TRADEABLE EMISSIONS PERMITS, EMISSIONS TAXES AND GROWTH*
byBERTRAND CRETTEZ†
University of Franche-Comté, France
This paper uses a dynamic general equilibrium model with overlappinggenerations in order to analyse and to compare emissions taxes andtradeable emissions permits. Even in the context of a perfect environ-ment, i.e. with perfect information, perfect competition . . . , it is shownthat privately owned emissions permits have some disadvantages. Anequilibrium with emissions permits would certainly be better than alaissez-faire equilibrium since it would entail a lower pollution level.However, it is far from clear that an economy with pollution permitswould be preferable over an economy with emissions taxes. While in bothcases pollution would be lower, growth would be higher in an economywith emissions taxes. This is because emissions permits divert saving from ‘productive’ resources and have a negative impact on capital accumulation. This happens whatever the way emissions taxes are redistributed.
1 I
There is an increasing support for the view that tradeable emissions permitsare an efficient way to curb pollution in the years to come. This reflects agrowing consensus on the superior performance of price-based instrumentsover command-and-control measures in environmental policy.
In the literature, it has been shown that emissions permits and emissionsfees are equivalent instruments. Nonetheless, such a result holds under somestrong conditions, in particular under the assumption of perfect information(see Hammond (1999) for a general and critical discussion of the relativemerits of both instruments). When a planner lacks information, emissionspermits are generally superior (see Baumol and Oates, 1988; Cropper andOates, 1992).
Tradeable permits have three main advantages.1 The market in tradeablepermits discourages emissions since it increases the cost of polluting. It gives
The Manchester School Vol 72 No. 4 July 20041463–6786 443–462
*Manuscript received 22.11.02; final version received 8.10.03.† The author is grateful to a patient anonymous referee who helped him to improve both the
presentation of the model and the discussion of several results of this paper.1There may also be disadvantages. Analysing the Kyoto protocol, Wooders and Zissimos (2001)
criticize pollution permits from the point of view of developing countries. They show thatemissions permits may harm these countries. Indeed, the South may be unable to purchaseboth polluting plants and the right to pollute. This paper does not address internationalequity and trade issues.
companies incentives to reduce emissions for less than it would cost to buya permit. It allows companies to decide how and when to reduce emissions.The emissions market can be seen as a flexible tax system because, unliketaxes, it determines the costs of emissions according to supply and demand.
Tradeable emissions permits are a means to securitizing the biosphere,i.e. in capitalizing certain types of biosphere investment projects (Chi-chilnisky and Heal, 2000). As suggested by Chichilnisky and Heal (2000)these investments can yield higher returns than investments in physical capitalor even in human capital.
Tradeable emissions permits are then increasingly considered as a newkind of asset. Indeed, they endow their owners with the right to emit a givenamount of pollution for a certain amount of time. One can buy permits andsell them with a capital gain. One could also devise options and futures inorder to enhance the flexibility and the liquidity of markets for pollutionrights.
There has been a recent attempt at analysing emissions permits withinthe context of a dynamic general equilibrium model. A dynamic context isindeed needed: this is because emissions permits are assets and the workingof their markets necessarily involves the expectations of agents about thefuture fundamental variables. Beltratti (1995) gives the first analysis of emis-sions permits in a dynamic model with overlapping generations (see also Bel-tratti, 1998). He studied the market prices of emissions permits pointing outthe role of current and future ‘fundamental’ variables such as technology,preferences toward the public (environmental goods) and environmentalpolicy.
This paper contributes to the debate on the superiority of emissionspermits over emissions taxes using a model that is slightly different from that of Beltratti. This model is similar to the one used by Fisher and VanMarrewijk (1998). It is a two-sector overlapping generations model in whichclean air (a pure public good) is (over) used as a private input in production.
Even in the context of a perfect environment, i.e. with perfect informa-tion, perfect competition . . . , it is shown that privately owned emissionspermits have some disadvantages.
Certainly, an equilibrium with pollution permits would be better than alaissez-faire equilibrium, since it would entail a lower pollution level. But asfar as growth is concerned, it is better to rely upon pollutions taxes. Indeed,we will show that the switch from a laissez-faire economy to an economy withpermits will have no impact on growth, whereas introducing emissions taxeswould always enhance growth. The gist of the argument is as follows.
In a laissez-faire economy, the free use of pollution generates pureprofits. These profits arise since pollution is a private input, which is not paidfor. In an economy with permits, there is a price for polluting; hence firmsdo not make pure profits any more. On the other hand, the rents accrue toagents and are equal to the value of the pollution used by the firms.
In both economies there is a system of property rights that enablesagents to get pure profits or rents. In a laissez-faire economy, there are equi-ties or shares that give the property of firms (and hence the right to receivetheir profits). In an economy with permits, an owner of a permit receives therents.
In both systems, these rights may be sold and resold and savings is inpart invested in shares or in permits. Hence, savings has a financial compo-nent that, in equilibrium, is equal to the discounted value of the pollutionper unit of time. It turns out that this value is the same at any date in botheconomies.
In the presence of pollution taxes, there remains only one asset in theeconomy and it is productive. There are no permits, and as pure profits arenil, so is the value of shares. The whole savings is now invested in a produc-tive way. This savings may be more or less important, depending upon theway pollution taxes are rebated to private agents. Even if the proceeds of thetaxes were redistributed to old agents, hence discouraging savings, growthwould still be higher than in an economy with permits. Again, this is becausethere is only one asset (there is no more diversion of savings from productiveuses).
The analysis presented in this paper uses special assumptions (log-linearutility functions, Cobb–Douglas functions and linear production functions).This is done in order to simplify the analysis (which is restricted to the studyof steady-state equilibrium growth rates). But the relevance of the argumentdoes not seem to be restricted by these assumptions. In so far as a new kindof asset is introduced, it is important to ask if it will mainly redistributewealth across agents, or if it will mainly affect the amount and the uses ofavailable resources. Here, the important thing is that (1) pollution permits arenot a productive asset per se, and (2) pollution taxes may be redistributed ina way that favours growth (for instance, redistributing the proceeds of thetaxes to young generations will always enhance growth). In our opinion, thesefacts would still prevail in a more general model.
The paper is organized as follows. Section 2 presents the model for thisstudy. Section 3 focuses on the laissez-faire equilibrium. In Section 4 we studyequilibria with various redistribution schemes of emissions taxes using threedifferent cases. In the first case, emissions taxes finance a public good; in thesecond, emissions taxes are redistributed to old consumers, and in the third,they are redistributed to young consumers. Section 5 provides a comparisonof the various equilibria. It is shown that the growth rates in a laissez-faireeconomy and an economy with emissions permits are equal (albeit in thelatter economy, emissions levels are lower). It is also shown that the growthrate at the various equilibria with emissions taxes are always higher than thegrowth rate of an economy with emissions permits. Finally, the consequencesof the various schemes on pollution levels are examined. Section 6 providessome concluding remarks.
Tradeable Emissions Permits, Emissions Taxes and Growth 445
This paper makes use of a version of the overlapping generations model à laAllais (1947)–Diamond (1965). The economy consists of agents and firms.Agents are non-altruistic in the sense that they do not take care of the well-being of future generations.2
2.1 Agents
At each date t a constant number of identical agents are born. Without lossof generality, their number is normalized to one. Each individual lives for twoperiods, supplies, inelastically, his labour in youth and consumes in bothperiods.
An agent born in t has a lifetime utility function
(1)
with 0 < a < 1, b > 0.In the above equation ct and dt+1 are respectively the amounts of
consumption goods consumed during youth and old age; Et and Et+1 arerespectively the pollutions emitted at date t and t + 1. The parameter a is the propensity to save whereas b is a parameter of the disutility resultingfrom pollution, which affects agents during the course of their lives.
2.2 Firms
There are two sectors of production. In each sector firms act competitively.For simplicity let us consider that there are two representative firms, one ineach sector.
The representative firm in the first sector produces a capital good withcapital good (K1,t) and uses a Rebelo (1991) like production function:
(2)
where R > 1. Note that the equality Y1,t = Kt+1 implies that capital depreciatesfully within the period.
The representative firm in the second sector produces a consumptiongood according to a Cobb–Douglas function with three inputs: capital (K2,t),labour (Nt) and pollution emitted (Et):
(3)
Assume that 0 < a < 1, 0 < b < 1, a + b < 1. In this framework, -Et repre-sents the decrease in the aggregate clean air resulting from the production ofconsumption goods. Hence, pollution is a pure public bad that is a private
Y K N Et t t t2 21
, ,= - -a b a b
Y K RKt t t1 1 1, ,= =+
U a c bE a d bEt t t t t= -( ) -( ) + -( )+ +1 1 1ln ln
input in production. It is assumed that the total amount of clean air is inlimited supply in each period.3
The main reason for using the Rebelo production function is its tractability. Indeed this is the simplest setup in which there are constantreturns to scale with respect to all reproducible and non-reproducible factors.A more general setup could be used in which the inputs of the productionfunction of the reproducible factors would be precisely the reproduciblefactors. For simplicity, pollution would only be used in the production of theconsumption good. Using such a framework would make the analysis morecumbersome without changing the results qualitatively.
3 L- E W E P E T
This section focuses on a laissez-faire economy wherein there are neither emis-sions permits nor emissions taxes. Such an equilibrium has been studied byFisher and Van Marrewijk (1998) in a somewhat similar framework. In whatfollows their ‘simple example’ (see Fisher and Van Marrewijk, 1998, Section5) is adapted and restated in greater detail.
Since there are neither emissions permits nor emissions taxes, the free useof pollution generates pure profits. Pollution is indeed a pure public bad thatis a private input in production. In equilibrium, the values of the marginalproduct of capital and labour equal their prices. As the production functionin the consumption good sector displays constant returns to scale, and sincethere is an input (pollution) that is not paid for, there are pure profits (theamount of which will be a constant share of production: (1 - a - b)PtY2,t).
Following Laitner (1982) and Eaton (1989), let us assume that there is asystem of property rights, i.e. equities or shares, which allows private agentsto capture these pure profits (or rents): the owner of the firm gets pure profitsonce capital and labour have been paid for. It is assumed that the ownership of the firm may be bought and sold (through trade in equities),and that the old at date 0 are the initial owners of the firm. At this period,they sell the firm to young agents. In so far as there is no altruism, this seems to be a relevant assumption, at least in models of the overlapping gen-eration type (in order to buy assets, young agents must first save).4 This trade
Tradeable Emissions Permits, Emissions Taxes and Growth 447
3It is hence assumed that there is a maximum amount of pollution which can be physically sus-tained at each date. But details are not provided about the way clean air is produced natu-rally. The fact that clean air is limited at each date is justified for two reasons. First, it is notunrealistic; and second, without this assumption, pure profits or rents would be infinite.
4Laitner (1982) allows agents to save in order to buy shares whose values are the discountedsums of pure profits due to imperfect competition. Also, old agents are the sole owners ofnon-competitive firms. In the same spirit, in a model of public debt (e.g. Diamond, 1965),the initial owners of the debt are the old agents. However, when altruism prevails, it maybe relevant to assume that young agents inherit some assets in their youth. But this assump-tion is not made here.
in shares exists in all periods (i.e. at any date, old agents sell their equities toyoung agents).
From the preceding assumptions a firm’s equity is then the sum of thepresent value of rents (this stems from the fact that the actual value of anequity is the discounted value of next-period profits and next-period value ofthe firm).
The existence of firm’s equities is a very important assumption. Indeed,agents may invest their savings both in capital goods and in the firm’s equities. As a consequence, in equilibrium, equities crowd out investment in capital and have a negative impact on growth (this is because capital accumulation is the main (in fact the sole) engine of growth in our model).To put it differently, savings in equities does not increase the economy stockof productive resources. It just results in a transfer of resources directly from the young to the old generation.5
Let us now consider some notation. The capital good is the numéraire.Let Pt denote the price of the consumption good, and Wt, Pt, Qt be, respec-tively, the wage, the rent and the price of a firm. Sk,t is the part of savingsinvested in capital, and QtSQ,t is the remainder invested in equity.
3.1 Agents
An agent born at date t faces two budget constraints. In youth, the budgetconstraint is
(4)
The wage income finances consumption in youth and savings, which can beinvested in two assets.
In old age, the budget constraint is
(5)
Consumption in old age is financed by the returns on savings, the rents accruing to the owner of equities and the resale of the latter.
P d r S Q St t t k t t t Q t+ + + + += +( ) + +( )1 1 1 1 11 , ,P
5In this respect, the role of the stock market may appear to be strange. As noticed by a referee,in modern economies, ‘the stock market is the place for trading claims on productive capitalon the part of the owners’. It is certainly true that in order to increase their stock of capital,firms raise funds by issuing shares. But this is not always the case. The funds may be usedto buy an existing company (without increasing the capital stock at all). Also, when anagent buys a share, it is not always a new one. In both cases, there is a transfer of resourcesbetween agents (without affecting the capital stock of the economy). Hence, following themodelling strategy of Laitner (1982) (see also for example Eaton, 1989), it is possible tomake a peculiar distinction between productive and unproductive savings. But the distinc-tion is interesting in so far as there are pure profits. The idea is the following. If there isperfect competition, if all inputs are paid for, and if there are constants returns to scale,institutional details regarding the stock market are unimportant since pure profits are nil.
If agents are to invest in both capital and equity, it must be true that thetotal return in equity equals that on capital. Hence
(6)
Under (6), the two budget constraints therefore reduce to
(7)
The optimal life-cycle consumption choices are
(8)
(9)
and d0 is equal to
(10)
3.2 Firms
The profit of the representative firm producing the capital good is
(11)
It will produce any amount of capital whenever
(12)
The profit of the representative firm producing the consumption good is
(13)
The first-order optimality conditions are (under (12))
(14)
(15)
The rents accruing to the equity owners are then
P t t t tB PY= = - -( )2 21, ,a b
PY
NWt
t
ttb 2, =
PY
KRt
t
t
a 2
2
,
,
=
B PY r K W Nt t t t t t t2 2 21, , ,= - +( ) -
1+ =r Rt
B Y r K R r Kt t t t t t1 1 1 11 1, , , ,= - +( ) = - +( )[ ]
dr K Q
P00 0 0
0
1=
+( ) + + P 0
da r W
Ptt t
t+
+
+=
+( )1
1
1
1
ca WPt
t
t
=-( )1
PcP d
rWt t
t t
tt+
++
=+
+
1
11+1
QQ
rtt t
t
=+
++
+
1
11P +1
Tradeable Emissions Permits, Emissions Taxes and Growth 449
Definition 1: A perfect foresight laissez-faire equilibrium for an economy is asequence of prices
and a sequence of decisions
such that
(a) given the sequences of prices, decisions are given by equations (2), (8),(9), (14), (15);
(b) given the sequences of decisions, the sequences of prices clear all markets (consumption good, (new and old) capital goods, labour andequities):
(16)
(17)
(18)
(19)
(20)
(Note that (6) must also be satisfied.)
To compute a laissez-faire equilibrium we proceed as follows (one maywish to skip this technical study of the laissez-faire equilibrium). From theconsumption good market equilibrium condition (16), the budget constraintof old agents and the demands for the consumption good, one gets
Now defining Kt = YtK2,t, the above equation reduces to
(21)
From the assets markets equilibrium condition
W Pc K Qt t t t t- = ++1
Q a PY
aRK
aRK
t t t t
tt
tt
t
= -( ) +[ ]
= -( ) +[ ]
= -( ) +[ ]
a b
a ba
a ba
1
1
1
2
2
Y
Y
YY
,
,
1 122
22 2-( ) + + + - -( ) =a PY
PY K
KQ PY PYt t
t t t
tt t t t tb a a b,
,
,, ,
SQ t, = 1
K St k t+ =1 ,
Nt = 1
K K Kt t t= +1 2, ,
Y c dt t t2, = +
c d S S K K Nt t K t Q t t t t t, , , , , ,, , , ,1 2 0
By restricting ourselves to the case where Yt remains constant throughtime, the equilibrium value of Y can be found using the arbitrage equation(6). Indeed, from
and equation (21), it follows that
(23)
Rearranging, one obtains:
(24)
Solving for Y, one gets
(25)
And the laissez-faire equilibrium growth rate of the capital gl is6
(26)
One can recover the remaining laissez-faire equilibrium values byexpressing every variable as a function of Kt. Not surprisingly, the growthrate of the capital stock is an increasing function of the saving rate (a); it isan increasing function of the share of wages in the value of the consump-tion good (b) since savings is financed out of wages. Finally, the greater theproductivity in the capital good sector, the higher the growth rate of capital.
In a standard overlapping generations model à la Diamond (1965),competitive equilibria may not be Pareto-optimal. This prevails when theeconomy exhibits overaccumulation with respect to the golden rule capitalstock. In our economy, things are quite different. There is no golden rule ofcapital accumulation (since endogenous growth is possible). However, alaissez-faire equilibrium generally also fails to be efficient since clean air neednot be rationed efficiently. In contrast, since the values of all prices are given,profit is maximized when all available clean air is used (the amount of whichis assumed to be bounded at every date).
11 1
+ ( )gRa
al =- -
bb
Y = +1-
1ab
b
P Y
PYR a
aKK
Rt t
t t
t
t
+ + +=-( ) +[ ]
-( ) + + - -( ) = =1 2 1
2
111 1
,
,
a ba b a b
YY
YY
-1
a b a b a b1 1 12 1 2 1-( ) +[ ] = -( ) + + - -( )[ ] + +Y Ya RPY a P Yt t t t, ,
RQ Qt t t= ++ +P 1 1
K a PY a PY
RK
t t t t t t
t
tt
+ = - -( ) +[ ]
=-
1 2 21
1
b a b, ,YY
Y
Tradeable Emissions Permits, Emissions Taxes and Growth 451
Let us assume that a government implements a set of Pigovian pollutionemissions taxes, and that there is a well-defined sequence of pollution emis-sions that the government wants to achieve. In order to study such an issue,this paper is confined to three relevant cases.
In the first case, it is assumed that emissions taxes finance a public goodthat does not enter the utility function (or if it does, it enters in a separateway). In the second case, it is assumed that emissions taxes are rebated to oldagents at each date. In the third and last case, taxes are rebated to youngagents at each date.
Let us see the effects of redistributing the emissions taxes on the budgetconstraints of agents. For simplicity, let us assume that the emissions taxescould be redistributed to both old and young agents.
Formally, let Tty and Tt
o be respectively the lump-sum transfers redis-tributed to young and old agents at date t. Then, the budgetary constraintsof a newborn agent at date t are
and the intertemporal budget constraint faced by this agent is now
(27)
The way the government disposes of the proceeds of the taxes will be shownto be of some importance in the study of the growth rates. Note that wheneither T1
y or T ot+1 increases, the agent is wealthier. He may consume more in
both youth and old age. However, depending upon the time at which thetransfers occur, savings may be more or less increased.
4.1 The Emissions Taxes Finance a Public Good 7
Consider that the government implements a sequence of emissions taxes{pt}t≥0 (e.g. in order to implement a given sequence of Et). As a consequence,the input of clean air (pollution emitted) has an explicit price. As firms in the consumption good sector use a technology that exhibits constant returnsto scale, their profits are zero in equilibrium and so are the equities prices. The definition of a competitive equilibrium with non-redistributedemissions taxes is easily adapted from the definition of a laissez-faireequilibrium.
One can simply compute such an equilibrium. The consumption good
PcP d
rW T
Trt
t t
tt
t
t
++
= + ++
+ +
+
+
+
1 1
11
1
11 1y
o
Pc s W T P d R s Tt t t t t t t t t+ = + = ++ + + + +y oand 1 1 1 1 1
Using equation (14) and denoting Y1 the ratio Kt/K2,t (assumed to be constant), one has
(29)
Using the asset market equilibrium condition (Kt+1 = RK1,t = st) yields
(30)
So, the equilibrium growth rate g1 of the capital stock is
(31)
Again it is an increasing function of the saving rate (a), the share bof wages in the value of the production of the consumption good (from which savings is financed). It is a decreasing function of the share of capitalin the value of the production of the consumption good (the argument used to explain the positive effect of b runs here in the opposite way).The productivity of capital in the capital good sector has the same positiveinfluence.
4.2 The Emissions Taxes are Redistributed to Old Agents
In this case, one has to modify the budget constraints faced by agents. LetTt
o denote the (nominal) lump-sum transfers received by agents during oldage. The life-cycle budget constraint is now
(32)
The demand of the consumption good in youth is
(33)
where in equilibrium Tot+1 = (1 - a - b)Pt+1Y2,t+1.
Proceeding as in Section 4.1, the equilibrium growth rate of the capital stock is studied. The consumption good market equilibrium condition is
ca W T r
Ptt t t
t
=-( ) + +( )[ ]+ +1 11 1
o
PcP d
rW
Trt
t t
tt
t
t
++
= ++
+ +
+
+
+
1 1
1
1
11 1
o
1 1+ =+
gRa
ab
a b
KK
R Ra
aRa
at
t
+ =-
=+
=+
1 1 11
YY1
b ab a
ba b
Y1 1= +aba
1 2 2-( ) + = +( )a PY RK PYt t t t tb a b, ,
Tradeable Emissions Permits, Emissions Taxes and Growth 453
Using RKt = aPtY2,tKt/K2,t and Y2 = Kt/K2,t (assuming that Kt/K2,t is constantthrough time), one finally gets
(35)
As one also has
the equilibrium value of Y2 is the positive root of the following equation
(36)
As we shall see, the (positive) root of the above equation differs from theequilibrium growth rates of the economy wherein there is a public goodfinanced with emissions taxes. The other difference is that while emissionstaxes are of constant unit value, the pollution levels are not similar (more onthis below).
4.3 The Emissions Taxes are Redistributed to Young Agents
Let us now study the last case in which the emissions taxes are redistributedto young agents. One must take care to modify the budget constraint duringyouth. The latter is now simply
(37)
This affects the market of the consumption good equilibrium condition.Indeed, this equilibrium
(38)
is now
(39)1 2-( ) +( ) + =a W T RK PYt t t t ty
,
c d Yt t t+ = 2,
Pc s W Tt t t t t+ = + y
P a aY Y Y2 22
2( ) = - + - -( ) -( )[ ] - -( ) - -( ) =a a b a a b1 1 1 1 0
and the equilibrium growth rate of the capital stock is
(41)
5 E T, E P G
5.1 Emissions Taxes and Growth
An interesting issue is the consequences of emissions taxes on the growthrates. Indeed, the abatement of pollution through the use of emissions taxesresults in different growth rates depending on the way the latter are redistributed.
Comparing the growth rates is possible in the relatively simple frame-work used in this paper.
Proposition 1 (Comparison of the growth rates): One has
Proof: This ordering obtains after a simple computation. The only difficultpart of the proof consists in showing that gl < g2 and that g2 < g1 (see theAppendix for details).
To understand the result, let us first compare the growth rates obtainedwith emissions taxes.
Here what matters is the way these taxes are redistributed to agents (seeequation (32)). Indeed, when the proceeds of the taxes are redistributed toconsumers, they are wealthier. So consumptions in youth and old age mustincrease since they are normal goods.
When the taxes are redistributed to old agents, one gets a negative effecton savings. Indeed, agents foresee that they will be endowed with moreresources in old age so that they are less prone to accumulate assets (accord-ing to the life-cycle motive). Also, consumption in youth being a normalgood, it must increase. But since resources in youth do not increase, the risein consumption during youth must be financed by a decrease in savings.
This negative effect on savings does not hold when the proceeds of thetaxes finance a public good. Indeed, agents are not wealthier as a result oftransfers; accordingly, their consumptions are unaffected.
The redistribution scheme that is the more conducive to growth is theone that redistributes resources in favour of young agents. Since they aregiven more resources, agents feel wealthier and their consumption must
g g g gl < < <2 1 3
1113+ =-( )
+ -( )gRa
aa
a a
Y3 =- -( ) -( )1 1 1a a
a
Tradeable Emissions Permits, Emissions Taxes and Growth 455
increase as they are normal goods. But an increase in consumption duringold age must be financed by an increase in savings, the latter being financedout of the transfers.
It remains to be explained why in an economy with pollution taxes redis-tributed to old agents growth is higher than in a laissez-faire economy. In thecase of a laissez-faire equilibrium, a part of saving is invested in ‘unproduc-tive’ assets, namely shares or equities. ‘Unproductive’ is to be understoodfrom the point of view of the production of capital goods, i.e. the engine ofgrowth in the model. When a part of saving is diverted from the purchasesof capital goods, the latter accumulates at a slower rate and so does theeconomy. This effect is also present in an economy with imperfect competi-tion (e.g. Laitner, 1982) and private property of land (e.g. Crettez et al., 1997).However, this effect disappears in an economy with emissions taxes. Indeed,pure profits are simply nil, and so are the equilibrium values of equities.Capital is then the sole support of savings. Hence, although savings in aneconomy with emissions taxes could, in principle, be lower than in a laissez-faire economy, the productive savings would always be higher in the formerthan in the latter economy.
In order to be complete, one now has to study the case of the pollutionpermits.
5.2 Tradeable Pollution Permits and Growth
It is well known that under standard assumptions, emissions permits andemissions taxes are alike. However, a distinctive feature of tradeable emis-sions permits consists of the fact that, like other assets, they can be used asa store of value. Indeed, an emissions permit can be bought, sold and resoldat distant instants of time.
Let us make the important assumption that the number of permits isconstant through time and is equal to x. This is not strictly necessary but itsimplifies the analysis considerably. Assume that at date zero the permits areowned by old agents. This is not the only possible assumption. But this isconsistent with the idea that in order to own assets, excepting the case of afree distribution, one must save (so only old agents could have saved and gotthe permits8).
Each permit allows its owner to get the price of the right to pollute—ora rent—together with the proceeds of the resale of the permit. This price willtypically be paid by firms to the agents who own the permits.
8One could have distributed pollution permits to young agents. Certainly these permits cannotbe sold to old agents (since they do not save). As all young agents are the same, there is notrade in pollution permits. Only in the next period can these permits be sold to the newbornagents (which is the case considered here).
To be more concrete, let us consider the budget constraints of an agentborn at date t:
(42)
(43)
where Vt is the price of an emission permit at date t, et is the number of permitspurchased at date t, and Pt is the price of the right to pollute at date t.
There is a no-arbitrage condition, which is
(44)
Note that this no-arbitrage condition only holds if the number of permits isconstant.
To study the competitive equilibrium with emissions permits, one canbegin to analyse the consumption good market equilibrium condition.9 Theequilibrium can be written as
(45)
Expressing this condition using the (assumed time invariant) ratio Yp =Kt/K2,t, it follows that
(46)
Now, the assets markets equilibrium condition is
(47)
Using the expression of xVt, it follows that
(48)
We still have Kt+1 = R[(Yp - 1)/Yp]Kt.Using the arbitrage condition and solving for Yp one obtains
(49)
Therefore, the following result is obtained.
Proposition 2: The growth rate at a laissez-faire equilibrium equals that of anequilibrium with tradeable emissions permits.
What Proposition 2 tells us is that a ‘central planner’ could reduce the levelof pollution by introducing permits in the economy, but that this would leavethe growth rate unaffected.
Y p
a= +
-1
1bb
K a PYt p t t+ = - + - - -( ) -[ ]{ }1 21b a b a b aY ,
K xV a PYt t t t+ + =1 2b ,
xV a PYt p t t= + - -( ) -[ ]a b b a1 2Y ,
1 1 2-( ) + +( ) + +( ) =a W r K x V PYt t t t t t tP ,
VV
rtt t
t
=+
++ +
+
P 1 1
11
P d r S V et t t tk
t t t+ + + + += +( ) + +( )1 1 1 1 11 P
Pc eV S Wt t t t tk
t+ + =
Tradeable Emissions Permits, Emissions Taxes and Growth 457
9One may proceed exactly as for the study of the laissez-faire equilibrium.
The intuition of this result is as follows. In a laissez-faire economy, firmshave positive values since there are pure profits. Savings has two components,one which is physical, the other being financial. Introducing permits does notaffect this sharing. On the one hand, now that there is a price for polluting,firms do not make pure profits any more (the value of these profits were equalto the value of the pollution used by the firms). On the other hand, in aneconomy with permits, the rents that accrue to old agents are still equal tothe value of the pollution used by the firms. This value is equal to pollution times the unit value of pollution. The lower the level of pollutiondecided by a central planner, the higher the unit price of the pollution. Andalthough firms have no positive values, this is not the case for pollutionspermits. In both systems a part of savings has a financial component that isequal to the discounted value of the pollution per unit of time. This value isthe same at any date in both economies, although the pollution in theeconomy with permits could be chosen to be lower (but the price of the rightto pollute is then higher). Apparently, there is a similar amount of saving thatis diverted.
Let us now use Propositions 1 and 2 to compare the growth rates. Letus concentrate on the comparison between the growth rates in an economywith permits and in an economy with pollution taxes.
The fact that emissions permits are tradeable and privately owned makesthem different from emissions taxes. Emissions permits are a very peculiarway of redistributing the rents created by the regulation of pollution. Agentsinvest their savings in order to get the rents and so it is not entirely produc-tive. In this respect, leaving aside the amount of emissions, an economy withpollution permits is like a laissez-faire economy.
However, one could distribute pollutions permits in a way that is con-ducive to more growth. But for this, permits must not be privately owned.Suppose that an independent organization is given the permits (without theright of selling them). This organization receives at any date the value of therights to pollute, which are paid by the firms. It may redistribute these rights,finance a public good or even invest directly in physical capital. Then, depend-ing on the redistribution scheme of the proceeds of the rights to emit, onecould obtain the corresponding equilibria with emissions taxes. It suffices to redistribute to the generations what they would get if the taxes were redistributed.
Note that it is also possible to get a growth rate higher than the one thatarises in an economy wherein emissions taxes are redistributed to youngagents. To examine this further, suppose that the organization invests the totalof what it receives (hence this organization owns pollution permits and a partof the capital stock). This public savings plus the savings of individual agentswould always be higher than private savings in the economy where emissionstaxes are rebated to young agents. This is so since, in this economy, youngagents would not save the total of the transfers received from the state. Indeed,
consumption in youth being a normal good, it would increase as agents havemore wealth, and this increase would be paid for out of the transfers.
To sum up, the difference between emissions permits and emissions taxes stems from the fact that the former are assets that can be privatelyowned. The existence of permits implies a peculiar redistribution of resourcesacross generations (through the redistribution of rents and the sale of thepermits).
The importance of the notion of property rights is then highlighted aswas already done by Chichilnisky (1994). Chichilnisky (1994) argues that thelack of a well-defined property rights system could lead to an overuse ofnatural resources and specialization of nations in wrong activities. Here, awell-defined system of private property rights is an impediment to growth.
Let us now turn to the analysis of the consequences of different taxesschemes on pollution. As far, as the unit value of the pollution tax is fixed(i.e. we keep {pt}t≥0 unchanged), different redistribution schemes lead to dif-ferent pollution levels. Given any pollution tax pt the pollution level chosenby firms obeys
(50)
Using the previous equation together with the optimality condition (14)yields
(51)
Then the following proposition is obtained.
Proposition 3: Let the sequence of pollution taxes {pt}t≥0 be given, andsuppose that at any date the demand for pollution by the firm does not hitits upper bound. Let E1
t, Et2, Et
3 denote respectively the equilibrium levels ofpollution when there is a public good, when pollution taxes are rebated toold agents and when pollution taxes are rebated to young agents. Then, thereis a date t such that, "t ≥ t, Et
2 £ E1t £ Et
3.
Proof: Indeed, at date 0, one has K2,0(i) = K0/Yi (with obvious notation andfor i = 1, 2, 3). Then certainly E3
0 £ E10 £ E2
0. However, for t ≥ 1, one has (againwith obvious notation) K2,t(i) = Kt(i)/Yi. One also has
K iK i
K i
K i
tt
i
i
i
t
i
i
it
2
1
2 1
1
1
,
,
( ) =( )
=- ( )
=- ( )
-
-
YY
Y YY
Y
ER K
tt
t
=- -( )1 2a b
a p,
1 2- -( )=
a bp
PY
Et t
tt
,
Tradeable Emissions Permits, Emissions Taxes and Growth 459
The growth rates of the K2,t(i) are that of Kt(i); then Proposition 1 and (51)yield the above result.
The intuition of the proposition is simple. We know that different redis-tribution schemes lead to different growth rates of the capital stock. Whenthere are pollution taxes, pollution levels are proportional to the capital level.When the capital grows faster, production needs more pollution.
This result hinges on the assumption that the pollution taxes areunchanged across schemes. Varying pollution taxes could reverse the result.As for the case of emissions permits, if well designed, emissions taxes couldlead to lower pollution levels while ensuring faster growth.
6 C
This paper uses a dynamic general equilibrium model with overlapping gen-erations in order to analyse and to compare emissions taxes and emissionspermits.
Even in the context of a perfect environment, i.e. with perfect informa-tion, perfect competition . . . , it is shown that privately owned emissionspermits have some disadvantages. An equilibrium with emissions permitswould certainly be better than a laissez-faire equilibrium since it would entaila lower pollution level (growth would be the same in both equilibria).However, it is not clear that an economy with emissions permits would bepreferable over an economy with emissions taxes. While in both cases pollu-tion would be lower, growth would be higher in an economy with emissionstaxes. This is because emissions permits may divert savings from ‘productive’resources and have a negative impact on capital accumulation. This happenswhatever the way emissions taxes are redistributed.
The analysis contained in this paper relies on several simplifying assump-tions. We have already discussed the importance of assuming a linear pro-duction function in the capital good sector (see the last paragraph beforeSection 3). Two other important assumptions ensuring the tractability of themodel are the log-linear specification of the life-cycle utility function and theCobb–Douglas specification of the production function in the consumptiongood sector.
With regard to the life-cycle utility function, the specification does notseem to be a particularly important one. As long as the interest rate remainsconstant, using a homothetic utility function would yield the same conclu-sions. This is because the saving rate would still be constant and this is whatmatters for the study of the dynamics.
An important assumption may be that of a Cobb–Douglas productionfunction in the consumer good sector. This is important in order to get a con-stant equilibrium growth rate and it makes possible the substitution of capital
for pollution or labour (for instance, using a Leontieff production functionwill not permit growth as long as pollution is limited).
Let us conclude with an issue that characterizes the present analysis andthe study of which is on the agenda for future research. We have analysed therelative advantages and disadvantages of emissions permits and taxes in thecontext of a closed economy (or a world economy). It would be interesting,given the relevance of the present context (see for example Chichilnisky,1996), to study further (national and international) emissions taxes and emis-sions permits together with international trade.
A
Let us first prove that the growth rate with taxes redistributed to old agents is higherthan the laissez-faire growth rate. Recall that the former growth rate is the positiveroot of
We only have to show that P[1 + ab/(1 - b)] < 0.
Now, proceeding as above, we prove that the growth rate when the Pigovian taxesfinance a public good is higher than the growth rate when the taxes are rebated to oldagents (g2 > g1). One must then show that P(Y1) > 0.
Pa a a
a a
a aa
a
1 1 1 1 1 1 1
1 1 1 1 1 1
2
+ÊË
ˆ¯ = +Ê
ˈ¯ - +Ê
ˈ¯ + - -( ) -( )[ ]- -( ) - -( )
= +ÊË
ˆ¯ +Ê
ˈ¯ - +( )È
Î͢˚̇
+ -( ) +ÊË
ˆ¯ -( ) - -
ba
ab
ab
aa b a a b
ba
ab
aa b
ba
a a --( )ÈÎÍ
˘˚̇
= +ÊË
ˆ¯ -( ) + -( ) +
-( )ÈÎÍ
˘˚̇
b
ba
b bb a
a1 1 1
1aa a
a
Pa a a
a a
a aa
11
11
11
1 1 1 1
11
11
1
2
+-
ÊËÁ
ˆ¯̃
= +-
ÊËÁ
ˆ¯̃
- +-
ÊËÁ
ˆ¯̃
+ - -( ) -( )[ ]- -( ) - -( )
= +-
ÊËÁ
ˆ¯̃
+-
ÊËÁ
ˆ¯̃
- +( )ÈÎÍ
˘˚̇
+ -
bb
abb
bb
a b a a b
bb
abb
a b (( ) +-
ÊËÁ
ˆ¯̃
-( ) - - -( )ÈÎÍ
˘˚̇
= +-
ÊËÁ
ˆ¯̃ -
-ÊËÁ
ˆ¯̃
+ -( ) -( )-
+ÈÎÍ
˘˚̇
=-
+-
ÊËÁ
ˆ¯̃
+ -( ) -
11
1 1
11 1
11
1
11
11
1
a
a aa
a
a aa
a
bb
a a b
bb
abb
bb a
bb
bab
bb
ba(( )
-+È
Î͢˚̇
- +-
ÊËÁ
ˆ¯̃
ÏÌÓ
¸˝˛
=-
+-
ÊËÁ
ˆ¯̃
--
+ -( )[ ]
=- -
- -( )ÈÎÍ
˘˚̇
<
11 1
1
11
1 11
1 11 0
2
bbb
bab
bb
bb
a a
bb
abb
a
a
a a aa
a
P a aY Y Y2 1 1 1 1 0( ) = - + - -( ) -( )[ ] - -( ) - -( ) =a a b a a b22
2
Tradeable Emissions Permits, Emissions Taxes and Growth 461
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