Binary Participation in Public Goods Allocations 1 Paul J. Healy 2
Abstract An environment is studied in which mechanisms suggest public goods allocations and individuals then choose whether or not to submit their requested transfer to the central planner. The set of allocations such that unanimous participation is an equilibrium is shown to be sub-optimal in a wide variety of environments and shrinks to the endowment as the economy is replicated. Optimal public goods mechanisms therefore suffer from non-participation in their outcomes when agents cannot be coerced to contribute. JEL Classification Numbers: C62, C72, H41. Key words: Public goods, mechanism design, voluntary participation, individual rationality.
1
Introduction
As noted by Palfrey and Rosenthal (1984), the well known free-rider problem raises two key issues in the study of public goods economies. The first is whether or not mechanisms or processes exist in which rational agents are willing to truthfully reveal their demand for the public project. This question has been studied extensively in a line of research extending from the works of Gibbard (1973) and Satterthwaite (1975), who show that demand revelation is generally impossible in the face of strategic manipulation. Clarke (1971), Groves (1973), Green and Laffont (1977), and Walker (1980) demonstrate that optimal public good levels may in fact be obtained when preferences are quasi-linear, although full efficiency remains an impossibility. By relaxing the requirement of dominant strategy revelation, fully optimal solutions have been derived by Groves and Ledyard (1977), Hurwicz (1979), and Walker (1981). 1
This research has benefited from discussions with Chris Chambers, John Ledyard, and Matt Jackson. 2 Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125. Email:
[email protected]. Telephone: (626) 395-4052 Fax: (626) 405-9841
Draft copy. Do not cite without permission.
22 June 2004
The second issue is that of voluntary participation. Palfrey and Rosenthal (1984) show the strength of the free-riding incentive in a world of purely voluntary contributions and symmetric preferences. Here, no mechanism is used to suggest outcomes; instead, all unilateral contributions enter into public goods production. Typical equilibria feature only a single contributor, resulting in inefficiently small production levels. Even in the presence of an optimal mechanism, it is important to understand whether agents would prefer to participate in the mechanism and its proposed outcome, or to withdraw from the process in favor of some outside option. In different environments, different specifications of this outside option are appropriate. In the mechanism design framework, Groves and Ledyard (1977), following the work on private goods of Hurwicz (1972), have shown the inconsistency between incentive compatibility, Pareto optimality, and individual rationality in public goods environments. Under the individual rationality constraint, the outcome of the mechanism must be preferred to the initial endowment by all agents. While the justification for this requirement is clear in the private goods context, wherein agents who opt out of the allocation naturally revert to consuming their endowment, the applicability of this requirement to a public goods setting is less transparent. A normative justification for this welfare lower bound can be given: the central planner or mechanism designer wishes to unanimously improve the welfare of all agents, and thus would want to satisfy this requirement. However, in most public goods environments, the non-excludable nature of the public good implies that a withdrawing agent’s outside option should somehow include his autarkic production and/or the production of the remaining agents. The applicability of the individual rationality requirement in public goods environments has been questioned by Saijo (1991), who proposes an alternative notion of an outside option based on autarkic production of the public good. Specifically, an agent’s outside option is the best consumption bundle obtainable if that agent were the sole contributor to the public good technology. A mechanism satisfies the autarkic individual rationality constraint if every agent prefers the proposed outcome over that which they could have achieved in isolation. Again, this constraint has a normative appeal in defining a welfare lower bound, but is not generally appealing as a model of participation. More recently, Dixit and Olson (2000) and Saijo and Yamato (1999) have discussed the concept of voluntary participation in the mechanism itself. The model is one in which some mechanism or Coasian bargaining process is known to generate optimal outcomes for those who participate, but can neither tax nor accommodate those who do not. The participation decision therefore represents a trade-off between influencing the public good outcome decision and paying a participants’ share of the cost. A non-participating agent thus enjoys the public good level chosen by the participants and consumes only his initial endowment of the private good. In this setting, the authors show that as the size of the economy grows, agents choose to participate less frequently. Thus, the free rider problem and its link to the size of the economy (as noted by Olson (1965)) are confirmed by this result. The current paper proposes an alternative notion of participation. In the standard implementation methodology, it is implicitly assumed that once an allocation is identified by a mechanism or bargaining process, that outcome necessarily obtains. This assumption is only 2
valid if either the participants, upon learning the realized outcome, are willing to contribute their required transfers, or if there exists a government or central planner with the power to coerce agents to contribute as requested. In the latter environment, the importance of participation is minimal; if the government can force the final allocation to be realized, then it is likely to be able to force participation in the selection mechanism through punishments of non-participants. Therefore, the former environment is considered. The public goods decision proceeds as follows. A group of agents first participates in a mechanism which results in the identification of a public good allocation. This group of agents may represent the entire economy, or it may represent the set of agents who chose to participate in the mechanism. Upon learning the allocation identified by the mechanism, the agents then decide whether or not to contribute their requested transfer of the private good. If unanimous participation is a Nash equilibrium, then the chosen outcome will be realized voluntarily. If one or more agents prefers to withhold her transfer, then the mechanism outcome cannot be expected to obtain without coercion. In this environment, implementation becomes a significantly more difficult affair. If a central planner cannot coerce agents to pay their required transfers, then the chosen mechanism must not only implement the desired social choice correspondence, but must also guarantee that all agents will indeed prefer to submit their transfer. As will be demonstrated, Pareto optimality can no longer be implementable without coercion in a wide range of simple environments. Thus, the optimal mechanisms of Groves and Ledyard (1977), Hurwicz (1979), or Walker (1981) are necessarily vulnerable to non-participation by agents who may withhold payments at no pecuniary cost to themselves. Furthermore, the set of allocations that satisfy this new participation requirement is shown to shrink to the initial endowment as the economy becomes arbitrarily large, reconfirming the presence of a free rider problem whose effect becomes most severe in large numbers. This participation constraint is in fact a constraint applied to allocations, not necessarily mechanisms. The negative results of this paper are not a consequence of information rents, incentive compatibility, or optimality. Instead, they represent a fundamental impossibility inherent in certain proposed allocations. The analysis focuses on properties of a given allocation and identifies a set of allocations for which participation is an equilibrium. The suggestion is that while mechanisms ought to select outcomes in this set of participatory alternatives in order for their suggested outcomes to eventually obtain, this set is often inefficient and shrinks to the endowment as the economy grows. Thus, the outcomes that can be implemented without coercion are often undesirable from the point of view of a central planner. The notation and key definition are provided in section 2. General properties of the set of allocations satisfying this constraint are explored in section 3, followed by an analysis of the constraint in classical, quasi-concave economies with convex technology in section 4. The implementability of the set of participatory allocations is considered in section 5, and the convergence to the endowment in large economies is proven in section 6. Concluding comments and open questions are discussed in section 7. 3
2
2.1
Notation & Definitions
The General Environment
Consider an economy with I agents who consume a private commodity and a pure public good. Let I denote the set of individuals with elements i = 1, 2, . . . , I. Each consumer i ∈ I has complete and transitive preferences ºi over the set of feasible consumption bundles Z ⊂RI+1 + . A consumption bundle (or ‘allocation’) is an element z ∈ Z given by z = (x; y), where x = (x1 , . . . , xI ) represents the allocation of the private good to each individual and y represents the total level of the public good. Note that xi may vary by i since every individual may consume a different quantity of each private good, but the nature of the public good requires y to be single-valued. Initial endowments are given by ω = (ω1 , . . . , ωI ; 0) ∈ Z with ωi > 0 for each i, so that no public good exists initially. The good x is assumed to represent a private commodity (such as money) which may be put to use in the production of the public good. Agents’ preferences only vary in their consumption of the private good and in the public good. Therefore, xi = x0i implies that i is indifferent between (x; y) and (x0 ; y). This allows ºi to be projected onto R2+ so that statements like (xi , y) ºi (x0i , y) are well defined. The production technology is represented by a set of feasible production plans Y ⊆R− × R+ with typical element (−T ; y), where T ∈ R+ is a non-negative number representing the total input of x used to produce y units of the public good 1 . Assume that Y is a comprehensive and closed set such that Y∩R2+ = {0}. Let F be a function representing the boundary of Y, with (−T, y) ∈ Y if and only if F (T ) ≥ y. The production function F (T ) is thus defined as the maximal level of y that may be produced given an input of T units of x. If F is continuous at T , then the marginal cost of production is given by the inverse of the slope of F at T ; if F is differentiable, the marginal cost is 1/F 0 . In the tradition of the mechanism design literature, a committee, government, or mechanism is assumed to have the ability to collect and redistribute the private good x and produce the public good y using its net proceeds of x. Individuals are either not able or not authorized to transform x into y, except through their contributions to the committee. The set of all such that feasible allocations Z is thus restricted to the set of allocations z ∈ RI+1 + z = ω + (−t; y)
1
(1)
The production set is often modeled as having as many dimensions as there are commodities. However, by defining the production set in terms of only aggregate inputs and outputs, the extension of the same production set to economies of varying size is made simpler in terms of notation.
4
for some t ∈ RI satisfying X Ã
−
i∈I
X
ti ≤ ωi for each i ∈ I,
(2)
ti ≥ 0,
(3)
!
ti ; y ∈ Y,
(4)
i∈I
and y ≥ 0.
(5)
The vector t represents a list of transfers of the private good to be paid by each agent i. The transfers are restricted so that no agent is required to surrender more than his endowment of the private good, the total of the transfers is non-negative to avoid a deficit, and the surplus of x collected by the transfers can be used to feasibly produce the non-negative public good level y. Note that wasteful allocations in which a surplus of the private good is collected but not redistributed or used in production are included in Z, although no agent may consume a negative quantity of any good, private or public. By these restrictions, any feasible allocation z ∈ Z is necessarily representable by the ‘net trade’ vector (−t; y) = z − ω. An P allocation z ∈ Z (e) is said to be balanced if y = F ( i∈I ti ), so that no surplus is collected. Let Z + (e) = {(x; y) ∈ Z (e) : [∀i ∈ I] xi ≤ ωi } represent the feasible allocations in which P P no agent receives a transfer. For notational simplicity, let T = i∈I ti and T−i = j∈I\{i} tj . An economy is parameterized by the set of agents I, their preferences {ºi }i∈I , the endowment vector ω, and the production set Y. Let EI represent the class of all admissible economies ³ ´ containing I agents. A generic element of EI is represented by the vector e = {ºi }i∈I , Y, ω . In many general equilibrium models, the consumption set is given exogenously. Here, the set of feasible allocations depends crucially on the parameters of the economy and is therefore written as Z (e). A mechanism is defined as a pairing of a strategy space and an outcome function such that strategies map into possible allocations in the economy. Formally, a mechanism Γ consists Q of a product set S = i∈I Si and two functions η : S → R+ and τ : S → RI . The set S is the strategy space of the mechanism, η is the outcome function, and τ is the transfer function 2 . A mechanism is feasible if (ω − τ (s) ; η (s)) ∈ Z (e) for all s ∈ S and e ∈ EI . Note P that feasibility requires that τ (s) satisfies (2) and (3), (− i∈I τi , η) satisfies (4), and η (s) P satisfies (5). A mechanism is balanced if, in addition, η (s) = F ( i∈I τi (s)) for all s ∈ S and e ∈ EI . An equilibrium correspondence µ is a mapping that identifies a set of strategy profiles in S given a mechanism Γ and a preference profile {ºi }i∈I . For example, µ may select the set of dominant strategy profiles in Γ. Following the notation of Saijo and Yamato (1999), let µΓ (e) denote the equilibrium profiles that obtain in Γ under economy e. The set OΓµ (e) is 2
Mechanisms may be defined whose outcome and transfer mappings are not single-valued. Such mechanisms are not considered in the following analysis.
5
defined as the set of allocations in Z (e) that may obtain as an equilibrium outcome of Γ under µ for the economy e. Formally, OΓµ (e) = {(x; y) ∈ Z (e) : [∃s ∈ µΓ (e)] (ω − τ (s) , η (s)) = (x; y)} . Attention is focused only on mechanisms and equilibrium correspondences such that OΓµ (e) 6= ∅ for all e. An allocation z ∈ Z (e) is Pareto optimal if there does not exist another allocation z 0 ∈ Z (e) such that z 0 ºi z for all i ∈ I and z 0 Âj z for some j ∈ I. Let P O (e) represent the set of Pareto optimal allocations of economy e. A mechanism is said to be efficient under µ if OΓµ (e) ⊆ P O (e) for every e. If preferences are strictly monotonic, Pareto optimality requires that y = F (T ). A mechanism is outcome efficient under µ if, for every y ∈ η ◦ µΓ (e), there exists some x such that (x; y) ∈ P O (e). The pivotal mechanism, for example, is known to be outcome efficient under dominant strategy equilibrium, but is not fully efficient because it collects a surplus in some economies (see Groves (1973), Clarke (1971), or Moulin (1986).) 3 In general, if G is a social choice correspondence (SCC) mapping each economy e to a subset of the feasible allocations Z (e), then Γ implements G under µ if (ω − τ ◦ µΓ (e) , η ◦ µΓ (e)) ∈ G (e) for every e. For example, IR (e) = {(x; y) ∈ Z (e) : [∀i ∈ I] (x; y) ºi (ω; 0)}
(6)
is the SCC that selects all points in the economy that are unanimously preferred to the endowment. If Γ implements IR (e) under µ, then all agents are made weakly better off by participating in Γ and playing a strategy in µΓ (e). However, in the context of pure public goods, free-riding on the public goods production of other agents is necessarily a possibility. Simply guaranteeing that agents are made better off than their initial endowment may not be sufficient to induce participation in the mechanism. Saijo and Yamato (1999) demonstrate that when agents may choose whether or not to participate in an efficient mechanism, full participation is not possible in all environments due to the free-riding incentive. In this paper, the possibility of not participating in the suggested outcome of a mechanism is explored with similar results.
2.2
The Participation Decision
The motivation of this paper is to consider a situation in which agents currently endowed with ω receive a proposed final allocation z ∈ Z (e) through some committee decision or mechanism and must subsequently decide whether or not to participate in this allocation by surrendering their share of the private good necessary for z to obtain. Suppose a mechanism Γ generates an outcome z ∈ OΓµ (e) under some economy e and equilibrium concept µ, but the 3
In fact, Walker (1980) demonstrates that full efficiency is unobtainable under dominant strategy equilibrium.
6
mechanism designer or committee is unable to coerce agents to pay their required transfer τi . The participation decision facing each agent i is whether to agree to the transfer or to remain at the initial endowment ωi . If i decides to ‘opt-out’ of the allocation z by withholding her required transfer, the non-excludable nature of the public good guarantees that she will still be able to enjoy the level of the public good produced in her absence. In this framework, a second decision node is added to the typical mechanism design problem. Effectively, the outcome of the mechanism defines an I-player subgame with a binary strategy space in which agents choose whether or not to submit their requested transfer. If the proposal is to be successfully implemented without coercion, a minimal requirement is that the proposal be an equilibrium outcome of the binary participation game. Thus, the mechanism must select solutions that are impervious to agents withholding their transfers. Consider a simple example with two agents and a strictly increasing production function F . An allocation z is proposed such that ti > 0 for each i and y = F (t1 + t2 ) > 0. If both agents participate in the proposed allocation by submitting the requested transfers t, then the allocation z obtains. If agent 1 chooses not to participate while 2 submits t2 , then t1 is withdrawn and the resulting allocation is given by ³
´
z (−1) = (ω1 , x2 ) ; y (−1) , where y (−1) = F (t2 ) . Note that agent 1 returns to her initial endowment of the private goods, but the public good level consequently falls. A symmetric definition also applies for agent 2. Figure I graphically demonstrates this example when F (T ) = T . Note that the two-dimensional simplex of nonwasteful allocations in this case is the familiar Kolm triangle diagram (see Kolm (1970), or Thomson (1999) for a complete discussion.) For the proposal z to be successfully implemented without coercion, both agents must prefer z to their ‘opt-out’ points z (−1) and z (−2) , respectively. In the case where t1 > 0 while t2 < 0, then y (−1) := 0 since negative quantities of the public good are not admissible. If t1 < 0, then y (−1) := y since agent 1 is not asked to contribute any private good. In this case, it is assumed that the negative transfer rejected by agent 1 is either redistributed among the other agents or destroyed, rather than affecting the level of the public good 4 . If the private good is desirable, agent 1 will always prefer participation in the suggested allocation when t1 < 0. These two cases are demonstrated in Figure II. Generalizing the definitions provided by this two-player example provides the key definition of this paper. Definition 1 For any I = 1, 2, . . . and any economy e ∈ EI , a feasible allocation (x; y) ∈ 4
Whether the transfer is redistributed or destroyed will not affect the i’s participation decision since ºi depends only on xi and y.
7
Z (e) such that x = ω − t satisfies Binary Participation for agent i (BPi ) if and only if ³
´
(x; y) ºi x(−i) ; y (−i) , where
(−i)
F (T−i )
y (−i) = 0 y and
xi
= ωi ,
if
ti ≥ 0, T−i ≥ 0, and y ≥ F (T−i )
if
T−i < 0
,
(7)
otherwise ³
´
x(−i) ; y (−i) ∈ Z (e) .
The allocation (x; y) ∈ Z (e) satisfies Binary Participation (BP) if and only if it satisfies BPi for all i ∈ I. The definition of y (−i) is complicated by the need to consider several possible scenarios. The ‘standard’ case is where agent i is asked to pay a non-negative amount of the private good, the sum of other transfers is also positive, and the surplus of the allocation is limited so that the level of the public good is at least as large as the efficient production level restricted to T−i . In this case, the removal of ti negatively affects the public goods production, resulting in F (T−i ) < y. If it were the case that T−i were negative, then i’s contribution is necessarily the only private good entering into production, so y (−i) must be zero. The third case subsumes the cases where i’s transfer has no effect on production. Specifically, if ti < 0 or if y < F (T−i ), then the removal of ti has no effect since the proposed level y may still be produced using the remaining transfers. This list of possibilities is exhaustive given that (x; y) is assumed to be feasible, so the definition is complete. Note that equation (7) selects the value of y (−i) that is as close to y given the constraints of the production technology. Implicit in this construction are certain assumptions about how the committee or mechanism handles production of the public good when an agent fails to contribute her requested transfer. For example, if it were instead assumed that the committee selects y (−i) = 0 if any agent i fails to contribute ti , then each agent would participate if and only if (x; y) ºi (ω; 0) and the standard notion of individual rationality would apply. Or, in the wasteful case of y < F (T−i ), the committee could scale y (−i) down to y times T−i /T rather than simply producing y. In a more general model, the behavior of the committee in such situations could be endogenized and equilibrium choices of y (−i) could be derived under various assumptions. However, the focus of the present analysis is on the difficulty of implementing a desirable social choice correspondence when agents have the ability to withhold their transfers. If a mechanism selects an allocation z ∈ G (e) and one agent withholds her transfer, then the committee may be assumed to prefer allocations as ‘close’ to z as possible given the available transfers. Selecting a value of y (−i) that is lower than that which could feasibly be produced is not credible as such a choice arguably violates subgame perfection on the part of the committee. 8
(−i)
Finally, the´ choice of xj for j 6= i is unrestricted, other than the requirement that ³ (−i) (−i) x ;y ∈ Z (e). Specifying how negative or excess transfers would be redistributed among the remaining agents is both arbitrary and irrelevant to the participation decision of agent i 5 . For any economy e ∈ EI , let BPi (e) := {z ∈ Z (e) : z satisfies BPi } , and define BP (e) :=
\
BPi (e) .
i∈I
Returning to the two-player example with F (T ) = T , Figure III demonstrates cases where an allocation z both satisfies and fails the Binary Participation requirement. In the first panel, z ∈ BP ³ (e). In the ´ second, z ∈ BP2 (e) while z 6∈ BP1 (e). It is important to note (−i) (−i) (−i) that z := x , y is carefully constructed so that z (−i) ∈ Z (e), guaranteeing that the ‘outside option’ of any agent is indeed a feasible allocation.
3
General Properties of Binary Participation
The notation and definitions provided in section 2 describe a general class of economies EI . In the analysis that follows, it is often useful to consider additional assumptions that describe the class of ‘classical’ economic environments. Specifically, consider the following list of possible assumptions: A1 (Continuity) For every z ∈ Z (e), the sets {z 0 ∈ Z (e) : z 0 ºi z} and {z 0 ∈ Z (e) : z 0 ¹i z} are closed. A2 (Convexity) If z 0 ºi z, then αz 0 + (1 − α) z ºi z for all α ∈ (0, 1). A3 (Monotonicity) If x0i ≥ xi and y 0 ≥ y, then (x0 ; y 0 ) ºi (x; y). A4 (Increasing marginal cost) Y is a closed, comprehensive, convex set. Denote the set of classical economies satisfying A1 through A4 by EIC . The following two assumptions prove useful in discussing marginal trade-offs: A5 (Differentiable utility) Preferences ºi can be represented by a differentiable utility function ui . A6 (Differentiable cost) The boundary of Y is represented by a differentiable function F . 5
If the analysis were extended to consider the possibility of implementing some correspondence G by selecting some z not necessarily in G, but such that the equilibrium of the binary participation (−i) game was a point in G, then the choice of xj would be important. However, the ability to apply (−i)
this method of two-stage implementation would depend crucially on the arbitrary choice of xj
9
.
Note that A5 requires A1. Adding A2 and A3 to A5 results in the standard assumption of an increasing, quasi-concave utility function. Let EID denote the set of differentiable classical economies satisfying A1 through A6. There are a few basic facts about the set of allocations satisfying Binary Participation that follow immediately from the definition. For example, BP (e) is non-empty for every e ∈ EI and every I since (ω; 0) is always an element of BP (e). Furthermore, when e ∈ EIC , the set of balanced allocations satisfying BPi (e) for some i ∈ I is the union of three closed sets, two of which are convex. Figure IV demonstrates a typical Binary Participation set for agent 1. When t2 ≤ 0, all allocations preferred to the endowment satisfy BP1 . When t1 ≤ 0, all allocations satisfy Binary Participation. If both transfers are positive, the set is indeed closed by continuity and convexity of preferences, though not necessarily convex. The underlying model of Binary Participation assumes that if agents withdraw from the economy, they may still enjoy the public goods production of others, but must withdraw completely. If withdrawing agents are assumed to leave the economy or if the committee is able to credibly set y (−i) to zero, then the decision to withdraw is based on a comparison between the proposed allocation and the initial endowment. Here, allocations satisfying individual rationality are impervious to transfer withdraws. Since the set of individually rational points is usually quite large, this is not an excessive restriction on outcomes. On the other extreme, if agents are able to enjoy the production of others and are able to arbitrarily alter their contribution to the public good (so long as the contribution remains positive and feasible,) then the Nash equilibria of the ‘voluntary contribution’ game are the only allocations guaranteeing participation. In the voluntary contribution game, agents contribute any positive amount ti ∈ [0, ωi ] to the production of the public good, thus restricting the set of allowable allocations to Z + (e). Production is assumed to be non-wasteful, so that the resulting level of the public good is F (T ) ≥ 0. A Nash equilibrium allocation of the game is any point (x∗ ; y ∗ ) such that P y ∗ = F (T ∗ ) (where T ∗ = i∈I t∗i and t∗i = ωi − x∗i ) and at which no agent would prefer some other transfer t0i 6= t∗i . The set of all Nash equilibria is given by
n
³
³
∗ N E (e) = (x∗ ; y ∗ ) ∈ Z + (e) : [∀i ∈ I] [∀t0i ≥ 0] (x∗i , y ∗ ) ºi ωi − t0i , F T−i + t0i
´´o
.
(8)
The well-known free-rider result is that the Nash solution will, under very lax assumptions, yield suboptimal levels of the public good. See Bergstrom, Blume, and Varian (1986) for an analysis of this game. The notion of Binary Participation is now shown to be more stringent than individual rationality, but less restrictive than the Nash equilibrium requirement. The former result requires the mild assumption of monotone increasing preferences, while the latter follows immediately from the definitions. 10
Proposition 2 Under monotone increasing preferences (A3), all allocations satisfying Binary Participation also satisfy individual rationality (BP (e) ⊆ IR (e).) ³
´
Proof. Consider a point (x; y) such that (xi , y) ºi ωi , y (−i) for all i ∈ I. Note that y (−i) ≥ 0 ³
´
for each i, so A3 implies that ωi , y (−i) ºi (ωi , 0). By transitivity, (xi , y) ºi (ωi , 0) for every i, proving the result. Proposition 3 All Nash equilibria of the voluntary contributions game satisfy Binary Participation (N E (e) ⊆ BP (e).) Proof. From any Nash equilibrium point, ³ ³ ´´the ‘opt-out’ allocation for agent i in the binary ∗ participation game is simply ωi , F T−i . Since the definition requires that (x∗i , y ∗ ) ºi ³
³
∗ ωi , F T−i pation.
´´
for all i by considering t0i = 0, the point (x∗ ; y ∗ ) must satisfy Binary Partici-
In the public goods context, the basic question of mechanism design theory is whether or not mechanisms can be constructed in a given environment whose outcomes are Pareto optimal. Although fully optimal outcomes are known to be unobtainable when dominant strategy equilibrium is required, there exist mechanisms such as that of Groves and Ledyard (1977) whose Nash equilibria are guaranteed to be Pareto optimal when utility is transferable. Walker (1981) and others have developed mechanisms whose Nash equilibrium outcomes correspond to the Lindahl equilibria of the economy, at which transfers are in proportion to marginal benefits and individual rationality is satisfied. However, if the outcomes of these mechanism fail to satisfy Binary Participation, then their desirable properties are of little use in environments where agents cannot be coerced to submit their transfers. Consider the following simple class of economies. For any I ≥ 2, let ui (xi , y) = v (y) + xi be a continuous utility function with v (y) =
3y 2I
if
y≤1
1y+1 2I I
if
y≥1
.
If F (x) = x, then the marginal cost is constant at unity. The sum of the marginal benefits of y, given by Iv 0 (y), is 3/2 when y ≤ 1 and 1/2 when y ≥ 1. Thus, by the conditions P of Samuelson (1954), Pareto optimality is achieved when y o = i∈I ti = 1. Since this is a simple, transferable utility environment, the Groves-Ledyard, Walker, or any of the other optimal mechanisms may be used to determine this optimal level of the public good when preferences are private information. Now consider the question of whether this outcome will induce agents to submit their required transfers. Since the sum of transfers is positive, there must be at least one agent such that ti > 0. If P P ωi < 1, then j6=i tj must be also positive. Thus, y (−i) = j6=i tj . If agent i participates in 11
the optimal allocation, she receives v (1) + xi =
3 + ωi − ti . 2I
If she withholds her transfer, she receives v (1 − ti ) + ωi =
3 (1 − ti ) + ωi . 2I
Since ti > 0 and I ≥ 2, she prefers withholding her transfer. If ωj < 1 for all j ∈ I and P j∈I ωj ≥ 1, then there always exist such an agent, so an economy exists in which all Pareto optimal allocations fail to satisfy Binary Participation. When an economy such as this is admissible, the outcomes of optimal mechanisms are not guaranteed to obtain when agents are free to withhold their transfers. Thus, the following proposition is proven. Proposition 4 For every I ≥ 2, there exists an economy e ∈ EIC such that no Pareto optimal allocation satisfies Binary Participation (P O (e) ∩ BP (e) = ∅.) Note that the example does not represent a knife-edge case; a wide range of similar examples can be constructed. In general, Pareto optimality is inconsistent with Binary Participation in much the same way that the Nash equilibrium of the voluntary contributions game is generically inconsistent with optimality.
4 4.1
Quasi-Concave Economies Necessary and Sufficient Conditions for Binary Participation
The additional structure gained by adding assumptions A1 through A6 allows for the derivation of separate necessary and sufficient conditions for an allocation to satisfy Binary Participation. Although these conditions are not tight, they require only ‘local’ information about the gradients of utilities and derivative of the production function. Proposition 5 In economies with differentiable, quasiconcave utilities and a differentiable, concave production function, if Binary Participation is satisfied at a point (x; y) = (ω + t; y), then 1 ∂ui (ωi ; F (T−i )) /∂y ≥ 0 (9) ∂ui (ωi ; F (T−i )) /∂xi F (T−i ) for all i ∈ I such that ti , T−i ≥ 0 and y ≥ F (T−i ). Proof. Pick any agent i such that ti , T−i ≥ 0 and y ≥ F (T−i ). Binary Participation implies that ui (ωi − ti , F (T−i + ti )) ≥ ui (ωi , F (T−i )) . 12
By quasi-concavity of ui , ∇ui (ωi , F (T−i )) · (−ti , F (T−i + ti ) − F (T−i )) ≥ 0, or
F (T−i + ti ) − F (T−i ) ∂ui (ωi ; F (T−i )) /∂xi ≥ . ti ∂ui (ωi ; F (T−i )) /∂y Thus, by concavity of F , ∂ui (ωi ; F (T−i )) /∂xi ≤ F 0 (T−i ) . ∂ui (ωi ; F (T−i )) /∂y Inverting this inequality gives the necessary condition. A similar condition is now shown to be sufficient for a point to satisfy Binary Participation. Whereas the necessary condition compares the marginal rate of substitution to marginal costs at the drop-out point, the sufficient condition compares these quantities at the proposed allocation. Proposition 6 In an economy with differentiable, quasiconcave utilities and a differentiable, concave production function, if a point (x; y) = (ω + t; y) satisfies ∂ui (x; y) /∂y 1 ≥ 0 ∂ui (x; y) /∂xi F (T )
(10)
for all i such that ti ,T−i ≥ 0 and y ≥ F (T−i ) and uj (x; y) ≥ uj (ω; 0)
(11)
for all j such that T−j < 0, then Binary Participation is satisfied at (x; y). Proof. By monotonicity, Binary Participation is trivially satisfied for all j such that tj < 0 or y < F (T−j ). Equation (11) guarantees Binary Participation when T−j < 0. Now consider some i ∈ I such that ti , T−i ≥ 0 and y ≥ F (T−i ), but for whom Binary Participation fails. For this agent, ui (ωi , F (T−i )) > ui (ωi − ti , F (T−i + ti )) ,
(12)
so that ∇ui (x; y) · (ti , F (T−i ) − F (T−i + ti )) > 0. This is equivalent to
∂ui (x; y) /∂xi F (T−i + ti ) − F (T−i ) > , ∂ui (x; y) /∂y ti 13
(13)
so applying the concavity of F at T−i + ti and inverting the resulting relationship gives ∂ui (x; y) /∂y 1 < 0 . ∂ui (x; y) /∂xi F (T−i + ti ) Equation (10) implies that (12) cannot hold, so by the contrapositive of this argument, (x; y) must satisfy BPi . Thus, Binary Participation is satisfied for all i ∈ I. Unlike the necessary condition, equation (11) implies that information about the utilities of some agents at both the suggested allocation and the endowment is needed. This may be undesirable from the standpoint of mechanism design since additional information is necessary to determine that the condition is met 6 . However, equation (11) could be replaced by a modified version of (10) and would still provide sufficiency for agents with T−i < 0. For these agents, y (−i) = 0 while F (T−i ) < 0. By replacing F (T−i ) with zero in the above proof, the argument is identical through equation (13). At this point, the subsequent relationship with F 0 (T ) cannot be derived from F (T ) /ti when T−i < 0, so inverting (13) gives the alternative sufficient condition ∂ui (x; y) /∂y 1 ≥ ∂ui (x; y) /∂xi F (T ) /ti
(14)
for all i ∈ I. This alternative condition is stronger than (11), but requires only ‘local’ information about utilities and the production function. Therefore, the alternative condition may be useful in reducing the size of the message space in a decentralized mechanism. Figure V demonstrates the interpretation of these conditions. The quantity (∂ui /∂y) / (∂ui /∂xi ) represents the slope of the gradient of ui , while 1/F 0 represents the slope of the normal to the tangent of −F . In the figure, F is reflected around the y-axis and horizontally shifted so that the graph of F accurately represents the production possibilities set given the endowments. The point z equalizes the gradient of the sum of utilities and 1/F 0 , and is therefore Pareto optimal. If agent i withholds the requested ti , then the allocation z (−i) results. In this case, i will prefer z to z (−i) . The necessary condition for Binary Participation is satisfied since the gradient of utility has a steeper slope than the normal to F at the point z (−i) . The sufficient condition is satisfied at z 0 since the gradient of utility is steeper than the normal to F , but this condition fails at the optimal point z. In fact, any point along F between z ∗ and z (−i) satisfies the sufficient condition. The intuition for this result is as follows. The point z ∗ represents i’s most-preferred feasible allocation. At points such as z 0 , the ‘opt-out’ point z (−i) is necessarily farther from z ∗ than the allocation under consideration, so withholding the transfer will result in lower levels of utility. 6
Of course, there could exist mechanisms whose outcomes satisfy Binary Participation without satisfying this sufficient condition.
14
The point z is a Pareto optimum and therefore solves the program max
{ti }i∈I
s.t
P i∈I
ui (ωi − ti , F (T )) .
ti ≤ ωi ∀i T ≥ 0.
Thus,
∂ui (z) X ∂uj (z) 0 = F (T ) ∂xi ∂y j∈I
for all i ∈ I, which implies the more familiar form of the Samuelson conditions, X ∂ui (z) /∂y i∈I
∂ui (z) /∂xi
=
F0
1 , (T )
(15)
which equates the sum of the marginal rates of substitution with the marginal cost of production. The gradient of ui must be less steeply sloped than the normal to F at an optimum, but Binary Participation requires that the gradient of ui must be more steeply sloped than the normal to F at the ‘opt-out’ point z (−i) . This implies that i’s most-preferred point z ∗ must lie between the optimum and the opt-out point. Furthermore, the opt-out point must be sufficiently far from z ∗ so that it lies ‘below’ the indifference curve of the optimum. Thus, the transfer paid by each agent must be fairly large at the optimal point. In the example of Figure V, the transfer of agent i is notably larger than the total of the others’ transfers.
4.2
Quasi-Linear Preferences
The transferable utility environment is especially important in implementation theory as the absence of wealth effects is useful in guaranteeing the ability to satisfy incentive compatibility constraints through transfer payments. It also allows a more precise quantification of the minimal transfer needed to satisfy Binary Participation. Assume agents have utility functions ui (xi , y) = vi (y) + xi , where vi0 > 0 and v 00 ≤ 0, and let the production function be concave, so that the marginal cost is increasing in y. Binary participation requires that ´ ³ vi (y) − ti ≥ vi y (−i) , or simply that transfers be less than the difference in value between the proposed level of the public good and the ‘opt-out’ level for each agent. This can be re-written as ti ≤
Z y y (−i)
vi0 (ζ) dζ.
15
However, if y = F (T ), then it must be that transfers cover the additional cost of producing the public good. Letting c represent the cost function (which is the inverse of F ,) it must be that if ti is non-negative, Z y ti ≥ c0 (ζ) dζ, y (−i)
with equality if (x; y) is non-wasteful. Thus, in order for (x; y) to satisfy Binary Participation, the change in each agent’s value by participating must exceed the increased cost. Note that if yi∗ is agent i’s most-preferred level of the public good given that she pays the full cost, then vi0 (yi∗ ) = c0 (yi∗ ) , so that when y > yi∗ ,
Z y yi∗
(vi0 (ζ) − c0 (ζ)) dζ ≤ 0.
Thus, it must be the case that Z y∗ y (−i)
(vi0 (s) − c0 (s)) ds ≥
Z y yi∗
(c0 (ζ) − vi0 (ζ)) dζ,
(16)
both of which are non-negative quantities. P
If y o represents the Pareto optimal allocation that satisfies i∈I vi0 (y o ) = c0 (y o ), then vi0 (y o ) ≤ c0 (y o ) and equation (16) (with y = y o ) is necessary for Binary Participation. This is demonstrated in Figure VI, in which y (−i) is chosen to be as large as possible while satisfying (16). The total transfer paid by the the agent is the area under the marginal cost curve, which here equals her increased value of participating so that she is indifferent between participation and withholding her transfer. The necessary and sufficient conditions from equations (9) and (10) are also intuitive in this figure; if y (−i) > yi∗ , then the necessary condition fails because marginal costs are everywhere larger than the marginal benefit between y (−i) and y > y (−i) , and the sufficient condition is satisfied if y < yi∗ since marginal costs are everywhere less than the marginal benefit between y and y (−i) ≤ y. The general observation that Pareto optimality places a strong restriction on the necessary size of the transfers is illustrated by the following two examples. First, let vi (y) = by − ay 2 for every i and c (y) = κy with b > κ. Here, the Pareto optimum is y o = (b − κ/I) /2a, while yi∗ = (b − κ) /2a for each i. The reader can verify that y (−i) ≤
b − (2 − 1/I) κ 2a
for any contributing agent if Binary Participation is to be satisfied at the optimum, so that all non-negative transfers must satisfy toi ≥
I − 1 κ2 . I a
(17)
Since the total cost is κy o but transfers must satisfy (17), only a small number of agents 16
may be feasibly required to pay positive transfers. In fact, if the parameter b is less than twice the size of κ, then only one agent may contribute positive transfers whenever I ≥ 2. As long as b is less than four times κ, there will always exist some finite I such that only one agent may contribute positive taxes 7 . If no agent has a large enough endowment to unilaterally produce the optimal quantity of the public good, then in these economies, Pareto optimality is inconsistent with Binary Participation. As a second example, consider any economy in which all preferences are monotone, strictly convex, and symmetric about the 45-degree line, so that ui (xi , y) = ui (y, xi ) for all xi , y ≥ 0, and F (T ) = T . It is not necessary that preferences be identical across agents; only that all agents’ indifference curves are symmetric. Any Pareto optimum of this economy requires that half of the entire endowment enter into production of the public good, although the unique Nash equilibrium divides the total endowment equally among the I + 1 components of (x; y). Let (xo ; y o ) represent an optimal allocation and suppose temporarily that agents may withdraw any amount of the their transfer they please. For any agent i such that y o > xoi , withdrawing transfers t0i such that y o − t0i = xoi + t0i will maximize i’s utility by moving i’s consumption down to the 45-degree line. Thus, t0i = (y o − xo ) /2 is i’s most-preferred amount to withdraw. By symmetry of preferences, if agent i’s requested transfer is at least twice as large, so toi ≥ (y o − xo ), then withdrawing the requested transfer moves i’s consumption to a point farther from the 45-degree line than (xo ; y o ). Thus, for any agent with y o > xoi , the transfer toi must be larger than (y o − xoi ) for Binary Participation to be satisfied. Note that since y o equals half of the total endowment, feasibility requires that y o ≥ xoi for all i ∈ I and y o > xoi for any i holding less than half of the initial endowment. Thus, all agents i must pay either non-positive transfers or toi ≥
1X ωj − xoi . 2 j∈I
Since xoi + toi = ωi , then it must be true that ωi ≥
X
ωj .
j6=i
This can be satisfied by at most one agent who holds more than half of the total endowment. This agent must unilaterally fund the public good production. If no such agent exists, Binary Participation and Pareto optimality are inconsistent. Note that as the economy becomes larger, the inequity in endowments necessary for Binary Participation and Pareto optimality to be jointly satisfied becomes more extreme. 7
These facts come from dividing c (y o ) = κ (b − κ/n) /2a by the smallest allowable transfer under Binary Participation, κ2 (I − 1) / (Ia), which then determines the maximum number of contributing agents feasible.
17
These examples demonstrate that the difficulty in requiring optima to satisfy Binary Participation is intimately linked to the standard free-rider problem. If the equilibrium allocation is Pareto optimal, Binary Participation is trivial. When optimal allocations deviate significantly from the voluntary contributions equilibrium, strong and carefully-crafted assumptions about the convexity of preferences, the concavity of the production function, and the distribution of endowments are needed to guarantee Binary Participation using feasible transfers.
5
Implementation with Binary Participation
The examples of the previous section imply that no mechanism can guarantee optimal outcomes without some restrictive a priori assumption about the class of admissible economies. Since Binary Participation is a natural restriction to require in environments where agents may withhold transfers, it is important to understand whether there exist any non-trivial mechanisms that satisfy this constraint, regardless of optimality. The ‘non-trivial’ qualifier is added since the status-quo mechanism that always selects the initial endowment trivially satisfies the requirement, but is obviously of no interest since it results in no reallocation. Given a particular equilibrium concept µ, a mechanism Γ implements the set of Binary Participation outcomes BP (e) if OΓµ (e) ⊆ BP (e) for every admissible economy e. By the theorems of Gibbard (1973) and Satterthwaite (1975) dominant strategy implementation (where µ selects dominant strategy profiles) is known to be generally impossible. Even with ‘classical’ preferences, Zhou (1991) shows that the impossibility result still holds. However, with single-valued social choice correspondences and quasi-linear preferences, the characterization of Roberts (1979) demonstrates that a social choice correspondence may be implementable as long as it maximizes a weighted sum of agents’ utilities plus an arbitrary function of the allocation itself. Formally, Roberts’ characterization is as follows. Let ui (xi , y) = vi (y) + xi for each i ∈ I. Given some class of admissible quasi-linear economies EI , let V represent the set of all possible utility profiles v = (v1 , . . . , vI ). Since Z (e) depends only on Y and ω, the selection of v has no affect on the feasible outcomes. Since the committee or mechanism designer may be assumed to know Y and ω, abbreviate Z (e) by simply Z. A social choice correspondence in EI is therefore a mapping G : V ³ Z. Assume that G is single-valued and that for each z ∈ Z, there exists some v ∈ V such that G (v) = z. If G is implemented by some mechanism in dominant strategies, then there exists some function g : Z →R and vector of weights k = (k1 , . . . , kI ) ≥ 0 such that (
G (v) ∈ z ∈ Z : [∀z 0 ∈ Z]
X
ki vi (z) + g (z) ≥
X
)
ki vi (z 0 ) + g (z 0 ) .
i∈I
i∈I
However, it is clear to see that this characterization prevents the implementation of any single-valued choice rule that satisfies Binary Participation whose range is all of Z. Thus, dominant strategy implementation must require restrictions on the range of the social choice 18
rule, such as in the mechanism which only selects the initial endowment. More positive results may be obtained when the requirement of µ selecting dominant strategies is weakened to Nash equilibria. Here, the theorem of Maskin (1999) provides sufficient conditions on a social choice correspondence such that it may be implemented in Nash equilibrium. The following verifies that these conditions are met by BP (e), so that the set of Binary Participation allocations is indeed implementable through Maskin’s canonical mechanism. Maskin’s theorem requires that a social choice correspondence satisfy a monotonicity requirement along with the ‘No Veto Power’ property. Letting R represent the set of possible preference profiles R = {ºi }i∈I , the definition of monotonicity is given as follows: Definition 7 A social choice correspondence³ G : R ´³ Z satisfies monotonicity if, for all z ∈ Z and all {ºi }i∈I , {º0i }i∈I ∈ R, if z ∈ G {ºi }i∈I and [[∀i ∈ I] [∀z 0 ∈ Z] z ºi z 0 ⇒ z º0i z 0 ], ³
´
then z ∈ G {º0i }i∈I . Thus, this definition requires that if preferences are changed so that the lower contour set of any allocation z selected by G does not shrink, then z is also selected under the new preferences. Lemma 8 BP (e) satisfies monotonicity. ³
´
Proof. Pick a point z = (x, y) in the set BP (e). Note that (xi , y) ºi ωi , y (−i) for all i ∈ I, where y (−i) is as defined in Definition 1. Now consider a monotonic transformation of {ºi }i∈I´ ³ 0 0 0 0 0 into {ºi }i∈I such that z ºi z implies that z ºi z for i and all feasible z . Since ωi , y (−i) ³
´
is feasible, (xi , y) º0i ωi , y (−i) , showing that z satisfies BPi under {º0i }i∈I . The No Veto Power assumption only applies to situations where I − 1 agents prefer a single allocation over all others. If preferences in the private good are monotonic, then this condition is vacuously satisfied. Therefore, the following proposition is simply an application of Maskin’s theorem. Proposition 9 The set of allocations satisfying Binary Participation (BP (e)) can be nontrivially implemented in Nash equilibrium when I ≥ 3. Note that the canonical mechanism used in the proof of Maskin’s theorem enables any point z ∈ BP (e) to be a Nash equilibrium selection. However, the mechanism itself is not a ‘natural’ mechanism, and the question of whether there exist simpler non-trivial mechanisms that always satisfy Binary Participation remains an open question. 19
6
Binary Participation in Large Economies
The free-rider problem in public goods economies is known to be more severe in economies with more agents. Palfrey and Rosenthal (1984) and Dixit and Olson (2000) show that the equilibrium probability of contributing to a public good declines as the economy grows. Similarly, Saijo and Yamato (1999) demonstrate that if agents in a symmetric Cobb-Douglass economy can choose whether or not to participate in an optimal mechanism (rather than the outcome of an optimal mechanism,) the measure of economies in which all agents participate quickly shrinks to zero as the economy expands. For Binary Participation, the discussion in section 4 implies that Binary Participation restricts or eliminates the set of Pareto optimal allocations, and the friction between these two requirements is apparently exacerbated by the size of the economy, which raises the natural question of whether increasing the size of the economy necessarily leads to a contraction of the set of Binary Participation allocations. The suggested result is intuitive; as an economy grows, one individual’s effect on the production of the public good becomes negligible. In the limit, no agent should have an interest in sacrificing a positive proportion of their endowment in exchange for an infinitesimal increase in the public good level. The fundamental difficulty in working with replica public goods economies lies in the relative sizes of private and public goods. If the total endowment grows as the economy expands, identical proportions of production inputs result in successively larger and larger levels of the public good. If the total endowment is fixed and divided into smaller shares per replicate as the economy grows, then public goods levels remain fixed for a given proportion of private goods contributions, but private goods consumption necessarily tends to zero. In either scenario, the ratio of the two goods diverges to infinity as the economy is replicated. Muench (1972), Milleron (1972), and Conley (1994) discuss the difficulty of replicating public goods economies and offer various possible solutions 8 . Muench (1972) considers a limit economy in which there is a continuum of consumers whose private goods consumption is represented by a measurable function. Conley (1994) simply replicates the agents without adjustment and uses certain assumptions about preferences as the public good level becomes large to generate convergence results. Milleron (1972) provides an intriguing alternative notion of replication; by splitting a fixed endowment among the replicates and adjusting preferences so that agents’ concerns for the private good are relative to their own endowment, the fundamental difficulties of replication are mitigated. In essence, as the economy is replicated and agents are given a smaller share of the endowment, their preferences adjust proportionally to become more sensitive to the private goods holding. Thus, a very small shift in the holdings of the private good is more significant to an agent participating in an economy with many agents than an agent in a small economy. 8
These authors are examining the convergence of the core of the economy to the Lindahl equilibrium.See Foley (1970) for the appropriate definitions.
20
In order to formally analyze the notion of Binary Participation as an economy grows, a concept of a replica economy with public goods is needed. Muench (1972) and Milleron (1972) provide examples of replica economies with public goods in which the core of the economy fails to converge to the Lindahl equilibrium 9 . Simply replicating the economy by giving each new replicated agent the same endowment as its predecessors leads an indeterminacy in the limit as the level of the public good diverges. As Milleron (1972) notes, one may take an ‘asymptotic’ viewpoint by either scaling preferences or endowments as the economy is replicated, or one may work in the limit economy with a continuum of agents for whom the private goods consumption is given by a measurable function defined on the space of agents. In what follows, the ‘asymptotic’ approach is applied in which endowments of each type of agent are divided among the number of replicates evenly. An interpretation of this assumption is that there exists an environment with a fixed supply of the private good, and as agents are replicated, the fixed supply is necessarily divided among the agents in successively smaller quantities. In the limit, each agent possesses an infinitessimal endowment, while the aggregate of production inputs remains a non-trivial scalar. Since agents’ consumption of the private good necessarily vanishes as the economy grows, it is often useful to further assume that the preferences over private goods are determined by the proportion of the endowment consumed, rather than the total consumption level. If endowments shrink as the economy is replicated and if preferences of the replicates are determined by consumption levels (rather than proportions,) positive statements about large economies hinge entirely on the marginal rates of substitution where the private good consumption is infinitessimal. The approach of Milleron (1972) provides a way to make the infinitessimal changes in consumption of the private good for agents in the large economy non-trivial in terms of the effect on preferences. ³
´
Consider a base economy e ∈ EI with I unique agents such that e = {ºi }i∈I , Y, ω . A replica economy eR is defined by replicating R times each i ∈ I. Each replicate of consumer type i, denoted by the pair (i, r) for r = 1, . . . , R, is endowed with ωi /R units of the private good and a preference relation ºi,r such that, because of the scaling of endowments, (x, y) ºi,r (x0 , y 0 ) if and only if (Rx, y) ºi (Rx0 , y 0 ). This assumption on preferences of replicates mimics the approach of Milleron (1972) and guarantees that private good consumption trade-offs are significant, even as the magnitude of those trade-offs becomes arbitrarily small. Finally, the production technology of eR is assumed identical to that of e. This intuition that Binary Participation becomes oppressively restrictive as an economy is replicated is confirmed by the following theorem. Theorem 10 For any economy with continuous, monotone preferences and an increasing, continuous production technology, the set of allocations satisfying Binary Participation converges to the initial endowment as the economy is infinitely replicated. 9
In these articles, the notion of “blocking” used in defining the core excludes the possibility of free-riding on the public goods production of non-coalition members.
21
The proof of this theorem is appears in the appendix. This negative result implies that Binary Participation is indeed linked to the standard free-rider problem. When agents are allowed to withhold their transfers, individual influence on production becomes negligible in large economies. Rational agents will contribute no more than a proportion of their endowment that is shrinking at a rate faster than 1/R. In sufficiently large economies, the total input to the public good is restricted by the Binary Participation constraint to be negligible. This is true in economies where the set of Pareto optimal allocations remain far from the endowment as the economy grows, so that notions of approximate efficiency are of no substantial benefit. For large economies, it is necessary that the committee or government have the power of coercion in order to overcome the free-rider problem.
7
Conclusion
The definitions and results of this paper are simple and intuitive. The concept of Binary Participation (Definition 1) introduces a notion of participation in allocations rather than mechanisms, and therefore suggests that if a mechanism is to implement a desired social choice correspondence without coercion, it must select an allocation impervious to agents withdrawing their transfers. The incompatibility between Binary Participation and Pareto optimality is established through simple quasi-linear examples, indicating that under assumptions often applied in implementation theory, Pareto optimality is unfortunately an unobtainable goal in the face of individual incentives. Furthermore, the notion that freerider problems become more severe in large economies is confirmed in this setting since only the initial endowment satisfies the participation constraint in the limit economy. The above analysis leaves open important questions about participation in public goods allocations. No successful attempt has been made to characterize those economies for which there exists a Pareto optimal allocation satisfying Binary Participation. If the class of such economies is sufficiently broad and reasonable to assume as the support of possible economies, then the negative results may be avoided for small numbers of agents. Similarly, there may exist a wide range of economies for which Pareto optimality may be well approximated under Binary Participation. If such partially desirable outcomes could be identified, perhaps there exists a more natural mechanism that can implement these outcomes in Nash equilibrium. Given that the Binary Participation constraint is typically represented as an upper bound on transfers, which in turn depends on the transfers of others, it is conceivable that a transfermaximizing solution to this system of upper bound restrictions may be identified and used to maximize the total size of the public good in a given economy. This results of this research are negative, indeed. The inability to implement optimal solutions appears to favor a Pigouvian system of government, used to both incentivize demand revelation and require participation in the resulting allocations. However, empirical observations demonstrate that non-trivial quantities of public goods are provided regularly. Governments and other voluntarily established methods of coercion exist as enforcement 22
devices to guarantee that welfare improving allocations are attained. Fair and altruistic preferences, observable both in the field and in the laboratory, imply that the negative results implied by purely selfish motives may be mitigated or even cured by appropriate notions of other-regarding or distribution-minded preferences. Thus, although this type of free-rider problem has negative consequences for implementation, the door remains open for research into mechanisms that take advantage of psychological and behavioral regularities that differ from the standard model of economic behavior. This ‘behavioral mechanism design’ approach may provide solutions that circumvent the free-rider problem by leveraging known irrational behavior patterns and deviations from selfishness.
A A.1
Appendix Proof of Theorem 10
The goal is to show that for any economy e and any sequence of allocations ³
´
³
´
n
such that xR ; y R ∈ BP eR for all R ≥ 1, it must be that xR n
infinitely-replicated endowment and y R
o∞
R=1
o∞
R=1
n³
xR ; y R
´o∞ R=1
converges to the
converges to zero.
By appending an infinite string of zeros to any finite-dimensional private n o∞goods allocation x, to the infinitelythe sup-norm kxk∞ is well-defined and convergence of a sequence xR n
o∞ R=1
remains a sequence replicated endowment is shown to be in this norm. The sequence y R R=1 of scalars, and convergence to zero is in the standard Euclidean norm. The method of proof is by contradiction. Since feasibility (ti,r ≤ ωi /R)n¯ guarantees that ° ° ¯o ° R ¯ R¯ ∞ R° °x − ω ° → 0 as R → 0, a contradiction is constructed in which ¯y ¯ fails to ∞ R=1 converge to zero. Proof. By of contradiction, assume that there¯ exists some economy e and some sequence ¯ n³ ´oway ³ ´ ∞ ¯ ¯ in BP eR for each R such that ¯yˆR ¯ fails to converge to zero. xR ; yˆR R=1
³
´
³
´
R R R R For each (i, r), let tR ˆ ∈ BP eR , if yˆR < F i,r = ωi,r − xi,r . For any x ; y
³
R
R
´
³
R
´
³P
R
R i,r ti,r
´
³P
R i,r ti,r
´
, then
. In other words, if a wasteful by monotonicity, x ; y ∈ BP e , where y = F allocation (x; yˆ) satisfies Binary Participation, so does the transfer-equivalent non-wasteful ´o n³ ∞ R R satisfies Binary Participation for each allocation (x; y). Thus, the sequence x ; y ¯
n¯ o∞ R and ¯¯y R ¯¯
R=1 n³
R=1
also fails to converge to zero. This implies that there exists an infinite ´o∞
¯ ¯
¯ ¯
such that ¯y R(k) ¯ > ε for some ε > 0 all k ∈ N := {1, 2, . . .}. subsequence xR(k) ; y R(k) k=1 Letting c (y) represent the minimal ´cost of producing y (which is the inverse of F ,) non³ R(k) ≥ c (ε) > 0 for each k since c is an increasing function convergence guarantees that c y 23
and Y∩R2+ = {0}. For any k, if R (k) > maxi∈I (ωi /c (ε)), then no one agent (i, r) can unilaterally fund y R(k) R(k) using ti,r since R(k)
ti,r
≤ max ωi /R (k) i∈I
< c (ε) ³
´
≤ c y R(k) . Letting
½
¾
∗
k = max k ∈ N : R (k) ≤ max (ωi /c (ε)) , i∈I
R(k)
there exists at least one sequence of agents {(ik , rk )}∞ k=1 such that tik ,rk ≥ c (y) / (R (k) I) ∗ for all k, and T−(ik ,rk ) > 0 for all k > k . In other words, there exists a sequence of agents such that at each k, the identified agent is paying a transfer which is more than the average transfer of c (y) / (R (k) I) > 0, and the sum of the others’ transfers is eventually positive as the feasibility constraint becomes restrictive. For example, {(ik , rk )}∞ k=1 might identify the R(k) agent (i, r) in each k for whom ti,k is maximal among all agents (although no such agent may exist in the limit, so the any ‘above average’ selection is used.) ³
´
Since each xR(k) ; y R(k) satisfies Binary Participation for all (i, r), it must be the case that ³
or equivalently,
³
ωi,r −
ωi −
R(k) tik ,rk , y R(k)
´
µ
Note that for k > k ∗ , ³
y
´ R(k) −(ik ,rk )
´ R(k) −(ik ,rk )
³
´ R(k) −(ik ,rk )
ºik ,rk ωi,r , y
R(k) R (k) tik ,rk , y R(k)
´
µ
¶
³
ºik ωi , y
, ¶
.
X R(k) R(k) = F tj,s − tik ,rk . j,s
³
By continuity of the production function, y R(k) R(k)
´−(ik ,rk )
becomes arbitrarily close to y R(k) as
R(k)
k grows. However, since tik ,rk > c (ε) / (R (k) I), R (k) tik ,rk is bounded below by c (ε) /I > 0 at all k. By monotonicity of preferences, it must be the case that Ã
! µ ¶ ³ ´ ³ ´ c (ε) R(k) R(k) R(k) −(ik ,rk ) R(k) . ºi ωi , y ,y ºi ωi − R (k) tik ,rk , y ωi − I ³
´−(ik ,rk )
to y R(k) implies that for large By continuity of preferences, convergence of y R(k) enough k, Ã ! ³ ´ c (ε) R(k) ωi − ,y ºik ωi , y R(k) . I 24
However, this ¯ ¯ violates monotonicity. Since there cannot be an infinite subsequence allocations ¯ R(k) ¯ with ¯y ¯ > ε for any ε > 0, it must be the case that y R → 0 as R → ∞. °
°
Feasibility alone requires that °°xR − ω R °°
∞
→ 0, completing the proof.
References Bergstrom, T., L. Blume, and H. Varian (1986): “On the Private Provision of Public Goods,” Journal of Public Economics, 29, 25–49. Clarke, E. (1971): “Multipart Pricing of Public Goods,” Public Choice, 2, 19–33. Conley, J. P. (1994): “Convergence Theorems on the Core of a Public Goods Economy: Sufficient Conditions,” Journal of Economic Theory, 62, 161–185. Dixit, A., and M. Olson (2000): “Does voluntary participation undermine the Coase Theorem?,” Journal of Public Economics, 76, 309–335. Foley, D. K. (1970): “Lindahl’s Solution and the Core of an Economy with Public Goods,” Econometrica, 38, 66–72. Gibbard, A. (1973): “Manipulation of Voting Schemes: A General Result,” Econometrica, 41, 587–602. Green, J., and J.-J. Laffont (1977): “Characterization of Satisfactory Mechanisms for the Revelation of the Preferences for Public Goods,” Econometrica, 45, 783–809. Groves, T. (1973): “Incentives in Teams,” Econometrica, 41, 617–633. Groves, T., and J. O. Ledyard (1977): “Optimal Allocation of Public Goods: A Solution to the ‘Free-Rider’ Problem,” Econometrica, 45, 783–809. Hurwicz, L. (1972): “On Informationally Decentralized Systems,” in Decision and Organization: A Volume in Honor of Jacob Marschak, ed. by C. B. McGuire, and R. Radner. North-Holland, Amsterdam. (1979): “Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points,” Review of Economic Studies, 46, 217–225. Kolm, S.-C. (1970): La Valeur Publique. Dunod, Paris. Maskin, E. (1999): “Nash Equilibrium and Welfare Optimality,” Review of Economic Studies, 66, 23–38. Milleron, J.-C. (1972): “Theory of Value with Public Goods: A Survey Article,” Journal of Economic Theory, 5, 419–477. Moulin, H. (1986): “Characterizations of the Pivotal Mechanism,” Journal of Public Economics, 31, 53–78.
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Muench, T. J. (1972): “The Core and the Lindahl Equilibrium of an Economy with a Public Good: An Example,” Journal of Economic Theory, 4, 241–255. Olson, M. (1965): The Logic of Collective Action. Harvard University Press, Cambridge, MA. Palfrey, T. R., and H. Rosenthal (1984): “Participation and the Provision of Discrete Public Goods: A Strategic Analysis,” Journal of Public Economics, 24. Roberts, K. (1979): “The Characterization of Implementable Social Choice Rules,” in Aggregation and Revelation of Preferences, ed. by J.-J. Laffont. North-Holland, Amsterdam. Saijo, T. (1991): “Incentive Compatibility and Individual Rationality in Public Good Economies,” Journal of Economic Theory, 55, 203–212. Saijo, T., and T. Yamato (1999): “A Voluntary Participation Game with a Non-excludable Public Good,” Journal of Economic Theory, 84, 227–242. Samuelson, P. A. (1954): “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, 36, 387–389. Satterthwaite, M. (1975): “Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions,” Journal of Economic Theory, 10, 187–217. Thomson, W. (1999): “Economies with Public Goods: An Elementary Geometric Exposition,” Journal of Public Economic Theory, 1, 139–176. Walker, M. (1980): “On the Nonexistence of a Dominant Strategy Mechanism for Making Optimal Public Decisions,” Econometrica, 48, 1521–1540. (1981): “A Simple Incentive Compatible Scheme for Attaining Lindahl Allocations,” Econometrica, 49, 65–71. Zhou, L. (1991): “Impossibility of Strategy-Proof Mechanisms in Economies with Public Goods,” Review of Economic Studies, 58, 107–119.
26
Z
Z
y
Z(−1)
Z(−1)
(−2)
Z
O2
Z(−2)
ω
O1
X2 X1
O1
O2
ω
(a)
(b)
Fig. I. A two-player example with F (T ) = T and ωi = 1/2. (a) Points on the simplex represent non-wasteful allocations. The ‘opt-out’ points z (−1) and z (−2) are derived from the suggested allocation z. (b) The face of the simplex is the familiar and useful Kolm triangle diagram.
Z(−1)
O
Z
O
1
2
ω = Z(−2)
Fig. II. Agent 1 pays a negative transfer under z, so the ‘opt-out’ point z (−1) has the same public good level as z. For agent 2, the ‘opt-out’ point is the initial endowment ω.
27
u2
u
u
2
u1
1
Z
Z
Z(−1)
Z(−1)
Z(−2)
Z(−2)
O1
ω
O1
O2
O2
ω
(a)
(b)
Fig. III. (a) The point z satisfies Binary Participation for both agents. (b) Binary Participation is satisfied only for agent 1.
u1
O1
ω
O2
Fig. IV. The set of balanced allocations satisfying Binary Participation for a given agent is comprised of three closed subsets; where T−i ≤ 0, where ti ≤ 0, and where both T−i and ti are positive.
28
y
6
ui (x; y)
¤º¡ µ ¡ ¤s
¢¸
z
P
¢¸¢ ¢s
j∈I
uj (x; y)
z∗
z y
(−i)
¤ºµ ¤0 ¡ ¡ s ¤º * ¤© s ©
z (−i)
F 0
ti
ωi
³P
j tj
´
-
T−i
xi
Fig. V. An example with quasi-concave utilities and convex production sets. The Pareto optimal point z satisfies Binary Participation. z ∗ is agent i’s most-preferred feasible allocation. The ‘opt-out’ point z (−i) satisfies the necessary condition for Binary Participation. The points z ∗ and z 0 satisfy the sufficient condition.
29
VI. The R Pareto optimal point y o exactly satisfies Binary Participation since yo 0 0 v (ζ)dζ ≥ y(−i) c (ζ)dζ is met with equality. If y (−i) were any larger, the inequality would y (−i) i fail. Fig. R yo
30
