International Journal of Ecological Economics & Statistics (IJEES) Year 2011, Vol. 23 (Special Volume), No. F11; Int. J. Ecol. Econ. Stat.; ISSN 0973-1385 (Print), ISSN 0973-7537 (Online) Copyright © 2011 IJEES, CESER Publications

Environmental Identity and Intergenerational Equity Isabel Almudí1 and Julio Sánchez Chóliz2 1

Economics Department, University of Zaragoza, Spain E-mail (correspondence author): [email protected]

2

Economics Department, University of Zaragoza, Spain E-mail: [email protected]

ABSTRACT In this paper we analyze how the consideration of non-strictly economic factors (such as the incorporation of a "green identity” into consumers' preferences) can modify intertemporal optimal allocation paths, making possible to achieve more equitable intergenerational societies. To show the aforementioned, we construct what we call “green” identity preferences. Likewise, to define intergenerational equity, this work takes the concepts and results obtained in Chichilnisky (1996; 1997; 2009) for the case of renewable resources as its fundamental point of reference. The basic idea underlying this work is the following: if we assume the existence of an environmental identity that agents take into account when making their choices, this can motivate a change in the behaviour of agents. And, as we show in the paper, under certain conditions, this change can generate more equitable intergenerational societies, even if all generations are not treated equally (i.e. we use a positive discount rate in the analysis).

Keywords: Identity, Intergenerational Equity, Sustainable Preferences. JEL Codes: Q56. Mathematics Subject Classification Numbers: 49K, 93, 34H05.

INTRODUCTION

In this paper we analyze how the consideration of non-strictly economic motivations (such as the incorporation of a "green identity" into consumers' preferences) can modify intertemporal optimal allocation paths, making possible to achieve more equitable societies from an intergenerational point of view. Even if all generations are not treated equally (i.e. using a positive discount rate in problems with infinite horizon). To show the aforementioned, we construct what we call “green” identity preferences (Almudi, 2010; Almudi y Sanchez, 2006; 2009; 2011). Likewise, to define, intergenerational equity, we take the concepts and results obtained in Chichilnisky (1996; 1997; 2009) for the case of renewable resources.

The question of how to incorporate future generations into the decisions of present generations in a fair way (i.e. it has to be assured that the well-being of all generations is equally considered when www.ceserp.com/cp-jour www.ceser.res.in/ijees.html www.ceserpublications.com www.ceser.in

International Journal of Ecological Economics & Statistics

optimal dynamic problems are solved) has been looked at for many years in economic theory (see Ramsey, 1928, for the first formulation in an intertemporal analysis). The consideration of time in allocative problems has provided us with new concepts such as the intertemporal discount rate, but its application throws up serious difficulties when intergenerational justice is discussed. For instance, the use of a positive and constant value for the discount rate is problematic since this implies an unequal treatment for future generations. Likewise, using a nil value for it makes the mathematical problem unsolvable when an infinite horizon is considered. If the number of generations is finite and the discount rate is zero, the problem has a mathematical solution and the generations are treated equally, but the need to consider only a finite number of them for this solution to exist makes this criterion unacceptable from a normative perspective.1 Therefore, the use of the discount rate is not exempt from problems. On the other hand, the appearance of the term Sustainable Development2 (Brutland, 1987) makes it clear that any optimality criteria posed with the aim of allocating intertemporal paths in industrial economies should be both ethically defensible and useful on practical grounds (Chichilnisky, 1997). In this regard, interesting alternatives to discounted utilitarianism have been proposed - i.e. the maximin criterion (Rawls,1972), the overtaking criterion, the sum of discounted utilities, the Green Golden Rule (Beltratti et al.,1995), etc.- but so far, the most valuable attempt has been the sustainable preferences posed by Chichilnisky (1996; 1997; 2009). These preferences capture the idea of sustainability as they are drawn upon the two basic axioms of no dictatorship of the present, nor of the future.

However, even if sustainable preferences are a step forward, the question of how to properly incorporate future generations in the decisions of present generations is still an open issue. Especially, if we try to incorporate motivations, underlying consumers' choices, which are different from traditional economic motivations (as prices and income). The need to incorporate non-strictly economic factors as variables to explain the behavior of agents has been demanded by a growing number of authors. Scholars such as Akerlof (2007) or Tsakalotos (2005) have pointed out that explanatory biases can be developed if we do not recognize the fact that non-strictly economic motivations exist, underlying agent's choices. Regarding this, the seminal contribution of Landa (1981; 1994) in the realm of economics and identity represented a real advance. The basic idea of her pioneering work rests on the fact that agents do not make their choices as isolated entities, but rather as individuals embedded in social structures which provide them with norms and values, limiting their behavior. These findings have been reinforced by the experimental evidence found in the realm of 3 Social Psychology and in the realm of so-called Experimental Economics. Taking all the above into

account, can we seek to incorporate these new findings on identity into our analysis of intergenerational justice making the mathematical problem solvable? We try to shed new light on this question.

To do that, we define what we call “green” identity preferences (Almudi, 2010; Almudi y Sanchez, 2006; 2009; 2011), drawn upon some of the previous contributions of Landa (1981; 1994), Kevane

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(1994) and Akerlof and Kranton (2000). Then, we analyze if these preferences allow us to find a configuration of the economy which guarantees, in the sense defined by Chichilnisky (1996; 1997), an equal treatment for all generations. To find the conditions under which this is possible we use sustainable preferences (Chichilnisky, 1996; 1997) as our fundamental reference point. Our results show that if agents behave according to what we call “green” identity preferences, even when using an adequate positive discount rate, they will show an identical behavior (an optimal path) to that of other agents with sustainable preferences (Chichilnisky, 1996; 1997; 2009). Hence, we can say that “green” identity preferences allow us to achieve paths with intergenerational equity.

The paper is organized as follows: We define what we call “green” identity preferences in section 2. Section 3 is devoted to exploring how, under the assumption of “green” identity preferences, the optimal trajectories are modified. In section 4 we find the conditions under which, using a positive discount rate, it is possible to guarantee an equal treatment for both, present and future generations, deducing the links that exist between sustainable preferences and “green” identity preferences. Finally, we summarize our conclusions.

GREEN IDENTITY PREFERENCES

In this Section we construct what we call “green” identity preferences for renewable resources (like fisheries, forests, biodiversity, etc.). The basic idea of these new preferences rests on the works of Landa (1981; 1994), Kevane (1994) and Akerlof & Kranton (2000). Thus, Landa (1981; 1994) stated that agents do not make their choices as isolated entities, but rather as individuals embedded in social structures, which provide them with norms and values, limiting their behaviour. These social structures can be considered as social identities. Kevane (1994) studied the economic implications of considering identities as ‘imaged communities’ attached to individuals. Finally, Akerlof and Kranton (2000) have proposed the incorporation of these ideas and others, offered by social psychology, within a neoclassic utilitarian framework, which leads to a higher analytical tractability of the concept of identity.

Drawing on the aforementioned, let us assume the following to construct the “green” identity preferences: Firstly, a unique social identity exists, which we call "Environmentally Friendly Identity" (EFI). This identity can be assimilated with what most people in a specific society understand by 4 “being green”, “being respectful towards the environment”, etc. Secondly, for simplicity, we assume

that there is only one action (or if preferred, one norm or value) which allows individuals to materialize their level of identification with the EFI: the choice of a determined consumption level of a renewable natural resource. What means that we assume that “green” individuals perceive that overconsumption can lead to the resource becoming extinct.5 Thirdly, we consider that the degree of identification of each individual with this "green" social identity (EFI) is only determined by their intrinsic environmental sensitivity (see also Almudi, 2010; Almudi y Sanchez, 2006; 2009; 2011). Finally, we formally characterize the personal identity function from the framework proposed by Akerlof and Kranton

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(2000).

The following function allows us to represent identity formally.

Di where

D (c i )

D i represents the self-image that each agent i possesses with respect to the "green" identity i

(EFI) and c represents the level of consumption of a renewable resource. Likewise, we assume that the

D (˜) function verifies that:

D i (c i ) ! 0 D ci  0 and bounded; D cci  0 and bounded D i ( 0)

A i ; with A i ! 0

The first derivative indicates that the lower (higher) the consumption of the renewable resource, the greater (lower) the level of identification of agent i with (EFI). Note that lower consumption implies a lower possibility of the resource becoming extinct. Furthermore, as the second derivative indicates, each additional increase in consumption leads to an ever greater decrease in this identification, i

where, A , is the maximum identification that agent i can reach with respect to the "green" identity and corresponds to the case in which the consumption level is zero. We can state, therefore, that the less intense the “green” identity, the flatter function

D i is. Notice that, in the case where the agent

values the environment but is not sensitive to the impact their consumption has, their flat, constant, and equal to

D i (c i )

D i would be

Ai .

Finally we consider that the image each agent i has of themselves is a source of personal satisfaction. Therefore, function grow as

D (˜) i

D i (˜)

will be an argument of the consumer's utility function which will

6

grows. If we assume that preferences are represented by an additive-separable

function, the instantaneous utility function for each agent responds to:7

u i (c i , s, D i (c))

u 1i (c)  u i2 ( s)  wi u i3 (D (c)) U i (c, s), con w i t 0

[1]

i i where c , s, D (˜), represent, respectively, for each instant of time and each agent i, the resource

consumption, the stock level of the renewable resource and the level of identification with respect to i

(EFI). In addition, w will be the parameter measuring the intensity with which agents incorporate the “green” identity into their preference structure. We can note that if an agent does not value the i

environment, their w will be

0 and, for them, the utility function will be: u i (c i , s) u 1i (c)  u i2 ( s) 22

[2]

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Likewise, we assume that the utility functions u k (˜) with k

1, 2, 3 are continuous, twice

differentiable and verify that:

u 1 c ! 0; u 1 cc  0; lim u 1 c c o0

f; lim u 1 c c of

0;

u 2 s ! 0; u 2 ss  0 lim u 2 s s o0

f; lim u 2 s s of

0;

u 3 D ! 0 and bounded ; u 3 DD  0; Uc

u 1 c  w.u 3 D D c ! 0

which allows us to affirm that the U (c, s ) associated with [1] verifies the usual properties - that is:

U c ! 0;U s ! 0;U cc  0;U ss  0. We call [1] “green” identity preferences. Finally, for analytical convenience let us say that individual i is environmentally more sensitive than

j, if the following condition is fulfilled:8

wi u i3 D D ci  w j u 3jD D cj , c œ U ci  U cj , c We also point out that if they have the same utility intensity for EFI, w more sensitive than j , if this condition is fulfilled:

i

w j , i is environmentally

u i3 D D ci  u 3jD D cj , c .

DISCOUNTED UTILITARIAN SOLUTIONS FOR RENEWABLE RESOURCES

Let us deduce the discounted utilitarian solutions when we consider that agents are represented by “green” identity preferences, with the only good of the economy being a renewable source. The following definitions are useful for studying the implications of considering “green” identity preferences.

Definition 1: The discounted utilitarian problem for an infinite horizon is given by (see Chichilnisky, 1997, p. 476): f

max ³ u (c, s)e Gt dt; G ! 0

[3]

0

x

s.t . s t

r ( s)  c, s o given

r ( s) continuous, concave and twice differentiable Which, under usual conditions and assuming that u (c, s ) is given by [2] (that is, the agent has no environmental sensitivity), has a solution given by the set of equations:9

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O

u 1 c (c ) x

O  GO

u 2 s ( s )  Ors ( s )

x

r (s)  c

s

The solution is characterized by a stable path, described by the following equations obtained from the previous ones: x

c x

s

u 2 s (s) º u 1 c (c ) ª «G  rs ( s )  » u 1 c ( c ) ¼» u 1 cc ( c ) ¬« r (s)  c

and it has a steady state, which depends on the discount rate

G , located on the curve c

r ( s ) . The

steady state solution verifies the equations:

u 2 s (s)

G  rs ( s)

u 1 c (c ) r ( s)

c

We shall now see how the problem and its solution vary when all the agents have some environmental sensitivity - that is, when wi z 0, i .

Definition 2: The discounted “green” identity problem for an infinite horizon, if it has the same constraint as [3], is given by: f

>

@

max ³ u 1 (c)  u 2 ( s)  wu 3 (D (c)) e Gt dt ; G ! 0 ; w t 0 0

x

st

r ( s)  c, so given

And its solution is given by the set of equations:

u 1 c (c)  wu 3 D D c (c) x

O  GO x

s or by the derived dynamic equations:

24

O u 2 s ( s )  Ors ( s ) r ( s)  c

[4]

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u 1 c (c)  wu 3 D D c (c)

º ª u 2 s ( s) «G  rs ( s )  » u 1 cc (c)  wu 3 DD D cD c  wu 3 D D cc «¬ u 1 c (c)  wu 3 D D c (c) »¼

x

c x

r ( s)  c

s

and it has a steady state, defined by:

u 2 s (s)

G  rs ( s)

u 1 c (c)  wu 3 D D c (c) r (s)

[5]

[6]

c

Let us note that [4] is formally a particular case of [3] . For this reason we can affirm the existence of a stable optimal path and a steady point at c

r (s ) . However, the existence of an identity function

modifies preferences and, thus, the stable paths and the steady state.

Proposition 1: Problems of the type [4] verify that: (1) The stock level of the steady state is greater, the greater the environmental sensitivity that the agents have. (2) The elasticity of the marginal utility of an agent with environmental sensitivity, as an absolute value, is greater than that of an agent without sensitivity. Proof: To demonstrate that the steady state stock is greater, the greater the environmental sensitivity x

is, first we can see that the curve, c

0, is strictly increasing. This curve is defined by: u2s u 1 c  wu 3 D D c

G  rs , t

which we can also express as: U st  U ct [G  rst ]

0

As U st ct  U ct ct [G  rst ] z 0, s t , G ! rst , for the implicit function theorem we can confirm that the x

points verifying c

0, with G ! rst , are those of a curve ct

previous expression with respect to st , along curve ct

g ( st ) . What is more, deriving the

g ( st ), we obtain:

0 U st st  U ct ct g c( st )[G  rst ]  U ct r cc( st ) Ÿ g c( st )U ct ct [G  rst ] U st st  U ct r cc( st ) Ÿ g c( st ) ! 0

which demonstrates that g ( st ) is strictly growing. Furthermore, as

lim u 1 ct ct o0

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for the properties of the utility function, and remembering that u 3 D and

D c are bounded, we can t

affirm that lim

ct o 0, ct g ( st )

G  rst

0,

which means that the curves g ( st ) always start out from the point (0, sˆ) , with sˆ being the stock defined by r c( sˆ)

G . In the same way, taking into account how we have obtained ct

g ( st ) , from

equations [5] and [6] we can verify that:

u2s u 1 c  wu 3 D D c

! ()G  rst if ( st , ct ) is to the left (right) of ct

Seeing these properties of the curve ct

g ( s t ).

g ( st ), if we assume that the environmental sensitivity of i

is greater than that of j ( w u 3 D D  w u 3jD D cj ) and that g ( s t ) and g ( s t ) are the corresponding i

i

i c

1

j

2

2

curves in each case, it is immediate that for the points of g ( s t ) we verify that:

u12 s u11 c

 w u3D Dc 1 1

!

u 22 s u12c

 w2 u 32D D c

2

G  rst

1

Thus, all points of g ( s t ) are situated to the left of g ( s t ) , with both curves coinciding only at sˆ . Consequently, the steady state of the agent with a higher sensitivity will be situated further to the right and will have a greater level of resource stock. Section 2) of the proposition is demonstrated if:

K1



U c1t ct ct

!K2

1

U ct



U c2t ct ct U c2t

In this case, for simplicity, we use the superscript 2 when the environmental sensitivity for agent i is nil; that is to say, when w 2 u 32D D c2t

0 . On the other hand, we use the superscript 1 when the

environmental sensitivity depends on consumption, that is to say: w1u 13 D D c1t  0. Note that:

U c2t = u 12 ct and U c1t U c2t ct

u 11 ct  w1u 13 D D c1t ! 0; as w1u 13 D D c1t  0 o U c2t ! U c1t

u 12 ct ct and U c1t ct

u 11 ct ct  w1u 13 DD D c1t D c1t  w1u 13 D D c1t ct ;

as w1 ! 0; u 13 DD  0; D c1t ct  0 o U c2t ct ! U c1t ct Taking these inequalities and the signs of U ct and U ct ct into account, we obtain that:

U c1t ct U c2t ct

!

U c1t U c2t

œ K1 ! K 2

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The argument for the first point is straightforward. Agents, when identifying themselves with the "green" identity, are more willing to reduce their resource consumption. Likewise, the second point implies that we should expect individuals with environmental sensitivity to choose paths with lower levels of consumption growth for each instant of time, since a greater elasticity of the marginal utility indicates that their degree of satiation, via consumption, has increased. Therefore, we can affirm that the optimal trajectories obtained from solving [4] are more equitable intergenerationally than those obtained from solving [3].10 This is so even though we use a positive discount rate in both cases. The relationship between environmental sensitivity, steady state and the discount rate

G is clearly

established in the following proposition.

Proposition 2: The problem [4] has a steady state with greater stock and lower consumption, the greater the intensity of

w , the higher the environmental sensitivity, and the lower the value of G . This

relationship is determined by the following expression.

G

Proof: The first part, regarding

rs ( s ) 

u 2 s ( s )

u1 c (c )  wu 3 D D c (c )

w and environmental sensitivity, is a direct consequence of the first

point of proposition 1. To prove the affirmation regarding [5] to be verified on the curve c

G , it is sufficient to see that for the equation

r (s ) , the lower the value of G , the greater the value of stock, s,

must be - which represents a lower consumption.

This proposition shows that the greater the intensity with which agents incorporate the "green" identity into their preference structure, or the lower the discount rate, the greater the stock level in the steady state will be. This means that as

w grows, even with a given discount rate, the equilibrium stock will

grow and consumption in the steady state will be reduced.

Taking these results on board, we propose to study whether, using a positive discount rate, it is possible to obtain a configuration of the economy which guarantees an equal treatment for both present and future generations in some way. Its exact sense will be seen in the Proposition 3. We prove that this is possible if we characterize agents through “green” identity preferences.

DISCOUNTING AND INTERGENERATIONAL EQUITY To prove that intergenerational equity is possible using a positive discount rate: (i) we take the steady state and dynamic equations obtained from sustainable preferences (Chichilnisky, 1996; 1997; 2009) as our point of reference,12 (ii) we prove that both, the stable path and the steady state associated with “green” identity preferences, and a suitable discount rate

G , coincide with the stable path and

steady state associated with the sustainable preferences of those agents with no environmental sensitivity (see Definition 3 below), or, if preferred, with those of the same agents before they acquire 27

International Journal of Ecological Economics & Statistics

environmental identity; and (iii) we offer a numerical example.

Sustainable preferences, as defined by Chichilnisky (1996; 1997; 2009), draw upon two basic axioms: no dictatorship of the present, nor of the future, thus capturing the basic idea of sustainability. For an agent with nil environmental sensitivity, and sustainable preferences, the optimal solution is given by: Definition 3: The sustainable preferences problem is given by (see, Chichilnisky, 1997, p. 475, 478 and 479; see also Heal, 1998): f

max a ³ >u1 (c )  u 2 ( s ) @' (t ) dt  (1  a ) lim >u1 (c )  u 2 ( s ) @, 0  a  1; lim q (t ) t of

x

s.t . s

[7]

t of

0

x

' (t ) / ' (t )

0

r ( s )  c , s o given

The dynamics of [7] are described by the following equations:

u 2 s (s) º u 1 c (c ) ª « q (t )  rs ( s )  » u 1 c ( c ) »¼ u 1 cc ( c ) «¬

x

c x

r (s)  c

s

Which is a non-autonomous system, converging asymptotically, when lim q (t )

0, to the

t of

autonomous system defined by:

u 1 c (c ) x

O x

s

O u 2 s ( s )  Ors ( s ) r ( s)  c

with dynamic equations: x

c x

s

u 2 s (s) º u 1 c (c ) ª «  rs ( s )  » u 1 c ( c ) ¼» u 1 cc ( c ) ¬«

[7.1]

r (s)  c

and whose steady state coincides with the Green Golden Rule equilibrium (see, Beltratti et al. 1993; 1995; Chichilnisky, 1997; Heal, 1998). The steady state solution is expressed in the following equations:

u 2 s (s)

 rs ( s )

u1 c (c ) r (s)

[8]

c

In normative terms, the stable trajectories which emerge from solving [7] guarantee an equal 28

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treatment for both present and future generations. Likewise, the steady state associated with sustainable preferences achieves its maximum utility level which is sustainable forever. Additionally, we have proved in propositions 1 and 2 that the incorporation of identity issues into preference structures makes optimal trajectories more equitable intergenerationally. Thus, it seems reasonable to conclude that there must be a strong relationship between the trajectories obtained from “green” identity preferences and some of the trajectories achieved through sustainable preferences. The following proposition offers the conditions that must be fulfilled for the trajectories of problems [4] and [7] to coincide. Proposition 3: ( s , c ) is the Green Golden Rule of the utility function of [2]. Problems [4] and [7], corresponding, respectively, to agents with and without environmental sensitivity, verify the following properties: (1) The steady state of [7], which is the Green Golden Rule ( s , c ) , and the steady state of [4] coincide (for given w and

G ) if, and only if, the difference between the marginal rates of



substitution of both problems at ( s , c ) is equal to

G.

(2) There is always a pair of {w ! 0, G ! 0} for which the steady states of [4] and of [7] coincide that is, they are exactly the Green Golden Rule of the agent with no environmental sensitivity. What is more, if w is the higher level of

w defined by the last condition of the utility function of

problem [4], for each w  (0, w ), there is a

G given by [9] for which the steady states of both

systems coincide with this Green Golden Rule.



(3) If [4] has ( s , c ) as its steady state and {s (t ), c (t )} is its stable optimal trajectory, there is a discount rate q (t ) which verifies lim t of q (t )

0, and for which [7] has {s (t ), c(t )} as its stable

optimal trajectory. In this case, the steady states of both systems coincide with the Green Golden Rule of the non-environmentally-friendly agent. The rate q (t ) is given by:

q (t )

rs ( s (t )) 

u 2 s ( s (t )) u 1 c (c (t ))



u 1 cc (c (t )) u 1 c (c (t ))

u 1 c (c (t ))  wu 3 D (D c (c (t )))D c (c (t )) u 1 cc (c (t ))  wu 3 DD (D c (c (t )))D c (c (t ))D c (c (t ))  wu 3 D (D c (c (t )))D cc (c (t )) ª º u 2 s ( s (t )) «G  rs ( s (t ))  »  u ( c ( t )) wu ( D ( c ( t ))) D ( c ( t )) c c 1c 3D ¬« ¼»

with w  (0, w ) and G ( w) verifying [9]; Proof:

(1) To prove point 1, we compare [8] and [5], and it is immediate that both equilibriums will coincide if and only if:

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u 2 s ( s )

G



u 1 c (c )  wu 3 D D c (c )

(2) To prove point 2, we assume firstly that



u 2 s ( s )

[9]

u 1 c (c )

w is given and that G fulfils [9]. In the steady state

s ) of [7], u 1 c ( c )

 rs ( s ) is verified; hence, at this point the first equation of [8] is also fulfilled

for the chosen

w and G . To complete this proof, we merely have to observe that for any

u2 s (

accepted value of w - that is, for all w  (0, w ) - [9] defines the discount rate

G leading to

the coincidence of both steady states with the Green Golden Rule of the agent with no environmental sensitivity. x

(3) To prove point 3 we will compare the trajectory of c for both problems [4] and [7]. In both systems the equation of s is the same, so we shall see what the expression of q (t ) must be for the equations of c to coincide. For this, we must verify throughout {s (t ), c (t )}

u 1 c (c ) ª u 2 s ( s) º «q (t )  rs ( s )  » u 1 cc (c) «¬ u 1 c (c) »¼ u 1 c (c )  wu 3 D D c (c) u 1 cc (c)  wu 3 DD D cD c  wu 3 D D cc

º ª u 2 s (s) », «G  rs ( s )  u 1 c (c)  wu 3 D D c (c) »¼ «¬

which requires that q (t ) be:

q (t )

rs ( s (t )) 

u 2 s ( s (t )) u 1 c (c(t ))



u 1 cc (c(t )) u 1 c (c(t ))

u 1 c (c(t ))  wu 3 D (D c (c(t )))D c (c(t )) u 1 cc (c(t ))  wu 3 DD (D c (c(t )))D c (c(t ))D c (c(t ))  wu 3 D (D c (c(t )))D cc (c(t )) ª º u 2 s ( s (t )) «G  rs ( s (t ))  » u 1 c (c(t ))  wu 3 D (D c (c(t )))D c (c(t )) »¼ «¬ This q (t ) is well-defined as the functions appearing in it are already known and calculated on the path {s (t ), c(t )}. We can also see that it fulfils the condition lim q (t ) t of

[7] to exist. As

w and G verify [9], we obtain that:

30

0 required for a solution of

International Journal of Ecological Economics & Statistics

lim q (t ) t of

ª « « u ( c ( t )) lim «  u1 ccc( c ( t )) 1 t of « « ¬

º » u1 c ( c ( t ))  wu 3 D (D c ( c ( t )))D c ( c ( t )) » u1 cc ( c ( t ))  wu 3 DD (D c ( c ( t )))D c ( c ( t ))D c ( c ( t ))  wu 3 D (D c ( c ( t )))D cc ( c ( t )) » » ª rs ( s (t ))  u 2 s ( s ( t )) º » u1 c ( c ( t )) ¼ ¬ ¼ rs ( s (t )) 

u 2 s ( s ( t )) u1 c ( c ( t ))

ª u s ( s (t )) º lim « rs ( s (t ))  2 » t of u1 c (c (t )) ¼» ¬« u ( c ( t )) ª º 1  u1 ccc( c ( t )) 1 « » u1 c ( c ( t ))  wu 3 D (D c ( c ( t )))D c ( c ( t )) « » ¬ u1 cc ( c ( t ))  wu 3 DD (D c ( c ( t )))D c ( c ( t ))D c ( c ( t ))  wu 3 D (D c ( c ( t )))D cc ( c ( t )) ¼ 0

Firstly, we can see that this proposition completes the affirmation in the Proposition 2, that the greater

w is, the greater the stock level in the steady state. That proposition affirms that as w grows, the steady state of problem [4] approaches the Green Golden Rule of the non-environmentally-friendly agent (i.e. problem [7], with utility functions u (c, s )

u1 (c)  u 2 ( s ) ), and it may reach, and even pass it,

w is sufficiently large. However, we must not forget that the last of the conditions specified for the utility function of problem [4], sets a higher bound for parameter w . Hence, for given preferences, if

u (c, s )

u 1 (c )  u 2 ( s ) , and for a, u 3 (D (c )), it may be impossible to reach the steady state

associated to [7] for a given

G . However, point 2 of the proposition 3 affirms that, whatever the bound

may be, we can always find a small enough value of

G for this point to be attainable.

Finally, the last part of the previous proposition affirms that any trajectory with “green” identity preferences and for a suitable pair ( w, G ) resembles a trajectory with sustainable preferences, defined with the utility functions u (c, s ) opposite-

i.e.

that

any

trajectory

u1 (c)  u 2 ( s ). However, the proposition does not prove the associated

with

sustainable

preferences

of

the

kind

u (c, s ) u1 (c)  u 2 ( s ), is a trajectory obtained with “green” identity preferences. Proving that “green” identity preference trajectories resemble those of sustainable preferences allows us to affirm that, if we consider that agents incorporate a certain environmental sensitivity into their preference structure, it is possible to find resource assignation trajectories which are equitable intergenerationally, even when all generations are not treated equally (i.e. by using a positive discount rate). It should be pointed out that incorporating environmental identity does not alter the valuation current generations make regarding the utility of future generations. It is the change in behavior with respect to resource consumption, derived from the increased valuation of the resource, that generates more equitable intertemporal assignations. In other words, sustainable preferences cannot be substituted by “green” identity preferences, but “green” identity preferences may offer a more realistic way of achieving a greater intergenerational equity.

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International Journal of Ecological Economics & Statistics

For a better understanding of the previous proposition, we shall illustrate it with a numerical example. Let us assume that

u 1 (c )

s ) 15 log[1  c ]

u 2 (s)

3 log[1  s ]

r ( s)

4 s (1 

w 1 u 3 (D )

D (c ) Starting out from U (c, s ) corresponding

( s , c )

to

the

log[1  D ], and 0.35(c  20 )(c  20 ) / 400;

u 1 (c)  u 2 ( s ) and the curve r (s ) , it is possible to obtain the steady state Green

Golden

Rule,

defined

by

[8].

The

point

we

obtain

is

x

(12.03618,9.51281) . Then we can obtain the trajectory described by c in problem [7] for

the proper discount rate (q(t)), which would be the one obtained if the future and present were dealt with in an equitable way. Depending on the discount rate, this trajectory is closer or less close to the limiting one. In figure 1 we can see the graphical representation of the limiting trajectory (dashing line), as well as the Green Golden Rule, which corresponds with the tangency point between the curve of indifference and the curve c

r (s ) .

The step from the preferences given by [7], to the preferences [4], is determined by incorporating the identity function

D (c ) , that we define as: D (c )

0.35(c  20)(c  20) / 400.

Figure 1

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International Journal of Ecological Economics & Statistics

Knowing the “green” identity preferences, it is possible to obtain an optimal trajectory with the Green Golden Rule as its steady point, but, for this, the discount rate at this point must be known previously. This value can be obtained from the expression [9]. For our example, the

G

G obtained is

0.386385 . The trajectory of the agent with “green” identity preferences and this G can be seen

in figure 1. In this figure we can also see that the trajectory with sustainable preferences is completely superimposed to the trajectory obtained with “green” identity preferences. The discount rate, q(t), is represented in figure 2, where we can see it tends to 0 .

Figure 2 CONCLUDING REMARKS

How to incorporate future generations fairly when we deal with intertemporal allocative problems is a recurrent question in economic theory. Discounted utilitarianism has been criticized for the unequal treatment offered to future generations. However, we show that, using a positive discount rate, it is possible to offer an equal treatment for present and future generations if agents behave according to what we call “green” identity preferences. The basic idea underlying “green” identity preferences is the following: if we consider the existence of a “green” identity and agents can incorporate it into their preference structure, this can motivate a change in the behaviour of agents. And, as we show in the paper, this change can generate societies which are more equitable from an intergenerational point of view, even if all generations are not treated equally. The reason for this stems from the fact that when agents incorporate a certain environmental valuation into their preference structures, they voluntarily reduce their resource consumption. Obviously, this does not alter the valuation present generations make regarding the utility of future generations. It is the change in behaviour with respect to resource consumption, what generates fairer intertemporal allocations. In other words, sustainable preferences cannot be substituted by “green” identity preferences, but the latter may offer a more realistic way of achieving a 33

International Journal of Ecological Economics & Statistics

greater intergenerational equity.

ACKNOWLEDGEMENTS The authors are very grateful to Graciela Chichilnisky, Renan Goetz, Catarina Roseta-Palma, Santiago Rubio Jorge and Juan Perote Peña for their very valuable comments and suggestions. This work has been supported by the Fundación Ramón Areces and the research project SEJ2007-60960/ECON, financed by the Spanish Ministry of Education and Culture.

REFERENCES Akerlof, G. & Kranton, R. (2000). Economics and Identity. Quarterly Journal of Economics, 65 (3): 715- 753. Almudi, I. (2010). Preference Change and Its Role in Renewable Resource Use. Lambert Academic Publishing. Almudi, I. y Sánchez, J. (2006). Influencia Social y Sostenibilidad en el uso de Recursos Renovables. Revista de Economia Agraria y Recursos Naturales, 6 (11): 23-47 Almudi, I y Sanchez, J. (2009). Identity and the Optimal Exploitation of Renewable Resources. International Journal of Ecological Economics and Statistics, 15: 47-67. Almudi, I y Sanchez, J. (2011). Sustainable Use of Renewable Resources. Journal of Bioeconomics. Forthcoming. Beltratti, A., Chichilnisky, G. & Heal, G.M. (1993). Sustainable Growth and the Green Golden Rule. Goldin, I y Winters, L.A. (eds.). Approaches to Sustainable Economic Development. Cambridge University Press. Beltratti, A., Chichilnisky, G. & Heal, G.M. (1995). The Green Golden Rule. Economic Letters, 49: 175-179. Beltratti, A., Chichilnisky, G. & Heal, G.M. (1998). Sustainable Use of Renewable Resources. Graciela Chichilnisky, Geoffrey Heal and Alessandro Vercelli (eds.) Sustainability: Dynamics and Uncertainty. Kluwer Academic Publishers. Berck, P. (1981). Optimal Management of Renewable Resources with Growing Demand and Stock Externalities. Journal of Environmental Economics and Management, 8: 105-117. Brutland, G.H. (1987). The U.N. World Commission on Environment and Development. Our Common Future. Oxford University Press. Chichilnisky, G. (1977a). Economic Development and Efficiency Criteria in the Satisfaction of the Basic Needs. Applied Mathematical Modeling, 1(6): 290-97. Chichilnisky, G. (1977b). Development Patterns and the International Order. Journal of International Affairs, 31(2): 275-304. Chichilnisky, G. (1996). An Axiomatic Approach to Sustainable Development. Social Choice and Welfare, 13(2): 231-57. Chichilnisky, G. (1997). What is sustainable development? Land Economics, 73(4): 467-91. Chichilnisky, G. (2009). Avoiding Extinction: Equal Treatment of the Present and the Future. 34

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Economics: The Open-Access, Open-Assessment E-Journal, 3: 2009-32. Dasgupta, P. S. & Heal, G. M. (1979). Economic Theory and Exhaustible Resources. Oxford University Press. Heal, G. (1998). Valuing the Future. Economic Theory and Sustainability. Columbia University Press. Herrera, A., Scolnik, G., Chichilnisky, G. (1976). Catastrophe or New Society: A Latin American World Model (The Bariloche Model). Ottawa, Canada: International Development Research Center. Kevane, M. (1994). Can There Be an “Identity Economics”? Mimeo. Harvard Academy for International and Area Studies. Krautkraemer, J.A. (1985). Optimal Growth Resource Amenities and the Preservation of Natural Environments. Review of Economic Studies, 52: 153-170. Lafforgue, G. (2005). Uncertainty and Amenity Values in Renewable Resource Economics. Environmental and Resource Economics, 31: 369-383. Landa, J. T. (1981). A Theory of the Ethnically Homogeneous Middleman Group: An Institutional Alternative to Contract Law. Journal of Legal Studies, 10 (2): 349-362. Landa, J. T. (1994). Trust, ethnicity, and identity: The new institutional economics of ethnic trading networks, contract law, and gift-exchange. Ann Arbor, MI: University of Michigan Press. Ramsey, F. (1928). A Mathematical Theory of Saving. Economic Journal, 38: 543-559. Rawls, J.(1972). A Theory of Justice. Oxford University Press. Wirl, F. (1999). Complex, dynamic environmental policies. Resource and Energy Economics, 21: 1941. Wirl, F. (2004). Thresholds in concave renewable resource models. Ecological Economics, 48: 259267.

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NOTES  Isabel Almudi is Assistant Professor at the Economics Department of the University of Zaragoza (Spain). Julio Sánchez Chóliz is Professor at the Economics Department of the University of Zaragoza (Spain). For correspondence: Isabel Almudi. Address: Dep. Analisis Economico. Gran Via, 2-4. Universidad de Zaragoza (Spain). E-mail: [email protected]

[1] For critical arguments see, among others, Ramsey (1928); Beltratti et al. (1995); Chichilnisky (1997); Heal (1998). [2] Sustainable Development definition was based on “the satisfaction of basic needs” concept, introduced by Chichilnisky (1977a, 1977b) and Herrera, Scolnik and Chichilnisky (1976) in the Bariloche Model. [3] See in this issue the paper of Barrera et al. as an example for natural resources. [4] In the realm of natural resources, it is unquestionable that new concepts related to the environment have appeared over the last 40 years: "green" products, renewable energy, responsible consumption, etc. This suggests the parallel emergence of a more or less generalized common collective identity relating to the environment. [5] The authors thank Graciela Chichilnisky for this comment. [6] As Akerlof and Kranton (2000) assume. [7] We incorporate the personal identity function within the tradition of Berck (1981); Krautkraemer (1985); Chichilnisky (1997); Beltratti et al. (1993); (1995); (1998); Lafforgue (2005); Wirl (1999); (2004). [8] The greater environmental sensitivity is expressed with i

i

u 1 (c )

j

j

i

j

D (˜)

u 1 (c ) and u 2 ( s )

and u3 (˜), because we assume that:

u 2 (s j ) i

[9] Then, the problem is the same as in Beltratti et al. (1998); Heal (1998). [10] Dasgupta and Heal (1979) and the references quoted therein, offer a detailed demonstration of this result.

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