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Contest for power in organizations
Economics Letters 114 (2012) 280–283
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Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Contest for power in organizations Martin Gregor ∗ Institute of Economic Studies, Faculty of Social Sciences, Charles University in Prague, Opletalova 26, Prague, CZ 110 00, Czech Republic
article
info
Article history: Received 22 July 2010 Received in revised form 12 October 2011 Accepted 20 October 2011 Available online 28 October 2011
abstract This paper explains the provision of private rent to powerful members in an organization as an outcome of a contest for power that raises the total contributions to the organization. A necessary condition for a socially efficient contest scheme with reimbursements is characterized. © 2011 Elsevier B.V. All rights reserved.
JEL classification: D23 D70 H41 Keywords: Contest Reimbursement Rent-seeking Collective good
1. Introduction The market provision of collective goods often goes through intermediary organizations that bundle members’ contributions, coordinate their efforts, and save on economies in scale. Executive discretion within the organization, nevertheless, creates an option to leak a part of the organization’s budget into private use. Aware of this possibility, the members compete for control over the organization, seeking a private rent that effectively rebates a portion of their contributions. For the organization, rebating a part of the contributions is equivalent to not receiving the rebates at all. For an individual, the presence of a contested rebate implies a dual role for a marginal individual contribution: It increases the amount of the collective good, and it also generates extra influence within the organization that improves the chance of obtaining the contested private rent. Thereby, the contest for this private rent artificially drops the effective price of the collective good to an optimal level and even below. Competition for power in an organization can thus be seen as a decentralized mechanism to improve the collective action of the organization. In the founding book on incentives for collective action, Olson (1965) highlighted that successful interest groups use ‘selective incentives’ in the form of private goods that motivate membership; our paper shows that mixing the provision of a
∗
Tel.: +420 222 112 306; fax: +420 222 112 304. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.10.027
private and collective good may improve the provision of the collective good as long as the private good is offered as the prize of a contest, with a sufficient level of discrimination. This paper finds that a contest is efficient only if it applies a sufficiently high, but not excessively large, marginal return. It also characterizes the specific share of the budget that an organization must convert into private rents in order to be able to provide an optimal amount of the collective good. These results are derived under an n-person symmetric pure strategy equilibria with complete information, for a general utility over the private and collective good. The idea of bundling a collective good with a private good is closely related to literature on the use of charity lotteries and charity auctions (Bos, 2010; Elfenbein and McManus, 2010). In a seminal article, Morgan (2000) examined fixed-prize and pari-mutuel-prize charity lotteries in the context of separable utility functions. Goerree et al. (2005) found a first-best scheme to raise donations for a public good, but only under linear spillovers and k-th price auctions, which are rather unrealistic in organizations. Bos (2011) and Faravelli (2011) extended their results for heterogeneous endowments and a fixed multiple-prize contest. Revenue comparisons of all-pay and first-price auctions for a public good are subject to extensive experimental testing, inter alia, in Lange et al. (2007), Carpenter et al. (2008), Schram and Onderstal (2009) and Corazzini et al. (2010). In contest literature, the effect of the positive spillovers of irreversible payments is well examined for linear parameterizations (Baye et al., 2009). In our setup, the marginal rate of substitution decreases in its arguments, hence a positive spillover of the
M. Gregor / Economics Letters 114 (2012) 280–283
collective good decreases in the amount of contributions, which is similar to Chung (1996), where non-linearity had a particular functional form. Contests with reimbursements have been investigated in relatively few papers. Matros and Armanios (2009) study winner- and loser-reimbursement contests, but do not consider rebates related to the opponents’ contributions. Chowdhury and Sheremeta (2010) offer a general setup with spillovers where some parameterizations of spillovers can be interpreted as reimbursements. Spencer et al. (2009) provide experimental support for the superior performance of proportional rebates in threshold public good games. The simple scheme analyzed here effectively combines three spillovers: (i) A positive spillover through the joint collective good consumption, (ii) a negative spillover on private good consumption caused by a lower share of the prize, and, if the prize is an increasing function of total contributions, also (iii) a positive spillover on private good consumption associated with an increase in the prize. A contest for power balances results in uncompensated positive spillovers (larger collective good, larger prize) with an uncompensated negative spillover of the competition (lower chance to obtain the prize). The construction of an efficient scheme can thus be viewed as a problem of the optimal design of winners’ and losers’ reimbursements in the presence of spillovers (cf. Matros and Armanios, 2009). 2. Linear schemes Let i = 1, . . . , n be identical agents endowed with income m > 0 and with a well-behaving utility u(ci , G), where ci ≥ 0 denotes i-th private good consumption and G ≥ 0 the collective good consumption. This guarantees a unique symmetric social optimum, characterized by G = G∗ . Each agent voluntarily contributes an amount xi ≥ 0 to a single organization, and the organization uses a share γ ∈ [0, 1] of the total contributions X ≡ x for the i i collective good, G = γ X . The remaining 1 −γ share of the collected revenues is used as rent allocated to the contributors based on the relative shares/power (σ1 , . . . , σn ), where we assume the logitform contest–success function with the elasticity of the payoff to the marginal contribution r > 0,
σi =
xri xri
+
xrj
j̸=i
if X > 0, and σi = 1n otherwise. A pair (γ , r ) fully characterizes a linear scheme. To avoid the issue of risk aversion, the rent attained is considered divisible, hence we effectively have n-tuple multiple prizes, (σ1 X , . . . , σn X ). Assuming for simplicity of unit prices, the consumption levels are:
(ci ; G) = (m − xi + σi (1 − γ )X ; γ X ) .
dci
= 1. dG With reimbursement (γ < 1), the relative price of the public good is:
Π (xi ; x−i ) =
∂σ
1 − (1 − γ ) σi + ∂ x i i
γ
k
Below, we investigate equilibria in pure strategies, hence we must avoid the full dissipation associated with mixed strategies.1 Without a collective good (or positive spillover) but with a variable prize dc ∂σ X , this amounts to dxi = −1 + σi + ∂ x i k xk < 0. Under symmei i try and for logit-form contest success function, r < 1. In our case dc with the collective good, the condition dxi < 0 is equivalent to a i positive relative price of the collective good, Π (x; (n − 1)x) > 0. γn We may introduce an upper boundary r˜ (γ ) ≡ 1 + (n−1)(1−γ ) , which is increasing, r˜ ′ (γ ) > 0, and unbounded, r˜ (γ ) ∈ [1, +∞). Thus, even largely competitive contests r ≫ 1, r < r˜ (γ ), are sustained in pure strategies as long as γ → 1. Alternatively, define a (n−1)(r −1) lower boundary γ˜ (r ) ≡ (n−1)(r −1)+n = r −11 , which is increasing, r + n−1
γ˜ (r )′ > 0, and bounded, γ˜ (r ) ∈ [1 − n, 1). To sum up, the parametric support for pure-strategy equilibria is written S = {(γ , r ) : max{0; sup γ˜ (r )} ≤ γ ≤ 1; r > 0}. Proposition 1 delivers a necessary condition for a scheme to be collectively efficient. Proposition 1 (Efficient Linear Contest). A symmetric social optimum is the Nash equilibrium in pure strategies under a linear scheme (γ ∗ , r ) only if γ ∗ = r −r 1 ∈ (0, 1) and r > 1. For symmetric contribution profiles, comparative statics over the contest parameters (γ , r ) ∈ S reveal that the public good is relatively cheaper if the contest is very competitive, ∂∂Πr > 0. In the case of the marginal contribution rate γ , we observe that the effect of reimbursement is conditional upon r Q 1:
(r − 1)(n − 1) < 0, ∂Π = 0, = ∂γ nγ 2 > 0,
r <1 r =1 r > 1.
The largest relative price occurs for r < 1 and γ → 0: Π → +∞. This is a case of a largely unattractive scheme, with almost nothing left for the collective good. All contests with Π > 1 are, in fact, dominated by a direct purchase of the collective good (if feasible). For r < 1, the price falls with a large γ to a no-contest Nash level Π = 1. At r = 1, the price remains Π = 1 irrespective of γ , which is equivalent to the analysis of a pari-mutuel raffle in Morgan (2000). Only at r > 1 do we achieve Π < 1, if and only if γ < 1. In addition, consider the sign of spillovers. A contribution dxi > 0 implies: (i) a decrease in its own private consumption, dci < 0, (ii) an increase in public consumption, dG > 0, and also (iii) an increase in the opponent’s consumption, dc−i . The latter is straightforward since ci + c−i = (1 − γ )X , and: dc−i dxi
=
d(1 − γ )(xi + x−i ) dxi
−
dci dxi
dci
=1−γ − > 0. dxi + −
The scheme affects the relative price of the private and collective good. In a classic public good game without any contest (equivalent to γ = 1, ci + xi = m), the marginal rate of the transformation of the private good and the public good consumption is one,
Π (xi ; x−i ) ≡ −
281
xk
.
Paradoxically, although the introduction of the contest scheme generates an extra positive externality upon the opponent (if compared to the case without any contest), it improves efficiency as it reduces the wedge between the optimal and equilibrium allocation. The explanation is that the contest scheme becomes a reimbursement scheme where the size of the positive collective good externality substantially diminishes, since only a γ -share of the marginal contribution translates into the collective good. Furthermore, notice that the supply side of the market with the collective good may in fact consist of competing providers who differ in (γ , r ). Each generic pair represents a particular relative price.2 A contributor selecting from a-organization and borganization favors a-organization if Π a < Π b , or:
In any symmetric profile,
Π (x; (n − 1)x) =
=
n − (1 − γ )[1 + r (n − 1)] 1
1 For a general condition characterizing full dissipation see Alcalde and Dahm (2010). 2 The relative price also falls with the number of contributors (increasing returns
n
to scale), but we may neglect this effect and assume a large n, where a change
nγ n−1 nγ
(1 − r (1 − γ )) + .
282
M. Gregor / Economics Letters 114 (2012) 280–283
1 − r (1 − γ ) a
a
γ ∂Π ∂γ
a
<
1 − r (1 − γ ) b
b
γ
b
n−1
.
n
Recall that for r > 1, the relative price satisfies ∂∂Πr < 0 and
> 0, and that γ > γ˜ (r ). Thus, for any r > 1, all schemes (γˆ , r ) ∗
such that γ ∗ > γˆ > γ˜ are both feasible, (γ , r ) ∈ S, and preferred ˆ > 0. To sum up, organizato the efficient scheme, Π ∗ = 1n > Π tions with efficient contests (γ = γ ∗ ) outcompete organizations without contests (γ = 1), but are crowded out by organizations with excessive (overly aggressive) contests for power (γ < γ ∗ ). In a similar fashion, for r ≤ 1, we would observe that organizations with large marginal contribution rates provide a better price, hence the market winner for insufficiently competitive contests would be an organization that prohibits contests for rent, γ = 1. 3. Non-linear schemes For r = 1, Morgan (2000) established that a fixed-prize raffle welfare-dominates a pari-mutuel raffle. In the previous section, we nevertheless found that any linear scheme (γ ∗ , r ) achieves the collective optimum if r > 1. Thus, we may ask if non-linear schemes such as a fixed-prized scheme can achieve the optimum as well. A non-linear contest scheme is characterized by a function Γ (X ) : R+ → R and the marginal return r > 0. Consumption levels are written:
(ci ; G) = (m − xi + σi [X − Γ (X )] ; Γ (X )) .
X
r + Γ ′ (X ) > 0.
Like in the linear case, the condition is more likely satisfied if the marginal return r is small or the marginal contribution rate Γ ′ (X ) is high. 4. Conclusion This paper analyzes whether a contest for power associated with private rents may improve collective action in an organization. The organization allocates a portion of the members’ contributions into the collective good, and a portion into a private rent that is subject to a contest between the members. Then, a necessary condition for an efficient contest is the convexity in payoffs (elasticity to marginal contribution r exceeding one), and a prize at the 1r share of the total revenues implemented at the social optimum. To both provide the efficient amount of the collective good and minimize the overall level of private rents, the organization must apply a highly competitive contest for power. We also found that although organizations with efficient contest schemes outcompete organizations without contests, they are less attractive to contributors than organizations with excessively competitive contests.
Comments by Peter Katuščák and an anonymous referee were appreciated. Financial support by the Czech National Science Council (GACR 402/08/0501) and the Czech Ministry of Education (MSM 0021620841) is gratefully acknowledged.
∂σ
1 − σi 1 − Γ ′ (X ) − ∂ x i [X − Γ (X )] i Π (xi ; x−i ) = . Γ ′ (X )
X − Γ (X )
Acknowledgments
The relative price is:
−
In a symmetric case, after rearranging,
Π (x; (n − 1)x) =
n−1 nΓ ′ (X )
1−r +r
Γ (X )
X
1
Appendix
+ . n
Proposition 2 (Efficient Non-Linear Contest). A symmetric social optimum is the Nash equilibrium in pure strategies under a non-linear scheme Γ (X ) only if Γ ( r −r 1 · G∗ ) = G∗ and r > 1. Proposition 2 brings in four interesting observations: (i) The marginal contribution rate at the optimal allocation is irrelevant as long as it is positive, Γ ′ (X ∗ ) > 0. (ii) Linear schemes as a subset of non-linear schemes satisfying Γ (X ) = γ X , γ ∈ [0, 1] achieve the social optimum for γ = γ ∗ , hence the average contribution rate Γ (X ) Γ (X ) is bound to γ ∗ everywhere. In contrast, the condition X = X ∗ γ binds in a non-linear scheme only in the social optimum. (iii) The contested prize X − Γ (X ) must be in optimum for exactly a 1r ∈ (0, 1) share of the total revenues X . In other words, a high marginal return allows the use of an efficient contest with low total private rents. (iv) A fixed-prize contest may achieve the optimum, ∗ but with pre-determined r we must impose a prize Xr ; such a non∗
∗
linear scheme is written Γ (X ) = X − Xr = X − rG−1 (i.e., it becomes a deficit scheme for low levels of X ). If the organization can manipulate r but needs to provide a fixed prize at maximal value V (e.g., due to regulatory constraints), it should impose V = ∗ G∗ , hence r = 1 + GV . r −1 For linear schemes, we discussed the restriction γ ≥ γ˜ (r ) that helped us to avoid full dissipation. A similar condition exists in the generalized non-linear case. We rewrite Π (xi ; x−i ) > 0 in a symmetric version as follows:
in the organization membership of a single consumer does not affect the relative prices. Otherwise, the shape of a stable partitioning of the system depends on extra assumptions.
Proof of Proposition 1. A social optimum is self-sustaining only if each contributor internalizes positive spillovers of their own contribution, hence pays only 1n of the marginal cost in the social optimum. There, the relative price should satisfy Π (x; (n − 1)x) = 1 , which amounts to 1 − r (1 − γ ) = 0, or γ ∗ = r −r 1 . Clearly, the n social optimum is not restricted by the lower bound, γ ∗ = r −r 1 > r −1 1 r + n− 1
= γ˜ . Yet, it is restricted by non-negativity to γ = 0 if r ≤ 1.
Thus, for r ≤ 1, the equilibrium amount is G = 2γ ∗ x = 0 < G∗ , which is inefficient. We must require r > 1. Proof of Proposition 2. We impose Π (x; (n − 1)x) = simplifies to 1 − r + r Γ (X )
Γ (X ) X
= 0, or
Γ (X ) X
=
r −1 . r
1 , n
which
(Notice the
equivalence to X = γ ∗ .) This condition must hold if evaluated in the social optimum, where G∗ = Γ (X ∗ ), hence X ∗ = r −r 1 G∗ .
Imposing into the condition, we have Γ −1 (G∗ ) = X ∗ = r −r 1 G∗ . The ∗ feasibility of the scheme dictates that XG∗ = r −r 1 ≤ 1, hence r ≥ 1. Moreover, r ̸= 1 since this would imply which is untrue.
Γ (X ) X
= 0 = Γ (X ) = G∗ ,
References Alcalde, J., Dahm, M., 2010. Rent seeking and rent dissipation: a neutrality result. Journal of Public Economics 94 (1), 1–7. Baye, M., Kovenock, D., de-Vries, C.G., 2009 Contests with rank-order spillovers. Economic Theory doi:10.1007/s00199-009-0489-2. Bos, O., 2010. Charity auctions for the happy few. Working Paper Series in Economics 45. University of Cologne, Department of Economics. Bos, O., 2011. How lotteries outperform auctions. Economics Letters 110 (3), 262–264. Carpenter, J., Holmes, J., Matthews, P.H., 2008. Charity auctions: a field experiment. The Economic Journal 118, 92–113.
M. Gregor / Economics Letters 114 (2012) 280–283 Chowdhury, S.M., Sheremeta, R.M., 2010. A generalized Tullock contest. Public Choice 147, 413–420. Chung, T.Y., 1996. Rent-seeking contest when the prize increases with aggregate efforts. Public Choice 87, 55–66. Corazzini, L., Faravelli, M., Stanca, L., 2010. A prize to give for: an experiment on public good funding mechanisms. The Economic Journal 120, 944–967. Elfenbein, D.W., McManus, B., 2010. Last-minute bidding in eBay charity auctions. Economics Letters 107 (1), 42–45. Faravelli, M., 2011. The important thing is not (always) winning but taking part: funding public goods with contests. Journal of Public Economic Theory 13 (1), 1–22. Goerree, J.K., Maasland, E., Onderstal, S., 2005. How (Not) to raise money. Journal of Political Economy 113 (4), 897–918.
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Lange, A., List, J.A., Price, M.K., 2007. Using lotteries to finance public goods: theory and experimental evidence. International Economic Review 48 (3), 901–927. Matros, A., Armanios, D., 2009. Tullock’s contest with reimbursements. Public Choice 141, 49–63. Morgan, J., 2000. Financing public goods by means of lotteries. The Review of Economic Studies 67 (4), 761–784. Olson, M., 1965. The Logic of Collective Action: Public Goods and the Theory of Groups. Harvard University Press. Schram, A., Onderstal, S., 2009. Bidding to give: an experimental comparison of auctions for charity. International Economic Review 50 (2), 431–457. Spencer, M.A., Swallow, S.K., Shogren, J.F., List, J.A., 2009. Rebate rules in threshold public good provision. Journal of Public Economics 93, 798–806.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Contest for power in organizations Martin Gregor ∗ Institute of Economic Studies, Faculty of Social Sciences, Charles University in Prague, Opletalova 26, Prague, CZ 110 00, Czech Republic
article
info
Article history: Received 22 July 2010 Received in revised form 12 October 2011 Accepted 20 October 2011 Available online 28 October 2011
abstract This paper explains the provision of private rent to powerful members in an organization as an outcome of a contest for power that raises the total contributions to the organization. A necessary condition for a socially efficient contest scheme with reimbursements is characterized. © 2011 Elsevier B.V. All rights reserved.
JEL classification: D23 D70 H41 Keywords: Contest Reimbursement Rent-seeking Collective good
1. Introduction The market provision of collective goods often goes through intermediary organizations that bundle members’ contributions, coordinate their efforts, and save on economies in scale. Executive discretion within the organization, nevertheless, creates an option to leak a part of the organization’s budget into private use. Aware of this possibility, the members compete for control over the organization, seeking a private rent that effectively rebates a portion of their contributions. For the organization, rebating a part of the contributions is equivalent to not receiving the rebates at all. For an individual, the presence of a contested rebate implies a dual role for a marginal individual contribution: It increases the amount of the collective good, and it also generates extra influence within the organization that improves the chance of obtaining the contested private rent. Thereby, the contest for this private rent artificially drops the effective price of the collective good to an optimal level and even below. Competition for power in an organization can thus be seen as a decentralized mechanism to improve the collective action of the organization. In the founding book on incentives for collective action, Olson (1965) highlighted that successful interest groups use ‘selective incentives’ in the form of private goods that motivate membership; our paper shows that mixing the provision of a
∗
Tel.: +420 222 112 306; fax: +420 222 112 304. E-mail address: [email protected].
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.10.027
private and collective good may improve the provision of the collective good as long as the private good is offered as the prize of a contest, with a sufficient level of discrimination. This paper finds that a contest is efficient only if it applies a sufficiently high, but not excessively large, marginal return. It also characterizes the specific share of the budget that an organization must convert into private rents in order to be able to provide an optimal amount of the collective good. These results are derived under an n-person symmetric pure strategy equilibria with complete information, for a general utility over the private and collective good. The idea of bundling a collective good with a private good is closely related to literature on the use of charity lotteries and charity auctions (Bos, 2010; Elfenbein and McManus, 2010). In a seminal article, Morgan (2000) examined fixed-prize and pari-mutuel-prize charity lotteries in the context of separable utility functions. Goerree et al. (2005) found a first-best scheme to raise donations for a public good, but only under linear spillovers and k-th price auctions, which are rather unrealistic in organizations. Bos (2011) and Faravelli (2011) extended their results for heterogeneous endowments and a fixed multiple-prize contest. Revenue comparisons of all-pay and first-price auctions for a public good are subject to extensive experimental testing, inter alia, in Lange et al. (2007), Carpenter et al. (2008), Schram and Onderstal (2009) and Corazzini et al. (2010). In contest literature, the effect of the positive spillovers of irreversible payments is well examined for linear parameterizations (Baye et al., 2009). In our setup, the marginal rate of substitution decreases in its arguments, hence a positive spillover of the
M. Gregor / Economics Letters 114 (2012) 280–283
collective good decreases in the amount of contributions, which is similar to Chung (1996), where non-linearity had a particular functional form. Contests with reimbursements have been investigated in relatively few papers. Matros and Armanios (2009) study winner- and loser-reimbursement contests, but do not consider rebates related to the opponents’ contributions. Chowdhury and Sheremeta (2010) offer a general setup with spillovers where some parameterizations of spillovers can be interpreted as reimbursements. Spencer et al. (2009) provide experimental support for the superior performance of proportional rebates in threshold public good games. The simple scheme analyzed here effectively combines three spillovers: (i) A positive spillover through the joint collective good consumption, (ii) a negative spillover on private good consumption caused by a lower share of the prize, and, if the prize is an increasing function of total contributions, also (iii) a positive spillover on private good consumption associated with an increase in the prize. A contest for power balances results in uncompensated positive spillovers (larger collective good, larger prize) with an uncompensated negative spillover of the competition (lower chance to obtain the prize). The construction of an efficient scheme can thus be viewed as a problem of the optimal design of winners’ and losers’ reimbursements in the presence of spillovers (cf. Matros and Armanios, 2009). 2. Linear schemes Let i = 1, . . . , n be identical agents endowed with income m > 0 and with a well-behaving utility u(ci , G), where ci ≥ 0 denotes i-th private good consumption and G ≥ 0 the collective good consumption. This guarantees a unique symmetric social optimum, characterized by G = G∗ . Each agent voluntarily contributes an amount xi ≥ 0 to a single organization, and the organization uses a share γ ∈ [0, 1] of the total contributions X ≡ x for the i i collective good, G = γ X . The remaining 1 −γ share of the collected revenues is used as rent allocated to the contributors based on the relative shares/power (σ1 , . . . , σn ), where we assume the logitform contest–success function with the elasticity of the payoff to the marginal contribution r > 0,
σi =
xri xri
+
xrj
j̸=i
if X > 0, and σi = 1n otherwise. A pair (γ , r ) fully characterizes a linear scheme. To avoid the issue of risk aversion, the rent attained is considered divisible, hence we effectively have n-tuple multiple prizes, (σ1 X , . . . , σn X ). Assuming for simplicity of unit prices, the consumption levels are:
(ci ; G) = (m − xi + σi (1 − γ )X ; γ X ) .
dci
= 1. dG With reimbursement (γ < 1), the relative price of the public good is:
Π (xi ; x−i ) =
∂σ
1 − (1 − γ ) σi + ∂ x i i
γ
k
Below, we investigate equilibria in pure strategies, hence we must avoid the full dissipation associated with mixed strategies.1 Without a collective good (or positive spillover) but with a variable prize dc ∂σ X , this amounts to dxi = −1 + σi + ∂ x i k xk < 0. Under symmei i try and for logit-form contest success function, r < 1. In our case dc with the collective good, the condition dxi < 0 is equivalent to a i positive relative price of the collective good, Π (x; (n − 1)x) > 0. γn We may introduce an upper boundary r˜ (γ ) ≡ 1 + (n−1)(1−γ ) , which is increasing, r˜ ′ (γ ) > 0, and unbounded, r˜ (γ ) ∈ [1, +∞). Thus, even largely competitive contests r ≫ 1, r < r˜ (γ ), are sustained in pure strategies as long as γ → 1. Alternatively, define a (n−1)(r −1) lower boundary γ˜ (r ) ≡ (n−1)(r −1)+n = r −11 , which is increasing, r + n−1
γ˜ (r )′ > 0, and bounded, γ˜ (r ) ∈ [1 − n, 1). To sum up, the parametric support for pure-strategy equilibria is written S = {(γ , r ) : max{0; sup γ˜ (r )} ≤ γ ≤ 1; r > 0}. Proposition 1 delivers a necessary condition for a scheme to be collectively efficient. Proposition 1 (Efficient Linear Contest). A symmetric social optimum is the Nash equilibrium in pure strategies under a linear scheme (γ ∗ , r ) only if γ ∗ = r −r 1 ∈ (0, 1) and r > 1. For symmetric contribution profiles, comparative statics over the contest parameters (γ , r ) ∈ S reveal that the public good is relatively cheaper if the contest is very competitive, ∂∂Πr > 0. In the case of the marginal contribution rate γ , we observe that the effect of reimbursement is conditional upon r Q 1:
(r − 1)(n − 1) < 0, ∂Π = 0, = ∂γ nγ 2 > 0,
r <1 r =1 r > 1.
The largest relative price occurs for r < 1 and γ → 0: Π → +∞. This is a case of a largely unattractive scheme, with almost nothing left for the collective good. All contests with Π > 1 are, in fact, dominated by a direct purchase of the collective good (if feasible). For r < 1, the price falls with a large γ to a no-contest Nash level Π = 1. At r = 1, the price remains Π = 1 irrespective of γ , which is equivalent to the analysis of a pari-mutuel raffle in Morgan (2000). Only at r > 1 do we achieve Π < 1, if and only if γ < 1. In addition, consider the sign of spillovers. A contribution dxi > 0 implies: (i) a decrease in its own private consumption, dci < 0, (ii) an increase in public consumption, dG > 0, and also (iii) an increase in the opponent’s consumption, dc−i . The latter is straightforward since ci + c−i = (1 − γ )X , and: dc−i dxi
=
d(1 − γ )(xi + x−i ) dxi
−
dci dxi
dci
=1−γ − > 0. dxi + −
The scheme affects the relative price of the private and collective good. In a classic public good game without any contest (equivalent to γ = 1, ci + xi = m), the marginal rate of the transformation of the private good and the public good consumption is one,
Π (xi ; x−i ) ≡ −
281
xk
.
Paradoxically, although the introduction of the contest scheme generates an extra positive externality upon the opponent (if compared to the case without any contest), it improves efficiency as it reduces the wedge between the optimal and equilibrium allocation. The explanation is that the contest scheme becomes a reimbursement scheme where the size of the positive collective good externality substantially diminishes, since only a γ -share of the marginal contribution translates into the collective good. Furthermore, notice that the supply side of the market with the collective good may in fact consist of competing providers who differ in (γ , r ). Each generic pair represents a particular relative price.2 A contributor selecting from a-organization and borganization favors a-organization if Π a < Π b , or:
In any symmetric profile,
Π (x; (n − 1)x) =
=
n − (1 − γ )[1 + r (n − 1)] 1
1 For a general condition characterizing full dissipation see Alcalde and Dahm (2010). 2 The relative price also falls with the number of contributors (increasing returns
n
to scale), but we may neglect this effect and assume a large n, where a change
nγ n−1 nγ
(1 − r (1 − γ )) + .
282
M. Gregor / Economics Letters 114 (2012) 280–283
1 − r (1 − γ ) a
a
γ ∂Π ∂γ
a
<
1 − r (1 − γ ) b
b
γ
b
n−1
.
n
Recall that for r > 1, the relative price satisfies ∂∂Πr < 0 and
> 0, and that γ > γ˜ (r ). Thus, for any r > 1, all schemes (γˆ , r ) ∗
such that γ ∗ > γˆ > γ˜ are both feasible, (γ , r ) ∈ S, and preferred ˆ > 0. To sum up, organizato the efficient scheme, Π ∗ = 1n > Π tions with efficient contests (γ = γ ∗ ) outcompete organizations without contests (γ = 1), but are crowded out by organizations with excessive (overly aggressive) contests for power (γ < γ ∗ ). In a similar fashion, for r ≤ 1, we would observe that organizations with large marginal contribution rates provide a better price, hence the market winner for insufficiently competitive contests would be an organization that prohibits contests for rent, γ = 1. 3. Non-linear schemes For r = 1, Morgan (2000) established that a fixed-prize raffle welfare-dominates a pari-mutuel raffle. In the previous section, we nevertheless found that any linear scheme (γ ∗ , r ) achieves the collective optimum if r > 1. Thus, we may ask if non-linear schemes such as a fixed-prized scheme can achieve the optimum as well. A non-linear contest scheme is characterized by a function Γ (X ) : R+ → R and the marginal return r > 0. Consumption levels are written:
(ci ; G) = (m − xi + σi [X − Γ (X )] ; Γ (X )) .
X
r + Γ ′ (X ) > 0.
Like in the linear case, the condition is more likely satisfied if the marginal return r is small or the marginal contribution rate Γ ′ (X ) is high. 4. Conclusion This paper analyzes whether a contest for power associated with private rents may improve collective action in an organization. The organization allocates a portion of the members’ contributions into the collective good, and a portion into a private rent that is subject to a contest between the members. Then, a necessary condition for an efficient contest is the convexity in payoffs (elasticity to marginal contribution r exceeding one), and a prize at the 1r share of the total revenues implemented at the social optimum. To both provide the efficient amount of the collective good and minimize the overall level of private rents, the organization must apply a highly competitive contest for power. We also found that although organizations with efficient contest schemes outcompete organizations without contests, they are less attractive to contributors than organizations with excessively competitive contests.
Comments by Peter Katuščák and an anonymous referee were appreciated. Financial support by the Czech National Science Council (GACR 402/08/0501) and the Czech Ministry of Education (MSM 0021620841) is gratefully acknowledged.
∂σ
1 − σi 1 − Γ ′ (X ) − ∂ x i [X − Γ (X )] i Π (xi ; x−i ) = . Γ ′ (X )
X − Γ (X )
Acknowledgments
The relative price is:
−
In a symmetric case, after rearranging,
Π (x; (n − 1)x) =
n−1 nΓ ′ (X )
1−r +r
Γ (X )
X
1
Appendix
+ . n
Proposition 2 (Efficient Non-Linear Contest). A symmetric social optimum is the Nash equilibrium in pure strategies under a non-linear scheme Γ (X ) only if Γ ( r −r 1 · G∗ ) = G∗ and r > 1. Proposition 2 brings in four interesting observations: (i) The marginal contribution rate at the optimal allocation is irrelevant as long as it is positive, Γ ′ (X ∗ ) > 0. (ii) Linear schemes as a subset of non-linear schemes satisfying Γ (X ) = γ X , γ ∈ [0, 1] achieve the social optimum for γ = γ ∗ , hence the average contribution rate Γ (X ) Γ (X ) is bound to γ ∗ everywhere. In contrast, the condition X = X ∗ γ binds in a non-linear scheme only in the social optimum. (iii) The contested prize X − Γ (X ) must be in optimum for exactly a 1r ∈ (0, 1) share of the total revenues X . In other words, a high marginal return allows the use of an efficient contest with low total private rents. (iv) A fixed-prize contest may achieve the optimum, ∗ but with pre-determined r we must impose a prize Xr ; such a non∗
∗
linear scheme is written Γ (X ) = X − Xr = X − rG−1 (i.e., it becomes a deficit scheme for low levels of X ). If the organization can manipulate r but needs to provide a fixed prize at maximal value V (e.g., due to regulatory constraints), it should impose V = ∗ G∗ , hence r = 1 + GV . r −1 For linear schemes, we discussed the restriction γ ≥ γ˜ (r ) that helped us to avoid full dissipation. A similar condition exists in the generalized non-linear case. We rewrite Π (xi ; x−i ) > 0 in a symmetric version as follows:
in the organization membership of a single consumer does not affect the relative prices. Otherwise, the shape of a stable partitioning of the system depends on extra assumptions.
Proof of Proposition 1. A social optimum is self-sustaining only if each contributor internalizes positive spillovers of their own contribution, hence pays only 1n of the marginal cost in the social optimum. There, the relative price should satisfy Π (x; (n − 1)x) = 1 , which amounts to 1 − r (1 − γ ) = 0, or γ ∗ = r −r 1 . Clearly, the n social optimum is not restricted by the lower bound, γ ∗ = r −r 1 > r −1 1 r + n− 1
= γ˜ . Yet, it is restricted by non-negativity to γ = 0 if r ≤ 1.
Thus, for r ≤ 1, the equilibrium amount is G = 2γ ∗ x = 0 < G∗ , which is inefficient. We must require r > 1. Proof of Proposition 2. We impose Π (x; (n − 1)x) = simplifies to 1 − r + r Γ (X )
Γ (X ) X
= 0, or
Γ (X ) X
=
r −1 . r
1 , n
which
(Notice the
equivalence to X = γ ∗ .) This condition must hold if evaluated in the social optimum, where G∗ = Γ (X ∗ ), hence X ∗ = r −r 1 G∗ .
Imposing into the condition, we have Γ −1 (G∗ ) = X ∗ = r −r 1 G∗ . The ∗ feasibility of the scheme dictates that XG∗ = r −r 1 ≤ 1, hence r ≥ 1. Moreover, r ̸= 1 since this would imply which is untrue.
Γ (X ) X
= 0 = Γ (X ) = G∗ ,
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