Download The trade-off between intra- and intergenerational equity in climate

11 May 2012
Preliminary draft
The trade-off between intra- and intergenerational equity
in climate policy1
by
Snorre Kverndokk 2, Eric Nævdal 3 and Linda Nøstbakken 4
Abstract
This paper focus on two equity aspects of climate policy, intra- and intergenerational equity,
that both have an important impact on the optimal response to global warming. We set up an
intertemporal two country model where each country suffers from global warming, but also
has inequality aversion within the generation. Preliminary results show that inequality
aversion within a generation leads to lower consumption in the rich country and higher
consumption in the poor country. This also gives higher investments in clean energy in the
rich region and higher investments in dirty energy in the poor region, giving a higher
abatement burden to the rich region in the baseline. These effects are reinforced in the social
optimum. However, consumption levels will converge in the long run. Finally, we study how
total emissions will depend on inequality aversion within a generation.
JEL codes: D31, Q54,
Keywords: Equity, inequality aversion, climate policy
1
This paper is funded by the program MILJØ2015 at the Research Council of Norway. The authors are
associated with CREE — Oslo Centre for Research on Environmentally friendly Energy. CREE is supported by
the Research Council of Norway.
2
The Ragnar Frisch Centre for Economic Research, Gaustadallèen 21, 0349 Oslo, Norway. E-mail:
[email protected].
3
The Ragnar Frisch Centre for Economic Research, Gaustadallèen 21, 0349 Oslo, Norway. E-mail:
[email protected].
4
Alberta School of Business, University of Alberta, 2-32D Business, Edmonton, Alberta, Canada T6G 2R6. Email: [email protected].
1
1. Introduction
While climate change for many years has been recognized as a threat to the future by most
scientists and politicians, there is still an ongoing debate on what to do about this. Researchers
may not agree on the optimal emission reductions even if they would agree on the natural
science background as well as the costs to abate greenhouse gas emissions. The reason is that
optimal emission reductions to a large degree depend on equity issues, and how we discount
future climate impacts is especially important, see Dasgupta (2008) for an overview, and
Anthoff and Tol (2009) for illustrations. However, ethical issues have not been fully explored
in economic analyzes as greenhouse gas (GHG) abatement not only affects the welfare
distribution between present and future generations, but also the distribution within a
generation such as between rich and poor countries. These two equity aspects are important
when studying optimal emissions reductions, and as we explain below, they may work in
different directions.
The two mentioned dimensions of equity in climate change polices are named
intragenerational and intergenerational. The first is mainly about how we should distribute
the burdens within a generation, either within the generation living today or in the future, see
Kverndokk and Rose (2008). Examples of this can be; who would suffer from climate change
(inaction) and how should the burdens of mitigation (action) be distributed? Over the next
decades, the world may face large climatic changes in form of increased temperature, sea
level rise, changed wind and precipitation patterns, more extreme weather, etc. (IPCC,
2007a). However, the damage of climate change will not be evenly distributed among
countries or within a country. Studies such as Tol et al., 2000; Tol, 2002a,b, and Yohe et al.,
2007 show that some sectors will lose from climate change while others will benefit, and poor
countries are likely to face relatively higher negative impacts than rich countries. Several
economic studies also show that the costs of action will vary among countries and sectors, and
that it is generally more expensive to abate the more energy efficient the economy is (see,
e.g., IPCC, 2007b). Policy instruments implemented to reduce greenhouse gas emissions will
also impose different burdens on people, and in general, economic instruments such as carbon
taxes, will often be regressive. I.e., the burden will be highest for the poorest, unless special
circumstances prevail (see, e.g., Bye et al, 2002).
2
While intragenerational equity is important, most of the equity debate on climate change
issues in the economic literature has been on intergenerational equity issues, i.e., distributions
across generations, focusing on how much emissions reduction we should aim for, or how
large the atmospheric GHG concentration or global mean temperature ceiling should be. This
involves the distribution of burdens between the current generation and future generations as
the burdens of mitigation - the costs - have to be taken by the present generation while the
benefits of mitigation will be felt by future generations.
There are several reasons for extensive mitigation today such as attitudes to risk as well as
concerns about catastrophic events (Weitzman, 2007a), but most of the discussion has been
about the appropriate discount rate to use for climate change policy decisions, as the optimal
level of abatement is very sensitive to the choice of discount rate. Discount rates are weights
put on future benefits of climate change policies in order to compare the future benefits to the
present costs. If we measure costs and benefits in consumption units, the main reasons for
discounting are that we may treat different generations differently (the pure rate of time
preferences, PRTP), and that the benefit of a consumption unit differs depending of the
consumption level. While a positive number on the PRTP means a discrimination of future
generations, the second argument is that a high level of consumption gives a low marginal
utility of an additional unit, represented by the elasticity of the marginal utility of
consumption (EMUC). Thus, a high consumption level for future generations may be an
argument for paying less attention to these generations, e.g., mitigate less as the benefits of
this in consumption units have low impact on future welfare levels. In this way the EMUC
represents intergenerational equity preferences. 5 Both the choice of the PRTP and the EMUC
then represent ethical choices. The consumption discount rate (CDR) that is used in economic
analyses combines these two arguments.
The choice of the appropriate discount rate has been a controversial issue for years. For
instance, the Stern Review (Stern, 2007) used a quite low CDR and, therefore, found a high
level of optimal abatement compared to other studies such as Nordhaus (1993) and Nordhaus
and Boyer (2000). This created a lively debate where, e.g., Nordhaus (2007) and Weitzman
(2007b) argued against the choice of the low discount rate based on observed values of the
5
Note that the EMUC is often referred to as the coefficient of risk aversion or the inequality aversion coefficient,
in addition to being the elasticity of the marginal utility of consumption. Thus, it plays a triple role. An attempt
to separate inequality aversion from risk aversion and to quantify them using experiments, is found in Carlsson
et al. (2005).
3
long-run return to capital. However, Heal (2009) shows that equality between CDR and the
return to capital only holds under idealized conditions, such as a one-good model and perfect
market assumptions such as lack of external effects, which obviously do not hold in the
presence of climate change. The one-good model has also been criticized by Hoel and Sterner
(2007) and Sterner and Persson (2008) who pointed out that if there is consumption of an
environmental good in addition to a produced consumption good, the environmental good will
become more scarce over time relative to the produced good as we experience more climate
change. This could be an argument for using a low and even negative CDR for environmental
services.
The problem is that most studies choose to focus on either an intragenerational or an
intergenerational distributive justice problem, and then assume that these problems are
autonomous, i.e., that they can be treated separately. However, choices that affect the
intergenerational distribution also have impacts on the intragenerational distribution between
rich and poor countries. As Heal (2009) points out there are two ways preferences for equality
affect the choice of action when it comes to climate change. First, a high EMUC will lead to
less aggressive action if we believe that consumption is growing over time. The reason is that
this makes future generations richer, and if we care about inequality between the present and
future generations, we place a lower value to the richer future generations (intergenerational
equity). But there is also another effect. The rich countries are mainly responsible for the
aggregate level of greenhouse gases in the atmosphere, while, as mentioned above, the poor
countries are likely to suffer most from climate change. Thus, if we put a low weight on future
outcomes, climate change is more likely to happen and poor countries may suffer more
(intragenerational equity). The gap between the welfare levels of the rich and the poor may,
therefore, be higher. Based on the latter effect, stronger preferences for equality should,
therefore, go in the direction of more action to prevent climate change.
These two effects of inequality aversion go in different directions and the impacts of stronger
preferences for equity on the level of greenhouse gas abatement are ambiguous. The problem
is that the global models that are used to determine the optimal level of greenhouse gas
emissions only take the first effect into account. This means that a greater preference for
equality in these models actually induces a low abatement level as found in, e.g., Nordhaus
and Boyer (2000).
4
So far we are not aware of any studies that take both inequality aversions into account when
finding the optimal abatement level for greenhouse gas emissions. Thus, this paper closes this
gap, by showing how inequality aversion may affect the optimal policy response to address
climate change. We ask the following question: is climate policy a good thing when policy
makers have preferences for both inter- and intragenerational equity? A simple model of two
regions, a rich and a poor, are set up that explicitly takes into account preferences for the two
aspects of equality, before finding the optimal climate policy. Preferences for
intragenerational equality will go in the direction of reducing consumption in the rich region
and increasing consumption in the poor region, which may both have impacts on emissions
that follow from production of an aggregate macro good. This socially optimal policy will be
compared to business-as-usual where there is no coordinated action, as well as to the optimal
policy under standard preferences in the economics of climate change, i.e., no explicit
modeling of inequality aversion within a generation. Finally, we study how inequality
aversion within a generation affects greenhouse gas emissions, i.e., the distribution of welfare
across generations.
2. A model of inequality aversion
2.1 The starting point
Following Rawls (1971), we consider a veil of ignorance situation where representative
individuals from prospective countries around the world meet to design the future
organization of the society. All individuals know that some countries are going to be rich and
some poor, and that economic activity will result in greenhouse gas emissions that will have a
negative impact on them in terms of climate change. However, they do not know if they are
going to live in a rich or in a poor country.
The question they would like to answer is: Is climate policy a good thing? The reason why we
ask this question is that climate policies may have different effects on distribution in an
intragenerational and an intergenerational perspective. Widespread mitigation today may
affect developing countries more negatively than industrialized countries as it may harm
economic growth and their possibility to develop, but may be good in the future as it reduces
future damages.
5
All the representative individuals behind the veil of ignorance care about inequality, but some
more than others; preferences are heterogeneous, but they do not change even when they learn
their position in the society. In addition to this, individuals have preferences about future
welfare (welfare of future generations) compared to the welfare of the present generation.
These preferences are represented by the concavity of the utility function; the elasticity of the
marginal utility of consumption, and the pure rate of time preference.
Behind the veil of ignorance, the representative individuals agree on the social planner’s
decision problem: Since climate change is a global problem, the representative individuals or
countries will not have the incentive to pursue the global social optimum. Thus, they decide to
follow the policy that comes out of the social planner’s optimization problem. Further, they
agree on the rate of pure time preferences: Since the social planner has to make decisions
across generations, they need to agree on a proper time preference rate.
In addition to this, the distribution of the different inequality parameters of the representative
individuals is known behind the veil of ignorance, and submitted as input to the social
planner. The social planner also takes into account other preferences, the development of the
environment, and the available technologies.
In the following we set up a simple model to illustrate the problems described above. We
simplify the set up by a considering a representative consumer or country of the poor and the
rich region and study their choices without the social contract before comparing those to the
decisions under the contract.
2.1 The basics of the model
The utility of a representative consumer/country in region i at time t is:
U i=
u (ci ,t , St ) − α i max ( c j ,t − ci ,t , 0 ) − βi max ( ci ,t − c j ,t , 0 )
,t
where c is consumption and S is the state of the environment. Preferences for equality are
modeled as inequality aversion following Fehr and Schmidt (1999). Based on this, consumers
have inequality aversion in such a way that they dislike having a higher consumption/income
than others, but they dislike even more having a lower consumption/income than other
consumers they compare themselves to. Let α be a parameter representing the negative feeling
6
(1)
of being worse off than other individuals, while β is the parameter representing the negative
feeling of being better off than other individuals. We then have that α ≥ β. We ignore strategic
interactions by assuming that each region, North and South, consists of many identical
countries that do not have any market power and cannot individually affect environmental
quality.
Without loss of generality, let us assume that the population sizes of the two regions are equal
and normalized to unity. Thus, ci is also per capita consumption in region i.
Further, we model each region's representative country's production of an aggregate macro
good, Yr,t, with two types of inputs, clean and dirty that are perfect substitutes,
=
Yr ,t Yr ,c ,t + Yr ,d ,t
(2)
Yr ,c ,t = Ac ,t K rγ,cc ,t
(3)
Yr ,d ,t = Ad ,t K rγ,dd ,t
(4)
where subscript c and d denote clean and dirty. Thus, the clean input is produced using
clean (green) capital, while the dirty input is produced using dirty (brown) capital. We assume
diminishing returns to scale so that ϒc and ϒd have values less than one. The productivity of
dirty capital is higher than for clean capital, i.e., Ad,t > Ac,t for all t . Further, we assume that
the technologies are exogenously given and constant over time ( Ajt = Aj , j = c, d ).
A country can invest in clean and dirty capital, with capital dynamics given by:
K r , j ,t +1 =
n, s, j =
c, d ,
(1 − δ j ) K r , j ,t + I r , j ,t , r =
where δ j is the capital depreciation rate .
The environment is modeled as a stock variable. It deteriorates with the aggregate use of the
dirty capital and regenerates naturally at a rate σ :
7
(5)
St +1 = σ S + (1 − σ ) St − ηYd ,t ,
(6)
Yd ,t is aggregated dirty input over the two regions, and η is emissions per unit of dirty input.
The environmental quality is constrained as follows: St ∈ 0, S  , where S is the level of
environmental quality in absence of pollution (dirty production). Note that in the absence of
pollution, S will converge asymptotically to S . Thus we treat climate change as a reversible
process in the long run.
Let n denote a representative consumer in the developed region (North) and s a
representative consumer in the developing region (South), where consumption initially is
larger in the North than in the South, i.e., cn ,1 > cs ,1 ., as North is endowed with more initial
productive capital than the South. Thus, we can write the utility functions as:
U n ,1= u ( cn ,1 , S1 ) − β n ( cn ,1 − cs ,1 )
(7)
U s ,1= u ( cs ,1 , S1 ) − α s ( cn ,1 − cs ,1 )
(8)
2.2 Business as usual
Now, assume that the policy makers in the different regions want to maximize the utility of a
representative consumer. Let us start with the business as usual problem (BAU), i.e., the
optimization problem of the policy makers when there is no coordinated action or
environmental agreement. This would be the situation without the social contract (social
optimum).
As mentioned above, consumption in n is larger than in s at the starting point, period 1, i.e.,
cn ,1 > cs ,1 . Would consumption in s ever be higher than consumption in n?
Assume first that α s = α n and β s = β n , i.e., all inequality preferences are equal. In this case
consumption in s will never exceed consumption in n in the non-cooperative Nash equilibria
in BAU, as otherwise cn ,1 > cs ,1 would not follow from utility optimization.
8
Then consider the case where α s ≠ α n and β s ≠ β n . Assume that environmental quality S is
considered exogenous, i.e., each country disregards its impact on the environment as it is so
small that its impact is approximately zero. In this case, if consumption in the initial period is
found as the non-cooperative Nash equilibria in BAU, we will reach a steady state BAU path
where consumption in s never exceeds consumption in n. If that was the case, n would not
initially consume more than s.
Can we achieve convergence of consumption in the two regions? Inequality aversion gives an
incentive to equalize consumption in the two regions, and we will discuss this further below.
Based on this, we can set up the BAU optimization problem as consumers maximizing
t=
t=
∞
∞
t −1
t −1
n ,t
n ,t
=t 1 =t 1
(9)
t=
t=
∞
∞
t −1
t −1
s ,t
s ,t
=t 1 =t 1
(10)
∑ρ
∑ρ
U
U
= ∑ρ
= ∑ρ
u ( c , St ) − β n ( cn ,t − cs ,t ) 


u ( c , St ) − α s ( cn ,t − cs ,t ) 


respectively, given the technology (2)-(5), environmental constraint (6) and the goods market
constraints cr ,t =Yr ,t − I r ,c ,t − I r ,d ,t , for r = n, s . ρ is the time preference rate, which is set equal
for both consumers in order not to have the effects of inequality aversion confounded by the
effects of discounting. Note that by substituting in for investment, I r ,c ,t and I r ,d ,t , using
equation (5), the goods market constraint for a country in region r can be rewritten as:
=
Yr ,t  K r ,c ,t +1 − (1 − δ c ) K r ,c ,t  +  K r ,d ,t +1 − (1 − δ d ) K r ,d ,t  + cr ,t .
(11)
To find the non-cooperative Nash equilibria for the two regions, we first define the
Lagrangian for the two regions. For region n the Lagrangian is
BAU
=
∞
∑ρ
t =1
t −1
{u(c
n ,t
, St ) − β n ( cn ,t − cs ,t ) 
}
+λn ,t Yn ,t ( K n ,c ,t , K n ,d ,t ) − K n ,c ,t +1 + (1 − δ c ) K n ,c ,t − K n ,d ,t +1 + (1 − δ d ) K n ,d ,t − cn ,t 
9
(12)
where λn,t > 0 is the shadow price on capital. Recall that we made the assumption of each
representative country being so small that its impact on the dynamics of environmental quality
is approximately zero. Hence, (12) is maximized over consumption, and dirty and clean
capital stocks, but not over environmental quality.
First order conditions are:
∂BAU
= ρ t −1 u 'ct − β n  − λ=
0
n ,t
∂cn ,t
{
}
(13)
 ∂Y
 
∂BAU

=
− δ c   − ρ t −1λnt 0
ρ t λn ,t +1  n ,t +1 + 1=
∂K n ,c ,t

 ∂K n ,c ,t +1
 
(14)
∂Yn ,t +1
∂BAU


ρ t λn ,t +1
δ d  − ρ t −1λn ,t 0
=
+1−
=
∂K n ,d ,t +1
∂K n ,d ,t +1


(15)
These first order conditions can also be written as:
[cn ,t ] :
u 'ct − β n =
λn ,t
(16)
 ∂Yn ,t +1

:
1
λ
ρλ
δ
+
−
[ K r ,c ,t=
]

+1
n ,t
n ,t +1
c
 ∂K n ,c ,t +1

(17)
 ∂Y

[ K r ,d ,=
λn ,t ρλn ,t +1  n ,t +1 + 1 − δ d 
t +1 ] :
 ∂K n ,d ,t +1

(18)
The first order conditions for region s are similar apart from the determination of consumption
which is found from:
[cs ,t ] :
u 'ct + α s =
λs ,t
(19)
Equations (16) and (19) show that in optimum, the present marginal benefit from increased
consumption should equal the opportunity cost of increasing the consumption, i.e., the
shadow price of capital.
10
Compared to standard preferences, i.e., without any inequality aversion (αr = βr = 0, r = n,s),
the marginal benefits of consumption would be lower in n in our case, while it will be higher
in s, indicating a lower level of consumption in n and a higher level of consumption in s. This
gives us the first proposition:
Proposition 1: Inequality aversion within one generation will reduce consumption in the
North in Business-As-Usual. For South we find the opposite result.
Equations (17) and (18) determine the investments in clean and dirty capital, respectively. The
interpretations of these equations are that the benefits of increased capital in period t should
equal the social costs of these capital investments. For both types of capital, the cost of
investment is lower consumption in the previous period.
Using equations (17) and (18), we can eliminate λn ,t , λn ,t +1 , and ρ . This yields the following
relationship between clean and dirty capital in region r = n, s :
∂Yr ,t
∂K r ,d ,t
−
∂Yr ,t
∂K r ,c ,t
δd − δc .
=
(20)
Equation (20) states that the country should adjusts its stock of clean and dirty capital so that
the difference in the marginal productivities of clean and dirty capital equals the difference in
capital depreciation rates. Hence, if clean and dirty capital depreciates at the same rate, the
country should adjust its capital stocks so that the marginal productivity of each type of
capital is the same. Given that dirty capital is more productive than clean capital, this implies
that all countries will have more dirty than clean capital in the business-as-usual case,
regardless of equality preferences.
How would inequality aversion affect investments in clean and dirty capital in the two regions
respectively? Consider first region n. In this region inequality aversion leads to a lower
consumption. This will further lead to lower investments in dirty capital and higher
investments in clean capital. The intuition behind this is that a lower consumption would not
require investments in the most effective capital (dirty) and also that investments in clean
capital will reduce emissions of greenhouse gases and thereby improve the environment. A
better environment will increase utility and therefore work as a substitute to lower
consumption in the utility function in n. In s, the intuition works the other way around and
11
investments in dirty capital will increase while investments in clean capital will be lower. In
general we cannot say whether this improves the environment compared to the case without
inequality aversion. [Note that the argument assumes that the consumers are aware of their
impacts on the environment. If they take the environment as exogenous as specified above, the
only way to reduce consumption without increasing consumption in the future is to invest in
the most ineffective technology, i.e., the clean technology].
This gives the following proposition:
Proposition 2: Inequality aversion within one generation will increase investments in clean
capital and reduce investments in dirty capital in the North in Business-As-Usual. For South
we find the opposite result.
2.3 The social contract
Within this modeling framework and with a social planner determining production,
consumption, and capital investment in every country (region), consumption in the North
would never exceed consumption in the South (per capita). Instead, we expect to see
convergence over time. Hence, we can use the simplified utility functions (7) and (8).
The social planner seeks to maximize the sum of discounted welfare across all countries,
where welfare in period t is defined as:
=
Wt u (cn ,t , St ) + u (cs ,t , St ) − ( β n + α s ) ( cn ,t − cs ,t )
(21)
The maximization problem can then be expressed as:
∞
max
{cnt ,cst , K nct +1 , K sct +1 , K ndt +1 , K sdt +1 , St +1 }

∑ ρ  ∑ u (c
t
r ,t
t =0
r

, St ) − φ ( cn ,t − cs ,t )  ,

where φ ≡ ( β n + α s ) , which is a constant, subject to (2), (5), (6), and the goods market
constraints (11).
The Lagrangian of the maximization problem (22) can now be expressed as follows:
12
(22)
SO
=
∞

∑ ρ ∑ u(c
t =1
t −1
r ,t
, St ) − φ ( cn ,t − cs ,t )
r
+ ∑ λr ,t Yr ,t ( K r ,c ,t , K r ,d ,t ) − K r ,c ,t +1 + (1 − δ c ) K r ,c ,t − K r ,d ,t +1 + (1 − δ d ) K r ,d ,t − cr ,t 
(23)
r


+ µt  St +1 − σ S − (1 − σ ) St + η ∑ Yr ,d ,t ( K r ,d ,t )  
r


First order conditions include:
[cn ,t ] :t = 0, 1, 2,⋅⋅⋅
 ∂u (cn ,t , St )

λn ,t
−φ  =


 ∂cn ,t

(24)
[cs ,t ] : t = 0, 1, 2,⋅⋅⋅
 ∂u (cs ,t , St )

λs ,t
+φ  =


 ∂cs ,t

(25)
[ K r ,d ,t ] : t = 1, 2,⋅⋅⋅
[ K r ,c ,t ] : t = 1, 2,⋅⋅⋅
[ St ] : t = 1, 2,⋅⋅⋅
 ∂Yr ,t
λr ,t 
 ∂K r ,d ,t

 ∂Yr ,d ,t 
−1
λr ,t −1 , r n, s
+ 1 − δ d  + µt η=
 ρ=

 ∂K r ,d ,t 
 ∂Yr ,t
(26)
λr ,t 

1
=
+ 1 − δ c  ρ −=
λr ,t −1 , r n, s
 ∂K r ,c ,t

(27)
 ∂u (cr ,t , St ) 
−1
∑
 = µt (1 − σ ) − ρ µt −1
∂
S
t
 r

(28)
Clearly, an important question when analyzing the optimal solution is whether consumption in
the two regions will in fact converge to the same level in the long run. We therefore start by
investigating this.
2.4 Will consumption levels converge in the long run?
In the long run, the economy will enter a steady state. In steady state there will be no growth
in any of the model variables since there is no technological progress. We can therefore
rewrite the first order conditions (24)-(28) for the steady-state levels of the variables:
 ∂u (cn , S )

λn
−φ  =

 ∂cn

(29)

 ∂u (cs , S )
+ φ  = λs

 ∂cs

(30)
13
 ∂Yr

 ∂Yr ,d 
−1
λr , r n, s
+ 1 − δ d  + µ η =
 ρ=
K
K
∂
∂

r ,d 
 r ,d


(31)
λr 
 ∂Yr

1
λr , r n, s
+ 1=
− δ c  ρ −=
 ∂K r ,c

(32)
 ∂u (cr , S ) 
−1
 ∑ ∂S = µ (1 − σ ) − ρ 
 r

(33)
λr 
From equation (32) it is clear that the long-run capital level must be the same in the two
regions. Since ρ and δ c are the same for both countries, the marginal product of clean capital
must be the same in the two regions, and with a production function that is increasing
monotonically in clean capital, this implies that K=
K s ,c ≡ K c* in equilibrium.
n ,c
Let us next consider the long-run levels of dirty capital, given by equation (31). Equation (31)
can be rewritten as follows:
 ∂Y
µ  ∂Y  1 
r
+ 1 − δ d + η r ,d  − =
r n, s
 0, =
 ∂K r ,d
λr  ∂K r ,d  ρ 

λr 
(34)
Hence, we can write:

λn  MPn ,d + 1 − δ d +

where MPr , j ≡

1
1
µ
µ
η MPn ,d =
 −  λs  MPs ,d + 1 − δ d + η MPs ,d  −  ,
λn
ρ
λs
ρ

(35)
∂Yr
is the marginal productivity of capital j in region r .
∂K r , j
Let us first assume that the south never catches up with the north, so that cn > cs in steady
state. From (29) and (30) we know that the shadow price of capital then must be lower in the
in the north than in the south ( λn < λs ), since north has a higher consumption level and thus a
lower marginal utility from goods consumption. Next, by introducing this into condition (35)
we see that the marginal productivity of capital must be higher in the north than in the south
for the condition to hold. However, if cn > cs it must also be the case that K nd > K sd , since we
already established that the level of clean capital is the same in the two regions in the long
run. Since the marginal productivity is decreasing in the level of dirty capital, this implies that
14
MPnd < MPsd when cn > cs . Hence, condition (35) cannot hold unless cn = cs , which means
that the consumption level of the two regions must converge in the long run. This gives us
Proposition 3:
Proposition 3: In the long run, the social optimal consumption levels of North and South will
converge.
2.5 Optimal policy
We have established that the optimal policy requires the consumption levels of the regions to
converge toward the same level in the long run. This implies that also the levels of dirty
capital in the two countries converge in the long run.
Let us now consider how inequality aversion affects the optimal consumption and capital
paths in the two regions. We start out by looking at optimal consumption. From and we
know that optimality requires that the marginal utility from goods consumption and inequality
( φ ) must equals the shadow value of capital. For the North, this implies that an additional
unit of consumption today yields higher utility from consumption but disutility from more
inequality. In the South, we have the opposite effect of inequality aversion. An additional unit
of consumption today increases utility both from consumption and from reduced inequality.
Consequently, introducing inequality aversion shifts the consumption path of the North
downwards compared to the case without inequality aversion, while the consumption path of
the South shifts upwards.
Next, by eliminating φ from the two equations, we find that: λn + λs = uc′n + uc′s . Hence, the
sum of the two regions' shadow values of capital must equal the sum of their marginal utility
of goods consumption. Hence, we do not need the shadow price of capital to equal the
marginal utility from consumption in each region, as long as the condition holds at the
aggregate level. This result holds regardless of preferences for equality.
Let us now compare the optimal consumption levels under the social contract with the BAU
levels of consumption by comparing equations (24) and (25) to equations (16) and (19). The
difference in the optimality conditions for consumption is that each country under the social
contract takes into account both countries disutility from inequality, not just its own disutility.
15
Hence, the North reduces (and the South increases) consumption more under the social
contract than in the business-as-usual case, all else equal. We summarize this result in
proposition 4:
Proposition 4: Inequality aversion within one generation will lead to a larger reduction in
consumption in the North and increase in consumption in the South under the social contract
than in the business-as-usual case.
For the North to reduce consumption and the South to increase consumption relative to the
case without inequality aversion, North invests relatively more in clean capital while South
invests relatively more in dirty capital. It follows that the North is taking a bigger hit to
improve the environment, which can be thought of as an indirect transfer of welfare from
North to South. This also implies that a stronger concern for equality causes the two regions'
consumption levels to converge faster.
Next, consider the optimal capital paths. By eliminating common terms from the two regions'
optimality conditions for clean capital (27) and rearranging, we obtain the following
condition:
∂Ys ,t
∂K s ,c ,t
−
∂Yn ,t
∂K n ,c ,t
=
λs ,t
λ
− n ,t .
ρλs ,t +1 ρλn ,t +1
(36)
In the short run, before the two regions converge at time t = t * , we know that the North has
more capital and a higher share of clean capital than the South. Hence, it must be that
K n ,c ,t > K s ,c ,t for t < t * , which implies that the marginal product of clean capital is higher in
the South than in the North over this time period. It follows that the left hand side of (36) is
positive. This implies that:
λs ,t λs ,t +1
>
,
λn ,t λn ,t +1
which means that the difference in the shadow value of capital between the south and the
north decreases over time until convergence is reached in the long run (at time t * ).
16
(37)
In a similar way, we can also find that the south has more dirty capital than the north, but that
there will be a convergence over time as the consumption levels also converges.
We can also do the same comparison between the social contract and BAU for regional
capital stocks. This involves comparing optimality conditions (27) and (28) to conditions (14)
and (15). The optimality condition for clean capital is the same in the two cases. However, the
optimality condition for dirty capital under the social contract includes an additional term
compared to the BAU case. This term represents the marginal effect of more dirty capital on
environmental quality. The inclusion of this term in the social contract case implies lower
investment in dirty capital under the social contract compared to the BAU case, since the
environmental externality is internalized.
2.6 The impact on the environment
How does inequality aversion across a generation affect greenhouse gas emissions? The
steady-state condition for environmental quality can be rearranged and expressed in terms of
the shadow prices of the environment:
µ=
uS′ ( cn , S ) + uS′ ( cs , S )
1 − σ −1 ρ
(38)
It follows that the steady-state value of the shadow price of the environment ( µ ) increases in
the marginal utility of environmental quality, given by the numerator in equation (38), and
increases in the replenishment rate of the environment and decreases in the discount factor.
Equation (38) also reveals that the steady-state level of environmental quality does not depend
directly on the regions’ preferences for equality ( φ ), but is indirectly affected by inequality
preferences through other variables, such as consumption.
As seen above inequality aversion will reduce emissions in North, but increase them in South
before the steady-state solution is reached. If there is a trade-off between inequality across and
within a generation, the introduction of the Fehr-Schmidt utility functions will increase
emissions, i.e.,
∂S
∂S
< 0 . Thus, to study the conditions for this to happen, we need to find
∂φ
∂φ
in the steady-state.
The steady state level of environmental quality is given by:
17
S∞ =
σ S − ∑η prYr ,d ,t ( K r ,d ,t )
r
(39)
σ
It is however hard to pin down exactly what the effect of inequality aversion is, so a
numerical experiment is performed. We choose a utility function of the form
U r = crθ1,t Stθ2
For numerical simplicity we also slightly modify preferences for inequality to be given by
−φ ( cn ,t − cs ,t ) . Solving for the social contract when φ = 0 and φ > 0 gives very different paths
2
for St. These are illustrated in Figure 1.
100
S without inequality aversion
S with inequality aversion
90
80
70
60
50
40
30
20
0
50
100
150
200
250
Although S has the same initial value, the case where there is inequality aversion clearly leads
to significantly lower levels of environmental quality.
3.6 Extensions
So far we have assumed constant population and technology. Would more realistic
assumptions change the conclusions above? Let us first start with BAU. The following can be
shown within the model:
Assume that population is increasing exogenously, but that the growth is higher in s than in n.
If we further assume that population also affects production as an input factor, and that we
have diminishing returns to scale, production per capita will go down, and the distance
18
between North and South when it comes to per capita production will widen. This will
reinforce the arguments above.
Now assume an exogenous technological change, i.e., an exogenous increase in Aj,t, j=c,d. If
the growth is equal for both technologies, the results above will not change. If there is higher
growth in the productivity of the green technology, the incentive to invest in this will be
higher.
Finally consider an endogenous technological change such as learning by doing, i.e., Ac,t and
Ad,t increase as a function of aggregated production of the clean or dirty input in the respective
region. In this case, the productivity of clean capital will be higher in North than in South and
the difference will increase over time. For dirty capital it will be the other way around.
3. Conclusions
While the results are only preliminary, they still show some interesting aspects about
inequality aversion and climate policy. Based on our modeling framework, inequality
aversion within a generation will reduce consumption and greenhouse gas emissions in the
rich region of the world, while consumption and emissions will increase in the poor region.
This follows from a transfer from dirty to clean capital in the rich region, while we have the
opposite transfer in the south. Whether total emissions will increase depends on the relative
size of the emissions reductions in the north compared to the increase in the south, which has
to be further analyzed. If total emissions increase, there is a trade-off between inequality
across and within generations.
In section 2 above, we asked the question: Is climate policy a good thing? In our model with
constant population and technology, a reduction in total greenhouse gas emissions will reduce
the burdens on future generations. If this increases the overall welfare in the model, climate
policies is good from an intertemporal equity perspective. This is the case in the model as the
social contract gives lower emissions than in BAU. The reason is that the shadow price of the
environment is taken into account in the social optimum and not in the optimization problems
for the single countries. However, whether climate policy is good or not in a world with
inequality aversion depends on how it is designed. We have shown that if the rich countries
take the bulk of abatement, we may achieve equity both within a generation and across
generations.
19
20
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