Burcu Afyonoglu Fazlıoglu

Agustín Pérez-Barahona

C ¸agrı Saglam

The dynamic implications of energy-intensive capital accumulation

Keywords: natural resources, energy prices, overlapping generations

We study the implications of assuming different energy intensities for physical capital accumulation and final good production in an overlapping generations (OLG) resource economy. Differing from the standard OLG literature, but consistently with the empirical evidence, physical capital accumulation is assumed to be relatively more energy-intensive than consumption. Focusing on exhaustible resources, we find that OLG equilibria can exhibit a "non-classical behaviour": our model can generate complex dynamics where extraction may increase during some periods and decrease afterwards. As a consequence, in contrast to the classical response predicted by the standard approach, resource prices may not increase monotonically. This result points out the importance of the assumptions about energyintensity considered in the literature.

Introduction

How does the scarcity of natural resources limit economic growth and to what extent physical capital accumulation offsets this constraint? These questions have been the research focus of many papers in the literature of resource economics dating back to Dasgupta andHeal (1974, 1979), [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF], and [START_REF] Stiglitz | Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths[END_REF]. In these studies, resources are assumed to be extracted in order to provide energy that will be used in the final good production. The limit to economic growth is then directly imposed by the usage of scarce resources in production. Greater physical capital accumulation is suggested (unless non-renewable resources are substituted with renewables) to overcome this constraint. However, the overwhelming majority of the literature, assuming the same technology for both consumption and physical capital accumulation, tends to contradict the empirical evidence about the energy intensity of these sectors. Data suggests that physical capital production is relatively more energy intensive than consumption, so that non-renewable resources could limit growth through the equipment production sector. 1 In this paper we analyze the effect of differentiating the energy intensities of physical capital and final good production in an overlapping generations (OLG) exhaustible resource economy. We study how the standard results regarding the dynamics of resources and prices are modified if physical capital accumulation is assumed to be relatively more energy-intensive than consumption.

To our knowledge, differing energy intensities has only been studied in Pérez-Barahona (2011), considering an infinitely lived agents general equilibrium model. 1 Azomahou et al. (2004 and 2006) builds an energy intensity measurement (ratio between energy consumption and value added) of 14 sectors of the economy from the Structural Analysis Database of OECD and the Energy Balances and Energy Prices and Taxes of IEA. Their empirical analysis shows that energy intensity is high in sectors closely involved in production of physical capital (e.g., iron and steel sector, 0.809; transport and storage, 0.85; non-ferrous metals, 0.599; and non-metallic minerals, 0.507). However, energy intensity is lower for consumption goods related sectors (e.g., food and tobacco, 0.134; textile and leather, 0.082; and construction, 0.018).

In the same direction, [START_REF] Williams | Energy Intensity of Computer Manufacturing: Hybrid Assessment Combining Process and Economic Input-Output Methods[END_REF] points out that the "life cycle energy use of a computer is dominated by production (81%) as opposed to operation (19%)".

1

This paper shows that the monotonicity properties of consumption and resource extraction change if physical capital accumulation is assumed to be relatively more energy-intensive than consumption. The study is limited however to infinitely lived agents models, while numerous papers claim that the OLG framework offers a better explanatory power for the discussion of natural resource problems. 2 As in Pérez-Barahona (2011), we assume that the physical capital sector is more energy intensive than consumption, and energy is obtained from the extraction of natural resources. Thus, going beyond the standard literature, in our paper the accumulation of physical capital is assumed to be determined not only by the savings but also by the energy that it requires. In addition, instead of taking resources as the only way of saving (Krautkraemer and Batina, 1999;[START_REF] Koskela | Renewable resources in an overlapping generations economy without capital[END_REF], both resource stock and man-made capital are considered to be alternative assets [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF][START_REF] Farmer | Intergenerational natural-capital equality in an overlappinggenerations model with logistic regeneration[END_REF]Agnani et al., 2005;[START_REF] Bednar-Friedl | Time consuming resource extraction in an OLG economy with capital[END_REF].

In the literature of natural resources and OLG models, the equilibrium trajectories converge to a single steady-state (or single balanced growth path) with saddlepath dynamics under linear/no regeneration of resources and with exogenous/no technological progress. The standard results on growth and dynamics in OLG exhaustible resource economies are mainly built upon the studies of [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF], and [START_REF] Olson | Exhaustible Resource Allocation in an Overlapping Generations Economy[END_REF]. It is shown that with exhaustible resources, no technical progress, and standard assumptions about preferences and technology, extraction trajectories monotonically decrease (and therefore, resources price monotonically increases) into the trivial steady-state in the limit. [START_REF] Olson | Exhaustible Resource Allocation in an Overlapping Generations Economy[END_REF] find however that under non-standard assumptions (finite horizon quadratic utility), the convergence to the zero steady-state does not need to be monotone. Extraction could transitory rise, with the corresponding decrease of 2 Three main reasons are frequently quoted. First, resources are a store of value between generations (see [START_REF] Koskela | Renewable resources in an overlapping generations economy without capital[END_REF][START_REF] Valente | Sustainable development, renewable resources and technological progress[END_REF][START_REF] Bednar-Friedl | Time consuming resource extraction in an OLG economy with capital[END_REF]2013). Second, intergenerational aspects should be taken into account when analyzing environmental issues and/or natural resource economies [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF][START_REF] Padilla | Intergenerational equity and sustainability[END_REF]and Agnani et al., 2005). Finally, there exists empirical evidence showing that agents are not perfect altruistically linked (among others, [START_REF] Altonji | Is the extended family altruistically linked? Direct tests using micro data[END_REF]and Balestra, 2003). 2 resources price. Moreover, cycle extraction is shown to be numerically possible under infinite horizon, CES preferences, and for a very particular set of parameters. [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF] studies the interaction between capital accumulation and natural resources exploitation under linear/no regeneration of resources. Besides from how physical capital accumulates, this model is similar to ours but the equilibrium dynamics are saddle-path stable.3 Differing from this approach, our paper contributes to the literature by showing that, without taking linearizing assumptions, our model can generate complex dynamics and the subsequent nonmonotonic behaviour of resources extraction and prices.

In absence of technological progress, we fully characterized the global dynamic response of our economy with respect to the share of exhaustible resources in the production of physical capital. The main contribution of our work comes from this characterization. In contrast to [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF], local indeterminacy and hopf bifurcations can arise if the share of energy resources in the equipment production is low enough. Moreover, a non-monotonic behavior of extraction is possible: the stock of (exhaustible) resources always decreases, yet extraction may increase during some periods and decrease afterwards. As a consequence, in contrast to the "classical behavior" predicted by the standard approach building on [START_REF] Hotelling | The economics of exhaustible resources[END_REF] seminal work, resource prices may not increase in the short-run.

Non-renewable resources literature widely discussed the discrepancy between the famous Hotelling prediction and the empirical evidence on resource prices. In fact, the prediction of monotonically rising prices has not been consistent with the empirical dynamic behavior of non-renewable resource prices and their in situ values. 4Investigation of mineral commodity prices shows relative declines and fluctuation around time trends, rather than persistent increases, for long periods of time (see figures 1-11 in Krautkraemer, 1998). Considering a large data set of U.S. mineral prices, Barnett and Morse (1963) were the first to systematically identify this pattern. [START_REF] Slade | Trends in natural-resource commodity prices: an analysis of the time domain[END_REF] further reveals that a U-shaped price evolution is consis-tent with the observed prices of many non-renewable resources over the period (for a recent study, see Ferraro and Peretto, 2003). Several modifications of Hotelling's model were then introduced in order to explain the behavior of prices over time. Environmental constraints and the natural resource abundance (Ahrens and Sharma, 1997), backstop technologies [START_REF] Heal | The relationship between price and extraction cost for resource with a backstop technology[END_REF], technical change [START_REF] Slade | Trends in natural-resource commodity prices: an analysis of the time domain[END_REF] and informational asymmetries [START_REF] Pindyck | Uncertainty and exhaustible resource markets[END_REF] are possible known channels for inducing resources price decrease. In this regard, our paper identifies an additional channel for this "non-classical response" of prices, which is based on the energy intensity characteristics of the economy. We provide at the same time a justification for the emergence of hopf cycles and indeterminacy, i.e., multiple equilibria. The interaction between energy intensity, discount rate and the share of physical capital in the final good production induces cyclic behavior in the economy via hopf bifurcations. Hopf bifurcations are economically important as they provide a powerful and easy tool to detect limit cycles and justify the emergence of cycles endogenously (for further details, see [START_REF] Benhabib | Indeterminacy and sunspots in macroeconomics[END_REF][START_REF] Kind | Remarks on the economic interpretation of hopf bifurcations[END_REF]. 5 Finally, another important feature of the paper is that our results do not rely on particular parameterizations of the exogenous functions involved in the model.

It rather provides a flexible set-up to study the role played by the energy intensity, keeping the model tractable, together with general and plausible qualitative properties. The results are in fact analytical, and the dynamics are fully characterized. The paper proceeds as follows. The model is presented in Section 1. The competitive equilibrium is defined in Section 2. Section 3 presents the equilibrium dynamics and examines the stability of the long-run response. Conclusions and broader theoretical implications are discussed in Section 4.

5 Hopf cycles appear when a fixed point loses or gains stability due to a change in a parameter and meanwhile a cycle either emerges from or collapses into the fixed point [START_REF] Asea | Time-to-build and cycles[END_REF]. Depending on the stability of the cycles a stylized business cycle (attracting/stable cycle) or corridor stability (repelling/unstable cycle) can appear [START_REF] Kind | Remarks on the economic interpretation of hopf bifurcations[END_REF][START_REF] Benhabib | Some new results on the dynamics of the generalized Tobin model[END_REF].

The model

We study a perfect foresight overlapping generations economy, without population growth, in discrete time with infinite horizon. In contrast to the standard OLG approach, our model differentiates the energy intensity of the physical capital sector and the corresponding of the final good production.6 There are three sectors in the economy: final good production, equipment (investment) good production and extraction sector. A single final good, which can be either consumed or invested, is produced in the economy by combining physical capital and labor. The physical capital that is used in this production process is provided by the equipment sector.

The physical capital is obtained by means of the already installed equipment and the energy supplied by the extraction sector. Finally, the extraction sector gets the energy directly from the natural resource extracted. All agents have rational expectations and each generation consists of a single representative agent. Moreover, they are price-takers and all the markets are assumed to be competitive.

Extraction sector

The purpose of natural resources in the model is double. They are both a store of value, as an asset, and an input in the production of investment good as energy.

Following Agnani et al. (2005), we assume a grandfathering economy (due to the ownership of natural resources) so that the initially old generation possesses the stock of natural resources. At the beginning of each period t, the old agents own the resource stock E t = e t N t , where e t is the amount of resources per-worker and N t is the number of workers. They choose how much to extract from the resource stock in order to sell it as energy to the equipment good sector, X t . The remaining part of the natural resource is sold to the young agents as resource assets, A t (= E t -X t ). From period t to t + 1, the resource stock regenerates at a linear rate Π [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF], where Π 1. 7 The transition dynamics of energy resources in per-worker terms can be formalized as follows:

e t+1 = Π(e t -x t ) = Πa t , (1)
where x t ≡ X t /N t and a t ≡ A t /N t .

Consumers

The representative individual receives an income equal to the real wage w t from supplying her one unit of labor to the firms when young. She allocates her income among current consumption, c t , savings related to investment, s t , and the purchase of natural resource ownership rights, a t . In her last period of life (when old at period t + 1), the agent is retired and consumes d t+1 out of her entire income, not leaving any bequests. Her income is then generated from the return on her savings made when young, R t+1 s t , from extracting the demanded portion of the energy resources and selling it to the firms, Q t+1 x t+1 , and from selling the rest to the young, P t+1 a t+1 , where R t is the interest rate, and Q t and P t are the corresponding prices. Accordingly, the budget constraints facing generation t is as follows:

c t + s t + P t a t = w t , (2)
d t+1 = R t+1 s t + Q t+1 x t+1 + P t+1 a t+1 . (3)
Generations derive utility from consumption, where their two-period intertemporal utility function dependens on the level of consumption when young c t and when old d t+1 . We assume an additively separable life-cycle utility function

U (c t , d t+1 ) = u (c t ) + βu (d t+1 )
, where β ∈ (0, 1) is the subjective discount factor. In particular, we adopt a logarithmic instantaneous utility function u since we are mainly concerned with the existence of the competitive equilibrium and its qualitative properties.

Taking the prices of the energy resource and wages as given, the representative agent born at time t maximizes her utility by choosing young and old periods' consumption, and the ownership rights of the energy resource. The corresponding optimization problem of the representative consumer is formalized as follows:

max {ct,d t+1 ,st,e t+1 } ln c t + β ln d t+1 subject to c t + s t + P t a t = w t , d t+1 = R t+1 s t + Q t+1 x t+1 + P t+1 a t+1 , e t+1 = Π(e t -x t ) = a t c t ≥ 0, d t+1 ≥ 0, e t+1 ≥ 0, E 0 > 0 given.
We get the following first-order conditions:

d t+1 c t = βR t+1 , (4)
P t+1 P t = R t+1 Π , (5)
P t+1 = Q t+1 . (6)
Equation ( 4) gives the equalization of discounted marginal utilities, where the marginal rate of substitution between current and future consumption is equal to their relative prices. Equation ( 5) is the non-arbitrage condition between the different types of savings (savings related to investment and ownership rights). Finally equation ( 6) is a non-arbitrage condition, which implies that in the equilibrium the asset price and the extracted energy price are the same. This last result is indeed consistent with [START_REF] Olson | Exhaustible Resource Allocation in an Overlapping Generations Economy[END_REF] and [START_REF] Valente | Sustainable development, renewable resources and technological progress[END_REF]. Moreover, notice that equations ( 5) and ( 6) are equivalent to the Hotelling rule presented in [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF].

Firms

Final good sector

Firms operating in the final good sector are owned by the old households. Firms produce the final good with a Cobb-Douglas constant returns to scale technology. Equation ( 7) presents this production function at any date t. The exogenous disembodied total factor productivity is represented by Z t (equation 8). With constant returns to scale, the number of firms does not matter and the production is independent of the number of firms that use the same technology. We are so concerned with the problem of a representative firm. Under this perfectly competitive environment, at each period t, taking the prices of inputs, the initial technology level, and the initial level of capital stock as given, the representative firm maximizes its profit by choosing the amount of labor and physical capital (equipment) inputs:

max {Kt,Nt} ∞ t=0 π t = Y t -P K t K t -w t N t , s.t. Y t = Z t K α t N 1-α t 0 < α < 1, (7)
Z t+1 = (1 + z)Z t z ≥ 0. ( 8
)
At an interior solution of the firm's optimization problem, where all variables are expressed in per-worker terms (k t ≡ K t /N t ), the following first-order conditions are satisfied equating the price of the inputs to their marginal benefits:

αy t = P K t k t , (9)
(1 -α)y t = w t .

(10) Equation ( 11) summarizes the market clearing condition of the economy. The final good is either consumed by young agents, C t , by old agents (generation t-1), D t , or invested for the production of the future capital stock, S t :

Y t = C t + D t + S t . (11)

Physical capital sector

In the standard OLG literature, the new capital stock at time t + 1 is fully determined by the savings made at time t, which are equal to the investments.

However, following Pérez-Barahona (2011), since the physical capital production is relatively more energy-intensive than consumption we model the accumulation of capital stock to be determined not only by the savings made at time t, but also by the energy that it requires. 8 The usage of natural resource introduces then a new constraint on economic growth through the capital accumulation sector.

In equation ( 12) the new capital at t + 1, K t+1 , is produced from the natural energy resources X t+1 and the investment made at time t, I t , with the following Cobb-Douglas technology:

K t+1 = B θ t+1 X θ t+1 I 1-θ t 0 < θ < 1, ( 12
)
S t = I t , (13)
B ṫ+1 = (1 + b)B t , b ≥ 0. ( 14
)
It is worthwhile to mention that the savings S t are still equal to the investments (equation 13). However, only a fraction of the investments can generate the new capital stock. B t is the technological progress in the equipment good sector.

In contrast to Z t , B t is energy-saving and specific to the accumulation of physical capital. If B t increases, the productivity of the natural resources in the production of equipment rises. Consequently, a lower amount of energy would be necessary to produce the same amount of new physical capital. Changes in B t also represent investment-specific technological change, which is assumed to affect exclusively the equipment sector.

In the equipment sector, at each period t, the representative firm maximizes its profit by choosing the amount of energy resources that will be used in the production process:

max {Xt} ∞ t=0 π t = P K t K t -Q t X t , s.t. K t = B θ t X θ t I 1-θ t-1 ,
taking as given the prices of capital (P K t ) and resource input (Q t ), and the initial level of capital stock. At an interior solution of the equipment firm's optimization 8 We should observe that in our economy energy is not a direct input to produce the final good. As in Pérez-Barahona (2011), this simplification is considered in order to highlight the importance of the energy intensity differences between consumption and equipment sectors.

problem the following first-order condition is satisfied:

Q t = θαY t X -1 t . ( 15
)
The profit of investing in capital R t S t-1 at time t should be equal to the profit of producing new capital (1 -θ)P K t K t to prevent arbitrage opportunities. 9 We then obtain the following condition:

R t = (1 -θ)P K t B θ 1-θ t X θ 1-θ t K θ θ-1 t .
(16)

The competitive equilibrium

The dynamic competitive equilibrium for this OLG natural resource economy is defined as a sequence of prices w t , R t , P K t , P t , Q t ∞ t=0 and feasible allocations {c t , d t , s t , e t , a t , x t , i t , y t , k t+1 } ∞ t=0 , given the positive initial values for S -1 , E 0 , Z 0 , B 0 , N 0 , and the law of motion of exogenous technological progresses Z t and B t , such that the consumers maximize their life-time utility, firms maximize their profits, and all markets clear at every period t. This equilibrium is therefore a solution of the system of equations ( 1)-( 16).

Equilibrium dynamics

The main focus of our paper is to study the dynamics around a steady-state equilibrium. We therefore assume no technological progress, i.e., b = z = 0. As in [START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF], [START_REF] Farmer | Intergenerational natural-capital equality in an overlappinggenerations model with logistic regeneration[END_REF] and [START_REF] Bednar-Friedl | Time consuming resource extraction in an OLG economy with capital[END_REF], the intertemporal equilibrium dynamics can be reduced to a two-dimensional system, which represents the law of motion of the resource stock, E t , and the extraction, X t .

Using the equipment technology (12), the condition (13), the budget constraints 9 Choosing the optimal resource stock (equation 15), the maximum profit that the representative firm can obtain is given by

π * t = P K t K t -θP K t B θ t X θ-1 t I 1-θ t-1 X t = (1 -θ)P K t K t .
and first-order conditions (2)-( 5) of the households' problem, and the first-order conditions (10) of the firms' maximization problem, we obtain the equation below for k t+1 :

k 1 1-θ t+1 = B θ 1-θ t+1 x θ 1-θ t+1 σ(1 -α) + θα(1 - e t x t ) y t , (17)
with σ ≡ β/(1 + β). In addition, taking equations ( 5), ( 6), ( 9), ( 15) and ( 16), the following difference equation for k t+1 arises:

Πk

1 1-θ t+1 x t y t x t+1 = α(1 -θ)B θ 1-θ t+1 x θ 1-θ t+1 . ( 18
)
Substituting for k t+1 in the above equations yields (19) below, which describes the dynamics of resource extraction. Since it also depends on the resource stock e t , we rewrite (1), the law of motion of the natural resource, as (20).

x t+1 = Π β (1+β) (1 -α) + αθ α(1 -θ) x t - Πθ (1 -θ) e t , (19)
e t+1 = Π(e t -x t ). (20)
The linear planar system ( 19) and ( 20) describes the dynamics of our economy, which can be rewritten in matrix form as

x t+1 e t+1 = ψ 1 ψ 2 ψ 3 ψ 4 x t e t ≡ Ψ x t e t with ψ 1 ≡ Π (1 -θ) β (1 + β) (1 -α) α + Πθ (1 -θ) , ψ 2 ≡ - Πθ (1 -θ) , ψ 3 ≡ -Π, ψ 4 ≡ Π.
Lemma 1 (steady-states) The steady-state equilibria (x * , e * ) of our economy are characterized by the following equations:

x * = ψ 1 x * + ψ 2 e * , (21)
e * = ψ 3 x * + ψ 4 e * . ( 22
)
These equations have two sets of steady-states:

(i) If {ψ 1 = 1} or {ψ 1 = 1 and ψ 4 = 1} or {ψ 1 = 1, ψ 4 = 1 and (1 -ψ 4 ) = ψ 2 ψ 3
(1-ψ 1 ) } there is a unique steady-state (x * , e * ) = (0, 0) .

(ii) If {ψ 1 = 1, ψ 4 = 1 and (1 -ψ 4 ) = ψ 2 ψ 3 (1-ψ 1 )
} there is a continuum of steadystates such that x * = ψ 2 1-ψ 1 e * .

Proof. A steady-state (x * , e * ) in this economy is a fixed point of ( 19) and ( 20), where x t+1 = x t = x * and e t+1 = e t = e * . Since Π ≥ 1, ψ 1 , ψ 4 > 0, and ψ 2 , ψ 3 < 0, the sets (i) and (ii) are easily determined from ( 21) and ( 22)

From the linear system ( 21) and ( 22), one can verify that there is either a unique steady-state (x * , e * ) = (0, 0), or a continuum of them where natural resources are not necessarily drained. If resources are non-renewable (Π = 1) the only possible steady-state, without technical progress, is (x * , e * ) = (0, 0) because ψ 4 = 1 (set i).

However, the natural replenishing property of renewable resources (Π > 1, thus, ψ 4 = 1) allows the economy to avoid depletion (set ii).

Stability and exhaustible resources

As pointed out in the introduction, the objective of this work is to study the dynamic implications of the energy intensity assumptions of the economy. In particular, our intention is to provide an alternative explanation for the non-classical behavior of exhaustible resources, which was empirically identified in the literature. Even if Section 2 offers a general framework for both exhaustible and renewable resources, from now on, we will concentrate on non-renewable resources and the dynamics around the corresponding unique steady-state (x * , e * ) = (0, 0).

We analyze below the stability of the system, the occurrence of indeterminacy and bifurcations, and the subsequent possibility of non-monotone convergence. In this regard, we find out that the share of energy in the equipment production plays a fundamental role, together with the discount rate and the share of physical capital in the final good production. Notice finally that our conclusions will be global since the dynamics of the economy are described by a linear system.10

We denote by λ 1 and λ 2 the eigenvalues of Ψ in the matrix form of the dynamical system ( 19) and ( 20). Let us define β ≡ β

(1-α) α > 0. The next proposition fully describes the dynamics of our economy by means of studying λ 1 and λ 2 . In particular, we identify cases where complex dynamics arise, allowing the economy to reproduce the non-classical response described before. These conditions are indeed deeply related to the assumption of considering that the equipment sector is more energy intensive than the final good production.

Proposition 1 (stability) For non-renewable resources and different parameter combinations, the stability of the zero steady-state changes such that:

1. For 1 2 ≤ θ < 1, the dynamics are non-complex and the eigenvalues are on the different side of one (λ 1 > 1 and 0 < λ 2 < 1) so that the steady-state is a saddle.

2. For 0 < θ < 1 2 and 2(1 -2θ) ≤ β, the dynamics are non-complex and the eigenvalues are on the different side of one (λ 1 > 1 and 0 < λ 2 < 1) so that the steady-state is a saddle.

For 1

4 ≤ θ < 1 2 and β < 2(1 -2θ), the dynamics are non-complex and the eigenvalues are on the different side of one (λ 1 > 1 and 0 < λ 2 < 1) so that the steady-state is a saddle. 4. For 0 < θ < 1 4 , β < 2(1 -2θ) and β + 1 β > 2(1 -2θ), the dynamics are non-complex and the eigenvalues are on the different side of one (λ 1 > 1 and 0 < λ 2 < 1) so that the steady-state is a saddle. 5. For 0 < θ < 1 4 and β + 1 β < 2(1 -2θ) the dynamics are complex:

(a) If θ < 1 -β both eigenvalues (in absolute value) are smaller than one, so indeterminacy occurs and the steady-state is stable.

(b) If θ > 1 -β both eigenvalues (in absolute value) are greater than one, so the equilibrium dynamics are monotone unstable.

Proof. For exhaustible resources Π = 1. If (1 + ψ 1 ) 2 ≥ 4(ψ 1 + ψ 2 ) the two eigenvalues of Ψ are real and given by λ 1,2 = (1+ψ 1 ) 2

± 1 2 (1 + ψ 1 ) 2 -4(ψ 1 + ψ 2 ).
Moreover, λ 1,2 > 0 since the det(Ψ) > 0 and Tr(Ψ) > 1. 11 This implies that nonmonotonic dynamics is only possible if the eigenvalues are complex: if

(1 + ψ 1 ) 2 < 4(ψ 1 + ψ 2 ), λ 1,2 = (1+ψ 1 ) 2 ± i 1 2 4(ψ 1 + ψ 2 ) -(1 + ψ 1 ) 2 .
We identify the cases of complex dynamics in the Appendix. Let us directly consider the stability analysis:

1. For 1 2 ≤ θ < 1, the dynamics are non-complex as the discriminant ∆ ≡ (1 + ψ 1 ) 2 -4(ψ 1 + ψ 2 ) > 0 (see Claim 2 in Appendix). Since λ 1,2 > 0, comparing λ 1 and λ 2 with 1, one can observe that the dynamics are stable iff both eigenvalues are smaller than one. This is equivalent to

(1+ψ 1 ) 2 < 1 - √ ∆ 2 . This is however impossible because (1+ψ 1 ) 2 > 1 - √ ∆
2 (see Claim 3 in Appendix). Therefore, stability cannot occur.

Let us study the possibility of saddle equilibrium or monotone unstable dynamics. The steady-state is saddle iff the two eigenvalues are on the different side of one. As λ 1,2 > 0, this condition is equivalent to λ 1 > 1 and λ 2 < 1 (notice that, from Claim 1, λ 1 > λ 2 ). Taking the expression of λ 1,2 , the condition reduces to 1 -

√ ∆ 2 < (1+ψ 1 ) 2 < 1 + √ ∆ 2 .
In contrast to that, the equilibrium dynamics are monotone unstable iff λ 1 > 1 and λ 2 ≥ 1. This

is equivalent to check that (1+ψ 1 ) 2 ≥ 1 + √ ∆ 2 .
Since the dynamics are noncomplex for 1 2 ≤ θ < 1, we can conclude from Claim 3 that

(1+ψ 1 ) 2 < 1 + √ ∆ 2 . We have also proved in Claim 3 that (1+ψ 1 ) 2 > 1 - √ ∆ 2 . Therefore, 1 - √ ∆ 2 < (1+ψ 1 ) 2 < 1 + √ ∆
2 and, consequently, the steady-state is a saddle.

11 Notice that, λ 1 λ 2 = det(Ψ) = ψ 1 + ψ 2 > 0. Since det(Ψ) > 0, λ 1,2 = 0 and their sign coincide. Moreover, λ 1 + λ 2 = Tr(Ψ) = 1 + ψ 1 > 1. We can then conclude that both eigenvalues are strictly positive.

2. For 0 < θ < 1 2 and β ≥ 2(1 -2θ) the dynamics are non-complex (Claim 2). Therefore, from Claim 3, (1+ψ 1 )

2 < 1 + √ ∆ 2 .
From the same claim we also know that, if 0 < θ < 1 2 and β ≥ 2(1 -2θ),

(1+ψ 1 ) 2 > 1 - √ ∆ 2 . We can then conclude that 1 - √ ∆ 2 < (1+ψ 1 ) 2 < 1 + √ ∆
2 , thus the steady-state is a saddle.

3. We know from Claim 3 that, if

1 4 ≤ θ < 1 2 and β < 2(1 -2θ), 1 - √ ∆ 2 < (1+ψ 1 ) 2 . Moreover, (1+ψ 1 ) 2 < 1 + √ ∆
2 because we are in a case on non-complex dynamics (see Claim 3). Therefore, as above, the steady-state is a saddle. 4. It is easy to very that statements 5 and 6 in Claim 3 are also valid for 0 < θ < 1 4 and β < 2(1 -2θ) if the dynamics are non-complex. Following Claim 2, we are in a case of non-complex dynamics. We can then conclude that 1 -

√ ∆ 2 < (1+ψ 1 ) 2 < 1 + √ ∆
2 , thus the steady-state is a saddle.

5. From Claim 2, we know that this is the case of complex dynamics. Taking the formulas for λ 1,2 above, one can verify that

|λ 1 | = |λ 2 | = β (1-θ) . Comparing |λ 1 | = |λ 2 |
with 1, we can directly conclude that the dynamics are stable iff the modulus of both eigenvalues is smaller than one. This condition is in fact equivalent to β < (1 -θ). Similarly, the dynamics are unstable iff the modulus of both eigenvalues is larger than one, which corresponds to β > (1 -θ).

The proposition clearly shows that energy intensity differences among economic sectors has important dynamic implications. In contrast to the standard approach, where a monotone converge of resource stock and extraction is predicted, our economy can also exhibit complex dynamics. This outcome is deeply related to the assumption that the equipment sector is more energy intensive than the final good production. Our property of non-monotone converge actually provides an alternative explanation to the empirically supported fact of the non-classical behaviour of exhaustible resources: resource stock and extraction can transitory increase (thus energy prices can temporally decrease) before converging to the zero steady-state.

Proposition 1 points out the role played by the share of resources in the accumulation of physical capital, θ. When the share of energy in the production of equipment is high ( 1 2 ≤ θ < 1), we get the usual saddle-path dynamics. However, for a lower share (0 < θ < 1 2 ) the economy can reproduce situations of complex dynamics and, in particular, non-monotone convergence. In this respect, two additional elements become fundamental through the parameter β: the share of physical capital in the final good production (α) on the one hand, and the individuals' discount rate (β) on the other. For combinations of (α, β) such that β would be low enough (i.e., lower than 2(1 -2θ)), complex dynamics arise. In this case, an additional upper-bound for θ appears in order to ensure the stability of the steady-state. This is statement 5 (a) in the proposition, which corresponds to the non-classical response of exhaustible resources. Since energy resources are non-renewable in this economy, the variable stock of resources e t in the system ( 19)-( 20) cannot increase. However, differing from non-complex dynamics, the extraction of resources (variable x t ) can increase during some periods, eventually converging (asymptotically) to the steady-state. Replacing Y t in ( 15) by ( 7), and K t by ( 12) afterwards, we get the usual negative relationship between extraction and energy prices Q t . Therefore, energy prices can transitory decrease, rising later on as extraction approaches to the steady-state. This possibility of non-monotonic convergence is an important property of our framework. It highlights in fact the influence of the energy intensity assumptions on the dynamic predictions of the model.

The parameters described above are related to the importance that each generation gives to energy resources. In particular, they affect the value of the ownership rights for the natural resources a t , which are sold from one generation to the next.

In economies where θ is small (lower than 1/4 in our model), energy resources are relatively unimportant with respect to the other inputs in the equipment technology. Therefore, future generations give little value to the ownership rights of resources. The current generation would then not have many incentives to leave resources to the next generation, increasing thus the extraction when old. This will continue until energy becomes relative scarce. At this moment, the value of the ownership rights raises, creating new incentives to reduce extraction and, consequently, increasing the amount of resources left to the next generation. 12 We should observe that the non-classical behaviour also depends on the share of equipment in the technology of final good and the discount rate through the parameter β. Proposition 1 (statement 5) identifies boundaries for β. The condition in (a) can be rewritten as β < 1 -θ, providing an upper limit. Moreover, it also

implies that β < 2(1 -2θ) since (1 -θ) < 2(1 -2θ) for 0 < θ < 1/4. The second condition β + 1/ β < 2(1 -2θ
) concludes then that β cannot be too small either.

Let us consider a reference economy for 0 < θ < 1/4 where α and β are such the conditions of statement 5 (a), and so the boundaries for β, hold. For a higher discount rate β, the parameter β will increase and, therefore, the upper limit may not be respected anymore. 13 However, this effect can be counterbalanced if the economy has a higher share of physical capital in the technology of final goods (α).

The interpretation is the following. A higher discount rate implies greater concern of consumers for their old age and, consequently, for the amount of resources they are going to extract or leave (sell) to the next generation. Since energy is not very important in this economy (0 < θ < 1/4), the next generation will not value so much the ownership rights. Therefore, as above, the current generation may increase the resource extraction for some periods. Nevertheless, a larger share of equipment in final good technology would increase how much future generations value energy resources and the corresponding ownership rights, ensuring then that extraction will eventually decreases.

Notice finally that the upper bound for β allow us to identify limit cycles. We show in the next corollary that the critical value (1-θ) reveals indeed the existence of a hopf-type bifurcation. Hopf cycles appear when a steady-state loses or gains stability due to a change in a parameter, and meanwhile a cycle either emerges

12 Since consumers are non-altruistic and live for two periods, ownership rights prevent the old generation from extracting all resources.

13 From the definition of β, it is easy to see that ∂ β/∂β > 0 and ∂ β/∂α < 0.

from or collapses into the steady-state (for further details, see among others, [START_REF] Asea | Time-to-build and cycles[END_REF][START_REF] Kind | Remarks on the economic interpretation of hopf bifurcations[END_REF][START_REF] Yüksel | Capital dependent population growth induces cycles[END_REF].

Corollary 1 (hopf bifurcation) Assume that 0 < θ < 1 4 and β + 1 β < 2(1 -2θ). Indeterminacy occurs for β < 1 -θ; a hopf bifurcation arises for β = 1 -θ; and the steady-state is unstable for β > 1 -θ.

For small values of the share of energy in the equipment production, depending on its interaction with the capital share in the final good technology and the discount rate, the equilibrium trajectories could be indeterminate, implying the possibility of multiple equilibria (see [START_REF] Benhabib | On growth and indeterminacy: some theory and evidence[END_REF], for a survey). If β < 1 -θ, in contrast to saddle stability, there would be multiple paths tending to the (unique) steady-state since many values of x 0 ≥ 0 would be compatible with this equilibrium. Moreover, as showed before, the convergence in this case will be non-monotonic. If β > 1 -θ cycles arise, but the economy will not converge to the steady-state. These unstable dynamics can be interpreted considering again the importance that each generation gives to energy resources. Assume an economy with (α, β) such that it converges, non-monotonically, to the steady-state. If we sufficiently increase β and reduce α, the subsequent larger β may induce cycles that will not converge to the steady-state (if β is greater than 1 -θ). As before, a greater discount rate (β) implies that generations are more concerned about their old age and, therefore, about how much they will extract or leave to the next generation. If at the same time the importance of equipment input in the final good production (α) lowers, energy would not be central either. Therefore, the next generation will not value natural resources much, reducing the incentives of the old generation to replace extraction by ownership rights. Let us lastly mention that since resources are non-renewable in this paper, the non-monotonic trajectories corresponding to β ≥ 1 -θ will end up in the steady-state but in a finite time, i.e., non-asymptotically. Because the quantity of resources cannot increase, the current stock e t is an upper-limit to how much the economy can extract in that period. Since the stock of resources always shrinks with extraction, the economy will completely deplete the resources in a finite time if β ≥ 1 -θ.

Although the literature widely assumes the same technology for consumption and capital accumulation sector, data suggest that physical capital production is relatively more energy-intensive than consumption and, thus, the usage of nonrenewable resources can limit growth through the equipment production. Using an overlapping generations (OLG) natural resources model, we examine the dynamic implications of differentiating the energy intensities of physical capital sector and final good production. The model assumes that the equipment sector is more energy intensive than consumption, where energy is obtained from the extraction of natural resources. In contrast to the standard approach, we model the accumulation of physical capital to be determined not only by the savings but also by the energy that it requires.

We find that the introduction of energy-intensity differentiation among sectors has important implications for the standard results in the area. Richer dynamics other than saddle arise. In particular, energy extraction and prices can follow a non-monotonic trend that is consistent with the empirical finding about the nonclassical behaviour of exhaustible resources. From an economic policy perspective, we believe that this paper provides a meaningful contribution too. On the one hand, we have identified a new component that improves our understanding about the puzzling issue of non-monotonic response of exhaustible resources. Our work makes compatible the theoretical predictions with the empirics in this regard. On the other hand, since proper policymaking depends on economic forecasts, "appropriately modeling the nature of the time series can be invaluable to forecasters" [START_REF] Lee | Non-renewable resource prices: deterministic or stochastic trends[END_REF]. In this line, a general message of our paper is that the assumptions about the energy-intensity differences among sectors should be taken with caution in the design of economic policies.

Several issues can be considered in future research. As pointed out before, our framework can be also applied to renewable resources by considering Π > 1.

This would potentially create even richer dynamics. In particular, convergence to steady-states without exhaustibility may be possible since resources naturally reproduce. Extraction, moreover, would not necessarily induce lower future stock of resources. This could allow then, in contrast to the corner solutions observed before, the emergence of the long-run cycles. Following the same direction, another interesting hypothesis to examine is the role played by the regeneration law of natural resources. One could indeed consider alternative regeneration specifications to the usual linear replenishing of resources. In this respect, a logistic reproduction law has been often recommended to study forestry problems [START_REF] Farmer | Intergenerational natural-capital equality in an overlappinggenerations model with logistic regeneration[END_REF]and Bednar-Friedl and Farmer, 2013, among others).

Appendix

Proof of Proposition 1

Claim 1 For non-renewable resources, the discriminant ∆ ≡ (1+ψ 1 ) 2 -4(ψ 1 +ψ 2 ) cannot be zero. In fact,

∆ > (<)0 ⇔ 2 (1 -2θ) -β < (>) 1 β . (23)
Moreover, for real eigenvalues, λ 1 > λ 2 .

Proof. Let us prove that ∆ = 0. By contradiction, we assume that ∆ = 0.

Therefore

(1 + ψ 1 ) 2 = 4(ψ 1 + ψ 2 ). Since ψ 1 = β+θ 1-θ and ψ 2 = -θ (1-θ) , the pre- vious condition is equivalent to 1+ β 1-θ 2 = 4 β 1-θ . This is only possible iff β = (1 -2θ) ± (1 -2θ) 2 -1. Since (1 -2θ) 2 < 1, β would be a complex number.
This contradicts however our definition of β, which is a real number. We indeed conclude that ∆ > (<)0 iff 2 (1 -2θ) -β < (>) 1 β . Finally, from the expression of

1. If 1 2 ≤ θ < 1 then 1 - √ ∆ 2 < 0.
2. For non-complex dynamics, 1 +

√ ∆ 2 > (1+ψ 1 ) 2 . 3. If 1 2 ≤ θ < 1 then (1+ψ 1 ) 2 > 1 - √ ∆ 2 .
4. If 0 < θ < 1 2 and β 2(1 -2θ) then (1+ψ 1 ) 2 > 1 -√ ∆ 2 .

5. If 1 4 ≤ θ < 1 2 and β < 2(1 -2θ) then 1 -√ ∆ 2 > 0.

6. If 1 4 ≤ θ < 1 2 and β < 2(1 -2θ) then (1+ψ 1 ) 2 > 1 -

√ ∆ 2 .
Proof.

1. Let us first show that 1 -√ ∆ 2 = 0. Suppose the contrary, 1 = √ ∆ 2 . Then β = (1-2θ)±2 √ 2θ 2 -3θ + 1. For θ = 1/2, β = 0 and we find a contradiction because β > 0. Let us study 1/2 < θ < 1. It can be verified that in this case 2θ 2 -3θ + 1 = 0: 2θ 2 -3θ + 1 = 0 for θ = 0 or θ = 1/2. We can actually proof that 2θ 2 -3θ + 1 < 0: assuming that 2θ 2 -3θ + 1 > 0, 3 < 1 θ + 2θ; however, for 1/2 < θ < 1, it is easy to check that 3 > 1 θ + 2θ. Nevertheless, this result implies that β is complex, which is a contradiction and, consequently, 1 -√ ∆ 2 = 0. In fact, we can show that 1 -√ ∆ 2 < 0. Let us assume the contrary, 1 -√ ∆ 2 > 0. This would imply that 4(1 -θ) 2 -1 > β β -2(1 -2θ) . However, for 1 2 ≤ θ < 1, the RHS> 0 while LHS< 0. So we get a contradiction. . This is equivalent to (ψ 1 -1) 2 ≥ ∆(> 0). Since ∆ = (1+ψ 1 ) 2 -4(ψ 1 +ψ 2 ), the previous condition would imply that 0 ≥ -4ψ 2 . But this is impossible because ψ 2 < 0.

3. From statement 1 of this claim, we already know that 1-√ ∆ 2 < 0 if 1 2 ≤ θ < 1. Our result is then verified since ψ 1 > 0. , where we can identify two cases for β < 2(1 -2θ): (i) (1 -2θ) ≤ β < 2(1 -2θ) and (ii) β < (1 -2θ) < 2(1 -2θ). Case (i) yields a contradiction since (ψ 1 -1) > 0.

Case (ii) would imply (ψ 1 -1) 2 ≥ ∆, which is impossible for non-complex dynamics (see proof of Claim 3, statement 2).

6 References

  can verify that 2θ 2 -3θ+1 > 0 and (1 -2θ) > 0. We can show that β = (1-2θ) + 2 √ 2θ 2 -3θ + 1 is impossible: let us rewrite β = (1 -2θ) + (1 -2θ) 2 + [4(1 -θ) 2 -1]; it is easy to check that 0 < β < 2(1 -2θ) < (1 -2θ) + (1 -2θ) 2 + a for a > 0; recalling a = 4(1 -θ) 2 -1, since 4(1 -θ) 2 -1 > 0 for 0 < θ < 1/2, the previous expression provides the contradiction β < β. Similarly, β = (1 -2θ) -2 √ 2θ 2 -3θ + 1 cannot hold either: since β > 0, this would imply that (1 -2θ) > (1 -2θ) 2 + [4(1 -θ) 2 -1]; however, (1 -2θ) < (1 -2θ) 2 + afor a > 0; as above, recalling a = 4(1 -θ) 2 -1(> 0) gives us a contradiction.We know now that 1 -√ ∆ 2 = 0. In order to show that 1 -This would imply that β[ β -2(1 -2θ)] > 4(1 -θ) 2 -1. But this is not possible since the LHS< 0 and the RHS> 0. 6. By contradiction. We assume (1+ψ 1 the one hand, -√ ∆ < 0 because we are in a situation of noncomplex dynamics. On the other hand, (ψ 1 -1) = β-(1-2θ)1-θ

  Agnani B., M.Gutierrez and A. Iza (2005), "Growth in overlapping generation economies with non-renewable resources", Journal of Environmental Economics and Management, 50, 387-407.Ahrens W. and V. Sharma (1997), "Trends in natural resource commodity prices: deterministic or stochastic?", Journal of Environmental Economics and 33,[59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]

For a detailed analysis of the dynamics in[START_REF] Mourmouras | Competitive equilibria and sustainable growth in a lifecycle model with natural resources[END_REF] see[START_REF] Farmer | Intertemporal resource economics: an introduction to the overlapping generations approach[END_REF].

See Krautkramer (1998) for a discussion of this literature.

For the discussion of standard OLG models see de laCroix and Michel (2004).

Note that the resource is non-renewable as long as Π = 1.

The results of this paper correspond to global dynamics: they are valid regardless how far away the economy is from the steady-state. Therefore, in contrast to linearization and local dynamics, we provide a more general outcome.

Notice that stable (unstable) cycles can be sustained, for β = (>)1 -θ, if the stock of resources were not subject to the positivity constrain e t ≥ 0.

The zero steady-state then emerges, but as a corner solution. 14 the real eigenvalues, we can directly observe that λ 1 > λ 2 since ψ 1 > 0 and ∆ = 0 Claim 2 Whether the dynamics are complex or not depends on the following parameter combinations:

1. For 1 2 ≤ θ < 1 or {0 < θ < 1 2 and β 2(1 -2θ)} the dynamics are non-complex.

2. For 1 4 ≤ θ < 1 2 and β < 2(1 -2θ) the dynamics are non-complex.

3. For 0 < θ < 1 4 and β + 1 β > 2(1 -2θ) the dynamics are non-complex.

4. For 0 < θ < 1 4 and β + 1 β < 2(1 -2θ) the dynamics are complex.

Proof. The dynamics are (non-)complex iff ∆(>) < 0.

2. By contradiction. Let us assume, for 1 4 ≤ θ < 1 2 and β < 2(1 -2θ), that ∆ < 0. Claim 1 would then imply that 2 (1 -2θ) > 1 β + β. Therefore, since all terms are non-negative, we conclude that 2 (1 -2θ) > 1 β and 2 (1 -2θ) > β. This can be rewritten as 2 (1 -2θ) > β > 1 2(1-2θ) . The condition holds iff 4 (1 -2θ) 2 > 1. This reduces to 2 (1 -2θ) > 1 since (1 -2θ) > 0. Hence θ < 1 4 , which contradicts our initial assumption on θ.

3. It follows from equation (23).

It follows from equation (23).

Finally notice that the case β + 1 β = 2(1 -2θ) is not possible since ∆ = 0 (see Claim 1) Claim 3 Considering the definition of ∆ in Claim 1, we can establish the following results:


Anderson Luis Szejka
email: anderson-luis.szejka@univ-lorraine.fr

Alexis Aubry
email: alexis.aubry@univ-lorraine.fr

Hervé Panetto
email: herve.panetto@univ-lorraine.fr

Osiris Canciglieri Júnior

Eduardo Rocha Loures
email: eduardo.loures@pucpr.br

Towards a conceptual framework for requirements interoperability in complex systems engineering

Keywords: Requirements Engineering, Requirements Interoperability, Complex Systems Engineering, Requirements Consistency

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Introduction

Enterprises have been specializing in specific domains and establishing partnerships with other companies to complement their initial skills to face globalization and consequently its intensified competition. This approach resulted in the so-called collaborative network that allows the development of complex systems and collaborative activities in many industrial domains like aeronautics, nanotechnology, aerospace, bioengineering, etc. According to [START_REF] Mallek | The application of interoperability requirement specification and verification to collaborative process in industry[END_REF], for succeeding in these collaborative engineering processes, it is important to formalize how different partners can work together and, through their interactions, how a common objective can be achieved within different perspectives. These engineering processes follow best practices generally defined in the so-called systems engineering domain. System Engineering (SE) is "an interdisciplinary approach and means to enable the realization of successful systems" [START_REF]INCOSE Systems Engineering Handbook: A Guide for Life Cycle Processes and Activities[END_REF]. It focuses on holistically and concurrently understanding stakeholder needs; exploring opportunities; documenting requirements; and, synthesizing, verifying, validating and evolving solutions while considering the complete problem, from system concept exploration throughout all phases until system disposal. One of the SE processes is dedicated to analysing users and systems requirements, denominated Requirement Engineering (RE). RE refers to activities of formulating, documenting and maintaining systems requirements [START_REF] Young | T he R equirem ents E ngineering H andbook[END_REF] to produce, from the users' needs, a set of specification related to what the final system must be.

Requirements provide the basis for all phases of the system development and must be controlled inside all these phases and domains to avoid misinterpretation and mistakes that would compromise the final results [START_REF]INCOSE Systems Engineering Handbook: A Guide for Life Cycle Processes and Activities[END_REF][START_REF] Afis | Guide Bonnes Pratiques en Ingénierie des Exigences[END_REF][START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF]. While approaches such as model-based systems engineering (MBSE) have been studied in [START_REF]The International Council on Systems Engineering[END_REF][START_REF] Holt | Model-Based Requirement Engineering[END_REF][START_REF][END_REF], for improving the definition of requirements based on models, there is still a semantic gap between all requirements definitions when they are defined in different domains for the same engineering project and requirement consistency management in different systems life cycle phases. In order to cope with this challenge, we are working to define a conceptual framework that aims to formally model requirements interoperation in term of impact and semantic equivalence or subsumption. This formal definition will facilitate the verification of the system requirements coherence taking into account the technical constraints defined by appropriate experts along the systems life cycle phases.

The paper is structured as follow: Section 2 addresses the problem statement regarding to the management of requirements when multiple information come from multiple stakeholders' needs during the system life cycle phases. Section 3 presents a literature review concerning the main issues on system requirements considering the cross-domains dimension, the cross-systems life cycle dimension and the crossrequirements dimension. Section 4 is devoted to discuss the main drawbacks and existing gaps in related works. Finally, section 5 concludes and presents perspectives for the research continuation.

Problem Statement

RE is a key activity in the process of engineering a system. Indeed, complex systems with multidisciplinary perspectives require special attention to ensure that all requirements are fulfilled and misinterpretation and mistakes do not occur during phase's evolution of the system life cycle [START_REF] Colombo | Bridging the gap between requirements and design: An approach based on Problem Frames and SysML[END_REF]. In fact, the misinterpretation and mistakes may cause significant a posteriori system refactoring, which result in scheduling overruns and increasing the projects costs [START_REF] Flanigan | System of Systems Requirements Capacity Allocation[END_REF].

The traditional system requirement approach does not support [START_REF] Brooks | System of System Integration and Test[END_REF][START_REF] Ratchev | Knowledge based requirement engineering for one-of-a-kind complex systems[END_REF][START_REF] Bernard | Requirements management within a full model-based engineering approach[END_REF]]:

• the cascading impacts of frequent changes or updates of requirements;

• the dispersion of responsibility and the risk of non-consistency of requirements due to the number of stakeholders involved in the development process.

Specialists normally define each requirement using different expertise from heterogeneous domains focused on a single domain and a single life cycle phase. This fact leads to risk of misunderstanding among specialists due to semantic gaps. However, it is important to enhance the system requirements engineering activity that identifies the potential risk for the system-of-interest if one requirement is not satisfied. RE standards, approaches and tools are not able to deal with the risk if the non-satisfied requirement affects other life cycle phases and/or others domains.

For analysing these issues, the authors intend to consider three dimensions of the requirements analysis process as illustrated in Figure 1: (i) the domains dimension; (ii) the system lifecycle phase's dimension; and (iii) the requirements dimension. The first dimension concerns the set of domains involved in system engineering process, for instance mechanical domain, electrical domain, computer science domain. For this particular case, each expert in these domains must define specific requirements based on their particular skills. The second dimension is related to different phases of the systems life cycle, where each phase has its proper constraints represented by specific requirements. The last dimension represents different requirements as basic elements defined by the requirements analysis process, which this requirement will represent accurately the stakeholder's needs. Each one of these requirements is associated to a single domain and a single life cycle phase. For each dimension, an interoperation issue can be identified. Within the requirements dimension there are problems of completeness, coherency, uniqueness, univocity, feasibility, traceability and verifiability (Detail A -Figure 1). The dimension related to the systems life cycle phases may have some issues concerning

(C) Cross-domains interoperation issue ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

System Requirements

Requirement "a" the impact analysis between all phases (Detail B -Figure 1). Finally, the main scientific issue comes from the heterogeneity of the domains, which imposes some knowledge representation and analysis for managing requirements and their semantic relationships (Detail C -Figure 1).

D e f i n i t i o n D e v e l o p m e n t P r o d u c t i o n U t i l i z a t i o n S u p p o r t R e t i r e m e n t

S y s t e m L i f e c y c l e P h a s e s Domains

The authors also identified a fourth issue that illustrates the interrelationships among the three dimensions presented (Detail D -Figure 1). This last issue, which takes also into account dynamical, interactive and recursive properties of the requirement analysis process, is the most critical one. For example, if a specific requirement in one particular domain for a single phase is added or updated, it may impact other requirements already defined in other domains and/or other phases.

Related Works

The related works were structured according to the three issues of this research: (i) cross-domains requirement interoperation, (ii) cross-systems life cycle requirement interoperation and (iii) cross-requirement interoperation in a single domain/systems life cycle phases.

Cross-domains requirements interoperation issue

The complex systems development requires the involvement of specialists from multiples domains to capture the system's overview as well as the overviews within the domains and their interactions [START_REF] Bjørner | From Domains to Requirements in Concurrency[END_REF]. This generates a multi-heterogeneous information environment from different groups of stakeholders, suppliers, analysts' engineers, etc., to define complex systems. However, the heterogeneity of information from different domains has generated divergences with requirements like misinterpretation and mistakes due to a lack of requirements formalism and impacting in different system life cycle phases [START_REF] Moneva | A Design Framework for Model-Based Development of Complex Systems[END_REF]. According to [START_REF] Vitharana | A knowledge based component/service repository to enhance analysts' domain knowledge for requirements analysis[END_REF], the requirement analysts have expertise in systems development, but their knowledge remains restricted to their domains. On the other hand, the stakeholders and other customers involved in the project have different expertise and knowledge that creates a semantic problem, which reduces the chances of success of the systems development. Additionally, in [START_REF]The International Council on Systems Engineering[END_REF] was verified an increasing in complex systems development and in systems related with other system. It occurs because simple system does not support all stakeholders' needs and different expertise involved in stakeholders' requirements, resulting in the intensification of heterogeneous domains issues. Thus, the crossdomains requirements interoperation issue is to manage the complexity of this heterogeneous knowledge in different systems life cycle phases, ensuring the requirement coherence and compromising the final outcomes.

Thereby, in [START_REF] Canciglieri | Information sharing in multiviewpoint injection moulding design and manufacturing[END_REF], the authors designed a conceptual multiple view approach model using object oriented model and UML (Unified Modelling Language) to structure information relationships between mechanical and manufacturing domain. Translating mechanisms propitiated the relationship between different domains. Each mechanism dealt with a specific knowledge, which is responsible for translating the information from product view to manufacturing view. Despite the cross-domain approach/solution presented in this research, the mechanisms were restricted to specific domains. In [START_REF] Schmidt | Model-Driven Engineering[END_REF], it was proposed the integration between Model-Driven Engineering (MDE) and Domain Specific Language (DSL), creating a common language and a reasoner to analyse information in multiple domains. DSL formalizes the application structure, behaviour and requirements in a single domain and MDE structures the link between information through reasoning mechanisms in multiples domains. This allowed the exchanging information between heterogeneous domains. However, this approach did not present how to model the domains knowledge in different phases of the system life cycle and the impact of environment changes, which the domain is associated.

In [START_REF] Strasunskas | Domain model-driven software engineering: A method for discovery of dependency links[END_REF], a model-driven domain was proposed and described as part of ontology without axioms and rules. This model provides a common reference point and is used to manage objects development of the system and automatically supports the discovery dependency link. It was limited to early system development life cycle phases (definition and concept) and did not have a mechanism to ensure the consistency of the requirements after the automatic discovery of dependency links. [START_REF] Haveman | Requirements for High Level Models Supporting Design Space Exploration in Model-based Systems Engineering[END_REF] employed MBSE to structure requirements from multiples domains during the system life cycle phases to ensure the requirement consistency. This approach adapted the Vee-model to specific driven to MBSE models supporting the system building. However, this approach did not address the model performance in systems that suffer from frequent requirement changes. In [START_REF] Barbieri | A model-based design methodology for the development of mechatronic systems[END_REF], the authors proposed a model-based design (MBD) methodology adapted from MBSE, integrated to the W model proposed by [START_REF] Nattermann | Approach for a data-management-system and a proceedingmodel for the development of adaptronic systems[END_REF] to support the complex system development in multiples domains. For each domain the methodology created a model with their requirements and specific information allowing in a SysML environment the interaction between different domains. The information follows the W model that ensures the consistency of requirements, verification and validation. However, this methodology worked with early phases of the systems life cycle and did not address the requirements control and management in different phases of the systems life cycle.

Cross-systems life cycle requirements interoperation issue

The systems life cycle phases relate all activities of engineering of system, from definition until retirement as well as rules or verifications to confirm the system maturity [START_REF]INCOSE Systems Engineering Handbook: A Guide for Life Cycle Processes and Activities[END_REF]. According to [START_REF] Schneider | A Literature Survey on International Standards for Systems Requirements Engineering[END_REF], there are standards and models (ISO/IEC/IEEE 29148:2011 [START_REF]ISO/IEC/IEEE 29148:2008 -Systems and software engineering -Life cycle processes and Requirement Engineering[END_REF], ISO/IEC/IEEE 15288:2002 [START_REF]International Organization for Standardization: ISO/IEC/IEEE 15288:2002 -Systems and software engineering -System life cycle processes[END_REF], etc.) that standardize each phase of systems life cycle and rules that define the evolution and verification of the system. However, for [START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF] a single view model of the system does not explicitly fit all situation of the system life cycle. According to [START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF][START_REF] Holt | Model-Based Requirement Engineering[END_REF][START_REF] Chandler | Through-Life Systems Engineering Design & Support with SysML[END_REF], these models or standards can be used to determine all phases of the life cycle, but they contain particular characteristics that make them more suitable for specific phases. For instance, the waterfall is suitable for defining phases, because this model uses the feedback concept ensuring and revising the information integrity during a single phase [START_REF] Ruparelia | Software Development Lifecycle Models[END_REF]. Moreover, the traditional models ensure the information consistency in the direct flow according to representative life cycle model proposed by ISO/IEC TR 24748-1:2010 [START_REF]International Organization for Standardization: ISO/IEC TR 24748:2010 -Systems and software engineering -Life cycle management -Part 1: Guide for life cycle management[END_REF], i.e., if it is necessary to change some information in previous phases, these models are not able to manage the new information [START_REF] Gausemeier | System Engineering Management Based on a Discipline-Spanning System Model[END_REF]. Nevertheless, the systems life cycle does not follow a linear progression, i.e., iteration and recursion will occur modifying the life cycle flow as illustrate in Figure 2. In this context, [START_REF] Holt | Model-Based Requirement Engineering[END_REF] proposed a framework for a model-based requirements engineering to structure the system requirements in a SysML modelling. Moreover, they extend this approach to system life cycle, proposing solutions to integrate requirement in different phases of the system life cycle. However, this approach did not depict the evolution of the requirement in different phases and if the framework is able to analysis the change impact in different phases of the system life cycle. In [START_REF] Schamai | Virtual Verification of System Designs against System Requirements[END_REF] a methodology to verify the requirement consistency during the system life cycle phases in a dynamic manner is proposed. The methodology, named vVDR (virtual Verification of System Design against System Requirement), contributed to three main steps in the system life cycle phases: system requirement analysis, system design and system testing. Although, the methodology covered different system life cycle phases and analysed the requirements consistency, it did not report if there are consistency checking when the requirements are replaced or if there are impact analyses in different requirements occasioned by their replacement.

In [START_REF] Liao | Formalization of Semantic Annotation for System Interoperability in a PLM environment[END_REF], the authors proposed a formalization of semantic annotation for system interoperability from different domains views in a Product Life Cycle Management (PLM) environment. The formalization made explicit the tacit knowledge intrinsic in application models and act to support all activities during the product life cycle. Nevertheless, this approach did not depict the annotations in requirements that change frequently along the life cycle and how to ensure the semantic of these requirements. [START_REF] Chungoora | A model-driven ontology approach for manufacturing system interoperability and knowledge sharing[END_REF] proposed a model-driven ontology, which integrate the model-driven architecture (MDA) and an ontology, to create a manufacturing system interoperable between design domain and manufacturing domain. The solution emphasized the need of designing the knowledge in a common-logic-based ontology language to allow information exchange between domains. But this solution was limited to two heterogeneous domains and there was no evidence of possibilities to expand information exchange to multi-domains and integrate them.

Concept Development

Systems'life'cycle'with'some'possible'progression'

3.3

Cross-requirements interoperation issue in a single domain/system life cycle phases

Requirements are necessary attribute in a system, a statement that identifies capabilities, characteristics and quality factor of a system to ensure its value and utility for a customer or a user [START_REF]INCOSE Systems Engineering Handbook: A Guide for Life Cycle Processes and Activities[END_REF][START_REF] Afis | Guide Bonnes Pratiques en Ingénierie des Exigences[END_REF][START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF]. Based on literature and standards, there are two main types of requirements: Functional Requirements and Non-Functional Requirements (Quality requirements, Constraints, etc.) [START_REF]INCOSE Systems Engineering Handbook: A Guide for Life Cycle Processes and Activities[END_REF][START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF][START_REF]IEEE Std 830-1998 -Recommended Practice for Software Requirements Specifications[END_REF]. It is necessary to certify that among requirements will not have problems with completeness, coherency, uniqueness and univocity, as well as the traceability between requirements.

In [START_REF] Ratchev | Knowledge based requirement engineering for one-of-a-kind complex systems[END_REF], the complexity of translating customer needs in functional or non-functional requirements is demonstrated, because the customers or stakeholders environment was associated with a different requirement environment than the requirements analyst (RA). Thus, RE emerges as a cooperative, interactive and incremental process to elicitation, negotiation and documentation of the requirements and constraints of complex systems. The RE aims to solve the requirements problem in early stages of the requirements process [START_REF] Young | T he R equirem ents E ngineering H andbook[END_REF][START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF][START_REF] Pohl | The three Dimensions of Requirements Engineering: A framework and its applications[END_REF]. Within RE, beyond elicitation and negotiation, the traceability stands for a relevant problem. Requirements traceability is responsible for tracking information from stakeholder to all level of the engineering of the system as well as providing an understanding about any requirement change [START_REF] Sebok | Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF]. However, requirements traceability relations are not automatic generated and maintained [START_REF] Ramesh | Toward reference models for requirements traceability[END_REF][START_REF] Egyed | Automating requirements traceability: Beyond the record & replay paradigm[END_REF] and the identification typically occurs manually [START_REF] Chang | Automating speculative queries through event-based requirements traceability[END_REF][START_REF] Kececi | Modelling functional requirements to support traceability analysis[END_REF] making traceability relations susceptible to errors, if changes occur during the engineering of the system [START_REF] Spanoudakis | Rule-based generation of requirements traceability relations[END_REF].

According to [START_REF] Spanoudakis | Rule-based generation of requirements traceability relations[END_REF], the lacks of automated traceability become a prominent problem in complex systems once there is a need to establish traceability between large collections of requirements and other systems documentation. To [START_REF] Welsh | Requirements tracing to support change in dynamically adaptive systems[END_REF], changes can be required in any phases of the system life cycle (design, implementation or use). However, in Dynamic Adaptive Systems (DAS) a large numbers of requirements are faced changes of environment. Thus, the traditional traceability approach, which works with static and simple system, does not support this new system development once it is necessary to analyse simultaneously the changed requirements, identifying them and tracing the impact of the change in other requirements.

In [START_REF] Cmyrev | Consistency Checking of Feature Mapping between Requirements and Test Artefacts[END_REF], the authors advocate that if the traceability is consistently maintained it would prevent a dissemination of potential requirements inconsistencies into different system life cycle phases. Thus, according to the authors further researches are necessary to ensure the requirement traceability, making sure that the requirements information is complete, coherent, unique and univocal. According to [START_REF] Oertel | Interoperable Requirements Engineering: Tool Independent Specification, Validation and Impact Analysis[END_REF], consistency of requirements can be ensured through validation and verification methods. In this context, [START_REF] Bouffaron | Revisiting the interoperation relationships between System Engineering collaborative processes[END_REF] proposed an interoperation meta-model to structure the information transforming from stakeholder requirements (problem space) to specifications (solution space). This meta-model was responsible to control the exchange information in collaborative domains, ensuring their consistency and traceability during all this process. However, this meta-model was limited to early phases of the system life cycle and did not ensure the requirement exchange in different phases.

In [START_REF] Ramesh | Toward reference models for requirements traceability[END_REF], the authors presented a systematization approach to ensure the requirement consistency in different phases of system life cycle. This approach consists of some mechanisms: (i) mechanism to formalize the requirements and its features, (ii) mechanisms to consistency checking and (iii) mechanisms to correct the inconsistency problems. Moreover, the authors proposed a mechanism to manage the variability of information in different phases of the system life cycle, its consistency in all system life cycle. The authors did not depict if there are consistency impacts with requirements changing during the system life cycle and if this systematization is able to identify these impacts. In [START_REF] Ahmad | Using RELAX, SysML and KAOS for Ambient Systems Requirements Modeling[END_REF], the authors proposed a model to integrate the goal-oriented approach to RELAX, based on KAOS (Knowledge Acquisition in automated specification) and DSL (Domain Specific Language). This model supports the constant requirements changing, but it does not support the requirements evolution during different life cycle phases.

Discussion

This research is working to evidence the relevant issues to the requirements engineering in order to ensure all requirements coherency and consistency in all systems life cycle phases. These issues provided support to a conceptual framework proposal that aims to formally model requirements interoperation in term of impact and semantic equivalence or subsuming. Therefore, the authors proposed three dimensions to investigate the related issues: the cross-domain, the cross-systems life cycle phases' and the cross-requirements dimensions. Related works were found for each dimension regarding requirements engineering and particular solutions proposals. Thus, based on the related works issues/solutions, the Table 1 is proposed, which shows specific analysis by categorization, positioning each paper according to their subjects and degree of importance for the research. The adopted classification criteria were: ─ (D1) Particular cases -Papers/articles concerning the requirements exchange limited to two specific domains; ─ (D2) Ability to be generic -Papers/articles concerning the requirements exchange among different domains and that can be adapted to other domains; ─ (D3) Generality of the approach -Papers/articles concerning the requirements exchange among different domains whose approaches do not need any adaptation; ─ (LC4) Yes -For papers/articles that concerns the requirement exchange among one or more phases of the system life cycle; ─ (LC5) No -For papers/articles that do not concern the requirement exchange among one or more phases of the system life cycle; ─ (R6) Requirements Traceability -Papers/articles regarding the requirements traceability in one or more system life cycle phases and different domains; ─ (R7) Requirements Interoperability -Papers/articles regarding the exchange of requirements between one or more systems lifecycle phases and different domains. This interoperability issue does not consider any requirements changes during the systems life cycle phases; ─ (R8) Requirements Impacts -Papers/articles regarding the exchange of requirements between one or more systems lifecycle phases and different domains. This interoperability issue considers the impacts caused by any requirements changes during the systems life cycle phases. It is observed in Table 1 that there are some poorly explored gaps: in cross-domain issue, items (D2) and (D3) and in cross-requirement issue, item (R8). In cross-domain issue, it was verified that existing approaches proposed by the literature solve specific information exchange between domains. But, when this approach is extended to multiples domains (more than three), there are strict and/or limited solutions. This issue makes evident the problem with the semantic gap in multiples domains as well as the risk of mistakes and misinterpretation. In cross-requirement issue was noticed that there are researches addressing the requirement traceability and interoperability. Nevertheless, these researches did not consider the impact, which frequents requirements improvements and variations may cause to the consistency and coherency among requirements as well as ensuring the requirement consistency during different systems life cycle phases.

These results represent a preliminary evaluation about the models, frameworks and methodologies found in the literature, concerning the three dimensions. However, it is important to consider that requirements are not static, i.e. requirements' variations, advances and improvements may occur during the system life cycle phases' evolution. Although, the three dimensions consider the inherent relationship to each of them, it is important address all three issues simultaneously. Therefore, it is necessary explore or develop methodologies to support the systems-of-system, focusing on the three dimensions concurrently. The authors consider this approach as the fourth issue in order to ensuring the system requirements consistency and coherence.

Conclusion

This research points towards a conceptual framework for requirements interoperability in complex system engineering in order ensure the system requirements consistency and coherence in all life cycle phases. Requirements are not static, i.e., they may suffer changes, updates or removals during the system life cycle phase's evolution. Thus, it is necessary to manage these relationships to avoid misinterpretation and mistakes with requirements.

The authors proposed four issues to be investigated. Three of them are directly generated from the different presented dimensions (cross-domain, cross-systems life cycle phases and cross-requirements). The last one is the interrelationship among these issues. From these issues, an extensive literature review has been provided and the related works has been classified in order to identify the gaps that were not explored and/or need further researches. Whilst, the literature review and its classification highlighted the gaps in the same issue and/or the relationship among them such as: the need of requirement's language formalization or standardization in order to avoid misinterpretation and mistakes in multiples domains and the impact that frequent requirement changes/updates cause in the system requirements during system life cycle.

The continuity of the research should therefore identify and determine scientific methods identification and determination that are able to conceptually represent these 3 dimensions and the dependencies existing among them. It should also explore how to cope with the impact of requirements changes can cause among requirements in multiples domains during the system life cycle phases.

Fig. 1 .

 1 Fig. 1. Requirements analysis dimensions and issues.

Fig. 2 .

 2 Fig. 2. Life cycle model representation adapted from ISO/IEC TR 24748-1:2010 [28].

Table 1 .

 1 Related works classification according to each research issue.

	Authors and Publication Year	Cross-Domains issue	Cross-Systems Life Cycle issue	Cross-Requirements issue
		(D1)	(D2)	(D3) (LC4) (LC5)	(R6)	(R7)	(R8)
	ADELSON and SOLOWAY, 1985 [45]	✔			✔
	RAMESH and JARK, 2001 [35]				✔	✔
	EGYED and GRÜNBACHER, 2002 [44]				✔	✔
	CLELAND-HUANG et al., 2002 [36]				✔	✔
	CANCIGLIERI JR. and YOUNG, 2003 [17]	✔	✔		✔		✔
	SPANOUDAKIS et al., 2004 [37]				✔	✔
	RATCHEV, URWIN, MULLER, PAWAR and MOULEK, 2003 [12]				✔		✔
	KECECI, GARBAJOSA and BOURQUE, 2006 [43]				✔	✔	✔
	SCHMIDT, 2006 [18]	✔	✔		✔
	STECHERT and FRANKE, 2008 [46]	✔	✔		✔	✔	✔
	WELSH AND SAWYER, 2009 [38]				✔	✔	✔
	HOLT and PIERRY, 2010 [7]				✔	✔
	SCHAMAI et al., 2010 [30]				✔	✔	✔
	MONEVA, HAMBERG AND PUNTER, 2011 [15]	✔	✔		✔
	AHMAD and BRUEL, 2012 [42]				✔	✔	✔	✔
	BOUFFARON et al., 2012 [41]	✔			✔		✔
	CMYREV et al., 2012 [39]				✔	✔	✔
	STRASUNSKAS and HAKKARAINEN, 2012 [19]	✔	✔		✔	✔
	OERTEL and JOSKO, 2012 [40]				✔	✔	✔
	LIAO et al., 2012 [31]	✔	✔		✔
	CHANDLER and MATTHEWS, 2013 [26]	✔	✔		✔
	CHUNGOORA et al., 2013 [32]	✔	✔		✔		✔
	HAVEMAN and BONNEMA, 2013 [20]	✔	✔	✔	✔
	BARBIERI et al., 2014 [21]	✔	✔		✔