investment project is more likely to be deferred. The possibility of higher project values in the future increase, while the downside risk is unchanged. This is a standard result in the real options literature. Note that the combination of low competitor abatement investment activity, high volatility in green goodwill, and a project which is elastic with respect to green goodwill, generates thresholds much larger than the traditional NPV-rule, i.e. V∗1,2 = cost of investing (in Table I, cost is equal to 1). For example, if µ = −0.01, θ = 1.50, and σ = 0.25, the threshold is 3.56, or 256% larger than the project cost.


25

Figure 1. The effect on cost and threshold values in regime 1 and 2 when varying discount rate, ρ, and regulatory intensity, λ.

In Figure 1(a–d), we plot the regime 1 and regime 2 investment threshold values and costs against the discount rate and the regulatory intensity. The discount rate and regulatory intensity are varied between 0.00–0.15 and 0.00–1.00 respectively. Base case parameters: λ = 0.10 (when ρ varies), ρ = 0.10 (when λ varies), γ = 1.00, µ = −0.05, θ = 1.00, σ = 0.20, C1 = 1.00, and C2 = C1 [ρ/(λ + ρ)]. A higher discount rate promotes investment sooner (Figure 1a). This is a pure manifestation of the option idea. However, the result also stands in sharp contrast to neoclassical investment theory. In the standard investment model, an increase in the discount rate/interest rate reduces investment by raising the user cost of capital. In this case, however, a lower discount rate makes the future more important, and therefore it increases the value of waiting (the option value), which tends to discourage investment in the actual project. If we would keep the difference ρ − µ constant (i.e., as the firm discount rate increases, the project value drift term is increased by the same amount) as ρ increases, we would get the opposite result.18 That is, as the discount rate increases, the project threshold value increases, and thus real investment is deferred. In Figure 1c, we see that if regulation becomes more likely (λ goes up) the abatement investment project is more likely to be launched. The interpretation of this result is simply that if the expected time until regulation decreases, there is an

26

TOMMY LUNDGREN

incentive to act fast – since time is running out – and reap the “extra profits” that green goodwill may generate. Since we know that V∗1 /C1 = V∗2 /C2 , the interpretation of the results when changing ρ and λ in regime 1 carries over to regime 2. However, at first glance, Figure 1b reveals an interesting result: as the discount rate is increased, the regime 2 threshold also increases. Neglecting the regime 2 cost, we would conclude that investment is deferred with higher discount rate, which is the opposite result in comparison with regime 1. But since regime 2 cost also rises with ρ, this result is not relevant. The rise in the regime 2 cost perfectly offsets the rise in the threshold value so that V∗1 /C1 = V∗2 /C2 , meaning the relationship between threshold and cost are equal in both regimes when changing the discount rate. Figure 1d shows that the regime 2 threshold value drops fast with regulatory intensity. This particular result reflects the sensitivity of the regime 2 investment cost to changes in regulatory intensity. As λ is increased, the “true” cost of investing drops significantly, and s a consequence the threshold also drops significantly.

4. Conclusions This paper has shown how voluntary investments in abatement equipment can be modeled using a real options framework. Assuming that the firm operates under two different regulatory regimes, we investigate the effect upon the abatement investment decision of changes in uncertainty about future evolution of green goodwill, regulations, competitor abatement investment behavior, etc. Numerical examples indicate that increased intensity of regulation and/or similar investments by competitors will encourage voluntary investment in abatement equipment. Uncertainty about future evolution of green goodwill discourages abatement investment. Furthermore, the model suggests that increased elasticity of the project value w.r.t. green goodwill tends to decrease abatement investment. This particular result is mainly due to the amplifying effect that higher elasticity has upon the uncertainty of the project value. Also, according to the model, the benefit-cost ratio of the abatement investment project, at the point where it is optimal to invest, is not dependent upon which regulatory regime the firm is operating in. The numerical illustrations show that option value considerations, caused by irreversibility and uncertainty, lead to more conservative investments in abatement capital. However, in many industries today one can observe that a large fraction of firms are overcomplying. According to Brännlund et al. (1996), 27 out of 30 pulp plants in Sweden emitted less than the allowed levels of certain pollutants in 1990. A valid question is then: is there room for environmental policy to regulate such an industry? For example, consider a firm that, at the time of the investment opportunity, is in a state of overcompliance. If this firm chooses not to invest, since there exists a value of waiting caused by irreversibility and uncertainty, it would still be compliant and there is no need to impose a regulatory measure on this particular firm. On the other hand, if a firm starts off emitting above the regulatory standard


27

and chooses not to invest in green capital19 because of option value considerations, then there is still room for environmental policy. How environmental policy should be designed, considering option values and voluntary abatement investments spurred by green demand, is an important issue which we leave for future research. Adopting the green goodwill-argument developed in Kriström and Lundgren (2001), we show that irreversibility and uncertainty may have considerable impact on the decision to voluntarily invest in abatement investment. Firms invest voluntarily in abatement equipment due to potential gains from green goodwill. That is, the market internalizes externalities, completely or to some degree. Without consumer preferences towards green products these externalities would have to be internalized with regulatory measures. Acknowledgement The author is grateful to Bengt Kriström, Peichen Gong, Anni Huhtala, Roger Craine, Kaj Nystróm, Matthew Auer and two anonymous referees for valuable input on earlier versions of the manuscript. Comments and suggestions from seminar participants at the lunch seminar at the Department of Agricultural and Resource Economics (October 1999, University of California Berkeley, U.S.), and at the environmental economics session at the Young Economist Conference (March 2000, Oxford U.K.), are acknowledged. The usual disclaimer applies. Financial support was generously provided by the Tore Browaldh Research Foundation (Handelsbanken, Sweden). Notes 1. By voluntary abatement investments we refer to those projects that are launched without the obligation to comply with regulatory standards. See for example Smart (1992), or Prakash (2000) for a general discussion on overcompliance and several possible explanations of this phenomenon. The latter reference also contains case studies. 2. This type of overcompliance is not created by the fact that some firms are trying to be green and other firms do not care about the environment. It is simply a result of suboptimal regulatory design. 3. Their model is based on empirical findings in Arora and Cason (1995) and Arora and Cason (1996). 4. See also ch. 5, Dixit and Pindyck (1994), which builds on McDonald and Siegel (1986). 5. It is, however, straightforward to allow for a stochastic operation cost. See, for example, McDonald and Siegel (1986). 6. This assumption is admittedly crucial for the results presented in this paper. We do not know if the results carry over to other assumptions about the underlying stochastic process for green goodwill. The literature on geometric Brownian motion is well-developed and comprehensive (even to an economist). Also, it is easily mixed with Poisson jump processes. Future research may include other assumptions about the stochastic process for green goodwill. 7. For a heuristic introduction to Ito’s lemma applied to an Ito process see Dixit and Pindyck (1994), ch. 3. For a technical derivation and discussion of Ito’s lemma see Malliairis and Brock (1982), section 5, ch. 2.

28

TOMMY LUNDGREN

8. the incorporation of a Poisson process to account for the threat of regulation makes our investment problem basically the same as the one analyzed in Dixit and Pindyck (1994) at the end of ch. 5. The main difference is that our trend parameter, µ , and the uncertainty parameter, σ , are functions of other model parameters. 9. The expected time until the value of the investment takes a Poisson jump downwards because of regulation is 1/λ. For example, if dt equals 1 year, then λ = 0.25 would imply that the expected time until a regulatory intervention arrives is 4 years. 10. Firm discount rate equals the market risk free rate, ρ = r. Note that the dynamic programming solution derived here is equivalent to the options pricing or contingent claims solution as long as risk neutrality is assumed. See Pindyck (1991) or Dixit and Pindyck (1994) for a discussion. 11. This particular kind of Bellman equation is, for example, derived and discussed in Dixit and Pindyck (1994). Note that µ < ρ must hold, otherwise no optimum exists to this investment problem. 12. Merton (1971) or Merton (1976) are the standard references for applications of Ito’s lemma when there are mixed Brownian/Wiener and Poisson jump dynamics. 13. See for example Dixit (1993) for a non-technical discussion of these concepts. 14. Note that a general solution to the differential equation is O(V) = A1 Vb1 + A2 Vb2 , but the second term vanishes since the boundary condition O(0) = 0 implies A2 = 0. 15. However, in most regulatory regimes, these subsidies were probably a part of industrial policy rather than environmental policy. 16. Note that abatement investments of other firms, represented by µ, will affect the focal firm’s abatement investment, but not vice versa. 17. The expected rate of change is (1dt)E[dV]/V = µ − λγ , and the variance is Var[dV] = (σ )2 V2 dt + λγ 2 V2 dt. This implies that ∂{E[dV]}/∂θ and ∂{Var[dV]}/∂θ > 0 for reasonable parameter values. 18. See Dixit and Pindyck (1994: 154–157) for a more detailed discussion of the effects of changes in the discount rate. 19. This would put the firm in compliance or overcompliance.

References Arora, S. and N. Cason (1995), ‘An Experiment in Voluntary Environmental Regulation: Participation in EPA’s 33/50 Program’, Journal Environmental Economics and Management 28, 271–286. Arora, S. and N. Cason (1996), ‘Why Do Firms Volunteer to Exceed Environmental Regulations? Understanding Participation in EPA’s 33/50 Program’, Land Economics 72, 413–432. Arora, S. and S. Gangopadhyay (1995), ‘Toward a Theoretical Model of Voluntary Overcomplicance’, Journal of Economic Behavior and Organization 28, 289–309. Beavis, B. and I. M. Dobbs (1986), ‘The Dynamics of Optimal Environmental Regulation’, Journal of Economic Dynamics and Control 10, 415–423. Björk, T. (1998), Arbitrage Theory in Continuous Time. Oxford, New York: Oxford University Press. Brännlund, R. and B. Kriström B. (1997), ‘Taxing Pollution in an Open Economy – An Illustration from the Nordic Pulp Industry’, Journal of Forest Economics 3, 189–207. Brännlund, R. and K. G. Löfgren (1996), ‘Emission Standards and Stochastic Waste Load’, Land Economics 72. Brännlund, R., L. Hetemäki, B. Kriström and E. Romstad (1996), ‘Command and Control with a Gentle Hand – The Nordic Experience’, Research Paper 115, Department of Forest Economics, Swedish University of Agricultural Sciences. Dixit, A. K. (1993) The Art of Smooth Pasting. Amsterdam: Harwood Academic Publishers. Dixit, A. K. and R. Pindyck (1994) Investment Under Uncertainty. Princeton, N.J.: Princeton University Press.


29

Farzin, Y. H. and P. M. Kort (1998), ‘Pollution Abatement Investment when Environmental Regulation is Uncertain’, Fondazione Eni Enrico Mattei, Economics-Energy-Environment 75. Feldman, S. J., P. A. Soyka and P. G. Ameer (1997), ‘Does Improving a Firm’s Environmental System and Environmental Performance Result in a Higher Stock Price?’, The Journal of Investing, 87– 97. Forster, S. (1988), ‘The Optimal Acquisition of Pollution Control Equipment with an Uncertain Date of Compliance’, Working Paper Department of Economics, University of Wyoming. Hamilton, J. T. (1995), ‘Pollution and News: Media and Stock Market Reactions to the Toxics Release Inventory Data’, Journal of Environmental Economics and Management 28, 98–113. Hart, S. L. and G. Ahuja (1996), ‘Does it Pay to be Green? An Empirical Examination of the Relationship between Emission Reduction and Firm Performance’, Business Strategy and the Environment 5, 30–37. Hartl, R. F. (1992), ‘Optimal Acquistion of Pollution Control Equipment Under Uncertainty’, Management Science 38, 609–622. Hartl, R. F. and P. M. Kort (1996), ‘Capital Accumulation of a Firm Facing an Emissions Tax’, Journal of Economics 63, 1–23. Klassen, R. D. and C. P. McLaughlin (1996), ‘The Impact of Environmental Management on Firm Performance’, Management Science 42, 1199–1214. Kriström, B. and T. Lundgren (2001), ‘Abatement Investments and Green Good-Will’, Manuscript presented at the (i) 7th Ulvön Conference on Environmental Economics, June 2000 (ii) the EAERE Annual Meeting Crete, July 2000, and (iii) Southern Economic Association Annual Meeting in Tampa, FL, USA, 2001. Malliairis, A. G. and R. S. Brock (1982), Stochastic Methods in Economics and Finance. New York: North-Holland. McDonald, R. and D. Siegel (1986), ‘The Value of Waiting to Invest’, Quarterly Journal of Economics 101, 707–727. Merton, R. C. (1971), ‘Optimum Consumption and Portfolio Rules in a Continous-Time Model’, Journal of Economic Theory 3, 37–413. Merton, R. C. (1976), ‘Option Pricing when Underlying Stock Returns are Discontinuous’, Journal of Financial Economics 3, 125–144. Pindyck, R. (1991), ‘Irreversibility, Uncertainty, and Investment’, Journal of Economic Literature 29, 1110–1152. Prakash, A. (2000), Greening the Firm – The Politics of Corporate Environmentalism. Cambridge: University press (U.K.). Smart, B. (1992), Beyond Compliance. World Resources Institute.

Appendix Note that partial derivatives are indicated with subindex. Applying Ito’s lemma to equation (3), f(G) = Gθ , yields 1 dV = df = ft dt + fG dG + fGG (dG), 2

(A1)

where ft = 0, fG = θ Gθ−1 , and fGG = (θ − 1)θ Gθ−2 . Substituting these partials into A1, we can write 1 df = 0dt + θ Gθ−1 dG + (θ − 1)θ Gθ−2 (dG)2 2 1 Gθ Gθ dG + (θ − 1)θ 2 (dG)2 . = θ (A2) G 2 G

30

TOMMY LUNDGREN

Recall that the stochastic process governing green goodwill is defined as dG = µGdt + σ GdW. Substituting this into A2, we can write 1 Gθ Gθ (µGdt + σ GdW ) + (θ − 1)θ 2 (µGdt + σ GdW )2 , G 2 G Gθ = θ (µGdt + σ GdW ) G Gθ 1 + (θ − 1)θ 2 (µ2 G2 dt 2 + σ 2 G2 dW 2 + 2µσ G2 dtdW ). 2 G

df = θ

(A3)

To the last term of A3, we apply the following formal multiplication table: dt 2 = 0, dW 2 = dt, and dtdW = 0. This multiplication table is well-known in stochastic calculus and can be found in, for example, Björk (1998). The result is 1 df = θ Gθ (µdt + σ dW ) + (θ − 1)θ Gθ σ 2 dt, 2 1 θ θ = µθ G dt + σ θ G dW + (θ − 1)θ Gθ σ 2 dt, 2 1 2 = θ µ + (θ − 1)θ σ Gθ dt + θ σ Gθ dW. 2

(A4)

Since V = f(G) = Gθ , we can write the stochastic differential for the value of the abatement investment, without the threat of regulation, as in equation (4), i.e. 1 2 (A5) dV = θ µ + (θ − 1)θ σ V dt + θ σ V dW. 2 Equation (10) is a second-order differential equation in the option value O(V). Here we will go through the derivation heuristically. The underlying stochastic process governing the option value is a geometric Brownian motion combined with a jump process, dV = µ V dt + σ V dW − V dP ,

(A6)

where µ = θ µ + 12 (θ − 1)θ σ 2 , and σ = θ σ . We want to find the expected value of the stochastic differential dO(V) (see equation 9). Using the Ito method when we have mixed Wiener and Poisson processes, is briefly discussed and outlined in Dixit and Pindyck (1994: 85–86). we follow their presentation here. Let us start off in general terms. Assume that we are working with the following general specification of a stochastic process: dX = a(X, t)dt + b(X, t)dW + g(X, t)dP . We want to calculate the expected stochastic differential of a function H which takes X and t as arguments, i.e., H = H(X, t). The expected change in the function H(X, t) is given by 1 2 (A7) E[dH ] = Ht + a(X, t)HX + b (X, t)HXX dt + E[HX g(X, t)dP ]. 2


31

The second term of A7 reflects the possibility, in each time interval dt, that a Poisson event may occur; if it does, X will change by the, possibly random, amount γ g(X, t), and H(X, t) will change accordingly. Following Dixit and Pindyck (1994) the second term can thus be rewritten as D[HX g(X, t)dP ] = Eγ {λ[H (X + γ g(X, t), t) − H (X, t)]}dt,

(A8)

where the expectation operator on the right hand side (rhs), Eγ , is with respect to the size of the jump γ . If γ is given and non-stochastic, as in the problem analyzed in this paper, then Eγ {λ[H (X + γ g(X, t)X, t) − H (X, t)]} = λ[H (X + γ g(X, t), t) − H (X, t)].

(A9)

If the rhs of A9 is substituted into A7 we get 1 E[dH ] = Ht + a(X, t)HX + b 2 (X, t)HXX dt + 2 λ[H (X + γ g(X, t), t) − H (X, t)]dt.

(A10)

To find E[dO] we must modify A10 by substituting H(X, t), a(X, t), b(X, t), and g(X, t) for the functions relevant to our problem; that is with O(V), µ V, σ V, and −V respectively. We then get the following expression for the expected rate of change in the option value: 1 E[dO] = µ V OV + σ 2 V 2 OV V dt + λ[O(V − γ V ) − O(V )]dt. (A11) 2 Finally, by substituting 21 into the Bellman equation defined in (9), and rearranging we have derived equation (10).