DBJ Research Center on Global Warming
Discussion Paper Series No. 46 (8/2013)

Emission Trading or Proportional Carbon Tax:
A Quest for More Efficacious Emission 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.

白紙

Emission Trading or Proportional Carbon Tax:
A Quest for More Efficacious Emission Control
Masayuki Otaki
Institute of Social Science, University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
Abstract
Background: Two alternative procedures currently exist for efficient carbon dioxide(CO2) emissions control: carbon emission trading and proportional carbon tax. This
article explores which of the two is more desirable. A prominently distinct feature ofthese schemes is unilateral, bilateral or universal trade. Carbon emission trading isbilateral in the sense that a country purchases the emission right from its specificcounterpart. If planned optimally, the proportional carbon tax is universal because eachemitting country is equally levied a fixed rate depending on its CO2 emission weight,
although the tax rate should be determined consummately after referring to mutual
negative externalities. This article examines how these differences affect the efficacy of
the CO2 emission control.
Results: Compared to emission trading, the equilibrium under a universal proportional
carbon tax achieves Pareto superior allocation. In addition, the price of a unit of CO2
emissions is much higher in the case of a universal proportional carbon tax than forbilateral emissions trading. This is because although the bilateral emissions trading
scheme can attain the Pareto efficient allocation between the two concerned countries,
the mechanism does not consider the negative externalities of these countries’ emissions
to the rest of the world. Indeed, such trading does not curb emissions; rather, it
generates net additional emissions.
On the other hand, as the name suggests, a universal proportional carbon tax ratewould be applied globally, and as far as it is obeyed by the modified Samuelson (1954)
rule, the secondary negative externalities arising from the emissions trading can be
entirely excluded. Thus, the first best allocation is achieved.
Conclusions: This paper reveals the superiority of a universal proportional carbon tax
over the emissions trading in the efficacy of CO2 emissions control. Although deciding
and abiding by a universal tax rate is likely to require great political will, any efforts inthis direction will be worthwhile. Unlike emissions trading which relies solely on
private economic incentives, a universal proportional carbon tax is the urgent need of1the hour for limiting the advance of global warming.
Keywords: CO2 emissions, Proportional carbon tax, Emissions trading, Modified
Samuelson Rule, Nonlinear carbon tax

2

Background
Carbon markets are jerry-built in nature. They cannot be sufficiently sustained
without government assistance and intervention. According to Lovins and Cohen (2011,
p.227), ``In the wake of the world’s failure to agree on a new trading legislation in
Copenhagen and the U.S. Senate failure to pass binding legislation, the price of carbon
fell drastically. In the United States it fell almost to nothing. In Europe, the price ofcarbon fell from a high of € 25 to € 8 a ton. By October 2010, prices had started torebound , hitting € 12. Subsequent measures have only strengthened the price per ton.’’First, this article explores the source of such fragility. We find that the decision of theeconomic agents emitting carbon dioxide (CO2) is isolated from direct and indirect
damages due to such emissions, owing to the genuine property of their external
diseconomy of the emission, therefore, at the very least, the market cannot remainsustainable without expectations of strengthened regulation in the future.
Second, we examine the properties of bilateral offset carbon trading, which is currently
the most popular trading scheme and is not subject to regulation concerning total
emission amounts. Unlike Otaki (2013), although the analysis is static, the extension to
dynamic analysis is not impossible, since the latter is an application of cap trading
which regulates total emissions. Moreover, analysis pertaining to cap trading, which
seems to be popular within the European market, requires the consideration of only one
additional constraint concerning total emissions in the model.
Finally, we compare the function of bilateral carbon trading with that of a universal
proportional carbon tax. The article concludes that a wide-ranged universal
proportional carbon tax is superior to bilateral carbon trading, although the procedureof income redistribution between developed and developing countries becomes rather
complicated for a proportional tax, because some progressive or digressive tax rateshould be adopted.
Results and Discussions
The Model
Assume that two types of countries exist. One of the countries is an advanced
industrialized or developed country which emits CO2. The other country is a developing
country that does not emit CO2. There are n pairs of a developed and a developing
country d , u  within which the emission right is traded, where d and u are the
j

j

j

j

j th developed and developing country respectively, comprising the j th block of emission
trading.
3

iEach country has the same utility function U j : 
U ij  c ij   e1d , , e dj , , end , e1u , , euj , , enu  c ij   e d , eu ,





(1)

i

where c j denotes the consumption level of the j th country that belongs to type i i  d , u  .  represents the disutility from the CO2 emission via the production processof a consumption good. We assume that  is linear homogenous, quasi concave andsymmetric in the following sense. That is,









 , elk , , elk' '    , elk' ' , , elk  , k , k ', l , l '.

(2)This symmetric assumption implies that the disutility derived from the emission does
not depend on where it is emitted. It is a plausible assumption when we consider the
diffusion speed of CO2 in the atmosphere.
The semi-reduced form production function, which represents the relationship between
the consumption c and the adjoined emission e , isc   F  e  ,  is some positive constant.









F '  0, F ''  0, e  e .F '  F ''  0, e  e .

(3)

We assume that only developed countries possess the production technology. No
developing country can access such an opportunity until the bilateral emission trading
is settled or some type of proportional carbon tax is levied.
The Complete Laissez-Faire Situation
Assume that, in the laissez-faire situation, the decision of the production sectors
within a country are separated from the consumer sector; they do not consider thenuisance incurred by their emission. Hence, they produce goods which amount to themaximum level F e . In such a case, the unit price of carbon is zero.
As noted by Lovins and Cohen (2011, ch.8) and evidenced by history, this fact implies
that without government intervention, the carbon market will surely collapse and theunit price of carbon will fall to become negligibly.
Unilateral Proportional Carbon Tax:

Implementation of the Nash Equilibrium4Formatting a Unilateral Proportional Carbon Tax
Moving from the complete laissez-faire situation, each country is separately
incentivized to levy a carbon tax to suppress its emissions for its own wellbeing, andthus, a unilateral proportional carbon tax is formatted by the solution of the following
optimization problem:  
max ek  F elk   elk , elk *  , elk *  elk '* , e2* , , el*1 , el*1 , , en* ,

l 

*
e j  e dj * , euj * .



 










Since the solution of this problem satisfies

 F '  elk *  | 1,c1d  lk  n(4)

we find that the implemented allocation by a unilateral carbon tax is identical to that
implemented by the Nash equilibrium of an international game concerning CO2
emission. In this sense, a unilateral proportional carbon tax belongs to a kind ofinternational laissez-faire scheme and does not require any cooperative behavior
between countries. However, it is evident that the resource allocation is improved
compared with the complete laissez-faire situation, because the equilibrium emissions
are determined by weaving the social disutility from excess emissions.
Bilateral Emissions Trading

Formatting the Bilateral Emission Trading Scheme
Bilateral emissions trading between the developed and the developing country in the jth pair is defined by the following optimization problem. That is,

 

 





max ed ,eu , P { F edj   F euj  P   edj , euj , e1d * , e1u*  end * , enu* 
j

j





  P   euj , edj , e1d * , e1u*  end * , enu*  U },where P is the total payment for the carbon emissions. * denotes the optimal contract
emissions of the other pairs.  is the Lagrangean multiplier of this problem. U denotes
the reservation utility of the corresponding developing country. The first term of (4) isthe developed-country’s utility derived from this emission trading, and the term within
the square brackets of the second term is the net welfare gain of the developing country.
Let us denote

 
l
j

elj
n

e

l , k 1

5

l
k



elj
E

.

By using  j and E ,  can be transformed intol    d , u , E    E.

Then by the symmetry of  , the optimality condition for the above-mentioned contract
problem under perfect information and symmetric equilibrium can be represented as  2 c |F ' elk * l1

 lj* 1n

, j , l , P*  U  2ne* | l* 1 .
j 

(5)

n

Universal Proportional Carbon Tax

Formatting a Universal Proportional Carbon Tax Scheme
A universal proportional carbon tax is formatted by the following optimization
problem:max   F e1d    E   lk  F elk      E  U m   .
( l , k )  1, d 



 



 

It is clear that the attained allocation under such a scheme is Pareto efficient bydefinition. By the symmetry of the problem, it is also clear that every Lagrangean
multiplier under optimal planning takes the value unityi. Thus, we obtain the following
formula concerning the optimal emission:F '  e*  n | c1d  lj* 1n(7)This is the modified Samuelson (1954) rule concerning the optimal public good (bad)provision: The marginal benefit accrued from the country’s emissions should beequalized to the sum of the marginal disutility diffused all over the world. The
right-hand side of (7) is the optimal tax rate common to all constituents.
Welfare Ordering for Various Emission Suppressing Measures
Although it is clear that a proportional carbon tax, with the rate as is expressed by (7),is the first-best policy, how are the other two measures ordered in terms of Pareto
efficiency? We can deal with this problem by using the symmetry and linear
homogeneity of  .
The utility of each country U lk can be written as c |U lk  j    F e*  j  

*
l
k* 1
1 l  nE *  j    F e*  j   n

6


*
|
* e  j  .
c1l  lk*  1n

(8)

Employing the envelop theorem,dU lk  j 
dj
dU lk  j 
dj
dU lk  j 
dj




e*  j   0, 1  j  n |
l k 1c1 l n

 0,  j  n e*  j   0,  n  j |
l k 1c1 l nholds. Thus, it is clear from the above equations that the bilateral emission trading
scheme Pareto-dominates a unilateral carbon tax scheme, whereby a country can set a
proportional carbon tax rate at its discretion.
This fact implies that although a proportional carbon tax possibly attains the first-best
allocation, emission trading is the second-best measure, unless all counties concur
about the seriousness of global warming, in which case a much higher carbon tax rate
than that in the unilateral case can be adopted. In addition, since n is likely far exceed
two, the suppression effect of emission trading is estimated to be rather restrictive fromthe view point of the first-best allocation.
Welfare Analysis of Emission-Saving Technological Progress
Consider the effect of emission-saving technological progress to the world economy as awhole. This progress is expressed by an increase in  in this model. Before proceeding to
the general equilibrium analysis, we must note that every trading pair increases
emissions in conjunction with technological progress. Although it seems to be
counterintuitive, if we note the fact that technological progress makes the imputed priceof CO2 cheaper as shown by the right-hand side of the left-half of (4), (5), and (7), it isnatural that emission-saving technological progress conversely heightens theaccumulation of CO2.
With this precaution in mind, we shall proceed with the general equilibrium analysis,
into which the mutual negative externalities between trading pairs are woven. Then,from (8) and the envelop theorem, we obtain

de* *

1
de
e , j , F e*   n  j  d | k 1
  F e*    , de1 l n d 
d
(9)


dU dj

*

 

 

dU nd
 0,
d
where is the elasticity of the emission volume to the unit of the technological progress.
7 

Since  F e*    0, if is small enough and the increase in the emission generated
by the technological progress is not so seriousii, the advance in the emission-saving
technology improves worldwide utility, although this advance also increases the total
amount of CO2 emissions. This fact is underscored in that when we extend the scope ofanalysis to dynamic and intergenerational emission allocation (e.g., Otaki 2013), we
may have to modify the obtained result, because the acceleration in emissions diffuses
the negative externality to future generations.
In other words, although the emission-saving technological progress lowers theimputed price of CO2 and stimulates the current generation’s consumption, suchcurrent prosperity may conversely worsen the descendants’ utility via the resulting
massive emissions. However, such a dynamic prospect is beyond of the scope of thisarticle, and it requires itself solving the simultaneous optimization concerningintertemporal and international emission problems.
On the Income Distribution between Countries: The Possibility of Nonlinear Pricing
Thus far, this article has assumed that a developed country directly invests in thecorresponding developing country and that it receives revenues after deducting thecarbon tax. Thus, F  e*  j    j 

| k 1 e*  j    F  e*     j   e*  j  ,le1  l  nwhere  j is the carbon tax rate, which is identical to the unit carbon price in emission
trading. Hence, the developing country obtains tax revenues R  j  , which amounts toR  j     j   e*  j  ,from the investing developed country.
Since every constraint concerning the joint utility from such a trading scheme binds
whenever planning is optimal, the net surplus from the trading in terms of consumption
becomes U

m iii.

Although we have not yet analyzed the possibility of additional

lump-sum transfer from the investing developed country to its counterpart developing
country (or the transfer inverted direction, which is possible if the tax payment is tooheavy for the developed country), one cannot envisage a universal proportional carbon
tax without some fair division of the surplus earned by direct investment through thislump-sum transfer, specifically because the standard of living of the remitting /recipient8

country decisively depends on its share of this surplus (Uzawa 2003).
Hereafter this article analyzes both directions of the transfer and clarifies how thedirection affects the effective tax rate. First, consider the transfer from the developed
country to the developing country. This is an application of nonlinear pricing, whichappears in basic microeconomics (e.g., see Tirole 1988). Let the sum of the transfer be s .
Then, the total payment of a developed country to her counterpart T becomess 
T    j  e  j   s    j    e  j .
e  j  The term within the square brackets is the effective tax rate, which is illustrated by
Figure 1. Thus, the effective tax rate is digressive although such a transfer enriches thedeveloping country. This is owing to the economy of scale from the de-facto massive
purchase of the right of emission.
Figure 2 illustrates the locus of the effective tax rate for the inverted transfer from thedeveloping country to the developed country ( s  0) . It is apparent that the effective tax
rate becomes progressive despite impoverishing the developing country. The progressive

effective tax rate owes its existence entirely to the diseconomy of scale concerningemissions. However, we must still note that whether the effective tax rate is progressiveor otherwise does not affect the efficacy of emission allocation. Moreover, this discussion
exemplifies that the progressive tax rate is not necessarily advantageous to thedeveloping country.

Conclusions
This article compared the static efficiency of consumption/emission allocation of threealternative emission control measures (proportional carbon tax (unilateral or universal)
and emission trading)iv. A unilateral carbon tax, akin to symmetric Nash equilibrium of
the worldwide consumption/emission game, is less efficient than bilateral carbon
emission trading. Although a universal proportional carbon tax, into which the entire
negative externalities of emissions of a country are woven, achieves the Pareto efficient
allocation, realizing such an ideal tax system at this time appears very difficult. That is
because the tax rate would become extremely high, much higher than the incumbent
tax rate or price of emission right. As such, gradualism seems inevitable. The transition
from bilateral to multilateral emission trading is desirable. Thus, a universal
proportional carbon tax should be considered as the ultimate solution and politicalarena are needed to gradually establish the system.
9

Finally, it is inevitable that an emission-saving technological progress stimulates not
only consumption but also emissions. Such technological progress cheapens the imputed
price of CO2. Although this possibly bodes well the current generation, it accelerates
CO2 accumulation and translates into a negative inheritance for our descendents.

Methods
Three measures for suppressing CO2 emission are compared from the viewpoint of
Pareto efficiency: a unilateral proportional carbon tax, a bilateral emission trading, anda universal proportional carbon tax. Heightened consumption within a country
increases its utility at the cost of scattering more voluminous CO2 all over the world andaccelerating global warming. In this sense, there is a serious trade-off between
consumption and CO2 emissions, and thus this induces excess consumption/emission
owing to the negative externalities, which are inherent to emissions.
The above three measures are modeled using a basic social planning theory undercertainty, and are developed for suppressing excessive economic activity. The
characteristics of these measures are classified by the extent of the negative
externalities which a developed country cares to address when it emits CO2. A unilateral
proportional carbon tax scheme only concerns with the country’s own disutility from its
emissions. A bilateral emission trading scheme limits the country’s concern to itself andits counterpart developing country, while a universal proportional carbon tax requests
each affiliate to calculate the worldwide disutility caused by the country’s emissions.
Competing interests
The author declares that he has no competing interests.
Author’s contribution
MO carried out the modeling and analyzed the model. He also drafted the manuscript.
MO read and approved the manuscript.
Acknowledgement
The author is thankful to Susumu Cato for his incisive and constructive comments.
References
Lovins, L. and Cohen, B. (2011) Climate Capitalism: Capitalism in the Age of Climate
Change, Hill and Wang, NY, USA.
10

Otaki, M. (2013) Endogenous social discount rate, proportional carbon tax, andsustainability: do we have the right to discount future generations’ utility?
Environmental Systems Research 2.
Samuelson, P. (1954) The pure theory of public expenditure. Review of Economic
Statistics 36: 387-389.
Tirole, J. (1988) The Theory of Industrial Organization. MIT Press, Massachusetts,
USA.
Uzawa, H. (2003) Economic Theory and Global Warming, Cambridge University Press,
Cambridge, UK.

11

Endnotesi

The first-order condition requires that



 

lk F '  elk *   1    jk j l ,k 1|d k 1jn.

Accordingly,  j  1, j , k satisfies this condition.
k

If  j '  1 for some p pairs of  j '', k '' ,k'for  j '', k '' .
Otherwise
1 
1    jk   1 
k '' 
s ''  j ,k 

n p



 j ', k '   j '', k ''

sk''''

 jk''holds


1 
1    jk    n  p    sk' ' holds.
k l  j l ,k  l ', k 'Let us define S p 1  . Then, the symmetric assumption requires l '', k ''

k ''
s ''

n  p  S p  sk''''



k ''
s ''

  n  p   S p  sk''''  1.

This is a contradiction. Therefore, lk  1, l , k .
Otherwise, the welfare is conversely aggravated by the emission-saving technologicalprogress under the second-best emission control systems. This is a fallacy ofcomposition.
iii Since our utility function is quasi linear, the surplus is equivalent to that in terms of
the utility. In addition, since we presume that the equilibrium is symmetric, theiireservation utility U is endogenously determined. This result comes from the required
property that all optimal Lagrangean multipliers should take the value unity.
iv This article studies how far emissions of CO2 are permissible with the remaining
given volume of CO2 in the atmosphere. As such, the dynamic and cumulative effects of
CO2 are beyond the scope of this article. While Otaki (2013) considered this problem, heneglected the problem of the international efficient allocation of emissions.
m12τ (j) +

s
e(j)

τ (j)

e(j)
O

Figure1: The Eﬀective Tax Rate (Digressive Case: s > 0)1τ (j) +

s
e(j)

τ (j)

e(j)

O

Figure 2: The Eﬀective Tax Rate (Progressive Case: s < 0)1