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Environ Resource EconDOI 10.1007/s10640-017-0155-2

Self-Enforcing Intergenerational Social Contractsfor Pareto Improving Pollution Mitigation

Nguyen Thang Dao1 · Kerstin Burghaus1 ·Ottmar Edenhofer1,2,3

Accepted: 5 April 2017© Springer Science+Business Media Dordrecht 2017

Abstract We consider, in an overlapping generations model with an environmental exter-nality, a scheme of contracts between any two successive generations. Under each contract,agents of the young generation invest a share of their labor income in pollution mitigationin exchange for a transfer in the second period of their lives. The transfer is financed ina pay-as-you-go manner by the next young generation. Different from previous work weassume that the transfer is granted as a subsidy to capital income rather than lump sum. Weshow that the existence of a contract which is Pareto improving over the situation withoutcontract for any two generations requires a sufficiently high level of income. In a steady statewith social contracts in each period, the pollution stock is lower compared to a steady statewithout contracts. Analytical and numerical analysis of the dynamics under Nash bargainingsuggests that under reasonable conditions, also steady state income and welfare are higher.Delaying the implementation of a social contract for too long or imposing a contract withtoo low mitigation can be costly: Net income may inevitably fall below the threshold in finitetime so that Pareto improving mitigation is no longer possible and the economy converges

We would like to thank Gerard van der Meijden, Rick van der Ploeg, Cees Withagen (guest editors), andthree anonymous referees for their constructive comments and suggestions. We are grateful to Niko Jaakkolafor his insightful discussion and to Claudine Chen and Quoc Trung Bui for their support in simulation. Thecomments by participants of several conferences and workshops in Berlin, Helsinki, Dublin, Münster,Strasbourg, and Amsterdam are also highly appreciated. The scientific responsibility lies with the authors.

B Nguyen Thang [email protected]

Kerstin [email protected]

Ottmar [email protected]

1 Mercator Research Institute on Global Commons and Climate Change, Torgauer Str. 12 - 15,10829 Berlin, Germany

2 Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany

3 Department of Economics of Climate Change, Technical University of Berlin, Berlin, Germany

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to a steady state with high pollution stock and low income and welfare. In the second part ofthe paper, we study a game theoretic setup, taking into account that credibly committing toa contract might not be possible. We show that with transfers granted as a subsidy to capitalincome, there exist mitigation transfer schemes which are both Pareto improving and give nogeneration an incentive to deviate from any of its contracts even in a dynamically efficienteconomy. Social contracts coexist with private savings.

Keywords OLG models · Pollution · Mitigation · Social contract · Pareto improvement ·Self-enforcing

JEL Classification D62 · D64 · E21 · Q54

1 Introduction

There is a sizable literature studying the role of (second best) optimal government interventionin overlapping generation (OLG) economieswith environmental externalities, focusing on thelong run steady state.1 But correcting the environmental externalities is often seen to imposecosts on current generations while benefits accrue largely to future generations. This makesit difficult for policymakers to make environmental policy measures appealing to their votersand has led to a discussion about appropriate social discount rates to weigh the welfare ofdifferent generations against each other (Nordhaus 2008; Nordhaus and Boyer 2000; Stern2007). Still, it has been pointed out e.g. by John and Pecchenino (1994), Bovenberg andHeijdra (1998, 2002), Gerlagh and Keyzer (2001), Foley (2009) and Rezai et al. (2012) thatthe discussion about social discount rates is partially misled, as environmental protection canbe achieved in a Pareto improving way, making also those alive today better off. The reason isthat the efficiency gain from correcting environmental externalities can be distributed amongcurrent and future generations such that each generation enjoys higher welfare.

We study the scope for achieving Pareto improving mitigation in an OLG model withnon-altruistic agents with and without commitment. Our work differs from existing literaturein two respects: (1) We propose establishing a scheme of bilateral social contracts betweenany two successive generations for mitigation and redistribution, rather than using fiscalpolicy. Under a social contract, agents of the current young generation give up a share oftheir labor incomewhich is invested in abatement technology or abatement capital and lowersthe pollution stock in the next period.2 In exchange, they receive a transfer to their old-agecapital income when retired. Each contract is thus characterized by a mitigation share and atransfer rate which can be determined through Nash bargaining. The transfer is financed in apay-as-you-gomanner from the labor income of the next young generation who benefits fromthe improved environmental stock. (2) We assume that transfers are granted as a subsidy tocapital income, rather than lump sum.

In this framework, we show that it is possible to determine contracts which make thegenerations involved better off compared to the situation without contracts. We prove thatwith transfers granted as a subsidy to capital income, schemes of period-by-period contractscan be designed such that they can be sustained over time without external enforcement by

1 An incomplete list of papers includes John andPecchenino (1994),Ono (1996), Jouvet et al. (2000),Gutiérrez(2008), Goenka et al. (2012), Fodha and Seegmuller (2014), Dao and Dávila (2014), and Dao and Edenhofer(2014).2 In practice, it could be centrally collected e.g. joint with taxes, like the German solidarity surcharge.

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government institutions. Different from results in previous literature, this is true even in adynamically efficient economy.

We survey related literature in Sect. 2, before describing the setup of our model in Sect. 3.Into a dynamic OLG-model, we incorporate an environmental externality in the sense thatthe stock of pollution in a period reduces next period’s total factor productivity. Any twosuccessive generations of agents have the option to sign bilateral contracts as we describedthem above. In Sects. 4 and 5, we prove the existence of contracts which are Pareto improv-ing over an equilibrium without contract for sufficiently high net income. We then analyzedynamics when generations choose the mitigation share and transfer rate of the contract viaNash bargaining. In a steady state with social contracts in each period, the pollution stock islower compared to a steady state without contracts. Analytical and numerical analysis of thedynamics under Nash bargaining suggests that under reasonable conditions, also steady statewelfare is higher. These are not trivial results because in our model, the long run effects of asystem of bilateral social contracts on the pollution stock and welfare are a priory ambigu-ous. Delaying the implementation of a social contract for too long or choosing contractswith too low mitigation may be costly: If the income threshold for a Pareto improvementis comparatively high, disposable income inevitably falls below the threshold in finite timeso that Pareto improving mitigation is no longer possible and the economy converges to asteady state with high pollution stock and low welfare.

Naturally, intergenerational contracts raise the question of commitment problems. The firstpart of the paper neglects any commitment problems of agents, reflecting the idea that the legalframework and institutions allow to enforce compliance. Legal enforcement may however beproblematic for intergenerational contracts as we consider them here, as they usually involvegenerations not legally of age or not even born. In Sect. 6 of the paper, we explicitly accountfor commitment problems and allow for strategic interaction between generations in a gametheoretic setup. We first characterize conditions under which a scheme of contracts betweensuccessive generations in every period is a subgame perfect equilibrium of a repeated gamewith overlapping generations. Such a scheme is self-enforcing in the sense that no generationhas an incentive to deviate.We then prove the existence of self-enforcing contract schemes forreasonable parameter constellations. Finally, we show that for sufficiently high income, thereexist schemes of period-by-period social contracts between any two successive generationswhich are simultaneously Pareto improving and self-enforcing.

2 Related Literature

The central finding of a seminal paper by Samuelson (1958) is that a system of perpetualintergenerational young to old (pay-as-you-go) transfers can lead to a Pareto improvementover the market equilibrium in OLG-economies with a low equilibrium interest rate, belowthe population growth rate. In these economies, the intergenerational transfer scheme curesdynamically inefficient oversaving by the young generation.3 As Weil (2008) points out,intergenerational transfers can also be Pareto improving for the participating generationsin dynamically efficient economies, if they cure some other inefficiency, e.g. the absence ofappropriate intergenerational risk-sharing. In our paper, there is an environmental externality.

Most of the existing literature on Pareto improving climate protection considers a combi-nation of different government policies, such as taxes or public abatement spending and debt

3 While Samuelson (1958) considers generations which live for two or three periods, Gale (1973) extends hisanalysis to the case of generations living n periods.

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policy.While taxes or public abatement internalize the environmental externality, debt policyis used to redistribute the gains from the internalization of the environmental externality overtime. Seminal contributions in this respect are the OLG models with polluting physical capi-tal by Bovenberg and Heijdra (1998, 2002). Foley (2009) and Rezai et al. (2012) put forwarda similar argument, but without explicitly modeling savings behavior and investment in anOLG-context. They point out that consumption of current generations need not be reduced ifan increase in mitigation investment is met by a decrease in conventional investment. Futuregenerations are thus left with more environmental but less man-made capital. As a way toinduce such a re-balancing of the investment portfolio, Foley (2009) suggests using a carbontax combined with debt policy. However, many countries face constraints on their debt levelsso that debt policy for redistribution purposes may not be an option. Further, as Gerlaghand Keyzer (2001, p 317) point out, when seeking to redistribute between all current andfuture generations, the informational requirement to appropriately adjust the debt level overtime may be quite strong. Social contracts between successive generations as we proposehere are advantageous in both respects and sufficient to make all generations better off withmitigation.

The literature above, and also this paper, ignores any effects of environmental protectionon the price of capital. Karp and Rezai (2014, 2015) point out that if there are adjustmentcosts in capital investment (or if the capital stock is fixed), environmental protection in formof a tax on resource-intensive production may, by raising future productivity, lead to anappreciation in the price of capital. This is a complementary channel for distributive effectsof environmental policy by which it benefits current old generations. The authors show thatredistributing a share of the tax revenue from the current old to the current young can bePareto improving.4

Instead of relying on fiscal policy, Gerlagh and Keyzer (2001) propose to transfer thevalue of natural resources to a trust fund and distribute equal claims to all current and futuregenerations. Distributing property rights equally not only leads to an efficient intertemporalallocation but ensures, contrary to debt policy, that polluters pay for the damage they cause.Welfare of every generation increases relative to a scenario where extraction of the resource isforbidden. However, current generations would be better off by not distributing any propertyrights to the future, which makes implementation of a trust fund dependent on altruism ofcurrent towards future generations.

Different from Gerlagh and Keyzer (2001), we do not aim to find an intertemporallyefficient allocation. In fact, contracts between two successive generations internalize onlya fraction of the future benefits from mitigation, i.e. those which arise to the next gener-ation. Mitigation will therefore be suboptimally low. While it would make the expositionmore complicated, our model could be extended to capture also the more distant benefitsof mitigation: A recent and closely related contribution by von Below et al. (2016) showsthat bilateral contracts between successive generations can always be Pareto improved uponby including further, more distant generations in the redistribution scheme, which leads tomore mitigation. The authors study redistribution through intergenerational pay-as-you-gotransfers as we do. They assume however that transfers are granted lump sum, not as a sub-sidy to capital income. This difference in the way of modeling transfers leads to a differencein saving behavior, hence capital accumulation, between our paper and von Below et al.

4 Our focus and the focus of the literature we review here is on redistribution between generations. Recently,Williams III et al. (2014) developed a quantitative paper to evaluate the near-term effects of a carbon tax andalternative redistribution schemes on welfare of income-heterogeneous households within each generation.Intragenerational redistribution is the topic of a large strand of literature, which we will not survey in detailin this paper. Further quantitative analyses have been carried out e.g. by Rausch et al. (2010, 2011).

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(2016). Indeed, in our analytical framework, the transfer to old-age income does not changethe saving behavior of agents, while a lump-sum transfer erodes saving motives.

Our modeling of transfers is crucial for the second part of our paper, where we take intoaccount that at the time the current young generation makes its mitigation investment, cred-ibly committing the still unborn or not legally mature future young generation to paying thetransfer might be impossible. Commitment problems are an issue in most of the aforemen-tioned models which consider redistribution between current and future generations but notstudied in either of them.5 Hammond (1975), building on Samuelson (1958) with a pureexchange economy, shows that perpetual Pareto improving intergenerational transfers can besustained as the equilibrium of a dynamic game without commitment between non-altruisticagents if the young generation expects punishment in case of not paying transfers to the old.Hammond neglects any possibility to save and thus transfer a part of one’s endowment fromone period to the next. Moreover, there is no public good which has to be financed. Kotlikoffet al. (1988) do model savings in an economy with fixed interest rate. Different from ourpaper, they do not link transfer payments to intergenerational public good financing, butshow that making transfer payments conditional on the ex-ante optimal capital tax can solvethe intragenerational time-inconsistency problem in capital taxation. Similar to our paper,the transfer scheme in their model does not discourage private saving but even prevents inef-ficient undersaving. Rangel (2003), Cigno (1993, 2006a, b) and Anderberg and Balestrino(2003) analyze the scope for private provision of an intergenerational public good (education,environmental quality) in a game-theoretic setup with non-altruistic agents. Contributionsto financing of the public good are compensated by lump sum pay-as-you go transfers. Theauthors show that a necessary condition to sustain the provision of the public good whencommitment is not possible is that the return to the compensating intergenerational transfermust be larger than the return to private savings. With a lump sum pay-as-you-go transfer,Cigno (1993, 2006a, b) and Anderberg and Balestrino (2003) prove that this requires suf-ficiently fast population growth, i.e. dynamic inefficiency of the economy. With a transferscheme in place, there is then no longer a motive to save and invest in capital. Rangel (2003)assumes that in a constant population, transfer-payers outnumber transfer receivers. He illus-trates that intergenerational transfers to finance a public good can be self-enforcing also indynamically efficient economies, where some generations will suffer a welfare loss, if con-tinuation is decided by majority vote. Sustainability then requires that generations who gainoutnumber those who lose. In our setup, every generation gains from upholding the schemeof social contracts. We contribute to the literature by showing that with transfers subsidiz-ing capital income, nevertheless dynamic inefficiency is not required. Moreover, while theaforementioned models typically consider a partial equilibrium or small open economy withfixed interest rate, we carry out our analysis in a model with endogenous interest rate andsavings. In this respect, it is also important that our modeling of transfers avoids the completecrowding out of private capital investment encountered with lump sum transfers.

3 The Benchmark Model

In this section, we set up our benchmark model. We discuss the robustness of the results tochanges in functional forms in the appendix, section “Robustness”.

5 Karp and Rezai (2014, 2015) derive time-consistent solutions for redistribution. However, as in their modelsboth generations are alive when the tax is collected and redistribution occurs, commitment problems as in ourmodel do not arise in theirs.

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3.1 Dynamics of the Pollution Stock and Production

The stock of pollution, e.g. the carbon concentration in the atmosphere, accumulates due tothe pollution flow released into the atmosphere by human economic activities (productionand mitigation). We assume that the pollution flow in period t, Pt , is given by

Pt = ξKt − γ Mt ,

where ξKt is the pollution flow from capital6 Kt and γ Mt is pollution abatement resultingfrom mitigation effort Mt . The parameters ξ, γ > 0 are the pollution coefficient of physicalcapital and the mitigation coefficient, respectively. Without loss of generality, we normalizeξ = 1. We assume that the dynamics of the pollution stock are described by7

Et = (1 − δ)Et−1 + Kt − γ Mt

where Et ∈ R is the pollution stock index at the end of period t . The decay rate δ ∈ [0, 1] ofthe pollution stock measures the convergence speed of the pollution stock to a natural state Ewhich would prevail without any human activity. For simplicity, we have normalized E = 0.

Polluting physical capital together with labor Lt is an input to the production of an aggre-gate final good Yt . The production technology is of the Cobb–Douglas type

Yt = z(Et−1)Kαt L

1−αt ; α ∈ (0, 1)

where z(Et−1) is total factor productivity in period t . Note that last period’s pollution stockaffects current productivity, reflecting the long-run effect of pollution. For simplification andfor the purpose of exposition, we specify the following functional form of z(E):

z(E) = Ae−|E |; A > 0

Total factor productivity gets maximal when the carbon concentration in the atmosphere isat the natural state.

We assume that capital fully depreciates during each period of use. Capital and laborrespectively are chosen so that factor prizes in t equal marginal productivities in production

Rt = z(Et−1)αkα−1t (1)

wt = z(Et−1)(1 − α)kαt (2)

where kt = Kt/Lt is capital per capita.

6 Environmental economists conventionally adopt the assumption that pollution is (constantly) proportionalto output. We assume instead in this paper that pollution is proportional to capital, which is a composite factorof production including machines but also polluting inputs like fuels. This assumption is a monotonic trans-formation of the standard assumption which facilitates analytical computations. The assumption of pollutingcapital has been adopted in the literature amongst others byWang et al. (2015) in an OLG context or by Gradusand Smulders (1993) with infinitely-lived agents.7 The mitigation coefficient γ , as well as the decay rate of pollution δ, may depend on the state of theenvironment, i.e. the stock of pollution, or on mitigation. To keep the model simple, we treat γ and also δ asconstants, adopting a standard approach in related literature (see John and Pecchenino 1994; Ono 1996; Jouvetet al. 2000 and recently Dao and Dávila 2014). Empirically, there seems to be some evidence of increasingreturns to mitigation, i.e. ∂γ /∂m > 0 (i.e. Andreoni and Levinson 2001, pp 278–281). Assuming increasingreturns to mitigation in our model would still preserve the positive relation between income and abatementγ Mt . On the other hand, one could well imagine that returns to mitigation are decreasing for very highmitigation levels. As pointed out by an anonymous referee, our assumption of a constant γ can be justified bya focus on a range of mitigation levels which are not too high.

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3.2 Agents and the Intergenerational Social Contract

We assume that each generation consists of a constant number of L identical agents livingfor two periods t, t + 1. In the first period (young age), each agent is endowed with oneunit of labor which he supplies inelastically to the market to earn labor income. During thisperiod, he allocates his disposable income between young-age consumption cyt and savingskt+1. Savings are lent to production and used for consumption cot+1 in the second period oflife (old age).

Hereafter, we denote by agent t and generation t the agent and generation born in periodt . We assume in line with Shell (1971) that before the beginning of time, all generationsmeet and each generation t negotiates with generation t + 1 for pollution mitigation and anintergenerational transfer: Generation t offers generation t + 1 to sign a contract accordingto which each agent t leave a portion mt ∈ [0, 1) of his working-age income for mitigation.In exchange, each agent t will receive a transfer at rate τ ot+1 ≥ 0 to his gross capital incomewhen old. We do not assume altruism between generations, so that each generation seeks tomaximize its own welfare. If generations t and t +1 reach an agreement, they sign a contract(mt , τ

ot+1). Otherwise there will be neither mitigation nor transfer payments: (mt , τ

ot+1) =

(0, 0). Contracts are then carried out as time proceeds, so that the mitigation share mt ispaid in period t , and the transfer τ ot+1 in period t + 1 (Fig. 1). The mitigation investmentmt It = Mt is used for mitigation/becomes effective at the beginning of period t + 1.

Before exploring in Sect. 4 when signing a contract would be beneficial, we now firstdescribe the consumption-savings decision of a representative agent t given a social contract(mt , τ

ot+1) and determine the equilibrium. Each agent is assumed to be negligible among

his generation. He does not internalize the impact of his action (i.e. engagement or non-engagement in the contract) on the aggregate capital and pollution stock. When he decidesabout consumption and savings so as to maximize his life-time utility, he treats the return tocapital, on which he has perfect foresight, as given. The maximization problem of an agentt given the social contract (mt , τ

ot+1) is

maxcyt ,kt+1,cot+1

ln cyt + β ln cot+1 (3)

subject to : cyt + kt+1 ≤ It (1 − mt ) (4)

cot+1 ≤ Ret+1kt+1

(1 + τ ot+1

)(5)

given net income It and the perfectly foreseen return to capital Ret+1 = Rt+1. The parameter

β ∈ (0, 1) is the agent’s time preference parameter. Net income It = wt (1 − τyt ) is defined

as wage income wt net of transfer payments at rate τyt ∈ (0, 1) to generation t − 1.

3.3 Equilibrium

Given an intergenerational social contract (mt , τot+1), the competitive equilibrium in the

economy is characterized by: (i) utility maximization of each agent according to (3) underthe budget constraints (4) and (5); (ii) the law of motion of capital Kt+1 = kt+1; (iii)

Fig. 1 Intergenerational social contracts

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the maximization problem of the final good producing firm determining the returns to theproduction factors (1) and (2); (iv) the dynamics of the pollution stock; and (v) a balancedbudget of intergenerational transfers [see (11)]. Therefore, the set {cyt , kt+1, cot+1, Et }t , whichfully characterizes the competitive equilibriumof the economy, is the solution to the followingsystem of equations:

cyt = 1

1 + βIt (1 − mt ) (6)

kt+1 = β

1 + βIt (1 − mt ) (7)

cot+1 = αz(Et )

(β It (1 − mt )

1 + β

)α (1 + τ ot+1

)(8)

Et = (1 − δ)Et−1 + kt − γ Mt (9)

given kt , Et−1, and Mt , where

It = z(Et−1)(1 − α)kαt

(1 − τ

yt)

(10)

The transfer paid by generation t must equal the transfer generation t − 1 receives underthe contract (mt−1, τ

ot ). The rates τ ot and τ

yt thus satisfy the balanced budget condition

wtτyt = Rtktτ ot at equilibrium:

τyt z(Et−1)(1 − α)kα

t = τ ot z(Et−1)αkαt ; i.e. τ

yt = α

1 − ατ ot (11)

Mitigation effort Mt in t is mitigation investment by generation t − 1, i.e. Mt = mt−1 It−1.The first three equations reflect the lifetime consumption-savings decision of the repre-

sentative agent t at equilibrium. Due to our assumption of logarithmic utility, consumption inboth stages of life and savings are proportional to disposable income It (1−mt ), which is netincome minus mitigation expenses, with β

1+βdenoting the savings rate. A rise in the transfer

τyt to the previous generation t −1 or in the mitigation sharemt thus reduces consumption cytand savings kt+1 proportionally. The decline in savings has a negative effect on consumptioncot+1 when old. There is however also a positive impact of a higher transfer or mitigationshare on cot+1, as the reduction in capital accumulation raises the equilibrium interest rate

Rt+1 = αz(Et )kα−1t+1 . The latter increase is more pronounced if the production elasticity of

capital, α, is smaller. Still, as (8) shows, the overall effect of a decline in disposable incomeIt (1−mt ) is to decrease cot+1. From (6) and (8), it is obvious that a higher transfer received,τ ot+1, means agents consume more in old age for any given level of savings kt+1.

4 Existence of a Pareto Improving Intergenerational Social Contract

In this subsection, we derive a condition for the existence of a Pareto improving intergener-ational social contract (mt , τ

ot+1) between generations t and t + 1. Such a contract leaves at

least one generation better off without creating a welfare loss for the other. Existence requiresthat starting from a situation without contracts, i.e. mt = τ ot+1 = 0, the marginal increasein the transfer rate τ ot+1 generation t + 1 can accept without suffering a welfare loss mustexceed the minimum increase the old generation t requires as a compensation for leavinga marginal share of its income for mitigation. We prove that this is the case for sufficiently

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high net income. To do so, we first derive the changes in lifetime indirect utilities for bothgenerations compared to a situation where the contract is not signed.8

Let �V t+1t and �V t+1

t+1 be the foreseen surpluses of agents t and t + 1 from the socialcontract (mt , τ

ot+1) � (0, 0) compared to the situation (mt , τ

ot+1) = (0, 0).9 Agent t does

not suffer a utility loss from the social contract (mt , τot+1) if and only if

�V t+1t = (1 + αβ) ln(1 − mt ) + β ln

(1 + τ ot+1

) ≥ 0 (12)

The first term, (1 + αβ) ln(1 − mt ), describes the lifetime utility loss from the decline inconsumption cyt and c

ot+1 which, as described in Sect. 3.3, follows the reduction in disposable

income by mitigation: The term (1 + αβ) adds up the direct negative effect of a marginalreduction in disposable income through the reduction of consumption cyt and savings kt+1

and the positive effect through a higher equilibrium interest rate. The second term in Eq. (12)is the welfare gain from receiving a transfer in period t + 1.

Agent t + 1 does not suffer a welfare loss from the contract (mt , τot+1) if and only if

�V t+1t+1 = (1 + αβ) ln

[(1 − α

(1 + τ ot+1

)

1 − α

)

(1 − mt )α

]

+ β(β + γ + γβ)

1 + βmt It ≥ 0

(13)

The first term in (13) is again the lifetime utility loss from the reduction in disposable incomeunder the contract. The second term is the welfare gain from a reduction in the stock ofpollution due to higher mitigation. More precisely, mitigation lowers pollution in t + 1 bothdirectly and because it reduces the capital stock. A lower pollution stock in t + 1 increasesproductivity z and thereby the return to savings in period t + 2. Because for any given mt ,

total mitigation Mt = mt It increases in income It , this second effect is stronger if It is higherand may dominate the effect of the reduction in disposable income if It is sufficiently large.

Consider Fig. 2: Denoted by S t+1t and S t+1

t+1 are the sets of contracts which do not makegenerations t and t + 1 worse off, i.e. for which (12) and (13) hold, and P t+1 is the setof Pareto improving contracts.10 We further specify the following indifference curves forgenerations t and t + 1 respectively:

τ ot+1 =(

1

1 − mt

) 1β

+α

− 1 ≡ (mt ) (14)

τ ot+1 = 1 − α

α

[1 − e− (β+γ+γβ)βmt It

(1+β)(1+αβ) (1 − mt )−α

]≡ ψ(mt , It ) (15)

8 Since the damage effect is larger (lower z(Et )) when the carbon concentration index is farther from thenatural state, rational agents will never reach a contract under which the carbon concentration is negative.Hence, we always impose implicitly or explicitly a non-negativity constraint Et+1 ≥ 0 ∀t on the carbonconcentration index. This is equivalent to

mt ≤ (1 − δ)(1 + β)Et/It + β

β + γ + γβ≡ m(It , Et ) = mt ∀t.

9 Throughout, we use the superscript t + 1 to refer to the contract between generations t and t + 1, while thesubscript refers to the generation. For the derivation of the surpluses, see the appendix, section “Utility Gainsfrom a Social Contract (mt , τ

ot+1)”.

10 See the appendix, section “Definition of the Pareto Set and Proof of Compactness” for a formal definitionof the sets.

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Fig. 2 The existence and nonexistence of P t+1: a the case It > I ; and b the case It ≤ I

The function (mt ) (ψ(mt , It )) returns for any given mitigation share mt the minimumtransfer τ ot+1 required (maximum acceptable transfer) to make generation t (t + 1) just as

well off with as without a contract (the transfer for which �V t+1t = 0 (�V t+1

t+1 = 0)).The slope ′(0+) (ψm(0+, It )) at the origin represents the minimum required (maximumacceptable) increase in the transfer which allows to marginally raise mitigation above zerowithout a welfare loss. The slopes equal the ratios of themarginal loss (gain) from an increaseinmt to the marginal gain (loss) from an increase in τ ot+1 for generation t (t+1). As the utilitygain from mitigation for generation t + 1 increases in income It , it is intuitive that

Proposition 1 There exists a threshold I of net income such that P t+1 = � if, and only if,net income It of agent t exceeds that threshold.

Proof See the appendix, section “Proof of Proposition1”. � As can be seen from (12) to (15), for income above the threshold

I = α(1 + αβ)(1 + β)2

(1 − α)(β + γ + γβ)β2 (16)

the marginal gain (β+γ+γβ)β It1+β

of generation t + 1 from increasing mitigation above zero netof its marginal losses α(1+αβ) due to slower capital accumulation in period t is sufficientlylarge relative to itsmarginal losses α

1−α(1+αβ) from a higher transfer τ ot+1. Generation t+1’s

relative gain exceeds the marginal loss 1+αβ of generation t from mitigation relative to thatgeneration’s marginal gain β from the increase in τ ot+1. Therefore the maximum marginalincrease in the transfer generation t + 1 can accept and be just as well off with and withoutthe contract exceeds what is required by generation t , i.e. ψm(0+, It ) > ′(0+).

Generation t + 1 benefits from mitigation because it leads to less pollution which hasthe positive effect on productivity and the expected return to savings described earlier. Themarginal gain (β+γ+γβ)β It

1+βis the larger, the more weight is put on the second period of life

(higher β), the greater the efficiency of the mitigation technology as measured by γ and thelarger net income It . The higher γ and/or income, the greater the reduction in the pollutionflow for a given income share mt invested in mitigation. Note that even if mitigation isperfectly ineffective, i.e. for γ = 0, so that the direct effect of mitigation on the pollutionstock is zero, the positive productivity effect prevails. This is due to the indirect effect of

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mitigation on the pollution stock through a reduction in the capital stock. For infinitelyeffective mitigation, i.e. for γ → ∞, the income threshold in (16) converges to zero, so thata Pareto improving social contract exists for any positive level of net income It . The incomethreshold therefore decreases in γ . Further, the threshold increases in the capital share α,because the marginal utility losses to both generations from reductions in disposable incomeincrease in α. The rate of time preference β raises both marginal losses and gains but it canbe shown that the overall effect is to reduce the income threshold.

5 Bargaining, Dynamics and Steady State

The previous section shows that at a given point in time and for given pollution and cap-ital stock at the beginning of period t , two generations t and t + 1 may find it beneficialto sign a social contract. This does not mean, however, that future generations might notbe better off without that contract. More generally, it is not straightforward to concludethat under a sequence of Pareto improving bigenerational social contracts, the economywill eventually converge to a steady state with higher stationary welfare than in the case ofno social contracts. We prove that if contracts are Pareto improving, steady state welfarewill be higher compared to the situation without contracts if both net (before-mitigation)income is higher and the pollution stock is lower. But due to positive feedbacks of pollu-tion reduction on productivity and polluting capital accumulation, it is not clear that theyare. Therefore, studying the full dynamics and the existence, number and characteristics ofsteady states under a scheme of period-by-period social contracts and without contracts isimportant to gain a better understanding of the welfare effects of establishing such contractschemes. We prove in this section that the steady state pollution stock is always lower undersocial contracts. It depends on the timing of signing the contracts, on the size of the incomethreshold for Pareto improvements and on the technology level whether steady state incomeand welfare are higher. If the income threshold is rather high, the economy may fall backto a ‘no-contract’ steady state even after contracts have been signed in several successiveperiods.

5.1 Bargaining

To analyze the dynamics of our model under period-by-period social contracts, we need toknow which (mt , τ

ot+1) will be the outcome of the negotiations between generations. Any

contract which is Pareto improving, i.e. any pair (mt , τot+1) ∈ P t+1, makes both generations

t and t + 1 better off. Even if we focus on Pareto efficiency, there are infinitely many Paretoefficient combinations of mt and τ ot+1 in each period. The agreement (mt , τ

ot+1) will in

general depend on the bargaining power of each generation, the determinants of which areunfortunately still largely unclear from bargaining theory. In the following, we will considerthe so-called Nash bargaining problem, introduced by Nash (1950). Note that the Nashbargaining solution is Pareto efficient. In our context, the Nash solution (m∗

t , τo∗t+1), given it

exists, is

(m∗

t , τo∗t+1

) ∈ argmax(mt ,τ

ot+1

)∈P t+1

{�V t+1

t �V t+1t+1

}(17)

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We prove that

Proposition 2 For any given net income It > I , there exists a unique (m∗t , τ

o∗t+1) ∈ P t+1

solving the Nash bargaining problem (17). If (m∗t , τ

o∗t+1) is an interior point in P t+1, then it

is the unique stationary point for the function �V t+1t �V t+1

t+1 defined over the set P t+1.11

Proof See the appendix, section “Proof of Proposition 2”. � The Nash bargaining solution we propose here is an open-loop Nash equilibrium (OLNE).Agents take future contracts as given independent of the negotiation outcome for the currentcontract. While this assumption may be a simplification, we use the OLNE concept merelyto select any particular contract scheme from the the set of Pareto improving contracts tobe able to study the dynamics of our model. Several of the key results in Proposition4, inparticular the conclusion that the steady state pollution stock is lower under social contracts,do not depend on the OLNE framework.

As the surplus of generation t + 1 from a contract (mt , τot+1) depends on net income It ,

it is unsurprising that also the Nash bargaining solution (m∗t , τ

o∗t+1) is a function of It . For

empirically plausible parameter values (α = 0.3, β = 0.7, γ ∈ (0, 1]), the mitigation sharem∗

t increases in It , reflecting the influence of generation t + 1, whose marginal gain froma higher mt increases in income. Contracts would thus lead to a ratcheting up of climateprotection efforts in growing economies. For the results in the next subsection, this is notessential.

5.2 Dynamics and Steady States

This subsection studies the dynamics of the economy with and without the sequence ofcontracts determined by Nash bargaining. We focus on the case m∗

t < mt for which thepollution stock always remains positive. The opposite case is analyzed in the appendix,section “Steady State with Zero Pollution”. For any given mitigation share mt and transferτ ot+1, the dynamics are characterized by the two equations

It+1 = Ae−Et

[β It (1 − mt )

1 + β

]α [1 − α

(1 + τ ot+1

)](18)

Et+1 = (1 − δ)Et + β − (β + γ + γβ)mt

1 + βIt (19)

describing the evolution of net income and the pollution stock, given I0, E0 > 0. FromEqs. (18) and (19), we can derive the dynamics in the two cases of no social con-tracts and period-by-period social contracts determined by Nash bargaining by substituting(mt , τ

ot+1) = (0, 0) for all t or (mt , τ

ot+1) = (m∗

t , τo∗t+1) for all t , respectively. We find that

Proposition 3 Without intergenerational social contracts in all periods, the economy con-verges to a globally stable steady state ( I , E).

Proof See the appendix, section “Proof of Proposition 3”. � While straightforward, the global stability of the steady state without social contracts willturn out to be important for the dynamics of the economy under period-by-period Nash

11 The uniqueness of the solution and the stationary point is important because it helps us to rule out the caseof bifurcation when we study the dynamic system.

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Fig. 3 Dynamics and steady states in the case I < I ,m′(It ) > 0

bargaining. We show in the following that if steady state income I without contracts is lowcompared to the income threshold I for the existence of Pareto improving social contracts,the economymay become attracted by the ‘no-contract’ steady state even after contracts havebeen signed in several successive periods.

We define I and E as the steady state income and pollution stock under social contracts. Anecessary condition for the existence of a steady state with social contracts is that the technol-ogy level A of the economy is large enough. We therefore make the following assumption:12

Assumption 1 A >[

α(1+αβ)β+γ+γβ

]1−α [ 1+ββ(1−α)

]2−α

.

Figures3 and 4 show the phase diagramsof the dynamic system in the empirically plausiblecase when the mitigation share m increases in income for all It ∈ ( I , I ).13 The dashedlines I I (0, 0) and EE(0, 0) are the loci I I and EE where net income and the pollutionstock remain constant over time when social contracts are never implemented. The dasheddirections of motion present the dynamics in the case of no contracts. Similarly, the bold linesI I (m∗

t , τo∗t+1), EE(m∗

t , τo∗t+1) and the bold directions of motion represent the corresponding

12 If Assumption1 is violated, the economy always converges to the steady state without social contracts( I , E). In this case, even the upper bound for steady state income in our model, denoted by I in Figs. 3 and 4and derived from (18) by setting It+1 = It , Et = 0, and mt = τot+1 = 0, is below the income threshold I forthe existence of Pareto improving social contracts.13 Relaxing this assumption would allow for different functional forms of the EE-curve but not affect ourmain results in Proposition4, which focuses on mitigation shares m∗

t ≤ mt for which the pollution stockalways remains positive. The results in Proposition4 mainly rely on the EE-curve with contracts lying belowthe one without contracts, which is the case independent of the assumption m′(It ) > 0 for all It ∈ ( I , I ). Theassumption is however relevant for the case m∗

t > mt analyzed in the appendix, section “Steady State withZero Pollution”, where it guarantees the existence of a zero-pollution steady state.

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Fig. 4 Dynamics and steady states in the case I > I ,m′(It ) > 0

loci and dynamics in the case of having social contracts (m∗t , τ

o∗t+1).

14 While the dynamicscan be cyclical, we assume convergence in the figures, as indicated by the black arrows.

If the income threshold is below steady state income without contracts, I < I , there existsat least one steady state with social contracts. Figure3 is drawn for the case of exactly onesteady state (I, E). When implementing the intergenerational social contracts (m∗

t , τo∗t+1)t

derived through Nash bargaining in every period with income above the threshold, the econ-omy has a chance to converge to this steady state. At the steady state (I, E) with socialcontracts, the pollution stock is lower than in the steady state without social contracts, i.e.E < E . While Fig. 3 shows a case where at the same time steady state net income I undersocial contracts exceeds steady state income I without contracts, we cannot exclude theopposite case, i.e. a steady state with net income I ∈ ( I , I ). Mitigation, on the one hand,lowers net income for a given pollution stock by slowing capital accumulation. But pollu-tion reduction, on the other hand, raises total factor productivity, which boosts income andcapital accumulation. If the latter slows the decline of the pollution stock sufficiently or evenincreases the stock despite mitigation (if the slope of the EE-curve is large for net incomein a medium income range above I ), the initial productivity effect may be diminished toan extent that steady state net income I under social contracts is lower. The high-pollutionsteady state without contracts ( I , E), however, disappears.

In the case of a relatively high income threshold ( I > I ) shown in Fig. 4, the existenceof a steady state with social contracts is not guaranteed. But at any steady state (I, E) withsocial contracts, given it exists, not only is the pollution stock lower, i.e. E < E , but alsonet income is higher, i.e. I > I , than at the steady state without social contracts. A steadystate with Pareto improving social contracts therefore unambiguously yields higher welfare.

14 For a description of the isoclines, see the appendix, section “Isoclines and Directions of Motion”.

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Interestingly though, the steady state ( I , E)without social contracts does not vanish. Because( I , E) is stable, as proved in Proposition3, there always exists a “dangerous area” (the shadedarea in Fig. 4) in which the economy will be attracted by the stable steady state ( I , E). Socialcontracts can then not help the economy to escape this trap. Figure4 shows that an economyrisks entering the dangerous area if its pollution stock is large relative to its net income. Itthus reveals an important economic implication: If the economy lets pollution increase toomuch by delaying the signing of intergenerational social contracts when its net income is stillhigh, or by choosing contracts with too low mitigation, the costs may be substantial since itmay converge to a steady state with high pollution and low income.

In the appendix, section “Simulation”, we provide simulation exercises to illustrate thequalitative results stated in this subsection. We summarize our main insights in the followingproposition:

Proposition 4 If I > I so that P = � and

(i) I < I and Assumption1 holds, then there exists at least one steady state with Paretoimproving social contracts (m, τ o) ∈ P . The steady state pollution stock is lower, butnet income I at steady state may be higher or lower compared to the steady state ( I , E)

without social contracts. The economy can no longer converge to ( I , E).(ii) I > I , then if

∥∥EE(m∗t , τ

o∗t+1) ∩ I I (m∗

t , τo∗t+1)

∥∥ ≥ 1 so that there exists a steady statewith Pareto improving social contracts (m, τ o) ∈ P , the steady state pollution stockis lower and steady state net income I and welfare are higher compared to the steadystate ( I , E) without social contracts. The steady state ( I , E) does not vanish so that theeconomy is at risk of converging to this low-income, high-pollution steady state.

Proof See the appendix, section “Proof of Proposition4”. � Which case seems to be themore relevant? Ifwe allowed for sustained total factor productivity(TFP) growth (rising A) in our model, I would eventually cross the threshold I . Dynamicswould be best described by Fig. 3. For high-income countries, this appears to be a validscenario. However, for the development of the global carbon concentration, future emissionsfrom low- and middle-income countries are crucial. Several such countries e.g. in Sub-Saharan Africa experienced a significant slowdown in their GDP growth in recent years,driven bydecliningTFPgrowth (Abdychev et al. 2015).Without structural reforms to enhanceTFP growth, such countries might be better represented by Fig. 4 for years to come.

6 Incentive Compatibility in a Strategic Game

So far, it has been assumed that once two generations t and t + 1 have determined a contract(m∗

t , τo∗t+1), the generation investing a sharem

∗t of its income in mitigation in t can be certain

to receive the transfer τ o∗t+1 in t + 1. This setup ignores any commitment problems of agents,reflecting the idea that the legal framework and institutions allow to enforce compliance. Legalenforcementmay however be problematic for intergenerational contracts aswe consider themhere, as they involve generations not legally of age or not even born. If agents of the working-age generation have an incentive to default on their obligations from the contract once themitigation level is set, a contract between generations—even if Pareto improving for both—will not be sustainable over time. In this section, we explicitly account for commitmentproblems and characterize ‘self-enforcing’ contracts which can persist in the absence ofexternal enforcement mechanisms.

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A commitment problem arises in our model because from the perspective of generationt+1, the mitigation levelmt is given and independent of whether or not it pays the transfer togeneration t . Generation t +1 thus has an incentive to default on the contract with generationt unless there is some kind of return to compliance. The latter can only exist if generationt + 1 expects the next generation to sign a similar contract so that agents of generation t + 1will themselves receive a transfer if, but only if, they comply. It has been shown e.g. byRangel (2003), Cigno (1993, 2006a, b) and Anderberg and Balestrino (2003) that such asystem of intergenerational transfers can ensure the financing of an intergenerational publicgood like education or environmental quality without external enforcement if the return tothe transfer system, i.e. the difference between the transfer paid and the transfer received,exceeds the return to private savings. In the aforementionedmodels, this is only the case if theeconomy is dynamically inefficient, with an interest rate below the population growth rate.In our setup, the return arises also in a dynamically efficient economy, from our assumptionof giving transfers as a subsidy to capital income. This assumption also reduces crowding outof private savings, as investment in the transfer scheme and in physical capital are to someextent complementary. In the models cited above, as far as savings are considered, agentswill chose either the transfer scheme or private savings, while in our model both can coexist.

In the following, we first transform the problem of finding period-by-period contractsbetween successive overlapping generations into a strategic game. We set up conditionsunder which a generation t will not default on its social contracts with generations t − 1 andt + 1. We then prove the existence of equilibrium paths with social contracts in each periodwhich satisfy these conditions for reasonable parameter settings. Such schemes of period-by-period contracts are subgame perfect equilibria of the dynamic game between generations. Inthe next step, we show that for sufficiently high income, it is possible to find contract schemeswhich are simultaneously Pareto improving (between any two successive generations) andself-enforcing.

6.1 Strategies and Incentive Constraints

WefollowRangel (2003) in formulating the problemoffinding incentive compatible intergen-erational contracts as a strategic game between two successive generations, which is repeatedover time with alternating players.15 The following definition clarifies the terminology:

Definition 1 The history ht , t ≥ T of a game starting in a period T denotes thevector of actions ((mt−1, τ

yt−1), . . . , (mT , τ

yT )) chosen up to time t . A strategy st for

generation t is a plan which assigns a choice (mt , τyt ) to every given history ht =

((mt−1, τyt−1), . . . , (mT , τ

yT )). A strategy profile is a set {st }∞t=T containing the strategies

chosen in each period t .

If a series of contracts (mt , τot+1)

∞t=T from some date T onward gives no generation an

incentive to default on any of the contracts it is involved in, it is sustainable without externalenforcement (by a government). We call such a contract scheme self-enforcing. Clearly, acontract scheme is self-enforcing if, for each generation and given the history of actions takenby previous generations (in particular given their mitigation investment) and expectationsabout future generations’ actions, complying with the contracts yields higher lifetime-utilitythan defaulting. Such a scheme (mt , τ

ot+1)

∞t=T constitutes a subgame perfect equilibrium of

the dynamic game starting at T :

15 Our assumption that each generation consists of a large number of identical agents rules out free-ridingwithin a generation.

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Definition 2 A contract scheme (mt , τot+1)

∞t=T starting in some period 0 < T < ∞ is self-

enforcing if and only if it is a subgame perfect equilibrium of the infinitely repeated gamebetween overlapping generations.

To make the existence of self-enforcing schemes possible, two ingredients are crucial: (1)There must not be a known terminal date: If it were known in a period t that there will not beanother contract after period t + s, the contract scheme will break down:16 Generation t + swill always want to default on its contract with generation t + s − 1. Anticipating that it willnot receive a transfer, generation t + s − 1 will neither pay for mitigation, nor supply thetransfer to generation t + s−2, which defaults on the contract with t + s−3 and so forth. (2)There has to be credible punishment for deviation from a contract (see Rangel (2003), i.a.):Generation t’s reward for compliance is the transfer it receives from generation t + 1 whenold. But it will only have an incentive to comply if it does not receive this transfer also if itdeviates. Generation t + 1 must thus have an incentive to punish generation t for defaulting.

As Rangel (2003) proves in his paper, a path of forward- and backward looking intergen-erational investments, in our case mitigation investment and transfers, can be sustained as asubgame perfect equilibrium in an infinitely repeated game if and only if it can be sustainedby ‘simple trigger strategies’.17 These are strategies under which each deviation is punishedin the next—but only in the next—period. The following simple trigger strategy st (ht ) is aslight modification of the one in Rangel (2003), where (mt , τ

yt ) � (0, 0) denotes some given

combination of mitigation investment and transfer payment to the current old:

st (ht ) ={(

mt , τyt)if p (ht ) = C

(mt , 0

)if p (ht ) = P

;

p (ht ) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

C if(mt−1, τ

yt−1

) = (mt−1, τ

yt−1

)and p (ht−1) = C

C if(mt−1, τ

yt−1

) = (mt−1, 0

)and p (ht−1) = P

P if t = T

P otherwise

The function value p(ht ) = C indicates that the game is in a compliance phase. This phaseoccurs if generation t − 1 has adhered to strategy s by (i) investing in mitigation and payingthe transfer to generation t − 2 if generation t − 2 has itself complied with the contracts it isinvolved in and (ii) punishing generation t − 2 by investing in mitigation but not paying thetransfer if generation t − 2 has defaulted on one of its contracts. Any deviation of generationt − 1 from strategy s will be followed by a punishment phase p(ht ) = P .

We need to show now that under strategy st (ht ), compliance is a subgame perfect equilib-rium of the game. If a generation t wants to deviate from either of the contracts it is involvedin, the best possible deviation is to choose (mt , τ

yt ) = (0, 0), thus defaulting on the obliga-

tions from both the contract with generation t − 1 and the one with generation t + 1. Thisis because it should expect to be punished by generation t + 1 and not to receive a transferwhen old (τ

o,et+1 = 0) in any case. If generation t complies, on the contrary, it can expect to

obtain the transfer it settles in the contract with generation t + 1, i.e. τ o,et+1 = τ ot+1.Generation t compares indirect utilities from compliance and deviation for any given

history ht , that is for the case that the previous generation complied and for the case that

16 As Boldrin and Rustichini (2000) point out, if the time is not known with certainty, generations may choosecompliance even if they are aware that the mitigation-transfer scheme will not be sustained forever.17 We refer the interested reader to Proposition 1 in Rangel (2003).

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it deviated. Denote the difference in expected indirect utility by �V (mt , τyt | p(ht ), τ ot+1),

where τyt ∈ [0, τ y

t]depends on the state p(ht ) of the game. There is no incentive to deviate

if and only if the difference �V (mt , τyt | p(ht ), τ ot+1) in indirect utilities is non-negative.

When the game is in compliance phase, generation t must pay (mt , τyt ) under the

contract. We show in the appendix, section “Incentive Constraints” that the condition�V (mt , τ

yt | p(ht ) = C, τ ot+1) ≥ 0 is equivalent to

(1 + αβ) ln

[(1 − α

1 − ατ ot

)(1 − mt

)]

+ β ln(1 + τ ot+1

) ≥ 0. (20)

The condition simply states that generation t needs a sufficiently large transfer τ ot+1 to becompensated for the losses in disposable income in both periods of life arising frommitigationand the transfer to generation t − 1. Note that the constraint does not depend on the size ofthe transfer or mitigation shares before period t , nor on any stock variable.

Now consider the decision of generation t when the game is in punishment phase. In thiscase, according to the strategy s defined above, generation t should punish generation t − 1with the choice (mt , 0). There is no incentive for deviation if and only if�V (mt , 0 | p(ht ) =P, τ ot+1) ≥ 0 which yields the condition

(1 + αβ) ln(1 − mt

)+ β ln(1 + τ ot+1

) ≥ 0. (21)

Not surprisingly, theminimum expected transfer τ ot+1 required tomake generation t followstrategy s is smaller when the game is in punishment rather than compliance phase wheneverτ ot > 0: To receive a transfer in period t + 1 when the history of the game is compliance,generation t has to honor both the contract with generation t +1, whereby it must pay mt formitigation, and also the contract with generation t − 1, whereby every agent of generationt has to give up a share τ

yt = α

1−ατ ot of his income for the intergenerational transfer. In the

punishment phase, the transfer does not have to be paid.Now suppose generations are contemplating to initiate a scheme of social contracts at

time T > 0. As the generation born in T − 1 has not invested in mitigation, the game startsin a state of punishment. The working-age generation in T will play strategy s and pay mT

for mitigation to generation T +1 but no transfer to generation T −1 if and only if condition(21) holds. If generation T does so, the game enters compliance phase and will remain therewith every generation paying mt to its successor and τ

yt to its predecessor if and only if

condition (20) holds. In this case, playing strategy s in each period constitutes a subgameperfect Nash equilibrium of the dynamic game. The insights of this paragraph lead to thefollowing lemma:

Lemma 1 A contract scheme with positive mitigation investment mt > 0 compensated forby a positive transfer τ ot+1 > 0 in every period t = T, . . . ,∞ can be sustained as a subgameperfect equilibrium of the game with initial state p(hT ) = P if and only if condition (21) issatisfied in period t = T and (20) is satisfied in every period t > T , with τ

o,et+1 = τ ot+1.

Proof The proof is contained in the text. � 6.2 Existence of Self-Enforcing Contract Schemes

We have defined the conditions for a path with period-by-period social contracts to be self-enforcing. Now the question arises whether there exist contract schemes (mt , τ

ot+1)

∞t=T with

mt > 0, τ ot+1 > 0 which satisfy the conditions in every period t ≥ T .

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The first conclusion in this regard must be that along a path with self-enforcing contracts,the intergenerational transfer and the mitigation share must converge to or stay in the vicinityof some constant values τ o,m. Neither can they decrease persistently while still satisfyingmt > 0, τ ot+1 > 0. Nor can they increase unboundedly: At the latest if all income has to begiven up to pay the transfer or invest in mitigation, the contract scheme will break down.The existence of a non-empty set of stationary pairs (m, τ o) with m > 0, τ o > 0 whichsatisfy the incentive constraints (20) and (21) is therefore both necessary and sufficient forthe existence of self-enforcing contract schemes.

The set of stationary self-enforcing contract schemes is the set henceforth denoted by SIC

which is enclosed by the vertical axis and the ‘stationary’ incentive constraint

mIC = 1 − 1(1 − α

1−ατ o)

(1 + τ o)β

1+αβ

∀t (22)

in Fig. 5 below. The stationary incentive constraint follows straightforwardly from the stricterof the two incentive constraints, condition (20) by setting mt = m and τ ot+1 = τ ot = τ o forall t .

A necessary and sufficient condition for the existence of a non-empty stationary set SIC

and thus the existence of self-enforcing contracts is the following: In a situation without asocial contract (so thatmt = τ ot = τ ot+1 = 0) the marginal utility gain from an increase in thetransfer received must exceed the marginal utility loss from an equal increase in the transferpaid.18 Graphically, the slope of the stationary incentive constraint at m = τ o = 0 must bepositive. Formally, this is guaranteed by the following condition:

β >α

1 − α(1 + αβ) (23)

Proposition 5 There exist contract schemes (mt , τot+1)

∞t=T with τ

o,et+1 = τ ot+1 and mt >

0, τ ot+1 > 0 for every t = T, . . . ,∞ which are sustainable as a subgame perfect equilibriumof the game between overlapping generations if and only if condition (23) is satisfied. Themaximum mitigation share which can be sustained is strictly smaller than one.

Proof See the appendix, section “Proof of Proposition5”. � The left hand side of condition (23) gives the marginal gain β in indirect utility of marginallyraising the transfer received, τ o, from zero. On the right hand side, 1 + αβ is the alreadydescribed net marginal decrease in lifetime utility due to an increase in the transfer paid,τ y, by one marginal unit and the according reduction in net income. Finally, the ratio α

1−α

states by how much the transfer paid, τ y, has to rise to finance a one unit increase in thetransfer rate τ o to the current old. For the usual value α = 1/3 of the capital elasticity in theproduction function, condition (23) can be satisfied for sufficiently patient households withβ > 3/5. Note that for the existence of a sustainable path, it is crucial that the ratio α

1−α,

which is the ratio of capital to labor income at mt = τ ot = τ ot+1 = 0, be strictly smallerthan one. As long as α

1−α< 1, the transfer rate paid is lower than the transfer rate received,

i.e. τ y = α1−α

τ o < τ o. This gives rise to a positive net return of investing in a system ofintergenerational transfers even in a dynamically efficient economy, which can make thefinancing of mitigation attractive for generation t .

Instead ofmotivating condition (23) looking atmarginal utility effects of a contract, we canalso derive it by comparing real returns from pure capital investment and capital investment

18 This requirement is equivalent to the condition B′(0) > V ′(w) on page 819 in Rangel (2003).

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combined with the transfer scheme:19 As in the previously cited literature with lump sumintergenerational transfers, investment in the transfer system from the viewpoint of an agentof a young generation competes with the accumulation of physical capital for given income.Our assumption that transfers are granted as subsidy to the return to savings in physicalcapital ensures however that there cannot be a return to the intergenerational transfer withoutcapital investment. Further, the return to an investment in the transfer system is the higher,the higher capital investment and vice versa. This complementarity is absent in models withlump sum transfers and avoids the complete crowding out of private investment by the transferscheme. Condition (23) then demands that in a situation without a social contract (so thatmt = τ ot = τ ot+1 = 0) the net return of marginally increasing τ ot to invest in a transfer schemein addition to investing in capital is strictly larger than the return that could be obtained byinvesting solely in capital.

6.3 Self-Enforcing Contract Schemes and Pareto Improvement

We have pointed out in Sect. 4 that for sufficiently high income, there exist combinations ofmitigation investment and transfers which, when implemented through a social contract, leadto a Pareto improvement between any two successive generations.We have also shown, in theprevious subsection, that there exist contract schemeswhich are self-enforcing if commitmentproblems are taken into account. We are now interested in the question if and under whichcondition(s) there exist series of Pareto improving contracts between successive generationswhich are also self-enforcing.

The crux here is that there exist pairs (mt , τot+1) within the set of self-enforcing contracts

which benefit generation t but make generation t + 1 worse off. The reason generation t + 1will still not deviate from these contracts is that it expects gains from the contract (mt+1, τ

ot+2)

with generation t + 2, which it only receives if it complies with the contract (mt , τot+1). If

there exists an equilibrium path with a scheme of contracts (mt , τot+1)

∞t=T from some period

T > 0 onward such that the stricter of the two incentive constraints, i.e. condition (20), canbe satisfied at the same time as condition (13), which prevents welfare losses for generationt+1 in every period t ≥ T , then the contract scheme (mt , τ

ot+1)

∞t=T is both Pareto improving,

in the sense that no generation suffers a welfare loss from any contract it is involved in, andself-enforcing.20

Conditions (13) and (20) have to hold for all t ≥ T . Understanding if ever and whenthis is the case is not easy because condition (13) changes over time with net income It .We know, however, from the last section that every self-enforcing contract scheme has tobe in the vicinity of some stationary pair (m, τ o) in the long run. A sufficient condition forthe existence of a Pareto improving and self-enforcing contract scheme starting in a periodT > 0 is then that for all t ≥ T , there exist Pareto improving contracts which lie also withinthe set SIC of stationary incentive compatible contracts and this is correctly anticipated.21

Figure5 shows the setP t+1 of Pareto improving combinations (mt , τot+1) in period t together

with the stationary incentive compatible set SIC , in(mt , τ

ot+1

)-space:

19 The comparison can be found in the appendix, section “Heuristic Derivation of Condition (23)”.20 Note that we demand every single contract (mt , τ

ot+1) of the series to be a Pareto improvement. As every

generation signs two contracts, a weaker requirement for the series of contracts to be Pareto improving wouldbe to allow welfare gains in one contract to compensate welfare losses in the other. Every self-enforcingcontract series would then also be Pareto improving.21 Because the contract scheme has to be stationary only in the long run, a necessary but not sufficient conditionis that this is true for all t ≥ T + s, 0 < s < ∞, that is, from some period T + s onward.

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Fig. 5 Pareto improvement and incentive compatibility

Formally, we require SIC ∩ P t+1 = �,∀t ≥ T . We prove below that in a period t , thisis guaranteed by the following condition on gross income wt ≡ I gt :

I gt >(1 − α)

β1+β

(1 + αβ + β)

β − α1−α

(1 + αβ)I where β >

α

1 − α(1 + αβ) (24)

Recall that β > α1−α

(1 + αβ) [condition (23)] guarantees that the set SIC of stationaryself-enforcing contracts is non-empty. The new condition (24) on income states that startingfrom (mt , τ

ot+1) = (0, 0), a slight increase in mt going along with an equal increase dτ o

both in the transfer rate paid to the current old, τ ot , and the transfer received τ ot+1, such thatagents of the transfer-providing generation are just kept from deviating, will also improvethese agents’ welfare compared to the situation without a contract.

Proposition 6 There exist contract schemes (mt , τot+1)

∞t=T which are both Pareto improving

compared to an equilibrium path without contracts and self-enforcing if along with condition(23), condition (24) holds and is anticipated to hold by the players in every period t ≥ T .

Proof See the appendix, section “Proof of Proposition6”. �

Condition (24) is satisfied the easier, the larger the net utility gain β − α1−α

(1 + αβ) of the

transfer system and the smaller the income threshold I for a Pareto improvement. Due to theeffects we described in detail in the previous section and Sect. 4, a smaller capital share α, alarger weight attached to future consumption as measured by β, and greater efficiency of themitigation technology γ increase the scope for Pareto improving, self-enforcing contracts.

It would be interesting to study how accounting for commitment problems alters thedynamics of the model compared to Sect. 5. Unfortunately, as the incentive constraints forthe bargaining problem in a given period are dependent on both past and future choices of mand τ o, even the numerical solution of theNash bargaining problemwith incentive constraintsis demanding and beyond the scope of the present paper.

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N. T. Dao et al.

7 Conclusion

While the burden of climate change mitigation mainly falls on current generations, bene-fits largely accrue to generations not yet born. In this paper, we have studied the scope forPareto improving mitigation in an OLG model with non-altruistic agents, with and withoutcommitment. We have considered a scheme of social contracts between any two successivegenerations according to which young generations invest a share of their labor income inmit-igation in return for a subsidy to their old-age capital incomewhich is paid by the next younggeneration. We have shown that contracts which are Pareto improving over the equilibriumwithout contract exist for sufficiently high net income. In a steady state with social contractsin each period, the pollution stock is lower compared to a steady state without contracts.Analytical and numerical analysis of the dynamics under Nash bargaining suggests that atthe same time, under reasonable conditions, steady state welfare is higher.

Both commitment and enforcement may be difficult in the context of intergenerationalcontracts.Wehave therefore derived a condition for the existence of a series of contractswhichis self-enforcing in the sense that no generation has an incentive to deviate from any contractit is involved in in a non-cooperative setting. Different from findings in previous literature,this is possible even in a dynamically efficient economy if transfers are given as subsidies tocapital income rather than lump sum. Finally, we have shown that for high enough income,it is possible to find combinations of the mitigation share and transfer for which a contractscheme is simultaneously Pareto improving and self-enforcing. With such contract schemesin place, institutions would still be needed to provide information, coordinate payments andsurveil compliance with the contracts. But climate change mitigation would be beneficial forand supported by all generations.

Our model could be extended in several ways. In particular, studies on how technicalchange and population growth affect the results of our paper are under way.

Appendix

Utility gains from a social contract (mt, τ ot+1)

Generation t : By substituting the optimal choices cyt and cot+1 as characterized in (6) and (8)respectively into the utility function, we derive the perfectly foreseen indirect utility of agentt under the contract (mt , τ

ot+1),

V t+1t

(It ,mt , τ

ot+1

) = � + ln[z (Et ) αkα−1

t+1

(1 + τ ot+1

)]β + ln [It (1 − mt )]1+β (25)

where � = ln 11+β

+ β ln β1+β

. We denote by kt+1 capital per capita if there is no socialcontract. The foreseen surplus that agent t gains from the social contract (mt , τ

ot+1) is

�V t+1t = (α − 1) β ln

kt+1

kt+1+ (1 + β) ln(1 − mt ) + β ln

(1 + τ ot+1

)

From Eq. (7), we find that the relation between capital per capita under the social contract

and without social contract is kt+1kt+1

= 11−mt

. The surplus �V t+1t+1 can then straightforwardly

be found to equal the expression in (12).

123


Generation t + 1: The perfectly foreseen indirect utility of generation t + 1 under thecontract (mt , τ

ot+1) is

V t+1t+1

(It+1,m

et+1, τ

o,et+2

) = � + ln

⎡

⎣z (Et+1) α

(β It+1

(1 − me

t+1

)

1 + β

)α−1(1 + τ

o,et+2

)⎤

⎦

β

+ ln[It+1

(1 − me

t+1

)]1+β (26)

where (met+1, τ

o,et+2) is the foreseen social contract that generation t + 1 will sign with the

succeeding generation t +2. It+1 = z(Et )(1−α)kαt+1(1− τ

yt+1) is net income of agent t +1

in period t + 1, i.e. the income after paying the intergenerational transfer τ ot+1Rt+1kt+1 =z(Et )(1 − α)kα

t+1τyt+1 under the social contract (mt , τ

ot+1).

The foreseen surplus that agent t + 1 gains from the social contract (mt , τot+1) with the

preceding generation is

�V t+1t+1 = (1 + αβ) ln

It+1

It+1+ β ln

z(Et+1)

z(Et+1)(27)

where It+1 = z(Et )(1 − α)kαt+1 and Et+1 are net income of agent t + 1 and the stock of

pollution, respectively, in period t + 1 in the case of no contract (mt , τot+1).

We know that

It+1

It+1= kα

t+1

(1 − τ

yt+1

)

kαt+1

where kt+1 = kt+1(1 − mt ) = β It1+β

(1 − mt ), Pt+1 = kt+1, Pt+1 = kt+1 − γmt It , and

τyt+1 = α

1−ατ ot+1. Upon substitution, we obtain:

It+1

It+1=(1 − ατ ot+1

1 − α

)(1 − mt )

α

We also know that

z(Et+1)

z(Et+1)= eEt+1−Et+1 = e Pt+1−Pt+1 = e

β+γ+γβ1+β

mt It

Upon substitution in Eq. (27), we obtain the expression in Eq. (13).

Definition of the Pareto Set and Proof of Compactness

Consider the set of contracts (mt , τot+1)which do not make generation t worse off. We define

this set S t+1t as

S t+1t = {(

mt , τot+1

) ∈ (0, m(It , Et )] × R, mt < 1 : �V t+1t ≥ 0

}(28)

And we define the indifference curve of agent t , i.e. the set of contracts (mt , τot+1) for which

agent t is indifferent between signing and not signing, as

S t+1t = {(

mt , τot+1

) ∈ [0,min{1, mt }] × R, mt < 1 : �V t+1t = 0

}

which can be expressed as

τ ot+1 =(

1

1 − mt

) 1β

+α

− 1 ≡ (mt ) with mt ∈ [0,min{1, mt }], mt < 1 (29)

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N. T. Dao et al.

It is straightforward from the last equation that (mt ) is strictly convex in mt ∈[0,min{1, mt }] and mt < 1.

Now consider the set of all pairs (mt , τot+1) which do not make generation t + 1 worse

off. We define the set S t+1t+1 as

S t+1t+1 =

{(mt , τ

ot+1

) ∈ (0, m(It , Et )] × R, mt < 1 : �V t+1t+1 ≥ 0

}(30)

And we define the indifference curve of agent t+1 as

S t+1t+1 =

{(mt , τ

ot+1

) ∈ [0,min{1, mt }] × R, mt < 1 : �V t+1t+1 = 0

}

which can be expressed as

τ ot+1 = 1 − α

α


(1+β)(1+αβ) (1 − mt )−α

]

≡ ψ(mt , It ) with mt ∈ [0,min{1, mt }], mt < 1 (31)

We prove that ψ(mt , It ) is concave in mt ∈ [0,min{1, mt }]. Indeed, we have

ψmm(mt , It ) = (α − 1)e−amt It

(1 − m)α

[1

(1 − mt )2+(aIt√

α−

√α

1 − mt

)2]

< 0 ∀mt ∈ [0, 1)

where a = β(β+γ+γβ)α(1+β)(1+αβ)

.The set of all contracts (mt , τ

ot+1) between generations t and t + 1 which lead to a Pareto

improvement over (0, 0) is defined as

P t+1 = S t+1t ∩ S t+1

t+1 \(S t+1t ∩ S t+1

t+1

)

Further, define the union of P t+1 with S t+1t ∩ S t+1

t+1, or the set of all contracts which do not

make generations t and t + 1 worse off compared to (0, 0), as P t+1:

P t+1 = P t+1 ∪(S t+1t ∩ S t+1

t+1

)

Lemma 2 P t+1is a compact set.

Proof The statement is trivially true for the case P t+1 = �. If P t+1 = �,

P t+1 = P t+1 ∪(S t+1t ∩ S t+1

t+1

)= S t+1

t ∩S t+1t+1 ∪ {(0, 0)}

It is obvious that S t+1t ∩S t+1

t+1 ∪ {(0, 0)} contains its boundary. Therefore P t+1is

closed. Moreover, since ψ(mt , It ) is continuous over mt ∈ [0, 1) and ψ(0, It ) =0, lim

mt→1−ψ(mt , It ) = −∞ for all It < +∞, it is true that ∃N < +∞ such that

ψ(mt , It ) < N for all mt ∈ [0, 1) and It < +∞. Thus P t+1is always bounded by the

ball BR(0, 0) of center (0, 0) and some finite radius R ≥ √1 + N 2. Therefore, P t+1

iscompact. �

123


Proof of Proposition1

We see from (29) and (31) that

(0) = ψ(0, It ) = 0 (32)

Now we take into account the slopes of (mt ) and ψ(mt , It ) as mt approaches 0 fromthe right. We have

′(0+) = limmt→0+

1 + αβ

β

(1

1 − mt

)1+α+ 1β = 1 + αβ

β

and

ψm(0+, It ) = 1 − α

αlim

mt→0+

[(β + γ + γβ)β It(1 + β)(1 + αβ)

− α

1 − mt

]e− (β+γ+γβ)βmt It

(1+β)(1+αβ) (1 − mt )−α

= 1α

1−α(1 + αβ)

[(β + γ + γβ)β It

1 + β− α(1 + αβ)

]

Due to the convexity of (mt ), the concavity of ψ(mt , It ), and (32), P t+1 = � if, andonly if,

′(0+) = 1 + αβ

β<

1α

1−α(1 + αβ)


1 + β− α(1 + αβ)

]= ψm(0+, It )

⇐⇒ It >α(1 + αβ)(1 + β)2

(1 − α)(β + γ + γβ)β2 = I (33)


Since both�V t+1t and�V t+1

t+1 are well-defined and continuous over the setPt+1

, the product

�V t+1t �V t+1

t+1 is continuous over P t+1. So the existence of (m∗

t , τo∗t+1) is guaranteed by the

compactness of P t+1which has been proven in Lemma2 in appendix section “Definition of

the Pareto Set and Proof of Compactness”. It is trivial to rule out the case where at least oneof the constraints �V t+1

t ≥ 0 and �V t+1t+1 ≥ 0 is binding: In this case the objective function

�V t+1t �V t+1

t+1 = 0 while any other point (mt , τot+1) ∈ P t+1 \ (S t+1

t ∪ S t+1t+1) in the interior

of the Pareto improvement set yields �V t+1t �V t+1

t+1 > 0. So at the maximum none of thesetwo constraints is binding. Hence, the Lagrangian for the optimization is

L (mt , τot+1, μ

) = �V t+1t �V t+1

t+1 − μ(mt − mt )

where μ ≥ 0 is the Lagrangian multiplier for the constraint mt − mt ≤ 0. The first-orderKuhn-Tucker conditions at the optimal point (m∗

t , τo∗t+1) ∈ P t+1 are

Lm(m∗

t , τo∗t+1, μ

) = μ− 1+αβ

1−m∗t�V t+1,∗

t+1 +�V t+1,∗t

[β + γ + γβ

1 + ββ It− α (1+αβ)

1 − m∗t

]= 0

(34)

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N. T. Dao et al.

Lτ

(m∗

t , τo∗t+1, μ

) = β

1 + τ o∗t+1�V t+1,∗

t+1 − �V t+1,∗t

α (1 + αβ)

1 − α(1 + τ o∗t+1

) = 0

μ(m∗

t − mt) = 0 (35)

(i) If m∗t = mt = (1−δ)(1+β)Et/It+β

β+γ+γβ, then from (35)

Q(τ o∗t+1

) ≡ β

1 + τ o∗t+1

{

(1 + αβ) ln

[(1 − α

(1 + τ o∗t+1

)

1 − α

)

(1 − mt )α

]

+ β + γ + γβ

1 + ββmt It

}

− {(1 + αβ) ln (1 − mt ) + β ln

(1 + τ o∗t+1

)} α (1 + αβ)

1 − α(1 + τ o∗t+1

) = 0 (36)

where Q(τ o∗t+1) is decreasing in τ o∗t+1, and

Q (0) = β2 β + γ + γβ

1 + βmt It − α (1 + αβ)2

1 − αln (1 − mt ) > 0 and

limτ o∗t+1→

(1−αα

)−Q(τ o∗t+1

) = −∞

Hence, there always exists a unique τ o∗t+1 ∈ (0, 1−αα

) solving (36), implying that there alwaysexits a unique (mt , τ

ot+1) = (mt , τ

o∗t+1) solving the Nash bargaining problem in case the

constraint mt ≤ mt is binding.(ii) If m∗

t < mt , then μ = 0 and from (34) and (35), we find that

1 + τ o∗t+1

1 − m∗t

1 + αβ

β=[β

β + γ + γβ

1 + βIt − α (1 + αβ)

1 − m∗t

]1 − α

(1 + τ o∗t+1

)

α (1 + αβ)

�⇒ 1 + τ o∗t+1 = β2 β+γ+γβ1+β

It(1 − m∗

t

)− αβ (1 + αβ)

α(β2 β+γ+γβ

1+βIt(1 − m∗

t)+ 1 + αβ

) (37)

It can be shown that Eq. (37) denotes the Pareto frontier. Since τ o∗t+1 > 0, it follows from (37)that

1 − m∗t >

α(1 + αβ)(1 + β)2

(1 − α)(β + γ + γβ)β2 Iti.e. m∗

t < 1 − α(1 + αβ)(1 + β)2

(1 − α)(β + γ + γβ)β2 It= mt

(38)

Substituting (37) into (34), we find m∗t as the solution to

Q(mt ; It ) = ln

[

(1 − mt )1+αβ

(βbIt (1 − mt ) − αβ(1 + αβ)

α (βbIt (1 − mt ) + 1 + αβ)

)β]bIt− α(1+αβ)

1−m

−1 + αβ

1 − mt

{

ln

[(1 − α2β2)(1 − mt )

α

(1 − α) (βbIt (1 − mt ) + 1 + αβ)

]1+αβ

+ bmt It

}

= 0

(39)

where b = β(β+γ+γβ)1+β

.The existence of a solution to (39) is proven by the existence of a solution (m∗

t , τo∗t+1) to

the Nash bargaining problem. The uniqueness of m∗t is guaranteed by the monotonicity of

the function Q(m∗t ; It ). We now prove that for all It > I , Q(m∗

t ; It ) is decreasing in m∗t .

123


From (39), it is sufficient to prove that (1−mt )α

βbIt (1−mt )+1+αβis increasing in mt for all It > I and

mt < mt . In effect, its derivative with respect to mt is

(1 − α)(1 − mt )βbIt − α(1 + αβ)

(1 − mt )1−α [βbIt (1 − mt ) + 1 + αβ]2> 0 ∀mt < mt , ∀It > I

Hence, there exists a unique m∗t = m(It ) ∈ (0, mt ) that solves (39), i.e. there exists a unique

interior pair (m∗t , τ

o∗t+1) ≡ (m(It ), τ (It )) ∈ P t+1 that solves the Nash bargaining problem.

Since (m∗t , τ

o∗t+1) is the unique pair solving the first order conditions (34) and (35), it is the

unique stationary point of the function �V t+1t �V t+1

t+1 defined over the set P t+1.

Steady State with Zero Pollution

In this section, we study the dynamics of the model assuming that parameters satisfy thefollowing sufficient condition for the mitigation share to be increasing in It for all It ∈ ( I , I ):

Assumption 2 β+γ+γβα(1+β)

[β(1−α)1+β

] 2−α1−α

< 1.

Under Assumption2, we can prove the existence of an area in the North-East quadrant of theI − E space such that under period-by-period social contracts an economy starting from anypoint in this area may converge to a steady state with zero pollution stock E∗ = 0, and highnet income I∗. We call this area the “ideal area”. Under Assumption2, for sufficiently highincome, we may find

m∗t = (1 − δ)(1 + β)Et + β It

(β + γ + γβ)It≡ mt (40)

where

m∗t ∈ argmax

mt

{�V t+1

t �V t+1t+1

}(41)

subject to�V t+1t ≥ 0 and�V t+1

t+1 ≥ 0.Note that in this optimization, we ignore the constraintmt ≤ mt in order to find the condition under which this constraint is just binding. And notealso that we substitute τ ot+1 as a function of mt from (37) in the appendix section ”Proof ofProposition2”.

Similar to the proof of Proposition2, the optimization problem (41) has a unique interiorsolution and it holds that22

Q(m∗t ; It ) = ln

⎡

⎣(1 − m∗t )

1+αβ

(βbIt (1 − m∗

t ) − αβ(1 + αβ)

α(βbIt (1 − m∗

t ) + 1 + αβ)

)β⎤

⎦

bIt− α(1+αβ)

1−m∗t

− 1 + αβ

1 − m∗t

⎧⎨

⎩ln

[(1 − α2β2)(1 − m∗

t )α

(1 − α)(βbIt (1 − m∗

t ) + 1 + αβ)

]1+αβ

+ bm∗t It

⎫⎬

⎭= 0

(42)

with b = β(β+γ+γβ)1+β

. We find that Qm(m∗t ; It ) < 0 and, under Assumption2, Q I (m∗

t ; It ) >

0 for all It ∈ ( I , I ), where I = [A(1 − α)]1

1−α

(β

1+β

) α1−α

is conditional steady state income

22 The Eq. (42) is indeed the FOC of the optimization problem (41).

123

N. T. Dao et al.

in the case of no social contract with the pollution stock set at E = 0. By applying the implicitfunction theorem we have

∀It ∈ ( I , I ), m∗t = m(It ) and m′(It ) > 0

We can now prove the following lemma:

Lemma 3 Under Assumption2, if m( I ) >β

β+γ+γβ, there exist I ∈ ( I , I ) and Et = E(It )

for It ∈ (I , I ) such that m∗t = mt (or equivalently Et+1 = 0) if, and only if, Et ≤ E(It ).

Moreover, E(I ) = 0, and E ′(It ) > 0 for It ∈ [I , I ].Proof Since m( I ) = 0,m( I ) >

ββ+γ+γβ

and m′(It ) > 0 for all It ∈ ( I , I ) , there exists a

unique I ∈ ( I , I ) such that m(I ) = ββ+γ+γβ

. From (40), it follows that

m(It ) − (1 − δ)(1 + β)Et + β It(β + γ + γβ)It

= 0 (43)

For all It ∈ [I , I ), there exists a unique Et solving (43), and

Et = (β + γ + γβ)m(It ) − β

(1 − δ)(1 + β)It ≡ E(It )

where E(I ) = 0 and E ′(It ) > 0 for all It ∈ [I , I ]. � The proof of existence of this steady state is fairly straightforward because starting from anypoint (It , Et ) in the ideal area leads to a social contract with mt = (1−δ)(1+β)Et+β It

(β+γ+γβ)It. Hence,

from t + 1 onward E = 0 and m = ββ+γ+γβ

. The existence and uniqueness of the transfer

τ o∗t+1 are proved in Proposition2. Since It+1 = z(Et )(

β(1−mt )It1+β

)α [1 − α(1 + τ ot+1)

], the

steady state is characterized by

I 1−α∗ − A

(γβ

β + γ + γβ

)α

[1 − α(1 + τ(I∗))] = 0

which always guarantees the existence and uniqueness of the steady state.


From Eqs. (18) and (19), we can derive the steady state of the dynamic system in the caseof no social contracts by setting Et+1 = Et = E and It+1 = It = I and substituting(mt , τ

ot+1) = (0, 0) for all t . The steady state is characterized by the following function

ϕ(E) ≡ E − 1

δ

[β(1 − α)

1 + βAe−E

] 11−α = 0 (44)

with

ϕ′(E) = 1 + 1

(1 − α)δ

[β(1 − α)

1 + βAe−E

] 11−α

> 0

ϕ(0) = −1

δ

[β(1 − α)

1 + β

] 11−α

< 0 and limE→+∞ϕ(E) = +∞

123


Hence, there exists a unique steady state which is characterized by (44). The steady stateincome of the agent in this case is

I = A1

1−α e− E1−α

(β

1 + β

) α1−α

(1 − α)1

1−α

The Jacobian matrix associated with the dynamic system (18) and (19) with (mt , τot+1) =

(0, 0) for all t and evaluated at the steady state ( I , E) is

J =⎛

⎜⎝

α − 1+ββ

δ E

β1+β

1 − δ

⎞

⎟⎠

Its determinant and trace are

det( J ) = α(1 − δ) + δ E > 0; Tr( J ) = α + 1 − δ > 0

and the characteristic function is

C(λ) = λ2 − Tr( J )λ + det( J )

(i) If Tr( J )2 > 4 det( J ), we have

C(−1) = 1 + Tr( J ) + det( J ) > C(1) = δ(1 − α) + δ E > 0

So, we have two distinct eigenvalues λ1, λ2 ∈ (0, 1). The steady state ( I , E) is a stable node.(ii) If Tr( J )2 = 4 det( J ), we have a pair of repeated real eigenvalues λ = α+1−δ

2 ∈ (0, 1).The steady state ( I , E) is stable.

(iii) If Tr( J )2 < 4 det( J ), we have two complex eigenvalues. It is obvious that

ϕ

(1 − α(1 − δ)

δ

)> 0 �⇒ E <

1 − α(1 − δ)

δ

Hence

det( J ) = α(1 − δ) + δ E < 1

Therefore in this case, the steady state ( I , E) is a spiral sink.

Isoclines and Directions of Motion

To study the dynamics with social contracts, we substitute the Nash bargaining solution(mt , τ

ot+1) = (m∗

t , τo∗t+1) = (m(It ), τ (It )) into the dynamic system (18)–(19) in every

period t . Note that whenever net income falls short of the threshold (16), i.e. It ≤ I , weset (m∗

t , τo∗t+1) = (0, 0). From the dynamic system, we then define the sets I I and EE where

income and the pollution stock, respectively, do not change over time, as follows:

I I ≡ {(It , Et ) ∈ �2+ : It+1 = It

}

i.e. Et = ln A + α ln

[β(1 − m(It ))

1 + β

]+ ln [1 − α(1 + τ(It ))] − (1 − α) ln It ≡ �(It )

(45)

123

N. T. Dao et al.

and

EE ≡ {(It , Et ) ∈ �2+ : Et+1 = Et

}

i.e. Et = β − (β + γ + γβ)m(It )

δ(1 + β)It ≡ �(It ) (46)

Without social contracts,�(It ) ismonotonously decreasing and�(It )monotonously increas-ing in income It . We can easily see this by substituting (0, 0) for (m(It ), τ (It )) into (45) and(46). In the case of social contracts, withm(It ), τ (It ) > 0, the functional relation between thepollution stock and income along the curves depends also on the derivatives m′(It ), τ ′(It ).The curves in Figs. 3 and 4 are drawn under Assumption2 that the mitigation share m(It )increases in income for all It ∈ ( I , I ). For any m(It ), τ (It ) > 0, both the I I -curve andthe EE-curve lie below their respective counterparts without contracts for all It > I . Thedynamics on and offside the I I and EE loci are described by the following lemma:

Lemma 4 For the dynamic system (It , Et )t characterized by Eqs. (18)–(19), it holds that:

(i) It+1− It

⎧⎪⎨

⎪⎩

> 0 if 0 < Et < �(It )

= 0 if Et = �(It )

< 0 if Et > �(It )

and (ii) Et+1 − Et

⎧⎪⎨

⎪⎩

> 0 if 0 < Et < �(It )

= 0 if Et = �(It )

< 0 if Et > �(It )

Proof The proof for this lemma is fairly straightforward. � Proof of Proposition4

(i) Existence We prove existence for the case shown in Fig. 3, where m(I ) <β

β+γ+γβ. For

the casem(I ) >β

β+γ+γβ, existence of a steady state (I∗, 0) is proved in the appendix section

“Steady Statewith Zero Pollution”. Ifm(I ) <β

β+γ+γβ, it is straightforward that�(I ) > 0 >

�(I ): The EE-curve lies above the I I -curve for large incomes. Now define the EE-curveand I I -curve without social contracts as EE(0, 0) = �NC (It ) and I I (0, 0) = �NC (It ).At the income threshold I ,m( I ) = 0, so that �( I ) = β

δ(1+β)I = �NC ( I ) <

βδ(1+β)

I =�NC ( I ) =�NC ( I ) < �NC ( I ) = �( I ). The equality �NC ( I ) =�NC ( I ) uses that the EE-and I I -loci intersect in the steady state without contracts. The inequality �NC ( I ) < �NC ( I )holds because �′(It ) < 0. Finally, �NC ( I ) = �( I ) because m( I ) = 0. From the continuityof the EE-curve and I I -curve, it follows that they intersect at least once for It ∈ ( I , I ).

Steady state pollution stock, income and welfare From the dynamic system (18)–(19), asteady state with social contracts is characterized by

I 1−α = Ae−E[

β(1 − m)

1 + β

]α [1 − α(1 + τ o)

]> I 1−α and

E = β − (β + γ + γβ)m

δ(1 + β)I ≥ 0

where (m, τ o) ∈ P = � is the social contract at steady state. Since I > 0, the two inequalitysigns imply that τ o < 1−α

αand m ≤ β

β+γ+γβ. The stock of pollution at the steady state with

social contracts is characterized by

eE

1−α E = A1

1−αβ − (β + γ + γβ)m

δ(1 + β)

[β(1 − m)

1 + β

] α1−α [

1 − α(1 + τ o)] 11−α (47)

123


It is follows from (47) that E is decreasing in both m and τ o so that

E = E(m, τ o) < E(0, 0) = E

As a steady state (I, E) with social contracts may fall in the area It ∈ ( I , I ), steady stateincome I and therefore welfare may be lower with than without social contracts.

Non-existence of the no-contract steady state ( I , E) The result follows straightforwardlybecause � decreases in both m and τ o, so that �(It ) < �NC (It ) for all It ∈ ( I , I ).

(ii) Steady state pollution stock, income and welfare The proof that E < E from (i) stillapplies. Now as I > I and I > I , it follows straightforwardly that I > I . From (25), we canderive lifetime indirect utility of a generation at a steady state (I, E) under social contracts.It is given by

V (I, E,m, τ o) = �′ + ln z(E) + (1 + αβ) ln I + (1 + αβ) ln(1 − m) + β ln(1 + τ o)

where �′ = � + β ln α + β (α − 1) ln β1+β

. Similarly, lifetime utility at the steady state

( I , E) without social contracts is given by

V ( I , E, 0, 0) = �′ + ln z(E) + (1 + αβ) ln I

Now as E < E and I > I , ln z(E) > ln z(E) and (1 + αβ) ln I > (1 + αβ) ln I . It followsthat V (I, E,m, τ o) > V ( I , E, 0, 0) if (1 + αβ) ln(1 − m) + β ln(1 + τ o) ≥ 0, which isguaranteed as it is the condition �V t+1

t ≥ 0 at steady state.Existence of the no-contract steady state ( I , E) Proof follows directly from �(It ) =

�NC (It ) and �(It ) = �NC (It ) for I∈(0, I ).

Incentive Constraints

Denote expected indirect utility from compliance with the strategy as VC (mt ,

τyt | p(ht ), τ o,et+1 = τ ot+1), where τ

yt ∈ [0, τ y

t]depending on the state p(ht ) of the game.

Expected indirect utility from deviation is V D(0, 0 | p(ht ), τ o,et+1 = 0).

Incentive constraint when the game is in compliance phaseWhen the game is in a compliance phase (p(ht ) = C) and generation t follows strategy

s and complies with the contracts with generations t − 1 and t + 1 it is involved in, indirectutility is:

VC (mt , τyt | p(ht

) = C, τo,et+1 = τ ot+1) = � + β ln

[

z(Et )α

(β I Ct (1 − mt )

1 + β

)α−1]

+ (1 + β)[ln I Ct + ln(1 − mt )

]+ β ln

(1 + τ ot+1

)

where � = ln 11+β

+ β ln β1+β

is a constant and I Ct = z(Et−1)(1 − α)kαt (1 − τ

yt ) is net

income of agent t in period t after paying z(Et−1)(1−α)kαt τ

yt = τ ot Rt kt to the old generation

t − 1 in t . When complying with the contracts, generation t expects to receive the perfectlyforeseen transfer τ ot+1 from generation t + 1. Recall that the pollution stocks Et−1 and Et aswell as the capital stock kt are given in period t .23

23 The stock variables E and k and therefore also income I and the interest rate of course depend on thecomplete history of the game. We will see below, however, that they do not affect generation t’s decision tocomply. Formeans of simplification, we therefore neglect the history dependence of E ,k and I in the following.

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N. T. Dao et al.

If generation t deviates, indirect utility is

V D (0, 0 | p(ht ) = C, τo,et+1 = 0

) = � + β ln

[

z(Et )α

(β I Dt1 + β

)α−1]

+ (1 + β) ln I Dt

where I Dt = z(Et−1)(1 − α)kαt is net income if generation t does not pay the transfer

(τ yt = 0).There is no incentive to deviate if and only if the difference�V (mt , τ

yt | p(ht ) = C, τ ot+1)

in indirect utilities is non-negative. Using the relation z(Et−1)(1−α)kαt τ

yt = τ ot Rt kt between

the transfer received by the old generation in t and the transfer paid by the young generationin t , the condition �V (mt , τ

yt | p(ht ) = C, τ ot+1) ≥ 0 can be shown to be (20).

Incentive constraint when the game is in punishment phaseIndirect utility from following strategy s when the game is in punishment phase (p(ht ) = P)is

VC (mt , 0 | p(ht ) = P, τo,et+1 = τ ot+1) = � + β ln

[

z(Et )α

(β I Ct (1 − mt )

1 + β

)α−1]

+ (1 + β)[ln I Ct + ln(1 − mt )

]+ β ln(1 + τ ot+1)

where I Ct = z(Et−1)(1 − α)kαt is net income if the agent born in t chooses compliance and

punishes generation t − 1.Indirect utility from deviation is

V D(0, 0 | p(ht ) = P, τo,et+1 = 0) = � + β ln

[

z(Et )α

(β I Dt1 + β

)α−1]

+ (1 + β) ln I Dt

where I Dt = z(Et−1)(1 − α)kαt = I Ct . Note that net income is the same whether generation

t follows its strategy or deviates because in a punishment phase, generation t will not paythe transfer in any case.

There is no incentive for deviation if and only if�V (mt , 0 | p(ht ) = Pt , τ ot+1) ≥ 0 whichyields (21).


The set of stationary self-enforcing contracts is given by:

SIC =⎧⎨

⎩(m, τ o) ∈ [0, 1) × [0, τ o,max) : m ≤ 1 − 1

(1 − α1−α

τ o) (1 + τ o)β

1+αβ

⎫⎬

⎭(48)

We define as τ o,max ≡ 1−αα

the maximum transfer reconcilable with non-negative net laborincome (i.e. τ y

t ≤ 1) of agent t . In the text, we motivated the following lemma:

Lemma 5 A self-enforcing contract scheme (mt , τot+1)

∞t=T must converge to or fluctuate

without trend around a constant (m, τ o). The set of stationary self-enforcing pairs (m, τ o)

is characterized by (48).

Proof The proof is contained in the text. �

123


It can be concluded fromLemma5 that a self-enforcing contract schemewithmt > 0, τ ot+1 >

0 for every t = T, . . . ,∞ exists if and only if the stationary set SIC is non-empty, SIC = �,with SIC given by (48).

As can be seen from (22), the function mIC (τ o) delineating the boundary of SIC isconcave, with mIC (0) = 0. SIC = � if and only if the slope of the IC-curve at the

origin is positive. Formally, this is expressed by the condition ∂mIC

∂τ o(0) = β− α

1−α (1+αβ)

1+αβ> 0

which yields condition (23). If condition (23) does not hold, only negative values of τ o

satisfy the incentive constraint. To see that condition (23) is also sufficient, note that the IC-curve is continuous in τ o for τ o∈ [0, τ o,max). A sustainable equilibrium path with positivetransfer payments and positivemitigation investment in each period will therefore exist undercondition (23) for sufficiently small mitigation levels mt .

Under condition (23), mIC = 0 not only at the origin but also for some positive τ o <

τ o,max and limτ o→τ o,max

mIC (τ o) = −∞ . Hence, a positive mitigation share m > 0 can only

be sustained for 0 < τ o < τ o. Further, there exists a maximum sustainable mitigation sharem, as the function mIC (τ o) is increasing for small but decreasing for large values of τ o. The

maximum sustainable m is derived by setting ∂mIC

∂τ o= 0. Solving for τ o and substituting the

solution back into (22), we obtain:

mmax = 1 −(

α

1 − α

1 + αβ

β

) 1+αββ(

(1 − α)β + 1 + αβ

1 + αβ

)1+ 1+αββ

< 1 (49)

mmax is strictly smaller than one.

Heuristic Derivation of Condition (23)

Condition (23) can be derived by explicitly comparing returns from pure capital investmentand capital investment combined with the transfer scheme: Assume that there is no socialcontract, i.e. mt = τ ot = τ ot+1 = 0. Marginally increasing the transfer τ ot to the currentold generation yields a perfectly foreseen return of Re

t+1kt+1dτo,et+1 = Rt+1kt+1dτ ot+1 in

period t + 1. Further, an agent of generation t takes into account the equilibrium effect ofslower capital accumulation due to lower income on the perfectly foreseen return to capital,∂Rt+1∂kt+1

∂kt+1∂τ ot

dτ ot > 0, which is associatedwith a total change in return of ∂Rt+1∂kt+1

∂kt+1∂τ ot

dτ ot kt+1 =(1 − α)

β1+β

Rt+1Rtktdτ ot . On the other hand, increasing the transfer τ ot reduces income inperiod t by Rtktdτ ot . Investing this income in physical capital insteadwould yield an expectedreturn of Rt+1Rtktdτ ot (as τ

o,et+1 = 0without social contract). The agentwill prefer to combine

capital investment with investing in the transfer system and the social contract over investingonly in capital if and only if

Rt+1kt+1dτ ot+1 + (1 − α)β

1 + βRt+1Rtktdτ ot − Rt+1Rtktdτ ot > 0

Assuming a stationary transfer system, so that dτ ot = dτ ot+1, this condition becomes

Rt+1kt+1 + (1 − α)β

1 + βRt+1Rtkt − Rt+1Rtkt > 0.

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N. T. Dao et al.

After some simplification, taking into account τyt = 0, so that Rt = αz(Et−1)k

α−1t and

kt+1 = β1+β

z(Et−1)(1 − α)kαt , the condition is equivalent to

1 − α

(1 − α)β

1+β

[1 − (1 − α)

β

1 + β

]> 0 ⇐⇒ β >

α

1 − α(1 + αβ)

which is condition (23).


Fulfillment of condition (24) in period T is necessary and sufficient for the set SIC ∩ PT+1

to be non-empty in the starting period of the contract scheme. If the condition holds also forevery t > T , then there exists some pair (m, τ o), m > 0, τ o > 0 in the set SIC ∩ PT+1

which will also lie in every set SIC ∩ P t+1 for t > T . Such a pair can be maintained asincentive compatible, Pareto improving contract for t → ∞.

Condition (24) is derived as follows: The boundary of the set SIC is the stationary incentiveconstraint mIC (τ o) in (22), while the boundaries of the Pareto improvement set P t+1 aregiven by (mt ) and ψ(mt ; It ) in (31) and (29). Because the stationary incentive constraintis defined as mIC (τ o) while the boundaries of the Pareto improvement set are defined asfunctions of m, we first invert the function mIC (τ o) over the interval τ oε

[0, τ o(mmax)

],

where mmax is defined in (49). Denote the inverse by τ o,IC (m).In (mt , τ

ot+1)-space, τ

o,IC (m) lies above the lower boundary(mt ) of the Pareto improve-ment area for mt , τ

ot+1 > 0,∀t . τ o,IC (m) and the upper boundary ψ(mt ; It ) of the Pareto

improvement area intersect at the origin (mt , τot+1) = (0, 0) for all t . Given the functional

forms of the two curves, they enclose a non-empty set of pairs (mt , τot+1) ,mt , τ

ot+1 > 0 (so

that SIC ∩P t+1 = �) in a period t if and only if at the origin, the slope of τ o,IC (m) is flatterthan the slope of ψ(mt ; It ), i.e. τ o,ICm (0) < ψm(0+, I gt ).

From (22), it follows that the derivative τo,ICm (0) is:

τ o,ICm (0) = 1 + αβ

β − α1−α

(1 + αβ)

The numerator gives the absolute value of the marginal utility loss from an increase in themitigation share m from zero. The denominator contains the marginal utility gain from anincrease in τ o net of the marginal utility loss incurred by an increase in the transfer paid, τ y .Under condition (23), the denominator and therefore τ

o,ICm (0) is strictly positive.

The slope of the upper boundary ψ(mt ; It ) at the origin was shown to be

ψm(0+, It ) = 1α

1−α(1 + αβ)


1 + β− α(1 + αβ)

]

which, setting τ ot = 0, becomes ψm(0+, I gt ) = 1α

1−α(1+αβ)

[(β+γ+γβ)β I gt

1+β− α(1 + αβ)

].

The relation

τ o,ICm (0) < ψm(0+, I gt

)

123


is satisfied if and only if

1 + αβ

β − α1−α

(1 + αβ)<

1α

1−α(1 + αβ)

[(β + γ + γβ)β I gt

1 + β− α(1 + αβ)

]

⇐⇒ I gt >(1 − α)

β1+β

(1 + αβ + β)

β − α1−α

(1 + αβ)

α (1 + αβ) (1 + β)2

(1 − α) (β + γ + γβ) β2︸︷︷︸

I

> I

(50)

which is condition (24). The new threshold is larger than I .

Simulation

We illustrate numerically that through Pareto improving intergenerational social contracts,an economy may have a chance to converge to a better steady state with lower stationarypollution stock and higher stationary income compared to the case without social contracts.We set the share of capital, α, to 0.3. For the rate of time preference, β, we choose thevalue 0.7 which yields a plausible savings rate of households around 40%. Without loss ofgenerality we set the effectiveness of mitigation to γ = 1. Note that these parameter valuesguarantee that the condition in Assumption2 holds. With the above parameter values we

compute I = α(1+αβ)(1+β)2

(1−α)(β+γ+γβ)β2 � 1.2744. We run the simulation with different levels of

technology A corresponding to the two distinct cases I < I and I > I in the followingsubsections (Figs. 6, 7, 8).

Robustness

Although functional forms used in this paper are standard in the literature, they are specific—in particular when they are combined. We adopt these standard functional forms in order tokeep our model tractable. They allow us to characterize conditions for the existence ofPareto improving, self-enforcing social contracts. In this section, we discuss the impli-cations of changes in functional forms for the existence of such social contracts. Wefirst consider the following changes: (i) a rate of capital depreciation λ ∈ (0, 1); (ii)a production function of the constant elasticity of substitution (CES) form with capi-tal intensive externality on labor productivity; and (iii) a utility function of the ConstantRelative Risk Aversion (CRRA) form combined with a CES production function asin (ii).

We can derive sufficient conditions on income for the existence of Pareto improvingand self-enforcing social contracts which are in general similar to the benchmark model.Differences arising from changes in functional forms will be pointed out in this section.

Capital Depreciation Rate λ ∈ (0, 1)

We define the aggregate per capita production function as follows

F(kt , Et−1) = z(Et−1)[kαt + (1 − λ)kt

] ; kt = Kt/Lt

123

N. T. Dao et al.

Fig. 6 Dynamics in the case I < I . a Income dynamics with and without social contracts, b Pollution stockdynamics with and without social contracts, cMitigation and transfer

123


Fig. 7 Dynamics in the case I > I with early signing of contracts. a Income dynamics with and withoutsocial contracts, b Pollution stock dynamics with and without social contracts, c Mitigation and transfer

123

N. T. Dao et al.

Fig. 8 Dynamics in the case I > I with delayed signing of contracts. a Income dynamics with and withoutsocial contracts, b Pollution stock dynamics with and without social contracts, c Mitigation and transfer

123


The disposable income and gross return to capital that an agent t receives in period t andt + 1, respectively, under the social contracts (mt−1, τ

ot ) and (mt , τ

ot+1) are

It (1 − mt ) = (1 − α) z (Et−1) kαt

(1 − ατ ot

1 − α

)(1 − mt )

[Rt+1(1 + τ ot+1

)+ z (Et ) (1 − λ)]kt+1 = z (Et )[αkα

t+1

(1 + τ ot+1

)+ (1 − λ) kt+1]

The welfare surplus under the social contract (mt , τot+1) of agent t is

�V t+1t = (1 + αβ) ln(1 − mt ) + β ln

α(1 + τ ot+1) + (1 − λ)[

β1+β

It (1 − mt )]1−α

α + (1 − λ)[

β1+β

It]1−α

Setting �V t+1t = 0 allows us to determine the indifference curve of agent t

τ ot+1 =1 + 1−λ

α

(β It1+β

)1−α

(1 − mt )α+ 1

β

− 1 − λ

α

(β It1 + β

(1 − mt )

)1−α

− 1 ≡ (mt , It )

where m(mt , It ) > 0, mm(mt , It ) < 0, (0, It ) = 0 and

m(0+, It ) = 1 + αβ

β+ 1 − λ

α

(1 + β

β

)α

I 1−αt

Different from the indifference curve (mt ) of agent t in the benchmark model, thecorresponding curve (mt , It ) now, due to the appearance of vintage capital, depends on netincome It , and so does its slope at the origin m(0+, It ).

Similarly, we obtain the welfare surplus of agent t+1 under the social contract (mt , τot+1)

�V t+1t+1 = (1 + αβ) ln

It+1

It+1+ β ln

z (Et+1)

z(Et+1

)

+β lnα(1 + τ

o,et+2

)+ (1 − λ)

[β It+1

(1−me

t+1

)

1+β

]1−α

α(1 + τ

o,et+2

)+ (1 − λ)

[β It+1

(1−me

t+1

)

1+β

]1−α

where It+1 and Et+1, as in the benchmark model, are respectively net income and the stockof pollution in the case of no social contract, i.e. (mt , τ

ot+1) = (0, 0). We can prove that:

�V t+1t+1 ∈

((1 + β) ln

It+1

It+1+ β ln

z(Et+1)

z(Et+1), (1 + αβ) ln

It+1

It+1+ β ln

z(Et+1)

z(Et+1)

)

This allows us to determine the lower bound and upper bound for the indifference curve ofagent t + 1 respectively:

τ ot+1 = 1 − α

α

[1 − e

− β(β+γ+γβ)

(1+β)2mt It

(1 − mt )−α

]≡ ψ(mt , It )

τ ot+1 = 1 − α

α


(1+β)(1+αβ) (1 − mt )−α

]≡ ψ(mt , It )

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N. T. Dao et al.

The slopes at the origin are respectively:

ψm(0+, It ) = 1 − α

α

[β(β + γ + γβ)

(1 + β)2It − α

](51)

ψm(0+, It ) = 1 − α

α

[β(β + γ + γβ)

(1 + β)(1 + αβ)It − α

](52)

It is fairly straightforward to show that ψ(mt , It ) and ψ(mt , It ) are strictly concave

and it is trivial that ψ(0, It ) = ψ(0, It ) = 0. By comparing the slopes ψm(0+, It ) and

m(0+, It ) at the origin, we can derive a sufficient condition on net income, I ∗ > 0, abovewhich P t+1 = �. Similarly, by comparing ψm(0+, It ) and m(0+, It ) we also can derive asufficient condition on net income, I ∗∗ ∈ (0, I ∗), below which P t+1 = �.

Contrary to the benchmark model which allows us to derive necessary and sufficientconditions on net income for the existence of the set of Pareto improving social contracts,we can now only derive sufficient conditions on net income which guarantee the existenceof this set. Nevertheless, the qualitative results for the existence of this set in this extensiondo not change crucially compared to those from the benchmark model.

Pareto improving and self-enforcing social contractsCrucial for the existence of a scheme of self-enforcing contracts is the incentive constraint

for generation t in compliance phase, given by the following condition

�VC(mt , τ

ot+1; I Dt

)= (1 + αβ) ln

I Ct (1 − mt )

I Dt

+β lnα(1 + τ ot+1

)+ (1 − λ)[

β ICt (1−mt )

1+β

]1−α

α + (1 − λ)[

β I Dt1+β

]1−α≥ 0 (53)

with I Ct = (1− α1−α

τ ot )I Dt , where I Ct and I Dt = I gt , respectively, denote the income of agentt in the case of complying with and deviating from strategy s when the game is in compliancephase.24

Unlike in the benchmark model, the incentive constraint now depends on gross incomeI Dt = I gt of agent t . The gain from a given self-enforcing social contract (mt , τ

ot+1), when

it exists, decreases in gross income I gt . From equation (53) we can find that, for any given(mt , τ

ot+1) ∈ (0, 1) × (

0, 1−αα

)and λ ∈ (0, 1),�VC (mt , τ

ot+1; I gt ) < 0 whenever gross

income I gt is sufficiently high. In this case self-enforcing social contracts cannot exist. Thatis because the appearance of vintage capital increases the agent’s old-age consumption morein the case of defaulting on the contract than in the case of complying. The remaining amountof capital, indeed, depends linearly and positively on gross income. So higher gross incomeI gt makes the existence of self-enforcing social contracts less likely.

We can derive a condition on gross income I g for the existence of a non-empty set SIC

of stationary self-enforcing contracts by requiring the slope of the boundary of this set at theorigin to be strictly positive. Applying the implicit function theorem, we find that:

s IC|(0,0) = −�VCm (0, 0; I g)

�VCτ (0, 0; I g) =

(1 − α)

[1 + αβ + 1+β

α(1 − λ)

(β

1+βI g)1−α

]

(1 − α)β − α(1 + αβ) − (1 + β)(1 − λ)(

β1+β

I g)1−α

> 0

24 The strategy s is defined in Sect. 6.1.

123


For the condition to be satisfied, the denominator must be strictly positive, hence it must holdthat

β >α

1 − α(1 + αβ) + (1 + β)(1 − λ)

1 − α

(β

1 + βI g)1−α

(54)

Comparing condition (54) with condition (23) from the benchmarkmodel, we can identifythe effect of slower capital depreciation. Given that β > α

1−α(1 + αβ), we can derive a

condition on gross income guaranteeing the existence of the set of stationary self-enforcingcontracts SIC :

I g <

[β(1 − α) − α(1 + αβ)

(1 + β + αβ)(1 − λ)

] 11−α 1 + β

β(55)

The right hand side increases unboundedly when λ increases and approaches 1. Thereforethe higher the depreciation rate of capital λ, the more likely it is that SIC ∩ P = �. That isbecause the higher λ decreases the slope of the stationary incentive constraint at the origin andrelaxes the condition on income (55) for the existence of the set of stationary self-enforcingcontracts SIC . So it can be predicted that when λ is sufficiently high and gross income I gt issufficiently high as well, so that ψ

m(0+, I g) > s IC|(0,0) then SIC ∩ P = �.

However, one could argue that in a persistently growing economy, when gross incomeI g exceeds some high threshold, condition (55) cannot be satisfied any longer and thus self-enforcing social contract schemesmay not exist. Notwithstanding, persistent growth is drivenby technological progress which also enhances the longevity of agents through its positiveeffects on medical technology and nutrition, for example. It thus lengthens the working-ageand old-age periods. The longer working-age period means capital is used longer, making thedepreciation rate of capital over the whole lengthened period higher as well. So as is obviousfrom (55), the condition on gross income for the existence of self-enforcing contract schemesis relaxed unboundedly when technological progress enhances longevity. Intergenerationalsocial contracts which are simultaneously Pareto improving and self-enforcing can thus existeven in a growing economy.

Production with Capital Intensive Externality

We argue that the qualitative results of the benchmark model do not change crucially whenwe adopt logarithmic preferences and the following CES production function featuring anexternality of capital on labor productivity:

Yt = z(Et−1)[αK ρ

t + (1 − α)(kt Lt )ρ] 1

ρ ; ρ ∈ (−∞, 1], kt = Kt/Lt (56)

For this production function, the marginal returns to labor and capital become

wt = (1 − α)z(Et−1)kt and Rt = αz(Et−1)

In this case the computations to obtain sufficient conditions on net income that guaranteethe existence of a non-empty intersection between the Pareto improvement set and the set ofstationary self-enforcing contracts are similar to what we have done in the benchmark model.Along with the CES production function, we can also take into account less than full capitaldepreciation at rate λ ∈ (0, 1) in characterizing the sufficient conditions for the existence

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N. T. Dao et al.

of a non-empty intersection between the Pareto improvement set and the set of stationaryself-enforcing contracts. The procedures and mathematical structures are similar to those inthe previous subsection.

Constant Relative Risk Aversion (CRRA) Utility Function

With a CRRA utility function,25

ut =(cyt)1−θ − 1

1 − θ+ β

(cot+1

)1−θ − 1

1 − θ; θ > 0

there are several challenges in deriving the sufficient conditions on income for the existenceof the Pareto improvement set as well as the set of stationary self-enforcing contracts. It istherefore not straightforward to reach a conclusion about the exact difference between thisgeneralized model and the benchmark one. That is because the conditions now depend notonly on income but also on the stock of pollution through complex exponential functions.In order to make useful analyses, we adopt the production function in (56) and assumethat capital fully depreciates after a period of use. With this utility function, the optimalconsumption plans of agent t under a contract (mt , τ

ot+1) are

cyt = Rt+1(1 + τ ot+1

)

Rt+1(1 + τ ot+1

)+ [βRt+1

(1 + τ ot+1

)]1/θ It (1 − mt ) ≡ cy(mt , τ

ot+1, It , Rt+1

)

(57)

cot+1 =[βRt+1

(1 + τ ot+1

)]1/θRt+1

(1 + τ ot+1

)

Rt+1(1 + τ ot+1

)+ [βRt+1

(1 + τ ot+1

)]1/θ It (1 − mt ) ≡ co(mt , τ

ot+1, It , Rt+1

)

(58)

The condition for agent t not to suffer a welfare loss from a social contract (mt , τot+1) is

�V t+1t = cy

(mt , τ

ot+1, It , Rt+1

)1−θ − (cyt)1−θ

1 − θ

+βco(mt , τ

ot+1, It , Rt+1

)1−θ − (cot+1

)1−θ

1 − θ≥ 0 (59)

where cyt = cy(0, 0, It , Rt+1) and cot+1 = co(0, 0, It , Rt+1).The existence of such a contract (mt , τ

ot+1) is guaranteed by the strictly positive slope

at the origin (0, 0) of the indifference curve �V t+1t = 0, st+1

t |(0,0). By applying the implicitfunction theorem, we find:

st+1t |(0,0) = − ∂�V t+1

t /∂mt

∂�V t+1t /∂τ ot+1 |(0,0)

=(1 + β(βRt+1)

1−θθ

) (Rt+1 + (βRt+1)

1θ

)

θ−1θ

(βRt+1)1θ Rt+1 + β(βRt+1)

1−θθ

(1θRt+1 + (βRt+1)

1θ

)

25 Note that when θ = 1, we have a logarithmic utility function as in the benchmark model.

123


It is trivial that st+1t |(0,0) > 0 for all θ > 1. For θ ∈ (0, 1), it is easy to find a condition on Rt+1

making the denominator of the last equation strictly positive to guarantee st+1t |(0,0) > 0.

Similarly, we determine the condition for agent t + 1 not to suffer a welfare loss from thesocial contract (mt , τ

ot+1), given the perfectly foreseen social contract (me

t+1, τo,et+2), as

�V t+1t+1 = cy

(me

t+1, τo,et+2, It+1, Rt+2

)1−θ − cy(met+1, τ

o,et+2, It+1, Rt+2)

1−θ

1 − θ

+βco(me

t+1, τo,et+2, It+1, Rt+2)

1−θ − co(met+1, τ

o,et+2, It+1, Rt+2)

1−θ

1 − θ≥ 0

where It+1 and Rt+2 are respectively net income in period t + 1 and the return to capital inperiod t + 2 in the case of no social contract (mt , τ

ot+1) between generations t and t + 1.

The slope of the indifference curve �V t+1t+1 = 0 at the origin (0, 0) is

st+1t+1|(0,0) = − ∂�V t+1

t+1/∂mt

∂�V t+1t+1/∂τ ot+1 |(0,0)

= 1 − α

α

⎛

⎜⎜⎜⎝

β

[(βRt+1)

1θ

Rt+1+(βRt+1)1θ

+ γ

]It

[β Rt+2

(1 + τ

o,et+2

)] θ−1θ + β

− 1

⎞

⎟⎟⎟⎠

(60)

It is easy to find a condition on income under which st+1t+1|(0,0) > 0. Moreover, comparing

the two slopes st+1t+1|(0,0) and st+1

t |(0,0), we can see that st+1t |(0,0) is independent of net income It

while st+1t+1|(0,0) increases unboundedly in It . That is to say when It is sufficiently high, then

st+1t+1|(0,0) > st+1

t |(0,0)which guarantees the existence of Pareto improving social contracts, i.e. P t+1 = �.

Different from the benchmark model, the threshold of net income now not only dependson the parameters of the model but also on state variables, i.e. the stock of pollution (notethat Rt+1 = αz(Et )), and on the forseen next social contract (me

t+1, τo,et+2). That is because

unlike with logarithmic preferences, the savings decision with CRRA preferences dependson the return to capital. Hence, the variables above appear in the welfare gain of agent t + 1.

Pareto improving and self-enforcing social contractsThe incentive constraint for generation t in compliance phase is

�VC(mt , τ

ot+1; I Dt

)= cy

(mt , τ

ot+1, I

Ct , Rt+1

)1−θ − cy(0, 0, I Dt , Rt+1

)1−θ

1 − θ

+βco(mt , τ

ot+1, I

Ct , Rt+1

)1−θ − co(0, 0, I Dt , Rt+1

)1−θ

1 − θ≥ 0

(61)

with I Ct = (1 − α1−α

τ ot )I Dt , where I Ct and I Dt = I g are respectively the income of agent tin the case of complying with and deviating from strategy s when the game is in compliancephase.

We focus on the set of stationary self-enforcing social contracts as characterized by

�VC (m, τ o, I g) = cy(m, τ o, I C , R)1−θ − cy(0, 0, I g, R)1−θ

1 − θ

+βco(m, τ o, I C , R)1−θ − co(0, 0, I g, R)1−θ

1 − θ≥ 0

123

N. T. Dao et al.

The existence of this set is guaranteed by the condition that the slope of the stationaryincentive constraint �VC (m, τ o, I g) = 0 at the origin be strictly positive, i.e.

s IC|(0,0) = −VCm (m, τ o, I g)

VCτ (m, τ o, I g)

> 0 (62)

where

VCm (m, τ o, I g) = − [

cy(0, 0, I g, R)1−θ + βco(0, 0, I g, R)1−θ]

< 0

VCτ (m, τ o, I g) = cy(0, 0, I g, R)−θ

[cyτ (0, 0, I g, R) − α I g

1 − αcyI (0, 0, I

g, R)

]

+βco(0, 0, I g, R)−θ

[coτ (0, 0, I

g, R) − α I g

1 − αcoI (0, 0, I

g, R)

]

Condition (62) holds if and only if VCτ (m, τ o, I g) > 0, which boils down to

1 − (1 − θ)(βR)1θ − αθ

1 − α

[R + (βR)

1θ

]> 0

We next find a condition under which there exist stationary social contracts that are bothPareto improving and self-enforcing, i.e. under which SIC ∩P = �. We do so by comparingthe slope of the “stationary” incentive constraint, �VC (m, τ o, I g) = 0, at the origin withthe slope defined in (60) at the origin. Using the property that consumption in both periodsof life is linear in income, as is obvious from (57) and (58), we can easily prove that theslope s IC|(0,0) is independent of gross income I g . Further, the slope s IC|(0,0) is bounded, whilethe slope st+1

t+1|(0,0), as defined in (60), increases unboundedly in income I . This suggests that

whenever gross income I g is sufficiently high, then SIC ∩ P = �.

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