Rent-Seeking Contests, Externalities
and Group Formation
Abstract
In this thesis we study the influence of externalities on the outcomes of rent-seeking contests. We consider two new methods to introduce externalities in rent-seeking contest models. In addition we investigate the effect of group formation on these models. Contrary to most rent-seeking contest models with externalities, we find that total rent-seeking outlays will decrease if the externality is positive. Furthermore, we find that if one group of players competes collectively the degree of rent-dissipation will be strictly lower.
Key Words: Externalities; Group rent seeking; Rent-dissipation rate; Rent-seeking contests
Table of Contents
1.
Introduction
4
2.
Literature Review
7
2.1 Seminal models on Rent-Seeking Contests 7 2.2 Rent-Seeking Contests and Externalities 8
2.3 Group Rent-Seeking Contests 12
2.3.1 Group Rent-Seeking with Competitive-Share Group 12 2.3.2 Group Rent-Seeking with Strategic Groups 17 2.3.3 The Equivalence of Rent-Seeking Outcomes for
Competitive-share and Strategic groups 18 2.4 Group Rent-Seeking Contests with Externalities 19
3.
Extensions and the Effect of Group Formation
21
3.1 Rent-Seeking Contests with Group Specific Externalities 21
3.1.1 Introduction 21
3.1.2 Model 23
3.1.3 Comparative Statics 26
3.2 Rent-Seeking Contest with Location Dependent Externalities 31
3.2.1. Introduction 31
3.2.2 Model 31
3.2.3 Comparative Statics 33
3.3 Rent-Seeking Contests with Group Specific Externalities and
Strategic Groups 35
3.3.1 Introduction 35
3.3.2 Model 35
3.4 Rent-Seeking Contests with Location Dependent Externalities and
Competitive- Share Groups 46
3.4.1 Introduction 46
3.4.2 Model 47
3.4.3 Comparative Statics 56
4.
Conclusions
58
4.1 Policy Implications and Future Research 60
References
61
1. Introduction
In economics, rent seeking refers to a situation in which agents try to make money, but where their rent-seeking efforts do not contribute to productivity. In other words, where the rent is not matched by corresponding labour or investment, but obtained through manipulation of the economic environment. Such situations are quite common in most economies. Rent seeking can be brought about by government regulations putting restrictions on economic activity. In this way, firms lobby to acquire the right to be a monopolist in a market, obtain import quotas or to get a government procurement contract. However, government restrictions on economic activity can also provoke illegal rent seeking activities like corruption, bribery, smuggling and black markets.
The purpose of this thesis is to investigate rent-seeking situations that take the form of a contest, see Tullock (1980), Nitzan (1994) and Lockard & Tullock (2001). Examples include R&D contests/patent contests, political campaigning, eliminations tournaments and many more. We are especially interested in rent-seeking contests that produce externalities on participating players (for a treatment of rent-seeking contests with externalities on none participating players, see Congleton (1989)). Externalities can be triggered for different reasons, for example, because efforts affect the rent itself or the costs of rent seeking, the rent has public characteristics, or individuals are not indifferent to who wins the rent if they do not themselves.
Rent seeking is not only practiced by individuals but also often by groups. Players may seek collectively for a rent because they share common interests. Companies, for example, may lobby for mutual beneficial legislative arrangements like protectionist trade policies. Moreover, incentives to compete as a group can also be strategic in nature. Through group formation, players can share the risk of losing the contest and possibly influence the behaviour of their opponents in their favour.
Given our interest in rent-seeking contests with externalities and understanding that rent seeking is not necessarily practiced by individuals, we come to the following research question:
With the following sub questions:
o In which way are externalities introduced in the rent-seeking contest literature? o Are there new ways to introduce externalities in rent-seeking contests?
o How does group formation influence rent-seeking contests with externalities? The objective of this thesis is to give detailed answers to these questions and relate our findings to the rent-seeking contest literature.
In this thesis we explore two new ways to introduce externalities in rent-seeking contests. In these models we discuss rent-seeking contests with externalities that are dependent on the rent. These kind of rent-seeking contests have been previously investigated by Linster (1993). However, our models differ in several important ways. In the general model that Linster (1993) considers, the focus is on the effect that these kinds of externalities have on the outcomes of rent-seeking contests. In our first model we consider a rent-seeking contest with two types of players. The first type of player does receive externalities, but the other type of player does not. Our focus is in particular on the effect of this asymmetry. Investigating this is interesting because externalities frequently have a limited public character and as a result are not received by all players.
Our second model shares some similarities with an example considered by Linster (1993) concerning the placement of a streetlight. In this example, a finite number of players are located along one side of a street and participate in a rent-seeking contest to get the streetlight at their preferred position. The main drawback of this example is that since players are located along a Hotelling line there is no symmetry. For this reason, generalisations to a case of many players are not possible. In our model we assume that players are located along a Salop circle. In this way, we are able to impose symmetry and we can easily make generalisations to a case with n players.
Investigating this extension is important, however, where as noted before rent seeking is also often practiced by groups.
Group rent-seeking contests and externalities have been previously investigated by Lee and Kang (1998). The important difference with our analysis is that the externalities they consider are very different from ours. In the model of Lee and Kang (1998), externalities are dependent on aggregate effort and affect the costs of rent seeking. In our models, externalities depend on the rent and affect the revenues associated with rent seeking. Other papers that have investigated rent-seeking contests with externalities are Chung (1996), Schaffer (2006) and Mikkelsen (2007). These papers have all one thing in common. The externalities they introduce are in some way dependent on effort. In our analyses, we do not consider any form of externalities that are dependent on effort levels.
2. Literature Review
In this chapter we will review and critically analyse the most important and relevant articles in the field of rent-seeking contests. In section 2.1 we start with a brief review of some of the seminal articles on rent-seeking contests. In section 2.2 we will discuss the articles that have incorporated externalities into rent-seeking contests. Next, in section 2.3 we consider the articles that have investigated group rent-seeking contests. Finally, in section 2.4, we examine a model that integrates externalities and group rent-seeking contests. Note that in the appendix we give by means of a diagram an overview of the different strands in which the literature discussed in this chapter can be divided.
2.1 Seminal Models on Rent-Seeking Contests
Tullock (1967) was the first to identify the phenomenon of rent seeking in relation to monopolies. The term rent seeking was introduced by Krueger (1974). Another early paper on rent seeking is Posner (1975). The first paper to model rent seeking as a contest, and actually the basic framework for all rent-seeking contest models in this chapter, is the seminal work by Tullock (1980). In this rent-seeking contest model, we have N identical agents who have the opportunity of winning a fixed rent R with R> . The rent is an 0 indivisible private good and the model takes the form of a one period “winner takes all” contest. The effort that players i spends to win the contest is xi. The probability that player i wins the contest takes the following symmetric logit form:
1 i p N = if xj = for all j 0 1 1 ( ) ( ) i i i N N i j i j f x x p f x x = = = =
¦
¦
otherwise (2.1.1)1 i i i i N i j j x V p R x R x x = = − = −
¦
(2.1.2)Maximizing this function over x , leads to the following first order condition: i
1 1 0 j j i i N i j j x V R x x ≠ = ∂ = − = ∂
¦
¦
(2.1.3)Imposing symmetry, gives us the following Nash equilibrium effort level:
2 (N 1) x R N − = (2.1.4) Using (2.1.4) we can deduce that the rent-dissipation rate is equal to:
1 ( 1) N j j x N t R N = − = =
¦
(2.1.5) One of the most controversial results of Tullock’s (1980) model is that if the number of players approaches infinity the rent-dissipation rate is equal to one.lim 1
N→∞t= (2.1.6)
This means that, if there is a large number of players the rent will be fully dissipated in equilibrium, which is of course from an economic perspective a rather unsatisfying result.
2.2 Rent-Seeking Contests and Externalities
general model and shows that most classic rent-seeking models can be written as special cases of this model. A direct consequence of his notation is that one can write the players’ valuations of the rent as vectors. Linster (1993) uses this property to look into models were players are, opposed to standard rent-seeking contest models, not indifferent to who wins the contest if they do not.
To illustrate this idea, Linster (1993) considers the example of the placement of a street light. A finite number of players are located alone a Hotelling line without a streetlight. To help these players the government will place one lantern. Every player obtains his maximum payoff if the lantern is placed in front of his own house. However, if the government places the lantern in front of the house of one his neighbours, a player will also partially profit because some of the light will still come through to his place. If the lantern is placed further away, its light will come through less clearly and the payoff will be lower. To capture this idea Linster (1993) assumes that the externalities associated with the lantern are exponentially decreasing in distance. The lantern features public characteristics and therefore there are externalities to nearby located players.
To demonstrate the richness of his model, Linster (1993) considers various other examples. These examples all have in common that the externalities are dependent on the rent. The main quality of the Linster (1993) model is that it explicitly states a matrix describing the valuations a player attributes to other players winning the contest. As a result, this valuation matrix can be specified any way one would like. This is particular useful if players, for what ever reason, are concerned with who wins the rent if they do not themselves.
A different way to introduce externalities into rent-seeking contests is proposed by Chung (1996). The author considers an individual rent-seeking contest model in which effort is productive. In this model, the rent is an increasing function of aggregate efforts and thereby endogenous. Because effort has a positive effect on the rent, players will increase their effort levels compared to standard rent-seeking contest and aggregate effort is strictly higher.
increase social welfare and if the rent-dissipation effect dominates the externality effect, the contest will decrease social welfare. Yet the first case, although feasible, is not likely to occur. It is only possible if there are very strong decreasing marginal returns to effort. In that case, aggregate effort will be below the social optimum. The socially optimal level of aggregate efforts is reached at the point were the externality and rent-dissipation effect just outweigh each other. The main result is that Chung (1996) shows that, even though the rent is endogenous and increasing in effort, the classic result that rent-seeking contests produce social waste is likely to hold.
A related paper to that of Chung (1996) is Schaffer (2006). This paper also makes use of the assumption that the rent is dependent on effort. A minor extension that Schaffer (2006) makes is that he also allows for the possibility of a negative relation between effort and rent. Furthermore, where Chung (1996) considered this relation in general terms, Schaffer (2006) uses explicit functions. He considers two different cases, linear and non-linear externalities. In the linear case Schaffer (2006) finds the “striking” result that equilibrium net payoffs are unaffected by the externality. As pointed out by Mikkelsen (2007) this result is not that surprising, because this linear “externality” does not affect the rent. In order to show this, note that in the two-player case, Schaffer (2006) comes to the following net payoff functions for respectively players 1 and 2:
1 (1 ) x x y V R x x y γ γ − − = − + (2.2.1) 2 (1 ) y x y V R y x y γ γ − − = − + (2.2.2) Player 1 spends x on rent-seeking activities and player 2 spends y. Furthermoreγ is the externality parameter, which can be either positive or negative. Following Mikkelsen (2007), the payoff functions can be transformed in:
This proves Mikkelsen’s (2007) point that the linear “externality” does not affect the rent, as suggested by Schaffer (2006), but in fact the unit costs of rent seeking, which are independent of effort.
Next Schaffer (2006) investigates the case of non-linear externalities. He finds that the equilibrium net payoffs are always higher when effort affects the rent than when the rent is exogenous, independent of the direction of this relation. Moreover, equilibrium net payoffs are always higher with positive externalities than with negative externalities, i.e. when given the choice, players strictly prefer positive to negative externalities.
Mikkelsen (2007) also considers a two-player rent-seeking contest dealing with the sharing of a natural resource with a linear externality on effort. The difference with Schaffer (2006) is that Mikkelsen (2007) considers an asymmetric externality, where the effort of player 2 affects the payoff of player 1, but not the other way around. Because the externality is asymmetric, the argument that it only affects the unit costs of rent seeking is no longer valid.
Mikkelsen (2007) finds that player 1 will always decrease his effort level if the externality increases. Lower effort decreases the probability of winning the rent, but this is exactly offset by the increase in the externality. The response of player 2 on the decrease in effort of player 1 depends on the sign of the externality. For negative values of the externality, player 2 will increase his effort level but for positive values, he decreases his effort level. If the externality is negative, player 2 is the favourite and he reacts tough at a decrease in effort by player 1, however, if the externality is positive player 2 is the underdog and he reacts soft.
negotiate on who has to move first, they will both agree on an arrangement in which the underdog should move first.
2.3 Group Rent-Seeking Contests
2.3.1 Group Rent-Seeking with Competitive-Share Groups
In an influential paper, Nitzan (1991a) considers the possibility of group rent-seeking contests. In his model, players form so called competitive-share groups and compete as groups for the rent. In a competitive-share group, members divide the rent partially on egalitarian grounds and they distribute the residual based on relative effort ( a player’s effort relative to the total effort of the group). A competitive-share group uses the following distributing rule:
( ) 1 (1 ) ( ) a ki ki n i ki k a x a f n i x = − = +
¦
(2.3.1)With the restriction that a∈[0,1]. We have that xki is the effort made on rent seeking by
individual k of group i. Members of group i are indexed by k=1,..., ( )n i and the number of groups by i=1,...,n. This means that ( )n i is the number of members of group i. Furthermore, the size of the total population is given by
1 ( ) n i N n i = =
¦
. Finally, a is the intragroup sharing rule parameter, which means that a proportion a is divided on egalitarian grounds and the rest based on relative effort.individual share in the rent. Therefore, in the case of a= , an individual is indifferent 0 between joining a competitive-share group or not.
The second case is where competitive-share groups completely divide the rent based on egalitarian grounds, i.e. a= . In this case, aggregate effort and the rent-1 dissipation rate will be lower than in standard rent-seeking contest. The free-riding effect dominates the positive effects and group effort is decreasing in group size. Thus contrary to the previous case, the smallest group has the highest probability of winning the contest. Obviously, in this case, the rent-dissipation rate is decreasing in N. In the case of a= , 1 player ki prefers competing individually to joining any competitive-share group.
The third case is the intermediate case, i.e. a∈(0,1). In this case, aggregate effort and the rent-dissipation rate will be strictly lower compared to individual rent seeking contests. The value of the intragroup sharing rule parameter is negatively related to the degree of rent-dissipation. This means the more emphasis the group puts on equal sharing the less the rent-dissipation will be. The effect of an increase in N on the rent-dissipation rate is ambiguous. For a<1/n, an increase in total population increases the rate of rent-dissipation. For a>1/n, an increase in N reduces the rate of rent-dissipation and for
1/
a= n the rent-dissipation is independent of N. As noted by Nitzan (1991a), an important underlying implicit assumption is that relative effort can be observed and rewarded costlessly. If such costs are explicitly introduced, see for example Ueda (2002) who introduces monitoring costs, effort levels are likely to be lower and related to the parameters of the model in a more complex way.
In another paper, Nitzan (1991b) makes a point about the possibility of non-existence of equilibrium in the case of competitive share groups using different sharing rules. Nitzan (1991b) considers a two-group collective rent-seeking contest where one group entirely divides the rent based on aggregate effort and the other group fully distributes the rent on egalitarian grounds. Nitzan (1991b) shows that there can only be an equilibrium if the following condition holds:
1 2 1 2
show that if this assumption is relaxed, allowing for zero effort by one group, condition (2.3.2) is no longer necessary for equilibrium. In addition, Davis and Reilly (1999) demonstrate that in this case there is a unique equilibrium. In this equilibrium, members of group 1, with a distribution rule totally based on egalitarian grounds, will contribute zero effort (the free-riding effect totally dominates) and members of group 2 will contribute 2 2 2 ( 1) ( ) n R n − .
Extensions
As noted before, the paper of Nitzan (1991a) has greatly influenced the literature on group rent-seeking contests. In the following, we will discuss some articles that have used his ideas and extended it into various directions.
Lee (1993) presents a model that can be seen as a special case of the Nitzan (1991a) model. There are some differences however. First, the number of groups is exogeneous and equal to two. Second, players completely share the rent on egalitarian grounds, i.e.
1
a= in (2.3.1). The extension is that the author allows for the possibility of increasing or decreasing marginal returns to effort. The most interesting result of this paper concerns the relation between the degree of rent dissipation and the difference in group sizes. In the case of increasing (decreasing) marginal returns to effort, the extent of rent dissipation is decreasing (increasing) in the disparity of group sizes, and with constant marginal return to effort, the rent dissipation is independent of the difference in group sizes.
The main weakness of the model of Nizan (1991a) is that all groups use the same sharing rule independent of their size. Lee (1995) solves this drawback by endogenizing the intra-group sharing rule parameter a. The distribution rule now reads:
( ) 1 (1 ) ( ) a i ki i ki n i ki k i a x a f n i x = − = +
¦
(2.3.3)In this way, each group has his own unique intra-group sharing rule parameter. Lee (1995) considers a two period model. In stage one, members choose their intra-group sharing rule parameter and in stage two all players simultaneously decide on their effort levels.
In the first part of his analysis, Lee (1995) restricts his attention to a model with two competitive-share groups. The author distinguishes two different cases. First, he considers a model with two unequal sized groups. In this case, the smaller group will select a distribution rule purely based on aggregate effort, i.e. ai = , and the larger group 0 will choose an intermediate distribution rule, i.e. ai∈(0,1). Both groups spend the same amount on rent seeking and logically have an equal change of winning the contest. The rent-dissipation rate is lower compared to individual rent seeking and decreasing in the disparity of group sizes. The second case Lee (1995) considers, is one with two equal sized competitive-share groups. In this case, both groups will choose a distribution rule solely based on aggregate effort and the contest will be equal to one of individual rent seeking.
Another debatable assumption of the model of Nitzan (1991a) is that all players are by assumption member of a competitive-share group. Baik and Shogren (1995) investigate a group rent-seeking contest model in which this assumption is relaxed. They consider a three-period model in which the size of the group and the distribution rule are endogenous, but the number of groups is exogenous and equal to one. In stage one, each player decides whether or not to join the competitive share group. In stage two, members of the competitive share group choose their distribution rule and in the third stage, all players simultaneously choose their effort levels. First, Baik and Shogren (1995) show that forming a competitive-share group is profitable and the players will therefore always form a competitive-share group voluntarily. Next, they determine the number of members of the competitive-share group in equilibrium. All players are members of the competitive-share group for n= and 2 n= , furthermore the group consist of three or 3 four players if n= or 4 n= . For 5 n>6 the equilibrium number of members of the competitive-share group is the smallest integer greater than half of the total number of players. Baik and Shogren (1995) show that aggregate effort and thereby the rent-dissipation rate are strictly lower with one competitive-share group compared to individual rent seeking contest.
effort. This means that the degree of rent-dissipation will be sizeable and close to that of individual rent-seeking contests.
2.3.2 Group Rent-Seeking with Strategic Groups
Baik (1994) developed the idea of a winner-help-loser group formation in rent-seeking contests. In later literature, such groups are called strategic groups and as from now, we will use this name as well. In a strategic group, players compete individually for the rent, but they agree to share the rent if one of them wins the contest. The winner keeps a prespecified percentage of the rent and the remaining rent will be divided equally among the other members. Baik (1994) considers a rent-seeking model with one strategic group. The expected payoff of individual i of the strategic group is:
(1 ) 1 i i i i x X x V R R x X Y n X Y σ σ − − = + − + − + (2.3.4) Where we have that xi is the effort of member i of the strategic group and n is the number of members of the strategic group. Furthermore X is the total effort by the members of the strategic group and Y the total effort by non-members. Finally, σ is the winner’s fractional share parameter. Baik (1994) finds that the expected payoff of a member of the strategic group will never be lower than the expected payoff of individual rent seeking, i.e. the formation of a strategic group is always profitable.
In section four, Baik (1994) investigates a (n+2)-period model were the size of the strategic group is endogenous. In the first n periods, players decide sequentially whether to join the strategic group or not. In period (n+1), players choose the winner’s fractional share parameter σ and in period (n+2) all players simultaneously choose their effort levels. Regarding the equilibrium group size, the results are exactly similar to those of Baik and Shogren (1995). All players are members of the strategic group for n= and 2
3
n= furthermore the group consist of three or four players if n= or 4 n= and for 5 6
degree of rent- dissipation is also similar, i.e. the rent dissipation will always be lower if a strategic group is formed.
Baik and Lee (2001) extend the model of Baik (1994) by making the number of strategic groups endogenous. They consider a three-period rent-seeking contest model. In period one, players decide whether to form a strategic group or not. In period two, players in a strategic group select the winner’s fractional share σi and in period three, players
choose their effort levels. The expected payoff of member k of group i now reads:
[
]
( ) 1 ( ) ( ) 1 1 1 1 (1 ) ( ) 1 n i i ki ki k i ki ki n i n n i n ki ki ki k i k i x x x V R R x x n i x σ σ = = = = = § · − ¨ − ¸ © ¹ = + − −¦
¦¦
¦¦
(2.3.5)Where we have that xki is the effort made by individual k of group i. Members of group i
are indexed by k=1,..., ( )n i and the number of groups by i=1,...,n . This means that ( )
n i is the number of members of group i. Furthermore σi is winner’s profit share
parameter of group i.
Baik and Lee (2001) compare their outcomes to individual rent seeking. They show that if in equilibrium there is just one strategic group this will be advantageous to the members and the non-members of the strategic group. If there are more than two strategic groups in equilibrium, this will be disadvantageous to both the members and the non-members of the strategic groups. Rent-dissipation will be less with one strategic group, whereas rent-dissipation will strictly greater with two or more strategic groups.
2.3.3 The Equivalence of Rent-Seeking Outcomes for Competitive-Share
and Strategic Groups
with competitive-share and strategic groups face the same optimization problem. Using (2.3.3), we find that the expected payoff of member k belonging to competitive-share group i is: ( ) 1 ( ) ( ) 1 1 1 1 (1 ) ( ) ( ) n i i ki i ki k ki n i n n i n ki ki ki k i k i a x a x V c R R x x n i x = = = = = − = + −
¦
¦¦
¦¦
(2.3.6)The expected payoff of member k belonging to strategic group i is equal to (2.3.5):
[
]
( ) 1 ( ) ( ) 1 1 1 1 (1 ) ( ) ( ) 1 n i i ki ki k i ki ki n i n n i n ki ki ki k i k i x x x V s R R x x n i x σ σ = = = = = § · − ¨ − ¸ © ¹ = + − −¦
¦¦
¦¦
(2.3.5)Non-members solve the standard individual rent-seeking problem, which is obviously equivalent for both cases. Baik et al (2006) show that (2.3.5) and (2.3.6) are equivalent for:
[
]
( ) 1 ( ) 1 i i n i a n i σ − = − (2.3.7) This means that rent-seeking contests with competitive-share groups can be rewritten to one with strategic groups and the other way around using (2.3.7). This result proves the strategic equivalence between the two forms of group rent-seeking contests.2.4 Group Rent-Seeking Contests with Externalities
Kang (1998) also allow for the possibility of the externality being negative. In the final part of their article, Lee and Kang (1998) consider a group rent-seeking contest model. This model is similar to that of Lee (1995), the important difference, however, is the presence of externalities.
Since the externality, denoted by β, affects real costs, Lee and Kang (1998) argue that the conventional rent-dissipation rate is no longer a good measure for social waste. Therefore, they introduce a new measure, the real rent-dissipation rate. This is defined as the sum of real costs, i.e. nominal costs minus the externality, divided by the rent. An increase in N strictly increases the nominal rent-dissipation rate, but it does not per se increase the real rent-dissipation rate. For β<1/ N, an increase in N leads to an increase in the real rent-dissipation rate, whereas for β >1/ N it leads to a decrease. Furthermore Lee and Kang (1998) show that the nominal rent-dissipation rate is increasing in β, but the real rent-dissipation rate turns out to be decreasing inβ.
The group rent-seeking contest model considered by Lee and Kang (1998) is a two-stage model. The model consists of two unequal sized competitive-share groups with an exogenous number of players and endogenous intra-group sharing rules. Where we have that group 1 is larger than group 2. In stage one, competitive-share groups decide on their intra-group sharing rule parameter and in stage two, all players simultaneously choose their effort levels. Due to an arithmetic error, the remainder of the article is flawed. Luckily, Gürtler (2005) corrected these errors and we can use his corrigendum for the correct continuation of the analysis.
3. Extensions and the Effect of Group Formation
In this section, we will investigate new ways to introduce externalities in rent-seeking contests. Furthermore, we will explore the effect of group rent seeking on these models. In section 3.1, we introduce an individual rent-seeking contest model with two types of players and externalities. The externality is rent dependent and type specific. Next, in section 3.2 we consider an individual rent-seeking contest model with externalities that are dependent on the rent and location. Subsequently, in section 3.3, we repeat our analysis of section 3.1, but we extend the model by giving one type of player the opportunity to form a strategic group. Finally, in section 3.4 we investigate group rent-seeking contests with competitive-share groups for the model presented in section 3.2. Note that due to the strategic equivalence of group rent-seeking contests with competitive-share and strategic groups (Baik et al, 2006), it does not matter for the outcomes of our analyses which form of group rent seeking we use in sections 3.3 and 3.4. We have used both forms just to keep the analysis a bit more diverse.
3.1 Rent-Seeking Contests with Type Specific Externalities
3.1.1 Introduction
An example of a group of players that share communal facilities can be found at our own university. The faculties of Management and Organisation, Economics and Spatial Sciences share one library. Every year these faculties compete in a rent-seeking contest to persuade the university to give them more money. The university only has a limited budget to support these faculties, so more money for one faculty will come at the expense of another. If one of these faculties, for example the faculty of Economics, wins the contest, it may invest some of the rent in upgrading the library. More places to study, new furniture, new books, more computers etc. Professors and students of the faculties of Management and Organisation and Spatial Sciences also profit from this upgrading, because they are allowed to use these new facilities as well. Therefore, the faculties of Management and Organisation, Economics and Spatial Sciences prefer one of the other two faculties to win the contest if they do not. In this example, the faculties of Management and Organisation, Economics and Spatial Sciences are the Insiders. Other faculties who do not share facilities with Insiders or with each other correspond to a group of Outsiders. For example, the faculties of Philosophy, Theology and Religion Studies and Medical Sciences.
3.1.2 Model
In this one-period model, there are n players competing for a fixed rent R. We have m Insiders and k Outsiders and m + k = n. To ensure sensible outcomes we assume m≥ 2 and k≥ . We define 1 I=
{
1,...,m}
as the set of Insiders and O={
m+1,....,n}
as the set of Outsiders. The total set of players is N ={
1,...,n}
. If an Insider wins the contest, all other Insiders benefit from it. Therefore, there is an externality of a winning Insider on all other Insiders. We denote this externality αR with α∈[ ]
1, 0 . An Outsider does not benefit from this externality nor does it have an externality from any other Outsider winning the contest. The effort of player i to win the contest is x and the expected ipayoff of player i of competing in the contest is Vi. Following Tullock (1980), the probability that player i wins the contest takes the following symmetric logit form:
1 i p n = if xj = for all j 0 1 i i n j j x p x = =
¦
otherwise (3.1.1)Taking this into account we come to the following.
The expected payoff of Insider i is:
1 i j j i i i j i n i j i j j j I j I x x V p R p R x R x x α α ≠ ≠ = ∈ ∈ § + · ¨ ¸ ¨ ¸ = + − = − ¨ ¸ ¨ ¸ © ¹
¦
¦
¦
(3.1.2)The first term is the Insider’s expected payoff of winning the contest himself. The second term is the expected payoff of receiving an externality if another Insider wins the contest and the third term is the effort/costs of rent seeking.
1 2 1 (1 ) 1 0 n j j j i j m i n i j j j I x x V R x x α ≠ = + = ∈ − + ∂ = − = ∂ § · ¨ ¸ © ¹
¦
¦
¦
(3.1.3)We define the effort levels of respectively the Insiders and the Outsiders as xI and xO. Rewriting (3.1.3) and imposing symmetry gives us:
2
(mxI +kxO) =ª¬(1−α)(m−1)xI +kxOº¼ (3.1.4) R
The expected payoff of Outsider i is:
1 i i i i n i j j x V p R x R x x = = − = −
¦
(3.1.5)Maximise Vi over xi results in the following first order condition:
1 2 1 1 0 m j j j j i i n i j j j O x x V R x x = ≠ = ∈ + ∂ = − = ∂ § · ¨ ¸ © ¹
¦
¦
¦
(3.1.6)Rearranging and using symmetry gives:
2 0
( I O) I ( 1)
mx +kx =ª¬mx + k− x º¼ (3.1.7) R
Equating the first order conditions gives us that:
[
1 ( 1)]
O I
x = + m− α x (3.1.8) Substituting (3.1.8) back into (3.1.7) and solving forxI gives:
[
]
2 1 ( 1)( 1) ( 1) I n k m x R n k m α α − + − − = + − (3.1.9) Using (3.1.9) and (3.1.8) we can determine that xO is:[
][
]
[
]
2 1 ( 1)( 1) 1 ( 1) ( 1) O n k m m x R n k m α α α − + − − + − = + − (3.1.10) Total effort is equal to:The probabilities of winning the contest for respectively the Insiders and the Outsiders are p and I p . Using (3.1.9)-(3.1.11) we can calculate O p and I p as: O
1 ( 1) I p n k m α = + − (3.1.12) 1 ( 1) ( 1) O m p n k m α α + − = + − (3.1.13) The expected payoffs of winning the contest for the Insiders and Outsiders are
respectively VI and VO. Inserting (3.1.9) and (3.1.12) into (3.1.2) determines VI. We have that:
[
]
2 2 1 ( 1) ( 1) ( 1) ( 1) I m n k m V R n k m α α α ª º + − ¬ + + − ¼ = + − (3.1.14) Substitution of (3.1.10) and (3.1.13) into (3.1.5) leads to V . We obtain: O2 1 ( 1) ( 1) O m V R n k m α α § + − · = ¨ ¸ + − © ¹ (3.1.15) Note that O I x >x and O I p > p , but O I
V <V . Outsiders make more effort to win the contest and as a result, their probability of winning is higher. However, more effort also means more costs and therefore the expected payoff of the Outsiders is strictly lower than that of the Insiders, for positive values ofα. We come to the following proposition:
Proposition 1:
In an individual rent-seeking contest model with m Insiders, k Outsiders and type specific externalities the respective equilibrium effort levels are:
[
]
2 1 ( 1)( 1) ( 1) I n k m x R n k m α α − + − − = + − and[
][
]
[
]
2 1 ( 1)( 1) 1 ( 1) ( 1) O n k m m x R n k m α α α − + − − + − = + − .Furthermore, the equilibrium expected payoffs are respectively:
3.1.3 Comparative Statics
In this section we will investigate the comparative statics properties of the model of section 3.1.2. We will examine the effect of changes in the number of players on the equilibrium effort levels, expected payoffs and rent-dissipation rate. Furthermore, we will also look into the effect of changes in the externality parameter α on the equilibrium outcomes of the model.
Effect of a change in the number of players
First, we will investigate how variations in the number of players affect effort levels. We derive that:
[
]
[
]
2 3 2 ( 2) ( 1) 2 ( 1)( 1) 0 ( 1) I n k n m k k k m x R m n k m α α α − + − + − + + − − ∂ = − < ∂ + − (3.1.16)[
][
]
[
]
3 1 ( 1) 2 ( 2)( 1) 0 ( 1) I m n k m x R k n k m α α α + − − + − − ∂ = − < ∂ + − (3.1.17)[
]
[
]
3 (1 ) 2 ( 2)( 1) 0 ( 1) O n k m x R m n k m α α α − − + − − ∂ = − < ∂ + − (3.1.18)[
] [
]
[
]
2 3 1 ( 1) 2 ( 1)( 2) 0 ( 1) O m n m k x R k n k m α α α + − − + − − ∂ = − < ∂ + − (3.1.19) An increase in m or k decreases the effort made by the Insiders and Outsiders to win the contest. An increase in the number of players lowers the probability of winning the contest and makes the players less aggressive i.e. they spend less effort.We can do the same to find out what the effect of a change in the number of Insiders has on the expected payoff of the Insiders. We find that:
0 I V m ∂ < ∂ if α α ∗ < (3.1.21) 0 I V m ∂ > ∂ if α α ∗ > (3.1.22) With 2 2 2 2 ( 2) 8 ( 1) 2 ( 1) kn kn k m k m α∗ − − + + + − −
=
(3.1.23)We also investigate the effect of variations in the number of Outsiders on V . We get: I
[
]
2 2 3 3 3 2 ( 1)( 4) ( 1) ( 2) ( 1) 0 ( 1) I V m n m n k k m R k n k m α α α α ∂ + − + + − + + + − = − < ∂ + − (3.1.24)Similarly, we can examine the effect of a change in the number of Insiders or Outsiders on O V . We have that:
[
]
[
]
2 3 2 1 ( 1) 1 ( 1) ( 1) 0 ( 1) O m k k m V R m n k m α α α α ª º + − + − + − ∂ ¬ ¼ = − < ∂ + − (3.1.25) 3 0 1 ( 1) 2 0 ( 1) V m R k n k m α α ª º ∂ + − = − « » < ∂ ¬ + − ¼ (3.1.26) An increase in m will decrease the expected payoff of the Insiders for low values of α, but increase the expected payoff of the Insiders for high values of α. An increase in m has two opposite effects. There is a positive effect through an increase in the ratio of Insiders over Outsiders. This increases the probability that another Insider will win the contest, which leads to an increased chance of receiving a positive externality. The negative effect is that an increase in the number of players lowers the probability that an Insider will win the contest himself. For high values of α the first effect dominates thesecond and for low values of α the second effect dominates the first. An increase in m decreases the expected payoff of the Outsiders and an increase in k
decreases the expected payoff for both the Insiders and the Outsiders. In these cases, there is only a negative effect through a lower probability of winning the contest.
t = 1 n j j x R =
¦
1 ( 1)( 1) ( 1) n k m n k m α α − + − − = + − (3.1.27) Now we will investigate how an increase in the number of players influences the rent-dissipation rate. It turns out that:[
]
2 1 0 ( 1) t m n k m α α ∂ − = > ∂ + − (3.1.28) 2 1 ( 1) 0 ( 1) t m k n k m α α ª º ∂ + − =« » > ∂ ¬ + − ¼ (3.1.29) Both derivatives are positive, which means that, an increase in the number of both Insiders and Outsiders always increases the rent-dissipation rate.The next thing we look into is the effect of an infinite number of players on the rent-dissipation rate. We find that:
lim 1 1 m t k α α →∞ = − + (3.1.30) lim 1 k→∞t= (3.1.31)
One of the most controversial results of the seminal model of Tullock (1980) is that there is full rent-dissipation if the number of players goes to infinity. In our model, for a given number of Insiders, there will be full rent-dissipation as well, if the number of Outsiders goes to infinity (see (3.1.31)). However, for a given number of Outsiders the rent-dissipation rate is smaller than one, if the number of Insiders goes to infinity (see (3.1.30)). The reason for this result is that, Insiders suffer from free-riding behaviour. This makes them spend less effort and as a result, the rent will not be fully dissipated. Summarising we have obtained that:
Proposition 2:
(a) The effort levels of both types of players are decreasing in the number of Insiders and
Outsiders.
(b) An increase in the number of Insiders will increase VI for α >α∗, but will decrease I
of Insiders. An increase in the number of Outsiders decreases the expected payoff of both types of players.
(c) The rent-dissipation rate is strictly increasing in the number of Insiders and Outsiders.
There is full rent-dissipation for an infinite number of Outsiders, but for an infinite number of Insiders the rent is not fully dissipated.
Effect of a change in the externality parameter
We will start by investigating the effect of changes in α on the effort levels. We obtain:
[
]
[
]
3 ( 1) ( 1)( 1) ( 1) 0 ( 1) I m k k m k n m x R n k m α α α − − − + − + ∂ = − < ∂ + − (3.1.32)[
]
[
]
3 ( 1) 2 ( 2)( 1) 0 ( 1) O m m n k m x R n k m α α α − − + − − ∂ = > ∂ + − (3.1.33) An increase inα makes Insiders spend less effort, because there is an increase in thefree-rider effect. High values of α make it more alluring for an Insider to spend less and reap the fruits of efforts made by other Insiders. Contrary to Insiders, increases in the externality parameter make Outsiders spend more effort. Outsiders have a strategic advantage over Insiders, because they do not suffer from the free-rider behaviour. Increases in α makes Outsiders increase their effort levels in an attempt to exploit this strategic advantage even further.
Next, we investigate the effect of variations in α on the probabilities of winning the contest. We derive that:
Similarly we can investigate how an increase in α affects the expected payoffs of the Insiders and Outsiders. We find that:
[
]
3 ( 1) ( 1) 2 ( 1)( 1) 0 ( 1) I V m m k k km k n m n k m α α α ∂ + + − + + − − = > ∂ + − (3.1.36)[
]
[
]
3 2 ( 1) 1 ( 1) 0 ( 1) O m m m V R n k m α α α − + − ∂ = > ∂ + − (3.1.37) For the Insiders the decrease in rent-seeking effort/costs, see (3.1.32), outweighs the decrease in expected revenues. Expected revenues decrease due to a decreased probability of winning the contest, see (3.1.34). The opposite holds for the Outsiders. We have that the increase in effort/costs, see (3.1.33), are outweighed by the increase inexpected revenues, see (3.1.35).
Another interesting thing to have a look at is how the externality parameter α influences the rent-dissipation rate. We obtain:
[
]
2 ( 1) 0 ( 1) t m m n k m α ∂ − = − < ∂ + − (3.1.38) An increase in the externality parameter decreases the rent-dissipation rate. An increase in the externality parameter makes Insiders less aggressive, but Outsiders more aggressive (see (3.1.32) and (3.1.33)). Apparently, the increased free-rider behaviour among Insiders outweighs the desire of Outsiders to exploit their strategic advantage even further and therefore the rent-dissipation rate is decreasing in α. This result is opposite to that of the model of Lee and Kang (1998). In their model, an increase in the externality parameter makes all players more aggressive and strictly increases the rent-dissipation rate. We conclude with the following proposition:Proposition 3:
3.2
Rent-Seeking
Contests
with
Location
Dependent
Externalities
3.2.1 Introduction
In this section we will discuss a rent-seeking contest model with externalities that are location dependent. In this model, players are distributed along a circle. For example, let us assume that n cities are located around a big circular lake. A large company is planning to build a factory in one of these cities. A new plant will create many new jobs and increases social welfare. Not only new people will be hired, but also supporting companies like suppliers, law firms, and notaries will profit from this new factory. Cities are aware of these positive effects and engage in a rent-seeking contest to persuade the company to build the plant in their city. The city that wins the contest will profit the most of this new factory, but cities located nearby will also partially profit from the new plant. The idea is that citizens of nearby located cities can travel to this new factory. People located in far away cities will not do so, because travelling costs will become too high.
The important difference with the previous model is that the externality is location dependent, whereas in the previous model it was dependent on the type of player. In present model, all players are concerned with whoever wins the contest. All players prefer one of their neighbours to win the contest if they do not themselves. In the model of section 3.1, only the Insiders were concerned with however won the contest if they did not. This model can be seen as an extension of Linster’s (1993) placement of a streetlight example. The main weakness of this example is that one can not impose symmetry and make generalisations because players are distributed alone a Hotelling line. We solve this drawback by assuming players are distributed along a Salop circle. In the next section we formalise this idea into a rent-seeking contest model.
3.2.2 Model
decreasing in distance. If player i wins the contest, players i− and 1 i+ receive R1 α and players i−2 and i+2 receive 2
R
α . For players that are located further away, we assume that the externality is equal to zero. The probability that player i wins the contest has the usual symmetric logit form (see (3.1.1)). Notation for effort levels and expected payoffs are similar to those in section 3.1.2.
City i City i−1 City i+1 R α R α 2 R α 2 R α City i+2 City i−2
Figure 1: Individual rent-seeking contest with cities located along a circle and location dependent externalities. If city i wins the contest cities i− and 1 i+ receive R1 α and cities i− and 2 i+ receive 2 2
R
α .
The expected payoff of player i is:
2 1 1 2 2 ( ) ( ) i i i i i i i V =p R+ p− + p+ αR+ p− +p+ α R−x 2 1 1 2 2 1 ( ) ( ) i i i i i i n j j x x x x x R x x α α − + − + = + + + + = −
¦
(3.2.1)The first term is the expected payoff of winning the contest. The second and third terms are the expected payoffs of receiving an externality if a nearby-located player wins the contest.
2 1 1 2 2 1 2 1 ( ) ( ) 1 0 n j i i i i i j i n i j j x x x x x x V R x x α α − + − + = = ª º −¬ + + + + ¼ ∂ = − = ∂ § · ¨ ¸ © ¹
¦
¦
(3.2.2)Imposing symmetry and solving for x gives:
2 1 2 (1 ) n x R n α α − − + = (3.2.3) Total effort is:
1 1 2 (1 ) n j j n x nx R n α α = − − + = =
¦
(3.2.4) The probability of winning the contest is:1
x p
nx n
= = (3.2.5) Substituting (3.2.3) and (3.2.5) into (3.2.1) gives the expected payoff for player i:
2 1 2 (1 )(1 n) V R n α α + + + = (3.2.6) Summarizing we come to the following proposition:
Proposition 4:
In an individual rent-seeking contest model with n players and location dependent externalities, the equilibrium effort level and expected payoff are, x n 1 2 (12 )R
n α α − − + = and V 1 2 (1 2 )(1 n)R n α α + + + = .
3.2.3 Comparative Statics
First, we will investigate how an increase in the number of the players affects the expected payoff of competing in the contest.
present model an increase in the number of players only has a negative effect. In this model the externality depends on location therefore an increase in the number of players has no benefits and only lowers the probability of winning the contest for all players.
Next, we will investigate how changes in α affect effort levels and expected payoffs. We can derive that:
2 2(1 2 ) 0 x R n α α ∂ + = − < ∂ (3.2.8) 2 2(1 )(1 2 ) 0 V n R n α α ∂ + + = > ∂ (3.2.9) An increase in the externality parameter makes the players less aggressive, because there
is an increase in the free-rider effect. This result is similar to that of section 3.1.2, where an increase in α decreased the effort made by Insiders. Since the probability of winning the contest is independent of α , a decrease in effort increases expected payoffs.
Next, we look into the effect of an increase in α on the rent-dissipation rate. The rent-dissipation rate is:
1 1 2 (1 ) n j j x n t R n α α = − − + = =
¦
(3.2.10) From this we derive that:2(1 2 ) 0 t n α α ∂ + = − < ∂ (3.2.11) Equivalently to the model in 3.1.2, the rent-dissipation rate is strictly decreasing inα . An increase in the externality parameter makes all players cut back their effort levels (see (3.2.8)) and lessens the degree of rent-dissipation.
Similar to the model of Tullock (1980) there will be full rent-dissipation if the number of players goes to infinity. We obtain:
lim 1
n→∞t= (3.2.12)
Proposition 5:
(a) The expected payoff of competing in the contest is strictly decreasing in the number of
players.
(a) The effort level and the rent dissipation rate are decreasing in α. The expected payoff is increasing in α.
(b) There is full rent-dissipation for an infinite number of players.
3.3 Rent-Seeking Contests with Group Specific Externalities
and Strategic Groups
3.3.1 Introduction
In this section we extend the model analysed in section 3.1.2 and give Outsiders the opportunity to form a strategic group. Outsiders do not have to form a strategic group, but if they decide to do so, then all Outsiders join. Insiders do not form a strategic group and compete individually. Following Baik (1994), in a strategic group, players compete individually for the rent, but they agree to share the rent if one of them wins the contest. The winner keeps a prespecified percentage of the rent and the remaining rent will be divided equally among the other members.
In light of our library example of section 3.1 it might be hard to imagine this kind of collaboration. Therefore, in this section we will make use of the charity groups with overlapping interests example, see section 3.1. This means that the charity groups without overlapping interests, i.e. the Outsiders, form a strategic group and the charity groups with overlapping interests, i.e. the Insiders, do not organize themselves. An incentive for forming a strategic group is that in this way Outsiders share the risk of not winning the rent. A second benefit is that group formation might influence the behaviour of non-members in their favour.
3.3.2 Model
exogenously determined and equal to one. In period two, Outsiders decide on the value of the winner’s fractional share parameter σ, if they have decided to form a strategic group. We do not allow for the possibility of bounty payments (Baik and Lee, 2001), therefore,
[0,1]
σ∈ . In period three, all players simultaneously decide on their effort levels. To solve this problem we will use backward induction and start with the last period of the game. Notation is equal to that of section 3.1.2.
Period t = 3:
The expected payoff of Insider i is:
1 i j j i i i j i n i j i j j j I j I x x V p R p R x R x x α α ≠ ≠ = ∈ ∈ § + · ¨ ¸ ¨ ¸ = + − = − ¨ ¸ ¨ ¸ © ¹
¦
¦
¦
(3.1.2)Note that this expression is equal to the expected payoff of Insider i in section 3.1.2. The optimization problem for the Insiders has not changed. Therefore, we get the same best-response function as in section 3.1.2.
2
(mxI +kxO) =ª¬(1−α)(m−1)xI +kxOº¼ (3.1.4) R
The expected payoff of Outsider i reads:
1 1 (1 ) (1 ) 1 1 j j i i i i j i n n i j i j j j j j O j O x x R R V p R p x R x k k x x σ σ σ σ ≠ ≠ = = ∈ ∈ = + − − = + − − − −
¦
¦
¦
¦
(3.3.1)The first term is the Outsider’s expected payoff of winning the contest himself. The second term is the expected payoff of receiving a share if another Outsider wins the contest.
The first order condition for maximising expected payoff is:
1 2 2 1 1 (1 ) 1 0 1 n j j i j i j i n n i j j j j i O x x x V R R x k x x σ = σ ≠ = = ∈ − ∂ = − − − = ∂ § · § · − ¨ ¸ ¨ ¸ © ¹ © ¹
¦
¦
¦
¦
(3.3.2)(
)
2(
)
1 ( 1) (1 )( 1) 1 I O I O O mx kx mx k x k x R k σ σ ª º + =« + − − − − » − ¬ ¼ =ª¬σmxI +(σk−1)xOº¼ (3.3.3) REquating the first order conditions and solving for xO gives us: (1 )( 1) 1 (1 ) O m m I x x k σ α σ − − − = + − (3.3.4) Substitution of (3.3.4) into (3.1.4) and solving for I
x leads to:
[
][
]
[
]
2 1 (1 ) (1 )( 1)( 1) ( 1) I k km m k x R n k m σ σ α σ + − − − − − = + − (3.3.5) Using (3.3.4) and (3.3.5) we can determine x . We have that: O[
][
]
[
]
2 (1 )( 1) (1 )( 1)( 1) ( 1) O m m km m k x R n k m σ α σ α σ α − − − − − − − = + − (3.3.6) Total effort is:1 (1 )( 1)( 1) ( 1) n I O j j km m k x mx kx R n k m σ α σ = − − − − = + = + −
¦
(3.3.7) By making use of the effort levels we can calculate Ip and O p as: 1 (1 ) ( 1) I k p n k m σ α + − = + − (3.3.8) (1 )( 1) ( 1) O m m p n k m σ α α − − − = + − (3.3.9) Inserting (3.3.5) and (3.3.8) into (3.1.2) leads to I
V :
[
] [
{
][
]
[
]
}
[
]
2 1 (1 ) 1 (1 ) 1 ( 1) ( 1) ( 1) ( 1) I k k m m n k m V R n k m σ σ α α α α + − + − + − + − + − = + − (3.3.10)Substitution of (3.3.6) and (3.3.9) into (3.3.1) gives us V : O
Period t = 2:
In the second period, the Outsiders, if organised in a group, choose the winner’s fractional share parameter σ to maximize their expected payoffs. Because all Outsiders are identical, they choose one representative Outsider among them to select the winner’s fractional share parameter.Taking the derivative with respect to σ leads to the following first order condition:
[
] [
{
]
[
]
}
[
]
2 1 ( 1) 1 (1 ) (1 )( 1) 0 ( 1) O m m k k m m V R n k m σ σ α σ α + − + − − − − − ∂ = = ∂ + − (3.3.12)Solving this equation for σ gives:
2 ( 1) 2 m k km k m km α σ∗= − + − − for 1 ( 1) m k m α > + − (3.3.13) 1 σ∗= for 1 ( 1) m k m α ≤ + − (3.3.14) Where the second equation follows from the restriction on σ. For σ = the model is 1 equal to that of section 3.1, therefore in the remainder of this analysis we assume that
1 ( 1) m k m α > + − .
Substituting (3.3.13) into (3.3.5) and (3.3.6) gives the effort levels:
[
]
2 2 1 ( 1) 4 I m m x R m α − + − = (3.3.15)[
]
2 1 ( 1) 4 O m m x R km α − + − = (3.3.16) Inserting (3.3.13) into (3.3.7) determines total effort:[
]
1 2 1 ( 1) 2 n j j m m x R m α = − + − =¦
(3.3.17) Substitution of (3.3.13) into (3.3.8) and (3.3.9) and gives us the respective probabilities ofSubstitution of (3.3.13) into (3.3.10) and (3.3.11) leads to the relevant payoffs: 2 1 ( 1)(1 2 ) 4 I m m V R m α + − + = (3.3.20) 1 ( 1) 4 O m V R km α + − = (3.3.21) Period t = 1:
To decide whether or not to form a strategic group Outsiders need to compare (3.1.15) and (3.3.20). Forming a strategic group is profitable if the following condition holds:
2 1 ( 1) 1 ( 1) 4 ( 1) m m km n k m α α α § · + − + − ≥ ¨ ¸ + − © ¹ (3.3.22) Rewriting this condition gives us:
[
]
2 2 2 2
1 2(m 1)α (m 1) α k 2m1 (m 1)α k m 0
ª + − + − º − + − + ≥
¬ ¼ (3.3.23) Solving the parabola on the left hand sight shows that equation (3.3.22) holds with equality for:
[
]
2 2 1 ( 1) 1 2( 1) ( 1) m m k m m α α + − = + − + − (3.3.24) For this unique value of k, the Outsiders are indifferent between forming a strategic group or not. This value of k also coincides with the minimum of the parabolic function. Therefore, for any other value of k, the Outsiders’ payoff will be strictly larger if they form a strategic group. Since Outsiders are never worse off by forming a strategic group they will choose to do so.The rent-dissipation rate
Now we will investigate the rent dissipation rate. Notice that:
[
]
2 1 ( 1) 1 ( 1)( 1) 2 ( 1) m m n k m m n k m α α α − + − − + − − < + − (3.3.26) Rewriting this as a condition on k gives:1 ( 1) m k m α > + − (3.3.27) This condition is equal to the restriction on k that followed from (3.3.12). Therefore, the rent-dissipation rate will always be lower with a strategic group. For σ < , Outsiders 1 suffer from free-riding behaviour and this makes them less aggressive. Therefore, a profit-share parameter smaller than one, ensures that the rent-dissipation rate will be strictly lower with a strategic group
To see that this result is related to Lee (1995) notice that for low values of α , (3.3.27) comes down to a condition that the number of Outsiders must be larger than the number of Insiders. We have that:
0 lim 1 ( 1) m m m α→ + − α = (3.3.28)
