The Theory and Empirics of the Green Paradox

Organiser: Karen Pittel, Rick van der Ploeg and Cees Withagen

The Green Paradox under Imperfect Substitutability between Clean and Dirty

Fuels

Ngo Van Long

The Green Paradox under Imperfect Substitutability

between Clean and Dirty Fuels

Ngo Van Long∗

CESifo and Department of Economics, McGill University

23 June 2011

∗For helpful discussions on the Green Paradox, I am indebted to Reyer Gerlagh, Michael Hoel, Chuck

Mason, Karen Pittel, Rick van der Ploeg, Rüdiger Pethig, Hans-Werner Sinn, Sjak Smulders, Frank Stähler,

and Cees Withagen. I am grateful to Ifo Institute for providing a stimulating research environment.

1

The Green Paradox under Imperfect Substitutability betweenClean and Dirty Fuels

Abstract: This paper shows that a technological change that increases marginally the

degree of substitutability of non-fossil fuels for fossil fuels may cause fossil fuels producers

to anticipate lower demand in the future, and to react by increasing current extraction,

leading to higher near-term emissions and accelerating climate change damages. Such a

Green Paradox outcome is more likely to occur if the existing degree of substitutability is

moderate or high. In fact, if the current degree of subsitutability is near zero, then there

will be no Green Paradox outcome associated with a marginal increase in substitutability.

JEL-Classification: Q54, Q42, Q30

Keywords: Imperfect Substitution, Green Paradox, Climate Change.

2

1 Introduction

The burning of fossil fuels generates emissions that harm the environment not only in the

present but also in the future, because emissions add to a pollution stock which decays

only very slowly. As has been pointed out by many authors, the first best policy measure

is to impose at each point of time a tax on emissions that equals the capitalized value of

the stream of marginal damages of emissions (see e.g. Hoel, 2011, Ploeg and Withagen,

2011). When the first best measure cannot be implemented because of political constraints,

there are a variety of policy measures that at first sight might seem to approximate the first

best measure. However, in some cases, a careful analysis would reveal that some policies

that seemingly would do the job will actually turn out to have an adverse effect on the

environment, contrary to the good intention of the policy makers. This outcome is known

as the Green Paradox (Sinn, 2008a,b, 2012).

The possibility of a Green Paradox outcome has been shown to exist under a wide varieties

of circumstances. Sinn (2008) pointed out that the announcement of a steeply rising path of

carbon tax can induce owners of oil and coal reserves to extract their resources more quickly,

resulting in a worse climate outcome in the shorter and the medium term.1 Hoel (2008)

demonstrated that technical progress in the backstop technology that produces at constant

cost a clean energy which is a perfect substitute for fossil fuels would have a similar effect

on the supply behavior of resource-extracting firms.2 Grafton, Kompas, and Long (2012)

showed that an increase in a time-independent ad valorem rate of subsidy on biofuels can

result in a Green Paradox outcome. Ploeg and Withagen (2012) showed that whether a

Green Paradox arises may depend on whether extraction costs increase sharply as the size of

the remaining stock diminishes. Long and Stähler (2012) demonstrated that, if both fossil

1For some early analyses of responses of intertemporal extraction plan to anticipations of taxation or

expropriation, see Long (1975), Long and Sinn (1985).2See also Strand (2007). Dasgupta and Heal (1979) and Pearce and Turner (1990) provide a good

exposition of substitute production under a constant cost technology. Earlier models of substitute production

include Heal (1976) and Hoel (1978,1983), which, though not dealing with 2 emissions, contain all the

ingredients from which one can deduce a Green Paradox result. See also Welsch and Stähler (1990) for an

early treatment of the dynamic supply response of the Green-Paradox type.

3

fuels and non-fossil fuels are being used concurrently, a fall in production cost of non-fossil

fuels may result in greater extraction rate of fossil fuels. All the above models assume that

if several types of fuels are available, they are perfect substitutes.(For an overview of recent

contributions to the Green Paradox literature, see van der Werf and di Maria (2011)).

Perfect substitution is often a useful assumption to simplify the analysis. One must

admit, however, that at the present level of technology, biofuels cannot entirely replace

petroleum in a number of uses, e.g., in aviation. Efforts are being made to improve the

substitutability of biofuels for petroleum. This paper therefore relaxes the assumption of

perfect substitutability.3 This allows us to ask a number of interesting questions. Does a

technological change that makes biofuels a closer substitute to petroleum benefit or harm the

environment in the near term? If it harms the environment, we call this a technology-induced

Green Paradox (as distinct from tax/subsidy induced Green Paradox).4 A related question

is whether imperfect substitutability increase or reduce the likelihood of a Green Paradox

outcome induced by raising the subsidy rate on biofuels or by increasing the base-year rate

of an advalorem tax on fossil fuels.

When the prices for fossil fuels and non-fossil fuels are not identical because of imperfect

substitutability, the analysis can become tedious, because now there are several prices to

consider, and they will be changing over time.To facilitate the analysis, this paper defines

3Our model is consistent with the current state of play in terms of biofuel production. The main producing

countries for transport biofuels are the U.S., Brazil and the EU. Brazil and the U.S. combined produced

55 and 35 percent, respectively, of the world’s ethanol in 2009 while the EU produced 60 percent of the

total biodiesel output. The main stimulus to the use of biofuels are policies that encourage the substitution

from fossil fuels, especially for road transportation. Government mandates for blending biofuels into vehicle

fuels have also been enacted in at least 17 countries, and many states and provinces within these countries.

Typical mandates require blending 10—15 percent ethanol with gasoline or blending 2—5 percent biodiesel

with diesel fuel. Recent targets have encouraged higher levels of biofuel use in various countries (UNEP,

2009, page 15-16).

The range of policies that have stimulated biofuel demand by setting targets and blending quotas has been

aided by supporting mechanisms, such as subsidies and tax exemptions. In the US, the total biofuels support

encompasses the total value of all government supports to the biofuels industry, including consumption

mandates, tax credits, import barriers, investment subsidies and general support to the sector such as public

research investment. A report by Koplow (2007, pp. 29, 31) for the Global Subsidies Initiative indicates

that the total support estimates for the US alone, in 2008, was between $9.2 and 11.07 billion.4There is a large literature on technological changes in the context of exhaustible resources. See for

example Pittel and Bretschger (2011), and Acemoglu et al. (2012).

4

the concept of a “reduced-form demand function” for fossil fuels. This function incorporates

the parameters of the demand and supply functions of the clean energy. Using this reduced

form demand function, we are able to analyse the possibility of a Green Paradox outcome

caused by a technological change by identifying its direct effect (usually “pro-green”), and

its indirect effect (usually “anti-green”). The direct effect is the change in the quantity

demanded, keeping the price of fossil fuels constant (while allowing the price of non-fossil fuels

to change to clear that market). The indirect effect arises from intertemporal optimization

behavior of owners of fossil fuel stocks. It works through the change in the equilibrium price

path of fossil fuels.

We assume that the non-fossil energy is produced under increasing marginal cost to reflect

the reality that renewable substitutes such as biofuels are produced using different grades of

land. As pointed out in Chakravorty et al. (2011), increased use of biofuel may, by moving up

the supply curve of land, increase the unit cost of the renewable resource. Using an dynamic

empirical model firmly grounded on the Hotelling framework, they estimated the effect of

biofuel mandates on food prices. They did not address the issue of the Green Paradox. With

a few exceptions, the bulk of the existing literature on biofuel subsidies uses static analysis.

Many authors have found, in the static context, mechanisms for increased carbon emissions

(a Green Paradox outcome) with biofuel subsidies.5 It has also been pointed out that the

production process of biofuels is not “green” because it involves the use of many inputs with

high carbon contents. Moreover, because the first generation of biofuels displace land for

food production it may increase food prices.

The present paper complements the existing literature with a dynamic mechanism that

arises even if technical change in favor of substitutability occurs only once. The contribution

is a much richer understanding of the Green Paradox, delimiting cases when it occurs and

when it does not.6

5For static analysis of energy substitution, see Hill et al. (2006), Steenblik (2007), Koplow (2007), Lapan

and Moschini (2009), Bandyopadhyay et al. (2009), Moschini et al. (2010). Bahel et al. (2011) developed a

dynamic model but their focus was on food prices.6Fischer and Salant (2012) examined the Green Paradox in the presence of a subsidy for renewable

5

We show that for the case of a system of linear demand functions, an increase in substi-

tutability can result in a Green Paradox outcome, if the existing degree of substitutability

is already high, and the supply of non-fossil fuels is sufficiently elastic. Intuitively, if the

supply response of non-fossil fuel producers is strong enough (i.e. a small price change in-

duces a large increase in supply), the fear of fiercer future competition (a substantial increase

in supply) from non-fossil fuel producers in the future, as the price of fossil fuels rises ac-

cording to the Hotelling Rule, will make owners of fossil fuels stocks extract more earlier

on. This increases the accumulated extraction at each point of time, until the stocks are

exhausted. On the other hand, starting from a very low degree of substitutability, a small

increase in substitutability cannot generate a Green Paradox outcome. We show that there

is a U-shaped relationship between the current degree of substitutability and the change in

near-term emissions that results from a small technical progress in substitutability.

We also examine the case where the fossil fuel producers form a cartel and act as a

Stackelberg leader. We show that for certain range of parameter values, there is a Green

Paradox outcome induced by an increase in substitutability.

2 A brief review of the related literature

There is a large literature that connects the dynamic analysis of non-renewable

fossil resources with climate change damages. Sinclair (1992, 1994) pointed out that climate

change policies must aim at delaying the extraction of oil, and argued that a carbon tax

must decline over time to encourage owners of fossil fuels to defer extraction. This result,

however, depends in part on the assumptions that damages appear multiplicatively in the

production function, and that capital and oil are substitutable inputs in a Cobb-Douglas

production function.7

resources. However, unlike our model, they assumed that the unit cost of renewable is constant in any given

period. Their formulation allows unit cost to fall over time through knowledge accumulation. They found

several cases where the Green Paradox may hold.7Heal (1985) and Sinn (2008a,b) also model damages from GHGs emissions as a negative externality in

production. Most papers however specify damages as an additive term in the social welfare function.

6

Recently, Groth and Schou (2007) confirm Sinclair’s declining tax result using a similar

model, but allowing endogenous growth. Ulph and Ulph (1994), specifying damages as an

additive term in the social welfare function, and assuming exponential decay of pollution,

find that the optimal carbon tax time profile has an inverted U shape. Hoel and Kverndokk

(1996) specify a model of economic exhaustion that includes rising extraction costs. They

show that carbon tax peaks before the peak in atmospheric carbon. Consistent with Ulph

and Ulph (1994), they find that in the long run the carbon tax approaches zero. They

also consider the case where there is a backstop technology that produces a substitute at

constant cost.8 Farzin and Tahvonen (1996), by assuming that the decay rate of pollution

is non-linear in its stock, show that the optimal carbon tax can take a variety of shapes.

Similarly, Tahvonen (1997) obtains eleven different tax regimes, depending on initial sizes of

the stock of 2 concentration and the stock of fossil fuels.

The key point of the Green Paradox literature is that the optimal carbon tax cannot be

implemented given the political economy that exists in most countries. In particular, gov-

ernments adopt policies that are conceived as second-best measures, but may cause greater

environmental damages, if owners of fossil fuel stocks hasten their extraction to avoid future

carbon taxes. Models that depict this adverse response to anticipation of taxes or substitute

production include Sinn (2008a,b), Hoel (2008), Di Maria et al. (2008), Gerlagh and Liski

(2008), Smulders et al. (2009), and Eichner and Pethig (2010).9Strand (2007) shows that

a technological agreement that makes carbon dedundant in the future may increase current

emissions. Hoel (2008) assumes that carbon resources remain cheaper than the substitute

and analyses the situation where different countries have climate policies of different ambi-

tion levels. He shows that, in the absence of an efficient global climate agreement, climate

costs may increase as a consequence of improved technology of substitute production.

8Hoel and Kverndokk (1996, section 4) consider an alternative specification of damages, where the envi-

ronmental damage is a function of the rate of change in the atmospheric stock of carbon.9As pointed out in Hoel (2008), prior to 2008, “there is little work making the link between climate policies

and exhaustible resources when policies are non-optimal or international agreements are incomplete”. He

mentioned a few exceptions: Bohm (1994), Hoel (1994) in a static framework.

7

A number of authors have investigated the possibility of a Green Paradox in the context

of policies that facilitate the availability of a substitute for fossil fuels. Hoel (2010, 2011)

uses a two-period model where firms invest in capacity of producing a substitute. A number

of key parameters are considered in his model (e.g. a parameter to capture how rapidly

extraction costs increase with increasing total extraction, and a parameter affecting the time

profile of the returns to investments in the substitute). Whether an investment subsidy

results in greater environmental damages (a Green Paradox) depends on the relationship

among these parameters.

Gerlagh (2011) also considers a model where extraction is at a constant cost, and a back-

stop technology can produce unlimited amount of a renewable substitute, at a constant cost

per unit. He defines a Weak Green Paradox as an increase in the current emissions in re-

sponse to an improvement in the backstop technology, and a Strong Green Paradox when the

net present value of damages increases as a result of an improvement in the backstop technol-

ogy.10 He shows that both the Weak and the Strong Green Paradox arise in this benchmark

model. Assuming linear demand, he finds that increasing extraction costs counteract the

Strong Green Paradox, while an imperfect energy substitute may reduce the likelihood of

both the Weak and the Strong Green Paradox. Ploeg and Withagen (JEEM,2012) focus on

the case where marginal extraction costs of the exhaustible resource depends on the existing

stock. They assume that the substitute is available in unlimited supply at a constant mar-

ginal cost. After characterizing the social optimum, they turn to the case where first-best

policies are not feasible, and show that the Green Paradox prevails if the cost of backstop de-

creases, provided that the backstop remains expensive such that the non-renewable resource

stock is eventually exhausted. By contrast, if the backstop becomes so cheap that physical

exhaustion will not take place, the Green Paradox no longer holds.

Grafton, Kompas and Long (JEEM, 2012) emphasised the facts that biofuels are already

available, but the expansion of biofuel output is possible only with increasing costs. They

10His specification of net present value of damages is rather non-standard: it depends on a shadow price

of emissions which is not derived from the model.

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assumed that biofuels and fossil fuels are perfect substitutes. Using a framework where both

types of fuels are simultaneously consumed in the first phase phase, they found conditions

under which a Green Paradox outcome will not occur, as well as conditions under which it

will occur. Their main focus was on the effect of a biofuel subsidy on the date of exhaustion

of the fossil fuel resources. A striking result of that paper was that in the case of a linear

demand for energy, together with (i) a zero extraction cost for fossil fuels and (ii) an upward-

sloping linear marginal cost of biofuels, a biofuel subsidy will have no effect on the amount

of fossil fuels extracted and consumed at each point of time, even though the subsidy does

result in a lowering of the price of both fuels at each point of time, and in a higher quantity

of energy demanded at each point of time. Under the stated assumptions they showed that

that the increase in quantity of fuel demanded is exactly matched by an increased output

of biofuels, leaving the extraction rate unchanged. Thus, the time at which the stock of

fossil fuels is exhausted is unchanged, irrespective of the rate of subsidy. Their result stood

in sharp constrast to the inevitable Green Paradox in the model of Hoel (2008) where the

supply curve of the renewable resource is horizontal. An increase in subsidy does not give

rise to a Green Paradox outcome in their model in the presence of a linear and increasing

marginal cost of the renewable substitute to fossil fuels.

A second and important result of Grafton, Kompas and Long (JEEM, 2012) was that

when the assumption of zero extraction cost of the fossil fuel is replaced by the assumption

of a positive and constant marginal extraction cost, a biofuel subsidy will result in a longer

time over which the fossil fuel stock is exhausted. In this second case the Green Paradox

outcome does not arise when the subsidy on biofuel is increased. Indeed, there is no paradox

because the subsidy works as intended because it extends the extraction period.

When the marginal cost of biofuels increases at an increasing, or a decreasing rate as

biofuels supply increases, the results turned out to be quite different. In particular, Grafton,

Kompas and Long (JEEM, 2012) considered the case where the marginal cost of biofuel

production is strictly increasing and strictly concave. In this situation along the equilibrium

9

price path, as the price of energy rises gradually along the optimal extraction path of fossil

fuels, biofuel output will increase over time, but the rate of the supply increase (per dollar

increase in energy price) is greater when the price is higher. Consequently, fossil fuel firms,

in anticipation of this greater expansion of the substitute in the later stage, respond by

increasing their extraction at an earlier date. In this case, a Green Paradox outcome results

.There is a large literature on the costs and benefits of developing biofuels as a means to

reduce green house gas emissions, but most authors focus on static analysis of energy substi-

tution and do not take into account the intertemporal response of fossil fuel producers (e.g.

Hill et al. 2006, Steenblik, 2007, Koplow, 2007, Lapan and Moschini, 2009, Bandyopadhyay

et al. 2009, Moschini et al. 2010).11

3 A model of imperfect substitutability between fossil

fuels and renewable energy

We consider an economy with three goods: fossil fuels, denoted by , renewable energy,

denoted by , and a numeraire good, denoted by . Assume that is a perfectly clean

source of energy. In contrast, the consumption of fossil fuels generates emissions which

contribute to a stock of pollution, denoted by .

For simplicity, we assume that the demands for these goods come from the utility max-

imization of a representative consumer. Goods and are imperfect substitutes and thus

command different prices.

The stock of pollution at time is denoted by The rate of change in is assumed to

be equal to . This simplifying assumption may be justified on the ground that the rate of

natural decay of GHG pollution is very slow. Then

= 0 +

Z

0

11Chakravorty et al. (2009) provide a comprehensive survey. A dynamic model is developed in Bahel et

al. (2011) but their focus is on the evolution of food prices, not on 2 emissions.

10

Fossil fuels are extracted from a resource stock :

= −, (0) = 0, ≥ 0. (1)

Then cumulative emissions from time zero to time isZ

0

= (0)−

and the stock of pollution is linearly related to the stock of exhaustible resources,

= 0 +(0)−

Note that will be falling over time, causing to rise over time, until the resource stock

is exhausted.

We assume that the representative consumer has a separable net utility function

( )− () (2)

where () represents the damages caused by pollution. Because of the pollution externali-

ties embodied in the net utility function (2), it is widely thought that policies that encourage

replacing fossil fuels by a clean energy should be promoted. One often hears arguments in

favour of the subsidization of public and private R&D activities that would increase the

degree of substitutability of clean energy for fossil energy.

In this paper, we do not model R&D activities. Instead, we wish to find out whether

an exogenous technical progress that increases the substitutability is good or bad for the

environment, given that first best policies are not available.

3.1 Assumptions on demand

For simplicity, we abstract from the income effect, and assume that is quasi-linear:

( ) = ( ) + . This assumption implies that any income change will impact

only the demand for the numeraire good. As usual, it is assumed that income is sufficiently

large such that the consumption of the numeraire good is strictly positive. The function

( ) is strictly concave.

11

We assume that the marginal utility of any good is finite even when its consumption is

zero.

Let 1 and 2 denote the consumer prices of good and respectively. Consumer’s

maximization then leads to the following first order conditions that characterize an interior

maximum,

1(

) = 1

2(

) = 2

where 1 and 2 stand for the partial derivatives of with respect to and respectively.

The superscript in and indicate that these are quantities demanded. From the FOCs,

we obtain the demand functions

= (1 2 ) (3)

= (1 2 ) (4)

where we have used as a parameter representing the degree of substitutablity. We assume

that each of these demand functions is decreasing in its own price, and increasing in the

price of the other good (its imperfect substitute). Furthermore, we assume that an increase

in substitutability will reduce the demand for a good if its price is higher than the price of

the substitute:

0 if 1 2

0 if 2 1

These assumptions are satisfied for the standard linear quadratic preferences.

3.2 Assumptions on the supply of renewable energy

Let denote the quantity of renewable energy supplied at time . We assume that marginal

cost is positive and increasing in output level, . The producers of renewables are price

takers. They produce at the level that equates their marginal cost to the producer price 2:

( ) = 2

12

The superscript indicates that this is the price the firms receive per unit sold. The difference

between the producer price and the consumer price, 2−2, is the subsidy on renewables.

For the time being, let us assume that there is a constant ad valorem subsidy rate,

denoted by ≥ 0, so that2 = (1 + )2 ≡ 2

We call the “subsidy factor”. Note that ≥ 1.We can then derive the supply function for renewables:

= (2)

where in the inverse of the marginal cost function . Clearly, 0 0 because we

assumed an upward sloping function.

3.3 Supply of fossil fuels and choke price

We assume that there are a large number of resource-extracting firms, each owning a deposit

of fossil fuels. The deposits are homogeneous and are of identical size. The aggregate fossil

resource stock at the present moment is denoted by 0. Resource-extracting firms operate

under perfect competition. They perfectly forecast the price path of the fossil fuel. Assume

the marginal cost of extraction is constant, ≥ 0. Define the net price of the fossil fuel as1− . As long as the net price rises at the rate of interest , individual fossil fuel producersare willing to supply any amount. In equilibrium, however, the industry’s supply of the fossil

fuel at any time must equal the demand for it. There will be a time when the price of

fossil fuel is so high that the demand for fossil fuel becomes zero, and from that time onwards

only renewable energy is consumed, in quantity and at the price 2 Such a high price for

fossil fuel is called the “choke price” for fossil fuel, and we denote it by 1. The price 1 is

implicitly defined by the following three equations, which determined¡ 1 2

¢:

1(0 ) = 1

2(0 ) = 2

13

= ( 2)

Let be the time at which the aggregate fossil resource stock is exhausted. Then

1 = 1. From Hotelling Rule, the present value of the net price is the same at all points

of time in the interval [0 ]:

(1 − )− =¡ 1 −

¢− for all ≤

This means that if we know , we can calculate 1 as follows

1 = +¡ 1 −

¢−(−) ≡ ( 1 )

Then the time of exhaustion must satisfy the following equation, which requires that total

consumption of fossil fuel from time 0 to time must be equal to the total resource stock:Z

0

(1 2 ) = 0

where, in this equation, 1 is given by the function ( 1 ) specified above. But what

about 2? Since the market must clear, the demand for the renewable energy at any time

must equal its supply. Thus

(( 1 ) 2 ) = (2)

This equation shows that 2 can be expressed as a function of ( 1 ), which is the

equilibrium 1 along the Hotelling path. This observation leads us to a simple reformulation,

using the concept of the “reduced form demand function for fossil fuel.” This will be made

clear in the next section.

3.4 The reduced-form demand function for fossil fuels

We assume that the market for renewables clears at each point of time: the quantity de-

manded equals the quantity supplied. Then

(1 2 ) = (2)

14

This relationship allows us to solve for the equilibrium consumer price function for renewable

energy, which we denote by 2()

2 = 2(1 ) (5)

In other words, given the subsidy factor and given the price 1, we can deduce the price

2 that would clear the market for renewables. Clearly,

2

1 0 and

2

0

where the first inequality reflects the fact that the two goods are substitutes rather than

complments. The second inequality simply means that an increase in the subsidy factor for

renewables will reduce the equilibrium consumer price for renewables, at any given price of

fossil fuels.

Now, substituting (5) into eq (3), we obtain the demand function for fossil fuels, given

that the market for renewables clears:

= (1 2(1 ) ) ≡ (1 ) (6)

We call (1 ) the reduced form demand for good . Notice that the function does

not contain the price 2 as an argument. This does not mean that the demand for fossil

fuels is independent of the price of non-fossil fuels. Rather, the market-clearing 2 which is

conditional on 1, has been used.

Clearly is decreasing in 1:

1=

1+

2

2

1 0

We assume that is decreasing in , at least for those values of 1 high enough so

that 1 2 = 2(1 ). In particular,assume that at any 1 near the choke price 1,

a small increase in substitutability will reduce the demand for fossil fuels:

( 1 )

0 (7)

15

This assumption can be verified in the linear quadratic utility function case.12

From our earlier definition of the choke price for fossil fuels. 1, it holds that

0 = ¡ 1

¢From the properties of the function , we deduce that an increase in will reduce the

choke price:

1

= −

( 1)

1

0

3.5 The two phases

We consider an equilibrium path consisting of two phases:

Phase 1: Both fossil fuels and renewable energy are simultaneously supplied.

Phase 2: Only renewable energy is supplied, because the fossil fuel stock has been ex-

hausted.

Let be the time at which Phase 1 ends and Phase 2 begins. At time , the fossil fuel

stock is just exhausted. We must determine endogeneously. At time , we have ( ) = 0

Thus must satisfy the equationZ

0

(1 )−0 = 0

where 1 = ( 1 ).

We wish to show that there is a range of value for such that, for any given in this

range, a small increase in it will lead to an earlier exhaustion date. The following fact is

obvious:

Fact 1: If there exists a real interval (∗ ∗∗) such that the exhaustion time is de-

creasing in for all in this interval, then a small increase in will cause the pollution

stock to be higher for all .

12It can also be shown that in increasing in . This implies that a subsidy for renewables will, at

constant 1, reduce the demand for fossil fuels.

16

4 Effect of an increase in substitutability on the ex-

haustion time

What are the forces that determine the net effect of an increase in the substitutability

parameter on the time of exhaustion ? We have assumed (and this can be verified for

the linear quadratic case) that an increase in will lower the fossil fuel choke price 1.

At the same time, the inequality (7) indicates that an increase in rotates the (reduced

form) demand curve for fossil fuel in the anti-clockwise direction. Does an increase in

generate, on average, greater demand for fossil fuels over the time interval ? To answer this

question, it is useful to decompose the change in demand into a direct effect and an indirect

effect.

Direct Effect: The anti-clockwise rotation of the (reduced form) demand curve for fossil

fuels implies that any given price 1 sufficiently high, the quantity demanded is smaller than

before. This direct effect is captured by the term which is generally negative. The direct

effect is “pro-Green”: an increase in substitutability reduces demand for fossil fuels, at any

given sufficiently high price 1.

Indirect Effect: The increase in (substitutability) lowers the fossil fuel choke price

1. This implies that, holding constant, the price 1 must fall for all . A fall in 1

increases the quantity demanded. The indirect effect, captured by the term³

1

´³1

´,

is positive, i.e. it is “anti-Green”.

The total effect on fossil fuels consumption at any time is the sum of the direct effect

and the indirect effect at that time. In general, if the function ( ) is not restricted

beyond the usual assumption that the demand curve is downward sloping, the total effect

can be positive at some points of time and negative at some other points of time. Therefore,

to find the effect of an increase in substitutability, one has to compute the cumulative total

effect, over the interval [0 ]. If this cumulative total effect is positive, it means that the

exhaustion time is brought closer to the present, which is a Green Paradox outcome. A more

formal analysis follows.

17

To find the net effect of a change in on the exhaustion time , we define the function

Ω( ) =

Z

0

(1 ) −0 = 0 (8)

Then

= −Ω

Ω

=− R

0

h

11+

i

( 1 ) +R 0

£

11

¤

(9)

The denominator is positive. Thus we can state the following Proposition.

Proposition 1: (Necessary and sufficient condition for a Green Paradox). A

small increase in substitutability will bring the resource exhaustion date closer to the present

if and only if the cumulative sum of the indirect effect (anti-Green) outweighs the cumulative

sum of the direct effect (pro-Green):Z

0

∙

1

1

+

¸ 0 (10)

Without a more explicit specification of the reduced form demand function, we cannot

determine whether a Green Paradox outcome will arise from an increase in substitutability.

4.1 Parameterizing substitutability: the linear quadratic case

For illustrative purpose, consider the following linear-quadratic formulation.

Assume that

( ) = −

22 + −

22 − +

where ≥ 0 0 and 0. To ensure that is concave, we assume that . In the

limiting case where = , the two types of fuels are perfect substitutes.

The above utility function implies that for given ( ), if the goods become closer substi-

tute ( increases) then the utility decreases.13 On the other hand, for environmental reasons,

an increase in substitutability gives the the economy the potential to increase welfare by a

well designed switch in composition of demand to reduce environmental damages.

13We could add a scale parameter that depends on , so that when increases, the scale parameter also

change in such a way that, keeping current consumption constant, utility rises. We refrain from doing this

in order to keep the analysis simple.

18

At each point of time , the consumer faces the budget constraint

1 + 2 + =

where is the total expenditure allocated to period . Assume that is sufficiently great,

so that is always positive. Then the consumer’s FOCs with respect to and are

− − − 1 ≤ 0, ≥ 0, [ − − − 1] = 0 (11)

− − − 2 ≤ 0, ≥ 0, [ − − − 2] = 0 (12)

It is straight forward to derive the demand functions

=( − 1)− ( − 2)

2 − 2(13)

=( − 2)− ( − 1)

2 − 2

Note that if 1 ≥ 2 an increase in substitutability will reduce the demand for fossil fuels:14

=−(− )2( − 2)− 2(1 − 2)

(2 − 2)2

Renewable energy is produced by perfectly competitive firms, under increasing marginal

cost. Its supply is denoted by . Assume that the marginal cost of producing is +,

where 0 ≤ and 0.Then the supply of renewable energy satisfies the condition

that marginal revenue, 2 , is equated to marginal production cost, +,

2 = + (14)

which gives

=2 −

0 if 2 ≥

Recall our assumption that the market for renweables clears at each point of time. Equat-

ing with , we obtain the equilibrium 2 for a given 1

( − 2)− ( − 1)

2 − 2=

2 −

14Note, however, that if 1 is much lower than 2, then an increase in will increase .

19

Thus, if we define

=

1

Then

2 =(− ) ( + (+ ))

+ (2 − 2)+

1

+ (2 − 2)≡ 2(1 ) (15)

where 2 is increasing in 1, decreasing in , and decreasing in :

2

= −(− )2 + 22

( + (2 − 2))2

0 (16)

Thus, for a given 1, an increase in substitutability reduces the market clearing price 2.

Remark: Along the Hotelling path, 1 will be rising, and so will 2, as equation (15)

indicates. In particular, this response is greater the higher is degree of substitutability:

2

1=

+ (2 − 2)(17)

We record this result as Fact 2:

Fact 2: Along the Hotelling path, the increase in the price of rewables in response to

the increase in the price of the exhaustible resource is itself an incresasing function of the

substitutability parameter

Notice that, given market clearance, the gap between renewable energy price and fossil

fuel price can be expressed as follows,

2 − 1 =(− ) (1 + (+ ))

+ (2 − 2)−∙(− )2 + (− )

+ (2 − 2)

¸1 (18)

Thus the price gap 2 − 1 decreases as 1 increases.

Substituting (15) into (13) we obtain the “reduced form demand function” for fossil fuels:

= − 1 ≡ (1 ) (19)

where

( ) ≡ +

(2 − 2)+ 0

( ) ≡ (− ) + +

(2 − 2)+ 0

20

Let us find the choke price for fossil fuels, given that non-fossil fuels are available. Setting

(1 ) = 0, we obtain

1 =

=(− ) + +

+=

µ1− (− )

+

¶Note that 1 is increasing in and decreasing in .

Let us use eq (19) to draw the reduced form demand curve in a diagram where 1 is

measured along the vertical axis, and on the horizontal axis. We note that the vertical

intercept is the “choke price” for fossil fuels.

Since ≥ , an increase in substitutability will lower the choke price 1, as expected.

When 1 equals the choke price 1, then 1 = 0, and the equilibrium price of non-fossil

fuels then attains its steady-state price 2 as defined below:15

2 = +

+

How does an increase in substitutability affect the reduced form demand function for

fossil fuels?

Let us determine the effect of an increase in substitutability on the slope of the “reduced

form demand curve” for fossil fuel:

=2 (+)

[(2 − 2) + ]2≥ 0

Thus, an increase in substitutability makes the residual demand curve flatter.

We find that an increase in will move the quantity intercept of the reduced form demand

curve to the left if ∈ (0 e) and to the right if ∈ (e ) =

−( − ) ((2 − 2) + ) + 2 ((− ) + +)

[(2 − 2) + ]2

The sign of the numerator is the sign of the expression

() ≡ −(− )¡(2 − 2) +

¢+ 2 ((− )+ +) (20)

15Interestingly, 2 is independent of .

21

or

() = −(− )2 + 2(+)− (+)(− )

Since , the function () is quadratic and concave in . It is negative at = 0

and positive at = . Consequently, there exists a unique value e in (0 ) such that for all ∈ (0 e), a marginal increase in will shift the quantity intercept to the left, and for all

∈ (e ), a marginal increase in will shift the quantity intercept to the right.16

In fact,

e = (+)−q(+)(+ − (−)2

)

− (21)

Notice that e is decreasing in . Thus the range of values of such that 0 is wider,

the greater is .

It follows that keeping 1 constant, a marginal increase in will unambiguously reduce

the quantity of fossil fuels demanded (the usual pro-green effect) if is in (0 e). In contrast,if is in (e ), then 0 and a marginal increase in will increase the quantity of fossil

fuels demanded if 1 is low, and reduce it if 1 is high. Therefore, for any given ∈ (e ),there exists a corresponding positive “pivot price” e1() such that for any given price 1 isbelow this pivot price, a marginal increase in will increase the demand for fossil fuels (at

that given price).17

For ∈ (e ), the pivot price ise1() ≡ ()

()=

()

2 (+)(22)

It can be seen that 1 e1() 1 − e1() = (− ) ((2 − 2) + )

+ 0

That is, for all given 1 in the interval³ e1() 1

´, an increase in will reduce the demand

for fossil fuels, at the given price.

16Since and () = −(− )2 + 2(+ )− (+ )(− ) it is monotone increasing over

the relevant range (0 ).17It can also be shown that for any given 1 ≥ 0, an increase in the subsidy rate for renewables will result

in a lower demand for fossil fuels. It follows that an increase in the subsidy rate for renewables will reduce

the demand for fossil fuels at any given price, and make the slope of the demand curve flatter.

22

5 Sufficient Conditions for a Green Paradox outcome

caused by an increase in substitutability

Let us investigate the possibility of a Green Paradox outcome when substitutability increases,

under the assumption of linear demand arising from a linear quadratic utility function.There

are two ways to proceed with the analyisis. The first method makes use of Proposition 1 (in

section 4) and thus consists of determining whether the necessary and sufficient condition

(10) is satisfied. The second method consists of evaluating the market clearing equation

(8) directly, which allows us to solve for as a function of . The first approach is useful

because it can be applicable also to the case of non-linear reduced form demand functions.

5.1 Approach 1: evaluating the direct effect and the indirect effect

Let us first evaluate the direct effect at any given point of time. For the case of linear reduced

form demand, the direct effect is

= − 1

= − £+

¡ 1()−

¢−(−)

¤This term is negative if e.If e, this term is positive for low values of 1 and negativefor high values of 1.

The indirect is anti-Green:

1

1

= −(−)

(− )

(2 − 2)+ 0

Combining the direct effect and the indirect effect, we obtain the following Proposition.

Proposition 2: Under the linear quadratic formulation, the necessary and sufficient

condition for a Green Paradox outcome (a marginal increase in substitutability hurts the

environment) is that is sufficiently high such that

−( −)£(2 − 2) +

¤+ 2 ((− ) + +)− 2 (+) 0

23

It will be convenient to define = and = . Then we obtain the following

corollary

Corollary: Under the linear quadratic formulation, the necessary and sufficient condi-

tion for a Green Paradox outcome (a marginal increase in substitutability hurts the environ-

ment) is that is sufficiently high such that

() ≡ −(− )£(2 − 2) +

¤+ 2 ((− )+ + − (+) ) 0

Remark: The function () is quadratic and concave in .

() = −(− )2 + 2(+)(1− )− (+)(− )

It is negative at = 0 and positive at = provided that is sufficiently small such

that

(+ ) + 2 2(+) (23)

Therefore, if this condition holds, there exists a unique value ∗ in (0 ) such that for all

∈ (0 ∗), a marginal increase in will reduce the exhaustion time, bringing climate changedamages closer to the present. Note that 0 e ∗. In fact,18

∗ =(+)(1− )−

q(+)((+)(1− )2 − (−)2

)

−

A large value of will decrease ∗and thus widens the range (∗ ) over which a Green

Paradox outcome occurs. Thus large values of facilitate a Green Paradox outcome.

Numerical Examples

For the base line scenario, we set = 1 = 1 = 05 = 01 = 01. Let the rate

of interest be = 005. We found that a a Green Paradox will occur if ∈ (∗ 1), where∗ ' 030.1918The condition that () 0 is sufficient for the root ∗ to be real.19We find that if = 050, then competitive firms will take 500 years to exhaust the stock, if 0 = 417

88. Now suppose there is a technological progress so that substitutability increases by 4%, i.e. the new is

052. We find that the time of exhaustion falls by 12%, i.e. is now 494 years (making exhaustion occur 6

years earlier).

24

Let us consider a smaller . Say = 002. Then ∗ ' 027 Thus a smaller extractioncost facilitates a Green Paradox outcome

What about the subsidy factor ? Keeping all other parameters as specified in the base

line scenatio, but let = 15. Then ∗ ' 043, i.e. a Green Paradox outcome is less likely.Consider now a lower marginal cost intercept of renewables, = . Let = 06. Then

∗ ' 023, i.e.a Green Paradox outcome is more likely.An increase in (the steepness of the supply curve) also makes a Green Paradox outcome

is more likely. If = 5, then ∗ ' 028

5.2 Approach 2: computing exhaustion time

Using the linear reduced form demand function , we can compute the exhaustion time

directly from the equation Z

0

( − 1) = 0

where

1 = +¡ 1 −

¢−(−) = +

µ

−

¶−(−)

Then the condition that total demand for fossil fuels over the time interval [0 ] must equal

the total supply reduces to a simple equation:Z

0

( − 1) = ( − )

µ − 1− −

¶= 0 (24)

So we can compute 0 directly, given the assumption that extraction cost is lower than

the the choke price, 1, i.e. .

Comparative statics

Define

() = ()− ()

( ) = − 1− −

25

Note that () 0 because we have assumed that is lower than the choke price.Then the

exhaustion time can be obtained from

() ( ) = 0

Proposition 3: The effect of a marginal increase in substitutability on the exhaustion

time is

=− ( )0()() 0( )

and the necessary and sufficient condition for a marginal increase in substitutability to

reduce exhaustion time is 0() 0, i.e.

− 0 (25)

Proof : Clearly,

0( ) = 1− − 0

and

() 0( ) + ( )0() = 0

Remark: The condition (25) is identical to () 0. Since 0, a necessary condition

for a Green Paradox outcome is 0.

Condition − 0 is equivalent to

i.e.

−( − ) ((2 − 2) + ) + 2 ((− ) + +)

2 (+)

Corollary 3: If condition (25) holds, the stock of pollution at each point of time is

increasing in the subsitutability parameter for ∈ (∗ ).Having computed the exhaustion time, the equilibrium time path of price can then be

calculated

= +¡ ()−

¢−( ()−)

26

Next we can compute () = − 1() and

= 0 +

Z

0

Then

()

=

Z

0

where

=

1()

= − 1()

= −∙−( ()−)

+¡ ()−

¢µ−

¶¸ 0

So the stock increases with if ∈ (∗ ).

6 Monopoly

What happens if the fossil firm is a monopolist? The monopolist chooses the price path of

price 1 to maximize its stream of discounted profits, knowing the reduced form function

= (1)

This formulation implies that the monopolist is a leader in the market for fuels. He knows

that the price of non-foosil fuels will adjust to his announced price of fossil fuel so that

the market clears. Formally,let denote the monopolist’s exhaustion time.The monopolist

seeks the terminat time and the price path 1 defined over [0 ] to maximizeZ

0

(1 − )(1)−

subject to

= −(1)

and

≥ 0.

27

Let denote the co-state variable. The Hamiltonian is

= (1 − )(1)− (1)

The necessary conditions are

(1 − − ) + = 0

and

= .

Simple manipulation yields

( − − 0)(− ) + ( − ) = 0

+ ¡+ 0

¢= 2

Then

=

2+1

2

¡+ 0

¢

When = = , we have

+ ¡+ 0

¢= 2

So

+ 0 =

0 =

µ

−

¶−

=

2+1

2

¡+ 0

¢=

2+1

2

µ+

µ

−

¶−

¶Then exhaustion impliesZ

0

( − ) =

Z

0

µ −

2−

2

µ+

µ

−

¶−

¶¶ = 0

28

From this condition, we obtain the monopolist’s planned exhaustion time:

1

2( − )

µ − 1− −

¶= 0 (26)

Comparing equation (24) with equation (26), we can see that the relationship between the

monopolist’s exhaustion time and the competitive exhaustion time satisfies the fol-

lowing equation:

≡

³ − 1−−

´³ − 1−−

´ = 2

Proposition 4: The monopolist takes a longer time to exhaust the stock 0, and the

response of to an increase in subsitutability when ∈ (∗ ) is of the same sign as theresponse of to an increase in substitutability. In absolute value, a given increase in

decreases the monopolist’s exhaustion time by more than under perfect competition.

Let us compare the magnitude of the responses.

Define

≡µ − 1− −

¶and

≡µ − 1− −

¶Then

ln

ln=

ln

ln

The response eslasticity of to an increase in is

ln

ln=

ln

ln

ln

ln

=

(1− −) ln

ln

Therefore the relative magnitude of the two elasticities is

ln ln

ln ln

= 2

µ(1− −)(1− −)

¶

29

A Numerical Example

We set = 1 = 1 = 05 = 01 = 01. Let the rate of interest be = 005. We

found that a a Green Paradox will occur if ∈ (∗ 1), where ∗ ' 030. We find that if

= 050, then under perfect competition, competitive firms will take 500 years to exhaust

the stock, if 0 = 417 88. Under the same parameter values, the monopolist will exhaust

the stock in 980 years. Now suppose there is a technological progress so that substitutability

increases by 4%, i.e. the new is 052. We find that under perfect competition, the time of

exhaustion falls by 12%, i.e. is now 494 years (making exhaustion occur 6 years earlier).

Under monopoly, the time of exhaustion falls by 122% ( falls from 980 to 968, making

exhaustion occur 12 years earlier).

7 Concluding Remarks

This paper explores the possibility of a Green Paradox associated with an increase in the

extent to which non-fossil fuels can be substituted for fossil fuels. We have shown that

a technological change that increases marginally the degree of substitutability may cause

fossil fuels producers to anticipate lower demand in the future, and to react by increasing

current extraction, leading to higher near-term emissions and accelerating climate change

damages. Such a Green Paradox outcome is more likely to occur if the existing degree of

substitutability is moderate or high. In fact, if the current degree of subsitutability is near

zero, then there will be no Green Paradox outcome associated with a marginal increase in

substitutability.

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