Teddy Virin

Jonas Koko

Emmanuel Piron

Philippe Martinet

Michel Berducat

OPTIMIZATION-BASED APPROACH FOR A BETTER CENTRIFUGAL SPREADING

Keywords: Parameter optimization, Reference variables, Mathematical programming, Agriculture, Environmental engineering

The use of centrifugal spreaders for application of granular fertilizers raises concern about application accuracy. These spreaders enable to distribute almost uniform deposits with regularly spaced parallel tramlines but lead over and under fertilizer application when distances separating the successive tractor trajectories are not constant or if paths are not parallel between them. The aim of this study is to propose an optimization-based method to compute optimal variables for uniform fertilizer application. Mechanical constraints are also introduced so that the calculated parameters can be afterwards used as reference variables for the control of the spreader. The resulting improvements are illustrated by numerical examples for a field with geometrical singularities.

INTRODUCTION

Fertilization practice is one of the most important operations in agricultural production. Indeed, this task is necessary to apply nutrients satisfying the demands of the plants according to yield objectives. To carry out inputs application, centrifugal spreaders are very popular thanks to their low cost and simplicity of use. Unfortunately, because of an inappropriate strategy, they lead to application errors. These errors involve negative consequences such as fertilizers waste, dramatic drop in production and harmful environmental effects like eutrophication phenomenon for example [START_REF] Isherwood | Mineral Fertilizer Use and the Environment[END_REF]. Faced with these statements, american and european governments decided to impose strict rules concerning the fertilization practice. These economic and environmental pres-sure cause then a significant increase in investigations for better nutrients applications. In this paper, we develop a new approach to improve fertization accuracy. The method is based on an optimization strategy where the cost function is formalized from a spatial distribution model previously studied in [START_REF] Colin | Etude du procede d'epandage centrifuge d'engrais mineraux[END_REF][START_REF] Olieslagers | Fertilizer distribution modelling for centrifugal spreader design[END_REF]. In order to not untimely solicit actuators and respect mechanical limits of the spreader, we are led to consider an optimization problem subject to constraints. Furthermore, thanks to this approach, optimal parameters should be used as reference variables for the future control of the machine. This document is organized as follows. The next section deals with centrifugal spreading principles and drawbacks. In section 3, the criterion function is formalized from the existing fertilizers spatial distribution model. Faced with a large scale problem, a decomposition of this one is proposed in section 4. The last section illustrates simulation results obtained after optimization algorithm execution in the main field body with parallel and non parallel tractor paths.

CENTRIFUGAL SPREADING CONTEXT

According to the soil and crops characteristics, agricultural engineers specify precise application rate to permit a correct growth of plants. These recommended doses often take the form of prescribed dose maps. These ones depicts the considered field which is virtually gridded and where each mesh corresponds to the previously specified doses. The operation consisting in applying nutrients with respect to the map represents fertilization itself. To get the distributed doses close to the desired ones, centrifugal spreaders with dual spinning discs are the most used applicators. During the spreading process, while the tractor progresses, fertilizers granulars contained in the hopper pour onto each disc and are ejected by centrifugal effect. With precision farming technologies, in order to apply inputs according to the machine location and a prescribed dose map, tractor-implement combination is equipped with a GPS antenna, a radar speed sensor and an actuator. The two first tools enable to know the tractor position and speed. Concerning the actuator, it permits to control the fertilizer mass flow rate. The actual amount of applied fertilizers currently called spread pattern has an irregular distribution which is often underlined by the transverse distribution curve obtained by summing the amounts along each travel direction. An example of spread pattern and its related transverse distribution for only one disc are illustrated in Fig. 1. As depicted in Fig. 2, this spatial distribution heterogeneousness lead then the tractor driver to follow outward and return paths in order to obtain an uniform deposit from transverse distributions summation for each successive travels within the field. The spreading regularity is achieved when the distance between two consecutive overlapping lines, currently called working width, is equal to the distance separating two successive tramlines. Besides, these overlapping lines correspond to symmetry axis which make two successive paths coincide. So, it is clear that fertilization strategy is today based above all on best overlappings of the transverse distributions with respect to the different tractor paths. Most of the experiments and simulations undertaken in order to assess fertilizer application accuracy or study device settings are only carried out by using this method as specified in test procedures defined by (ASAE, 2001). Some studies relying on these tests are also led to assess performance of applicators like in [START_REF] Yule | Performance of fertiliser spreading equipment for precision agriculture applications[END_REF]. Unfortunately, this reasoning do not take into account the actual phenomenon occurring during the spreading process. Indeed, in fact, the global deposit of nutrients in the farmland is due to spread pattern overlappings and thus is the result of the heterogeneous spatial distribution summation at each position of the spreader.

Then, though the application results are correct for parallel paths, some local application errors are unavoidable when geometrical singularities are met in the field like non parallel tramlines, start and end of spreading...Some errors examples are illustrated in Fig. 3 where results are obtained by simulating spread patterns overlappings with settings imposed by manufacturers rules. To minimize these issues, a solution could be to look for optimal paths for the machine as in [START_REF] Dillon | Optimal path nutrient application using variable rate technology[END_REF]. However, this type of solution cannot be applied when tramlines are already fixed by other agricultural operations like sowing. So, it is obvious that some efforts must be done to achieve better spread patterns arrangement according to geometrical constraints met in farmland during fertilization practice. The computed adjustments should be continuously achieved for each position of applicator by modifying its settings. In this work, we study then a method which permits to calculate optimal parameters to have the best spread patterns arrangement within the arable land in the presence of imposed tractor trajectories.

OPTIMIZATION CRITERION

Let us first precise some notations used to define the spread pattern model. This model uses different variables such as the time (t ∈ R), the spatial domain, in other words the field, (Ω ∈ R 2 ), the path (s(t) ∈ Ω), the coordinates of points (x ∈ Ω), the distance between s(t) and x (r(x, t) ∈ R) and the angle between ---→ s(t)x and s(t) (θ(x, t) ∈ R). The spread pattern is currently defined by its medium radius and medium angle. The first parameter, varying with the speed of disc, corresponds to the distance between the disc centre and the spread pattern one while the second, modifiable with the fertilizers dropping point on the disc, states the angle between the travel direction and the straight line passing through the disc centre and the spatial distribution one. The respective mass flow rates for the right and left discs are defined by m(t) and d(t). ρ(t) and ξ(t) stand for the medium radius related to the right and left discs respectively. At last, the right and left discs medium angles are defined by ξ(t) and ψ(t). Furthermore, if we assume σ r and σ θ to be the respective constant standard deviations for the medium radius and the medium angle, we can calculate the right and left spatial distributions q r and q l , according to [START_REF] Colin | Etude du procede d'epandage centrifuge d'engrais mineraux[END_REF], as:

q r (x, m(t), ρ(t), ϕ(t)) = τ • exp(-A(x, t) 2 /a) • exp(-B(x, t) 2 /b) (1) q l (x, d(t), ξ(t), ψ(t)) = κ • exp(-C(x, t) 2 /a) • exp(-D(x, t) 2 /b) (2) with A(x, t) = r(x, t) -ρ(t) (3) B(x, t) = θ(x, t) -ϕ(t) (4) C(x, t) = r(x, t) -ξ(t) (5) D(x, t) = θ(x, t) -ψ(t) (6)
and where a = 2σ 2 r , b = 2σ 2 θ , τ = m(t)/(2πσ r σ θ ) and κ=d(t)/(2πσ r σ θ ). To simplify notations, we define

M (t) = (m(t), d(t)) ∈R 2 , R(t) = (ρ(t), ξ(t))
∈R 2 and Φ(t) = (ϕ(t), ψ(t)) ∈R 2 . The global distribution is then equal to the summation of right and left contributions:

q tot (x, M (t), R(t), Φ(t)) = q r (x, m(t), ρ(t), ϕ(t)) +q l (x, d(t), ξ(t), ψ(t)) (7)
Therefore, the actual distributed dose Q ∈ R 2 during the interval of time (0, T ) for single tramline can be evaluated as:

Q(x, M, R, Φ)= T 0 q tot (x, M (t), R(t), Φ(t)) dt (8) If Q * ∈ R 2 stands
for the prescribed dose, to reduce harmful fertilization effects, the following functional must be minimized:

F (M, R, Φ) = Ω [Q(x, M, R, Φ) -Q * ] 2 dx (9)
Given that (9) cannot be calculated in an analytical way, discretization is carried out. Then, Ω is gridded so that Q and Q * can be computed with bilinear approximations. A temporal discretization is also performed by dividing the interval (0, T ) into n elements with equal length δ = T /n. Thus, we can define t j = jδ with j = 0, 1, ..., n.

Consequently, we can assume M j = M (t j ), R j = R(t j ) and Φ j = Φ(t j ). The corresponding vectors are defined as

M = [M 0 • • • M n ] T , R = [R 0 • • • R n ] T and Φ = [Φ 0 • • • Φ n ]
T . Moreover, me-chanical limits are taken into account, by assuming the functions M , R and Φ and their time derivative to be subject to bound constraints. The set of solutions is then S = (M , R, Φ) ∈ R 6(n+1) so that:

                   M min ≤ M ≤ M max R min ≤ R ≤ R max Φ min ≤ Φ ≤ Φ max |M i+1 -M i | ≤ αδ |R i+1 -R i | ≤ βδ |Φ i+1 -Φ i | ≤ γδ, (10)
whith α, β and γ are known parameters fixed according to the spreader characteristics. Consequently, we are led to consider the nonlinear programming problem given by: (P) min

(M ,R,Φ)∈S F (M , R, Φ) (11)
Given that S is bounded closed, according to the Weierstrass theorem, there exists at least one solution to (P). The field mostly contains several tramlines and thus the actual distributed dose is the result of the summation of the applied dose for each k-indexed tractor trajectory: 13) and where U = (M, R, Φ). w is, here, the number of paths and the trajectories s k (t) are assumed to be defined in the interval (t k i , t k f ). By considering the definitions of M k j = M (t k j ), R k j = R(t k j ), and Φ k j = Φ(t k j ) we can also use the discretization techniques as before. Therefore, from these definitions, optimization in the whole field considering all paths can be carried out by solving the problem (P).

Q(x, U ) = w k=1 Q k (x, U ) (12) with Q k (x, U ) = t k f t k i q tot (x, M (t), R(t), Φ(t))dt(

PROBLEM DECOMPOSITION

Like for the main prescribed dose map depicted in Fig. 3(a), farmland are virtually 1 m-gridded. By applying the discretization method previously detailed, we are led to consider a large scale problem. Indeed,to lose informations as little as possible, 2 samples of parameters per elementary mesh are computed. If we assume only 3 paths 100 m long in the field, the number of parameters reaches then 3600. So, it is clear that this large scale problem cannot be solved directly by an optimization algorithm in view of the high computational time which would be involved. Consequently, a problem decomposition is necessary. First of all, let us define the following notations:

K 1 = {k ∈ N| 1 ≤ k ≤ w} , K 2 = {k ∈ N| 1 ≤ k ≤ w -1} , K 3 = {k ∈ N| 2 ≤ k ≤ w} , L 1 = {l ∈ N| ∀z ≥ 2 ∈ N, 1 ≤ l ≤ z} , L 2 = {l ∈ N| ∀z ≥ 2 ∈ N, 2 ≤ l ≤ z} , Ω = k∈K1 Ω k , Ω k = l∈L1 Ω k l ,
where Ω k ∈R 2 the k th subdomain of Ω, and Ω k l ∈R 2 the l th subdomain of Ω k . To deal with each path s k (t) apart from the others, the subdomains Ω k are defined so that:

∂Ω k ∩ Ω k+1 = s k+1 (t), ∀(k, t) ∈ K 2 × (t k+1 i , t k+1 f ) ∂Ω k ∩ Ω k-1 = s k-1 (t), ∀(k, t) ∈ K 3 × (t k-1 i , t k-1 f )
In order to make easier to understand the spatial decomposition, Fig. 4 illustrates the example of three parallel tramlines in a domain Ω with a rectangular geometry. Furthermore, by assuming the vectors M k l , R k l , Φ k l to be the respective restrictions of M , R and Φ in the subdomain Ω k l , we can also define the set S k l as the restriction of S in the same subdomain. By considering the symmetries conditions exposed in the first section, (P) is naturally decomposed in the following way:

(P ′ )      min z l=1 w k=1 J k l (x, M k l , R k l , Φ k l ) (m k l , R k l , Φ k l )∈S k l , (l, k) ∈ L 1 ×K 1 (14) whith J k l (x, M k l , R k l , Φ k l ) = Ω k l [Q k l (x) -Q * ] 2 dx (15)
where Q k l (x) is the actual applied dose within Ω k l taking into account not only the amounts already distributed in Ω k-1 l and Ω k l-1 but also the future ejected dose along the path s k+1 (t) which is predicted so that it respects the symmetries properties previously underlined. The problem (P ′ ) is an optimization problem subject to inequality constraints. For (l, k) ∈ L 1 × K 1 , to minimize the functional J k l , the problem (P ineq ) is considered and defined as:

(P ineq )      min J k l (M k l , R k l , Φ k l ) u j ≤ h j (M k l , R k l , Φ k l ) ≤ v j , j = 1, 2, ...dim(M k l ) (16) Fig. 4. Rectangular domain Ω divided into 9 subdomains Ω k l , 1 ≤ l ≤ 3, 1 ≤ k ≤ 3.
where h j stands for the j th double inequality, u j and v j its lower and upper bound. To obtain an acceptable solution after algorithm execution and avoid solving the problem which consists in determining saturated constraints, we decide to apply an augmented lagrangian algorithm [START_REF] Bertsekas | Constrained Optimization and Lagrange Multipliers Methods[END_REF] which severely penalizes unacceptable parameters. Moreover, we also use a lbfgs technique shown to be efficient with large scale optimization problem [START_REF] Byrd | Representations of quasi-newton matrices and their use in limited memory methods[END_REF].

NUMERICAL RESULTS

In our case, we are only interested in application optimization in the main field body. The prescribed dose is fixed at 100 Kh/Ha. Indeed, nowadays, even in the case of fixed desired dose, farmers are unable to obtain satisfying results.

The speed of tractor is also constant and equal to 10 Km/h. The studied field is illustrated in Fig. 5. The paths 2 to 6 are parallel. The default working width is defined as being 24 m. However, the distance between the 4 th and 5 th tramlines is equal to 25 m. Besides, the 6 th path is 19 m from the 5 th one. At last, the first path includes a travel direction change causing a bottleneck at the headland. Nowadays, for this kind of farmland, the settings imposed by manufacturers do not change along the process and correspond to the rules established for a 24 m working width. As shown in Fig. 6, these settings lead to over-application between the 5 th and 6 th tramlines and also at the end of the first trajectory. An under-dosage zone appears slightly below the break-point marking the beginning of the travel direction change for the first path. Everywhere else, error is included between -8% and +6% and are then acceptable. Before applying our optimization algorithm, mechanical constraints are imposed by taking into account the characteristics of the most used applicators. From these constraints, after algorithm execution, we succeed in reducing fertilization errors between -8.7% and +6.2% as illustrated in Fig. 7 which is very satisfying. We obtain little Here for successive parallel tramlines, optimal parameters are computed so that they are time independent. Thus for the trajectories 3 to 6, these ones are gathered in Table 1. As expected, for the path 5 the mass flow rate for the right disc slumps because of the narrow pass occurring in this case. The corresponding medium radius and angle are also slightly modified to obtain uniform overlappings. Unlike the parallel tramlines, the optimal parameters are time dependent for the two first trajectories. The optimal variables for the first path are shown in Fig. 8. For the left disc, after the travel direction shift, the medium radius slumps. Concerning the medium angle and mass flow rate, slight variations occur around the break-point marking the travel direction shift. Regarding the right disc, the medium angle increases before the break-point and drops straight away after this one. The mass flow rate displays the same trends for this path. Considering the optimal decision variables during the second path in Fig. 9, we can note that all parameters increase at the beginning and stabilize after a while. This phenomenon is inherent to the narrowing existing when the tractor comes in the arable land and the pass widening occurring after some distance.

Here, for the different studied cases, the computed parameters, respecting the overall specified con- straints, can then used as reference variables in a future control strategy. Thus, the machine settings (speed of discs, fertilizers dropping point position and mass flow rate) can be continuously modified along each tramline as advised by [START_REF] Olieslagers | Fertilizer distribution modelling for centrifugal spreader design[END_REF].

CONCLUSION

A new method relying on optimization techniques has been proposed in order to fulfill the economic and environmental requirements inherent to fertilization practice. From the spread pattern analytical model and the mechanical characteristics of the spreader, a cost function subject to constraints has been formalized. Faced with a large scale problem after discretization, a decomposition of this one has been proposed. In order to deal with the constraints and the large size of the problem, an augmented lagrangian method associated with a l-bfgs technique have been implemented. Thanks to this strategy, fertilization errors have been significantly reduced and the optimal parameters should be considered as reference values to control the machine. In order to achieve a spatial optimization within the whole farmland, works are necessary to deal with the boundaries

Fig. 1 .

 1 Fig. 1. Spread pattern (spatial distribution) and transverse distribution

Fig. 2 .

 2 Fig. 2. Fertilization strategy based on transverse distribution summation.

  Application errors map calculated from the difference between the prescribed dose map and actual dose map.

Fig. 3 .

 3 Fig. 3. Application errors resulting from the reasoning based on the best transverse distribution investigation and not on spread patterns overlappings.

Fig. 5 .

 5 Fig. 5. Field with parallel and non parallel tramlines

Fig. 7 .

 7 Fig. 7. Application errors obtained after optimization

Fig. 8 .

 8 Fig. 8. Optimal parameters for the first path

Fig. 9 .

 9 Fig. 9. Optimal parameters for the second path where modified spread patterns are often applied. Combined with the use of information technology and recent advances in agricultural vehicle guidance, these studies should efficiently improve fertilization application for the future.

Table 1 .

 1 Optimal values for successive parallel tramlines (M

			Left Disc
		Path 3	Path 4	Path 5	Path 6
	M f	20.17	22.99	24.94	21.11
	Rm	15.33	16.45	16.63	15.11
	θm	-19.68	-19.13	-17.28	-19.48
			Right Disc
		Path 3	Path 4	Path 5	Path 6
	M f	19.84	18.67	11.74	13.89
	Rm	15.24	15.18	12.51	12.72
	θm	20.12	21	18.74	17.22

f : Mass Flow Rate (Kg/min); R m : Medium Radius; θ m : Medium Angle ( • ))


Axel Pierru Ifp

D Perruchet

J.-P Cueille

Ceg-7 I Cadoret

J.-L Karnik

Ceg-9 A Fauveau

P Chollet

F Lantz

Allocating the CO 2 emissions of an oil refinery with Aumann-Shapley prices

Keywords: . CADORET, P. RENOU, Élasticités et substitutions énergétiques : difficultés méthodologiques. Avril 1991

Les cahiers de l'économie -n° 58

Série Recherche

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Introduction

In automotive transport, Life Cycle Assessment (LCA) aims at quantifying and evaluating the environmental performance of a fuel from "well to wheel" in order to help decision-makers to choose between alternative fuels and processes. In the case of conventional petroleum fuels, this means allocating among the different fuels the greenhouse gas and pollutant emissions released by the production, transport and refining of oil, and by conversion to mechanical energy by engines.

Petroleum refining is by its very nature a multiproduct activity. Therefore, there are innumerable ways to allocate the greenhouse gas emissions produced by a refinery among the various refined products. However, linear programming is widely used in short and medium term planning of refinery operations (see for instance [START_REF] Griffin | The process analysis alternative to statistical cost functions: an application to petroleum refining[END_REF][START_REF] Palmer | A model-management framework for mathematical programming -An Exxon monograph[END_REF]. Decisions supported by such models are typically the selection of the types of crude oil to be refined and the operating modes of processes like distillation, catalytic cracking and reforming. Linear programming is also routinely used to determine optimal capacities for new units. For an industrial process that can be modelled using linear programming, Azapagic andClift (1998, 1999) suggest that a marginal contribution to total CO 2 emissions should be defined for each finished product. This marginal contribution may be used in a well-to-wheel LCA, instead of allocating the refinery's emissions in proportion (in terms of mass or volume) to the produced quantities of finished products, as had been done so far. This approach is entirely consistent with the economic theory that advocates making decisions on the basis of a marginal signal. The work of these two authors is, however, open to discussion on two points:

-they consider it inevitable that a formulation must rely on multi-objective programming (since the objective function to be minimized is a vector with two components: an economic function and the environmental burden); -they do not study the relevance of these marginal contributions in the LCA context (i.e. if the total volume of CO 2 emissions can be allocated on the marginal basis).

Concerning the first point, [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] stresses that the establishment of a market for CO 2 emission rights in Europe leads refining companies to integrate a cost associated with carbon emissions in the objective function of linear programming models used to manage refineries. European refineries thus minimize a scalar cost function including the purchase (or sale) of CO 2 emission permits, under the constraint of satisfying an exogenous demand for refined products. These models can be used to determine a marginal cost associated with the refinery's emissions for each of its finished products. Dividing this marginal cost by the price of an emission permit gives the marginal contribution of the product under consideration to the refinery's CO 2 emissions.

As regards the second point, [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] - [START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF] for a more formal presentation of this result -shows that, under certain conditions, this marginal contribution behaves as an average contribution: multiplying the quantity produced for each finished product by its marginal contribution to CO 2 emissions, and adding the resulting figures for all products, gives the total quantity of carbon emitted by the refinery. Then, under certain conditions, the marginal contribution to CO 2 emissions of each product is the relevant means of allocating all the refinery's CO 2 emissions in the LCA context.

In fact, this result holds if the demand equations are the only binding constraints with a nonzero right-hand side coefficient . The refiner's cost function is then a homogenous function of degree 1. It is possible to identify two instances in which the marginal contribution of the various refined products to the refinery's total emissions does not behave as an average contribution:

-when the refiner uses a short-run model in which the capacities of the various units are fixed;

-when CO 2 emissions exceeding a given volume have a higher cost.

In Europe, refiners must, in the short term, deal with an expected increase in the market share of diesel fuel to the detriment of gasoline. The use of short-run models can therefore supply relevant results for decision-making purposes by public authorities. In these models, capacity constraints with a non-zero right-hand-side coefficient are quite likely to be binding at the optimum. The second instance referred to above is the result of the regulatory framework recently established in Europe: industrial facilities exceeding their emission quotas, which no longer have the option of purchasing emission permits, must pay for additional tons of CO 2 discharged at a higher price set by the authorities. The linear programming model then includes a constraint, pertaining to the limitation in quantity of less-expensive CO 2 , with a non-zero right-hand side coefficient. Thus, even long-run models, in which capacities are variables, can result in the determination of marginal contributions which do not behave as average contributions. To conclude, in both cases, the refiner's cost function is no longer homogenous. Allocating CO 2 emissions on a marginal basis would tend to overvalue (or undervalue) the total volume of emissions.

There are a variety of approaches in economic literature that can help to resolve this type of problem. For instance, the by-now classical literature on second-best pricing for public enterprises suggests cost-sharing prices, such as Ramsey prices. This type of approach, however, requires information on demand (price elasticity of demand, at the very least). Furthermore, two finished products that affect the cost function of the multiproduct firm in exactly the same way will generally have distinct prices. Thus, in petroleum refining, the kerosene cut is sold as jet fuel, dilutants and, in some countries, illuminating kerosene. The production cost (and process) is the same for the refiner, since the individual product specifications can be attained by the later addition of additives, but the three markets are very different. A Ramsey-type price system would consequently lead to a specific allocation of CO 2 emissions for each of these three products.

To solve the problem studied here, we have chosen to adopt a more-easily implemented approach which requires that two finished products that affect refining costs in exactly the same way must have the same price, independently of demand. For this reason, we suggest that the Aumann-Shapley cost sharing method be applied to the problem of allocating the refinery's emissions. Derived from the theory of values in non-atomic cooperative games elaborated by [START_REF] Aumann | Values of non-atomic games[END_REF], this method currently holds an important place in the literature on axiomatic cost sharing. It generalizes the concept of average cost to the multiproduct case, while satisfying the principle of marginality.

In the context of the linear programming models used in refining, the Aumann-Shapley cost sharing method broadens the approach proposed by [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] since:

-when demand constraints are the only ones with a non-zero right-hand side coefficient, using this method amounts to allocating a contribution to each refined product equal to its marginal contribution;

-in other cases, this method determines a rule for the distribution of total emissions among the different products.

In the first section, we briefly review the methodology proposed by [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] for allocating a petroleum refinery's CO 2 emissions and its limitations. Notations are introduced.

We then present the Aumann-Shapley method and its implementation in the context of the problem studied. A certain number of elements justifying the choice of this approach are presented. In the last section, we apply this approach to a simplified model of an oil refinery. [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] provides a precise description of refining models, of which we will only present the main features here. Our objective is to introduce the refiner's cost function and the properties verified by it under certain conditions. The notations introduced are used in the rest of the article.

Notations and overview of the problem

The refiner's cost function

The refinery's objective is to meet a given demand for finished products at minimum cost. We In a refining program, the main endogenous variables are the flows of crude oil to be processed, intermediate products and finished products. In addition to the demand constraints for finished products, these models take into account three main types of constraints:

-material balance equations, which express the equality between an available quantity of a given intermediate product and the quantities used for the different possible destinations of this product;

-quality constraints, which express each finished product's obligation to meet legal specifications;

-capacity constraints, which reflect the capacity limitations of existing units (in the long-run models used to analyze investment decisions, the capacities of units to be built are considered as variables).

The objective function to be minimized (subject to constraints) is the sum of two "elementary"1 functions:

-the operating cost (plus the investment cost of units to be built in the long-run models used to analyze investment decisions);

-the cost associated with CO 2 emissions (assumed to be equal to the quantities released times the price of an emission permit).

Where C is the refiner's cost function, for a given vector b of demand for finished products, at the optimum, the objective function takes the value ( ) ( )

1 2 ( ) C b C C b = +

Properties of the cost function and allocation of CO 2 emissions

In the absence of a quota on CO 2 emissions, in long-run models, demand constraints are often the only constraints with a non-zero right-hand side coefficient. In this case, if the optimal basic solution is nondegenerate, the sum of the products of every marginal cost by the corresponding quantity demanded is equal to the total cost:

( ) ( )

1 m i i i C b b C b b = ∂ = ∂
The marginal cost is then the relevant means of allocating the refinery's cost among the finished products. [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] has shown that this property was true for each elementary function:

( ) ( ) 1 1 1 m i i i C b b C b b = ∂ = ∂ ( ) ( ) 2 2 1 m i i i C b b C b b = ∂ = ∂ ( ) 1 i C b b ∂ ∂ and ( ) 2 i C b b ∂ ∂
are called "elementary dual variables". It is thus possible to allocate the cost associated with the refinery's emissions to the various finished products, using the corresponding elementary dual variable ( )

2 i C b b ∂ ∂ . The ratio of ( ) 2 i C b b ∂ ∂
to the emission permit price gives the marginal contribution of product i to the CO 2 emissions of the refinery.

Need for a more general approach

Under certain assumptions, the method presented above meets two requirements:

-emissions are allocated on the marginal basis, thus facilitating decision-making (in accordance with economic theory);

-the total quantity of the refinery's emissions is allocated to the various products, which is consistent with an LCA-based approach.

There are nonetheless situations where this method does not satisfy the second requirement. When demand constraints are not the only binding constraints with a non-zero right-hand side coefficient, it is no longer possible to allocate costs (or emissions) on the marginal basis. Thus, in short-run models, the capacity constraints of existing units have a right-hand side coefficient other than zero. These models can also take into account the availability constraints of certain types of crude oil.

In Europe, the emission permits market covers all of the emission quotas allocated to the various industrial facilities concerned. If global limits are exceeded, certain industrial facilities would be unable to purchase a sufficient number of permits to cover their total emissions. They would then need to pay for each additional ton of CO 2 released at a price set by the authorities. Consequently, long-run models can include a constraint on the availability of CO 2 emission permits at the market price (with a non-zero right-hand side coefficient).

It is therefore necessary to suggest a more general approach, which can best meet the two preceding requirements. We suggest using the Aumann-Shapley cost-sharing method.

Allocation of emissions according to the Aumann-Shapley method

Presentation of the method

The Aumann-Shapley (A-S) method is a cost sharing method in which agents demand arbitrary quantities of possibly different goods (no other information than the cost function is used). The A-S method computes the Shapley value (1953) of the game where each unit demanded by an agent is regarded as a separate player. The vast literature on the A-S method, rooted in Aumann and Shapley's theory of value for non-atomic games, focuses on the case of perfectly divisible goods and a continuum of players. Let us denote ( ) C b as the cost function of the demand vector b. Under suitable differentiability assumptions, the per-unit cost share (also called "A-S price") imputed to product i with this method, denoted ( )

, i s b C , is then: ( ) ( ) 1 0 , i i C s b C b d b λ λ ∂ = ∂ (1)
The per-unit cost share imputed to product i is the integral of the marginal cost of product i along the ray to b. The per-unit cost shares thus defined allow us to allocate the total cost:

( ) ( ) ( ) 1 1 1 0 , m m i i i i i i C b s b C b b d C b b λ λ = = ∂ = = ∂
In the literature on axiomatic cost sharing, the A-S cost-sharing method is considered a natural extension of average cost pricing from a single product to a finite number of products with non-separable production cost functions. [START_REF] Samet | An application of the Aumann-Shapley prices for cost allocation in transportation problems[END_REF] and [START_REF] Haimanko | Cost sharing: the nondifferentiable case[END_REF] have shown that the A-S cost-sharing method can be extended to non-differentiable cost functions such as piecewise linear functions. This comment is important since the refiner's cost function is piecewise linear and nondifferentiable in each degenerate optimal solution (with two one-sided derivatives for the marginal cost).

Implementation of the method in the context of the problem studied

The literature (e.g., [START_REF] Billera | Internal telephone billing rates: a novel application of nonatomic game theory[END_REF] proposes various applications of the A-S method. [START_REF] Samet | An application of the Aumann-Shapley prices for cost allocation in transportation problems[END_REF] consider a cost function derived from a linear program, as is the case with the function C studied here. As this function is piecewise linear, the A-S cost vector is a sum of the gradients of the linear "pieces" of C along the ray to b, where each of these is weighted by the normalized length of the subinterval in which C has a constant gradient. In other words, the formula (1) is a sum of areas of rectangles, with as many rectangles as there are basic solutions, successively determined along the line from the origin to b.

If there are 1 nsuccessive basis changes, where k b λ denotes the output value at which the k th basis change occurs, we have:

( ) ( ) ( ) 1 1 0 , n i k k k k i C s b C b b λ λ λ - + = ∂ = - ∂ (2)
Where:

0 0 λ = , 1 n λ = ( ) k i C b b λ ∂ ∂
stands for the right-hand derivative.

Consider for example a long-run model that takes into account a limitation constraint on the quantity of emission permits that can be purchased by the refinery (as mentioned in the introduction). In such a model, except for demand constraints, the only constraint with a nonzero right-hand side coefficient is the constraint pertaining to the emission-permits quota. For a certain level of demand b, a basis change will take place along the ray to b. The A-S cost then results from a weighted average of two marginal costs.

Our objective here is not to determine an average cost per finished product, but to calculate the contribution of each product to the refinery's total CO 2 emissions. Emission costs must first be allocated to the various products. To that end, we use the breakdown of the objective function into elementary functions suggested by [START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF]. The basic solutions successively determined when demand moves along the line from the origin to b are found by minimizing the total cost (including operating costs and costs associated with CO 2 emissions). On the other hand, for each of these basic solutions, only the elementary dual variable ( )

2 k i C b b λ ∂ ∂
associated with the cost of the emissions is taken into account in the formula (2). Where ( ) 2 , i s b C denotes the contribution (per-unit) of the product i to the cost associated with carbon emissions: The contribution of the product i to the refinery's emissions is obtained by dividing ( )

( ) ( ) ( ) 1 2 2 1 0 , n i k k k k i C s b C b b λ λ λ - + = ∂ = - ∂ (
2
, i s b C by the price of the emissions permit. When the price per ton of CO 2 can take two values (in the presence of a constraint on emissions-permit availability), we suggest that the emissions cost be considered as resulting from the sum of two elementary functions: that corresponding to CO 2 at the market price and that corresponding to CO 2 at the price established by the authorities. The elementary A-S cost for each elementary function can then be divided by a single price per ton of CO 2 . For each finished product, both contributions to the emissions are simply added.

Justification of the choice of method

We will not deal here with certain properties (e.g. dummy, scale invariance) verified by this cost-sharing method. These are extensively dealt with in the literature on axiomatic cost sharing (e.g., [START_REF] Friedman | Three methods to share joint costs or surplus[END_REF]. We will however present four types of considerations that seem relevant to us in the context of the issues studied. [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] When demand constraints are the only constraints with a non-zero right-hand side coefficient, the refiner's cost function is a homogenous function (of degree 1). The use of the A-S method then results in allocating to the various products a contribution equal to their marginal contribution.

Generalization of the method proposed by

More precisely, by varying the quantities produced along the ray extending from the origin to the vector of the quantities demanded, the basic solution remains the same (with the same binding equations) and the marginal cost of each product remains constant. The elementary dual variables obviously also remain constant. The allocation of the refinery's CO 2 emissions to the various products on the basis of their marginal contribution is therefore a particular application of our approach.

Additivity (in the cost function)

This property is crucial in the context of the problems studied here. First of all, from a methodological point of view, the elementary dual variables are calculated by considering the objective function as resulting from the addition of two elementary functions. As proved by [START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF], the elementary dual variables verify a certain number of properties. Obviously, these properties are also verified by elementary A-S costs. If we denote for a given product i:

( ) 1 , i s b C : elementary A-S cost associated with operating costs, where:

( ) ( ) ( ) 1 1 1 1 0 , n i k k k k i C s b C b b λ λ λ - + = ∂ = - ∂ ( ) 2 ,
i s b C : elementary A-S cost associated with CO 2 emissions, where:

( ) ( ) ( ) 1 2 2 1 0 , n i k k k k i C s b C b b λ λ λ - + = ∂ = - ∂
We have the following three equations:

( ) ( ) ( ) 1 2 , , , i i i s b C s b C s b C = + ( ) ( ) 1 1 1 , m i i i b s b C C b = = ( ) ( ) 2 2 1 , m i i i b s b C C b = =
Thus, the A-S cost of a given product is equal to the sum of the elementary A-S cost associated with operating costs and of the elementary A-S cost associated with CO 2 emissions. All results relating to elementary dual variables can therefore be generalized by using the A-S cost-sharing method.

Furthermore, the elementary A-S costs are calculated in the same way for all products (with the same coefficients weighting the successively determined marginal costs). Thus, if the objective function also includes the cost associated with SO 2 emissions, then the calculation of the contribution of each product to the refinery's sulphur emissions would be entirely consistent with the calculation of their contribution to carbon emissions. This is a strong argument in favour of the use of this method. Conversely, this argument led us to reject other methods, such as the "proportionally adjusted marginal pricing method" developed by [START_REF] Wang | Proportionally adjusted marginal pricing method to share joint costs[END_REF], which are nonadditive in the cost function.

Marginality property

A marginal analysis constitutes a key element of the decision-making process. In the general case, the A-S method does not lead to the determination of a contribution equal to the marginal cost. This method nevertheless satisfies a marginality property, as emphasised by [START_REF] Friedman | Three methods to share joint costs or surplus[END_REF]: the cost share imputed to product i depends only on the marginal cost function with respect to product i.

A complementary observation can also be made when one considers, as is the case here, a cost function derived from a linear optimization problem. If, apart from demand constraints, one other constraint (e.g. a capacity constraint) presents a non-zero right-hand side coefficient, the A-S method allocates to a given product a contribution lying between the two successive marginal costs of the product. In linear programming, this contribution is similar in some ways to a long-run marginal cost (calculated by considering the capacity variable) necessarily lying between the two short-run marginal costs (calculated by considering the fixed capacity) that are determined successively (before and after the capacity constraint is binding). [START_REF] Mirman | Demand compatible equitable cost sharing prices[END_REF] place special emphasis on the economic aspects of cost allocation. They deal with a partial equilibrium model. Taking into account consumers' utilities, they prove the existence of an equilibrium under the A-S price mechanism for a general class of cost functions. Thus, although this price mechanism depends only on the cost structure and not on consumers' utilities, standard assumptions on preferences and weak assumptions on the cost function guarantee the existence of a supply such that the corresponding A-S prices lead to demands that match supply. If the cost function is homogenous of degree 1, A-S prices and marginal prices are the same and the solution coincides with the standard competitive equilibrium. [START_REF] Samet | An application of the Aumann-Shapley prices for cost allocation in transportation problems[END_REF] consider a piecewiselinear cost function and show that there exists an A-S price vector compatible with consumer demands.

Demand compatibility

This result gives an economic dimension to the price system (and therefore to the emissions allocations) determined using the A-S method at the refinery level. One simply assumes that the refinery has a monopoly and that it anticipates the demand for finished products corresponding to a situation of equilibrium.

Case study

Presentation of a simplified short-run refining model

We give a simplified numerical example for a topping/reforming/cracking refinery, which includes a distillation unit, reforming unit, fluid catalytic cracking (FCC) unit and one or more hydroprocessing and hydrodesulphurization units. We consider a single type of distillation cut and a single operating severity for the FCC unit. The FCC unit's processing capacity is limited to 850,000 tons of distillate annually. We are therefore dealing with a short-run model. The refinery's distillation unit processes two types of crude oil to produce five main types of product: liquefied petroleum gas, gasoline, gas oil, distillate and residue. For a given crude, the yields obtained for each of these products, as well as the operating costs (purchase and processing) and the quantities of CO 2 released, are given in Table 1. The FCC unit converts a portion of the distillate into gasoline (and other products) as shown in Table 2. We assume that each ton of CO 2 released by the refinery represents a cost of US$30 (through the purchase of an emission permit at this price or, equivalently, the loss of the opportunity of reselling a permit at same price).

Over a typical year, the refinery's output must satisfy the demand forecast for each of four finished products: gasoline, automotive diesel fuel, heating oil and heavy fuel oil. The refiner's aim is to satisfy demand constraints while minimizing total annual costs (operating costs plus cost associated with CO 2 emissions).

When the capacity constraint for the FCC unit is binding, it is no longer possible to allocate the CO 2 emissions on the marginal basis. We will therefore proceed with an allocation according to the A-S method.

Construction of the LP model

The model includes 15 variables (including four slack variables), which are listed in Table 3. Block flow diagrams of the topping/reforming/cracking refinery are given in Figure 1.

Since the straight-run gas oils (from distillation) can be incorporated into automotive diesel fuel or heating oil, we have:

0.37 0 FC FCGODI FCGOHO - - = 0.41 0 SC SCGODI SCGOHO - - =
It is assumed that the cetane number for the automotive diesel fuel must be higher than 46. As in the simplified model proposed by [START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF], the cetane number of the light cycle oil produced by the FCC unit is too low for this product to be incorporated into the automotive diesel fuel. The refinery produces this automotive diesel fuel by blending the straight-run gas oils. The gas oil produced by distilling the first type of crude oil has a cetane number of 51 and a density of 0.83, while the gas oil produced by distilling the second type of crude oil has a cetane number of 42 and a density of 0.84. Since the cetane number for a volumetric blend is supposed to follow a linear law, we must have:

5 4 0 0.83 0.84 FCGODI SCGODI - ≥
The distillate, obtained from distilling either type of crude oil, can either be processed using the catalytic cracker or incorporated into heating oil: 0.24 0.2 0 FC SC DCC DHO + --=

The amount of distillate processed by the FCC unit cannot exceed 850,000 tons per year. The model therefore includes the following capacity constraint:

DCC ≤

The following equation gives the quantity of liquefied petroleum gas (LPG) produced: 0.02 0.03 0.12 0

FC SC DCC LPG + + - =
The refiner anticipates that, each year, the refinery should produce 1.5 million tons of gasoline, 1.7 million tons of automotive diesel fuel, 700,000 tons of heating oil and 800,000 tons of heavy fuel oil (these four values represent the demand vector). It is anticipated that excess production of heating oil and heavy fuel oil can be exported. The refiner assesses the value of liquefied petroleum gas production at US$300 per ton. It is assumed that the excess production of heating oil and heavy fuel oil can be exported at the FOB price of US$40.50 per ton. A portion of this price (US$0.50 per ton) represents the cost associated with CO 2 emissions that would have to be borne by the importer if it had to resort to an alternative source of heating oil or heavy fuel oil. In other words, we estimate that these alternative sources emit 0.017 tons of CO 2 per ton of manufactured product.

We therefore have:

-operating cost, i.e., the cost of purchasing crude oil and processing it minus the assessed value of the liquefied petroleum gas and of the exported heating oil and heavy fuel oil (excluding the portion of the export price associated with carbon emissions): Along the ray to the demand vector, two optimal basic solutions (and therefore two marginal cost vectors) are successively determined. Adopting the notations introduced in section 3.2, the first basic solution is associated with 0 0 λ = . The capacity constraint of the FCC unit is not binding and no heating oil is exported (HOEX is non-basic). The second basic solution is associated with 1 0.8413 λ =

. The capacity constraint of the FCC unit is binding and some heating oil is exported. These two basic solutions are described in table 4. The elementary dual variables associated with the CO 2 emission costs are given in table 5.

Allocating the emissions on the basis of the marginal costs calculated at the optimum would lead to an over-estimation of the quantity discharged by the refinery (in thousands of dollars):

( ) ( ) ( ) ( )

3.84 1500 1.17 1700 0.5 700 0.5 800 8, 499

× + × + × + × =
The elementary A-S cost associated with CO 2 emissions is calculated according to the formula (3) for each finished product. In the case of gasoline for instance, this formula is written (in dollars per ton):

( ) ( )

Conclusion

Petroleum refining entails the simultaneous production of various finished products. The refiner's cost function is thus equal to the value of the linear program minimizing the sum of the variable costs of the refinery under a set of constraints. If the demand constraints are the only binding constraints with a non-zero right-hand side coefficient, the refiner's cost function is a homogenous function of degree 1 and the refinery's total variable costs (including those associated with CO 2 emissions) can be allocated to the various finished products on the marginal basis. In the case of short-run models (with binding capacity constraints) or long-run models incorporating a constraint on emission-permits availability, this type of allocation is not possible. The A-S method can then be used to allocate the refinery's CO 2 emissions (as well as the different variable costs) to the finished products. This has proved to be an easy-touse method. We simply applied the formula (3), which only takes into account the elementary dual variables associated with the emissions cost. Generally speaking, it is possible to calculate an elementary A-S cost for each elementary objective function. All elementary A-S costs are determined in the same way and can be added together, due to the property of additivity of the method (and of the elementary dual variables). Thus, if an emission permits market for SO 2 emissions were established, the contributions of the various finished products to sulphur emissions would be calculated in a manner consistent with the calculation of their contributions to carbon emissions. When the refiner's cost function is homogeneous, the use of the A-S method amounts to allocating emissions to the finished products on the basis of their marginal contribution. In this regard, our approach generalizes the suggestions made by [START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF].

In the literature on axiomatic cost sharing, the A-S method is considered as an extension of average cost pricing, which respects the principle of marginality. We have not discussed the interpretation of this method in the light of cooperative games theory. The A-S method is based on the calculation of the Shapley value when demand is shared by an infinite number of consumers. In a first approach, 0.093 thus represents in our example the expected additional refinery's emissions generated by the consumer of a ton of gasoline when this consumer adds his demand to that of an already existing coalition of consumers (all possible sizes of coalitions are considered and assumed to be equally probable).

Finally, we would stress that, given certain assumptions, one can always determine a production level (and its related system of A-S prices) compatible with consumer demand (i.e. a situation of equilibrium).

  representing the quantities demanded for m finished products.

  the value taken by the elementary function representing the operating cost and 2 ( ) C b the value taken by the elementary function representing the cost associated with CO 2 emissions. We have:

  as the underlying problem falls within the scope of linear programming. Breaking down a marginal cost into two marginal costs then has meaning. For this raison, will also be called the "elementary A-S cost associated with CO 2 emissions".

  -the costs of the refinery's CO 2 emissions (less the cost of the CO 2 that would have been discharged by alternative sources of heating oil and heavy fuel oil if they had not been exported):

	150	FC	+	160	3 SC DCC +	-	300	LPG	-	40	HOEX	-	40	HFEX
	1.5	FC	+	1.2	SC	+	0.6	DCC	-	0,5	HOEX	-	0,5	HFEX
	The objective function to be minimized is the sum of these two elementary functions:
								151.5	FC	+	161.2	SC	+	3.6	DCC	-	300	LPG	-	40.5	HOEX	-	40.5	HFEX
	4.3 Allocation of CO 2 emissions with the Aumann-Shapley formula
	At the optimum, the total cost amounts to US$798,738 thousand. This cost results from the
	sum of the operating costs (US$791,069 thousand) and the cost associated with CO 2
	emissions (US$7,669 thousand). The refinery discharges 255.6 thousand tons of CO 2 .

  The results of the calculation are given in table 5. A finished product's contribution to refinery's emissions is obtained by dividing the corresponding elementary A-S cost by US$30 (emission-permit price). Table6gives the contribution of each of the finished products. Rounded off to the nearest thousand, we get the refinery's total emissions (in thousands of tons):

	(	0.093 1500 ×	) ( +	0.048 1700 ×	) ( +	0.029 700 ×	) ( +	) 0.017 800 255 × =
	A similar calculation allows us to determine the elementary A-S cost associated with
	operating costs (table 7), giving us (in millions of dollars):
	(	) ( 253.80 1.5 161.03 1.7 × + ×	) ( 149.45 0.7 + ×	) ( +	) 40 0.8 791.07 × =
			0.8413 0 2.60 1 0.8413 3.84 2.80 -× + -× =

Table 1 : Yields, operating costs and CO 2 emissions for the distillation unit

 1

	Type of crude oil	First crude	Second crude
	Liquefied petroleum gas	2%	3%
	(Yield per ton of processed
	crude oil)
	Gasoline	14%	22%
	(Yield per ton of processed
	crude oil)
	Gas oil	37%	41%
	(Yield per ton of processed
	crude oil)
	Distillate	24%	20%
	(Yield per ton of processed
	crude oil)
	Residue	19%	10%
	(Yield per ton of processed
	crude oil)
	Purchase and processing costs	150	160
	(US$ per ton of processed crude
	oil)
	CO 2 emissions	0.05	0.04
	(ton per ton of processed crude
	oil)

Table 2 : Yields, operating costs and CO 2 emissions for the FCC unit

 2

	Liquefied petroleum gas	12%
	(Yield per ton of processed
	distillate)
	Gasoline	50%
	(Yield per ton of processed
	distillate)
	Light Cycle Oil	20%
	(Yield per ton of processed
	distillate)
	Residue	12%
	(Yield per ton of processed
	distillate)
	Operating costs	3
	(US$ per ton of processed
	distillate)
	CO 2 emissions	0.02
	(ton per ton of processed
	distillate)

Table 3 : Variables in the model

 3 Each represents a quantity expressed in thousands of tons, with the exception of the slack variable associated with the cetane number constraint for automotive diesel fuel. Processed crude oil FC Processed quantity of the first type of crude oil SC Processed quantity of the second type of crude oil Intermediate products DCC Distillate processed by the FCC FCGODI Gas oil that results from distillation of the first type of crude oil, incorporated into automotive diesel fuel FCGOHO Gas oil that results from distillation of the first type of crude oil, incorporated into heating oil SCGODI Gas oil that results from distillation of the second type of crude oil, incorporated into automotive diesel fuel SCGOHO Gas oil that results from distillation of the second type of crude oil, incorporated into heating oil DHO Distillate incorporated into heating oil Slack variable associated with the cetane number constraint for automotive diesel fuel b Slack variable associated with the demand constraint for gasoline c Slack variable associated with the demand constraint for automotive diesel fuel d Slack variable associated with the cracker's capacity constraint

	Finished products	LPG Total production of liquefied petroleum gas
	and slack variables	HOEX Heating oil exported
		HFEX Heavy fuel oil exported
		a

Table 4 : description of the two successive basic solutions

 4

Table 6 : contribution of each finished product to the refinery's total emissions

 6

	Gasoline	2.60		3.84	2.80
	Automotive diesel	1.50		1.17	1.45
	Heating oil	0.95		0.50	0.88
	Heavy fuel oil	0.50		0.50	0.50
		Gasoline	Automotive	Heating oil	Heavy fuel oil
			diesel
	Contribution to	0.093	0.048	0.029	0.017
	CO 2 emissions
	(ton)

Table 7 : elementary dual variables and A-S costs associated with operating costs (US$ per ton)

 7

	Elementary dual variable associated with
	operating cost:

block flow diagrams of the topping/reforming/cracking refinery

  Evaluation de projets d' investissement par une firme multinationale : généralisation du concept de coût moyen pondéré du capital et conséquences sur la valeur de la firme. février 2004 ECO 57. N. BRET-ROUZAUT, M. THOM, Technology Strategy in the Upstream Petroleum Supply Chain. mars 2005 * une version anglaise de cet article est disponible sur demande

					Liquefied
	ECO 56. A. PIERRU, D. BABUSIAUX,			petroleum gas
	Reforming Yields Second Crude (SC) LPG 3 Gasoline 22 Gasoil 41 Distillate 20 Residue 10 Figure 1: Topping First Crude (FC) LPG 2 Gasoline Gasoil 37 Distillate Residue 19 24 14	DCC	FCC Yields Residue LCO Gasoline 50 12 20 LPG 12	DHO SCGOHO FCGODI FCGOHO SCGODI	Gasoline Oil Heavy Fuel Oil Automotive Diesel Heating
	Gasoline	196.55	527.27	253.80
	Automotive diesel	176.21	80.55	161.03
	Heating oil	170.10	40	149.45
	Heavy fuel oil	40	40		40

We were adopt the term used by[START_REF] Babusiaux | Allocation of the CO 2 and pollutant emissions of a refinery to petroleum finished products[END_REF] and[START_REF] Pierru | Breaking down a long-run marginal cost of an LP investment model into a marginal operating cost and a marginal equivalent investment cost[END_REF].


François Lescaroux

Ceg-1 D Perruchet

Ceg-7 I Cadoret

J.-L Karnik

Ceg-9 A Fauveau

P Chollet

F Lantz

The Economic Consequences of Rising Oil Prices

Keywords: . CADORET, P. RENOU, Élasticités et substitutions énergétiques : difficultés méthodologiques. Avril 1991

Les cahiers de l'économie -n° 59

Série Analyses et synthèses

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à l'Institut français du pétrole, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés.

Les opinions émises dans les textes publiés dans cette collection doivent être considérées comme propres à leurs auteurs et ne reflètent pas nécessairement le point de vue de l'École du pétrole et des moteurs ou de l'IFP.

Pour toute information complémentaire, prière de

Summary

In the context of rising crude oil prices observed in the last five years, this paper attempts to shed light on the possible consequences of a costlier barrel. We shall begin with a brief presentation of the main results of the analyses conducted in the last 30 years, concerning the impact of energy prices on economic activity. We shall then interpret these analyses and their conclusions, and try to draw a number of lessons about the anticipated effects of the recent trend in energy prices.

THE INFLUENCE OF OIL PRICE VARIATIONS: THEORY AND EMPIRICAL

RESULTS

The question of the macroeconomic impact of oil price movements comprises a significant political component, and all sorts of opinions have currency. Some of them, held by a small minority, claim that the price increases are in fact favorable to growth [START_REF] Andrew | A counterintuitive notion: economic growth bolstered by high oil prices, strong oil demand[END_REF] while others find in the same increases premonitory signs of the apocalypse, or at least of a recession. From the quantitative standpoint, the estimates of the impact of an oil price variation on macroeconomic activity fail to agree. As to the United States, for example, elasticities between real GDP and oil price (i.e. the ratio of the rate of change of real GDP to the rate of change of the oil price) evaluated using econometric models, form a wide spectrum, stretching from a value close to -1% for the OECD/IEA (with the INTERLINK model) to -11.62% for James D. Hamilton. Thus from one extreme to the other, a $10 rise in the price per barrel, from $20 to $30, causes an American GDP deviation ranging between about 0.5% to over 5.5%, from the reference level.

The differences between empirical estimates reflect a temporal instability of the relationship, as well as theoretical disagreements about the mechanism by which a fluctuation in the crude price propagates through the economic system and affects the economic situation.

STATIC MACROECONOMIC ANALYSIS

Studies conducted immediately after the first oil shock explained a set of "standard" effects that enjoy a relative consensus.

For any economy, an oil price increase represents an exogenous inflationary shock.

International trade is affected thereby, because the rising price of one of its most widely traded commodities upsets the terms of trade between the net exporting and importing countries. Through a price effect, the latter first suffer a deterioration of their trade balances; over the longer term, exchange rate adjustments and demand of the exporting countries for goods, services and savings, determine subsequent developments.

In terms of national economies, a rise in the price of crude oil is passed on to the price of petroleum products and, from the consumer standpoint, the energy bill of the agents (households, industry and government) grows, whereas from the production standpoint, companies have to contend with a rise in unit costs. As to demand, this slows down consumption expenditures, unless the price change is perceived as short-lived and the agents prefer to maintain their standard of living by reducing their savings or by borrowing (which would exert

The econometric estimates of the macroeconomic consequences of variations in the price of oil disagree… …although the direct effects are well known…

upward pressure on interest rates). In terms of supply of goods and services, a rise in the energy price causes a drop in productivity, which is passed on to 1) real wages and employment; 2) selling prices and core inflation; 3) profits and investment, as well as stock market capitalization.

The theoretical analysis of the mechanisms whereby an oil price increase propagates through the economic system is based on a "decision tree" type of structure: depending on the decisions made by the economic agents, the detriment caused by a costlier barrel is supported in variable proportions by the various types of players (employees, entrepreneurs, shareholders, consumers, investors, etc.).

Figure 1 shows the initial chainings of some of the possible trajectories.

Regardless of the decisions made, real economic growth will be smaller than what it would have been in the absence of an oil price increase. On the other hand, the impact on the other macroeconomic indicators, particularly on value added distribution and real interest rates, is ambiguous and conditioned by the way in which the price increase propagates through the economic system (Figure 1).

Figure 1: Propagation of an oil price increase: Decisions and their implications However, the inclusion of these effects in standard econometric models -systems of structural equations or compact models of the VAR family -has rapidly revealed that the impact of the crude oil price on economic activity is exerted in an unstable manner, both in the short term and over a long period. As to this second form of instability, the statistical equations estimated until the end of the 1970s appear to weaken from the onset of the 1980s, until they become non-significant during the 1990s [START_REF] Mork Knut | Oil and the macroeconomy when prices go up and down: An extension of Hamilton' s results[END_REF], Hooker, 1996a).

…because the temporal instability of the models constructed to quantify these effects has led to opposing interpretations.

This finding has led to various interpretations. Economists who worked on the subject have suggested many explanations, sometimes complementary, often contradictory. Some of them have tried to understand why we observed this progressive attenuation and short term variability of the effects of the oil price movements.

Others, on the contrary, have interpreted the insufficient quality of the estimated equations as the sign that the impact of the oil price had hitherto been improperly construed. They have accordingly attempted to identify the consequences specific to the sudden and striking changes in prices, which would legitimize a nonlinear influence on the aggregate indicators of economic activity, and they try to identify stable relations between "oil shock" measurements, expressing such effects, and the variations in GDP or unemployment rate.

SOURCES OF TEMPORAL INSTABILITY

Apart from the reduced energy intensity of the industrialized countries in the last 30 years, many sources of variability can be found in the relationship between the oil price and economic activity. A higher cost per barrel does not propagate in a predetermined way through the economic system, and the choices of the various agents (individuals, companies, governments and, above all, the monetary authorities) condition development. These choices are obviously strongly influenced by the current situation and by anticipations to various time horizons.

As to the second of these decision making determinants, a learning mechanism has been active in the field of monetary policy. In the long term, the Central Banks no longer rely on the arbitrage between inflation and unemployment expressed by the first version of the "Phillips Curve". The consideration of adaptative expectations has led to long range rectification of the curve (Cf. Inset: the "Phillips Curve").

The economic situation and anticipations play a leading role

On the one hand, the transmission of production costs to consumer prices depends on monetary policy…

The "Phillips Curve"

The economist Alban William Phillips (1958) showed a negative empirical relationship between the unemployment rate and the variation in nominal wages. This correlation suggested that a decrease in the unemployment rate was possible provided a price increase could be tolerated: it largely inspired the Keynesian economics of the 1960s, which were aimed to maintain the economy artificially in a "full employment" situation by boosting demand through an increase in the money supply.

In 1968 however, Milton Friedman warned against the effects of such an interventionist policy, suggesting that a monetary illusion did not exist for a long time and that the agents would soon adjust their expectations to the new context, which would cause the Phillips Curve to shift. Let us assume that the economy is positioned in the situation corresponding to point U, characterized by zero inflation. If the government decides to promote demand and employment by an expansionist money policy, a positive effect will be temporarily obtained if the agents are subject to a monetary illusion (situation V). However, once the companies realize that real demand has not grown and the workers realize that their purchasing power has decreased, labor demand and supply will shrink (situation W). If the government repeats the same policy, the trend would be identical (from W to X and from X to Y), resulting in a commensurately higher inflation than the agents had anticipated. For Friedman, the unemployment rate characterizing point U corresponds to the "natural unemployment rate", i.e; the rate prevailing when real income rises in a natural progression, determined by the savings rate and capital productivity. Any attempt to lower the unemployment rate below this threshold can only be beneficial over a short period, and will culminate in higher inflation: the Phillips Curve is vertical in the long range.

In the early 1980s, during the term of Paul Volcker, controlling inflation became the Fed's top priority. This reversal in the ranking of objectives meant that the monetary authority would react far more drastically to rising price indexes (in a "Taylor Rule" type of equation, this development was reflected by an increase in the g factor, Cf. inset: the "Taylor Rule") and it led to a durable rise in base lending rates.

…which has been conducted differently since the 1980s…

Since then, there is a correlation between oil "shocks" and monetary "shocks". According to Bernanke, Gertler and Watson (1997), the major share of GDP losses following rising crude prices was due to the restrictive policies adopted by the Fed to fight inflation, and particularly to the increases in the Federal Fund Rate, tending to suggest that the new governor of the Federal Reserve Bank, Ben Bernanke, is unlikely to react too vigorously to increasing crude oil prices, as long as inflationary risks are under control.

The "Taylor Rule" Roughly speaking, the behavior of the Federal Fund Rate set by the Federal Reserve Bank can be modelled by a Taylor Rule (1993):

0 g y g g i t y t t + ⋅ + ⋅ = π π
where i t is the nominal short term interest rate, t is the inflation rate and y t is the deviation, in percentage, of the real GDP from its potential. The coefficient g y (negative) determines the response of the Federal Reserve Bank to the change in GDP and g (positive) and g 0 (negative) determine the "target" set by the monetary authority pertaining to the inflation rate and its response to the differences observed. This formulation expresses the arbitrage of the Federal Reserve Bank between its two objectives: stabilize prices and promote growth.

Due to this greater vigilance of the Central Banks concerning price movements, it now appears certain that the increases in prices of petroleum products are no longer passed on to the core inflation (Hooker, 1999b). In particular, inflationary spirals (rising prices/rising wages) like those associated with the first oil shock, need no longer be feared.

In these conditions, however, the empirical analysis of the impact of oil price variations is complicated by the changing practices in the monetary area.

Whether too accommodating or too restrictive, monetary policy is undoubtedly an art of fine tuning, where neutrality is difficult to define objectively.

A final current of monetarist inspiration also considers that the stagflation episodes following the first two oils shocks resulted from the too erratic measures adopted by the Federal Reserve Banks and the "stop and go" policies implemented [START_REF] Barsky Robert | Do we really know that oil caused the great stagflation? A monetary alternative[END_REF].

Monetary policy decisions, and economic decisions in general, also depend on the present context. Raymond and Rich (1997) specifically analyzed the impact of the oil price on the American economy according to the phase of the economic cycle during which the price

…and which proves to pay close attention to changes in oil prices… ...on the other hand, the phase of the economic cycle…

variation occurred. They concluded that a rise is detrimental when it occurs in a period of weak growth or recession, but has no impact in periods of strong growth.

This statistical result can be explained by the changing power relationships between the groups of economic agents during the economic cycle, and by the existence of broader leeway when economic activity accelerates than when it slows down.

The first two oil shocks occurred during deceleration phases of economic activity. In contrast, the present rise is accompanied by a period of acceleration.

As noted by the Federal Open Market Committee, it appears that companies today do not enjoy sufficient market power to raise their selling prices because of the pressure of international competition which deters them from passing on the increases in production cost to their consumers.

For the time being, therefore, costlier oil is primarily reflected in company unit profits.

OIL PRICE VARIATIONS VS "OIL SHOCK"

Empirical research on the impact of oil price movements has benefited by the work done in other areas of economic research. Two patterns have specifically aroused strong interest.

According to the "dispersion hypothesis" [START_REF] Lilien David | Sectoral shifts and cyclical unemployment[END_REF], an exogenous shock can cause inter-and intra-sectoral imbalances (between demand and supply of production factors) which can lead to a durable underutilization of labor and capital resources in certain industries. With respect to employment, in particular, the rise in unemployment would be greater and longer if the agents wrongly expected that the impact of the shock would be short-lived, and if switching from one branch of activity to another incurs a cost.

According to the hypothesis of deferred investment in the presence of uncertainty, the value associated with the option of postponing an investment decision sharply increases when the useful information concerning the future trend in prices of an input is awaited [START_REF] Robert | Irreversibility, uncertainty and investment[END_REF]. Consequently, if the agents have this choice, they will tend to defer their investment decisions in "putty-clay" 1 technologies until the information is known.

Thus, a downward drop in the price of oil could, if it causes sectoral imbalances or greater uncertainty, exert a detrimental "indirect" influence that would offset the favorable "direct" effects. If prices rise, 1 A "putty-clay" production technology displays broad possibilities of substitution before being implemented, but limited possibilities once installed.

… influences value added sharing

Rigidities, market imperfections and anticipations of the agents appear to exacerbate the direct impact of price variations, and make way for indirect influences associated with the absolute value of the variations…

the effects would be magnified. imperfections, particularly employment rigidities and imperfect information, could exert a multiplying effect on the increases and neutralize the beneficial effects of decreases.

The empirical validation of these hypotheses and the quantification of the effects is difficult at the macroeconomic level, because of the need to construct a measurement of the "oil shock" expressing the aggregate, asymmetric and nonlinear influence of price movements. The measurements proposed reflect a wide variety of opinions, such as there are "not any significant effects of oil price declines" [START_REF] Mork Knut | Oil and the macroeconomy when prices go up and down: An extension of Hamilton' s results[END_REF], "it seems more appropriate to compare the current price of oil with where it has been over the previous year rather than during the previous quarter alone" [START_REF] Hamilton James | This is what happened to the oil price-macroeconomy relationship[END_REF]; Figures 6 and7), or "an oil shock is likely to have greater impact in an environment where oil prices have been stable than in an environment where oil price movement has been frequent and erratic" (Lee, Ni and Ratti, 1995; Figure 5). Generally speaking, empirical analyses exploiting "oil shock" measurements display two weaknesses that limit their practical value: from the theoretical standpoint, the measurements used are not justified except by an opinion as expressed in the quotations above (Hooker, 1996b); empirically, the equations estimated using these measurements generally prove to be unstable and non-significant after a few years (Hooker, 1997(Hooker, , 1999a)).

Estimates of equations between the price of oil and economic indicators depend heavily on the effects taken into account in the model and the empirical approach used to construct it. The differences between the elasticities obtained reflect the differences between the assumptions introduced into the models, and particularly in the transformation applied to the price of oil.

The IMF, FRB and OECD (2004) models treat oil price variations as any shocks on supply and obtain elasticities close to -1% for the United States (when monetary policy is not expansionist). Using the OECD model, the IEA estimated ( 2004) that a $25 to $35 increase in the barrel price causes a two-year drop in GDP of 0.3 percentage points in the United States, 0.4 points in Japan and 0.5 points in the Euro zone countries considered as a whole.

Most of the other researches and empirical studies conducted on the subject use American data. For example, Mory (1993) and Mork et al.

(1994) obtained estimates of GDP elasticity to price increases of -5.5% and -5.4% respectively from autoregressive, log-linear regressions of GDP. With the "oil shock" measurements of Hamilton (2003; Figure 7) and of Lee, Ni and Ratti (1995; Figure 5), the results obtained are -11.62% and -5.35% respectively (after 8 quarters). Dotsey and Reid (1992) obtained a cumulative elasticity of -9.4% at the time that the reaction is strongest (after 7 quarters) for a specification using the Federal Fund rate.

…but such mechanisms are difficult to account for in an aggregate model except by constructing an "oil shock" measurement… …and the estimated elasticities are very sensitive to the price or "shock" measurements selected

Jones, [START_REF] Jones Donald | Oil price shocks and the macroeconomy: What has been learned since 1996[END_REF] report that the US Department of Energy (DOE), in its analyses of energy policy, used a value of between -2.5% and -5.5% over the last 15 years.

As to the other countries, and particularly the European countries, far fewer studies have been produced. Worth mentioning are [START_REF] Mork Knut | Macroeconomic responses to oil price increases and decreases in seven OECD countries[END_REF] and [START_REF] Papapetrou | Oil price shocks, stock market, economic activity and employment in Greece[END_REF]. The former obtained estimates of GDP elasticity to price increases of -2.3% for Japan, -8.1% for West Germany, -9.8% for France, -6.4% for Canada, -3.8% for the United Kingdom and 5.1% for Norway. As to Papapetrou, she analyzes the impact of the consumer price index of petroleum products on the Greek economy between 1981MI and 1999M6, and estimates for industrial production and employment elasticities of -2.7% and -0.8% respectively.

More recently, Jimenez-Rodriguez and Sanchez ( 2004) compared the elasticities estimated (for a 100% increase in the leading industrialized countries) using a conventional measurement of the price of oil with those obtained using an "oil shock" measurement: in the former case, the GDP losses are 3.5% in the United States and about 2% for the European countries (including the United Kingdom, an exporter), while the GDP gains of Norway are 1%; in the latter case, the GDP losses are up to 5% in the United States, ranging from 3 to 5% for the European countries (2.28% in the United Kingdom) and the gains of Norway amount to 2.6%.

DISAGGREGATE ANALYSES AND MICROECONOMIC EFFECTS

Some economists have analyzed the influence of oil price on the American economy using disaggregate data, regional [START_REF] Davis Steven | Regional labor fluctuations: oil shocks, military spending, and other driving forces, Board of Governors of the Federal Reserve System[END_REF] or sectoral (Keane and Prasad, 1995[START_REF] Davis Steven | Sectoral job creation and destruction responses to oil price changes[END_REF][START_REF] Lee Kiseok | On the dynamic effects of oil price shocks: A study using industry level data[END_REF][START_REF] Lescaroux François | Le prix du pétrole et les cycles conjoncturels américains[END_REF]. The extreme complexity of these analyses makes it difficult to synthesize their results. In particular, the aggregate impact (on GDP or employment at the national level) of a change in the price of oil depends on the shares of the various regions or the various sectors in the economy, and therefore evolves over time.

In the short term, the price shift, more or less abrupt, vitiates the allocation of production factors (capital and labor), and can also cause a disequilibrium between worker skills and capital characteristics on the one hand, and company needs on the other. Thus, the American automotive industry ran on undercapacity for several years after the first oil shock [START_REF] Bresnahan Timothy | Segment shifts and capacity utilization in the U.S. automobile industry[END_REF] because demand for large vehicles, the models in which the automakers were specialized, fell in favor of compact, less fuel-greedy cars, for which they had neither the technical skills nor the associated commercial skills, particularly aesthetic expertise. The Japanese automotive industry largely exploited this switch in demand by gaining market share.

As to disaggregate analyses, they supply results that are more interesting but are generally not easy to summarize

On average, Davis and Haltiwanger that a positive shock of one standard deviation on the price of oil leads to the destruction of 290 000 jobs and the creation of 30 000 jobs in the first two years following the price hike (for comparison, the 1973-74 shock corresponded to a shock of 1.7 standard deviations and the 1979-81 shock to 2 standard deviations). After four years, the net response consists of the disappearance of 60 000 jobs and the reassignment of 414 000 workers, representing more than 3% of the total industry payroll. Besides, the impact of an oil shock is more pronounced for companies with a high capital to labor ratio, which produce durable goods and which have strong needs for energy in their production systems (these criteria are classed by order of importance).

Moreover, costlier oil has an unequal impact on skilled and other workers (Keane and Prasad, 1995): in the short term, the former suffer a smaller drop in their real wages; in the long term, the likelihood of finding themselves out of a job is lesser.

In the conditions of 1996 for the economic structure (according to the SIC classification into 10 super-sectors), the sectoral energy intensities and the value shares of imports in the GDP and of petroleum products in imports, Lescaroux estimates a short term elasticity of -3.1% between real US GDP and the production price index for petroleum products. Most of the analyses mentioned could turn out to be of limited use in the present situation. We are in fact faced with a rise in the equilibrium

WHAT CAN WE EXPECT FROM THE RECENT RISE IN THE PRICE OF OIL?

It is important to distinguish between the impact of an upward disequilibrium and an upturn in the equilibrium

price of oil (assimilated to the low frequency component2 in Figure 2). The published results are generally obtained using models that do not differentiate between the variations in the high frequency component and those in the low frequency component, and in which the greatest confusion appears to surround this distinction.

THE CONTRASTED EFFECTS OF CYCLES AND BREAKS IN THE PRICE OF

OIL

The "standard" effects thus identified are clearly exerted in the short term in case of an upward rise in prices.

On the other hand the appearance of inter-and intra-sectoral imbalances or the deferral of investment decisions can only result from a durable price increase like the one observed today.

For the sake of simplification, we can therefore consider that the analyses that use price variations as a petroleum variable (Figure 3 and 4), quantify the "standard" effects associated with the fluctuations in crude prices about its equilibrium level, while those based on "oil shock" measurements (Figures 3 and 5 to 7) try to evaluate the consequences of a break in the equilibrium level. This is not, strictly speaking, true because the former do not completely eliminate the low frequency component of the time signal representing the price of oil by considering it as a range of variation and, above all, the latter do not perfectly neutralize the high frequency component with the filters they apply in order to construct their "oil shock" measurements (particularly those of [START_REF] Lee Kiseok | Oil shocks and the macroeconomy: The role of price variability[END_REF][START_REF] Hamilton James | This is what happened to the oil price-macroeconomy relationship[END_REF].

QUANTIFYING THE IMPACT OF CYCLES AND BREAKS IN THE OIL PRICE

Accepting this reservation, the elasticities of real American GDP with respect to an upward imbalance and with respect to a rise in the equilibrium price would accordingly lie approximately in the ranges extending, respectively, from -1% to -5.5% and from -5% to -11.5%.

The interval in the former case reflects the variability of the impact of an oil price fluctuation according to its shape and to a set of short-and long-run parameters. [START_REF] Lescaroux François | Le prix du pétrole et les cycles conjoncturels américains[END_REF] explicitly models the relationships between the high frequency components of the economic variables and the high frequency component of the petroleum product price index. The elasticity of real American GDP and the oil price is especially higher when the imbalance is longer and sharper, and increases very slowly in absolute value, with the maximum amplitude. The elasticity of -3.1% mentioned earlier corresponds to the reaction, in 2 The low frequency component of crude expresses its long run variations. It has been extracted with a "window" low-pass filter. This kind of filter consumes data at the extremities of the sample; therefore, we also approximated it with a Hodrick-Prescott filter with a parameter of 14400.

The extent of the range of estimated elasticities is partly explained by the fact that the analyses address distinct

problems.

The fluctuations in the oil price have a moderate impact on economic activity with a decreasing

trend, despite some variability the 1996 situation, of the GDP to a rectangular imbalance lasting six quarters and with an amplitude of 10%. For a period of one quarter (respectively 8 quarters), the elasticity becomes lower (resp. higher) in absolute value, than -1% (resp. -3.5%). These elasticities are obtained by imposing an oil price variation to the economic system considered at its long-run steady-state equilibrium; they therefore correspond to a mean response. The recessionary impact is deeper (resp., less deep) when the price rise occurs in a decelerating (resp., accelerating) phase of the business cycle. The relationships are stronger when the model is simulated in the structural conditions of 1973, in terms of the relative shares of the industrial sectors, the share of importations in GDP and the share of petroleum products in total importations.

The interval covered by the estimates of the impact of a break in the price of oil, which is the one of interest to us in the current situation, must be regarded with extreme caution.

As we have already pointed out, the numerical values of the elasticities obtained depend heavily on the choice of the oil shock measurement used, whereas the main functional forms proposed are not theoretically justified and empirically lead to regressions which turn out to be unsatisfactory only a few years after their publication. This search for an ad hoc filter that would help to reveal a stable and significant relationship between the oil price and the GDP or the unemployment rate, has been compared to data mining by [START_REF] Bernanke Ben | Systematic monetary policy and the effects of oil price shocks[END_REF].

Generally speaking, we can question the relevance of an econometric approach when quantifying the consequences of exceptional events. In fact, the breaks in the record of the price of oil are relatively rare and have, so far, coincided with shocks of other natures (end of the Bretton Woods monetary system in the early 1970s, tightening of American monetary policy and rise of the dollar in the early 1980s). It is therefore difficult to distinguish statistically between the respective consequences of these various influences. Moreover, due to their violence, these events encourage a vast number of researches and analyses aimed at a closer understanding and better identification of the least appropriate decisions, in order the avoid the mistakes of the past. Thus for example, the greater vigilance of the Federal Reserve Banks with respect to inflation will no doubt help in the future to avoid price/wage spirals. Similarly, the price control policies implemented in most of the western countries in the early 1970s revealed their ineffectiveness and their dangers. Thus the oil shocks appear to be not enough frequent and their consequences appear to be too variable to lend themselves satisfactorily to econometric analysis.

Without even going as far as the use of a statistical model to predict the consequences of costlier oil, it could be a mistake to refer to the experiences of the 1970s and 1980s. In fact, shifts in the equilibrium

The consequences of a change in regime like the one observed today are more difficult to analyze… …and by other types of analysis

level of the price of oil bring structural adjustments of the economic system and of society as a whole in their wake.

After the first two oil shocks, the rise in prices encouraged users to control their consumption and to optimize it to enable marginal energy products to adjust to their prices. This was achieved thanks to efficiency gains in plant and equipment. Often, the industrial structure also refocused on sectors offering the lowest energy intensities. And energy consumption became more efficient, with the allocation of the various forms of energy to the uses for which they offer the best yields: the energy mix diversified and, for instance, oil is more and more consumed in the transport sector, where its specific characteristics are exploited to the maximum.

Moreover, the globalization and the accelerated development of some countries has induced changes in the international trade in goods and services. Oil revenues have predominantly benefited to western countries after the first two oil shocks; the share of emerging countries is actually close to 50% in the total imports of the principal oil exporting countries (I.M.F., 2006). The I.M.F. also note that these revenues are, till now, recycled less rapidly than during the 70s and 80s. However the growing perception of a rise in the long-run equilibrium level of oil prices might soon lead to an acceleration in consumption expenditures of many important exporters who face a rise of their needs resulting from the strong demographic pressure.

Thus, the analysis of the past can just shed a little light on the forthcoming evolutions for two reasons: the structural conditions of the rise are not the same; and the efforts made to adapt can be continued but not repeated. The scale of the energy savings achievable without contracting production capacity is difficult to assess.

One of the chief factors of uncertainty associated with the current rise in oil prices concerns the forthcoming demand pattern in the developing countries. During the first two oil shocks, the increase in consumption essentially occurred in the OECD countries, and after the fleeting attempts to establish price controls, the rise in consumer prices allowed a self-corrective mechanism to act by a levelling off in demand. In recent years, incremental demand comes mainly from the emerging countries, where energy price policies are governed by social rather than economic considerations. Olivier Rech (2005) drew attention to the multiplication by a factor of two to five of the cost associated with the subsidy systems in a group of big oil consuming countries, like China, India, Indonesia, Thailand and Egypt. Despite the impact on the standard of living of their population and the attendant political risks, the governments of these countries cannot allow their public finances to deteriorate indefinitely. At their rates, they will have to let consumer prices rise. Here and there, we are starting to observe such developments (in Thailand for example) as well as their early effects on demand for petroleum products.

Figure 2 :

 2 Figure 2: Brent spot price and its high and low frequency components.

Figure 3 :Figure 4 :

 34 Figure 3 : Real price oil and its spectral high and low frequency components.

Figure 5 :

 5 Figure 5 : "Oil shock" variable of Lee et al. (1995).

  Figure 6 : "Oil shock" variable of Hamilton (1996).

Figure

  Figure 7 : "Oil shock" variable of Hamilton (2003).