In entives for Firms to Share Abatement
Te hnology under a Proportional Liability Rule
Yusen Sung∗
We analyze in this paper the technology diffusion decisions of two polluting firms. Firm 1, armed with superior pollution abatement technology, must decide on how much technology to share with firm 2. Meanwhile, firm 2 must determine how much of the available technology to adopt. Though inspired by Endres and Friehe (2011), which investigates a related issue under environmental laws of strict liability and negligence rules, this paper differs crucially in two fun-damental aspects: we consider a proportional liability rule and costly technology diffusion. We find that the decision to release or adopt more advanced technology hinges on the form of external damage as well as the existence of diffusion costs. Only when external damage is linear and technology diffusion is free will the market equilibrium be efficient.
Keywords:pollution emission, technology transfer, proportional
liability
JEL lassi ation:K13, Q52, Q55
1 Introdu tion
Polluting firms in an industry usually have different pollution abatement technologies at their disposal. The technology state of a firm depends nat-urally on changes in technology and the time the firm makes purchases of technology. Milliman and Prince (1989) pioneered the analysis of a firm’s incentives to promote changes in technology, and was followed by Jaffe and
∗Department of Economics, National Taiwan University.
經濟論文叢刊(Taiwan Economic Review), 42:3 (2014), 315–331。
Stavins (1995), Requate and Unold (2003), and Coria (2009). These au-thors considered various policy instruments, including taxes, subsidies and permits, and compared their welfare effects. However, no victim compensa-tion requirements are included in any of this analysis. An up-to-date survey of this literature is provided by Allan, Jaffe, and Sin (2014).
Endres and Friehe (2011) were the first to embody environmental lia-bility laws in their model. They compared firm diffusion decisions under strict liability, which requires polluting firms to compensate harm irrespec-tive of their behavior, and a negligence rule, which holds firms liable for harm only when they breach certain behavioral norms (for example, excess emission beyond an announced pre-set level). They argued that firm dif-fusion incentives are socially optimal under strict liability, but suboptimal under negligence, if advancement in technology lowers marginal abatement costs for all levels of abatement.
The problem we consider here differs from that of Endres and Friehe (2011) in two important aspects. Firstly, they assumed an exogenous split of compensation responsibility for their strict liability scenario, but offer no explanation or justification as to how the split ratio is determined. We assume instead that firms’ payments are proportional to their emission levels. A firm discharging twice the pollutants of another firm will accordingly face a compensation fine double that of the other.
Secondly, Endres and Friehe (2011) did not take transfer costs of tech-nology into account. There is thus no cost for the leading firm to diffuse its superior technology, and there is similarly no cost for the sluggish firm to upgrade its technology. This is certainly un-realistic. We incorporate tech-nology diffusion costs in our model, and find that the extent of techtech-nology sharing by firms depends critically on these costs.
This paper follows the literature on voluntary technology diffusion (e.g., Endres and Friehe (2011), Coria (2009), and Requate and Unold (2003)) and rules out the possibility of side-payments by firms to acquire superior technology from other firms. We acknowledge that there is indeed a sub-stantial literature on technology licensing (e.g., Fosfuri (2006), Arora and Ceccagnoli (2006), Lach and Schankerman (2004), Sine, Shane, and Gre-gorio (2003), and Arora (1996)) that focuses on paid technology transfer. It would be interesting to compare the market outcomes of these two scenar-ios, but this is outside the scope of the current paper and will be dealt with in follow-up work.
out in section 2. In the third section we analyze the socially efficient levels of technology diffusion with and without diffusion costs as benchmark cases for comparison. The fourth section then focuses on market interaction be-tween the firms. Two dimensions are considered: constant versus increasing marginal damage, and the presence/absence of diffusion costs. We compare market outcomes in these settings with corresponding social optima to check on their efficiency. The final section concludes and summarizes our findings.
2 The Model
Consider two polluting firms (1 and 2) generating identical pollutants in a metropolitan area. Assume that firm 1 possesses a more advanced pollution abatement technology than firm 2. Let their emission levels be denoted by:
(x1, x2) , 0 ≤ xi ≤ ¯xi,
with ¯xi being firm i’s natural uncontrolled emission level. The external pollution damage caused by the two firms depends on the aggregate level:
D(X), X ≡ x1+ x2,
which may exhibit constant or increasing marginal damage (MD): D′(X) > 0, D′′(X) ≥ 0.
To reduce the emission of pollution below its initial unregulated level ¯xi, firm i has to incur a cost:
Ci(xi, Ti)
for installing the end-of-pipe emission control equipment, Ti(∈ [0, 1]) de-noting firm i’s state of technology. The abatement cost function has the usual properties of increasing marginal cost and diminishing returns to tech-nology:
Cxi < 0, Cxxi > 0, ∀xi ∈ (0, ¯xi) , CTi < 0, CT Ti > 0, ∀Ti ∈ (0, 1).
It is also plausible that an advancement in technology will imply a downward shift in the marginal abatement cost (MAC) for all firm emission levels:
−Ci
Further, for notational simplicity, we normalize firm 1’s superior tech-nology state to 1:
T1 ≡ 1.
Firm 2’s level of technology depends on how much technology is actually transferred from firm 1:
T2= t1· t2
with t1 ∈ [0, 1] representing firm 1’s rate of technology sharing, and t2 ∈ [0, 1] being firm 2’s rate of adoption.
3 So ially E ientTe hnology Diusion
We first derive the necessary conditions for the efficient diffusion of technol-ogy when there are no costs associated with sharing or adopting technolo-gies. In this case, the EPA aims to minimize the total social cost:
min T2,x1,x2
TSC0 ≡ C1(x1, 1) + C2(x2, T2) + D(X).
It is obvious that the total social cost can always be reduced if firm 2 possesses a better technology for abatement:
∂TSC0 ∂T2
= CT2 (x2, T2) < 0.
As such, free diffusion of technology prescribes total technology diffusion: T2∗ = t
∗ 1 = t
∗ 2 = 1,
and firm 2 should always embrace the best available technology. This re-quirement for full transfer comes as no surprise because improved tech-nology can lower firms’ marginal abatement costs for all emission levels, and hence is necessary for social cost-effectiveness. Meanwhile, the interior abatement choice x∗ i ∈ (0, ¯xi)must satisfy: x1∗ Cx1(x1, 1) + D′(X) = 0, x2∗ Cx2(x2, 1) + D′(X) = 0, (2) which has the intuitive interpretation of the marginal abatement cost being equal to the marginal social damage.
If, instead, diffusion costs have to be incurred by both firms to bring about the transfer of technology, complete diffusion may no longer be so-cially desirable. Let cibe the constant marginal cost that firm i has to expend to share (or adopt) a superior technology. The EPA goal, after incorporating these diffusion costs, becomes:
min t1,t2,x1,x2
TSCc ≡ C1(x1, 1) + C2(x2, t1t2) + D(X) + c1t1+ c2t2. The first-order necessary conditions thus become:
(t1) − t2· CT2 (x2, t1t2) = c1, (t2) − t1· CT2 (x2, t1t2) = c2, (x1) Cx1(x1, 1) + D′(X) = 0,
(x2) Cx2(x2, t1t2) + D′(X) = 0. (3) Apparently, unless ci is extremely low, only partial diffusion ti∗∗ ∈ (0, 1) is desired in an interior solution. The EPA now has to balance the costs of transferring technology with the abatement costs saved due to the improve-ment in firm 2’s technology. To understand how the diffusion of technology affects the firms’ emissions, we totally differentiate the two equations
Cx1(x1, 1) + D′(x1+ x2) = 0, Cx2(x2, T2) + D′(x1+ x2) = 0, with respect to three variables (x1, x2, T2), and find that:
dx1 dT2 > 0, dx2 dT 2 < 0, d [x1+ x2] dT 2 < 0. (4)
The intuition behind the comparative statics results (4) runs as follows: since firm 2 possesses better abatement technology, its marginal cleanup cost (−C2
x(x2, T2)) declines (by assumption (1)), and hence it should cause less pollution, leading to lower x2. Meanwhile, as x2 goes down, firm 1 faces a lower marginal damage D′(X)for all levels of x1, and hence should increase its pollution emission x1. In aggregate, however, a better technology owned by firm 2 still leads to a reduction in total emission.
Further, total differentiation of the complete set of first-order conditions (3) yields:
dti dcj
which indicates that higher diffusion costs for either firm will lower the ef-ficient rate of diffusion, and ultimately lead to a higher aggregate level of pollution X. Therefore, the government is advised to provide platforms to facilitate inter-firm sharing of technology.
4 The Two-stageDe entralized Market Equilibrium
We next analyze the market interaction between the firms under a propor-tional liability rule. Contrary to the strict liability rule with the exogenous responsibility split as in Endres and Friehe (2011), we assume a more eq-uitable rule, which dictates that polluting firms be obligated to compensate victims an amount proportional to their emission share:
ηi ≡ xi x1+ x2
, i = 1, 2. In this case, the individual firm’s liability will be:
Fi(x1, x2) = ηi· D(X).
First proposed by John Makdisi in Makdisi (1988), the proportional liability rule has been much discussed in the past two decades by legal scholars (for example, Fischer (1993), Boyd and Ingberman (1996), and Green (2004)) as a way to incorporate ‘causation’ in tort case judgements. In Makdisi’s words, “damage should be allocated in proportion to the probability of causation." Simply put, the idea is to make liability depend on the cause of the damage. In reality, proportional liability laws have already been adopted in many countries. For example, in Taiwan, parties involved in traffic accidents will be held responsible according to the level of their negligent behavior.
The interaction between firms can be modeled as a two-stage game. In Stage 1, both firms set their individual diffusion rate
(t1, t2)
simultaneously and independently. Subsequently in Stage 2, given (t1, t2), each firm chooses its emission level to minimize costs, including the costs of pollution abatement and victim compensation, again simultaneously and
independently: min x1 C1(x1, 1) + F1(x1, x2) = C1(x1, 1) + x1 x1 + x2 · D (x1+ x2) , (5) min x2 C2(x1, T2) + F2(x1, x2) = C2(x2, t1t2) + x2 x1+ x2 · D (x1+ x2) . (6) The reason for a Nash setting in Stage 2 is pure symmetry: both firms gen-erate the same pollutants and jointly cause environmental damage, so there is no particular reason to employ a sequential setting.1
To solve for the subgame-perfect Stackelberg equilibrium backwards, we begin with the second stage. The interior stage-2 firm choices of emission xi(t1, t2)are implicitly defined by the first-order conditions:
(x1) Cx1(x1, 1) + x1 x1+ x2 · D′(X) + x2 [x1+ x2]2 · D(X) = 0, (7) (x2) Cx2(x2, t1t2) + x2 x1+ x2 · D′(X) + x1 [x1+ x2]2 · D(X) = 0. (8) Back in Stage 1, firm choices (t1, t2) apparently will depend on the cost of technology diffusion.
With costless technology adoption, firm 2 will certainly set the maximal rate of adoption to reduce its abatement costs:
t2= 1. As for firm 1, it would simply like to choose t1 to
min t1 TC1 ≡ C1(x1(t1, 1) , 1) + x1(t1, 1) x1(t1, 1) + x2(t1, 1) × D (x1(t1, 1) + x2(t1, 1)) . (9) If instead the diffusion of technology is costly with marginal diffusion cost ci, the firms will now engage in a Nash game in Stage 1 to minimize respective 1For a sequential emission game in stage 2, firm 1 would have to pollute before firm 2,
costs: min t1 TC1c≡ C 1 (x1(t1, t2) , 1) + x1(t1, t2) X (t1, t2) × D (X (t1, t2)) + c1t1, (10) min t2 TC2c≡ C2(x2(t1, t2) , t1t2) + x2(t1, t2) X (t1, t2) × D (X (t1, t2)) + c2t2. (11) Solving first-order conditions (10) and (11) simultaneously, the extent to which the firms are willing to share/adopt new technology can be found.
For concrete results, we consider in the following sub-sections first the standard case of linear pollution damage and then the more complicated case of increasing marginal damage.
4.1 ConstantMarginalDamage
In reality, many pollutants, such as SO2(sulfur dioxide, the major industrial discharge causing acid rain) and CO2 (carbon dioxide, the major cause of global warming) demonstrate the property of constant marginal damage to victims, including detrimental health effects as well as material losses. For these pollutants, the damage function is rightfully linear:
D(X) = kX, k > 0,
with k being the constant marginal damage. In what follows in this section, we derive equilibrium conditions for both cases of free and costly diffusion.
4.1.1 Free DiusionofTe hnology
In the absence of diffusion costs, firm total costs (5) (6) become simply: C1(x1, 1) + x1 x1+ x2 · kX = C1(x 1, 1) + kx1, C2(x2, T2) + x2 x1+ x2 · kX = C2(x 2, T2) + kx2.
We know already that, when adoption of technology is free, firm 2 will choose the maximal technology adoption rate t2 = 1. Given this, both
firms, in Stage 2, intend to minimize respective costs: min x1 C1(x1, 1) + kx1, min x2 C2(x2, t1) + kx2,
which yields the necessary MAC=MD conditions: −Cx1(x1, 1) = k, −Cx2(x2, t1) = k. It can be easily verified that:
dx1 dt1 = 0, dx2 dt1 = −C 2 xT C2 xx < 0.
Therefore, the more firm 1 is willing to share its advanced technology (as manifested by a higher t1), the better technology firm 2 will have at its dis-posal (and hence the less pollutants it will generate).
Back in Stage 1, with no diffusion costs, firm 1 is indifferent about its diffusion rate t1. We might as well suppose good will and assume maximal release of technology:
t1= 1.
Thus, in the current case of free diffusion, the market outcome is efficient with complete technology diffusion:
T2∗ = t ∗ 1 = t
∗ 2 = 1.
This efficient result is intuitively straightforward, yet interestingly it is the only case in which the market outcome coincides with the social goal.
4.1.2 Costly DiusionofTe hnology
Now consider the case in which diffusion of technology is costly for both firms, with marginal cost ci. Firm 1, trying to minimize
has no incentive to share its superior technology with firm 2, and will set t1 = 0,
leading to:
T2 = t1· t2= 0.
As a result, with no new technology available, firm 2’s goal min TC2c ≡ C2(x2, 0) + kx2+ c2t2 dictates zero adoption efforts:
t2 = 0.
Contrary to the efficient no-cost case analyzed in the previous subsection, the market outcome with costly diffusion is utterly inefficient, with abso-lutely no technology transfer at all. That is, no firm has an incentive to share, and no firm has an incentive to learn.
Note that, when pollution damage is linear, a firm’s compensation liabil-ity depends only on its own emissions under the proportional liabilliabil-ity rule. Consequently firm 1 cares nothing about firm 2’s emission level, and has no incentive to share its superior technology as it is costly to do so. It is there-fore the government’s responsibility to promote the sharing of technology by providing economic incentives such as tax relief or diffusion subsidies to the leading firms.
4.2 In reasingMarginalDamage
We next turn to the situations in which pollution damage grows at increasing rates. Unlike SO2 or CO2 that cause linear harm, many pollutants, such as heavy metals, often inflict marginally increasing health hazards on pollution victims:
D′(X) > 0, D′′(X) > 0.
For analytical tractability, we consider the standard quadratic damage case: D(X) = kX +sX
2
2 , k, s > 0.
In the following subsections, we again consider and compare two possible cases of diffusion cost: free versus costly diffusion.
4.2.1 FreeTe hnologyDiusion
In the absence of diffusion costs, firm 2 has no reason not to adopt more advanced technologies available to it, and hence must set
˜t2= 1.
Therefore in Stage 2, firm objectives (5) (6) can be simplified as: min x1 C1(x1, 1) + kx1+ sx1[x1+ x2] 2 , (12) min x2 C2(x2, t1) + kx2+ sx2[x1+ x2] 2 . (13)
And the firm-optimal emissions must satisfy the following first-order condi-tions, given t1: Cx1(x1, 1) + k + sx1+ sx2 2 = 0, (14) Cx2(x2, t1) + k + sx2+ sx1 2 = 0, (15)
from which we can derive some interesting comparative statics: dx1 dt1 = s 2C 2 xT C1 xxCxx2 + sCxx1 + Cxx2 + s2− s2 4 > 0, dx2 dt1 = −C 1 xx+ s CxT2 C1 xxCxx2 + sCxx1 + Cxx2 + s2− s2 4 < 0, dX dt1 = −C 1 xx+ s 2 C 2 xT C1 xxCxx2 + sCxx1 + Cxx2 + s2− s2 4 < 0, and dD(X) dt1 = D′(X) ·dX dt1 < 0.
In words, these equations state that, as firm 1 renders more superior tech-nology to firm 2, the latter will be induced to abate more pollution (x2 goes down). In the mean time, however, firm 1 will find itself polluting more (x1 goes up). The aggregate emissions X and the external damage D(X) will nevertheless become lower.
As for firm 1’s diffusion rate choice ˜t1 in Stage 1, we find that dTC1 dt1 = C1 x· dx1 dt1 +d [ηi(t1) · D (X (t1))] dt1 R 0.
With a larger t1, firm 1 can save on pollution control costs (due to the larger x1). Change in its compensation payment, however, is ambiguous: though total damage D(X) becomes smaller, its payment share η1(= x1/X) becomes larger. The result is partial diffusion on the part of firm 1:
˜t1 ∈ (0, 1), ˜t2= 1, T˜2 = ˜t1· ˜t2 ∈ (0, 1),
which obviously deviates from the socially efficient full diffusion of T2∗ = 1. Contrary to what we might expect from the free transfer assumption, firm 1 is still unwilling to fully share its technology even if doing so incurs no extra cost. As explained in the previous paragraph, helping sluggish firms will adversely raise the superior firms’ liability as the former firms abate more emissions.
4.2.2 Costly Diusionof Te hnology
We have shown in Section 3 that the social optimum in the presence of diffusion costs requires only a partial diffusion that satisfies conditions (3). Now with positive diffusion costs, firm objectives (12)(13) become:
min x1 C1(x1, 1) + kx1+ sx1[x1+ x2] 2 , (16) min x2 C2(x2, t1t2) + kx2+ sx2[x1+ x2] 2 . (17)
And the first-order conditions (14) (15) require only minor modification now with t1replaced with t1t2 in (15):
Cx1(x1, 1) + k + sx1+ sx2 2 = 0, (18) Cx2(x2, t1t2) + k + sx2+ sx1 2 = 0, (19)
which define the firm emission choices as functions of the firm diffusion rates (t1, t2) set in Stage 1:
ˆ
It can be noted, by comparing (18) (19) with (14) (15), that: d ˆx1 dt1 = t2 t1 ·d ˆx1 dt2 > 0, d ˆx2 dt1 = t2 t1 ·d ˆx2 dt2 < 0, and d ˆX dt1 = t2 t1 ·d ˆX dt2 < 0, dD ˆX dt1 = t2 t1 · dD ˆX dt2 < 0.
In Stage 1, the firms choose their diffusion rates by carrying out the calcula-tions (10) and (11), respectively:
min t1 TC1c ≡ C1 xˆ1, 1 + ( k ˆx1+ s ˆx1 ˆx1+ ˆx2 2 ) + c1t1, (20) min t2 TC2c ≡ C2 xˆ2, t1t2 + ( k ˆx2+ s ˆx2 ˆx1+ ˆx2 2 ) + c2t2. (21)
The corresponding first-order conditions are: c1x+ k + s ˆx1+ s ˆx2 2 ·d ˆx1 dt1 +s ˆx1 2 · d ˆx2 dt1 = −c1, (22) c2x+ k + s ˆx2 + s ˆx1 2 ·d ˆx2 dt2 +s ˆx2 2 · d ˆx1 dt2 + t1CT2 = −c2. (23)
To see how diffusion costs may affect firm incentives with regard to technol-ogy sharing/adoption, we conduct comparative statics on (22) and (23) and find: d ˆt1 dc1 < 0, d ˆt2 dc1 < 0, ∀c1> 0, d ˆt1 dc2 < 0, d ˆt2 dc2 < 0, ∀c2> 0.
It is intuitively clear, for the interior case, that higher diffusion costs will dampen firm enthusiasm for transferring technology. Therefore, the tech-nology diffusion process can only be partial when it is costly, and will be even less than in the free diffusion case:
ˆt1 < ˜t1 < 1, ˆt2 < ˜t2= 1.
As diffusion costs lessen the firms’ incentives to share technology, it will be up to the EPA to provide proper channels for firms to easily transfer abate-ment technology. One possibility is to have the EPA act as an information and technology intermediary.
5 Con lusions and Poli y Impli ations
We investigate the interaction between firms with different states of technol-ogy under the proportional liability rule. Our findings can be summarized in the table below:
Market Equilibrium
Tech Diffusion Social Optimum Linear Damage Increasing Damage
Free Complete Complete Partial
(t1∗= t2∗= 1) (t1= t2= 1) (˜t1< 1, ˜t2 = 1)
Costly Partial Zero Partial
(t1∗∗, t2∗∗< 1) (t1= t2= 0) (ˆt1, ˆt2< 1) It is found that, when pollution damage is linear, the market outcome is always at a corner. We have either full technology transfer (in the absence of diffusion costs), or no transfer at all (in the presence of diffusion costs). The former coincides with social efficiency, whereas the latter does not.
If instead marginal pollution damage is increasing, market equilibrium will always be an interior solution (diffusion rate ti ∈ (0, 1)), whether dif-fusion is costly or not. The transfer is only partial, and the extent of the transfer falls below the socially efficient level. Firms do have some incen-tive to diffuse/adopt advanced technology, but the incenincen-tive diminishes as its cost (ci) rises.
With the above findings, it is clear that appropriate government poli-cies will depend on the functional form of the damage. With linear damage
and free transfer, market diffusion is complete and efficient, and hence there is no need for EPA intervention. In other cases especially with increasing damage, market outcomes are always inefficient; so it is up to the EPA to provide economic incentives to promote social efficiency. The government should either force firms to diffuse superior technology for pollution abate-ment (i.e., direct control) or provide economic incentives (e.g., a subsidy for diffusion, or a tax for insufficient diffusion).
Though inspired by Endres and Friehe (2011), this current paper estab-lishes two results fundamentally different from theirs. Firstly, our findings are applicable to the proportional liability rule, as opposed to their strict liability and negligence rules, and hence provide a different perspective on the issue of voluntary technology transfer. Secondly, the efficiency of market equilibrium in our model depends crucially on the structure of the pollution damage function as well as the existence/absence of the cost of technology transfer, which is lacking in Endres and Friehe (2011). It is therefore our hope that this paper offers a more rounded, if not full-fledged, model of environmental technology diffusion.
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