1February 1, 2005

GLOBAL WARMING AND OTHER

TRANSBOUNDARY ENVIRONMENTAL PROBLEMS a

by

Michael Hoel

Department of Economics, University of Oslo b

Abstract One of the most serious environmental problems in this century will most likely be climate change caused by emissions of greenhouse gases. From an economic point of view, the climate problem is a special case of a transboundary environmental problem. This article therefore discusses some important features of a general transboundary problem, giving particular emphasis to the special case of global warming. Transboundary environmental problems require some kind of international cooperation in order to obtain a socially efficient outcome. With a suitably designed international agreement, it is in principle possible to make all countries better off when they cooperate than if no countries cooperate. An agreement of the Kyoto type for the climate problem has some desirable properties: Correctly designed, it will give a cost-effective allocation of emissions across countries, and all countries can be made better off under such an agreement than they are without any cooperation. However, if an agreement focuses only on emissions and not on the development of new technologies, as is the case with the Kyoto agreement, the outcome may be inefficient due to international technology spillovers. Keywords: transboundary pollution, global warming, international environmental agreements, technology spillovers.

JEL classification: O30; Q54; Q55 a This article draws heavily on my previous work, in particular on Hoel (1999) and Golombek and Hoel (2004a; 2005a,b). Most of this work has been financially supported by the Research Council of Norway, under the programmes SAMRAM and SAMSTEMT. b Address: P.O Box 1095, N-0317 Oslo, Norway; email: mihoel@econ.uio.no; web: http://folk.uio.no/mihoel/.

2 1. Introduction

One of the most serious environmental problems in this century will most likely be climate

change caused by emissions of greenhouse gases. Increased atmospheric concentration of

greenhouse gases, of which CO2 is the most important, may increase the average temperature

by 2-5.5 degrees (Celsius) by the end of this century, see IPCC (2001)1. Not only will

temperatures increase in most areas of the world, but it is also likely that there will be large

changes in the patterns of precipitation and large increases in the frequencies of dramatic

events such as hurricanes, floods and draughts.

From an economic point of view, the climate problem is a special case of what economists

call a transboundary environmental problem. This article therefore discusses some important

features of a general transboundary problem, giving particular emphasis to the special case of

global warming. The article is organized as follows: Sections 2-7 discuss why it is necessary

to have international cooperation to handle the climate problem and other transboundary

environmental problems in an efficient manner, what form such cooperation might take, and

why such cooperation might be difficult. Section 8 gives an explicit analysis of international

climate agreements. This discussion is expanded in Sections 9-13, where it is shown that

climate agreements of the Kyoto type typically give too weak incentives for the development

of new technologies. Finally, some concluding remarks are given in Section 14.

2. Transboundary Environmental Problems

We have a transboundary environmental problem whenever the environment in one country

is directly affected by actions taken in one or more other countries.2 Typical examples of

transboundary environmental problems are (i) several countries polluting a river, a lake, or an

ocean, (ii) acid rain caused by emissions of SO2 and NOx, (iii) global warming caused by

emissions of CO2 and other greenhouse gases, (iv) depletion of the ozone layer caused by

emissions of CFC's and other ozone depleting substances. Throughout this article particular

emphasis is given to the problem of global warming.

1 Various summaries of the large four volume report from IPCC can be found at http://www.ipcc.ch/. 2 Notice that the term, "directly affected," excludes any indirect effects via prices, incomes, etc., making actions in one country affect actions in other countries.

3To formalize the analysis of transboundary environmental problems, consider n countries

with emissions (e1, ...,en). For each country j there is an environmental variable zj that

depends on emissions from all the n countries. The variable zj is defined so that an increase in

zj gives a deterioration of the environment in country j. For pollution of a river or a lake, zj

would be a measure of the pollution in that part of the river or lake which country j is

concerned about. For acid rain, zj could measure the amount of sulfur and nitrogen deposited

in country j. For global environmental problems such as climate change and ozone depletion,

zj could be a measure of the quality of the atmosphere. As will be seen soon, all zj's will be

equal for the latter case.

For simplicity we assume that the relationships between emissions and depositions are linear,

i.e.

j i ijiz e a=∑ (1)

where the element aij gives the amount of depositions in country j per unit emission in

country i.

For many environmental problems, it is not the flow of depositions of the pollutant that

matters for the environment, but the accumulated stock. When this is the case, one should

specify the link between the zj's and the stocks of pollutants in each country. However, for the

purpose of the present discussion the distinction between flows and stocks are not a major

concern, we therefore stick to the simple specification in which the zj's measure the

environmental quality in the countries.

The general description (1) includes several special cases. Consider, for example, the pure

unidirectional case in which a river runs through several countries which all pollute the river.

Clearly, a country that is further downstream cannot pollute an upstream country. For the

three-country case, this gives the following transportation matrix, when the countries are

indexed so that their index is higher the further downstream the country is

4

11 12 13

22 23

33

00 0

a a aA a a

a

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

(2)

Climate change and depletion of the ozone layer are examples of environmental problems for

which it is only the sum of emissions from all countries that matters for the environment. For

this special case we thus have aij = 1 for all i,j, implying from (1) that

1 ... n iiz z e= = =∑ (3)

As mentioned above, it is only the sum of emissions that is of importance in the case of the

climate problem. However, the relationship between emissions and the environmental

variable z (equal for all countries) is slightly more complex than (3). The environmental

variable z is some measure of the global climate, and is non-indexed since the global climate

is common to all countries. This variable changes gradually over time, depending (with quite

a long time lag) on the development of the atmospheric concentrations of a large number of

greenhouse gases (of which CO2 is the most important). The development of each of these

atmospheric concentrations in turn depends on the emissions of all of the greenhouse gases

from all countries. Any applied analysis of the climate problem must start with a model of

these two first steps, in order to obtain a specification of the connection between emissions

(Σiei for all greenhouse gases) and a description of the climate (z). There have been a number

of analyses of this relationship, perhaps the most well known are the IPCC studies. Simpler

versions of these scientific models are sometimes used as parts of economic models, where

also the development of production for all countries (or group of countries) is modeled (see

e.g. Nordhaus and Yang (1996) as an important example).

Consider next the problem of acid rain. Acid rain is caused by the discharge of sulfur and

nitrogen oxides into the air. The problem of acid rain is transboundary because these oxides

remain in the air long enough to be transported across national boundaries. Unlike the climate

problem, the transportation matrix A in the problem of acid rain for a particular region (e.g.

Europe) does not have a specific and simple structure. Typically, all or most coefficients in

the matrix differ, and are positive. An early economic analysis of the acid rain problem in

5Europe was given by Mäler (1989), and Downing et al. (1997) have given an analysis of the

acid rain problem in Asia.

3. Abatement costs and environmental costs

We assume that each country’s income is increasing – up to a limit – in its own emissions.

Put differently, each country has some emission level that would follow from its production

decisions if these decisions were made without considering the environmental impacts of the

emissions. This emission level is often called the country’s “business as usual” emission

level. Reducing emission below the business as usual level is costly, i.e. reduces the country’s

income. We formalize this cost of reducing emissions – often called abatement costs –by the

income function Rj(ej). This function is assumed to be increasing (up to the business as usual

level of emissions) and strictly concave. Denoting the business as usual level of emission by

ej0 , we thus assume Rj’>0 (for ej<ej

0) and Rj’’<0. Since Rj’ tells us by how much income

declines as emissions are reduced, we often call Rj’ the marginal abatement cost.

A large number of studies have tried to give numerical estimations of the functions Rj(ej) for

greenhouse gas emissions. Overviews of some of these studies are given in the articles in a

special issue of The Energy Journal, see Weyant (1999).

As mentioned previously, country j considers the environmental variable zj as something

negative. We assume that each country j has some monetary valuation of how harmful the

depositions zj are. These valuations are typically called “environmental cost” by economists.

Formally, the environmental cost of country j is given by the environmental cost function

Dj(zj), which is larger the larger zj is. Notice that the valuation of the environmental damage

may differ between countries even if the environmental variable z is the same for all

countries, as is the case for the climate problem. There have been a number of studies trying

to estimate the environmental damage functions Dj(z) for the climate problem, although it is

broadly agreed that the exact nature of this function is very uncertain (see Tol et al. (2000) for

a recent discussion). Notice that this latter uncertainty comes in addition the scientific

uncertainty regarding the relationship between emissions and the development of the climate.

4. The non-cooperative equilibrium3

3 The analysis in this and the next section is closely related to the analysis by Markusen (1975).

6In the absence of any coordination between countries, it is usually assumed that each country

chooses its own level of emissions so that its own net benefits are maximized, taking

emission levels of other countries as given. Formally, country j maximizes ( ) ( )j j j jR e D z−

subject to (1) and taking the emission levels of other countries as given. Solving these

maximization problems, we find that the Nash equilibrium is given by emission levels that

satisfy4

'( ) '( )j j jj j jR e a D z= for all j (4)

The Nash equilibrium (4) is illustrated in Figure 1 for the two-country case in which all aij>0.

In this Figure, ICj' and ICj'' are iso-welfare curves for country j, i.e. curves for which net

benefits Rj(ej)-Dj(zj) are constant. ICj'' represents a higher welfare level than ICj' (since

Dj’>0). The line r1(e2) is country 1's response function, giving its optimal emission level for

any given emission level of country 2. It will be downward sloping as in Figure 1 provided

R1 is strictly concave and D1 is strictly convex. The interpretation of this is that an increase in

emissions from country 2 will increase z1, which will increase the marginal environmental

cost in country 1, thus making it optimal to reduce emissions from country 1. Notice that this

reason for a downward sloping response function (discussed in more detail in e.g. Hoel

(1991)) is different from the reason often given for “carbon leakage” for the climate problem.

Carbon leakage is the result of changes in world market prices of energy and energy intensive

goods caused by one or several countries introducing a carbon tax or other policies reducing

the demand for of fossil fuels in these countries (for a further discussion see e.g. Golombek et

al. (1995), Hoel (1996) and Hoel (2001)).

The line r2(e1) in Figure 1 is country 2's response function. It will be downward sloping

under corresponding assumptions as for country 1. The Nash equilibrium is given at point N

in Figure 1.

5. Pareto optimality with and without side payments

4 Throughout the article, we make the “standard” assumption that Dj’’≥0 for all j. Moreover, we assume that functions have properties that rule out corner solutions (i.e. zero emissions, and in sections 9-13, zero R&D expenditures).

Figure 1 approximately here

7It is clear from Figure 1 that N is not a Pareto-optimal point. Ignoring any side payments, the

Pareto-optimal points in Figure 1 are all points along the line L connecting the tangency

points of iso-welfare curves. Starting from N or any other point that is not on the line L it is

possible to change emissions in both countries in a way that increases the net benefits Rj(ej)-

Dj(zj) for both countries. Among the Pareto-optimal points, all points on the heavily drawn

portion of L are Pareto-preferred (i.e., preferred by both countries) to the Nash equilibrium,

N.

Formally, the Pareto-optimal emission levels follow from maximizing a weighted avereage of

the net benefits in all countries. Letting the weights be α1,...,αn (all non-negative and adding

to one), we maximize

( ) ( )k k k k i ikk iW R e D e aα ⎡ ⎤= −⎣ ⎦∑ ∑ (5)

with respect to all emission levels. The outcome, given by

'( ) '( )kj j jk k kk

j

R e a D zαα

=∑ for all j (6)

of course depends on the weights (α1,...,αn). The set of all Pareto outcomes are given by (6)

for the set of non-negative αi's satisfying Σiαi=1.

If side payments are permitted, the set of Pareto-optimal outcome changes. In this case, we

can set all αi's in (5) equal to 1 before maximizing, since distributional considerations in this

case are taken care of through transfers between countries. Maximization of (5) in this case

gives the following conditions:

'( ) '( )j j jk k kkR e a D z=∑ for all j (7)

The term on the left-hand side of (7) is the marginal cost of emission abatement, and the

terms on the right-hand side of (7) are the marginal environmental costs. The interpretation

of (7) is thus that the marginal abatement cost in country j should be equal to the sum of

marginal environmental costs its emissions causes in all countries.

8

Our specification of environmental costs disregards possible income effects, i.e. the valuation

of the environment is independent of the country’s income. The outcome given by (7)

therefore gives a unique level of emissions for each country. In Figure 1, this unique level of

emissions is the point on the line L in Figure 1 that maximizes W for all αj=1. Notice that the

Pareto-optimal point in this case may very well be given by a point such as Q in Figure 1.

Without side payments, country 1 would be worse off under this equilibrium than it was

under the Nash equilibrium. With side payments, however, any distribution of welfare

between the countries is possible to achieve. In particular, a continuum of equilibria with

welfare levels higher than under the Nash equilibrium are possible to achieve with suitable

side payments.

Mäler's (1989) study of acid rain in Europe is an example of the case in which a unique

Pareto optimal point Q is worse for some countries than the Nash equilibrium (in the absence

of side payments). Mäler's study includes 26 European countries, with linear transportation

functions as in (1), and with net benefits in each country given as in (5). The Pareto optimal

allocation of emissions is thus given by (7). Although the total gain to the countries is

substantial, four of the countries are worse off than under the Nash equilibrium (in the

absence of transfers).

It is useful also to consider the case of a pure unidirectional transboundary problem.

Consider the two-country case, and assume that country 1 is the upstream country, i.e., a21=0.

This case is illustrated in Figure 2. Country 2's iso-welfare curves and response function is

the same as in Figure 1. However, for country 1, net benefits are independent of the

emissions from country 2, and are maximized at e1*, defined by 1 1 11 1 11 1'( *) '( *)R e a D a e= . This

implies that the iso-welfare curves for country 1 are now vertical, forming a ridge at e1*. The

iso-welfare curve on the top of this ridge is also the response function of country 1. The

Nash equilibrium is now given by N, and in the absence of side payments the Pareto-optimal

outcomes are on the line L. Notice in particular that for this case the Nash equilibrium is

Pareto optimal and that no other Pareto-optimal outcomes are Pareto preferred to the Nash

equilibrium. These properties of the Pareto-optimal outcomes are of course only valid when

side payments are ignored. With side payments, the Pareto optimal outcomes must satisfy

9(7), which gives a particular level of emissions somewhere on the line L to the left of N

(since (7) implies that 1 1 11 1 11 1'( ) '( )R e a D a e> ).

Consider next the climate problem. As mentioned previously, this special case corresponds to

all aij being equal to one. When side payments are permitted, it thus follows from (7) that

'( ) '( )j j kkR e D z=∑ for all j (8)

where z in this case simply is the sum of emissions.

That important property of (8) is that the right-hand side is independent of j. This implies

that marginal abatement costs should be equalized across countries. This equalization of

marginal abatement costs is necessary and sufficient for cost-effectiveness for this type of

environmental problem. Cost-effectiveness means that the environmental goals, whatever

they are, are reached at as low costs as possible. When side payments are permitted, cost-

effectiveness is obviously a necessary (but not sufficient) condition for Pareto optimality5. If

side payments are ruled out, cost-effectiveness as it is defined above is not a particularly

interesting concept, since total costs are of limited interest when there are restrictions on the

distribution of the costs between countries.

6. International cooperation

It is clear from the discussion above that without cooperation between countries, the outcome

will generally not be Pareto optimal. The only exception is the case of a unilateral

transboundary environmental problem. In this case, the Nash equilibrium is Pareto optimal,

provided side payments are ruled out. It is, however, difficult to find good reasons for ruling

out side payments. When side payments are allowed, there always exist outcomes that are

Pareto preferred to the Nash equilibrium. This follows directly from (4) and (7): Emission

levels satisfying (4) cannot satisfy (7), as long as aij > 0 for some i≠j.

5 Cost-effectiveness means that all Rj’ are equal. This common value of Rj’ must equal the right hand side of (8) in order to have full Pareto optimality.

Figure 2 approximately here

10Even if there were only two countries, cooperation may be difficult to achieve. The reason is

that there are several outcomes that are Pareto optimal and Pareto preferred to the Nash

equilibrium. One of the problems of reaching an agreement between the countries is that it

may be difficult to agree on which of these outcomes to implement. If the two countries were

equal in all respects, it is likely that the countries would quickly agree upon a symmetric

agreement, with equal emissions and no transfer payments. When countries differ, as they do

in reality, it is likely to be more difficult to reach an agreement.

For most transboundary environmental problems, there are more than two countries involved.

This is likely to increase the difficulties of reaching an agreement, due to the free rider

problem: If a country stays outside an agreement between all other countries, it can enjoy

(almost) the same benefits of reduced emissions as if it participates in the agreement, while it

doesn't bear any of the costs of reducing emissions. This free rider incentive remains even if

the agreement is such that all countries are better off with the agreement than without: A

country may be better off participating in an agreement than it would be without any

agreement. But it will usually be even better off if the other countries cooperate, while it itself

stays outside the agreement and pursues its self-interest.

The issue of free riding, and the possibility of creating stable coalitions, has been extensively

discussed during the last couple of decades. For a recent thorough discussion of this and

related issues, see Barrett (2003).

7. The design of an international environmental agreement

Assume now that the difficulties of reaching an agreement between countries affected by a

transboundary environmental problem have been “solved”, in the sense that some or all

countries involved agree to cooperate. The next issue is how an international environmental

agreement between these countries ought to be designed.

There are two main types of international environmental agreements for transboundary

environmental problems. The first, and probably most common, focuses directly on emissions

in each country. The second type of agreements focuses on environmental policies in each

country. An obvious example of the second type would be an agreement that specified the

emission tax rates to be used by each country.

11

An international agreement focusing directly on emissions from each country would have to

specify emissions according to (7) in order to be Pareto optimal. More or less arbitrary

emission allocations, such as e.g. cutting back emissions by some uniform percent rate in all

countries (compared with a specified base year) would not be Pareto optimal. Moreover, even

if one succeeded in designing an agreement specifying emissions in accordance with (7), it

would in many cases be necessary to supplement this agreement with a set of transfer in order

to make all countries better off with the agreement than without (cf. the discussion above of

Mäler's (1989) study on acid rain).

The Sulfur Protocols of several European countries are examples of international

environmental agreements. The aim of these agreements has been to reduce acid rain cased

by the emissions of SO2. The First Sulfur Protocol is from 1985, and was signed by 20

countries. This Protocol was an example of a simple “uniform percent reduction” type: Each

country was required to reduce their annual emissions of SO2 by 30 percent as rapidly as

possible, and no later than 1993, using 1980 as a base year. This protocol was followed up

with the Second Sulfur Protocol in 1994, which was signed by 26 countries. This protocol

had a more sophisticated design than the First Protocol. In the Second Protocol, the required

emission reductions differ between countries. A starting point for the emission reductions

agreed upon is the cost-effective emissions, given specific limits on depositions in various

regions.6

8. International climate agreements

In this section we consider an international climate agreement that has strong similarities with

the Kyoto agreement, which came into force in February 2005.7 The agreement specifies the

initial distribution of emission quotas between countries, but allows countries to buy or sell

quotas from/to other countries. The agreement imposes no restrictions on how a country sets

its domestic policy, as long as its emissions does not exceed its quotas (i.e. initial endowment

adjusted for quotas purchased or sold).

Consider a particular country j and assume that it initially is given emission quotas equal to

6 See Klaassen (1996, ch. 8) for a further discussion of the details of the Second Sulphur Protocol. 7 See http://unfccc.int/essential_background/kyoto_protocol/items/2830.php for details of this agreement.

12je . Quotas can be traded in a competitive international market where the price of quotas is

termed p.8 The country faces costs of abatement and costs of its net purchase of quotas. The

latter cost equals the price of quotas (p) times net purchase of quotas, which is the difference

between the countries emissions (ej) and endowment ( je ). Country j therefore

maximizes ( ) [ ]j j j jR e p e e− − . Notice that the environmental cost function does not appear in

this expression, since the sum of emissions is given from the climate agreement (equal to the

sum of quotas initially allocated to all countries). The first order condition for the

maximization problem of country j is

'( )jR e p= (9)

The price of quotas is determined in the international market so that the total supply of quotas

equals the total demand of quotas. The larger is the amount of quotas initially allocated, the

lower is the equilibrium price of quotas. Hence, the group of cooperating countries

determines the price of quotas through the total number of quotas agreed upon. It follows

from (9) that we get cost-effectiveness as required by (8), i.e. marginal abatement costs are

equalized across countries. Moreover, this common marginal abatement cost can be made to

equalize the sum of marginal environmental costs as required by (8) by a suitable choice of

the total amount of quotas initially allocated. These desirable properties of this type of

agreement hold no matter how the emission quotas initially are distributed among countries.

This distribution can therefore be made with a focus on the distribution of the net benefits

across countries. With a suitable distribution, all countries will achieve higher net benefits

with the agreement than under the non-cooperative equilibrium. Notice that this is true even

if some of the countries are of the opinion that the climate change following from increased

concentrations of greenhouse gases will not adversely affect them. For such a country we will

have Dj(z)=0, so that this country must have an initial quota that is so high that the

maximized value of ( ) [ ]j j j jR e p e e− − is higher than the maximized value of ( )j jR e . In

words, this means that the country must earn more from selling excess emission quotas than it

looses from reducing its emissions.

8 Assuming there are many countries, each country will be small, and hence each country will consider the market price as given. For discussions of the properties of this type of agreement if some countries have market power on the emission market, see e.g. Hoel (1997) and Westskog (1996).

13

In spite of the fact that an agreement may be designed so that all countries are better off with

the agreement than without, the free rider problem nevertheless remains: Any single country

will typically be better off without cooperating, given the decision of the other countries to

cooperate.

As mentioned above, an alternative to agreements focusing directly on emissions is an

agreement specifying the use of policy instruments, e.g. emission taxes. For the climate, an

emission tax which is equalized across countries is an obvious candidate for this type of

agreement. Given that households and producers respond to the tax so that marginal

abatement costs in equilibrium are equal to the emission tax rate, marginal abatement costs

will be equalized across countries. There are, however problems with this type of agreement.

Consider e.g. a carbon tax which is equalized across countries. One problem is the associated

distribution of cost between countries. Even if marginal costs are equalized across countries,

total costs of reducing emissions will generally differ between countries. An analysis by

Kverndokk (1993) suggests that the cost as percent of GDP differs sharply between countries

when CO2 emissions are allocated in a cost efficient manner. Moreover, Kverndokk's analysis

suggests that it is the richest countries in the world that would have the smallest total costs of

reducing emissions (relative to GDP). An international climate agreement with such

distributional properties will be unacceptable to a large group of countries, and will therefore

in practice be infeasible unless it is supplemented with some kind of transfer payments

between countries.

Another problem associated with harmonizing carbon taxes is related to the free rider issue.

The free rider incentive implies that it is in each country's interest to have little or no

restrictions on its own CO2 emissions, given the emissions from other countries, or given the

policies of other countries. If a country is required to have a specific carbon tax through an

international agreement, it is therefore in the interest of that country to try to render this tax as

ineffective as possible. As argued in e.g. Hoel (1992) there are several ways in which a

country can reduce the effect of an imposed carbon tax on the country's consumption and

production pattern, and thereby reduce the cost for the country, even though it in a formal

sense is adhering to the international agreement to tax CO2 emissions. To eliminate evasions

of this type, the agreement would have to be more complex than simply specifying a uniform

14carbon tax to be used by all countries. Even if one restricted oneself to existing fossil fuel

taxes in the narrow sense, it would be difficult to specify exactly what each country can and

can not do with these domestic taxes. Expanding an agreement to include more or less

detailed instructions on how each country can use other policy instruments which strongly

affect CO2 emissions would make the agreement very complicated. Moreover, it seems likely

that most countries would find detailed specifications and restrictions on their use of various

domestic policy instruments over time as an unacceptable restriction on their sovereignty.

9. Technology development

The rest of this article focuses exclusively on global warming. For this particular

transboundary environmental problem, we shall expand our previous analysis to include

endogenous development of improved technology. The motivation for this is that a

significant reduction in global greenhouse gas emissions will require development of new

technologies if such reductions are to be achieved without excessive costs. An important

question is whether an agreement of the Kyoto type will give sufficient incentives to develop

such new technologies. On the one hand, since greenhouse gas emissions will become costly

for countries and private producers, countries and individual producers will have incentives

to undertake effort and costs, e.g. research and development (R&D) expenditures, to develop

new technologies. On the other hand, R&D in one country is not only advantageous for this

country, but usually also for other countries. The reason for this is that producers in these

countries in many cases will learn from the R&D project, for example, through formal and

informal networks, journals, and in some cases through the import of goods from the country

where the new technology is developed. Such technology spillovers imply that without any

additional elements in a climate agreement (for example, the Kyoto agreement) there will be

insufficient incentives to develop new technologies. An international climate agreement

should therefore ideally include some elements making countries undertake more R&D than

they would choose in the absence of such elements.

An obvious question is why the Kyoto agreement does not include elements related to R&D

of new technologies. One reason could be the problems of designing an agreement to include

such elements, as the magnitude of R&D expenditures in a country is difficult to verify by

other countries. If a country is required - through an international agreement - to have more

R&D expenditures than what is individually rational for the country, it will be relatively easy

15for the country to have less R&D than required by the agreement, but to report other

expenditures as R&D activities. The same problems apply with respect to including policy

instruments that affect R&D expenditures in an international agreement. In general, policies

aimed at influencing R&D investments by private firms will be an integrated part of a

country’s tax system and to some extent other domestic policies. As tax systems and other

policies vary significantly across countries, it will in practice hardly be feasible for a country

(or some international agency) to verify all aspects of R&D policies of other countries.

For the reasons given above we shall analyze an international climate agreement that does not

contain elements related to R&D expenditures. A key issue will be to examine whether a

Kyoto type of agreement can provide the correct social amount of aggregate emissions and

R&D investments in new technologies. We argue that the outcome of a Kyoto type

agreement will differ from the social optimum. In particular, for a given level of abatement a

Kyoto type agreement provides too little R&D investments relative to the social optimum.

10. Technology spillovers

In order to keep the analysis as simple as possible, we continue to use a static framework,

thus neglecting the dynamic aspects of R&D (and as before ignoring the fact that greenhouse

gases are stock pollutants). We also consider only one type of greenhouse gas, namely CO2.

None of our results are affected by this simplification. Moreover, all types of uncertainties –

like the rate of return on R&D investments - are disregarded. Finally, all countries are

assumed identical. While these assumptions of course are drastic simplifications, the analysis

nevertheless gives insight that is relevant also in the real world.

There are n identical countries, and we assume that the technology level in a specific country

depends on the amount of R&D expenditures in that country, x, and also the R&D

expenditures in all other countries. However, technological diffusion is not perfect, only part

of the R&D expenditures investments undertaken in one country ( 0 1γ< < ) is beneficial for

other countries. Hence, the technology level of a particular country investing x in R&D is

given by:

( 1) *y x n xγ= + − (10)

16assuming that the R&D expenditures in each of the other n-1 countries are x*. In (10) we

have assumed an additive structure of technology spillovers, that is, the technology level of a

country depends on the sum of R&D investments undertaken in all countries, corrected by the

technology diffusion parameter γ . This way of modeling spillovers can be found in a wide

range of theoretical and empirical contributions, and goes back at least to Spence (1984).

Although spillovers often are modeled as in (10), it is not obvious that this is the best way of

modeling technology spillovers. Cohen and Levinthal (1989) have argued that the ability of a

firm and a country to learn from other agents may depend on its own R&D effort. For an

application of the ideas of Cohen and Levinthal on climate policy, see Golombek and Hoel

(2004b). We shall stick to the “standard” formulation (10).

The income of each country is as before assumed to depend on emissions. It is now also

assumed to depend on the technology level of the country. Income is thus given by ( , )R e y .

As before we assume Re>0 (for e<e0) and Ree<0. We also assume that R is increasing in y and

strictly concave, i.e. Ry>0, Ryy<0 and ReeRyy-(Rey)2>0. Finally, we make the important

assumption that technology development reduces marginal abatement costs. This means that

Rey<0. As technology improves, the slope of the curve Re thus becomes flatter. This is

illustrated in Figure 3, where the solid line represents the old technology and the dashed line

represents the new technology (higher y). The assumption Rey<0 implies that the business as

usual emission level goes down as in Figure 3 (from e0 to e1).

11. The non-cooperative equilibrium with endogenous technology

To simplify the discussion we now assume that environmental costs are linear, i.e. that the

marginal environmental costs are constant, denoted by b. In other words, D(z)=bz=bne,

since total emissions in the present case are simply ne. When there is no international climate

agreement, each country chooses its abatement and R&D expenditures in order to maximize

its net benefits, which now are given by [ ]( , ) ( 1) *R e y x b e n e− − + − . Formally, we find the

non-cooperative equilibrium by maximizing this expression (subject to (10)) assuming that

each country takes R&D expenditures and emission levels in all other countries as given (for

each country equal to x* and e*, respectively). The first order conditions for this problem are

given by:

Figure 3 approximately here

17 ( , )eR e y b= (11)

1 ( , )yR e y= (12)

Equations (11) and (12) determine emission levels and R&D expenditures in all countries.

Because countries are identical, they will choose identical amounts of abatement and R&D

expenditures. Equation (11) corresponds completely to (4). The interpretation of (12) is that

increased costs of one additional unit of R&D in a country should at the margin be balanced

against the benefits to the country from the R&D expenditure. Since R&D expenditures are

measured in money, the cost of one unit of R&D is by definition equal to one. The benefits of

increased R&D investments in a country are reduced abatement costs for the country, i.e. yR .

12. The social optimum with endogenous technology

The social optimum is defined as the outcome (given by emission levels and R&D

expenditures) that maximizes net benefits aggregated over all countries. Because countries

are identical, all countries must have the same emission levels (e) and R&D expenditures ( x )

in the social optimum. For each country net benefits now equal income minus R&D

expenditures and environmental costs. Aggregating net benefits over all over all countries

gives

[ ( , ) ]n R e y x bne− − (13)

Maximizing (13) subject to the relationship between technology level and R&D expenditures

(eqation (10) with x*=x) gives the first order conditions

( , )eR e y nb= (14)

1 (1 ( 1) ) ( , ))yn R e yγ= + − (15)

Equation (14) corresponds completely to (8). The interpretation of (15) is similar to the

interpretation of (12). The difference is that the benefits of R&D expenditure in a country

now not only are the reduced abatement costs for the country itself ( yR ), but also the reduced

abatement costs for other countries through technological diffusion ( ( 1) yn Rγ − ).

18

Comparing the non-cooperative equilibrium (given by (11) and (12)) with the social optimum

(given by (14) and (15)), we immediately see that there are two differences. First, in the case

of no international agreement each country valuates changes in emissions only through its

own marginal environmental cost b and not the total marginal environmental costs nb as in

the social optimum (this is equivalent to what we found when (4) was compared to (7)). This

tends to yield lower abatement in the case of no international agreement than in the social

optimum.

The second difference is that in the case of no agreement each country ignores the impact of

its R&D investment on abatement costs in other countries through technological diffusion

( ( 1) ( , )yn R e yγ − ). This difference suggests that R&D expenditures will be lower in the case

of no international agreement than in the social optimum.

13. The inefficiency of a Kyoto type agreement when technology is endogenous.

We now consider the same type of agreement as we did in Section 8. The present case is

practically identical to the case treated in Section 8, except that we now have the additional

technology variable. In the present case, country j now maximizes

( , ) [ ]R e y x p e e− − − subject to (10). The first order conditions for the minimization problem

are

( , )eR e y p= (16)

and

1 ( , )yR e y= (17)

Equation (16) is basically the same as (9), while equation (17) is identical to what we had in

the non-cooperative equilibrium.

Let us compare the Kyoto type agreement with the social optimum, given by (14) and (15).

Notice that in the first-best optimum both abatement levels and R&D expenditures are

uniquely determined. This is not the case for the Kyoto type agreement. For this agreement

the outcome depends on the quota price p, which in turn depends on the total amount of

emissions that the agreement specifies. There are therefore several ways one could compare

19these two cases. One alternative would be to compare them under the assumption that the

total amount of quotas in the Kyoto type of agreement is set equal to the total amount of

emissions that follows from the social optimum (often called the first-best optimum). A

second alternative would be to set the total amount of quotas in the Kyoto agreement in order

to minimize total social costs (given by (13)), subject to the constraints given by the

behaviour of the individual countries, that is, (16) and (17). Such a minimization will give a

second-best optimal outcome for abatement levels and R&D expenditures under the Kyoto

agreement, which can be compared with the abatement levels and R&D expenditures in the

first-best optimum.9

Consider the first of the alternatives above. When the variable e is the same in the two cases,

one important difference between the Kyoto type agreement and the first-best optimum is

given by the difference between equations (15) and (17). Since e is given, it immediately

follows from the properties of the abatement cost function and the fact that 0γ > that the

technology level y is lower under the quota agreement than in the first-best optimum. The

reason for this is that the positive spillover effects of R&D expenditures to other countries are

ignored under the Kyoto type of climate agreement. We can thus conclude that there will be

too little R&D expenditures in the Kyoto type agreement even if total emissions are set equal

to what they are in the first-best optimum.

Consider next a Kyoto type agreement where the total number of quotas (which determines

total emissions, and thus also determines abatement) are set in order to maximize the sum of

net benefits. Whatever this number of quotas is, R&D expenditures will be set so that (17) is

satisfied. This equation defines y as an declining function of e; y(e). The group of cooperating

countries takes into account the function y(e) when choosing the optimal number of quotas.

Moreover, the group takes into account that with identical countries, R&D expenditures will

be the same in all countries. Inserting y(e) and (10) for *x x= into the expression (13) for

total net benefits aggregated over all countries, and maximizing with respect to e gives

1( , ) 1 '( )1 ( 1)eR e y nb y e

n γ⎡ ⎤

= − −⎢ ⎥+ −⎣ ⎦ (18)

9 In a second-best optimum an optimization problem (maximizing total net benefits in the present article) is solved by taking into account the behaviour of agents (relations (16) and (17)), which restricts the set of possible outcomes. In the first-best optimum, there are no behavioural restrictions.

20

where we have used (17). The term in square brackets is positive and '( )y e is negative. The

right hand side of (18) is therefore larger than nb, indicating the additional benefit from

abatement that comes from inducing countries to spend more on R&D through a high quota

price (that is, by choosing a low value of total number of quotas). In the second-best optimum

one should therefore set total emissions so low that the marginal abatement costs in

equilibrium ( ( , )eR e y ) become higher than the marginal environmental costs (nb).

Let us now compare the second-best Kyoto type agreement (where emissions are determined

by (18)) with the first-best optimum (where emissions are determined by (14)). From the

discussion above we know that marginal costs ( ( , )eR e y ) are higher in the Kyoto type

agreement than in the first-best optimum. One might therefore expect total emissions to be

lower in the second-best Kyoto type agreement than in the first-best optimum. This is

however not obvious, since R&D expenditures may be different in the two cases. To formally

compare the two outcomes, use superscripts K and F for Kyoto type agreement and the first-

best optimum, respectively. From (14) and (18) we have ( , ) ( , )K K F Fe eR e y R e y> . Since eeR

and eyR are both negative, this inequality implies that if K Fy y≥ then K Fe e< , i.e. emissions

will be lower in the Kyoto type agreement than in the first-best optimum. However, since

there are no elements directed towards R&D expenditures in the Kyoto type agreement, it

may very well be the case that K Fy y< . If this is the case we cannot conclude from

( , ) ( , )K K F Fe eR e y R e y> whether the Kyoto agreement will give lower or higher emission

levels than the fist-best optimum. In Golombek and Hoel (2005b) it is shown that the sign of K Fy y− is generally ambiguous, and sufficient conditions for this sigh to be negative are

also derived. Finally, it is shown that if K Fy y< , we generally do not know whether the

Kyoto agreement will give lower or higher emission levels than the first-best optimum.

14. Concluding remarks

The climate problem is a special case of general transboundary environmental problems.

Such problems require some kind of international cooperation in order to obtain a socially

efficient outcome. With a suitably designed international agreement, it is in principle possible

to make all countries better off when they cooperate than if no countries cooperate. However,

21international cooperation may be undermined by free riding: Although a potential agreement

may make all countries better off than a situation without any agreement, it will typically be

the case that each county is even better off if it does not join the agreement but lets the other

countries cooperate to reduce emissions. Since all countries have this free rider incentive, it

may be difficult to obtain voluntary participation.

Ignoring the free rider issue, an agreement of the Kyoto type for the climate problem has

some desirable properties: Correctly designed, it will give a cost-effective allocation of

emissions across countries, and all countries can be made better off under such an agreement

than they are without any cooperation. However, a weakness with a Kyoto type agreement

focusing only on emission levels is that it may give an inefficient outcome due to

international technology spillovers. A Kyoto type of agreement will not necessarily imply

lower abatement levels than what is socially optimal, but there will typically be too little

R&D activities relative to abatement efforts.

As mentioned previously, one reason for not including R&D policies in an agreement is

difficulties in monitoring compliance of this element of an agreement. However, even if it is

difficult or impossible to design a first-best optimal agreement, agreements where R&D

policies are included in an imperfect manner may be superior to agreements that ignore R&D

policies. There is also another reason why it might be desirable to include elements related to

technology development in an international climate agreement. The strong free rider

incentives present in agreements of the Kyoto type may be one of the reasons why the present

Kyoto agreement that regulates aggregate emissions for the period 2008-2012 only includes

countries that totally stand for less than 30% of aggregate world emissions of greenhouse

gases. The weakness of agreements focusing only on emission reductions has lead several

economists to argue that agreements ought to have more focus on the development of new

technology (see e.g. Barrett (2003), Buchner and Carraro (2004), Carraro and Marchiori

(2003)). Such technology based agreements might have weaker free rider incentives than

traditional agreements of the Kyoto type, and therefore be more likely to be successful.

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25

Fig. 1. Nash equilibrium (point N) for the two-country case in which 0ijα >

Fig. 2. Nash equilibrium (point N) for the two-country case in which 21 0α =

( )1 2r e

( )2 1r e

1e

2e

1e∗

L

N

( )1 2r e

1e

( )2 1r e

L

N

1IC′

1IC′′

2IC′′

2IC′

2e

Q

26

Fig. 3. Improved technology, ( )R e changes from _____ to _ _ _ _

( )R e

e0e1e