Nicolas Quérou
email: nicolas.querou@umontpellier.fr

Agnes Tomini
email: agnes.tomini@univ-amu.fr

Christopher Costello
email: costello@bren.ucsb.edu

Limited-tenure concessions for collective goods *

Keywords: Concessions, public goods, cooperation, natural resources, spatial externalities, dynamic games JEL classification: C7, D62, H00, Q20

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Introduction

The impediments to private provision of public goods, carefully described over half a century ago [START_REF] Samuelson | The pure theory of public expenditure[END_REF][START_REF] Buchanan | An economic theory of clubs[END_REF], remain pervasive and economically relevant today [START_REF] Kotchen | Green markets and private provision of public goods[END_REF][START_REF] Marx | Dynamic voluntary contribution to a public project[END_REF][START_REF] Cornes | The Theory of Externalities, Public Goods, and Club Goods[END_REF]. While governments can (and do) provide these goods, there is a perpetual quest to develop institutions and incentives that give rise to efficient private provision. In this paper we formalize and analyze an institution called a "limited-tenure concession" for its ability to induce the efficient private provision of public goods.

While we will analyze a general model of public goods, we are motivated by the tragedy of the commons, a particular public goods setting in which natural resources are over-extracted by agents who fail to internalize the consequences of their extraction on others' payoffs. In this way, agents' over-extraction can be viewed as under-provision of the public good. Even today many natural resources including forests, fisheries, and irrigation water are over-extracted and are thus inefficiently provided. 1 One increasingly common approach is to devolve ownership of these resources to individuals, communities or cooperatives; the idea being that this assignment of property rights creates a sole-owner-like incentive to steward the resource. But even in that setting, because these resources often move in space, one owners' extraction affects other owners' future payoffs, and the externality persists. In other words, while perfectly delineated spatial property rights almost certainly improves upon open access, it cannot solve the tragedy of the commons for spatially-connected natural resources.

We find that over-extraction of natural resources in a common pool is a particular public goods setting that is ripe for limited-tenure concessions. Here, a concession is a limited-duration assignment of property rights, under which the temporary owner is completely autonomous and can behave in any manner she sees fit -that is, she is free to extract as much or as little as she wishes over the duration of her tenure within her concession. In most real-world settings, trade in concession contracts is prohibited because the extracted resource is often held in public trust. However, allowing trade in concession contracts is innocuous so long as the buyer is bound by the same terms as was the seller. Trade in the underlying property right (for example via land markets or fishing area markets) may also be allowed in this setup, so long as a single owner cannot consolidate all of the properties and therefore internalize all spatial externalities. Full consolidation is unlikely, for example, in developing countries where financial constraints may bind. Within this patchy spatial setting, two parameters of a concession contract will turn out to be pivotal for our analysis. First, the duration of tenure plays an important role in incentives and can make, or break, the ability of the concession to induce efficient provision of the public good. Second, we allow for the possibility of renewal of the concession contract, provided that certain conditions have been met. We will show that the enticement of renewal can induce efficient private provision of a public good, even from completely selfinterested parties.

Limited-tenure concessions are also applicable across a much broader set of public goods, whose efficient provision is hindered by free-riding incentives. [START_REF] Bergstrom | On the private provision of public goods[END_REF] provides the seminal paper on this issue, which has received attention in many areas, including environmental applications [START_REF] Vicary | Donations to a public good in a large economy[END_REF][START_REF] Kotchen | Green markets and private provision of public goods[END_REF][START_REF] Kotchen | Voluntary provision of public goods for bads: A theory of environmental offsets[END_REF]). In the general public goods setting, a concession can be thought of as a limited-duration assignment of a property right to a temporary owner. Over her tenure she enjoys all the benefits of the public good and she may also decide to contribute to the public good. At the conclusion of her tenure there is the possibility of renewal, which is contingent on her private contributions over the preceding tenure block. If her tenure is not renewed, then she is excluded from enjoying the future benefits of the public good. It is intuitive to see how the ability to exclude the player from enjoying the future benefits of the public good could induce private provision in the present. We begin the analysis with a simple, stylized repeated public goods contribution game. That simple analysis highlights the important tradeoffs and incentives engendered by a limited-tenure concession.

After illustrating the principal incentives of this intervention in a stylized setting, we turn to a more substantial application to common pool natural resources. Natural resources generalize the simple case in important dimensions including natural resource growth (i.e. a production function in which next year's resource stock depends on this year's resource stock), mobility (water flows and fish swim), and heterogeneity (e.g. growth or movement can differ over space) in incentives across users. These features may exacerbate the tragedy of the commons (see [START_REF] Cornes | On commons and tragedies[END_REF]) and we examine whether, and how, these features undermine the ability of limited-tenure concessions to induce efficient private provision. Our analysis also shares similarities with [START_REF] Copeland | Trade, tragedy, and the commons[END_REF], who model resource extraction under incomplete monitoring and enforcement. They embrace the idea that countries differ in their ability to monitor and enforce regulations, and they show how this heterogeneity gives rise to different predictions about natural resource overexploitation, even when all countries are well-intentioned in their regulations. Their main objective is to show how trade, technology, and population growth affect resource dynamics when regulatory efficacy is endogenous. While our model can in some ways be viewed as an application of theirs, rather than explicitly modeling ineffective regulations, we ask whether a particular instrument (the limited-tenure concession) can be designed to incentivize the same harvesting behavior as would be achieved by an infallible regulator. 2 In our model, since concessions are awarded over a fixed geographical area, the resources they are meant to encapsulate can disperse beyond the concessionaire's domain; this could significantly alter incentives for efficient resource use, since this implies a spatial externality. For instance, the world's oceans consist of about 200 property right assignments (exclusive economic zones) traversed by species such as tuna, sharks, and whales [START_REF] White | Close the high seas to fishing?[END_REF]. The mismatch between the scales of property rights and of the resource is emphasized as a limitation [START_REF] Aburto-Oropeza | Property rights for fishing cooperatives: How (and how well) do they work?[END_REF] for mobile resources [START_REF] Costello | Partial enclosure of the commons[END_REF] and [START_REF] Kapaun | Does the optimal size of a fish stock increase with environmental uncertainties?[END_REF]). We thus amend the model to account for these characteristics. Introducing a set of spatially-distinct property right owners, we consider three management regimes: (i) the socially optimal regime, (ii) the decentralized regime and (iii) the concession regime. The last regime involves assigning limited-duration tenure of each patch to a concessionaire, with conditional renewal. The grantor of the concession (which we call a "regulator") announces for each patch a minimum stock below which the concessionaire should never extract. This is a stylized version of how many concessions are implemented. For instance, TURF systems in Japan, Mexico and Chile contain maximum harvest provisions, whose adherence is required for renewal. As a yearly stock assessment is carried out by consultants approved by the government to determine a total allowable catch (TAC) for each TURF, such a requirement may translate into a minimum stock requirement [START_REF] Hilborn | Institutions, incentives and the future of fisheries[END_REF]). Wildlife management areas in developing countries rely on ownership devolution to local communities and also require coordination from governments [START_REF] Pailler | Impacts of community-based natural resource management on wealth, food security and child health in tanzania[END_REF]). Groundwater is increasingly managed by property rights, where an adjudication process relies on a watermaster to enforce the terms of the property right. Because groundwater migrates spatially according to geological features, the 2 Other features of the setup also distinguish our model from that in [START_REF] Copeland | Trade, tragedy, and the commons[END_REF]:

(1) Our resource dynamics allow for movement of the resource across space, which gives rise to spatial externalities, not just an externality of harvest; (2) We have explicit spatial property rights;

(3) Our concession instrument allows for ownership over a block of time, which alters incentives relative to the case where regulation occurs at every instant (as in [START_REF] Copeland | Trade, tragedy, and the commons[END_REF]); Finally (4) [START_REF] Copeland | Trade, tragedy, and the commons[END_REF] explicitly model a fine for non-compliance (as in traditional regulation). Instead, we do not impose a fine, but rather induce a sort of punishment through non-renewal of the concession contract. In practice, though, the explicit fine derived by [START_REF] Copeland | Trade, tragedy, and the commons[END_REF] is similar to our implicit penalty for defection because they both hinge on the difference between potential profit earned in the resource sector compared to an outside option.

groundwater management setting shares the basic features of our concession system (see [START_REF] Ayres | How transaction costs obstruct collective action: The case of california's groundwater[END_REF] for real-world cases). In all of these settings, each concessionaire must decide whether to comply or to defect, given that her payoff will depend on others' strategies. Complying guarantees renewal, which raises future payoffs, while driving the stock below the requirement returns large current payoffs, but ensures the contract will not be renewed. One special case of this model is when all agents have perpetual decentralized property rights, and we show that limited-tenure concessions outperform this oft-touted benchmark.

We show that limited-tenure concessions can induce the first best (that is, socially optimal) behavior in this setting, and analyze the properties ensuring cooperation. We find an interesting result: longer tenure is more likely to lead to defection from the first best. This result has crucial implications for policy design, and it seems to contradict the intuition that more secure property rights (here, the longer the tenure duration) give rise to more efficient resource use. Indeed, [START_REF] Costello | Natural resource use with limited-tenure property rights[END_REF] show that any tenure length may induce efficient resource use, provided the renewal probability is high enough. In our paper, under a long tenure period the regulator loses the ability to affect an agent's incentives via the promise of tenure renewal. Thus, for sufficiently long tenure length, concessionaires always defect: tenure must not be too long. Finally, we discuss how concessions may still induce first-best behavior even when monitoring and enforcement are imperfect and costly.

This discussion highlights the shortcomings of short tenure equal to, say, a single period, which is effectively traditional command and control regulation. With limited-tenure concessions, compliance is incentivized by the promise of renewal (a carrot), rather than punished with a monetary penalty (a stick, as in [START_REF] Copeland | Trade, tragedy, and the commons[END_REF]. Short tenure can induce efficient behavior, but would be costly to implement if more frequent monitoring brings higher costs. Thus, shorter tenure may induce stronger incentives to comply, but could increase the expected monitoring costs. We analytically solve for the tenure length ensuring compliance at least cost.

Overall this paper makes three primary contributions. First, we show in a repeated public good contribution game that limited-tenure concessions can induce efficient provision of public goods. Second, we extend the model to account for characteristics of common-pool resources more typical in natural resource settings: spatially-connected resources, and growth dynamics. We show that the limitedtenure concession can incentivize the first best. Finally, in a setting with costly monitoring, we discuss the features that ensure first-best behavior, but at least cost to implement. All results are analytically derived, allowing us to draw general conclusions.

The paper is structured as follows: Section 2 introduces a motivating model of the private contribution to a public good and highlights how a concession alters incentives for private provision. In Section 3 the model is generalized to allow for heterogeneity and complex resource dynamics. In Section 4 we highlight the conditions for cooperation with an emphasis on spatial characteristics of the model and the tenure length. Various extensions are discussed in Section 5. A comparison with other potential policies is provided in Section 6 and Section 7 concludes the paper. Proofs are provided in an Appendix.

A simple model of public good contributions

To motivate our main contribution and build intuition, we begin with a simple model of individual behavior with both private and public consequences. Consider initially a static game in which (exogenous) N agents interact, where each agent takes other agents' actions as given. Agent i chooses action (or "contribution") z i , which confers a public benefit but comes at a private cost. Her utility in this static game is given by:

W l z l + u i (Φ -z i ) (1)
Here, the function W (•) represents the public component of utility and u i (•) represents the private component. The fixed parameter Φ denotes an agent's endowment (or maximum effort level). Assuming that both functions are increasing and strictly concave together with W ((N -1)Φ) > u i (Φ) and W (Φ) < u i (0) (for all i) ensures an interior equilibrium. This setting is a version of the canonical public goods model popularized by [START_REF] Samuelson | The pure theory of public expenditure[END_REF]. We slightly depart from this canonical model by assuming that the public and private components of utility are bundled. This public good is also more aptly referred to as an excludable public good, because we restrict its consumption to a limited set of N agents. In this way our model departs from a pure public good setting (see for instance [START_REF] Wang | Non-monotonic group-size effect in repeated provision of public goods[END_REF]).

Here, increasing contribution z i comes at the cost of decreasing the private benefits u i (Φ -z i ). It is straightforward to show that agent i under-provides this excludable public good because she fails to consider the beneficial effect of a larger contribution on other players' utilities. That is, agent i maximizes Equation 1 by setting u i (Φẑi ) = W ( l ẑl ), while the social planner would like to maximize the sum of utility across all agents, so she sets

u i (Φ-z * i ) = N W ( l z * l )
. Straightforward comparative statics reveals that private agents will contribute too little: ẑi < z * i . Attempting to incentivize efficient provision, consider a new institution under which this game is repeated every period for a limited duration tenure. That is, each period, all N members contribute to, and enjoy the benefits from, public good provision, but any player i's tenure lasts only for a limited duration. For example, tenure may extend for a period of T = 10 years. A manager has the ability to renew tenure to agent i, and agrees to do so if and only if agent i has acted responsibly, for example if and only if she has chosen z * i in every preceding period (up to T ). This limited-duration tenure with the possibility of renewal is the focus of the rest of this paper, and in this section we use this simple setup to illustrate how this institution can induce efficient provision of the public good.

Clearly, the enticement of renewal induces a tension in agent i's decision about her contribution. On one hand, if she chooses to defect from the renewal rule stated by the manager (while all other agents choose to comply and thus collectively contribute l =i z * l ), she maximizes her payoff in any given period by choosing a contribution to the public good that is lower than the socially optimal level (z D i instead of z * i , where defection strategy

z D i is, implicitly, u i (Φ -z D i ) = W l =i z * l + z D i .
). She is permitted do so each period for the duration of her tenure (which lasts T periods). On the other hand, the revocation rule ensures that by doing so, she will obtain zero benefit after T periods. Instead, by contributing z * i in all T periods, she is ensured renewal for another tenure block. This tradeoff -of large current period benefits from defection vs. infinite, though lower, benefits from cooperation -is similar to the tradeoff in a Nash Reversion punishment strategy (see, e.g., [START_REF] Mas-Colell | Microeconomic Theory[END_REF]), except that: (1) the punishment happens at date T (not immediately upon defection), (2) the punishment payoff is zero (rather than the Nash equilibrium payoff), and (3) under this setup, other players besides i are not required to play Nash upon defection. Here, it is the limited-tenure institution that is designed to induce efficient contributions and punishment comes in the form of the failure to renew tenure. We refer to this institution as a "limited-tenure concession".

We now sketch why this type of concession contract can be designed to maintain cooperation around z * i , and that there is a Folk-theorem-like result that ensures cooperation [START_REF] Mailath | Repeated games and reputations: long-run relationships[END_REF], and see [START_REF] Dutta | A folk theorem for stochastic games[END_REF] for a generalization to stochastic games). If other agents comply, how will i respond? In that case, agent i's infinite horizon cooperation payoff is given by:

Π C i = u i (Φ -z * i ) + W ( l z * l ) 1 -δ (2)
where δ is the discount factor. Instead, if agent i defects, it can be shown that she will do so in the first tenure block, so her defection payoff is:

Π D = 1 -δ T +1 1 -δ   u i (Φ -z D i ) + W   l =i z * l + z D i     + 0 (3)
which is just the defection payoff for a total of T periods and zero thereafter. The defection payoff is zero thereafter because she is excluded from all public and private components. In this simple situation the agent compares Π C i ≶ Π D i . Straightforward algebraic manipulation implies that a necessary and sufficient condition ensuring that the limited-tenure instrument induces the first-best outcome as an equilibrium is the following, for any i ∈ I:

δ T +1 > u i (Φ -z D i ) + W ( l =i z * l + z D i ) -(u i (Φ -z * i ) + W ( l z * l )) u i (Φ -z D i ) + W ( l =i z * l + z D i ) (4)
The right hand side is the percentage loss in single-period utility to agent i from cooperating, rather than defecting. If the discount factor is sufficiently large, so agents are sufficiently patient, then cooperation will always be supported as an equilibrium outcome. Notice that, depending on the fundamentals, the actual value of the bound defined in condition 4 might not be very high. One interesting consequence of Condition 4 is that longer tenure blocks (i.e. larger T ) require higher discount factors (i.e. lower discount rates) to sustain cooperation -sustaining cooperation under a long tenure period requires more patience on the part of the agents. The model used here is kept simple to describe the main result and intuition in the simplest way, though extensions are possible. 3Even the simple repeated game presented here provides some useful and interesting insights about the ability of a limited-tenure concession to induce socially optimal provision of a public good. Versions of limited-tenure concessions are employed commonly in real-world settings in which club members are expected to regularly contribute to an excludable public good. Most municipally-or publicly-owned social clubs require contributions of time, money, and expertise to sustain membership and enjoy club benefits. Formal charters ensure that the failure to contribute leads to exclusion from the club's future benefits. They might be related to different activities (culture, education, sport) and are usually managed by a (non-member) representative acting on behalf of the municipality or public body. In natural resources, some clubs come in the form of international environmental agreements. For example, to govern fisheries on the high seas, regional fishery management organizations function as clubs where member countries contribute resources and data to the group, and agree to uphold certain sustainable practices. Failure to meet those requirements leads to future exclusion of public goods such as scientific information and management coordination. Governance (and thus enforcement) is still externally managed: indeed, the agents here are the fishers, while the countries' representatives (in charge of governance and enforcement) are not fishers but government delegates. Thus, the limited-tenure concession concept may even be applicable to international settings, provided there is a mechanism to exclude defecting members from public goods.

This paper is motivated by a class of public goods challenges that has historically led to the tragedy of the commons. We are interested in whether a limited-tenure concession can help reverse over-extraction incentives for complex, spatially-connected natural resources, so the simple model presented above will require some elaboration. In what follows, we maintain the basic idea behind this simple model, but allow for a sophisticated array of economic and ecological interactions including spatially owned natural resource patches, natural resource growth and dispersal across space: we thus move from a repeated game to a spatially dynamic game setting. Modeling this richer environment allows us to draw conclusions about the features of a natural resource setting in which limited-tenure concessions can be designed to achieve socially efficient outcomes.

Model & strategies

Here we introduce a model of natural resource exploitation with spatially-connected property owners. 4 We then home-in on the incentives for harvest strategies corresponding to three property right regimes: a social planner optimizing resource extraction over space and time; decentralized perpetual property right holders; the case of decentralized limited-tenure concessions, on which we focus. The social planner's benchmark and the case of perpetual property right holders have been analyzed previously in the literature: we briefly summarize the most important results for our analysis.

The model

We follow the basic setup of [START_REF] Costello | Partial enclosure of the commons[END_REF] where a natural resource stock is distributed heterogeneously across a discrete spatial domain consisting of N patches or properties. Patches may be heterogeneous in size, shape, economic, and environmental characteristics, and resource extraction can occur in each patch. Using a discrete-time model, the stock residing in property i at the beginning of time period t is given by x it , and harvest undertaken in that property, h it , reduces the stock over the course of that time period. This leaves a "residual stock" at the end of the period of e it ≡ x it -h it . The residual stock may grow, and the growth conditions may be patch-specific denoted by the parameter α i . Finally, as the resource is mobile and can migrate around this system, we follow the natural science literature (see, e.g., [START_REF] Nathan | Mechanisms of long-distance dispersal of seeds by wind[END_REF], or [START_REF] Siegel | Lagrangian descriptions of marine larval dispersion[END_REF]) who denote dispersion by D ij ≥ 0 the fraction of the resource stock in patch i that migrates to patch j in a single time period. The parameter D ii denotes the fraction of the resource stock in patch i that remains in i (i.e. self-dispersal). This model assumes density-independent dispersal parameters, D ij , which allows us to analyze the comparative statics of dispersal on cooperation vs. defection incentives. In Section 5.4 we consider the case of densitydependent dispersal (as in [START_REF] Sanchirico | Optimal spatial management of renewable resources: matching policy scope to ecosystem scale[END_REF]). Since some fraction of the resource may indeed flow out of the system entirely, the dispersal fractions need not sum to one: i D ji ≤ 1. The equation of motion in patch i is thus given as follows:

x it+1 = N j=1 D ji g(e jt , α j ).
(5)

Here g(e jt , α j ) is the period-t resource growth in patch j. As usual we require that ∂g(e,α) ∂e > 0, ∂g(e,α) ∂α > 0, ∂ 2 g(e,α) ∂e 2 < 0, and ∂ 2 g(e,α) ∂e∂α > 0. These assumptions must be satisfied within the relevant range of variable e. The logistic growth function, among many others, is consistent with them. We assume that extinction is absorbing, g(0; α j ) = 0, and that the growth rate is finite, ∂g(e,α) ∂e | e=0 < ∞. For expositional ease, we suppress the growth-related parameter except briefly in Section 4.2, where its effect will be analyzed. Thus, we will use the notation g i (e) and g i (e) instead of (respectively) ∂g(e,α i ) ∂e and ∂ 2 g(e,α i ) ∂e 2 in most parts of the paper. All standard biological production functions are special cases of g(e, α).

We assume that both price and marginal harvest cost are constant in a patch, though they can differ across patches. The resulting net price is given by p i . This assumption is fairly common in resource models and is consistent with cases where the market price is the same in all patches, while marginal costs are patch-specific (e.g. due to different costs of access from different locations). For many natural resources the number of implemented concession systems is large so production from any one concession system will have negligible effects on price. The current profit from harvesting h it ≡ x it -e it in patch i at time t is:

Π it = p i (x it -e it ) .
(6)

We will employ this framework to compare the outcome and welfare implications of three alternative property right systems. At this stage it is important to make the following observation. Real world natural resource management is more complex than the setting depicted here. For instance, there could be more complicated cost structures. We propose a relatively simple, analytically tractable model to gain insights on the performance of our concession instrument, while keeping the most relevant features when studying this issue. This model allows for dynamic and spatial externalities, and for strategic behavior between patch owners: we thus consider a spatial dynamic game, instead of a repeated game as in Section 2. It allows us to derive insights on the effects of ecological and economic fundamentals and of features of the instrument (tenure length and target stock requirements) on its performance. We will derive closed form expressions of the owners' optimal payoffs when committing to the instrument, and when following their best defection strategies. This is necessary to analytically assess the performance of the instrument. We discuss several extensions later in the paper, including cases where instantaneous payoffs are not constant.

Social Planner's Problem

Our benchmark is the case of the social planner who maximizes the net present value of profit across the entire domain given the discount factor δ. Her objective is:

max {e 1t ,...,e N t } ∞ t=0 N i=1 δ t p i (x it -e it ) , (7)
subject to the equation of motion (5) for each patch i = 1, 2, ..., N . Focusing on interior solutions, in any patch i, the planner should achieve the residual stock level:

g i (e * it ) = p i δ j D ij p j (8)
The optimal residual stock results from the trade-off between the present profits from harvest and the discounted sum of future benefits given growth and dispersal to all patches. Note, by inspection, that these optimal residual stock levels are time and state independent. Thus, each patch has a single optimal residual stock level that should be achieved every period into perpetuity satisfying, for any period t:

e * it = e * i . (9)
Since biological growth, dispersal, and economic returns are patch-specific, the optimal policy will vary across patches. Equation 8emphasizes that this policy depends on patch-specific net prices, growth, and dispersal and self-retention parameters. We focus on policies with an interior solution -that is, those that are consistent with the sustainable management of the resource. This emphasizes the importance of ecological and economic fundamentals on the performance of the instrument, and is formally equivalent to assuming that g i (0) > 0) and initial stock level x i0 lie above minimum threshold values. In Section 3.1.2 we characterize the decentralized property rights case, and show that the resulting extraction levels are higher than the socially optimal levels: thus, in the decentralized reference setting, the resource is over-exploited. We then show that limited-tenure concessions can overcome this over-exploitation problem.

p i δ j D ij p j and x i0 > (g i ) -1 p i δ j D ij p j , that is, marginal growth g i (

Decentralized Perpetual Property Right Holders

The second regime is the case in which each patch is owned in perpetuity by a different owner who seeks to maximize the net economic value of harvest from his patch, with complete information about the stock, growth characteristics, and economic conditions present throughout the system.5 In that case owner i solves:

max {e it } ∞ t=0 δ t p i (x it -e it ) .
(10) subject to the equation of motion (5). Following Lemma 1 in [START_REF] Kaffine | Unitization of spatially connected renewable resources[END_REF], in this uncoordinated benchmark setting, owner i will always harvest down to a residual stock level ēit that satisfies:

g i (ē it ) = 1 δD ii . ( 11
)
We assume that this corresponds to an interior equilibrium outcome, requiring that g i (0) > 1 δD ii and x i0 > (g i ) -1 1 δD ii . Thus, marginal growth g i (0) lies above a minimum threshold value. This keeps the exposition as simple as possible, but our instrument can address more general cases. Moreover, as efficient policies are interior (see Section 3.1.1), the condition on the initial stock level is satisfied.

At the equilibrium outcome, the owner takes other owners' behaviors as given and realizes that he will not be the residual claimant of any conservative harvesting behavior. Thus, he behaves as if any additional resource that disperses out of his patch will be lost (indeed it will be harvested by his competitors). This is why the only dispersal term to enter the optimal residual stock term is D ii , the fraction of the resource that remains in his patch. We have ēit ≤ e * it (with strict inequality as long as D ii = 1): achieving social efficiency requires some kind of intervention or cooperation. Moreover, Equation ( 11) implies that ēit = ēi for any time period. [START_REF] Kaffine | Unitization of spatially connected renewable resources[END_REF] show that the open loop and feedback control rules are identical in this setting.

In our specification of decentralized property rights, we implicitly assume no trade in spatial property rights. This assumption accords with many real-world cases in which concessions are used: For example, in fisheries, temporary spatial concessions (often 20-30 years with possible renewal) are typically granted to communities or cooperatives and are non-transferable. Yet, some elaboration is warranted. With spatially-connected resources, extraction in one location imposes an externality on other locations, and tends to result in over-extraction over the entire area, as we have just shown. Many of these resources, such as fisheries, water, and game, are traditionally held in public trust. One possible solution to this spatial-dynamic externality market failure would be to allocate spatial property rights in perpetuity and to allow the trade of those rights. If trade were allowed, but consolidation of those rights was not, then the inefficiency result would maintain -ownership of an area may change hands, but this would not dilute the incentive to overextract. However, if consolidation is allowed, and if one owner was to buy up the entire spatial domain, she could, in principle, internalize all of these spatial externalities. But governments are often unwilling to relinquish all control of these publicly-held resources in perpetuity. The approach we are going to study, limited-tenure concessions, honors the public trust idea without granting private ownership in perpetuity, but still provides incentives for the concessionaires to extract at a socially-optimal rate. For the remainder of this analysis we thus focus on the case in which consolidation is not allowed, so each property is managed by a different owner.

Unlike in Section 2, the game setting here is dynamic, owing to the stock growth over time. This raises several conceptual differences from a standard repeated game. First, the usual equilibrium concept for dynamic games is Markov perfect nash equilibrium (MPNE): there is no implicit assumption of agents' binding commitment about actions they will take at future dates [START_REF] Reinganum | Oligopoly extraction of a common property natural resource: the importance of the period of commitment in dynamic games[END_REF] and the outcome is subgame perfect. Since the perpetual decentralized property rights setting can be considered a wholly uncoordinated setting, it makes sense to consider that the benchmark assumes no binding commitment and, as such, that the appropriate equilibrium concept in the benchmark is MPNE. Second, the trigger strategy equi-librium (another solution concept) is often criticized as it implicitly assumes that non-deviating agents can credibly commit to punish a deviating agent. This also implies that these agents would also punish themselves by using trigger strategies. But this is often regarded as not credible because the trigger strategy equilibrium is not renegotiation-proof [START_REF] Heitzig | Self-enforcing strategies to deter free-riding in the climate change mitigation game and other repeated public good games[END_REF]. This likely also rules out the use of financial transfers between agents. Such transfers, at least at the level that would be required, might also be infeasible in the presence of credit constraints.

Decentralized, Limited-Tenure Property Rights

In the final regime, and the one on which we focus in this paper, we assume that ownership over patch i is granted to a private concessionaire for a duration of T i periods, to which we will refer as the "tenure block" for the concession. All concessionaires have the possibility of renewal provided that certain conditions are met. Indeed, it is the possibility of renewal that will ultimately incentivize the concessionaire to deviate from her (excessively high) privately-optimal harvest rate; we will leverage this fact to design concession contracts to induce efficient outcomes. 6 We begin by defining an arbitrary set of instrument parameters.

Definition 1. The Limited-Tenure Concession Instrument is defined by, for any concessionaire i: a per-period "target stock," S i , a tenure period, T i , and a renewal probability 0 ≤ f i ≤ 1 which is the scalar probability of renewal conditional upon meeting the announced target stock.

The concessionaire is allowed to extract as much of the resource as she wishes over her tenure block, and the regulator imposes only one rule on the concessionaire: At time T i -1 (since the block starts at t = 0) the concession will be renewed (under terms identical to those of the first tenure block) with probability f i if and only if the stock is maintained at or above the target stock (S i ) in every period. So, concession i will be renewed with probability f i if and only if:

e it ≥ S i ∀t ≤ T i -1. ( 12
)
The renewal requirement is defined with respect to the stock level at the end of any given time period: the residual stock level in patch i at time period t (e it ) must lie above the target stock S i . We allow for this instrument to be explicitly spatial (S i = S j ). If the announced target stock S i is not met then the probability of renewal is zero. In that event, to whom will the subsequent concession be granted?

We assume that it is granted to a new concessionaire who is not currently part of the game. It is intuitive, and we will later show formally, that being granted a concession confers significant value (this is true whether the concessionaire expects to comply or to defect), which raises the possibility of selling the concession right. The simplest approach would be to allocate the concession for free, for example via a lottery. Since in our model all concessionaires are equally efficient at harvesting the resource, this would entail no change in incentives, though rent seeking could arise if lobbying played a role in allocation. Another approach would be to auction the concession to the highest bidder. Since the bid is a lump sum, paid upon receipt of the concession, it can be viewed as a fixed cost, which again would have no marginal extraction incentives. Because neither of these approaches affects extraction incentives, for the remainder of the paper we remain agnostic about to whom a new concession is allocated.

Beyond the enforcement of the concession contract, the regulator plays no role: all harvest decisions are made privately by the concessionaire. Because the regulator would like to replicate the social planner's solution (see Section 3.1.1), she must determine a set of target stocks in each area {S 1 , S 2 , ..., S N }, tenure lengths {T 1 , T 2 , ...T N } and renewal probabilities {f 1 , f 2 , ...f N } (i.e., a {S i , T i , f i } triple to offer concessionaire i) that will incentivize all concessionaires to simultaneously, and in every period, deliver the efficient level of harvest in all patches. We will restrict attention to tenure lengths satisfying T i = T , ∀i. In principle, though, since concessionaires are heterogeneous, tenure lengths could be heterogeneous. In order to limit complexity of the model, and because the use of a uniform tenure length seems to be the norm for real-world cases of concessions-regulated resources, we consider the longest tenure that is compatible with concessionaires' incentives to cooperate (see expression 18 in Section 4).

We will show that, if designed properly, limited-tenure concessions can be used to induce concessionaires to manage resources in an efficient manner. Agents may, or may not, comply with the terms of the concession contract. If all N concessionaires choose to comply with the target stocks in every period of every tenure block, we refer to this as compliance. All owners will then earn an income stream in perpetuity (though in principle this could eventually end if f i < 1). Instead, if a particular owner i fails to meet the target stock requirement (i.e, in some period she harvests the stock below S i ), then, while she will retain ownership for the remainder of her tenure block (and thus be able to choose any harvest over that period), she will certainly not have her tenure renewed. In that case, owner i's payoff will be zero every period after her current tenure block expires. Thus, the instrument raises a trade-off for each concessionaire who chooses whether to comply or to defect. Since an owner's payoff depends on others' actions, we assume that if concessionaire i defects, then the concession is granted to a new concessionaire in the subsequent tenure block. If all initial owners decide to defect and are not renewed at the end of the current tenure, then the game ends. In our setup, this rule turns out to be moot because if everyone defects, the resource is driven extinct.

Compliance vs. Defection

How will a concessionaire decide whether to comply with the terms of the concession or to defect? We begin by characterizing the payoffs that each concessionaire achieves under compliance, and the best defection strategy, should they choose to defect. We first consider the case when all N concessionaires comply with the target stocks in every period of every tenure block. Provided they do not exceed the target stock then concessionaire i's expected payoff is:

Π c i = p i   x i0 -S i + T -1 t=1 δ t (x * i -S i ) + ∞ l=1 (f i ) l   (l+1)T -1 t=lT δ t (x * i -S i )     . ( 13
)
where x i0 is the (given) starting stock and x * i = j D ji g(S j ). Because it is fundamental to the externality we examine, the role of resource migration deserves a few remarks. First, whether a player can achieve e it ≥ S i depends on what the other players do. Of course, in (at least pure strategy) equilibrium, agents rationally expect others to follow the equilibrium strategy, and this dependence is not an issue. Second, a deviating agent considers only D ii . This is because, agent i knows that agent j will make sure that e jt ≥ S j regardless of the amount migrating from patch i to patch j. Hence, no matter what agent i does, the amount migrating from patch j to patch i remains at D ji g(S j ).

We next turn to the characterization of the concessionaires' best defection strategies. If concessionaire i defects during an arbitrary tenure block k and all other concessionaires follow their equilibrium strategies (that is, they comply), we have: Proposition 1.

1. First assume that

p i δ j D ij p j < g i (0) ≤ 1 δD ii .
Then the best defection strategy of concessionaire i in tenure block k is given by ēit = 0 for any period (k -1)T ≤ t ≤ kT -1.

Second, assume that g

i (0) > 1 δD ii .
Then the best defection strategy of concessionaire i in tenure block k is characterized by ēikT-1 = 0 and, for any period (k -1)T ≤ t ≤ kT -2, we have ēit = ēi > 0 where:

g i (ē i ) = 1 δD ii with xi = D ii g(ē i ) + j =i D ji g(S j ) > ēi .
When marginal growth g i (0) is sufficiently low in area i, a concessionaire who decides to defect sometime during tenure block k, will completely mine the resource in his patch at every period of the tenure block. By contrast, when marginal growth is high enough, this defecting concessionaire will (1) choose the non-cooperative level of harvest (see Section 3.1.2) up until the final period of the tenure block and (2) then completely mine the resource. Note that if only one concessionaire defects, the entire stock will not (in general) be driven extinct because patch i can be restocked via dispersal from patches with owners who cooperated. Either way, the resource is completely mined in that patch by the end of the tenure block. The best defection strategy depends neither on the tenure block, k, nor on the renewal probabilities, and so we have:

Lemma 1. Suppose the regulator can choose the renewal probabilities to maximize the likelihood of compliance. Then she chooses f i = 1 for any concessionaire i.

Lemma 1 simply says that the regulator designs the concession such that compliance guarantees renewal. Combined with Proposition 1, this feature simplifies the characterization of equilibrium strategies. We thus consider now that f i = 1 for any concessionaire i. The present value of owner i's defection payoffs is:

Π d i = p i   x i0 -S i + (k-1)T -1 t=1 δ t (x * i -S i ) + δ (k-1)T (x * i -ēi ) + kT -2 t=(k-1)T +1 δ t (x i -ēi ) + δ kT -1 xi   (14
) Patch owner i's defection payoff during tenure block k is given by (1) the profit obtained while abiding by the target stock prior to the k th tenure block (first two terms on the right-hand side of ( 14)), and (2) the profit from non-cooperative harvesting during tenure block k (the third and fourth terms on the RHS of ( 14)), until finally extracting all the stock in the final period of the k th tenure block, kT -1 (the fifth and final term on the RHS of ( 14)). We will extensively use the defection strategy in what follows. We next turn to the conditions ensuring compliance.

Conditions for Cooperation

Here we derive the conditions under which all concessionaires willingly choose to comply in perpetuity. First, we derive the target stocks (S 1 , ..., S N ) that must be announced by the regulator to replicate the efficient level of extraction in every patch at every time, and we derive necessary and sufficient conditions for cooperation to be sustained. Henceforth, when all concessionaires choose to comply, we will refer to this as cooperation, since all agents choose extraction levels corresponding to the cooperative (that is, socially efficient) outcome. Second we discuss the effects of the patch-level parameters. Finally, we assess the influence of the tenure duration T on the emergence of cooperation.

The emergence of cooperation

Our interest here is to design the concession instrument to replicate the sociallyoptimal harvest in each patch at every time. We first prove that the regulator must announce, as a patch-i target stock, the socially-optimal residual stock for that patch.

Lemma 2. A necessary condition for social optimality is that the regulator announces:

S 1 = e * 1 , S 2 = e * 2 ,..., S N = e * N
, where e * i is given in Equation 8.

Lemma 2 relies on two main results from above. First, because ēi ≤ e * i , if the regulator announces any S i < e * i , then the concessionaire will optimally drive the stock below e * i , which is not socially optimal. Second, if the regulator sets a high target, so S i > e * i , then the concessionaire either complies with the target (and the stock is inefficiently high) or defects and reaches an inefficiently low target stock. Thus, Lemma 2 provides the target stocks that must be announced. Thus, we can now restrict attention to the target stocks S i = e * i ∀i: concessionaire i's compliance requires that e it ≥ e * i ∀t, so she must never harvest below that level. We show that, while concessionaire i is free to choose e it > e * i she will never do so.

Proposition 2. If concessionaire i chooses to cooperate, she sets e it = e * i ∀i, t.

Proposition 2 establishes that, if it can be achieved, cooperation involves each concessionaire leaving precisely the socially-optimal residual stock in each period. We proceed as follows. We characterize the conditions ensuring that any given concessionaire i lacks incentives to defect from the strategy characterized by Proposition 2 when all other concessionaires follow this strategy. These conditions ensure that the socially optimal outcome constitutes an equilibrium outcome. In any tenure block, the decision facing concessionaire i is whether or not to comply with the target stock requirement in each period. When all other concessionaires follow the strategy characterized by Proposition 2, one simply calculates her payoff from the best defection strategy (see Proposition 1) and compares it to her payoff from the cooperation strategy. We define concessionaire i's willingness-to-cooperate by:

W i ≡ Π c i -Π d i . ( 15
)
To be consistent with our equilibrium concept, we compute W i assuming that all other agents follow their equilibrium strategies. It can be shown7 that concessionaire i's incentives to cooperate are lower if her neighbor defects. This result arises due to stock dynamics because defection entails driving the stock to a low level, which takes time to rebuild. Reminiscent of Folk-Theorem results in repeated games (see [START_REF] Mailath | Repeated games and reputations: long-run relationships[END_REF]), each concessionaire must trade off between a mining effect, in which she achieves high short-run payoffs from defection during the current tenure block, and a renewal effect, in which she abides by the regulator's announced target stock, and thus receives lower short-run payoff, but ensures renewal in perpetuity.

Proposition 3. Cooperation emerges as an equilibrium outcome if and only if, for any concessionaire i, the following condition holds:

δx * i -e * i > 1 -δ T -1 (δx i -ēi ) . ( 16
)
Condition 16 is the analog to Condition 4, which was derived in the simple case of private provision of public goods. Proposition 3 shows that the gains from cooperation to concessionaire i (δx * i -e * i ) must be large enough compared to defection gains (δx i -ēi ) to sustain cooperation forever. The proof of Proposition 1 relies on the fact that defection entails at least some harvest (there are no corner solutions). Consider that concessionaires are patient, thus the discount factor δ is high: The right-hand side of Condition 16 gets close to zero, and the left-hand side to x * i -e * i , so as long as the solution to the optimal spatial problem is interior, the condition holds. By contrast, when concessionaires are impatient (the discount factor gets close to zero), as e * i > ēi , cooperation never arises. These cases are just examples: there are cases (depending on spatial parameters) where Condition 16 holds without assuming sufficiently patient concessionaires.

While the stock dynamics make our model more complicated than a repeated game, insights from the repeated game literature can still provide intuition for our results. Here, setting T i = 1 relaxes the incentive constraint the most. This is comparable to the perfect monitoring setup in repeated games, so the optimal penal code applies [START_REF] Abreu | On the theory of infinitely repeated games with discounting[END_REF]. Hence, as soon as one concessionaire deviates, she should be punished as severely as possible. In the current setup, the most severe punishment is to kick the concessionaire out of the game for the rest of time.

We show that our concession instrument can lead to efficient extraction across space and time in perpetuity.8 But this relies on a relatively strict enforcement system (an owner who defects is not renewed). Because the welfare gains from cooperation vs. non-cooperation are potentially large, less stringent systems might also lead to efficient behavior. Yet, the renewal process adopted here is consistent with the main characteristics of real-world cases of concessions-regulated resources. Our analysis highlights that, even without accounting for additional incentives (financial penalties), limited-tenure concessions have attractive practical appeal. Indeed, financial penalties may be infeasible in some settings, where financial or bankruptcy constraints prevail. One practical advantage of the concession instrument is that it uses financial incentives, rather than financial punishments, to achieve cooperation, and so it sidesteps potential complications of credit constraints or bankruptcy.

How does limited-tenure compare to alternatives that rely on self-punishment? The trigger strategy equilibrium implicitly assumes that non-deviating agents can credibly commit to punish a deviating agent, and that these agents will also punish themselves by using trigger strategies. But this might be non-credible, as a trigger strategy equilibrium is not renegotiation-proof. Under our proposed instrument, the non-deviating agents are not required to punish themselves; that is, rather than being punished by other players, a deviating player is simply kicked out of the game. Moreover, we can show that there is a set of tenure durations for which a wider set of conditions ensure that the social optimum is implemented as an equilibrium outcome under our instrument (compared to under the use of trigger strategies, formal details are available upon request).

While our purpose is not to exhaustively compare concessions to self-enforced systems, it is helpful to examine the conditions for a successful self enforced system that are made apparent by the Folk Theorem. These include: (1) agents must be sufficiently patient, (2) a self punishment system must be feasible, and (3) upon defection, the non-defecting agents must agree to revert permanently to non-cooperative extraction strategies, yielding potentially low payoffs to themselves. A self enforced system is thus unlikely to be appropriate if any of these conditions fails (see e.g. [START_REF] Engelmann | In the long run we are all dead: On the benefits of peer punishment in rich environments[END_REF] when there is a fear of counter-punishment and [START_REF] Heitzig | Self-enforcing strategies to deter free-riding in the climate change mitigation game and other repeated public good games[END_REF] for a discussion of non-credible systems). Self enforced systems will thus be appropriate in some settings, while limited-tenure concessions, which has different characteristics, will be appropriate in other settings. This implies that there are two ways to induce Folk Theorem like cooperative outcomes. Self enforced systems may do so by insisting on perpetual punishment by all parties inflicted on all parties upon defection of a single party; applicability of this approach may be limited by credibility or feasibility issues. Instead, concessions induce cooperation by wielding the incentive of renewal but require external monitoring; applicability of this approach may be limited by higher transaction costs.

We conclude this section by discussing some salient cases that can be addressed by our instrument. First, the polar case where social efficiency requires e * it = 0 ∀t ≥ 0 (that is, harvesting the entire stock) in some patches can be addressed by our instrument: the marginal incentives for these patches in the decentralized situation also correspond to this case. Second, the other polar case, where e * it = x it ∀t ≥ 0 (where it is optimal to forbid extraction) in at least one patch i, cannot be addressed by our instrument or by any concession instrument. To implement the socially optimal path, this would require combining our instrument with a sidepayment scheme. However, in cases where there exists a time period t 0 such that e * it = x it ∀t ≤ t 0 and e * it < x it thereafter for any patch i, one can design a concession instrument inducing the socially optimal path starting at t = t 0 + 1. It would be defined as: ∀t ≤ t 0 we have S it = Si < x it (where this target level is characterized depending on the fundamentals of the setting) and ∀t ≥ t 0 + 1 we have S it = e * i .

Effects of Patch-Level Characteristics

Patch-level characteristics affect concessionaires' payoffs and therefore play a role in the decision of whether to defect or cooperate. The fact that these characteristics may also affect the announced target stocks further complicates the analysis. We next examine the effects of price, growth, and dispersal on the concessionaire i's willingness-to-cooperate, defined by Condition (15). As a parameter changes, we trace its effects through the entire system, including how it alters others' decisions.

Assuming that the willingness to cooperate is initially positive, the impact of prices {p i , p j } is as follows: Concessionaire i's willingness-to-cooperate, W i , is increasing in p i , but is ambiguous in the price of the adjacent area, p j , and depends on the degree of the connection between patches. The effect of productivity of connected patches is also nuanced. Agent i will have higher incentives to cooperate with a higher growth rate of the adjacent property, α j . Since defection implies harvesting one's entire stock, there is little opportunity (under defection) to take advantage of one's neighbor's high productivity. But under cooperation, a larger α j implies larger immigration, which translates into higher profit. The impact of own growth (α i ) is negative when the self-retention rate, D ii , is small, and is positive for sufficiently large D ii . The direct impact on the residual stock in patch i offsets all other impacts, but as a small proportion of the resource stays in the area; this decreases the gains from cooperation.

Finally, we provide cases in the Appendix where the cooperation incentives are increasing in self-retention, D ii , but its impact is mixed as it affects the resource stock under defection and cooperation. W i is increasing in D ji for reasons similar to those driving comparative statics on α j . In contrast, a higher emigration rate (D ij ) reduces the incentives to cooperate: defection incentives are not altered much (since concessionaire i harvests the entire stock under defection), but cooperation incentives are reduced because the regulator will instruct concessionaire i to reduce her harvest under a larger D ij .

Table 1: Effect of patch-specific parameters on willingness-to-cooperate.

θ p i p j α i α j D ii D ij D ji ∂W i ∂θ + +/-+/-+ +/-- +
These results provide insight about how the strength of i's cooperation incentive depends on parameters. Whether this incentive is sufficiently strong to induce cooperation (i.e. whether W i > 0) remains to be seen. We focus on resource dispersal. If the resource was immobile, the patches would not be interconnected, and private property owners with secure property rights would harvest at a socially optimal level in perpetuity. Dispersal undermines this outcome and induces a spatial externality which leads to overexploitation. Thus, the nature and degree of dispersal will play an important role in each concessionaire's cooperation decision.

Dispersal is characterized by the N xN matrix whose rows sum to something less than or equal to 1 ( j D ij ≤ 1). There are N 2 free parameters describing dispersal, so at first glance it seems difficult to get general traction on how dispersal affects cooperation. But Proposition 1 provides a useful insight: If concessionaire i defects, she will optimally do so by considering only D ii , thus ignoring all other N 2 -1 elements of the dispersal matrix. We can thus assess the effect of spatial parameters on the emergence of cooperation. We show that a high degree of self-retention in all patches -that is a situation with low migration rates -is sufficient to ensure cooperation. Proposition 4. Let patch i be the patch with smallest self-retention parameter. For sufficiently large D ii , cooperation over all N concessions can be sustained as an equilibrium outcome.

Intuitively, if all patches have high enough self-retention, then the externality across patches is relatively small, which implies that the renewal effect outweighs the mining effect in all patches. When spatial externalities are not too large, the concession instrument overcomes the externality caused by strategic interaction. If self-retention is very low a large externality exists, and it may be more difficult to sustain cooperation. The formal result is not quite as straightforward because D ii also plays a role in e * j for all patches j, and affects defection incentives in all patches.

Proposition 5. Let patch i be the patch with the largest self-retention parameter.

For sufficiently small D ii , cooperation will not emerge as an equilibrium outcome provided the following condition is satisfied:

p i j =i D ji g(e * j ) < j =i D ij p j g (e * i )e * i . ( 17
)
Proposition 5 establishes that if the resource is highly mobile (sufficiently low selfretention rates), then cooperation might be destroyed. This result relies on the fact that economic benefits mainly depend on resource immigration. Condition (17) compares concessionaire i's cooperation benefits due to incoming resources and the sum of benefits others may get from the resource migrating from patch i.

Effect of tenure duration

Thus far we have focused on inherent features of the system as a whole that affect a concessionaire's incentives to cooperate or defect. But Condition (16) also depends on the tenure length T . This is a policy issue for a concession regime to be successful. We now focus on the optimal determination of T .

A basic tenet of property rights and resource exploitation is that more secure property rights lead to more efficient resource use. [START_REF] Costello | Natural resource use with limited-tenure property rights[END_REF] found that longer tenure duration indeed increased the likelihood of sustainable resource extraction in limited-tenure (though aspatial) concessions. We here find the opposite: Proposition 6. For sufficiently long tenure duration, T , cooperation cannot be sustained as an equilibrium outcome.

Proposition 6 seems to contradict basic intuition: if tenure duration is long, it is impossible to achieve efficient extraction of a spatially-connected resource by using our instrument. Yet this result still accords with economic principles, due to defection incentives driven by spatial externalities. Consider the case of very long tenure duration -in the extreme, when tenure is infinite, gains from defection always outweigh gains from cooperation. The promise of renewal has no effect on incentives, so each concessionaire acts in his own best interest, which involves the defection path identified in Proposition 1. Proposition 6 also holds in an extended version of the instrument, where the regulator can (with some probability Ψ < 1) terminate tenure immediately upon defection (rather than waiting until the end of the tenure block in which defection occurs). This extended version is described in sub-section 5.1.

Short tenure duration harbors two incentives for cooperation: First, when tenure is short, the defection payoff is small because the concessionaire has few periods in which to defect. Second, the renewal promise is significant because it involves a much longer future horizon than does the current tenure block. This result obtains because the spatial externality drives a wedge between the privately optimal decision and the socially optimal one. In fact, we can characterize a threshold tenure length for which concessionaire i will defect if T i > Ti , and owner i will cooperate otherwise:

Ti = 1 + ln δ(x i -x * i )+e * i -ē i δ xi -ē i ln(δ) (18)
Thus, cooperation is sustained by assigning to all N concessionaires a threshold value, which we summarize as follows:

Proposition 7. Assume the following holds for concessionaire i:

δx * i -e * i > (1 -δ) (δx i -ēi ) ; ( 19
)
Then there exists a threshold value T = min i { Ti } > 1 such that cooperation is sustained as an equilibrium outcome if and only if T ≤ T .

Condition ( 19) is a restatement of Proposition 3 for a tenure period T = 2. Thus, we know that a tenure period of 1 will guarantee cooperation. Since T = min i { Ti } depends on patch level characteristics, we briefly examine its dependence on patch, and system-level characteristics in Section B of the Appendix.

Robustness checks

To maintain analytical tractability, and to sharpen the analysis, we have relied on a number of simplifications. Here we discuss the issue of stock assessment and monitoring, then examine the consequences of three noteworthy assumptions. Specifically, we discuss the effect of a finite horizon on incentives to cooperate, then the cases of, respectively, stock-dependent costs and density-dependent dispersal.

Imperfect stock assessment, monitoring, and other types of uncertainty

We assume that the regulator can monitor the stock to verify compliance with the terms of the concession contract. In practice, stock assessment may be difficult to implement, and the cost of monitoring may thus prove important. Several points are worth highlighting. First, Proposition 6 also holds in an extended version of the instrument, where the regulator can (with some probability Ψ < 1) terminate tenure immediately upon defection (rather than waiting until the end of the tenure block in which defection occurs). We maintain the assumption that, in the last period of the tenure block, the regulator can terminate tenure immediately upon defection with probability one. If tenure could be terminated immediately upon defection with probability Ψ < 1, the best defection strategy will retain many of the features of Proposition 1: ēit = ēi (Ψ) > 0 at every period but the last one, and ēikT-1 = 0 (as long as 1 -Ψ is large enough so that ēi (Ψ) > 0 holds). Since cooperation payoffs remain unchanged, results in Proposition 3 and thus Proposition 6 remain valid under this extension. Parameter Ψ could also reflect stock assessment uncertainty (so Ψ is the probability of correct assessment). Then the instrument is robust to imperfect stock assessment (when Ψ is large enough). On the other hand, if it denotes the probability that stock assessment is actually implemented (maintaining the assumption that it is always implemented at the last period of the tenure block), then the expected monitoring cost would decrease as the tenure length increases. Closer examination of the interpretation where Ψ is the probability of implementing a stock assessment, we can examine the defection incentives formally. As explained above, we obtain ēit = ēi (Ψ) = (g ) -1 ( 1 δD ii (1-Ψ) ) > 0 in every period but the last one, and ēikT-1 = 0. Without loss of generality assume defection occurs during the first tenure block, which yields payoffs

Π d i = x i0 -ēi (Ψ) + T -2 t=1 δ t (1 -Ψ) t [x i (Ψ) -ēi (Ψ)] + δ T -1 (1 -Ψ) T -1 xi (Ψ) with xi (Ψ) = l =i D li g(e * l ) + D ii g (ē i (Ψ)). The cooperation payoffs are still equal to Π c i = x i0 -e * i + ∞ t=1 δ t [x * i -e * i ]
and simplifying gives:

Π c i -Π d i = δx * i -e * i 1 -δ -[δ(1 -Ψ)x i (Ψ) -ēi (Ψ)] 1 -δ T -1 (1 -Ψ) T -1 1 -δ(1 -Ψ)
Thus cooperation emerges as an equilibrium outcome if and only if, for any concessionaire i, the following condition holds:

δx * i -e * i > (1 -δ)[1 -δ T -1 (1 -Ψ) T -1 ] 1 -δ(1 -Ψ) [δ(1 -Ψ)x i (Ψ) -ēi (Ψ)]
The expected monitoring cost can be written

Ec T = T -2 t=0 δ t (1 -Ψ) t Ψc + δ T -1 (1 -Ψ) T -1 c
where c > 0 denotes the cost of implementing the stock assessment. We thus have

Ec T -1 -Ec T = δ T -2 (1 -Ψ) T -2 c(1 -Ψ)(1 -δ)
> 0 as long as Ψ < 1 and δ < 1 are satisfied: the expected monitoring cost decreases as the tenure duration increases.

Thus, when it is costly to frequently monitor users' actions, and to revoke and reallocate rights upon defection, there is a trade-off for tenure duration: Long tenure duration might result in defection, while short duration might entail higher monitoring costs. The same conclusion holds in the excludable public good setting when monitoring is costly.

Several contributions suggest that regular stock assessment is a mandatory part of a well-designed concession system, even if it is based on extraction levels. In many real-world settings, an annual stock assessment is carried out by technical consultants approved by the government and paid by concession members (See [START_REF] Wilen | The economics of territorial use rights fisheries, or turfs[END_REF] for a discussion). This requirement is further supported by [START_REF] Hilborn | Institutions, incentives and the future of fisheries[END_REF]: successful concession systems based on extraction levels tend to engage in active research programs funding stock assessments. Thus, for a system to be effective, proper stock assessment is mandatory, whether the system is based on extraction or on (residual) stock requirements.

Moreover, endogenous enforcement might be strengthened by parameters inducing persistent cooperation over time, particularly when monitoring involves capital expenditures. Concession rights might strengthen endogenous enforcement, and this could be rewarded via management certification, which may in turn provide improvements in market access. Thus, certifications might decrease transaction costs and strengthen agents' monitoring activities; both mechanisms would plausibly ease the conditions under which our instrument induces the efficient outcome. Enforcement issues may be driven by lack of legitimacy or the "need" for profit versus risk of deterrence. In developing countries this motivation might be greater than in developed ones; this might underscore enforcement issues. Yet, initiatives like communitybased concessions might improve legitimacy while reducing monitoring costs. Indeed, monitoring costs are likely to be lower compared to the case of state monitoring. Legitimacy may increase because of active and engaged leadership [START_REF] Crona | The importance of interplay between leadership and social capital in shaping outcomes of rights-based fisheries governance[END_REF]. These institutional arrangements are receiving increasing attention in developing countries. Since participation in the organization of the concession instrument can contribute to its legitimacy, such concessions might be interesting to increase enforcement in such areas. Finally, real-world cases suggest that sciencebased stock assessment is an integral part of the property rights system, which makes it less onerous for managers to monitor stocks and assess patch-specific characteristics. Cooperation between communities and government might help to decrease the cost of stock assessment, providing incentives for engagement in assessment practices [START_REF] Hilborn | Institutions, incentives and the future of fisheries[END_REF]). Indeed, it allows increasing interactions between concession owners and public-sector scientists, who might contribute to stock assessment, thus decreasing the assessment cost in return for access to the data collected.

A last point is that, if stock assessments require a fixed cost each year, they also influence the planner's optimized payoff, but will not affect her optimal choice of residual stock (Section 3.1.1). This will also be the case for concessionaires under our proposed instrument: their optimized payoffs will be affected, but their optimal choice to cooperate vs. defect will not. Monitoring costs will affect the agents' optimized payoff, but they will not affect the ability of the instrument to act as an effective cooperation device.

Our discussion of monitoring gives rise to an important question: Given the seemingly high information costs of implementing a limited-tenure concession, perhaps the public authority would be better off doing the harvesting itself? We make two brief comments on this possibility. First, we note that the informational requirements are similar between the private sector concession instrument vs. government harvesting. For example, knowledge of resource dynamics, prices, and movement patterns are required in both settings, so this confers no inherent advantage to government harvesting, though some savings could accrue as foregone costs of monitoring compliance with the terms of the concession. Second, we note that in many real world settings, the government is responsible for monitoring, while the private sector engages in the harvesting, but this need not be the case. If the government is more efficient at both monitoring and harvesting, then it can do so without the need for concessions (assuming it can internalize all spatial and dynamic externalities). If the private sector is more efficient at harvesting but not monitoring, the concession operates as we have discussed. If the private sector is more efficient at harvesting and monitoring, then monitoring could be delegated to an industry group, though the incentives would need to be carefully thought through (for example, the individuals responsible for monitoring could not be among the pool of potential new entrants).

While this section has shed some light on how costly monitoring might affect the viability of concessions, we refer the reader to [START_REF] Copeland | Trade, tragedy, and the commons[END_REF] for an explicit analysis of how endogenous enforcement affects natural resource outcomes.

Before concluding this section, we briefly discuss the robustness of our findings under imperfect information or stochastic resource growth. As long as patches are symmetric regarding the anticipated effects, no significant change is expected in the qualitative results. It would be interesting in future research to extend the analysis to allow for asymmetries in expectations.

The case of a finite horizon

In this analysis, concessionaires must trade off a finite single tenure block against an infinite number of renewed tenure blocks. It raises the question of whether the instrument is still effective at inducing cooperation when the horizon is finite. Suppose time ends after K tenure blocks where 1 < K < ∞ after which all concessionaires' payoffs are zero. We briefly explain here why cooperation is subgame perfect under the finite horizon problem, and that this requires more stringent conditions than under an infinite horizon. Specifically, it can be proved that the instrument then induces cooperation for the first K -1 tenure blocks (formal details are available upon request). Thus, the time horizon need not be infinitely long for our instrument to be effective: yet this requires more stringent conditions. Indeed, a condition equivalent to the one provided in Proposition 3 characterizes the incentive constraint. The gains from defection remain the same than in the infinite horizon setting, while the gains from cooperation become more complex. Concessionaires anticipate that they will not be renewed at the end of the final tenure block: they thus follow the cooperative strategy during the first tenure blocks, then they all defect in a similar manner than in the infinite-horizon case (they choose a positive residual stock before mining the resource in the final period). The cooperation payoffs during the entire process are now lower due to the increase in the defection payoffs in the final period. In other words, shorter time horizons require more stringent conditions for cooperation to be effective: longer time horizons (not to be confused with longer tenure durations) are most effective.

Stock-dependent costs

So far, we assume that extraction costs are linear in the amount extracted. Here we discuss how this assumption can be relaxed to be stock-dependent. Concessionaire i's period-t payoffs then become:

Π it = p i (x it -e it ) - x it e it c i (s)ds
where c i (s) < 0 is continuously differentiable (see [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF] for an early treatment of stock-dependent costs). We now explain briefly why the logic of Proposition 3 remains valid here. The proof relies mainly on two arguments (Full details are available upon request). First, the best defection strategy does not depend on the tenure block considered. Second, for the tenure block during which defection occurs, patch owner i's best defection strategy remains qualitatively the same as in Proposition 1: He chooses the non-cooperative level of harvest up until the final period of the block, and he then mines the resource by eventually harvesting down to level c -1 i (p i ). Thus, even though the characterization of the best defection strategy differs, and so the conditions ensuring the emergence of cooperation differ from Conditions ( 16), the qualitative conclusion of Proposition 3 still remains valid.

There is one interesting difference, though. When costs are stock independent, defecting agent i eventually extirpates the stock in his patch. With stock-dependent costs, defection does not entail extirpation, rather c -1 i (p i ) > 0. This implies that even if everyone defects, the resource will not go extinct (whereas extinction is ensured if everyone defects in the stock-independent case). This important difference has implications for defection incentives, though stock-dependent costs also affect the socially-optimal target stock, so it is not straightforward to draw general conclusions about how stock-dependent costs affect the willingness to cooperate.

Stock dependent dispersal

We have thus far assumed stock-independent dispersal. Here we show that this assumption can easily be relaxed. We can re-define for any patch i the law of motion as x it+1 = j D(e jt )g j (e jt ) which models density-dependent dispersal, and where D(e jt ) denotes the difference between self-retention and dispersal rate, and assumes that dispersal out of patch j depends on the residual stock in j.

In this amended version of the model, following [START_REF] Costello | Optimal harvesting of stochastic spatial resources[END_REF] it is easily confirmed that the socially-optimal policy still remains time and state independent. Moreover, the characterization of the best defection strategy follows from backward induction arguments as in Section 3.2: assuming defection occurs at tenure k + 1, we obtain e i(k+1)T -1 = 0 and then, for any preceding time period t in tenure k + 1, we have e it = êi satisfying

-1 + δ [D (e it )g i (e it ) + D(e it )g i (e it )] = 0
This condition highlights two effects: a direct effect on marginal productivity, which might result in higher or lower defection incentives, and an indirect effect on dispersal, which tends to increase benefits from higher in-migration if one assumes negative density-dependent dispersal. Specifically, we conclude:

g i (ê i ) = 1 -δD (ê i )g i (ê i ) δD(ê i ) (20)
Compared to the case of density-independent dispersal, it is more difficult to induce cooperation under negative density-dependent dispersal.

Proposition 8. Denote D ii ≡ D(ê i ) and consider êi the solution to condition (20). Then êi < ēi where ēi denotes agent i's best defection strategy under densityindependent dispersal when self-retention rate in patch i is given by D ii . The defection payoff increases, and the conditions for cooperation becomes more stringent under density-dependent dispersal.

Since the optimal defection strategy yields higher payoffs under density-dependent dispersal, it becomes more difficult to sustain cooperation.

Comparison with other potential policies

Our paper explicitly compares three alternative policies. First, we examine the social planner's problem: externalities are completely internalized and the result is Equation 8 in each and every patch, which yields the highest possible present value of the spatially-connected resource. Second, we examine the completely decentralized policy where property rights are allocated, but without coordination across properties. This leads to over-extraction in all patches (as shown in Equation 11). Finally, we examine a range of possible concession instruments (longer and shorter tenure duration, higher and lower target stocks). We derive the parameters of the concession contract that guarantee that the efficient extraction level will take place every period.

It is natural to consider alternative concession approaches (though a full comparison is beyond the scope of this paper). One candidate is to consider concessions with renewal based on maximum total extraction, where a concession is renewed only if a maximum amount of harvest over the tenure block is not exceeded. The characterization of the socially optimal paths obtained in Section 3.1.1, together with the characterization of the best defection path in Proposition 1, suggest that this instrument would not achieve the socially optimal outcome. Even if total extraction requirements are satisfied by the end of the tenure, it will induce over-harvest in certain time periods. Thus, it cannot ensure that the socially optimal outcome is implemented in all time periods.

A second possibility is to base renewal on the maximum total extraction in any time period, similar to a non-transferable patch-specific quota each period. This is similar to our proposed system, except that renewal is based on extraction rather than residual stock. This quota-based alternative induces the same incentives to concessionaire i as does our instrument, provided that all other concessionaires comply. This can be seen by the identity h it = x it -e it , so one could choose either extraction or residual stock as the main defining variable (given the state of the system (x t ) one derives from the other). As discussed in Section 5.1, both instruments require regular stock assessment. However, if some other concessionaires defect, the incentives for compliance differ for concessionaire i between the quota-based instrument and the target stock instrument. This occurs because the stock in patch i will be lower upon defection of some j = i, which implies a different payoff to i from abiding by the welfare-maximizing target stock vs. quota approach.9 We briefly also comment on the viability of a transferable quota approach, where a patch-specific quota is set, but trade is allowed across patches, and renewal is granted only if i's actual extraction does not exceed the quota shares owned by i in each period. Even if all patches comply with this approach, it will generally not lead to a socially optimal outcome because trade will drive an inefficient spatial allocation of extraction, even though the total extraction is socially optimal. To get an intuitive idea of why this could occur, consider a source patch j that delivers fish to all other patches, so has high D ji for all i. The social planner would like j to leave a very high target stock, so may set a very low quota for j (or even a zero quota, see [START_REF] Costello | Optimal harvesting of stochastic spatial resources[END_REF]), but j would like to extract in excess of this amount, so may purchase quota rights from others. The resulting total extraction will be socially efficient, but the spatial allocation will be inefficient.

Third, consider policies that employ property rights over the resource rather than over space. This approach induces challenges for spatial resources because biological growth, dispersal, and economic returns are patch-specific, and the optimal policy will thus vary across patches. Equation 8 reveals that the optimal policy depends on patch-specific net prices, growth, and dispersal and self-retention parameters. So the efficient outcome is spatially explicit, while using property rights over the resource results in a non-spatial instrument. Thus, it cannot achieve the first best, unlike our proposed instrument. Furthermore, as explained in Section 5.1 it is not clear that such system would be less demanding in terms of the related monitoring costs if the manager wants this policy to be as effective as possible (See [START_REF] Wilen | The economics of territorial use rights fisheries, or turfs[END_REF] for other advantages of spatially explicit instruments).

While the size of concessions is not endogenously chosen here, this dimension might be part of the manager's decision. If size is somehow related to biological productivity, then the findings from Section 4.2 suggest that variations in the size of connected patches may have complex effects. Indeed, agent i's willingness to cooperate increases as the size of an adjacent property increases, but the effect of an increase in the size of agent i's own property on his incentives to do so is ambiguous. Such a policy would then have to account for a variety of direct and indirect effects. This raises many new and interesting questions about design and effectiveness.

Conclusion

We have analyzed the ability of limited-tenure concessions to induce socially-efficient private provision of a class of public goods. We first analyzed a stylized public goods contribution setting and then turned to a more complicated natural resource extraction setting with resource growth, mobility, and heterogeneity across space. That limited-tenure concessions can achieve first-best private contributions may be surprising, as it does not rely on any transfers, side-payments, or explicit punishment, though it does accord with many real-world institutions for managing natural resources and public goods. The instrument works effectively by offering the promise of concession renewal, and therefore the promise of future benefits of the public good, but only if socially-optimal behavior has been undertaken in the past. Thus, if well-designed, concessionaires will be incentivized to adhere to efficient contribution, and to thus achieve renewal, rather than to under-contribute in the short run, and thus fail to achieve renewal. Contrary to an initial intuition, longer tenure actually induces underprovision. This implies that there is an optimal tenure length, which we derive in the paper.

We have used as a benchmark the case of perpetual spatial property rights. In that setting, resource owners over-extract the resource because they fail to internalize the spatial externality of their extraction on other property owners, and this over-extraction is the motivation for the limited-tenure concession instrument we analyze. We showed that a well designed limited-tenure concession always outperforms the perpetual spatial property rights approach, and can even reproduce the socially optimal spatial allocation of extraction. As an alternative, consider the complete absence of any regulation. In our model, this would quickly lead to extinction (indeed, in a single period), and it is straightforward to show that both the perpetual spatial property rights approach and the limited-tenure concession approach would outper-form this alternative on both welfare and environmental grounds. This implies that simply assigning spatial property rights (in perpetuity) leads to some level of public goods provision above what would have been achieved without regulation, even in the presence of spatial externalities. Therefore, as an alternative to concessions, one could consider assigning perpetual spatial property rights in different ways to induce different amounts of public good provision. For example, assigning separate property rights to each patch leads to a relatively low level of public good provision (because the externality is large), but assigning larger-scale property rights to closely-connected patches (i.e. those with large values of D ij ) will lead to greater provision of the public good because the externality is mostly internalized. Ultimately, this could help determine the optimal spatial allocation of rights. To our knowledge this kind of question has not been addressed.10 However, our analysis does provide some insights on this matter. Specifically, if the resource is highly mobile, then the gains from moving from spatial property rights (SPRs) to the limited tenure regime are larger than the gains from moving from open access to SPRs. This result arises because when the resource is highly mobile, even SPRs are still highly inefficient.

Several additional extensions remain. The regulator's incentives in offering concessions may also be an interesting issue; in this setting, the regulator could be viewed as a Stackelberg leader. While this paper has focused on identifying design parameters that induce cooperation, a next step could introduce different objectives for the regulator. Also, depending on the ecosystem dynamics, there could be different timing of growth, extraction, and dispersal. This reduces model tractability and neither renders our results moot nor obviously makes the analysis more realistic. Finally, we note that this basic framework applies across multiple case-studies, where one timely example is groundwater management. In many basins, a regulator's only control over pumping is in issuing groundwater well permits. In many agricultural areas, the overlying land parcels are large enough to allow some attribution of drawdown of the stock to the overlying landowner. When aquifers are heterogeneous, the recharge (growth) and conductivity (dispersal functions) may differ over different patches. Our model could be applied to this setting to evaluate whether threatening to revoke a pumping permit could induce more individual conservation behavior. 11Overall, the results suggest that limited-tenure concessions may be one important institutional tool to help achieve socially-optimal private provision of public goods. It sustains cooperation by relying on future financial incentives (via the promise of renewal), as opposed to financial penalties (as with traditional regulation), which could increase its practicality and appeal. The analysis in this paper suggests that concessions may not only have attractive intuitive appeal for managing natural resources and other collective goods, but if designed with care, they could be theoretically grounded in economic efficiency.

Parameters λ i and ε i denote the lagrangian parameters associated to the problem (knowing that 0 ≤ z i ≤ Φ must be satisfied). If λ i > 0 (so z i = 0) then we must have W ( l =i z l ) < u i (Φ) but this would contradict our assumptions as we have W ( l =i ) ≥ W ((N -1)Φ) ≥ u i (Φ) (the first inequality holds as W is concave). Thus λ i = 0 must hold. Now, if ε i > 0 (so z i = Φ) then we must have W ( l =i z l + Φ) > u i (0) but this would contradict our assumptions as we have W ( l =i z l + Φ) ≤ W (Φ) ≤ u i (0) (the first inequality holds as W is concave). We conclude that, for any i ∈ I, we must have:

W ( l∈I z l ) = u i (Φ -z i )
Due to our assumptions, there is no agent such that this condition is satisfied by either zero or full contribution: equilibria do exist and are necessarily interior. To prove uniqueness, let us assume that there exist at least two equilibria (z 1 , ..., z N ) and (x 1 , ..., x N ) such that there is i ∈ I satisfying z i > x i . This implies that u i (Φ -z i ) > u i (Φ -x i ) and so W ( l∈I x l ) < W ( l∈I z l ) which in turn implies l∈I x l > l∈I z l is satisfied. We conclude that there must exist j = i such that x j > z j and this implies u j (Φ -x j ) > u j (Φ -z j ) is satisfied: using the optimality condition again we conclude that W ( l∈I x l ) > W ( l∈I z l ) and thus l∈I x l < l∈I z l , which is a contradiction. The equilibrium is thus interior and unique.

Proof of Proposition 1

We first consider the case where g

i (0) > 1 δDii . At final period kT -1, concessionaire i's problem is max e ikT -1 ≥0 p i (x ikT -1 -e ikT -1 )
Using the first order condition yields ēikT -1 = 0 and, moving backward, at period T -2, this concessionaire's problem becomes:

max e ikT -2 ≥0 p i   x ikT -2 -e ikT -2 + δ   j =i D ji g(ē jkT -2 ) + D ii g(e ikT -2 ) -ēikT -1     .
Using the first order condition and ēikT -1 = 0, we have δD ii g (ē ikT -2 ) = 1 since ēikT -2 = 0 is ruled out by the lower bound on g (0), and ēikT

-2 = x ikT -2 is ruled out if x ikT -2 > (g ) -1
1 δDii holds. Using again backward induction highlights that any ēit ((k -1)T ≤ t ≤ kT -3) is characterized by the same condition provided that x it > (g ) -1 1 δDii = ēi . We have, by definition of ēi and concavity of g(.), that g(ē i ) > ēi g (ē i ) = ēi δDii which implies D ii g(ē i ) > ēi δ ≥ ēi for δ ∈]0, 1], and we deduce that x it > ēi for any tenure block but the first one. Even if concessionaire i defects initially,

since x i0 > (g ) -1 pi δ j Dij pj > (g ) -1
1 δDii by assumption, the same conclusion follows. The second case follows from similar arguments because of the upper bound on g (0).

Proof of Lemma 1

Let us consider the case of concessionaire i. If this concessionaire chooses to defect in tenure block k, then his payoff from cooperation and his payoff from defection are the same during the first k -1 tenure blocks. Then, during tenure block k, his payoff from defection does not depend on the renewal probability. By contrast, from tenure block k on, his payoff from cooperation does increase as f i increases (due to expression 13). Thus, a direct implication is that raising f i always increases the likelihood that concessionaire i chooses to cooperate.

Proof of Proposition 2

If there is t during which concessionaire i chooses e it > e * i : e it is strictly profitable only if:

p i (1 + δ) (x * i -e * i ) < p i   (x * i -e it ) + δ   j =i D ji g e * j + D ii g (e it )     .
Simplifying this inequality, we obtain:

δD ii (g (e it ) -g (e * i )) > e it -e * i . (21)
Since g(.) is continuously differentiable and increasing, there exists e i ∈]e * i , e it [ such that g (e it )g (e * i ) = (e it -e * i ) g (e i ) and we rewrite expression 21 as follows:

δD ii (e it -e * i ) g (e i ) > e it -e * i ⇔ g (e i ) > 1 δD ii = g (ē i ).
Since g(.) is strictly concave we have e * i < e i < ēi , which is a contradiction (since e * i ≥ ēi as explained in subsection 3.1.2). This implies that e it = e * i for any time period t.

Proof of Proposition 3

If concessionaire i deviates during tenure k + 1 then his payoff is Π d i = p i A, where :

A = x i0 -e * i + δ(1 -δ kT -1 ) 1 -δ (x * i -e * i ) + δ kT (x * i -ēi ) + δ kT +1 (1 -δ T -2 ) 1 -δ (x i -ēi ) + δ (k+1)T -1 xi . Now, using Condition (13), we compute Π c i -Π d i = p i B, with: B = δ kT p i 1 -δ δx * i -e * i -(1 -δ T -1 ) (δ xi -ēi ) (22)
The conclusion follows from Equality (22).

Proof of Proposition 4

We prove that the concessionaire does not defect from the initial period until the end of the first tenure. From the proof of Proposition 3 (using the expression ( 22) when k = 0) we know that:

Π c i -Π d i = p i 1 -δ δx * i -e * i -1 -δ T -1 (δ xi -ēi )
When D ii gets arbitrarily close to one, we deduce that ēi gets arbitrarily close to e * i , so that xi gets arbitrarily close to x * i . We can deduce that Π c i -Π d i gets arbitrarily close to:

p i 1 -δ δx * i -e * i -1 -δ T -1 (δx * i -e * i ) > 0 (23)
Thus, for D ii = 1 we know that Π c i -Π d i > 0 which, by a continuity argument, implies that this deviation is not profitable for sufficiently large (but less than one) values of self retention.

Proof of Proposition 5

Using Proposition 3, we know that concessionaire i would defect if δx

* i -e * i < 1 -δ T -1 (δ xi -ēi ) holds. The right hand side increases as T increases: its derivative is -δ T -1 ln(δ) (δ xi -ēi ) > 0 since ln(δ) < 0 and δ xi -ēi is positive. Indeed, δ xi -ēi = δ j =i D ji g(e * j ) + δD ii g(ē i ) -δD ii g (ē i )ē i = δ j =i D ji g(e * j ) + δD ii (g(ē i ) -g (ē i )ē i ) > 0

Proof of Proposition 6

Assume that any concessionaire j = i follows the cooperation path; we analyze concessionaire i's incentives to defect. According to Proposition 1, his payoff from the best defection strategy is equal to Π d i and we prove that Π c i -Π d i ≤ 0 for large enough values of T . Using the proof of proposition 3 we have:

Π c i -Π d i = δ kT p i 1 -δ δx * i -e * i -1 -δ T -1 (δ xi -ēi ) .
When T gets large, the term between brackets gets close to

[δx * i -e * i -(δ xi -ēi )] . (24)
Now, we obtain x * i -xi = D ii (g(e * i ) -g(ē i )) < D ii g (ē i )(e * i -ēi ) and [δD ii (g(e * i ) -g(ē i )) -(e * i -ēi )] < pi 1-δ [δD ii g (ē i ) -1](e * i -ēi ) = 0 and we conclude that (24) is negative. A continuity argument implies that Π c i -Π d i ≤ 0 for sufficiently large values of T .

Proof of Proposition 7

For a given concessionaire i, consider Ti defined implicitly by:

ēi

-e * i + δ 1 -δ (x * i -e * i ) - δ(1 -δ Ti-1 ) 1 -δ (x i -ēi ) -δ Ti-1 ēi = 0.
Since ēi and e * i do not depend on the value of Ti we can differentiate with respect to T and we obtain δ T -1 ln (δ) 1-δ (δ xiēi ) < 0 since ln(δ) < 0 as 0 < δ ≤ 1 and δ xiēi is positive (as shown in the proof of Proposition 5). Thus this function is decreasing and continuous in T (where T is assumed to take continuous values). Since this function takes on negative values as T becomes large, if it has a positive value when T = 2 then Ti is uniquely defined and Ti > 1. As Ti actually takes on discrete values, the proof implies that Ti is at least equal to 2. Then, the proof of Proposition 4 implies that concessionaire i will have incentives to defect as soon as the renewal time horizon is larger than Ti . For T = 2 the value of the function is given by the following expression:

ēi

-e * i + δ 1 -δ (x * i -e * i ) -δ xi = 1 1 -δ [δx * i -e * i -(1 -δ) (δ xi -ēi )] .
It is positive by Assumption ( 19), which implies Ti = 1 +

ln δ xi -ē i -(δx * i -e * i ) δ xi -ē i ln(δ)
is well defined. This concludes the proof since T = min i Ti qualifies as the appropriate threshold value.

Proof of Proposition 8

Negative density-dependent dispersal implies that D (.) < 0 holds, and we conclude from (20) that g i (ê i ) > g i (ē i ) which, due to concavity of the growth function, allows to conclude the proof.

Sections 4.2 and 4.3

We have the following stocks, respectively, when patch i defects and when all patches cooperate: xi = D ii g (ē i , α i ) + j =i D ji g e * j , α j ; x * i = j D ji g e * j , α j

We assume that one parameter, θ i = {p i , α i , D ii , D ij } or θ j = {p j , α j , D ji }, is elevated. We obtain:

dx i dθ i = ∂

A. Impact on the emergence of cooperation

Impact of net price, p

Impact of p i

We first analyze the impact of p i on concessionaire i's willingness to cooperate, and we obtain:

Impact of growth, α

Effect of α i

We have

d(Π c i -Π d i ) dαi = δ kT pi 1-δ ∂e * i ∂αi δD ii g e * i -1 + δD ii g α * i -(1 -δ T -1
)g ᾱi so, if D ii is small while ēi > 0, then

Impact of dispersal rate, D

Effect of D ii

We have

d(Π c i -Π d i ) dDii = δ kT pi 1-δ δ[g(e * i , α i ) -g(ē i , α i )] + δ T g(ē i , α i ) -1 -δD ii g e * i ∂e * i ∂Dii .
The RHS term is the sum of two terms of opposite signs, and is thus ambiguous.

Effect of D ij

We have

d(Π c i -Π d i ) dDij = δ kT pi 1-δ δ ∂x * i ∂e * i ∂e * i ∂Dij - ∂e * i ∂Dij = -δ kT pi 1-δ • ∂e * i ∂Dij 1 -δD ii g e * i < 0.

Effect of D ji

We have

d(Π c i -Π d i ) dDji = δ (k+1)T pi
1-δ ∂e * j ∂Dji D ji g e * j + g(e * j , α j ) > 0.

B. Impact on the time threshold Ti

We have

If D ij is small enough, then Ti increases when p j increases. Second, if D ji and l =i,j D li D lj are small enough, then expression 28 is positive: Ti decreases when p j increases. Indeed, this leads to a small value of the last term between brackets. Thus, the sign of d Ti dpj depends only on that of ∂e * i ∂pj (1 -δD ii g ei ), which is positive. We thus conclude that ∂ Ti ∂pj is negative. If δ is small enough while ēi > 0 then the sign of the expression is that of ∂e * i ∂Dii , which is positive.

Effect of α

  . If D ii = 1, then 1 -δD ii g e * i g α * i -(1 -δ T -1 )g ᾱi > 0. By a continuity argument, this conclusion remains valid when D ii is large.Effect of α j , j = i

  since the second term is positive by concavity of g. If D ii = 0 then δ xiēi = δ xi is positive too. As such, for any tenure length T there will be defection if δx * i -e * i is negative. Now, if D ii is sufficiently small, then ēi = 0 and we focus on cases where e * i is still positive. We examine the extreme case where e * i > 0 even when D ii is equal to zero. Using the characterization of e * i , we can rewrite δx *

	i -e * i = δ	j =i D ji g(e * j ) -j =i D ij	pj pi g (e * i )e * i and
	thus, when Condition 17 holds, then δx * i -e * i is negative.

Table 2 :

 2 Computations of derivatives> 0 D ii g α * i > 0 D ii g ᾱi > 0 ifthe condition on concessionaire i's willingness-to-cooperate is satisfied. So, an increase in p i results in a larger value of First, if both dispersal rates D ij and D ji are small, then the first and second term between brackets on the RHS of the equality are small, which implies that ∂pj D li g e * l , which is positive. Second, if D ii + D ij and D jj + D ji are large, then l =i,j

	θ				∂e * i ∂θ						∂ē i ∂θ	∂x * i ∂θ	∂ xi ∂θ
	p i		1-δD ii ge i N j=1 δD ij p j ge i e i	< 0		0	0	0
	p j	-	D ij ge i l=1 D il p l ge i e i N	> 0		0	0	0
	α i		-	ge i α i ge i e i	> 0			-	ge i α i ge i e i
	α j				0						0	D ji g α * j > 0 D ji g α * j > 0
	D ii -	p i ge i j=1 D ij p j ge i e i N	> 0 -	ge i ge i e i	> 0 g(e * i ) > 0	g(ē i ) > 0
	D ij -						> 0		0	0	0
	D ji				0						0	g(e * j )	g(e * j )
		d Π c i -Π d i dp i	=	δ kT 1 -δ	δx * i -e * i -(1 -δ T -1 )(δ xi -ēi )
					+	δ kT p i 1 -δ	  -	∂e * i ∂p i	1 -δD ii g e * i	+	j =i	∂e * j ∂p i	δ T D ji g e * j	 
	The second term between brackets is positive as	∂e * i ∂pi < 0, 1 -δD ii g e * i > 0 and	∂e * j ∂pi > 0. Thus
	d(Π c i -Π d i )
									d(Π c i -Π d i )

p j ge i N j=1 D ij p j ge i e i dpi > 0 dpi

, thus an increase in the willingness-to-cooperate.

Effect of p j , j = i dpj is negative.

  -x * i ) + e * i -ēi > 0, the first term in this equality is always negative. Using (25)-(26) and Table1, we haveδ dxi dθ -∂ ēi = {p i ; α j ; D ji } > 0 if θ = {D ij } and d Ti dθ > 0 for θ = {p i ; α j ; D ji } ( d Ti dDij < 0).The sign is ambiguous for θ = {p j ; α i ; D ii }.

	d Ti dθ =	1 ln(δ)[δ(xi-x * i )+e * i -ēi]	∂e * i ∂θ -δ	dx * i dθ +	δx * i -e * i δ xi-ēi	δ dxi dθ -∂ ēi ∂θ	. Since δ ∈ (0, 1) and
	δ(x i ∂θ	0. Notice that:
	∂e * i ∂θ  < 0 if θ Effect of p j , j = i -δ dx * i dθ = ∂e * i ∂θ (1 -δD ii g e * i ) -δ   ∂x * i ∂θ + j =i D ji g e * j  ∂e * j ∂θ
		∂e * i ∂p j	1 -δ	∂x * i ∂e * i	-δ	l =i	∂x * i ∂e * l	∂e * l ∂p j	+ δ	δx * i -e * i δ xi -ēi l =i	∂ xi ∂e * l	∂e * l ∂p j	(27)
	⇔	∂e * i ∂p j	(1 -δD ii g ei ) + δ	δ(x * i -xi ) -e * i + ēi δ xi -ēi	  D ji g ej	∂e * j ∂p j	+	l =i,j	D li g e l	∂e * l ∂p j	 

  If δD ii is small enough while ēi remains positive, then Ti decreases as α i increases.

		i
			∂e * i ∂α i	1 -δ	∂x * i ∂e * i		-δ	∂x * i ∂α i	+	δx * i -e * i δ xi -ēi	δ	∂ xi ∂α i	+	∂ xi ∂ē i	∂ē i ∂α i	-	∂ē i ∂α i	(29)
		⇔	∂e * i ∂α i	1 -δD ii g e * i	-δD ii g α * i -g ᾱi	δx * i -e * i δ xi -ēi	(30)
	Effect of D ii
		∂e * i ∂D ii	1 -δ	∂x * i ∂e * i	-δ	∂x * i ∂D ii	+	δx * i -e * i δ xi -ēi	δ	∂ xi ∂D ii	+	∂ xi ∂ē i	∂ē i ∂D ii	-	∂ē i ∂D ii
	⇔	∂e * i ∂D ii	1 -δD ii g e * i		-δ g(e * i , α i ) -	δx * i -e * i δ xi -ēi	g(ē i , α i )
													>0

Other relevant examples include green goods and climate protection infrastructure, provided their collective benefits may be potentially excludable.

The main result holds when each agent i's utility function is v i (Z, Φ -z i ) with Z = l∈I z l denoting the aggregate contribution to the collective good (all formal details are available upon request).

This model contains many elaborations on the model in Section 2, including that this common pool model contains subtractability.

Note here that we take the spatial delineation of properties as given. An interesting extension of this work would be to derive the socially optimal spatial delineation of property rights, which would presumably attempt to internalize spatial externalities.

While we continue to implicitly rule out consolidation of properties (and thus, we rule out the possibility that a sole owner will buy up all properties and solve the externality problem), it is innocuous to allow trade in concession contracts (again, without consolidation) because the incentives facing the buyer would be identical to those facing the seller.

Formal details are available upon request.

Under our instrument, there might exist equilibria in which all agents extract more than the amount specified by the concession contract. It is straightforward to show that there is a set of tenure durations for which such candidates cannot constitute an equilibrium provided there is at least one patch i which growth-related parameter α i is large enough (formal proof available upon request).

While his analysis was not explicitly spatial, we owe this intuition to[START_REF] Weitzman | Landing fees vs harvest quotas with uncertain fish stocks[END_REF].

[START_REF] Edwards | What lies beneath? aquifer heterogeneity and the economics of groundwater management[END_REF] suggests that the extent to which the resource is shared plays an important role in the benefits of regulator intervention.

We are grateful to a referee for this suggestion.

The authors are grateful for useful comments from a co-editor, an anonymous associate editor, two anonymous referees, and seminar participants at various institutions. N. Quérou acknowledges the ANR project GREEN-Econ (Grant ANR-16-CE03-0005). A. Tomini acknowledges the ANR financial support (Grant ANR-17-EURE-0020). C. Costello acknowledges the Bren Chair in Environmental Management and the Environmental Markets Lab.

Appendix Proof of existence and uniqueness in Section 2

Concavity ensures that agent i's maximization problem is well defined, and the related optimality condition is: