DBJ Research Center on Global Warming
Discussion Paper Series No. 50 (10/2014)

Local Altruism as an Environmental Ethic
in CO2 Emissions Control
Masayuki Otaki

This paper is a preliminary material in the draft form to stimulate discussion
and comments from academics and policy makers.Views expressed in thispaper are those of the author(s) and do not necessarily reflect the official
views of the Development Bank of Japan.

Local Altruism as an Environmental Ethic in CO2 Emissions Control

Masayuki Otaki
Institute of Social Sciences
University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
7-3-1 Hongo Bunkyo
Tokyo, 113-0033, Japan
Phone: +81-42-575-8663

Abstract
When considering emissions control problems associated with carbon dioxide (CO2),social planning over quite a long-term horizon is usually considered to be necessary
because it takes much time for the full absorption of CO2 by oceans and forests.
Sometimes the required time horizon even becomes infinite. In addition, as Otaki(2013a) reveals, the optimal social discount rate is zero at the stationary state. These
facts seem to impose patience beyond the limits of human cognition. However, thisstudy proves that the first-best emissions scenario can be achieved only by local
altruism, which is dubbed parentage. Parentage is defined as the action of applying zerosocial discount rate to its subsequent generation, and discounting the utility ofgenerations thereafter infinitely. In this sense, the nearly first-best emissions scenario
is feasible within the ordinal cognition and benevolence of human beings.

Keywords： Parentage as Local Altruism; Social Discount Rate;
Environmental Ethics
JEL Classifications: D61, D62, D64, H23, H43,

1

1. Introduction
CO2 emissions control is a difficult problem since it requires consistent decisions acrossgenerations. There are serious conflicts between the generations. For example, as longas individuals’ economic concerns are limited to their own lives, they ruthlessly discountfuture generations’ wellbeing. Their resultant excessive consumption is certainly
connected to excessive emissions of CO2, which have become a serious cause of global
warming.
Nevertheless, even though we can overcome such a difficulty in principle, to achieve
the precise control requires exorbitant information. By virtue of dynamic programming,
we must determine beforehand the terminal condition that corresponds to the wellbeingof generations belonging to the far future in the context of emissions control. Obviously,
this is beyond the cognitive ability of human beings.
This study provides an effective control method presume that the time horizon of anindividual is far shorter than the whole length of the time horizon of the history ofhuman beings. Parentage, which implies that love for children does not contradict
parents’ own economic concern, plays a crucial role. That is, individuals are assumed tobe myopic in the sense that their economic concern is limited to themselves and theirchildren. Parents are called devoted to their children whenever they apply the zero
discount rate to the children’s wellbeing. As precisely analyzed below, this is a crucial
condition for the global stability of CO2 emissions that requires no information
concerning the economic situations of far descendants. That is, the concept of devotion,
which is a stronger concept than parentage yet still remains within the cognitive abilityof people with common sense, can ultimately hinder excess consumption and emissions
that stem from selfish economic motives.
The rest of paper is constructed as follows. Section 2 deals with a laissez faire economy
in which there is no emissions control by using the method of sequential equilibrium
proposed by Kreps and Wilson (1982). Section 3 defines the first-best emissions control
under the stationary state originated by Otaki (2013a). In addition, this section reveals
the extent of divergence between the stationary state of the laissez faire economy andthat of the properly controlled economy. This fact acutely conveys the importance of CO2
emissions control. Section 4 considers how the economy can reach the first-best
allocation without imposing transcendent and stringent morals beyond the cognitive
ability of human beings. The concept of parentage and devotion play crucial roles.
Section 5 provides brief concluding remarks.

22. The Basic Structure of the Model and the Laissez Faire Economy
2.1 The Basic Structure of the Model
It is assumed that every individual lives in one period, and his or her utility function

U  ct , et  is defined as2
U  ct , et    ct  c    et2  , 0   .



(1)

where  represents how much importance individuals, who belong to generation t , puton the direct disutility from the accumulated emission of CO2, et , relatively to current
consumption, ct . While such a quadratic function seems quite a naïve formulation, it
can exclude inessential phenomena, which are peculiar to nonlinear difference
equations, such as limit cycle and chaos, completely. In terms of economics, thenonlinearity, which stems from the complexity of the utility function, is regarded as aless relevant problem compared with the problem that shall be deal with hereafter.
By the same token, the emission dynamics is assumed to obey the following simple
linear first order difference equation.

et   et 1  ct ,0    1 .

(2)

2.2 The Laissez Faire Economy as a Sequential Equilibrium
This subsection deals with the consumption/emission dynamics within the laisses faireeconomy. The laissez faire economy is defined as a sequential equilibrium in the sense of
Kreps and Wilson (1982). That is, it is assumed that generation t maximizes its utility
for a given previously accumulated CO2, et 1 . The first-order condition, which is derived
from (1) and (2), implies the following linear difference equation.

et 


c
.
et 1 
1
1

(3)

It is clear that this equation is stable and monotonously converges to the stationary
state:

 c , e    1 1   c , 1  1   c  .
*
L

(4)

*
L

3

The values in (4) comprise the pivotal point for acknowledging the acute necessity forthe emissions control.

3. The First-Best Allocation in the Stationary State
This section calculates the first-best allocation of consumption/accumulated emissionsin accordance with the method developed by Otaki (2013a). By assuming a proportional
carbon tax under the stationary state, it is straightforward from Figure 1 that themarginal substitution rate must be equal to the correct effective relative price of CO2,

1
, to the consumption good for achieving the first-best allocation1. This implies the1
following formula that the optimal planning must satisfy:c*  c e*
1c** FB *   1    eFB
 c   FB  eFB

.

 eFB
1
1
1 
1

(5)

Thus, the fist-best allocation at the stationary equilibrium is





1
c
*
*

.
c
,
e

c
,
 FB FB  

 
1 
2
 1
1   
1






(6)

Compared with (4), the consumption level of the lasses faire economy exceeds that of thefirst-best allocation by


2
1 

c
1




1

c
1 
1
1
*
L
*
FB

1 



1

(7)

times. As discussed below, it should be noted that  , which is the remaining ratio of
CO2 carried over from the previous generation, takes a positive value not far from unity2.

1

In addition, evaluating the correct (or socially justified) price of CO2 as

1
implies1that the optimal social discount is unity in the stationary state. For more detail, see
Otaki (2013a).
2 According to Tanaka (1993), CO2 emissions from fossil fuel combustion are estimatedat 5.4  0.5 giga-ton and the current absorption ability of oceans is generally estimated
2.0  0.8 giga-ton. However, Houghton et al. (1990) report that there serious
discrepancy exists in the emission/abruption of CO2 of the order of 1.6  1.4 giga-ton.
4Hence, as long as the emission problem is precarious and  takes a significant value,
one must acutely recognize the importance of emissions control.

4. Parentage as the Minimum Environmental Ethic
The first-best allocation shown in Section 3 imposes quite stringent and transcendent
ethics on human beings. Every generation must have deep sympathy for theirunforeseen far descendants in order to achieve the idealistic allocation. This criterion is
too strict and unfeasible in reality. Instead this section introduces the concept ofparentage, which implies that the concern of an individual with wellbeing is limited tothose of his/herself and their children. This concept is realistic and coheres with humanbeings’ cognitive limits in the sense that people can hold sympathy only with the next
generation, with whom they can communicate directly. This section analyzes how suchparentage contributes to emissions problem.
By using the utility function (1), parentage can be represented as the fact that eachindividual possesses the following utility function, V , which is contrastive to thesequential equilibrium case in Section 2. That is,V  ct , et |  , et 1 , et 1   U  ct , et    et 1   et  c  ,2(8)where  is the discount rate that is applied to the utility of the next generation’s
consumption. The utility function (8) implies the emission decision is diversified acrossgenerations. It is the current generation’s due to determine the current accumulated
CO2 emission, et , while future decisions are reserved for future generations. In addition,(8) implies that a generation is not at all directly concerned with the wellbeing of its
grandchildren and descendants thereafter. In this sense, such a decision process ismyopic. Thus, in some case, the adjustment process towards the stationary state
possibly becomes roundabout even though the process is stable.
The maximization of (8) under the constraint of (2) yields the following second-orderdifference equation:

et 1   2  1    et   et 1  1    c  0 .

(9)The eigen value,  , of the corresponding characteristic equation is  2 
24

 ,    1 


(10)

5





The corresponding stationary state cP* , eP* is





1


c


c
,
 cP* , eP*     1  1 
1  
1   (11)

It should be noted that, by comparing (11) with (6), the first-best allocation in thestationary state is achieved if parents possess deep parentage enough that they are

egalitarians to their children (   1 ), as long as the adjustment process is stable.
Accordingly, the stability of the economy with such deep parentage as an environmental
ethic is a quite important problem. Hereafter, the stability of the economy is defined asfollows:
Definition 1

The economy is stable if and only if it converges to the stationary state cP* , eP* for any

arbitrarily given initial condition, e1 .

Such a definition of stability implies that even though the initial parents faceunchangeable past accumulation of CO2 within the rational expectations equilibrium,
the economy converges to its stationary state if sufficient parentage is embedded to themind of an individual. In this sense, we, hereafter, search for the minimum ethic thatenables the economy to stabilize CO2 emissions autonomously.
Mathematically, Definition 1 is equivalent to the property that the smaller eigen
values,  , in (10) should be located within the interval  0,1 3. Thus, from (10), the3

By an elementary calculus, it can be ascertained that  takes real values. If people did

not abandon the larger eigen value,    2 
24

 , the value of which possibly

exceeds unity, the following vicious cycle would emerge in the economy: higher
emissions aggravate the initial condition of the next period. The next generation would
be compensated by increasing their consumption further, and this makes the excess
emissions problem more serious and so on. It is assumed that individuals are notunwise to be allured by such a devastating consumption explosion.
6

necessary and sufficient condition of stability is that  satisfies the following
inequality:  2 
24

  1.

(12)(12) is equivalent to0 1  
.
 1   (13)In addition, the following condition is necessary for keeping the stationary state (11) is
well defined.

0    1 .

(14)

The reason why   0 is contained within the above inequality is that the economy,
which corresponds to such case has been already analyzed in Section 2. It should be
noted that the egalitarian parentage   1 is located within this range. This induces thefollowing theorem concerning the role of parentage in the stability of the economy.
Theorem 1

Parents should be devoted to their children in the sense that they should apply zerosocial discount rate (i.e.,  =1) for stabilizing CO2 emission.
Theorem 1 implies that it is an acute environmental ethic for parents to have as muchconcern for their children’s wellbeing as that for themselves to stabilize emissions ofCO2, and this achieves the first-best allocation in the stationary equilibrium althoughsuch a long-run problem might be out of their scope.
In addition, some discussions are necessary about the properties of the social discountrate. First, excess devotion is harmful conversely in the sense that parents apply thenegative discount rate to their children’s wellbeing at least in the long run. This isbecause such self-sacrifice thwarts consumption excessively, even though emissions of
CO2 are controlled stringently.
Second, although a reliable value of the crucial parameter,  , is not yet obtained, thelocally-optimal social discount rate is zero independent of this value. This suggests thateven though precise knowledge concerning the circulation mechanism of CO2 is inprogress, it is social justice for parents to apply zero discount rate to their children’s
wellbeing.
Third, as exhibited in (6) and (11), the allocation approaches the first best together
7with a decrease in the social discount rate1
within the range 1,  . This implies thatas parents become more benevolent to their children, although not perfectly, more
efficient allocation is achieved in the long run.
Finally, although this is the most serious problem, even though the laissez faireeconomy converges to the stationary state, it should be emphasized that there is no
guarantee that such a stationary state is harmonious with the viability of human beings.
This implies that not only the parameter of the remaining ratio,  , but also that ofawareness of the environment,  , play crucial roles for the stability of the atmosphere.
In this sense, proper education on the environment is an acute political issue.

5. Concluding Remarks
This study considers how CO2 emissions should be effectively controlled within the
cognitive abilities of human beings. The role of parentage, which is defined as parents’partial altruism to their children, plays a crucial role. If parentage is perfect, parents
are devoted to their children, which means that parents apply zero local social discountrate limited to their children. Then, the first-best resource allocation is achieved.in thestable stationary state. Otherwise, some incentive schemes should be constructed forthe efficient control of CO2 emissions.
This theorem advocated that artificial carbon tax schemes and/or emissions trading,properties of which are analyzed by Otaki (2013a, 2013b), play only subsidiary roles asmeasures of emissions control. The most important role should be ascribed to theestablishment of environmental ethics that are deeply rooted in love for children.

References[1] Kreps, D. M. and R. Wilson (1982). Sequential equilibrium. Econometrica 50,
pp.863-894.
[2] Houghton, J., et al. (1990). Climate Change. Intergovernmental Panel on Climate
Change. Cambridge University Press, Cambridge, UK.
[3] Otaki, M. (2013a). The endogenous social discount rate, proportional carbon tax,
and sustainability: do we have the right to discount future generations’ utility?
8

Environmental Systems Research 2-1. doi:10.1186/2193-2697-2-1.
[4] Otaki, M. (2013b). Emission trading or proportional carbon tax: a quest for more
efficacious

emissioncontrol.

Environmental

Systems

Research2-8,

doi:10.1186/2193-2697-2-8.
[5] Tanaka, M. (1993). The mechanism of global warming (in Japanese). In Uzawa H.
and Kuninori M. (eds) Economic Analysis of Global Warming. University of Tokyo
Press, Tokyo, Japan.

9

e=

e

c
1−α

Indiﬀerence Curve

Increasing UtilitycOFigure 11

10Supplement
1. Egalitarian Allocation Does Not Imply Justice as Fairness
Sustainability in the sense of Pezzy (1997) and Vanderheiden (2008) does not imply
non-discounting future generations’ utility.
Such sustainability is defined by the following egalitarian inequalities:

U  ct  j , et  j   U , j .

(1)

Since under the initial conditione1  e ,and the feasibility conditionet  j   et  j 1  ct  j ,j ,(1) requires that the economy should stay at the initial position. That is,j .

et  j  e ,(2)

It is assumed that the allocation of the economic is initially located at an excess
consumption/emission stationary equilibrium, and thus,de c  1    e 1 c  1    e      c  1    e  ,
dc

1

(3)

holds.
The definition of sustainability (1) is advantageous for the initial generation andconservative in the sense that its incumbent interests are reflected as much as possible
when the intertemporal emissions allocation is considered. Since the optimized utility ofeach generation must satisfies

t  j









2
2
d 
d
c   et  j   et  j 1    et2 j   t  j 1c   et  j 1   et  j   0 ,
det  j 
det  j(4)

and the conditions (2) and (3), the Lagrangean multipliers satisfyt  j 1 1 c  1    e   1.
t  j  c  1    e(5)(5) implies that the seemingly egalitarian allocation represented by (1), does not meanthat the current generation, who plans the emissions control, takes future generations’utility fairly. Since (1) implies that it makes possible for the current generation, who
11faces excess consumption/emission, to advocate the status quo even though moredesirable allocations exist in the future, the applied social discount rate becomes
positive as shown by (5). As such, the progress of emission control is hindered if we
accept the Pezzy-Vanderheiden’s definition of sustainability.

2. Priorities of the Value Judgment Concerning CO2 Emissions Control and the Role of
Parentage
We assume that value judgments concerning CO2 emissions control obey the following
order.
I. An economy must converge to stable stationary state.
II. The path of CO2 emissions must satisfy the sustainability in the following
progressive sense. That is,

Ut  Ut 1    Ut  j    U * ,(6)where U is the utility of a generation in the stationary state. We must note that*

Pezzy-Vanderheiden’s definition of sustainability is a special case of our definition.
III. As long as I and II are satisfied, the planning that achieves higher U is desirable.
*

The first value judgment is imperative for stabilizing the atmosphere and the climatechange. The second order judgment concerning sustainability implies that theemissions control should progress incessantly as long as excess consumption/emission
prevails in an economy. The third order judgment asserts the importance of theefficiency of the long-run emissions control. An economy will reach the vicinity of thestationary state sooner or later, and stays there during all time thereafter. Accordingly,
for the future generations’ wellbeing, this condition should be entailed.
It is already shown the local altruism, which we call parentage, satisfies the conditions
I and III. What is left is to check whether such an ingenious emissions control satisfies
Condition II. Thus, the following theorem is obtained.
Theorem 2

The emissions control due to parentage, which is represented by (9) and   1 , satisfies
Condition 2.

12

ProofBy using the eigen value  , the utility of generation t can be written as c        t 1    2

Let us denote x  t 1

2 t 1.

(7), and differentiating (7) with respect to x , 4      2  x       c
is obtained. It is enough for completing the proof to show    . Then, Let f be definedasf  y    y 2   2  1    y   2 .
By the definition of the eigen value, f     0 . On the other hand,f     2   2  1       2   1   1        0 ,holds. Figure 2 illustrates the locus of f  y  . Thus, it is clear that    .

References
Pezzy, J. (1997). Sustainability constraint versus “optimality’’ versus intertemporalconcern, and axiom versus data. Land Economics 73: 448-446.
Vanderheiden, S. (2008). Atmospheric Justice: A Political Theory of Climate Change.
Oxford University Press, Oxford.

13

f

α
O

y

µ

Figure 2 Characteristic Equation

2

14