Stefano Bosi
email: stefano.bosi@universite-paris-saclay.fr.

Carmen Camacho
email: carmen.camacho@psemail.eu.

David Desmarchelier
email: david.desmarchelier@gmail.com.

Human capital and welfare

Keywords: human capital, unbalanced growth, transversality condition I00, O11, C61 human capital, unbalanced growth, transversality condition

, where human capital has an additional positive effect on social welfare through the quality of individual health and education. In a simple economy with a Cobb-Douglas technology and logarithmic preferences, we provide the explicit trajectories for human capital, consumption and the HDI, which correspond to the Balanced Growth Path (BGP). Using a two-step maximization strategy, we compute the optimal initial value of the control variable, in this case, the initial optimal labor supply. In other words, we prove the optimality of the BGP. We highlight a HDI crossing property: the propensity to consume has a positive effect on the HDI in the short run, but negative in the long run. Finally, both the growth rate and overall welfare are proven to decrease in the propensity to consume.

Introduction

When Mahbub ul Haq introduced human development as a compound process which included education and life expectancy, he paved the way for a new avenue of research in development economics. The introduction of the Human Development Index (HDI) in ul [START_REF] Haq | Reflections on Human Development[END_REF] led to a new paradigm and to a modern theory of human development. However, although his works have influenced prominent economists like Amartia Sen, most growth theorists still focus on the utility of consumption. Stemming from the common belief that educated and healthy people enjoy life differently, the present paper considers that human capital is made of both education and health. Then we incorporate the HDI into the utility function defining a new composite good, which generalizes the consumption good. The new good embodies the notion that human capital is crucial to appreciate consumption, and it allows us to study the cross effect of human capital on the marginal utility of consumption.

While the seminal notion of human capital was introduced by Adam Smith in 1776 and later by Arthur Cecil Pigou in 1928, the modern theory of human capital can be traced back to [START_REF] Schultz | Investment in human capital[END_REF] and [START_REF] Becker | Human Capital: A Theoretical and Empirical Analysis, with Special Reference to Education[END_REF]. [START_REF] Uzawa | Optimum technical change in an aggregative model of economic growth[END_REF] was first to incorporate human capital as an engine of growth in a theoretical model. The emergence of a new endogenous growth literature stimulated the interest of economists in the role of human capital. Building on [START_REF] Rosen | A theory of life earnings[END_REF], [START_REF] Lucas | On the mechanics of economic development[END_REF] underlined that the accumulation of human capital can trigger a mechanism of perpetual growth. In particular, he shows that the growth rate of income per capita depends on the growth rate of human capital, which in turn depends on the time individuals use to acquire skills and to protect their health. After [START_REF] Lucas | On the mechanics of economic development[END_REF], the literature developed fast, scrutinizing all factors underlining the formation of human capital: education, transmission, health, culture,... 1 Nevertheless, the literature becomes thinner regarding the other various roles of human capital in an economy. Although sociologists report that as a matter of fact, education does affect life enjoyment, research in economics has for the most part neglected the role of human capital in welfare. Among the studies in sociology let us mention [START_REF] Ross | The links between education and health[END_REF], who find that well educated individuals have a more fulfilling job and a better control over their lives. Moreover, they are in a healthier condition since they smoke and drink less. Similarly, [START_REF] Finkelstein | What good is wealth without health? The effect of health on the marginal utility of consumption[END_REF] point out a complementarity between health and consumption demand: an increase of 1% in the number of chronic diseases reduces the marginal utility of consumption from 10% to 25%. From an economist's viewpoint, this empirical evidence suggests that human capital does increase the marginal utility of consumption.

In order to consider the effect of education and health on wellbeing, it is necessary to enlarge the standard view of household's preferences. Nevertheless, introducing human capital in the utility function requires careful attention. Indeed, from a theoretical perspective, the effects of human capital on the marginal utility of consumption are potentially ambiguous. On the one hand, human capital increases the marginal utility of consumption when, for instance, a well educated individual watches a movie and fully grasps all cultural references; on the other, human capital decreases the marginal utility of consumption when she becomes aware about the environmental damages of consumption. Empirical studies are not definite about the cross effect of human capital on consumption demand.

To our knowledge, [START_REF] Chakraborty | A note on the inclusion of human capital in the Lucas model[END_REF] is the only model à la [START_REF] Lucas | On the mechanics of economic development[END_REF] where human capital has the two essential roles we have underlined. There, human capital is, at the same time, a production factor and a source of wellbeing, and it enters both in the production and in the utility functions.

When human capital affects more the agent's preferences than consumption, then they find that the growth rate is higher along the Balanced Growth Path (BGP). When it affects less the production function, then multiple steady states coexist. Note however that the authors do not study the optimality of their BGP solution. Our objective is to address the important question of the role of human capital in wellbeing by considering a growth model where, as in [START_REF] Lucas | On the mechanics of economic development[END_REF], the household chooses the working time and the time devoted to acquire skills. Following Ben-Porath (1967), there is no physical capital, but human capital affects both the production and the utility function as in [START_REF] Chakraborty | A note on the inclusion of human capital in the Lucas model[END_REF].

Despite the absence of an optimality analysis in [START_REF] Chakraborty | A note on the inclusion of human capital in the Lucas model[END_REF], intuition suggests that the optimality of the BGP is a relevant question when human capital enters preferences. Indeed, along the BGP, human capital grows at a constant rate by definition. Now, suppose human capital enters the utility function with standard properties, that is, a decreasing marginal utility. Then, depending on preferences, there may exist a critical level of human capital beyond which a further increase does not confer any extra utility. Beyond the threshold, the household stops acquiring skills and devotes all her time to work, in contradiction with the perpetual growth of human capital along a BGP. This paper aims precisely at questioning the optimality of the BGP when human capital is an argument of the utility function. Unlike [START_REF] Chakraborty | A note on the inclusion of human capital in the Lucas model[END_REF], we reinterpret here the composite good as the mixture of consumption and human capital, in the form of a Human Development Index. Under a Cobb-Douglas technology and logarithmic preferences, we are able to provide the explicit trajectories for capital, consumption and HDI not only because we obtain the analytical solution to the system of differential equations, but also because we compute the initial control (labor supply) in a two-step maximization, which is new in growth literature. We actually prove that the optimal initial value of labor supply belongs to the BGP, demonstrating that the BGP is actually optimal from the start. Interestingly we highlight a HDI crossing property: the propensity to consume has a positive effect on consumption and, thus, on the HDI in the short run, but a negative effect in the long run. Finally, both the growth rate and welfare are proven to be decreasing in the propensity to consume, while, interestingly, the impact of time preference on welfare is positive when the initial human capital is below a threshold because of a scale effect.

The rest of the paper is organized as follows. The model is presented in Section 2. Section 3 considers the optimal solution when both utility and technology are isoelastic functions. Assuming a unit elasticity of substitution both in preferences and in the HDI, Section 4 provides the explicit optimal solutions for human capital and consumption. Finally, Section 5 concludes. All the proofs are gathered in the Appendix.

The model

In the spirit of [START_REF] Chakraborty | A note on the inclusion of human capital in the Lucas model[END_REF], we assume that human capital (health and education) increases workers' productivity and households' utility. There exists a representative household, who lives forever and maximizes the infinite-horizon utility:

max ! ∞ 0 e -θt u (x (t)) dt (1)
choosing the optimal trajectories of consumption, c, and labor, l, and where the HDI2 , x (t) ≡ x (c (t) , h (t)), is an increasing function of consumption and human capital. Optimal decisions are subject to the human capital accumulation law:

ḣ (t) h (t) ≤ B [1 -l (t)] (2)
the resource constraint:

c (t) ≤ y (t) ≡ h (t) l (t) (3)
and the initial condition h(0) ≡ h 0 . Output y (t) is a linear function of labor, whose productivity is precisely human capital. At each date in time, the representative household is endowed by one unit of time that she arbitrates between the working time in the firm, l (t) ∈ [0, 1], and the time spent to acquire skills, 1 -l (t). As in [START_REF] Lucas | On the mechanics of economic development[END_REF], the growth rate of human capital is a linear function of 1 -l (t), and it attains its maximum value B when l(t) = 0, that is, when the household's labor is zero.

For simplicity, we will omit the time argument in the following.

Assumption 1 Function x : R 2 + → R + is C 2 , strictly increasing and homogeneous of degree 1. u is C 2 , strictly increasing and strictly concave.

From (3) and the principle of non-satiation, c is uniquely determined by l for a given h. That is, the representative household chooses a unique control l to maximize her discounted intertemporal utility. In this context, an admissible control l is a locally integrable function l : [0, +∞) → [0, 1] satisfying (2) with h(0) ≡ h 0 > 0.

In order to solve the household's program, let us introduce the Hamiltonian function:

H (h, l, λ, t) ≡ e -θt u (x (hl, h)) + λB (1 -l) h (4)
The following proposition characterizes the optimal behavior of human capital, h, and of µ ≡ λe θt , its shadow price.

Proposition 1 (necessary conditions) Consider program (1) under Assumption 1, and assume that there exists an interior continuous solution (l, h) ∈ [0, 1] × [0, +∞). Then, there exists a continuously differentiable function µ > 0 such that the optimal control l and the corresponding state variable h satisfy the following necessary conditions:

µ = u ′ (x) B ∂x ∂c (5)
μ µ = θ -B (1 -l) - u ′ (x) µ " ∂x ∂c l + ∂x ∂h # (6) ḣ h = B (1 -l) (7)
lim t→∞ e -θt µh ∈ R (8)
Proof. See the Appendix.

Applying the implicit function theorem to the optimal condition (5), l can be written a function of h, that is l ≡ l * (h, µ, t) ≡ l * (h). This observation allows us to prove that the set of necessary conditions in Proposition 1 are not only necessary but also sufficient. 3 Proposition 2 (sufficient conditions) Under Assumption 1, conditions (5) to [START_REF] Borissov | Heterogeneous human capital, inequality and growth: The role of patience and skills[END_REF] are necessary and sufficient optimal conditions for (1) if the Arrow-Kurz sufficient condition holds:

∂ 2 x ∂c 2 $ l * (h) + h l * ′ (h) % 2 + 2 ∂ 2 x ∂c∂h $ l * (h) + h l * ′ (h) % + ∂ 2 x ∂h 2 < 0 (9)
together with a stronger transversality condition:

lim t→∞ e -θt µ * h * = 0
Proof. See the Appendix. The following proposition provides the optimal trajectories for human capital, labor supply and consumption demand.

Proposition 4 (dynamic system) The optimal solution (h, l) * satisfies the following dynamical system:

ḣ h = B (1 -l) (10)
ll = θ -B -B ∂x/∂h ∂x/∂c -B (1 -l) $ xu ′′ (x) u ′ (x) & c x ∂x ∂c + h x ∂x ∂h ' + c ∂x/∂c ∂ 2 x ∂c 2 + h ∂x/∂c ∂ 2 x ∂c∂h % c x ∂x ∂c xu ′′ (x) u ′ (x) + c ∂x/∂c ∂ 2 x ∂c 2 (11)
where c = hl.

Proof. See the Appendix.

The computation of the optimal solution in the general case presented in Proposition 4 turns out to be impossible. However, in the case of isoelastic functional forms, the two-dimensional system ( 10)-( 11) boils down to a single differential equation, easier to study.

The case of isoelastic functional forms

Consider the isoelastic functions:

u (x) ≡ ε ε -1 x ε-1 ε (12)
x (c, h) ≡

( ac σ-1 σ + bh σ-1 σ ) σ σ-1 ( 13
)
where ε is the (dynamic) elasticity of substitution between the HDI x today and tomorrow, and σ is the (static) elasticity of substitution in the HDI between the consumption c and the human capital h today. Notice that there is no loss in generality if we assume

a + b = 1 (14) Indeed, if a + b ) = 0, then ! ∞ 0 e -θt ε ε -1 * ( ac σ-1 σ + bh σ-1 σ ) σ σ-1 + ε-1 ε dt = $ (a + b) σ σ-1 % ε-1 ε ! ∞ 0 e -θt ε ε -1 * ( ãc σ-1 σ + bh σ-1 σ ) σ σ-1 + ε-1 ε dt where ã ≡ a a + b , b ≡ b a + b
, and ã + b = 1 and, thus,

arg max ! ∞ 0 e -θt ε ε -1 * ( ac σ-1 σ + bh σ-1 σ ) σ σ-1 + ε-1 ε dt = argmax ! ∞ 0 e -θt ε ε -1 * ( ãc σ-1 σ + bh σ-1 σ ) σ σ-1 + ε-1 ε dt
Proposition 5 (constant elasticities of substitution) If the utility function and the HDI are defined by ( 12) and ( 13), then the Arrow-Kurz criterion ( 9) is satisfied. As a result, the first-order conditions ( 5) to ( 7) jointly with the transversality condition are necessary and sufficient for utility maximization.

Optimal labor supply is driven by a single differential equation:

l = Bl " l + ε b a l 1 σ -1 -ε θ -B B # al σ-1 σ + b al σ-1 σ + b ε σ ≡ ϕ (l) (15)
Proof. See the Appendix.

Proposition 6 (Balanced Growth Path) There exists a non-zero stationary solution l to [START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF], which is a solution to the following equation:

l + ε b a l 1 σ = 1 + ε θ -B B (16)
The steady state l for labor supply determines the human capital growth rate

g = B & 1 - l'
. Furthermore, human capital and consumption grow at the same rate, that is, the economy follows a BGP:

h (t) = e B(1-l)t h 0 (17) c (t) = e B(1-l)t h 0 l (18)
The BGP in ( 17)-( 18) is globally unstable.

Proof. See the Appendix. One of the main issues of this paper is the analysis of the impact of human capital on consumption demand. More precisely, we are interested in the impact of human capital on the marginal utility of consumption. As it follows from [START_REF] De La | Life expectancy and endogenous growth[END_REF], this effect is negative (positive) if consumption and human capital are substitutes (complements). Hence, the elasticity of consumption-human capital substitution affects the optimal solution to the initial program, which generically differs from the BGP.

According to [START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF], labor supply l (t) decreases when the initial condition l 0 < l, and it increases otherwise. Note that the dynamics of labor are not necessarily monotonic nor symmetric on either side of l. l (t) can decrease to zero if l 0 < l, while, if l 0 > l, it reaches 1 in a finite lapse of time T . After date T , human capital stops growing.

The elasticity of consumption-human capital substitution, σ, affects the critical value l and, as such, it plays a key role in the dynamics of labor supply and human capital accumulation. Indeed, totally differentiating ( 16), we get

σ l ∂ l ∂σ = B & 1 - l' + ε (θ -B) σaB l + B & 1 - l' + ε (θ -B) ln l = εb l 1 σ σa l + εb l 1 σ ln l < 0
Hence, σ has a clear negative impact on l:

(1) the more substitutable are consumption and human capital, the lower is l and, hence, the more likely is that l 0 > l. That is, the more plausible is the situation in which labor supply increases over time until the critical date T beyond which the household stops investing in human capital;

(2) the less substitutable are consumption and human capital, the larger is l. Thus, the more likely is that l 0 < l, that is, the more plausible a decrease of labor supply to zero ensuring, asymptotically, the largest human capital (and consumption) growth rate g = B.

In other words, when human capital lowers the marginal utility of consumption (substitutability case), the individual no longer invests in human capital in the long run, or more precisely, beyond the critical date T . Conversely, when human capital raises the marginal utility of consumption (complementarity case), the individual wants to invest in human capital at the maximal rate (B) in the long run.

For the sake of precision, let us observe that the optimal starting point l * 0 is endogenous. Consequently, even a very narrow (large) interval & 0, l' does not ensure that l * 0 > l (neither that l * 0 < l). In order to avoid any heuristic interpretation and to determine unambiguously whether the optimal initial labor supply l * 0 is lower or higher than l, in the following example, we fix σ = 1.

Optimal BGP in a simple economy

In order to obtain a complete description of the optimal solution, let us consider logarithmic preferences: u (x) ≡ ln x, with a Cobb-Douglas HDI: x (c, h) ≡ c α h 1-α . These functions have unit elasticities of substitution: ε = σ = 1. Indeed, taking the limit of the logarithm of ( 13) and applying the de l'Hôpital's rule, we get

lim σ→1 ln x (c, h) = lim σ→1 ln ( ae σ-1 σ ln c + be σ-1 σ ln h ) σ-1 σ = a ln c + b ln h a + b = ln & c a h b '
because a + b = 1 according to [START_REF] De La | Life expectancy and endogenous growth[END_REF]. Thus, x (c, h) = c a h b in the limit. Setting α = a = 1 -b, we recover the Cobb-Douglas HDI function x (c, h) = c α h 1-α .

In the Cobb-Douglas case, we are able to compute explicitly the optimal initial value for labor supply and to prove that the optimal initial value for labor supply (l * 0 ) is precisely αθ/B. This value corresponds to the BGP. In other words, in the Cobb-Douglas case, the optimal trajectory is unique and coincides with the BGP.

To prove the optimality of the BGP, we adopt a constructive strategy based on the explicit solutions of the dynamic system. More precisely, following Borissov et al. (2018), we remaximize (1) with respect to l 0 under the system of constraints ( 10)- [START_REF] Dechert | A complete characterization of optimal growth paths in an aggregated model with a non-concave production function[END_REF].

The following proposition provides the optimal paths for human capital and consumption demand depending on the value of α, B and θ.

Proposition 7 (optimal path) If ε = σ = 1, the optimal labor supply is given by l * (t) = αθ/B and the optimal growth path is the BGP:

h * (t) = h 0 e (B-αθ)t (19) c * (t) = α θ B h 0 e (B-αθ)t (20)
x * (t) =

" α θ B # α h 0 e (B-αθ)t (21)
for any t ≥ 0. Along the BGP, the transversality condition is zero:

lim t→∞ e -θt µ (t) h (t) = e -θt µ * 0 h 0 = 0
Proof. See the Appendix.

In light of Proposition 7, we can easily compute the impact of the three main parameters α, θ and B on the BGP in terms of the elasticities of h * (t), c * (t), x * (t). Furthermore, their impact on the balanced growth rate and on the welfare functional also obtain straightforwardly.

Let us introduce the following critical dates:

T x,α ≡ 1 + ln l * θ , T x,θ ≡ 1 θ , T c,α ≡ 1 αθ , T x,B ≡ α B , T c,B ≡ 1
B
where l * = αθ/B is the optimal labor supply. We observe that T x,α < T x,θ < T c,α . Moreover, T x,B < T c,B and T x,B < T x,θ .

The following four corollaries directly stem from Proposition 7. Let us start by considering the effects of propensity to consume on the BGP.

Corollary 8 (propensity to consume) The impact of α on human capital is negative at any date. Its impact on consumption demand is positive in the short run, when t < T c,α , and it becomes negative thereafter. Similarly, the impact of α on the HDI is positive in the short run, while t < T x,α (< T c,α ), and negative in the long run, when t > T x,α . Note that if T x,α < t < T c,α , the impact of α is positive on consumption but negative on the HDI. In addition, the impact on the HDI is always negative if T x,α < 0, that is if α < B/ (θe).

Proof. See the Appendix.

As Corollary underlines, α is a new important incoming parameter. When the HDI is given by ( 13) and σ = 1, the HDI is defined as a geometric average of income, which equals consumption, and human capital with weights α and 1 -α, respectively. Hence, α captures the propensity to consume by construction, while 1 -α represents the propensity to invest in education and health. Consequently, the larger the propensity to consume, the higher the labor supply and the lower the capital accumulation at any time.

There exists a critical moment in time, T c,α , such that the reaction of c * to α overturns. Indeed, in the short run, the higher labor supply increases consumption, while the lower human capital accumulation reduces both labor productivity and consumption in the long run. The following figures depict the crossing property plotting consumption demand and HDI for two different values of α, namely 1/3 and 2/3. For this exercise, h 0 = 1, B = θ = 0.01 (quarterly discounting). The solid lines correspond to the lower propensity to consume, i.e. to α = 1/3. η c,α and η x,α respectively stand for the elasticity of c and x with respect to α.

Recall that the impact of α on the HDI is positive in the short run and negative in the long run if α > B/ (θe). Under our calibration this is the case when α = 1/3. However, α always affects negatively the HDI when α = 2/3. Here again, the higher labor supply increases consumption in the short run. In the long run, the lower human capital accumulation reduces both labor productivity and consumption. Next, let us consider the effects of time preference on the BGP.

Corollary 9 (time preference) The impact θ on the human capital is negative at any date. The impact on consumption is positive in the short run, when t < T c,α , and negative thereafter. θ makes the HDI increase in the short run, when t < T x,θ , and decrease in the long run, when t > T x,θ . Since T x,θ < T c,α , if T x,θ < t < T c,α , then the impact of time preference is positive on consumption but negative on the HDI.

Proof. See the Appendix.

The effect of θ on both h * (t) and c * (t) mimics exactly the effect of α in strength and sign. Note in particular that the elasticity of c * with respect to θ also changes at T c,α .

Since returns to human capital investment are not instantaneous, a more impatient household's reduces the time to accumulate human capital and increases her labour supply. Production and consumption increase in the short run. However, the lack of human capital accumulation reduces production in the long run, lowering consumption in turn.

In the short run, the impact of θ on the HDI is positive because the positive effect on consumption is sufficiently large to compensate for the negative effect on capital. However, the impact becomes negative because either the negative effect on capital dominates the positive effect on consumption, or because both effects become negative.

Finally, let us consider the impact of the productivity of human capital formation on the BGP.

Corollary 10 (productivity of human capital formation) The impact of B on human capital is always positive. B affects negatively consumption demand in the short run, when t < T c,B , and positively in the long run, when t > T c,B . Similarly, the impact of B on the HDI is negative in the short run, when t < T x,B (< T c,B ), and positive in the long run, when t > T x,B . If T x,B < t < T c,B , then the impact of time preference is negative on consumption demand but positive on the HDI.

Proof. See the Appendix.

B has opposite effects on h * (t) and c * (t) with respect to θ. The higher the productivity of human capital formation, the more households will invest in human capital. Then the higher B, the lower the labor supply in the short run, and as a consequence, the lower the consumption. In the long run, when t > T c,B , the induced larger stock of human capital allows for higher production and consumption.

In the short run, when t < T x,B , the impact of B on the HDI is negative because the negative effect on consumption is sufficiently large to compensate for the positive effect on capital. However, the impact of B becomes positive in the long run, when t > T x,θ , because either the positive effect on capital dominates the negative effect on consumption, or because both effects become positive.

It is important to compare the effects on the optimal growth rate with those on the optimal welfare functional. In this regard, let us introduce a threshold for the initial human capital:

h0 ≡ e -2(1-l * ) B θ l * α
Corollary 11 (growth and welfare) (1) The effects of α and θ on the balanced growth rate are always negative, while the impact of B is always positive.

(2) The impact of α on the optimal welfare level W * ≡ , ∞ 0 e -θt ln x * (t) dt is non-positive at any date, while the effect of B is always positive. The impact of θ on W * is positive if the initial human capital is low (h 0 < h0 ) and negative if the initial capital is large (h 0 > h0 ).

Proof. See the Appendix.

The negative impact of the propensity to consume on growth is not surprising. Indeed, economic growth is driven by the growth rate of human capital, which decreases as α decreases.

The time preference θ and the productivity of human capital formation B have opposite effects. As already underlined, a less patient household reduces the time devoted to human capital accumulation. Conversely, a higher human capital productivity always fosters human capital accumulation, which increases economic growth along the BGP.

Interestingly, the changing-in-time effects that α has on consumption are not present for welfare. α always has a non-positive effect on welfare, even in the short run when consumption increases. In order to understand the role of each and all of the model's elements, note that a higher α reduces the length of the period during which α has a positive effect on consumption ∂T c,α /∂α = -1/ & θα 2 ' < 0. The permanent negative effect of α on human capital formation and on consumption from T c,α onwards always dominates the short-run positive effect on consumption.

Regarding B, a higher human capital productivity always increases human capital while it increases consumption only in the long run. Here, the positive effect always dominates the negative short-run effect leading to a higher overall welfare.

As discussed before, returns on human capital investment are not instantaneous and, as a result, a less patient household increases her labour supply, which increases consumption in the short run and lowers it in the long run. Moreover, an increase in impatience always lowers human capital. According to [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF], the drop in human capital induced by θ is amplified by the initial value h 0 . Therefore, overall welfare decreases with θ if h 0 is sufficiently high.

Conclusion

In this paper, we have extended the model of human capital accumulation published by Lucas in 1988 by considering instead of consumption a more general Human Development Index. Considering a Cobb-Douglas technology and logarithmic preferences, we have provided the explicit trajectories for human capital, consumption and the HDI, proving the optimality of the balanced growth path. Finally, we have highlighted a HDI crossing property: the propensity to consume has a positive effect on the HDI in the short run, but a negative impact in the long run. Finally, both growth rate and welfare decrease with the propensity to consume.

Appendix Proof of Proposition 1

We define a feasible solution to be a trajectory (h, l) which satisfies the initial condition for h, h(0) = h 0 and the law of motion (2).

To apply Ekeland's variational principle, we build a value function V (h, l, λ) as follows:

V (h, l, λ) ≡ ! ∞ 0 e -θt u (x (h (t) l (t) , h (t))) dt + ! ∞ 0 λ * (t) ( h (t) B [1 -l (t)] -ḣ (t) ) dt (22)
where λ * is the multiplier function associated to the optimal solution (h, l) * . We can rearrange the last term in the second integral using integration by parts:

! ∞ 0 λ * (t) ḣ (t) dt = [λ * (t) h (t)] ∞ 0 - ! ∞ 0 λ * (t) h (t) dt = lim T →∞ λ * (T ) h (T ) -λ * (0) h (0) - ! ∞ 0 λ * (t) h (t) dt
Replacing this into [START_REF] Pontryagin | Mathematical Theory of Optimal Processes[END_REF], we obtain

V (h, l, λ) = ! ∞ 0 e -θt u (x (h (t) l (t) , h (t))) dt + ! ∞ 0 ( λ * (t) h (t) + λ * (t) h (t) B [1 -l (t)] ) dt +λ * (0) h (0) -lim T →∞ λ * (T ) h (T ) (23)
According to Ekeland's variational principle, if there exists an optimal solution (h, l) * , then any other trajectory (h, l) can be written as a deviation from the optimal.

Hence, for every feasible pair (h, l) ) = (h, l) * and a constant ε ∈ R/ {0}, there exist functions H, L : R + → R such that

H (t) ≡ [h (t) -h * (t)] /ε L (t) ≡ [l (t) -l * (t)] /ε that is (h (t) , l (t)) = (h * (t) , l * (t)) + ε (H (t) , L (t))
(H, L) represents a given direction in the function space. Any feasible solution can be written as an ε-deviation from the optimal along a direction (H, L). We can find the optimal solution by maximizing the value function with respect to ε. Indeed, given (H, L) and (h, l)

* , the value V computed along the feasible trajectory (h, l) can be written as a function of ε. More precisely, we have

Ṽ (ε) ≡ V (h * + εH, l * + εL, λ * )
where (h, l) * , (H, L) and λ * are given.

Under Assumption 1, the necessary condition for (h, l) * to be an optimal solution is Ṽ ′ (0) = 0.

First, we observe that

λ * (0) h (0) -lim T →∞ λ * (T ) h (T ) = λ * (0) h * (0) -lim T →∞ λ * (T ) h * (T ) -ε lim T →∞ λ * (T ) H (T ) so that ∂ ∂ε $ λ * (0) h (0) -lim T →∞ λ * (T ) h (T ) % = -lim T →∞ λ * (T ) H (T ) because λ * (0) h (0) -lim T →∞ λ * (T ) h * (T ) does not depend on ε.
Next, let us obtain the optimal conditions. We obtain the derivative (23) w.r.t. ε:

Ṽ ′ (ε) = ! ∞ 0 e -θt u ′ (x (h (t) l (t) , h (t))) " ∂x ∂c [H (t) l (t) + h (t) L (t)] + ∂x ∂h H (t) # dt + ! ∞ 0 ( λ * (t) H (t) + λ * (t) H (t) B [1 -l (t)] -λ * (t) h (t) BL (t) ) dt -lim T →∞ λ * (T ) H (T ) (24)
The first-order condition is given by Ṽ ′ (0) = 0, that is by 4

! ∞ 0 e -θt u ′ (x (h * (t) l * (t) , h * (t))) " ∂x ∂c [H (t) l * (t) + h * (t) L (t)] + ∂x ∂h H (t) # dt + ! ∞ 0 ( λ * (t) H (t) + λ * (t) H (t) B [1 -l * (t)] -λ * (t) h * (t) BL (t) ) dt -lim T →∞ λ * (T ) H (T ) = 0 (25)
because the optimal solution corresponds to ε = 0:

(h (t) , l (t)) = (h * (t) , l * (t)) + 0 * (H (t) , L (t)) = (h * (t) , l * (t))
Focus again on [START_REF] Schultz | Investment in human capital[END_REF]. We gather the terms that multiply H(t) and L(t):

! ∞ 0 H (t) " e -θt u ′ (x (h * (t) l * (t) , h * (t))) * ∂x ∂c l * (t) + ∂x ∂h + + λ * (t) + λ * (t) B [1 -l * (t)] # dt + ! ∞ 0 L (t) * e -θt u ′ (x (h * (t) l * (t) , h * (t))) ∂x ∂c h * (t) -λ * (t) h * (t) B + dt -lim T →∞ λ * (T ) H (T ) = 0 (26)
Consider the subclass S of functions H such that lim

T →∞ λ * (T ) H (T ) = 0
or, equivalently, such that lim

T →∞ λ * (T ) h (T ) = lim T →∞ λ * (T ) h * (T ) ∈ R (27)
In this case, condition ( 26) is equivalent to

! ∞ 0 H (t) " e -θt u ′ (x (h * (t) l * (t) , h * (t))) * ∂x ∂c l * (t) + ∂x ∂h + + λ * (t) + λ * (t) B [1 -l * (t)] # dt + ! ∞ 0 L (t) * e -θt u ′ (x (h * (t) l * (t) , h * (t))) ∂x ∂c h * (t) -λ * (t) h * (t) B + dt = 0
4 With some notational abuse, we write

! ∂x ∂c , ∂x ∂h " instead of ! ∂x ∂c (h (t) l (t) , h (t)) , ∂x ∂h (h (t) l (t) , h (t))
" whatever the direction (H, L) with H ∈ S you consider. Because (H, L) identifies an arbitrary direction in the function space, we require

e -θt u ′ (x (h * (t) l * (t) , h * (t))) * ∂x ∂c l * (t) + ∂x ∂h + + λ * (t) + λ * (t) B [1 -l * (t)] = 0 e -θt u ′ (x (h * (t) l * (t) , h * (t))) ∂x ∂c h * (t) -λ * (t) h * (t) B = 0
These are necessary conditions for (h * , l * ) to be optimal. But, according to [START_REF] Smith | An Inquiry into the Nature and Causes of the Wealth of Nations[END_REF], these conditions also imply that lim T →∞ λ * (T ) H (T ) = 0, that is lim T →∞ λ * (T ) h (T ) ∈ R for whatever feasible h and, in particular, for h * . The (transversality) condition

lim T →∞ λ * (T ) h * (T ) ∈ R (28)
is also necessary. We obtain then the system of necessary optimal conditions:

λ * (t) = e -θt u ′ (x (h * (t) l * (t) , h * (t))) 1 B ∂x ∂c (29) λ * (t) + λ * (t) B [1 -l * (t)] + e -θt u ′ (x (h * (t) l * (t) , h * (t))) * ∂x ∂c l * (t) + ∂x ∂h + = 0 (30)
jointly with the law of motion, ḣ * (t) = h * (t) B [1 -l * (t)], and the transversality condition lim t→∞ λ * (t) h * (t) ∈ R.

Since µ ≡ λe θt and λ/λ = μ/µ -θ, (29) and (30) become:

µ * (t) = u ′ (x (h * (t) l * (t) , h * (t))) B ∂x ∂c μ * (t) µ * (t) = θ -B [1 -l * (t)] - u ′ (x (h * (t) l * (t) , h * (t))) µ * (t) * ∂x ∂c l * (t) + ∂x ∂h
+ that is ( 5) and [START_REF] Ben-Porath | The production of human capital and the life cycle of earnings[END_REF]. Since µ * (t) > 0, (2) holds with equality and we obtain [START_REF] Borissov | Heterogeneous human capital, inequality and growth: The role of patience and skills[END_REF]. Finally, ( 8) is given by [START_REF] Tamura | Teachers, growth, and convergence[END_REF].

Proof of Proposition 2

The Arrow-Kurz sufficiency theorem in [START_REF] Arrow | Optimal growth with irreversible investment in a Ramsey model[END_REF] states that the first-order conditions (5) to [START_REF] Borissov | Heterogeneous human capital, inequality and growth: The role of patience and skills[END_REF] are not only necessary but also sufficient if the Hamiltonian H (h, l, λ, t) maximized with respect to the control variable l (given h, λ and t), that is H * (h, λ, t) = H (h, l * (h, λ, t) , λ, t), is concave in h (state variable), given λ and t, that is

∂ 2 H * ∂h 2 (h * , λ, t) < 0
jointly with the transversality condition lim t→∞ λh * = 0.

In our case, (4) becomes H * (h, λ, t) = e -θt u (

x

( h l * (h) , h )) +λB $ 1 -l * (h) % h,
where l * (h) ≡ l * (h, λ, t) given λ and t. We compute the first derivative:

∂H * ∂h (h, λ, t) = e -θt u ′ ( x ( h l * (h) , h )) " ∂x ∂c $ l * (h) + h l * ′ (h) % + ∂x ∂h # +λB $ 1 -l * (h) % -λB l * ′ (h) h
then the second derivative:

∂ 2 H * ∂h 2 (h, λ, t) = e -θt u ′′ ( x ( h l * (h) , h )) " ∂x ∂c $ l * (h) + h l * ′ (h) % + ∂x ∂h # 2 +e -θt u ′ ( x ( h l * (h) , h )) * " ∂ 2 x ∂c 2 $ l * (h) + h l * ′ (h) % 2 + 2 ∂ 2 x ∂c∂h $ l * (h) + h l * ′ (h) % + ∂x ∂c $ 2 l * ′ (h) + h l * ′′ (h) % + ∂ 2 x ∂h 2 # -2λB l * ′ (h) -λB l * ′′ (h) h (31)
Using ( 29), we write

e -θt u ′ (x) = λB " ∂x ∂c # -1
and replacing it in (31), we obtain

∂ 2 H * ∂h 2 (h, λ, t) = n + λB " ∂x ∂c # -1 " ∂ 2 x ∂c 2 $ l * (h) + h l * ′ (h) % 2 + 2 ∂ 2 x ∂c∂h $ l * (h) + h l * ′ (h) % + ∂ 2 x ∂h 2 #
where n is given by

n ≡ e -θt u ′′ ( x ( h l * (h) , h )) " ∂x ∂c $ l * (h) + h l * ′ (h) % + ∂x ∂h # 2
Under Assumption 1, n < 0 and ∂x/∂c > 0. Then, (9), implies

∂ 2 H * ∂h 2 (h, λ, t) < 0 Proof of Proposition 4
Taking the logarithm of ( 5) and differentiating the result with respect to t, we obtain

μ µ = xu ′′ (x) u ′ (x) -" c x ∂x ∂c + h x ∂x ∂h # ḣ h + c x ∂x ∂c l l . + " c ∂x/∂c ∂ 2 x ∂c 2 + h ∂x/∂c ∂ 2 x ∂c∂h # ḣ h + c ∂x/∂c ∂ 2 x ∂c 2 ll (32)
Replacing ( 5) in ( 6), we get

μ µ = θ -B -B ∂x/∂h ∂x/∂c (33)
Substituting ( 7) and ( 33) in (32), we find system ( 10)- [START_REF] Dechert | A complete characterization of optimal growth paths in an aggregated model with a non-concave production function[END_REF].

Proof of Proposition 5

From (5), using the functional forms ( 12) and ( 13), and noticing that c = hl, we obtain

h = l -ε σ ( al σ-1 σ + b ) ε-σ σ-1 " a µB # ε
that is l * = l * (h) ≡ l * (h, λ, t), an implicit function of h with the following elasticity:

ω (h) ≡ h l * ′ (h) l * (h) = - σa l * (h) σ-1 σ + σb σa l * (h) σ-1 σ + εb (34)
Given λ and t, let H * (h) ≡ u (x * (h)) + µB $ 1 -l * (h) % h be the maximum of H with respect to l, with x * (h) ≡ x ( h l * (h) , h

) . According to [START_REF] Arrow | Optimal growth with irreversible investment in a Ramsey model[END_REF], we require H * to be strictly concave: H * ′′ (h) < 0.

In the isoelastic case, we obtain

( x c , x h ) = ( al σ-1 σ + b ) σ σ-1 " 1 l , 1
# (35) " ∂x ∂c , ∂x ∂h # = ( al σ-1 σ + b ) 1 σ-1 ( al -1 σ , b ) (36) - ∂ 2 x ∂c 2 ∂ 2 x ∂h∂c ∂ 2 x ∂c∂h ∂ 2 x ∂h 2 . = ab cσ ( al σ-1 σ + b ) σ σ-1 -2 l σ-1 σ * -1 l 1 1 -l + (37)
As discussed in the proof of Proposition 1, the first-order conditions of the maximization program are not only necessary but also sufficient if (9) holds. That is, if

( l * [1 + ω (h)] ) 2 ∂ 2 x ∂c 2 + 2 l * [1 + ω (h)] ∂ 2 x ∂c∂h + ∂ 2 x ∂h 2 < 0
Replacing (34) and (37), we obtain

- ab cσ * l * 2σ-1 σ ( a l * σ-1 σ + b ) σ σ-1 -2 + / σa l * σ-1 σ + σb σa l * σ-1 σ + εb 0 2 < 0
which is always true. Surprisingly, we do not need any restriction on σ or ε. Reconsidering ( 12) and ( 13), and using (35), ( 36) and (37), we find

∂x/∂h ∂x/∂c = b a l 1 σ (38)
and the elasticities

" c x ∂x ∂c , h x ∂x ∂h # = 1 al σ-1 σ + b ( al σ-1 σ , b ) (39) " c ∂x/∂c ∂ 2 x ∂c 2 , h ∂x/∂c ∂ 2 x ∂c∂h # = 1 σ b al σ-1 σ + b (-1, 1) (40)
Replacing c = hl, u ′ (x) / [xu ′′ (x)] = -ε and expressions (39) and ( 40) in [START_REF] Dechert | A complete characterization of optimal growth paths in an aggregated model with a non-concave production function[END_REF], we obtain equation [START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF].

Proof of Proposition 6

Let l be the nonzero steady state solution of ϕ (l) = 0. Thus, l is the (unique) solution of [START_REF] Kalemli-Ozcan | Mortality decline, human capital investment, and economic growth[END_REF].

Integrating ḣ/h = B & 1 - l'
over time, we get (17) and, replacing l and h in c = hl, we obtain [START_REF] Lucas | On the mechanics of economic development[END_REF]. Reconsidering [START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF], we find that l decreases if l 0 < l, while l increases if l 0 > l. Thus the BGP is globally unstable.

Proof of Proposition 7

Under the parametric specification ε = σ = 1, the differential equation ( 15

l (t) = αθl 0 Bl 0 + (αθ -Bl 0 ) e θt (41)
where

l 0 ≡ l (0) ∈ [0, 1], with l ′ (t) = - αθ 2 l 0 (αθ -Bl 0 ) e θt [Bl 0 + (αθ -Bl 0 ) e θt ] 2 > 0 ⇔ l 0 > α θ B
Then, depending on the initial labor allocation, three different cases arise:

(1) If 0 < l 0 < αθ/B, then l ′ (t) < 0 and lim t→∞ l (t) = 0.

(2) If l 0 = αθ/B, then l ′ (t) = 0 and l (t) = l = αθ/B forever: the economy grows at the balanced growth rate g = B (1 -αθ/B) (BGP).

(3) If αθ/B < l 0 ≤ 1, then l ′ (t) > 0 and

l (t) = αθl 0 Bl 0 + (αθ -Bl 0 ) e θt if 0 ≤ t < T (42) l (t) = 1 if t ≥ T where T ≡ 1 θ ln Bl 0 -αθl 0 Bl 0 -αθ (43)
Therefore, the steady state l = αθ/B is globally unstable. In addition, recall that h 0 is a given initial condition, while l 0 is a choice variable.

We compute the value of the functional , ∞ 0 e -θt u (x) dt in these three cases with u (x) ≡ ln x and x (c, h) ≡ c α h 1-α .

We observe that these functions have a unit elasticity of substitution: ε = σ = 1, and α = a = 1 -b.

We organize the proof in three parts, computing c (t) and h (t) in each of the three cases and evaluating the utility functional

! ∞ 0 e -θt ln & c α h 1-α ' dt (44)
Case (1): 0 < l 0 < αθ/B. Replacing (41) in ( 10):

(ln h) ′ = ḣ h = B * 1 - αθl 0 Bl 0 + (αθ -Bl 0 ) e θt +
Integrating both the sides, h(t) follows:

h (t) = h 0 e Bt " αθ -Bl 0 + Bl 0 e -θt αθ # α (45)
with h (0) = h 0 . Replacing (41) and (45) in c (t) = h (t) l (t), c (t) can be written as

c (t) = h 0 l 0 e (B-θ)t " αθ αθ -Bl 0 + Bl 0 e -θt # 1-α (46)
Using ( 45) and ( 46), the utility functional v 1 (l 0 ) is evaluated: Replacing l by l = l 0 = αθ/B in ( 17) and [START_REF] Lucas | On the mechanics of economic development[END_REF], and then introducing these expressions for c and h in (44), we find the value of the utility functional along the BGP. More explicitly, h (t) = h 0 e (B-αθ)t c (t) = h 0 l 0 e (B-αθ)t (1) Recall that output is defined as y (t) ≡ h (t) l (t). As we have proven in this paper, l (t) is constant along the optimal path, and growth is balanced: (2) Along the BGP, l (t) = l * = αθ/B and, according to (47), the welfare functional is given by

v 1 (l 0 ) ≡ ! ∞ 0 e -θt ln & c α h 1-α ' dt = ! ∞ 0 e -
W * = ! ∞

Fig. 1

 1 Fig. 1 Consumption and HDI crossing properties.

e

  -θt dt + (B -αθ) ! ∞ Case (2): l 0 = αθ/B.

and v 2 (eCase ( 3 ) 0 e -θt ln & c α h 1 = h 0 l α 0 "ce 1 e% = µ * 0 h 0 lim t→∞ e -θt = 0 Proof of Corollary 8 .

 23010108 -θt dt + (B -αθ) : αθ/B < l 0 ≤ 1. As seen above, if l 0 > αθ/B, then l ′ (t) > 0 and l increases over time. At time T , l reaches the upper bound: l (T ) = 1.Accordingly, the utility functional (44) is then evaluated considering the integrals before and after T :! ∞ ln & c α h 1-α ' dt(48) where T is given by (42).As in the first case, if 0 ≤ t < T, then (41) holds and the trajectories (45) and (46) for human capital and consumption hold as well. If t ≥ T , l (t) = 1 and using the definition of T in (43)h (t) = h (T ) = h 0 e BT " αθ -Bl 0 + Bl 0 e -θT αθ # α (t) = h (t) l (t) = h T(50)Let us consider (48). When t < T, let us replace c and h using (45) and (46). When t ≥ T , we use (50), (49) and (43) to substitute for c, h T and T .The utility functional becomes a function v 3 of l 0 :v 3 (l 0 ) -θt [ln (h 0 l α 0 ) + (B -αθ) t] dt + ln h T -θt µ * (t) h * (t) 2 = limt→∞ $ e -θt µ * 0 e (αθ-B)t h 0 e (B-αθ)t Using the results in Proposition 7, we can easily compute the elasticities of the BGP with respect to α:α h * (t) ∂h * (t) ∂α = -αθt < 0 α c * (t) ∂c * (t) ∂α = 1 -αθt > 0 ⇔ t < T c,α α x * (t) ∂ x * (t) t < T x,αProof of Corollary 9. As in the previous proof, the elasticities of the BGP with respect to θ follow:θ h * (t) ∂h * (t) ∂θ = -αθt < 0 θ c * (t) ∂c * (t) ∂θ = 1 -αθt > 0 ⇔ t < T c,α θ x * (t) ∂ x * (t) ∂θ = α (1 -θt) > 0 ⇔ t < T x,θProof of Corollary 10.The elasticities of the BGP with respect to B obtain as in the previous corollaries, using the description of the BGP provided in Proposition 7:B h * (t) ∂h * (t) ∂B = Bt > 0 B c * (t) ∂c * (t) ∂B = Bt -1 > 0 ⇔ t > T c,B B x * (t) ∂ x * (t) ∂B = Bt -α > 0 ⇔ t > T x,BProof of Corollary 11.

  = g * = B (1 -l * ) = B -αθ ≥ 0 because l * ∈ (0, 1].The impact on the optimal (balanced) growth rate of α, θ and B are the following:∂g * ∂α = -θ < 0, ∂g * ∂θ = -α < 0 and ∂g * ∂B = 1 > 0

  Remark 3The transversality condition lim t→∞ e -θt µ * h * ∈ R is necessary for optimality (see the proof of Proposition 1). Notice that lim t→∞ e -θt µ * h * = 0 is a sufficient condition in Pontryagin et al. (2018, page 49) because the arrival space is not full-dimensional, which is not the case in our unconstrained problem. Many economic papers introduce lim t→∞ e -θt µ

* h * = 0 as a sufficient condition without giving proof, while few show the sufficiency (see

[START_REF] Bosi | Rational land and housing bubbles in infinite-horizon economies[END_REF]

, among others). Halkin was first to provide a mathematical example of a nonzero transversality condition, but meaningless in economic terms (see the footnote at page 46 in Arrow and Kurz, 1970, and the counterexample on page

Let us mention some of the most notorious works. Regarding education and human capital formation, see De la Croix and Doepke (2003), Tamura (2001) or Cervellati and Sunde (2005). Concerning health and human capital formation, the interested reader can refer to De la Croix and Licandro (1999) or Kalemli-Ozcan et al. (2000).

If y (t) ≡ h (t) l (t) denotes the per capita income (production) and c (t) = y (t), we recover ul Haq's definition of HDI.

[START_REF] Halkin | Necessary conditions for optimal control problems with infinite horizons[END_REF]). Theorem 7.13 in[START_REF] Acemoglu | Introduction to modern economic growth[END_REF] demonstrates that the transversality condition is of the form lim t→∞ e -θt µ * h * = 0 when the economy converges towards a steady state or towards a balanced growth path. We recover this property in the next section.[START_REF] Gozzi | Internal habit formation and optimality[END_REF] The Arrow-Kurz sufficient conditions for optimality are not necessary and, in this sense, they are possibly too restrictive: an optimal path can exist even if these conditions are not verified. See for instance[START_REF] Dechert | A complete characterization of optimal growth paths in an aggregated model with a non-concave production function[END_REF], or Bambi and[START_REF] Gozzi | Internal habit formation and optimality[END_REF], where the first-order conditions lead to a unique optimal solution and the Arrow-Kurz conditions fail. However, we show in the following section that the Arrow-Kurz sufficient conditions are always verified in our problem when both u and x take isoelastic functional forms.

Therefore,

that is, the BGP is optimal. We find the optimal HDI path (21) by using [START_REF] Michel | On the transversality condition in infinite horizon optimal problems[END_REF] and [START_REF] Michel | Some clarifications on the transversality condition[END_REF] to replace c * and h * in x * (t) = c * (t) α h * (t) 1-α . It only remains to check the transversality condition along the BGP. From equations (33) and (38):

Integrating both sides we obtain that µ (t) = µ * 0 e (αθ-B)t , and using h (t) = h 0 e (B-αθ)t , it can be proven that the transversality condition is satisfied along


L Quéval
email: loic.queval@geeps.centralesupelec.fr

Estimation of the electricity production of a PV installation using PVGIS

Objectives 1) Understand the link between climate and intermittent renewable energy sources 2) Use climate models to assess the potential of solar PV rooftop residential 3) Learn some stuffs about PV (incidentally) and techno-economic analysis

II. Resources

loic.queval@geeps.centralesupelec.fr

  Photovoltaics2023-04-18

+ lecture notes "Position d'une surface inclinée" loic.queval@geeps.centralesupelec.fr Installation Photovoltaics 2023-04-18

Photovoltaics2023-04-18


Sylvain Béal
email: sylvain.beal@univ-fcomte.fr.

Marc Deschamps
email: marc.deschamps@univ-lorraine.fr.

On compensation schemes for data sharing within the european REACH legislation *

Keywords: REACH, Data sharing problem, Core, Nucleolus, Shapley value, Pooling, Compensation, Reasonableness, Invariance to deleting non-exclusive data, Equal treatment of Equals, Invariance JEL Classification number: C71, D71, K32, L65

 concerns the sharing of data between users of a chemical substance. We study this bargaining problem by means of a special class of games in coalitional form called data games (Dehez and Tellone [10]). For such problems, compensation schemes specify how the data owners should be compensated by the agents in needs of data. On the class of data games, the Core, the Nucleolus and the Shapley value provide relevant compensation schemes. We provide three comparable axiomatic characterizations of the set of all (additive) compensation schemes belonging to the Core, of the Nucleolus and of the Shapley value. The axioms reflects principles of various theories of justice.

Introduction

The chemical industry includes a wide range of products and activities covering upstream to downstream, base chemicals, specialty chemicals and pharmaceuticals. Nowadays, it appears very difficult to find products that do not contain chemicals in their production in most societies. For example, plastics derived from petrochemical processing bases are widely used in packaging, buildings and automobiles. The European Commission estimates that more than one hundred thousand chemicals are currently used within the European Union. In Europe, the chemical industry is a major economic sector. According to Cefic (European Chemical Industry Council), in 2012 it produced more than 21% of the chemicals sold in the world, is a net exporter, directly employs 1.2 million workers in 29,000 companies and contributes to the European economy by 673 billion euros.

As a consequence, european governments are confronted with two major challenges. The first is to ensure the health and environmental management of each chemical product as well as their combinations, especially in a context where European citizens are concerned about these issues. The second lies in the fact that the chemical industry plays a central role in European economies and is an asset for economic growth.

It is to meet these challenges simultaneously that the European institutions have decided, since December 13, 2006, the implementation of a new harmonized legislative framework in the field of chemical industry: the European Regulation (EC) No 1907/2006 REACH (Registration, Evaluation, Authorization and Restriction of Chemicals), which ultimately aims to ensure greater human and environmental safety (based on impact studies by the European Commission it will reduce the number of deaths due to cancer by a range of 2000 to 4000 per year and lead to a reduction in public health spending of up to 50 billion over thirty years), while preserving and enhancing the competitiveness of the European chemical industry. These objectives are explicitly set out in its Article 1: "This Regulation should ensure a high level of protection of human health and the environment as well as the free movement of substances, on their own, in preparations and in articles, while enhancing competitiveness and innovation". In other words, REACH aims to transform the European chemical industry into the world's first sustainable chemical industry. According to the European institutions, about 30,000 chemicals are directly affected by this legislation.

The REACH Regulation entered into force on June 1st, 2007 and sets out the principle that it is for manufacturers, importers and downstream users "to ensure that they manufacture, place on the market or use such substances that do not adversely affect human health or the environment. Its provisions are underpinned by the precautionary principle" (Article 1, §3). The effectiveness of this principle is guaranteed by the fact that all substances, preparations or articles which do not comply with the Regulation may not be marketed within the European Union. In practical terms, it is the ECHA (European Chemicals Agency), a European Agency established by the Regulation, which is responsible for implementing the REACH legislation. It is to ensure that all existing and future chemicals produced or imported in quantities over one ton per year in the European Union are registered, evaluated and authorized, except in cases that are expressly exempted (e.g radioactive substances).

The most innovative aspect of this Regulation lies in the fact that companies are forced to provide all existing data on the properties of the chemicals they use when registering with the ECHA. The procedure provides that, without data, companies cannot use the substances: "No data, no market", summarizes ECHA. To facilitate the exchange of toxicological and ecotoxicological information, to avoid the duplication of tests including those involving vertebrate animals, and to allow different registrants to agree on a classification and labeling of a substance, the Regulation created a transitional regime. During this period, any company wishing to declare a substance must participate in a SIEF (Substance Information Exchange Forum). Within a SIEF, members are free to decide on legal form, organization and communication modalities. The Regulation requires, however, that any member of a SIEF meet the demand for information from another member and that all members work collectively to identify and implement additional studies that may prove necessary. The purpose of SIEF, operational from January 2009 to June 2018 (when all substances will have been recorded), is to allow a joint submission for each substance with a lead registrant, other registrants needing only to refer to this registration and to provide data on the characteristics of its production or its use. In cases where such a joint submission involves too high costs for a company or where it requires it to disclose confidential information, the Regulation provides that it can voluntarily exclude the obligation to participate in the joint submission. Nevertheless, it remains a member of the SIEF and must continue to share its non-confidential data. After the transitional regime, the Regulation insists on the fact that registrants "shall make every effort to ensure that the costs of sharing the information are determined in a fair, transparent and nondiscriminatory way" (Article 27, §3), and ECHA decides in case of disagreement between the parties on the necessary compensations. The question of how to realize data sharing within a SIEF and to determine the above-mentioned compensations is therefore a critical issue until June 2018.

In this article, we use games in coalitional form, a classical model to study bargaining, to tackle this question. The coalitional function specifies for each coalition the total cost of acquiring the missing data available in the SIEF should its members cooperate. Such games in coalitional form are called data games and have been introduced in Dehez and Tellone [START_REF] Dehez | Data games: Sharing public goods with exclusion[END_REF]. Since all members of the SIEF have the obligation to cooperate, it is essential to find compensation vectors specifying how the data owners should be compensated by the other SIEF members. A compensation scheme assigns a compensation vector to any data game.

There exists infinitely many compensation schemes, several of which may be considered as intuitively appealing. In order to distinguish among them, we adopt the axiomatic approach and proceed in two steps. 1 Firstly, we list desirable properties for a compensation scheme on the class of data games. These properties conveys principles such as efficiency, equity and fairness. Secondly, various combinations of the properties are tried out so as to eventually retain a unique compensation scheme or a pertinent family of compensation schemes. We believe that the axiomatic method is of particu-lar interest in the field of Law and Economics. Judges and arbitrators often adhere to philosophical principles which reflects their political ideologies and serve as a basis for making their decisions. For instance, Kornhauser [START_REF] Kornhauser | Modeling collegial courts I: Path-dependence[END_REF], Spiller [START_REF] Spiller | Rationality, decision rules, and collegial courts[END_REF] and Fleurbaey and Roemer [START_REF] Fleurbaey | Judicial precedent as a dynamic rationale for axiomatic bargaining theory[END_REF] propose an axiomatic approach to the courts' decision making process.

In data sharing problems, the determination of fair compensation schemes also relies on philosophical principles which interpret the ethical vision of the bargainers within a SIEF. On these aspects, theories of justice offer an abundant literature along the lines of Rawls [START_REF]Rawls -A theory of justice[END_REF], Dworkin [START_REF] Dworkin | What is equality? part 2: Equality of resources[END_REF], Young [START_REF]Young -Equity in theory and practice[END_REF] and Fleurbaey and Maniquet [START_REF] Fleurbaey | Maniquet -Equality of opportunity: The economics of responsibility[END_REF], among others. These scholars underline that compensation schemes aim at restoring equality or diminishing individual inequalities. The formal statements of our axioms borrow to a great extent these theories as exemplified in section 2. While ECHA provides a detailed guide on data sharing2 , this guide only rests on very specific examples instead of offering a general and operational approach. Our axiomatic analysis remedies this weakness by formulating properties and investigating compensations schemes for all data sharing problems.

We study compensations schemes stemming from the main three solution concepts used in games in coalitional form. The Core (Gillies [START_REF] Gillies | Some theorems on n-person games[END_REF]) collects those compensation vectors which no coalition of SIEF members can improve upon in the Pareto sense. The Core of data games is always nonempty, but can contain many compensation vectors. It is therefore interesting to extract specific compensation schemes from the Core. The Nucleolus (Schmeidler [START_REF] Schmeidler | The Nucleolus of a characteristic function game[END_REF]) is probably the most-known such Coreselector. Roughly speaking, the Nucleolus incorporates Rawls [START_REF]Rawls -A theory of justice[END_REF] vision of distributive justice in order to be of the greatest benefit to the least-advantaged coalition of SIEF members. Besides Core compensation vectors, there exist other compensation schemes incorporating fair considerations, such as the Shapley value (Shapley [START_REF] Shapley | A value for n-person games[END_REF]). The Shapley value measures, for each SIEF member, his average contribution to all coalitions he can belong to. This article offers three comparable characterizations of the Core-selectors, of the Nucleolus and of the Shapley value in the sense that the combinations of axioms in these characterizations only differ with respect to one or two axioms. These results are outlined in detail in the next section.

Other articles dealing with the data sharing within the REACH legislation via games in coalitional form are Dehez and Tellone [START_REF] Dehez | Data games: Sharing public goods with exclusion[END_REF], which formulates the Core, the Nucleolus and the Shapley value on the class of data games, and Béal et al. [START_REF] Béal | Les informations exigées par la législation reach : Analyse du partage des coûts[END_REF], which provides an alternative axiomatic characterization of the Shapley value and characterizations of equal surplus division values Games in coalitional form have also been employed extensively to tackle problems in Law and Economics but in the very different perspective of measuring voting power. We refer the reader to Bindseil and Hantke [START_REF] Bindseil | The power distribution in decision making among EU member states[END_REF], Fedeli and Forte [START_REF] Fedeli | Voting powers and the efficiency of the decision-making process in the european council of ministers[END_REF] and Algaba, Bilbao and Fernández [START_REF] Algaba | The distribution of power in the European Constitution[END_REF] for the distribution of voting power within the various institutions of the European Union, and to Braham and Steffen [START_REF] Braham | Voting rules in insolvency law: a simple-game theoretic approach[END_REF], Leech and Manjón [START_REF] Leech | Corporate governance in Spain (with an application of the power indices approach)[END_REF] and Casajus, Labrenz and Hiller [START_REF] Casajus | Majority shareholder protection by variable qualified majority rules[END_REF] for studies of voting power in corporate finance.

We are also aware of some limits of our model. The simplicity of the data games comes at a cost, for instance because no reference is made to the underlying problem of reaching a consensus on the cost parameters. Moreover, the model is silent on situations in which the composition of the SIEF members evolves over time. As a consequence, even if our model can be adapted to take these features into account, it should be considered as a first step towards more sophisticated analysis. It is also true that we neglect the economic consequences of other facets of the REACH legislation, on which it is surprising to note the tenuousness of academic work, in particular by economists. 3The rest of the article is organized as follows. An informal presentation of the results is given in section 2. The data sharing problems modeling the bargaining within a SIEF is introduced in section 3. This section also presents the class of data games. Section 4 is devoted to the axiomatic study and contains the main results. Concluding remarks are provided in section 5. The logical independence of the axioms used in our results is demonstrated in the Appendix.

Outline of the results

The first principle that we impose is the unquestionable compensation principle, which stipulates that the compensations chosen by the SIEF members (often called agents henceforth) add up to zero. This principle is a restatement for data games of the axiom of (Pareto) efficiency for general games in coalitional form. Our second principle is called pooling and possesses a natural bargaining interpretation within the REACH legislation: the outcome of multi-data negotiations does not depend on the agenda chosen by the negotiators. Whether data are discussed separately or all at once does not affect the result of the negotiation. This principle is a variation for data games of the axiom of additivity for general games in coalitional form that Shapley himself supports as follows: "the third axiom ("law of aggregation") states that when two independent games are combined, their values must be added player by player" (Shapley [START_REF] Shapley | A value for n-person games[END_REF], page 309). All the compensation schemes mentioned in the introduction satisfy these first two basic principles. In order to distinguish among them, we invoke several other principles.

The principle of reasonableness has been introduced by Milnor [START_REF] Milnor | Reasonable outcomes for n-person games[END_REF] and requires that any agent's compensation lies between its worst and best (marginal) contributions to the coalitions he can belong to. These contributions are interpreted by Milnor [START_REF] Milnor | Reasonable outcomes for n-person games[END_REF] as the minimal and maximal compensations that an agent can legitimately claim. The principle of invariance to deleting non-exclusive data states that a compensation scheme should not be affected if a data held by more than one agent is removed and has a basic non-cooperative economic justification. Assume that a data is not exclusive in the sense that it is possessed by at least two agents. Then these agents somehow compete against each other in order to attract monetary compensations from the agents lacking the data. The situation is therefore similar to a Bertrand oligopoly. 4 The data owners set prices at which the agents lacking the data can buy it. The data owner with the lowest price gets the full demand, i.e. it sells to all agents lacking the data. Thus, prices are null at equilibrium since there is a zero marginal cost here. If one adheres to this interpretation, then it seems natural to require an absence of compensation for any non-exclusive data. Proposition 2 states that if a compensation scheme satisfies the pooling principle, then it is a Core-selector if and only if it further satisfies the principles of compensation, reasonableness and invariance to deleting non-exclusive data.

In Proposition 3, the Nucleolus is selected among the set of all Core-selectors. It is sufficient to replace in Proposition 2 the principle of reasonableness by the principle of equal concessions. The latter principle only deals with data sharing problems in which every data is exclusive to some agent. The best contribution of an agent to the coalitions he can belong may be interpreted as his utopia claim (especially because most of time the sum of the corresponding compensations is unrealistic, i.e. much less than zero). With this view in mind, equal concessions imposes that all agents abandon exactly the same amount from their utopia claims. The principle behind equal concessions has the flavor of the equity based on tangible claims in Young [START_REF]Young -Equity in theory and practice[END_REF], and one of its consequences is to compensate the lack of individual internal resources (i.e. the lack of data here) as suggested by Dworkin [START_REF] Dworkin | What is equality? part 2: Equality of resources[END_REF].

Finally, Proposition 4 characterizes the Shapley value by replacing invariance to deleting nonexclusive data in Proposition 3 by invariance to enlarging the owner set and adding equal treatment of equals. The principle of invariance to enlarging the owner set stipulates that the price necessary to acquire a data for an agent lacking it should not depend on how many agents initially own it. Invariance to enlarging the owner set and invariance to deleting non-exclusive data belong to the larger category of axioms of consistency, which require invariance of a solution if specific modifications are brought to the problem. Equal treatment of equals is the basic principle which imposes to treat equally two agents owning the same data. Since Aristotle, this principle appears under different forms in most of the theories of justice (see Fleurbaey [START_REF]Fleurbaey -Fairness, responsibility, and welfare[END_REF] for instance).

3 Data sharing problems and data games Let N = {1, . . . , n} is a finite set of at least three agents involved in a SIEF. We shall use subscripts i and j to refer to agents. A coalition of agents is denoted by S and we refer to 2 N as the set of all coalitions on N . A data sharing problem on N is described by a pair (D, O), where A compensation vector is a vector x ∈ R n which assigns a compensation x i ∈ R to each i ∈ N . If x i > 0, then agent i has to pay a compensation and if x i < 0, then agent i receives a compensation. For any coalition S ∈ 2 N with at least two members, we shall use the shorter notation x(S) instead of i∈S x i . A compensation solution is an application which assigns to each data game a set of compensation vectors. In this article, we are rather interested in compensation schemes, i.e. compensation solutions assigning a unique compensation vector to each data game. Formally, a compensation scheme is an application f that assigns to each

•
C (D,O) ∈ C(N ) a unique compensation vector f (C (D,O) ) ∈ R n
+ . The most well-known compensation scheme is the Shapley value (Shapley [START_REF] Shapley | A value for n-person games[END_REF]), which assigns to each data game C (D,O) ∈ C(N ), the compensation vector

Sh i (C (D,O) ) = S∈2 N :S i (n -|S|)!(|S| -1)! n! C (D,O) (S) -C (D,O) (S\{i}) , i ∈ N.
The Core (Gillies [START_REF] Gillies | Some theorems on n-person games[END_REF]) is the most-used compensation solution that assigns to each data game

C (D,O) ∈ C(N ) the set of compensation vectors Core(C (D,O) ) = {x ∈ R n : x(N ) = 0 and x(S) ≤ C (D,O) (S), ∀S ⊂ N }.
The Core of any data game is nonempty since it always contains the n-dimensional null vector 0 := (0, . . . , 0). Nonetheless, the Core often contains many compensation vectors. Therefore, it is interesting to look for a Core-selector, i.e. a compensation scheme f which selects a Core element f (C (D,O) ) in every data game C (D,O) ∈ C(N ). The most famous Core-selector is the Nucleolus (Schmeidler [START_REF] Schmeidler | The Nucleolus of a characteristic function game[END_REF]), denoted by η, which selects the unique Core compensation vector x that lexicographically minimizes the excesses x(S) -C (D,O) (S) of all coalitions. On the class of data games, the Shapley value, the Core and the Nucleolus have a specific structure as summarized by the following result. where z i (k) = 0, and

z j i (k) = -c k , z j j (k) = c k and z j r (k) = 0 for each r ∈ N \{i, j}. Furthermore, η(C (D,O) ) = 0 if D E (O) = ∅ and η i (C (D,O) ) = k∈D E (O) c k n - k∈D E i (O) c k , i ∈ N if D E (O) = ∅. Finally, Sh i (C (D,O) ) = k∈D c k n - k∈D i (O) c k o k , i ∈ N.
This proposition reveals that the Core and the Nucleolus possess an additive property: they can be computed by looking at elementary data games. The formulas of the Nucleolus and the Shapley value are also close to each other. The Shapley value contains two parts. In the first part, each agent contributes to a communal fund through the same fraction 1/n of the total cost of data. In the second part, the accumulated fund is redistributed data by data at an equal rate to each owner. It is easy to figure out that the Nucleolus implements the same principle, but only for the exclusive data.

Axiomatic studies

Since data games are build from data sharing problem, it makes sense to formulate axioms depending not only on the structure of data games but also on the primitive of the associated data sharing problems. Our approach consists of mixing these two types of axioms. More specifically, we invoke the following axioms for a compensation scheme f on C(N ), which have already been informally discussed in section 2.

Compensation. ∀C (D,O) ∈ C(N ), f (C (D,O) )(N ) = 0. Pooling. ∀C (D,O) ∈ C(N ) and D , D such that D ∪ D = D and D ∩ D = ∅, f (C (D,O) ) = f (C (D ,O) ) + f (C (D ,O) ). Equal treatment of equals. ∀C (D,O) ∈ C(N ) and all i, j ∈ N such that D i (O) = D j (O), f i (C (D,O) ) = f j (C (D,O) ).
Two of the next axioms rely on the following notation. For a data game C (D,O) ∈ C(N ) and an agent i ∈ N , we denote by

b max i (C (D,O) ) = max S∈2 N :i∈S C (D,O) (S) -C (D,O) (S\{i}) and b min i (C (D,O) ) = min S∈2 N :i∈S C (D,O) (S) -C (D,O) (S\{i})
agent i's maximal (or worst) and minimal (or best) marginal contributions to the coalitions he belongs to, respectively.

Reasonableness. ∀C (D,O) ∈ C(N ), ∀i ∈ N , b min i (C (D,O) ) ≤ f i (C (D,O) ) ≤ b max i (C (D,O) ).
Reasonableness prevents an agent to pay more than the total cost of the data he needs, and to be paid more than the total cost of his data.

Equal concessions. ∀C

(D,O) ∈ C(N ) such that D E (O) = D, ∀i, j ∈ N , f i (C (D,O) ) -b min i (C (D,O) ) = f j (C (D,O) ) -b min j (C (D,O) ).
Invariance to deleting non-exclusive data.

∀C (D,O) ∈ C(N ) such that |D| ≥ 2, ∀k ∈ D\D E (O), f (C (D,O) ) = f (C (D\{k},O) ).
So far, all the axioms have been stated for data games associated with the same ownership function. The last axiom that we invoke relies on interrelated ownership functions.

Invariance to enlarging the owner set.

∀C (D,O) , C (D,O ) ∈ C(N ) such that O h = O h for all h ∈ D\{k} and O k ⊂ O k , and all i ∈ N \O k , f i (C (D,O) ) = f i (C (D,O ) ).
This formal definition allows a more detailed description of the axiom than the one given in section 2. Invariance to enlarging the owner set considers two data games based on data sharing problems that only differ with respect to the owner set of data k. More specifically, the owner set in the second problem in a strict superset of the owner set in the first problem. Invariance to enlarging the owner set requires that this enlargement of the owner set has no influence on the compensation paid or received by an agent who does not hold the corresponding data in both problems. In a sense, the axioms says that the price necessary to acquire such data should not depend on how many agents initially own it.

Compensation, Pooling, Equal treatment of equals and Invariance to deleting non-exclusive data are already invoked in Béal et al. [START_REF] Béal | Les informations exigées par la législation reach : Analyse du partage des coûts[END_REF], Reasonableness is due to Milnor [START_REF] Milnor | Reasonable outcomes for n-person games[END_REF], and Equal concessions and Invariance to enlarging the owner set are newly introduced. We now have the material to study the consequences of various combinations of the above-defined axioms. As a first result, we show that if a compensation scheme satisfies Pooling, then it is a Core-selector if and only if it also satisfies Compensation, Reasonableness and Invariance to deleting non-exclusive data.

Proposition 2 Consider any compensation scheme f on C(N ) satisfying Pooling. Then f is a Coreselector if and only it further satisfies Compensation, Reasonableness and Invariance to deleting nonexclusive data.

Proof. Let f be any compensation scheme on C(N ) satisfying Pooling. The proof is split in two steps. Firstly, we show that if f is a Core-selector, then it satisfies Compensation, Reasonableness and Invariance to deleting non-exclusive data. By definition, the sum of all compensations on any element of the Core is equal to zero so that f trivially satisfies Compensation. It is known that the Core satisfies Reasonableness (stated for all compensations vector of a compensation solution) on the class of all games in coalitional form, which implies that any of its element also satisfies Reasonableness on C(N ), and in turn that f satisfies Reasonableness on C(N ). Finally, from Proposition 1, for each k ∈ D such that o k ≥ 2, it holds that Core(C ({k},O) ) = {0}. Thus, since f is a Core-selector, it must be that f (C ({k},O) ) = 0 for such a data game. By Pooling, for each

C (D,O) ∈ C(N ) and each k ∈ D such that o k ≥ 2, we get f (C (D,O) ) = f (C (D\{k},O) ) + f (C ({k},O) ) = f (C (D\{k},O) ) + 0 = f (C (D\{k},O) ),
proving that f satisfies Invariance to deleting non-exclusive data.

Secondly, we show if a compensation scheme f on C(N ) satisfies Compensation, Reasonableness and Invariance to deleting non-exclusive data (in addition to Pooling), then it is a Core-selector. Pick any C (D,O) ∈ C(N ). By Pooling and the expression of the Core given in Proposition 1, it is enough to show that f (C ({k},O) ) ∈ Core(C ({k},O) ) for each k ∈ D. We consider two cases. As a first case, suppose that o k ≥ 2. Consider any other data h and the data game C ({h,k},O ) such that O k = O k . On the one hand, by pooling, we get

f (C ({h,k},O ) ) = f (C ({h},O ) )+f (C ({k},O ) ) = f (C ({h},O ) )+f (C ).
On the other hand, applying Invariance to deleting non-exclusive data to k yields that f (C ({h,k},O ) ) = f (C ({h},O ) ). Combining these two observations implies that f (C ({k},O) ) = 0, and it turns out that 0 is the unique Core element according to Proposition 1. As a second case, suppose that o k = 1, and denote by i ∈ N the unique owner of data k. Choose any j ∈ N \{i}, which is possible since

n ≥ 2. It is easy to see that b min i ({k}, O) = C ({k},O) (N ) -C ({k},O) (N \{i}) = -c k and that b max i ({k}, O) = C ({k},O) ({i})-C ({k},O) (∅) = 0. By Reasonableness, we obtain f i (C ({k},O) ) ∈ [-c k , 0]. Similarly, observe that b min j ({k}, O) = C ({k},O) (N ) -C ({k},O) (N \{j}) = 0 and that b max j ({k}, O) = C ({k},O) ({j}) - C ({k},O) (∅) = c k . Another application of Reasonableness yields that f j (C ({k},O) ) ∈ [0, c k ] for each j ∈ N \{i}. Together with Compensation, it is easy to conclude that f (C ({k},O) ) ∈ conv({z j (k)} j∈N )
where z i (k) = (0, . . . , 0), and

z j i (k) = -c k , z j j (k) = c k and z j r (k) = 0 for each r ∈ N \{i, j} as in Proposition 1. As a consequence, f (C ({k},O) ) ∈ Core(C ({k},O) ), as desired.
Replacing Reasonableness in Proposition 2 by Equal concessions singles out the Nucleolus among all the Core-selectors.

Proposition 3

The Nucleolus is the unique compensation scheme on C(N ) that satisfies Compensation, Pooling, Invariance to deleting non-exclusive data and Equal concessions.

k∈D i (O) c k , i ∈ N. (1)
Since D only contains data held by a unique agent, we have, for each i ∈ N , that

b min i (C (D,O) ) = C (D,O) (N ) -C (D,O) (N \{i}) = - k∈D i (O) c k . (2)
Subtracting ( 2) to (1) yields k∈D c k n .

Since this expression does not depend on agent

i ∈ N , we get η i (C (D,O) ) -b min i (C (D,O) ) = η j (C (D,O) ) - b min j (C (D,O) ) for all i, j ∈ N ,
f i ({k}, O) + c k = f j ({k}, O) (3)
for each j ∈ N \{i}. Summing on all j ∈ N and using Compensation, we obtain

0 = f (({k}, O))(N ) = f i ({k}, O) + (n -1) f i ({k}, O) + c k so that f i ({k}, O) = c k /n -c k = η i (
Sh i (C (D,O ) ) = h∈D c h n - h∈D i (O ) c h o h = h∈D c h n - h∈D i (O) c h o h = Sh i (C (D,O) ).
f (C ({k},O) )(O k ) = -(n -o k ) c k n .
By Equal treatment of equals,

f i (C ({k},O) ) = 1 o k × -(n -o k ) c k n = c k n - c k o k = Sh i (C ({k},O) ).

Conclude by Pooling that

f (C (D,O) ) = Sh(C (D,O) )

Conclusion

The absence of available data on the types of compensation schemes that are actually used within the SIEF prevents us from concluding this article by a confrontation of our theory with the current practice. This important step is left for future research. Instead, we offer two concluding remarks by coming back to the properties mobilized throughout the article.

The set of axioms invoked in Proposition 3 and 4 are almost identical, in particular because the Nucleolus satisfies Equal treatment of equals even if this axiom is not needed in Proposition 3. Therefore, it makes sense to ask whether the separating axioms of Invariance to deleting non exclusive data and Invariance to enlarging the owner set are compatible. As a partial answer, it is possible to show that the price to have simultaneously both axioms is to rule out any form of compensation if the basic axioms of Compensation, Pooling and Equal treatment of equals are also imposed. In other words, it is possible to show that the null compensation scheme that assigns the null compensation vector 0 to any data game is the unique compensation scheme satisfying Invariance to deleting non exclusive data, Invariance to enlarging the owner set are compatible, Compensation, Pooling and Equal treatment of equals.

Such comparisons between axioms are a crucial facet of the axiomatic method. Applied to the decision problem faced by a judge or and arbitrator, this amounts to examine whether philosophical principles are not contradictory, or if a philosophical principle has stronger implications than another one. In order to have a clear overview of our results, we conclude this article by the following recap chart in which a "+" means that the compensation scheme satisfies the axiom, in which "-" has the converse meaning and in which the "⊕" symbols indicate the characterizing sets of axioms in Propositions 2, 3 and 4 respectively.

  D is a nonempty finite set of data. Data in D are relevant to the study of the chemical substance associated to the SIEF. Subscripts k and h are used to refer to data. Data k ∈ D is characterized by a cost c k ∈ R + , which is interpreted as the cost of replicating the data. • O : D -→ N is an ownership function assigning to each data k the nonempty set O k of agents owning the data, and o k := |O k |. In other words, function O specifies the distribution of property rights within the SIEF. Note that every data is owned by some agent. We denote by P(N ) the set of all data sharing problems on N . For a given data sharing problem (D, O) ∈ P(N ), the set of data held by an agent i ∈ N is denoted by D i (O) = {k ∈ D : i ∈ O k }. For each nonempty coalition of agents S ∈ 2 N , D S (O) = ∪ i∈S D i (O) stands for the set of all data held by the members of S. We shall keep the notations D i (O) instead of D {i} (O) and D instead of D N (O). For each i ∈ N , D E i (O) is the set of data in (D, O) that i exclusively holds, i.e. the set, possibly empty, of data held by i and by no other agent in N \{i}. Thus D E i (O) = D\D N \{i} (O) or equivalently D E i (O) = {k ∈ D : O k = {i}}. Define D E (O) = ∪ i∈N D E i (O) as the set of exclusive data in (D, O). To each data sharing problem (D, O) ∈ P(N ), it is useful to associate a data sharing game in coalitional form or simply a data game C (D,O) on N , where the characteristic function C (D,O) : 2 N -→ R + assigns to each coalition S ∈ 2 N the total cost C (D,O) (S) ∈ R + of the data in D that the members of S do not hold. Formally, for each S ∈ 2 N , S = ∅: C (D,O) (S) = k∈D\D S (O) c k , and by convention C (D,O) (∅) = 0. Observe that C (D,O) (N ) = 0. We denote by C(N ) the set of all data games that can be constructed from P(N ), i.e. C(N ) = {C (D,O) : (D, O) ∈ P(N )}. Each data k generates elementary data games C ({k},O) ∈ C(N ).

Proposition 1 (

 1 Dehez and Tellone [10]) Consider any data game C (D,O) ∈ C(N ). Then Core(C (D,O) ) = k∈M Core(C ({k},O ). Moreover, • For each k ∈ D such that o k ≥ 2, it holds that Core(C ({k},O) ) = {0}, • For each k ∈ D such that O k = {i} for some i ∈ N , it holds that Core(C ({k},O) ) = conv({z j (k)} j∈N ),

Proof.

  As a Core-selector, the Nucleolus satisfies Compensation and Invariance to deleting nonexclusive data. It also satisfies Pooling by its expression in Proposition 1. It remains to show that it satisfies equal concessions. So pick any C (D,O) such that D E (O) = D. By Proposition 1, this means that η i (C (D,O) ) = k∈D c k n -

  as desired. Next, consider any compensation scheme f on C(N ) that satisfies Compensation, Pooling, Invariance to deleting non-exclusive data and Equal concessions. Pick any C (D,O) ∈ C(N ), and any k ∈ D. From the proof of Proposition 2, we already know that Pooling and Invariance to deleting non-exclusive data imply that f (C ({k},O) ) = 0 = η(C ({k},O) ) if o k ≥ 2. So suppose that o k = 1 and denote by i ∈ N the unique owner of data k. In the elementary data game ({k}, O), recall from the proof of Proposition 2 that b min i ({k}, O) = -c k and that b min j ({k}, O) = 0. Thus, Equal concessions rewrites

Proposition 4

 4 {k}, O). By (3), conclude that f j ({k}, O) = c k /n = η j ({k}, O) for each j ∈ N \{i} as well. By Pooling, f (C (D,O) ) = η(C (D,O) ), which completes the proof. Replacing Invariance to deleting non-exclusive data in Proposition 3 by Invariance to enlarging the owner set and adding Equal treatment of equals yield a characterization of the Shapley value. It is known that the Nucleolus also satisfies Equal treatment of equals. As such Propositions 3 and 4 provide comparable characterizations of the Nucleolus and the Shapley value that only differ with respect to one axiom. The Shapley value is the unique compensation scheme on C(N ) that satisfies Compensation, Pooling, Equal treatment of equals, Invariance to enlarging the owner set and Equal concessions. Proof. It is obvious that the Shapley value satisfies Compensation, Pooling and Equal treatment of equals. To see that it also satisfies Invariance to enlarging the owner set, consider any pair of data games C (D,O) and C (D,O ) ∈ C(N ) such that O h = O h for all h ∈ D\{k} and O k ⊂ O k ⊂ N for some k ∈ D. Recall that O k = ∅ by assumption. Choose any i ∈ N \O k . Observe that D i (O ) = D i (O) and O h = O h for each h ∈ D i (O). As a consequence, the expression of the Shapley value given in Proposition 1 immediately yields

  Since the Nucleolus satisfies Equal concessions and η(C (D,O) ) = Sh(C (D,O) ) whenever D E (O) = D, the Shapley value satisfies Equal concessions as well. Next, consider any compensation scheme f on C(N ) that satisfies Compensation, Pooling, Equal treatment of equals, Invariance to enlarging the owner set and Equal concessions. Pick any C (D,O) ∈ C(N ), and any k ∈ D. Suppose that o k = 1 and once again denote by i ∈ N the unique owner of data k. In the elementary data game ({k}, O), from the proof of Proposition 3 and the fact that {k} E (O) = {k}, we know that the combination of Compensation and Equal concessions imply that f (C ({k},O) ) = η(C ({k},O) ) = Sh(C ({k},O) ). In particular, for each j ∈ N \{i}, we have f j (C ({k},O) ) = c k /n. Now, consider any k ∈ D such that o k ≥ 2. If o k = n, then Equal treatment of equals and Compensation yield f (C ({k},O) ) = 0. If 2 ≤ o k < n, we can choose an agent i ∈ O k and j ∈ N \O k . Then, construct the elementary data game ({k}, O ) such that O k = {i}. Invariance to enlarging the owner set and Equal concessions can be applied to data games ({k}, O) and ({k}, O ), and to agent j because j ∈ N \O k . Thus, since o k = 1, we obtain from the previous part of the proof that f j (C ({k},O) ) = f j (C ({k},O ) ) = c k /n. Since j was arbitrary chosen in N \O k , this equality holds for all agents in N \O k . By Compensation, we get

5

 5

Appendix 2 ••Proposition 3 •Proposition 4 ••••

 234 In order to demonstrate independence of the axioms imposed in Propositions 2, 3 and 4, we exhibit the following compensation schemes.Proposition The Shapley value Sh on C(N ) satisfies Pooling, Compensation, Reasonableness but not Invariance to deleting non exclusive data. The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) byf 1 (C (D,O) ) = k∈D E (O) c k and f i (C (D,O) ) = -1 n -1 k∈D E (O) c k , i ∈ N \{1},5 Recall that Proposition 2 only consider core-selectors satisfying the Pooling axiom. As such, Pooling is rather an assumption than a requirement in Proposition 2.ifD E (O) = ∅ and f (C (D,O) ) = 0 if D E (O) = ∅satisfies Pooling, Compensation and Invariance to deleting non exclusive data but not Reasonableness.• The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) byf i (C (D,O) ) = η i (C (D,O) ) + k∈D E (O) c k n , i ∈ N,satisfies Pooling, Reasonableness and Invariance to deleting non exclusive data but not Compensation. The compensation scheme f on C(N ) defined, for allC (D,O) ∈ C(N ) by f (C (D,O) ) = 0 satisfiesCompensation, Pooling and Invariance to deleting non exclusive data but not Equal concessions.• The Shapley value Sh satisfies Compensation, Pooling and Equal concessions but not Invariance to deleting non exclusive data.• The compensation scheme f on C(N ) defined, for allC (D,O) ∈ C(N ) by f (C (D,O) ) = η(C (D,O) ) if D E (O) = ∅ and f (C (D,O) ) = Sh(C (D,O) ) if D E (O) = ∅ satisfies Compensation, Equal concessionsand Invariance to deleting non exclusive data but not Pooling.• The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) byf i (C (D,O) ) = η i (C (D,O) ) + |D E (O)|, i ∈ N,satisfiesPooling, Equal concessions and Invariance to deleting non exclusive data but not Compensation. The Nucleolus η satisfies Compensation, Pooling, Equal treatment of equals and Equal concessions but not Invariance to enlarging the owner set. The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) by f (C (D,O) ) = 0 satisfies Compensation, Pooling, Equal treatment of equals and Invariance to enlarging the owner set but not Equal concessions.• The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) byf i (C (D,O) ) = Sh i (C (D,O) ) + |D|, i ∈ N,satisfies Pooling, Equal treatment of equals, Equal concessions, Invariance to enlarging the owner set but not Compensation. Consider any compensation vector a ∈ R n \{0}. The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) byf i (C (D,O) ) = k∈D:o k <n c k n -k∈D i (O):o k <n c k o k + a i × |{k ∈ D : o k = n}|, i ∈ N,satisfies Pooling, Compensation, Equal concessions and Invariance to enlarging the owner set but not Equal treatment of equals. Consider the data game C ({h,k},O ) ∈ C(N ) such that O h = {1, 3} and O k = {2, 3}. The compensation scheme f on C(N ) defined, for all C (D,O) ∈ C(N ) by f (C (D,O) ) = Sh(C (D,O) ) if C (D,O) = C ({h,k},O ) and such that f i (C ({h,k},O ) ) = Sh i (C ({h,k},O ) ) for each i ∈ N \{1, 2, 3}, f 1 (C ({h,k},O ) ) = Sh 1 (C ({h,k},O ) )+c h /2, f 2 (C ({h,k},O ) ) = Sh 2 (C ({h,k},O ) )+c k /2 and f 3 (C ({h,k},O ) ) = Sh 3 (C ({h,k},O ) )c h /2c k /2 satisfies Compensation, Equal treatment of equals, Equal concessions and Invariance to enlarging the owner set but not Pooling.

We refer to Thomson[START_REF] Thomson | On the axiomatic method and its recent applications to game theory and resource allocation[END_REF] for a detailed survey of the axiomatic method.

http://echa.europa.eu/documents/10162/13631/guidance_on_data_sharing_en.pdf

An attempt is made by Canton and Allen[START_REF] Canton | A microeconomic model to assess the economic impacts of the EU's new chemicals policy[END_REF]. The other few exceptions are Wolf and Delgado[START_REF] Wolf | The impact of REACH on innovation in the chemical industry[END_REF], Logomasini[START_REF] Logomasini | Strategic approach to international chemicals management: Lack of interest belies importance[END_REF], Ackerman, Stanton and Massey[START_REF] Ackerman | European chemical policy and the United States: The impacts of REACH[END_REF], Ackerman et al.[START_REF] Ackerman | Implications of REACH for developing countries[END_REF], Lorenz, Lebreton and van Wassenhove[START_REF] Lorenz | The REACH directive and its impact on the european chemical industry: a critical review[END_REF], Von Holleben, Rosenfeld and Scheidmann[START_REF] Von Holleben | Scheidmann -REACH consortia model agreements: And other forms of data sharing according to REACH[END_REF] and Bergkamp[START_REF] Bergkamp | The european union REACH regulation for chemicals: Law and practice[END_REF] but lack of theoretical foundations.

The Regulation reminds that data owners agreeing on the price of their data would be placed under the condemnation of the European Union competition law.

* Financial support by the National Agency for Research (ANR) -research programs "DynaMITE" and "DAMAGE"is gratefully acknowledged. We also thank Philippe Solal for constructive comments.


Macbeth, mis Jacquelyn Bessell

At the beginning of the play, four bodies are lying on the ground, like corpses. All four dressed in black, wearing sportswear clothes, black hoods, black kneepads. Three women, one man.

In this short introduction remains the meaning of the whole performance : acting amounts to an athletic challenge, and 88min (the lenght of the play) is almost the length of a marathon. In Jacquelyn Bessell's productions, people sweat, people play with guts and tense muscles. And when the play is ending, bodies are back to the ground, lying inert once again, completely exhausted.

The Performance Research Group, based at the Guilford School of Acting at Surrey University, is thus characterized by an approach to Shakespearean text entirely focused on the stage. Dixit Jaquelyn Bessell, « PRG performances incorporate 21st century actor training with verse and text work, and fuse physical theatre with elements of 'original practices' staging, in a approach framed with a strong and abiding punk aesthetic. ».

The principles related to the Performance Research Group were already present in its first production, Antony and Cleopatra: a cut and a box for two actors and a piano (2010), which premiered at the 2010 « International Shakespeare Conference », and were at work in this particular reading of Macbeth, performed within the International Congress ESRA, « Shakespeare and Myth », organized by the IRCL.

In this very physical and embodied nature of the staging of Macbeth lies Bessell's distinguishing feature, giving in this way an invigorating reading of Shakespearean theater. Contemporary in the same time, it goes back to the essential Shakespeare's features and of Elizabethan theater. Minimalist, evocative and, above all, with a great expressive and symbolic power.

It is not surprising that most of the sources claimed by this troupe of actors and English academics are contemporary music sources, including for instance John Cage and the punk movement.

The stage is thus constantly supported by sound experiments based on the use of live music and in particular the use of the so-called « prepared piano » (practice borrowed in part to John Cage). This piano, and also the other instruments or musical elements played during the performance, assure both the coherence of the play and its variety of tones and atmospheres. For instance, a single piano can be used in many ways, and by several musicians, and can suggest differents emotional landscapes or narrative frames. It can be played as a drum, to suggest a military scene, or as a punk rock guitar, or even as an harp.

But the most powerful and significant idea, from this point of view, lies in the use of crystal glasses to systematically stress the appearances of Banquo's ghost. Each time the actress playing the character is entering on stage, she and the four other actors, positioned in the corners of the room, clink their crystal glasses. The minimalist aspect of this choice of staging gives a powerful expressivity to this scene, while creating an eerie sound atmosphere, with Hitchcockian accents. At last, we have to notice the presence of the stage director behind the piano : this multiplies the possibilities of action and gives to the performance an improvisation dimension, as in a free-jazz concert.

Another important element of PRG's productions can be found in its particular conception of the relationship between the actor and the audience : it occurs that the spectator is not a passive observer but also an active member of what's happening on stage (this aspect is also stressed by the use of universal lightning). According to that, the performance of Macbeth in a small and intimate place (located in the room Dugès of the old Faculty of Medecine), by putting actors and audience in a common space involved sucn an aesthetic. Similarly, the absence of backstage theater increases the proximity between spectators and actors. These, when they are not playing, stay crouching in the corners of the stage, in front of the audience.

Similarly, the scenic area is characterized by the absence of backstage theater : the actors change, prepare, drink in front of the eyes of the spectator. The gap space between the theatrical action and neutral zone is minimal : in this edge can be measured the risk took by the actors. By the way, an essential component of this representation amounts, as it seems, in the fact that players do not perform in a raised area, but on the same stage plan than the spectators, who become part of the performance not only as passive witnesses. The viewer is thus placed in a « discomfort » area : however, this discomfort is relative because it is not a source of displeasure but just a consequence of the physicality of the staging.

From the beginning of the play, Duncan's character apostrophizes the audience, playing thus with this proximity. Jacquelyn Bessell's staging revives designs of theatrical performance such as those outlined in the scene of the architects in A Midsummer Night's Dream, for it relies less on a mimetic illusion, which would imply a passive public belief, than on a pact of complicity with the spectator. The representation of the plot can then be minimal: a boy is simply designated by a hand gesture of Banquo, but without being played by an actor ; and the baby held in Macbeth's arms is figured (or 'disfigured' to quote A Midsummer Night's Dream again) by a sweatshirt, while Macbeth mimics his moans.

In addition, this apparently non-theatrical space is used in all its dramatic potentialities and leads to bold choices : for example, the fight scene between the protagonist and Lady Macbeth on the other side of a door, hiding from the audience what is happening and suggesting the text with actors's loud voices.

Macbeth's role is held here by the comedian Jan Knightley, one of the RPG members since its foundation, and who gives a high contrast to the character's duality. His interpretation alterns moments of calm and moments of terror, during which the character leans toward a grotesque exuberance. By this way, the comedian proposes a new and astonishing interpretation of the character, whose tragic path becomes an infantile regression. The play moving towards its end, Macbeth curls up on himself more and more, sucking his thumb and hiding in Lady Macbeth's skirts.

With its choices of actor playing, the Performance Research Group therefore affirms its commitment to draw links between contemporary practice and a historical restitution of Elizabethan theater practices. As in Elizabethan and Jacobean troupes of actors, the PRG multiplies games of doubling. But far from being a simple effect of reconstitution, this choice involves in enriching the meaning effects produced by Shakespearean text. Jan Knightley thus embodies both Macbeth and one of the murderer character ; and the actress Helena Quinn plays both Lady Macbeth and one of the witches. As in the adaptation of Antony and Cleopatra, where the doubling of roles was extreme, with only two actors interpreting a total of sixteen characters, this process increases the meanings of the play, without losing the audience. Jacquelyn Bessel's staging is thus based on a high degree of adaptability to the conditions of representation, which are perceived less as constraints than expressive possibilities. In that, she recalls that the principles and the material conditions of performance during the Elizabethan era are transferable today, and may provide staging opportunities for staging that meet the most contemporary proposals.

The performance of Macbeth by the Jacquelyn Bessell's Performance Research Group can be considered, by the radicalism of its staging choices, as a bridge between an eminently contemporary theater practice and the numerous attempts to restitute the Elizabethan dramatic practices. The bridges built between those periods are obviously fragiles, buth however rich in potential.

Alban Déléris (IRCL -Université Montpellier III)