How Does a “Green” Good Affect Environmental Quality and Social Welfare?

Anja Brumme

doi:10.1515/jbnst-2021-0016

Published by De Gruyter Oldenbourg May 12, 2022

How Does a “Green” Good Affect Environmental Quality and Social Welfare?

Anja Brumme

From the journal Jahrbücher für Nationalökonomie und Statistik

https://doi.org/10.1515/jbnst-2021-0016

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Abstract

In this paper an impure public good model is applied to analyze the effects of introducing a green good, i.e. a consumption good that contributes to preservation of environmental quality. I distinguish three types of consumers: “grays”, who consume the green good and a private good; “greens”, who consume the green good and make additional donations to environmental organizations; and “edge” consumers, who consume only the green good. With respect to environmental quality I find that the gap between voluntary provision and optimal provision is unaffected by the introduction of the green good, no matter if consumers are gray or green. However, in the case of gray agents environmental quality is improved in absolute terms after the green good has been launched whereas it remains at the same level as before if consumers are green. The effect of the green good on the social welfare gap is more likely to be beneficial if agents are gray. The gaps with respect to environmental quality and social welfare are both closed if the agents are edge consumers.

Keywords: environmental quality; equivalent variation; green goods; impure public goods; index of easy riding; Pareto-optimum; social welfare; voluntary provision of public goods

JEL Classification: C72; D64; H41; Q50

Corresponding author: Anja Brumme, Technische Universität Bergakademie, Freiberg, Germany, E-mail: anja.brumme@vwl.tu-freiberg.de

Appendix A: Mathematical Derivations

Derivation of Pareto-Optima

For the calculation of Pareto-optima we employ the same technique as for the Nash equilibria and transfer the maximization problems entirely to characteristics space. Recall problem (6):

max c i , c j , D , q i , q j u i ( y i , Z ) = y i γ Z 1 − γ

s . t . ∑ i = 1 2 w i = ∑ i = 1 2 c i + D + ∑ i = 1 2 q i ,

y i = c i + α q i ,

Z = D + β ∑ i = 1 2 q i ,

u j ( y j , Z ) = y j γ Z 1 − γ = u ‾ j .

For the pure public good benchmark case set q _i = q _j = 0 to obtain

max c i , c j , D u i ( y i , Z ) = y i γ Z 1 − γ

s . t . ∑ i = 1 2 w i = ∑ i = 1 2 c i + D ,

y i = c i ,

Z = D ,

u j ( y j , Z ) = y j γ Z 1 − γ = u ‾ j .

Now substitute the second and third constraint for c _i and D in the first constraint and summarize. This gives the virtual problem (7) which is expressed in characteristics space:

max y i , y j , Z u i ( y i , Z ) = y i γ Z 1 − γ

s . t . ∑ i = 1 2 w i = ∑ i = 1 2 y i + Z ,

u j ( y j , Z ) = y j γ Z 1 − γ = u ‾ j .

The first-order conditions are

(i) γ y i γ − 1 Z 1 − γ − λ 1 = 0 ,

(ii) − λ 1 − λ 2 γ y j γ − 1 Z 1 − γ = 0 ,

(iii) ( 1 − γ ) y i γ Z − γ − λ 1 − λ 2 ( 1 − γ ) y j γ Z − γ = 0 ,

(iv) ∑ i = 1 2 w i − ∑ i = 1 2 y i − Z = 0 ,

(v) u ‾ j − y j γ Z 1 − γ = 0 .

Dividing (ii) by (iii) and canceling yields

− λ 1 ( 1 − γ ) y i γ Z − γ − λ 1 = γ 1 − γ Z y j .

Insert λ ₁ = γy _i ^γ − 1 Z ^1 − γ from (i):

− γ y i γ − 1 Z 1 − γ ( 1 − γ ) y i γ Z − γ − γ y i γ − 1 Z 1 − γ = γ 1 − γ Z y j .

After canceling and rearranging we obtain

y i + y j = ∑ i = 1 2 y i = γ 1 − γ Z .

From (iv) we know that

∑ i = 1 2 y i = ∑ i = 1 2 w i − Z ,

Z ∗ ∗ = ( 1 − γ ) ∑ i = 1 2 w i ,

∑ i = 1 2 y i ∗ ∗ = γ ∑ i = 1 2 w i .

The results for the gray, green, and edge consumers are derived analogously by setting D = 0 , c _i = c _j = 0, c _i = c _j = D = 0, respectively, in (6).

Derivation of conditions (10), (11) and (12) for the determination of consumer types

Recall the original maximization problem (1):

max c i , d i , q i u i ( y i , Z ) = y i γ Z 1 − γ

s . t . w i = c i + d i + q i ,

y i = c i + α q i ,

Z = ∑ i = 1 2 z i = ∑ i = 1 2 d i + β ∑ i = 1 2 q i = D + β ∑ i = 1 2 q i .

As we know that each consumer type will purchase at most two of the three goods, this problem can be simplified and transferred to characteristics space. Starting with the gray consumer, set d _i = 0 to obtain

max c i , q i u i ( y i , Z ) = y i γ Z 1 − γ

s . t . w i = c i + q i ,

y i = c i + α q i ,

Z = ∑ i = 1 2 z i = β ∑ i = 1 2 q i = β ∑ i = 1 2 q i .

Rearranging the third constraint for q _i = (1/β)z _i, the first constraint becomes c _i = w _i − (1/β)z _i, which in turn can be inserted in the second constraint to yield y _i = w _i − (1/β)z _i + (α/β)z _i. Expressed in characteristics space, the three constraints can now be summarized in one constraint:

w i = y i + 1 − α β z i .

Adding the value of spill-ins of the other agent’s public characteristic provision to both sides of this equation results in

w i + 1 − α β z j ≠ i = y i + 1 − α β Z ,

which is equivalent to (13a). (13b) and (13c) can be obtained in the same way by setting c _i = 0 and c _i = d _i = 0, respectively.

By solving the maximization problem in characteristics space we obtain the reaction function of the gray consumer

z i = β ( 1 − γ ) 1 − α w i − γ z j ≠ i ,

and equalizing the reaction functions of both agents i and j gives the Nash equilibrium values stated in Table 1 of Appendix B.

To find the condition for a consumer to be gray, as given by (10), imagine that this agent initially purchases only the green good. This means that she will receive y _i = αw _i of the private characteristic and contribute z _i = βw _i to environmental quality. However, as this type additionally consumes the conventional good, we can conclude that she must prefer a different ratio of the private and public characteristic, i.e. y _i ^∗/z _i ^∗ > α/β. Inserting the Nash equilibrium values of the gray consumer yields

γ ( 1 − γ ) 1 − γ 2 ∑ i = 1 2 w i β 1 − α 1 − γ 1 − γ 2 ( w i − γ w j ) > α β ,

and consequently,

γ ∑ i = 1 2 w i w i − γ w j > α 1 − α .

This is equivalent to expression (10) above.

A green consumer will, according to the same reasoning, prefer more of the public and less of the private characteristic than generated by the green good alone. Hence, y _i ^∗/z _i ^∗ < α/β:

α 1 − β γ ( 1 − γ ) 1 − γ 2 ∑ i = 1 2 w i 1 − γ 1 − γ 2 ( w i − γ w j ) < α β ,

i.e.

γ ∑ i = 1 2 w i w i − γ w j < 1 − β β ,

which gives condition (11).

As α + β > 1 in case (b), α/(1 − α) > (1 − β)/β and we are left with the interval

1 − β β ≤ γ ∑ i = 1 2 w i w i − γ w j ≤ α ( 1 − α ) ,

where consumers choose only the green good and are therefore of the edge type (b3).

Appendix B: Tables

Table 1:

Nash equilibrium levels of the private and public characteristics and Nash utility levels in the pure public good (case (a)) and impure public good (cases (b1)–(b3)) frameworks.

		y i ∗	z i ∗	Z ∗	u i ( y i ∗ , Z ∗ )
(a)	Benchmark	γ ( 1 − γ ) 1 − γ 2 ∑ i = 1 2 w i	1 − γ 1 − γ 2 ( w i − γ w j )	( 1 − γ ) 2 1 − γ 2 ∑ i = 1 2 w i	( γ 1 − γ ) γ ( 1 − γ ) 2 1 − γ 2 ∑ i = 1 2 w i
(b1)	Gray consumers	γ ( 1 − γ ) 1 − γ 2 ∑ i = 1 2 w i	β 1 − α 1 − γ 1 − γ 2 ( w i − γ w j )	β 1 − α ( 1 − γ ) 2 1 − γ 2 ∑ i = 1 2 w i	γ γ ( 1 − γ ) 1 − γ 2 ( β 1 − α ( 1 − γ ) ) 1 − γ ∑ i = 1 2 w i
(b2)	Green consumers	α 1 − β γ ( 1 − γ ) 1 − γ 2 ∑ i = 1 2 w i	1 − γ 1 − γ 2 ( w i − γ w j )	( 1 − γ ) 2 1 − γ 2 ∑ i = 1 2 w i	( 1 − γ ) 2 1 − γ 2 ( α 1 − β γ 1 − γ ) γ ∑ i = 1 2 w i
(b3)	Edge consumers	αw _i	βw _i	β ∑ i = 1 2 w i	( α w i ) γ ( β ∑ i = 1 2 w i ) 1 − γ

Table 2:

Pareto-optimal levels of the private and public characteristics and Pareto utility levels in the pure public good (case (a)) and impure public good (cases (b1)–(b3)) frameworks.

		∑ i = 1 2 y i ∗ ∗	y i ∗ ∗	z i ∗ ∗	Z ∗ ∗	u i ( y i ∗ ∗ , Z ∗ ∗ )
(a)	Benchmark	γ ∑ i = 1 2 w i	γw _j	w _i − γw _j	( 1 − γ ) ∑ i = 1 2 w i	( γ w j ) γ ( ( 1 − γ ) ∑ i = 1 2 w i ) 1 − γ
(b1)	Gray consumers	γ ∑ i = 1 2 w i	γw _j	β 1 − α ( w i − γ w j )	β 1 − α ( 1 − γ ) ∑ i = 1 2 w i	( γ w j ) γ ( β 1 − α ( 1 − γ ) ∑ i = 1 2 w i ) 1 − γ
(b2)	Green consumers	α γ 1 − β ∑ i = 1 2 w i	α γ 1 − β w j	w _i − γw _j	( 1 − γ ) ∑ i = 1 2 w i	( α γ 1 − β w j ) γ ( ( 1 − γ ) ∑ i = 1 2 w i ) 1 − γ
(b3)	Edge consumers	α ∑ i = 1 2 w i	α w i	β w i	β ∑ i = 1 2 w i	( α w i ) γ ( β ∑ i = 1 2 w i ) 1 − γ

Table 3:

Index of easy riding values and social welfare gaps in the pure public good (case (a)) and impure public good (cases (b1)–(b3)) frameworks.

		IER	EV
(a)	Benchmark	1 − γ 1 − γ 2	u ‾ i ∗ ∗ 1 1 − γ ( 1 − γ ) ( γ w j ) γ 1 − γ − u ‾ i ∗ ( γ 1 − γ ) γ ( 1 − γ ) 2 1 − γ 2
(b1)	Gray consumers	1 − γ 1 − γ 2	u ‾ i ∗ ∗ 1 1 − γ β ( 1 − γ ) 1 − α ( γ w j ) γ 1 − γ − u ‾ i ∗ ( β 1 − α ) 1 − γ ( γ 1 − γ ) γ ( 1 − γ ) 2 1 − γ 2
(b2)	Green consumers	1 − γ 1 − γ 2	u ‾ i ∗ ∗ 1 1 − γ ( α γ 1 − β w j ) γ 1 − γ − u ‾ i ∗ ( α 1 − β γ 1 − γ ) γ ( 1 − γ ) 2 1 − γ 2
(b3)	Edge consumers	1	0

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Received: 2021-04-30

Accepted: 2022-04-04

Published Online: 2022-05-12

Published in Print: 2022-06-27

How Does a “Green” Good Affect Environmental Quality and Social Welfare?

Abstract

Appendix A: Mathematical Derivations

Derivation of Pareto-Optima

Derivation of conditions (10), (11) and (12) for the determination of consumer types

Appendix B: Tables

References

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