Efficiency and Group Size in the Voluntary Provision of ... · retain the standard assumptions;...
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Efficiency and Group Size in the Voluntary Provision of Public Goods with Threshold Preference
Yoshito Funashima
Faculty of Economics, Tohoku Gakuin University
February 2020
Efficiency and group size in the voluntary provision of
public goods with threshold preference∗
Yoshito Funashima†
Faculty of Economics, Tohoku Gakuin University
February 5, 2020
Abstract
What is the optimal group size in the voluntary provision of public goods in a
purely altruistic economy? The popular consensus on this fundamental question is
that the free-rider problem worsens as group size increases. This study provides a
counterexample to the consensus, featuring threshold preferences that are plausible
for certain typical public goods. Under these threshold preferences, marginal utility
hardly diminishes below a threshold level, but declines significantly around the
threshold, becoming almost zero over the threshold. We show that the threshold
preferences significantly reduce inefficiency. We also show that if the marginal cost
increases, the threshold preferences are contrary to the consensus and lead to a
partly positive relationship between efficiency and group size, thereby detecting the
local efficient group size. Moreover, the local efficient group size is proportional to
the slope of marginal cost as well as the threshold of marginal utility.
Keywords: Threshold preferences, Voluntary provision of public goods, Group size
JEL classification: H41, D61
∗Financial support for this research from a Grant-in-Aid for Scientic Research by the Japan Societyfor the Promotion of Science (No. 17K03770) is gratefully acknowledged.
†1-3-1 Tsuchitoi, Aoba-ku, Sendai, Miyagi 980-8511, Japan; E-mail: [email protected]
1
1 Introduction
What effect the group size of a voluntary economy has on the degree of efficiency in the
private provision of public goods has been one of the fundamental questions in public eco-
nomics since the pioneering work of Olson (1965). To date, many researchers have studied
the issue and theoretical studies have reached a broad consensus that there is a mono-
tonically decreasing relationship between efficiency and group size in a purely altruistic
economy with noncooperative contributors, as shown by Andreoni (1988), among others.1
The theoretical consensus is robust regardless of the type of standard preferences.2
Although we often take such consensus for granted, in this study, we show that, in
particular cases, there is more to this issue than previously thought. In the early theo-
retical works, the possible preference features of public goods have not been sufficiently
addressed.3 Recognizing that there would be no appropriate preferences for all types
of public goods, it is all the more important to investigate how (in)efficient resource al-
locations can arise from every conceivable angle. Specifically, it cannot be ruled out a
priori that the diminishing degree of marginal utility depends to a greater extent on the
consumption level than usual. As part of the possible preferences, the present analysis
considers the case of individuals who have a threshold preference for public goods con-
sumption. Under this threshold, marginal utility hardly diminishes below a minimum
threshold, but declines significantly around the threshold, becoming close to zero over the
threshold; that is, unlike the standard case, utility increases in an approximately linear
fashion below the threshold and hardly increases over the threshold, as shown in Figure
1. We refer to this as individuals’ threshold preference (of public goods).
[Insert Figure 1 around here]
The threshold preference is in fact considered plausible in some cases of well-recognized
public goods such as charitable volunteer activities and clean environments. For example,
one could imagine a charity for natural-disaster victims as a social contribution issue. In
this case, the marginal utility of supportive activities could potentially decrease very little
1In this study, unless otherwise noted, we use “altruism” to mean the pure altruism in which individualshave a preference depending only on the total supply of the public goods. While beyond the scope of thisstudy, a branch of the literature considers impure altruism (warm-glow giving) in which individuals havea preference depending on their own contributions, including Harbaugh (1998) and Diamond (2006). SeeAndreoni (1990) for the details of an impure altruism case. In addition, while also beyond the scopeof this study, some research highlights the role of the cooperative behavior of contributors; see Wilson(1992), Pecorino (1999), Fehr and Gachter (2000), and Keser and Winden (2000), among others.
2See, e.g., Mueller (1989) for Cobb–Douglas preferences and Cornes and Sandler (1985) for quasi-linearpreferences. Isaac and Walker (1988) provide experimental support for the theoretical consensus.
3One important exception is Hayashi and Ohta (2007), who consider the satiation at some consumptionlevels of public goods, as detailed below.
2
at insufficient levels of total contributions, because the victims still tolerate the inconve-
niences of an unsettled lifestyle. Rather than assuming an immediately diminishing rate
of the marginal utility, it is more likely that the marginal utility substantially decreases
only after total contributions are sufficient to aid the victims. Once their standard of
living is sufficiently restored, the marginal utility could become virtually zero.
As another example, consider garbage-strewn beaches and a local beach clean-up effort
as an environmental issue. Irrespective of how much some people clear up the garbage,
the marginal utility of additional clean-up would remain high, as long as the remaining
garbage stands out, thereby spoiling an intrinsically beautiful landscape. Only after
beaches are restored to a satisfactory level of cleanliness, would the marginal utility finally
begin to noticeably decrease. Eventually, little marginal utility is derived from removing
the remaining very small and inconspicuous garbage.
In contrast to the preceding theoretical consensus, our analysis reveals that public
goods provision in the Nash equilibrium can lead to approximately efficient outcomes.
To this end, we consider a very simple and standard model in the literature assuming
an altruistic economy, except the utility function with the following three features: (a)
marginal utility is relatively slowly-diminishing when public goods are below the threshold,
(b) it is relatively sharply-diminishing only when public goods provision is in the vicinity
of a threshold value, and (c) it is almost zero when it exceeds the threshold. We emphasize
that these three features are involved with relative changes in the marginal utility and
retain the standard assumptions; that is, positive marginal utility and law of diminishing
marginal utility. Thus, our analysis is in line with the underlying framework in the
altruism literature, but it nevertheless reaches a conclusion that voluntary provision of
public goods can attain approximately efficient resource allocations.
In addition, we find that if the marginal cost of contributions by individuals is in-
creasing, the threshold preferences are contrary to the broad consensus that an increase
in group size inevitably makes efficiency worse. Although much of the previous litera-
ture assumes constant marginal cost, increasing marginal cost seems rather natural in
the provision of some public goods. For example, in the case in which the provision of
public goods involves some sort of physical tasks, contributors are gradually fatigued as
their contributions increase and they eventually become exhausted. The aforementioned
examples, charitable volunteer activities and environmental cleaning, apply to this case.4
On the basis of this natural assumption, we demonstrate that the threshold preferences
lead to a partly positive relationship between efficiency and group size, detecting the local
efficient group size. Within the confines of simple noncooperative behavior of contributors
in a purely altruistic economy, the present study is first to uncover the existence of the
4Other examples of the increasing marginal cost can be found in Hayashi and Ohta (2007).
3
local efficient group size.
This study is at the crossroads of two lines of research. One is the work raising the
issue of efficiency–group size nexus in the voluntary provision of public goods in a purely
altruistic economy, with certain representative studies already mentioned above. Among
others, Hayashi and Ohta (2007) deserves special mention as an important precursor to
this study. The authors took a step in this direction by positing two assumptions: in-
creasing marginal cost of voluntary provision and the existence of a finite satiety point
in utility. As a result, they claim that inefficiency is alleviated as group size increases,
and optimality is achieved when group size approaches infinity. While their work overlaps
with ours, there are notable differences in both assumptions and findings. Specifically,
we assume threshold preferences instead of a finite satiety point in utility and find sub-
stantial improvements in efficiency when group size is not infinite. The characteristics of
threshold preferences are similar to the assumption of satiation in that the threshold level
of provision could be interpreted as roughly corresponding to the satiation level; how-
ever, there are crucial differences between the two—in threshold preferences, marginal
utility is slowly-diminishing below the threshold level and sharply-diminishing around the
threshold level. In addition to these differences in preferences, we relax the assumption
of the increasing marginal cost, in which the marginal cost is required to approach zero
as contributions by individuals approach zero.
The second line of research is characterization of types of public goods (e.g., pure
and impure public goods). Traditionally, there are various types, such as congestible
goods and local public goods. In this line of research, this study is most closely related
to the recently growing body of literature on the so-called threshold public goods (e.g.,
Cadsby and Maynes, 1999; Spencer et al., 2009; Corazzini et al., 2015; Li et al., 2016;
Brekke et al., 2017). While much of the literature exploits experimental approaches to
investigate a public goods game, threshold public goods are characterized as those that
are consumable only when total contributions surpass a minimum threshold (provision
point). Thus, threshold public goods and the present threshold preferences are analogous
in that total contributions have meaningful threshold values; however, they critically differ
from each other in that marginal utility is positive only after total contributions exceed
a critical level in threshold public goods whereas it is sizable only when below a critical
level in threshold preferences.
The rest of this paper is organized as follows. In section 2, we formally define threshold
preferences and present our model. In section 3, we analytically examine the relationship
between efficiency and group size. In section 4, we provide further results using numerical
analysis. Concluding remarks are presented in section 5.
4
2 Basic theory
2.1 The model
Consider an economy in which n identical individuals exist, with n ∈ (1,∞). Let x and
G denote a private good (numeraire) and a public good, respectively. The individuals are
considered to have preferences that are represented as a quasilinear utility function
U = U(x,G) = x+ f(G), (1)
where f(G) satisfies the standard assumptions, and is strictly increasing and concave,
f ′ > 0 and f ′′ < 0. The marginal rate of substitution is given by
π(G) =∂U/∂G
∂U/∂x= f ′(G). (2)
As explained in our introduction, we assume threshold preferences in which the marginal
utility, ∂U/∂G = f ′(G), is rapidly diminishing around a threshold level of public goods
γ. Staying coordinated with the above standard features of utility function, threshold
preferences are defined in the following way.
Definition 1. The utility function U(x,G) is called a threshold utility function if the
marginal utility, ∂U/∂G = f ′(G) > 0, is a twice continuously differentiable function with
f ′′(G) < 0 for all G ∈ [0,∞), and if it satisfies the following properties:
(i) limG→∞ f ′(G) = 0, and limG→∞ f ′′(G) = 0.
(ii) f ′(G) has a unique inflection point (γ, f ′(γ)); that is, f ′′′(G) = 0 if and only if G = γ.
(iii) f ′(G) is a strictly concave (convex) function when G < γ (G > γ); that is, f ′′′(G) ≷ 0
if and only if G ≷ γ.
(iv) An arbitrary small neighborhood of γ, (γ − γl, γ + γh) exists, such that
0 ≲∣∣∣f ′′(G)
∣∣G/∈(γ−γl, γ+γh)
∣∣∣ ≪ 1 ≪∣∣∣f ′′(G)
∣∣G=γ
∣∣∣, (3)
where γl, γh > 0, and f ′′(G)∣∣G=γ
is a global minimum.
A public good is supplied by individuals’ voluntary contributions, and the total con-
tributions G are the sum of all individuals’ voluntary contribution g (i.e., G =∑
g). An
individual’s endowment w is divided into two parts: private consumption x and the cost
of producing their contribution to a public good c(g). Hence, each individual faces the
budget constraint
w = x+ c(g). (4)
5
As the property of cost function c(g), we consider the two cases of the marginal cost c′(g).
The first case is a standard assumption in the literature, that is, the marginal cost is
positive and constant (i.e., c′(g) = θ > 0). In the second case, as in Hayashi and Ohta
(2007), the marginal cost is assumed to be increasing (i.e., c′′(g) > 0). In what follows,
to examine the relationship between group size and efficiency, we assume the following
conditions for each case.
Assumption 1. If c′(g) is constant θ, then 0 < θ < f ′(0).
Assumption 2. If c′(g) is increasing and c′′(g) > 0, then 0 ≤ c′(0) < f ′(0).
While both assumptions are required to guarantee that solutions are interior, we make
some remarks on Assumption 2, which is related to Hayashi and Ohta (2007). To present a
counterexample to the inverse relationship between group size and efficiency, Hayashi and
Ohta (2007) impose a restriction on increasing marginal costs, such that they approach
zero as g → 0 (i.e., c′(0) = 0). Note that Assumption 2 encompasses this restriction and
allows for the positive values (i.e., c′(0) > 0).
2.2 Nash equilibrium and efficient resource allocation
The present problem with Nash behavior of an individual is to maximize (1) subject to (4)
by choosing x and g, given contributions by others. Noting that our model is an identical
economy, the following equation is satisfied in Nash equilibrium
f ′ (G)= c′ (g) , (5)
where G = ng, and upper bars represent the equilibrium values. In preparation for the
following analysis, it is useful to consider the group size elasticity. Totally differentiating
(5) with respect to G and n yields the group size elasticity of equilibrium provision:
dG
dn
n
G=
c′′ (g) /n
c′′ (g) /n− f ′′(G) . (6)
On the other hand, solving the problem to maximize (1) subject to (4) by choosing x
and G, we obtain the Samuelson condition:
f ′ (G∗) =c′ (g∗)
n, (7)
where G∗ = ng∗, and asterisks represent optimal values. As before, totally differentiating
(7) with respect to G∗ and n yields the group size elasticity of optimal provision:
dG∗
dn
n
G∗ =c′′ (g∗) /n+ c′ (g∗) /G∗
c′′ (g∗) /n− nf ′′ (G∗). (8)
6
3 Efficiency and group size
We now examine the discrepancies between Nash and optimal provisions under the thresh-
old preferences. We begin by examining the case in which marginal costs are constant.
3.1 Constant marginal costs
In the case of constant marginal costs, the right-hand side (RHS) of (5) is θ and that of (7)
is θ/n. Thus, G∗ increases as n increases, whereas G does not depend on n; accordingly,
an inverse relationship exists between group size and efficiency. This inverse relationship
can also be confirmed in terms of group size elasticity. Since c′′ = 0, the Nash group
size elasticity in (6) becomes zero. In contrast, the optimal group size elasticity in (8) is
−θ/ [nG∗f ′′ (G∗)], while it takes positive values and converges zero as n → ∞.
[Insert Figure 2 around here]
Thus, G∗−G inevitably increases as n increases due to the law of diminishing marginal
utility (f ′′(G) < 0); however, given the group size n, the threshold preferences could
alleviate the inefficiency of the private provision of a public good. To confirm our intuition,
we depict the representative shape of (5) and (7) as Figure 2, in which Panels A and B
respectively show the cases of standard and threshold preferences. Under the standard
preferences, marginal utility f ′(G) immediately decreases, and the left-hand side (LHS)
of (5) and (7) has a markedly negative slope even when G is small. By contrast, in the
case of the threshold preferences, the slope of marginal utility is flatter for small values
of G and steeper around G = γ than in the case of the standard preferences. As a result,
given the group size n, the Nash equilibrium provision G increases and approaches the
optimal provision G∗. In particular, when the curvature of marginal utility approaches
infinity, both G and G∗ converge to the threshold level γ, as shown by the dashed line in
Panel B of Figure 2.
In summary, the following proposition is obtained.
Proposition 1. Assume that marginal costs are constant and Assumption 1 holds. Then,
while G∗ − G increases as n increases, it decreases as the curvature of the threshold
preferences becomes large.
3.2 Increasing marginal costs
Turning to the case of increasing marginal costs, the slope of the RHS of (5) can be
represented as c′′(G/n)/n and decreases as n increases. It follows that G is strictly
7
increasing in n, unlike the case of constant marginal cost. The slope of the RHS of (7) is
c′′(G/n)/n2 and smaller than that of (5) for all G.
[Insert Figures 3 and 4 around here]
Even when assuming increasing marginal cost, the introduction of the threshold pref-
erences could make the voluntary provision less suboptimal. As before, Figure 3 shows
the representative shape and illustrates how the introduction of the threshold preferences
alleviates suboptimality. Panels A and B depict the standard and threshold preferences,
respectively. The figure exemplifies that if we assume the threshold preferences, then G
increases and G∗ decreases. It should be noted that in contrast to the case of constant
marginal costs, this exemplification is not universally applicable for any n. Specifically,
consider a case in which n is small and the slopes of the RHS of (5) and (7) are steep, as
shown in Figure 4. In the figure, a red line represents the LHS in threshold preferences.
In this situation, G∗ −G increases as the curvature of the threshold preferences becomes
large. The main results are summarized in the following proposition.
Proposition 2. Suppose that marginal costs are increasing and Assumption 2 holds.
Then, G∗ −G decreases as the curvature of the threshold preferences becomes large, pro-
vided that n is not small.
We next investigate how the inefficiency G∗ − G relates to the group size n. First,
noting that limn→1G = limn→1G∗, we denote the limits as Ginf . It is obvious that if we
consider a situation such that Ginf ≥ γ, then G∗ −G is monotonically increasing with n.
To exclude the monotonicity case, we assume the following condition:
Assumption 3. If c′(g) is increasing and c′′(g) > 0, then Ginf < γ.
Assumption 3 implies that the threshold level γ is not an extremely small amount.
[Insert Figure 5 around here]
Importantly, whileG∗ is larger thanG for any n, G∗−G is not monotonically increasing
with n, provided that Assumption 3 is satisfied. The reason behind the non-monotonicity
is straightforward, and we can understand this by considering the potential three stages
of group size. In the first stage, where group size is small (G ≪ γ) and f ′′ ≃ 0 as shown
in Panel A of Figure 5, we can respectively reformulate the group size elasticity in (6)
and (8) as
dG
dn
n
G≃ 1, (9)
dG∗
dn
n
G∗ ≃ 1 +nc′ (g∗)
G∗c′′ (g∗). (10)
8
It evidently follows that the optimal group size elasticity is larger than the Nash group
size elasticity, and G∗ −G is increasing with n when n is small.
In the second stage, where group size n is large to some extent and G∗ reaches the
values around the threshold level γ as shown in Panel B of Figure 5, the optimal group
size elasticity in (8) becomes approximately zero because f ′′(G∗) takes large negative
values. On the other hand, G is smaller than γ and the Nash group size elasticity in (6) is
positive. In this case, the Nash group size elasticity is larger than the optimal group size
elasticity, and G∗ − G is decreasing with n. In a limiting case, as shown by the dashed
line in Panel B of Figure 5, when the curvature of marginal utility approaches infinity,
the Nash equilibrium provision and the efficient provision converges.
In the last stage, as shown in Panel C of Figure 5, when group size n is sufficiently
large so that c′′ (g∗) /n ≃ 0, and f ′′ (G∗) ≲ 0 so that nf ′′ (G∗) takes finite values,
dG
dn
n
G≃ 0, (11)
dG∗
dn
n
G∗ ≃ − c′ (g∗)
G∗nf ′′ (G∗). (12)
This suggests that when group size n is sufficiently large, G∗ −G is increasing with n.
The second and third stages indicate that a U-shaped relationship exists between
G∗ − G and n, meaning the existence of local efficient group size n∗. Thus, we establish
the following proposition:
Proposition 3. Suppose that marginal costs are increasing and Assumptions 2 and 3
hold. Then, a unique n∗ that locally minimizes G∗ −G around G = γ exists.
This result is striking and has some important implications. A monotonically inverse
relationship between efficiency and group size in the private provision of public goods is
a widely accepted result; that is, a large group size aggravates the free-rider problem.
However, Proposition 3 indicates that even if group size becomes large, the efficiency
could be better. This implies, for example, that there is suboptimal population size n∗ of
nations or local municipalities, when some national or local public goods are voluntarily
provided.
[Insert Figures 6 and 7 around here]
It is intuitive that n∗ depends on γ. Figure 6 illustrates the effect of an increase in
γ. In this case, f ′(G) locus shifts right by the amount of the increase in γ from γ1 to γ2.
When γ = γ1, G∗ −G is G∗
1 −G1 and small. When γ = γ2, G∗ −G becomes G∗
2 −G2 and
larger; in this case, G∗ − G could substantially decrease by increasing n. This suggests
9
that n∗ becomes larger as γ increases. Moreover, n∗ depends on slopes of marginal costs,
c′′(g). Figure 7 exemplifies the case of increases in the slopes of marginal costs. Due to
the increases, both G∗ and G decrease, and there is room to decrease G∗−G by increasing
n. In other words, n∗ becomes larger as the slopes of marginal costs increase. Thus, from
Proposition 3, the following corollaries are immediately apparent.
Corollary 1. Suppose that marginal costs are increasing and Assumptions 2 and 3 hold.
Then, n∗ is proportional to γ.
Corollary 2. Suppose that marginal costs are increasing and Assumptions 2 and 3 hold.
Then, n∗ is proportional to c′′(g).
4 Quantitative analysis
The previous section analytically shows that threshold preferences could allow the allevi-
ation of inefficiency in the voluntary provision of public goods. In particular, if we assume
threshold preferences and increasing marginal cost, the general consensus undergoes mod-
ification. That is, the monotonical increasing relationship between inefficiency and group
size no longer holds.
In this section, we undertake several numerical analyses of the model, focusing on the
case of increasing marginal costs. Our purpose here is to illustrate the qualitative effects
of threshold preferences on inefficiency and to present further results that are analytically
ambiguous. For example, although the analysis thus far has explained the existence of
local efficient group size n∗, we still lack an understanding of the extent to which the local
efficient level deviates from the global efficient level. First, we consider the specification
of threshold preferences.
4.1 Parameterization
Up to this point, f(G) is not a specific function. To study this numerically, we now specify
threshold preferences. Note that, as shown below, the marginal utility of the following
specified utility function has the form whose graphs are seen to be mirror images of the
graphs of the logistic function with reference to the vertical axis. For this reason, we call
this an axisymmetric-logistic utility function, which has the following form.
Definition 2. An axisymmetric-logistic utility function is defined as
U(x,G) = x− α
βln (1 + exp [−β(G− γ)]), (13)
with α, β, γ > 0.
10
Lemma 1. Consider the axisymmetric-logistic utility function. Then, it holds that
(i) the marginal utility, ∂U/∂G, is a twice continuously differentiable function and satisfies
the law of diminishing marginal utility.
(ii) The marginal utility has a unique inflection point (γ, α/2).
(iii) If G < γ (G > γ), the marginal utility is a strictly concave (convex) function.
Proof. The law of diminishing marginal utility can be straightforwardly confirmed, such
that
∂U
∂G=
α exp [−β(G− γ)]
1 + exp [−β(G− γ)]> 0, (14)
∂2U
∂G2= − αβ exp [−β(G− γ)]
(1 + exp [−β(G− γ)])2< 0. (15)
Moreover, since we have
∂3U
∂G3=
αβ2 exp [−β(G− γ)] (1− exp [−β(G− γ)])
(1 + exp [−β(G− γ)])3⪌ 0 ⇔ G ⪌ γ, (16)
it is obvious that a unique inflection point of ∂U/∂G is (γ, α/2).
[Insert Figure 8 around here]
Lemma 1 means that this felicity function has the properties stated in Definition 1
of the threshold preferences. β and γ are key parameters characterizing the threshold
preferences. β determines the curvature of the marginal utility, and γ represents the
threshold levels. To confirm these, Figure 8 plots (13) and (14) when α = 1 and γ = 5
together with some values of β. When β = 1, the curvature of the utility function appears
to be relatively mild, the marginal utility declines across the board values of G. If we
examine the case of β = 2, the marginal utility declines at a narrower range centering
around G = γ = 5. When β = 10, the utility function is almost linear except the vicinity
of G = γ = 5, and accordingly, the marginal utility exhibits a sharp decline around the
threshold level. Thus, we have the next lemma.
Lemma 2. Consider the axisymmetric-logistic utility function. Then, when β is large,
this function is regarded as a form of threshold utility function.
To present quantitative analysis, specification of c(g) is also required. We then specify
the cost for the voluntary contribution to be quadratic
c(g) =ϕ
2g2 + ηg. (17)
11
From (5) and (7), the voluntary and optimal provision are respectively obtained by
solving the following equations
α exp[−β(G− γ)
]1 + exp
[−β(G− γ)
] = ϕg + η, (18)
α exp [−β(G∗ − γ)]
1 + exp [−β(G∗ − γ)]=
ϕg∗ + η
n. (19)
Since these cannot be solved analytically, we present the results on numerical solution
below.
There are five parameters characterizing the equilibria. From now on, unless otherwise
noted, we set α = 1, β = 1, and γ = 100 for the threshold preferences; similarly, ϕ = 1
and η = 0 for the cost function.
4.2 Effects of threshold preferences
Our first step is to provide an illustration of the basic result that the suboptimality is
improved when individuals have threshold preferences. Figure 9 shows the relationship
between group size n and inefficiency G∗ − G for various parameters of β. Note that
the figure is a double logarithmic plot, and the minimum of n is set to 2. Consistent
with Proposition 2, G∗ −G shrinks considerably when β is large, on the whole. When n
is small and less than approximately 10, G∗ − G rises. However, when β is large, such
rises appear much less pronounced. Furthermore, consistent with Proposition 3, when
β = 10−1, 100, and 101, a unique n∗ (local efficient group size) exists between 102 and
103. More importantly, in addition to these illustrations of Propositions 2 and 3, we find
that when β = 101, n∗ is not local minimum but a global minimum.
[Insert Figure 9 around here]
According to Corollary 1, higher threshold levels cause local efficient group size to
also be higher. Figure 10 illustrates this point by plotting the relationship between n and
G∗ − G for various γ. By definition, the degree of n∗ becomes higher as γ gets higher.
A striking pattern that emerges from the figure is that G∗ −G undergoes a much larger
change for higher γ; consequently, in contrast to the widespread consensus, a large group
size could substantially improve the efficiency of the private provision of public goods.
[Insert Figure 10 around here]
Similarly, according to Corollary 2, steeper slopes of marginal cost cause local efficient
group size to be also higher. In fact, Figure 11 shows that the degree of n∗ becomes higher
12
as ϕ increases, illustrating the result in Corollary 2. Compared with changes in γ (Figure
10), maximum values of G∗ −G are less influenced by changes in ϕ.
[Insert Figure 11 around here]
Finally, Figure 12 explores their relationships for various η. As expected, we notice
the qualitatively similar results to the case of ϕ, but the effects of η appear quantitatively
negligible. In comparison to Hayashi and Ohta (2007), what is more noteworthy is that
the outcomes are less sensitive to c′(0) (i.e., η in the present case). In Hayashi and Ohta’s
(2007) framework, the assumption that c′(0) = 0 is indispensable for achieving a notable
conclusion that the inefficiency converges zero as the group size approaches infinity. In
contrast, our results are robust to the situation in which c′(0) > 0.
[Insert Figure 12 around here]
Overall, the numerical results consistently exhibit that if individuals have threshold
preferences, the increasing relationship between group size and inefficiency is different
depending on group size; comparing the increasing phases, aggravation of efficiency is
more severe when group size is small than large. This offers a new insight that the
free-rider problem becomes less serious when group size is large rather than small.
5 Conclusion
Since Samuelson’s (1954) influential article, most undergraduate public economics text-
books state that public goods are underprovided in static games with voluntary contri-
butions and inefficiency arises in a general context. Moreover, there is now a general
consensus in the existing literature that the relationship between inefficiency and group
size is monotonically increasing. Although there is no doubt of the validity of such consen-
sus in general, this study has shed new light on this fundamental issue in particular cases.
In other words, there are plausible cases in which the inefficiency can be substantially
lessened and the monotonical increasing relationship is broken. To present the results, we
analyzed standard models except for a newly proposed preference of individuals, called
threshold preference. Although the proposed utility function seems plausible in some
types of public goods and satisfies standard assumptions such as the law of diminishing
marginal utility, it nevertheless alleviates inefficiency. Furthermore, if we additionally as-
sume increasing marginal cost, which also seems plausible in some cases, a local efficient
group size is shown to exist, in contrast to the general consensus.
13
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15
Utility
Threshold Public goods
Standard preference
Threshold preference
Figure 1: Threshold preference
16
A
(0) RHS (Nash)
RHS (Optimal)
LHS(0)
(0)
B
(0) RHS (Nash)
RHS (Optimal)
LHS(0)
(0)
Figure 3: Case of increasing marginal costs
18
A
(0)
RHS (Nash)
RHS (Optimal)
LHS
(0)
(0)
B
(0) RHS (Nash)
RHS (Optimal)
LHS
(0)
(0)
C
(0)
RHS (Nash)
RHS (Optimal)
LHS
(0)
(0)
Figure 5: Three phases in case of increasing marginal costs
20
(0) RHS (Nash)
RHS (Optimal)
LHS
(0)
(0)
Figure 6: The effects of increases in threshold levels
(0) RHS (Nash)
RHS (Optimal)
LHS
(0)
(0)
Figure 7: The effects of increases in slopes of marginal costs
21
0 2 4 6 8 10−5
−4
−3
−2
−1
0
U
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
∂U/∂
G
G
β = 1
β = 2
β = 10
Figure 8: Axisymmetric-logistic utility function
22
100
102
104
106
108
1010
1012
10−1
100
101
102
103
104
105
106
G∗−G
n
β = 10−4
β = 10−3
β = 10−2
β = 10−1
β = 100
β = 101
Figure 9: Group size and suboptimality for various β
100
102
104
106
108
1010
1012
100
101
102
103
104
105
106
G∗−G
n
γ = 101
γ = 102
γ = 103
γ = 104
γ = 105
γ = 106
Figure 10: Group size and suboptimality for various γ
23
100
102
104
106
108
1010
1012
10−3
10−2
10−1
100
101
102
103
G∗−G
n
φ = 10−2
φ = 10−1
φ = 100
φ = 101
φ = 102
φ = 103
Figure 11: Group size and suboptimality for various ϕ
100
102
104
106
108
1010
1012
100
101
102
G∗−G
n
η = 0
η = 0.1
η = 0.2
η = 0.3
η = 0.4
η = 0.5
Figure 12: Group size and suboptimality for various η
24