Pierre Cariou
email: pierre.cariou@euromed-management.com

Francois-Charles Wolff

Outsourcing ship management: Implications for the logistics chain

The purpose of this paper is to contribute to a better understanding of the probability that shipping companies outsource the management of vessels to ship management companies, a decision with many implications for the logistics chain. Data on 39,925 vessels are used to investigate to the extent to which 4,049 different ship-owners (each operating a fleet of at least two vessels) outsource to ship-management service providers. We rely on multinomial Logit random and fixed effects regressions to explain how the characteristics of the owners and vessels influence this decision. We find that the size of the firm in terms of the number of vessels and the number of different types of vessels influence the likelihood of outsourcing. Also, ship-owners frequently implement a mixed strategy and outsource only some of their vessels.

Introduction

With 80% of the volume of international trade transported by sea and with an annual growth rate of 4% over the last twenty years (UNCTAD, 2009), the maritime transport industry plays a crucial role in international supply chains. As in many other industries [START_REF] Hoffman | 3PLs maturing fast[END_REF][START_REF] Hoffman | 3PLs reach record revenue[END_REF][START_REF] Quinn | 3PLs hit their stride[END_REF][START_REF] Marasco | Third-party logistics: A literature review[END_REF], the increase in business complexity when companies move into foreign markets and globalize their supply chains and material sources [START_REF] Anderson | What drives the choice of third party logistics provider?[END_REF][START_REF] Patterson | How different is carrier for third party logistics companies?[END_REF] has led many ship-owners to specialize in value-adding activities and to outsource some portion of their activities. This is particularly prevalent for activities related to ship management. This decision strongly affects shippers and freight forwarders. When selecting a carrier, both actors have an interest in knowing whether shipping companies carrying their cargo are managing the vessels themselves or whether they outsource their management to professional ship managers. Indeed, such a decision is likely to impact the carrier's flexibility, its ability to fulfill shippers' expectations and the safety and security of the international logistics chain. Despite these consequences, few empirical studies have been conducted to improve our understanding of the degree of vessel management outsourcing to third parties and to study factors influencing decisions made by ship-owners.

In this paper, we perform an empirical analysis of this decision-making process using a unique dataset covering the world fleet. We study whether there are differences in outsourcing rates between various shipping sectors and the factors that explain the likelihood of outsourcing. We emphasize the role of covariates such as the size of a ship-owner's fleet, country of domicile and vessel characteristics in our econometric analysis. We find that the outsourcing of vessels is common to all shipping sectors, but that certain owners turn to thirdparty managers more often than others. Owners also tend to outsource only part of their fleet, using outsourcing as a means of benchmarking.

Literature review

The decision to use external companies for functions traditionally performed within a company can be explained by many reasons [START_REF] Lieb | The use of third-party logistics services by large American manufacturers[END_REF][START_REF] Silver | How 3PLs can help manage the increasing complexities of transportation[END_REF][START_REF] Tompkins | The business imperative of outsourcing[END_REF][START_REF] Hannon | Choose wisely: Not all third-party logistics are the same[END_REF][START_REF] Selviaridis | Benefits, risks, selection criteria and success factors for third-party logistics services[END_REF][START_REF] Lieb | The use of third-party logistics services by large American manufacturers[END_REF][START_REF] Peters | The use of third-party logistics services by European industry[END_REF][START_REF] Fugate | Operational collaboration between shippers and carriers in the transportation industry[END_REF] and has an effect on final customers [START_REF] Araujo | Managing interfaces with suppliers[END_REF][START_REF] Stank | Logistics service performance. Estimating its influence on market share[END_REF].

Outsourcing of a transport asset [START_REF] Cruijssen | Horizontal cooperation in transport and logistics: A literature review[END_REF] and in particular the management of a vessel is a common feature in international transport. Similarly, this decision is not neutral for shippers as this induces a loss of control for carriers, uncertainties about service levels and questions about the true costs of third party ship management companies [START_REF] Gourdin | Global logistics management. A competitive advantage for the 21 st century[END_REF], a decision shippers might not be aware of. The issue raised becomes clearer when marine incidents occur in dramatic circumstances. This was the case when the tanker Erika sunk in 1999. Months were required for investigators to identify the companies in charge of operations, months during which the shipper's image (Total) deteriorated.

Even without the occurrence of such extreme cases, outsourcing of ship management by transport operators or by Non Vessel Operating Common Carriers (NVOCC) also has an effect on the safety and security of the logistics chain. For instance, the service provider no longer chooses the flag of registry or the classification society of the vessel, two components often seen as indicators of substandard shipping [START_REF] Knapp | The econometrics of maritime safety -Recommendation to enhance safety at sea[END_REF][START_REF] Cariou | Do Port State Control inspections influence flag-and class-hopping phenomena in shipping?[END_REF].

Based on such examples, we then consider that it matters for a shipper to know whether the carrier transporting its cargo is likely to outsource the management of the vessel. This leads to a generic distinction between the registered ship-owners and ship managers, the latter being responsible of the day-to-day running of the ships (Lloyd's Register Fairplay, 2009).

Outsourcing ship management does not only affect the logistics chain negatively however. This decision is a common means for a shipping company to obtain a competitive advantage, an opportunity to focus on core skills and to access best-management practices and technologies developed by a third-party service provider [START_REF] Gourdin | Global logistics management. A competitive advantage for the 21 st century[END_REF][START_REF] Parashkevova | Logistics outsourcing -A means of assuring the competitive advantage for an organization[END_REF]. Redundant costs are hence reduced, quality and reliability improved, and speed and flexibility increased [START_REF] Selnes | Promoting relationship learning[END_REF][START_REF] Panayides | Logistics service providers-client relationships, relationships[END_REF][START_REF] Selviaridis | Benefits, risks, selection criteria and success factors for third-party logistics services[END_REF].

All these benefits are well-acknowledged in the shipping sector. Outsourcing ship management is further expected to produce technical and operational economies of scale, marketing advantages and bargaining power [START_REF] Sletmo | Shipping's Fourth Wave: Ship Management and Vernon's trade cycles[END_REF][START_REF] Schulte | Third party management moving forward and adapting to change[END_REF][START_REF] Richards | Reducing ship investments risks. The ship manager's role[END_REF][START_REF] Ebsworth | In-house ship managers[END_REF][START_REF] King | Globalisation of logistics management: present status and prospects[END_REF][START_REF] Gunton | Ship management[END_REF][START_REF] Willingale | Ship Management, Third Edition[END_REF][START_REF] Panayides | Professional Ship Management: Marketing and Strategy[END_REF][START_REF] Panayides | The vertical disintegration of ship management: choice criteria for third party selection and evaluation[END_REF][START_REF] King | Third Party Ship Management: A Greek Perspective[END_REF]. Interviews with the CEOs of the top-five professional ship management companies (Lloyd's Ship Manager, 2008a, 2008b[START_REF] Stopford | Maritime economics[END_REF] highlight the shortage of seamen, which may give specialized ship managers a competitive edge over shipping companies by providing flexibility and access to skilled crews. However, such access comes at a substantial cost, estimated at between USD 100,000 and USD 175,000 per year, per vessel, in 2004(Drewry Shipping Consultants, 2004), or 5% of a vessel's annual operating costs [START_REF] Justad | Taking delivery: how to achieve lasting value from outsourcing[END_REF].

According to previous studies, several firm-specific factors affect the likelihood of outsourcing. [START_REF] Mitroussi | Third party ship management: the case of separation of ownership and management in shipping context[END_REF]Mitroussi's ( , 2004a[START_REF] Mitroussi | The role of organisational characteristics of ship owning firms in the use of third party ship management[END_REF] survey of Greek and UK-based ship-owners indicates that the ownership structure and age of a company play significant roles. Firms controlled by at least third-generation managers are more likely to outsource when internationalizing their activities. In addition, the firm's country of domicile appears to be paramount. For instance, UK ship-owners, with an outsourcing rate of about 70%, outsource more frequently than owners from other countries. Conversely, outsourcing is seldom chosen by Indonesian (9.2%), Russian (19.2%) or Greek (24.5%) ship-owners [START_REF] Cariou | Understanding the choice by ship-owners of third-party managers[END_REF]. [START_REF] Panayides | The vertical disintegration of ship management: choice criteria for third party selection and evaluation[END_REF] found a higher likelihood of outsourcing for container vessels operated by large firms in need of flexibility. In contrast, Drewry Shipping Consultants (2004), and [START_REF] Cariou | Understanding the choice by ship-owners of third-party managers[END_REF] found that professional ship managers are more often involved in the tanker and offshore markets. Finally, small and large companies outsource approximately 16% of their vessels [START_REF] Mitroussi | The role of organisational characteristics of ship owning firms in the use of third party ship management[END_REF], with similar proportions explained by needs for expertise or flexibility, respectively [START_REF] Cariou | Understanding the choice by ship-owners of third-party managers[END_REF].

In summary, many factors play a role in operational partnerships between ship-owners and managers, these partnerships having an impact on logistics chain organization. However, no comprehensive econometric analysis has been carried out to study which factors influence a shipowner's decision to outsource the management of all or some vessels or how a company's fleet structure influences that decision.

Data and descriptive statistics

In this paper, we analyze the likelihood of a ship-owner outsourcing the management of a vessel using an initial database of 110,384 vessels. The data is gathered from Lloyd's Register Fairplay (June 2009). We then limit the sample to vessels of more than 1,000 gross tons for which both the names of the owner and the manager are known, which reduces the sample to 39,925 merchant vessels. In order to include an analysis of differing outsourcing levels (ranging from outsourcing of all vessels to the outsourcing no vessels), we then limit the sample to ship-owners operating at least two vessels. This leaves us with a sample of 4,049 owners of 39,925 vessels. The dataset includes information on each vessel's size in terms of gross tonnage, as well as data on each vessel's type, age and flag of registry (Table 1).

Insert Table 1 here

The largest ship management companies in 2009 are shown in Figure 1. Data on the firms' holdings was aggregated using Drewry Shipping Consultants' data (2004, p. 9) as well as the members' list provided by Intermanager, the International Association of Ship Management Companies (http://www.intermanager.org/). Thirty percent of all vessels managed by professional ship managers are tankers. V Ships' (#2 in Figure 1) main customer is CSL Group Inc., which accounts for 10.8% of vessels managed by V.Ships, while Schulte's (#1) biggest customer is Schulte Group with 27.2%.

A careful examination of data shows that NYK Management (#6) and Cosco HK Shipping Co Ltd (#10) mainly work for one company, NYK Group (87% of vessels under management by NYK Management) and COSCO Group (100% of vessels under management by Cosco HK Shipping). This could place some limitations on our research, as a relationship might exist between the owner and the manager in these cases. Therefore, given the legal and operational complexities of company structures in shipping, our choice of identifying cases of outsourcing in which the owner's name is different from the manager's name increases the apparent outsourcing rate. These figures may, therefore, include situations in which shipowners have outsourced to subsidiaries, making a direct comparison with former studies difficult.

Insert Figure 1 here Given these limitations, we define three configurations: i) the owner manages all vessels inhouse (owner's name = manager's name for all vessels); ii) the owner outsources the management of all vessels (owner a mixed strategy, in which only some vessels are outsourced. Table 2 reports preliminary statistics on these mutually exclusive configurations.

Insert Table 2 here

Of the 4,049 owners, 45.1% manage all vessels in-house, 29.3% outsource the management of all vessels and 25.5% have a mixed-management strategy. Owners registered in the UK or in Norway outsource more often, while Indonesian, Russian or Greek owners outsource less often. There are indications of a size effect, as 60% of owners with two vessels manage all vessels in house, while the corresponding figure is only 6% for firms with more than 50 vessels. Finally, a mixed strategy is common among ship-owners with a mean fleet size of 18 vessels, while in-house management is common among ship-owners with a mean fleet size of 5.

We also consider the variety of the vessel fleet in terms of three configurations: i) only one type of vessel in an owner's fleet (variety =1), ii) two types of vessels in an owner's fleet (variety =2) or iii) more than two types of vessels in an owner's fleet (variety 3). Figure 2 shows that variety increases with the size of ship-owner. Approximately 90% of owners with more than 50 vessels have at least two different types of vessel in their fleets, compared to 18% for owners with only two vessels. Fifty three percent of owners with vessels of one type keep vessel management in-house (Table 3), while the corresponding figure is 14.8% for owners with three or more types of vessels.

Insert Figure 2 here Insert Table 3 here

Figure 3 provides further insights into the various configurations. When a fleet has only one type of ship (Figure 3A), in-house management is the preferred option. This is most prevalent when the vessels are bulkers, dry cargo ships, passenger ships/ferries or reefers. When the fleet is composed of two (3B), or three or more (3C) distinct types, mixed strategies are more likely.

Insert Figure 3 here

Econometric analysis of the outsourcing decision

To identify the factors that influence the ship-owner's decision to outsource, we rely on a Logit multinomial model. Let 1 j when all vessels are managed in-house, 2 j when the management of all vessels is outsourced and 3 j when the mixed strategy is chosen. For each of these random alternatives, the owner's utility is

j j j X U
, with X representing a set of control factors, j representing the associate vector of coefficients for j ( 3 , 2 , 1 j

),

and j representing a random perturbation. The probability of an owner choosing an alternative j is ) , Pr( Pr

l j k j U U U U j
, with j l k, .

Assuming that stochastic utilities, independently and identically distributed with extreme value distribution apply, the corresponding model is a multinomial Logit model (MNL) with the probability of choosing j equal to

3 1 / ; Pr i X X i j e e X j
. The choice is only dependent on utility differences because ) 0 , 0 Pr( Pr

l j k j U U U U j
. Thus, identification is achieved by normalizing the utility of an alternative to zero, with all vessels managed in-house serving as the reference category.

Insert Table 4 here MNL estimates (Table 4) show that the owner's country of domicile is a significant covariate.

Ship-owners registered in Denmark, the UK and Japan choose to outsource more often, while

Indonesian and Greek ship-owners are less likely to do so. Furthermore, UK, Norwegian and US ship-owners are more likely to implement a mixed strategy. Size influences outsourcing decisions when the fleet consists of more than 10 vessels. The pattern is different for companies adopting a mixed strategy -in those situations, outsourcing continuously increases with size. Finally, the choice of a mixed strategy is 93% higher for companies with variety.

To measure the influence of vessel characteristics, we use the sample at the vessel level (39,925 vessels). As the focus is on inequalities, we first estimate the probability of a vessel having the same owner and manager, denoted by ) Pr( ji SAME , where ji SAME is a dummy variable equal to 1 when vessel i is owned and managed by the same company j , and 0 otherwise. This probability can be expressed as:

) 0 Pr( ) 1 Pr( ji j ji j ji u Z X SAME (1)
where j X and ji Z represent sets of characteristics for the owner and the vessel, respectively. and are vectors of coefficients to be estimated, j u is an owner-specific unobserved heterogeneity term and ji is a residual term. These perturbations are assumed to be normally distributed such that ) N(0; ~2 u j u and N(0;1) ĩt . The corresponding model, with estimates in column 1 of Table 5, is thus a random effect Probit model estimated by quadrature techniques [START_REF] Butler | A computationally efficient quadrature procedure for the one factor multinomial Probit model[END_REF].

Insert Table 5 here

Outsourcing is influenced by vessel and owner characteristics. The probability of having a vessel managed in-house decreases with vessel age, although it begins to increase again when a vessel is more than 35 years old (+11.6 points of probability). One possible explanation of this phenomenon is that older vessels bring in lower earnings, which might not be sufficient to cover the outsourcing costs. The probability is higher when the vessels are reefers (+13.4 points), but lower when they are tankers (-12.7 points) and offshore vessels (-16.0 points). Inhouse management is more likely for large vessels and when the owner is domiciled in Indonesia, Greece, South Korea or Turkey (in contrast to the UK, Norway, the US or Denmark). Fleet size has a positive effect when the fleet comprises more than ten vessels.

This effect peaks when the fleet numbers 31-50 vessels. Finally, variety matters since the probability of in-house management is reduced by 34.3 points for fleets with at least two types of vessels.

One shortcoming pertaining to the random-effects specification is that the unobserved ownerspecific component j u is, by definition, uncorrelated with covariates j X and ji Z . This assumption is relaxed with a fixed-effects framework that uses the conditional likelihood approach described in [START_REF] Chamberlain | Analysis of covariance with qualitative data[END_REF]. Two specific features must be considered. First, this specification explains the choice of managing only certain vessels in-house, meaning that owners with all vessels managed in-house or all vessels outsourced are not considered.

Second, invariant owner characteristics are captured by the owner fixed effect and are not included in the regressions. The results are reported in column 2 of Table 5 for 19,485 vessels belonging to 1,034 different owners in 2009.

Coefficients from the fixed effect Logit and random effect Probit regressions are fairly similar. Mixed strategies often mean that management is kept in-house for vessels less than five years old and for vessels that are more than 35 years old. Larger vessels (those with a size of more than 50,000gt), reefers and roros are generally managed in-house, in contrast to dry cargo ships, offshore vessels and tanker vessels. The fixed effects specification, therefore, confirms that a vessel's characteristics influence the probability of having a vessel outsourced, which in turn explains why variety produces an impact.

Discussion and concluding comments

The aim of this research was to investigate vessel management outsourcing patterns and to study the factors explaining differences in outsourcing rates. We rely on an econometric analysis of a large dataset covering the world fleet in 2009, with a focus on the roles played by the owner and vessel characteristics. Our findings confirm that although vessel management outsourcing is a feature of the shipping industry, only certain types of owners outsource and owners only outsource the management of some vessels with specific characteristics.

The owner's size is a key factor in decisions to outsource, especially when the owner's fleet encompasses more than 10 vessels. When owners have less than ten vessels, variety governs the decision. With respect to previous studies (Mitroussi, 2004a[START_REF] Mitroussi | The role of organisational characteristics of ship owning firms in the use of third party ship management[END_REF], our estimates suggest that size combined with variety creates the highest likelihood of outsourcing. An owner with many similar vessels will outsource only some of these vessels in order to benchmark its own management performance and to attain greater flexibility. Conversely, an owner with a small number of vessels, even if they are of different types, may not have all of the necessary knowledge available in-house.

Finally, our paper provides insights into a specific configuration called the "mixed strategy".

This type of configuration is relevant as, regardless of country, 17% to 40% of owners have implemented such a configuration. Our findings stress that this strategy often leads to the inhouse management of young and old vessels, and of reefer and roro vessels. It also leads to the outsourcing of middle-aged, dry cargo ships, offshore vessels and tanker vessels.

These empirical findings on diversity in outsourcing rates across maritime sectors highlight many open questions. Many of them have been discussed in previous research on partnership drivers and facilitators [START_REF] Lambert | Supply chain partnerships: model validation and implementation[END_REF], on types of operational partnerships within the transportation industry and their evolution over time [START_REF] Cruijssen | Horizontal cooperation in transport and logistics: A literature review[END_REF][START_REF] Fugate | Operational collaboration between shippers and carriers in the transportation industry[END_REF], and on the need for customized partnerships [START_REF] Lambert | Building successful logistics partnerships[END_REF]. Further investigations into these issues would require an in-depth analysis of the ownership structure of shipping companies with a classification system that makes a clear distinction between vessels outsourced to third-party professional managers and vessels outsourced to ship management companies with "shareholding ties" to the owning companies [START_REF] Mitroussi | Third party ship management: the case of separation of ownership and management in shipping context[END_REF].

Lastly, it raises questions on the transparency of supply chains relying heavily on shipping and for which supply chains safety and security are two hot issues. For shippers, the increasing trend of ship management outsourcing multiplies the number of stakeholders and raises questions on which carrier shippers should select. This article clearly contributes to this debate by shedding light on the complexity of the outsourcing decision-making process.

Figure 1 .

 1 Figure 1. Top 12 ship management companies managing more than 100 vessels (2009)*

Figure 2

 2 Figure 2. Variety and fleet size (2009)

Table 1 . Description of the sample

 1

	Fleet (number of vessels)	Ship-owners (%)	Vessels (%)
	2	30.4	6.2
	3-4	25.3	8.7
	5-7	16.8	9.8
	8-10	7.2	6.4
	11-15	7.3	9.4
	16-20	3.8	6.8
	21-30	3.7	9.3
	31-50	2.7	10.7
	50+	2.7	32.8
	Total number	4,049	39,925
	Source: Authors' calculations based on Lloyd's Register Fairplay -World Shipping Encyclopaedia (June 2009)

Table 2 . Outsourcing level by owner (N=4,049)* Variables

 2 The row proportion for all vessels comprises the remainder, i.e., cases when only the management of some vessels is outsourced. Source: Authors' calculations based onLloyd's Register Fairplay -World Shipping Encyclopaedia (June 2009)

	All vessels :	All vessels:	Distribution
	owner manager	owner = manager	(in %)

Table 3 . Outsourcing level by fleet size and variety (N=4,049 owners)*

 3

	Owner's size	Variety:	All vessels:	All vessels:
		number of types of	owner manager	owner = manager
		vessel (per owner)
	2 vessels	1	27.9	62.0
		2	32.1	50.7
	3-4 vessels	1	25.5	57.9
		2	28.8	48.2
		3	33.9	30.4
	5-7 vessels	1	25.7	49.6
		2	33.5	37.1
		3	27.5	26.1
	8-10 vessels	1	28.2	45.4
		2	25.9	25.9
		3	32.6	9.3
	11-15 vessels	1	29.9	30.6
		2	34.4	30.0
		3	35.2	15.5
	16-20 vessels	1	33.9	23.2
		2	38.3	8.3
		3	44.7	10.5
	21-30 vessels	1	32.5	22.5
		2	28.6	15.9
		3	35.6	15.6
	31-50 vessels	1	51.7	10.3
		2	38.5	11.5
		3	50.0	3.6
	More than 50 vessels	1	6.3	6.3
		2	28.1	9.4
		3	31.9	4.4
	All owners	1	27.3	53.9
		2	31.5	36.6
		3	35.8	14.8
	* The row proportion for all vessels comprises the remainder, i.e., cases when only the management of some vessels is
	outsourced.
	Source: Authors' calculations based on Lloyd's Register Fairplay -World Shipping Encyclopaedia (June 2009)

Table 4 . Outsourcing MNL estimates (Reference category: owner = manager for all vessels)

 4 Estimates from an MNL logit model. Significance levels are 1% ( *** ), 5% ( ** ) and 10% ( * ).

	Variables

Table 5 . Estimates of the probability of a vessel having the same owner and manager

 5 Authors' calculations based on Lloyd's Register Fairplay -World Shipping Encyclopaedia (June 2009) Note: Estimates from random effect probit models. Significance levels are 1% ( *** ), 5% ( ** ) and 10% ( * ).

	Variables		(1) Random effects probit	(2) Fixed effects logit
			coef.	t-value	coef.	t-value
	Constant		1.626 ***	11.07
	Age	0-4	Ref		Ref
		5-9	-0.139 ***	-3.71	-0.188 ***	-2.85
		10-14	-0.173 ***	-4.41	-0.230 ***	-3.28
		15-19	-0.238 ***	-5.15	-0.355 ***	-4.31
		20-24	-0.318 ***	-6.42	-0.531 ***	-5.95
		25-29	-0.183 ***	-3.90	-0.290 ***	-3.38
		30-34	-0.083	-1.51	-0.105	-1.04
		35 and more	0.319 ***	5.02	0.567 ***	4.66
	Size (gt)	1000-4999	Ref		Ref
		5000-9999	-0.036	-0.84	0.060	0.78
		10000-24999	-0.073	-1.58	0.019	0.23
		25000-49999	-0.104 **	-1.98	-0.047	-0.49
		More than 50000	0.143 **	2.51	0.427 ***	4.13
	Type of ship	Bulker / combination	Ref		Ref
		Container	-0.011	-0.20	-0.017	-0.18
		Dry cargo	-0.128 **	-2.43	-0.249 **	-2.56
		Offshore	-0.543 ***	-6.22	-0.886 ***	-5.53
		Pass./Ferry	0.153	1.60	0.163	0.86
		Reefer	0.363 ***	3.27	0.634 ***	2.91
		Roro	0.128 *	1.86	0.262 **	2.08
		Tanker	-0.397 ***	-8.27	-0.666 ***	-7.53
	Owner's country Japan	-0.547 **	-2.33
		Germany	-0.835 ***	-3.15
		China	0.159	0.44
		Greece	1.860 ***	8.51
		US	-1.457 ***	-4.68
		Norway	-1.712 ***	-6.43
		Russia	1.151 ***	4.79
		Turkey	0.912 ***	3.66
		Korea (South)	0.928 ***	3.23
		Singapore	-1.037 ***	-2.89
		Panama	-0.400	-0.87
		Indonesia	2.859 ***	9.08
		United Kingdom	-2.726 ***	-8.46
		Hong Kong	-0.530	-1.41
		Netherlands	-1.163 ***	-2.83
		Italy	-0.509	-1.20
		Denmark	-1.167 ***	-3.23
		Others	Ref
	Owner's size	2 vessels	Ref
		3-4 vessels	0.073	0.47
		5-7 vessels	-0.110	-0.61
		8-10 vessels	-0.281	-1.07
		11-15 vessels	-0.967 ***	-3.40
		16-20 vessels	-1.447 ***	-4.72
		21-30 vessels	-1.212 ***	-3.93
		31-50 vessels	-2.684 ***	-9.37
		More than 50 vessels	-1.508 ***	-5.70
	Variety (more than 2 types in fleet)	-0.969 ***	-7.21
	Number of vessels		39925		19485
	Number of owners		4049		1034
	Log likelihood		-12544.2		-6804.5
	Source:


Nguyen Thang Dao

Julio Dávila

Implementing steady state efficiency in overlapping generations economies with environmental externalities

Keywords: overlapping generations, environmental externality, tax and transfer policy générations imbriquées, externalités environementales, politiques fiscales. JEL Classification: D62, E21, H21, H41

We consider in this paper overlapping generations economies with pollution resulting from both consumption and production. The competitive equilibrium steady state is compared to the optimal steady state from the social planner's viewpoint. We show that any competitive equilibrium steady state whose capital-labor ratio exceeds the golden rule ratio is dynamically inefficient. Moreover, the range of dynamically efficient steady states capital ratios increases with the effectiveness of the environment maintainance technology, and decreases for more polluting production technologies. We characterize some tax and transfer policies that decentralize as a competitive equilibrium outcome the social planner's steady state.

Resumé On étudie dans ce papier des économies de générations imbriquées avec pollution provenant aussi bien de la consommation que de la production. L'état stationnaire d'équilibre concurrentiel est compare à l'état stationnaire optimal du point de vue du planificateur. On montre que lorsque le ratio capital-travail excède à l'état stationnaire de l'équilibre concurrentiel celui de la règle d'or, alors le premier est dynamiquement inefficace. De plus, l'intervalle d'états stationnaires dynamiquement efficaces s'accroît avec l'effectivité de la technologie de maintient de l'environnement, et diminue lorsque la technologie de production est plus polluante. On caractérise des politiques fiscales qui décentralisent l'état stationnaire du planificateur comme équilibre concurrentiel.

Introduction

Environmental externalities in economies with overlapping generations have been studied since at least the 1990s. In particular, the effects of environmental externalities on dynamic inefficiency, productivity, health and longevity of agents have been addressed, as well, as the policy interventions that may be needed. While in most papers pollution is assumed to come from production, and the environment is supposed to improve or degrade by itself at a constant rate (Marini and Scaramozzino 1995;Jouvet et al 2000;[START_REF] Jouvet | Longevity and Environmental Quality in an OLG Model[END_REF]Pautrel 2007;[START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF], other papers assume that pollution comes from consumption [START_REF] John | An Overlapping Generations Model of Growth and the Environment[END_REF][START_REF] John | Short-lived agents and the long-lived environment[END_REF][START_REF] Ono | Optimal tax schemes and the environmental externality[END_REF]. As a consequence of the differing assumptions, the effect of environmental externalities on capital accumulation vary widely across papers. Specifically, [START_REF] John | Short-lived agents and the long-lived environment[END_REF] showed that when only consumption pollutes, the economy accumulates less capital than that what would be optimal. Conversely, [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] showed that when only production pollutes, the economy accumulates instead more capital than the optimal level. This is so because in [START_REF] John | Short-lived agents and the long-lived environment[END_REF] agents pay taxes to maintain environment when young, so that an increased pollution reduces their savings; however, in [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] pollution increases health costs in old age, leading agents to save more to pay for them. The difference seems therefore to come from when the taxes are paid (when young or old) rather than from whether pollution comes from production or consumption. Another main difference between [START_REF] John | Short-lived agents and the long-lived environment[END_REF] and [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] is their different assumptions about the ability of environment to recover from pollution. [START_REF] John | Short-lived agents and the long-lived environment[END_REF] assumes that environment naturally degrades over time, while [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] assumes that environment recovers naturally.

This paper aims at disentangling the effects of both production and consumption on environment. Specifically, as in [START_REF] John | An Overlapping Generations Model of Growth and the Environment[END_REF][START_REF] John | Short-lived agents and the long-lived environment[END_REF], we assume that the environment degrades naturally over time at a constant rate and that young agents devote part of their income to maintain it. 1 In this setup, we characterize the range of dynamically inefficient capital-labor ratios. Next, we introduce taxes and transfer policies that decentralize the first-best steady state as a competitive equilibrium steady state.

The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 characterizes its competitive equilibria. Section 4 presents the problem of the social planner, defines the efficient allocation with and without discounting, and characterizes the range of dynamically inefficient capital ratios (Proposition 1). The competitive equilibrium steady state and the planner's steady state are compared in Section 5, where we introduce some tax and transfer schemes that decentralize the planner's steady state as market outcome (from Proposition 2 to Proposition 9). Section 6 concludes the paper.

The model

We consider the overlapping generations economy in [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] with a constant population of identical agents. The size of each generation is normalized to one. Each agent lives two periods, say young and old. When young, an agent is endowed with one unit of labor which he supplies inelastically. Agents born in period t divide their wage w t between consumption when young c t 0 , investment in maintaining the environment m t , and savings k t lent to firms to be used in t + 1 as capital for a return rate r t+1 . The return of savings r t+1 k t is used up as old age consumption. Agents born at date t have preferences over their consumptions when young and old (c t 0 , c t 1 ) ∈ R 2 + and the environmental quality when old,

E t+1 ∈ R, represented by u(c t 0 ) + v(c t 1 ) + φ(E t+1 ) with u , v , φ > 0, u , v , φ < 0, and u (0) = v (0) = +∞, u (+∞) = v (+∞) = 0.
Environmental quality evolves according to

E t+1 = (1 -b)E t -αF (K t+1 , L t+1 ) -β(c t+1 0 + c t 1 ) + γm t for some α, β, γ > 0 and b ∈ (0, 1], where F is a Cobb-Douglas pro- duction function F (K t , L t ) = AK θ t L 1-θ t .
Capital fully depreciates in each period. Under perfect competition, the representative firm maximizes profits solving

M ax Kt,Lt≥0 F (K t , L t ) -r t K t -w t L t
so that the wage rate and the rental rate of capital are, in each period t, the marginal productivity of labor and capital respectively. Since population is normalized to 1, period t aggregate savings (i.e. period t + 1 aggregate capital K t+1 ) and labor supply are k t and 1 respectively, and the wage and rental rate of capital faced by the agent born at period t are

r t+1 = F K (k t , 1) = θA(k t ) θ-1
(1)

w t = F L (k t-1 , 1) = (1 -θ)A(k t-1 ) θ (2)
Environmental quality converges autonomously to a natural level normalized to zero at a rate b that measures the speed of reversion to this level. Nonetheless, production and consumption degrade environmental quality by an amount αF (K t+1 , 1) and β(c t+1 0 +c t 1 ) respectively, while young agents can improve the environmental quality by an amount γm t if they devote a portion m t of their income to that end. 2 The life-time utility maximization problem of the representative agent is

M ax c t 0 ,c t 1 ,k t ,m t ≥0 E t , E e t+1 u(c t 0 ) + v(c t 1 ) + φ(E e t+1 ) (3)
subject to

c t 0 + k t + m t = w t (4) c t 1 = r t+1 k t (5) E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (6)
2 One can thus interpret environmental quality as including any characteristic that make the environment more apt for human life, like the cleanness of rivers and atmosphere, and the quality of soil or groundwater, etc. It also includes the state of forests, agricultural lands, parks and gardens, which left on their own naturally revert to wilderness, unless subject to regular maintenance.

E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (7) given E t-1 , c t-1 1 , k t-1 , m t-1
, w t , r t+1 as well as the expected consumption of the next generation young agent c t+1,e 0 . Since the presentative agent is assumed to be negligible within his own generation, he thinks of the impact of his savings k t on aggregate capital K t+1 to be negligible as well, ignoring that actually K t+1 = k t at equilibrium. This assumption implies that he does not internalize the impact of the savings decision on environment via production. Agent t's optimal choice

(c t 0 , c t 1 , k t , m t , E t , E e t+1
) is therefore characterized by the first-order conditions

u (c t 0 ) -[β(1 -b) + γ] φ (E e t+1 ) = 0 (8) v (c t 1 ) -β + γ r t+1 φ (E e t+1 ) = 0 (9) c t 0 + k t + m t -w t = 0 (10) c t 1 -r t+1 k t = 0 (11) E t -(1 -b)E t-1 + αF (k t-1 , 1) + β(c t 0 + c t-1 1 ) -γm t-1 = 0 (12) E e t+1 -(1 -b)E t + αF (K t+1 , 1) + β(c t+1,e 0 + c t 1 ) -γm t = 0 (13)
as an implicit function of E t-1 , c t-1 1 , k t-1 , m t-1 , w t , r t+1 and c t+1,e 0 as long as the Jacobian matrix of the left-hand-side of the system above with respect to c t

0 , c t 1 , k t , m t , E t , E e
t+1 is regular at the solution. The existence of the optimal solution is established in Appendix A1 and the regularity of the Jacobian matrix at equilibrium is established in Appendix A2. For these FOCs to be not only necessary but also sufficient for the solution to be a maximum, the second order conditions (SOCs) are shown to hold at equilibrium in Appendix A3.

Competitive equilibria

Perfect foresight competitive equilibria are characterized by (i) the agent's utility maximization under the budget constraints, with correct expectations, (ii) the firms' profit maximization determining factors' prices, and (iii) the dynamics of environment. Therefore, a competitive equilibrium allocation {c t 0 , c t 1 , k t , m t , E t+1 } t is solution to the system of equations

u (c t 0 ) -[β(1 -b) + γ] φ (E t+1 ) = 0 (14) v (c t 1 ) -β + γ F K (k t , 1) φ (E t+1 ) = 0 (15) c t 0 + k t + m t -F L (k t-1 , 1) = 0 (16) c t 1 -F K (k t , 1)k t = 0 (17) E t+1 -(1 -b)E t + αF (k t , 1) + β(c t+1 0 + c t 1 ) -γm t = 0 (18)
Note that the feasibility of the allocation of resources is guaranteed by the agent's budget constraints ( 16) and ( 17), since at t

c t 0 + c t-1 1 + k t + m t = F K (k t-1 , 1)k t-1 + F L (k t-1 , 1) = F (k t-1 , 1)
A perfect foresight competitive equilibrium steady state, in particular, is a (c 0 , c 1 , k, m, E) solution to the system of equations

u (c 0 ) -[β(1 -b) + γ] φ (E) = 0 v (c 1 ) -β + γ F K (k, 1) φ (E) = 0 c 0 + k + m -F L (k, 1) = 0 c 1 -F K (k, 1)k = 0 bE + αF (k, 1) + β(c 0 + c 1 ) -γm = 0
The perfect foresight competitive equilibria of this economy follow a dynamics represented by a first-order difference equation, because of the regularity of the associated Jacobian matrix of the left hand side of the system of equations above with respect to c t+1

0 , c t 1 , k t , m t , E t+1 (see Appendix A1).
4 The social planner's choice with and without discounting

In this section, we consider the optimal allocation from the viewpoint of a social planner that allocates resources in order to maximize a weighted sum of the welfare of all current and future generations. The allocation selected by the social planner, which is optimal in the Pareto sense, is a solution to the problem

M ax {c t 0 ,c t 1 ,k t ,m t ,E t+1} ∞ t=0 ∞ t=0 1 (1 + R) t u(c t 0 ) + v(c t 1 ) + φ(E t+1 ) (19)
subject to, ∀t = 0, 1, 2, ...,

c t 0 + c t-1 1 + k t + m t = F (k t-1 , 1) (20)
E t+1 = (1 -b)E t -αF (k t-1 , 1) -β(c t+1 0 + c t 1 ) + γm t (21)
given some initial conditions c -1 1 , k -1 , E 0 , where 0 ≤ R is the social planner's subjective discount rate. 3 The first constraint (20) of the problem is the resource constraint of the economy in period t requiring that the total output in that period is split into consumptions of the current young and old, savings for next period's capital, and environmental maintenance. The second constraint ( 21) is the dynamics of the environmental quality.

The social planner's choice of a steady state is a

(c 0 , c1 , m, k, Ē) satisfying (see Appendix A4) u (c 0 ) = (1 + R) γ + β(1 + R) b + R φ ( Ē) (22) v (c 1 ) = γ + β(1 + R) b + R φ ( Ē) (23)
F K ( k, 1) = γ(1 + R) γ -(1 + R)α (24) c0 + c1 + k + m = F ( k, 1) (25) b Ē + αF ( k, 1) + β(c 0 + c1 ) -γ m = 0 (26)
(the planner's discount rate R cannot be arbitrarily high for the optimal steady state to be characterized as above, specifically γ > (1 + R)α needs to hold, which requires γ > α, so that F K ( k, 1) > 0). More specifically, in the case of the social planner caring about all generations equally, i.e. R = 0, the planer's steady state is the socalled golden rule steady state {c * 0 , c * 1 , k * , m * , E * } that maximizes the utility of the representative agent and is characterized by being a solution to the system

u (c * 0 ) = γ + β b φ (E * ) (27) v (c * 1 ) = γ + β b φ (E * ) (28) F K (k * , 1) = γ γ -α (29) c * 0 + c * 1 + k * + m * = F (k * , 1) (30) bE * + αF (k * , 1) + β(c * 0 + c * 1 ) -γm * = 0 (31)
Note that, from ( 27) and ( 28), the marginal utility of consumption of the young agent must equal that of the consumption of the old agent. [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] shows that in the standard OLG model without pollution externalities, a competitive equilibrium steady state whose capital per worker exceeds the golden rule level is dynamically inefficient. Notwithstanding, [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] shows that, when the pollution externality is large enough, there are dynamically efficient competitive equilibrium steady state capital ratios that exceed the golden rule capital ratio. Specifically, [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] shows the existence of a "super golden rule" level of capital ratio, beyond the golden rule level, such that any economy with pollution externalities whose stationary capital ratio exceeds this level is necessarily dynamically inefficient. Nonetheless, it should be noted that in [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] (i) pollution externalities only from production are taken into account; (ii) the environment recovers itself overtime at a constant rate; (iii) no resource is devoted to maintaining the environment; and (iv) the pollution externality decreases the utility of the agents only indirectly by requiring each agent to pay for extra health costs in the old age. In this paper, we consider instead an economy with pollution externalities coming from both production and consumption, in which the environment degrades itself over time, and the quality of the environment can be improved through maintenance. Also the quality of environment directly affects the utility of the agents. As a consequence, this paper shows instead that, as in [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] and in contrast with [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF], in an economy with consumption and production pollution externalities, the golden rule capital ratio is still the highest level of capital ratio that is dynamically efficient.

Proposition 1: In a [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] overlapping generations economy with consumption and production pollution, for an efficient enough cleaning technology, compared to the marginal polluting impact of production (specifically, for γ > α in the model), the golden rule (i.e. the planner's steady state choice without discounting) is the highest dynamically efficient capital ratio.

Proof: Since F KK (k, 1) < 0 for all k, the planner's optimal capital ratio k is implicitly defined to be a differentiable function k(R) of R by the condition

F K ( k, 1) = γ(1 + R) γ -(1 + R)α
whose derivative, by the implicit function theorem, is

k (R) = 1 F KK ( k(R), 1) γ γ -(1 + R)α 2 < 0 (32)
So, k is decreasing in R. Hence, k(R) is maximal when R = 0, which is corresponds to the golden rule level of capital k *

Proposition 1 shows that any steady state capital ratio exceeding k * is dynamically inefficient. From (29) the golden rule capital ratio k * is decreasing in the production pollution parameter α. It is, however, increasing in the environment maintaining technology γ. Hence, the more polluting is production, the smaller the range of steady state allocations that are dynamically efficient for some discount factor R. Similarly, the more effective is the environment maintainance technology, the bigger the range of steady state allocations that are dynamically efficient for some discount factor R.

Policy implementation of the planner's optimal steady state

In this section, we provide tax and transfer policies allowing to implement the planner's optimal steady state. [START_REF] Ono | Optimal tax schemes and the environmental externality[END_REF] and [START_REF] Gutiérrez | Dynamic inefficiency in an Overlapping Generation Economy with Pollution and Health Cost[END_REF] introduced also tax and transfer schemes to decentralize the golden rule steady state in the context of the pollution externalities they consider (from consumption and production only, respectively). However, their schemes uphold the golden rule once the economy is already at that steady state. Nevertheless, in this section we provide policies that lead the economy towards the social planer's steady state and will keep it there once reached (for the golden rule, the social planner's discount rate just needs to be set to R = 0). The policies fulfill this in two stages. In the first stage, in the period t, taxes and transfers are set in order to make the agent born in period t choose his savings and consumption when old to be equal to the optimal steady state capital ratio and optimal steady state old agent's consumption, respectively, from the viewpoint of the social planner. Then in the second stage, taxes and transfers are reset to uphold the planner's steady state. The first scheme based on the taxation of consumption is presented next in detail. The subsequent schemes work analogously.

Taxes on consumptions

Suppose that, at the begining of period t, the social planner announces proportional taxes and lump-sum transfers. Letting τ t 0 and τ t 1 be the tax rate on agent t's consumption when young and old respectively, T t 0 a lump-sum tax (if positive) levied on agent t's income when young, and T t 1 a lump-sum transfer (if positive) to the same agent when old at date t + 1, the problem of agent t is then

M ax c t 0 ,c t 1 ,k t ,m t ≥0 E t ,E e t+1 u(c t 0 ) + v(c t 1 ) + φ(E e t+1 ) (33)
subject to . Note again that in equation ( 37), the agent, being negligible within his generation, ignores the fact that K t+1 = k t and hence is unable to internalize the effect of the savings decisions on environment through the aggregate output. Hence, the first-order conditions characterizing agent t's optimal choice are

(1 + τ t 0 )c t 0 + k t + m t = w t -T t 0 (34) (1 + τ t 1 )c t 1 = r t+1 k t + T t 1 (35) E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (36)
E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (37) given c t-1 1 , k t-1 , m t-1 , E t-1 , w t ,
u (c t 0 ) = β(1 -b) + γ(1 + τ t 0 ) φ (E e t+1 ) (38) v (c t 1 ) = β + γ(1 + τ t 1 ) F K (k t , 1) φ (E e t+1 ) (39)
(1 + τ t 0 )c t 0 + k t + m t = w t -T t 0 (40) (1 + τ t 1 )c t 1 = r t+1 k t + T t 1 ( 41
)
11 Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (42) E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (43)
At a perfect foresight equilibrium the wage rate and capital return are given by the labor and capital productivities respectively, and forecasts coincide with actual values, i.e. E e t+1 = E t+1 and c t+1,e 0 = c t+1 0 .

The next proposition shows that the tax rates and lump-sum transfers can be set at levels that make agent t choose the planner's steady state capital ratio and consumption when old.

Proposition 2: In a Diamond (1965) overlapping generations economy with pollution from both consumption and production, for any given period t, there exists a period by period budget balanced policy of consumption taxes, and lump-sum taxes and transfers, (τ t 0 , τ t 1 , T t 0 , T t 1 ), that implements at t + 1 the planner's steady state capital ratio k and consumption when old c1 at a competitive equilibrium.

Proof: Let the tax rates be

τ 0 = [γ + β(1 + R)](1 + R) -[γ + β(1 -b)](b + R) (b + R)γ > 0 τ 1 = 1 + R b + R • γ + β(1 -b) γ -α(1 + R) -1 > 0
then equations ( 38), (39) become equations ( 22), ( 23) when the equation ( 24) characterizing the planner's steady state capital ratio,

F K ( k, 1) = 1+R
1-(1+R)α/γ , holds. Therefore agent t's chooses ice the planner's steady state capital ratio k and consumption when old c1 under the consumption tax rates τ 0 , τ 1 above if, and only if,

u (c t 0 ) = (1 + R) γ + β(1 + R) b + R φ (E t+1 ) (44) v (c 1 ) = γ + β(1 + R) b + R φ (E t+1 ) (45) (1 + τ 0 )c t 0 + k + m t = F L (k t-1 , 1) -T t 0 (46) (1 + τ 1 )c 1 = F K ( k, 1) k + T t 1 ( 47
)
E t = (1-b)E t-1 -αF (k t-1 , 1)-β(c t 0 +c t-1 1 +τ 0 c t 0 +T t 0 )+γm t-1 (48) E t+1 = (1 -b)E t -αF ( k, 1) -β(c t+1 0 + c1 ) + γm t (49) given E t-1 , c t-1 1 , k t-1 , m t-1 , c t+1 0 = c t+1,e 0
. Note that this is a system in {c t 0 , T t 0 , T t 1 , m t , E t , E t+1 } (note also that in the equation ( 48) the consumption when old of agent t -1 is now ct-1

1 = c t-1 1 + τ 0 c t 0 + T t 0
since he receives as a lump-sum transfer the tax raised from the young agent t, τ 0 c t 0 +T t 0 , so that the government's budget is balanced. The solution in {c t 0 , T t 0 , T t 1 , m t , E t , E t+1 } to the system of equations ( 44)-( 49) is the following. Given that φ < 0, from ( 45) and ( 23)

v (c 1 ) = γ+β(1+R) b+R φ (E t+1 ) v (c 1 ) = γ+β(1+R) b+R φ ( Ē) ⇒ E t+1 = Ē
then from (44), ( 22) and

E t+1 = Ē, u (c t 0 ) = (1 + R) γ+β(1+R) b+R φ (E t+1 ) u (c 0 ) = (1 + R) γ+β(1+R) b+R φ ( Ē) E t+1 = Ē ⇒ c t 0 = c0
Also from (47) it follows

T t 1 = (1 + τ 1 )c 1 -F K ( k, 1) k
and from (46) and c t 0 = c0 we have

m t = F L (k t-1 , 1) -(1 + τ 0 )c 0 -k -T t 0
It can also be easily checked that, substituting where

E t+1 = Ē, c t 0 = c0 , m t = F L (k t-1 , 1)-(1+τ 0 )c 0 -k -T
I = (1 -b) (1 -b)E t-1 -αF (k t-1 , 1) -βc t-1 1 + γm t-1 -αF ( k, 1) -[β(1 -b) + γ] (1 + τ 0 )c 0 -β(c t+1 0 + c1 ) + γ F L (k t-1 , 1) -k so that T t 0 = I - Ē β(1 -b) + γ
and hence, from equation ( 48),

E t = (1-b)E t-1 -αF (k t-1 , 1)-β(c t-1 1 +(1+τ 0 )c 0 + I - Ē β(1 -b) + γ )+γm t-1
(50) Therefore, the solution to the system of equations ( 44)-( 49) is

0 B B B B B B @ c t 0 T t 0 T t 1 m t Et Et+1 1 C C C C C C A = 0 B B B B B B B @ c0 I- Ē β(1-b)+γ (1 + τ1)c1 -FK ( k, 1) k FL(k t-1 , 1) -(1 + τ0)c0 -k -I- Ē β(1-b)+γ (1 -b)Et-1 -αF (k t-1 , 1) -β(c t-1 1 + (1 + τ0)c0 + I- Ē β(1-b)+γ ) + γm t-1 Ē 1 C C C C C C C A
so that under the policy (τ 0 , τ 1 , T t 0 , T t 1 ) agent t's chooses ice the planner's steady state capital ratio k and consumption when old c1 .

The next proposition shows that there exist lump-sum transfers T t+1 0 , T t+1 1 that make agent t + 1 choose, under the same tax rates τ 0 , τ 1 , the planner's steady state c0 , c0 1 , k, m, Ē, Ē .

Proposition 3: In a [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] overlapping generations economy with pollution from both consumption and production the policy (τ 0 , τ 1 , T 0 , T 1 ) such that

τ 0 = [γ + β(1 + R)](1 + R) -[γ + β(1 -b)](b + R) (b + R)γ > 0 (51) τ 1 = 1 + R b + R • γ + β(1 -b) γ -α(1 + R) -1 > 0 (52) T 0 = F L ( k, 1) -(1 + τ0 )c 0 -k -m (53)
T 1 = (1 + τ 1 )c 1 -F K ( k, 1) k (54)
implements at t + 1 the planner's steady state, after the first stage of taxation in period t, and keeps the government budget balanced.

Proof:

Given E t , c t 1 = c1 , k t = k, m t , c t+2 0 = c t+2,e 0
and the planner's steady state capital ratio, k, the planner's steady state consumption of the old, c1 , and the consumption tax rates τ 0 , τ 1 , then at the perfect foresight equilibrium, c t+1 0 , T t+1 0 , T t+1 1 , m t+1 , E t+1 , E t+2 is characterized by the following system of equation Agent t + 1 will also choose the planner's capital ratio and consumption when old under a policy (τ 0 , τ 1 , T t+1 0 , T t+1 1 ) -with the same tax rates as before-if

u (c t+1 0 ) = (1 + R) γ + β(1 + R) b + R φ (E e t+2 ) (55) v (c 1 ) = γ + β(1 + R) b + R φ (E e t+2 ) (56)
(1 + τ 0 )c t+1 0 + k + m t+1 = F L ( k, 1) -T t+1 0 (57) (1 + τ 1 )c 1 = F K ( k, 1) k + T t+1 1 (58) E t+1 (= Ē) = (1 -b)E t -αF ( k, 1) -β(c t+1 0 + c1 ) + γm t (59) E e t+2 = (1 -b) Ē -αF ( k, 1) -β(c t+2 0 + c1 ) + γm t+1 (60)
The solution in c t+1 0 , T t+1 0 , T t+1 1 , m t+1 , E t+1 , E e t+2 to this system is the following. Given that φ < 0, from ( 56) and ( 23) it follows

v (c 1 ) = γ+β(1+R) b+R φ (E e t+2 ) v (c 1 ) = γ+β(1+R) b+R φ ( Ē) ⇒ E e t+2 = Ē
From ( 55), ( 22) and

E t+2 = Ē u (c t+1 0 ) = (1 + R) γ+β(1+R) b+R φ (E e t+2 ) u (c 0 ) = (1 + R) γ+β(1+R) b+R φ ( Ē) E e t+2 = Ē ⇒ c t+1 0 = c0
At a perfect foresight equilibrium, c t+2,e 0 = c t+2 0 = c0 holds. Substituting c t+2 0 = c0 and E t+2 = Ē into equation ( 60) and comparing with equation ( 26) we have

b Ē + αF ( k, 1) + β(c 0 + c1 ) -γm t+1 = 0 b Ē + αF ( k, 1) + β(c 0 + c1 ) -γ m = 0 ⇒ m t+1 = m
Substituting c t+1 0 = c0 and m t+1 = m into (57) we have

T t+1 0 = F L ( k, 1) -(1 + τ 0 )c 0 -k -m = T 0
Equation ( 58) gives us

T t+1 1 = (1 + τ 1 )c 1 -F K ( k, 1) k = T 1
So, by implementing the taxes and transfers policy, (τ 0 , τ 1 , T 0 , T 1 ), the optimal choice of agent born at date t + 1 will coincide with the planner's optimal steady state.

We know the old agent in period t + 1 also receives a transfer

T 1 = (1 + τ 1 )c 1 -F K ( k, 1) k. It is obvious that T 1 = τ 0 c0 + τ 1 c1 + T 0
So, at such steady state, under this taxes and transfers policy, the government budget is kept balanced every period.

Taxes on consumptions and capital income

In the section 5.1, we introduced taxes on consumptions in which the tax rates differ between consumptions of the old and the young. In the reality, however, this tax scheme seems to be difficult to apply because it may violate the equity among generations. In order to avoid the discrimination between the old and the young, a unique rate of consumption tax τ t should be applied. Beside that, a capital income tax τ t k and a system of lump-sum tax T t 0 (if positive) and lump-sum transfer T t 1 (if positive), levied on agent t's incomes, are introduced to show that the best steady state allocation can be achieved. The problem of agent t is then

M ax c t 0 ,c t 1 ,k t ,m t ≥0 E t ,E e t+1 u(c t 0 ) + v(c t 1 ) + φ(E e t+1 ) (61)
subject to

(1 + τ t )c t 0 + k t + m t = w t -T t 0 (62) (1 + τ t )c t 1 = (1 -τ t k )r t+1 k t + T t 1 ( 63
)
E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (64) E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (65) given E t-1 , c t-1 1 , k t-1 , m t-1 , w t , c t+1,e
0 and r t+1 . Note again that in equation ( 65), the agent, being negligible within his generation, ignores the fact that K t+1 = k t and hence is unable to internalize the effect of the savings decisions on environment through the aggregate output. Hence, the first-order conditions characterizing agent t's optimal choice are

u (c t 0 ) = β(1 -b) + γ(1 + τ t ) φ (E e t+1 ) (66) v (c t 1 ) = β + γ(1 + τ t ) (1 -τ t k )F K (k t , 1)
φ (E e t+1 ) (67)

(1 + τ t )c t 0 + k t + m t = w t -T t 0 (68) (1 + τ t )c t 1 = (1 -τ t k )r t+1 k t + T t 1 (69) E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (70) E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (71)
At a perfect foresight equilibrium the wage rate and capital return are given by the labor and capital productivities respectively, and forecasts coincide with actual values, i.e. E e t+1 = E t+1 and c t+1,e 0 = c t+1 0 .

Proposition 4: In a Diamond (1965) overlapping generations economy with pollution from both consumption and production, in any period t, there exists a period by period budget balanced policy of consumption tax, capital income tax and lump-sum taxes and transfers, (τ t , τ t k , T t 0 , T t 1 ), that supports to attain the planner's steady state capital (saving) ratio, k, and the planner's steady state consumption of the old, c1 , at the competitive equilibrium.

Proof : The proof for this proposition is similar to the proof for proposition 2.

Proposition 5: In a [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] overlapping generations economy with pollution from both consumption and production, after finishing period t (the first stage of taxation), the economy can achieve the planner's steady state from period t + 1 onward by implementing the following combination

τ = [γ + β(1 + R)](1 + R) -[γ + β(1 -b)](b + R) (b + R)γ > 0 τ k = 1 - (b + R)(γ -(1 + R)α)(1 + τ ) (1 + R)(γ + β -βb) T 0 = F L ( k, 1) -(1 + τ )c 0 -k - m T 1 = (1 + τ )c 1 -(1 -τ k )F K ( k, 1) k
At such the steady state the goverment's budget is kept balanced every period.

Proof: The proof for this proposition is similar to the proof for proposition 3.

Taxes on consumptions and production

We still keep the non-discriminatory tax rate τ t on consumptions and the system of lump-sum tax T t 0 (if positive) and lump-sum transfer T t 1 (if positive). However, we now introduce a Pigouvian tax on production instead of tax on capital income. In any period, let τ p be the tax paid by firms per one unit of output produced. We will design taxes and transfers policy ensuring the government's budget to be balanced and achieving the planner's steady state through competitive markets.

Under the production tax, the problem that the firm much solve is

M ax Kt (1 -τ p )F (K t , 1) -r t k t -w t (72)
The return of capital and the return of labor at the equilibrium are

r t = (1 -τ p )F K (k t-1 , 1) (73)
w t = (1 -τ p )F L (k t-1 , 1) (74)
Under the taxes and transfers policy, the agent t's problem is

M ax c t 0 ,c t 1 ,k t ,m t ≥0 E t , E t+1 u(c t 0 ) + v(c t 1 ) + φ(E e t+1 ) (75)
subject to

(1 + τ t )c t 0 + k t + m t = w t -T t 0 (76) (1 + τ t )c t 1 = r t+1 k t + T t 1 ( 77
)
E t = (1 -b)E t-1 -αF (k t , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (78)
E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (79) given E t-1 , c t-1 1 , k t-1 , m t-1 , w t , c t+1,e
0 and r t+1 . Note again that in equation ( 79), the agent, being negligible within his generation, ignores the fact that K t+1 = k t and hence is unable to internalize the effect of the savings decisions on environment through the aggregate output. Hence, the first-order conditions characterizing agent t's optimal choice are

u (c t 0 ) = β(1 -b) + γ(1 + τ t ) φ (E e t+1 ) (80)
v (c t 1 ) = β + γ(1 + τ t ) (1 -τ t p )F K (k t , 1) φ (E e t+1 ) (81)
(1 + τ t )c t 0 + k t + m t = w t -T t 0 (82) (1 + τ t )c t 1 = (1 -τ t p )r t+1 k t + T t 1 ( 83
)
E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (84)
E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (85)
At a perfect foresight equilibrium the wage rate and capital return are given by the labor and capital productivities respectively, and forecasts coincide with actual values, i.e. E e t+1 = E t+1 and c t+1,e 0 = c t+1 0 .

Proposition 6: In a Diamond (1965) overlapping generations economy with pollution from both consumption and production, in any period t, there exists a period by period budget balanced policy of consumption tax, production tax, lump-sum taxes and transfers, (τ t , τ t p , T t 0 , T t 1 ), that supports to attain the planner's steady state capital (saving) ratio, k, and the planner's steady state consumption of the old, c1 , at the competitive equilibrium.

Proof: The proof for this proposition is similar to the proof for proposition 2.

Proposition 7: In a [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] overlapping generations economy with pollution from both consumption and production, after finishing period t (the first stage of taxation), the economy can achieve the planner's steady state from period t + 1 onward by implementing the following combination

τ = [γ + β(1 + R)](1 + R) -[γ + β(1 -b)](b + R) (b + R)γ > 0 20
τ p = 1 - (b + R)(γ -(1 + R)α)(1 + τ ) (1 + R)(γ + β -βb) T 0 = F L ( k, 1) -(1 + τ )c 0 -k - m T 1 = (1 + τ )c 1 -(1 -τ p )F K ( k, 1) k
At such the steady state the goverment's budget is kept balanced every period.

Proof: The proof for this proposition is similar to the proof for proposition 3.

Taxes on consumption, production and labor income

We now modify the tax and transfer policy introduced in section 5.3 by using the labor income tax rate τ t w to replace the lump-sum tax T t 0 on wage. All other things are kept the same as in the section 5.3. Under this policy, the agent t's problem is

M ax c t 0 ,c t 1 ,k t ,m t ≥0 E t , E t+1 u(c t t ) + v(c t t+1 ) + φ(E t+1 ) (86)
subject to

(1 + τ c )c t 0 + k t + m t = (1 -τ t w )w t (87) (1 + τ )c t 1 = r t+1 k t + T t 0 (88) E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (89) E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (90) given E t-1 , c t-1 1 , k t-1 , m t-1 , w t , c t+1,e 0
and r t+1 . Note again that in equation ( 90), the agent, being negligible within his generation, ignores the fact that K t+1 = k t and hence is unable to internalize the effect of the savings decisions on environment through the aggregate output. Hence, the first-order conditions characterizing agent t's optimal choice are

u (c t 0 ) = β(1 -b) + γ(1 + τ t ) φ (E e t+1 ) (91) v (c t 1 ) = β + γ(1 + τ t ) (1 -τ t p )F K (k t , 1) φ (E e t+1 ) (92)
(1 + τ t )c t 0 + k t + m t = (1 -τ t w )w t (93) (1 + τ t )c t 1 = (1 -τ t p )r t+1 k t + T t 1 ( 94
)
E t = (1 -b)E t-1 -αF (k t-1 , 1) -β(c t 0 + c t-1 1 ) + γm t-1 (95) E e t+1 = (1 -b)E t -αF (K t+1 , 1) -β(c t+1,e 0 + c t 1 ) + γm t (96)
At a perfect foresight equilibrium the wage rate and capital return are given by the labor and capital productivities respectively, and forecasts coincide with actual values, i.e. E e t+1 = E t+1 and c t+1,e 0 = c t+1 0 .

Proposition 8: In a [START_REF] Diamond | National Debt in a Neoclassical Growth Model[END_REF] overlapping generations economy with pollution from both consumption and production, in any period t, there exists a period by period budget balanced policy of consumption tax, production tax, lump-sum taxes and transfers, (τ t , τ t p , τ t w , T t 1 ), that supports to attain the planner's steady state capital (saving) ratio, k, and the planner's steady state consumption of the old, c1 , at the competitive equilibrium.

Proof: The proof for this proposition is similar to the proof for proposition 2.

Proposition 9: In a Diamond (1965) overlapping generations economy with pollution from both consumption and production, after finishing period t (the first stage of taxation), the economy can achieve the planner's steady state from period t + 1 onward by implementing the following combination

τ = β + (1 -b)(γ -βb) + βR(1 + b + R) (b + R)γ τ p = 1 - (b + R)(γ -(1 + R)α)(1 + τ ) (1 + R)(γ + β -βb) τ w = 1 - (1 + τ )c 0 + k + m (1 -τ p )F L ( k, 1) T 1 = (1 + τ )c 1 -(1 -τ p )F K ( k, 1) k
At such the steady state the goverment's budget is kept balanced every period.

Proof: The proof for this proposition is similar to the proof for proposition 3.

Conclusion

We have presented a general equilibrium overlapping generations model with environmental externalities from both production and consumption. For such a model we proved that the competitive equilibrium steady state is not the efficient steady state, for any discount rate the social planner may use. The pollution externality from consumption does not affect the range of dynamically inefficient capital ratios, whereas the pollution externality from production does. The higher the production pollution parameter α, the larger the inefficient range. The environmental maintaining technology γ also plays a role in determining the best steady state capital ratio k. The cleaner the environment maintaining technology, the smaller the range of the dynamically inefficient allocations. By comparing the competitive steady state and the best steady state, we designed a balanced budget taxes and transfer policy that decentralizes the planner's steady state.

This paper makes many simplifying assumptions, such as the technology being exogenous, the population growth rate being zero and there being only one production sector. Further developments including endogenous technology and fertility, as well as the impact of human capital accumulation are left for future research. We also find that is a monotone function of c t 0 since

∂ ∂c t 0 = u (c t 0 ) + [β(1 -b) + γ] β(1 -b) + γ + (β + γ r t+1 )ϕ (c t 0 ) φ (E e t+1 ) < 0
So there exists a unique solution c t 0 > 0 to (99), meaning that there exists a unique optimal solution of the agent.

A2. Competitive equilibrium dynamics

The competitive equilibrium conditions impose on (c t+1 0 , c t 1 , k t , m t , E t+1 ) a dynamics described by a first-order difference equation, since

u (c t 0 ) -[β(1 -b) + γ] φ (E t+1 ) = 0 v (c t 1 ) -β + γ F K (k t , 1) φ (E t+1 ) = 0 c t 0 + k t + m t -F L (k t-1 , 1) = 0 c t 1 -F K (k t , 1)k t = 0 E t+1 -(1 -b)E t + αF (k t , 1) + β(c t+1 0 + c t 1 ) -γm t = 0
implicitly define it to be a function of its lagged value (c t 0 , c t-1 1 , k t-1 , m t-1 , E t ). In effect, the associated Jacobian matrix with respect to where

(c t+1 0 , c t 1 , k t , m t , E t+1 ) J =       0 0 0 0 G 0 v (c t 1 ) D 0 H 0 0 1 1 0 0 1 -C 0 0 β β αF K (k t , 1) -γ 1       25
C = F K (k t , 1) + F KK (k t , 1)k t = θ 2 A(k t ) θ-1 > 0 D = γF KK (k t , 1) F K (k t , 1) 2 φ (E t+1 ) < 0 G = -[β(1 -b) + γ] φ (E t+1 ) > 0 H = -β + γ F K (k t , 1)
φ (E t+1 ) > 0 is regular, since

det(J) = G 0 v (c t 1 ) D 0 0 0 1 1 0 1 -C 0 β β αF K (k t , 1) -γ = -Gβ v (c t 1 ) D 0 0 1 1 1 -C 0 = -Gβ D + Cv (c t 1
) > 0 Since, the Jacobian matrix is regular for all (c t+1 0 , c t 1 , k t , m t , E t+1 ) then it is evidently regular at the solution. This implies that for all competitive equilibrium (c t+1 0 , c t 1 , k t , m t , E t+1 ) t there exists, for all t, a function ψ : R 5 → R 5 such that

      c t+1 0 c t 1 k t m t E t+1       = ψ       c t 0 c t-1 1 k t-1 m t-1 E t      

A3. Checking the SOCs for the maximization problem of the agent

For the FOCs to be sufficient conditions to characterize a (local) maximum to the optimization problem, we have to check the sufficient SOCs. The Lagrangian of the maximization problem is which guarantees that the solution to the agent's problem is a maximum indeed.

Z = u(c t 0 ) + v(c t 1 ) + φ(E t+1 ) + λ 1 (c t 0 + k t + m t -w t ) + λ 2 (c t 1 -r t+1 k t ) +λ 3 E t -(1 -b)E t-1 + αF (k t-

A4. Solving the problem of the social planner

The Lagrange function for this problem is

L = +∞ t=0 1 (1 + R) t u(c t 0 ) + u(c t 1 ) + φ(E t+1 ) + +∞ t=0 µ t (1 + R) t F (k t-1 , 1) -c t 0 -c t-1 1 -k t -m t + +∞ t=0 η t (1 + R) t E t+1 -(1 -b)E t + αF (k t , 1) + β(c t+1 0 + c t 1 ) -γm t
The FOCs of the maximization problem are

∂L ∂c t t = u (c t 0 ) (1 + R) t - µ t (1 + R) t + βη t-1 (1 + R) t-1 = 0 ∂L ∂c t 1 = v (c t 1 ) (1 + R) t - µ t+1 (1 + R) t+1 + βη t (1 + R) t = 0 ∂L ∂E t+1 = φ (E t+1 ) (1 + R) t + η t (1 + R) t - η t+1 (1 -b) (1 + R) t+1 = 0 ∂L ∂k t = - µ t (1 + R) t + µ t+1 F K (k t , 1) (1 + R) t+1 + η t αF K (k t , 1) (1 + R) t = 0

  r t+1 and c t+1,e 0

13

  t 0 and equation (48) into equation (49) we have Ē = I -[β(1 -b) + γ] T t 0 Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

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  Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

  φ (E t+1 ) H5 > 0

  1 , 1) + β(c t 0 + c t-1 1 ) -γm t-1 +λ 4 E t+1 -(1 -b)E t + αF (k t , 1) + β(c t+1,e

0 + c t 1 ) -γm t

whose bordered Hessian will appear as

[START_REF] John | An Overlapping Generations Model of Growth and the Environment[END_REF][START_REF] John | Short-lived agents and the long-lived environment[END_REF], only the consumption of old agents pollutes, young agents do not consume. In[START_REF] Ono | Optimal tax schemes and the environmental externality[END_REF], it is assumed that consumption of both young and old agents degrade the environment but with a period lag. Here, we assume also that consumptions of both old and young agents and production pollute without decay.

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

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Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

The discount rate R is strictly positive when the social planner cares less about a generation's welfare the further away in the future that generation is, while R equals to zero when she cares about all generations equally, no matter how far in the fuure they may be.

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Documents de Travail du Centre d'Economie de la Sorbonne -2010.104

Appendix

A1. Existence of the agent's optimal solution By substituting (10), ( 11), ( 12) and ( 13) into ( 8) and ( 9) the existence of solution to the system of the first order conditions (8)-( 13) is equivalent to the existence of solution to the system of two following equations

where

From ( 98), by implicit function theorem we can treat c t 1 as a function of c t 0 , c t 1 = ϕ(c t 0 ) where ϕ (•) > 0, ϕ(0) = 0 and ϕ(+∞) = +∞. We rewrite

where

0 + γw t . Now the system of equations ( 97) and (98) leads to the following equation

The existence of the agent's optimal solution is equivalent to the existence of solution to equation (99). In effect, set

is a continuous function of c t 0 . We have,

At the steady state,