Ecological Economics 80 (2012) 25–37

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Ecological Economics

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Methodological and Ideological Options

The value and applicability of bargaining in an intergenerational setting☆

Gregory Howard ⁎Department of Agricultural, Environmental, and Development Economics, Ohio State University, 2120 Fyffe Road, Columbus, OH, 43210, United States

☆ Portions of this paper are not to be duplicatedwithoutproper citation is present.⁎ Tel.: +1 614 292 9424.

E-mail address: howard.761@buckeyemail.osu.edu.

0921-8009/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.ecolecon.2012.05.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 May 2011Received in revised form 17 April 2012Accepted 14 May 2012Available online 8 June 2012

Keywords:Intergenerational externalityCoasian bargainingWelfare analysis

This paper considers the effectiveness of a variation of Coasian bargaining as a policy instrument for internalizingone or multiple intergenerational externalities. The variation involves appointing a contemporary party to rep-resent the interests of the affected parties who are currently unable to represent themselves. Potential criticismsof such a policy are considered and addressed, and precedents to such a policy are put forth. To test the value ofsuch a policy, a two-period model in which two externalities exist in the production/consumption decisions andwith representative agents is used to compare the welfare effects of four scenarios: 1) Agents in each period op-timize their period-specific utility, 2) a benevolent social planner chooses allocations in both periods tomaximizea social welfare function, 3) the externalities are internalized using a bargaining policy, and 4) the externalitiesare internalized using a tax and subsidy policy. I find that, contrary to general consensus in the literature, Coasianbargaining can be adapted in such a way as to make it applicable in an intergenerational framework.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

When dealing with externalities that exist between two or morecontemporary agents, the environmental economic literature has rec-ognized many different methods of internalizing the external cost inorder to achieve efficient outcomes (for a comparison of these differentpolicies under uncertainty, see Weitzman (1974)). These methodsrange from regulating prices through taxes and subsidies (Cornes andSandler, 1985; Pigou, 1952; Wittman, 1985), to regulating quantitiesthrough quotas and cap and trade, to more “decentralized” methodslike Coasian bargaining (Coase, 1960). However, when attempting toresolve issues such as global warming, the use of nuclear power, andbiodiversity loss, the relevant externalities are not limited to the currentpopulation, but instead extend to agents in future generations.

It is not the purpose of this paper to suggest that intragenerationalexternalities (those for which the affected parties are all contemporary)are simple to solve. Indeed, the very reason there is an expansive liter-ature on the subject can be traced to its difficulty. This is partly due tothe fact that “efficient” outcomes are often easy to demonstrate in amodel and always nigh impossible to implement in real policy. In addi-tion, the very concepts of efficiency and optimality in policy must carrywith them strong assumptions regarding our evaluation of utility andour perception of rights. The most basic form of utilitarianism wouldbe to argue that nobody has the inherent right to anything, but thissurely would meet strong objections if it were actually introduced inthe public sphere as such, even though the assumption is often implicit

the author's permission, unless

rights reserved.

in economic modeling, usually for the sake of simplicity. What I willassert herein is that intergenerational externalities are, for variousreasons, even tougher nuts to crack than their intragenerationalcounterparts.

Coasian bargaining is generally considered an infeasible policy op-tionwhen dealingwith intergenerational externalities as it relies on ne-gotiations between parties separated by decades (or even centuries).Padilla (2002) said, in reference to market solutions to inter-generational externalities, “In particular, the ‘Coasian’ analysis is out ofplace: there is no possible agreement between the parts because futuregenerations are not present nor represented either.” (p. 72). This is a le-gitimate objection, and it focuses on the decentralized nature of Coasianbargaining. We face a permanent and inflexible constraint in which weare unable to move freely through time as we do space. This constraintseems to irrevocably damn decentralized bargaining, as well as anyother potential decentralized policies, when the problem is inter-generational in nature. However, decentralization is only half of theCoasian bargaining story. The feasibility of a decentralized bargainingpolicy is dependent on not just the possibility of decentralization(meaning the “policy” is implemented by the effected parties and notthe government), but also on the ability of these parties to bargain.This bargaining aspect is the baby that is thrown outwith the decentral-ization bath water. Because decentralization and bargaining are tied soclosely in the minds of economists, the profession may fail to considerthe possibility of having one without the other.

One might be incredulous toward any bargaining scheme in whichthe two affected parties have no chance to negotiate. The future cannotsit opposite the present at any negotiating table. However, if a contem-porary party to represents the interests of the future, bargaining couldonce again be viable. If this party was the government, we would notbe engaging in Coasian bargaining per se, but instead in centralized

26 G. Howard / Ecological Economics 80 (2012) 25–37

bargaining. Having today's government represent a faceless future thatdoes not yet exist is enough to raise further objections (for example,how can today's government presume to know the preferences of fu-ture peoples?), but it would be erroneous to assume that centralizedbargaining is unique among policy instruments in such weaknesses.For example, when calibrating an optimal tax aimed at internalizingan intergenerational externality, the government must be aware ofthe preferences and endowments of people who have yet to be born.If they do not know these preferences and endowments, or if their esti-mates are wildly inaccurate, the chosen tax rate is likely to be far fromoptimal. Thus, for any policy, bargaining or otherwise, the governmentmust have a good estimate of future preferences to have any chanceof implementing effective policy. It is reasonable to be skeptical aboutour estimates of future preferences, but such skepticism is equallydamning to policies across the board.

In addition to possessing accurate estimates of future preferences andendowments, the government (or any party representing the future)must also be committed to representing the best interests of the future.This assumption is likewise fair game for skepticism and credulity. Itseems unlikely that politicians will forgo the interests of their constitu-ents in order to negotiate forcefully and effectively for parties whohave no say in their reelection. This is the “missing voter” problemaddressed by Doeleman and Sandler (1998). If this is true, centralizedbargainingwill not be effective. However, assuming that the governmentdoesn't actually care about future citizens causes ALL policies aimed atinternalizing intergenerational externalities to be ineffective, not justbargaining. If politicians care only about the interests of their constitu-ents, not only would bargaining policies fail, but any regulation of quan-tity or price will fail in similar fashion. What hope is there for effectiveCap and Trade legislation or carbon taxes that accurately reflect the socialcost of carbon when the government is unconcerned about the interestsof future generations, who are the beneficiaries of such action1? There isnone. In addition, more restrictive policies that require weak or strongsustainability criteria to bemetwill likewise fail, as politicianswill simplyrefuse to enforce them.

It is also reasonable to suggest that transaction or information costscan be prohibitive when it comes to bargaining outcomes. In the case ofmany heterogeneous parties negotiating on either or both sides, it is likelythat the costs of negotiating are high. Similarly, the existence of private in-formation can lead to parties “holding out” in negotiations and failing toachieve efficient outcomes (Farrell, 1987). Clearly, the existence of thesereal world frictions relegates any bargaining policy to theworld of secondbest outcomes. However, the same is undoubtedly true for price andquantity regulations, as the cost of acquiring information on preferences,as well as the cost of monitoring and enforcement, is certainly nonzeroand in many cases substantial. The question, then, is not whetherbargaining policies are in all cases efficient or superior to price and quan-tity alternatives, but ratherwhether bargaining policies are comparable orsuperior to these alternatives under the right circumstances.

The argument put forth in this paper is not a blanket endorsement ofbargaining outcomes or even one of utilitarian economics in general. Iam not asserting that bargaining will achieve 1st best outcomes whenapplied to the complexity of real world issues, just as price adjustmentsand quantity controls will not achieve 1st best outcomes. Furthermore,it would be erroneous to say that any policymeeting some Potential Pa-reto Criterion is advisable, regardless of distributional consequences orrights infringement. A referee noted that the tone and tenor of thispaper is strictly utilitarian, in the tradition of Bentham and Coase,while the rich and insightful area of rights-based ethics is ignored.

1 One could argue that governments could appear to care about future generation,without actually doing so, by levying a tax to correct a contemporaneous externalityif said tax has the unintended consequence of being beneficial to future generations.If, however, a tax was purely detrimental to the present and beneficial to the future,this caveat would not hold. In addition, even if the externality affects both contempo-rary and future parties, a tax that takes into account only the interests of contemporaryparties will not be an optimal tax.

This is no doubt true; rights-based ethics should have some (probablysubstantial) standing when making any economic decision, but espe-cially economic decisions with large and lasting intergenerational con-sequences. In some ways, this may be built into utilitarian models inthe form of constraints, although this area is indeed nuanced anddoing the topic justice is beyond the scope of this paper.

With that being said, arguing in favor of bargaining outcomes is notequivalent to asserting that future generations have no rights. It is entire-ly reasonable to believe that one can bargain over some resources andamenities while refusing to do so for others (drinkable water, for in-stance), just as allowing the government to tax the extraction of some re-source does not imply that this generation should have carte blanche todo what they please with no consideration for future generations. Thepurpose of this paper is not to denigrate rights-based ethics in anyway. However, I see the discussion of rights based vs. utilitarian ethicsas being a parallel argument to the argument addressed here, in whichthe resolution the former does not change the qualitative results of thelatter, but instead speaks to the set of circumstances to which thispaper applies. Should a discussion on rights vs. utilitarianism concludethat utilitarianism has no bearing, this paper would provide little guid-ance in resolving conflict. Should the discussion reject right-based ethicsentirely and fully endorse utilitarianism, the assertions of this paperwould have bearing in many more intergenerational allocation issues.

In fact both of these extremes are likely problematic. Just as there areallocation issues for which rights and ethics should bear heavily on ourdecision-making, there are other choices we make that likely do not in-fringe upon anyone's rights. Clearly it is hard to know which is which,but again that is an argument that is parallel to this one.

This paper will consider the impact of assigning property rightsand allowing bargaining between generations, through the aid of anegotiator or ombudsman that represents the interests of futureparties in current negotiations. The concept of assigning propertyrights to future generations in order to protect their interests is notnew. Pasqual and Souto (2003) argue that the assignment of resourcerights can help alleviate externalities between generations. They alsoconsider various property rights assignments and their respective im-plications for sustainability. The work of this paper deviates fromthese previous efforts in several ways. First, this paper does not con-sider sustainability, though the extension of the model presentedherein to infinite periods would produce interesting implications forsustainability. Second, Pasqual and Souto consider the property rightsassignment of a single resource that generates intergenerational ex-ternalities, while this paper presents a model in whichmultiple exter-nalities exist from multiple sources, each of which can have rightsassigned to it. Third, while previous work allows for the assignmentand enforcement of property or resource rights to the future, theydo not consider the prospect of bargaining between the present andthe contemporary party representing the interests of the future.

Padilla (2002) advocates the assignment of property rights to futuregenerations in the interest of sustainability, arguing that failing to do soimplicitly assigns property rights to the present and is still “valueladen.” (p. 75). This is undoubtedly true. Padilla eschews bargaining so-lutions as well as “Pigouvian” solutions on the grounds that market val-uations assigned to the future are unreliable and somewhat arbitrary. Inaddition, he acknowledges the “compensation rule,” in which the pre-sent generation's use of resources in a manner that violates the rightsof future generations should bear with it compulsory compensationfrom present to future (how to determine appropriate compensation inthe face of unreliable and arbitrary information is left unclear). This issimilar to Bromley's (1989) liability rule. Padilla also advocates the de-velopment of institutions “acting as representatives, defenders, and tu-tors,” of future generations' rights (2002, p. 80).

While there is no clear precedent in favor of representing the inter-ests of future generations as a matter of policy, it is standard practice toappoint parties to represent the interests of children in both custody(Parley, 1993) and neglect/abuse proceedings (Elrod, 1995–1996). In

27G. Howard / Ecological Economics 80 (2012) 25–37

this context, the policy tool suggested in this paper is akin to appointinga class-action guardian for future generations. In addition, it is commonpractice among many Native American tribes, both historically andtoday, to consider the impact of any decision on the next sevengenerations of the tribe (Trosper, 1995). While such a system is notidentical to the one proposed in this paper, as it involves no bargaining,it is similar in its intention of internalizing externalities affecting unborngenerations.

The purpose of this paper is to demonstrate the potential usefulnessof bargaining in the case of multiple intergenerational externalities. Forthe remainder of this paper, the term Coasian bargaining refers tobargaining in which a contemporary agency represents the interestsof the future. This agency could be the government, in which caseCoasian Bargaining is a misnomer. However, this agency can also bedecentralized. For instance, in an attempt to protect biodiversity, orga-nizations like the Sierra Club or the WWF could potentially representthe interests of the future in negotiations. I am not endorsing theseagencies for the job; I'm merely providing them as possible exampleswithout justification or defense. I want to emphasize that this is not tosuggest that the Sierra Club knows what is best for the future, orknows the preferences of future people, or even that they care aboutthe welfare of future generations more than the average person oragency. Nobody knows these things, not even the government, econo-mists, or philosophers; we can have educated guesses at best. Similarly,nobody knows exactly what rights future people should be entitled to,let alone what actions might potentially infringe upon these rights.

The model that follows considers an economy with multiple exter-nalities (Key and Kaplan, 2007). This is done for two reasons. First,this ismore realistic.Whetherwe like it or not, externalities abound, es-pecially when taking an intergenerational outlook. Just as GHG emis-sions, biodiversity loss, and deforestation generate negativeexternalities, R&D research, carbon sequestration, and investing in edu-cation generate positive externalities. In addition, multiple externalitiesare examined because in a model with a single externality, the methodof payment in bargaining situations is purely monetary. For all intentsand purposes (especially if we are considering centralized bargaining),this is not much different from a tax. A model with multiple externali-ties, on the other hand, provides additional methods of compensationbetween the parties.

I develop a two periodmodel (period 1 being the present and period2 being the future) inwhich the choices of the present generate two ex-ternalities, one positive and one negative, for the future. I then solve themodel under four different scenarios: 1) The representative agent ineach period chooses allocations to optimize his own utility, 2) a benev-olent social planner chooses allocations to maximize a social welfarefunction, 3) a Coasian bargaining scenario in which the present and fu-ture negotiate, through a government intermediary, to determine allo-cations, and 4) a tax-and-subsidy regime in which the governmenttaxes the source of the negative externality and subsidizes the sourceof the positive externality. I compare the equilibria and comparativestatics of these four scenarios, analytically where possible and usingsimulations when the problem becomes analytically intractable. I addi-tionally extend the model to allow for some inequality aversion andconsider how themodel changeswith the introduction of a third period.

One obvious conclusion from this paper is that the planner's optimi-zation produces significantly greater socialwelfare thandoes the individ-ual period optimizations. In general, the Coasian bargaining solution doesnot achieve thefirst-best outcome, but is still significantly superior to theindividual optimization scenario. I find that for some parameter valuesthe tax-and-subsidy policy fares better than the Coasian bargainingpolicy, while for other values they are effectively equivalent, suggestingthat bargaining policies could indeed be as viable as other policies inproper circumstances.

The rest of the paper proceeds as follows. Section 2 presents the el-ements of the model. Section 3 solves the model under four differentscenarios: 1) individual periods maximizing utility separately, 2) a

social planner choosing allocations in order tomaximize a socialwelfarefunction, 3) the government assigns and enforces property rights andacts as an intermediary in negotiations to determine allocations ineach period, and 4) agents choose allocations to maximize period-specific utility subject to a tax-and-subsidy regime imposed by the gov-ernment. In addition, Section 3 presents a numerical analysis withwhich to compare the four scenarios. This section also provides sensitiv-ity analyses for the parameters of themodel, adjusts the functional formof the social welfare function, and extends themodel to three periods inorder to test for model robustness. Section 4 concludes and suggestspossible extensions.

2. Model

The model consists of a closed economy where agents produce acomposite consumption good over two periods, t=1, 2. Each periodhas one representative agent who exists for that period only. Whileeach period has its own representative agent, all agents across pe-riods are identical in their preferences, production function, andtotal labor endowment L. For simplicity I assume agents do notvalue leisure. They allocate their labor between resource extraction,Lr, and final good production, Lf. Resource extraction is a linear func-tion of labor and follows:

rt ¼ r Lrtð Þ ¼ Lrt; ð1Þ

where rt is the resources extracted for use in period t, so one unit of laborextracts oneunit of resource. The resource stock available in the followingperiod is defined as Rt+1=Rt−rt. The use of resources in the productionprocess generates pollution, which has a deleterious effect on welfare infuture periods. The use of the resource in final good production generatespollution at a constant one-to-one rate, while pollution already in exis-tence dissipates, giving rise to the pollution growth function

Ptþ1 ¼ δPt þ rt; ð2Þ

where Pt is pollution in period t and 0≤δ≤1. Thismay seem like a strongassumption, but it is based on a reasonable argument. Clearly an agentwill not commit labor to extract resources and then not use the resourcesin production, so resources extracted will always equal resources used inproduction. Assuming the resource is homogeneous, the first unit usedshould not producemore pollution than thenth unit. The one assumptionthat is unrealistic is the assumption that the marginal productivity oflabor in resource extraction is constant. This assumption was made forsimplicity and could be relaxed in future manifestations of this model.Production in each period is Cobb–Douglass and exhibits constant returnsto scale. Production takes the form

Yt ¼ F Lft; rtð Þ ¼ TtLftαrt

1–α ¼ TtLftαLrt

1–αð Þ: ð3Þ

In this formulation 0bαb1 and Tt represents total factor produc-tivity in period t. Output can be used as either current consumptionor invested in research and development (R&D), Yt=Ct+At, whereCt is consumption and At is R&D spending in time period t. Consump-tion represents the benefit of production for the current population,while R&D reflects production that improves total factor productivityfor future periods. In this way, TFP evolves according to the rule

Ttþ1 ¼ γTt þ ρAt; ð4Þ

where γ>1 reflects advances in TFP that occur even in the absence ofR&D and ρ reflects the marginal contribution of research and develop-ment to TFP. While I acknowledge that assuming R&D spending ben-efits only the future is at best tenuous and more likely highlyinaccurate, such an assumption is made for the sake of model simplic-ity and exists without loss of generality. The important (and certainly

28 G. Howard / Ecological Economics 80 (2012) 25–37

not groundbreaking) finding regarding R&D spending is that the so-cial planner will choose a level of R&D spending higher than thelevel chosen by an individual optimizer. If the social planner allocatesX units to R&D, it makes no real difference whether the private opti-mizer allocates zero or X/2, the basic qualitative finding is the same.

Utility in each period is given by the utility function

Ut Ct;Ptð Þ ¼ log Ctð Þ–D Ptð Þ ¼ log Yt–Atð Þ–φPt1þθ; ð5Þ

where D(.) is the pollution damage function and U(.) is monotonicallyincreasing and concave in its first argument. θ>0, which implies thatdamages are convex in Pt. Convexity of the environmental damagefunction is a common assumption of the literature (see, for example,Babu et al., 1997). Often environmental quality enters positively andconcavely into the utility function. This is equivalent to a convex envi-ronmental quality damage function. I also considered themodel assum-ing values of θ between −1 and 0, reflecting strict concavity of theenvironmental damage function, and the results were not significantlydifferent. Additionally and without loss of generality, we assume thatP1=0.We also assume that R1 is sufficiently large so that resource con-straints will not bind in either period.

3. Results

3.1. Individual Optimization

In this section, I examine four possible outcomes: the agent in eachperiod optimizing their individual utility function, a social planner opti-mizing an additive combination of welfare for both periods, a Coasianbargaining solution, and a tax and subsidy solution. In the individual op-timization scenario, the first period faces the following problem:

maxLf1 ; Lr1 ; A1

log T1Lf1αLr1

1 –αð Þ–A1

� �n o;

subject to Lf1 þ Lr1 ≤ L;0 ≤ A1; Lf1; Lr1:

ð6Þ

In this scenario, the constraint on A1 is binding, so no R&D spendingoccurs in equilibrium. The first order conditions for Lf1 and Lr1 are as fol-lows:

Lf1 :αT1Lf1

α–1Lr11–αð Þ

T1Lf1αLr1

1–αð Þ–A1¼ μ ; ð7Þ

Lr1 :1�αð ÞT1Lf1αLr1–αT1Lf1

αLr11–αð Þ–A1

¼ μ ; ð8Þ

where μ is the Lagrange multiplier for the labor constraint. CombiningEqs. (7) and (8), we derive the following equality:

1−αð ÞT1Lf1αLr1–α ¼ αT1Lf1α–1Lr1

1–αð Þ: ð9Þ

Explicit values for Lf1 and Lr1 are easily found and shown inAppendix A, but the important result is that period 1 allocateslabor so as to equate the marginal products of the two labor choices,MPLf1=MPLr1. Period 2 is faced with a similar, but not identical, op-timization problem:

maxLf2; Lr2;A2

log γT1Lf2αLr2

1–αð Þ–A2

� �– φLr1

1þθð Þn o;

subject to Lf2 þ Lr2 ≤ L;0 ≤ A2; Lf1; Lr1:

ð10Þ

Once again, I find that the R&D constraint binds, leaving A2 equalto zero. In addition, period 2 will also allocate labor in a manner

that equates the marginal product of Lf2 with the marginal productof Lr2.

3.2. Planner Problem

I turn next to the central planner's problem. The goal of the centralplanner is defined here as maximizing the weighted sum of period-specific utilities. This can be formally written as

maxLf1; Lr1;A1;Lf2; Lr2;A2

(log T1Lf1

αLr11–αð Þ

–A1

� �þ β log γT1 þ ρA1ð ÞLf2αLr2 1–αð Þ

–A2

� �–φLr1

1þθð Þh i)

Subject to Lf1 þ Lr1 ≤ L;

Lf2 þ Lr2 ≤ L;

0 ≤ A1;A2; Lf1; Lr1; Lf2; Lr2: ð11Þ

In this formulation, β is the weight assigned to period 2 and theweight assigned to period 1 is normalized to 1. I use the term“weighted sum,” as opposed to “discounted sum” because I put no re-strictions on the value of β. The typical restriction in the literature isβb1. This is sometimes due to mathematical necessity and sometimesstems from the observation that individuals tend to discount futureconsumption. In either case, I find it an unnecessary assumption forthe purposes of this model.

As is demonstrated in Appendix A, the planner's allocation of laborand R&D in period 2 is identical to the allocations achieved using the in-dividual optimization method. There is, however, an important differ-ence in period 1 allocations between the two solution methods,assuming β≠0 (If β=0, the planner's problem becomes equivalentto the problem faced by period 1 in the individual optimization prob-lem). This divergence can be seen by considering the first order condi-tion with respect to Lr1,

1� αð ÞT1Lf1αLr1–αT1Lf1

αLr11–αð Þ–A1

¼ μ1þ βφ 1þ θð ÞLr1θ: ð12Þ

In this formulation, μ1 is the Lagrangemultiplier associated with theperiod 1 labor constraint. Combining Eq. (12) with the first order condi-tion with respect to Lf1 (which is identical to the result in the individualoptimization example, Eq. (7)) yields a new maximization condition:

1� αð ÞT1Lf1αLr1–αT1Lf1

αLr11–αð Þ–A1

¼ αT1Lf1α–1Lr1 1–αð Þ

T1Lf1αLr1

1–αð Þ–A1þ βφ 1þ θð ÞLr1θ: ð13Þ

A rather intuitive and well documented result can be seen from thisequation. Individual welfare maximization yields different allocationsfor labor thandoes the social plannermaximization, so long as the socialplanner uses a nonzero weight β. In this way, intergenerational exter-nalities are no different than externalities imposed upon contemporaryagents. In addition to this finding, we can derive an optimal level of A1

from the first order condition as shown below:

A1 ¼ ρβTLf1αLr1

1–αð Þ–γTρ 1þβð Þ : ð14Þ

Analysis of this finding enables us to state the following proposition:

Proposition 1. The equilibrium value of A1 in the social planner's prob-lem will be strictly positive (and therefore different from the individualoptimization scenario) provided γbρβ Lf1αLr1(1–α). In addition, the equilib-rium value of A1 is negatively correlated with the parameter γ and pos-itively correlated with β and ρ.

The derivations of the comparative statics that support Proposition 1are in Appendix A. These findings are supported by economic intuition,

2 In a world where the resource constraint is binding, Period 1 must set the totalbenefits of R&D equal to the total damages in period 2 resulting from period 1 usingthe resource plus the disutility from lost production in period 2 due to the transfer ofresources to period 1.

29G. Howard / Ecological Economics 80 (2012) 25–37

as one would expect that increasing the relative weight assigned to pe-riod 2,β, placesmore emphasis on futurewelfare and results in a higherequilibrium A1. Similarly, increasing ρ (the marginal increase in TFPfrom R&D spending) increases the marginal future gains from researchand development, and as such increases in ρ result in greater equilibri-um A1. Lastly, increasing γ (themultiplicative increase in TFP in the ab-sence of R&D) while keeping ρ constant decreases the productivitygains from R&D relative to the baseline productivity gains achievedwithout R&D spending. In other words, increasing γ makes the futurebetter off, and as such the need for R&D spending is lessened and equi-librium A1 falls. In addition to the externality produced by the resource,the planner's scenario corrects the externality related to research anddevelopment. When optimizing individually, the first period under-invests in R&D, just as they overexploit the resource.

As mentioned earlier, setting β=0 collapses the planner's probleminto the period 1 individual optimization problem. Setting β=0 inEq. (14) returns a negative number. Combining this finding with thenonnegativity constraint presented in Eq. (11) yields an equilibriumlevel of R&D spending of zero, the same result found in the individualoptimization problem.

Eqs. (13), (14), and the labor constraint give us three equationswhich we can use to find equilibrium solutions for the three variablesin question: A1, Lf1, and Lr1. The equilibrium values cannot be obtainedanalytically, but instead must be derived using numerical methods.This will be done for a variety of parameter values in Section 3.5.

3.3. Coasian Bargaining

I now assume the existence of a societal agency that can assign andenforce property rights between different generations. Such an agencyis not far-fetched, as governments routinely protect natural resources,such as forestry andwildlife, in order to ensure their existence for futuregenerations. This, in essence, this is equivalent to the assignment ofproperty rights to agents in the future. We will also assume that thisagency performs its duties at no cost.

This agency could assign the resource rights to either period 1 or pe-riod 2. The pattern of property rights assignment has important welfareand equity implications for the model (for a more detailed elaboration,see Howarth and Norgaard, 1990). Throughout the paper, all of therights to the resource will be assigned to period 2. In the context ofour current model, it makes more sense to assign the resource rightsto period 2 for several reasons. First, the model is composed in such amanner as to leave period 2 no means to pay period 1 in exchange forthe right to use resources (There are, of course, viable methods of com-pensation from the future to the present, foremost among them beingthe ability of the present to acquire deficits that must be paid off bythe future). In addition, there is reason to believe that compensationshould be paid by the party that is imposing the externality, in thiscase period 1, rather than by the “victim.” This is known as the polluterpays principle (Conversely, itmay also be argued that productivity gainswill make the future more affluent than their present counterparts, andas such the future is in a better position to bear the weight of compen-sation). Lastly, a positive externality exists in the model in the extent towhich period 1 engages in R&D. By assuming that period 1 is under noobligation to generate research and development, I implicitly assignproperty rights for R&D investment decisions to period 1 (i.e. the rightto not engage in R&D). By assigning the resource rights to period 2, Iam assigning one set of rights to each period.

Once property rights have been accordingly assigned, I model nego-tiations between periods in the form of an ultimatum game similar toSchweizer (1988) in which the period 1 agent makes an offer (A1, r1)and the agency, representing the interests of the period 2 agent, caneither accept or reject. In the offer, compensation will be paid to period2 in the form of increased levels of period 1 R&D. This will be accompa-nied by period 2 granting the right of period 1 to extract and use a spec-ified amount of the resource, r1. Because period 2 controls the resource

property rights and the resource stock is sufficiently large, the optimalallocation for period 2 will be identical to the allocations found in theprevious section of the paper. Period 2 will accept any offer so long asthe resultant utility from accepting the offer is greater than or equal tothe utility achieved without the offer. Formally, the offer must satisfythe following condition:

log γT1 þ ρA1ð ÞLf2αLr2 1–αð Þ� �– φr1

1þθð Þ ≥ log γT1Lf2αLr2

1–αð Þ� �: ð15Þ

Rearranging this condition, period 2 will accept any offer (r1, A1)that satisfies

r1 ≤ φ�1 logγT1þ ρA1ð Þ

γT1

� �� �1= 1þ θð Þ: ð16Þ

This equation can be rewritten to solve for A1 and combined withEq. (1) to yield

A1 ≥ γT1 exp φLr11þ θð Þ� �

–γT1h i

=ρ: ð17Þ

Because period 1 gets no direct benefit from A1, this will hold withequality.

The comparative statics (see Appendix A for derivations) forEq. (17) allow me to make the following proposition:

Proposition 2. The equilibriumvalue A1 in the Coasian bargaining scenariois increasing in Lr1 and γ (the multiplicative increase in TFP in the absence ofR&D) and is decreasing in ρ (the marginal increase in TFP from R&Dspending).

This is an interesting finding, as the comparative statics for γ and ρin the social planner's problem were decreasing and increasing, re-spectively. This difference can be explained by considering the differ-ent goals of the decision makers in each problem. In the planner'sproblem, the planner chooses a level of R&D in which the marginalbenefits of research and development equal the marginal costs. In-creasing ρ in this case increases the marginal benefits without affect-ing the marginal costs to the present, and as such A1 increases with ρ.Conversely, increasing γ decreases the marginal effectiveness of R&Dspending due to the concavity of the utility function, and as such A1

decreases with increases in γ. In the Coasian bargaining solution theequilibrium equation is set by the agent in period 1. Her goal is toequalize the total benefits of R&D with the total damages incurredin period 2 from resource use in period 1.2 In this case, increasing ρincreases the total benefits of A1, so for a given level Lr1 the amountof R&D spending needed to equalize total benefits and damages islower. Increasing γ, on the other hand, effectively decreases the util-ity gained from a given level of R&D spending. As a result, period 1must offer a greater level of A1 to equalize utility gains with utilitylosses for period 2. Ergo, increasing γ increases equilibrium A1.

Using Eq. (17), which I will call the acceptance constraint, I cannow articulate the maximization problem faced by period 1:

maxLf1; Lr1

log T1Lf1αLr1

1–αð Þ– γT1 exp φLr1

1þθð Þ� �–γT1

� �=ρ

h in oSubject to Lf1 þ Lr1 ≤ L;

0 ≤ Lf1; Lr1:

ð18Þ

Table 1Parameter ranges for numerical solutions.

Parameter Description Baselinevalue

Range

α Output elasticity of Lfi, i=1,2; Exponentialparameter for Lf in the production function

0.65 0.2–0.8

β Weight assigned to 2nd period utility inSWF

0.95 0.1–1.1

γ Multiplicative increase in TFP in the absenceof R&D

1.4 1–4

φ Multiplicative parameter in the pollutiondamagefunction

0.2 0–1

ρ Marginal increase in TFP from R&D spending 1.3 0.3–2.3θ Exponential parameter (1+θ) in the pollution

damage function0.2 0–2

L Total endowment of labor in each period 5 NoneT1 TFP in period 1 1 None

30 G. Howard / Ecological Economics 80 (2012) 25–37

The first order conditions are as follows:

Lf1 :ραT1Lf1

α–1Lr11–αð Þ

ρT1Lf1αLr1

1–αð Þ– γT1 exp φLr11þ θð Þ�

–γT1� ¼ μ ; ð19Þ

Lr1 :ρ 1�αð ÞT1Lf1αLr1–α–γTφ 1þ θð ÞLr1θ exp φLr1

1þ θð Þ� �ρT1Lf1

αLr11–αð Þ– γT1 exp φLr1

1þ θð Þ� –γT1

� ¼ μ : ð20Þ

Solving for μ leaves two variables (Lr1 and Lf1) and two equations (acombination of Eqs. (19) and (20) as well as the labor constraint). Theequilibrium solutions for these variables cannot be solved analytically,but they are obtained through numerical methods in Section 3.5.

3.4. Tax and Subsidy Regime

I will now consider the equilibrium allocation of labor and R&Dwhen a different policy instrument is being used, specifically a taxand subsidy regime. In this policy, the government will levy a per-unit tax τ on resource use in the present. The revenue from this taxwill then be invested in research and development. The agent ineach time period optimizes individually, but takes the government'stax/subsidy as given. We assume that the government can see the op-timal response to the tax from both periods and can use this knowl-edge to choose a tax level that maximizes the government's socialwelfare function (SWF), which we take to be equivalent to the socialplanner's SWF outlined in Section 3.2. In this case, period 1 faces thefollowing optimization problem:

maxLf1 ;Lr1 ;A1Log TLf1

αLr11–α–τLr1–A1

� �Subjectto Lf1 þ Lr1 ≤ L;

0 ≤ A1; Lf1; Lr1:

ð21Þ

As in the individual optimization scenario, the chosen level of A1

will be zero. The other two first order conditions can be combinedto yield the following equality:

αTLf1α–1Lr1

1–α¼ 1–αð ÞTLf1αLr1�α–τ: ð22Þ

This equation shows that the optimal distribution of labor in period1 depends on the tax rate τ. The allocations of Lf2, Lr2, and A2 are notinfluenced by τ. A demonstration of this is included in Appendix A.

The government is then faced with the following optimizationproblem:

maxτ

Log TLf1 τð ÞαLr1 τð Þ1–α–τLr1 τð Þ

h iþ

β Log γTþ ρτLr1 τð Þð ÞLf2 τð ÞαLr2 τð Þ1–αh i

–φLr1 τð Þ1þθ� �

;

Subject to 0 ≤ τ:ð23Þ

This problem can equivalently be written:

maxLf1;Lr1;τ Log TLf1αLr1

1–α–τLr1h i

þβ Log γTþ ρτLr1ð ÞLf2αLr21–α

h i–φLr1

1þ θ� �

;

Subject to Lf1 þ Lr1≤L;αTLf1

α–1Lr11–α¼ 1–αð ÞTLf1αLr1�α–τ:

ð24Þ

This optimization problem cannot be solved analytically. Howev-er, using the second constraint in Eq. (24) I can eliminate τ in the

maximization problem. After doing so, the first order conditions areas follows:

Lf1 :αTLf1

α–1Lr11–α–α 1–αð ÞTLf1α–1Lr11–αþ α α–1ð ÞTLf1α–2Lr1

2–α

αTLf1αLr1

1–αþ αTLf1α–1Lr1

2–α þ

β ρ α 1–αð ÞTLf1α–1Lr11–α

–α α–1ð ÞTLf1α–2Lr12–α

� �h iγTþ ρ 1–αð ÞTLf1αLr11–α–αTLf1α–1Lr1

2–α� � ¼ μ ;

ð25Þ

Lr1 :1–αð ÞTLf1αLr1�α– 1–αð Þ2TLf1αLr1�αþ α 2–αð ÞTLf1α–1Lr11–α

αTLf1αLr1

1–αþ αTLf1α–1Lr1

2–α þ

βρ 1–αð Þ2TLf1αLr1�α–α 2–αð ÞTLf1α–1Lr1

1–αh iγTþ ρ 1–αð ÞTLf1αLr11–α–αTLf1α–1Lr1

2–α� � −φ 1þ θð ÞLr1θ

24

35 ¼ μ ;

ð26Þ

where μ is the Lagrange multiplier for the labor constraint. Solving forμ leaves two variables and two equations (a combination of Eqs. (25)and (26) as well as the labor constraint). The equilibrium allocationscannot be solved analytically, but may be solved using numericalmethods.

3.5. Numerical Methods and Robustness

In order to compare thewelfare effects of the four scenarios outlinedin this section, a numerical analysis is required. Welfare is evaluatedusing the weighted sum social welfare function (SWF) optimized inthe social planner's problem. The baseline values used in the numericalanalysis, as well as parameter ranges used in sensitivity analysis and abrief description of the parameter, are presented in Table 1.

Figs. 1 through 6 are sensitivity analyses for six different parametersin themodel. They dealwithα, β, γ,φ, ρ, and θ, respectively. Each graphillustrates welfare achieved in the four scenarios outlined in this paper:The individual optimization solution (IO), the planner's solution (P), theCoasian bargaining solution (C), and the tax-and-subsidy solution (TS).Three conclusions can be drawn from the sensitivity analyses.

Proposition 3. Both the TS regime and the C regime are significantimprovements over the IO solution. The welfare achieved in IO rivalsthat of TS and C only in two instances: very low values of ρ and very lowvalues of β.

This outcome makes intuitive sense. The individual optimizationscenario leads to no R&D spending and heavily favors utility in thepresent over utility in the future. Decreasing the value of ρ (the mar-ginal increase in TFP from R&D spending) decreases the marginalbenefit of R&D spending, while decreasing β (the weight assigned

Fig. 1. Alpha sensitivity analysis. Fig. 3. Gamma sensitivity analysis.

31G. Howard / Ecological Economics 80 (2012) 25–37

to 2nd period utility in SWF) puts less importance on the future in thesocial welfare function.

Proposition 4. Each figure, with the exception of Fig. 6 (θ SensitivityAnalysis), has a parameter value at which both C and TS achieve thefirst-best welfare level of the planner problem. The parameter values atwhich this convergence occurs for C and TS are similar.

The exception to this finding is the sensitivity analyses for θ (the ex-ponential parameter in the pollution damage function). As illustrated inFig. 6, adjusting θ has an identical impact on welfare in both C and P.Therefore, if the parameter values are such that there is a disparity inwelfare between C and P, adjusting θ cannot close the gap.

Proposition 5. As each parameter moves away from the range whereboth policy instruments achieve their first-best outcomes, TS tends tooutperform C. The one exception is Fig. 1 (α Sensitivity Analysis).

The difference is modest for all parameters except ρ (the marginalincrease in TFP from R&D spending), where the divergence in welfarebetween a tax-and subsidy regime and the Coasian bargaining regimeis significant. My discussion of the comparative statics for ρ inSection 4 may explain this result, as changing ρ has different marginaleffects in C than it does in P and TS. This difference is also highlightedin Fig. 7, which shows that the Coasian bargaining solution has

Fig. 2. Beta sensitivity analysis.

equilibrium R&D decreasing in ρ while the Planner and Tax-and-subsidy solutions have equilibrium R&D increasing in ρ. In addition,Fig. 1 shows that C outperforms TS for low values of α (the exponen-tial parameter for Lf in the production function), while the converse istrue for high values of α.

It is reasonable to question how robust these results are, giventhat they assume a specific SWF when evaluating welfare outcomesin each scenario. The specific social welfare function used throughoutthis paper is as follows:

W ¼ U C1;P1ð Þ þ βU C2;P2ð Þ; ð28Þ

where U(Ci, Pi)=log(Ci)−φPi1+ θ, i=1,2. To test the robustness ofmy results, I reevaluate the four scenarios in the model and performsensitivity analysis using a social welfare function that displays in-equality aversion. This inequality averse (IA) SWF is as follows:

W ¼ U C1;P1ð Þ þ βU C2;P2ð Þ–κ U C1;P1ð Þ–U C2;P2ð Þ½ �2: ð29Þ

Sensitivity analyses for both social welfare functions are shownside-by-side in Appendix B. The propositions put forth earlier inthis section all still hold using a social welfare function that dis-plays inequality aversion with a few small caveats. First, C doessignificantly worse for high values of γ (the multiplicative increase

Fig. 4. Phi sensitivity analysis.

Fig. 5. Rho sensitivity analysis. Fig. 7. Sensitivity analysis between rho and equilibrium R&D.

32 G. Howard / Ecological Economics 80 (2012) 25–37

in TFP in the absence of R&D) using the IA SWF, even achievingless welfare than IO. Second, unlike with the normal SWF, C doesnot outperform TS for low values of α using the IA SWF. Lastly,C does outperform TS (though not by much) in the sensitivityanalysis of θ (the exponential parameter in the pollution damagefunction) using the IA SWF. With these small exceptions, the find-ings of this paper are borne out even under changes in the func-tional form of the social welfare function, which suggests thatthe results and conclusions herein are robust.

It is also reasonable to questionwhether the results presented aboveare due to the 2 period nature of themodel. In order to test whether theabove propositions hold when the number of periods is increased, themodel was extended by adding a third period. In doing so, relativelyfew changes were necessary for the individual optimization and plan-ner scenarios. The SWF used in the planner and tax scenarios becomesW=U(C1)+βU(C2)+β2U(C3). For the Coasian bargaining scenario,joint ownership of the resource in period 1 is given to the period 2and 3 agents. This means that an offer must be made by the period 1agent which is accepted by the agents of both subsequent periods inorder for the period 1 agent to have access to the resource. However,this additional complication does not affect the resultant offer. This ismade evident by the following proposition:

Proposition 6. Any offer accepted by the period 2 agent will be acceptedby the period 3 agent so long as δ≤1 (Proof in Appendix B).

Fig. 6. Theta sensitivity analysis.

In period 2, ownership of the resource is given to the period 3agent, so the period 2 and 3 agents must likewise bargain over alloca-tion of the resource and R&D spending.

In the tax and subsidy scenario, two distinct taxes and subsidies arelevied. The government chooses the optimal tax rates, and by extensionthe optimal subsidies, by maximizing the planner SWF subject to re-source and feasability constraints as well as the constraints that outlinehow agents in periods 1 and 2will react to taxes. For simplicity I assumethat all of the period 1 tax is immediately spent on the period 1 subsidy.The assumption is trivial in the present model, as period 1 R&D spend-ing is clearly more valuable to the social planner/government than pe-riod 2 R&D spending. This is a result of period 1 R&D being beneficialto production on both periods 2 and 3, while period 2 R&D is beneficialto production in period 3 alone. With the introduction of additional av-enues of investmentwith somepositive rate of return, however, this as-sumptionmay bemore restrictive. While it is clear that one unit of A1 ispreferred to one unit of A2, it is possible that (1+r) units of A2 is pre-ferred to one unit of A1, where r is the rate of return on the alternativeinvestment. For the purposes of the present model, however, this isnot an issue.

As can be seen in Appendix B, including a third period to the modeldoes not significantly impact any of the findings presented above, andin fact the sensitivity analyses performed on the main parameters arestrikingly similar between the 2- and 3-period models. This suggeststhat the results are not an artifact of the small number of periods, butin fact are robust to variations of model length.

4. Conclusions

Intergenerational externalities add a layer of uncertainty that is gen-erally absent in their intragenerational counterparts. In order to proper-ly internalize such an externality, one must make assumptions aboutthe endowments and preferences of a party that has not yet beenborn. This is true no matter what policy instrument one uses. It has,however, been argued that Coasian Bargaining is particularly ill-equipped to handle intergenerational externalities, as negotiations be-tween the present and future are seen as impossible. However, the ap-pointment of a contemporary party to represent the interests of thefuture in present-day negotiations allows bargaining policies to circum-vent such difficulties. There are additional objections thatmay be raisedregarding the ability and willingness of any contemporary party toknow and represent the interests of the future, but I argue that thesesame problems also exist for any alternate policy option. Moreover, all-owing a governing body to represent the future in property rights nego-tiations is appealing in the sense that it will drastically diminish

3 Proposition 6 shows that, in doing so, the later future generations will agree to anyoffer deemed acceptable by more proximate future generations so long as δ≤1, whicheffectively means so long as the natural environment does not act as a net source ofpollution. Counterexamples to this can be made (CO2, for instance, in the event thatnatural sinks become saturated and positive feedback loops develop), but in generalthe natural environment tends to act as a net sink for man-made pollutants. As such,extending the current model to many generations will most likely not affect the resultsof the bargaining scenario

33G. Howard / Ecological Economics 80 (2012) 25–37

transaction costs, one of the main inhibitors of bargaining theory. De-clining negotiation costs is a common outcome whenever a governingbody is able to adequately represent the interests of a large, diverse,and widespread group.

One could reasonably ask, “What qualities would be required of arepresentative in order for them to do an admirable job of representingthe future in negotiations?” These qualitieswere already highlighted, tosome extent, in the introduction. He or she would need a good estimateabout the preferences of future generations. He or she would need toknow the endowments of future generations and how actions todaytranslate into impacts on his clientele. He or she would clearly need tohave an active desire to ably represent these interests, even at the riskof reducing the welfare of his or her own generation. Is there anyagentwith such a resume? In all likelihood there is not. I amnot arguingthat Person A or Organization B or Government Agency C is necessarilyup to the task.My point is that the resumeneeded to qualify some partyto ably represent the future in bargaining is the same resume needed toqualify our government to formulate appropriate quantity or price con-trols or to set up appropriate rights-based rules. Bargaining outcomesare no different from any other form of intervention, and yet they aretreated as such. To do this is to miss a potential opportunity.

In this paper I consider a two periodmodel inwhich two externalitiesexist. One externality concerns pollution created in resource extractionand production, while the other stems from underinvestment in re-search and development. I show that each time period maximizingtheir personal welfare leads to allocations that are different fromthe first-best allocations derived using the planner's problem.Next, I consider a Coasian bargaining situation in which the pre-sent has the right to invest nothing in R&D while the future con-trols the property rights of the resource needed for production. Iderive the condition under which the equilibrium bargaining out-come achieves allocations equivalent to the planner's problem, aswell as how deviations from this condition affect both exter-nalities. I then compare the effectiveness of Coasian bargainingpolicy to a tax-and-subsidy policy. I find that for some parametervalues the tax-and-subsidy policy fares better than the Coasianbargaining policy, while for other values they are effectively equiv-alent. Furthermore, for low values of α (corresponding to labor-intensive goods), the bargaining policy outperforms the tax-and-subsidy regime.

While it is true that the tax-and-subsidy regime marginally outper-forms the Coasian bargaining regime for certain values in my model, Ido not find this to be a discouraging result. The theme and importantmessage of this paper is not that Coasian bargaining is at all times a su-perior policy instrumentwhen dealingwith intergenerational external-ities. Rather, this paper demonstrates that the traditional notion ofCoasian bargaining,whichmakes no intuitive sense in intergenerationalissues, may be adapted and adjusted into a form that addresses the ex-ternality problem and produces at least as second-best type outcome.Clearly there will be issues for which the cost of enforcing propertyrights will be immense, or transaction cost will be prohibitive. Forthese issues, Coasian bargaining will not be feasible. However, if issuesexist for which these problems are not as pronounced and a tax orquantity regulation has proven exceedingly difficult, either economical-ly or politically, Coasian bargaining may be an effective and welcomealternative.

As an example, suppose a large quantity of fossil fuel was discov-ered on a tract of land owned by the government. Several companiesare vying for the right to extract and sell the resource. The govern-ment is mindful of the intergenerational externalities that exist in fos-sil fuel consumption, and as such wish to increase the price in anattempt to optimize the extraction path of the resource. One methodthey could use to this end is levy a tax on the extracted resource.However, they could also allow companies to submit bids and acceptthe bid that is deemed the best for the future, provided accepting thebid is considered to improve future welfare over letting the resource

remain in the ground. These bids could contain multiple compensa-tion methods, from money payments to R&D spending to emissionsdecreases in other areas of the company, so long as these compensa-tions have a positive impact on future welfare. In this way, the adjust-ed Coasian Bargaining tool has more potential flexibility than a tax inthe right situation.

This is a simple model that can be enriched in many differentways. Possible future extensions of the model include allowing theresource constraint to bind, increasing the number of periods inthe model to some arbitrary number N or to infinity,3 using moregeneral forms or comparing additional different forms of both theindividual period utility functions and the social welfare function,and adding altruism to the period utility function. In addition to util-ity functions, several other functional forms were specified, fromproduction and extraction functions to environmental damage func-tions. The model could be extended to treat these functional formsmore generally. The model could also be changed to include someform of cash or output transfers, with transfers from the future tothe present being in the form of debt accumulated by the presentand paid by the future. The model could introduce additional fric-tions in the form of transaction and property rights enforcementcosts for Coasian bargaining, as well as implementation and moni-toring costs for other policy instruments. The negotiations thattake place in the model could be expanded in complexity beyondthat of a very basic ultimatum game. Lastly, the basic concept under-lying this model could be applied to an overlapping generations(OLG) model form, either over a discrete or infinite time period. Iwould venture to guess that the main difference between themodel as presented here and as an overlapping generations modelwill be that individual optimization will more closely resembleplanner allocations, with the size of the disparity depending on therelative size of the damage function, social planner discount rate(β), and the parameter capturing the marginal increase in TFPfrom R&D spending (ρ). Each of these extensions would introducean additional layer of complexity that could prove informative,both in our theoretical understanding of intergenerational external-ities and in our comprehension of the policy options we possesswhen addressing these externalities. Beyond adjustments to themodel, this line of questioning could be advanced as well. Themost pressing issue as I see it is identifying what situations wouldbe conducive to bargaining outcomes. Setting up a rubric thatcould identify first which intergenerational allocation issues do notpresent much threat of rights violation for those involved and sec-ond whether these issues are best suited for bargaining solutionswould be a useful extension of this work.

Acknowledgments

I thank Tim Haab and Brian Roe for helpful guidance and sugges-tions. I also thank Benjamin Anderson, Carolina Castilla, Afif Naeem,Carlianne Patrick, and two referees for helpful comments, as well asseminar participants at Ohio State University and the 2011 MidwestEconomic Association Annual Conference. Financial support fromthe Environmental Policy Initiative is gratefully acknowledged. TheEnvironmental Policy Initiative had no active involvement in deter-mining the content of this manuscript.

34 G. Howard / Ecological Economics 80 (2012) 25–37

Appendix A. Equilibrium Solutions and Comparative Statics

A1). Equilibrium Labor Allocations When Each Period OptimizesIndividually

Using Eq. (9), we can solve for Lf1 in terms of Lr1:

Lf1 ¼ α1−αð Þ Lr1: ðA1Þ

Combining this with the resource constraint yields the equilibri-um allocations

Lf1 ¼ αL; ðA2Þ

Lr1 ¼ 1–αð ÞL: ðA3Þ

Similarly, we can solve for Lf2 in terms of Lr2:

Lf2 ¼ α1−αð Þ Lr2: ðA4Þ

Combining this with the resource constraint yields the equilibriumallocations

Lf2 ¼ αL; ðA5Þ

Lr2 ¼ 1–αð ÞL: ðA6Þ

A2). Equilibrium Allocations under the Social Planner

The first order conditions, in addition to Eq. (12), are

Lf1 :αT1Lf1

α–1Lr11–αð Þ

T1Lf1αLr1

1–αð Þ–A1¼ μ1; ðA7Þ

A1 :1

T1Lf1αLr1

1–αð Þ–A1¼ βρLf2

αLr21–αð Þ

γT1þ ρA1ð ÞLf2αLr2 1–αð Þ–A2; ðA8Þ

Lf2 :βα γT1þ ρA1ð ÞLf2α–1Lr2 1–αð Þ

γT1þ A1ð ÞLf2αLr2 1–αð Þ–A2

¼ μ2; ðA9Þ

Lr2 :β 1−αð Þ γT1 þ ρ A1ð ÞLf2αLr2–α

γT1þ ρA1ð ÞLf2αLr2 1–αð Þ–A2¼ μ2; ðA10Þ

where, as noted in the body of the paper, μ1 is the Lagrangianmultiplierassociated with the period 1 labor constraint. In addition, μ2 is theLagrangianmultiplier in the period 2 labor constraint. As in the individ-ual optimization problem, A2=0 in equilibrium. Combining Eqs. (A9)and (A10) yields the same equilibrium labor allocations of period 2 asin the individual optimization problem.

A3). Comparative Statics for the Social Planner Equilibrium Level of A1

To determine the comparative statics, we take the partial derivativeof Eq. (14) with respect to the parameters of interest (γ, ρ and β):

∂A1=∂γ ¼ − Tρ1þ βð Þb0; ðA11Þ

∂A1=∂ρ ¼ γT 1þ βð Þρ 1þ βð Þ½ �2 > 0; ðA12Þ

∂A1=∂β ¼ ρ2T1Lf1αLr1

1–α 1þβð Þ–ρ2βT1Lf1αLr11–αþγTρρ 1þ βð Þ½ �2 : ðA13Þ

This can be simplified to,

∂A1=∂β ¼ ρ2T1Lf1αLr1

1–αþ γTρρ 1þ βð Þ½ �2 > 0: ðA14Þ

A4). Comparative Statics for the Coasian Bargaining Equilibrium Level of A1

To determine the comparative statics, we take the partial deriv-ative of Eq. (17) with respect to the parameters of interest (γ, ρand Lr1):

∂A1=∂γ ¼ Tρ

exp φLr11þθ

� �–1

h i≥0; ðA15Þ

∂A1=∂ρ ¼ −γTρ2

exp φLr11þθ

� �–1

h i≤0; ðA16Þ

∂A1=∂Lr1 ¼ ρ−1γTφ 1þ θð ÞLr1θ exp φLr11þ θ

� �> 0: ðA17Þ

In addition, Eqs. (A15) and (A16) are strictly greater than zero ifLr1 is strictly positive.

A5). Evaluating the Period 2 Allocations using the Tax and SubsidyRegime

When the government initiates a tax and subsidy regime as de-scribed in Section 3.4, period 2 faces the following optimizationproblem:

maxLf2;Lr2;A2 Log γTþ ρτLr1ð ÞLf2αLr21–α–A2

h i–φLr1

1þ θ;

Subject to Lf2 þ Lr2 ≤ L;0 ≤ Lf2; Lr2;A2:

ðA18Þ

It is clear that A2=0. Using the first order conditions of theLagrangian with respect to Lf2 and Lr2 and combining them yieldsthe equation:

Lf2 ¼ α1�αð Þ Lr2: ðA19Þ

This demonstrates that period 2's optimization decisions are inde-pendent of the level of τ.

Appendix B. Robustness Analyses and Proofs

B1). SWF comparison

The following figures present welfare outcomes using the SWFthat exhibits inequality aversion outlined in Section 3.5 (IA SWF)(Figs. B1–B6):

B2). Proof of Proposition 6

The period 2 agent will accept any offer in which U*(C*2,P*2)≥U(C2,P2), where U*, C*2, and P*2 are the period 2 utility, consumption,and pollution achieved if the offer is accepted and U, C2, and P2 arethe period 2 values achieved if the offer is rejected. This is capturedby the following inequality:

Log γTþ ρA1ð ÞLf2αLr21–α� �

–φLr11þθ ≥ Log γTLf2

αLr21–α

� �: ðB1Þ

Fig. B4. Phi sensitivity analysis IA SWF.

Fig. B5. Rho sensitivity analysis IA SWF.Fig. B2. Beta sensitivity analysis IA SWF.

Fig. B3. Gamma sensitivity analysis IA SWF.

Fig. B1. Alpha sensitivity analysis IA SWF.

Fig. B6. Theta sensitivity analysis IA SWF.

35G. Howard / Ecological Economics 80 (2012) 25–37

Fig. B7. Alpha sensitivity analysis 3 period.

Fig. B8. Beta sensitivity analysis 3 period.

Fig. B9. Gamma sensitivity analysis 3 period.

Fig. B10. Phi sensitivity analysis 3 period.

Fig. B11. Rho sensitivity analysis 3 period.

Fig. B12. Theta sensitivity analysis 3 period.

36 G. Howard / Ecological Economics 80 (2012) 25–37

37G. Howard / Ecological Economics 80 (2012) 25–37

Similarly, the period 3 agent will accept any offer in whichU*(C*3,P*3)≥U(C3,P3), where U*, C*3, and P*3 are the period 3 utility,consumption, and pollution achieved if the offer is accepted and U, C3,and P3 are the period 3 values achieved if the offer is rejected. This iscaptured by the following inequality:

Log γ γTþ ρA1ð ÞLf3αLr31–α� �

–φ δLr1ð Þ1þθ≥ Log γ2TLf3αLr3

1–α� �

: ðB2Þ

In order for Proposition 6 to hold, any offer (A1, Lr1) for which (B1)holds with equality, inequality (B2) must hold as well. Assuming (B1)holds with equality and rearranging yields,

LogγTþ ρA1

γT

� �¼ φLr1

1þθ; ðB3Þ

Which can be further simplified to

A1 ¼ γTexp φLr1

1þ θ� �

–1

ρ

0@

1A: ðB4Þ

Inequality (A21) can also be rearranged to yield,

A1 ≥ γTexp φ δLr1ð Þ1þ θð Þ–1

ρ

� �: ðB5Þ

The right sides of (B4) and (B5) are identical except for the δ in (B5).So long as δ≤1, the right side of (B5) will be less than or equal to theright side of (B4). This means that any offer (A1, Lr1) that satisfies (B1)with equality must satisfy (B2) so for any δ≤1. QED

B3). 3-Period Model Comparison

The following figures present welfare outcomes using the 3-periodmodel outlined in Section 3.5 (Figs. B7–B12).

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