Audrey Desbonnet

Sumudu Kankanamge
email: kankanamge@univ-paris1.fr

Yann Algan

Antoine D'autume

Fabrice Collard

Jean-Olivier Hairault

Public debt and aggregate risk ⋆

Keywords: dette publique, risque agrégé, épargne de précaution, contrainte de crédit public debt, aggregate risk, precautionary saving, credit constraints JEL classification : E32, E62, H31

Résumé : Cet article reconsidère l'impact de l'introduction de la dette publique dans le cadre de marchés incomplets et de fluctuations macroéconomiques. Aiyagari et Mc Grattan (1998) montrent que l'incomplétude des marchés et l'existence d'une contrainte d'endettement créent un motif d'épargne de précaution qui réaffirme le rôle de la dette publique. Parce qu'elle offre un moyen additionnel aux ménages de lisser leur consommation, l'introduction de la dette publique est bénéfique. Cependant, cette analyse ignore une source non négligeable d'exacerbation du risque idiosyncrasique de revenu non assurable qu'est le risque agrégé et risque donc d'avoir sous estimé le rôle de la dette. L'accroissement du taux et de la durée du chômage lors des phases de récession entame la capacité assurantielle de l'épargne de précaution. En outre, l'occurrence de faibles taux d'intérêt durant les périodes de récession rend plus coûteuse la constitution d'une épargne de précaution. L'introduction de la dette publique accroît le taux d'intérêt. L'épargne de précaution devient moins coûteuse. C'est la raison pour laquelle le niveau optimal de dette publique est égal à 5% du produit annuel et s'établit en moyenne à un niveau plus élevé que dans un environnement dépourvu de fluctuations macroéconomiques.

Abstract : This paper assesses the long-run optimal level of public debt in a framework where aggregate fluctuations are taken into account. Households are subject to both aggregate and idiosyncratic shocks and the market structure prevents them from perfectly insuring against risk. We find that the long-run optimal level of public debt is generally higher in a setting embedding aggregate fluctuations than in a setting without. Aggregate fluctuations modify both the cost and the motive for precautionary saving. Higher levels of public debt, by effectively reducing the cost of precautionary saving, help agents to smooth consumption when they face price and employment fluctuations.

Introduction

The introduction of uninsurable risk was a major step in the recent literature on public debt. [START_REF] Woodford | Public Debt as Private Liquidity[END_REF] is the first author to depart from the complete market framework to consider credit constraints or as he phrases it "imperfect financial intermediation". [START_REF] Woodford | Public Debt as Private Liquidity[END_REF] builds a simple economy embedding credit constraints and finds that public debt can be efficient because it keeps interest rates higher and closer to time preference rates. Aiyagari and McGrattan (1998) quantitatively address the question of the optimal level of public debt in a heterogeneous agent, incomplete market model. Calibrating on the U.S. economy they find the annual positive debt over GDP ratio of 2/3 to be optimal. Finally, [START_REF] Floden | The Effectiveness of Government Debt and Transfers as Insurance[END_REF] uses a similar framework to look at public debt/transfers optimal combinations. This author underlines the strong uncertainty and inequality implications behind public debt policies. Very much in the same way, the introduction of uninsurable risk in the literature on the cost of business cycles has generated rich implications. In an effort to reconsider the small welfare effect of business cycles found by [START_REF] Lucas | Models of Business Cycles[END_REF], both [START_REF] Krusell | Revisiting the Welfare Effects of Eliminating Business Cycles[END_REF] and [START_REF] Storesletten | The welfare cost of business cycles revisited : finite lives and cyclical variation in idiosyncratic risk[END_REF] draw a link between the aggregate risk and the cross sectional distribution. In other words, the aggregate productivity shock is correlated to the individual specific shocks to produce strong distributional effects of aggregate fluctuations. In such a framework, these authors find a greater cost of business cycles than [START_REF] Lucas | Models of Business Cycles[END_REF]. Even more interestingly, Imrohoroglu (1989) briefly suggests that economic policies could be used to help individuals reduce the cost of business cycles. In this paper, we explore this latter suggestion and its implications for the optimal level of public debt. This paper's main objective is to introduce a framework exhibiting aggregate fluctuations à la Den [START_REF] Den Haan | Heterogeneity, Aggregate uncertainty and the Short Term Interest Rate[END_REF] or [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF] to quantify the long-run optimal level of public debt. As aggregate fluctuations impact the saving behavior of agents, they are an important aspect to take into account for public debt policies that has been ignored so far. Unlike previous literature, this paper also takes some steps to reproduce the wealth distribution of the targeted U.S. economy as it has been documented that the need for public debt is very different across the population. Thus our framework can be used to decompose the effects of public debt along the cycle and also across the population. Following Bewley-Huggett-Aiyagari type models, we build a production economy with capital market imperfections where a large number of ex-ante identical infinitely-lived agents face idiosyncratic income shocks and aggregate productivity shocks. Households' saving behavior is influenced by precautionary saving motives and borrowing constraints. Private capital and government bonds, both yielding the same interest, can be claimed to insure against future risk. Government levies proportional taxes on households and issues debt in order to finance its consumption.

In this setup, our benchmark calibration yields a long-run optimal level of public debt of 5% of GDP on an annual basis. This is to be compared to a model without aggregate fluctuations. We consider several methods of removing aggregate fluctuations to obtain a comparable long-run idiosyncratic risk model. The level of public debt obtained in those idiosyncratic models are generally lower although with one method the optimal level of debt is almost similar to the aggregate risk model. Thus we argue that the effect of aggregate risk on the optimal level of public debt is moderate. In the benchmark economy, we find that the gains of being at the optimal level of debt instead of the benchmark debt to GDP ratio of 66% amounts to 0.257% of consumption. Also, the consumption gains of being at the optimal level of debt are higher in recessions than in expansions and higher for the poorest percentile of the population. The intuition behind our results is the following. Credit constraints and uncertainty lead agents to engage in precautionary saving. The result of precautionary saving is a higher level of capital that in turn lowers the interest rate away from the time preference rate. Aggregate fluctuations, as it is correlated with the labor market process, exacerbate the level of risk faced by agents. An employment fluctuation effect increase both the risk of losing one's job and the time one could spend in unemployment in recessions. This strengthens households' precautionary saving motive. A price fluctuation effect changes the level of prices between recessions and booms making it more costly to accumulate precautionary saving in recessions. As a result of these effects, the capital stock rises and the interest rates move further away from the time preference rate. A higher level of public debt has two opposing effects here. A crowding out effect crowds out private capital and reduces welfare. A cost of precautionary saving effect increases the interest rate, reduces the cost of precautionary saving and enhances welfare. Because the employment fluctuation effect and the price fluctuation effect change the precautionary saving motive and cost, the cost of precautionary saving effect is stronger in an aggregate fluctuations setting than in an idiosyncratic risk setting. Thus the optimal level of public debt is higher in a setting with aggregate fluctuations. Finally, we emphasize that the optimal level of debt can be significantly higher if we modify the parameters governing the employment fluctuation effect so as to strengthen this effects.

The rest of the paper is organized as follows. Next section describes the benchmark economy. Section 3 details the results. In section 3.5 we characterize the optimal level of public debt for the European labor market. The last section concludes.

The Benchmark Model

Our benchmark economy is a Bewley-Huggett-Aiyagari type dynamic stochastic general equilibrium model augmented to allow aggregate fluctuations à la Den [START_REF] Den Haan | Heterogeneity, Aggregate uncertainty and the Short Term Interest Rate[END_REF] or [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF] and public debt. Insurance markets are incomplete. Agents face idiosyncratic and aggregate risks and are borrowing constrained. These three assumptions lead agents into precautionary saving [START_REF] Aiyagari | Uninsured Idiosyncratic Risk and Aggregate Saving[END_REF]). Our benchmark model can be related to the model in [START_REF] Aiyagari | The Optimum Quantity of Debt[END_REF] but present three deviations of importance apart from its aggregate fluctuations feature. First, the productivity process here is simpler as agents can only be employed or unemployed. Second, leisure is not valued. Those two deviations greatly simplify the model with aggregate fluctuations. Finally, there is no exogenous growth in our benchmark economy. [START_REF] Aiyagari | The Optimum Quantity of Debt[END_REF] specify an exogenous annual growth rate of 1.85%. This assumption exogenously reduces the cost of public debt for individuals since public debt interest repayments are diminished by this exogenous growth factor. It is noteworthy that making this assumption lead to a higher optimal level of public debt.

Firms

We assume that there is a continuum of firms which have a neoclassical production technology and behave competitively in product and factor markets. The output is given by :

Y t = z t F (K t , N t )
where K is aggregate capital and N aggregate labor used in production. The function F exhibits constant returns to scale with respect to K and N, has positive and strictly diminishing marginal products, and satisfies the Inada conditions. Capital depreciates at a constant rate δ. The economy is subject to an exogenous aggregate shock noted z. There are two possible aggregate states : a good state where z = z g and a bad state where z = z b . The aggregate shock follows a first-order Markov process with transition probability η z|z ′ = Pr(z t+1 = z ′ /z t = z). Thus η z|z ′ is the probability that the aggregate state tomorrow is z ′ given that it is z today. We note η the matrix that describes the transition from one aggregate state to another such that :

η =    η gg η gb η bg η bb   
Finally, our setting assumes that inputs market are competitive. The wage w and the interest rate r verify :

r t + δ = z t F K (K t , N t ) (1) w t = z t F N (K t , N t ) (2)

The government

The government issues public debt and levies taxes to finance public expenses.

Both the revenue of capital and labor are taxed proportionally at an identical rate τ . The government's budget constraint verifies :

G t + r t B t + T R t = B t+1 -B t + T t with T t = τ (w t N t + r t A t )
G t is the level of public expenses, B t the level of public debt, T t tax revenues and T R t a lump sum transfer to households that amounts to zero at the equilibrium. Because of the aggregate fluctuations property of the model, we have to be cautious about how we close the model with respect to the government budget constraint. We proceed as follows. As there is no formal steady state in this economy because of the aggregate fluctuations property, we approximate a pseudo steady state by averaging aggregate variables over long periods of time. We first guess an interest rate in this pseudo steady state and derive a tax rate by assuming that public expenses and public debt are a constant fraction of the associated long run GDP. As this tax rate does not necessarily balance the government budget constraint along the cycle, we use the lump sum transfer T R to make sure the budget will be balanced. Finally, we update our guess on the interest rate in the pseudo steady state until we reach a fix point. When this final step is completed, the model is closed with respect to the government budget constraint and T R amounts to zero at the equilibrium1 .

A t accounts for total average wealth in the economy. It is the sum of average physical capital K and public debt B such that :

A t = K t + B t (3)

Households

The economy is populated by a continuum of ex ante identical infinitely lived households of unit mass. Their preferences are summarized by the function V :

V = E 0    ∞ t=0   t j=0 β j u(c t )      ( 4
)
where β is the discount factor. We assume that this discount factor is random.

Thus the discount factor β can differ across agents and can vary over time. We specify that the latter follows a three-states first-order Markov process. This assumption on the discount factor helps to reproduce the wealth distribution as shown in [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF]. The discount factor verifies :

     β 0 = 1 β j≥1 ∈]0; 1[ c t is the household level consumption.
The utility function we use has a standard CRRA specification and writes :

u(c) =      c 1-σ 1-σ if σ = 1 log(c) if σ = 1
Agents are subject to idiosyncratic unemployment shocks. Let s be the household's labor market status. A household can either be unemployed (s = u) or employed (s = e). Households are also subject to shocks at the aggregate level. Aggregate shocks exacerbate idiosyncratic unemployment risks. The unemployment rate and the unemployment duration are higher in recessions than in booms. Therefore, transitions on the labor market are correlated to the aggregate state. We note Π zz ′ |ss ′ the joint transition probability to a state (s ′ , z ′ ) conditional on a state (s, z). The matrix that jointly describes the transition from a state (s, z) to a state (s ′ , z ′ ) is the following :

Π =           Π bbuu Π bbue Π bguu Π bgue Π bbeu Π bbee Π bgeu Π bgee Π gbuu Π gbue Π gguu Π ggue Π gbeu Π gbee Π ggeu Π ggee          
where Π ggee = Pr(z t+1 = z g , s t+1 = e|z t = z g , s t = e).

When agents are in an employed state, they receive the wage w. However when agents are unemployed their income corresponds to their home production that we note θ. Insurance markets are incomplete so that agents can only partially self-insure against idiosyncratic risk. Following Aiyagari and McGrattan (1998) no borrowing is allowed. The only way for households to self-insure against idiosyncratic risk is to accumulate physical capital and government bonds both yielding the same return r. Their overall holding in the later assets is noted a. Therefore a typical household solves the following problem :

max ct, a t+1 E 0 { ∞ t=0 ( t j=0 β j )u(c t )}, subject to : a t+1 + c t = (1 + (1 -τ )r t )a t + χ(s t )w t + T R t c t ≥ 0 a t+1 ≥ 0 with χ (s t ) =      θ if s t = u, (1 -τ ) if s t = e
The existence of aggregate risk leads us to distinguish between individual state variables and aggregate state variables. The individual state variables are given by the vector (a, s, β). The aggregate state variables are summarized by the vector (z, Γ) where Γ(a, s, β) is a distribution of agents over asset holdings, employment status and preferences. To determine the wage and the interest rate, households need to forecast the aggregate stock of physical capital. Therefore, they need to know the wealth distribution. That is why wage and interest rate depend on that wealth distribution. In the computational appendix, we explain how we avoid manipulating the wealth distribution by approximating it with some of its moments using the methodology developed in Den Haan (1996) and [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF]. We detail the computational strategy we used to solve the model in appendix A.

Equilibrium

The recursive equilibrium consists of a set of decision rules for consumption and asset holding {c(a, s, β; z, Γ), a ′ (a, s, β; z, Γ)}, aggregate capital K(z, Γ), factor prices {r(z, Γ), w(z, Γ)}, tax rate τ and a law of motion for the distribution Γ ′ = H(Γ, z, z ′ ) which satisfy these conditions :

(i) Given the aggregate states, {z, Γ}, prices {r(z, Γ), w(z, Γ)} and the law of motion for the distribution Γ ′ = H(Γ, z, z ′ ), the decision rules {c(a, s, β; z, Γ), a ′ (a, ǫ, β; z, Γ)} solve the following dynamic programming problem :

v(a, s, β; z, Γ) = max c, a ′ {u(c) + βE [v(a ′ , s ′ , β ′ ; z ′ , Γ ′ )|(s, β; z, Γ)]} subject to : c + a ′ = (1 + r(z, Γ)(1 -τ ))a + w(z, Γ)χ(s) + T R c ≥ 0 a ′ ≥ 0 and Γ ′ = H(Γ, z, z ′ ) (ii) Market price arrangements are : r(z, Γ) = zF K (K, N) -δ w(z, Γ) = zF N (K, N) (iii) Government budget constraint holds.
(iv) Capital market verify :

K + B = a ′ (a, ǫ, β; Γ, z)dΓ
(v) Consistency : agents' optimization problem is satisfied given the law of motion H and the law of motion is consistent with individual behavior.

Calibration

For the sake of comparison, the model economy is calibrated to match certain observations in the U.S. data. We let one period in the model be one quarter in the data. To avoid confusion we have converted all values to their annual equivalents in the results section. To remain simple and allow comparisons, we closely follow Krusell and Smith (1998) when calibrating the characteristics of the labor market and the aggregate risk.

Technology

We choose the production function to be Cobb-Douglas :

Y t = z t F (K t , N t ) = z t K α t N 1-α t 0 < α < 1
Technology parameters are standard. The capital share of output α is set to 0.36 and the capital depreciation rate δ is 0.025. As Krusell and Smith (1998) we assume that the value of the aggregate shock z is equal to 0.99 in recessions (z b ) and 1.01 in booms (z g ).

Preferences and discount factor

In the benchmark economy we assume a logarithmic utility function. We now detail the calibration steps to generate a realistic wealth distribution, the observed U.S. wealth Gini index and the capital-output ratio. Here, our calibration differs from Krusell and Smith (1998). In their economy agents can borrow whereas here, for the sake of comparison and simplicity, we follow Aiyagari and McGrattan (1998) and no borrowing is allowed. To reproduce the shape of the U.S. wealth distribution we first assume that unemployed agents receive income too and fix the home production income θ to be 0.102 . This assumption produces a large group of poor agents. Next we use the preference heterogeneity setting discussed in Krusell and Smith (1998) to generate a long thick right tail 3 . We impose that the discount factor β takes on three values {β l , β m , β h } where

β l < β m < β h :        β l β m β h        =        0.9750 0.9880 0.9985       
Thus an agent with a discount factor β m is more patient than an agent with a discount factor β l . To calibrate the transition matrix, we impose that the invariant distribution for discount factors has 10% of the population at the lowest discount rate β l , 70% at the medium discount factor β m and 20% at the highest discount factor β h . As Krussell and Smith (1998) we assume that there is no immediate transition between extreme values of the discount factors. Finally, we set the average duration of the lowest discount factor and the highest discount factor to be 50 years (200 quarters). These assumptions yield the following transition matrix4 :

Υ =       
0.9950 0.005 0.0000 0.0007 0.9979 0.0014 0.0000 0.0050 0.9950

      
The conditions that we have described so far specify 10 parameters so 10 targets are needed. When we account for the Gini index and the capitaloutput ratio, we need 8 additional targets. Given our calibration strategy, those 8 targets would be 8 points from the U.S. wealth distribution. In practice, instead of targeting 8 specific points, we searched for a set of parameters so that the wealth distribution in the model economy is as similar as possible to its U.S. counterpart.

As shown in Table 1 this calibration does a fairly good job at approximating the shape and the skewness in the U.S. wealth distribution5 and yields a Gini index of 0.82 and a capital-output ratio of 10.66 .

Labor market processes

For the sake of simplicity our calibration of the aggregate shock and the labor market process follows Krusell and Smith (1998). The process for z is set so that the average duration of good and bad times is 8 quarters. Therefore, the transition matrix η for aggregate state changes is defined by :

η =   
0.8750 0.1250 0.1250 0.8750

  
The average duration of an unemployment spell is 1.5 quarters in good times and 2.5 quarters in bad times. We also set the unemployment rate accordingly : in good periods it is 4% and in bad periods it is 10%. These assumptions enable us to define the transition matrixes for labor market status for each aggregate state change : Π gg for a transition from a good period to a good period, Π bb for a transition from a bad period to a bad period, Π gb for a transition from a good period to a bad period and Π bg for a transition from a bad period to a good period 7 : Finally the joint transition matrix Π for labor market statuses and aggregate states can be defined as :

Π bb =    0
Π =    η bb Π bb η bg Π bg η gb Π gb η gg Π gg    =           0.
         

Government

We fix the ratio of government purchases to GDP to 0.217. The debt over GDP ratio, noted b is set to the quarterly value of 8 3 which is equivalent to an 7 Further details on this calibration can be found in Appendix B.

annual value of 2 3 . Those values are the observed ratios in the U.S. as reported by [START_REF] Aiyagari | The Optimum Quantity of Debt[END_REF].

Results

We now present the results obtained with our benchmark economy. A first section reports the aggregate behavior of the model. A second examines the long-run welfare effects of public debt in an aggregate fluctuations setting. In a third, we move on to the business cycle and distributional effects of public debt. In the next to last section we compare the benchmark economy to simpler idiosyncratic risk only models. Finally, we explore an alternative calibration that changes the labor market process. We start by discussing the aggregate behavior of our benchmark model. Our computations are reported in Figure 1. Increasing the level of public debt increases the supply of safe assets in the economy. Consequently, the before tax interest rates increases. Because the repayment of debt interests are higher, the income tax rate increases. Nevertheless, the after tax interest rate unambiguously increases. In turn, public debt has a crowding out effect on private capital : higher levels of debt decrease the aggregate amount of private capital in the economy. The crowding out of capital induces the observed decline in output 8 . However, the decline in physical capital is smaller than the increase in public debt. The increase in the after-tax interest rate reduces the gap between the after-tax interest rate and the rate of time preference. The cost of postponing consumption to build up a buffer stock of saving is then reduced. Households choose to hold more assets at the steady state equilibrium. That is why the overall wealth level A, which is the combination of private capital and public debt, is higher.

Welfare analysis and optimal level of debt

The welfare analysis we conduct below apply to the long run optimal level of public debt with aggregate fluctuations. As there is no formal steady-state in a model with aggregate fluctuations, we consider for our welfare analysis the values of aggregate variables averaged over long periods of time for a given debt to GDP ratio 9 . We define the optimal level of public debt as the debt over GDP ratio that maximizes the traditional utilitarian welfare criterion µ. As explained in [START_REF] Lucas | Models of Business Cycles[END_REF] and [START_REF] Mukoyama | Cost of business cycles for unskilled workers[END_REF], this criterion measures the amount of consumption that one would have to remove or add in order to make the agent indifferent between the benchmark debt over GDP ratio and some other level of public debt. It verifies :

E 0   ∞ t=0   t j=0 β j   log((1 + µ)c bench. t )   = E 0   ∞ t=0   t j=0 β j   log(c t )   with c bench. t ∞ t=0
the consumption stream in the benchmark model when the debt over GDP ratio is equal to 8 3 . {c t } ∞ t=0 is the consumption stream when the debt over GDP ratio is some other level than the benchmark level. For logarithmic utility we can show that :

µ = exp V -V bench. /S -1, where V bench. = E 0 ∞ t=0 t j=0 β j log(c bench. t ) , V = E 0 ∞ t=0 t j=0
β j log(c t ) and 8 the increase in wealth would have been the same as the increase in public debt. The steady state consumption would have been higher. 9 More details can be found in Appendix A.

S = E 0 ∞ t=0 t j=0 β j .
The result of the introduction of public debt on welfare is a priori undetermined because of two opposing effects. The first effect is a crowding out effect. The crowding out of physical capital clearly reduces consumption and then welfare. Moreover the increase in the income tax rate tends to amplify the negative impact of public debt on welfare. The second effect is a cost of precautionary saving effect : the increase in the after-tax interest rate makes it less costly to accumulate precautionary saving in order to smooth consumption as the interest rate gets closer to the time preference rate. This second effect is welfare enhancing 10 . It is difficult to predict which effect overcomes the other analytically.

Figure 2 depicts the long-run optimal level of debt in the benchmark economy. In a setting embedding aggregate risk and calibrated on the U.S. economy, the optimal public debt level is 5% of output on an annual basis. Aggregate fluctuations, idiosyncratic risk and credit constraints lead agents to engage in precautionary saving in order to smooth consumption. Without public debt, the cost of precautionary saving is higher because when households accumulate, the interest rate lowers. Any level of public debt, raises the interest rate and reduces the cost of precautionary saving. As a result welfare is enhanced. However any level of public debt also crowds out private capital and reduces welfare. Here, a lower level of debt than the benchmark level increases welfare out of reducing the crowding out of private capital. At the same time, a lower level of debt than the benchmark level decreases welfare out of increasing the cost of precautionary saving. As long as the first effect dominates the second, a lower level of debt is optimal. At the optimal level, one effect exactly balances the other. For a debt over GDP ratio smaller than 5%, the consumption loss out of increasing the cost of precautionary saving is higher than the consumption gain out of reducing the crowding out of private capital.

As illustrated in Table 2, going from the benchmark level of debt to the optimal level of debt is welfare enhancing. The consumption gain of being at the optimal level of debt instead of the benchmark level is 0.257%. On the contrary, it is welfare decreasing to go to a higher level of public debt than the benchmark level. In an economy where the level of debt is 75%, the consumption loss not to be at the benchmark level (resp. optimal level) would be on average 0.072% (resp. 0.329%). These results suggest that the role the interest rate plays in reducing the cost of precautionary saving is central to understand why agents settle for the optimal level of public debt. If we block any price movements by considering a price fixed small open economy, it appears that there is no consumption gains or losses out of the cost of precautionary saving effect. In the case where all prices are set to their average benchmark level for any level of public debt, there is no consumption loss out of increasing the cost of precautionary saving for a lower level of public debt than the benchmark level. In this case, there are only consumption gains out of reducing the crowding out of private capital.

Business cycles and distributional effects

In this section we move on to the effects of public debt along the cycle and across the distribution of the population. To generate statistics along the cycle, we compute separately the average value of our aggregate variables either when the economy is in boom or when it is in recession over the whole sequence of simulation. The last two columns of Table 2 illustrate the consumption gain or loss of being at the optimal level of debt instead of the benchmark level. In recession, the consumption gain of being at the optimal level of debt is higher than on average. On the contrary, in expansion this consumption gain is lower. This is due to a price fluctuation effect : in recessions, the cost of postponing consumption to build up a buffer stock of savings is higher because the interest rate is lower as shown in Table 3. In expansion, we have the opposite : interest rates are higher and the cost of precautionary saving is lower. The fluctuation of prices is not the only relevant effect we need to account for to explain the optimal level of public debt. When we take aggregate risk into consideration, an employment fluctuation effect modifies the precautionary saving motive. Along the cycle, the unemployment rate and duration increase in recessions and reduce in booms. In recessions, the precautionary motive becomes stronger : employed agents face a higher risk of loosing their job and unemployed agents find it more difficult to find a job.

In recessions agents want to save more for precautionary motives because of the employment fluctuation effect but at the same time the price fluctuation effect raises the cost of saving. Thus public debt helps reducing the cost of precautionary saving in recessions. In booms, agents want to save less because of the employment fluctuation effect. At the same time the price fluctuation effect makes it less costly to save. Thus public debt is less useful in booms. The overall consumption gains or losses of a change in the level of public debt is shared very differently in the population. To show that we decompose the welfare gains of going from the benchmark level of public debt to the optimal level across the population. The last two rows of Table 2 show this decomposition. For instance, the row Bottom 10% shows the welfare gap between the 10% least fortunate people living in an economy with the benchmark level of public debt and those living in an economy with the optimal level of public debt. This decomposition closely matches a decomposition by wealth levels as the lowest (resp. highest) expected utilities refer to people who have experienced the highest unemployment (resp. employment) spells and who end up with the lowest (resp. highest) level of assets. The poorest agents are better off with a lower level of public debt than the optimal level. Going from the benchmark level of debt to the optimal debt level leads to an increase in consumption of 0.885% for the poorest 10% of the population. In the meantime, the richest 10% of the population would loose as much as 10.401% of consumption. This is explained by the fact that rich people's income is mainly capital income whereas poor people's income is mostly labor income. Thus when the level of public debt is higher, poor people suffers from the reduction in output caused by higher tax rates, lower wages and crowding out of capital. On the contrary, as interest rates raise with higher level of public debt, rich people are better off. The same type of effects are discussed in [START_REF] Ball | What Do Budget Deficits Do ?, in Budget Deficits and Debt : Issues and Options[END_REF] and [START_REF] Floden | The Effectiveness of Government Debt and Transfers as Insurance[END_REF].

Tab. 3 Macro variables along the cycle in the benchmark economy

Optimal level of debt without aggregate fluctuations

In this section we look at setups without aggregate fluctuations and compare them with our benchmark economy. There are several ways of eliminating business cycles 11 to derive a comparable long run idiosyncratic model. We look at three methods.

Imrohoroglu (1989) method

In the spirit of [START_REF] Lucas | Models of Business Cycles[END_REF] we replace the aggregate stochastic process with its conditional mean. We then follow Imrohoroglu (1989) to derive the labor market process. The transition probabilities of this economy are set so that the average rate of unemployment and the average duration of unemployment are the same between this economy and the benchmark economy. All other calibrated parameters are kept to their benchmark values, especially time preference rates and the risk aversion parameter12 . This model is similar to the benchmark model in many ways : higher levels of debt raise interest rates and crowd out private capital, overall wealth increases with debt and taxes are higher. But as the risk faced by agents is lower, agents save less and interest rates are higher in this model.

Figure 3 (top) depicts the welfare profile we find in this idiosyncratic risk model. The optimal level of public debt is 2.5% of output on an annual basis. This level is lower than the level we found in the benchmark economy. This is because there is no price fluctuation effect and employment fluctuation effect in this economy. In the benchmark economy, as the unemployment rate and the unemployment duration increase during recessions, the precautionary motive is stronger. Moreover, in the benchmark economy the interest rate and the physical capital are smaller in recessions. Therefore, it is more costly to save for precautionary motive. That is why the need for public debt is more important in the benchmark economy. In the idiosyncratic model the risk is lower because agents are subject to a less risky labor market and at the same time they no longer face price fluctuations. The precautionary motive is weaker here than in the benchmark model and agents save less. As a consequence interest rates are at a higher level in this economy.

Brute-force averaging

This method is related to the first one above. We again set the aggregate process to its conditional mean. The transition probabilities on the labor markets are generated as follows. We take the exact same transition process as in the benchmark economy and repeat it over very long periods of time. Thus we create a history of transition on the labor market conditional on the aggregate risk faced by agents. In the end of this simulation, we are left with a sequence of aggregate shocks and corresponding labor market situations for individuals. We then brute-force average the sequence of labor market transitions out. To do so, we count the numbers of transitions from unemployed to unemployed, from unemployed to employed, from employed to employed and from employed to unemployed over the entire simulated sequence and then take the mean. As we simulate over very long periods of time, we respect the law of large numbers. After brute-force averaging over the simulated series, we are left with a transition matrix on the labor market. The matrix we get by brute-force averaging is very similar to the matrix obtained by the Imrohoroglu method above. Figure 3 (middle) depicts the welfare profile we find in this idiosyncratic risk model. The optimal level of public debt is 2.5% of output on an annual basis. We can characterize this optimal level in the same way as we did above.

Conditional approach

In this section we consider an alternative method of removing business cycles to obtain a long-term idiosyncratic model by conditioning on the aggregate state. First, as above, we equate the aggregate process to its conditional mean. To obtain the labor market process, we condition on the aggregate state. For instance, we compute the probability of being employed tomorrow conditional on being employed today π ee in the idiosyncratic model as follows :

π ee = Pr(z = z g |s = e) * (Π bgee +Π ggee )+(1-Pr(z = z g |s = e)) * (Π gbee +Π bbee ),
where for instance Π ggee = Pr(z ′ = z g , s ′ = e|z = z g , s = e).

This means that to find who will remain employed in the next period conditional on being employed today, we consider all occurrences of being employed tomorrow conditional on being employed today in every aggregate state of the benchmark calibration and we weight them accordingly by the probability of being in either good or bad state conditional on being employed. We apply the same method to derive π uu . The remaining probabilities are found by applying the raws summing to unity property of the idiosyncratic transition matrix. This method of removing aggregate risk yield a more persistent labor process than the methods above. Notably, the probability of remaining unemployed tomorrow conditional on being unemployed today is higher. Thus the unemployment rate is not equal to its average value in the benchmark model. Here the unemployment rate is higher and amounts to 7.33% instead of an average value of 7% in the benchmark model. We find that the optimal level of debt is higher here than with the methods above. The annual optimal debt to GDP ratio is only slightly lower than 5% and is almost identical to the optimal level of debt found is the benchmark model with aggregate risk. This result is depicted in Figure 3 (bottom). It is straightforward to see why the optimal level of debt is higher with this methods. As both the persistence of the labor market process and the unemployment rate are higher, public debt plays a greater role in reducing the cost of precautionary saving here than with the methods above. We illustrate this in the next section with the benchmark economy.

Higher unemployment rate and longer unemployment spells

Our benchmark calibration reflects the behavior of the U.S. labor market. In this section we consider an alternative calibration of the labor market, that is a step towards reproducing European labor market features although we do not take into consideration or reproduce the employment benefit system found in Europe. We follow the methodology used in [START_REF] Algan | Incomplete Unemployment Insurance under Aggregate Fluctuations[END_REF] and the data set of [START_REF] Blanchard | The Role of Shocks and Institutions in the rise of European Unemployment : The Aggregate Evidence[END_REF]. We only modify the labor market features and leave the rest of the calibration unchanged. We fix the unemployment rate to be 13% in recessions and 7% in booms. We also set the duration of an unemployment spell to be 6 quarters in recessions and 4 quarters in booms. We now find the optimal level of debt to be 30% on an annual basis as depicted in Figure 4. Longer unemployment spells and higher unemployment rates tend to raise the optimal level of debt. This calibration purposely strengthens the employment fluctuation effect : unemployed agents have a harder time finding a job in this economy when compared to the benchmark economy and employed agents face a higher risk of loosing their jobs. In recessions, this is amplified. The precautionary motive is stronger here than in the benchmark economy. The harder it is for households to smooth consumption and the higher is the need for public debt. Here a higher level of public debt is needed to help households effectively smooth consumption.

Conclusion

This paper reconsidered the optimal level of public debt in an environment of aggregate fluctuations. Our benchmark model calibrated on the U.S. economy finds that a positive public debt level of 5% of output on a annual basis is optimal. This level is generally higher than in an economy without aggregate fluctuations. Our benchmark economy shows that in an aggregate fluctuations setting, households are subject to an employment fluctuation effect and to a price fluctuation effect. These effects make the precautionary saving motive stronger and at the same time the cost of saving higher. Our results suggest that the higher supply of safe assets induced by higher levels of public debt tends to raise the interest rate. This helps households to reduce the cost of precautionary saving and make smoothing of consumption easier. As a result there are welfare gains in the economy. However for the poorest agents, higher levels of debt are not optimal. Poor agents rely mainly on labor income and are dependent on higher wages and lower taxes. Public debt decreases wages and increases tax level thus poor agents prefer lower levels of public debt. We also emphasize thatlonger unemployment spells tend to raise the optimal level of public debt.

for this pseudo steady state. Also, the amount of public debt in the economy is defined according to this pseudo steady state by taking the appropriate ratio of the GDP in the pseudo steady state. Given the tax rate we just defined, we execute the above steps 1, 2 and 3. When those steps have converged, we update our guess of the interest rate in the pseudo steady state until a fix point is found.

When step 4 is complete, the pseudo steady state interest rate coincide with the average interest rate in the actual economy with aggregate fluctuations, the government budget constraint is balanced and the lump sum transfer to individuals amounts to zero at the equilibrium.

B Calibration

We now show how we derived the transition matrices for aggregate state changes (η), for joint transition between aggregate states and labor market statuses (Π) and for discount factor changes (Υ).

B.1 Aggregate state change transition matrix

To deduce the aggregate state change transition matrix η, we solve the following system :

η gg = η bb η bg = η gb η gg + η gb = 1 η bb + η bg = 1 η bg = 1 8                            =⇒    0

B.2 Matrix for joint transition between aggregate states and labor market statuses

The determination of the matrix Π that describes the transition between unemployment and employment requires the identification of the aggregate shock (whether we are in a recession or in a boom). The transition matrix Π is built thanks to the matrix η and to the transition matrixes Π gg , Π bb ,Π gb and Π bg . Π verifies :

Π =    η bb Π bb η bg Π bg η gb Π gb η gg Π gg   
We assumed that in recessions the duration of the unemployment, that we note durub, amounts to 2.5 quarters and the unemployment rate u b is set to 10%. In booms, the duration of unemployment, durug, is equal to 1.5 quarters and the unemployment rate, u g , is set to 4%. From this information, we can deduce the matrices Π gg and Π bb .

The transition matrix Π gg corresponds to the case (z, z ′ ) = (g, g). It verifies :

Π gg =    Π gg uu Π gg ue Π gg eu Π gg ee   
Solving the system below gives the values of Π gg ee , Π gg eu , Π gg ue and Π gg uu :

                       Π gg ee + Π gg eu = 1 Π gg ue + Π gg uu = 1 Π gg ue = 1 durug Π gg ee = 1 - u g Ψ gg ue 1 -u g =⇒ Π gg =    0.3333 0.6667 0.0278 0.9722   
The transition matrix Π bb corresponds to the case (z, z ′ ) = (b, b). It verifies :

Π bb =    Π bb uu Π bb ue Π bb eu Π bb ee   
Solving the system below gives the values of Π bb ee , Π bb eu , Π bb ue and Π bb uu :

                       Π bb ee + Π bb eu = 1 Π bb ue + Π bb uu = 1 Π bb ue = 1 durub Π bb ee = 1 - u b Π bb ue 1 -u b =⇒ Π bb =    0.6 0.4 0.0445 0.9555   
When the cycle changes the unemployment rate changes. The transitions between unemployment and employment get modified. We make the same assumptions as Krusell and Smith (1998) :

     Π bg uu = Pr(ǫ t+1 = u g /ǫ t = u b ) = 0.75Π gg uu Π gb uu = Pr(ǫ t+1 = u b /ǫ t = u g ) = 1.25Π bb uu
The probability to remain unemployed when the next period is a recession (resp. boom), increases (resp. decreases) since by assumption the unemployment rate is higher in recession than in boom.

The transition matrix Π bg corresponds to the case (z, z ′ ) = (b, g). It verifies :

Π bg =    Π bg uu Π bg ue Π bg eu Π bg ee   
The system below gives us Π bg ee , Π bg eu , Π bg ue and Π bg uu :

                  

B.3 Matrix for discount factor changes

We assumed that the discount factors follow a three-states first-order Markov process. Therefore, the matrix describing the transition from the discount factor β i to the discount factor β j is the following :

Υ =        Υ ll Υ lm Υ lh Υ ml Υ mm Υ mh Υ hl Υ hm Υ hh       
As we assumed that there is no immediate transition between β l and β h as in [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF], it involves that Υ lh = Υ hl = 0. Moreover, as we set the duration of the extreme states (β l and β h ) to 50 years namely 200 quarters, we have Υ lm = 1 200 = Υ hm . Solving the following system gives us the transition matrix Υ :

Υ ll + Υ lm + Υ lh = 1 Υ ml + Υ mm + Υ mh = 1 Υ hl + Υ hm + Υ hh = 1 Υ lh = Υ hl = 0 Υ lm = 1 200 = Υ hm Υ ml = Pr(βt=β l )Υ lm Pr(βt=βm) Υ mh = Pr(βt=β h )Υ hm Pr(βt=βm)                                          =⇒ Υ =

C Model without aggregate risk

We now briefly detail the model without aggregate risk and its calibration. Most of this model is similar to the benchmark model, thus we underline only differences.

Model

As this model serves comparison purposes most of the benchmark assumptions remain unchanged. The assumptions about the representative firm are similar with the exception of the technical progress z. In the absence of aggregate risk, z is fixed to its average value. In the absence of aggregate risk, the budget constraint of the government is :

G t + r t B t = B t+1 -B t + T t
Household preferences are unchanged. In the absence of aggregate risk, the matrix that describes the transition on the labor market becomes :

π =    π uu π ue π eu π ee   
with π uu = Pr(s t+1 = u|s t = u). When there is no aggregate risk, it is no longer necessary to distinguish the nature of the cycle. In the absence of aggregate risk, the state variables are summarized by the vector (a, s, β). The program the household solves is : In the brute-force averaging method, we first simulate the transition process and find the transition probabilities. This yield the following transition matrix : With this matrix, we derive an unemployment rate of 7%.

In the conditional approach, we use our definitions of transition probabilities to derive the following transition matrix : With this matrix, we derive an unemployment rate of 7.33%.

Fig. 1 .

 1 Fig. 1. Aggregate behavior of the benchmark model

Fig. 3 .

 3 Fig. 3. Welfare gain versus debt/GDP ratio in idiosyncratic models

Fig. 4 .

 4 Fig. 4. Welfare gain versus debt/GDP ratio in the model with higher unemployment rate and longer unemployment spells.

  As we assumed that the duration of a boom or a recession is the same, we deduce the two first equations. Moreover, the duration of a cycle is set to 8 quarters, it follows that η bg = P r(z t+1 = g/z t = b) = 1 8 and η gb = P r(z t+1 = b/z t = g) = 1 8 .

1 - 1 -

 11 u g ) -u b Π bg ue ) 1 -u b =⇒ Π bg =The transition matrix Π gb corresponds to the case (z, z ′ ) = (g, b).The system below gives us Π gb ee , Π gb eu , Π gb ue and Π gb uu :u b ) -u g Π gb ue ) 1 -u g =⇒ Π gb =

θπ uu + π ue = 1 π

 1 v(a, s, β) = max c, a ′ {u(c) + βE [v(a ′ , s ′ , β ′ )|(s, β)]} (C.1)subject to :c + a ′ = (1 + r(1 -τ ))a + wχ(s) if s = u, (1 -τ ) if s = e eu + π ee = 1

  Fig. 2. Welfare gain versus debt/GDP ratio in the benchmark model Tab. 2 Consumption gain (%) of going to the optimal level of debt in the benchmark economy

		0.3
		0.25
		0.2
	Consumption Gain (%)	0.05 0.1 0.15
		0
		-0.05
		-0.1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
						Debt to GDP Ratio
								Business Cycle
	Population				Average		Recessions		Booms
	All				0.257			0.267		0.247
	Bottom 10%				0.885			0.901		0.868
	Top 10%				-10.401		-10.535		-10.264

More details can be found in Appendix A.

This corresponds to about 12% of the average wage at the equilibrium.

There are several ways to reproduce a thick right tail. One would be to give rich agents higher propensity to save or higher returns on saving for instance by introducing entrepreneurs (e.g.,[START_REF] Quadrini | Entrepreneurship, Saving ans Social Mobility[END_REF]). For the sake of simplicity we explore here the preference heterogeneity setting introduced by Krusell and Smith (1998).

For further details on the calibration of this matrix, see appendix B.

The data we report on the U.S. distribution comes from Krusell and Smith (1998) and Budria-Rodriguez, Diaz-Gimenez, Quadrini and Rios-Rull (2002).

The value of the capital-output ratio can change with the definition of capital.Here we adopt the definition in[START_REF] Quadrini | Entrepreneurship, Saving ans Social Mobility[END_REF]. Thus aggregate capital results from the aggregation of plant and equipment, inventories, land at market value, and residential structures. This definition is close to the findings of[START_REF] Prescott | Theory Ahead of Business Cycle Measurement[END_REF] and is also used for instance in[START_REF] Floden | Idiosyncratic Risk in the United States and Sweden : Is There a Role for Government Insurance[END_REF]. This yields a capitaloutput ratio of 2.65 on an annual basis that we convert to its quarterly equivalent of 10.6.

[START_REF] Woodford | Public Debt as Private Liquidity[END_REF] argues that welfare can be enhanced if the interest rates are kept high enough, that is, closer to time preference rates in a liquidity-constrained economy. We find the same effect here.

For a survey, see[START_REF] Barlevy | The Cost of Business Cycles and the Benefits of Stabilization : A Survey[END_REF].

For greater details on the models without aggregate risk and their calibration, see appendix C.

Appendix A Computational strategy

We solve the model using the methodology developed by Den Haan (1996) and [START_REF] Krusell | Income and Wealth Heterogeneity in the Macroeconomy[END_REF]. They show that agents need only a restrictive set of statistics about the wealth distribution to determine prices. This set includes the mean of the wealth distribution and the aggregate productivity shock. A linear prediction rule based only on the average level of capital provides an accurate prediction. This result comes from the near linearity of the decision rule a ′ (a, s, β; z, Γ). As the aggregate capital stock is mainly held by rich people who have approximately the same propensity to save, next period's aggregate capital is accurately predicted by current period's aggregate capital. In our model, we assume the following law of motion :

where K′ and K denote respectively the average stock of capital of the next period and of the current period. Thus the strategy is the following :

(Step 1) : Given a set of parameter values (a 0 , a 1 , a 2 ) for the law of motion, we solve the individual problem. To solve the individual problem, we iterate on the Euler equation :

on a discrete grid until a fix point is found. When the borrowing constraint bind, the solution can be deduced from the budget constraint. (Step 4) : Because of the aggregate fluctuations property and the government budget constraint, on more step is necessary. As there is no formal steady state in an economy with aggregate fluctuations, we approximate a pseudo steady state by averaging aggregate variables over long periods of time. We first make a guess for the interest rate in this pseudo steady state, and with the long run budget constraint of the government, we derive a valid tax rate

Equilibrium

The recursive equilibrium consists of a set of decision rules for consumption and asset holding {c(a, s, ), a ′ (a, s, β; z, Γ)}, aggregate capital K, factor prices {r, w}, tax rate τ satisfying these conditions :

(1) Given the prices {r, w}, the decision rules {c(a, s, β), a ′ (a, ǫ, β)} solve the dynamic programming problem (C.1) subject to the constraints (C.2), (C.3) and (C.4)

(2) Market price arrangements are :

(3) Government budget constraint is balanced.

(4) Capital Market clears when :

with Γ(a, s, β) the distribution of agents over asset holdings, employment status and preference.

Calibration

We present here the calibration strategy in an economy without aggregate risk. The calibration of the preferences, the discount factors and the behavior of the government are unmodified. The calibration of z and the characteristics of the labor market differ in the absence of aggregate risk. In an economy without aggregate risk, z is constant and set to the average value of the aggregate shock, namely the unit value. As we consider three alternative calibration of the idiosyncratic labor process, we detail them sequentially.

In the Imrohoroglu method, the unemployment rate u and the unemployment spells duru are respectively set to 7% and 2 quarters (we average over good and bad periods in the benchmark model). These two assumptions define the transition matrix π :


J.-P Gattuso
email: gattuso@obs-vlfr.fr

H Lavigne

Technical Note: Approaches and software tools to investigate the impact of ocean acidification

Although future changes in the seawater carbonate chemistry are well constrained, their impact on marine organisms and ecosystems remains poorly known. The biological response to ocean acidification is a recent field of research as most purposeful experiments have only been carried out in the late 1990s. The potentially dire consequences of ocean acidification attract scientists and students with a limited knowledge of the carbonate chemistry and its experimental manipulation. Hence, some guidelines on carbonate chemistry manipulations may be helpful for the growing ocean acidification community to maintain comparability. Perturbation experiments are one of the key approaches used to investigate the biological response to elevated pCO 2 . They are based on measurements of physiological or metabolic processes in organisms and communities exposed to seawater with normal or altered carbonate chemistry. Seawater chemistry can be manipulated in different ways depending on the facilities available and on the question being addressed. The goal of this paper is (1) to examine the benefits and drawbacks of various manipulation techniques and (2) to describe a new version of the R software package seacarb which includes new functions aimed at assisting the design of ocean acidification perturbation experiments. Three approaches closely mimic the on-going and future changes in the seawater carbonate chemistry: gas bubbling, addition of high-CO 2 seawater as well as combined additions of acid and bicarbonate and/or carbonate.

Introduction

Ocean acidification refers to the on-going decrease in ocean pH as a result of the uptake of anthropogenic carbon dioxide (CO 2 ) in the ocean. The partial pressure of CO 2 (pCO 2 ) increases in the atmosphere due to anthropogenic inputs. It has increased by about 40% (267 to 384 µatm) since the beginning of the industrial revolution leading to changes in the Earth's climate and in the functioning of terrestrial ecosystems. The world's oceans currently absorb on average about one metric ton of CO 2 produced by each person every year. It is estimated that the surface waters of the oceans have taken up 118 Pg C, or about 25% of the carbon generated by human activities since 1800 [START_REF] Sabine | The oceanic sink for anthropogenic CO 2[END_REF]. The increase of pCO 2 profoundly affects the seawater carbonate system (Table 1). It increases the concentrations of bicarbonate and dissolved inorganic carbon, and lowers pH, the concentration of carbonate ions (CO 2- 3 ) and the saturation state of the major shell-forming carbonate minerals:

CO 2 + CO 2- 3 + H 2 O → 2HCO - 3
Surface ocean pH is estimated to have decreased from approximately 8.20 to 8.07 between preindustrial time and the 1990s and may reach 7.79 in 2100 (Table 1). The expression "ocean acidification" refers to the decrease in pH but does not imply that the pH of surface ocean waters will become acidic (below 7.0) any time soon.

Although the chemistry of ocean acidification is very well known, the associated impacts on marine organisms and ecosystems remain poorly known. The impact of low pH on marine organisms started to be investigated several decades ago (e.g. [START_REF] Bouxin | Action des acides sur le squelette des larves de l'oursin Paracentrotus lividus. Influence du pH[END_REF]), yet research into the biological response to ocean acidification is still in its infancy: the first Published by Copernicus Publications on behalf of the European Geosciences Union.

Table 1. Changes in the carbonate chemistry of surface seawater from the Last Glacial Maximum (LGM; 18 ky BP) to 2100. Total alkalinity, CO 2 partial pressure (pCO 2 ), salinity and temperature were fixed and used to derive all other parameters using the seacarb software [START_REF] Lavigne | seacarb 2.0, an R package to calculate parameters of the seawater carbonate system[END_REF] and the dissociation constant of carbonic acid of [START_REF] Lueker | Ocean pCO 2 calculated from dissolved inorganic carbon, alkalinity, and equations for K1 and K2: validation based on laboratory measurements of CO 2 in gas and seawater at equilibrium[END_REF]. It is assumed that the ocean and atmosphere are in equilibrium with respect to CO 2 . Values of temperature, salinity, total alkalinity and total phosphate in 1766, 2007 and 2100 are from [START_REF] Plattner | Feedback mechanisms and sensitivities of ocean carbon uptake under global warming[END_REF] prescribing historical CO 2 records and non-CO 2 radiative forcing from 1766 to 1990 and using the A2 IPCC SRES emissions scenario (Nakićenović and Swart, 2000) thereafter. Temperature during the LGM was set 1.9 • C colder than in 2007 (MARGO Project Members, 2009). The LGM salinity was set 1.07 higher than during pre-industrial time [START_REF] Paul | Modeling the water masses of the Atlantic Ocean at the Last Glacial Maximum[END_REF]. Total alkalinity in the LGM was scaled to salinity while the total phosphate concentration was assumed to be the same as in 1766. pCO 2 in 1766 and 2100 are from [START_REF] Plattner | Feedback mechanisms and sensitivities of ocean carbon uptake under global warming[END_REF], while values in the LGM and 2007 are from [START_REF] Petit | Vostok ice core data for 420[END_REF] and [START_REF] Keeling | Atmospheric CO 2 records from sites in the SIO air sampling network[END_REF], respectively. The concentration of total silicate is assumed to have remained constant from the LGM to 2100. It was calculated using the gridded data reported by [START_REF] Garcia | Nutrients (phosphate, nitrate, silicate)[END_REF] between 0 and 10 m and weighing the averages using the surface area of each grid cell. pH is expressed on the total scale.

Parameter

Unit LGM 1766 2007 2100 Temperature purposeful manipulative experiment was carried out only in the 1980s [START_REF] Agegian | The biogeochemical ecology of Porolithon gardineri (Foslie)[END_REF] and others began only in the late 1990s. The potentially dire consequences of ocean acidification have attracted the interest of scientists and students with a limited knowledge of the carbonate chemistry and its experimental manipulation. Hence, some guidelines on carbonate chemistry and its manipulation may be helpful for the growing ocean acidification community in order to ensure that the most adequate approaches are used and that results and subsequent analyses are comparable. The need to standardize measurement of parameters of the carbonate system has been identified [START_REF] Kleypas | Impacts of ocean acidification on coral reefs and other marine calcifiers: a guide for future research[END_REF], a gap which was recently filled [START_REF] Dickson | Guide to best practices for ocean CO 2 measurements[END_REF]. Perturbation experiments are one of the key approaches used to investigate the biological response to elevated pCO 2 . They are based on measurements of physiological or metabolic processes in organisms and communities exposed to seawater with normal or altered chemistry. Seawater chemistry can be manipulated using different ways depending on the facilities available and on the question being addressed. [START_REF] Langdon | Review of experimental evidence for effects of CO 2 on calcification of reef builders[END_REF] summarized the state-of-the-art techniques used with reef builders. Here we examine the benefits and drawbacks of current methods and provide a software package to assist the design of perturbation experiments.

Method

Detailed information on the carbonate chemistry, including the merit of units and scales, as well as on the measurement methods can be found in [START_REF] Dickson | Guide to best practices for ocean CO 2 measurements[END_REF], [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF] and in the Guide for Best Practices in Ocean Acidification Research and Data Reporting in preparation (see http://www.epoca-project.eu/index.php/ Best-Practices-Guide/). The R package seacarb was used to calculate the carbonate chemistry parameters. R is a free cross-platform language and environment for statistical computing and graphics (R Development Core Team, 2008). The previous version, seacarb 1.2.3 [START_REF] Proye | seacarb, an R package to calculate parameters of the seawater carbonate system[END_REF] is superseded by the recently released version 2.0 [START_REF] Lavigne | seacarb 2.0, an R package to calculate parameters of the seawater carbonate system[END_REF]. The new version has been updated in order to be fully consistent with the most recent guidelines to best practices for ocean CO 2 measurements [START_REF] Dickson | Guide to best practices for ocean CO 2 measurements[END_REF]. All equations are solved analytically except when total alkalinity is one of the two input parameters of the carbonate system that are required. In that case, the concentration of H + or dissolved inorganic carbon are solved numerically using the R function uniroot. Five new functions have been added to seacarb 2.0 to assist the design of ocean acidification perturbation experiments. The syntax of these commands is briefly described in Appendices A1 to A5. Additional information is available in the help files of the seacarb package.

Main approaches

The seawater chemistry can be manipulated in various ways in order to alter parameters of the carbonate system. The following sections examine the five techniques that are most useful in the context of ocean acidification. Calculations were carried out using the first and second dissociation constants of carbonic acid given by [START_REF] Lueker | Ocean pCO 2 calculated from dissolved inorganic carbon, alkalinity, and equations for K1 and K2: validation based on laboratory measurements of CO 2 in gas and seawater at equilibrium[END_REF]. In systems open to the atmosphere, it is assumed that the seawater and atmosphere are in equilibrium with respect to CO 2 . It must be pointed out that the methods described below enable to set the carbonate chemistry at the beginning of a perturbation experiment. However, initial values can be altered by biological (e.g. photosynthesis, respiration and calcification) and physical processes (e.g. air-seawater exchange of CO 2 and temperature changes), which must also be taken into account in the experimental design.

Gas bubbling

Bubbling seawater with gases is an efficient way to manipulate its carbonate chemistry. The seacarb function pgas estimates the changes in the carbonate chemistry resulting from changes in pCO 2 generated by bubbling gases.

Example: seawater with a pCO 2 of 384 µatm and total alkalinity (TA) of 2325 µmol kg -1 is bubbled with a mixture of CO 2 and air with a pCO 2 of 793 µatm. Salinity is 34.9, temperature is 18.9 • C and calculations are made for surface waters. The command needed is: pgas(flag=24, var1=384, var2=2325e-6, pCO2g=793, S=34.9, T=18.9) This approach exactly reproduces the changes of all parameters of the carbonate system expected in the year 2100 (Fig. 1 andTable 2). [START_REF] Smith | Effect of carbon dioxide concentration on calcification in the red coralline alga Bossiella orbigniana[END_REF] is perhaps the first publication which manipulated the carbonate chemistry via gas bubbling within a range compatible with that of the on-going ocean acidification.

Two techniques have been used in the context of ocean acidification: pH-stat and bubbling with premixed gases. In pH-stat systems, pH is monitored continuously and a controller opens or closes valves when the pH goes above or below a set value. Gases are then delivered until the pH reaches the set value again. Different combination of gases have been used: (1) air and pure CO 2 , (2) CO 2 -free air and pure CO 2 , and (3) air, CO 2 -free air and pure CO 2 . CO 2 -free air can be produced easily using CO 2 scrubbers such as soda lime or NaOH and Ca(OH) 2 (C. J. Hintz, personal communication, 2008). The pH threshold is calculated using the desired pCO 2 and total alkalinity which must either be constant or frequently measured. This method allows one to compensate for any change in the carbonate chemistry due to photosynthesis and respiration or, in the case of open culture systems, to changes in the source water. However, it does not compensate for changes in total alkalinity that would result from precipitation and dissolution of CaCO 3 which could occur between measurements of total alkalinity. Hence, the carbonate chemistry can be maintained with good efficiency in the culture vessel, e.g. with variability of pCO 2 better than ±10 µatm. The main drawback of this technique is that the pH electrode must be frequently calibrated in order to account for drift. Hence, the second technique of bubbling with premixed gases may be attractive. Air with the desired pCO 2 can be produced using gas mixing pumps or purchased.

Another approach would be to control atmospheric pCO 2 to the desired level in the laboratory or growth cabinet in which the experiments are carried out. A simple air pump can then be used to bubble the experimental seawater. To the best of our knowledge, this technique has yet to be used in the context of ocean acidification.

Seawater aeration by bubbling may lead to difficulties in phytoplankton cultures [START_REF] Shi | Effects of the pH/pCO 2 control method on medium chemistry and phytoplankton growth[END_REF]. It may also enhance the coagulation of organic matter [START_REF] Engel | Transparent exopolymer particles and dissolved organic carbon production by Emiliania huxleyi exposed to different CO 2 concentrations: a mesocosm experiment[END_REF]. This coagulation may be critical for studies investigating the response of microbial communities because their metabolism depends on the respective abundance of dissolved and particulate organic matter. This drawback may be avoided by enclosing the community in a dialysis bag, which is itself enclosed in a container bubbled with a gas having the desired pCO 2 (M. Weinbauer, personal communication, 2008). Such bags are permeable to gases and small molecules in solution but impermeable to larger molecules and particles.

Addition of high-CO 2 sea water

Dissolved inorganic carbon (DIC) and total alkalinity (TA) are conservative quantities with respect to mixing [START_REF] Wolf-Gladrow | Total alkalinity: The explicit conservative expression and its application to biogeochemical processes[END_REF]. Hence, when two water parcels are mixed, the amount of a solute in the mixture equals the sum of the amounts of this solute in the two initial water parcels. Hence:

[DIC] mixed ×M mixed = [DIC] sw ×M sw + [DIC] high CO 2 ×M high CO 2 [TA] mixed ×M mixed = [TA] sw ×M sw + [TA] high CO 2 ×M high CO 2
Where the subscripts "sw" and "high CO 2 ", respectively indicate concentrations in the normal and high CO 2 seawater. Concentrations are in µmol kg -1 and M is the mass in kg. The seacarb function pmix estimates the carbonate chemistry after mixing of two water samples.

Example: one mixes, in a closed system, 0.99623 kg of seawater having a TA of 2325 µmol kg -1 , and pCO 2 of 384 µatm with 0.00377 kg of seawater having a TA of 2325 µmol kg -1 and saturated with CO 2 (pCO 2 =1×10 6 µatm).

The weight fraction, wf, is 3.76×10 -3 . Salinity is 34.9, temperature is 18.9 ), salinity and temperature were used to derive all other parameters using the seacarb package [START_REF] Lavigne | seacarb 2.0, an R package to calculate parameters of the seawater carbonate system[END_REF] except for manipulations of the calcium concentration for which DIC was used rather than pCO 2 sw . For simplicity, it is assumed that temperature, salinity and total alkalinity in 2100 are identical to their 2007 values, respectively 18.9 • C, 34.9 and 2325×10 -6 mol kg -1 , and the concentrations of total phosphate and silicate were set to 0. The seawater pCO 2 was set to 384 µatm in 2007 [START_REF] Keeling | Atmospheric CO 2 records from sites in the SIO air sampling network[END_REF] and 793 µatm in 2100 (according to the A2 SRES emission scenario; [START_REF] Plattner | Feedback mechanisms and sensitivities of ocean carbon uptake under global warming[END_REF]. The main body of the text provides further information.

(a): ×10 -9 mol kg -1 , (b): and calculations are made for surface waters. The command needed is: pmix(flag=24, var1=384, var2=2325e-6, pCO2s=1e6, wf=3.76e-3, S=34.9, T=18.9)

×10 -6 mol kg -1 . pCO 2 sw pH T [H + ] TA DIC [CO 2 ] [HCO - 3 ] [CO 2- 3 ] c a (µatm) (-) (a) (b) (b) (b) (b) (b) (-) (-)
This produces seawater with a final pCO 2 of 792 µatm (the 1 µatm difference with the target value is due to rounding errors) and all parameters of the carbonate chemistry in 2100 are perfectly reproduced. To the best of our knowledge, this approach has only been used twice: in mesocosms by Schulz et al. (unpublished data) and in the laboratory by C. McGraw (personal communication, 2009). It is therefore not possible to evaluate its pros and cons at this stage.

Addition of CO 2-

3 and/or HCO -

3

Dissolved inorganic carbon and total alkalinity can be increased by adding CO 2- 3 in the form of Na 2 CO 3 and/or by adding HCO - 3 in the form of NaHCO 3 . In closed systems, the change in DIC generated by these additions is proportional to the changes in concentration: 1× CO 2- 3 and 1× HCO - 3 . The contribution of these anions to TA is proportional to the product of their charge and concentration. Thus: TA increases by 2× CO 2- 3 and 1× HCO - 3 . The changes in the carbonate chemistry generated by manipulations of total alkalinity therefore depend on the proportion of CO 2- 3 and HCO - 3 added (Fig. 3). This approach can be used to keep the pH constant or combined with addition of acid to keep TA constant (see Sect. 3.5). In an open system, the carbonate system re-equilibrates through air-sea CO 2 gas exchange after adding chemicals. At equilibrium:

CO 2- 3 + H + → HCO - 3 (R1) HCO - 3 + H + → CO 2 + H 2 O (R2)
The seacarb function pTA computes the changes in the carbonate chemistry due to addition of CO 2- 3 and/or HCO - 3 . Example: HCO - 3 (1081×10 -6 mol kg -1 ) is added to seawater for which pCO 2 (384 µatm) and TA (2325 µmol kg -1 ) are known. No CO 2- 3 is added. The atmospheric pCO 2 is 384 µatm, salinity is 34.9, temperature is 18.9 • C and calculations are made for surface waters. The respective commands in a closed and open system are: pTA(flag=24, sys=0, var1=384, var2=2325e-6, pCO2a=384, co3=0e-6, hco3=1081e-6, S=34.9, T=18.9) pTA(flag=24, sys=1, var1=384, var2=2325e-6, pCO2a=384, co3=0e-6, hco3=1081e-6, S=34.9, T=18.9) Results are shown in Fig. 3 andTable 2. In a closed system, the target pCO 2 of 793 µatm is reached but all other parameters of the carbonate system are very different from their values expected in 2100. pH is higher than it should be (7.942 instead of 7. saturation states of aragonite and calcite ( a and c ) are higher than the target values. They are even higher than the values of the initial seawater. Those differences are magnified in open systems. Manipulation of total alkalinity has often been used to investigate the physiological response to changes in the carbonate chemistry (e.g. [START_REF] Swift | The effect of pH on the division rate of the coccolithophorid Cricosphaera elongata[END_REF]. [START_REF] Marubini | Bicarbonate addition promotes coral growth[END_REF] used bicarbonate additions to test the hypothesis that the supply of DIC limits coral calcification. [START_REF] Langdon | Effect of calcium carbonate saturation state on the rate of calcification of an experimental coral reef[END_REF] used this approach in the Biosphere 2 coral reef mesocosm. They added NaHCO 3 and Na 2 CO 3 to increase the CaCO 3 saturation state. Since the experiment lasted several months, additions of CaCl 2 were used to restore a normal Ca 2+ concentration. The manipulations of the Ca 2+ concentration is described in Sect. 3.6.

Addition of strong acids and bases

The carbonate chemistry can be manipulated by addition of a strong acid or a strong base, such as HCl and NaOH, which completely dissociate in seawater. Adding of a strong acid or base to a system closed to the atmosphere does not alter the concentration of dissolved inorganic carbon but modifies total alkalinity. TA decreases when adding an acid and increases after addition of a base.

The change of total alkalinity after addition of a strong acid or base in a system open to the atmosphere is identical to that described above for a closed system. However, the concentration of DIC is modified through CO 2 exchange at the air-water interface because the partial pressures of CO 2 in the atmosphere and seawater equilibrate.

The seacarb function ppH calculates the changes in the carbonate chemistry during pH manipulations. The change in salinity due to the addition of acid or base is negligible. Example: a volume of 14.08 ml of 0.01 N HCl is added to 1 kg of seawater having a known pCO 2 (384 µatm) and TA (2325 µmol kg -1 ). The atmospheric pCO 2 is 384 µatm, salinity is 34.9 and temperature is 18.9 • C. Calculations are made for surface waters. The commands needed in a closed and open system are: ppH(flag=24, sys=0, var1=384, var2=2325e-6, pCO2a=384, vol=-14.08e-3, N=0.01, S=34.9, T=18.9) ppH(flag=24, sys=1, var1=384, var2=2325e-6, pCO2a=384, vol=-14.08e-3, N=0.01, S=34.9, T=18.9) Results are shown in Fig. 4 and Table 2. The target pCO 2 of 793 µatm is reached in a closed system but the pH is lower than the value expected in 2100 (7.768 instead of 7.793, corresponding to an increase of 2.9% in [H + ] that resulting from the decrease in total alkalinity generated by the addition of acid). This is an undesirable effect of the direct manipulation of pH as TA will not change significantly during the course of this century (Table 1). As a result, DIC, HCO - 3 , CO 2- 3 , a and c are lower than their target values. However, it is possible to restore TA to its initial level by adding CO 2- 3 and/or HCO - 3 , an approach that is described in Sect. 3.5. Obviously, due to gas exchange at the air-water interface, pCO 2 and other parameters of the carbonate chemistry cannot be successfully adjusted by a one-time addition of acid in an open system.

Addition of acid has often been used in studies of the impact of ocean acidification on organisms. The first purposeful experiment is that of [START_REF] Agegian | The biogeochemical ecology of Porolithon gardineri (Foslie)[END_REF] who looked at the response of coralline algae.

Addition of strong acid as well as CO 2-

3 and/or HCO -

3

As mentioned in Sect. 3.4, the addition of acid does not allow to fully mimic the changes in carbonate chemistry expected during the present century but the addition of CO 2- problem. The first addition elevates DIC to the desired level while adding acid (at constant DIC) precisely cancels out the increase in TA resulting from the addition of CO 2- 3 and/or HCO - 3 . There is no specific seacarb function to handle this approach. The changes in the carbonate chemistry can be calculated using successively the functions pTA and ppH. tmp=pTA(flag=24, sys=0, var1=384, var2=2325e-6, pCO2a=384, co3=15.3e-6, hco3=111.2e-6, S=34.9, T=18.9) ppH(flag=24, sys=0, var1=tmp$pCO2[2], var2=tmp$ALK[2], pCO2a=384, vol=-14.18e-3, N=0.01, S=34.9, T=18.9)

Results are shown in Table 2. The first addition raises DIC to the desired level of 2191×10 -6 mol kg -1 but increases TA to a value higher than the target (2467 vs. 2325×10 -6 mol kg -1 ). The subsequent addition of HCl restores TA to the desired value without affecting DIC. After both additions, all carbonate parameters are at the target values (Table 2).

To our knowledge, the first paper which used addition of acid, base and NaHCO 3 to manipulate the carbonate chemistry within a range comparable to ocean acidification is that of [START_REF] Borowitzka | Photosynthesis and calcification in the articulated coralline red algae Amphiroa anceps and A. foliacea[END_REF] who investigated, in the laboratory, the response of photosynthesis and calcification of an articulated coralline red algae.

Manipulation of the Ca 2+ concentration

Although manipulating the calcium concentration does not technically alter the carbonate chemistry per se, this approach has been used in the context of ocean acidification. The reason is that some calcifying organisms, such as corals, respond to the calcium carbonate saturation state of seawater ( ) which is expressed as:

= [Ca 2+ ] sw ×[CO 2- 3 ] sw K * sp
where [Ca 2+ ] sw and [CO 2- 3 ] sw are the concentrations of calcium and carbonate in seawater, respectively, and K * sp is the solubility product at the in situ conditions of temperature, salinity and pressure [START_REF] Zeebe | CO 2 in seawater: equilibrium, kinetics, isotopes[END_REF]. It can readily be appreciated that the changes in resulting from the decrease in [CO 2- 3 ] sw driven by ocean acidification can be mimicked by altering [CO 2- 3 ] sw . This approach involves the use of artificial seawater [START_REF] Gattuso | Effect of calcium carbonate saturation of seawater on coral calcification[END_REF]. It is recommended to use the simplied synthetic seawater recipe based on DOE (1994) as described by [START_REF] Gattuso | Effect of calcium carbonate saturation of seawater on coral calcification[END_REF] and because it is the basis of the synthetic seawater that has been used to determine a variety of equilibrium constants for use in seawater.

The seacarb function pCa estimates the changes in c and a resulting from the manipulation of the [Ca 2+ ] sw concentration. Note that this function does not account for the effect of the changes on the calcium concentration of the dissociation constants of carbonic acid and on the solubility product of CaCO 3 [START_REF] Ben-Yaakov | The influence of sea water composition on the apparent constants of the carbonate system[END_REF].

Example: artificial seawater is made with a known TA (2325 µmol kg -1 ) and DIC (2064 µmol kg -1 ). The calcium concentration is set to 6.03×10 -6 mol kg -1 . The command needed is: pCa(flag=15, var1=2325e-6, var2=2064e-6, Ca=6.03e-3, S=34.9, T=18.9) Results are shown in Fig. 5 and Table 2. The saturation states of aragonite and calcite expected in 2100 are well reproduced without affecting any of the other parameters of the carbonate system which remain at their 2008 values.

This approach has been used to investigate the response of corals [START_REF] Gattuso | Effect of calcium carbonate saturation of seawater on coral calcification[END_REF] and coccolithophores [START_REF] Trimborn | Effect of varying calcium concentrations and light intensities on calcification and photosynthesis in Emiliania huxleyi[END_REF] to changes in the saturation state of calcium carbonate. It has also been used to replenish calcium when its concentration decreases below its natural levels during longterm experiments with calcifiers (e.g. [START_REF] Langdon | Effect of calcium carbonate saturation state on the rate of calcification of an experimental coral reef[END_REF]. The letters "b" and "a" indicate, respectively, the value of the calcium carbonate saturation state before and after perturbation for the example described in the text.

Discussion and conclusion

Three approaches closely mimic the on-going and future changes in the seawater carbonate chemistry: gas bubbling, addition of high-CO 2 seawater, and combined addition of acid and bicarbonate and/or carbonate. All three methods increase dissolved inorganic carbon at constant total alkalinity, a situation that closely resembles the changes in the carbonate chemistry that have occurred during the last 200 years and are expected to continue during the next few hundreds years. All three approaches therefore offer precise control of carbonate chemistry variables, that reach target values at the beginning of a perturbation experiment. Gas bubbling is the easiest to implement and can be used to maintain constant conditions over long periods of time. However, during bubbling, biological processes (e.g., calcification, dissolution of CaCO 3 , nutrient uptake and release) can alter the carbonate chemistry by changing total alkalinity even though the concentration of pCO 2 remains constant. Also, in all three approaches, calcification can deplete the concentration of Ca 2+ when the organism to volume ratio is high or the incubation time long. The approach of adding CO 2- 3 and/or HCO - 3 appears to be of limited practical use because most carbonate variables reach values that are far from target values. Several studies have added acid to manipulate the carbonate chemistry. Although this technique offers precise control of pCO 2 in systems closed to the atmosphere, it also alters total alkalinity which drives other carbonate system variables to levels far from those expected in the future. That is, pH is lower than its target value, DIC remains unchanged (whereas it increases with natural ocean acidification), and the CaCO 3 saturation states as well as the concentrations of bicarbonate and carbonate are higher than the target values. To avoid this drawback, it is recommended to first add bicarbonate or carbonate, or both, to increase DIC, before adding acid. Iglesias-Rodriguez et al. (2008a) have recently reported that, in contrast to previous reports, calcification of the coccolithophore Emiliania huxleyi increases at elevated pCO 2 . They argued that this is due to the different approaches used to manipulate the carbonate chemistry and that the approach of gas bubbling (changing DIC at constant TA) is superior to the approach of adding acid (changing TA at constant DIC) (Iglesias-Rodriguez et al., 2008a,b). The argument is that the addition of acid does not reproduce the increase in the HCO - 3 concentration occurring from natural ocean acidification whereas gas bubbling does reproduce it. The authors claim that HCO - 3 may stimulate photosynthesis and, in turn, calcification. They conclude that experiments using addition of acid demonstrating that calcification declines at lower pH, confounded the issue. This statement is misleading, although there is no doubt that gas bubbling better mimics the future carbonate chemistry (see above as well as Table 2). In contrast to the claims by Iglesias-Rodriguez et al. (2008a,b), several previous perturbation experiments were also carried out with gas bubbling and reported lower rates of calcification of coccolithophores at lower pH and higher pCO 2 [START_REF] Sciandra | Response of the coccolithophorid Emiliania huxleyi to elevated partial pressure of CO 2 under nitrate limitation[END_REF][START_REF] Delille | Response of primary production and calcification to changes of pCO 2 during experimental blooms of the coccolithophorid Emiliania huxleyi[END_REF][START_REF] Feng | Interactive effects of increased pCO 2 , temperature and irradiance on the marine coccolithophore Emiliania huxleyi (Prymnesiophyceae)[END_REF]. Additionally, the concentrations of CO 2 , HCO - 3 , CO 2- 3 , and H + all change in the same direction with both approaches: gas bubbling and acid addition (compare Fig. 1a-c with Fig. 4a-c; Table 2). Hence, the magnitude of biological responses may be different, but should at least tend to go in the same direction. Differences in the two approaches cannot explain biological responses going in opposite directions. Finally, perturbation experiments carried out using gas bubbling can also lead to poor control of the carbonate chemistry, for example, when the experiments last too long or when the biomass-to -volume ratio is too large (see above). Although the cause of the discrepancy in the response of calcification by the coccolithophore E. huxleyi to ocean acidification remains a matter of debate (e.g., [START_REF] Riebesell | Comment on "Phytoplankton calcification in a high-CO 2 world[END_REF][START_REF] Shi | Effects of the pH/pCO 2 control method on medium chemistry and phytoplankton growth[END_REF], it does not appear to be related to the approach used to perform the manipulation of the carbonate chemistry in perturbation experiments.

Fortunately, the situation seems clearer in zooxanthellate scleractinian corals. [START_REF] Schneider | The effect of carbonate chemistry on calcification and photosynthesis in the hermatypic coral Acropora eurystoma[END_REF] measured the rate of calcification under constant DIC, constant pH and constant pCO 2 and showed that calcification is controlled by the concentration of CO 2- 3 , the future value of which is relatively well mimicked by adding acid.

Our goal here has been to provide the appropriate computational tools and to highlight the positive and negative aspects of the different approaches that are used to investigate ocean acidification by perturbing the carbonate chemistry. We have not addressed practical issues that must be taken into account to design and conduct perturbation experiments. To mention just a few: response time, bubbling rate, impact of autoclaving, evaporation, heat flux, impact of nutrients and temperature. These issues, as well as others, will be covered in a Guide for Best Practices in Ocean Acidification Research and Data Reporting in preparation (see http://www.epoca-project.eu/index.php/ Home/Guide-to-OA-Research/).

Appendix A A1 Gas bubbling: syntax of pgas

The function pgas calculates the carbonate chemistry after perturbation using the CO 2 partial pressure of the gas used to bubble the medium and the initial total alkalinity which is not altered by gas bubbling. The syntax is: pgas(flag, var1, var2, pCO2g, S, T, P, Pt, Sit, k1k2, Kf, pHscale)

The input parameters below are required:

-flag: a number (1 to 24) indicating which couple of parameters of the carbonate system is known. For example, flag is 8 when pH T and total alkalinity are known.

-var1 and var2: the values of the two parameters of the carbonate system prior to the perturbation which are known (pH T and alkalinity in the example above). See the seacarb manual for more details, especially regarding the units.

-pCO2g is the CO 2 partial pressure (in µatm) of the gas used to bubble the medium.

-S: salinity.

-T: temperature in • C -P: hydrostatic pressure (P; 0 by default). Note that this function was only tested at P=0, that is for surface waters.

-Pt: concentration of total phosphate (mol kg -1 ; 0 by default).

-Sit: concentration of total silicate (mol kg -1 ; 0 by default).

-k1k2: "l" for using K 1 and K 2 from [START_REF] Lueker | Ocean pCO 2 calculated from dissolved inorganic carbon, alkalinity, and equations for K1 and K2: validation based on laboratory measurements of CO 2 in gas and seawater at equilibrium[END_REF] and "r" for using K 1 and K 2 from [START_REF] Roy | The dissociation constants of carbonic acid in seawater at salinities 5 to 45 and temperatures 0 to 45 • C[END_REF], default is "l" -Kf: "pf" for using K f from [START_REF] Perez | Association constant of fluoride and hydrogen ions in seawater[END_REF] and "dg" for using K f from [START_REF] Dickson | The estimation of acid dissociation constants in seawater media from potentiometric titrations with strong base. I. The ionic product of water[END_REF], default is "pf" -pHscale: "T" for using the total scale, "F" for using the free scale and "SWS" for using the seawater scale, default is "T" A2 Addition of high-CO 2 seawater: syntax of pmix

It is important to note that pmix assumes that the initial and high-CO 2 seawater have the same total alkalinity. pmix calculates the carbonate chemistry after perturbation using the final DIC concentration calculated as described in Sect. 3.2 and the initial total alkalinity which is not altered by the mixing. The syntax is: pmix(flag, var1, var2, pCO2s, wf, S, T, P, Pt, Sit, k1k2, Kf, pHscale) flag, var1, var2, S, T, P, Pt, Sit, k1k2, Kf and pHscale were defined in Appendix A1. The other input parameters required are:

-pCO2s: pCO 2 of the high-CO 2 water in µatm -wf: weight fraction of the high-CO 2 seawater vs. final weight A3 Addition of CO 2- 3 and/or HCO - 3 : syntax of pTA For closed systems, the function pTA calculates the carbonate chemistry after perturbation using the final DIC concentration and total alkalinity calculated as described in Sect. 3.3. For open systems, the carbonate chemistry is calculated using atmospheric pCO 2 , therefore assuming equilibrium of pCO 2 between air and seawater, and total alkalinity calculated as described in Sect. 3.3 (as is not further altered by air-sea CO 2 exchange). The syntax is:

pTA (flag,sys,var1,var2,pCO2a,co3,hco3,S,T,P,Pt,Sit,k1k2,Kf,pHscale) flag, var1, var2, S, T, P, Pt, Sit, k1k2, Kf and pHscale where defined above. The other input parameters below are required:

-sys: 0 if the manipulation is carried out in a system closed to the atmosphere or 1 if it is carried out in a system open to the atmosphere (in the latter case, full equilibrium between air and water is assumed).

-pCO2a: the value of atmospheric pCO 2 in µatm. It is only used in systems open to the atmosphere (i.e. sys=1).

-co3: amount of CO 2- 3 added in mol kg -1 . -hco3: amount of HCO - 3 added in mol kg -1 .

A4 Addition of strong acids and bases: syntax of ppH

For closed systems, the function ppH calculates the carbonate chemistry after perturbation using the DIC concentration (which is unchanged by the addition of acid or base) and the total alkalinity which can easily be estimated as strong acids and bases completely dissociate in seawater. For open systems, the carbonate chemistry is calculated using atmospheric pCO 2 , therefore assuming equilibrium of pCO 2 between air and seawater, and the total alkalinity after addition of acid or base as it is not further altered by air-sea CO 2 exchange. The syntax is:

ppH(flag, sys, var1, var2, pCO2a, vol, N, S, T, P, Pt, Sit, k1k2, Kf, pHscale) flag, sys, var1, var2, pCO 2 a, S, T, P, Pt, Sit, k1k2, Kf and pHscale were defined above.

The other input parameters required are:

-vol: the volume of acid or base added in liter. By convention, it is given a negative sign for additions of acid and a positive sign for additions of base.

-N: the normality of the acid or base (mol kg -1 ).

A5 Manipulation of the Ca 2+ concentration: syntax of pCa

The function pCa calculates the carbonate chemistry using the standard equations of [START_REF] Dickson | Guide to best practices for ocean CO 2 measurements[END_REF] except that the actual calcium concentration is used rather than the standard concentration of 0.01028×(S/35) mol kg -1 . The syntax is: pCa(flag, sys, var1, var2, Ca, S, T, P, Pt, Sit, k1k2, Kf, pHscale) flag, var1, var2, S, T, P, Pt, Sit, k1k2, Kf and pHscale were defined above. Ca is the calcium concentration in mol kg -1 .

Fig. 1 .

 1 Fig.1. Effect of bubbling seawater with gases of varying pCO 2 on the carbonate chemistry of seawater as computed with the seacarb function pgas. The letters "b" and "a" indicate, respectively, the value of a parameter before and after perturbation for the example described in the text.

Fig. 2 .

 2 Fig. 2.Effect of mixing with various weight fractions of high-pCO 2 seawater on the carbonate chemistry of seawater in a closed system as computed with the seacarb function pmix. The letters "b" and "a" indicate, respectively, the value of a parameter before and after perturbation for the example described in the text.

Fig. 3 .

 3 Fig. 3. Effect of adding CO 2- 3 and/or HCO - 3 on the carbonate chemistry of seawater in closed (A, B and C) and open systems (Dand E) as computed with the seacarb function pTA. The "b" and "a" indicate, respectively, the value of a parameter before and after perturbation for the example described in the text.

Fig. 4 .

 4 Fig. 4. Effect of adding a strong acid or a strong base on the carbonate chemistry of seawater in closed (A, B and C) and open systems (D, E and F) as computed with the seacarb function pgas. By convention, addition of acid is negative whereas addition of base is positive. The letters "b" and "a" indicate, respectively, the value of a parameter before and after perturbation for the example described in the text.

Fig. 5 .

 5 Fig. 5. Calcium carbonate saturation state as a function of the calcium concentration as computed with the seacarb function pCa.The letters "b" and "a" indicate, respectively, the value of the calcium carbonate saturation state before and after perturbation for the example described in the text.

Table 2 .

 2 Seawater carbonate chemistry in 2007, 2100, and after perturbation. The total alkalinity (TA), partial pressure of CO 2 in seawater (pCO 2 sw

Biogeosciences, 6, 2121Biogeosciences, 6, -2133Biogeosciences, 6, , 2009 www.biogeosciences.net/6/2121/2009/

www.biogeosciences.net/6/2121/2009/ Biogeosciences, 6, 2121-2133, 2009

and/or HCO - 3 followed by addition of acid circumvents this Biogeosciences, 6, 2121-2133, 2009 www.biogeosciences.net/6/2121/2009/

(15.3×10 -6 mol kg -1 ) are added to seawater for which pCO 2 (384 µatm) and TA (2325 µmol kg -1 ) are known. The atmospheric pCO 2 is 384 µatm, salinity is 34.9, temperature is 18.9 • C and calculations are made for surface waters. Then, 14.18 ml of 0.01 N HCl is added. The commands needed are: www.biogeosciences.net/6/2121/2009/ Biogeosciences, 6, 2121-2133, 2009

Acknowledgements. Thanks are due to G.-K. Plattner for providing model outputs, to P. Grosjean, J. C. Orr, U. Riebesell, K. Schulz and R. E. Zeebe for useful comments on an early draft of this paper, as well as to A. Dickson, A. Engel, C. Langdon, H.-O. Pörtner and the participants of the "Workshop on Best Practices for Ocean Acidification Research and Data Reporting" for helpful discussions. This work is a contribution to the "European Project on Ocean Acidification" (EPOCA) which receives funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement 211384, the BOOM project funded by the Agence Nationale de la Recherche (ANR-05-BIODIV-004) and CARBOOCEAN (FP6/511176-2). Edited by: A. Boetius www.biogeosciences.net/6/2121/2009/ Biogeosciences, 6, 2121-2133, 2009