Common Pool Resource with Free Mobility Voting with Their Herds Dolgorsuren Dorj

∗ †

January 14, 2009

Abstract This work considers multiple localities to capture essential aspects of the traditional extensive grazing. We study the effects of the different institutions on the spatial distribution of herders in a twocommunity economy. Each herder chooses the locality in which he wants to reside and appropriate resource. We ask whether a sanctioning system can support cooperation, when free mobility allows individuals to choose between communities that differ by institutions governing them. We obtain following results: (i) when both pastures are unregulated a unique equilibrium always exists in which a homogeneous population is divided equally between two localities and both pastures are overgrazed; (ii) under a sanctioning system a unique equilibrium always exists in which the identical agents reside in both localities, the population in each community is of an equal size, and there is no overgrazing; (iii) under sanctioning mechanism in one locality and unregulated pasture in the second locality, the equilibrium yields more herders in locality with the sanctioning system because sanctions prevent overgrazing. This migration process improves the appropriation efficiency in unregulated pastures. Moreover, herders “vote by ∗

Corresponding author. Department of Economics, University of Hawaii at Manoa, 2424 Maile Way, Honolulu, HI 96822. Email: [email protected] † Thanks go without implication to Katerina Sherstyuk, Arnaud Dellis, Nori Tarui, John Lynham. I would also like to thank conference participants at Western Economic Association International meeting 2008, Economic Science Association North-American meeting 2007, Ronald Coase Institute 8th Workshop in Institutional Economics, 2006.

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the feet” for the sanctioning system if and only if the monitoring cost is low enough relative to the benefit of adopting sanctioning structure. JEL classification codes: C70, Q2, R5.

1

Introduction

We study a common pool resource (CPR hereafter) problem relevant in natural resource extraction situations. We explore the community based property regime in response to the problem of commons. Moreover, the emphasis is given to a multiple locality environment. The focus of this paper is to develop a theoretical model of CPR, where citizens in multiple localities are free to choose the place to live. We examine the comparative properties of alternative sanctioning institutions assuming a group ownership or communal tenure system. The basic question to deal with is: How can the CPR (overgrazing) problem be solved when community management structure and multiple localities are in place? Similar types of questions that consider efficiency of the resource use were asked in the literature (Ostrom et al. 1992, Falk Fehr, and Fischbacher 2002, Casari and Plott 2003), however, they were concerned about a one locality problem rather than multiple locality context.

1.1

Context and Relevance of the Research Background

This research is motivated by the land use practice in Mongolia, where pastures have been shared between herders for decades and open access is allowed even today. As a primary economic activity, traditional extensive livestock production involves approximately half of the entire population. Pastures occupying 76.5 percent of the country’s territory have been shared in common between herdsmen for centuries. Further, geographic location and climatic conditions related to heavy snow, ice disasters, or summer droughts create a high risk to the livestock production. Hence, traditional extensive grazing systems require high mobility and flexibility in the presence of natural disorders and risks. In this regard, rangeland can be shared between neighboring herding households in the case of a bad state of nature and lack of available forage to prevent unpredictable losses. Therefore, extensive grazing system is characterized by the seasonal movement of herders from one location to 2

another within a large territory of grassland. We have to distinguish between internal movement within the community and external migration, where the latter is of our concern. The question of overgrazing became most visible in Mongolia in the 1990s. An increase in the number of herding households due to a privatization of livestock, concentration of herding families in areas nearby the markets, lack of a migration control, and open access into the common pastures brought the overgrazing problem in Mongolia. Under open access every one has a right to graze as much as one can. As the process of adding more animals into the land continues, beyond a certain level, the quality and quantity of the land available reduces. It is in the best interest for the individual to add more animals, since increase in the herd size increases one’s revenue. Here the marginal private benefits outweighs the marginal social costs of putting an extra animal. As a result pastures near the urban areas face the tragedy of commons. It has been shown that private ranching systems are unable to cope with the risk when extensive methods of pastoralism are used (Hanstad and Duncan 2001, Deininger and Binswanger 1999). Under the communal system, the pooling of the resource allows risk to be shared under unpredictable conditions. Consequently, when herders can move freely within the larger territory, they are able to manage the risk successfully. However the communal grazing system suffers from an overgrazing problem. The dilemma between risk sharing and the common pool resource problem suggests that with the highrisk weather conditions prevalent in Mongolia, it is advantageous to pool the land into commons. According to the Constitution of Mongolia all pastures are owned by the state. However, possession rights to winter campsites and customary rules allows to manage the resource as a common property. Hence, we develop the model based on the notion of community management structure.

1.2

Theoretical overview

This research is located at the intersection of two broad bodies of literature: topics on common pool resource, particularly sanctions, and theory of local public good.

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1.2.1

Common Pool Resource Problem Revisited

Since the publication of Hardin’s famous article ”The Tragedy of Commons” social dilemmas has been studied intensively. Social dilemmas are defined as situations where in equilibrium everyone receives a lower payoff compared to the social optimum, however all individuals are better off if they cooperate than if they all defect (Dawes 1980). Hence, the tragedy of the commons has been formally represented in a Prisoner’s Dilemma non-cooperative game (Dawes 1973, 1975) with complete information, in which a rational player’s dominant strategy is to defect rather than cooperate. The equilibrium resulting from individual strategy choice (or self interested behavior) is not a Pareto-optimal outcome. According to the property rights school, the tragedy of the commons can be resolved by creating private property rights (Demsetz 1967, Cheung 1970, Johnson 1972, and Smith 1981). Other scholars advocate that an external agency such as the government would better regulate the commons; state property regimes can reduce over-exploitation of common pool resources (Hardin 1968). Recently, field studies and laboratory experiments show that in many cases people are able to sustain cooperative behavior and improve their social outcome (Berkes 1989, Wade 1987, Jodha 1986, and Chopra et al. 1989). Theory predicts an overuse of CPR under the assumption of selfish preferences. However, establishing institutional arrangements such as the possibility to sanction each other in community could improve the cooperative outcome. Case studies of Swiss village communal land tenure, common lands in Japanese villages; Huerta irrigation institutions and Zanjera irrigation communities in the Philippines point the importance behavioral factors, institutions, and motivations to have successful result. As an alternative to the market and state controlled institutional arrangements, the communal property regime among local users has been highlighted (Ostrom 1990). There are different approaches to the problem of commons. Classical prisoners’ dilemma game assumes only selfish players (Dawes 1973, 1975). Note that a selfish player cares only about her own income; her utility does not tie to the income of others. Social preference models that incorporate reciprocity (Rabin 1993, Falk and Fischbacher 1998), difference aversion (Bolton and Ockenfels 2000), inequality/inequity aversion (Falk, Fehr and Fischbacher 2002, Fehr and Schmidt 1999), altruism (Andreoni 1990), social welfare (Charness and Rabin 2002), other-regarding preferences (Casari and Plott 2003) extend classical model, explain cooperation in the social 4

dilemmas and induce higher level of efficiency. However, without sanctions it is difficult to restore efficiency even with reciprocal preferences (Fehr and Schmidt 1999). We do not add the assumption of social interest; here the emphasis is given to institution structure and enforcement mechanism such as sanctions. Case studies suggest that graduated sanctions more often used against the excessive use of the resource are helpful tool in CPR setting (Ostrom 1990). However, laboratory experiments report that sanctions do not alter the result of classical PD game and the appropriation level above a selfish Nash equilibrium because monitoring is a costly activity for the selfish players (Walker, Gardner and Ostrom 1990, WGO hereafter). It is different with the strong sanctioning structure that produces higher level of cooperation (Casari and Plott 2003). Similarly, in voluntary contribution mechanisms punishment increases efficiency (e.g. Fehr and Gachter, 2000, Visser, 2006). Thus sanctions encourage herders to be more cooperative and harvest less than the classical model predicts. The point is that herders having possession rights and who are organized into a community of users would realize their common problem and act in a way so that their behavior would benefit, rather than hurt, their production. The most important advantage of having co-management is that herders would have private information that the state would not. If the community introduces an enforcement mechanism, such as mutual monitoring and sanctioning, the harvesting level is less and closer to the socially optimal level. One approach to sanctions described in Falk, Fehr and Feschbacher (2002) relaxes the assumption on selfishness, incorporating reciprocal preferences based on the notion of inequality-aversion. However, in a CPR game with inequity-averse players the possibility of achieving a more efficient outcome is weak if enforcement mechanisms are absent. We could obtain a better result by introducing sanctioning mechanism into the model. The sanctioning equalizes the payoff of those who inspect and those who deviate from the common rule. The intuition is that when some players get a higher payoff by deviating from the rules, those who do not deviate have a disadvantageous position and are willing to punish the defector by inspecting. There are other types of player, who feel uncomfortable receiving higher payoffs than others. This type of player who has an favorable position by simply not punishing the defectors and therefore not incurring extra costs would be better off if he joins the inspectors. Hence, mutual monitoring restores efficiency in commons. A second approach, exemplified in Casari and Plott (2003) considers a 5

special system called ”Carte de Regola” for managing the common properties in Alpine villages and demonstrates the cooperative outcome is achievable with the sanctioning mechanism. Casari and Plott (2003) paper introduces sanctions that are not just pure cost, but a revenue-generating device in mutual monitoring. They show that self-governance structure built among resource users has the outcome close to the social optimum. A decentralized sanctioning system of ”Carte de Regola” has advantage over other sanctioning systems because it reduces the deadweight loss to the society. Fines imposed on violators are not a pure cost to the system; it is a transfer (revenue) to monitors. In addition, sanctions are imposed on the violator only once, so there are no multiple fines on the same action. The only loss in a system of mutual monitoring is the inspection cost. Also, the punishment level is not subjective; it depends on the deviation of players from the publicly known level and is proportional to the amount of excessive use. Behavior studied in laboratory experiments of ”Carte de Regola” fits better to the other regarding model as compared to classical model. Observe that Falk, Fehr and Feschbacher (2002) provide social preferences model, based on the notion of fairness and inequity toward the other player’s payoff. On the other hand Casari and Plott (2003) model with heterogeneous agents assumes other regarding preferences when players preferences are tied to the payoff of the group. When players display other regarding preferences, the sanctioning opportunity will discipline selfish players and improves efficiency of the system. Again, we are not concerned with other regarding preferences model in Casari and Plott 2003; we focus on the classical model and adopt sanctioning mechanism (Casari and Plott 2003), where mutual monitoring among players in a selfish environment brings efficiency back to commons. Our work extends the literature on CPR discussed above. Extensive grazing requires a high mobility among localities; hence to reflect a free mobility, we use tools that are common in a local public good literature. Much of the previous CPR research deals with incentive problem within one community. Here we extend existing research to allow for two communities in which citizens of each area are free to choose the place to live. In each community, there is a land parcel with pastures for grazing activity. We consider two possible community management regimes for each locality. In one locality, pastures are regulated by the collective action such as the adoption of a sanctioning mechanism where mutual inspection opportunities let herders keep meadows in good condition. In the second locality there are no rules towards the grazing activity. Facing multiple localities with different institu6

tional structures, which community will the herder choose? Particularly, we test the survival of different institutions under the migration pressure condition. Which type of locality will stand pressure from outsiders when mobility is costless? We address these questions in a multi-community environment CPR model, where players make the decision regarding the location to graze and harvesting level. Our model is an example of Tiebout-type economy with free mobility where each locality faces CPR problem rather than the local public good problem. 1.2.2

Free Mobility and Local Public Goods literature

Two equilibrium concepts, Nash equilibrium and the core of coalition structure, are used in local public good models (see Greenberg and Weber 1986). The Nash equilibrium defined as partition of agents into jurisdictions, where no single agent wants to move from the current position to join other existing jurisdiction. That is, concept of Nash equilibrium often referred, as a free mobility equilibrium or Tiebout equilibrium, does not allow any unilateral deviation by agents. Formation of new jurisdiction by individual agents is not considered as well. Examples of work on a Tiebout Nash equilibrium are in Westhoff (1977), Greenberg (1983), Epple et al. (1984, 1993), Dunz (1989), Konishi (1996), Nechyba (1997). The Nash equilibrium concept does not address incentives to deviate by coalitions. For this reason, it is often that equilibrium is suboptimal (Buchanan and Goetz 1972, Flatter, Henderson and Mieszkowski 1974). In contrast, the core of coalition structure (Strong Tiebout Nash equilibrium, coalition-proof Nash equilibria) allows coalitional deviations and free entry of new as well as existing jurisdictions (see McGuire 1974, Wooders 1978, Berglas and Pines 1981). The core of coalition structure is an alternative interpretation of the Tiebout’s equilibrium, where no group of agents wants to form a new jurisdiction or join existing one to make members better off. This approach generates a set of efficient equilibria; however it turns out that equilibrium may not exist if the population is not the integer multiple of the optimal coalition size. The leftover individuals create instability and the equilibrium may fail. To answer this question Wooders (1980) introduces e-equilibria as an approximation of exact core equilibrium; so the fraction of leftovers is approaching zero as economies gets large. Cole and Prescott 1997, Conley and Wooders 1997, Ellickson et al. 1999 showed that exact core equilibrium exists in a continuum type of economy. Conley and Konishi (2002) present Migration-proof Tiebout equilibrium where equi7

librium is stable against unilateral deviations; however the problem of too many Pareto inefficient equilibria has been avoided by the notion of credible coalitional deviations; hence equilibrium appears as refinement of the core concept. The idea of migration-proofness requires leftover agents not to follow the defecting coalition, such that no outsiders would chase coalitional deviation. In other words outsiders from the previous partition are excluded in the next partition process so that no outsiders would join a new jurisdiction and within the coalitional deviation nobody wants to move to any other jurisdiction. We use the first concept, Nash equilibrium, in our two-community CPR model, for the following reasons. First, extensive animal production is an individual family based activity, which does not require group formation that yields economies of scale. Second, the distance between families and lack of a comprehensive communication tool make the group formation unavailable. Third, seasonal migration from one place to other prevalent in nomadic culture (unsettled life) stands as an obstacle to any stable coalition structure. Fourth, the core equilibrium may not exist. Another feature in many recent local public good studies is to bring in the voting procedure, where individuals vote for the preferred jurisdiction to move in or vote for redistributive taxation for each region. Depending on the two types of voting process, Epple and Romer (1991) identify external (voting with the feet) and internal (voting with the ballot) equilibrium. We focus on external equilibrium and it is required that (i) all the localities be inhabited, and (ii) no agent wishes to migrate to another locality. Differing from the local public good literature, another line of research appeared since Buchanan (1965) called ”club theory” (Pauly 1967, 1970 a,b; Shapley and Shubik 1966, Anderson 1985, Wooders 1980; Scotchmer 1985a,b; Cole and Prescott 1997, Ellickson et al. 2001). The club theory considers the sharing of the excludable public good, where there is no free entry from other coalitions. In contrast, we study a non-excludable congestible good; hence clubs are not relevant to our research. In addition, in club theory a formation of new jurisdictions has no limit; however this is not the case in my study due to limited land resource. Finally, our study excludes appropriation in multiple sites; hence joining of more than one jurisdiction, which is available in club modeling, is of no interest. Geographic location (e.g. distance, transportation cost) was not a focus in LPG. However, it has been discussed in ”spatial clubs” (see Starrett 1988) and ”neighborhood goods” (Fujita 1989). See also Arnott and Stiglitz (1979), Thisse and Wildasin (1992), Thisse and Zoller (1982), and Hochman, Pines 8

and Thisse (1995). If theory of spatial clubs considers only fixed costs (without congestion costs), then it is very similar to the theory of firm location (Hotelling 1929, Lerner and Singer 1937, Eaton and Lipsey 1975, Graitson 1982, Shaked 1986, ReVelle 1997, Collins and Sherstyuk 2000). Location theory applied to the public goods has focus on a social planning problem such as where to locate the facilities and how the costs are to be covered. In comparison, location theory applied to the private firms concerned on the existence of the equilibrium, if it does exist on its efficiency and pricing policies for firms (Scotchmer 2002). At this point we ignore geographic location as a source of transportation cost, because mobility has been part of the extensive grazing activity of herders for decades and we assume in our model that mobility is free with no cost. Herder involved in animal production activity has no other alternative choice to explore; hence, no opportunity cost of production.

1.3

Research Methods and Questions

As we discussed in the previous section sanctions were proposed to solve the overgrazing problem, however our case is of specific inquiry. The Casari and Plott (2003) model with sanctions solves one locality problem, where players restore efficiency because of the sanctioning mechanism. Taking into account the fact that grazing in Mongolia requires high mobility within a large area of grassland, our scope is to consider the model with many localities each having CPR problem. First, we adopt standard CPR model as a baseline. Second, we allow free mobility across different communities and define existence of equilibrium with migration. Consider free mobility equilibrium as a game in which the population in each community is endogenously determined. Each herder is free to move from one location to other, so that free mobility is allowed between communities. Each family chooses the locality (Si ) to live and decides on a grazing level (xi ) within the chosen locality. Further, the herder observing total group use within the community may or may not inspect (Ii ) other herders in locality with sanctions. Note that the community with a sanctioning mechanism is able to set rules among herders within the locality in such a way that total grazing level is at the social optimum. We solve the game using the notion of external equilibrium where each locality is inhabited and no one wants to move. The question I am addressing in this study is: How does free mobility 9

affect the efficiency of a sanctioning system? Does it disturb the efficiency of the mutual monitoring established among the resource users within a community? We assume here that herders have the same preferences. In addition, localities may differ by the institutions serving a particular area. Thus, we consider three cases of two-community institutions: (i) both localities are unregulated; (ii) both localities adopt sanctioning systems; and (iii) one of localities adopts a sanctioning system, while the other locality has no regulation. We define efficiency as the percentage of maximum surplus extracted by the group. Maximum surplus is obtainable when the resource is used at the socially optimal level; hence the efficiency is equal to one hundred percent. We find that with multiple localities, free mobility leads to redistribution of population between localities such that more people live in regulated locality with sanctions compared to unregulated locality. Interestingly, efficiency in unregulated locality is improved. We believe that the results given here are interesting because in particular they shed new light for the community management structures which are of great interest in the study of CPR. Our main focus is to characterize free mobility equilibrium. We review the results in one locality as a benchmark when different institutions are present. Communities have distinct mechanisms for allocating common pool resource. Institutions explored throughout the paper are (1) unregulated pasture and (2) sanctioning mechanism.

2 2.1

Common Pool Resource Model Basic CPR Model

In a standard common pool resource game (e.g. Falk, Fehr and Feschbacher 2002) all herders with endowment, e, simultaneously decide on the amount of appropriation xi , where subscript refers to herder, i ∈ 1, ..., N . The population, N , consists of a finite number of herders. In this paper we assume that all herders are identical in our setting. We focus on symmetric equilibria where herder’s strategies are the same within the locality, xi = x. The cost of grazing is independent of every other member’s decision while the revenue for each member will depend on grazing choice of all members. Let X be the amount of total appropriation and f (X) is total revenue of all herders. The share of each individual in total grazing is, xi /X. Each herding family 10

obtains the amount of revenue equal to [xi /X]f (X). Then total monetary payoff received by the herder is given by πi = e − cxi + [xi /X]f (X), where c indicates the cost of maintaining one animal. Note that the classical model assumes a selfish player; thus, utility function, Ui (πi ), has no parameter related to the payoff of others. Each animal needs a certain amount of grass in order to survive; and there is a maximum number of animals that the meadow can carry, Xmax . The revenue per animal, f (X)/X > 0, is positive for X < Xmax , but f (X)/X = 0 for the X > Xmax . In a standard CPR model the production function is increasing in X until reaching a certain value where it declines. Specifically, production function takes the following form: f (X) = aX − bX 2 , where a > 0 and b > 0 are the parameters determining the production function. Note that the first few animals have plenty of grasses (X < X0 )1 and adding one more animal has little negative impact on those already grazing, but when the total grazing level is close to the Xmax level, additional grazing does harm the rest. Formally, for X < X0 , f 0 (X) > 0 and f 00 (X) < 0, revenue increasing at decreasing rate. For Xmax > X > X0 , marginal product is decreasing, f 0 (X) < 0 and f 00 (X) < 0. Thus, each herder has a negative externality from the others. Proposition 1 Social optimal appropriation level is determined by X opt = a−b , where we maximize the sum of profits across all herders. 2b See Appendix 6.1 for details. When the number of users is infinite, complete destruction of positive revenues that the community could have made from extraction is in place and the game can be called an open access, where efficiency level is at zero because resources are used up to the point where average costs is equal to average benefits. However, we focus on Nash equilibrium concept where meadow has positive revenue from grazing and strategic interaction between herders holds. If all herders are selfish the outcome is inefficient and there is overgrazing. Proposition 2 Selfish Nash equilibrium in one locality with homogeneous types and complete information determines the appropriation by each herder: a−c . Total appropriation in selfish Nash equilibrium is given by X ∗ = x∗i = b(n+1) n · a−c , which is higher than the social optimum. n+1 b 1

Production function takes maximum value at X0 and equals zero at Xmax .

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See Appendix 6.2. Too many animals are grazed in Nash equilibrium compared to the social optimum, X ∗ > X opt . Here we have conflict of individual’s interests with a society’s best interest.

2.2

Sanctioning mechanism

Community rules The sanctioning mechanism2 restricts grazing level to the certain amount of animals (threshold). In addition all herders are free to monitor each other with respect to the threshold. Thus, herders establish rules among themselves, monitor each other’s behavior and impose sanctions on those who break the rules. Monitoring is a costly activity, but also it generates revenue in terms of fees paid to the inspector from the noncompliance side. Therefore, herders in the inspection game are motivated to induce monitoring costs to extract revenue from the monitoring activity. Fines, as a fixed proportion of the overuse paid by the violators of the rule, are not deadweight loss to the system since it is the transfer to other herder who inspects. Game Consider an inspection game in which herders decide on two issues: the appropriation level from the common pool resource, xi , and the inspection activity, Iij . Recall that herders are identical in our setting. Each herder decides on how much he is going to extract from the common in the chosen locality. This is the number of animals he is putting in a meadow. Further, the herder makes an inspection decision by selecting the subset of herders for which he wants to pay for having inspected. The inspection involves mutual monitoring with information discovery and punishment. Before requesting the inspection, herders know only the total group use. Each herder can infer the degradation of the resource, but he cannot identify which herder violated the rule and has excessive use. Herding families are located far away from each other, so monitoring is costly and each herder’s action is not observable. After the inspection, the use levels of inspected herders become the public information. Following the inspection game presented in Casari and Plott (2003) suppose when each herder has chosen his community and his appropriation level, each of them decide whether to inspect at a certain cost, k, any other herder who violated the threshold, λ. The penalty mj would be imposed to each herder, if one violates the rule and appropriates above the threshold, λ. Let 2

See Casari and Plott 2003.

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mi be the fine that herder i pays if inspected by herder j and violates the rule, where i 6= j. Taking into account the use level and inspection activity, the payoff function is determined by: πi =

X xi Iij rij f (X) + e − cxi − Ii mi + X i6=j

(1)

where, rij = mj − k, is the revenue generated by agent i from monitoring of agent j. Clearly in this situation inspection occurs if monitoring activity is profitable for herder i. Formally, Iij = 1 if i inspects j and Iij = 0 otherwise. If the agent i has been monitored by more than one inspector, j 6= i, a person randomly selected out from requested agents receives the direct transfer from P monitoring. Thus, Ii = 1 if j6=i Iij ≥ 1, and Ii = 0 otherwise. Note that inspection will take place if agent j inspects other agent i and i violated the rules. Formally, Ii = 1 if i is inspected by j and xi > λ. If none of these conditions hold then inspection will not occur, hence Ii = 0. Agent i makes a profit from monitoring agent j when monitoring fine collected from all extra animals of agent j is greater than the cost of inspection required to detect one herder, rij = (mj − k) > 0. The fine mj is imposed and paid to the inspector i, if agent j exceeded his grazing level, and i is selected from inspectors. This fine is proportional to the overuse of the resource:  0 if i does not inspect    

0 if agent i who inspected j is not randomly selected from inspectors mij =   0 if i inspects j and xj ≤ λ   h · (xj − λ) if xj > λ and i is randomly selected from inspectors (2) Here λ is the maximum number of animals allowed to each herder, h is a unitary fine for each extra animal, and xj is the number of animals (appropriation) put in a meadow by herder j. As from the model, herder i will inspect other herder j, if the fine imposed to the violator is greater than the monitoring cost incurred, mij = h · (xj − λ) > k, where k is a unitary cost of inspection. Herders will inspect each other if the grazing level of the suspected person j is more than xe ≡ (k/h) + λ = ∆x + λ. From the equation there is some level of violation that might occur depending on the ratio between unitary monitoring costs, k, and unitary monitoring revenue, h. The higher the ratio of k/h, the higher the violation and vice versa, the lower the cost of enforcement relative to the revenue from monitoring. The lower the violation (extra animals) denoted by ∆x. The inspection might 13

occur or not depending on the probability that total grazing level exceeds the community’s proposed grazing level, E(X ∗ ) > N xe. Sanctions serve as a threat for herder; hence incentives to overuse the resource are limited. Note P that the total fine paid by j to the inspectors is equal to mj , i6=j mij = mj , because herder i inspects, if xe > k/h + λ. When in equilibrium everyone inspects each other and multiple fines are excluded, the payoff from inspection activity for herder i who inspects j is mj , since Iij = 1. Since in one round herder i will be inspected only by one inspector, she does pay a fee mi only once. In order to induce proper level of grazing, community sets the threshold level, λ, in such a way that equilibrium appropriation for each herder is equal to the social optimal level. We can show that to have X ∗ = X opt community sets the threshold level equal to λ = xopt − k/h − ε, where ε is arbitrary small number. The higher the cost of enforcement relative to the monitoring revenue, the lower the threshold must be. Each herder choice ∗) . Here p∗ on how much to extract from common is given by: x∗i = a−(c+hp b(n+1) is the perceived probability that herder inspects. See Appendix 6.3 for the derivation of sanctioning mechanism equilibrium in one locality. Therefore, equilibrium of the sanctioning system is characterized by:  ∗) n  · a−(c+hp X ∗ =n+1   b   ∗ e   0 if E(X ) < nx (X ∗ , p∗ ) = ∗ ∗  p = 1 if E(X ) > nxe       ∈ [0, 1] if E(X ∗ ) = nxe

(3)

Three types of equilibria are possible: 1. X ∗ = X opt , p∗ = 1 if E(X ∗ ) > nxe. This is possible with strong sanctions3 ; 2. X ∗ = X N E > X opt , p∗ = 0 if E(X ∗ ) < nxe. This equilibria arises when weak sanctions are introduced; 3

In Casari and Plott (2003), strong sanctions induce full efficiency outcome, while weak sanctions designed to have no effect on subgame perfect Nash equilibrium such that no inspection is strictly profitable when individual use is at or lower than x e = 16. Strong sanctions environment was available with parameters k = 7, λ = 7, h = 4, weak sanctions were with parameters k = 7, λ = 9, h = 1. Symmetric mixed strategies equilibria is possible with parameters in range: k = 7, λ = 7, 1 < h < 4.

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3. X opt < X ∗ < X N E , 0 < p∗ < 1 if E(X ∗ ) = nxe. This is symmetric mixed strategies equilibria. We are interested only in the outcome where sanctions lead to optimal use of resources, i.e. the equilibrium, where X ∗ = X opt , p∗ = 1. From Appendix 6.3 this means λopt = xopt − k/h − ε. In turn the fine for each extra animals must be set to h = a − c − xopt · (n + 1) · b, where xopt = X opt /n. Then, i nxe = X opt − nε < E(X ∗ ). This implies that p∗ = 1. Here xe ≡ k/h + λ. Following is established in Casari and Plott (2003) Proposition 3 Suppose locality has a sanctioning mechanism, where the threshold and punishment levels are set as λopt = xopt − k/h − ε and h = a − c − xopt (n + 1) · b, respectively. Then this sanctioning mechanism supports the socially optimal level of harvesting as subgame perfect Nash4 equilibrium, X ∗ = X opt . Also in this equilibrium every one inspects each other, p∗ = 1. Proof: See Appendix 6.3. Assumption 1 Grazing level in regulated locality is restricted to λopt , where this threshold sets so that to provide optimal level of appropriation, XR∗ = X opt , in equilibrium. If the perceived probability is positive, the best response for the agent is to use the less resource than in the no inspection case. Prediction of this game depends on the assumption of identical players (endowment, cost, and production). Hence, we get x∗i = X ∗ /n, p∗i = p∗ for ∀i. All player are homogeneous in term of preferences; they are all selfish players motivated by the reward (mj ) from the inspection activity 5 . The monitoring is a costly activity in Mongolia for neighbors who are far away from each other. However, 4

In the original paper by Casari and Plott (2003) equilibrium of the inspection game referred as Nash equilibrium since any SPNE is a Nash, and in the text the two stage game is solved by backward induction 5 Given some level of monitoring cost per person, k > 0, total monitoring cost increases with the probability of inspection. The higher is the probability of inspection, the higher is the total cost of enforcement for the society, nk. We have perfect monitoring, when p = 1, everyone inspects. In other words, for the society it would be optimal to have monitoring level less than perfect monitoring level, 0 < p < 1, which eliminate any incentives to violate the rules. However, our objective is to see the effect of multiple localities on the performance of sanctioning system when everyone inspects each other, p = 1. Hence, probabilistic inspection can be thought as a separate project beyond the scope of this paper.

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neighborhood can be monitored with negligible cost since extensive grazing by itself requires each herder to keep his own animals separate from other herds; he can distinguish own herd from others; he observes violation level by others while taking care of own animals on common pasture. However, it is too costly to monitor actions taken in sites where he does not live. However, positive monitoring cost, k > 0, is a sunk cost that reduces efficiency of sanctioning system. Hence the choice between institutions must be done based on the cost-benefit analysis.

2.3

The Choice of Institution

From Proposition 2 we learn that unregulated locality has an advantage of having no sunk cost related to institution building, however disadvantage of the no regulation characterized by the exploitation of the resource beyond socially optimal level. Here the outcome is determined by the fact that private marginal benefits outweigh the social marginal costs. Proposition 3 states sanctioning mechanism induces the proper level of grazing by equalizing the social marginal benefits to the social marginal costs. We replicated in previous section that there exists a possibility to set up rules (threshold, monitoring possibility, fines) among the members of the community that have common property (grazing land) such that the appropriation is at the socially optimal level. The advantage of the sanctioning mechanism essentially is that the maximum socially optimum yield is captured from the resource. Also in formal regulation such as sanctioning mechanism the punishment targeted toward the free rider, hence multiple fines are not considered, while in informal sanctioning mechanism such as in infinitely repeated game solution the punishment occurs at the group level. For example, in repeated game, player may punish free rider by selecting the defecting choice in the next period. Clearly, this player punishes free rider and himself at the same time, because the payoff from defecting choice is smaller than the payoff from cooperating choice for both players. Moreover, fees collected from the violators are not deadweight loss anymore, these fees go to inspectors as a revenue from the inspection activity. The fees collected from the violators finance the incentive to inspect for selfish individual. On the other hand the monitoring activity is a costly for individual and it is a deadweight loss to the society. The choice of appropriate institution depends on the relative cost benefit analysis of adopting alternative institutions. The benefit of the sanctioning mechanism outweigh total cost of adopting this institution, if 16

f (X opt ) − f (X N E ) − c(X opt − X N E ) > nk, where k stands for the monitoring cost per person. Recall from previous section that efficiency of the sanctioning system excluding the monitoring cost is one hundred percent, while including the monitoring cost it will run into lower number. However, this is much higher than the unregulated locality efficiency. From here we conclude that inspection mechanism is welfare improving. Proposition 4 If the inspection cost is low enough for the society and parameters determining the rules within the community are such that the benefits of sanctioning mechanism outweigh the costs of adopting of it, f (X opt ) − f (X N E )−c(X opt −X N E ) > nk, the sanctioning mechanism is welfare improving as compared to unregulated use. The community should adopt sanctioning system if and only if the benefit of adopting sanctioning system per head is greater or equal to the monitoring cost required to inspect one additional person: k ≤ (X opt ) − f (X U ) − c(X opt − X U )]/n If locality adopts it, then it’s welfare improves. The benefit of adopting a sanctioning system outweighs the monitoring cost, where n indicates total population. If this condition does not hold, then unregulated locality regime prevails. The net social benefit between regulated and unregulated regimes is defined by: N SB = f (X opt ) − f (X N E ) − c(X opt − X N E ) − nk, where net social benefit per person decreases in n, ∂(N SB) = −X 0 [f 0 (X U ) − c] − k, ∂n

(4)

Since f 0 (X) < 0 for X0 < X U < X max , X 0 = ∂X/∂n > 0, if k > X 0 [f 0 (X U ) − c], we have ∂(N∂nSB) < 0. Therefore, the benefit of adopting the sanctioning system is decreasing in n. See Appendix 6.4 for the details.

3

Two-Community and Free Mobility Equilibrium

Suppose there are two communities of herding families located in neighboring areas denoted by Si ∈ {1, 2}. We allow free mobility across herding communities, which means that migration is a costless decision. All herders are perfectly mobile between localities, but can live in only one. Herders choose 17

the location for grazing depending on local enforcement mechanisms. From the herder’s point of view, all communities in different localities are identical except for the local enforcement mechanism, which all herders take as given. Consider two localities with different institutions: unregulated pasture and sanctioning system. In an unregulated environment, basically no rules exist concerning the maximum number of animals allowed to put in a meadow by herders. In contrast, the community with the sanctioning system restricts grazing level to the certain amount of animals and keeps an efficient level of resource use. The herder decides which of two communities to choose for, depending on the rules present in each locality. We allow free mobility between communities. In the unregulated locality herder i decision consists of a pair, (Si , xi ), where Si is herder’s location and xi is the size of the herd, while in the locality with sanctioning mechanism herder has inspection decision in addition to the location and appropriation choices (Si , xi , Ii ). Again we assume that rules and enforcement mechanisms are exogenous. Thus, we assume that a community with sanctions is able to set up threshold, λopt Si , that leads to the social optimal grazing, X ∗ = X opt , p∗ = 1. Also, we assume that the threshold is fully adjustable to the population change of the locality. The location choice and appropriation level depend upon herder’s preferences regarding his profit from living in one of localities. At the same time the profit for herder will depend on the rules governing a particular area. See Appendix 6.1 for the social optimal level of grazing as an equilibrium outcome when the social planner is in place. Free mobility equilibrium is stable against any unilateral deviations. This means that no individual wants either move into currently existing jurisdictions or form a new jurisdiction. The core of coalition structure is stable against group deviations to the existing jurisdiction or new jurisdictions. However, Nash concept is more appropriate for our setting; because formation of stable group of people becomes questionable with the mobile structure in extensive grazing. Moreover, limited land resource prevents formation of new jurisdictions; hence we exclude the second requirement of free mobility equilibrium (stable against unilateral deviations to new jurisdictions). Our definition of free mobility equilibrium 6 in a two-community economy reduces 6

In general a partition is a free mobility equilibrium, if and only if (i) for all k = 1, ..., KS k 6= 0 and i ∈ S k , it holds that πi (S k ) ≥ πi ({i}), and (ii) for all k, k¯ = 1, ..., K ¯ ¯ and all i ∈ S k it holds that πi (S k ) ≥ πi (S k ∪ {i}) where k 6= k.

18

to no individual can be better off by moving to other existing jurisdiction: s = l only if πil (xi , Si ) ≥ πik (xi , Si ), l 6= k, l = 1, 2; k = 1, 2. In addition, all individuals choose optimal appropriation level: ∂πi /∂xi = 0 for ∀i. We define equilibrium as a bundle (S ∗ , X ∗ , I ∗ ) consisting of location strategy profiles, grazing decision profile, and monitoring decision profiles in communities with sanctions. In equilibrium each herder chooses grazing level, xi , that maximizes his preferences. Second, no herder has incentive to move. Recall that we explore economy with finite number of citizens. We use equal revenue sharing mechanism in CPR because of the assumption on identical preferences: f (X) · xi /X = f (X)/n. Also the utility function depends on profit, which is a non-linear function of the resource use. Our model presents the negative externality (congestion) effect from the beginning 7 ; one individual moving to other existing jurisdiction has welfare reducing effect to the citizen-receiving jurisdiction. There are two identical communities in terms of land productivity; however, rules governing the two localities may differ. Also grazing in one locality excludes the grazing in other locality. Herders can graze in only one of two localities; they have production activity in the site where they live. Thus, our model excludes grazing activity in multiple sites. Consider free mobility equilibrium when localities adopt variety of institution structures.

3.1

Social Optimum

Suppose benevolent government allocates resources such that the whole society is better off. The social planner solves per each locality: P maxSi ,Xs ,Ns πs = Ns · e + f (Xs ) − cXs . Note that index s here refers to the locality. Hence, total grazing level in each locality is given by: Xsopt = (a − c)/2b.

(5)

Population in each community is denoted by Ns . Thus, social optimal grazing level for herder is defined by xopt s = (a − c)/2bNs . Proposition 5 In two-community economy with homogeneous agents, socially optimal level of grazing in each community is the same as the social 7

In public good economy either proportional or equal sharing tax schemes are considered; thus one additional member in jurisdiction is extending the tax base and has welfare improving effect. In order to realize the congestion effect new term is introduced in the utility function.

19

optimal level where only one locality is present, Xsopt = (a − c)/2b, and it is independent of number of people in the locality. Per person-grazing level in each locality is determined by xs = X opt /Ns depending on the number of people within the locality. Proof: See Appendix 7.1 for the details.

3.2

Unregulated pasture’s Nash equilibrium and free mobility

Suppose herders in both communities face no rules concerning the appropriation. Herders in both localities have two choice variables: the appropriation and the location for grazing activity. With costless mobility, herder’s location and harvesting choices must maximize: max πis = e − cxis + [xis /Xs ] · f (Xs ) Si ,xi

(6)

Necessary FOC are: ∂πis /∂xis = 0;

(7)

sj = l only if πil (xi , si ) ≥ πik (xi , si ), l 6= k, l = {1, 2}, k = {1, 2}

(8)

Given identical herders condition following must hold: πi1 = πi2

(9)

Here N is the total number of herders in both localities together, N1 and N2 are the population in each locality respectively. Condition in (7) states, each herder chooses his level of extraction to maximize his utility given by equation (6) dependent on each other herder’s appropriation level. Condition (8) states that if the utility from being in locality one is greater than the utility of grazing in locality two, the herder will choose locality one and otherwise choose locality two. Equation (9) serves stability of equilibrium assuming that all herders are identical. The equilibrium is characterized by (N1 , x1 , N2 , x2 ) so that all herders are indifferent between two communities. Each person’s grazing level is xs , where subscript denotes the locality, total grazing level in each locality is given by Xs , where subscript again denotes the locality. The symmetric Nash equilibrium in each locality is given by: a−c , Xs∗ = a−c · Ns , where two localities denoted by S = {1, 2}. See x∗s = b·(N b Ns +1 s +1) 20

Appendix 7.2 for details. In equilibrium with homogeneous herders stability requires the payoff for herder from grazing activity in locality one to be equal to his payoff from grazing in locality two. This is true if the number of people in each locality is equal. Thus, equilibrium grazing level for herder is defined N/2 by x∗s = b·(a−c , total grazing in each locality is given by Xs∗ = a−c · N/2+1 = N b +1) a−c b

·

N , N +2

2

where N = N1 + N2 , N1∗ = N2∗ , X1∗ = X2∗ .

Proposition 6 Free mobility equilibrium in two neighboring, unregulated pastures with identical herders is characterized by the symmetry of outcomes. Half of herders graze in locality one and half of them choose locality two, N1∗ = N2∗ . Also total grazing level in each locality is the same as the selfish ∗ Nash equilibrium in one locality, Xoneloc. = X1∗ = X2∗ . Both localities’ grazing levels are above the social optimal level, X1∗ = X2∗ > X opt . Thus, free mobility equilibrium efficiency level in unregulated pastures is lower than the efficiency at social optimal level. Proof: See Appendix 7.2. For example the parameters 8 used in Casari and Plott (2003), when free mobility is allowed the results of the model are the same as the case with ∗ one locality, Xoneloc. = X1∗ = X2∗ = 128 . Given the parameters, we calculate Q P socially optimal surplus as opt = π − N e = 324. Surplus extracted in P Q unregulated pastures is unreg = π − N e = 128. Efficiency is evaluated as the ratio of obtained surplus at equilibrium to the socially optimal surplus, Q Q E ∗ = ∗ ·100/ opt . Hence efficiency of the unregulated pastures with free mobility at 39.5 percent. With homogeneous agents and symmetry of rules in localities, unregulated pastures free mobility equilibrium efficiency for one locality is the same as the unregulated locality equilibrium efficiency with no mobility. See Appendix 7.2 for the detailed derivation of the results. Observe that equilibrium grazing level is equal to the social optimum when each locality has one herder. 8

As an example we illustrate free mobility equilibrium given parameters used in Casari and Plott(2003) for one community: N = 8, a = 23/2, b = 1/16, c = 5/2, h = 4, k = 7, λ = 7. We will use this example throughout the paper.

21

3.3

Identical Sanctioning systems (symmetric equilibria) and free mobility

Suppose now both communities adopt sanctioning mechanism as in Casari and Plott (2003) where each community set λopt so that X ∗ = X opt , p∗ = 1. Therefore, each herder has three choice variables: max πis = e − cxis + [xis /Xs ] · f (Xs ) − Ii mi +

Si ,xi ,Ii

X

Iij (mj − k)

(10)

j6=i

The payoff of a herder in locality S with a sanctioning system differs from the payoff of his counterpart in a locality with unregulated pasture by the last two terms in equation (10). Fourth term in equation (10) is the amount of total fines imposed to herder i, if he violates the rules concerning the grazing level. Herder i pays fine in proportion to his exceeded animals. Fifth term determines the revenue generating possibility from inspection activity, when herder i inspects other herder j and j violated the threshold denoted by λ. The following conditions must hold for the maximization problem: ∂πis /∂xis = 0;

(11)

Si = l only if πil (xi , si ) ≥ πik (xi , si ), l 6= k, l = {1, 2}, k = {1, 2}; i = 1, ..., N ; (12) ( P 1 if i6=j Iij ≥ 1 (13) Ii = 0 if otherwise Given identical herders, above conditions imply: πi1 = πi2 for ∀i ∈ I = (1, ..., N )

(14)

The condition (11) is immediate from the maximization problem. Given the other herder’s appropriation level each herder chooses his level of extraction to maximize his utility given by equation (10). The condition in (12) also states that if the utility from being in locality one is greater than the utility of grazing in locality two, herder will choose locality one to graze, and otherwise locality two. By stability profits of being in either of localities must be equal for each homogeneous herder, πi1 = πi2 . Assuming symmetry of localities in terms of production functions, and identical herders this is true if the following condition is satisfied: Ns = N/2, where s indicates the locality. The sanctioning mechanism sets the threshold, λ, to achieve 22

efficient use of the resource. Thus, the grazing level for each herder is de∗ · N2+2 , where p∗ = 1 as characterized in Proposition 3. fined by x∗s = a−c−hp b ∗) Hence, total appropriation level is determined by X ∗ = a−(c+hp · NN+2 . See b Appendix 7.3. In equilibrium, given that all herders are identical, communities have same thresholds and grazing levels, and population divided between communities in symmetric manner and: X1∗ = X2∗ = X opt , N1∗ = N2∗ . In equilibrium every one inspects each other. Also, inspection excludes multiple penalties, P therefore, Ii = 1. Hence, we have mi = Iij mj . Observe that last two terms in equation 10 are equal to k = h(xi − λ). Proposition 7 In two localities with the identical sanctioning mechanisms, the symmetric free mobility equilibrium with identical herders is characterized by the following outcomes: X1∗ = X2∗ = X opt , N1∗ = N2∗ . Both communities appropriate at the social optimal level, X opt . Efficiency level in both communities is equal to 100 percent 9 if the monitoring cost equals zero. However, it is less than 100 percent if localities have a positive monitoring cost. Proof: See Appendix 7.3. Using the numerical example from Casari and Plott (2003), in equilibrium grazing level is equal to the social optimal defined without inspection activity, X ∗ = X opt = 72. Note that the social optimal level of appropriation does not take into account monitoring costs. However, in a sanctioning mechanism the monitoring cost is the loss, but fines are not anymore deadweight loss to the system. Thus, maximal attainable surplus subtracting the monitoring cost for the society yields overall efficiency: Q Q Q P Q E ∗∗ = ∗∗ / opt ·100 = ( ∗ − k)/ opt ·100 = 82.7 Community with lower monitoring costs has higher efficiency.

3.4

Asymmetric institutions (regulated vs. lated) and free mobility

unregu-

Consider the case of asymmetry in institutions governing both localities. Suppose locality one adopts the sanctioning mechanism (regulated denoted Q∗ Qopt Q Qopt E = / ·100 = ( −N e)/ ·100. One can also say sanctions restore efficiency, yet not without a loss, a monitoring cost is a waste of the resources. We characterize an outcome in terms of efficiency as an optimality of appropriation level and overall efficiency, which includes a monitoring cost. Efficiency and an overall efficiency coincide when a monitoring cost is equal to zero. 9

23

by subscript R) and locality two be unregulated (denoted by subscript U). In equilibrium herder’s grazing level in locality with inspection game determined a−c−hp∗ by x∗R = b·(N , while herders in unregulated locality chose the grazing level R +1) ∗ . Again stability of equilibrium requires no herder to be equal to xU = b·(Na−c U +1) wants to move; by the identical herders condition we have πi1 = πi2 . This condition is satisfied only when the number of herders in a locality with sanctioning mechanism is higher than the population in locality with unregulated pastures, NR > NU . See Appendix 7.4. Results of the free mobility asymmetric institutions model suggest that herders prefer organized community with a sanctioning system to unregulated pastures. Recall that proposition 4 states given the monitoring cost structure the regulated community has a higher welfare, hence higher per person welfare or profit. Convergence to the equilibrium occurs in such a way that herders move from unregulated community to regulated one until the profits are equalized. In equilibrium the locality with sanctioning system can accommodate more people than unregulated locality, because sanctioning mechanism keeps the efficient use of the resource. Again how long a sanctioning system can stand migratory pressure? How far it can accept new entries? We assume that regulated community adjusts its rules to the migration process. Community with sanctions may have new comers from neighborhood area, who are willing to accept the rules governing this locality and join the community. Hence, when population changes community decides on the new rules adjusted to the number of people within the locality to keep overall grazing level at the efficient level. Community decides on two issues such as threshold and the fines if someone violates. Optimal level of grazing for each herder is equal to xopt = X opt /Ns . In order to induce this grazing level, the fines are redefined as: h = a − c − (xopt ) · (Ns + 1) · b. To this end set up legal amount of animals as follows: λopt = xopt − k/h − ε. This adjustment to the migration process allows community with the sanctioning institution to keep the optimal level of grazing within the community no matter how many people it can serve. Equilibrium total appropriation in a community with sanctions equals to the appropriation at the socially optimal level. In contrast, appropriation in unregulated locality is above social optimal level. Proposition 8 1. In free mobility equilibrium with asymmetric institutions and identical herders, the locality with the sanctioning system has a higher number of individuals than the unregulated locality, NR∗ > NU∗ . The appropriation levels are as follows, X opt = XR∗ < XU∗ . 24

2. Assuming the community with sanctioning system adjusts its rules on the threshold for each herder and the punishment level for violators, in response to the migration process, total appropriation level in this community will remain the same at the social optimal level and equal to XR∗ = X opt . Efficiency of sanctioning system is at maximum. Efficiency of unregulated pasture improves as compared to the case where there is no sanctioning system neighboring the unregulated pasture. Here migration to regulated locality reduces the population in unregulated locality, hence the externality, yet appropriation is above social optimal level, XU∗ > X opt . See Table 1, and Figures 1 − 6 prepared for this case numerical example. When community adjusts the sanctioning structure, free mobility does not distort efficiency. In equilibrium efficiency of the sanctioning system is 100 percent excluding monitoring costs and appropriation level at XR∗ ≈ 72. In equilibrium total appropriation in unregulated locality is XU∗ ≈ 123.4. Efficiency of unregulated pasture improves from 39.5 to 49 percent because migration to locality with sanctions reduces the population in unregulated locality, hence the externality. From the Proposition 8 there is an equilibrium in which herders reside both in community with sanctions and community with unregulated pastures. Sanctioning system keeps efficiency at the social optimal, even if free mobility is in place. However, the community has to adjust the rules according the migration. Migration continues until the payoffs in both localities are equalized at equilibrium. When community one does not adjust rules according to the number of people in community, there is a deviation. On the contrary, when the threshold is adjustable the outcome of the equilibrium is at the socially optimal level. Observe that we draw this conclusion assuming that total population in the economy is fixed. Note that the locality with sanctioning system accommodates new entries until the equilibrium point where herders are indifferent between two localities. Therefore, we have existence of both types of institutions in two-locality economy analyzed in this section. In this and previous sections we assume that localities have the same size of the pasture while in the next section we relax this assumption.

25

3.5

Equilibria in localities with asymmetric sanctioning mechanisms

Communities may have distinct mechanisms for allocating a common pool resource due to the differences in the amount of grazing land contained within their boundaries or the monitoring costs. Localities with the larger amount of land (or lower monitoring cost) may impose higher threshold, while locality with limited grazing land (or higher monitoring cost) has to set low level of legally allowed animals. We could think about this case as either one of two scenarios: (i) due to the higher monitoring cost locality decides to set lower threshold; (ii) a higher productivity of land in the second locality lead to higher threshold in it. Therefore, we have two effects affecting herder’s profit in the second locality: increase in revenue due to increase of productivity of land and lower monitoring cost to detect violation. Differences in monitoring costs: Now we assume that localities with sanctioning mechanisms have distinct monitoring costs. Suppose localities are identical in terms of productivity and punishment level if someone violates. However, due to different monitoring cost structure in equilibrium herder in one locality has lower profit than herder in the second locality with lower monitoring cost. Stability condition requires the herder to be indifferent between two sanctioning systems, πi1 = πi2 , ∀i. This is true when number of herders in locality one that has higher monitoring cost is smaller than the population in locality two, N1∗ < N2∗ . See Appendix 7.5, Table 2. Corollary 1 : If all herders are identical in two asymmetric localities with distinct cost structure, k1 > k2 related to the enforcement of the sanctioning rules, the community 1 with higher monitoring cost can accommodate smaller number of people compared with the second community that has lower enforcement cost, N1∗ < N2∗ . Social optimal is obtainable, X1∗ = X2∗ = X opt , by setting different threshold levels, λ1 > λ2 . Along with, an adjustment of the sanctions to the migration is important to obtain social optimal level of appropriation. Communities may decide on the optimal threshold each time when the population changes in terms of number. Different size of pastures: Suppose now the productivity of land differ, a1 < a2 10 . Hence, to achieve the optimal grazing levels, two localities set different thresholds, 10

Recall that our production function is of type f (X) = aX − bX 2 .

26

opt λopt 1 < λ2 . Then, in equilibrium population in locality one is smaller than the population in the second locality, such that, N1 < N2 . See Appendix 7.6 and Table 3.

Corollary 2 : Free mobility equilibrium in two asymmetric localities with distinct productions, a1 < a2 , is characterized by the asymmetry of outcomes, X1opt < X2opt , N1∗ < N2∗ . The number of herders in locality with lower land productivity is smaller compared to other locality. Both communities have appropriation efficiency of 100 percent. Social optimal is obtainable by setting opt different level of threshold for each community, λopt 1 > λ2 . When production functions differ, the locality with higher productivity can accommodate more population in equilibrium other things equal. Notice that under free mobility conditions in order to induce optimal level of grazing, both communities must adjust their existing rules regarding the threshold and inspection, in response to their population change induced by the migration. Thus, free mobility leads one more condition to be satisfied in order to get social optimal under asymmetric sanctions. When both factors such as productivity of land (a1 6= a2 ) and monitoring cost (k1 6= k2 ) affect the pastures, the prevalence of one factor over other will determine the equilibrium outcome. If both effects have the same power, the outcome stays the same. Conjecture 1 In equilibrium of asymmetric sanctioning systems the size of population positively related to the productivity of land; however, negatively related to characteristics of community such as inspection cost.

3.6

The choice of institutions is endogenous

We define the benefit of adopting sanctioning system by: B(n) = f (X opt ) − f (X N E ) − c(X opt − X N E );

(15)

Per person benefit is b(n) = B(n)/n. Note that f (X opt ) − cX opt [f 0 X U 0 − cX U 0 ] · n − [f (X U ) − cX U ] ∂b(n) = −[ ] − [ ]; (16) ∂n n2 n2 Therefore, ∂b(n) πR + k πU X U 0 [f 0 (X U ) − c] = −[ ]+[ ]−[ ]; ∂n n n n 27

(17)

Recall that b(n) = π R +k −π U > 0 for ∀n > 1. We know that f 0 (X U ) < 0 when X0 < X U < Xmax , where f 0 (X0 ) = 0 and the production function takes the maximum value. Also X 0 = ∂X/∂n > 0. The benefit of adopting the sanctioning system is decreasing in population level, ∂b(n) < 0. See Appendix ∂n 7.7. In previous sections we assume that sanctioning system is welfare improving for community that adopts sanctions. In particular, per person benefit of sanctioning system always exceeds the costs, b(n1 ) > k for ∀n1 ∈ [0, N ]. However there are cases where monitoring cost is too large such that sanctioning system is welfare improving until the population size reaches some critical level beyond which it is better to remove the regulation. In locality one herders choose sanctions if and only if b(n1 ) ≥ k. Define c b(n1 ) = k if n1 = N 1 c k, for ∀n1 < N 1 c b(n1 ) < k for ∀n1 > N .

(18) (19) (20)

c > N , then trivial. See Appendix 7.7 for: If N 1

• case (A), where benefit of sanctions always exceeds the cost of adopting sanctions and • case (B), where benefit of sanctions always below the cost of adopting sanctions. We also consider cases: • case (C) The benefit of adopting sanctions equals the cost of sanctions when the population size in regulated community is greater than the half of entire population, • case (D) The benefit of sanctions equals the cost of sanctions when the population in regulated locality is smaller than half of total population, and • case(E) The benefit of sanctions equals the cost of sanctions exactly c < N. with the half of population. All three cases occur when 0 < N 1 We start with two symmetric localities. In free mobility equilibrium suppose herders in community one decide on the regime by majority voting: sanctions or no regulation. Locality two is unregulated. There are two possible outcomes depending on the monitoring cost, k relative to the benefits of adopting sanctions. 28

Proposition 9 In two-community economy with homogeneous agents, where community one decides the appropriate regime the choice of institutions changes depending on the monitoring cost structure for the locality. Community with c >N ¯ /2 chooses sanctions and in equilibrium the cost structure such that N 1 ∗ ∗ c ≤ N ¯ /2 chooses n1 > n2 , while locality with the cost structure such that N 1 ∗ ∗ no regulation and in equilibrium n1 = n2 . Proof: See Appendix 7.7 for the details.

4

Discussion

Unregulated pasture use presents a problem of overgrazing in which the short-term benefits lead rational herders to over-appropriate the resource. The issue on best governance of common pool resources has been studied in theoretical literature, field research and laboratory experiments since the influential work of Hardin (1968). There are three solutions for the common problem: state control, privatization and community based management. Previous literature finds that sanctions can prevent overuse of resource. In particular, Casari and Plott (2003) examined the inspection game that can improve efficiency close to the social optimal level. We extend this finding by free mobility condition allowing more localities. Standard common pool resource model considers the over-use problem within one community. This paper explores the importance of institutions such as sanctioning in a two-community, identical herders economy. Our framework extends existing research allowing free mobility between different localities. Our results of the CPR with free mobility model shows a locality that has a sanctioning system is able to maintain the social optimal level even under high migratory pressure if it can adjust the rules within it in accordance to migration. We obtain following results: (i) asymmetric equilibrium with sanctioning mechanism in one locality and unregulated pastures in the second locality yields an unequal distribution of population; more herders chose to live in locality with sanctioning system. We find that appropriation in unregulated locality is higher than in locality with sanctions and efficiency level of the unregulated pastures is lower despite fewer herders. However, efficiency of unregulated pasture improves when neighbor community adopts sanctions compared to the efficiency of the isolated unregulated locality; (ii) when 29

herders decide on the regulation, the outcome depends on monitoring cost structure such that locality with a lower monitoring cost adopt sanctions while locality with higher monitoring cost refrain from doing so. When the two-locality economy is explored total appropriation level in community with sanctioning system will remain at the social optimal level if community with sanctions is able to adjust rules to the population change induced by the migration. This analysis of institutions under free mobility conditions allows us to draw several conclusions. Identical herders facing multiple localities with different institutions choose both, community with sanctioning mechanism as well as unregulated locality. This is due to the assumption of a fixed population in the economy. Returning to the research question, identical herders’ two-community model predicts co-existence of both institutions. Thus, we have allocation in unregulated pasture and allocation of the resource in sanctioning system too. Second, sanctioning system stands pressure from outsiders when mobility is costless and monitoring cost is low enough. In equilibrium, locality with sanctions has a larger population; efficiency is higher compared to unregulated pasture. Therefore, when mobility is costless, which is in fact true in our context, sanctioning system favors over the unregulated Nash allocation. Third, sanctioning mechanism developed by Casari and Plott (2003) survives free mobility condition without any loss to the system if community adjusts to the migration process, restricts membership or moves to other locality. Moreover, it improves efficiency of the unregulated pasture, because fewer herders are left in the unregulated locality. We also note that free mobility may distort equilibrium of the sanctioning system if monitoring costs are high; sanctions alone without adjustment are not sufficient to prevent CPR problem when multiple localities with free mobility are examined. Threshold needs to be adjusted optimally to the population change and monitoring cost has to be low.

5

Further research

This model of identical herders allows us to separate the effect of cost heterogeneity in our subsequent results. In the next paper we investigate how heterogeneity changes our results. There are two types of problems associated with the common pool resource problem considered under the free mobility conditions. Rational herders have incentive to over-appropriate the resource. This problem can be avoided by the sanctioning mechanism, one 30

of which we explored in the identical herders setting in previous sections. When herders are heterogeneous, there is a concern on whether the community can be agreed upon the threshold within the community, which is chosen by voting rule. There is a conflict of interest regarding the legal threshold, some herders prefer to have large herd size while others do not. This second type of problem leads us to examine different types of herders’ model in the presence of free mobility condition. We may consider the extension of the model incorporating types of herders’ model. Suppose herders differ in terms of ability to manage the production activity. Since their effort cost is higher, herders with low abilities prefer to put less effort into production in less crowded areas based on the extensive method of grazing. On the other hand, there are herders with high abilities, who are willing to put more effort with low cost into the production, and therefore are willing to extract as much as possible from the commons. We are interested in the effect of cost (type) distribution on the location choice pattern across communities. More specifically, we explore a complete information case followed by the incomplete information. We will consider the game of incomplete information when herders differ by their cost of maintaining one animal and the types of herders are not observable. The distribution of types of players and the rules governing two localities are common knowledge. Herders can observe overall meadow’s situation, but can’t distinguish between types of players either defector, who has high grazing level that is above the threshold, and cooperator, who keeps harvesting level at the legally allowed level. Bayesian game of incomplete information and social planner’s mechanism design approach to misrepresentation by types of herders are in our research target. Next, we may study the cases when trespassing is in place. In previous sections grazing in one locality excludes the grazing in another locality. Herders may belong to only one of two localities. However, users can appropriate resources both in their own locality where they do live and in the other locality. Suppose locality one has sanctioning mechanism and locality two is unregulated. Then herder who chooses locality two and grazes in locality two will have no punishment. In contrast, the herder who lives in site one with sanctioning mechanism and appropriate in the same locality can be spotted and punished. If a user from site two appropriates in site one he also can be inspected and punished for any appropriation above zero. However, when user from site with sanctions appropriates resource in unregulated locality he cannot be punished. We can explore this condition when variety of institutions occupies the localities. 31

Also, we may study costly mobility when migration decision creates additional costs in comparison with free mobility. Also assumption of fixed population can be relaxed. We may add voting into the model such that NE can be sustained by majority voting. We may relax assumption of exogenous institutions such that community chooses the institutions. Finally, we may test theoretical predictions using laboratory experiments (work in progress). Subjects will be recruited from the Academy of Management campus (Mongolia). Proposed treatments are: (i) Symmetric institutions (no sanctions) with free mobility setting investigate the effect of free mobility; (ii) Asymmetric institutions (sanctions in one locality and no regulation in other) setting with free mobility investigate effect of institutions. We compare the result to the benchmark with no free mobility. The goal is to see: (a) How free mobility affects efficiency of the resource use and monitoring behavior in sanctioning mechanism; (b) How localities are populated when mobility is costless; (c) What is effect on the total grazing level; (d) How the institutions evolved change when people decide on the preferred regulation within the locality.

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6

6.1

Appendix A: IDENTICAL HERDERS AND ONE LOCALITY PROBLEM Social optimal outcome with identical herders

For one locality social planner solves: max xi

X

πi = n · [e + xi · f (X)/X − c · X];

(21)

Where f (X) is a concave function. max x i

X

πi = n · e + f (X) − c · n · xi = n · e + f (X) − c · X;

(22)

Where X = nx, since all herders are identical. First order condition is given by: f 0 (X) − c = 0, or f 0 = c. The level where marginal revenue from extraction is equal to the marginal cost of extraction determines socially optimal extraction level. From above, when f (X) = aX − bX 2 , the social optimal for each herder and for the locality is characterized by respectively: xopt = (a − c)/2bn; X opt = (a − c)/2b. i

6.2

Selfish Nash equilibrium

In a standard CPR game equilibrium as in Falk, Fehr and Fischbacher (2002), each herder maximizes his profit given by: xi max πi = f (X) + e − cxi , (23) xi X Where f (X) is a concave function. Suppose f (X) = aX − bX 2 , X = Substitute the production function into the profit function: xi πi = e − cxi + [aX − bX 2 ] X 38

Pn

i=1

xi .

(24)

First order condition dictates : ∂πi ∂πi X − xi xi 2 =0⇒ =[ ][aX − bX ] + [a − 2bX] − c = 0; xi xi X2 X xi xi + bxi + a − 2bxi − c = 0; X X a − c − b(X − xi ) − 2bxi = 0;

a − bX − a

a − c − bX−i = 2bxi , where X−i =

Pn

j6=i

(25) (26) (27) (28)

xj ;

a−c 1 − X−i . (29) 2b 2 Note that X−i = (n − 1)xi by identical herders condition. Therefore, equilibrium grazing level for each herder is determined by: x∗i =

x∗i =

a−c b(n + 1)

(30)

Total appropriation at Nash equilibrium is given by: X∗ =

n a−c · n+1 b

(31)

If n > 1, then this is higher than the social optimal defined in A.1: X∗ =

6.3

n a−c a−c · > = X opt . n+1 b 2b

(32)

Equilibrium of the Inspection game in one locality

Proposition 3: Suppose locality has a sanctioning mechanism, where the threshold and punishment levels are set as λ = xopt − k/h − ε and h = a−c−xopt (n+1)b, respectively. Then the sanctioning mechanism can support the socially optimal level of harvesting as subgame perfect Nash equilibrium, X ∗ = X opt . Also in this equilibrium every one inspects each other, p∗ = 1. Following Casari and Plott (2003), when locality is regulated by a sanctioning mechanism each herder solves: max πi = xi ,Ii

X xi f (X) + e − cxi − Ii mi + Iij (mj − k) X j6=i

39

(33)

FOC for the maximization: ∂πi /∂xi = 0 (

Ii =

(34)

P

1 if i6=j Iij ≥ 1 0 if otherwise

∂πi X − xi xi =[ ][aX − bX 2 ] + [a − 2bX] − c − hp∗ = 0 2 ∂xi X X xi xi + bxi + a − 2bxi − c − hp∗ = 0 X X Where h is the unitary fine for each extra animal. a − bX − a

(35) (36) (37)

a − c − hp∗ − b(X − xi ) − 2bxi = 0

(38)

a − c − hp∗ − bX−i = 2bxi

(39)

a − c − hp∗ 1 − X−i 2b 2

(40)

x∗i = Where

X−i =

n X

xj

j6=i

Each herder choice on how much to extract from common is given by: x∗i = a−(c+hp∗ ) . We know that equilibria are characterized by 11 : b(n+1)  ∗) n  X ∗ =n+1 · a−(c+hp   b   ∗ e   0 if E(X ) < nx (X ∗ , p∗ ) = ∗ ∗  p = 1 if E(X ) > nxe       ∈ [0, 1] if E(X ∗ ) = nxe

(41)

Here x˜ ≡ k/h + λ. If the perceived probability is positive, the best response for the agent is to use the resource less than in the no inspection case. Prediction of this game depends on the assumption of identical players 11

If in Casari and Plott (2003), we actually have p∗ = 0 if E(X ∗ ) ≤ n˜ x, p∗ = 1 if ∗ E(x ) > n˜ x. However, in general, if E(X ) = n˜ x, each agent is indifferent between inspecting or not, and hence we may have p∗ ∈ [0, 1] be part of equilibrium ∗

40

(endowment, cost, and production). In symmetric equilibrium we get x∗i = X ∗ /n, p∗i = p∗ for ∀i since all players in the setting are homogeneous in term of preferences. We claim that under the appropriate parameter values on k, λ and h, there is equilibrium in inspection game with (X ∗ , p∗ ) = (X opt , 1). To show this, set λopt = xopt − k/h − ε, and h = a − c − xopt (n + 1)b, where ε denotes an arbitrary small positive number. Then we have x˜ ≡ k/h + λ = k/h + xopt − k/h − ε = xopt − ε; hence n˜ x = X opt − nε. Also, i ∗ opt X = n/(n + 1) · (a − c + [a − c − x · (n + 1) · b] · p)/b. Suppose p = 1, then X ∗ = nxopt = X opt . Hence from 41, we have X ∗ > n˜ x and therefore p∗ = 1 is ∗ ∗ opt consistent with 41. Hence, (X , p ) = (X , 1) is equilibrium. If ε = 0, then we have a weak equilibrium, where X ∗ = X opt , p∗ = 1. This equilibrium is weak because even though everyone is indifferent between inspecting or not, they all choose to inspect in Equilibrium. Set λopt = xopt − k/h such that x˜ = k/h + λ = k/h + xopt − k/h = xopt . Therefore, we have n˜ x = X opt . opt Also as before, set h = a − c − x (n + 1) · b. Then from 41 if p = 1, then X ∗ = nxopt = X opt = n˜ x; hence we have X ∗ = X opt = n˜ x. Therefore, ∗ ∗ opt (X , p ) = (X , 1) is equilibrium. However, by lowering the threshold a bit, ε > 0, we can ensure that the equilibrium is strong, and each herder’s unique best response is to inspect with probability one 12 , p = 1.

6.4

Appendix A.4. Choice of Institution

From Proposition 1 we know that the social optimum is derived from maximizing ne + f (X) − cX. We know from proposition 3, that the community can set λopt so that X R = X opt . Total surplus generated with Q sanctioning mechanism including the monitoring cost is: R = n · π R = n · e + f (X opt ) − cX opt − n · k. Q The group profit with no regulation is defined by U = n·e+f (X U )−cX U . 12

There may be equilibria in mixed strategies if 0 < p < 1. The higher is the probability of inspection; the closer is the equilibrium to the social optimum. However, recall that the monitoring is costly activity where society wasting some of the resources to detect violator, k > 0. The higher is the monitoring cost to detect one person; given level of violation the lower is the efficiency of the sanctioning system. Moreover, perfect monitoring when everyone is inspecting has higher deadweight losses than the equilibrium, where only fraction of actions is inspected. Hence, optimal policy involves symmetric inspection by players with some positive probability less than one. However, probabilistic inspection is the extension of this model and is not the objective of this project. It is useful to explore, especially in societies with high level of monitoring cost, where desired outcome is to have less than perfect monitoring.

41

By Proposition 3 we know that the resource is exploited optimally under the sanctioning system. Hence, f (X opt ) − cX opt > f (X U ) − cX U . Then the total profit captured from the resource in inspection game subtracting the cost of inspection is higher than the profit derived by the group with no regulation if and only if: f (X opt ) − cX opt − n · k ≥ f (X U ) − cX U . From here the choice of the alternative institution is determined by: choose sanctions if k ≤ [f (X opt )−f (X U )−c(X opt −X U ]/n ; choose unregulated otherwise, where n is the total population within the communities. We choose sanctioning system if and only if the per person total benefit of adopting sanctioning system is greater than equal the monitoring cost per person. Note that X opt is endogenous does not depend on population size while X U depends on n. We define N SB = f (X opt ) − f (X N E ) − c(X opt − X N E ) − nk ∂(N SB) = −[f 0 (X U ) · X U 0 − cX U 0 ] − k; ∂n

(42)

∂(N SB) = −X 0 [f 0 (X U ) − c] − k; ∂n

(43)

Since f 0 (X) = a − bX = a − b ·

(a − c)n ; b(n + 1)

a(1 − n) + 2cn < 0; n+1 because n > 1 and f 0 (X) =

for X0 < X < Xmax

X0 = if k > X 0 (f 0 − c), then

7

∂(N SB) ∂n

a−c > 0; b(n + 1)2

(44) (45)

(46)

< 0.

Appendix B IDENTICAL HERDERS AND TWO LOCALITIES PROBLEM

Suppose now there are two localities, indexed 1 and 2. We will denote Xs , Ns harvesting and population levels in locality, S; and xs per person harvesting level in locality, S. 42

7.1

Social optimal

Given Ns , S, in each locality social planner chooses to maximize sum of herders’ profits. Social planner problem is presented by: max

X

Si ,Xs ,Ns

πs = Ns e + f (Xs ) − cXs

(47)

For X opt FOC: f 0 (X) − c = 0. Hence, marginal benefit equals marginal cost: f 0 (X) = c. I.e., in each locality Xs = X opt , xs = X opt /Ns . Hence the total profits across localities are maximized: X

X

πi =

X

πi1 +

X

πi2

πi = N1 [e − cx1 − f (X1 )/N1 ] + N2 [e − cx2 − f (X2 )/N2 ];

X

πi = N e − c(a − c)/b + 2f (Xs ) where f (X1 ) = f (X2 )

(48) (49) (50)

Thus maximum profit does not depend on the distribution of herders between localities. Social optimal outcome in each locality is independent of the population distribution across localities, Xsopt = a−c . Per person grazing 2b level xs = X opt /Ns . Proof: When f (X) = aX − bX 2 social planner problem rewritten as: X

πis = Ns e + aX opt − b[X opt ]2 − cX opt

(51)

Here sj = {1, 2} stands for localities. By FOC: a − c − 2bX opt = 0; Total grazing level in each locality is given by Xsopt = a−c . Hence, we have 2b a−c . xopt = i 2bNs

7.2

Unregulated pastures and free mobility case

In both localities herder solves the following problem: max πis = e − cxis + [xis /Xs ] · f (Xs ) Si ,xi

(52)

Necessary conditions for the optimum: ∂πis /∂xis = 0; sj = l only if πil (xi , si ) ≥ πik (xi , si ), l 6= k, l = {1, 2}, k = {1, 2} 43

(53) (54)

Since all herders are identical following must hold: πi1 = πi2 for ∀i

(55)

First order condition states: a − c − bXs0 xis + bXs x0is = 0; a − c − b[xis + Xs ] = 0; a − c − b[xis + xis Ns ] = 0; a − c − b[xis (Ns + 1)] = 0 where

(56) (57) (58) 2 X

Ns = N

(59)

s=1

(60) Equilibrium grazing level for each herder and in total for both localities must a−c s satisfy: x∗s = b·(N ; Xs∗ = a−c · NNs +1 ; S ∈ {1, 2}. Herder chooses the locality b s +1) 1 for the residence if the profit from locality one is greater or equal to the profit in locality two, if πi1 ≥ πi2 , Si = 1. Therefore, in equilibrium with homogeneous herders free mobility stability condition must hold: πi1 = πi2 which is possible if and only if Ns = N1 = N2 = N/2. Thus equilibrium grazing level in both localities is the same, X1∗ = X2∗ , and given by: x∗s = a−c · N2+2 , Xs∗ = a−c · NN+2 , where N = N1 + N2 . Under the unregulated b b pasture use regime we have overgrazing in both localities and equal split of herders across localities. Proof: In equilibrium no herder wants to move if πi1 = πi2 . Hence [f (X1 )−cX1 ]/N1 = [f (X2 )−cX2 ]/N2 . Suppose N1 6= N2 , then f (X1 )−cX1 6= f (X2 )−cX2 . Then this is not equilibrium. Hence, migration takes place until N1∗ = N2∗ .

7.3

Symmetric inspection game and free mobility

As in Proposition 3, suppose each regulated community sets (λ, h) so that Xs = X opt . When both localities adopt sanctioning mechanism each herder problem is given by: X xi Iij (mj − k) (61) max πi = f (X) + e − cxi − Ii mi + Si ,xi ,Ii X i6=j Necessary FOC for the optimum: ∂πis /∂xis = 0; 44

(62)

sj = l only if πil (xi , si ) ≥ πik (xi , si ), l 6= k, l = {1, 2}, k = {1, 2}; i = 1, ..., N (63) ( P 1 if I ≥ 1 ij i6=j (64) Ii = 0 if otherwise Note that last two terms in (61) are equal to kp∗ = h(xi − λ)p∗ . From proposition 3 we know that the grazing level for each herder in one locality is given by: x∗s = a−(c+h) when p∗ = 1. By symmetry πi1 = πi2 must hold, then b(Ns +1) Ns = N/2 and x∗s = a−c−h · N 2+2 . Hence Xs∗ = (a−c−hp∗ )·N/b(N +2) = X opt . b Where x˜ ≡ k/h + λ. The threshold is set equal to λ = xopt − k/h − ε such that equilibrium appropriation in both localities is at the social optimal. Proof: To show that N1 = N2 , suppose on the contrary, that N1 > N2 , then ⇒ πi1 < πi2 . Therefore, herders will move to locality 2 where the profit is higher until the moment when population in two communities equalized. Hence, in equilibrium, N1∗ = N2∗ , no matter what is initial distribution of population, when communities are identical in terms of monitoring costs and other characteristics such as land productivity.

7.4

Asymmetric institutions (regulated vs. lated) and free mobility

unregu-

Proposition 8: 1. In free mobility equilibrium with asymmetric institutions and identical herders, the locality with the sanctioning system has a higher number of individuals than the unregulated locality, NR∗ > NU∗ . The appropriation levels are as follows, X opt = XR∗ < XU∗ . 2. Assuming the community with sanctioning system adjusts its rules on the threshold for each herder and the punishment level for violators, in response to the migration process, total appropriation level in this community will remain the same at the social optimal level and equal to XR∗ = X opt . Efficiency of sanctioning system is at maximum. Efficiency of unregulated pasture improves as compared to the case where there is no sanctioning system neighboring the unregulated pasture. Here migration to regulated locality reduces the population in unregulated locality, hence the externality, yet appropriation is above social optimal level, XU∗ > X opt . Proof: Proposition states in free mobility equilibrium with two distinct institutions population in regulated community with sanctioning mechanism exceed the population in unregulated locality: NR∗ > NU∗ . By contradiction, it is sufficient to show that if NR∗ ≤ NU∗ , and then we can’t have an equilibrium. 45

We know that in free mobility equilibrium, we have: ∂πiR /∂xiR = 0

(65)

∂πiU /∂xiU = 0

(66)

πiR = πiU

(67)

First two conditions in (65) and (66) are immediate from the optimization problems. Condition (67) hold since all agents are identical, if not, they would move. First suppose, by contradiction, that NR∗ = NU∗ = n. Proposition 3 and 1 combined together state that an equilibrium grazing level in regulated locality is socially optimal: XR∗ = X opt . Hence, x∗iR = X opt /n = a−c . From 2bn proposition 2 equilibrium grazing level by each player in locality with no rega−c ulation is x∗iU = b(n+1) = XnU . By free mobility equilibrium, given symmetry, the profit for herder who live in regulated locality must equal the profit level if he join unregulated locality. Hence condition (67) implies: πiR =

xopt xiU f (X opt ) − cxopt − k = f (XU ) − cxiU = πiU opt X XU

(68)

By symmetry we can rewrite the stability constraint (68) as: f (XU∗ ) f (X opt ) − cxopt − k = − cx∗iU n n

(69)

Where xopt and xU represent the grazing levels for herder in regulated and unregulated localities, respectively. From here: f (X opt ) f (XU∗ ) − cX opt /n − k = − cXU∗ /n n n

(70)

Now multiply both sides by n and get f (X opt ) − cX opt − kn = f (XU∗ ) − cXU∗

(71)

We know that the cost structure is the following: k ≤ [f (X opt ) − f (X U ) − c(X opt −X U )]/n; the sanctioning mechanism has higher benefit. We conclude that regulated community produces higher welfare, hence higher per person welfare, πiR > πiU . Selfish individuals will migrate to location with the sanctions to capture higher benefits. Therefore, it can’t be the case that the population in both communities is of equal size, NR∗ = NU∗ . Now consider 46

what happens if people start migrating from locality 2 to locality 1. Because ∂πiR < 0 and ∂π∂niU < 0. This implies that πiR decreases and πiU increases as ∂n people move from unregulated to regulated locality. Then, by continuity of profit functions πiR , πiU and by symmetry of localities exists NR , NU s.t. N > NR > NU > 0, NR + NU = N ;

(72)

πiR (NR ) = πiU (NU )

(73)

with This is the equilibrium. We know that f (X) − cX > 0, k > 0. Then, ∂πiR f (X opt ) − cX opt (a − c)2 =− = − < 0; ∂NR NR2 4bn2

(74)

2(a − c)2 ∂πiU X 0 (f 0 − c) − (f (X) − cX) = − = < 0. ∂NU NU2 b(n + 1)3

(75)

Per person benefit in regulated locality, [f (X opt ) − cX opt ]/NR , is decreasing in population size while the monitoring cost per person stays constant. Similarly, per person profit in unregulated community is decreasing in population size. Hence, in equilibrium we have NR∗ > NU∗ . The equilibrium partition of herders yields higher population in community with sanctioning mechanism compared to unregulated locality.

7.5

Asymmetric sanctioning systems with free mobility

Suppose we have the difference in monitoring cost between communities. Community 1 has higher monitoring cost, while community 2 has lower enforcement cost, k1 > k2 . We claim that social optimal is obtainable by setting different threshold levels, and in equilibrium the population in locality 2 is higher, N1∗ < N2∗ . Proof: The profits in equilibrium are: f (X opt )/N1 − cX opt /N1 − k1 = f (X opt )/N2 − cX opt /N2 − k2

(76)

By contradiction, suppose the population is of equal size, N1 = N2 = n. Then the locality with the higher monitoring cost has lower profit: f (X opt )/n − c/n − k1 < f (X opt )/n − c/n − k2 47

(77)

Hence the migration from locality 1 into locality 2 will take place until herders are indifferent. If N1 > N2 = n is true, then again, we have migration to locality 2 due to higher monitoring cost and lower per head yield in locality 1. Hence in equilibrium we have N1∗ < N2∗ = n.

7.6

Asymmetric localities with different size pasture

Claim: Suppose a1 < a2 . We show that X1opt > X2opt and N1∗ > N2∗ . Proof: Note that f (X1 ) < f (X2 ). Suppose the monitoring cost per person is the same across communities: k1 = k2 = k. By contradiction, let N1∗ = N2∗ = n. In equilibrium player is indifferent between two localities: πi1 = πi2 , thus π10 = π20 . Hence by stability condition we must have: f (X1 )/n − cX1 /n − k = f (X2 )/n − cX2 /n − k

(78)

Further, we have f 0 (X1∗ )/n − c/n = f 0 (X2∗ )/n − c/n

(79)

Marginal product in locality with lower productivity is lower: f 0 (X1∗ ) < f 0 (X2∗ )

(80)

f 0 (X1∗ )/n < f 0 (X2∗ )/n

(81)

Then, must hold. Hence this can’t be equilibrium when population is the same across localities. Thus N1∗ < N2∗ . Hence locality with the higher size of pastures available can accommodate more herders.

7.7

Endogenous institutions

Suppose community 1 decides on regulation and locality 2 has no regulation at all. We relax the assumption that benefit of regulation is always greater than the cost of adopting sanctions. As long as b(n1 ) > k, community 1 will continue with regulation regime. However, if too many people enter community 1, we may no longer have b(n1 ) > k, in which case community 1 would switch to no regulation. Here we explore how the equilibria between the two localities depend on the parameter values, assuming that ∃n1 < N such that b(n1 ) < k. In free mobility equilibrium π1 (n1 ) = π2 (n2 ). Let’s 48

consider whether we may have b(n1 ) = k in equilibrium, where b(n1 ) is a per person benefit of adopting sanctions in community 1: b(n1 ) =

f (X opt ) − cX opt f (X U (n1 )) − cX U (n1 ) − n1 n1

(82)

Define the per person net benefit of adopting sanctions 4 = b(n) − k = π1R − π1U 4=

f (X opt ) − cX opt f (X U (n1 )) − cX U (n1 ) −k− n1 n1

(83) (84)

and find that ∂4 f (X opt ) − cX opt [f 0 (X U )X 0U − cX 0U ] · n1 − [f (X U ) − cX U ] =− − ∂n1 n21 n21 (85) By definition f (X opt ) − cX opt πR = − k; (86) n1 then f (X opt ) − cX opt = π R + k; (87) n1 From here ∂4 [π R + k] π1U [n · π1U ]0 =− 1 + − ; (88) ∂n1 n1 n1 n1 ∂4 [π R + k] π1U X 0 · (f 0 − c) =− 1 + − ∂n1 n1 n1 n1

(89)

∂4 π R + k − π1U + X 0 · (f 0 − c) =− 1 ; ∂n1 n1

(90)

We know that π1R + k > π1U for ∀n > 1. Also X U 0 > 0, f 0 (X U ) < 0, c > 0, k ≥ 0 Then

∂4 ∂n

(91)

< 0 if b(n) > |X U 0 (f 0 (X U ) − c)|. Thus, we show that π1R + k − π1U > −X 0 · (f 0 − c)

f (X opt ) − cX opt − [f (X U (n1 ) − cX U (n1 )] > |X U 0 (f 0 (X U ) − c)| n1 49

(92) (93)

Rewrite equation f (X opt ) opt f (X U (n1 ) U ·x − ·x +[c(xU −xopt )] > |X U 0 f 0 ·(X U )−X U 0 ·c)| (94) X opt XU We know that xU − xopt > 0 and b(n) > 0. Also πR + k = πU = hence

f (X opt ) · xopt − cxopt > 0, X opt

(95)

f (X U ) · xU − cxU > 0, XU

(96)

f (X opt ) f (X U ) > >c X opt XU

(97)

X opt < X0 < X U hence by concavity , f (X opt ) > f (X U )

(98)

Recall,

Therefore,

Thus,

f (X opt ) f (X U ) − >0 n1 n1 f (X opt ) opt f (X U ) U ·x − ·x >0 X opt XU f (X U ) U f (X opt ) · xopt · X U · x · [ opt − 1] > 0 XU X · f (X U ) · xU

(99)

(100) (101)

Note that X U 0 < xU and f (X U )/X U > f 0 (X U ). This is follows from concavity, f 0 (X opt ) = c > 0, f 0 (X U ) < 0, hence f 0 (X U ) < c (102) From equation (97) follows f 0 (X U ) < c <

f (X U ) XU

(103)

Therefore, f (X U ) U · x [ζ − 1] + [c(xU − xopt ] > |X U 0 f 0 (X U ) − X U 0 c)| XU 50

(104)

where ζ =

f (X opt )·xopt ·X U X opt ·f (X U )·xU

> 1 and

f (X opt ) − cX opt − [f (X U (n1 ) − cX U (n1 )] > |X U 0 (f 0 (X U ) − c)| n1

(105)

Hence, b(n) > |X U 0 (f 0 (X U ) − c)| and ∂4 < 0 for any concave function with ∂n per unit cost in the range c ∈ [0, −f 0 (X U )]. Also, with f (X) = aX − bX 2 we have f (X U ) − cX U πU = (106) n1 (a − bX U − c)X U π = n1 a−c a−c π U = (a − b − c) · b(n1 + 1) b · n1 · (n1 + 1) U

and

πR =

(108)

(a − c)2 b · (n1 + 1)2

(109)

f (X opt ) − cX opt −k n1

(110)

πU = with regulation,

(107)

(a − bX opt − c)X opt −k n1 a−c a−c π R = (a − b − c) − k; 2b 2bn1 πR =

(111) (112)

(a − c)2 π = − k, 4bn1

(113)

∂4 (a − c)2 2(a − c)2 =− − <0 ∂n 4bn21 b(n1 + 1)3

(114)

R

Communities may have different monitoring costs 0 < kmin < k1/2 < kmax , ¯ , then k = kmin . Together, such that: If π R = π U at n = N ¯; π1R (n1 ) > π1U (n1 ) if n1 < N ¯; π1R (n1 ) = π1U (n1 ) if n1 = N ¯. π1R (n1 ) < π1U (n1 ) if n1 > N 51

(115) (116) (117)

¯ /2, then k = kmax . Together, If π R tangent to π U at n = nmax < N π1U (n1 ) > π1R (n1 ) if n1 < nmax ; π1U (n1 ) = π1R (n1 ) if n1 = nmax ; π1U (n1 ) > π1R (n1 ) if n1 > nmax .

(118) (119) (120)

¯ /2, then k = k1/2 . Together, If π R = π U at n = N ¯ /2; π1R (n1 ) > π1U (n1 ) if n1 < N ¯ /2; π1R (n1 ) = π1U (n1 ) if n1 = N ¯ /2. π1R (n1 ) < π1U (n1 ) if n1 > N

(121) (122) (123)

Now we check whether the profit schedules with and without regulation cross each other. From Proposition 3 if monitoring cost is low enough for ∀n > 1, 4 > 0. We know limn→∞ π R = −k; (124) limn→∞ π U = 0;

(125)

limn→∞ 4 = −k < 0.

(126)

Hence, When n1 = 1 we do not have overgrazing problem and no need for sanctions, hence we exclude this range from the analysis. When the population is small n1 = 2 and k is low enough, 4 > 0. With continuous profit functions on the interval [2, N ] and monotonically decreasing benefit per person, ∂4 < 0, if ∂n monitoring cost is low enough, k ∈ [0, kmax ], by intermediate value theorem profit schedules cross each other at least once.13 • Case (A) If b(n1 ) > k, where k ∈ [0, kmin ] is small enough for ∀n1 ≥ 0, then in equilibrium by Proposition 8 community 1 has sanctions. 2 Community 2 has no regulation. Therefore, n∗1 > n∗2 if k < (a−c) − 4bn (a−c)2 14 for ∀n > 1. b·(n+1)2 • Case (B) If b(n1 ) < k, where monitoring cost is too large to keep sanctions k > kmax for ∀n1 ≥ 0, then locality 1 never adopts sanctions 13

Parameters are a = 23/2, b = 1/16, c = 5/2. Note also that profits with and without regulation cross each other twice when k < kmax U U opt opt 14 − f (X )−cX indicates the benefit of sanctions per person Note b(n) = f (X )−cX n n while b denotes the parameter of production function f (X) = aX − bX 2

52

2

2

(a−c) and in equilibrium n∗1 = n∗2 . In this case k > (a−c) − b·(n+1) 2 for 4bn ¯ > n1 > 1 in case (A) and 4 < 0 for ∀n > 1. Note that 4 > 0 for ∀N ¯ ∀N > n1 > 1 in case (B).

f π1R (n1 ) if n1 < N 1 U R f π1 (n1 ) = π1 (n1 ) if n1 = N1 ; f. π1U (n1 ) < π1R (n1 ) if n1 > N 1

(127) (128) (129)

¯ >N ˆ1 > N ¯ /2 such that and there exists N ˆ1 ; π1R (n1 ) > π1U (n1 ) if n1 < N ˆ1 ; π1R (n1 ) = π1U (n1 ) if n1 = N ˆ1 . π1R (n1 ) < π1U (n1 ) if n1 > N

(130) (131) (132)

f , however π U (n ) > π U (N ¯− We know π1U (n1 ) > π1R (n1 ) when n1 < N 1 1 1 2 n1 ). Thus, migration will occur from locality 2 to locality 1. By that time π1R (n1 ) > π1U (n1 ) become true, community 1 adopts sanctions and f is not an equimore people migrate to community 1. Thus, n1 = N 1 ¯ /2) < π1R (N ¯ /2), then by symmetry and ∂π/∂n < 0 librium. If π1U (N ¯ −N ˆ1 ) > π1U (N ˆ1 ) if n1 = N ˆ1 . Therefore, people want to we have π2U (N ˆ1 ) = k is not an equilibswitch from locality 1 to locality 2. Hence, b(N rium. Thus, by continuity of profit functions population in locality 2 in¯ /2 < n∗ < N ˆ1 . Since π R (N ¯ /2) > π U (N ¯ /2), creases until n2 = n∗2 s.t. N 2 1 1 R ¯ U ¯ ¯ then by the symmetry π1 (N /2) > π2 (N /2) if n1 = N /2. Hence people move to locality 1 and population in locality 1 increases until n1 = n∗1 ¯ /2 < n∗1 < N ˆ1 by continuity of profit functions. Therefore, s.t. N ¯ /2) is not an equilibrium. Unique outcome is when community n1 = ( N 1 stays with sanctions and locality 2 is unregulated and n∗1 > n∗2 . f π1R (n1 ) if n1 < N 1 R U f π1 (n1 ) = π1 (n1 ) if n1 = N1 ; f. π1U (n1 ) < π1R (n1 ) if n1 > N 1

(133) (134) (135)

ˇ1 < N ¯ /2 s.t. if There exists 0 < N ˇ1 ; π1U (n1 ) < π1R (n1 ) if n1 < N 53

(136)

ˇ1 ; π1R (n1 ) = π1U (n1 ) if n1 = N ˇ1 . π1U (n1 ) > π1R (n1 ) if n1 > N

(137) (138)

f is not stable because π U (n ) > π U (N ¯ − n1 ). As in part (D) n1 = N 1 1 1 2 Thus, migration will occur from locality 2 to locality 1. Then π1R (n1 ) > π1U (n1 ) become true with migration, community 1 adopts sanctions and more people migrate to community 1. We know that π1R (n1 ) < π1U (n1 ) if ˇ1 . Hence community drops sanctions when n1 > N ˇ1 . However, n1 > N U ˇ U ¯ ˇ π1 (N1 ) > π2 (N − N1 ) and population in locality 1 increases until ˇ /2 in free mobility equilibrium n∗1 = n∗2 . n∗1 = N f π1R (n1 ) if n1 < N 1 U R f π1 (n1 ) = π1 (n1 ) if n1 = N1 ; f. π1U (n1 ) < π1R (n1 ) if n1 > N 1

(139) (140) (141)

˘1 = N ¯ /2 s.t. if Also, we have N ˘1 ; π1R (n1 ) > π1U (n1 ) if n1 < N ˘1 ; π1R (n1 ) = π1U (n1 ) if n1 = N ˘1 . π1R (n1 ) < π1U (n1 ) if n1 > N

(142) (143) (144)

f is not stable. However, n = N ˘1 is an equilibrium. Again n1 = N 1 1 We know that π1R = π1U by assumption and π1U = π2U by symmetry at f π U > π U . There exists n1 = N 1 1 1 1 2 ˘ is not equilibrium. migration from locality 2 to 1. Hence, any n1 < N ˘1 , then π1R < π1U < π2U . There exists migration from Suppose n1 > N ˘ is not equilibrium. Free mobility locality 1 to 2. Hence, any n1 > N ¯ /2 where community 1 may choose equilibrium yields n∗1 = n∗2 = N sanctions or no sanctions, hence indifferent between two.

Note k=

(a − c)2 (a − c)2 − 4bn (n + 1)2

(145)

(a − c)2 (n − 1)2 4bn(n + 1)2

(146)

k=

54

and

(a − c)2 (a − c)2 − N . 2bN ( 2 + 1)2

(147)

¯ +N ¯ 2) (a − c)2 · (4 − 4N = ¯ (4 + 4N ¯ +N ¯ 2) . 2bN

(148)

k = k1/2 = ¯ /2. Simplify to when n1 = N k1/2

Overall, if k ∈ [0, k1/2 ], then in equilibrium n∗1 > n∗2 and monitoring cost is characterized by k = ( a−c − λ − ²) · n. However, for community with 2bn relative high monitoring costs, k ∈ [k1/2 , ∞], free mobility equilibrium yields ¯ /2 and sanctions are dropped. Note that in cases C, D and E n∗1 = n∗2 = N profit schedules cross each other twice (one where population is very low and the other where population relatively higher). At the very small population level benefit per person with no regulation is greater than the profit with regulation if locality experiencing higher monitoring cost. However, this phenomena vanishes rapidly as the number of people increases.

55