Life Cycle of Products and Cycles

Transcription

1 Life Cyle of Produts and Cyles Jean De Beir, Mouez Fodha, Franeso Magris To ite this version: Jean De Beir, Mouez Fodha, Franeso Magris. Life Cyle of Produts and Cyles. Douments de travail du Centre d Eonomie de la Sorbonne ISSN : X <halshs > HAL Id: halshs Submitted on 22 De 2008 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Douments de Travail du Centre d Eonomie de la Sorbonne Life Cyle of Produts and Cyles Jean DE BEIR, Mouez FODHA, Franeso MAGRIS Maison des Sienes Éonomiques, boulevard de L'Hôpital, Paris Cedex 13 ISSN : X

3 Life Cyle of Produts and Cyles Jean De Beir EPEE, University Evry-Val-d Essonne. Bd. F. Mitterrand, Evry Cedex, Frane. Tel.:+33(0) Mouez Fodha Centre d Eonomie de la Sorbonne and Paris Shool of Eonomis. University Panthéon-Sorbonne. Maison des Sienes Eonomiques, Bld de l Hôpital, Paris Cedex, Frane. Tel.:+33(0) Fax.: +33(0) (fodha@univ-paris1.fr) Franeso Magris EPEE, University Evry-Val d Essonne. Bd. F. Mitterrand, Evry Cedex, Frane. (franeso.magris@univ-evry.fr) May

4 Life Cyle of Produts and Cyles Corresponding Author: Mouez Fodha Address: Centre d Eonomie de la Sorbonne, University Paris 1 Panthéon-Sorbonne. Maison des Sienes Eonomiques, Bld de l Hôpital, Paris Cedex, Frane. Telephone: +33 (0) Fax:+33(0) Mailing address: fodha@univ-paris1.fr 2

5 Abstrat The aim of this paper is to examine whether the development of waste reyling ativities an be a soure of eonomi flutuations. We assume that the reyling setor has four fundamental harateristis. (i) The prodution fators are restrited by the prodution of the last period. (ii) These prodution fators are waste for whih the prie determination is non-ompetitive. (iii) It produes a reyled good, whih is a perfet substitute to the primary good. (iv) It redues waste stream. We onsider the simplest eonomy with an infinitely lived agent and a life yle hypothesis for the goods. We show that the equilibrium is unique and is always determinate. In spite of the lak of indeterminay, however, our system an display ylial behavior, depending on some usual onditions on parameters. Namely, the steady-state may undergo a Flip and a Hopf bifuration. JEL lassifiation: E32; Q53. Keywords: yles; reyling; waste. 3

6 1 Introdution Waste reyling ativity is an instrument for a sustainable development poliy. Therefore, the European Union had adopted in 2002 a Diretive onerning the reovery and the reyling of eletrial and eletroni waste produts. The adoption of this Diretive onto national law statutes has reinfored finanial subsidies aimed at the reovery and reyling of the diverse onstituent waste of these goods at the end of their useful life. 1 In addition to environmental onerns, waste reyling ativity orresponds to an industrial reality and to tehnologial onstraints. In Frane, for example, more than 58% of the paper industry s input is from reyled paper and ardboard. The reovery rate 2 is as muh as 54%. 3 Reyling is a growing industry for a number of raw materials. Some European ountries, for example, ahieve reord numbers in the reyling of glass (96% in Switzerland, 92% in Sweden, 88% in Germany and in Belgium, but only 58% in Frane in 2003). 4 The reyling setor has four fundamental harateristis. First, the fators of prodution are restrited by the prodution of preeding periods. Seondly, these fators are waste for whih the prie determination is non-ompetitive. 5 Third, it produes a reyled good, whih is a perfet substitute to the primary good. Finally, it redues waste stream. Eonomi literature analyses reyling from three theoretial points of view. First, reyling is integrated into the analysis of industrial organization; following the Aloa affair in 1945, studies have measured the possible erosion of monopoly power by the ompetitive ativity applied by manufaturers of reyled produts. They showed that Aloa s ontrol of raw aluminium prodution allowed it to onserve its market power when faed by a ompetitive fringe of reylers. As a matter of fat, the growth of the aluminium market was to the advantage of the monopoly: it was able to maintain its rent through ontrolling its raw material (Gaskins; 1974, Swan; 1980). Furthermore, in the period between the two World Wars, there were tehnial problems in the reovery and reyling of aluminium whih hampered seondary prodution and redued the possibilities of substitutability of primary and reyled produts (Suslow; 1986, Grant; 1999). 6 Seondly, the eonomi interest of natural resoures allows us to understand the usefulness of reyling when faed with the diminution of availability of resoures in the future; reyling postpones the working-out of mining resoures and redues the intensity of forestry exploitation (Mäler; 1974, Dasgupta and Heal; 1979). Thirdly, environmental eonomis analyses reyling as an instrument to redue negative 4

7 externalities. The question of reyling is introdued into this researh via the problem of domesti waste (Lusky; 1976). Reovery and reyling are methods of treating waste whih omplement other proesses (Keeler and Renkow; 1994). Extended researh into industrial pollution and into the onsumer waste was arried out by Palmer and Walls (1997) within a partial equilibrium framework. Fullerton and Wu (1998) present a general equilibrium model in whih produers offer a good using raw and reyled materials, while hoosing their level of pakaging and reyling. The level of waste has an influene on the welfare of onsumers and these studies show that it is possible to introdue a ost inentive measure (Fullerton and Kinnaman; 1995, Choe and Fraser; 1999). These general or partial equilibrium models analyze eonomi instruments (rarely ommand and ontrol instruments) aimed at reduing negative externalities arising from the disposal of household waste, of illiit household behavior in terms of waste and the rewarding of effort for households sorting their waste. This artile takes into aount the four harateristis of the reyling setor highlighted by these studies, but we onsider these harateristis simultaneously in a general equilibrium framework. More preisely, we assume the following properties: (i) the quantity of inputs reovered by this setor is limited by virgin goods produed in the preeding period; (ii) waste pries are non ompetitive; (iii) primary and seondary goods are perfet substitute; (iv) reyling is an instrument for reduing waste linked to industrial prodution. The aim of this paper is to examine whether the development of waste reyling ativities an be a soure of eonomi flutuations. We onsider a reovery and reyling publi setor in a dynami general equilibrium framework for the simplest eonomy i.e. without externalities and no waste market. The framework is a Ramsey-Cass-Koopmans model with infinitely lived agents with size normalized to one and a life yle hypothesis for the goods (two periods of use). We show that, in this very simple eonomy, in spite of the lak of indeterminay, however, our system an display ylial behavior, depending on some weak onditions on the propensity to onsumption relative to that to investment and on three elastiities: the reyling prodution funtion elastiity, the elastiity of the interest rate and the elastiity of marginal utility. Namely, the steady state may undergo a flip and a Hopf bifuration, with the assoiated two-period yles and losed orbits arising near the (unique) steady state, respetively. This result is in ontrast with the standard one-setor models with inelasti labor supply, whih, as shown in Boldrin and Rustihini (1994), do not display neither indeterminay nor ylial behavior. On the other hand, our results onfirm those of Seegmuller and Verhère (2007) whih show that, in a ontext 5

8 of overlapping generations, the aount for environment an be a soure of ylial behavior and, in their speifi ase, endogenous flutuations. We finally argue that, if reovery - reyling ativity is defined by standards poliy, government may hoose the level of the reovery or/and reyling rate suh as yles are avoided. The paper is organized as follows. Setion 2 presents the model and haraterizes the ompetitive equilibrium. Setion 3 analyses the stability onditions and determines the flip and Hopf bifurations. The last setion onludes. 2 The model 2.1 Households Assume population is normalized to one, there is no growth in its size and individuals are all idential. The representative infinite-lived household seeks to maximize overall utility, as given by: X Max β t u ( t ) t=1 where t is onsumption at time t and β [0, 1] is the onstant disount fator. The individual s resoure onstraint is: t + k t+1 = w t +(1 δ + r t ) k t + π t where w t is the wage, r t is the interest rate paid on saving, δ [0, 1] is the depreiation rate of apital, k t represents apital stok and π t measures the profit of the representative firm operating in the reyling setor, whih we will disuss later on. The first order onditions yield to the Euler equation u 0 ( t )=β(1 δ + r t+1 ) u 0 ( t+1 ) This relation determines the optimal onsumption path of the representative household. 2.2 Virgin setor Virgin prodution satisfies the usual assumptions. The virgin prodution setor onsists of one ompetitive representative firm, haraterized by a prodution funtion f (.) whih has onstant returns to sale and satisfies the Inada onditions. Prodution is a funtion of apital. The firm s profit π V t writes: π V t = f(k t ) r t k t w t 6

9 Sine markets are ompetitive, apital and labor earn their marginal produts, i.e. r t = f 0 (k t ) and w t = f (k t ) f 0 (k t ) k t. 2.3 Reyled publi prodution A publi reyling setor ompetes diretly with the virgin prodution setor. Its output is y R. Seondary prodution and virgin prodution are perfet substitutes. In this setor, the publi firm inludes reovery and reyling ativities. As Swan (1980) and Martin (1982) stated, we suppose that the unique prodution fator is the reovered used virgin goods and that the returns are dereasing (a harateristi of the tehnology of reovery). reovery-reyling setor is: ½ πt = g(r t ) pr t w.r.t. R t = φy V t 1 The program of the where R t is the quantity of srap purhased. The reyling funtion yt R = g(r t ) satisfies the Inada onditions. g (.) is stritly onave. The onstraint shows that the virgin good lasts for oneperiodbeforeitisreyled;i.e. period t s virgin prodution is reovered in period t +1or it is definitely lost. Consequently, the stok of available waste in t depends on the amount of the virgin goods produed during the preeding period t 1: the quantity of waste available for reovery in t is supposed equal to φyt 1 V. By definition, φ [0, 1[ i.e. a onstant share of the virgin goods is lost eah period at the srap reovery stage (depreiation ost). The waste is olleted and supplied to the reovery-reyling publi setor by the government at the prie p. This prie is exogenous and set arbitrarily by the government. 7 p measures also some osts linked to the reyling ativity: a share is lost in the seondary prodution (shrinkage ost). 8 Assume that there is no profit maximization task. This assumption orresponds to the three main harateristis of this setor. (i) The reyling rate φ is defined by tehnial standards for some types of waste. In order to omply with the objetives of the European Diretive (2004/12/EC) on pakaging and pakaging waste, Member States shall take the neessary measures to attain the following target, among others, overing the whole of their territory: no later than 31 Deember 2008 between 55% as a minimum and 80% as a maximum by weight of pakaging will be reyled. (ii) Moreover, loal government ontrol reovers the reylable material in a waste stream, 7

10 respeting preferene order defined by EC environmental poliy. It orresponds to a goal ranking between reyling, energy reovery and landfilling. (iii) Finally, it exists in European ountries, private ompanies aredited by the publi authorities (for example DSD in Germany) whih have to sort all the available households waste pakaging. Then, the reovery - reyling setor profit is: π R t = g(φf (k t 1 )) Af (k t 1 ) (1) with A = pφ. Hene, there is reyling ativity at time t ifandonlyifπ t > 0, implying: g(φf (k t 1 )) f (k t 1 ) We assume that there exists p suh that p = g(φf) φf A. (2) >p 0. 3 Intertemporal equilibrium and steady state Consider (2) to be verified at the steady state of the model and nearby, and (1) to represent profits in the reyled goods prodution setor. Then, our dynami system inludes in eah period the variables t, t+1,k t 1,k t,k t+1.namely,thefirst formula defines the Euler equation, while the seond the equilibrium in goods market. Therefore we have the following definition. Definition 1 An intertemporal equilibrium of the eonomy is a sequene { t, t+1,k t 1,k t,k t+1 } t=0 satisfying the following dynami system: u 0 ( t )=β(1 δ + f 0 (k t+1 )) u 0 ( t+1 ) (3) t + k t+1 = f (k t )+(1 δ) k t + g (φf (k t 1 )) Af (k t 1 ) together with the initial onditions k 0,k 1 and the usual transversality one. One immediately obtains that the steady state of system (3) is unique and defined by: β (1 δ + f 0 (k)) = 1 = f 0 (k) = β 1 (1 δ) θ>0 = f (k)+(1 δ) k + g (φf (k)) Af (k) k (4) Fore sake of simpliity in notation, we will not label the steady state values of apital and onsumption, but when they are not indexed by the time it is supposed that we are referring to the stationary values of the variables under study. 8

11 Remark 1 System(3)isaseond-orderdifferene equation whih an be expressed, as we will see below, as a three-dimensional first-order differene system. Notie that (3) is not optimal sine when maximizing, agents do not take into aount the possibility of reyling previous-period output. However, the study of the optimal dynamis is quite diffiult sine it involves the sequene of the variables { t, t+1, +2,k t 1,k t,k t+1 } t=0. Indeed, in suh a ase, the Euler equation would inlude three onseutive marginal utilities, sine a derease of a unit of onsumption in period t allows an inrease in onsumption in the next period, by investing it, and also in period t +2, by means of the reyling setor. It follows that we would have to study a four-dimensional firstorder differene equation, whih turns out to be hard to handle. Nevertheless, we will present some optimal results and give intuitions in setion 6. 4 Stability analysis Setting z t k t 1 we an rewrite system (3) in the following way: k t+1 = f (k t )+(1 δ) k t t + g (φf (z t )) Af (z t ) u 0 ( t )=β(1 δ + f 0 (k t+1 )) u 0 ( t+1 ) z t+1 = k t (5) Then, (5) is a first-order three-dimensional system. In order to study the stability properties of suh a system, we linearize it around its stationary solution. Notie that the system inluded two predetermined variables, k t 1 and k t andonewhihtheagentsmusthoose, t. It follows that in order to be indeterminate, the Jaobian must inlude three stable eigenvalues; in the opposite ase it will be determinate, either a saddle (one or two unstable eigenvalues) or a soure (three unstable eigenvalues). If we linearize system (5) around its stationary solution and we assume that there are not eigenvalues lying on the unitary irle, we get the following Jaobian matrix: J = 1 θ k where ε g g0 (φf(k))φf(k) g(φf(k)) apital share in the virgin prodution, ε r = β 1 B ε r βθk ε r +1 βθk ε r B is the reyling prodution funtion elastiity, s f 0 (k)k f(k) is the gross dr r = f 00 (k)k f 0 (k) > 0 the elastiity of the interest dk k rate, = u00 () u 0 () > 0 the elastiity of marginal utility (i.e. inverse of the elastiity of intertemporal substitution in onsumption) and B ε g s k + δ +(A 1) θ s Aθ. The assoiated 9

12 harateristi polynomial is: P (λ) =λ 3 µ 1+ 1 β + k θ ε µ r 1 β λ 2 + β B λ + B (6) In order to study the stability of system (5) we evaluate the harateristi polynomial at, respetively, 1, 0, 1 whih gives P ( 1) = 2+ ³ β 2 + k θ εr β +2B, P (0) = B and P (1) = k θ εr β<0. Sine P (1) is always negative and lim λ + P (λ) =+ one immediately verifies that there always exists a real eigenvalue greater than one, as it is stated in the following Proposition. Atually, this follows from the fat that the steady state is unique, properties that generially rules out the existene of saddle node bifurations. Proposition 2 The Jaobian J possesses a real eigenvalue greater than one. Sine our system has two predefined variables, k t 1 and k t, the existene of an unstable eigenvalue implies that system (5) is always determinate. Corollary 3 System (5) is always determinate. It follows that it an be a saddle or a sink, aording to the number of stable eigenvalues. Remark 2 Let us observe that although J has at least a real positive eigenvalue (namely greater than one), however the sign of the other two ruially depends on the sign and size of B. Speifially, if P ( 1) and P (0) are negative, all three eigenvalues ould be positive. Therefore, the onvergene towards the steady-state will be monotoni. If P ( 1) is positive and P (0) negative, there ould be two negative eigenvalues with one of them lower than minus one and the other inluded in ( 1, 0), in the ase that P (λ) was not stritly inreasing in [0, ). Insuha ase, the system will move towards its steady-state following an osillatory path. In spite of the lak of indeterminay, however, system (5) an display ylial behavior; namely, as we will show, the steady state may undergo a flip and a Hopf bifuration. The impossibility of getting instead a saddle node bifuration, as we have already said, is due to the uniqueness of the steady state. We will arry out our bifuration analysis with respet to perturbation in the elastiity ε g [0, 1] of the funtion g. 4.1 Flip bifuration A flip bifuration ours when one eigenvalue goes through 1. Sine the roots of the harateristi polynomial orrespond to the eigenvalues, the ondition boils down to find a value of 10

13 ε g inluded between zero and one and solving P ( 1) = 0. Straightforward omputation shows ³ that P ( 1) = 0 if and only if 2+ 2 β + k θ ε r β +2B =0. Therefore the bifuration value for ε g is ε F g = 1+ β 1 + k θ ε r β 2 + Aθ s k + δ +(A 1) θ (7) s Sine ε g must belong to [0, 1] in order to get a flip bifuration the Right Hand Side (RHS) of (7) must be inluded in this interval; the previous inequality is respeted when 1+ 1 β sδ + θ s θ εr β 2 < k (8) under the onditions that A 1+ k + δ s εr β θ > 0 and s θ 2 > 0. Condition (8) is more likely to be respeted as soon as either the elastiity of intertemporal substitution in onsumption is high or the elastiity of the interest rate is low. In the limit ase ε r =0or =+, wehave that a flip bifuration does exist when 1 s + 1 βs δ + θ/s < k In other words, eonomies haraterized by a higher propensity to onsume are more likely to display a flip bifuration and therefore, as showed among the others in Grandmont (1988), two-period yles. All the previous results are summarized in the following Proposition. Proposition 4 Under ondition (8) the steady state of system (5) undergoes a flip bifuration for the elastiity of the reyling prodution funtion defined in (7). Then, there will exist near the state a stable or unstable two-period yle, aording to the diretion of the bifuration. We now provide an expliation of the mehanism at the base of the ourrene of a two-period yle. Let fix a given level of k, sayk 1. If it is relatively high and the propensity to onsume is high too, in the next period the stok of apital will be lower, say k 2 <k 1. If we are in the region of the reyling prodution funtion in whih the profits are dereasing, in the following period total prodution will be higher, and so investment will inrease by attaining again the level of apital k 1. A similar argument does hold in the ase in whih we are in the region of the reyling prodution funtion in whih profits are inreasing. The mehanism desribed explain why, in order to have yles, the propensity to onsume must be large enough. 11

14 4.2 The Hopf-Poinaré bifuration In order to verify if system (5) displays a Hopf bifuration, we must hek if there exists a onfiguration suh that two omplex eigenvalues have modulus equal to one. If this is true, it follows that there will exist around the steady state a losed urve, attrating or repulsing, aording to the diretion of the bifuration (Grandmont; 1988). harateristi polynomial an be written as As it is well known, the P (λ) =λ 3 Tλ 2 + Σλ D (9) where T is the Trae, Σ the Sum of Minors of order two and D the Determinant. If there exists a Hopf bifuration, the Determinant must be a real root of the polynomial, whih should be written as (λ D) λ 2 2aλ +1 (10) for some a suh that a < 1. In fat, in suh a onfiguration, a is the ommon real part of the two eigenvalues with unitary modulus. In the Appendix, we show that the bifuration value for the elastiity of the reyling prodution funtion is the following: ³ r ³ k θ ε r ε H β + β 1 k θ ε 2 ³ r β +2 k θ ε r + 1β Aθ g = 2s k + δ +(A 1) θ (11) s Now, we must prove that ε H g is inluded between 0 and 1. Sine the denominator is positive, there will exist a value for A large enough suh that ε H g > 0 is verified. Suh A is given by ³ r ³ k θ ε r β + 1 β + k θ ε 2 ³ r β +2 k θ ε r + 1 β 2 2 A> (12) 2θ Condition ε H g < 1 does not depend upon A. Atually, to ensure this last inequality, we obtain the following ondition: ³ r ³ k θ ε r β + 1 β k θ ε 2 ³ r β +2 k θ ε r + 1 β θ 2 <s (13) k + δ In view of the previous onsiderations, we an introdue the following Proposition. Proposition 5 The steady state undergoes a Hopf bifuration at ε H g. Suh a value is inluded between zero and one provided (12) and (13) are satisfied. Thus, there will be a losed urve around the stationary solution whih will be either stable or unstable aording to the diretion of the bifuration. 12

15 An explanation for the emergene of the Hopf is muh less intuitive than the flip ase and rests upon the role played by the elastiity of the reyling prodution funtion, although to reprodue the mehanism at work turns out to be quite ompliate. The analysis of the stability properties shows that under some usual onditions, steady-state may display a flip or a Hopf bifuration. These results imply stabilization publi poliy. One of the entral parameters, among others, is the A term, whih depends on the waste prie p and on the reovery rate φ (A = pφ). Hene, government may use market-based instruments (tax and subsidies) to determine p or ommand and ontrol instruments like φ, suh as eonomi flutuations are avoided. Atually, European governments mix these two instruments to ahieve environmental objetives. For example, in the ase of the pakaging waste poliy, sine 1994, a state-registered eo-organisation subsidizes the reovery setor 9 in eah EU ountry; moreover, the 1994 EU Pakaging Diretive 10 determines minimum reovery and reyling targets for pakaging waste. 5 Simulations In order to analyze the onsequenes of the values of the reyling parameters, we hoose the following simple funtional forms. Assume Cobb-Douglas prodution tehnology for the virgin setor f (k) =Fk s and log prodution funtion for the reyling setor tehnology g (R) = G ln (1 + R) where F and G are sale parameters. We suppose also a log funtion for the households welfare, i.e. u () =ln. We therefore have these given values for the elastiities: ε r =1 s and =1. Under these parameters onfiguration, we ompute the bifuration values of the elastiity ε g. Atually, we want to test the sensitivity of ε F g and ε H g with respet to the reovery rate φ and the waste total ost p. More in detail, we will plot for different realisti values of φ the values of the ritial elastiity as a funtion of p. Notie that p represents an eonomi tool hosen by the government and at the same time, it measures transation ost paid by a State-registered eo-organisation for the resoures R. Aninreaseofp means higher shrinkage ost and higher industrial pakaging ost of waste. p flutuates for material types and also for eah material of the same type. The reovery rate φ vary aording to the material types; for example in 2004 in Frane, φ =55%for metals and φ = 75% for paper and ardboards. This rate φ might vary also for eah material of the same type; for example, in the ase of the paper 13

16 Critial elastiity values 1 0,9 0,8 0,7 0,6 0,5 0, p Φ = 0.5 Φ = 0.6 Φ = 0.7 Φ = 0.8 Φ = 0.9 Figure 1: Flip bifuration and ardboards, φ = 55% in 1997 and φ = 75% in In the sequel, we will retain the following strutural parameters: β = 0.9, δ = 0.25, s = 0.3, F = 1,G = 100. For the flip bifuration, we get the following piture (see Fig. 1). Notie that the bifuration value of ε F g is inreasing with the waste total ost p and the reovery rate φ. Hene, for a given elastiity, the more the good is reylable, the less the waste total ost is. This result seems quite realisti sine the government has to fix minimum prie when the good is highly reylable. Indeed, for high reovery rate, the supply of waste that ould be reyled is high too. As we an see, these values of ε F g fall within standard intervals even when p is set large enough. Conerning the Hopf bifuration, we have the following piture (see Fig. 2). Again, we find the same main features: the ritial elastiity is inreasing with p and φ. However, in order to get a Hopf bifuration the ost should be large enough. One an easily show that in orrespondene with our simulations, the ondition of non-negativity of profits is respeted. 6 Extensions 6.1 The optimal ase It ould be interesting to analyze the soial planner problem of our eonomy. In suh a ase, the reyling ativity is taken into aount when maximizing. Atually, the soial planner onsiders 14

17 Critial elastiity values 1 0,8 0,6 0,4 0, p Φ = 0.5 Φ = 0.6 Φ = 0.7 Φ = 0.8 Φ = 0.9 Figure 2: Hopf bifuration the life-yle of investment whih lasts two periods: in the first one, it is used in the standard prodution funtion while in the seond, it represents the input of the reyling setor. Straightforward omputations give the following Euler equation: u 0 ( t )=βu 0 ( t+1 ) (1 δ)+f 0 (k t+1 ) +β 2 u 0 ( t+2 ) g 0 (φf (k t+1 ))φf 0 (k t+1 ) Af 0 (k t+1 ) (14) whih together the resoure onstraint desribes the dynamis of the model. The steady-state level of apital is obtained by solving: 1=β 1 δ + f 0 (k) + β 2 g 0 (φf (k))φf 0 (k) Af 0 (k) (15) By manipulating (15), we obtain: f 0 (k) = 1 β (1 δ) β (1 + β (g 0 (φf (k)) φ A)) (16) TheLeftHandSide(LHS)of(16) dereases from to 0, meanwhile the RHS starts from 0 andtheninreasesmonotoniallybyreahing 1 β(1 δ) β A when k beomes infinite. Notie that whatever the sign of β A, the LHS and the RHS interset only one. Therefore, the steady-state is unique. The dynamis of the optimal model involves now the profit maximization ativity of the reyling setor. Therefore, two regimes are possible. In the first one, profit is dereasing in k i.e. k>k max, where k max solves k max =argmaxg (φf (k) Af (k)), andthenthesoial planner hooses to use only k max and throws away k k max. Therefore, the model boils down 15

18 to the standard Ramsey-Cass-Koopmans one up to a onstant whih does not interfere with the dynamis. In the seond onfiguration, profit are inreasing in k and therefore, all of the amount of apital available is used in the reyling setor. In the latter ase, the system desribing the equilibrium dynamis is four dimensional and not easily tratable but ould be an interesting extension to study. Obviously, some nie analytial results would disappear sine the dimension of the system is higher. However, it would be possible to ompute numerial solutions in this ase and to ompare them to the non optimal ase. These results will imply to define the optimal poliy for the waste publi management but need also to introdue the pollution externalities generated by the waste and therefore the environmental benefits of the reyling setor. This is a topi for further researh. 6.2 Current reyling ativity Assume no lag in the olleting and the reyling of waste by the government, i.e. the reyling setor uses the apital stok during the same period than the standard setor. This ould happen for non durable goods like papers or glass. If we onsider the deentralized eonomy, the dynami system writes: ( P max β t u ( t ) (17) w.r.t. t + k t+1 = w t +(1 δ + r t ) k t + π t and the intertemporal equilibrium is: ½ u 0 ( t )=β (1 δ + f 0 (k t+1 )) u 0 ( t+1 ) t + k t+1 = f (k t )+(1 δ) k t + g (φf (k t )) Af (k t ). (18) Notie that system (18) involves one predetermined variable, k and one whih is not,. We an then determine the steady-state: ½ f 0 (k) = 1 β (1 δ) θ = f (k) δk + g (φf (k)) Af (k) = δk + g (φf (k)) + (1 A) f (k) Proposition 6 There exists a unique steady-state for the intertemporal equilibrium desribed by system (18). The linearization of system (18) gives the following Jaobian matrix: Z 1 J = βθ ε s k Z 1+βθ k where Z 1 β + ε gs B Aθ and B δ + k + θ s (A 1). The Jaobian matrix is haraterized by Determinant D = Z and Trae T = Z +1+βθ k 16 ε s (19) ε s. From the definition of the Trae, we an

19 draw the onlusion that there is no room for a saddle node bifuration nor a Hopf bifuration, and that the system is always determined. Conversely, there ould arise a flip bifuration when D is lower than 1. Namely, this ours when T = D +1+βθ k the following elastiity for the reyling prodution funtion: ε s = D 1, whih requires ε F g = 1 β θ ε s 2 k + Aθ 1 β sδ + k + θ s (A 1) (20) Proposition 7 System (18) is loally determinate either a saddle or a soure. In addition it undergoes a flip bifuration when the elastiity of the reyling prodution funtion goes through ε F g defined in (20). 7 Conlusion In this paper we have analyzed an infinite horizon model in whih there is a reyling setor produing a good whih is perfetly substitute to the one produed by a standard onstantreturns tehnology. The reyling prodution funtion is set to respet the parameters required by the European Diretive. After a disussion on the haraterization of the reyling setor, we have arried out a stability analysis. More indetails,wehaveshownthatthereyling tehnology may be responsible for the existene of deterministi yles through a flip bifuration and a losed urve surging through a Hopf-Poinaré bifuration. These bifurations our for reliable parameter values and for high enough propensity to onsume. These results ontradit some findings, as in Boldrin and Rustihini (1994), aording to whih omplex dynamis annot our in one-setor models when an inelasti labor supply is assumed. An interesting extension of the model ould be that of analyzing the role played by the reyling tehnology in an overlapping generations framework. Indeed, in suh a ase, there would be an interation between the agents life yle and the produt one. Another possible extension is that of inluding the environmental dynamis in the model. Aknowledgements The authors are indebted to two anonymous referees for their helpful suggestions and omments on an earlier version of this artile. We are also grateful for the omments of the partiipants of the CREE 2007 onferene and the GED seminar of CES. 17

20 8 Notes 1. In Frane, the Deree of July 2005 onerning the disposal of eletrial and eletroni waste results in the appliation of the European Diretive of 2002 (Diretive WEEE 2002/96/EC). Immediately from November 2006, produers of these goods ontributed funds to one of the four State-registered eo-organisations (Eo-Systèmes, ERP, Eologi and Reylum). 2. This is the ratio between the reovery of used papers and ardboard and the onsumption of virgin papers and ardboard. 3. Soure: Frenh Confederation of the Paper, Cardboard and Cellulose Industry, Soure: European Federation of Glassware Containers, This harateristi is a onsequene of the legal definition of waste. Aording to the law, waste orresponds to... all residues resulting from a proess of manufature, of transformation or of use of all substane, material, produt (...) abandoned or whih the produer intends to throw away.... It is therefore of no interest to the person, whih renders the waste a useless or negative value. But the notion of waste annot be preise as it depends on the tehnology available and on the struture of the eonomy: the development of the reovery and reyling setors transforms waste into a raw material. 6. Martin (1982) onduts an in-depth analysis of how the different forms of vertial integration of the market between Aloa, srap reovery and reyling firms affet monopoly profit. 7. This way of waste prie determination (fixed prie) is an industrial reality and orresponds, for example, to the appliation of the 2002 EU Diretive (Diretive WEEE 2002/96/EC) onerning eletrial and eletroni material waste produts, where used omputers were atually reovered free of harge in 2006; it orresponds also to the guarantee of the takingbak of Eo-Emballages between 1992 and 2005 in Frane. 8. This ost orresponds to an ieberg ost: a part of the goods disappears during transportation (Samuelson; 1954, Helpman and Krugman; 1985). 18

21 9. The onsumer pays a tax on pakaged goods; this eo-ontribution has to fill the gap between the re-sale prie of seondary materials and the marginal ost of their reovery. 10. The 1994 Diretive on Pakaging and Pakaging Waste (94/62/EC) and the 2004 Pakaging Diretive Amendments (2004/12/EC). Referenes Boldrin, M. and A. Rustihini (1994) Growth and Indeterminay in Dynami Models with Externalities. Eonometria 62, Choe, C. and I. M. Fraser (1999) An eonomi analysis of household waste management. Journal of Environmental Eonomis and Management 38, Dasgupta, P.S. and G. Heal (1979) Eonomi Theory of Exhaustible Resoures. James Nibset- Cambridge University Press: Cambridge. Fullerton, D. and T. C. Kinnaman (1995) Garbage, Reyling, and Illiit Burning or Dumping. Journal of Environmental Eonomis and Management 29, Fullerton, D. and W. Wu (1998) Poliies for Green Design. Journal of Environmental Eonomis and Management 36, Gaskins, D. W. (1974) Aloa Revisited: Welfare Impliations of a Seondhand Market. Journal of Eonomi Theory 7, Grandmont, J.M. (1988) Non linear Differene Equations, Bifurations and Chaos: an Introdution. CEPREMAP DP Grant, D. (1999) Reyling and market power: A more general model and re-evaluation of the evidene. International Journal of Industrial Organization 17, Helpman, E. and P. R. Krugman (1985) Market Struture and Foreign Trade. The MIT Press: Cambridge. Keeler, A. G. and M. Renkow (1994) Haul Trash or Haul Ash: Energy Reovery as a Component of Loal Solid Waste Management. Journal of Environmental Eonomis and Management 27, Lusky, R. (1976) A model of reyling and pollution ontrol. Canadian Journal of Eonomis 9, Mäler, K. G. (1974) Environmental Eonomis: A Theoretial Inquiry. JohnHopkinsUniversity Press: Baltimore. 19

22 Martin, R. E. (1982) Monopoly Power and the Reyling of Raw Materials. The Journal of Industrial Eonomis 30, Palmer, K. and M. Walls (1997) Optimal poliies for solid waste disposal: Taxes, subsidies and standards. Journal of Publi Eonomis 65, Samuelson, P. A. (1954) The transfer problem and transport ost, II: analysis of effets of trade impediments. Eonomi Journal 64, Seegmuller, T. and A. Verhère (2007) A note on indeterminay in overlapping generations eonomies with environment and endogenous labor supply. Maroeonomi Dynamis 11, Suslow, V. Y. (1986) Estimating monopoly behavior with ompetitive reyling: an appliation to Aloa. Rand Journal of Eonomis 17, Swan, P. L. (1980) Aloa: The Influene of Reyling on Monopoly Power. Journal of Politial Eonomy 88, Appendix The Hopf bifuration If we develop (10) and using (9), by identifiation we obtain: ½ a = T D 2 a = Σ 1 (21) 2D In order to find an a whih verifies both equalities, we exploit the expression of our polynomial whih is Again by identifiation, we have µ P (λ) =λ β + k θ ε µ r 1 β λ 2 + β B λ + B ³ T = 1+ β 1 + k θ εr ³ B Σ = 1 β D =( B) ³ Hene, using (21), we obtain a polynomial of seond degree in B suh that B 2 +B k θ εr β + β β 1=0. In order to get two real roots B 1,2, the disriminant must be positive or zero. The disriminant of suh a polynomial is µ = k θ ε 2 r β +2 k θ ε µ r β 2 > 0 20 β

23 The solutions of the polynomial are therefore: B 1 = ³ r ³ k θ ε r β + 1 β + k θ ε 2 ³ r β +2 k θ ε r + 1 β 2 2 ³ r ³ k θ εr β + β 1 2 ³ 2 k θ εr β +2 k θ εr + 1β 2 B 2 = 2 Notie now that the term a = T D 2 is suh that either a 1 = ³ 1+ 1 β + k θ ε r β + k θ εr ε β+ 1 + β 2 2 k θ εr ε β 2+2 k θ εr ε + 1 β (22) (23) or a 2 = ³ 1+ 1 β + k θ εr β + k θ εr ε β+ 1 β 2 k θ εr ε β 2+2 k θ εr ε + 1 β If we want now a < 1, it must be s µ 2+ 1 β + k θ ε r β ± k θ ε 2 r β +2 k θ ε µ r 1 + β 2 2 < 4 If the solution is a 1, sine all the terms are positive, we have 2+ 1 β + k θ ε r β + s µ k θ ε 2 r β +2 k θ ε µ r β 2 < 4 whih implies the ontradition 0 < 4 k θ εr β. In the ase instead in whih the solution is a 2 = r ³ 2+ 1 β + k θ ε r β k θ ε 2 ³ r β +2 k θ ε r + 1 β 2 2, the result an be either positive or negative. In r ³ the ase where a 2 is negative we have 2 1 β k θ ε r β+ k θ ε 2 ³ r β +2 k θ ε r + 1 β 2 2 < 4 and ³ +2 1 β therefore we obtain that the inequality 2 In the ase in whih a 2 is positive we have 2+ 1 β + k θ ε r β < 3 k θ ε r β is always verified; thus a 2 < 1. r ³ k θ ε 2 ³ r β +2 k θ ε r + 1 β 2 2 < 4 whih implies 0 > 2β, that is always satisfied. Therefore the system undergoes a Hopf bifuration when B = B 2 = ³ r ³ k θ εr β ³ β k θ εr β +2 k θ εr + 1 β (24)

24 sine in suh a ase there exists a 2 with a < 1. We need eventually to prove that ondition (24) is satisfied for some parameters onfiguration. Sine we have µ B ε g s k + δ +(A 1) θ Aθ s it must be ³ r ³ µ ε g s k + δ +(A 1) θ k θ ε r β + β 1 Aθ = s k θ ε r 2 ³ β +2 k θ ε r + 1β 2 whih is satisfied for every ε g provided that A is hosen appropriately (sine B = B 2 < 0, A must be large enough). Therefore, after tedious omputations, we find that the Hopf bifuration value for ε g is given by eq. (11)