Antoine D'autume

Katheline Schubert

Cees Withagen

Should the carbon price be the same in all countries ?

International di¤erences in fuel taxation are huge, and may be jus-ti…ed by di¤erent local negative externalities that taxes must correct, as well as by di¤erent preferences for public spending. In this context, should a worldwide unique carbon tax be added to these local taxes to correct the global warming externality? We address this question in a second best framework à la Ramsey, where public goods have to be …nanced through distortionary taxation and the cost of public funds has to be weighted against the utility of public goods. We show that when lump-sum transfers between countries are allowed for, the second best tax on the polluting good may be decomposed into three parts: one, country speci…c, dealing with the local negative externality, a second one, country speci…c, dealing with the cost of public funds, and a third one, global, dealing with the global externality and which can be interpreted as the carbon price. Our main contribution is to show that the uniqueness of the carbon price should still hold in this second best framework. Nevertheless, if lump-sum transfers between governments are impossible to implement, international differentiation of the carbon price is the only way to take care of equity concerns.

Introduction 1

The virtues of a unique carbon price, taking the form of a world carbon tax or a world emission permits market, are well recognized. A unique carbon price, re ‡ecting the true social cost of emissions, is the best incentive to curb all the negative externalities associated with fossil fuel consumption and global warming. Uniqueness of the price implies the equalization of marginal abatement costs and therefore minimizes the worldwide cost of abatement of emissions. The redistribution of tax receipts or the initial allocation of permits o¤ers then the possibility to accompany carbon taxation with an international redistribution scheme and to share equitably the burden of taxation between countries.

This optimistic picture is often questionned in the name of realism. Existing di¤erences in national energy taxation, especially fuel taxation, are important (see Table 1). The question that arises is whether we should consider them as implicit carbon taxes and therefore abstain from super-imposing a speci…c carbon tax. Moreover, is it really possible to use the allocation of tax receipts or emission quotas to alter substantially the world distribution of income? First of all, using a quota allocation mechanism clearly implies that the implied lumpsum transfers to local governments are restricted to be positive. Second, will governments even accept to depart from the simple rule that each country should be paid back exactly the amount of taxes it pays or the value of the permits it had to buy? If international transfers are so restricted, isn't it preferable to allow poor countries to face a lower carbon price? [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] put forward such an argument against the international equalization of abatement costs and suggested that a lower e¤ort should be requested from poor countries. Shiell [START_REF] Shiell | Equity and e¢ ciency in international markets for pollution permits[END_REF] follows on this idea and in particular characterizes the set of second best optimal allocations when intercountry transfers are restricted to be positive or to exacly equal the sums paid to the world regulator.

A proper analyzis of these issues requires a clari…cation of the purpose of existing fuel taxes in the …rst place. If no clear economic reason may be invoked for their existence, adding to them a unique carbon price has no chance to appear optimal. To the contrary, if existing taxes are, in some sense, already …xed at an optimum level, adding a common tax to curb global warming may make sense. Existing fuel taxation has two prime objectives. The …rst one is to counter local externalities independently of global warming. For example, burning of fossil fuel contributes to global warming through CO 2 , but also leads to SO 2 emissions which have a more local e¤ect. The second objective of fuel taxation is that it o¤ers governments an easy way to levy funds and …nance the provision of public goods. The French gasoline tax, the so-called TIPP (Taxe Intérieure sur les Produits Pétroliers), an excise tax currently …xed at 0.69 euros per liter, has indeed been created explicitely with the two objectives of reducing negative externalities associated with fuel consumption by cars and of providing means to …nance highway construction.

International di¤erences in fuel taxation may a priori be justi…ed by different local negative externalities that taxes must correct, as well as by di¤erent preferences for public spending in the di¤erent countries. In this context, should a worldwide unique carbon tax, or emission permit price, be added to these optimal local taxes? This is the question we consider.

Addressing seriously the …nancing of public goods provision imposes a second best approach. Public goods have to be …nanced through distortionary taxation and the cost of these distorsions, i.e., the cost of public funds, has to be weighted against the utility of public goods. On the other hand, Pigovian taxes, aiming at reducing negative externalities, and (distortionary) taxes, aiming at …nancing the provision of public goods, are to some extent substitutes. A Pigovian tax required to decrease emissions is also a means to …nance public spending. Inversely, a negative externality associated with the production of a commodity decreases the cost of public funds associated with the use of this good as a tax-base. The negative impact of the distortions associated with taxation are mitigated by the reduction of the negative externality. Sandmo [START_REF] Sandmo | Optimal Taxation in the Presence of Externalities[END_REF] initiated this kind of analysis in a one country case.

In this paper, we extend the [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] model by introducing local externalities and public goods. We consider a clean consumption good, a dirty consumption good, source of both local externalities and global warming, and a public good. The three goods may be produced in each country from a given endowment of a generic good. There is no international trade and countries interact only through global emissions.

We …rst examine …rst best optimal allocations and show that they can be implemented by a world government through country-speci…c emission taxes and lump-sum transfers to consumers, possibly negative. The pollution tax consists of two parts, one dealing with the local externality and another one dealing with the global externality. The latter part is uniform across countries. This suggests a more decentralized implementation. Local governments then levy a tax correcting the local externality, are in charge of the provision of the local public good, and tax or subsidize their consumers in a lump-sum way. The federal government collects the uniform global pollution tax and makes lump-sum (positive or negative) transfers to individual states, thus taking in charge international equity concerns. This implementation assumes passive local governments, who do not take advantage of the in ‡uence they might have on the world carbon price and the levels of international transfers. Basically, this preliminary analysis con…rms the optimistic view which we described above. In a …rst best setting, the uniqueness of the carbon tax should be the rule, provided local governmental decisions are taken in an optimal way.

We then consider a more realistic second best framework. We follow the standard Ramsey approach and assume that governments cannot tax consumers in a lump-sum way. They have to rely if necessary on distortionary taxation to …nance public good provision. We show that the second best taxes on the polluting good may be decomposed into three parts: one, country speci…c, dealing with the local pollution externality, a second one, country speci…c, dealing with the cost of public funds, and a third one, global, dealing with the global externality. Our main contribution is to show that uniqueness of the carbon price should still hold in this second best framework. This is a striking result as we know since Lipsey and Lancaster [1956-7] that rules which are optimal in a …rst best world fail to be so in a second best world.

To get a better understanding of this result we show that a true decentralization of the second best optimum may be obtained. The world regulator …xes a common carbon tax as well as lump-sum transfers to local governments. Each of these is then free to deal as it wishes with local externalities and distortionary taxation needed to …nance the provision of local public goods. This result supports, in a somewhat general setting, [START_REF] Tirole | Politique climatique, une nouvelle architecture internationale[END_REF]'s proposition to base climate policy upon a world permits market, leaving each country free to choose whatever instrument they want to meet their emission target.

This result relies on the feasibility of arbitrary lump-sum transfers between countries. This takes us back to the [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] analysis2 . Let us …rst point out that the issue of the non-negativity of transfers di¤ers when applied to consumers or to local governments.

The impossibility to levy lump-sum taxes on consumers is motivated by information problems. Personalized lump-sum transfers are impossible because governments lack the necessary information about individual characteristics. The Ramsey approach then simply assumes away the possibility to levy a lump-sum tax on consumers. The use of this approach in the case of a representative consumer may be, and has often been, criticized on the grounds that if all agents, in a given country, are identical, nothing prevents the government from levying the same lump-sum tax on all of them. The Ramsey approach may nevertheless be considered as a useful short-cut to analyze a situation where distortionary taxation is required. [START_REF] Sandmo | The Public Economics of the Environment, The Lindhal Lectures[END_REF] p. 97, among others, provides such a defense.

The impossibility to implement arbitrary lump-sum transfers between governments cannot be motivated by information problems. Political economy reasons are more convincing. Governments know that their citizens would probably object to sizable transfers to other governments. The assumption of no inter-governmental transfers may then appear realistic. The receipts of the world carbon tax, or the product of the sale of the permits, have to be redistributed however. In our setting, the no inter-governmental transfers assumption has to be rephrased as the assumption that each country should receive a positive lump-sum transfer precisely equal to the amount of carbon taxation supported by its citizens. We examine in a last section the second best Pareto optimum under this assumption. The non-negativity constraint on transfers to consumers still holds. We thus revert to the [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] result. International di¤erentiation of the carbon price is the only way to take care of equity concerns. Poorer countries should proba-bly pay a lower carbon tax. This result may be put in perspective however. First, the redistributive gains of such a di¤erentiation of carbon taxes may be small. A substantial amount of redistribution would then be at the cost of an important loss of e¢ ciency. Second, accepting to implement direct intergovernmental transfers would enlarge the domain of feasible allocations. A step in this direction has been made at the Cancun United Nations Framework Convention on Climate Change conference in December 2010. A Green Climate Fund has been established to assist developing countries in the …nancing of their climate policies. This new institution should receive by 2020 up to 100 billion dollars per year from developed countries.

We restrict our analysis to the case of local governments which are passive in the sense that they take as given the world carbon price and the transfers they receive from the world regulator. [START_REF] Verbon | Do permit allocation matters? CE-Sifo Working Paper Series[END_REF] consider on the contrary the case of active local governments who act stragically as they understand that their behavior a¤ects the world carbon price. In a rather general framework, they show that a proper initial allocation of permits may counter strategic behavior and thus lead to a Pareto e¢ cient allocation. [START_REF] Ogawa | Think Locally, Act Locally: Spillovers, Spillbacks, and E¢ cient Decentralized Policymaking[END_REF] follow a di¤erent route and show that local government decisions may, in a very special case, produce a Pareto-e¢ cient outcome.

The result that the world carbon tax is uniform brings to mind an important contribution by [START_REF] Gauthier | Separability and Public Finance[END_REF], who show that, in certain circumstances, …rst best rules prevail in a second best world. Our framework however is quite di¤erent and their result does not apply in our case. We have altogether di¤erent utility functions, whereas their main result, following [START_REF] Atkinson | The design of tax structure: direct versus indirect taxation[END_REF], assumes identical consumption tastes. Secondly, and more importantly, we follow the Ramsey approach rather than the Mirrlees one. Lastly, we have an international dimension and, in particular, distinguish between transfers to consumers and to countries Our argument is therefore of a very di¤erent kind.

The model

We consider n countries (n 2), indexed by i = 1; 2; :::; n: There is a representative consumer in each. Each country has an endowment Y i of a generic good. This good may be used to produce, one for one, a private clean good, a local non-polluting public good and a private polluting good. Pollution is both local and global and encompasses country speci…c pollutions and greenhouse emissions contributing to global warming. We denote by C i , G i and X i consumptions of the three goods by the representative consumer in each country. Pollution is denoted by Z: The polluting good is scaled in such a way that Z i = X i : World pollution is denoted by

Z w = Z 1 + Z 1 ::: + Z n = X 1 + X 2 ::: + X n = X w :
By X we denote the vector (X 1 ; X 2 ; :::; X n ): The vectors C; Y; G and Z are de…ned in a similar way.

There is no international trade. Each country consumes its production of the three goods and only interacts with others through its contribution to global warming. International transfers of the non-polluting consumption good will be possible, however. As we saw, the production side of the model is highly simpli…ed, all marginal costs of production being basically equal to one, in terms of the generic good. These simpli…cations do not a¤ect our main results.

The utility function of the representative consumer in jurisdiction i is

U i (C i ; X i ; G i ; Z i ; Z w ) (1)
We assume

U i C ; U i X ; U i G > 0; U i Z ; U i Zw < 0: Here U i C
is the partial derivative with respect to C i and the other expressions have a similar meaning. Moreover, we assume that the utility function satis…es the usual conditions such as di¤erentiability, concavity in (C; X; G); convexity in (Z; Z w ) and, in maximization processes, allows for interior solutions.

3 First best

Social optimum

A (…rst best) Pareto optimum may be characterized as a feasible allocation which maximizes a weighted sum of individual utilities, since we can identify states and their representative consumers. Under convexity assumptions, varying the weights provides an easy way to describe the whole set of Pareto optima. The weights reveal how much the planner cares about each individual agent. They may be interpreted as the derivatives of a social utility function having individual utility levels as arguments. Some care must be taken when interpreting the weights along these lines however, as this interpretation implicitly assumes comparability of individual utility levels. Ordinal transformations of individual utility functions do not change the set of social optimal, but do change the weights attached to each one of these optima. Let = ( 1 ; 2 ; :::; n ) be the weights attached to the utilities of the representative consumer in each jurisdiction. A (…rst best) Pareto optimum is a solution of the following problem:

max n X i=1 i U i (C i ; X i ; G i ; X i ; X w ) (2) subject to the resource constraint n X i=1 Y i n X i=1 (C i + X i + G i ) (3)
and non-negativity conditions. Under mild non-satiation conditions, the resource constraint ( 3) is binding at a Pareto optimum.

Proposition 1 A …rst best Pareto optimum is characterized by the following conditions:

1 U 1 C = ::: = i U i C = ::: = n U n C (4) U i G U i C
= 1; i = 1; 2; ::::; n (5)

U i X U i C + U i Z U i C + n X j=1 U j Zw U j C
= 1; i = 1; 2; ::::; n

The proof of this proposition is in the appendix, as well as the proofs of all other propositions.

The three sets of conditions and the resource constraint (3) jointly determine optimal levels of production of all goods.

Conditions ( 5) and ( 6) are e¢ ciency conditions which do not depend on the social weights attached to the various consumers. They state that the consumers' marginal rate of substitution between the three goods should be equal to one, namely their relative production costs. The MRS between the polluting good X i and the non-polluting good C i takes into account the negative damages it creates, both locally and worldwide, through global warming. These damages are evaluated in terms of the private non-polluting good.

Condition (4) selects a particular Pareto optimum, depending on the chosen social weights attached to individual consumers. It shows that weighted marginal utilities derived from clean consumption should be equalized across countries.

Let us now see how this Pareto optimum may be implemented. We assume that a world regulator is in charge of the control of local and global externalities and of public goods provision. Local governments have no role. Without loss of generality, we assume that clean consumption is not taxed. The central government imposes di¤erentiated pollution taxes i (i = 1; 2; :::; n) on consumers in all countries. It …nances all public spending G i and makes direct positive or negative transfers T ci to consumers.

Budget constraints for the consumers and the federal government are:

C i + (1 + i )X i = Y i + T ci ; i = 1; 2; :::; n (7) n X i=1 i X i = n X i=1 (G i + T ci ) (8)
The representative consumer of country i maximizes utility under the budget constraint (7), taking as given pollution levels Z i and Z w . The optimality condition is:

U i X U i C = 1 + i ; i = 1; 2; :::; n (9)
Comparison with (6) shows that optimality requires tax rates

i = U i Z U i C n X j=1 U j Zw U j C ; i = 1; 2; :::; n (10)
with quantities taken at the optimum. Each tax rate appears as the sum of a local component, equal to the local marginal damage, and a common global one, equal to the sum of all worldwide marginal damages resulting from global warming:

i = i + ; i = U i Z U i C ; = n X j=1 U j Xw U j C ( 11
)
With these Pigovian tax rates, condition (9) states that the price of the polluting good, for the consumer, is equal to its true marginal cost, including its own marginal production cost, equal to one, and marginal environmental costs. Optimality also requires proper levels of provision of public goods, according to condition (5). Finally, lump-sum transfers T ci are used to reach a speci…c Pareto optimum, characterized by the weights i s attached to each country. These transfers may be positive or negative. In the case where …rst best Pigovian tax receipts are su¢ cient to …nance all …rst best public spendings,

n X i=1 i X i X G i ;
the world regulator's budget constraint (8) shows that the sum P T ci of feasible transfers to consumers is positive. All of them may be positive. In the opposite case, at least one consumer must receive a negative transfer.

Decentralization

The analysis of tax rates suggests that the optimal allocation may also be implemented in a more decentralized setting, where a local government imposes a local tax, aimed at correcting the local externality, while a global (carbon) tax is implemented at the world level. We may then assume that the world regulator redistributes the receipts of the carbon tax to local governments rather than to consumers.

Consumer budget constraints are still (7), with i = i + . The consumer pays tax i to the local government and tax to the world regulator. Let T i be the transfer to government i. Local government budget constraints are

G i + T ci = i X i + T i (12)
while the world regulator budget constraint is

n X i=1 T i = n X i=1 X i (13)
Carbon tax receipts are redistributed to local governments. Each of them uses it as well as local tax receipts to …nance local public goods provision and lump-sum transfers to its consumers. Nothing prevents transfers T i and T ci from being negative however.

From ( 7) and ( 12) we obtain country budget constraints:

C i + G i + (1 + )X i = Y i + T i ; i = 1; 2; :::; n (14)
They can be written as

T i = X i + (C i + G i + X i Y i ) ; i = 1; 2; :::; n (15)
Each transfer to a government includes the redistribution of the world taxes levied on the country and an additional term taking care of the desired redistribution between countries.

The conclusion of this exercise is that di¤erences in local energy taxes do not imply to forsake the uniqueness of the carbon tax, provided that local taxes are optimally designed to counter local externalities. This result is natural in a …rst best setting where lump-sum transfers may be freely implemented between countries and consumers. The issue is to see whether this result still holds in a second best setting.

Second best

We now assume that governments are unable to …nance public good provision through lump-sum taxation of consumers and must instead resort to distortionary taxation. We follow the Ramsey optimal taxation approach. The new constraint is simply that lump-sum transfers T ci have to be non-negative.

As was …rst shown by Sandmo [START_REF] Sandmo | Optimal Taxation in the Presence of Externalities[END_REF], the presence of externalities mod-i…es the optimal tax scheme. If Pigovian taxation receipts are su¢ cient to …nance public good provision and the desired redistribution is limited, the …rst best allocation is attainable. Tax receipts are able to …nance public goods provision and some positive lump-sum transfers to consumers. If not, additional distortionary taxation is required and we switch to a second best situation.

In our framework, the non-negativity of transfers to consumers takes the following form, which follows from (7) :

C i + (1 + i )X i Y i ; i = 1; 2; :::; n (16)
The overall transfer to a country may be de…ned as in (15).

Second best optimum

A world regulator is again in charge of all public decisions. Formally, the problem is to maximize (2) under the resource constraint (3) and the addi- tional incentive constraint

U i C (C i ; X i ; G i ; X i ; X w ) (C i Y i )+X i U i X (C i ; X i ; G i ; X i ; X w
) 0; i = 1; 2; :::; n (17) This is a so-called primal version of constraint ( 16), where we use the consumer's optimality condition (9) to eliminate the tax rate.

Assuming that the resource constraint is global amounts to accepting that intercountry transfers C i + G i + X i Y i may be freely implemented. They sum to zero and may be positive or negative.

Proposition 2 A second best Pareto optimum with intercountry transfers is characterized by the following conditions:

1 (1 + 1 + 1 H 1 C )U 1 C = ::: = n (1 + n + n H n c )U n C (18) 1 + i H i G 1 + i + i H i C U i G U i C
= 1; i = 1; 2; ::::; n (19)

1 + i + i H i X 1 + i + i H i C U i X U i C + 1 + i H i Z 1 + i + i H i C U i Z U i C + n X j=1 1 + j H j Zw 1 + j + j H j C U j Zw U j C
= 1; i = 1; 2; ::::; n

where

H i C = (C i Y i ) U i CC =U i C + X i U i XC =U i C ( 21
)
H i G = (C i Y i ) U i CG =U i G + X i U i XG =U i G (22) H i X = (C i Y i ) U i CX =U i X + X i U i XX =U i X ( 23
)
H i Z = (C i Y i ) U i CZ =U i Z + X i U i XZ =U i Z ( 24
)
H i Zw = (C i Y i ) U i CZw =U i Zw + X j U i XZw =U i Zw (25)
The conditions for a Pareto optimum must be adjusted to take into account the impossibility to …nance public goods through a lump-sum tax on consumers. Variable i is the cost of public funds in country i or, more precisely, the cost of being unable to levy a lump-sum tax on the representative consumer in this country. It is zero if the world regulator wishes instead to make a positive transfer T ci to this consumer. It might be the case that the world regulator wishes to do so for all consumers. This occurs if, on the one hand, tax receipts from strictly Pigovian taxes (the …rst best taxes) prove su¢ cient to …nance all the desired public goods and, on the other hand, the world regulator is satis…ed with redistributing the excess as positive lumpsum taxes to all consumers, as he/she does not want to alter too much the world distribution of income. In such a case all i s are zero and the second best optimality conditions reduce to the …rst best ones. This is a very special case however. A more plausible case is the one where some, or all, i s are positive. First best conditions must then be adjusted to take into account the costs of public funds.

A particular case is the one where the utility functions (1) are separable and quasi-linear in C i . Then U i c equals unity and the cross derivatives are nil which implies

H i C = H i G = H i Z = H i Zw = 0 and H i X = X i U i XX =U i X .
The …rst two optimality conditions reduce to

1 (1 + 1 ) = ::: = n (1 + n ) U i G = 1 + i Public
goods provision must be pushed in each country to the level where the marginal utility of public goods equals their production cost, here equal to one, augmented by the cost i of public funds. Moreover, optimal international redistribution requires the equality of the weighted overall costs of public goods 1 + i :

In the general non-linear case, the conditions are modi…ed to take into account cross-e¤ects represented by the H i s, which are sums of second order elasticities and are standard in the Ramsey approach.

The three sets of conditions appearing in Proposition 2, together with resource constraint (3) and incentive constraints (17) determine the second best optimum. Conditions ( 19) and ( 20) are e¢ ciency conditions which do not depend on the social weights . Conditions (18) select a particular optimum, depending on .

From these necessary conditions we can retrieve the second best optimal pollution tax i and obtain a meaningful decomposition of this tax rate. It follows from ( 9) that U i X = (1 + i )U i C : From condition (20), we have the following.

Corollary 3 In a second best setting with intercountry transfers, the carbon tax is unique. The overall tax on the polluting good may be decomposed in the sum of two local taxes and one global tax:

i = i + i +
i = 1 + i H i Z 1 + i + i H i C U i Z U i C ; i = i H i C H i X 1 + i + i H i C U i X U i C = n X j=1 1 + j H j Xw 1 + j + j H j C U j Zw U j C
Tax i can be seen as a Pigovian tax on polluting the local environment. Tax i is a Ramsey tax designed to …nance local public good provision. Finally, is a world carbon tax needed to curb greenhouse gas emissions.

If the optimal second best allocation is such that country i consumers receive a positive transfer T ci , the incentive constraint ( 17) is not binding in this country and the cost of public funds i equal to zero. The Pigovian tax i is then equal to the value that yields the …rst best (see ( 11)), while the Ramsey tax i designed to provide additional …nancing for the public good is equal to zero.

As in the …rst best case, we may allow for some decentralization of taxation at the local government level. The budget constraints (12) which were relevant in the case of the …rst best, must be adapted to take into account the local Ramsey taxes i . Government i's budget constraint now is

G i + T ci = ( i + i ) X i + T i ; T ci 0 (26)
The world regulator collects the carbon tax on consumers and redistributes the receipts as lump-sum transfers T i to local governments. The sum of these transfers is positive but some of them may be negative. If local Pigovian taxation receipts i X i are large, or if redistribution favors country i, taking the form of a large positive T i , the country will not have to rely too much on the supplementary distortive tax to …nance the provision of public goods.

The cost i of public funds will be low. It may even be zero, in which case the supplementary Ramsey tax i becomes zero and the government may make a positive transfer T ci to its consumers. The striking thing here is that the carbon tax is unique: it is the same for all states involved. This result would seem natural in a …rst best setting. It is much more surprising in a second best framework, where optimal rules are usually quite di¤erent from what they would be in a …rst best. The virtues of a unique carbon tax and its ability to provide proper incentives are much less compelling when speci…c constraints preexist and unavoidably introduce many distortions in the economy. Di¤erentiated carbon prices might then appear as a useful compromise between e¢ ciency and equity. We do assume the feasibility of lump-sum international transfers, which allows to take into account equity concerns. This does not free us from the constraints of a second best world however.

Decentralization through a world carbon price and intercountry transfers

To put our result in perspective, we now show that the second best optimum may be obtained in a decentralized way. The world regulator sets the world carbon tax and lump-sum transfers to local governments. Each of these is then free to deal as it wishes with local externalities and distortionary taxation needed to …nance the provision of local public goods.

A close examination of the determination of a second best Pareto optimum suggests a two stage solution method. Let us rewrite the problem

max C;X;G;Zw n X i=1 i U i (C i ; X i ; G i ; X i ; Z w ) U i C (C i ; X i ; G i ; X i ; Z w ) (C i Y i )+X i U i X (C i ; X i ; G i ; X i ; Z w ) 0; i = 1; :::; n (27) n X i=1 X i = Z w (28) n X i=1 (C i + X i + G i ) n X i=1 Y i (29)
We may check that this problem is equivalent to a problem where we introduce new variables and T i , i = 1; :::; n and substitute constraints

C i + G i + (1 + )X i = Y i + T i ; i = 1; 2; :::; n (30) n X i=1 T i Z w : (31)
to constraint (29).

Taking into account (28), constraints (30) and ( 31) imply (29). Reciprocally, for any arbitrary , consider a triplet (C; X; G) satisfying (29). De…ne

T i = C i +G i +(1+ )X i Y i ; i = 1; 2; :::; n. Summing implies P T i Z w = P (C i + G i + X i Y i )
which is negative as (C; X; G) satis…es ( 29). Thus constraint (31) is satis…ed.

Let us now consider the maximization of social welfare under constraints (28), ( 30) and (31). We may solve this problem in two stages. For given Z w , and T , we …rst look, for all i, for C i , G i , X i which maximize U i (C i ; X i ; G i ; X i ; Z w ) under constraints ( 27) and ( 30). This yields functions

C i = i ( ; T i ; Z w ); X i = i ( ; T i ; Z w ); G i = i ( ; T i ; Z w )
and the indirect utility function

W i ( ; T i ; Z w )
In the second stage, we solve the following problem3 :

max ;T;Zw n X i=1 i W i ( ; T i ; Z w )
under constraints ( 29) and ( 31), ( 29) being now written as

n X i=1 i ( ; T i ; Z w ) = Z w
This procedure may be interpreted as a decentralized implementation of a second best optimum. The world regulator controls carbon emissions, through a permit market or a carbon tax. It is clearer to assume the former. The world regulator chooses a maximum amount Z w of emissions, sells o¤ the permits to local governments and distributes the proceeds as lump-sum transfers to local governments. These transfers may be negative in order to further subsidize other countries. A unique carbon price is thus imposed at the world level. Each local government is still unable to …nance its public goods provision through lump-sum taxation. But it is left free to choose its own way to levy funds and regulate local externalities. A subsidiarity principle is thus in e¤ect.

Local governments are passive in the sense that they take as given the price of permits as well as total emissions Z w and transfers T i s. Their problem is to maximize the utility of their representative consumer under their country budget constraint ( 14) and their incentive constraint (17). We do not make explicit the precise way they will implement this solution. The world regulator only takes into account the indirect utility function W i ( ; T i ; Z w ) of each country and the associated demand of permits i ( ; T i ; Z w ).

Proposition 4 A second best optimum with intercountry transfers may be implemented by a world regulator controlling the total amount of emissions, imposing a unique carbon price and implementing intercountry lump-sum transfers.

This result shows how e¤ective a unique carbon price remains in a framework where each country faces second best constraints. Control of total emissions and intercountry transfers are the sole international coordination required to reach a social optimum. We also note that the proof does not depend on the form of the incentive constraints and is therefore quite general.

This proof provides a deeper explanation of the uniqueness of the carbon tax. Our …rst proof relies on the inspection of optimality conditions. The decentralization described in this section shows that this uniqueness property is in fact the consequence of a subsidiarity principle. The world regulator does not have to take into account the precise way in which each country regulates local externalities and …nances the provision of its public goods. He may trust that this is achieved e¢ ciently and only care about the global climatic externality and the proper distribution of permits. In some sense, the international regulation of the climatic externality remains a …rst best problem.

Second best without intercountry transfers

We now assume, as in Shiell [START_REF] Shiell | Equity and e¢ ciency in international markets for pollution permits[END_REF], that for political economy reasons, local governments are not ready to accept a smaller transfer than the amount of carbon taxes they are paying to the world regulator. Each local government actually receives a positive transfer from the world regulator, as the country level of emissions is always positive. There are no other intercountry transfers. Each country then has to satisfy constraint

C i + X i + G i = Y i ; i = 1; 2; :::; n ( 32
)
The problem is to maximize (2) under the resource constraints (32) and the additional incentive constraints (17).

Proposition 5 A second best Pareto optimum without intercountry transfers is characterized by the following conditions:

i 1 + j + j H j X U j X + 1 + j H j Z U j Z 1 + j + j H j C U i C = X j 1 + j H j Z U j
Zw ; i = 1; 2; ::::; n (33)

1 + i H i G 1 + i + i H i C U i G U i C
= 1; i = 1; 2; ::::; n (34)

n X j=1 1 + j H j Zw U j Zw 1 + j + j H j C U j C 1 + j + j H j X U j X 1 + j H j Z U j Z = 1 (35)
The three sets of conditions appearing in Proposition 5, together with the resource constraints (3) and incentive constraints (17) determine the second best optimum without transfers. Condition (34) governing the provision of public goods is the same as the one which appears in the second best optimum with intercountry transfers (condition ( 19)). Condition ( 35) is implied by the set of similar conditions (20) appearing in the case with transfers. Indeed, (20) may be written as

(1 + i + i H i X ) U i X + (1 + i H i Z ) U i Z (1 + i + i H i C ) U i C (1 + i + i H i C ) U i C = n X j=1 1 + j H j Zw 1 + j + j H j C U j Zw U j C
; i = 1; 2; ::::; n (36)

Dividing through by the common value of the LHS yields (35). Consider now a particular allocation in the set of second best optima with intercountry transfers, where all transfers happen to be zero. Such an equilibrium usually exists and is unique. By de…nition it satis…es constraints (32) and therefore all constraints entering the de…nition of a second best optimum without intercountry transfers. We just showed that it satis…es all the optimality conditions for such an optimum. We thus con…rm the rather natural property that, in general, one of the second best optima without intercountry transfers is a particular second best optimum with intercountry transfers. Both hypersurfaces in the space of utility levels will be tangent. This property has been stressed by Shiell [START_REF] Shiell | Equity and e¢ ciency in international markets for pollution permits[END_REF].

Other second best optimum allocations without transfers are not second best optimum allocations with transfers. As we now show, this means that di¤erentiated carbon prices will be necessary to support them.

Corollary 6 In a second best setting without intercountry transfers, the carbon tax is generally not unique. The overall tax on the polluting good can be decomposed into three country-speci…c taxes:

i = i + i + i with i = 1 + i H i Z 1 + i + i H i C U i Z U i C ; i = i H i C H i X 1 + i + i H i C U i X U i C i = 1 i (1 + i + i H i C ) U i C X j 1 + j H j Xw U j Zw
The expressions of the Pigovian and the Ramsey tax rates i and i are the same than in the case with intercountry transfers. The carbon tax i is now country-speci…c. More precisely, we have

1 (1 + 1 + 1 H 1 C )U 1 C i = ::: = n (1 + n + n H n c )U n C n (37)
Carbon prices are not equalized worldwide. Only weighted carbon prices are, the weight of country i being i (1+ i + i H i C )U i C . This generalizes the similar relation which underlines the results of [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF]. As we take into account the provision of public goods, a new element 1 + i + i H i C appears which takes into account the cost of public funds in the country.

We may then restate the core of the [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] argument. In the case where intercountry transfers are feasible, condition (18) in Proposition 2 states that country weights are equalized. It follows that carbon prices, and abatement costs in the [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] set-up, are equalized. In the absence of intercountry transfers, the weights are not equalized and neither are the optimal carbon prices.

We may assume that social weights re ‡ect the world regulator's aversion to inequality, which would automatically follow if we deduced them from the maximization of a symmetric international social utility function. A poorer country is then characterized by a higher i . This leads to reduction of inequality, but probably not its disappearance. A poorer country presumably remains poorer in the double sense that it has both a lower level of consumption, and therefore a higher marginal utility U i C of consumption, and a higher cost of public funds, and therefore a higher 1 + i + i H i C coe¢ cient. We may thus safely assume that a poorer country is characterized by a larger weight i (1

+ i + i H i C )U i C .
The carbon price i must then be lower in a poorer country. A smaller environmental e¤ort must be required from poor countries, which is simply another part of the worldwide social optimization.

Conclusion

It has been shown that the presence of public goods, restrictions on the redistributions of local tax revenues to local consumers, and the addition of a local component to the damage caused by global emissions does in principle not aggravate the problem of addressing the design of a second best optimal global pollution tax. An appropriate decomposition of the pollution tax can deal with the local aspect of pollution and with the constraint that only non-negative transfers can be given to local consumers. Hence, the global externality can be dealt with at the global level, through a uniform tax or a system of tradable permits. Nevertheless it is required that at the government levels international transfers can still be implemented. And it cannot be hoped that this problem can be solved. Indeed, it is uncertain that governments will accept negative lump-sum transfers and even to receive less than what the country paid as carbon taxes. [START_REF] Chichilnisky | Who Should Abate Carbon Emissions? An International Perspective[END_REF] have stressed that the impossibility of international redistribution should lead to reject the principle of equalizing worldwide abatement costs, that is having a unique world carbon price. If we introduce constraints on intercountry transfers into our model, a unique carbon price would not allow to reach the desired second best optimum. International di¤erentiation of the price of carbon sometimes provides a useful instrument to reach a more equitable allocation. However, when it comes to climate change international transfers (whether monetary or technological or in terms of allocating permits) are in the picture already. The international community is aware of the fact that in reaching international agreements side payments are unavoidable. Dividing (43) by and choosing from (41), in each place, the appropriate expression for leads to (20) in Proposition 2. This equation may be written as

1 + i H i X H i C 1 + i + i H i C U i X U i C + 1 + i H i Z 1 + i + i H i C U i Z U i C + n X j=1 1 + j H j Zw 1 + j + j H j C U j Zw U j C = 1 or U i X U i C 1 = i H i X H i C 1 + i + i H i C U i X U i C 1 + i H i Z 1 + i + i H i C U i Z U i C n X j=1 1 + j H j Zw 1 + j + j H j C U j Zw U j C
which is Corollary 3.

C Proof of Proposition 5 and Corollary 6

Let i and e i = i i be the Lagrange multipliers associated to constraints (32) and ( 17). Optimality conditions are, for i = 1; 2; ::::; n,

i 1 + i + i H i C U i C = i (44) i 1 + i H i G U i G = i ( 45
) i 1 + i + i H i X U i X + 1 + i H i Z U i Z = i n X j=1 j 1 + j H j Z U j

Zw

(46) (44) and ( 45) imply (34) in Proposition 5. From ( 44) and ( 46) we have

i 1 + i + i H i X U i X + 1 + i H i Z U i Z 1 + i + i H i C U i C = n X j=1 j 1 + j H j Z U j
Zw ; i = 1; 2; ::::; n (47)

The RHS does not depend on i so that all the LHS are equal for all i. Dividing through by this common value, and in particular dividing each term of the sum by the j-indexed expression of the LHS yields which is (35) in Proposition 5. Condition (47) may be written as

1 = n X j=1 1 + j H j Zw U j Zw 1 + j + j H j C U j C 1 + j + j H j X U j X 1 + j H j Z U j
1 + i H i X H i C 1 + i + i H i C U i X U i C + 1 + i H i Z 1 + i + i H i C U i Z U i C 1 = 1 i (1 + i + i H i C ) U i C X j 1 + j H j Z U j Zw or U i X U i C 1 = 1 + i H i Z 1 + i + i H i C U i Z U i C i H i X H i C 1 + i + i H i C U i X U i C 1 i (1 + i + i H i C ) U i C X j 1 + j H j Z U j
Zw which is Corollary 6.

11

  Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

13

  Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

  with

20

  Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

Z22

  Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

Table 1 :

 1 Excise on unleaded gasoline in some OECD countries, 2008 (% of the consumer price)

	G e rm a ny	D e n m a rk	U n ite d S ta te s	F in la n d	Fra n c e	Ita ly	J a p a n	U n ite d K in g d o m	S w e d e n
	46.7	38.7	14.5	42.9 43.6 40.9 34.2	44.5	40.6

Source: IEA Statistics, Energy prices and taxes, 2009

We thank Julien Daubanes, Pierre Lasserre, Stéphane Gauthier, and especially Roger Guesnerie for useful discussions. A. d'Autume and K. Schubert acknowledge the support of the French National Research Agency (ANR) under the CLEANER project (ANR_NT09_505778).

Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

[START_REF] Sheeran | Who Should Abate Carbon Emissions? A Note[END_REF] reexamined their argument in a more general setting. Also, Sandmo[START_REF] Sheeran | Who Should Abate Carbon Emissions? A Note[END_REF] shows in a model without public spending and local externalities that correcting a global externality by an internationally uniform Pigovian tax is only optimal when international lump-sum transfers are possible. 5 Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

Documents de Travail du Centre d'Economie de la Sorbonne -2011.76

The carbon price is not a choice variable since it is determined endogenously through the equilibrium of the permits market. However, as stressed by Shiell[START_REF] Shiell | Equity and e¢ ciency in international markets for pollution permits[END_REF], it is mathematically equivalent to treat it as a choice variable subject to the constraint that the permits market clears.16Documents de Travail du Centre d'Economie de laSorbonne -2011.76

A Proof of Proposition 1

Let be the Lagrange multiplier associated to the resource constraint. Necessary conditions are:

i U i G = ; i = 1; 2; ::::; n (39)

Conditions ( 38) and (39) yield condition (5) in Proposition 1.

Dividing both sides of ( 40) by , and noting from (38) that may be taken equal to i U i C or to any of the j U j C yields

= 1; i = 1; 2; ::::; n which is condition (6) in Proposition 1.

B Proof of Proposition 2 and Corollary 3

Let and e i = i i be the Lagrange multipliers associated to constraints (3) and ( 17). i will be interpreted as the cost of public funds in country i.

Assuming an interior solution for (C i ; X i ; G i ; X i X w ) > 0, we can write the optimality conditions as follows:


Dr Charlotte E Bolton
email: charlotte.bolton@nottingham.ac.uk

Wiebke Schumacher

John R Cockcroft

Nicholas J Timpson

George Davey Smith

John Gallacher

Anne Rumley

Do Gordon

Shah Lowe

Dennis Ebrahim

Shale

Gordon D Lowe

Shah Ebrahim

Dennis J Shale

Yoav Ben-Shlomo

The CRP Genotype, Serum Levels and Lung Function in men: The Caerphilly Prospective Study

Keywords: C-reactive protein, systemic inflammation, lung function, FEV 1

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Introduction

Circulating C-reactive protein (CRP), an acute phase protein has been used widely in studies evaluating associations between low-grade systemic inflammation and chronic respiratory disease. Several studies have demonstrated the association of CRP to impaired lung function, expressed as forced expiratory volume in first second (FEV 1 ) in healthy subjects [START_REF] Shaaban | Change in C-reactive protein levels and FEV1 decline: a longitudinal populationbased study[END_REF][START_REF] Fogarty | Systemic inflammation and decline in lung function in a general population: a prospective study[END_REF] and additionally with greater decline in lung function, [START_REF] Shaaban | Change in C-reactive protein levels and FEV1 decline: a longitudinal populationbased study[END_REF] although there are conflicting results [START_REF] Fogarty | Systemic inflammation and decline in lung function in a general population: a prospective study[END_REF]. The association of CRP with lung function has been shown to be independent of confounders such as smoking. Systemic inflammation has also been implicated in chronic obstructive pulmonary disease (COPD), which is associated with a lower FEV 1 and higher serum CRP [START_REF] Gan | Association between chronic obstructive pulmonary disease and systemic inflammation: a systematic review and a metaanalysis[END_REF][START_REF] Eid | Inflammatory response and body composition in chronic obstructive pulmonary disease[END_REF].

Low-grade systemic inflammation has also been implicated in cardiovascular morbidity and mortality and hence an inflammatory link has been postulated in the observed relationship between lower lung function and more cardiovascular events [START_REF] Ridker | Inflammation, aspirin, and the risk of cardiovascular disease in apparently healthy men[END_REF][START_REF] Mendall | C-reactive protein: relation to total mortality, cardiovascular mortality and cardiovascular risk factors in men[END_REF][START_REF] Sin | The relationship between reduced lung function and cardiovascular mortality: a population-based study and a systematic review of the literature[END_REF] CRP is a central target for predictive and possibly therapeutic purposes in both COPD and cardiovascular disease. Indeed, some studies have suggested there may be benefits from pharmacological lowering of serum CRP levels in atrisk populations [START_REF] Ridker | Rosuvastatin to prevent vascular events in men and women with elevated C-reactive protein[END_REF]. However reverse causality or confounders may explain the associations [9,10].

In this study we hypothesised that a causal role for CRP in reducing lung function could be identified by the method of Mendelian Randomisation: the CRP gene contains single nucleotide polymorphisms (SNPs), which are known to determine baseline serum CRP levels. If higher serum CRP levels reduce lung function, then a CRP polymorphism that was associated with higher levels of serum CRP should also lead to impaired lung function [11]. By Mendelian laws of inheritance, alleles are inherited independently of each other and randomly at a population level tend not to be associated with confounding factors [12]. Furthermore, disease processes do not influence germline genetic variants, so reverse causation whereby disease processes influence apparent exposures, rather than vice versa, will also be avoided [12]. Our study would thereby mimic a randomised trial [13]. Sunyer et al employed this strategy in their small cross-sectional study of post-MI survivors in Italy [START_REF] Sunyer | Systemic inflammation, genetic susceptibility and lung function[END_REF]. Similarly, two studies demonstrate conflicting results for an association of CRP polymorphism with development or presence of COPD [15,[START_REF] Van Durme | C-reactive protein levels, haplotypes, and the risk of incident chronic obstructive pulmonary disease[END_REF].

Here we examine the association between three previously well-studied CRP polymorphisms (rs1130864, 1800947, rs1205) and haplotypes, circulating CRP levels and the continuous variables of lung function of FEV 1 and FVC and with lung function decline. These data allow us to explore whether the existing association between serum CRP and lung function is explained by residual confounding, reverse causation or implicates a genuine role for inflammation on adverse lung function. A c c e p t e d M a n u s c r i p t

Methods

Subjects

The Caerphilly Prospective Study is a population-based male cohort study of all men aged between 45 to 59 years who were resident in the small South Wales town of Caerphilly [START_REF] Caerphilly | Caerphilly and Speedwell collaborative heart disease studies[END_REF]. The initial examination (phase 1) took place between 1979 and 1983 where 2512 of the 2818 eligible men (89%) were recruited. An additional 447 patients were recruited at phase 2. Men were reinvited for further follow-up approximately every five years. The last follow-up (phase 5) occurred between 2002 and 2004. Subjects gave written informed consent and the study had the approval of the local research ethics committee. At commencement, a detailed questionnaire was administered and review of their personal and medical history was collected. Items included their occupational social class (grouped as non-manual/manual), any established diagnoses of ischaemic heart disease, diabetes mellitus or hypertension, and their smoking habit. Smoking status was categorised according to never, ex, pipe and cigar smokers, or current cigarette smokers (1-15, 15+ cigarettes/day). At phase 5 smoking status was re-established by classification into never, ex and current smokers. Clinic measures Height in bare feet was measured using a Holtain stadiometer and weight in light clothes using standardised scales. From this, body mass index (BMI) was calculatedkg/m 2 as a measure of obesity. Spirometry Spirometry at phase 2 was performed using the McDermott spirometer in a standing position. The FEV 1 , forced vital capacity (FVC) and their ratio (FEV 1 /FVC) were calculated. An acceptable result was noted if the two valid, best traces of the 3 performed had less than 100mls variability in FEV 1 . At Phase 5, spirometry was performed using the Vitalograph spirometer (Vitalograph, Bucks, UK) with the same strict criteria and methodology. All measurements were taken pre-bronchodilator. Venous blood samples At phase 2, subjects attended an early morning clinic having fasted overnight and a blood sample was taken with minimal venous stasis. CRP was measured in a citrated sample, which had been centrifuged within one hour and stored at -70°C until analysed using a high sensitivity nephelometric assay (Dade-Behring, Germany). Intra-and inter-assay cv's for CRP were 4.7% and 8.3% respectively. The samples were analysed in 2000. Values are reproducible, stable over time and methods highly-standardised [START_REF] Kaptoge | C-reactive protein concentration and risk of coronary heart disease, stroke, and mortality: an individual participant metaanalysis[END_REF].

Genotyping

Genetic variants of the CRP gene on chromosome 1q21-q23 were identified on blood samples taken at phase 4. Three SNPs were used in the analysis; rs1130864 (G-A base change in the 3' untranslated region, AF449713 position 3014), rs1800947 (C-G base change at codon 188, AF449713 position 2667), and rs1205 (C-T base change in the 3' flanking/untranslated region, AF449713 position 3872). These SNPs were chosen on the basis of available evidence showing their replicable and robust association with circulating CRP levels [START_REF] Russell | Polymorphism at the C-reactive protein locus influences gene expression and predisposes to systemic lupus erythematosus[END_REF][START_REF] Suk | Relation of polymorphism within the C-reactive protein gene and plasma CRP levels[END_REF][START_REF] Zee | Polymorphism in the human C-reactive protein (CRP) gene, plasma concentrations of CRP, and the risk of future arterial thrombosis[END_REF]. Genotyping was undertaken by Kbiosciences (Hertfordshire, UK), which designed and used assays based on its proprietary competitive allele-specific polymerase chain reaction system method (www.kbioscience.co.uk). All assays were validated before use with a standard 96-well validation plate used by Kbiosciences. All variants were checked for adherence to Hardy-Weinberg equilibrium to avoid gross genotyping error, and a small number of internal validations and non-DNA test controls were used to validate genotyping after assay. Hardy-Weinberg equilibrium was tested at each SNP locus on a contingency table of observed-versus-predicted genotypic frequencies using an exact test. Our allelic frequencies at rs1800947 represent the base on the antisense strand, and C and G frequencies are therefore reversed in our analyses.

Statistical analysis

Statistical analysis was conducted in Stata version 10. Circulating CRP was log 10 transformed as it was not normally distributed. We have expressed results as back-transformed geometric means for serum CRP levels. All analyses involving lung function were adjusted for age (at specific phase) and height. While examining the effects of adjusting for more curvilinear models with different age variables, we found that this did not improve the model significantly; hence the linear model for the age variable was used as the best simplest fit. For the analyses, smoking status was grouped into never-(non)smokers, ex-smokers at phase 2, smokers at phase 2 and smokers at both phases 2 and 5. Subjects with a heterozygous state for rs1800947 (ie.C:G) were grouped with the homozygous genotype (G:G) due to low numbers (6 (phase 2) and 4 (phase 5) subjects had the G:G genotype). All 3 polymorphisms were found to be in Hardy-Weinberg equilibrium. We used linear regression models to assess the association between CRP polymorphisms, serum CRP level and lung function tests. We then introduced various confounders in the model to assess any change in the association between serum CRP levels and lung function tests (BMI, smoking status, presence if ischaemic heart disease, diabetes or hypertension). Haplotypes were constructed using the genetic data analysis program SIMHAP (http://www.genepi.com.au). SIMHAP uses current estimation-maximization-based methods for the estimation of haplotypes from unphased genotype data [START_REF] Excoffier | Maximum-likelihood estimation of molecular haplotype frequencies in a diploid population[END_REF]. The current implementation of SIMHAP uses the statistical computing package R (www.r-project.org) to resolve haplotypes and provide their posterior probabilities. After the construction of haplotypes, SIMHAP is also able to examine relationships between common CRP haplotypes and circulating CRP concentration. All possible haplotype configurations are resolved for each individual within the program itself, and the posterior probability of each configuration is calculated. Association analyses within the generalized linear model framework then uses simulation to correctly deal with the uncertainty around imputed haplotypes. In the case of plasma CRP concentration, mean values by common CRP haplotype were derived from the regression coefficients and CIs estimated by SIMHAP. For these analyses, all common haplotypes were run in the model assuming additive effect of each haplotype and setting a baseline haplotype as the most common in this sample (1000 iterations were used).

Results

In phase 2 there were 2398 participants, of which 2173 individuals had valid lung function measures. By the time the men were rescreened for phase 5, only 1225 men were seen for whom we obtained valid lung function data on 1021 (83%). For the men not seen, 667 had died, 86 had moved out of the area, 125 could not be traced and 295 refused to participate. Lung function at phase 5 was measured on average 16.8 years later (interquartile range 16 to 18 years). Basic participant characteristics as well as baseline data on non-participants are shown in Table 1. There were differences in the mean values or percentages of participants in phase 2 who have follow-up lung function measures at phase 5 compared to participants who have been lost to follow-up, Table 1. These findings are largely consistent with the healthy survivor effect: participants who have been lost to follow-up (ie. likely deceased) are more likely to have had worse lung function, been heavier smokers and suffered from concomitant cardiovascular disease or diabetes.

Table 2 shows the relationship between alleles of the CRP polymorphism rs1800947 and serum CRP along with potential confounders. The geometric mean ratio (GMR) for rs1800947 was 0.72, 95% CI 0.61 to 0.84 p<0.001. We present data for this CRP polymorphism because it showed the strongest effects on serum CRP levels. The effects of rs1130864 and rs1205 polymorphisms on serum CRP were as follows: GMR for rs1130864 GMR=1.1, 95% CI 1.00 to 1.21, p=0.04; for rs1205 GMR=0.87, 95% CI 0.79 to 0.95, p=0.002. No association was found between confounding variables and the polymorphisms except the rs1205 where the heterozygotes had a reduced IHD risk of p=0.01 compared to either the wild or mutant homozygous, which is likely to represent a chance finding.

Phase 2 circulating CRP showed predictive value for both phase 2 and 5 lung function measures (p values all ≤0.02), Table 3. A one log-unit increase in Phase 2 CRP level is associated with reduced FEV 1 and FVC (β= -0.13 and -0.14 respectively), and reduced FEV 1 /FVC ratio (β= -0.007) in phase 2.

Repeat analysis adjusting for potential confounders (BMI, smoking, presence of diabetes, ischaemic heart disease and hypertension, occupational status) had only minor effects on the βcoefficients, with significant relationship maintained, except for FEV 1 /FVC ratio at phase 5 (p=0.104); hence the association with serum CRP could not be abolished by conventional confounders. The FEV 1 and FEV 1 /FVC decline, but not FVC, between phase 2 and 5 was associated with Phase 2 serum CRP, but was attenuated in the fully adjusted model.

Having found associations between lung function with serum CRP, we then explored lung function in relation to the CRP genotype, Table 4. In phase 2 and 5, the CRP polymorphisms were not associated with lung function measures. Looking at lung function decline, however, there were weak associations with FVC decline. The rare rs1205 genotype (T:T) is associated with lower CRP levels, but was associated with an increased decline in FVC (β= 8.8, 95% CI 0.89 to 16.79, p=0.03).

The haplotype frequency was as previously published [START_REF] Schumacher | Association between Creactive protein genotype, circulating levels, and aortic pulse wave velocity[END_REF]. There were 4 common haplotypes with a frequency >5%. Inclusion of the haplotype into the lung function model (either phase 2 or phase 5 or the longitudinal change) was not superior to excluding them (all p values for likelihood ratio test were all greater than a p value of 0.1). The lack of haplotype supporting data for the weak effect of polymorphism rs1205 on FVC decline reinforces the borderline p value and perhaps reflects multiple p values.

Discussion

This study demonstrates a robust association between CRP polymorphisms and serum CRP level but in this large male cohort found no clear association between the CRP polymorphisms and lung function at either phase 2 or 5 or longitudinally, which was supported further by the haplotype analysis [START_REF] Zee | Polymorphism in the human C-reactive protein (CRP) gene, plasma concentrations of CRP, and the risk of future arterial thrombosis[END_REF][START_REF] Hage | C-reactive protein gene polymorphisms, C-reactive protein blood levels, and cardiovascular disease risk[END_REF]. We have confirmed the inverse association between serum CRP and lung function parameters as previously described [START_REF] Shaaban | Change in C-reactive protein levels and FEV1 decline: a longitudinal populationbased study[END_REF][START_REF] Kony | Association of bronchial hyperresponsiveness and lung function with C-reactive protein (CRP): a population based study[END_REF][START_REF] Aronson | Inverse association between pulmonary function and Creactive protein in apparently healthy subjects[END_REF], which is not attenuated by adjustment for confounders such as cardiovascular disease, smoking and BMI. Additionally, in the longitudinal analysis, circulating CRP at phase 2 was predictive of decline in FEV 1 over 16+ years.

Genetic polymorphisms are useful tools in assessing causality and associations between established risk factors and chronic disease states [10,[START_REF] Smith | Association of C-reactive protein with blood pressure and hypertension: life course confounding and mendelian randomization tests of causality[END_REF]. Their random acquisition by Mendelian Inheritance laws equates to a randomised controlled trial, and relationships are not affected by confounders [13]. Van Durme et al. failed to demonstrate an association of CRP haplotype and polymorphisms with a risk of developing COPD, although, here, the definition of COPD appears to be either physician based or according to spirometric evaluation alone, with "probable COPD" potentially encompassing a wider range of diagnosis and not dependent necessarily on spirometry [START_REF] Van Durme | C-reactive protein levels, haplotypes, and the risk of incident chronic obstructive pulmonary disease[END_REF]. In parallel, spirometric definitions of COPD according to the method protocol presented could have encompassed other non-COPD airways obstruction as smoking history, diurnal variation and atopic features did not appear to be a requisite. Our work is consistent with this report, but we have extended into the natural history of lung function deterioration, the intermediary of developing disease such as COPD or another chronic respiratory disease.

In the small cross sectional study on a limited number of post-MI patients recruited into the Italian AIRGENE study, Sunyer et al explored polymorphisms within the CRP and other inflammatory marker genes, noting that the rs1205 CRP polymorphism and one haplotype were associated with better lung function, the rare rs1205 polymorphism being associated with lower circulating CRP levels [START_REF] Sunyer | Systemic inflammation, genetic susceptibility and lung function[END_REF]. However, by the nature of the study, the participants all had a cardiovascular history and were pre-selected by being the "survivor" subset of myocardial infarction. In our study, the rate of decline was clearly only possible in those that had survived the follow-up period but was strengthened by the cross-sectional data (at 2 time-points) and was not reliant on every subject surviving a major cardiovascular insult. Meanwhile, Yanbaeva et al. detected an increased allele frequency between patients with COPD and controls for the CRP SNP rs1800947, although its significance after adjusting for age, sex and tobacco consumption was lost [15]. The haplotype inclusion was consistent with chance and interestingly, the biggest difference in haplotype frequency was the H2 (24.2% in COPD patients and 30.3% in the controls), yet H2 showed only a modest effect on CRP, p=0.34. They did detect however a more clear association of a 4 fold increased COPD risk with IL-6 polymorphism yet this was associated with a lower CRP level.

The inverse association of circulating CRP with lung function measures may thus indicate that reverse causality is the link -with impaired lung function causing the elevated circulating CRP. In patients with COPD, there is an apparent persisting inflammatory state with further elevation in inflammatory mediators at times of exacerbations [START_REF] Gan | Association between chronic obstructive pulmonary disease and systemic inflammation: a systematic review and a metaanalysis[END_REF][START_REF] Wedzicha | Acute exacerbations of chronic obstructive pulmonary disease are accompanied by elevations of plasma fibrinogen and serum IL-6 levels[END_REF]. In addition, serum inflammatory status has been associated with lung function in apparently healthy subjects, even never smokers [START_REF] Wedzicha | Acute exacerbations of chronic obstructive pulmonary disease are accompanied by elevations of plasma fibrinogen and serum IL-6 levels[END_REF]. However, in patients with mild to moderate COPD, there was no parallel in inflammatory mediators in the sputum and serum [START_REF] Vernooy | Local and systemic inflammation in patients with chronic obstructive pulmonary disease: soluble tumor necrosis factor receptors are increased in sputum[END_REF]. Similarly, in healthy subjects, there was no set systemic response in subjects when exposed to inhalational lipopolysaccharide [START_REF] Michel | Healthy subjects express differences in clinical responses to inhaled lipopolysaccharide that are related with inflammation and with atopy[END_REF]. Both of these studies suggest the systemic inflammation is not simply overspill from the airways component.

The elevated serum CRP in patients with lower lung function may be mediated through other pathways for eg. IL-6 is an upstream mediator of CRP production. In deed, Yanbaeva et al. found a much stronger relationship of IL-6 polymorphism with a diagnosis of COPD than with the CRP polymorphisms however this polymorphism was actually related to lower circulating CRP levels [START_REF] Sunyer | Systemic inflammation, genetic susceptibility and lung function[END_REF]. He et al. have recently reported SNP's to IL-6 associated with lung function decline and susceptibility to COPD in smokers, [START_REF] He | Associations of IL6 polymorphisms with lung function decline and COPD[END_REF] however this requires replication. Alternatively, an, as yet, A c c e p t e d M a n u s c r i p t unknown common mediator may be the key. In a similar manner, whilst serum CRP has been associated with atherosclerosis and with the metabolic syndrome, CRP genotypes have failed to be associated with arterial stiffness, carotid-intima thickness, metabolic syndrome or coronary heart disease [10, [START_REF] Schumacher | Association between Creactive protein genotype, circulating levels, and aortic pulse wave velocity[END_REF][32][START_REF] Schnabel | Relations of inflammatory biomarkers and common genetic variants with arterial stiffness and wave reflection[END_REF][START_REF] Elliott | Genetic Loci Associated With C-Reactive Protein Levels and Risk of Coronary Heart Disease[END_REF] Our study has several strengths by being a large community based study of unselected male population. It examined the associations both cross-sectionally and with longitudinal measures of lung function decline over a long period of time. The latter is a much more sensitive phenotypic measure as cross-sectional measures cannot differentiate between developmental and degenerative trajectories over the life course [35]. For lung function decline, there was likely a healthy survivor effect but the lack of association of genotype with lung function at either cross sectional analysis strengthens the lack of association with decline. Our sample size cannot rule out moderate sized effects given our confidence intervals. However to avoid the problems of publication bias, it is important that such results are which can then be included in a meta-analysis to explore small or modest effect sizes. Unfortunately, from the data collected, smoking pack years could not be calculated. It must be borne in mind, that the Caerphilly Prospective study is limited to Welsh men only and therefore may not be generalisable to women. Although similar aetiology is likely, verification is required. Similarly it is not generalisable to other geographical areas as there are differences. Lung function measures were pre-bronchodilator and thus may include asthmatics. However, our aim a priori was not to diagnose disease states but to explore the lung function as a continuous variable and over time. Mendelian randomisation provides an opportunity to test causality. In general, limitations of this approach include confounding by polymorphisms in linkage disequilibrium with the polymorphism being studied, pleiotropy of the specific polymorphism, compensation during development of the effect of the polymorphism or if there was not a suitable functional polymorphism [12]. However, for this paper, we had recognised functional polymorphisms and we measured potential confounders in order to overcome these.

In conclusion, we have demonstrated that CRP is inversely associated with lung function parameters of FEV 1 and FVC. However, the CRP genotypes are not associated with lung function and for the first time we demonstrate they are not associated with lung function decline. This implies there may be no causal association though we cannot rule our small effect estimates. We believe that the association between CRP and lung function is more likely to reflect reverse causation or an unknown common mediator.

  Clinical Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504 A c c e p t e d M a n u s c r i p t

Clinical

  Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504 A c c e p t e d M a n u s c r i p t

Clinical

  Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504

  Clinical Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504

Table 1 . Baseline Characteristics of subjects at Phases 2 and 5

 1

		Clinical Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504
		Participants with phase 5	Participants without phase 5 data
			data
		N	Mean or percent (SD/IQR)	N	Mean or percent (SD/IQR)	P
	THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504	Phase 2 Age (yrs) Height (m) CRP (mg/L) * FEV 1 (litres) FVC (litres) FEV 1 /FVC ratio BMI (kg/m²) Hypertension (%) Diabetes mellitus (%) Ischaemic heart disease (%) 256 1022 1013 707 958 1022 958 1012 204 23 Smoking status (%) Never-smoker 238 Ex-smoker 415 1-14 cigs/day or cigar/pipe 220 15+ cigs/day 145 Phase 5 Age (yrs) 1022 CRP (mg/L) * 699 FEV 1 (litres) 1021 FVC (litres) 1020 FEV 1 /FVC ratio 1016 Smoking status (%) Never-smoker 283 Ex-smoker 491 1-14 cigs/day or cigar/pipe 107 15+ cigs/day 55 Lung function decline between phase 2 and 5 56.0 (4.4) 1.72 (0.06) 1.45 (0.73, 2.78) 3.01 (0.7) 3.81 (0.7) 0.79 (0.1) 26.6 (3.4) 20.0 2.6 25.1 23.4 40.8 21.6 14.2 72.8 (4.2) 2.80 (1.47, 5.58) 2.26 (0.6) 3.31 (0.8) 0.68 (0.1) 30.2 52.5 11.4 5.9 FEV 1 change (mls/yr) 957 -44.0 (21.7) FVC change (mls/yr) 1020 -29.8 (31.7) FEV 1 /FVC change (%/yr) 952 -0.64 (0.55) A c c e p t e d M a n u s c r i p t 1302 57.6 (4.5) <0.001 1283 1.71 (0.06) <0.001 825 2.01 (0.99, 4.22) <0.001 1215 2.63 (0.7) <0.001 1302 3.50 (0.8) <0.001 1215 0.75 (0.1) <0.001 1279 26.3 (3.8) 0.02 340 26.1 <0.001 65 5.0 0.001 456 35.0 <0.001 181 13.9 462 35.5 369 28.4 288 22.2 <0.001

Abbreviations : Standard deviation (SD) ; Interquartile range (IQR) ; Forced expiratory volume in 1 second (FEV 1 ), forced vital capacity (FVC); C-reactive protein (CRP); body mass index (BMI) * geometric mean for log-transformed variable

Table 2 . Mean values and coefficients for each allele of SNP 1800947

 2 Clinical Science Immediate Publication. Published on 16 Nov 2010 as manuscript CS20100504 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100504A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

		C:C	C:G / G:G	Difference (β-coeff or %)	95% CI		P
		n=1164	n=195
		(85.7%)	(14.3%)
	Phase 2
	Age (yrs)	56.8	56.5	-0.29	-0.96	, 0.38	0.40
	Height (mtr)	1.71	1.72	0.003	-0.007 , 0.01	0.55
	CRP (mg/L) *	1.72	1.23	0.72*	0.61	, 0.84	<0.001
	BMI (kg/m²)	26.6	26.5	-0.11	-0.67	, 0.44	0.69
	Smoking status %
	Never	15.6	3.5
	Ex-smoker	34.3	4.9
	Cigar, pipe or 1-14/day	21.5	3.6
	15+/day	14.2	2.4	-0.06	-0.21	, 0.09	0.41
	Ischaemic heart disease
	%	86.6	13.4	-0.02	-0.09	, 0.05	0.51
	Hypertension %	84.8	15.2	0.01	-0.05	, 0.08	0.64
	Diabetes Mellitus %	90.5	9.5	-0.01	-0.04	, 0.01	0.37
	Non-manual %	84.2	15.8
	Manual %	86.4	13.6				0.29
	Abbreviations : C-reactive protein (CRP); body mass index (BMI); confidence interval (CI);
	significance (P); ß regression coefficient
	* geometric mean ratios for log-transformed variables

Table 4 . Associations between CRP polymorphisms and lung function parameters FEV 1 FVC

 4

			N	β	95% CI		P	N	β	95% CI	P
	Phase 2
	rs1800947 CG/GG vs. CC	1224	0.01 -0.08 , 0.11	0.82	1305	0.008 -0.09 , 0.11	0.88
	rs1130864 AG vs. GG	1221 -0.02 -0.09 , 0.05	0.54	1302	0.01 -0.07 , 0.08	0.80
		AA vs. GG		-0.003 -0.11 , 0.10	0.95		0.02 -0.09 , 0.14	0.72
	P for trend						0.75			0.70
	rs1205	TC vs. CC	1221	0.04 -0.04 , 0.11	0.33	1301	0.01 -0.06 , 0.09	0.75
		TT vs. CC		-0.04 -0.15 , 0.07	0.45		-0.09 -0.20 , 0.03	0.13
	P for trend						0.90			0.31
	Phase 5
	rs1800947 CG/GG vs. CC	762	0.08 -0.04 , 0.20	0.21	763	0.1	-0.03 , 0.25	0.11
	rs1130864 AG vs. GG	763	-0.01 -0.1	, 0.08	0.80	764	-0.006 -0.1	, 0.09	0.90
		AA vs. GG		0.1	-0.03 , 0.24	0.13		0.1	-0.05 , 0.26	0.18
	P for trend						0.31			0.34
	rs1205	TC vs. CC	761	0.03 -0.06 , 0.12	0.49	761	0.04 -0.07 , 0.14	0.49
		TT vs. CC		-0.04 -0.18 , 0.09	0.52		-0.02 -0.17 , 0.14	0.84
	P for trend						0.86			0.88
	Lung function decline
	rs1800947 CG/GG vs. CC	511	-0.5 -5.71 , 4.65	0.84	544	4.3	-3.03 , 11.56 0.25
	rs1130864 AG vs. GG	515	-1.03 -4.72 , 2.65	0.58	547	-2.73 -7.96 , 2.51	0.31
		AA vs. GG		2.83 -3.05 , 8.72	0.35		0.57 -7.76 , 8.90	0.89
	P for trend						0.67			0.69
	rs1205	TC vs. CC	512	-0.9 -4.62 , 2.72	0.61	544	1.37 -3.89 , 6.63	0.61
		TT vs. CC		1.05 -4.53 , 6.63	0.71		8.8	0.89	, 16.79 0.03
	P for trend						0.96			0.06

Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.© 2010 The Authors Journal compilation © 2010 Portland Press Limited

A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law. © 2010 The Authors Journal compilation © 2010 Portland Press Limited

© 2010 The Authors Journal compilation © 2010 Portland Press Limited

YBS and JG were responsible for attaining the funding for Phase 5; YBS, JG and GDS sit on the steering committee for Phase 5 Caerphilly Prospective Study; CEB, JRC and YBS developed the hypothesis; CEB, WS, NT and YBS analysed the data; CEB wrote the first draft; all authors contributed to the interpretation of the data and writing the paper. Funding: The Caerphilly Prospective Study was funded by the Medical Research Council of the United Kingdom and was established by the former MRC Epidemiology Unit (Cardiff). The phase 5 follow-up was funded by the Alzheimer's Society. The CRP genotype was funded by grant RD1634. Charlotte Bolton is funded by the NIHR Nottingham Respiratory Biomedical Research Unit.