Vol. 44 (2001) 169–176
A model of institutional formation within a rent
seeking environment
Kevin Sylwester
∗Department of Economics, Southern Illinois University Carbondale, Mailcode 4515, Carbondale, IL 62901, USA
Received 9 November 1998; received in revised form 10 January 2000; accepted 18 January 2000
Abstract
This paper presents a game theoretic model in which some fraction of output is appropriated from entrepreneurs. Entrepreneurs are able to form a league to prevent this appropriation, but this might not be individually rational because of either the free rider problem or coordination failure. The model also shows that poorer countries are less able to form this league and so might not be able to develop institutions establishing property rights. © 2001 Elsevier Science B.V. All rights reserved.
JEL classification: O10; H41
Keywords: Rent Seeking; Institutions; Property rights
1. Introduction
Although most economic models assume that agents can costlessly retain their property, this assumption has seldom held. Merchants and property owners have often been subject to theft or to other external threats to their possessions. In addition to illegal transfers of property, legal appropriation is also common. Agents engaged in rent seeking hire lawyers and lobbyists to redistribute wealth through laws and government regulations or to protect their property from government confiscation. North (1990) and Eggertsson (1990) focus upon how various institutions can promote or deter productive activities. But why should a society have institutions that lead to rent seeking and redistributive as opposed to productive activities? Why have some communities been able to form protective leagues or institutions that promote growth whereas others have not been as fortunate?
This paper attempts to answer these questions by presenting a model in which producers lose a fraction of their output to rent seekers. (See Tollison (1992) and Tollison (1982) for
∗Tel.:+1-618-453-5347; fax:+1-618-453-2717. E-mail address: [email protected] (K. Sylwester).
surveys on the rent seeking literature.) Agents can form an institutional structure (which is denoted as a coalition in the paper) to protect their property, but the free rider problem and coordination failure can prevent the coalition from forming. Thus, this paper focuses upon some of the problems of acting collectively as discussed in Olson (1965) and Olson (1982). This coalition can be viewed not only as a physical fortification but as the emergence of a social contract or constitution as in Buchanan (1975). In this setting, the formation of the coalition is contractarian in approach in that agents come together to form a collective and contrasts that of Olson (1993) in which property rights are either established by an autocrat or by several individuals when no one agent is strong enough to dominate the others. Of course, other collective institutions such as guilds and cartels seek to diminish competition in order to protect rents and are more formally considered in Olson (1982). The coalition in this paper is viewed as more encompassing of an entire community or society.
This model differs from other models of predation in two respects. First, there is no common property or tragedy of the commons as in Tornell (1997), Benhabib and Rustichini (1996), Hirshleifer (1991), and De Meza and Gould (1992). A more important difference is that agents have the opportunity to work collectively although this is not assumed. In Grossman and Kim (1995,1996a); Grossman (1991,1994), and Usher (1989), protecting property is a private endeavor. These models do not include a technology in which agents can band together for mutual protection. Grossman and Kim (1996b); Freeman (1993), and Shavell (1991) present models in which agents act collectively to protect property but these models do not consider any deterrents to act collectively. Collective action is assumed. Finally, Skogh and Stuart (1982) consider how the rise of a criminal code can lower theft and promote productive activity relative to anarchy but they also do not consider issues such as the free rider problem and coordination failure as to why this code fails to develop.
The model is presented in section two and equilibria are determined in section three. Section four presents comparative statics and implications. A conclusion follows.
2. The Model
There is a two stage game involving two sets of agents.1 These agents take actions consistent with rational beliefs as to what the other agents will do. All parameters are common knowledge. Part one of this section describes the agents. Part two describes the two stages.
2.1. Agents
The first set comprises an identical group of agents which I denote as entrepreneurs. An extension with heterogeneous agents is available from the author. Each agent has density equal to one and the total density of the group isN > 0. Letg > 0 denote the level of output that each agent produces.
1The model could be changed so as to contain only one set of agents and such a version is available from the
The second set consists of a continuum of heterogeneous agents. These agents also produce output but in a different sector (e.g. a traditional sector) of the economy than do entrepreneurs. This continuum lies along some distribution B within the range [0, bH] where 0 < bH <∞. Each agent, j, has density one. b is an element of B and denotes agent j’s level of production where the “j” is understood. This group of agents has the option to appropriate output from entrepreneurs. If an agent chooses to appropriate output, he does not produce output himself. I denote an agent who appropriates output as a bandit although this should not be interpreted to mean that appropriation is necessarily an illegal activity.
The payoff for each agent is the amount of output that he acquires.
2.2. Stages
2.2.1. Stage 1: Agents choose whether or not to become bandits
Each agent in the traditional sector decides whether or not to become a bandit. Each bandit appropriates the fraction z of output from all entrepreneurs with 0< z <1.2The parameter
z denotes the return to rent seeking. If appropriating output is not an illegal activity, a low z
could denote a political system where those in the traditional sector have little power and so cannot acquire transfers from other agents. The return to banditry (Ban) with no coalition is Ban=zNg. With mass S bandits, the return for an entrepreneur is (1−Sz)g. I only consider
cases whereS <1/zso that an entrepreneur does not lose all of his output.
2.2.2. Stage 2: Forming the coalition
Each entrepreneur has the option of joining a coalition which collects contributions from its members in order to provide protection from bandits. Agents noncooperatively decide whether or not to join the coalition. For expositional purposes, I assume that not more than one coalition exists.3 The coalition cannot exclude any agent who wants to join. The total cost of the coalition is F[Ec, S,ψ] which can be thought of as the cost of building fortifications or as enforcement costs under a legal system. Ecdenotes the mass of coalition members. F[∗,∗,∗] is increasing with both Ecand S. As the mass of coalition members increases, coordination costs increase. With more bandits, more resources are required to protect output. The parameterψcaptures factors such as the efficiency of the bureaucracy in fighting rent seeking or the technology of protecting coalition members from theft. I assume F[1,S >0,∗]> gwhich implies that no agent can afford to pay for the coalition if he is the lone member of the coalition. I also assume that F[Ec, S,ψ]/S is increasing with S. Each member of the coalition pays an equal share of the cost and so makes a payment (P) of
P =F[Ec, S,ψ]/Ec. The coalition cannot price discriminate among agents. An Appendix A shows that the conclusions of the model continue to hold under a voluntary payment scheme.
Provided a coalition forms, a coalition member retains all of his output and so is com-pletely protected. For those entrepreneurs not in the coalition, each retains g(1−Szδ). The
2The model does not assume that predation is costly in that no output is destroyed during the transfer from
entrepreneur to bandit, but the model could be extended so as to allow for lost output and still retain its general conclusions.
parameter 0≤δ ≤1 measures the degree of exclusion. Ifδ=1, agents outside the coali-tion are completely excluded from the benefits provided by the coalicoali-tion. Ifδ =0, there is no exclusion and the coalition is a public good. The parameterδdenotes such factors as the ability of the society to ostracize nonparticipants as well as the technology of protec-tion. For example,δ may be “high” if the coalition denotes a protective league in which noncontributors can be kept outside the walls of a fort or castle.
As stated, the coalition cannot deny entry to any agent nor can it price discriminate. The technology of the coalition is discrete in that no protection arises if it is not fully funded. Under these assumptions, the objective of the coalition becomes trivial and is to fully protect all agents who make a payment of P and join. The extensions mentioned above are interesting issues but are beyond the scope of this paper.
3. Decision rules and equilibria
In this section, I use backward induction to solve the model for pure strategy, sub-game perfect equilibria. Agents act noncooperatively and view their actions as negligible in determining outcomes.
In stage two, S is given. An agent joins the coalition if the return to joining outweighs the return from free riding:g−P ≥g(1−Szδ). Dividing by g, 1−F[Ec, S, ψ]/gEc≥(1−Szδ). Given Ecand S, this inequality holds for all entrepreneurs if and only if the inequality holds for any entrepreneur. Thus, either all entrepreneurs join the coalition or no one joins and only one of these two extremes is realized. For a coalition to exist in which all entrepreneurs join, the above inequality must hold forEc=Ngiven S. As assumed above, this inequality does not hold forEc=1 for anyS >0.
In stage one, the return to banditry is dependent upon the mass of coalition members: Ban = zδ(N −Ec). From above, either all entrepreneurs join the coalition or no en-trepreneurs join. If all enen-trepreneurs join, thenN =Ecand Ban=0. The only agents that become bandits are those for whichb=0 since they are indifferent to becoming bandits or to remaining in the traditional sector.4 No agent for whichb >0 becomes a bandit. Instead, these agents produce output. If no entrepreneurs join the coalition, then all entrepreneurs are unprotected. Agents in the traditional sector become bandits until the return to banditry equals the return to working in the traditional sector. For the marginal bandit,b=zNg.
Therefore, there are potentially two equilibria. An equilibrium that always exists is one in which no entrepreneurs join the coalition and in which there are mass S∗∗bandits where
S∗∗is determined by the mass of agents for whichb≤zNg. Entrepreneurs do not join the
coalition because a coalition withEc=Nis not individually optimal given S∗∗or because of coordination failure. Since no agent can afford to pay for the coalition himself, no agent wants to be the lone member of the coalition and so will never join by himself. The other equilibrium consists of all entrepreneurs joining the coalition and for S to equal the mass of agents for whichb = 0. I denote this mass as S∗ andS∗ < S∗∗since 0 < zNg. The
following proposition summarizes the possible equilibria of the model.
4I assume that all agents for whichb=0 become bandits even though they are indifferent when Ec=N. These
Proposition 1. Consider the following inequality:
1−F[N, S, ψ]
Ng ≥1−zδSa (1)
Regardless of (1), an equilibrium always exists whereEc =0 andS =S∗∗. If (1) holds
forS =S∗, then an equilibrium with a coalition exists whereEc=NandS =S∗. If (1)
does not hold forS =S∗, then (1) does not hold forS =S∗∗and the only equilibrium is forEc=0 andS=S∗∗.
Given a coalition, there are S∗bandits. Eq. (1) says that joining the coalition is individually optimal if all other entrepreneurs join given S bandits. Hence, an equilibrium with a coalition exists provided (1) holds withS=S∗. Since F[N, S,ψ]/NgS is increasing with S, then (1) does not hold forS =S∗∗if (1) does not hold forS=S∗. Therefore, if (1) does not hold forS =S∗, the only equilibrium is forS = S∗∗ andEc =0. A coalition having only a subset of entrepreneurs does not exist in equilibrium since all agents join the coalition if any agent finds it optimal to join.
A special case occurs ifδ=0. No coalition forms in equilibrium since the payoff from not joining the coalition (g) is higher than the payoff from joining the coalition (g−P) for all agents. No agent joins the coalition since nonmembers receive the same protection as members. It is this case that most resembles the argument from Olson (1965). Moreover, no agent can afford to be the lone member of the coalition withg < F[1,S >0,∗]. Thus, an equilibrium with a coalition does not exist withδ=0.
4. Comparative statics and implications
The parameters of the model are g, B, z, N, F,ψ, andδ. The first four parameters determine
S∗∗, the mass of bandits in the absence of a coalition. S∗∗is increasing with z since the return to banditry increases as bandits take a higher fraction of output from entrepreneurs. S∗∗is also increasing with Ng, the aggregate income of entrepreneurs. Multiplying each element of B by some constant greater than one lowers S∗∗since there is less incentive to become a bandit when the return in the traditional sector increases. However, multiplying g and each element of B by the same constant does not affect S∗∗because it does not affect the relative return of producing output to become a bandit. S∗∗is only affected when g and B grow at unequal rates. If g grows faster than the elements of B, S∗∗increases.
Given S, the parameters g, N, F[∗,∗,ψ], andδdetermine whether or not a coalition with an equilibrium exists. Asψ increases, the cost of the coalition rises implying that there are fewer values of S for which (1) holds. The effect of N upon the per agent cost of the coalition is ambiguous. A large entrepreneurial sector raises the aggregate cost of forming a coalition but also lowers the per agent cost given F[N,∗,ψ].
to see countries with better institutions grow faster over time (as argued elsewhere), but one would also expect to see that at a point in time higher income countries are more likely to develop institutions to protect property. This might be one reason why poor countries are not converging to high income nations as implied by many neoclassical growth models (see Quah (1993)). In fact, using the same aggregate production function for both high and low income nations (so that the only difference between the two is the capital to labor ratio) may not be reasonable if various institutional environments hold different implications for the marginal returns to inputs.
A lowerδmakes free riding more attractive and creates a disincentive for agents to join the coalition thereby further separating socially optimal and individually optimal outcomes. Suppose (1) holds withδ =1. Then, entrepreneurs will form a coalition given S bandits if they could act collectively. This is the outcome in Grossman and Kim (1996b) where collective action is assumed to occur. If (1) does not hold withδ=1, then entrepreneurs as a group tolerate banditry and concerns such as coordination failure or the free rider problem become irrelevant. However, suppose (1) holds in country A withδA =1 but does not hold
in country B withδB<1 but which is otherwise identical to A. Without agents being able to
act collectively, the free rider problem prevents the coalition from forming in B even though an entrepreneur would prefer to join a coalition as opposed to having no coalition at all.
A limiting case occurs whenδB=0. Then, all entrepreneurs in B can be protected by the
coalition even if the coalition is small. Consequently, the aggregate cost of the coalition is lower in B and so it is less expensive for B to protect all of its entrepreneurs than can A where nonmembers are excluded. In this sense, country B has a superior “protection technology” since it entails lower aggregate costs for society. However, it is not individually optimal for an entrepreneur in B to “use the superior technology” (i.e. join the coalition) and so having a superior protection technology may not lead to a better outcome for society. This implies that when designing a system of property rights or of physical protection, avoiding the free rider problem by excluding or punishing noncontributors might entail higher costs for society along other dimensions.
However, not all agents are better off with a coalition. Agent j in the traditional sector receives b from producing. Ifb <zNg, then his income is higher without a coalition. Since
the poorest agents have higher income and richer agents lose income, income inequality declines without a coalition. In this case, a trade-off exists between maximizing output and lowering income inequality. 5 Moreover, a fall of this institutional structure (possibly brought about by some political upheaval) would result in a loss of aggregate output as well as a fall in the level of income inequality between the two groups.
5. Conclusion
Spending resources to appropriate property negatively affects aggregate production although agents find it in their private interest to engage in rent seeking. By forming a
5This result stems from the assumption that a coalition only comprises entrepreneurs and is designed to protect
coalition, a community can deter predation and thereby increase aggregate output (although this might not benefit all agents). However, coordination failure or the free rider problem can prevent the creation of a coalition. In addition, the model implies that poor countries are less able to develop institutional structures that protect property than are higher income communities and so causality between the institutional and economic environments runs in both directions.
Acknowledgements
I would like to thank Rody Manuelli, Steve Durlauf, Larry Samuelson and four anony-mous referees for their suggestions. All errors are mine.
Appendix A
In the text, the cost of joining the coalition is the same for each agent. This appendix considers an environment in which each agent voluntarily contributes an amount to fund the cost of the coalition thereby showing that the results of this paper do not rely upon the specific payment scheme constructed above.
Consider the game of deciding how much to contribute to the coalition. A strategy for each coalition member, denoted by c, is a fraction p(c) (where 0≤p(c)≥1) of his revenue to contribute to paying for the coalition. For a coalition to be funded,
F[Ec, S, ψ]≤gXp(c)
where the summation runs across all members of the coalition. But an agent should decrease his contribution if there are more than enough funds so,
F[Ec, S, ψ]=gXp(c)
in equilibrium. For an agent to join the coalition, the return from doing so must be greater than the return from free riding:g−gp(c)≥g(1−Szδ)→1−p(c)≥1−Szδ. Provided that the coalition is funded, all entrepreneurs join if δ > 0 since p(c) is allowed to be arbitrarily close to zero and each agent takes F[∗,∗,∗] as given. So withδ >0, all agents join the coalition provided the costs of the coalition are met.
When can a coalition exist? A coalition can form given S if there exists some set
{p(c)}c∈[0,N]with 1−p(c)≥1−Szδfor allc∈[0, N] such that,
gXp(c)=F[N,∗,∗]
Suppose all agents choose the same rate,p(c) = p for all c ∈ [0, N] where p solves
On the other hand, if there exists a voluntary payment scheme{p(c)}c∈[0,N]such that
1−p(c)≥1−zδS∗for allc∈[0, N] and
gXp(c)=F[N, S∗,∗]
then a coalition exists withp(c)=pfor all agents, which is the case in the text. This is true because p also solvesF[∗,∗,∗]=gNp. Moreover,p ≤ max{p(c)}with equality only if
p(c)=pfor all agents. So if joining the coalition is individually optimal when contributing max{p(c)}, it is individually optimal for an agent contributingp≤max{p(c)}. If all agents contributed less than p, then the coalition would be underfunded and would not exist.
In summary, a coalition exists in equilibrium givenp(c)=pfor all agents if and only if there is at least one voluntary payment scheme which supports a coalition.
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