Common Pool Resource with Free Mobility: Voting with Their Herds Dolgorsuren Dorj∗ Department of Economics, University of Hawaii at Manoa 2424 Maile Way, Saunders 521, Honolulu, HI, 96822 tel: (808)-956-8169, fax: (808)-956-4347 email: [email protected] April 22 2009

Abstract We characterize free mobility equilibrium in a common pool resource setting with two localities. We find that adopting a community management regime in just one locality increases agents’ welfare not only in the regulated locality but in the unregulated locality as well. JEL classification codes: C70, Q2, R1. Keywords: Common dilemmas; Sanctions; Free mobility.

Many thanks to Katerina Sherstyuk, Arnaud Dellis, Nori Tarui, John Lynham, Sean D’Evelyn. This work benefited from discussions at Western Economic Association International meeting 2008 in Honolulu, Economic Science Association North-American meeting 2007 in Tucson, Ronald Coase Institute 8th Workshop in Institutional Economics 2006 in Boulder.




We study the common pool resource (CPR hereafter) problem with free mobility. Existing CPR literature (e.g., Ostrom et al., 1992) examines one locality problems rather than multiple locality contexts. The focus of this paper is to extend the CPR model to situations where agents in multiple localities are free to choose their place to live and extract resources. Examples of commons with multiple locations include pastureland under extensive grazing or international fisheries. We consider two possible management regimes for each locality, both in exogenous and endogenous institutional settings. In some localities common resources may be unregulated. Alternatively, resources may be regulated by a community based sanctioning mechanism as in Casari and Plott (2003), where mutual inspection and sanctioning opportunities allow agents to obtain socially optimal use level.1 We consider this sanctioning system as an example of institution that has a historical precedence and that has been shown to restore the efficiency in closed communities. The research question is how does free mobility affect the performance of the sanctioning system? If one locality is regulated and the other is not, would the sanctioning institution withstand the migratory pressure from the neighboring unregulated locality? Would it be possible to prevent “the tragedy of the commons” (e.g., Hardin, 1968)? To characterize a free mobility equilibrium we use the Tiebout equilibrium concept defined as a partition of agents into localities, where no single agent wants to move from the current position to join the other existing localities (e.g., Tiebout, 1956, Westhoff, 1977). We find that a locality with sanctions can sustain efficient use level and prevent over-use under free mobility as long as the community adjusts its harvesting target in response to migratory pressure from the unregulated locality. Moreover, the locality with no regulation experiences a positive externality from the regulated locality because the latter accommodates more people. We also show that if the institutions are endogenous, i.e. agents in each locality vote for the regulatory regime, the sanctioning system is adopted as long as it is Pareto-improving. 1

Sanctions have been shown to resolve social problems in CPR, public good, and truthtelling settings (e.g., Casari and Plott, 2003, Fehr and G¨ achter, 2000, S´ anchez-Pag´ es and Vorsatz, 2007).



Basic CPR Model

As a benchmark, we review the basic CPR model with one locality. A finite number of identical agents N with endowment e each, simultaneously decide on the amount of harvest xi from the common pool, where i ≤ N is the P agent’s index. Let X = N i=1 xi be the total appropriation, and f (X) be a concave production function. For simplicity, we assume f (X) = a·X −b·X 2 . The cost per unit of harvest is denoted by c. Then each agent i0 s profit is given by πi = e − c · xi + xXi · f (X).


Unregulated locality

If the locality is unregulated, denoted by U , then the total appropriation in the symmetric Nash equilibrium is given by X U = NN+1 · a−c , which is higher b (a−c) opt than the social optimum, X = 2·b if N > 1 (e.g., Falk et al., 2002). The (a−c)2 . Nash equilibrium per person profit is given by πiU = e + b·(N +1)2


Locality with sanctions

The model with sanctions (e.g., Casari and Plott, 2003) assumes that the community restricts per agent harvesting to the threshold amount, λ, which targets the socially optimal harvesting level. After observing the total appropriation in their locality, all agents are free to monitor each other. By paying inspection fee, k, the inspector may obtain exact information about the harvesting decision of one other member. If monitored, a violator pays to the inspector a unitary fine h for each unit of excess harvest. We use R for the regulation, i.e sanctions. Casari and Plott (2003) establish the following: Proposition 1 Suppose a locality has a sanctioning mechanism, where the threshold is set as λ = xopt − k/h − ε, with ε > 0 small enough and the unit fine is set as h = a − c − xopt · (N + 1) · b, with xopt = X opt /N = (a−c) . 2·b·N Then this mechanism supports the socially optimal level of harvesting as a Nash equilibrium, X R = X opt . In this equilibrium, every agent inspects and is being inspected with certainty. 2

− k. Note The equilibrium per person profit is given by πiR = e + (a−c) 4·b·N that for any given population size, N > 1, πiU (N ) < πiR (N ) as long as k is 3

small enough2 . Therefore, each agent prefers the sanctioning regime to no regulation.


CPR free mobility equilibrium

Now consider two communities with identical production functions, f (X). Let the total population in two localities be N . We assume free mobility and examine two cases: exogenous and endogenous institutions.


Exogenous institutions and free mobility

First, we assume that regulatory regimes in each locality are exogenous. Also, we assume that the harvesting threshold in a regulated locality is set optimally and instantaneously to the population level in the locality: λ = λ(N R ), defined as in Proposition 1. Further, the inspection cost is low enough so that the benefits of sanctioning mechanism outweigh the costs of adopting it in a regulated locality for any population level n ≤ N . Consider free mobility equilibrium in a game, where agents are free to move from one location to another. Each agent chooses the locality (S j ), j = 1, 2 to live. Once harvesting thresholds are adjusted according to the population levels, agents decide on a harvesting level (xi ) within the chosen locality. Further, each agent in a regulated locality may inspect other agents after observing total group use. We solve the game using the backward induction technique and the notion of Tiebout equilibrium classified as external in the sense that agents “vote with the feet” by choosing their preferred locality (e.g., Epple and Romer, 1991). In the free mobility equilibrium, two conditions must hold for each locality: (i) agents’ actions constitute a Nash equilibrium within each locality; (ii) no agent wants to move, i.e. each agent’s profit in a chosen locality is at least as high as the profit in the other locality. It is straightforward to show that given all agents are identical, under symmetric regimes agents split equally between two localities. With no regulation, both communities over-use the resource. With the sanctions, both communities obtain social optimum. Interesting results are derived for the asymmetric institutions case, where 2

The monitoring cost has to satisfy k ≤

(a−c)2 b


1 · ( 4·N −

1 (1+N )2 ).

one locality adopts the sanctioning mechanism and the other locality is unregulated. The free mobility equilibrium is characterized as follows: Proposition 2 In the free mobility equilibrium with asymmetric institutions and identical agents, the locality with the sanctioning system accommodates more individuals than the unregulated locality, N R > N U . The appropriation levels satisfy X opt = X R < X U . The sanctioning system introduced in one locality improves the welfare of all agents in both localities as compared to no regulation. Proof: Note that in equilibrium, by the identical agents assumption πiR = To prove that N R > N U , we need to show that cases N R = N U and N < N U both lead to contradiction. Let N be the total population, N R + NU = N. First, assume N R = N U = N2 . Refer to section 2.2 to see that if N R = N U , then πiR ( N2 ) > πiU ( N2 ), which contradicts the equilibrium condition of “no one wants to move” (πiR = πiU ). To show that N R < N2 < N U also cannot be an equilibrium, note πiU . R

that πiR (n) and πiU (n) are both strictly decreasing in population n.

πiR ∂n N ) 2


2 πU 2(a−c)2 R R − (a−c) < 0 and ∂ni = − b·(n+1) > < 2 < 0. This implies that πi (N 4·b·n2 N R N U N U R U U πi ( 2 ) > πi ( 2 ) > πi (N > 2 ). Contradiction. Hence, N > N . To show that πiR and πiU both increase as compared to no regulation,

note that without regulation the population in each community would be N with profits πiU ( N2 ). Then by the monotonicity of the profit schedule in 2 n, π U (N U < N2 ) > π U ( N2 ), and by the equal profits condition, πiR (N R ) = πiU (N U ) > πiU ( N2 ). Done. This result indicates the importance of decentralized regulation and its impact on the neighborhood locality. We can show that the results are easily generalizable to any number of localities. As the number of localities with sanctions grow, the whole system Pareto improves. This is because in the regulated locality per person profits increase due to the sanctioning mechanism, while in the unregulated localities per person profits increase due to smaller population, hence lower appropriation.


Endogenous institutions and free mobility

Now consider internal equilibrium (voting with the ballot) where each community decides on the regulatory structure by majority voting (e.g., Fiorina 5

and Plott, 1978). We show that the regulatory regime can be sustained for both localities in a subgame perfect Nash equilibrium. Again, each individual chooses a locality first. Next, each member of the community votes either for sanctions or no sanctions in their locality, and the outcome is determined by majority voting. Third, each agent decides on his/her harvesting level. In the communities with the sanctioning regime, monitoring decisions follow. Proposition 3 In the voting equilibrium, agents vote for sanctions in both localities. The appropriation levels in both localities are at the social optimum, X R1 = X R2 = X opt , and the population sizes are identical, N R1 = N R2 = N2 . The result easily follows from the identical agents assumption. By the median voter theorem (e.g., Downs, 1957) the outcome of the majority voting is the median voter’s preferred institution. By assumption, πiR (n) > πiU (n) for any n ≤ N since monitoring costs are low. Therefore, sanctions are preferred to no sanctions.



Our analysis of the CPR problem with free mobility and exogenous institutions shows that the locality with the sanctioning system maintains the socially optimal use level even under migratory pressure as long as harvesting targets are adjustable according to the population level. Moreover, a positive externality is captured if we introduce sanctions in only one locality. This suggests that regulatory mechanisms for managing commons such as fisheries, grassland, forests, irrigation systems may be introduced gradually. With endogenous choice of institutions between no regulation and sanctioning system, equilibrium yields institutions with sanctions as long as monitoring costs are low.

References [1] Casari, M. and C. R. Plott, 2003, Decentralized management of common property resources: experiments with a centuries-old institution, Journal of Economic Behavior and Organization 51, 217-247. [2] Downs, A., 1957, An Economic Theory of Democracy. (Harper Collins). 6

[3] Epple, D. and T. Romer, 1991, Mobility and redistribution, Journal of Political Economy 99 (4), 828-858. [4] Falk A., E. Fehr and U. Fischbacher, 2002, Appropriating the commons: a theoretical explanation, in: E. Ostrom, T. Dietz, N. Dolsak, P. C. Stern, S. Stonich and E. U. Weber, eds., Drama of the Commons, (National Academy Press) 157-192. [5] Fehr, E. and S. G¨achter, 2000, Cooperation and punishment in public goods experiments, American Economic Review 90 (4), 980-994. [6] Fiorina, M. P. and C. P. Plott, 1978, Committee decisions under majority rule: an experimental study, The American Political Science Review 72 (2), 575-598. [7] Hardin, G., 1968, The tragedy of the commons, Science 162, 1243-48. [8] Ostrom, E., J. Walker and R. Gadner, 1992, Covenants with and without a sword: self-governance is possible, The American Political Science Review 86, 404-417. [9] S´ anchez-Pag´ es, S. and M. Vorsatz, 2007, An experimental study of truth-telling in a sender receiver game, Games and Economic Behavior 61, 86-112. [10] Tiebout, C., 1956, A pure theory of local expenditure, Journal of Political Economy 64, 416-424. [11] Westhoff, F., 1977, Existence of equilibria in economies with a local public good, Journal of Economic Theory 14, 84-112.


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