structure of the existing models of the provision of international public goods is identical to that ... on the amount of contributions to international public goods.
Chapter 1
The Voluntary Provision of Public Goods: A Refined Synthesis Introduction Since the seminal study by Olson and Zeckhauser (1966). the provision of internationd public goods has been. bot h t heoretically and empirically, modelled as voluntary contributions by individual countries to the *societyo'of nations.'
This is n a t u r d because. unlike within a
domestic society. there is no world authority which can legitimately coerce national governments to contribute. Moreover. there are several examples that would seem to fit the voluntary contribution model. including multilateral foreiga aid, military expenditures in defense alliances and pollution abatements in trans-border environmental issues. This chapter provides a refined synthesis of the literature and sets the stage for the following two chapters in Part 1 which study the voluntary provision of international public goods. Note that the analysis in this chapter also bears a direct relevance to the standard model of the private provision of public goods where households are the decision units which voluntarily contribute to (domestic) public goods (e.g.. Bergstrom et a l , 1984; Cornes and Sandler, 1985). The formd structure of the existing models of the provision of international public goods is identical to that of the standard model: the contributing national governments and/or economies are typicdy treated as if they were single representative agents like households in a domestic society. As 'Se, for example, the referred studies in Sandler (1992, 1993).
such. this chapter in effect concem the literature on the private provision of public goods in general. It would be useful to articulate what the present chapter is not concerned with. First. we
are ignoring the fact that nations are comprised of populations of citizens. and presumably a national government takes account of the joint benefits ôccruing to their citizens when deciding on the amount of contributions to international public goods. Chapter 2 will tackle this issue by constructing a mode1 which assumes more-than-one identical individuals in a country and a government which contributes to an international publie good so as to rnaximize the welfare of its citizens. Second. throughout Part 1, we o d y consider the cases where a public good in question is "pure". That is. the good in question is perfectly non-rival and nonexcludable, and the source of contributions is irrelevant in enjoying its benefits. We thus exclude two cases which have been discussed in the related literat ure. (i) where a country obtains additionai private benefits out of the very act of donating its own resources (Andreoni, 1988;Cornes and Sandler. 1984.
1994). and (ii) where a unit of contributions by others is less than a perfect substitute for a unit of own contribution (Ihori. 1992. 1994). Third. maay of the analytical results in this chapter apply only to countries who are contributors in an equilibrium. However. it should be noted that there are cases where there is an unexpected large number of non-contributing countries, and that such a pattern of public good contributions may require further anaiysis. While we consider this issue important and beIieve that some implications for it will be obtained in Chapters 2 and 3, we leave the issue as a topic for our future research.
Meanw hile. the current st udy adopts the standard assumpt ion t hat countries behave as
single representative agents and that public goods are pure, since it is intended to be a refined synthesis of the existing literature. Given this objective of this chapter, our aim is largely expositional. and we claim no great novelty for the results presented here. However, we do refine some neglected aspects of the literature and derive several resdts which are instructive by themselves and will be instrumental in the ensuing chapters in Part 1. Among others, the following points are worth mentioning. First, we formally establish that equilibrium contributions will be disproportionate. That is.
a contributor. say. with a large income will bear a disproportionately larger share of the cost of public good contributions. It is rather surprising that this result had not been formally treated for decades until Weber and Wiesmeth (1991) tackled the issue, since it has been considerd to be one of the important themes in the studies of international public goods (Sandler. 1992). However. given the fact that Weber and Wiesmeth (1991) draws on a rather specific setup to show the result .* it should still merit demonstrating that the disproportionate contribution still holds in a more generai e n ~ i r o n m e n t . ~ Second. we provide a novel analytical tool which we cal1 an aggregate response function. This enables us to translate a heterogeneous n-agent interaction into a pseudo two-agent interaction. i.e.. interaction between an agent in question and the aggregate of n - 1 ot hers. By empioying this device. we will demonstrate that comparative static analysis can compactly be conducted in a quite general n-agent environment. Its merit will also be demonstrated when we extend the tweagent Stackelberg model of public-good contributions by Bruce (1991) and Varian (1994). This extended model will constitute one of the theoretical models to be used in our empirical analysis in Chapter 3. Third. we delineate precise factors which influence what could be called the eficiency para-
dox. In a linear-technology environment that we will consider, an improvement in efficiency is a reduction in the .price" or constant marginal cost of contributing one unit of a public good. This sort of efficiency consideration is especiaily relevant to multi-jurisdictional settings since decision-making units. Say countries, are likely to have dzerent technologies or prices" of public goods. It can be shown that. when a country improves its efficiency in contributing to a public good. it may or rnay not be better off in a new equilibrium. This yields a prospect of the efficiency paradox: a welfare decrease in the country where contribution efficiency improves. Since Schulyer (1982) 6 r s t pointed out the paradox, Jack (1991) and Ihori (1996) have made some advancement. However, they are not able to precisely provide ail of the important factors which constitute the paradox. We will both complete the analysis of the paradox and provide an accompaqing intuitive explanation by taking advant age of the aggregat e response function.
The composition of this chapter is as follows. In Section 2, we pedagogicaily delineate the 'For the setup of Weber and Wiesmeth (19911,refer to their Assumption 2.1 (p.185). 'We wil1 further eiaborate on the disproportionate burden sharing in Chapter 2.
standard mode1 of the private provision of a pure public good, and highlight three important propositions which will be instrumental in the analysis that follows. Section 3 re-examines the properties of the Nash equilibriun of voluntary contributions, and provides further andysis on the disproportionate contributions and income transfer among contributing countries. Section 4 synthesizes the comparative statics analysis. We tidy up the results in a very general form by
introducing the device of aa aggregate response function. We also provide an elaborate analysis of the efficiency paradox. Section 5. again utilizing the aggregate response function. ofFers a generalized version of Brucevarian andysis of public-good contributions under the Stackelberg allocation process. Section 6 concludes this chapter.
While Olson (1965) is regarded as the first substantial contribution to the theory of voluntary provision of public goods as Bergstrom et al. (1986) note, Olson and Zeckhauser (1966). an application of Olson (1965) to the analysis of international public goods. have founded the standard formulation of the model we are familiar with. F'urther theoretical developments on the provision of pure public goods have b e n seen in a series of studies. Representative studies include Shibata (1971). Camberlin (1974. 2976). Schulyer (l982), Warr (1983). Cornes and SandIer (1985). Bergstrom et al. (1986). Andreoni (1988), Bruce (1991). Jack (1991). Weber and Wiesmeth (1991). Dion (1994). Varian (1994), Buchholz and Konrad (1994. 1995) and Ihori ( 1996).'
In this section. we set up the model of the voluntary provision of a pure public good. which encompasses a variety of issues developed in the studies above. As mentioned. we note that the following model has the formal structure w hich is applicable not only to national governments in an international **society"of countries but also to agents/members of any society :group. As such. to make the analysis more general. we thus drop "international," and use "agent" in place of "government" in the discussion t hat ensues in this chapter. ' ~ o t ethat these do not include the studies on the voluntary provision of impure public goods. Notable studies on impure public goods inctudes -4ndreoni (1990), Cornes and Sander (1984, 1994). Ihori (1992. 1994) and V i c q (1997).
1.2.1
The basic setup
Suppose an n-agent society where agent i has a utility function:
where xi and G respectively denote a private good and a public good. AS usual. we assume t hat Liz(-) is twice dserentiable and t hat its underlying preferences are strict ly convex. We
assume the standard summation technology for public good c o n s ~ m p t i o n . ~
where gi is agent i's provision of or contribution to the public good. Let us define G-i
xjFi
gj.
Then. by definition:
G = gi + G-i. Agent i has a constraint: wi = Xi f pigi
where xi serves as a numeraire. We may interpret this constraint as (i) a budget constraint with
wi being incorne and pi being the purchasing price of the pubLic good or as (ii) a production c0nstra.int with
wi
being a single resource endowment and pi being the constant marginal
transformation rate between xi and gi. We assume that pi's are generally different across the economic agents. as in the recent studies which focus on the effects of different +prices.'* This is especially important when we apply the mode1 to multi-jurisdictiond settings. where decision-making units. say countries. are likely to have different public-good technologies or 'The summation technology is not the only public-good consumption technology considered in the literature. Hirshleifer (1983) hst introduced two interesting specifications of consumption technotogy. One is cdled Weakest Link and defined as:
G= Another is termed as Best Shot and defined as:
See also Cornes (1991) who provides a further analysis of the former consumption technology. %ee Buchholz and Konrad (1994, 1995). Dion (1994), Ihori (1996), Jack (1991) and Schulyer (1982).
individual "prices." Zn a related note. we foilow Buchholz and Konrad (1994, 1995). and regard the reciprocal of pi as contribution eficiency.
In the case of international public goods. it may be usefd to distinguish between contribu-
tion in Ciind and contribution in cash, when considering the "price" of contribution. Typicai examples of the former includes military buildup of an alliance and pollution abatement. where an individual country directly provides the collective goods or services in question. Thus. in
the context of contribution in kind. pi is the (constant) marginal transformation rate in the usual sense which reflects difEerent production technology of a contributing country. The latter. contribution in cash. will be relevant when countries voluntarily send money as a fund to a cornmon agency which produces international public goods. A good example is the subscription of funds to multinational foreign aid or to the services of international organizations. In this instance. p, is interpreted to reflect the marginal costs of public funds. or simpIy tax prices.
in a contributing country since its national governrnent taxes their econornies and sends the proceeds as the fund for the provision of international public goods.
1.2.2 Individual optimization and the reaction function The standard mode1 assumes Nash behaviour of every agent. Under the Nash allocation process. by definition, agent i is assurned to believe that G-i is independent of his or her own contribution gi . His maximizat ion problem is t hen:
for a fixed value of G-,. The first-order conditions for this problem are: (
where ri(xi. G)
G
)p
i
(T'(z~,G)-~~)~~=O.
U;(xi, G)/u:(x~. G). For a contributor.
Pi = x8(xi,G)
must hold in an equilibrium. R o m the k t - o r d e r conditions, we derive the demand for the private good: xi = xi b i . wi. G - i ) , and the contribution function: gi = gi (pi.wi . G - i ) . If pi and
are held fVted while G d i is d o w e d to vary, gi (-) is considered to be i T s reaction function.
wi
The early studies focused on the properties of the reaction c w e . Chamberlin (1974) diagrammat ically establis hes the following proPo& ion.' Proposition 1 i f and only if both the private good and the public good are normal goods. the slope of agent i's reaction function is such that - 1
< dgi/dG-i 5 O .
Taking a different analytical root £rom Cornes and Sandler (1984), we prove the above. by establishing the following proposition that has not been articdated in the literature.
Proposition 2 The slope of the m c t i o n finction is equal to the negative of the income egect for the private good: agi/aG-i = -axi/duri-
Proof. Consider the ..full income' versions of maximization problem (P): max U1(xi. G) subject to I,.G
Rom this. the demand for G is obtained as
xi
+ piG
= wi
G-i
2 O-
r i(pi, wi + piG-i).
+ piG-*
On the other hand the de-
mand for G fiom the original problem is written as g i b i , wi, G - i ) g i ( P i . wi. G - i ) + G - i = r i ( p i .wi
+ G-i
which results in
Partid dserentiation of the equation with respect to
+ 6'gi/dG-i = piaI'i/dMi. Therefore. = piagi/awi. Moreover. f?om agent Z'S budget constra,.int. dxi/i/awi+piag,/dwi =
wi and G-i respectively yield d g i / h i = â r i / d M i and 1 1+
1 . which dong with the above results in: dgi/aG-i = -axi/awi. Then. Proposition 1 follows
Proposition 2. if both goods are normd (Le.. dxi/dwi > O and dgi/dwi > 0 ) . Another fundamental proposition concerns the characteristics of the marginal rate of substitution. Although the following proposition is fiequently referred to (e.g.. Cornes and Sandler.
1985). its formal elaboration is quite rare. We thus substantiate it as follows. ' ~ e ealso Cornes and Sandler (1984) for an analyticd proof. ' ~ h i sformulation is to Our knowedge attributable to Bergstrom et ai. (1986).
Proposition 3 If and only if both xi and G are normal.
a ~ ' / >aO ~and~ & r Ï / a G i< o.
Proof. Rewrite the 6rst-order condition for a contributor as u&(M~ - pic,G ) = p i ( l i ( -~ i
piG. G ) .where Mi
xi + p i G We then dinerentiate this condition with respect to G to obtain:
Noting X; = M i- p i e . we obtain:
By the second-order condition.
u,&
- 2piUC, + pi>^:, < O. Then. if âG/ôM and a x / a M are
> O and that (I&-piU& < 0. It is then straightforward to show that a * ' / d ~>~ O and d r i / a c i < O since air'/a~= (u&,positive (Le.. both goocis are normal). it follows that
p i U & = ) / ~and : aîri/azi = (U;, - p i U : , ) / ~ : .H
1.3
Nash Equilibrium and Its Properties
We consider a Nash docation process in an n agent interaction. A Nash equilibrium in the public good contributions is defined as an allocation {gi}:=l which simultaneously solves problem
(P) for each of the n agents.
1.3.1
Existence and uniqueness
Proposition 4 If both goods are normal. there is a unique Nash equiLzb7ium {gi}:=l with a unique quantity of a public good G and a unique set of contributing agents C.
The existence of a unique Nash equilibrium when prices are identical among agents (i.e.. pi = p V
2)
is originally proved in Bergstrom et al. (1986). Improved proofs are also provided
by F'raser (1992) and by Bergstrom et al. (1992). Therefore, there are no needs to replicate the proof here. For cases where prices p:s are different, Buchholz and Konrad (1994) provides the proof in a two-agent economy. However, such an additional proof is unnecessary and so is the assumption of a two-agent economy, since we can translate a different-price case into an
identical-price analysis. To see this. pick a p,. the price the r-th agent faces. as an arbitrary reference price. and defme gi
pr/pi,
ci
kxi. d i
ikwi
and v2(ci.gi
+ G-i) G Ui(ci/ki.
+ G-i) where constant ki is suppressed. Then, another quivalent expression for (P) follows:
Note that p, is cornmon for every agent. The demand functions are now
ci = ci(pp.
di.G-i)
and gi =gi(pT.ai.G - i ) . Given the definitions of the new variables. it is straightforward to show that aCi/dwi
> 0.
dgi/dwi
> O and O < ptdgi/8wi < 1. Hence, we can validly apply Bergstrom
et al. (1986) to prove the existence of a unique Nash equilibrium under different prices.
1.3.2
Welfare comparison in a Nash equilibrium
The propositions concerning welfare comparison in a subscription equiiibrium are rather striking. Note that. to make welfare comparison meaningful. we are assuming for Propositions 5 and
6 that the preferences axe identical among the agents. The f i s t result which assumes identical prices is by Bergstrom e t al. (1986).
Proposition 5 Even if their wealth endowments are diflerent. ail t h e contributors end u p with the s a m e level of welfare when p n c e s are identical.
Proof. The identical prices and the first order condition ensure that rk(z*. G) = r' (fr . G) for any d s e r e n t contributors k and 1 in an equilibrium. It is clear that Fi = Z V i . as preferences and
G are identical for al1 agents. which results in u ~ (G) ( I= , u'(z.G) H
Note t hat our proof above is much simpler t han the original one. Notice also t hat Proposition
5 implies that wk - wl = pijk -pgi: daerences in incomes will be reflected in equal differences in expenditures on the contributions. Therefore, incorne redistribution between the two wiU not change the total provision of the public good. nor wili it change the equilibrium utility levels. This is what has corne to be known as the neutrality theorem. a more generd proof of which will be explored in 1.3.4. The second proposition concerns the cases where contributing prices are different.
Proposition 6 A more e s c i e n t contributor ends up with the lower level of welfare than a less eficient contributor. regardless of the Ievel of indiviàtial wealth.
.
< pi. Then. (1.1) irnplies that f (zk, G) < $ (Zi. G) in an equilibrium. Proposition 3 then implies that Zk < Tl.Therefore. uk(tk, G) < U'(z~. G).
Proof. Assume
pk
Proposition 6 is found in Schulyer (1982) and Ihori (1986). both of which assume a two-agent economy. On the other hand. our proof is general enough to embrace an n-agent interaction and. above dl. is quite easy as shown above. Note that our proof does not reguire a specific assumption on individual wealth: the result holds even when a more efficient contributor has more income than the Iess efficient one. This is the point previous studies have not articulated.
1.3.3
Disproportionate contributions
Another interesting feature of the equilibrium is that equilibrium-contribution patterns are disproportionate (which is to be defined soon). As we will see. this implies that when each
agent is identical except his weaith. the burden of voluntary provision of a public good is in effect progressive. which has not been articulated in the literature. Disproportionate contributions have been considered to be one of the important themes in the literature (e-g.. Sandler 1992). exemplified by the cerebrated phrases like '-a surprising tendency for exploitation of the great by the s m a l r in OIson (1965) and "disproportionate burden sharing' in Oison and Zeckhauser (1966)~' Nonetheless, the literature has not provided
a complete and formal analysis of the disproportionate burden sharing. A partial exception is Weber and Wiesmeth (1991). who develop a rigorous restatement of Olson and Zeckhauser
(1966). Still. their anaiysis is somewhat incomplete. In their model, a special assumption is made on agent 's utility function which incorporates what they c d sire parumeter. which needs some further explanation why it is made a s such. Here, we demonstrate that disproportionate burden sharing can formally be established under a ceteris-paribus condition but without resorting to some special assumptions. Disproportionate burden sharing is usuaily characterized in terrns of one of the foilowing two measures. The first measure is the percentage of an individual's wealth devoted to his public' ~ ewill elaborate on this topic in Chapter 2.
good contribution. We c d this ratio the contribution ratio and denote it by y i
p i g i / w i for
agent i. The second measure is the ratio of an individual's share of total public-good expenditure to his share of the total wealth-endowment of the economy. We refer to this as the shure ratio and denote that for agent i by
ai
bigi/
C jp j g j ) / ( w i / C jwj).ùi the empirical literature
on military burden sharing. for example. Oison and Zeckhauser (1966) and Sancller and Forbes
(1980) employ the former. while Oneal (1990) utilizes the latter. The former
&O
accords with
Weber and Wiesmeth ( 1 9 91 ) . ln fact. it suf6ces to discuss only one of the two measures. This is simply because:
which explicit ly shows t hat a higher contribut ion ratio implies and is implied by a higher share ratio. We thus only employ the contribution ratio and Say that agent k contributes disproportionately more than agent 1 if y k > 71.
Proposition 7 In an economy characterized by ( P ) and at least one of the agents being compared as a contributor i n a Nash equilibriurn,
fi) a wealthier agent, ceteris paràbus. contn'butes
to the public good disproportionately more than a less wealthy agent; (ii) a n agent with a lower price for the public good. ceteris paribus. contributes to the public good disproportionately more than an agent with a higher price: und (iii) an agent with stronger preferences for the public good. ceteris paribus. contributes t o the public good disproportion~telymore than an agent with weaker preferences.
Proof. Assume identical preferences and prices for ( i ) . By Proposition 5 , Zk = Zl. if k and 1 are contributors. The budget constraint shows that wk - pgk = ut, w l ( l - p g l / w i ) . Therefore. if
ulk
> w l . pgk/wk = yk > pgl/wi
- pgl
or w k ( l - p g k / w k ) =
= yl. If one of them is a non-
contributor. the less wealthy must be the non-contributor since & = ( w i- Z)/p,which shows that the resuit still holds.
< pl, Proposition 6 shows that for contributors k and 1. then it follows that w - pkgk < w or .uk > yl. For Zk < the case where one of them is a non-contributor, assume that ( a ) pk < pl and that ( b ) k is a non-contributor while 1 is a contributor. (b) implies that pk > 7 r k ( 4 ,G) and pi = 7r1(zb C) Assume identical preferences and wealths for ( i i ) . If
pk
and that Zk > Zi since wk = wi. But the latter implies irk(zk. G) > d(zi, G) by Proposition 3. which results in pl < pk. contradicting (a). Therefore, 1 must be the non-contributor. Assume identical w and p for (iii). We characterize preferences in terms of the marginal rate of substitution: k prefers a public good more than 1, if *(xi, G)> T'(x~.G) for the same set of (xi.G). When both are contributors. p =
nk(zc,G)= ir'(~~,G). If k prefers G mcre
K~(Z~> . Gd) ( ~ ~ , ëThen. ). 9 yl. For the case where one of them is a non-contributor, assurne that (d) rk(ri.G) > pl ( z ~ G) . and t hat (e) k is a non-contributor while 1 is a contributor. (e) implies . and that. since wk = wi. l k > 21. But these two imply that rrk(zk,G) < p = ? r ' ( ~ ~G) that ?~*(T~.G) < rr'(~~.G) by Proposition 3 . which contradicts (d). Therefore. 1 must be the than 1.
non-contribut or. Notice that Proposition 5 and Proposition 7-(i) show that if the preferences of agents are identical and ail of the agents are contributors in an equilibrium. (1) everyone enjoys the same level of welfare Le.. the voluntary contribution is enuy-free in Foley's sense (Foley. 1966) and
(2) the ratio of payment for the contribution to income is increasing in wealth level. Le.. the voluntary contribution is progressive.
Indeed. it is highly progressive in the sense that the
difference in provisions by two contributors with different wealth levels is precisely equal to the difference in wealth. This seems not to have been emphasized in the üterature. These two results toget her show a striking feature of the subscription economy: when preferences and prices are identicai. the voluntnry provision of a public good is envy-free and progressive.
1.3.4
Income transfer and the neutrality theorem
Before we consider the effects of income transfers. it is convenient to note the following fundamental result which helps account for some of our new results:
Proposition 8 When an equilibr-ium public-good provision G increases (decreases) and prices each agent faces are held constant. the equilibrium priuate-good consumption f also increases (decreases) for each contributor. thereby makang al1 contributors better (worsej off.
Proof. Note that pi = rri ( z i .G ) must hold in an equilibrium for a contributor. With pi being held constant. it is straightforward to see:
By Proposition 3. it then follows that d%i/dG > 0.
.
Now we derive an instructive formula for examining the transfer problem. Consider the demand functions for G. r(pi.M i ) . For contributhg agents. î(.),being strictly increasing in
M i . can be inverted to yield h/li = 4; ( p i .G), or:
where C is the set of contributors. For non-contributors. G-i = G. Since G = r ( p i . @ , ( p i .G ) ) . we have: &bi/3G = (iX'i/a~%fi)-l = (agi/awi)-'. Assuming that a Nash equilibrium is in effect. summing the equations in (1.2) for d l contribut ors yields:
where c is the number of contributors. Differentiating this with respect to wi and G results in:
The coefficient of dG is positive. as we have established that O < p i a g i / a w i < 1. This expression should be instructive in examining the following propositions concerning the effects of income transfer. The first is the classic neutrality theorem by Warr (1983).
Proposition 9 If the prices of public good contributions are identical, the redistribution of individual wealth among contributors does not change either the total provision of the public good o r the equilibrium weijare of contributors.
Proof. Let pi = p Vi E C. Then. the right-hand side of (1.3) is now p-l tributing wealth results in
Cdui.
As redis-
duti = 0. (1.3) clearly shows that the post-tramfer equilibrium is
accompanied by no changes in the total provision (Le., d c = O). In addition, since p is h e d . the FOC t hat p = ai ensures that identical Z for any contributors after the redistribution. The second concerns cases where agents have different contribution efficiencies.
Proposition 10 If the pn'ces are not identical, redistribution o f wealth from a less eficient contributor to a more eficient contràbutor not only increases the total provision but it also produces a Pareto superior post-redistribution allocation.
Proof. Consider two contributors. k and 1 with pk < pl. Given the fact that pkldw-p;'dw > 0. ( 1.3) clearly
shows t hat redistributing wealt h from a less efficient contributor to a more efficient
contributor is accompanied by d c > O. Thus. by Proposition 8. we see that this transfer makes everyone better off. I Proposition 10 was 6rst infonnally suggested by Olson and Zeckhauser (1967. 1970) and Sandler and Shelton (1972). Later. Dion (1994) and Buchholz and Konrad (1995) formally demonstrated the resdt. Notice that these studies only deal with a two-agent model. while our proof is general enough to allow for a generai n-agent case.
1.4
Comparative Statics
In t his section. we will extend and generalize the comparative statics results in the literature. -4s almost always the case in any economic analysis. the literature shows a seemingly inevitable
trade-off between t heoret ical generaiity and analytical tractability. For example. Bergstrom et al. (1986) thoroughly examine the effects of difTerent private wealths but limit themselves by assuming identical preferences and prices. Jack (1991) and Ihori (1996) analyze the effects
of different prices on individual contributions with heterogeneous preferences but Iimit their analysis to a two-agent setup. although the latter includes a simplified version of an n-person case.''
On the other hand. a very general analysis of Dasgputa and Itaya (1992) requires so
elaborate an analysis that some might not find it very tractable. These studies exemplifj a typical example of trade-off between a theoretical generality and analytical tractability. This section. in the hope of reconciling this trade-off, introduces what we call an -'aggregate response function." This enables us to formally translate a heterogeneous n-agent interaction "hori (1996) assumes that al1 agents except the one whose pnce changes are identical.
15
into a pseudo two-agent interaction, Le.. an interaction between an agent in question and the aggregate of n - 1 others. As such. this enables us to conduct a quite general analysis without complicating the andytics of the rnodel. However, we hasten to note that we are excluding the cases where the set of contributors changes before and after relevant changes.
1.4.1
The aggregate response function
We now int roduce the aggregate response function, which shows how t o t d equilibrium contributions in the rest of the group respond to changes in contribution by agent k. First. aggregate (1.2) over all other contributors ( j # k ) to give:
This equat ion wiil be satisfied in a Nash equilibrium among all non-k countries for a given level
G-r. The aggregate response function G-k(9k)is then defined implicitly by (1.4) for given values of wj and pj V j # k E CC. of gk. Denote the solution to (1.4) by
Proposition 11 If an agent changes its contribution. the rest of the contributors in aggregate changes their contributions in the opposite direction but by less in absolute amount: -1 < dC-k/dgk < 0.
Proof. Differentiating (1.4) with respect to G - and ~ gk and using i%$,/aG =
(agj/&u,)-l
y ields:
In the proof of Proposition 2. we have s h o w that O < pjdgi/awi < 1 . Therefore. it follows immediately that -1 < d ~ - k / d< ~ 0.~ U
This property turns out t o be an important property as we see in the following analysis. Also, the following is wort h not ing:
Proposition 12 1j the nurnber of contributors is large enough, a change in an agent's contribution is perjectly offset by the change in the contributions by the rest of the economy: as c
+ 00.d
~ - ~ -% / -1. d ~ ~
Proof. This is evident fkom (1- 5 ) . 1 1.4.2
The effects of wealth changes
The following sub-sections conduct comparative statics analysis and demonstrates the rnerit of the aggregate response function. First. we examine the effects of incorne changes. Although its results are what we would easily expect. we List the propositions and provide the proofs thereof. for the completion of the analysis and for the demonstration of the merit of our method.
Proposition 13 After an increase (decrease) in wk, ( i ) the total provision level of the public good increases (decreases); ( i i ) every agent 2s better (worse)
08b y consuming more (less) of
every good: (iia') agent k contributes more (less): and (ui) each of the other agents contributes less (more). Bergstrorn et al. (1986) and Andreoni (1988) show (i) by assiiming identicaI preferences
and prices. The general analysis of Dasgputa and Itaya (1992) which also d o w s for conjectural variation shows (i) and (vi) in an n-agent model. But, with a straightforward use of matrix operation, it does not tend to be very tractable. If we are equipped with the aggregate response function. however. the above results can concisely be demonstrated even in a general setting (i.e.. n agents. heterogeneous preferences. different prices and uneven individual wealths).
Proof. In a Nash equilibrium. it must be the case for agent k that:
where gi, is the Nash equilibrium contribution of agent k. Totally dxerentiating (1.6) with
dpk = O and rearranging the remaining terms will yield (iii):
d G Y k / d g k> O . implying (i): d ~ = d dgk dwk dgk dwk
L
I I -
> 0.
.
Proposition 8 then immediately shows (ü). Last. Proposition 8 &O proves (vi). since for agent j
# k. it shows that d Z j / d w k = ( d i E j / d ~ ) ( d ~ / d w > kO )yielding d g j / d w k < O with
hed.
1.4.3
Wj
being
The effect of price changes
Proposition 14 After a decrease (increase) in pk, (i) the total provision level of the public
08;(iii) agent k's rueifare ( v ) agent j # k contributes less
good increases (decreases); ( i i ) agent j # k is always better (worse) change is uncertain: (iv) agent k contributes more (less): and (more).
Proof. We andogously differentiate ( 1 . 6 ) with respect to pk
since dgk/dpk may be assumed to be negative and the second factor is positive. This proves (iv). R e c d that d ~ / d &> O. which implies (i):
For agent j # k. Proposition 8 shows that dZj/dpk = ( d z j / d C ) ( d c / d p k )< O .
dÜ1/dpr < O (ii). Also. since
Wj
is fixed, dgj/dpk
>O
Hence.
( v ) . For agent k. the FOC results
in:
Note that Q k > O.
s i g n { d Ü k / d p k }(iii).
.
We thus cannot generaily determine s à g n { d z k / d p k } and accordingly
1.4.4 The efficiency paradox As shown above. for contributor k whose price pk changes, the direction of its welfare change is not unarnbiguous. There is thus a prospect of what may be called the eficiency pamdox,
a welfare decrease for a contributor with an efficiency irnprovement (Le., a decrease in price).
In what foilows. we improve upon Schulyer (1982), Jack (1991) and Ihori (1996) and provide
detailed conditions for the paradox, as weU as accompanying intuitive interpretations. in our n-agent set t ing.
Now consider agent k's equilibrium welfare. which can be expressed as:
where g k ( p k ) is the relationship between gk and
U k with respect
pk
impiicitly defined by (1.6). Differentiating
to pk and using the first-order conditions for a contributor yields:
This expression gives the change in welfare of agent k measured in terms of the numeraire private good. Substituting fkom (1.7) for dgkldpk. (1.8) can be rewritten:
Xoting that the Slutsky relation concerning the demand of G results in:"
and recalling t hat d g k 1aG-k = pk3gk /8wk - 1 . the expression for welfare change can compact ly be written as:
where e c
- ( a G / a p k l L ~ ) ( p k / G is ) the compensateci elasticity of demands for the public
good G with respect to price. s r g k / G is the agent's contribution share. and A
= (1 -
(agk/a~-k)(d~-k/dgk))/(gkv,) > O. Since O < ) d ~ - ~1 < / d1, ~the~ sign of (1.9) is ambiguous. Thus. an agent experiencing an efficiency improvement may be better off or worse off in the new Nash equilibrium. confirming Proposition l 4 ( i i ) .
However. our formula of ( 1.9) is more than a confirmation. It also establishes three limiting cases, which highiight the conditions when the efficiency paradox is Likely to occur. First. as I d ~ - ~ / + d ~1.~ the l right-hand side of (1.9) equals -eG/s
< O. It then follows. when the
aggregate contribution by the rest of the economy is responsive enough. the welfare of the efficiency improving agent wiil fall. R e c d . as c
+ m,
1
I d ~ - ~ /+d 1.~ ~ This suggest that
when the number of contributors is large enough. the efficiency paradox is Likely to occur. Second. as s
+ 0 . dÜk/dpk< O . This suggests that when the efficiency-improving agent
initially contributes very lit t le relative to the total, the efficiency improvement will reduce its welfare. The previous comparative statics results have established that. ceteris paribus. the more poor and inefficient an agent is, the less it contributes to the public good. This implies that the paradox is iikely to occur to a poor and inefficient contributor. Last. as EC + 0 . dÜk/dpk> O . Thus. if there is no substitutability between the public good
and the private good (i.e.. indifference curves were kinked), an efficiency irnprovement results in what we normaliy expect, a welfare improvement. In other words, welfare wiil be more likely to f d . the geater is the compensated elasticity of demand for the public good with respect to
a change in prices.
The above limit ing cases help us understand the intuition behind the paradox. Consider an decrease in pn. which is an improvement in k's efficiency. Initially there will be a substitution effect and an income effect due to this decreased effective price. With the contributions by the others being held constant. the substitution effect induces an increase in his contribut ion
gk
and a decrease in consumption of x t . The income effect a x k / d w k further increases gk whereas it also stimulates the consumption of xk which counteracts the substitution effect on xp. Then. consider the response of the others, d ~ - ~ / If d it ~ is~ very . sensitive. the initial increase in gk is offset by a decrease in G-k: G almost stays at the initial level however large gk may change (Le.. if dG-*/dgk = -1.
G is always held at the initial level). This forces the
pchanging actor to adjust to the new marginal condition almost only through the variation in xk. AS pk is now lower, he has to reduce xk in a new equilibrium under the normality assumption. Given the same amount of G . his equilibrium welfare is now reduced. hirthermore. if the agent has a very small share of the initial total provision (Le.. very small
.
value of sk ) the increase in gp would d e c t the total G very lit tle in the fùst place. Then again.
he in effect has to adjust to the new marginal condition almost only by changing x k : with being lower. must
&O
xk
pk
has to be reduced. With G being changed very Little, his equilibrium welfare
be reduced.
Last. if the compensated price elasticity rc is large enough. the initial substitution effect due to a price decrease overwhelms the concurrent income effect and thus results in an increase in his contribution gk and a decrease in his private good consumption xk. This aIso makes the paradox more likely to occur. On the other hand, if income effect axk/awkwhich increases both
gk
EG
is very close to zero. ail rnatter is the
and xk. Recail that the rest of the economy
does not reduce G V ka s much as the increase in g k . Hence. the net income effect is positive in
a new equilibrium, making the agent better off (no paradox o c c ~ r s ) . ' ~
1.5
The Stackelberg Allocation Process
In t his last section. we consider public-good contributions under a Stackelberg allocation process. While Bruce (1990) and Varian (1994) andyze a two-agent interaction with one leader and one foliower, the following analysis generalizes t heir studies by considering an n-agent interaction which consists of one leader and n
- 1 followers. In so doing, we demonstrate another merit of
the aggregate response function in extending the existing model to a more general one which allows for n - 1 followers. Such an extended model will lay a theoretical foundation for the empiricd analysis of Chapter 3. where we will conduct statisticd tests to distinguish among several strategic assumptions of pubiic-good contributions, including the Stackelberg allocation process in a more than one follower environment. Consider an n-agent econorny which consists of one leader and n
- 1 foIlowers. Under the
Stackelberg process. the timing of the interaction is: (i) the leader (with subscript 1) chooses his public-good contribution level at gf and credibly cornmits himself to that level: (ii) the foIlowers (agent j . j = 2. -
- - . n). d e r observing the leader's cornmitment level. choose their -
-
" ~ s i nthe ~ opposite logic. we can also demonstrate when an agent experiencing an efficiency improvement is Likely to be better off. Recall that the other agents are better off. Thus, this leads to Pareto improvement in the subscription economy. The Pareto improvement is likely to occur when (i) the aggregate contribution by the rest of the economy is inelastic enough, (ü) the efficiency-improving agent initially contributes a large portion of the total provision, or (iii) there is no substitutability between z and G (as stated in the text). The intuition follows anaiogously to the one in the text.
contribution levels at g;. Notice that n - 1 followers are assumed to be Nash cornpetitors. who take the leader's contribution level fixed (Le.. credibly committed). It is also assumed that the leader can anticipate how the foIlowers will respond to changes in his contribution. The leader's maximization problem is now
where
c- (gi ) is the aggregate response function for the leader. The FOC is now
We assume an interior solution. A Stackelberg equiiïbrium may be defined as an equilibrium {gi}F=l. where the condition (1.10) must be satisfied given the aggregate response function
G - (g1) . Comparing t his equilibrium wit h the Nash alternat ive yields the foIlowing proposition: Proposition 15 In a Stackefberg equilibrium being compared to a Nash equilibrium: ( i ) the leader contributes less: ( i i ) each of the /ollowers contributes more: (iii) the total provision of public good is reduced: ( i v ) the leader is betler os: and ( v ) each of the followers is worse
08.
Proof. In a Nash equilibrium. it m u t be the case that
where G- 1 (31 ) = Cjil G j . On the other hand. in a Stackelberg equilibrium (g,b}:=l.
where G-'(9;) = Cjtlg;. Define v l ( g i )
by Propositions 3 and 11. As pl
< pl/(l
id ( w 1
-pigl, gl
+ ~ - ~ ( g l )which ). yields:
+ d ~ - l / d ~ l v1(ij1) ). < ~ ' ( ~ by f )(1.11) and (1.12). 22
< O yields (i) iji > 9:. Note that, given G = gl d G / d g i = 1 + d ~ - l / dor ~d G~/ d s l > O. Then, (i) gi > g: irnplies (iii) G > Gs. Thus. the result that d v l / d g l
+G - ~ ( ~ ~ ) .
For a follower in either of the equilibria (Nash or Stackelberg), it must be the case that pj =
x J ( w j - p j g j . G) where gj and G can be either of the equilibrium values. Taking total
differentials results in dgj /dG = p - L (&rj/aG) /&'/axj
which is negative by Proposition 3.
Then. (iii) G > GS implies that (ii) g,? > g j . Findy. because the leader did not choose the latter which he could have, (iv) the leader's maxirnized utility is higher in a Stackelberg equilibrium than a Nash counterpart. On the other
hand. by (i). (ii) and (iii). the welfare of a follower is now lower in a Stackelberg equilibrium than in a Nash equilibrium (v).
1.6
Concluding Remarks
It is our hope that this chapter has successfully put in perspective the main results concerning the voluntary provision of a public good and provided further refinements and accompanying insights to the iiterature. We believe that our main contributions should be found in the
foiiowing. First. we have formally established that an equilibrium contribution will be disproportionate in a generd environment and explicitly related it to the concept of progressiveness. Second. we have introduced the analytic device of aggregate response function. which translates a heterogeneous n-agent interaction into a pseudo twwagent interaction. The merit of doing so was demonstrated in simpiifiing the comparative-static analysis of the standard mode1 and the
extended analysis of the Stackelberg interaction in an n-agent economy. Third. we have delineated precise factors which influence the egiciency pamdox and also provided an accompanying intuitive explanat ion t hereof. Recall that we have modeled the contributing economies as if they consisted of single representative agents. This conceptualization is taken since a majority of the studies on international public goods treats national economies as such and this chapter was intended as a synthesis and extension of the existing Literature. However. this masks the fact that nations are comprised of populations of citizens. and presumably a national government takes account of the joint benefits accruing to their citizens when deciding on the amount of contributions to interna-
tional public goods. The next chapter tackles this issue and provides further analysis when populations are explicitly taken into account in the formulation of the model.
Bibliography 111 Anderson. S.. 1958. An Introduction to Multiuariate Statistical Anahjszs (Wiley. New York). [2j Andreoni. J., 1988. Privately provided public goods in a large economy: the limits of aitruism. Journal of Public Economics 35. 57-73.
[3] Andreoni. J.. 1990. Impure altruism and donations to public goods: A theory of warmglow giving, Econornic Journal 100, 464477. [41 Arnott. R.and R. Grieson. 1981. Optimal fiscal policy for a state and local government. Journal of Urban Economics 9 . 23-48. 151 At kinson. A. and J. Stiglitz. 1987. Lectures on Public Economics ( McGraw-Hill. London).
[6] Atkinson. S. and P. Wilson. 1992. The bias of bootstrapped versus conventional standard errors in the general linear and SUR models. Econometric Theory 8. 258-275. 171 Baldwin. D..1985. Economic Statecmft (Princeton University Press, Princeton). 181 Barro. R. and C. Sahasakul. 1983. Measuring the average marginal tax rate from the individual income tax. Journal of Business 56, 419-452. 191 Barro. R. and C. Sahasakul, 1986, Average marginal tax rates hom social security and
the individual income tax. Journal of Business 59, 555-566. [IO] Bart let t. M.. 1937. Properties of sufficiently and statistical tests. Proceedzngs of the Royal Society of London. Ser. A. 160. 268-282.
[lll Besley, T.and
H. Rosen. 1996, States' responses to federal tax setting: Evidence fkom
gasoline and cigarettes. mineo. Princeton University.
R.and E. Slack. 1990. Equalization: The representative tax system revisited. Canadian Tax Journal 38. 913-027.
1121 Bird.
[131 Boadway.
R. and F. Flatters. 1982. Efficiency and equalization payrnents in a federal
system of goverment: A synthesis and extension of recent results. Canadian Journal of Economics 1 5 . 613-633.
1141 Boadway, R. and M. Keen. 1994. Efficiency and the fiscal gap in federal systems. Queen's h t it ute for Economic Research Discussion Paper, No. 915 [151 Boadway.
R. and M. Keen. 1996. Efficiency and the optimal direction of federal-state
transfers. International Tax and Public Finance 3, 137-155. 1161 Boadway.
R.. P. Pestieau and D. Wildasin. 1989, Tau-transfer policies and the voluntary
provision of public goods. Journal of Public Economics 39, 157-176. [171 Boadway. R.. P. Pestieau and D. Wildasin. 1989. Non-cooperative behaviour and efficient
provision of public goods. Public Finance 44. 1-7. [181 Bergstrom. T..L. Blume and H. Varian. 1986, On the private provision of public goods. Journal of Public Econornics 2 9 . 25-49.
[191 Bergstrom. T..L. Blume and H. Varian. 1992. Uniqueness of Nash equilibrium in private provision of public goods: An improved proof, Journal of Public Economics 49. 391-392. [201 Brown, B.. 1981. Sample size requirements in full information maximum likelihood estimation. International Economic Review 47, 239-53.
[211 Bruce, N.. 1990. Defense expenditures by countries in allied and adversarial relationships. Defense Economics 1 , 179-195. [22] Buchholz. W. and K. Konrad. 1994. Global environment problems and the strategic choice of technology. Journal of Economics 60. 299-32 1.
1231 Buchholz. W. and K. Konrad. 1995. Strategic transfers and private provision of public
goods, Journal of Public Economics 57, 489-505. [241 Bureau of Economic Analysis, 1997. Suruey of C u m n t Business (US Government Printing
Office. Washington D.C.) . 1251 Cassing, J. and A. HiUman, 1982, State-federal resource tax rivalry: The Queensland
railway and the federal export tax, Economic Record, 235-241. [261 Chamberlin. J.. 1974. Provision of collective goods as a function of group size, American Political Science Review 68. 707-716. 1271 Cornes, R.. 1993. Dyke maintenance and other stories: Some neglected types of public
goods, Quarterly Journal of Econornics 107, 259-27'1. [281 Cornes. R. and T . Sandler. 1984. Easy riders. joint production and pubIic goods. Econornic
Journal 94. 580-598. [291 Cornes. R. and T. Sandler. 1985. The simple analytics of pure public good provision. Econonaica 52.103-1 16.
[301 Cornes. R. and T.Sandier. 1994. The comparative static properties of the impure public good model. Journal of Public Econornics 5 4 , 403-421. [311 Courchene. T. and D. Beavis. 1973. Federal-provincial tax equalizat ion: An evaluat ion. Canadian Journal of Economzcs 6 . 483-502. 1321 Dahlby. B.. 1996. Fiscal externalities and the design of intergovernmentai grants. International T m and Public Finance 3 . 397-412. [331 Dasgupta. D. and J. Itaya. 1992. Comparative statics for the private provision of public
goods in a conjectural variations model with heterogeneous agents. Public Finance 47. 17-31. 1341 Davidson. R. and J. MacKinnon, 1981. Several tests for model specification in the presence
of alternative hypotheses, Econometrica 49, 781-793.
1351 Davidson, R. and J. MacKinnon, 1993. Estimation and Inference in Econometrics (Oxford
University Press. New York). 1361 Davidson. R. and J. MacKinnon. 1996, The size of distortion of bootstrap tests. Queen's Instit ute for Economic Research Discussion Paper, 936. [37] Deaton, A. and J. Meullbauer. 1980, Economics and Consumer Behavior (Cambridge University Press. New York). 1381 De Boer. P. and R. Harkema. 1989. Some evidence on the performance of size correction
factors in test ing consumer demand models, Economics Letters 29. 31 1-315. [39] Development Assistance Cornmittee. 1998. httg: . www. oecd. o ~ d a c . ' i n d e xhtm. . [401 Dion. P., 1994, A Theoretical and Econometric Analyszs of the Economzc Theory of Alliances: Nash-Cournot Behavior. Oligarchie Decision Making and Heterogeneous Marginal Contribution Costs (Ph.D. dissertation. Washington State University). [41) Dudley, L.. 1979. Foreign aid and the theory of alliances, Rewiew of Economics and Statzstics 61. 564579. 1421 Dudley, L. and C. Montmarquette. 1981. The demand for military expenditures: an international cornparison. Public Choice 37. 5-3 1. (431 Edwards. J. and M. K e n . 1996. Tax cornpetition and Leviathan. European Economic Revieur 40. 113-134. [441 Feenberg, D. and H. Rosen. 1986. State personal income and sales taxes. 1977-1983. in:
H. Rosen ed. Studies in State and Local Public Finance (University of Chicago Press. Chicago). 1451 Fisher, P.. 1981. State equalizing aids and metropolitan tax base sharing: A comparative analysis. Public Finance Quarterly 9 , 449-470.
[461 Foley, D.. 1967. Resource ailocation and the public sector, Yale Economic Essays 7. 45-98. [471 Flowers, M. 1988. Shared tax sources in a Leviathan mode1 of federalism. Public Finance Quarterly 16, 67-77.
[48j Raser, C.,1992, The uniqueness of Nash equilibrium in the private provision of public goods: An alternative proof. Journal of Public Economics 49, 389-390. 1491 hdenberg. D.and J. Tirole, 1991, Game Theory
(MITPress, Cambridge).
[50] Hartwick. J.. 1980. The Henry George rule. optimal population, and interregional equity. Canadian Journal of Economics 13. 695-700. [511 Hilton. B. and A. Vu. 1991. T h e McGuire model aod the economics of the NATO alliance.
Defense Economics 2. 105-121. 1521 Hirshleifer. J.. 1583. Fkom weakest-link to bat-shot : The voluntary provision of public
goods. Public Choice 41. 371-386. [531 Hoel. M.. 1991. Global environmental problems: The effects of unilateral actions taken by one country, Journal of Environmental Economics and Management 20. 55-70. [54] Ihori. T.. 1992. Impure public goods and transfers in a three-agent model. Journal of
Public Economics 48. 385-401. [551 Ihori. T.. 1994. Irnrniserizing growt h wit h interregional externalities of public goods. Re-
gional Science and Urban Economics 24. 485-496.
1561 Ihori. T.. 1996. International public goods and contribution productivity difFerentials. Journal of Public Economics 61. 139-154. [571 International Monetary Fund (1997). International Financial Statistics Yearbook (International Monetary Fund. Washington. D.C.) . [581 Itdianer. A.. 1985. X small-sample correction for the likelihood ratio test. Economics
Letters 19. 315-317. [591 Jack, B..1991. International Public Goods: The Economics of Their Provision and CostControl Incentiues under the Cournot-Nash Hypothesis (Ph.D. dissertation. University of
Maxy land, College Park). [601 Keen, M.. 1995. Pursuing Leviathan: Fiscal federalism and international tax cornpetition. mineo, University of Essex.
[611 Keen. M. and C. Kotsogiannis, 1996. Federalism and tax cornpetition. mineo. University of Essex. (621 Krelove.
R.. 1992. Efficient tax exporting, Canadian Journal
of Economics 25. 145-155.
(631 Laitinen. K.. 1978. Why is demand homogeneity so ofken rejected? Economics Letters 1. 187-191. 1641 Lindblom. C.. 1990. The Policy- Making Pmcess, 2nd ed.. ( Prentice-Hall. Englewood
C m )-
W.and A. Tassonyi. 1993. Shared tax bases and local public sector expenditure decisions. Canadian Tax Journal 4 1 , 941-957.
1651 Lock.
[66] MacKinnon. J.. 1997. Economics 950: Lecture Notes 5 (Department of Economics.
Queen's University). 1671 MacKinnon. J.. 1983. Model specificat ion tests against non-nested alternatives. Econometric Reviews 2. 85-1 10. [68] MacKimon. J..
H. White and R. Davidson. 1983, Tests for mode1 specification in the
presence of alternative hypothesis: Some further resdts. Journal of Econometrics 29. 305-251691 McDermott. A.. 1995. An Empirical Analyszs of Cigarette Tazotion in Canada (Undergraduate Thesis. Depart ment of Economics. Queen's University). [701 McGuire. M. and C. Groth.. 1985. A method for identifiing the public good allocation process within a group, Quarterly Journal of Economics 100, 915-235. [711 Meisner, J., 1979, The sad fate of the asymptotic Slutsky symmetry test for large systems. Economics Letters 2 . 231-233.
1721 Mendoza. E., A. Razin and L. Tesar. 1994. Effective tax rates in macroeconomics: Crosscountry estimates of tax rates on factor incomes and consumption, Journal of Moneta y Economics 34, 297-327.
[731 Mintz. J. and H. Tulkens, 1986, Commodity tax cornpetition between member states of
a federation: Equilibrium and efficiency, Journal of Public Economics 29, 133-172. [741 Mosley, P.. 1987. The political economy of foreign aid: A mode1 of the market for a public
good, Economic Development und Cultuml Change 33, 373-393. (751 Murdoch. J. and T. Sander, 1982. A theoretical and empirical analysis of NATO. Journal of Conflict Resolution 2 6 , 237-263.
[761 Murdoch, J. and T. Sandler, 1984. Complemantarity, free riding, and the military expenditures of NATO ailies, Journal of Public Economics 25, 83-101. [77] Murdoch. J. and T. Sandler. 1997. The voluntary provision of a pure public good: The
case of reduced CFC emissions and the Montreai Protocol, Journal of Public Economics 63, 331-349.
[?81 Okamura. M.. 1991. Estimating the impact of the Soviet Union's threat on the United
State-Japan alliance: A demand system approach, Review of Economics and Statistics 73. 200-206. [791 Okamura. M.. 1996, The w e h e effects of disarmament on the United States under NATO
and the Warsaw Pact. Review of Econdmics and Statistics 7 8 , 277-285. [SOI Olson. M.. 1965. The Logic of Collective Action (Harvard University Press. Boston). 1811 Oison. M. and R. Zeckhauser. 1966. An economic theory of alliances. Review of Economics and Statistics 48. 266-279. 1821 Olson. M. and R. Zeckhauser. 1967. Collective goods, comparative advantage, and alliance
efficiency. In R. McKean (ed.) Issues in Defense Economics (Columbia University Press. New York). [83] Olçon, M. and R. Zeckhauser, 1970, The efficient production of external economies. American Economic Review 60, 512-5 17. 1841 Oned, J.. 1990. The theory of collective action and burden sharing in NATO, International Organization 44. 379-402.
Organizat ion for Economic Ceoperation and Development. 1964. The Flow of Financial
Resources to Countries in Course of Economic Development: 1956-1963 (OECD.Paris). Organization for Economic Co-operation and Development . 1973, The Flow of Financial
Resources to Developing Countn'es (OECD. Paris). Organization for Economic Co-operation and Development, various years. Development
Co-opemtion (OECD. Paris). Organization for Economic Co-operation and Development. 1997a. Development Co-
operation (OECD.Paris). Organizat ion for Economic Cwoperat ion and Development, 1997b, National Accoun ts (OECD. Paris). Rilstone. P. and M. Veall. 1996. Using bootstrapped confidence i n t e d for improved inferences with seemingly unrelated regression equations, Econometn'c Theory 12. 569-
580. Rocke, D.. 1989.Bootstrap Bartlett adjustment in seemingly unrelated regression. Journal
of the American Statistical Association 84, 589-601. Rothenberg, T., 1984. Hypothesis testing in Lnear modek when the error covariance matrix is nonscalar. Econometrica 52. 827-842. Samuelson. P.. 1954. The pure theory of public expenditure, Review of Economics and
Statzstics 36. 387-89. Sandler. T.. 1992. Collective Action: Theory and Application
o f Michigan
Press. Ann Arbor). Sandler, T., 1993. The economic theory of alliances: A survey, Journal of Conj?ict Reso-
lution 37, 446-483. Sandler, T . and J. Forbes, 1980, Burden sharing, strategy, and the design of NATO,
Economic Inauiru 18. 425-444.
[971 Sandler. T. and K. Hartley. 1994. The Economics of Defense (Cambridge University Press. Cambridge). 1981 Sandler. T.and J. Murdodi. 1990. Nash-Cournot or Lindahl behaviour?: An empirical test for the NATO allies. Quarterly Journal of Economics 105. 875-894.
1991 Sandler. T. and R. Shelton. 1972. Fiscal federalism, spillovers and the export of taxes. Kyk10s 25. 736-753.
[100] Sagan. J.. 1975. Asymptotic theory and large models. International Economic Review
16,75-91. [IO11 Schulyer. M..1982. Essays on Burden Sharing and Eficiency in Alliances (Ph.D. dissertation. University of M q l a n d . College Park). [IO21 Smart. M..1998. Taxation and deadweight l o s in a system of intergovenimental transfers. Canadian Journal of Economics 31. 189-206.
[IO31 Sims. C..1980. Macroeconomics and reaiity, Econornetrim 48, 1-48. [IO41 Smith. R.. 1989. Models of military expenditure. Journal of Applied Econometrics 4 . 345-359. [IO51 Smith. R.. 1995. The demand for rnilitary expenditure, In K. Kartley and T. Sandler (eds. ) Handbook of Defense Econornics (Elsevier Science. Amsterdam). 69-87.
11061 Usher. D.. 1995. The Uneasy Case for Equalization Payments (Fraser Institute. Vancouver). 11071 Varian. H.,1994. Sequential contributions to public goods, Journal of Public Economics
53. 165-186. [108] Vicary. S.. 1997. Joint production and the private provision of public goods. Journal of Public Economics 63. 429-445.
[log] Warr,
P..1983. The private provision of a public good is independent of the distribution
of income, Econornics Letters 13, 207-21 1.
[Il01 Weber. S. and H. Wiesmeth. 1991. Economic models of NATO, Journal of Public Eco-
nornics 46,181-197. [llll Wildasin,
D.,1987. The demand for public goods in the presence of tax exporting, Na-
tional T m Journal 40. 591-601.
[1121 Zodrwo.
G..and P. Mieszkowski. 1986. Pigou. Tiebout. Property taxation. and the un-
derprovision of local public goods. Journal of Urban Economics 19,356-370.
