Relational Voluntary Environmental Agreements with Unverifiable Emissions

Berardino Cesi; Alessio D’Amato

doi:10.1515/bejeap-2022-0464

Article

Relational Voluntary Environmental Agreements with Unverifiable Emissions

Berardino Cesi and Alessio D’Amato

Published/Copyright: September 29, 2023

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 23 Issue 4

Abstract

Environmental regulation and pollution control may clash against the presence of unverifiable tasks, like source-specific emissions. To tackle this issue, we reshape a voluntary agreement instrument, already available in the existing literature, from a dynamic perspective by means of a relational contracting approach. We define a Relational Voluntary Environmental Agreement (RVEA) in an N firms symmetric context, and show that even if emissions are not contractible across firms, and therefore enforcement cannot be delegated to a third party, if firms are sufficiently patient, a self-enforcing RVEA induces the achievement of the environmental objective. Finally, our welfare analysis reveals a notable result: our RVEA can imply less free riding and be welfare-improving with respect to a Voluntary Environmental Agreement enforced by a third party (along the lines of McEvoy, D. M., and J. K. Stranlund. 2010. “Costly Enforcement of Voluntary Environmental Agreements.” Environmental and Resource Economics 47: 45–63).

Keywords: relational contracts; environmental policy; unverifiability; Voluntary Environmental Agreement

JEL Classification: D62; H23; Q58

Corresponding author: Berardino Cesi, Department of Political Sciences, La Sapienza University of Rome, p.zza Aldo Moro, 5, Rome, Italy, E-mail: berardino.cesi@uniroma1.it

Acknowledgments

The authors would like to thank Andrea Vecchi for Research Assistance, Karine Nyborg and audience at EAERE 2021, IAERE 2021 and SIE 2020 Conferences, as well as seminar participants at Department of Political Sciences, Università di Roma La Sapienza, for insightful comments and suggestions. The usual disclaimer applies.

Conflict of interest: The authors have no conflicts of interest to declare that are relevant to the content of this article.

Appendix A

Proof of Proposition 2

Each member chooses its level of emissions by solving the following maximization problem:

M a x e P π ( e P )

Subject to

(11) S e P + ( N − S ) e N P ≤ N e T

and

(12) 1 1 − δ π ( e P ) ≥ π ( e u ) + δ 1 − δ π ( e T ) − τ e T

The Lagrangian function is

L = π ( e P ) + λ ( N e T − S e P − ( N − S ) e u ) + μ 1 1 − δ π ( e P ) − π ( e u ) − δ 1 − δ π ( e T ) − τ e T

The agreement participants FOC are:

(13) ∂ L ∂ e P = π ′ ( e P ) − S λ + μ 1 1 − δ π ′ ( e P ) ≤ 0

with e ^P ≥ 0 and ∂ L ∂ e P e P = 0

(14) ∂ L ∂ λ = N e T − S e P − ( N − S ) e u ≥ 0

with λ ≥ 0 and ∂ L ∂ λ λ = 0

(15) ∂ L ∂ μ = 1 1 − δ π ( e P ) − π ( e u ) − δ 1 − δ π ( e T ) − τ e T ≥ 0

with μ ≥ 0 and ∂ L ∂ μ μ = 0 .

We ignore the solution such that λ, μ = 0, as in this case we would get e ^P = e ^u = b/b″. We also exclude λ = 0 and μ > 0, as also in this case (13) would imply e ^P = e ^u = b/b″. We focus on cases where λ > 0. Finally, we ignore corner solutions where e ^P = 0.□

From (14) and from (3) we get e P = S b − N τ S b ″ , that implies e ^P > 0, assuming b > N τ S > τ . (13) implies π′(e ^P) − Sλ = 0, that gives e P = b − λ S b ″ , with the corresponding multiplier λ = N τ S 2 ; from (15) we have:

δ ≥ π ( e u ) − π ( e P ) π ( e u ) − π ( e T ) − τ e T = N 2 τ S 2 ( 2 b − τ )

which is the minimum value of δ such that s ^P and s ^R characterize an equilibrium of the repeated game.

Similar conclusions are possible when μ > 0. We have

(16) ∂ L ∂ e P = π ′ ( e P ) + μ 1 1 − δ π ′ ( e P ) − S λ = 0

(17) ∂ L ∂ μ = 1 1 − δ π ( e P ) − π ( e u ) − δ 1 − δ π ( e T ) − τ e T = 0

(18) ∂ L ∂ λ = N e T − S e P − ( N − S ) e u = 0

after substituting for e ^T and e ^u, (17) and (18) imply:

δ = N 2 τ S 2 ( 2 b − τ ) ,

as, clearly, when μ > 0 the IC constraint is binding, and

(19) e P = S b − N τ S b ″

Proof of Proposition 3

Profit of being part of an agreement and delivering e ^P from Proposition 2 increase with S; since static profits in the punishment phase and in the unregulated scenario do not depend on S, we can conclude that the incentive to cooperate increases in S. The lowest possible cooperative profit, for S = S _min, is:

π ( e P ) S = S min = β + 1 2 b 2 + τ δ τ − 2 b b ″

Focusing first on internal stability, when the number of participants in the RVEA is exactly S = S _min, exiting the agreement leads to the imposition of the emissions taxation policy, and the internal stability is satisfied, as π ( e P ) S = S min ≥ π ( e P ) S = S min − 1 , due to the fact that for S = S _min − 1 the REA does not form and the tax is applied, inducing a lower profit for each firm. On the other hand, internal stability does not hold for S > S _min: each member of the REA can indeed leave the agreement, free ride on emissions reduction by other firms without inducing the RVEA to break up (recall that IC is satisfied at values of S equal or larger than S _min).

Consider external stability. For S = S _min the RVEA exists and is internally stable. As non-members receive the “business as usual” profit π(e ^u), and as cooperative profits increase with S, we can show that even for S ≃ N, we have:

π ( e P ) ≃ β + b 2 − τ 2 2 b ″ < π ( e u )

Thus, the REA is externally stable for any S ≥ S _min. We can therefore conclude that internal and external stability are satisfied at S = S _min and for π ( e P ) S = S min .□

Proof of Corollary 1

First of all, define Ω = S _m − S _min. In order to have S _min < N, we need to assume δ > τ 2 b − τ .Clearly, Ω is monotonically increasing in δ for δ ∈ (0, 1). Notice that Ω δ = τ 2 b − τ = S m − N < 0 . Also, we can easily rewrite

Ω δ = 1 = N ( 2 b − τ ) b ″ + ( b ″ ) 2 + α m 2 f m 2 τ 2 b − τ − α m 2 f m 2 τ 2 b − τ α m f m ( 2 b − τ ) .

which, given b″ > 0, is positive. The cutoff value δ* where S _m = S _min can be defined as follows:

δ * = α m f m 2 τ 2 b − τ b ″ + ( b ″ ) 2 + α m 2 f m 2 τ 2 b − τ 2

Assume now that b − τ b ″ > S m α m f m > 1 α m f m , then δ * ∈ τ 2 b − τ , 1 and we can conclude that Ω is negative (positive) for δ < δ* (δ > δ*). In this case, our parameter values are such that also an equilibrium featuring a static VEA a la McEvoy and Stranlund (2010) exists (see their Proposition 2).□

Proof of Proposition 4

In the static VEA in McEvoy and Stranlund (2010) the enforcement cost is paid by firms taking part in the agreement whose enforcement is ensured by a costly a third party. Also, label emissions arising under the static VEA as e m P = S m b − N τ S m b ″ borrowed from McEvoy and Stranlund (2010); we can rewrite the corresponding welfare (defined by W _m) as follows:

W m = S m π e m P S = S m + ( N − S m ) π ( e u ) − N τ S m α m f m

where the last term is the total payment by firms under third party enforcement (See McEvoy and Stranlund 2010, equation (12)). The welfare differential is:

(20) Δ = W RV EA − W m = N 2 τ 2 2 b ″ S min − S m S min S m + N τ S m α m f m

First focus on the conditions guaranteeing that welfare in McEvoy and Stranlund (2010) and in our RVEA is larger than under the emission tax. This holds:

in the case of the static VEA in McEvoy and Stranlund (2010) if
α f < α f 0 = 2 b ″ S m α m f m ( N − S m ) τ α m f m + 2 b ″ S m with α f 0 ∈ ( 0 , α m f m )
for the RVEA, from (8), if α f ≤ α f min = 2 S min b ″ τ N − S min . Thus, in order for a VEA (both static and relational) to be preferred to a standard emission tax, we assume α f ≤ min { α f 0 , α f min } .

From (20), we can conclude, after some manipulation, that Δ > 0 if

N τ 2 b ″ S min − S m S min S m + 1 S m α m f m > 0

When S _m < S _min (i.e. δ < δ*) we have thus that Δ > 0. When δ = δ*, i.e. S _m = S _min, then Δ = N τ S m α m f m > 0 . Consider, S _m ≥ S _min for δ ≥ δ*. This case is cumbersome and the results depend on the parameter values of the model. However, we can characterize a scenario in which Δ < 0 as follows. Let:

S ̂ m = 2 + 2 b ″ N α m f m τ

δ 0 = α m f m + 2 b ″ N τ τ 2 b − τ 2 b ″ + ( b ″ ) 2 + α m f m 2 τ 2 b − τ 2

Since S _min decreases with δ then have that S min − S m S m S min = 1 S m − 1 S min decreases with δ. Note from (20) that Δ < 0 requires, after some manipulation, δ > δ ₀ > δ*, where δ ₀ > δ* as

δ 0 − δ * = 4 b ″ b ″ + N α m f m τ 2 b − τ N 2 τ 2 b ″ b ″ + ( b ″ ) 2 + α m f m 2 τ 2 b − τ + α m f m 2 τ 2 b − τ > 0

On the other hand we also need to make sure that δ ₀ < 1 for an equilibrium where Δ < 0 to be possible. After some manipulation, we can conclude that δ ₀ < 1 if:

N b ″ + ( 2 b − τ ) ( α m f m ) 2 + ( b ″ ) 2 ( α m f m ) 2 b − τ > 2 + 2 b ″ N ( α m f m ) τ ,

i.e. S m > S ̂ m . As a result, we can conclude that for any ( S m ≤ S ̂ m ), then it is always δ ₀ ≥ 1 and therefore Δ > 0 for any δ ∈ τ 2 b − τ , 1 , while when S m > S ̂ m , then for δ ∈ τ 2 b − τ , δ 0 we have Δ > 0, while for δ ∈ δ 0 , 1 , then Δ < 0.□

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Supplementary Material

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Received: 2022-12-29

Accepted: 2023-09-11

Published Online: 2023-09-29

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Keywords for this article

relational contracts; environmental policy; unverifiability; Voluntary Environmental Agreement