David Martimort
email: david.martimort@psemail.eu.

Jerome Pouyet
email: pouyet@essec.edu.

Francesco Ricci
email: francesco.ricci@umontpellier.fr.

Contracts for the Management of a Non -Renewable Resource under Asymmetric Information and Structural Price Breaks *

Keywords: Non-Renewable Resource Management, Delegated Management, Optimal Contract, Asymmetric Information. JEL codes. Q31; D82

HAL is

Introduction

The shale boom. According the U.S. Energy Information Administration (EIA (2015)), fossil fuels will still meet 87% of energy needs in 2025 and will remain the dominant source of energy supply for the 21st century. Natural gas is an alternative to other sources due to its abundance and its lower environmental damages.

In the U.S. gas trapped in shale rocks started to be exploited on an industrial scale in the 2000s. At that time, soaring oil and gas prices and increasing scarcity of conventional gas in North America made the exploitation of new plays protable. An eort in research and development targeting access to unconventional hydrocarbon sources since the 1980s was supported by the U.S. government, including partnerships with vanguard companies such as Mitchell Energy. As breakthrough innovation in production technology emerged in the early 2000s, the exploitation of shale gas exploded. The improved extraction techniques combined hydraulic fracturing and horizontal drilling.

1 Such developments lead to a massive increase in the production of, rst, gas and, then, oil as from 2006 in North America. Shale gas completely ood the local market, especially after 2004, osetting the depletion of conventional natural gas. In 2017, shale gas represented 57% of total domestic gas production. This positive supply shock has led North America to be ranked as the second region having the lowest costs for energy and raw materials in the world after the Middle East (EIA (2015)).

Before the mid-2000s, natural gas prices were tightly linked to those of oil and their price dynamics were accordingly quite similar. However, from 2004 on, prices in North America deviated from the global trajectory of rising prices: leaving aside the 2005 and 2008 spikes due to exceptional climate, prices reached a plateau in 2004. Since 2008 U.S. natural gas prices fell, as a result of a combination of demand and supply factors. On the one hand, the abrupt and persistent negative demand shock followed from the subprime crisis. On the other hand, the tremendous expansion of the natural gas supply reected the shale gas boom: the U.S. natural gas production in April 2016 is 40% higher than in April 2006(EIA (2015)). This led natural gas prices in North America to a 15-years low by 2016.

Shale oil boom is also playing a role in oil price dynamics. Although the latter reects worldwide shocks due to the global nature of the oil market, the potential supply from the exploitation of shale oil became clear in the last few years. Shale oil production underpinned growth of U.S. crude oil output by over 65% in the 7 years following April 2009. [START_REF] Joskow | The shale gas revolution: introduction[END_REF] observes retrospectively that the complementary development of 1 Additional specicities of the U.S. contribute to explain the emergence of this new technology. The country hosts massive shale plays. American academic and industrial organizations host much of the expertise on geology and hydrogeology. Domestic production of conventional gas and oil attained a plateau and were expected to decline, boosting eorts to make unconventional resources competitive.

The gas market was competitive and deregulated, relying on a dense network of pipelines, a feature that reduced the risk of downstream bottlenecks for successful innovative entrants. Federal and State regulations were also relatively favorable, in particular reducing investors' responsibility over potential damages on water resources, and protecting secrecy of rm's specic hydraulic fracturing technology.

These features combined with the institutional and social context. Oil and gas production is customary in much of the production sites, reducing local political opposition to drilling activity. From the institutional point of view, some States had very liberal regulation of oil and gas production and related regulation of industrial and environmental hazards (see [START_REF] Krupnick | Heterogeneity of state shale gas regulations[END_REF]). shale oil has also been dramatic and unexpected. According to [START_REF] Baumeister | Understanding the decline in the price of oil since june 2014[END_REF] over half of the predictable drop in oil prices in the 3rd and 4th quarter of 2014 is attributable to positive supply shocks, which include information on the ongoing expansion of shale oil production in the U.S. Beyond shale oil, it is noteworthy that oil prices have been regularly subject to unexpected shocks as explained by Baumeister and Kilian (2016a). This feature, common to non-renewable resources markets, including minerals, is closely scrutinized in this article.

The regulation of production. Mineral rights in the U.S. are attached to surface property, which may be private or public. Lease contracts have developed for oil and gas exploration and production. They include an upfront payment from the oil and gas company to the land owner, a deadline for beginning production, and royalty payments to the land owner proportional to production or value of sales. A part for the deadline for beginning production, the oil and gas company chooses the quantity to be produced at each date. The management of the resource is de facto delegated from the resource owner to the oil and gas company for the duration of the lease. Similar contracts rule the exploration and production over federal lands. State regulations apply to oil and gas activity, on top of Federal and State regulation on industrial and environmental hazards.

The shale oil and gas activity is spreading to other countries, including the Canada, Argentina, China and the U.K. In several cases foreign companies are involved in exploration and production programs. These rms are often large multinational corporations (hereafter IOC for International Oil Companies). In most cases, mineral rights are exclusively in the hand of public authorities, governments are involved in agreements with IOC over the activity. Historically, concession contracts were rst used, transposing abroad the American contractual relationships whereby the IOC has to pay an upfront fee to become the residual claimant over the production. Governments seeking to improve their bargaining power developed and increasingly adopted an alternative form of contract, namely production sharing agreements (PSA). Under a PSA the resource owner keeps all the rights on the resource, which formally allows it to choose the quantity produced at each date. The government accepts to hand a share of production to the IOC, to compensate it for the cost of exploration and development (cost-oil) over the rst years of production, and then a smaller share of production to grant some protability of the operation to the IOC (prot-oil).

2 Notice that despite the formal dierences between concession contracts and PSAs, in practice most governments take part to the choices concerning the scale and rhythm of production also under concession contracts, either because their national oil company are directly involved or because contractual clauses include government approval of production plans.

When it comes to the assessment of the amount of resource in a given eld, the IOC or the oil and gas company certainly benet from superior information and knowledge with respect to governments or private landowners. As [START_REF] Osmundsen | Time Consistency in Petroleum Taxation[END_REF] argues, the superior expertise of IOC as well as the complexity of petroleum operations open the door to the strategic reporting of costs or reservoir sizes at the dierent stages of the life cycle of the petroleum eld (such as resource estimates at the licensing stage and cost reports at the production stage). Therefore, the contract established between the IOC and the 2 Scholars distinguish a third type of contract, less common, that corresponds to a delayed xed payment from the government to the IOC for exploration and development, possibly for the production operations, called service contract.

resource owner should take into account the incentives of the rm to report strategically its information in its best interest.

Model and main results. The questions we address can thus be stated as follows:

First, how does asymmetric information shape the contractual relationship between the public authority, which owns the resource, and the IOC which is in charge of its production? Second, how does a structural break in the price of the resource aect the optimal extraction path and how should the contract take into account such a break?

To tackle those issues, we consider a model of regulation in which a government delegates to a rm the extraction of non-renewable resource through a contract. We consider that the initial stock of the resource is private information of the rm and aects the marginal cost of extraction: the higher the stock of resource, the lower is the marginal cost of extraction. We bear a particular attention on how the quantity of resource extracted varies over time. In practice, the contract concerns a concession area (a so-called block), over which the IOC can decide at which speed to develop new wells, if any. This makes it possible for the rm to manage production over time. As compared to the conventional oil and gas production, shale gas is characterized by greater exibility.

3

The price of the resource is assumed to follow an exogenous dynamics. This captures the fact that the resource is sold on a world market and that the rm has no inuence on the world price, as it is the case for the oil and most mineral markets. To begin with, we consider a smooth raising price dynamics according to which, loosely speaking, the world price is increasing smoothly over time, traducing the progressive scarcity of the resource at the world level.

Our framework allows us to obtain a clear benchmark for the case of complete information. The rst best extraction path has two interesting features. First, independently of the size of the reserve, the rm is asked to follow a unique extraction path, and to sell at the beginning of the contract the quantity of resource in excess of the stock that is worth selling later in light of the anticipated price path. Second, extraction always ends up in nite time: as price increases over time, the rm extracts less and less resource up to the point where the reserve is fully exhausted. We also show that a possible implementation of the optimal regulation under complete information consists in making the rm pay a bond ex ante and in leaving the rm with the discretion to choose the extraction path. This is tantamount to a concession contract with no royalty and the maximum license fee acceptable by the IOC.

When the rm possesses private information, the extraction path must be designed in a way that is compatible with the rm's incentives to manipulate its information. The delegated management of resource extraction should share some of the general features 3 Abstracting the stochastic nature of the activity, the relative exibility in the production schedule is due to the following features: rst, for a given potential productivity of the shale play, production increases proportionally to the number and length of horizontal fracking wells; second, each single well produces most of the output over the rst 24 months; third, reserves are embedded in shale plays that extend over very large areas. Hence, a rm that wishes to scale upward (downward) its production, can durably drill more or longer (less or shorter) wells that what would be required to replace declining production from aging wells.

found with other forms of procurement.

4 The literature has highlighted that the design of optimal contracts under informational constraints results from a trade-o between eciency and rent extraction.

5 Choosing to operate at the ecient level might also necessitate to leave excessive information rents, which might be viewed as socially costly by public authorities. Reducing information rents instead calls for lower output, lowpowered incentives and more generally procurement policies that are less sensitive to information.

We use the analysis undertaken in [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF] to analyze the requirement of incentive compatibility on the extraction path. As in that paper, we assume that the government can commit to a contract. In line with our companion paper, we show that the Hotelling rule has to be modied and becomes a virtual Hotelling rule where the marginal cost of extraction is replaced by the virtual marginal cost. Hence, because the marginal costs of extraction dier now across elds with dierent reserves, the extraction paths as well as the termination dates of those elds dier as well.

We then introduce downward structural breaks in the price dynamics. Such durable reduction of the price characterize natural resource markets and result from demand shocks, such as the economic slowdown in the aftermath of the nancial crisis, or from supply shocks following, for instance, the discovery of new reserves or of more ecient extraction technologies. We consider that the price jumps downwards and then follows the same dynamics as before the break. When considering price breaks, one issue concerns the capacity to anticipate those breaks. When a price break can be fully anticipated by the regulator and the rm, then the optimal extraction path is such that production must be stopped before the break. How long production is stopped depends only on the characteristics of the price break and not on the extraction cost. Therefore, a similar pattern emerges under incomplete information.

The analysis concludes with the perhaps more convincing case where the price break cannot be perfectly anticipated by the parties. This has consequences both on the extraction path and the design of the contract. Under complete information, and quite intuitively, granting ownership of the resource to the rm with the rst best license fee is no longer feasible, for the rm would run a loss if it had paid an upfront price to become the owner which does not take into account the possibility of a price break. A better alternative is for the principal to keep ownership of the eld but use a contract where at any date the rm keeps all control rights on production but pays a rental price equal to the current prot. This contract is rst clearly immune to the possibility of an unexpected break since it does not require any upfront payment that eventually could not be covered ex post with realized prots in sharp contrast with the ownership solution. Second, with this scheme, the rm always makes the right production decision even following an unexpected price break. Such a contract resembles to a PSA, where costs are observable. Under symmetric information, PSAs allow to insure a minimum return on investment to the rm, avoiding to leave any extra surplus.

Under asymmetric information, PSAs no longer fare as well as under complete information. Indeed, the principal cannot extract the realized prot at each given date since the cost function of the rm is private information ex ante and thus cannot be specied into a contract. Here, we should throw a word of caution by noticing that, although the rm's stock is unknown ex ante, the earlier path of extraction has already revealed this parameter. Ex post, when the unexpected break arises the cost function is thus perfectly known by all parties. Yet, an arrangement where those costs are compensated by the principal cannot be written ex ante and thus cannot be enforced.

Literature review. Our paper belongs to the literature on resource extraction and incentives. [START_REF] Osmundsen | Dynamic Taxation of Non-renewable Natural Resources Under Asymmetric Information About Reserves[END_REF], [START_REF] Poudou | Concession minière et asymétrie d'information[END_REF] and [START_REF] Hung | Optimal resource extraction contract with adverse selection[END_REF] have analyzed the design of a mining concession contract in a (nite) multi-period adverse selection setting with a nite number of periods and a persistent shock on cost which is private information of the concessionaire. Reducing information rents requires downward distortions of production early in the relationship. With a nite number of periods, this implies that more resource can be extracted in a terminal phase of extraction. In a two-period model, [START_REF] Gaudet | Optimal resource royalties with unknown and temporally independent extraction cost structures[END_REF] studied optimal royalty contracts for a nonrenewable resource under asymmetric cost information with intertemporally independent cost parameters, and where the government cannot commit.

that the cost of extraction depends both on current production and on the remaining stock as in [START_REF] Solow | Extraction Costs in the Theory of Exhaustible Resources[END_REF], [START_REF] Pindyck | The Optimal Exploration and Production of Nonrenewable Resources[END_REF] and [START_REF] Pindyck | On monopoly power in extractive resource markets[END_REF], [START_REF] Swierzbinski | Exploration and Exhaustible Resources: The Microfoundations of Aggregate Models[END_REF].

Our analysis builds on a companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF]), which provides a theoretical framework addressing the question of the optimal regulation of a nonrenewable resource. There, we develop a framework where a regulator contracts with a rm for the extraction of a resource. Key to our analysis is the presence of asymmetric information between the regulator and the rm. We depart from that analysis in several important ways. First, the price of the resource, and its dynamics, are taken as exogenous. Second, we analyze the impact of structural breaks on the regulation.

The Model

Resource Dynamics. We consider the dynamic management by a rm of a stock S(θ, t) of resources over time and how this management can be optimally regulated through a concession contract designed by a public authority (referred to as the principal in the sequel). The initial stock of resource at date t = 0 is (2.1) S(θ, 0) = θ.

The resource stock evolves according to the following dierential equation

(2.2) ∂S ∂t (θ, t) = -q(θ, t),
where q(θ, t) denotes the non-negative amount of resources extracted at date t starting from an initial stock θ. For the rest of the analysis, it turns out that making this nonnegativity assumption explicit turns out to be useful

(2.3) q(θ, t) ≥ 0.
Production technology and preferences. Following the existing literature, we assume that extracting resources is all the more dicult that the remaining stock is low.

Formally, we write the cost of extracting q units when the remaining stock is S as C(S) q.

Therefore, we assume constant returns-to-scale in extraction but we also posit that the marginal cost of production C(S) is decreasing and convex in the level of that stock (C < 0 ≤ C ). For technical reasons, we assume also that this marginal cost remains nite, C(0) < +∞ and that C ≤ 0.

The revenue from exploiting q units of resource at date t is denoted by R(q, t) = p(t) q, where p(t) is the world price of resource which is xed on a world market and thus taken as given for the rest of our analysis. The dynamics of the price is here taken as exogenous.

7

In the sequel, it will be particularly useful to dene the net present value u(t) of one unit of resource at date t as u(t) = p(t) exp(-rt)

where r is the interest rate. We will assume that u(t) is once dierentiable and decreasing almost everywhere for most of this presentation, although later on, we will give specic attention to the case of a break in the price dynamics with the possibility of a discontinuous downward jump in the dynamics of u(t) (see the details below for a denition). That u(t) decreases captures the idea that, over time, suppliers may have invested in alternative technologies and thus other sources of resources may become available for consumption, reducing de facto the value of the resource under scrutiny.

As we will see below, some of the properties of optimal extraction paths depend of course on the properties imposed on u(t) which are themselves inherited from the price dynamics. To get later on a sharp writing of the Hotelling rule, we also dene the function

z(t) as z(t) = -u(t) exp(rt).
Observe that z(t) is positive whenever u(t) is decreasing.

Asymmetric Information. We will be particularly interested in the sequel in the scenario where the initial stock of resource θ may be private information to the rm.

We then suppose that this parameter is drawn from the set Θ = θ, θ according to the cumulative (atomless) distribution F (•) with (strictly positive) density f (•) = F (•).

Following the screening literature, we also assume that the so called Monotonicity of 7 In our companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF]), we fully endogenize those price dynamics.

the Hazard Rate Property holds:

8

(2.4)

d dθ 1 -F (θ) f (θ) ≤ 0 for all θ ∈ Θ.
That the rm has better information on the stock of resources than the regulator is in line with casual evidences that operators have more precise signals than outsiders (regulators, nanciers, local communities) on the value of reserves.

Smooth Raising Price Dynamics

We start with the simple case where the price dynamics is such that z(t) is continuously increasing over time. We rst analyze the case of complete information before moving to the case where the rm holds private information on the initial level of resources it may extract. Beforehand, we propose the following denition.

Definition 1. A smooth raising price dynamics is such that z(t) is continuous and everywhere increasing.

Complete Information

When θ is common knowledge, the principal can let the rm enjoy all revenues from exploiting the resource and then fully extract the rm's prots through a lump-sum upfront payment equal to the discounted value of those prots. The only important question is to determine the optimal path of resource extraction. The next proposition describes carefully the optimal dynamics of extraction. We denote respectively by q * (θ, t) and S * (θ, t) the optimal quantity extracted at a time t and the optimal remaining stock of resources at that date. This optimal path maximizes the intertemporal discounted prot subject to the constraints imposed by resource dynamics, that is

(P * (θ)) : max q,S,T T 0 (p(t) -C(S(θ, t))) q(θ, t)exp(-rt)dt
subject to (2.1), (2.2) and (2.3).

The maximand is taken on the resource trajectories with an extra optimization variable being a priori the end-point date T at which extraction ceases. In the above constrained set, we make explicit the non-negativity constraint on the amount of resource extracted at any point in time. This solution is easily characterized if we further impose the following assumption:

Assumption 1.

z(0) > rC(θ) and lim t→+∞ z(t) < rC(0)

As the analysis below will demonstrate, the condition z(0) > rC(θ) ensures that extraction is valuable at date t = 0 even for the lowest possible level of initial stock. Because C(S) is decreasing, it follows that z(0) ≥ rC(θ) for all θ ∈ Θ. The second inequality guarantees that complete extraction takes place in nite time.

8 [START_REF] Bagnoli | Log-concave probability and its applications[END_REF].

Proposition 1. Suppose that Assumption 1 holds and consider a smooth raising price dynamics (i.e., ż(t) > 0 for all t). The optimal path of resource extraction under complete information (S * (t), q * (t)) follows the same standard Hotelling Rule whatever the initial level of resource θ. This path is dened as:

(3.1) rC(S * (t)) = z(t) ∀t ∈ (0, T * ) and (3.2) q * (t) = - ż(t) rC (S * (t)) > 0 ∀t ∈ (0, T * )
where T * , which is the nite date at which all elds are fully extracted, is dened as

(3.3) rC(0) = z(T * ).
This proposition showcases two important features. First, Assumption 1 ensures that at date t = 0, all elds start extraction and do so maybe with a large output extraction at that date so that, right after that date, all elds jump on the same path of extraction (S * (t), q * (t)). That all trajectories follow the same path of extraction even though they may start at dierent levels of resource at date t = 0 is rather intuitive. After some initial period of extraction moving an initial stock θ to a lower value θ , the continuation path must obviously be the same as if the stock had started from this lower value. This just follows from the Bellman Principle for optimization of a time-consistent objective.

This common path follows the usual Hotelling rule. Indeed, at the optimum, the rm (or alternatively the principal, remember that the former can be made residual claimant by the latter) must be just indierent at any date t between consuming some extra amount dq today or delaying for an extra length of time that resource in the ground to facilitate extraction later on. Producing and consuming dq more units beyond q * (t) for a length of time dt around date t increases overall prot at date t by approximately

(3.4) (p(t) -C(S * (t))) dqdt.
However, consuming those extra units dq reduces the stock of resources from date t+dt onward to approximately S * (t)-q * (t)dt+dqdt. Future costs of extraction are thus increased from date t + dt onwards by an amount whose discounted value is worth approximately

(3.5) - +∞ t C (S * (τ ))q * (τ )exp(-r(τ -t))dτ dqdt.
At the optimal path, (3.4) and (3.5) must be equal so as to leave the rm indierent between extracting that extra unit or not. This indierence can be written as:

p(t) -C(S * (t)) = - +∞ t C (S * (τ ))q * (τ )exp(-r(τ -t))dτ .
Dierentiating with respect to t and simplifying nally gives us the Hotelling Rule (3.1).

This rule can be reinterpreted as viewing z(t)/r as the current discounted value of extracting one more unit of output at date t. Condition (3.1) thus equals the marginal benet of extracting one more unit with its costs.

The second important feature of the solution is that extraction always ends up in nite time when the rm faces a smooth raising price dynamics. As price increases (more precisely, as z(t) increases) over time, the rm extracts less and less of the resource up to the point of exhausting all reserves in nite time T * .

Implementation. As suggested above, a possible implementation of the optimal contract under complete information consists in leaving all revenues to the rm and extract a lump-sum payment equal to the net present value of its prots. Being residual claimant for optimal resource management, the rm chooses the optimal extracting path. This contractual solution is akin to leaving full ownership to the contractor at a type-dependent price P * (θ) such that:

P * (θ) ≡ T * 0 p(t) -C θ - t 0 + q * (τ )dτ
q * (t)exp(-rt)dt.

Asymmetric Information

We now turn to the case of asymmetric information, still assuming that the price dynamics remains smooth. The analysis in this section borrows from our companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF] and we refer the reader to that paper for details of the analysis.

Incentive Compatibility. A long-term regulatory contract is written so as to regulate the relationship between the public authority and the rm. This contract stipulates how much quantity should be extracted and possibly a payment for such service at any point in time. From the Revelation Principle, 9 we can without loss of generality view such contract as a direct and truthful revelation mechanism that stipulates the output q( θ, t) and the payment ω( θ, t) proles over time as a function of the rm's announcement θ on the resource stock it starts with. Because we assume that the principal commits to the mechanism, this announcement takes place once for all at date 0 -. 10

Henceforth, we may denote by U (θ) the rm's informational rent (or intertemporal payo ) when adopting such optimal truthful announcement strategy as

(3.6) U (θ) = +∞ 0 (ω(θ, t) -C(S(θ, t))q(θ, t))exp(-rt)dt
where the trajectory S(θ, t) is dened through (2.1) and (2.2).

Incentive compatibility requires that this payo is greater than what the rm with type θ may obtain by adopting the extraction patterns of a type θ while still starting from the initial stock θ. Incentive compatibility can thus be written as

(3.7) U (θ) = max θ∈Θ +∞ 0 ω( θ, t) -C θ - t 0
q( θ, τ )dτ q( θ, t) exp(-rt)dt.

9 [START_REF] Myerson | Optimal coordination mechanisms in generalized principal-agent problems[END_REF].

10 Observe also that there is no loss of generality in making the convention of an innite planning horizon for the rm because the contract can just stipulates zero output and zero payment from a given date on if extraction takes place over a nite period of time.

A contract ((ω( θ, t), q( θ, t)) t≥0 ) θ∈Θ thus induces an allocation (U (θ), (q(θ, t)) t≥0 ) θ∈Θ .

Next Lemma, which is directly taken from our companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF], characterizes incentive compatible allocations, i.e., allocations that can be implemented by a particular contract. We postpone below the analysis of contracts that implement the optimal allocation. Lemma 1. An allocation (U (θ), (q(θ, t)) t≥0 ) θ∈Θ is incentive compatible if and only if:

1. U (θ) is absolutely continuous and thus a.e. dierentiable with at any point of differentiability θ:

(3.8)

U (θ) = - +∞ 0 C (S(θ, t))q(θ, t))exp(-rt)dt;
2. The amounts extracted satises the following conditions

(3.9) +∞ 0 θ θ Q( θ,t) Q( θ,t) C ( θ -Q)d Q d θ exp(-rt)dt ≥ 0 ∀(θ, θ) ∈ Θ 2 .
In particular, (3.9) holds if the following sucient condition is satised

(3.10) Q(θ, t) non-decreasing in θ.
Proof. See [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF].

The Envelope Condition (3.8), a standard condition for incentive compatibility repeatedly found in the mechanism design literature, 11 shows that the information rent of the rm is non-decreasing in its initial stock of resources. Hence, any type of rm accepts the contract if the rm with the lowest reserve θ already does so. We will thus impose on top of the incentive compatibility conditions (3.8) and (3.10) the following participation constraint (3.11) U (θ) ≥ 0.

Together incentive and participation constraints dene the incentive-feasible set.

Virtual Hotelling Rule. The principal's objective is to maximize his expected intertemporal payo dened as (3.12)

E θ +∞ 0 (p(t)q(θ, t) -ω(θ, t))exp(-rt)dt
subject to the equations (2.1), (2.2) and (2.3) that describe how the stock of resource evolves on path and subject to the rm's incentive and participation constraints (3.8),

(3.9) and (3.11).

Taken together (3.8) and (3.11) (which is clearly binding at the optimum since the principal wants to reduce as much as possible the rm's payments) allows us to rewrite 11 [START_REF] Laont | The theory of incentives: the principal-agent model[END_REF].

a rm θ's information rent as

U (θ) = - θ θ +∞ 0 C (S( θ, t))q( θ, t))exp(-rt)dt d θ (3.13) = +∞ 0 ω(θ, t) -C θ - t 0
q(θ, τ )dτ q(θ, t) exp(-rt)dt.

From this, we may derive the value of the discounted payment +∞ 0 ω(θ, t)exp(-rt)dt made to the rm, and insert into the principal's objective function (3.12) to obtain the new expression of this objective as

E θ +∞ 0 (p(t)q(θ, t) -C(S(θ, t)q(θ, t))exp(-rt)dt -U (θ) .
Further integrating by parts using (3.13) yields a last rewriting of this objective

E θ +∞ 0
(p(t)q(θ, t) -C(θ, S(θ, t)q(θ, t))exp(-rt)dt .

The above maximand showcases that, under asymmetric information, the marginal cost of extraction is now replaced by the virtual marginal cost of extraction that is now dened as

(3.14) C(θ, S) = C(S) - 1 -F (θ) f (θ) C (S) ∀(S, θ).
This expression accounts for the extra cost of information rent that has to be given to the rm to induce information revelation on the value of the stock. As a result, the virtual marginal cost of extraction is always above the true marginal cost of extraction C(θ, S) ≥ C(S) ∀(S, θ)

with an equality only for the highest possible type θ.

The virtual marginal cost of extraction is increasing and convex in S under the assumptions made earlier on for C. Importantly, the marginal cost of extraction can also be ranked with respect to the initial stock θ since

(3.15) ∂ C ∂θ (θ, S) ≤ 0 ∀(S, θ).
A type with a higher initial stock has thus a lower virtual marginal cost of extraction and is more willing to extract, at any level of stock along the trajectory.

The analysis made in our companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF] carries over in the present environment mutatis mutandis. In particular, the solution to this problem is obtained by solving separately a collection of maximization problems for each possible realization of θ. Each of those problems can then be written respectively as (P sb (θ)) : max q,S,T T 0 (p(t)q(θ, t) -C(θ, S(θ, t))q(θ, t))exp(-rt)dt subject to (2.1), (2.2) and (2.3) .

The cost of extraction at any given date now depends not only on the current stock but also on the initial stock. Hence, rms become heterogeneous in terms of their costs of extraction while under symmetric information they all had the same technology and just diered by their initial stocks. Such heterogeneity will imply that the optimal paths of extraction under asymmetric information are now all dierent and depend on the initial stock of resources.

It is straightforward to extend mutatis mutandis the analysis undertaken under complete information to this new scenario. The next proposition does so, stressing that now rms with dierent initial stocks follow dierent paths of extraction. Before that, we impose a new assumption that is the counterpart of Assumption 1 under asymmetric information.

Assumption 2.

z(0) > r C(θ, θ) and lim t→+∞ z(t) < r C(θ, 0).

The monotonicity of the hazard rate (2.4) together with the monotonicity properties of C and C imply that C(θ, θ) is a decreasing function of θ. In other words, the rst inequality implies that all rms, whatever their initial stock, start to produce and actually face a downward jump of their remaining stock at date 0 + since z(0) > r C(θ, θ) for all θ ∈ Θ. After this jump, the extraction path follows the smooth path described in Proposition 2 below. The second inequality also holds at all θ thanks again to the monotonicity properties of C, C and C . This condition implies that extraction always ends in nite time whatever the initial level of resource.

We are now ready to describe optimal extraction paths under asymmetric information.

Proposition 2. Suppose that Assumption 2 holds and consider a smooth raising price dynamics (i.e., ż(t) > 0 for all t), the optimal path of resource extraction (S sb (θ, t), q sb (θ, t)) under asymmetric information evolves according to a virtual Hotelling rule that depends on the initial level of resource θ. This path is dened as follows:

(3.16) r C(θ, S sb (θ, t)) = z(t) ∀t ∈ (0, T * (θ))
and

(3.17)

q sb (θ, t) = - ż(t) r ∂ C ∂S (θ, S sb (θ, t)) > 0 ∀t ∈ (0, T * (θ))
where T * (θ), which is the nite date at which a eld with initial reserves θ is fully extracted, is dened as

(3.18) r C(θ, 0) = z(T * (θ)).
Proposition 2 is much alike Proposition 1. The dierence is that the costs of extraction are now replaced by the virtual costs of extraction. Because those costs dier across elds with dierent reserves, the extraction paths of those elds dier as well.

For a given initial stock of resource, the dynamics looks quite similar to the case of complete information. To understand this eect, it is important to come back on (3.8). This formula shows that a type-θ rm enjoys some information rent by extracting the same amount as a rm with a lower initial stock but doing so at a lower marginal cost. This rent is reduced when the regulator calls for lower quantities. Asking for less extraction by a given type θ indeed reduces the information rent of all supramarginal types θ ≥ θ. In other words, doing as if more of the resource had already been exploited by a given type and pushing forward the extraction path for that type helps to extract rent from types endowed with more resource. The second-best path of extraction exhibits a trade-o between implementing ambitious extraction paths that come close to eciency and, to do so, leaving excessive information rent to the rm. Extracting more of this information rent calls for extracting the resource at a slower pace.

Next Lemma illustrates this insight by looking at the date at which dierent elds are exhausted.

Lemma 2. T * (θ) is an increasing function of θ with T (θ) = T * .

While under complete information extraction ends at the same date for all elds, this is no longer the case under asymmetric information. Fields with larger initial stocks extract during longer periods.

Implementation. We now address the question of what sort of instruments can be used to achieve the second-best patterns highlighted in Proposition 2. In a dynamic environment, there are many possible ways to structure incentive payments over time to provide the needed information rent. In our companion paper [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF],

we discuss how a royalty scheme can be implemented based on the rm's prediction on the actual extraction path it will follow. In that implementation, the operator is oered a menu of options involving dierent royalties and license fees, among which it chooses right at the start of extraction. This choice commits the rm on an extraction path.

For the purpose of the analysis undertaken in Section 4.2 below, we now propose a simple dynamic implementation where choices are repeated over time although those choices do not convey more information to the principal, i.e., all the operator's private information ends up being already revealed right at the start of the extraction phase.

Our procedure takes advantage of the dynamic structure of the rm's information rent.

Indeed, observe that we can write the intertemporal rent of the rm as a stream of per-period rents ũ(θ, t)

U (θ) = +∞ 0 ũ(θ, t)exp(-rt)dt
where the type-θ rent at date t ũ(θ, t) is

(3.19) ũ(θ, t) = - θ θ C (S( θ, t))q( θ, t)d θ.
This formulation suggests that we may implement those per-period rents with a sequence of nonlinear prices (Ω(Q, q, t)) t≥0 that determines the rm's payment at date t as a function of the past (observed) overall quantity that has been extracted Q, the current level of extraction q and calendar time t. Formally, let dene

(3.20) û(θ, Q, t) = max q≥0 Ω(Q, q, t) -C (θ -Q) q.
Assuming quasi-concavity in q of the rm's objective, the optimal quantity q(θ, Q, t) chosen at date t by a rm with initial stock θ and having already extracted an overall quantity Q can be written as

(3.21) ∂Ω ∂q (Q, q(θ, Q, t), t) = C (θ -Q) .
Condition (3.17) can be rewritten, at any interior point on the optimal trajectory, as

q sb (θ, t) = - ż(t) r C (θ -Q sb (θ, t)) -1-F (θ) f (θ) C (θ -Q sb (θ, t)) ∀t ∈ (0, T * (θ))
where Q sb (θ, t) = t 0 q sb (θ, τ )dτ is the second-best prole of cumulative extraction. We generalize this condition o-path, i.e., for any cumulative levels of extraction Q that could be dierent from Q sb (θ, t) = θ -S sb (θ, t) so as to get

q(θ, Q, t) = - ż(t) r C (θ -Q) -1-F (θ) f (θ) C (θ -Q) ∀t ∈ (0, T * (θ))
Provided that C ≤ 0 and thanks to the monotone hazard rate property, this condition can be inverted to get a function Θ(q, Q, t) that associates to each possible level of current production q and each possible level of cumulative extraction that the type-θ reaches at date t. Coming back on (3.21), we can thus write

(3.22) ∂Ω ∂q (Q, q, t) = C Θ(q, Q, t) -Q ∀(q, Q, t)
which can be integrated to get the shape of the nonlinear payment schedule Ω(Q, q, t) as

Ω(Q, q, t) = q 0 C Θ(q, Q, t) -Q dq + Ψ(Q, t) ∀(Q, t)
where Ψ(Q, t) is some arbitrary function of (Q, t) that is dened at the boundary condition by the zero-prot requirement for the θ type, namely ũ(θ, t) = 0 or

Ψ(Q sb (θ, t), t) = - q sb (θ,t) 0 C Θ(q, Q sb (θ, t), t) -Q sb (θ, t) dq+C θ -Q sb (θ, t) q sb (θ, t).
A standard application of the Envelope Theorem plus the imposed requirement that û(θ, Q, t) = 0 gives us

(3.23) û(θ, Q, t) = - θ θ C θ -Q q( θ, Q, t)d θ.
We thus have (3.24) ũ(θ, t) ≡ û(θ, Q(θ, t), t).

At any interior point on the optimal trajectory, the optimality condition (3.21) be-

comes ∂Ω ∂q (Q sb (θ, t), q sb (θ, t), t) = C S sb (θ, t) = z(t) r + 1 -F (θ) f (θ) C S sb (θ, t) < z(t) r .
This formula shows that, at the margin, the rm chooses to lower current extraction below the complete information level as required by the optimal trajectory under asymmetric information. In other words, this implementation is akin to using a time-dependent royalty

R(θ, t) = - 1 -F (θ) f (θ) C S sb (θ, t) ,
which, for any given type θ, increases over time independently of the resource price z(t).

We end this paragraph by noticing that leaving ownership to the rm is no longer optimal. Although it allows an ecient extraction of resource and all types whatever their initial stock of resources choose to move to that path at date 0 + , it also leaves too much rent to the rm. Indeed, the highest price that would be paid by the rm for getting ownership would be

P * (θ) ≡ T * 0 p(t) -C θ - t 0 + q * (τ )dτ q * (t)exp(-rt)dt.
Requesting such payment leaves extra benets to a type-θ rm

P * (θ) -P * (θ) ≡ T * 0 C θ - t 0 + q * (τ )dτ -C θ - t 0 +
q * (τ )dτ q * (t)exp(-rt)dt.

Price Breaks

We now analyze how contracts should optimally be designed to react to the possibility of a price break. We have here in mind the possibility that the world market be modied by the discoveries of new sources of resources which change discontinuously how the value of one extra unit of extracted resource changes. To make such statement more formal, we propose the following denition Definition 2. A raising price dynamics exhibits a price break at date T if and only if there exists a smooth raising price dynamics z(t) such that (4.1)

z(t) = z(t) if t ≤ T , z(t -∆) if t > T .
This denition of a price break bears on the properties of z(t). It is quite natural since only z(t) matters to characterize the Hotelling rule in our previous analysis. We require that z(t), which represents how the net present value of extraction evolves over time, has a downward jump at some date

T ∆z ≡ z( T ) -z( T + ) = z( T ) -z( T -∆).
We will assume in the sequel (and make this requirement implicit) that this jump is small enough so as to z(t) remains everywhere positive and that T + ∆ < T * ; the latter requirement will make it sure that the no extraction area (soon to be dened) ends before T * . Beyond the break at T , the net present value of extraction goes back to a dynamics that is similar to what was on course before the break but which is now delayed by ∆.

This denition is actually consistent with dierent patterns of price dynamics. The price prole p(t) might be a continuous function of t with a discontinuous jump at T ; or only its derivative may have such jump. For technical reasons, we will favor the rst possibility in the sequel, since the optimization problems remain then with continuous objectives, thereby avoiding the use of non-smooth calculus.

12 In any case, and although the shape of the z(t) proles that we consider look specic, the analysis we develop below could be readily extended to more general patterns. The key assumption is that z(t) is not everywhere increasing.

Implicit in our analysis of Section 3.1 and especially in the Hotelling rule (3.1) was the fact that when z(t) is continuously strictly increasing, extraction is always positive at any point in time before exhaustion of the stock of resource. The possibility that z(t) may decrease opens however the possibility that it becomes preferable to suspend extraction because the current value of doing so may fall short of the current stock.

Of course, the analysis changes depending on whether those breaks can be anticipated by parties to the contract or are just unexpected. We devote a separate section to each case.

Expected Breaks

Consider rst the scenario where the break can be anticipated. Next proposition shows how parties should design the optimal extraction path under those circumstances. Proposition 3. Suppose that Assumption 1 holds and consider a price dynamics with a break at T . The optimal path of resource extraction under complete information (S * b (t), q * b (t))

follows the same standard Hotelling Rule whatever the initial level of resource θ. This path is dened as follows:

(4.2) S * b (t) = S * (t) and q * b (t) = q * (t) ≥ 0 ∀t ∈ (0, t 1 ] ∪ [t 2 + ∆, T * ); (4.3) S * b (t) = S 0 and q * b (t) = 0 ∀t ∈ [t 1 , t 2 ] where (4.4) rC(S * (t)) = z(t) ∀t ∈ (0, t 1 ] ∪ [t 2 + ∆, T * ); (4.5) S 0 = S * (t 1 ) = S * (t 2 ); (4.6) t 2 -t 1 = ∆; and t 1 < T solves (4.7) u(t 1 + ∆) = u(t 1 ) + u(t 1 ) 1 -exp(-r∆) r .
The nal date T * b = T * + ∆ > T at which extraction stops solves: This proposition characterizes how the possibility of a price break can be managed when it is fully anticipated. The rm should stop production at a date t 1 that arises before the value of current extraction starts declining at T so as to start again production sooner in the future after a length of time worth ∆. Of course, in practice, it may be hard to disrupt production at all and the insights provided here is only indicative that production should be carefully slowed down.

What is remarkable is that the bunching area [t 1 , t 2 ] only depends of characteristics of the price dynamics. Costs play no role whatsoever. The immediate consequence of this nding is that the same pattern is found also under asymmetric information.

Proposition 4. Suppose that Assumption 2 holds and consider a price dynamics with a break at T . The optimal path of resource extraction under asymmetric information (S sb b (θ, t), q sb b (θ, t)) follows a Virtual Hotelling Rule that depends on the initial level of stock θ. This path is dened as follows:

(4.9)

S sb b (θ, t) = S sb 0 (θ) and q sb b (θ, t) = q sb (θ, t) ≥ 0 ∀t ∈ (0, t 1 ] ∪ [t 2 + ∆, T * ); (4.10) S sb b (θ, t) = S sb 0 (θ) and q sb b (θ, t) = 0 ∀t ∈ [t 1 , t 2 ]
where (4.11)

S sb 0 (θ) = S sb (θ, t 1 ) = S sb (θ, t 2 );
the boundaries of the no-production area t 2 and t 1 satisfy (A.17 The remarkable nding is that all types of rm, whatever the level of their initial stocks, stop extracting at the same date and for the same duration anticipating the price break; although, and in sharp contrast with complete information, the level of remaining stocks S sb 0 (θ) at which they do so dier.

Unexpected Price Breaks and the Costs and Benets of Concession Contracts

The possibility of a price break might be hard to anticipate for the parties. This has consequences both on the extraction path and on contract design.

Complete Information. Consider rst the case of complete information. Reaching date T , the value of extracting one more unit of output z( T + )/r discontinuously falls below its cost C(S * ( T )) and thus extraction should cease, waiting for a duration ∆ before which z( T + ∆) reaches again the value z( T -) so as to start extraction again.

The important point to notice is what sort of contracts or ownership structure allows to reach this outcome. Clearly, granting ownership to the rm is no longer feasible.

Indeed, if the rm has paid upfront a price P * (θ) to become an owner it would run a loss once the price break arises since from date T on the price becomes so low that production is no longer optimal and the optimal production plan is to depart from the (wrongly) expected output q * (t). Instead of reaching an expected prot +∞ T p(t) -C θ -t 0 + q * (τ )dτ q * (t)exp(-rt)dt

where p(t) and q * (t) are respectively the price and the optimal quantity associated to the (wrongly) expected dynamics z(t), the rm gets only the lower prot q * b (τ -∆)dτ q * b (t)exp(-rt)dt from date T on.

A better alternative for the principal is thus to keep ownership of the eld but use a concession contracts where the rm enjoys all revenues from managing the resource.

This solution also leaves to the rm all control rights on production, but it requests, at any possible date, a rental price equal to the current prot (p(t) -C (θ -Q(t)))q(t)

whatever the current extraction q(t) and the cumulative past extraction Q(t). This scheme is rst clearly immune to the possibility of an unexpected break since it does not require any upfront payment that eventually could not be covered ex post with realized prots in sharp contrast with the ownership solution. Second, with this scheme, the rm always makes the right production decision even following an unexpected price break.

Such a regulation corresponds to a PSA whereby the principal reimburses the cost, and leaves the responsibility of managing production to the rm. The contract can be signed ex ante, since the cost is known under full information.

Asymmetric Information. Asymmetric information makes things much more dicult. First, relinquishing control rights on production either through a concession contract or through ownership to the rm still allows a better adaptation of decisions to the unexpected price break. The rm still chooses to eschew production when its current return is too low. Unfortunately, at any point in time, the rm would be producing too much with those arrangements compared to the optimal second-best solution that, as we saw earlier, slows down extraction. Having the rm choosing freely its output ex post is bad from a rent extraction point of view. Second, granting ownership to the rm raises the same diculty as under complete information, i.e., the price paid upfront would not be recouped with future prots if some unexpected break were to arise. Hence, rm's ownership remains quite unattractive.

Concession contracts have their own curse as well and they no longer fare as well as under complete information. Indeed, and in sharp contrast with the case of complete information, the principal cannot commit to a rental price equal to the realized prot at each given date since the cost function of the rm is private information ex ante and thus cannot be specied into a contract. Here, we should throw a word of caution by noticing that, although the rm's stock is unknown ex ante, the earlier path of extraction has already revealed this parameter. Ex post, when the unexpected break arises at date T the cost function is thus perfectly known by all parties. Yet, an arrangement where those costs are compensated by the principal cannot be written ex ante and thus cannot be enforced. Technically, everything happens as if the stock θ was observable ex post although not contractible ex ante.

One possibility consistent with the information structure is for the principal to ask at any point in time for (p(t) -C (θ -Q(t)))q(t)

where the current extraction the cumulative past extraction are respectively q(t) and Q(t). Of course, this scheme fares badly in terms of extracting rent whenever q(t) is chosen to be positive but it preserves the rm from being asked to pay when the price has fallen too far to warrant production.

Conclusion

We develop a model in which a rm extracts a resource, has private information about the initial level of resource, and is regulated. We characterize the optimal contract and extraction paths under two dierent scenarios about the dynamics of the resource price:

rst, when the resource price increases smoothly across time; second, when the resource price can be aected by a price break. We discuss how the anticipation of the price break, or the impossibility to anticipate it, aects the optimal extraction path. Our analysis could be further extended to better understand how various elements of organizational design, such as control rights in unforeseen contingencies, may be adequately used in these contexts.

  ) and (4.7) and the nal date of extraction T * b (θ) is equal to T * (θ) + ∆ where T * (θ) still satises (3.18).

See Laont and Tirole (1993) and[START_REF] Armstrong | Recent developments in the theory of regulation[END_REF] for accounts of the so-called New Regulatory Economics.

[START_REF] Laont | The theory of incentives: the principal-agent model[END_REF].

A common nding of these papers is that asymmetric information shifts production to the future. We dier from these papers along several lines. First, our model has continuous time and innite horizon in order not to constrain the resource patterns. This means that all eects that stem from the existence of a terminal date disappear in our context. Second, we assume

[START_REF] Castonguay | Resource agency relationship with privately known exploration and extraction costs[END_REF] extend the latter analysis to consider a preliminary stage where reserves are developed.

It would be straightforward to extend the analysis to this case with just a little bit of extra burden; see[START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] for instance.

CEPREMAP, Paris for comments related to a companion paper. This project has beneted from the nancial support of CEPREMAP. All

Appendix A

Proof of Proposition 1. Problem (P * (θ)) is a maximization problem with free-end point to which we apply Pontryagin Principle:

(P * (θ)) : max T,q,S T 0 (p(t) -C(S(θ, t)) q(θ, t)exp(-rt)dt subject to (2.1) and (2.2).

We write the Hamiltonian for problem (P * (θ)) as:

H(S, q, t, λ) = (p(t) -C(S))qexp(-rt) -λq where λ is the costate variable for (2.2). This Hamiltonian is concave in (q, S). The necessary conditions for optimality are thus also sucient conditions of the Mangasarian type. 13 We write those conditions as follows:

Costate variable:

which amounts to (A.1)

Control variable: q * (θ, t) ∈ arg max q H(S * (θ, t), q, t, λ(t))

which amounts to

Free-end point and transversality condition:

1. If T * (θ) < +∞, we should have:

2. If T * (θ) = +∞, we should have:

Several facts follow from those optimality conditions.

Observe that dierentiating (A.2) with respect to t when q * (θ, t) > 0 yields:

Using (A.1) and (A.2) and simplifying yields:

Now observe that

Hence, (A.6) can be rewritten as the Hotelling rule (3.1). This condition fully determine the common path S * (t) for all t > 0 and all initial level S * (θ, 0) = θ. At t = 0 + , the optimal path jumps downward from θ to S * (0) which is lower from Assumption 1 and which is dened as:

Dierentiating with respect to t for t ∈ (0, T * ) gives us the common level of extraction q * (t)

given by (3.2). For t > T * , S * (t) = q * (t) = 0. Finally, the nal date of extraction

for all θ is such that S * (t) is positive on a left-neighborhood. It is of course given by (3.2).

Running Example. Observe that we may rewrite (3.2)

.

where T * is dened by (3.3).

Proof of Lemma 1. See [START_REF] Martimort | Extracting information or resource? The Hotelling rule revisited under asymmetric information[END_REF].

Proof of Lemma 2. Dierentiating (3.18) with respect to θ gives:

where the last inequality follows from (3.15). From this and the fact that ż(t) > 0, we deduce that Ṫ * (θ) > 0.

Proof of Proposition 2. This is the almost same proof as for Proposition 1 mutatis mutandis provided that C(S) and S * are respectively replaced by C(θ, S) and S(θ). Details are thus omitted. The only minor dierence comes from Step 3 which has to be rewritten with some care. Indeed, for the optimal path of resource, we may again rewrite (2.2) as (A.9)

Replacing with C(θ, S), Equation (A.10) becomes:

(A.10)

The system made of (A.9) and (A.10) admits a limit, which is necessarily the unique stationary point ( S(θ), 0). If convergence towards that point occurs in nite time T sb (θ) < +∞, we demonstrate that output is necessarily null along the whole trajectory. Indeed, by Cauchy-Lipschitz Theorem, there is a unique solution to the system now made of (A.9) and (A.10) with the initial condition (S sb (θ, T sb (θ)), q sb (θ, T sb (θ))) = ( S(θ), 0) and this is (S sb (θ, t), q sb (θ, t)) = ( S(θ), 0) itself. But this contradicts the initial condition S sb (θ, 0) = θ θ if θ > S(θ). Hence, the system cannot converge in nite time towards ( S(θ), 0) and q sb (θ, t) remains positive. Finally, gathering everything the system (A.9)-(A.10) converges towards ( S(θ), 0) in innite time when θ > S(θ).

Suppose instead that θ ≤ S(θ), or equivalently (because C(θ, S) is decreasing in S)

Then extraction is not even valuable for the rst unit and zero extraction takes place for those lower values of θ.

Proof of Proposition 3. We now make explicit the non-negativity constraint on output (2.3) and denote by µ the corresponding multiplier. We write the Hamiltonian and the Lagrangean for the corresponding optimization problem respectively as:

Proceeding as in the proof of Proposition 1, and looking for solutions such that µ(t) is continuously dierentiable, we get the following condition:

(A.11)

or (A.12)

We look for an interval [t 1 , t 2 ] where (2.3) is binding. On the complementary set ∈ (0, t * ] ∪ [t * + ∆, T * ), we have µ(t) = 0. The solution has the same features as in (4.2) and it does not depend on θ so that we omit this variable for brevity and write

where (A.14)

The optimality conditions requires that µ(t) is continuous at dates t 1 and t 2 which gives us:

Now, observe that (A.12) implies:

(A.15) We deduce from (A.13) and (4.1) that necessarily (A.17) holds.

(A.17)

Inserting into (A.16) and simplifying then yields (A.17).

Proof of Proposition 4. The proof borrows heavily from the proofs of Propositions 2 and 3 and is thus omitted.


Kalliopi Violaki
email: kalliopi.violaki@mio.osupytheas.fr

François Bourrin

Dominique Aubert

Giorgos Kouvarakis

Nicole Delsaut

Nikos Mihalopoulos

Organic phosphorus in atmospheric deposition over the Mediterranean Sea: An important missing piece of the phosphorus cycle

To obtain a comprehensive picture of the spatial distribution of atmospheric phosphorus (P) speciation in the Mediterranean Sea, deposition samples were collected from the eastern (E) and north-western (NW) Mediterranean and analyzed for both organic and inorganic P forms. Dissolved P, especially in organic form, is dominant during the wet season, while higher insoluble P fluxes are recorded during the dry period, mainly due to dust outbreaks. Wet deposition has a significant contribution to the atmospheric phosphorus flux; over the NW Mediterranean accounting for 80% of the total dissolved phosphorus (TDP). The seven-year average TDP flux during the wet season in the NW basin, of 1.78 mmol m -2 y -1 , is almost double that of the two-year average in the eastern Mediterranean. The difference is attributed mainly to the precipitation height being 2-3 times higher in the NW Mediterranean compared to the East. Dissolved organic phosphorus (DOP) is the dominant P fraction in both wet and dry deposition samples, presenting similar fluxes (1.16 and 0.94 mmol m -2 y -1 , in the eastern and north-western basins respectively). Finally, assuming that all TDP is bioavailable, atmospheric deposition of TDP could account for up to 38% of new production in the eastern Mediterranean, while during oligotrophic periods of the NW Mediterranean it could increase the new production by 14%.

Introduction

Phosphorus is an essential micronutrient for living organisms; it is a part of DNA and RNA molecules and of lipids in cell membranes. Phosphorus can be found on the earth in water, soil, sediments and the atmosphere in both dissolved and particulate pools, in organic and inorganic forms. Phosphorus in the atmosphere is predominantly associated with particulate phase [START_REF] Vet | A global assessment of precipitation chemistry and deposition of sulfur, nitrogen, sea salt, base cations, organic acids, acidity and pH, and phosphorus[END_REF]; however, the presence of gaseous phosphorous compounds such as phosphine and methylphosphine has also been reported in marine and terrestrial atmosphere [START_REF] Zhu | Phosphine in the marine atmosphere along a hemispheric course from China to Antarctica[END_REF][START_REF] Han | Production and emission of phosphine gas from wetland ecosystems[END_REF], while the contributions from gaseous organophosphate pesticides [START_REF] Degrendele | Pesticides in the atmosphere: a comparison of gasparticle partitioning and particle size distribution of legacy and current-use pesticides[END_REF] and gaseous organophosphate ester-flame retardants (OPE) [START_REF] Castro-Jiménez | Organophosphate ester flame retardants and plasticizers in the global oceanic atmosphere[END_REF] are not negligible. Inorganic P species (e.g. mono-or diprotonated orthophosphate) are considered the most bioavailable P forms, and the emphasis for many decades has been on the study of their abundance and dynamics. On the other hand, information regarding the chemical speciation of atmospheric organic phosphorus and its role in the biogeochemical cycle of P in the atmosphere is completely unknown.

In seawater, inorganic phosphorus consists mostly of orthophosphate (PO 4 3-and HPO 4 2-), pyrophosphate (H 2 P 2 O 7 2-

) and other condensed inorganic P compounds, whereas organic phosphorus compounds include mainly nucleic acids, phospholipids, inositol phosphates, phosphoamides, phosphoproteins, phosphonic acids, organophosphate pesticides, humic-associated organic phosphorus compounds, and organic condensed phosphates in dissolved, colloidal and particle-associated forms (Karl and [START_REF] David | Dynamics of Dissolved Organic Phosphorus[END_REF]. Despite the acknowledged abundance of the organic P fraction in aquatic systems, the importance of this fraction in the atmosphere is not widely recognized.

Aeolian transport of particulate matter from the continents to the open ocean is considered a major pathway for supply of various trace elements, contaminants and nutrients. Strong Sahara events can bring large amounts of particulate matter and associated nutrients (phosphorus and iron) over the Mediterranean Sea [START_REF] Group | Marine ecosystems' responses to climatic and anthropogenic forcings in the Mediterranean[END_REF][START_REF] Richon | Modeling the impacts of atmospheric deposition of nitrogen and desert dust-derived phosphorus on nutrients and biological budgets of the Mediterranean Sea[END_REF]. Phosphorus is the limiting factor of biological activity in many oligotrophic marine ecosystems, including the eastern Mediterranean Sea [START_REF] Krom | Nutrient budget for the Eastern Mediterranean: implications for P limitation[END_REF]. It is well known that the atmosphere is an important nutrient path, especially for the Levantine basin in the E. Mediterranean and moreover is the dominant nutrient path for PO 4 3-compared to the riverine fluxes [START_REF] Koçak | Atmospheric nutrient inputs to the northern Levantine basin from a long-term observation: sources and comparison with riverine inputs[END_REF].

The atmospheric deposition of organic P may further influence the primary productivity of the euphotic zone in the Mediterranean, especially in phosphate-limited waters, leading to a greater utilization of the DOP pool, maintaining the production [START_REF] Mather | Phosphorus cycling in the North and South Atlantic Ocean subtropical gyres[END_REF]. The assessment of bioavailability of the organic P pool depends on a number of factors, including the chemical composition, the diversity of the microbial community, the ambient conditions and exposure time (e.g. myo-inositol hexakisphosphate (phytic acid) was hydrolyzed to phosphate within days under typical aerobic marine conditions [START_REF] Suzumura | Isolation and determination of inositol hexaphosphate in sediments from Tokyo Bay[END_REF]).

In the literature, few studies have reported the presence of organic phosphorous compounds in atmospheric samples. For instance, in rainwater samples collected in New Jersey, 49 organo-phosphorus compounds with structures containing CHONP (29), CHONSP (8), CHOP (7), and CHOSP (5) have been identified [START_REF] Altieri | Oligomers, organosulfates, and nitrooxy organosulfates in rainwater identified by ultra-high resolution electrospray ionization FT-ICR mass spectrometry[END_REF]. Amato et al. ( 2007) determined an ATP concentration of about 0.40 pmol mL -1 in cloud water, showing that a large majority of cells remain alive in clouds. In PM 10 aerosol samples collected in Spain, the presence of organophosphate pesticides Chloropyrifos-e, Chloropyrifosm, Diazinon and Malathion, with mean concentrations of 122, 4, 58 and 12 pg m -3 , respectively, has been reported [START_REF] Coscollà | Determination of 40 currently used pesticides in airborne particulate matter (PM 10 ) by microwaveassisted extraction and gas chromatography coupled to triple quadrupole mass spectrometry[END_REF]. Finally, Castro-Jiménez et al. (2014) reported 14 different organophosphate ester flame retardant (OPE) compounds with concentration levels, given as P, ranged from 0.04 to 0.8 ng P m -3 (average value of 0.25 ng P m -3 ).

Since phosphorus is the limiting factor in many oligotrophic marine environments, valuable information could be gained by defining the relative contribution of organic and inorganic forms of phosphorus in atmospheric deposition. Few studies of atmospheric phosphorus deposition exist and several parts of its global cycle are poorly studied [START_REF] Vet | A global assessment of precipitation chemistry and deposition of sulfur, nitrogen, sea salt, base cations, organic acids, acidity and pH, and phosphorus[END_REF]. There are very few long-term records of phosphorus deposition, which are unevenly distributed in space and time. Besides, comprehensive studies that include all forms of P do not exist in the literature. In this study, P speciation was performed in deposition samples, collected from eastern (E) and north-western (NW) Mediterranean Sea, providing a more complete picture of atmospheric deposition of P species in the Mediterranean.

Experimental methods

Sampling

Wet and bulk deposition in East Mediterranean (Greece, Finokalia: 35°20′N 25°40′E)

Rainwater and bulk deposition samples were collected at the Finokalia monitoring station (see [START_REF] Mihalopoulos | Atmospheric aerosol composition above the Eastern Mediterranean region[END_REF] for more details) close to a small village on the northern coast of Crete (Fig. 1). Measurements of aerosols and gases performed simultaneously at this sampling site and at airborne and land-based stations during numerous studies [START_REF] Lelieveld | Global air pollution crossroads over the Mediterranean[END_REF][START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF] show that measurements at Finokalia are representative of the background eastern Mediterranean atmosphere. This is due both to the location of Crete in the eastern Mediterranean (at the crossroad of the main air masses and fronts influencing the area), as well as to the location of that sampling site (upwind and far from the main cities and tourist resorts of the island).

During a two-year period (2012-2013) rainwater samples (n = 43) were collected on an event basis using a wet-only collector installed at Finokalia monitoring station (Fig. 1). Bulk deposition samples (n = 28) were also collected in a two-week basis, during the dry period (April-September); hereafter termed dry deposition samples. The estimation of bulk deposition was based on the collection of particles on a flat surface covered by multiple layers of glass beads, which could trap larger particles and avoid resuspension. The glass beads were placed in a funnel installed 3 m above the ground and exposed to the atmosphere for one to two weeks (Fig. 1). Then the system was washed with ultrapure water (500 mL). A detailed description of the technique is provided in [START_REF] Kouvarakis | On the importance of atmospheric inputs of inorganic species on the productivity of the Eastern Mediterranean Sea[END_REF].

Immediately after collection the wet and bulk deposition samples were filtered through a pre-weighed 0.45 μm cellulose filter. The nonfilterable mass was regarded during this study as insoluble matter. From the filtrated solution three aliquots were obtained. The first was immediately used for pH determination; to the second, 50-100 μL of chloroform were added as a biocide and stored in the refrigerator at 4 °C until analysis of main anions and cations. The third was stored at -18 °C for P analysis.

2.1.2. Bulk sampling deposition in NW Mediterranean (France, Cap Béar: 42°52′Ν 03°3′E)

Monthly bulk deposition samples (n = 84) were collected according to the ADIOS protocol [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF] at Cap Béar (Fig. 1) over a seven-year period (2005)(2006)(2007)(2008)(2009)(2010)(2011). The sampling station is situated 20 km from the town of Perpignan and 158 m above the sea level. The sampling device used to collect the bulk samples consisted of a 4 L Nalgene high-density polyethylene bottle, with a polyethylene funnel (surface area = 113 cm 2 ) attached to the top; a polyester mesh (mesh 33 μm) at the base of the funnel prevented contamination of the sample by plant debris or insects. Thymol (200 mL of 2.5 g L -1 solution) was added to the sampling bottle to prevent any biological activity [START_REF] Ayers | Thymol as a biocide in Japanese rainwater[END_REF] during each sampling period, although the acidity of thymol might have influenced the dissolved/particulate equilibrium in the sample. The bottle was wrapped in aluminum foil to avoid direct sun exposure and photochemical degradation of the collected compounds. Whatman filters (nuclepore) with pore diameter of 0.45 µm were used for filtration; the filters were rinsed with alcohol in order to avoid residual thymol crystallization on the filter due to poor solubility in water, and then they were weighed. The filtered solutions were stored in the refrigerator (4 °C).

Chemical analysis

Analysis of P speciation

The speciation of P was determined according to the analytical protocol referred to in Standard Methods for the Examination of Water and Wastewater (20th Edition). Total phosphorus (TP) in both dissolved (TDP) and particulate fractions (TPP) was measured after acid digestion of samples according to the persulfate digestion method; TP was measured colorimetrically as PO 4 3-at 690 nm using the stannous chloride method. TP recoveries obtained by the use of certified reference materials (MESS-3) were found to be 98 ± 12%. Dissolved phosphate (DIP) was determined as HPO 4 2-by anion chromatography in samples from the eastern Mediterranean, while in samples from NW Mediterranean it was measured colorimetrically as PO 4 3-at 690 nm using the stannous chloride method [START_REF] Markaki | Atmospheric deposition of inorganic phosphorus in the Levantine Basin, Eastern Mediterranean: spatial and temporal variability and its role in seawater productivity[END_REF]. Total acid-hydrolyzed inorganic phosphorus was determined after acidification of sample (pH = 1.6) with a mixture of sulfuric acid (0.05 M) and nitric acid (0.5 μM), separately in filterable (TIP) and non-filterable (TIP insol ) samples. Organically bound phosphates were determined in both soluble (DOP) and insoluble matter (OP insol ) by subtracting TIP from TDP and TIP insol from TPP, respectively, while the condensed phosphates (pyro-, meta-, and other inorganic polyphosphates), hereafter termed CP, were determined only in the soluble fraction by subtracting HPO 4 2-from soluble TIP; see the abbreviations in Table S4 in Supplementary material.

The quantification of TDP was achieved by using a calibration curve. For this purpose, the TDP stock solutions were prepared by dissolving equal amounts (in terms of P) of analytical grade KH 2 PO 4 (Merck #6537.0500) and adenylic acid (AMP: C 10 H 12 N 5 Na 2 O 7 P, Merck #1.01145.1000). Blank solutions are prepared with Milli-Q water and they were treated as the deposition samples. The detection limit, estimated as three times the standard deviation (SD) of the blank, was found to be 10 ppb P for acid-hydrolyzable and 13 ppb P for TP.

Furthermore, we evaluated our choice to reduce by half the proposed concentration of acid referred to in the protocol of Standard Methods for the Examination of Water and Wastewater (20th Edition) for the determination of total acid-hydrolyzable P (Fig. 2). The main purpose of this test was to assess whether part of the organic P was recovered as part of the hydrolyzable fraction, by estimating the recoveries of both total inorganic and organic P. In order to achieve this, we used as standards (not a mixture, but separately) sodium pyrophosphate tetrabasic decahydrate (Na 2 O 7 P 2 •10H 2 O, Sigma-Aldrich: 30411) to represent the total hydrolysable P and adenylic acid to represent the organic P and then we followed the analytical protocol for the determination of total acid-hydrolyzed inorganic phosphorus as described above. The optimal average recovery of inorganic P (101 ± 7%) in the concentration range of 38-500 ppb was achieved, while the recovery of adenylic acid under the modified acidification was estimated at 22 ± 4% in the concentration range of 26-500 ppb. This evidence suggests that is possible to remove organic forms of P in the hydrolyzable fraction, resulting in an underestimation of OP and overestimation of TIP. The average concentrations of inorganic and organic P determined during this study were in the range of 26-500 ppb; similar to those achieved in our recovery tests (Fig. 2).

Analysis of main ions

The main ions were determined in order for them to be used as tracers of various sources. A Dionex AS4A-SC column with ASRS-300 4 mm suppressor in auto-suppression mode of operation was used for the analysis of anions (Cl -, Br -, NO 3

- , SO 4 2- , HPO 4 2- , C 2 O 4 2-) in
rain samples and aerosol extractions. All the anions were determined with isocratic elution at 1.5 mL min -1 of Na 2 CO 3 /NaHCO 3 eluent. For the cations (Na + , NH 4 + , K + , Mg 2+ and Ca 2+ ) a CS12-SC column was used with a CSRS-300 4 mm suppressor. Separation was achieved under isocratic conditions with MSA (20 mM) eluent and flow rate of 1.0 mL min -1 . Details for the IC analysis are reported in [START_REF] Bardouki | Gaseous (DMS, MSA, SO 2 , H 2 SO 4 and DMSO) and particulate (sulfate and methanesulfonate) sulfur species over the northeastern coast of Crete[END_REF]. The reproducibility of the measurements was better than 2% and the detection limit ranged from 1 to 5 ppb for the main anions and cations. The detection limit of HPO 4 2-was 3 ppb, while the average HPO 4 2-value of the whole data set was 80 times above the DL. Blanks were always below the detection limits.

Results

Phosphorus speciation in deposition samples over the eastern Mediterranean

Phosphorus speciation in rain deposition samples during wet season

In Table 1 the P speciation in the rain water samples collected during the wet season (October to March) at Finokalia sampling station for both years (2012 and 2013) is presented. The 2012-2013 average of annual deposited flux of TDP and TPP was estimated at 0.82 and 0.23 mmol P m -2 y -1 , respectively. Higher total dissolved phosphorus (TDP) concentrations were recorded in 2012 (1.02 mmol P m -2 y -1 ), while 2013 was characterized by higher total particulate phosphorus (TPP) values. TPP accounted for 11% (0.13 mmol P m -2 y -1 ) and 35% (0.33 mmol P m -2 y -1 ) of total deposited phosphorus (TP) for 2012 and 2013, respectively. The prevalence of SW air masses (Sahara dust outbreaks) during 2013 resulted in higher mass deposition (13.51 g m -2 ) compared to 2012 (7.40 g m -2 ), and may explain this increased level of TPP in 2013.

The significant contribution of organic P in both soluble and insoluble fractions during both years is characteristic. The average percentage contributions of phosphate ions (DIP), DOP and CP to TDP were found to be 3, 92 and 5%, and 15, 75 and 10%, for 2012 and 2013, respectively (Fig. 3). The percentage speciations of TPP for particulate inorganic P (TIP insol ) and particulate organic P (OP insol ), was found to be 54 and 46%, and 36 and 64%, for 2012 and 2013, respectively.

In terms of percentage contribution, wet deposition can contribute equally to the dry (see ection 3.1.2), since ∼50% of TP (Table 1; twoyear average) is deposited during the rainy season. This observation is in agreement with similar studies performed in other sites across the Mediterranean Sea [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF]. Monthly wet fluxes of TDP and TPP during the wet season 2012-13 with rain height and wet-deposited atmospheric particulate mass are depicted in Fig. 4. During wet seasons TDP and TPP follow well the particulate mass variability and rain height, especially in 2012.

A Spearman's correlation analysis was run in rain samples (n = 40), using concentration values (ppb) between the P species and various parameters including major ions and particulate mass (Table S1). Considering as significant the correlations with p < 0.05, information regarding the sources of P species could be retrieved. Insoluble inorganic phosphorus species were related both to particulate mass (r s = 0.41, p < 0.05) and to nss-Ca 2+ (r s = 0.39, p < 0.05) suggesting a contribution from dust. Furthermore, insoluble organic P species correlated significantly to oxalate, indicating a contribution from regional pollution and photochemistry (r s = 0.60, p < 0.00005). Oxalate is produced secondary by the oxidation of volatile biogenic organic compounds by OH and NO 3 radicals (79% of the global oxalate budget), while more than 60% of oxalate is removed from the atmosphere by wet deposition [START_REF] Myriokefalitakis | In-cloud oxalate formation in the global troposphere: a 3-D modeling study[END_REF]. Furthermore, it was also correlated with nss-K + (r s = 0.59, p < 0.0005) indicating contribution from wood burning or soil. Finally, DOP correlated well with nss-Ca 2+ (r s = 0.46, p < 0.005) suggesting a contribution from dust.

P speciation in dry deposition samples during dry season

Dry deposition samples (n = 29) were collected on a two-week basis during the dry season (April to September). Similar to the rainy season, an important contribution of organic P in both soluble and insoluble fractions was observed during both years (Fig. 5). Indeed, the two-year average percentage contributions of dissolved P fractions (DIP, DOP and CP) to TDP during the dry period were found to be 11% (0.08 mmol P m -2 ), 66% (0.46 mmol P m -2 ) and 23% (0.16 mmol P m -2 ), respectively highlighting the importance of organic P forms in the dissolved fraction. On the other hand, the average percentage speciation of TPP was 49% and 51% for particulate inorganic P (TIP insol ) and particulate organic P (OP insol ), respectively, indicating an equal contribution of inorganic and organic forms in the particulate fraction.

A Spearman's correlation analysis was run also in dry deposition samples data set (n = 29) using concentration values (ppb), and the results are presented in Table S2. TIP insol appears to be associated with more complex matrixes since significant correlations were found with different tracers. Furthermore, TIP insol was significantly correlated with nss-SO 4 2-(r s = 0.47, p < 0.05) and NH 4 + (r s = 0.64, p < 0.0005), suggesting a possible contribution from anthropogenic and agricultural activities, such as fertilizers. TIP insol was also correlated with nss-K + (r s = 0.43, p < 0.05) suggesting a possible contribution from wood burning or soil, while marine sources are not excluded, since significant correlation was found with Cl -(r s = 0.39, p < 0.05).

Insoluble organic P species were correlated well with nss-K + (r s = 0.39, p < 0.05) suggesting a possible contribution from wood burning or soil. Significant correlation was observed between DIP and NH 4 + (r s = 0.41, p < 0.05) indicating contribution from agriculture. Finally, CP was correlated well with the presence of the particulate mass (r s = 0.49, p < 0.05). a Bulk mass deposition in g m -2 year -1 .

Phosphorus speciation in bulk deposition samples over NW Mediterranean

In this study speciation analysis for P was only performed in the filterable part of bulk deposition samples (TDP) collected at the Cap Béar sampling station. In the seven years 2005-2011, the average of annual bulk deposited flux of TDP was estimated at 2.16 ± 0.56 mmol P m -2 y -1 . Annual fluxes were calculated from the monthly data by integration and extrapolated to 12 months in case of missing data. The average percentage contribution of the various P species to TDP was 44% for DOP (0.94 ± 0.44 mmol P m -2 y -1 ), 34% for CP (0.74 ± 0.69 mmol P m -2 y -1 ) and 22% for DIP (0.48 ± 0.35 mmol P m -2 y -1 ).

A Spearman's correlation analysis was run in bulk deposition samples data set (n = 70) using concentration values, and the results are presented in Table S3. DIP correlated with nss-Ca 2+ (r s = 0.34, p < 0.005) and NO 3 -(r s = 0.28, p < 0.05) suggesting contribution from a mixture of dust and anthropogenic sources. No significant correlation was found between both DOP and CP with the main ions, confirming the complex matrix of those P species over the NW Mediterranean.

The inter-annual variability of bulk deposition P species in parallel with particulate mass, nss-Ca 2+ and rainfall during the seven-year period (2005)(2006)(2007)(2008)(2009)(2010)(2011) is depicted in Fig. 6 and Table 3. Maximum values of nss-Ca 2+ were recorded in 2007 (32.8 mmol m -2 y -1 ) and 2008 (44.7 mmol m -2 y -1 ), but only during 2007 was followed by the maximum of DIP (1.03 mmol P m -2 y -1 ). [START_REF] Pey | African dust outbreaks over the Mediterranean Basin during 2001-2011: PM 10 concentrations, phenomenology and trends, and its relation with synoptic and mesoscale meteorology[END_REF] studied the occurrence of African dust outbreaks over different areas of the Mediterranean Basin over an 11-year period (2001-2011). According to that study the highest number of dust event days over the NW Mediterranean were recorded in 2007, which is in agreement with the high deposition flux of nss-Ca 2+ reported in this study for that year. Maximum DOP was observed during 2011 (1.55 mmol P m -2 y -1 ), parallel with the maximum rainfall.

The seven years of data were divided in two periods, depending on the occurrence of rain: the dry period (from June to August) with total average monthly rainfall less than 2.2 cm and the wet period (from September to May). The main deposition mechanism of P species in the NW Mediterranean is wet deposition, with about 80% of the TDP deposited during the wet period, in agreement with [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF]. Monthly average percentage speciation of P species to TDP is depicted in Fig. 7; with rain height, nss-SO 4 2-, and nss-Ca 2+ as monthly averages through the years. In the dry season, the maximum percentage contribution of DOP (0.10 mmol m -2 ) was reported in August, while for inorganic species in June and July. During the wet season, the two rainfall maxima occurred in May and October and followed the maximum monthly fluxes of nss-Ca 2+ (5.87 and 2.52 mmol m -2 , respectively). The maximum percentage contribution of DOP (59%) was reported for October while in May there was a prevalence of inorganic P.

Discussion

It is well known that straightforward comparisons within and between regions are difficult because of differences in the sample types, sampling period lengths, and sample handling procedures. For instance, bulk collectors, used in Cap Béar, could underestimate dry deposition collection [START_REF] Ridame | Trend in total atmospheric deposition fluxes of aluminium, iron and trace metals in the north-western Mediterranean over the past decade[END_REF]. However, the efficiency of this sampler was evaluated by comparing it with the glass bead collectors used in Finokalia sampling station, in [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF], and it was found that both sampling devices collected dry deposition of P effectively. Table 2 depicts the literature data regarding the P speciation of deposition samples around Mediterranean areas. TPP fluxes measured in this study for the E. Mediterranean are in agreement with the values reported by [START_REF] Izquierdo | Atmospheric phosphorus deposition in a near-coastal rural site in the NE Iberian Peninsula and its role in marine productivity[END_REF] for NE Spain, not only when yearly averaged values were compared, but also during wet and dry periods (Table 2). Phosphate estimated during this study for the E. Mediterranean was within the range of the values reported by [START_REF] Markaki | Atmospheric deposition of inorganic phosphorus in the Levantine Basin, Eastern Mediterranean: spatial and temporal variability and its role in seawater productivity[END_REF][START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF]. In addition, DIP fluxes calculated here for Cap Béar (0.48 ± 0.35 mmol m -2 y -1 ) are in agreement with the previous study performed in this area [START_REF] Markaki | Variability of atmospheric deposition of dissolved nitrogen and phosphorus in the Mediterranean and possible link to the anomalous seawater N/P ratio[END_REF] and are higher compared to the eastern basin.

Both wet and dry deposition fluxes make significant contributions to the atmospheric phosphorus flux over the E. Mediterranean, although their relative contributions vary depending on precipitation rates and the presence of mineral dust. Generally, soluble P, especially the organic species, are dominant during the wet season, contributing up to 89% and 65% of the total dissolved P for 2012 and 2013, respectively. Higher TPP/TP ratios are recorded during the dry season, which seems to be linked to dust outbreaks, since they follow well the variation of nss-Ca 2+ fluxes (Table 1). This was also observed by [START_REF] Baker | Trends in the solubility of iron, aluminium, manganese and phosphorus in aerosol collected over the Atlantic Ocean[END_REF], who predict lower solubility of P in samples of Saharan origin. In addition, results from P-NEXFS spectroscopy [START_REF] Longo | P-NEXFS analysis of aerosol phosphorus delivered to the Mediterranean Sea[END_REF] suggest that the inorganic P is mainly apatite and metal-P, confirming further that Saharan outbreaks enrich the E. Mediterranean with insoluble inorganic P.

The inter-annual variability in deposited particulate mass during the dry season (Table 1) results in an increase of the particulate mass flux during 2013 by 146% followed by increases of 50% in TP and 70% in total organic phosphorus compared to the dry season in 2012. It is difficult to attribute this increase exclusively to anthropogenic sources or dust events, since the two-week resolution time of sampling makes it impossible to categorize the samples into wind sectors. However, [START_REF] Mahowald | The global distribution of atmospheric phosphorus deposition and anthropogenic impacts[END_REF], in their global atmospheric P assessment, concluded that the largest source by far of atmospheric phosphorus is mineral dust from the wind erosion of soils, and suggested that this source accounts for 82% of the global aerosol P.

In contrast to the eastern Mediterranean, wet deposition makes a significant contribution to the atmospheric phosphorus flux over the NW Mediterranean, with about 80% of the TDP being deposited with rain. The seven-year average of TDP during the wet season (1.78 ± 0.44 mmol m -2 y -1 ; Table 3) was almost double compared to the two-year average TDP during the wet season in the eastern Mediterranean (0.82 mmol m -2 y -1 ; Table 1). The difference is attributed mainly to the precipitation level being 2-3 times higher in the NW Mediterranean compared to the Eastern. However, DOP is the dominant P fraction at both sites and presents similar fluxes (0.74 and 0.70 mmol m -2 y -1 , respectively), despite the difference in precipitation level, indicating more intensive sources of organic P species in the eastern basin. During dry season, the seven-year average of TDP in the NW Mediterranean (0.39 ± 0.17 mmol m -2 y -1 ; Table 3) was almost a half of that in the eastern basin (0.69 mmol m -2 y -1 ; Table 1), mainly because of the extended dry period at the Levantine basin. Dissolved organic P species are dominant at both sites, with fluxes of 0.20 mmol m -2 y -1 and 0.46 mmol m -2 y -1 , for the NW and E. Mediterranean, respectively.

The atmospheric deposition of P may influence the primary productivity (PP) of the euphotic zone in Mediterranean seawater, especially during the stratification period, which experiences a severe phosphate depletion after the spring bloom, leading to a P-limitation of biological activity [START_REF] Krom | Phosphorus limitation of primary productivity in the eastern Mediterranean Sea[END_REF][START_REF] Thingstad | P limitation of heterotrophic bacteria and phytoplankton in the northwest Mediterranean[END_REF]. It is well known that phosphate is the most soluble form of inorganic phosphorus that can be consumed by organisms and converted to organic or other inorganic forms of P [START_REF] Björkman | Bioavailability of inorganic and organic phosphorus compounds to natural assemblages of microorganisms in Hawaiian coastal waters[END_REF]. The different organic phosphorus compounds are potential sources of phosphorus for marine communities, since marine biota could hydrolyze them to orthophosphate [START_REF] Björkman | Bioavailability of inorganic and organic phosphorus compounds to natural assemblages of microorganisms in Hawaiian coastal waters[END_REF]. However, the fraction of atmospheric phosphorus that is bioavailable is uncertain at present [START_REF] Vet | A global assessment of precipitation chemistry and deposition of sulfur, nitrogen, sea salt, base cations, organic acids, acidity and pH, and phosphorus[END_REF].

In order to calculate the maximum impact of total phosphorus deposition in the marine ecosystems, we assumed that all TDP is bioavailable. However, the bioavailability of soluble organic phosphorus remains unknown, while non-phosphate inorganic phosphorus could be bioavailable due to atmospheric acidity [START_REF] Nenes | Atmospheric acidification of mineral aerosols: a source of bioavailable phosphorus for the oceans[END_REF][START_REF] Krom | Response of the Eastern Mediterranean microbial ecosystem to dust and dust affected by acid processing in the atmosphere[END_REF][START_REF] Stockdale | Understanding the nature of atmospheric acid processing of mineral dusts in supplying bioavailable phosphorus to the oceans[END_REF]. The average total deposition (wet + dry) of P during 2012 and 2013 was estimated at 1.51 mmol P m -2 y -1 in the eastern Mediterranean. However, this value is an underestimate, since dry deposition referred to during the dry season only, and not to the entire year. Converting the TDP flux into carbon uptake by using the Redfield C/P ratio of 106, the TDP is found to be responsible, on an annual basis, for the fixation of 1.92 g C m -2 y -1 . The annual new productivity for the eastern Mediterranean was estimated in previous studies to be 5 g C m -2 y -1 [START_REF] Kress | Spatial and seasonal evolution of dissolved oxygen and nutrients in the Southern Levantine Basin (Eastern Mediterranean Sea): chemical characterization of the water masses and inferences on the N:P ratios[END_REF], which means that the atmospheric deposition of TDP contributes 38% to new production. In the NW Mediterranean, the seven-year average (2005-11) of TDP was estimated at 2.16 mmol P m -2 y -1 (Table 3), and is responsible for the fixation of 2.75 g C m -2 y -1 . The mean value of new production in the area during 1993-1999 was estimated at 42 g C m -2 y -1 [START_REF] Marty | Seasonal and interannual variations in phytoplankton production at DYFAMED time-series station, northwestern Mediterranean Sea[END_REF], which means that the atmospheric input of TDP represents only about 7% of new production. However, in less productive periods in the NW Mediterranean, new production is about 19 g C m -2 y -1 [START_REF] Marty | Seasonal and interannual variations in phytoplankton production at DYFAMED time-series station, northwestern Mediterranean Sea[END_REF]. In this case, the atmospheric TDP input of 2.75 g C m -2 y -1 represents 14% of the new production.

Conclusions

It is now well recognized that organic P is not only a missing piece of atmospheric P but is also an important contributor to atmospheric deposition of P. During this study, wet and dry deposition samples from the eastern and NW Mediterranean were analyzed for P speciation. The wet deposition of TDP in the NW Mediterranean was almost double the two-year average of TDP during the wet season in the eastern Mediterranean, mainly due to the 2-3 times higher precipitation rate in the northwestern basin. However, it is important to stress that DOP is the dominant P fraction at both sites. Dry deposition of TDP in the NW Mediterranean is less than half of that in the Eastern basin, mainly because of the extended dry period in the Levantine basin. Similarly, to the wet deposition, the soluble organic P species in dry deposition are dominant in both sites, accounting for at least 50% of the TDP.

Although additional work is needed to evaluate P bioavailability, especially regarding the organic fraction, this study clearly shows that the atmospheric deposition of TDP could contribute up to 38% of new production in the eastern Mediterranean, while in oligotrophic zones of the NW Mediterranean, the atmospheric TDP input could represent up to 14% of the new production.

Fig. 1 .

 1 Fig. 1. Finokalia station (35°20′N 25°40′E) and Cap Béar (42°52′Ν 03°13′E).

Fig. 2 .

 2 Fig. 2. Recovery of adenylic acid (AMP) and sodium pyrophosphate tetrabasic decahydrate during the acid hydrolysis of a mixture of both forms of P.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Percentage contribution of P species in rain samples to (a) Total Dissolved Phosphorus (TDP) and (b) Total Particulate Phosphorus (TPP) during the wet season in 2012-13 at Finokalia sampling station.

Fig. 6 .Fig. 7 .

 67 Fig.6. Inter-annual bulk deposition of P species in parallel with nss-Ca 2+ , particulate mass & rain height (cm) for seven years(2005)(2006)(2007)(2008)(2009)(2010)(2011) in NW Mediterranean (Cap Béar).

Table 1

 1 Annual wet and dry deposition of P species during wet and dry season, respectively over the Eastern Mediterranean.

			Deposition (mmol m -2 year -1 )
			Rain season		Dry season
			2012	2013	2012	2013
	Soluble
	h (cm)	34.6	32.1
	TDP	1.02	0.61	0.58	0.80
	DOP	0.94	0.46	0.36	0.55
	CP		0.05	0.06	0.09	0.22
	PO 4	3-	0.03	0.09	0.13	0.03
	nss-Ca 2+	9.73	16.27	18.44	28.50
	nss-Sulfate	4.68	6.25	10.68	11.80
	Insoluble
	TPP	0.13	0.33	0.24	0.44
	OP insol	0.06	0.21	0.11	0.25
	TIP insol	0.07	0.12	0.13	0.19
	Particulate mass a	4.37	6.10	3.03	7.48
	Total P
	OPtotal	1.00	0.67	0.47	0.80
	TIPtotal	0.15	0.27	0.35	0.44
	TP		1.15	0.94	0.82	1.23

Table 2

 2 Annual deposition (wet & dry) of P species in around Mediterranean areas in mmol m -2 year -1 .

			Wet	Dry	Total	Period	Reference
	TP
	CapoCavallo (Corsica)			1.29	1986-87	Bergametti et al. (1992)
	CapFerrat, France	0.16	-		1997-98	Migon and Sandroni (1999)
	Sierra Nevada, Spain	0.18	0.45	0.64	2001	Moral-Baquero et al. (2006)
	All Medit. Sites			1.06	2001-02	Guieu et al. (2010)
	Crete		1.15	0.82	1.97	2012	This study
	Crete		0.94	1.23	2.17	2013	This study
	TPP
	Cap Ferrat, France	0.07	-			Migon and Sandroni (1999)
	Montseny (NE Spain)	0.29	0.29	0.58	2002-03	Izquierdo et al. (2012)
	Crete		0.13	0.24 a	0.37 a	2012	This study
	Crete		0.33	0.44 a	0.77 a	2013	This study
	TDP
	Cap Ferrat, France	0.10	-		1997-98	Migon and Sandroni (1999)
	Frioul Island			0.39	2015-16	Djaoudi et al. (this issue)
	Crete		1.02	0.58 a	1.60	2012	This study
	Crete		0.61	0.80 a	1.41	2013	This study
	Cap Béar				2.16	2005-11	This study
	TIP total
	Israel		0.29	1	1.29	1995-98	Herut et al. (2002)
	Crete		0.15	0.35 a	0.50	2012	This study
	Crete		0.27	0.44 a	0.71	2013	This study
	Cap Béar				1.22	2005-11	This study
	DOP
	Frioul Island			0.07	2015-16	Djaoudi et al. (this issue)
	Crete		0.94	0.36 a	1.30	2012	This study
	Crete		0.46	0.55 a	1.01	2013	This study
	Cap Béar				0.94	2005-2011	This study
	DIP or PO 4	3-
	Cap Ferrat, France	0.06				Migon and Sandroni (1999)
	Cap Ferrat, France	0.15	0.76		2010	Pasqueron de Fommervault et al. (2015)
	Cap Ferrat, France	0.10	0.40		2011	Pasqueron de Fommervault et al. (2015)
	Cap Ferrat, France	0.09	0.75		2013	Pasqueron de Fommervault et al. (2015)
	Crete		0.07	0.13	0.19	1999-2000	Markaki et al. (2003)
	Erdemli (Turkey)	-	0.17		1999	Markaki et al. (2003)
	Erdemli (Turkey)	0.31	-		1996-1997	Özsoy (2003)
	Cap Béar				0.57	2001-2002	Markaki et al. (2010)
	Cap Béar				0.67	2002-2003	Markaki et al. (2010)
	Ostriconi				0.46	2001-2002	Markaki et al. (2010)
	Ostriconi				0.43	2002-2003	Markaki et al. (2010)
	Crete				0.25	2001-2002	Markaki et al. (2010)
	Crete				0.33	2002-2003	Markaki et al. (2010)
	Montseny (NE Spain)	0.05-0.06	0.06-0.08		2002-03	Izquierdo et al. (2012)
	Crete		0.03	0.13 a	0.16	2012	This study
	Crete		0.09	0.03 a	0.12	2013	This study
	Cap Béar				0.48	2005-2011	This study

a The fluxes are referred to dry period (April-September) in Crete.

Acknowledgments

The research project was financed by the EU project Perseus (FP7) and was conducted as part of the WP4 MERMEX/MISTRALS project and is a contribution to the international SOLAS, IMBER and LOICZ program. KV acknowledges the financial support of PHOSPHOTRAC (Proposal Number: 707624).