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Transfers and the weakest-link
Journal
of Public
Economics
43 (1990) 375-394.
TRANSFERS
AND
North-Holland
THE
WEAKEST-LINK
An Extension of Hirshleifer’s Simon
Analysis
VICARY”
Received July 1989, revised version
received June 1990
With the weakest-link social composition function analysed by Hirshleifer it is shown that an incentive exists for some agents to transfer resources to others so as to achieve higher quantities of the public good. The characteristics of such an equilibrium with transfers are examined, together with the implications of differing income distributions for the quantity of the public good. Some observations are offered as to how well the model can capture co-operative behaviour.
1. Introduction In recent years much progress has been made in analysing the private provision of public goods. This literature has concentrated on the idea that the total quantity of a public good available to a community will equal the sum of the contributions made by individuals. The basis of much of this work has been the Nash model, and the results here have been well surveyed by Bergstrom, Blume and Varian (1986) (henceforth BBV). Another theme has been the different ways in which the overall provision of the public good is determined. Thus, Bliss and Nalebuff (1984) examine a case in which the good must be provided by one person, and where the cost of provision varies between individuals who do not know one another’s costs. Palfrey and Rosenthal (1984) examine the case of a discrete public good to be provided by individual subscriptions. Calling the ways in which public good allocations are determined from individual decisions social composition functions, Hirshleifer (1983) has listed three cases: (1) G =cgi, summation (the traditional view); (2) G = min {gi} weakest-link, (3) G =max {gi} best-shot,
*I am grateful to Professor Hirshleifer, as well as to the referees, for useful comments earlier version of this paper. Errors are my own responsibility, however. 0047-2727/90,‘$03.50
0
199GElsevier
Science Publishers
B.V. (North-Holland)
on an
376
S. Vicary, Transfers and the weakest-link
where G is the total quantity of the public good, and gi represents individual i’s contribution. The best-shot case has some similarity to the ‘dragon slaying’ of Bliss and Nalebuff, and is prone to serious underprovision. On the other hand, private provision of a weakest-link public good seems to be less inefficient, as well as being less vulnerable to the ‘Olson problem’ [Olson (1965)] of the greater ineficiency of larger groups. Weakest-link public goods also turn out to have another property not formally analysed by Hirshleifer, namely that under certain circumstances there will be an incentive for agents to make voluntary transfers so as to raise G. These transfers involve no altruism on the part of donors, and are made simply to maximise donors’ utilities. It is the purpose of this paper to analyse the consequences for the weakest-link model of allowing such transfers to take place. As an instance of a weakest-link or near weakest-link public good, Hirshleifer imagined a low-lying island, Anarchia, inhabited by individuals with an extreme aversion to collective action, or to making binding agreements with each other. Each built sea defences to his or her own requirement, and the overall protection provided was that given by the lowest standard along the coastline. (Once any part of the defences was breached the whole island would be flooded.) Similar examples would be the protection of a military front, taking a convoy across the ocean (going at the speed of the slowest ship), or maintaining an attractive village/landscape (one eyesore spoils the view). Many instances of teamwork involve weak-link elements, for example moving a pile of bricks by hand along a chain or providing a theatrical or orchestral performance (one bad individual effort spoils the whole effect).’ A slightly different type of weak-link good (involving discreteness) is that of keeping a (military) secret. In many of these cases the problem of underprovision can be alleviated through clubbing (professionals do not often act with amateurs), but even allowing for this some underprovision tends to occur (e.g. without the slowest ship the convoy could have gone faster). The weakest-link has also attracted attention as a possible explanation for co-operative behaviour, particularly during periods of adversity. As Hirshleifer writes (1987, p. 159); ‘The alliance we call society.. is threatened in time of disaster. In these circumstances alliance supportive activities, cooperativeness and self-sacrifice become an important public good.. . . But a public good in large part describable in terms of the weakest-link social composition function.’ As an illustration of this consider the following story told by Agnelli (1986): In 1977, six boys in Tonga, all friends, went fishing. Their boat was caught in a storm and after several terrifying days was wrecked on a
‘Some other examples
are given in Harrison
and Hirshleifer
(1989).
S. Vicary, Transfers
and the weakest-link
371
reef. The crew had just enough strength to scramble ashore, on to an unknown tropical island. They realised than it was totally uninhabited. Confronted with their predicament, they promised each other that as long as they were there they would never quarrel, because that would spell the end of them; that they would always go about in pairs, in case one had an accident or got lost; and that two of them would keep guard day and night. They kept their promises, and 15 months later were found and rescued.2 There is a strong hint here that the probability of survival of each boy depended on the active co-operation of all the boys. One way of modelling this situation is through the weakest-link, and there is a hint of this in the reason given for not quarrelling. However, an alternative interpretation is that what the boys faced was a critical muss, in that without some minimal amount of overall co-operation there was no possibility of survival.3 Both alternatives make the contribution of each individual of great importance, but their general implications for behaviour are not the same. In particular, as will be explained presently, a critical mass public good provides no incentive for voluntary transfers. It could be claimed that in most of the examples (excluding, perhaps, cooperative behaviour) we do not observe money transfers of the type to be analysed in this paper. This is true, and as we will explain in the conclusion, there are good reasons why this is so. Nevertheless, the analysis of transfers provides a good motivation for some behaviour that we do observe, as well as enabling us to pursue to the limit the implications of the pure individual action allowed by the inhabitants of Anarchia. In any case, if we take cooperative behaviour to be captured by a weakest-link public good, direct transfers are both feasible and, arguably, observed. We return to this point in the conclusion. After a brief exposition of Hirshleifer’s own analysis in section 2, there is a detailed analysis of transfers in sections 3 and 4. Here we find that public good theory of a more traditional kind becomes relevant. Indeed, we are able to replicate many of the results in BBV. Section 5 analyses the impact of a disaster on voluntary transfers, thus providing one possible way of developing predictions which can distinguish a weak link from a critical mass view of co-operative behaviour. Taking the model to represent ‘ordinary times’, the concluding section examines what predictions we might make about voluntary transfers, and compares these with some ‘stylised facts’.
‘It is not clear that Agnelli would accept the interpretations here. 3Hirshleifer mentioned this possibility in his original paper, referee.
of this story
that are discussed
and it was also mentioned
by a
S. Vicary, Transfers
378
and the weakest-link
2. Hirshleifer’s analysis To understand the Hirshleifer model we can imagine which each individual maximises the function:4
an n-person
model in
ui(xi, G) subject
to
wi = xi +g,,
i=1,2 )...) n,
and where
G=min{g,},
i=1,2 ,..., n
where Ui represents the utility of individual i, xi the quantity of the private good consumed by i, G the quantity of the public good available, wi the income level of i, and gi i’s contribution to the public good. We assume constant costs and normalise the units of each good such that all prices are unity. To solve this model we may simply note that no individual will buy more of the public good than anyone else. To do so would cause a loss of consumption of the private good, and no compensating increase in the public good. We can thus imagine that each person decides on an ‘ideal’ quantity of G, assuming this to be matched by all other members of the community. That is, we solve the above problem assuming that G=g,. If ii represents the optimal amount of G for individual i; then the actual amount enjoyed by the community will be min {g,). All those who wish for more G will be unable to have their wishes fulfilled, and what they might otherwise have spent on the public good will now be devoted to the private good. The solution for a two-person economy is depicted in fig. 1. Here we see the reaction curves R, and R, for the two individuals 1 and 2, together with a pair of illustrative indifference curves, IC, and IC,, one for each person. In this case 8,
4Hirshleifer allows the marginal complication in what follows.
cost
to differ
between
individuals.
We will ignore
this
S. Vicary,
“ramfen and the weakest-link
379
/
Fig. 1. Equilibrium without transfers. Individual l’s ideal point is at A, and 2’s at B. Nash Equilibria lie along the line OA, but the outcome is taken to be at A.
economy will be in equilibrium at point A.’ Along the line AB we have a locus of Pareto optima, indicating that A itself is Pareto optimal. There is nevertheless a sense in which the good is underprovided, as can be seen by examining the Samuelson (1955) condition for optimal provision, which in this case is 1 MRS’,,= n. At A, MRSh,= 1 and MRS&_> 1, so the level of G is suboptimal. There is, however, a problem. The way to achieve efficiency with the standard social composition function is to try to devise some form sAs pointed out by Hirshleifer, all points on the line OA are Nash Equilibria. We have here what Sen (1967) called an assurance game, the Pareto preferred point among them having stability properties absent from the prisoner’s dilemma game, which is usually taken as capturing the problem of ensuring co-operation. Harrison and Hirshleifer (1989) point out that the selection of point A can be justified as Selten’s (1975) sub-game-perfect equilibrium, if the game is played sequentially. This is not possible if strategies are played as sealed bids, and neither the perfectness concept nor Myerson’s (1978) notion of a proper equilibrium can justify the choice of A, despite its intuitive appeal. Nevertheless, the experimental results of Harrison and Hirshleifer support this outcome.
380
S. Vim-y, Transfers and the weakest-link
of collective provision of the public good. With Hirshleifer’s Anarchia parable (1983), this might be possible if the islanders conquered their aversion to collective action. However, where this model is being applied to co-operative behaviour under conditions of adversity, the public good is inextricably linked to individual action. A/I members of the community must contribute, and it does not seem possible, if we take the weakest-link idea seriously, to have the quantity decision delegated to some central agent, even through a voting device such as the Lindahl mechanism. In any case, in the above example individual 1 has already achieved an ideal outcome at point A. Collective provision will make him worse off, unless accompanied by a transfer. However, we have not yet exhausted the potential for individual action. Imagine we increase 2’s income in the above example. As the economy will remain at point A, the whole of any increase in 2’s income will be devoted to the private good, so that MRS& (= -dx/dG) will be increasing the whole time (we assume normality). Hence, there is likely to be some income level for 2 beyond which it will become worthwhile to give money to 1 even though only a fraction of the money donated will actually be spent on G. To ensure that such an income level exists we will assume that for all individuals lim,, o. MRSL,= GO. The next section is based on this insight.’ Whilst this paper concentrates on the weakest-link, the possibility of transfers of the nature analysed here extend wider than this basic model. Whenever the effect on the marginal provision of the public good of $1’s worth of expenditure varies as between individuals there is the potential for it being cheaper to acquire extra units by transfers rather than by direct purchase. Hirshleifer’s best-shot case is interesting for possibly inducing transfers to just one individual if an efficient collective provision mechanism can be derived. Another example is the social composition function G = f(c a,gJ, as used by Sugden (1984). In an unpublished paper Cornes has examined the social for transfers, in composition function [I gy] I”, which also has a potential this case along the lines analysed here. It should be noted that in the case of the critical mass public good there is no difference between the impact of $1’s worth of expenditure on provision as between individuals. Consequently, no transfers will take place, an important difference from the weakest-link.
3. Analysis of transfers We now consider the economy of n individuals described in section 2. We assume there exists a set of individuals with incomes sufliciently high to make them possibly interested in donating money so as to raise the amount ‘Gifts are impersonal, and 1 and 2 cannot spent. We comment on this in the conclusion. of recipients is ignored.
make a contract to determine how the money is The possibility of strategic behaviour on the part
S. Vicary, Transfers
and the weakest-link
381
of G. Clearly, any such money donated will go only to that individual or those individuals who have the minimum demand for G. Let us call this set R. The cost of increasing the amount of G by one unit for one individual can be written as follows:
iFR(~G/‘~Wi)-’ + 1,’ where the first term represents what has to be paid to the members of R in order to induce each one to raise consumption of G by one unit. The second term (unity) is simply the direct cost of purchasing one more unit of G. It is evident from this that if there exists a group of individuals interested in donating money to individuals in group R, then donations have the property of a conventional public good. If individual i donates some money to R, then individual j (also a potential donor) need only incur the direct cost of gaining extra G and can take a free ride on i’s donation. The benefits to an individual thus vary with the sum of the donations from others, and the optimum individual donation can be derived given this total. In short, we have for potential donors a conventional problem in the private allocation of public goods. To analyse this problem formally we will assume the existence of a charity which accepts donations, and distributes them to the members of R in such a way that G increases at minimum cost. Given the comments at the end of section 2, we can start from an initial solution of no donations, where for each agent i there is an income level wT below which i will not be interested in donating money. Let us now define three sets of individuals:
R=
T={il Wi>Wi*}. In considering total donations to the charity we need only focus our attention on 7: Members of S will never become donors, although they may become recipients. Each individual in T solves the following problem: ‘dD=xdwi, ieR, and dwi=dG/(3G/dw,), term in the expression.
igR,
imply dD/dG=~(dG/8w,)~‘,
which is the first
382
S. Vicary, Transfers
and the weakest-link
max Ui(Xi, G) X,.Qi subject
to
where d, represents i’s donation to the charity. Ui is assumed to be twice differentiable and preferences are assumed to be strictly convex (these assumptions hold for all agents). If we now define D=xd,
and
D-i=xdj
isT
jsT j#i
we can rewrite this problem
as:
max Ui(Xi, G) X,,D subject
to Wi+D_i=Xi+G(D)+D.
Here we take G=g, and use the function G(D) which determines the quantity of G given total donations. Our initial task is to prove the existence of a Nash equilibrium in donations. To this end we need to establish certain properties of the G(D) function. Firstly, given the differentiability properties of all utility functions, and the consequent continuity and differentiability of demand functions, G(D) is continuous. From the assumption of normality it is also monotonic strictly increasing and so has an inverse. Where there is no change in the set of recipients of transfers the function will also be differentiable, the value of the derivative being found from the following argument: dD=
1 dw,=dG ieR
c (aw,/~?G) iGR
so that
where (c~w~/LJG)is the inverse of i’s marginal propensity to consume G. However, as donations increase, individuals will gradually move from set S to set R, and when this occurs an extra term is added to the summation inside the square brackets. In short, the G(D) function is kinked at points at
S. Vicary, Transfers
383
and the weakest-link
which an extra individual becomes a recipient. Given normal goods, the derivative G’, where it exists, will be positive. Left-hand and right-hand derivatives will exist at all points, and at the kinks dG/dD 1,,>dG/dD ( RH. We can now form a picture of the budget constraint facing each individual in 7: Adopting the conventional assumption of fixed donations by the other members of T, we have, from the budget constraint: dx,+dG+D’dG=O for points where D’, the derivative follows that for such points: dxJdG=
of the inverse
function,
is defined,
It
+D’).
-(l
To ensure convexity of the budget set we assume that for all individuals aG/aw, is a non-increasing function of income. This guarantees a nonnegative sign for D” (where defined). This is obviously restrictive, but avoids many complications. Furthermore, it follows from the properties of G(D) that at all points with strict inequality at the kinks. This gives D’\,H4D’(RH convexity of the budget set. The individual’s choice can be visualised in fig. 2. Given strictly convex preferences there will be a unique choice of xi and G, these implying an optimal amount of donations for i. We can state the solution of the problem by the vector (x:,dT). This solution can be characterised as follows. With the differentiability properties of Ui, the MRS,, is defined at every point of the positive orthant. Strict convexity of preference and convexity of the choice set mean that the necessary and sufficient condition for i’s utility maximisation is (1 + D’),, >=MRSL, 2 (1 + D’)LH (the subscripts LH and RH here refer to the one-sided derivatives and (dxJdG),,], the first inequality of the budget constraint [i.e. (dx,/dG),, ensuring that the individual does not wish to increase G (or donations), the second that no decrease is desired. Our next step is to try to prove the existence of a Nash equilibrium. We have, first of all: Definition. A Nash (Xi, di) =(x), d:). Theorem
Proof.
1.
Equilibrium
A Nash
Define the following d:=fi(wi+D_J,
Given
Equilibrium
the assumptions
is a vector of donations
in donations
(dJ such that V in IT;
exists.
function: iE 7:
we have
made,
this function
is defined
and
conti-
384
Fig. 2. Individual drawn here. A Members of set members of T to
S. Vicary, Transfers and the weakest-link
utility maximisation. In principle, all agents will face the budget constraint as corresponds to some initial quantity of G before any receipt of transfers. R would locate themselves on MA, members of set S would be at A and the right of A. Points B and C represent points where extra individuals join set R as a result of the donation our individual makes.
JET), where t is the nuous. a Now define the set W={d~tR’~O~d~=
S. Vicary,
Transiers
and the weakest-link
385
In the light of Theorem 1 this needs some modification, because embedded in the weakest-link model is a conventional public goods problem. Thus, whilst underprovision is less of a problem as a result of transfers, to the extent to which transfers fail to respond as community size grows (income distribution remaining much the same), the problem re-emerges. Of course, this assumes that the Olson problem actually exists in this form, and, given that donations can be thought of as a normal good, this may not in fact be justified [see, for example, Cornes and Sandler (1985, pp. 82-84)]. For the rest of the paper it will be convenient to start any formal analysis from an initial equilibrium involving some positive donations. Hence, we need to redefine our sets. Given any equilibrium, for each individual i there must exist a ki such that i will, in the absence of any transfers, at that income, demand exactly the quantity of G determined. Thus we can define:
R={il Wi~ki}: S={iIWi>ivi,
di=O},
T=jild,>O}.
Our next Uniqueness Theorem
task is to establish the here refers to the quantity 2.
There
is
a
unique
uniqueness of G.
quantity
of
of the
G
Nash
consistent
equilibrium.
with
a
Nash
Equilibrium. Proof.
Suppose there are two different levels of G consistent with a Nash Equilibrium, G, and G,, with G, MRSgx. But from our assumption about dG/aw, we know that (1 +LY)lns( 1 + D’):,. Thus, using the condition for utility maximisation in case 1, we have (1 +D’);,,z(l +D’);nz MRS&> MRS& which implies (1 + o’):, > MRSgx which is inconsistent with utility maximisation in case 2 by i. This contradiction establishes the theorem. Q.E.D. It is worth noting that if equilibrium occurs at a point where the membership of set R is not changing then D’ is defined and individual donations are uniquely determined. However, where an individual is about to and equilibrium is consistent leave or enter the set R (1 +D’),,>( 1 +D’),,, with many different sets of contributions from the individuals in set T. We will now examine the model to establish the implications of various types of income redistributions for the quantity of the public good provided.
386
S. Vicary, Transfers and the weakest-link
As with conventional public goods we find a number of neutralities. However, whilst our results parallel those in BBV, there are some differences resulting from the changing membership of sets R, S and 7: Let us first consider ‘small’ redistributions of income, that is where the change in income is less than either donations made or received. Theorem 3. For the case of a small redistribution change in the quantity of G provided.
of income
there
is no
Proof. There are three cases to consider: (a) redistributions within the set T; (b) redistributions within the set R; and (c) redistributions between these two sets. We take each in turn. (a) Assume for each ig T Ad,= Awi. Total donations remain the same, so there is no change in G. Also, Ax,=0 for all ie 7: Hence, for all individuals there is no change in MRS,,. As AG=O there will be no change in the value of the one-sided derivatives of the budget constraint at this new position, which must therefore constitute a new (and unique) Nash Equilibrium. (b) In this case assume the charity adjusts contributions to each jE R so as to ensure that A(r,+ wj) =O, where rj represents transfers to j. The budget constraint of each j E R, after allowing for transfers, remains the same so that AG=O. If all agents in T maintain the same donations as before, then they all meet the conditions for utility maximisation. We have again constructed a new Nash Equilibrium with G unchanged. (c) Assume Awi = Adi V i E T and AWj= - Arj Vj E R. This is a feasible reaction as O=CAw,+c Awj=c Ad,-CAri. As A(wj+rj)=O VjE R, AG=O. Each iE T maintains the same consumption of X, so that there is no change in any marginal rate of substitution. Also there is no change in the one-sided derivatives of the budget constraint. Thus, in this new position all agents are maximising utility and we have again constructed the new (unique) Nash Equilibrium. Q.E.D. We now consider
‘large’ redistributions.
Theorem 4. If a redistribution of income occurs between the members then G cannot fall. Jf G rises, then the contributors after the redistribution a proper subset of 7:
of T form
Proof. Suppose, in contrast to the statement of the theorem, AG ~0. Then there must exist at least one individual iE T such that Axi>O. For this i, MRS& > MRS&, where the superscripts refer to the pre (1) and post (2) redistribution equilibria. As dG < 0, (1 + D’)&, s( 1 + 0’);“. Thus, MRSzx > for utility maximisation in case 1. This (I+ D’)iL, using the conditions contradicts the condition for utility maximisation in case 2. Hence, AG 2 0.
S. Vicary, Transfers
and the weakest-link
387
Now suppose AG >O, and that all previous donors continue to donate. There must exist at least one individual jE T who now has a lower consumption of the private good. Thus, MRS-&< MRS’,‘,. As AC>0 we have which, with the conditions for utility maximisation in (1 +D’);“5(1 +D’);,, case 1, yields MRS& <( 1 + D’)tn, contradicting the condition for utility maximisation by j in case 2. As it is obvious that with AG>O no member of Q.E.D. S or R will start to contribute, the theorem is proved. Theorem 4 produces results parallel to those in BBV, and represents an extension of Theorem 3 in that we do not restrict ourselves to ‘small’ redistributions. We may now consider what happens when redistribution raises the aggregate income of the members of set 7: As we have dealt with redistributions within 7: we can now focus on bilateral transfers involving sets R and S. It is easy to see that a redistribution from an individual in R to an individual in T has no effect on G. Furthermore, a redistribution from someone in set S to someone in set T cannot cause G to fall. This can be seen by supposing the ‘loser’ remains in set S and by pursuing the reasoning in the first part of the proof of Theorem 4 to set 7: Should the loser join R, then there will be no effect on G for that part of the redistribution which gives him an income before transfers of less than +, the borderline income for this individual (allowing for any change in G up to this point). Hence, for redistributions involving members of set T for which the aggregate wealth of the members of T rises G cannot fall, this result following from combining the results we have derived up to now. However, some complications arise from considering set S. Obviously ‘small’ redistributions within S which do not change the membership of S do not change G. But if such a redistribution causes someone to move into T the quantity of G will rise (by definition members of T make positive donations). If a loser joins R, then G may fa1k9 the presumption that this will be so if the redistribution is large enough. Similarly, a redistribution from a member of R to a member of S may cause G to fall, this being the presumption for a large enough enough change. Consequently, we are not able to say that AG 20 for a general redistribution in which zieT Aw,>O. 4. Identical tastes It will now be useful to make
the assumption
of identical
tastes
between
‘Initially we have (1+ D’),,, 2 MRS,,8( 1 + D’),,, V in 7: The consequence of bringing an individual in S down to the border with R is to raise the value of (1 + D’),,,. A further fall in this individual’s income will cause (1+ D’),,, to rise by an equal amount. If the initial equilibrium was not at a kink then, as fixing G causes MRScx to fall, dG=O is not possible, and, pursuing familiar reasoning, neither is dG>O. However, if the initial equilibrium is at a kink the rise in (1 +D’),, may not be enough to reverse the inequality with MRS’,,. For a large enough transfer this will occur, however, due to the effect on MRSL,, iE 7:
388
S. Vicary,
Transfers
and the weakest-link
individuals. This will enable us to gain some intuition about the pattern of transfers we might observe in an economy with a weakest-link public good, as well as helping us to analyse the consequence of greater equality for the provision of G. Given our previous work there will be a unique equilibrium level of G determined by the Nash process. This will in turn give us particular values and contributors will be maximising utility for (1 SD’),, and (1 +D’),,, when (1 + D’hus MRS,,s( 1 +D’),,. Let w’ and w” be the levels of income such that in the absence of donations MRS,, equals (1 +D’),, and i is definitely (not) a donor if wi > w’ (1 +D’)I.H, respectively. An individual (wi < ~3”). An individual i with w” < wil w’ may or may not become a donor, but no donor will have a post-transfer income of less than w”. As a certain quantity of G is voluntarily provided and as G is a normal good, it follows that all individuals with income below some level w* will receive donations in final equilibrium. Also, unlike the conventional case, individuals in set T do not necessarily end up with the same utility level unless w’=w”. We now examine the impact of equalising income redistributions G. Obviously, equalising redistributions within sets R and S have no effect on G. In the former case redistribution is anyway made redundant by the transfers from set 7: The case of equalising redistributions within set T is a special case of Theorem 3 and there is likewise no effect on G. If we consider a redistribution from a member of set T to a member of set S, then dG 50. In the light of the last paragraph we need only consider ‘small’ changes in initial income which do not change either individual’s status. Suppose dG>O. In the new equilibrium (2) there must be an individual iE T for whom xi has fallen so that MRSgx> MRS$. As AC>0 which implies (1 + LY);, > MRS& which we have (l+D’)~H~(l+D’)~H, contradicts the condition for utility maximisation in the conjectured new equilibrium. We cannot, however, eliminate the possibility that AC=0 as initially the economy may be at a kink in the G(D) function. For large enough redistributions, however, AC < 0. For the case of equalising redistributions between R and S we have the following. For ‘small’ (no change of status) redistributions, the effect of increasing the income of some i E R is to cause a local upward shift in the budget constraint facing the members of T around the initial level of G. This is illustrated in fig. 3. Given utility maximisation in the original position, if will rise. It follows from familiar AC ~0, then for some jg T MRS’,, reasoning that this is not possible and that AGZO. Again, the possibility that initially we were at a kink means we cannot eliminate AG=O. The case of equalising income between sets R and T follows immediately from what has just been said. Where there is no change in status there is the familiar neutrality result. When the ie T joins the set S we have AC 20, and where je R joins S we have AGSO.
S. Vicary, Transfers
and the weakest-link
389
X
, Fig. 3. The impact of increased donations on a donor’s budget constraint. After the redistribution to jE R, the output at which resources begin to be transferred to j rises from M to N.
Table Income
redistribution
1 and the impact
on
G.
XT
S
R
T
AC20
AGjO
AC=?
S
AC20
AC=?
AC20
R
AC=0
AC50
AC50
It may be useful to summarise our results by means of table 1. It is evident from this that there is no general presumption about equalising redistributions on the provision of G, particularly when we focus on redistributions from the very wealthy. On the other hand, redistributions from the middle income range to the poor may incrase the provision of G. It should, however,
390
S. Vicary, Trunsfers and the weakest-link
be noted that the weak inequalities in the table generally result from an initial equilibrium at a kink, and that for sufficiently large redistributions the strong inequality usually holds.”
5. Analysis of disasters Our final analytical section deals with a question posed by Hirshleifer. We consider how co-operative behaviour responds to a disaster. We interpret ‘disaster’ here as a widespread fall in income in the community. Co-operative behaviour here can mean two things: total provision of the public good G; or the total amount of transfers made. Two cases will capture the main likely consequences of a general disaster. (a) Suppose dwi
S. Vicary, Transfers and the weakest-link
391
Fig. 4. The change in the post-disaster equilibrium of a donor. The fall in the income of recipients causes a downward shift in the budget constraint of a donor. The slope of the budget constraint to the right of B does not change, and given the definition of R, the slope to the left of B cannot become gentler. This must induce a fall in G for a sulliciently large disaster.
indifference curve at point B must be steeper than at point A, all individual members of T would wish to increase donations. Using familiar reasoning, dG
392
S. Vicary, Transfers and the weakest-link
Fig. 5. The impact of a disaster on donations. If all donors maintain consumption of the private good, then movement is from B to A. At B, however, all donors wish to increase donations. B cannot be an equilibrium.
better way of understanding some adversity, and at other times as well.
aspects
of the
social
alliance
during
6. Conclusion In this concluding section we examine some of the predictions that the model seems to suggest once we accept that co-operative behaviour (not necessarily at a time of adversity) is in some way captured by the weakestlink idea. As it happens there is a general consistency with observations made by a number of writers, notably Tullock (1986). Firstly, transfers in the model are predominantly from the very wealthy to the very poor, and will be greater the more unequally income is distributed. This accords at least with the casual observation that people do not give money (for redistributional purposes) to those who have an income level similar to their own. As a consequence of this, the middle income groups are not greatly involved in
S. Vicary, Transfers and the weakest-link
393
voluntary transfers, although they benefit from the transferring that does takes place. One awkward piece of evidence is that charitable donations as a proportion of income do not seem to increase with income as the analysis of section 4 seems to suggest [see, for example, Collard (1978)]. However, many charities are not redistributive in the sense meant here, and often can be thought of as providing a conventional public good. Also, the existence of government policy on income distribution needs to be taken into account. Nevertheless a disaggregated study of charities may represent one way of testing the model. Secondly, it will be noted from our analysis that transferring is an expensive way to raise G. If donors could, relatively cheaply, force recipients to increase expenditure on G, then they will do so. Hence, legislation to fix certain levels of expenditure, or quantity restrictions on the way donations may be spent, are likely to be observed. Thus, this model provides one explanation as to why so much redistribution is in kind, and it does so without any paternalistic assumptions. It should be noted, however, that in our model it is not necessarily cheaper to provide recipients with G (assuming this is possible), because recipients could substitute x for G in their expenditure. To achieve a substantial fall in costs donors must directly interfere with the consumption quantity decisions of recipients. In the context of some of the examples listed at the beginning of the paper, this may not necessarily be a problem, however. If an actor is experiencing difficulty with his part, then the other players may consider time well spent in helping him to improve, and there are various ways in which compensation can be made for a poor individual performance on the night. It is possible, though, that there may be some moral hazard problem. Thirdly, when we consider the reasons for transferring it is obvious that transfers only take place to the extent that recipients provide extra G for donors. If we suppose the number of public goods thus provided varies inversely with the ‘distance’ (geographic, class or whatever) between the individuals involved we have a possible explanation as to why the bulk of the redistribution we observe is within rather than between communities (Tullock puts a lot of emphasis on this point). Fourthly, we may note that donations are determined by a process directly comparable to the private allocation of a standard public good. This means that there is underprovision of G after the transfer [( 1 + D’)Rr, > 11, even from the point of view of donors. Thus, the model provides an explanation of the fact that governments are involved in redistribution. Naturally, this will be limited if donors maintain control of the political machine. None of this denies the existence of other factors in income redistribution. Nevertheless the extent to which the predicted pattern of redistribution corresponds to observations is intriguing, and may by itself make the weakest-link model worthy of further scrutiny.
394
S. Vicary, Transfers and the weakest-link
References Agnelli, S., 1986, Street children (Weidenfeld). Bergstrom, T., L. Blume and H. Varian, 1986, On the private provision of public goods, Journal of Public Economics 29, 25549. Bliss, C. and B. Nalebuff, 1984, Dragon-slaying and ballroom dancing: The private supply of a public good, Journal of Public Economics 25. 1-l 2. Collard, D., 1978, Altruism and Economy (Martin Robertson). Cornes, R., Dyke maintenance, dragon-slaying and other stories: Some neglected types of public goods, Unpublished paper, Australian National University. Cornes. R. and T. Sandier, 1985, The theory of externalities, public goods and club goods (Cambridge University Press, Cambridge). Harrison, G.W. and J. Hirshleifer, 1989, An experimental evaluation of weakest-link/best-shot models of public goods, Journal of Political Economy 97, 201-225. Hirshleifer, J., 1983, From weakest-link to best-shot: The voluntary provision of public goods, Public Choice 41, 37 I -386. Hirshleifer, J., 1985, From weakest-link to best-shot: Correction, Public Choice 56, 221-223. Hirshleifer, J., 1987, Economic behaviour in adversity (Harvester Press). Myerson, R.B., 1978, Refinements of the Nash equilibrium concept, International Journal of Game Theory 7, 73-80. Olson, M., 1965, The logic of collective action (Harvard University, Cambridge). Palfrey, T.R. and H. Rosenthal, 1984, Participation and the provision of discrete public goods: A strategic analysis, Journal of Public Economics 24, 171-193. Samuelson, P.A., 1955, A diagrammatic exposition of a theory of public expenditure, Review of Economics and Statistics 37, 35&356. Selten, R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4, 25555. Sen, A.K., 1967, Isolation, assurance and the social rate of discount, Quarterly Journal of Economics 81, 112-124. Sugden, R., 1984, Reciprocity: The supply of public goods through voluntary contributions, Economic Journal 94, 772-787. Tullock, G., 1986, The economics of wealth and poverty (Harvester Press)
of Public
Economics
43 (1990) 375-394.
TRANSFERS
AND
North-Holland
THE
WEAKEST-LINK
An Extension of Hirshleifer’s Simon
Analysis
VICARY”
Received July 1989, revised version
received June 1990
With the weakest-link social composition function analysed by Hirshleifer it is shown that an incentive exists for some agents to transfer resources to others so as to achieve higher quantities of the public good. The characteristics of such an equilibrium with transfers are examined, together with the implications of differing income distributions for the quantity of the public good. Some observations are offered as to how well the model can capture co-operative behaviour.
1. Introduction In recent years much progress has been made in analysing the private provision of public goods. This literature has concentrated on the idea that the total quantity of a public good available to a community will equal the sum of the contributions made by individuals. The basis of much of this work has been the Nash model, and the results here have been well surveyed by Bergstrom, Blume and Varian (1986) (henceforth BBV). Another theme has been the different ways in which the overall provision of the public good is determined. Thus, Bliss and Nalebuff (1984) examine a case in which the good must be provided by one person, and where the cost of provision varies between individuals who do not know one another’s costs. Palfrey and Rosenthal (1984) examine the case of a discrete public good to be provided by individual subscriptions. Calling the ways in which public good allocations are determined from individual decisions social composition functions, Hirshleifer (1983) has listed three cases: (1) G =cgi, summation (the traditional view); (2) G = min {gi} weakest-link, (3) G =max {gi} best-shot,
*I am grateful to Professor Hirshleifer, as well as to the referees, for useful comments earlier version of this paper. Errors are my own responsibility, however. 0047-2727/90,‘$03.50
0
199GElsevier
Science Publishers
B.V. (North-Holland)
on an
376
S. Vicary, Transfers and the weakest-link
where G is the total quantity of the public good, and gi represents individual i’s contribution. The best-shot case has some similarity to the ‘dragon slaying’ of Bliss and Nalebuff, and is prone to serious underprovision. On the other hand, private provision of a weakest-link public good seems to be less inefficient, as well as being less vulnerable to the ‘Olson problem’ [Olson (1965)] of the greater ineficiency of larger groups. Weakest-link public goods also turn out to have another property not formally analysed by Hirshleifer, namely that under certain circumstances there will be an incentive for agents to make voluntary transfers so as to raise G. These transfers involve no altruism on the part of donors, and are made simply to maximise donors’ utilities. It is the purpose of this paper to analyse the consequences for the weakest-link model of allowing such transfers to take place. As an instance of a weakest-link or near weakest-link public good, Hirshleifer imagined a low-lying island, Anarchia, inhabited by individuals with an extreme aversion to collective action, or to making binding agreements with each other. Each built sea defences to his or her own requirement, and the overall protection provided was that given by the lowest standard along the coastline. (Once any part of the defences was breached the whole island would be flooded.) Similar examples would be the protection of a military front, taking a convoy across the ocean (going at the speed of the slowest ship), or maintaining an attractive village/landscape (one eyesore spoils the view). Many instances of teamwork involve weak-link elements, for example moving a pile of bricks by hand along a chain or providing a theatrical or orchestral performance (one bad individual effort spoils the whole effect).’ A slightly different type of weak-link good (involving discreteness) is that of keeping a (military) secret. In many of these cases the problem of underprovision can be alleviated through clubbing (professionals do not often act with amateurs), but even allowing for this some underprovision tends to occur (e.g. without the slowest ship the convoy could have gone faster). The weakest-link has also attracted attention as a possible explanation for co-operative behaviour, particularly during periods of adversity. As Hirshleifer writes (1987, p. 159); ‘The alliance we call society.. is threatened in time of disaster. In these circumstances alliance supportive activities, cooperativeness and self-sacrifice become an important public good.. . . But a public good in large part describable in terms of the weakest-link social composition function.’ As an illustration of this consider the following story told by Agnelli (1986): In 1977, six boys in Tonga, all friends, went fishing. Their boat was caught in a storm and after several terrifying days was wrecked on a
‘Some other examples
are given in Harrison
and Hirshleifer
(1989).
S. Vicary, Transfers
and the weakest-link
371
reef. The crew had just enough strength to scramble ashore, on to an unknown tropical island. They realised than it was totally uninhabited. Confronted with their predicament, they promised each other that as long as they were there they would never quarrel, because that would spell the end of them; that they would always go about in pairs, in case one had an accident or got lost; and that two of them would keep guard day and night. They kept their promises, and 15 months later were found and rescued.2 There is a strong hint here that the probability of survival of each boy depended on the active co-operation of all the boys. One way of modelling this situation is through the weakest-link, and there is a hint of this in the reason given for not quarrelling. However, an alternative interpretation is that what the boys faced was a critical muss, in that without some minimal amount of overall co-operation there was no possibility of survival.3 Both alternatives make the contribution of each individual of great importance, but their general implications for behaviour are not the same. In particular, as will be explained presently, a critical mass public good provides no incentive for voluntary transfers. It could be claimed that in most of the examples (excluding, perhaps, cooperative behaviour) we do not observe money transfers of the type to be analysed in this paper. This is true, and as we will explain in the conclusion, there are good reasons why this is so. Nevertheless, the analysis of transfers provides a good motivation for some behaviour that we do observe, as well as enabling us to pursue to the limit the implications of the pure individual action allowed by the inhabitants of Anarchia. In any case, if we take cooperative behaviour to be captured by a weakest-link public good, direct transfers are both feasible and, arguably, observed. We return to this point in the conclusion. After a brief exposition of Hirshleifer’s own analysis in section 2, there is a detailed analysis of transfers in sections 3 and 4. Here we find that public good theory of a more traditional kind becomes relevant. Indeed, we are able to replicate many of the results in BBV. Section 5 analyses the impact of a disaster on voluntary transfers, thus providing one possible way of developing predictions which can distinguish a weak link from a critical mass view of co-operative behaviour. Taking the model to represent ‘ordinary times’, the concluding section examines what predictions we might make about voluntary transfers, and compares these with some ‘stylised facts’.
‘It is not clear that Agnelli would accept the interpretations here. 3Hirshleifer mentioned this possibility in his original paper, referee.
of this story
that are discussed
and it was also mentioned
by a
S. Vicary, Transfers
378
and the weakest-link
2. Hirshleifer’s analysis To understand the Hirshleifer model we can imagine which each individual maximises the function:4
an n-person
model in
ui(xi, G) subject
to
wi = xi +g,,
i=1,2 )...) n,
and where
G=min{g,},
i=1,2 ,..., n
where Ui represents the utility of individual i, xi the quantity of the private good consumed by i, G the quantity of the public good available, wi the income level of i, and gi i’s contribution to the public good. We assume constant costs and normalise the units of each good such that all prices are unity. To solve this model we may simply note that no individual will buy more of the public good than anyone else. To do so would cause a loss of consumption of the private good, and no compensating increase in the public good. We can thus imagine that each person decides on an ‘ideal’ quantity of G, assuming this to be matched by all other members of the community. That is, we solve the above problem assuming that G=g,. If ii represents the optimal amount of G for individual i; then the actual amount enjoyed by the community will be min {g,). All those who wish for more G will be unable to have their wishes fulfilled, and what they might otherwise have spent on the public good will now be devoted to the private good. The solution for a two-person economy is depicted in fig. 1. Here we see the reaction curves R, and R, for the two individuals 1 and 2, together with a pair of illustrative indifference curves, IC, and IC,, one for each person. In this case 8,
4Hirshleifer allows the marginal complication in what follows.
cost
to differ
between
individuals.
We will ignore
this
S. Vicary,
“ramfen and the weakest-link
379
/
Fig. 1. Equilibrium without transfers. Individual l’s ideal point is at A, and 2’s at B. Nash Equilibria lie along the line OA, but the outcome is taken to be at A.
economy will be in equilibrium at point A.’ Along the line AB we have a locus of Pareto optima, indicating that A itself is Pareto optimal. There is nevertheless a sense in which the good is underprovided, as can be seen by examining the Samuelson (1955) condition for optimal provision, which in this case is 1 MRS’,,= n. At A, MRSh,= 1 and MRS&_> 1, so the level of G is suboptimal. There is, however, a problem. The way to achieve efficiency with the standard social composition function is to try to devise some form sAs pointed out by Hirshleifer, all points on the line OA are Nash Equilibria. We have here what Sen (1967) called an assurance game, the Pareto preferred point among them having stability properties absent from the prisoner’s dilemma game, which is usually taken as capturing the problem of ensuring co-operation. Harrison and Hirshleifer (1989) point out that the selection of point A can be justified as Selten’s (1975) sub-game-perfect equilibrium, if the game is played sequentially. This is not possible if strategies are played as sealed bids, and neither the perfectness concept nor Myerson’s (1978) notion of a proper equilibrium can justify the choice of A, despite its intuitive appeal. Nevertheless, the experimental results of Harrison and Hirshleifer support this outcome.
380
S. Vim-y, Transfers and the weakest-link
of collective provision of the public good. With Hirshleifer’s Anarchia parable (1983), this might be possible if the islanders conquered their aversion to collective action. However, where this model is being applied to co-operative behaviour under conditions of adversity, the public good is inextricably linked to individual action. A/I members of the community must contribute, and it does not seem possible, if we take the weakest-link idea seriously, to have the quantity decision delegated to some central agent, even through a voting device such as the Lindahl mechanism. In any case, in the above example individual 1 has already achieved an ideal outcome at point A. Collective provision will make him worse off, unless accompanied by a transfer. However, we have not yet exhausted the potential for individual action. Imagine we increase 2’s income in the above example. As the economy will remain at point A, the whole of any increase in 2’s income will be devoted to the private good, so that MRS& (= -dx/dG) will be increasing the whole time (we assume normality). Hence, there is likely to be some income level for 2 beyond which it will become worthwhile to give money to 1 even though only a fraction of the money donated will actually be spent on G. To ensure that such an income level exists we will assume that for all individuals lim,, o. MRSL,= GO. The next section is based on this insight.’ Whilst this paper concentrates on the weakest-link, the possibility of transfers of the nature analysed here extend wider than this basic model. Whenever the effect on the marginal provision of the public good of $1’s worth of expenditure varies as between individuals there is the potential for it being cheaper to acquire extra units by transfers rather than by direct purchase. Hirshleifer’s best-shot case is interesting for possibly inducing transfers to just one individual if an efficient collective provision mechanism can be derived. Another example is the social composition function G = f(c a,gJ, as used by Sugden (1984). In an unpublished paper Cornes has examined the social for transfers, in composition function [I gy] I”, which also has a potential this case along the lines analysed here. It should be noted that in the case of the critical mass public good there is no difference between the impact of $1’s worth of expenditure on provision as between individuals. Consequently, no transfers will take place, an important difference from the weakest-link.
3. Analysis of transfers We now consider the economy of n individuals described in section 2. We assume there exists a set of individuals with incomes sufliciently high to make them possibly interested in donating money so as to raise the amount ‘Gifts are impersonal, and 1 and 2 cannot spent. We comment on this in the conclusion. of recipients is ignored.
make a contract to determine how the money is The possibility of strategic behaviour on the part
S. Vicary, Transfers
and the weakest-link
381
of G. Clearly, any such money donated will go only to that individual or those individuals who have the minimum demand for G. Let us call this set R. The cost of increasing the amount of G by one unit for one individual can be written as follows:
iFR(~G/‘~Wi)-’ + 1,’ where the first term represents what has to be paid to the members of R in order to induce each one to raise consumption of G by one unit. The second term (unity) is simply the direct cost of purchasing one more unit of G. It is evident from this that if there exists a group of individuals interested in donating money to individuals in group R, then donations have the property of a conventional public good. If individual i donates some money to R, then individual j (also a potential donor) need only incur the direct cost of gaining extra G and can take a free ride on i’s donation. The benefits to an individual thus vary with the sum of the donations from others, and the optimum individual donation can be derived given this total. In short, we have for potential donors a conventional problem in the private allocation of public goods. To analyse this problem formally we will assume the existence of a charity which accepts donations, and distributes them to the members of R in such a way that G increases at minimum cost. Given the comments at the end of section 2, we can start from an initial solution of no donations, where for each agent i there is an income level wT below which i will not be interested in donating money. Let us now define three sets of individuals:
R=
T={il Wi>Wi*}. In considering total donations to the charity we need only focus our attention on 7: Members of S will never become donors, although they may become recipients. Each individual in T solves the following problem: ‘dD=xdwi, ieR, and dwi=dG/(3G/dw,), term in the expression.
igR,
imply dD/dG=~(dG/8w,)~‘,
which is the first
382
S. Vicary, Transfers
and the weakest-link
max Ui(Xi, G) X,.Qi subject
to
where d, represents i’s donation to the charity. Ui is assumed to be twice differentiable and preferences are assumed to be strictly convex (these assumptions hold for all agents). If we now define D=xd,
and
D-i=xdj
isT
jsT j#i
we can rewrite this problem
as:
max Ui(Xi, G) X,,D subject
to Wi+D_i=Xi+G(D)+D.
Here we take G=g, and use the function G(D) which determines the quantity of G given total donations. Our initial task is to prove the existence of a Nash equilibrium in donations. To this end we need to establish certain properties of the G(D) function. Firstly, given the differentiability properties of all utility functions, and the consequent continuity and differentiability of demand functions, G(D) is continuous. From the assumption of normality it is also monotonic strictly increasing and so has an inverse. Where there is no change in the set of recipients of transfers the function will also be differentiable, the value of the derivative being found from the following argument: dD=
1 dw,=dG ieR
c (aw,/~?G) iGR
so that
where (c~w~/LJG)is the inverse of i’s marginal propensity to consume G. However, as donations increase, individuals will gradually move from set S to set R, and when this occurs an extra term is added to the summation inside the square brackets. In short, the G(D) function is kinked at points at
S. Vicary, Transfers
383
and the weakest-link
which an extra individual becomes a recipient. Given normal goods, the derivative G’, where it exists, will be positive. Left-hand and right-hand derivatives will exist at all points, and at the kinks dG/dD 1,,>dG/dD ( RH. We can now form a picture of the budget constraint facing each individual in 7: Adopting the conventional assumption of fixed donations by the other members of T, we have, from the budget constraint: dx,+dG+D’dG=O for points where D’, the derivative follows that for such points: dxJdG=
of the inverse
function,
is defined,
It
+D’).
-(l
To ensure convexity of the budget set we assume that for all individuals aG/aw, is a non-increasing function of income. This guarantees a nonnegative sign for D” (where defined). This is obviously restrictive, but avoids many complications. Furthermore, it follows from the properties of G(D) that at all points with strict inequality at the kinks. This gives D’\,H4D’(RH convexity of the budget set. The individual’s choice can be visualised in fig. 2. Given strictly convex preferences there will be a unique choice of xi and G, these implying an optimal amount of donations for i. We can state the solution of the problem by the vector (x:,dT). This solution can be characterised as follows. With the differentiability properties of Ui, the MRS,, is defined at every point of the positive orthant. Strict convexity of preference and convexity of the choice set mean that the necessary and sufficient condition for i’s utility maximisation is (1 + D’),, >=MRSL, 2 (1 + D’)LH (the subscripts LH and RH here refer to the one-sided derivatives and (dxJdG),,], the first inequality of the budget constraint [i.e. (dx,/dG),, ensuring that the individual does not wish to increase G (or donations), the second that no decrease is desired. Our next step is to try to prove the existence of a Nash equilibrium. We have, first of all: Definition. A Nash (Xi, di) =(x), d:). Theorem
Proof.
1.
Equilibrium
A Nash
Define the following d:=fi(wi+D_J,
Given
Equilibrium
the assumptions
is a vector of donations
in donations
(dJ such that V in IT;
exists.
function: iE 7:
we have
made,
this function
is defined
and
conti-
384
Fig. 2. Individual drawn here. A Members of set members of T to
S. Vicary, Transfers and the weakest-link
utility maximisation. In principle, all agents will face the budget constraint as corresponds to some initial quantity of G before any receipt of transfers. R would locate themselves on MA, members of set S would be at A and the right of A. Points B and C represent points where extra individuals join set R as a result of the donation our individual makes.
JET), where t is the nuous. a Now define the set W={d~tR’~O~d~=
S. Vicary,
Transiers
and the weakest-link
385
In the light of Theorem 1 this needs some modification, because embedded in the weakest-link model is a conventional public goods problem. Thus, whilst underprovision is less of a problem as a result of transfers, to the extent to which transfers fail to respond as community size grows (income distribution remaining much the same), the problem re-emerges. Of course, this assumes that the Olson problem actually exists in this form, and, given that donations can be thought of as a normal good, this may not in fact be justified [see, for example, Cornes and Sandler (1985, pp. 82-84)]. For the rest of the paper it will be convenient to start any formal analysis from an initial equilibrium involving some positive donations. Hence, we need to redefine our sets. Given any equilibrium, for each individual i there must exist a ki such that i will, in the absence of any transfers, at that income, demand exactly the quantity of G determined. Thus we can define:
R={il Wi~ki}: S={iIWi>ivi,
di=O},
T=jild,>O}.
Our next Uniqueness Theorem
task is to establish the here refers to the quantity 2.
There
is
a
unique
uniqueness of G.
quantity
of
of the
G
Nash
consistent
equilibrium.
with
a
Nash
Equilibrium. Proof.
Suppose there are two different levels of G consistent with a Nash Equilibrium, G, and G,, with G,
386
S. Vicary, Transfers and the weakest-link
As with conventional public goods we find a number of neutralities. However, whilst our results parallel those in BBV, there are some differences resulting from the changing membership of sets R, S and 7: Let us first consider ‘small’ redistributions of income, that is where the change in income is less than either donations made or received. Theorem 3. For the case of a small redistribution change in the quantity of G provided.
of income
there
is no
Proof. There are three cases to consider: (a) redistributions within the set T; (b) redistributions within the set R; and (c) redistributions between these two sets. We take each in turn. (a) Assume for each ig T Ad,= Awi. Total donations remain the same, so there is no change in G. Also, Ax,=0 for all ie 7: Hence, for all individuals there is no change in MRS,,. As AG=O there will be no change in the value of the one-sided derivatives of the budget constraint at this new position, which must therefore constitute a new (and unique) Nash Equilibrium. (b) In this case assume the charity adjusts contributions to each jE R so as to ensure that A(r,+ wj) =O, where rj represents transfers to j. The budget constraint of each j E R, after allowing for transfers, remains the same so that AG=O. If all agents in T maintain the same donations as before, then they all meet the conditions for utility maximisation. We have again constructed a new Nash Equilibrium with G unchanged. (c) Assume Awi = Adi V i E T and AWj= - Arj Vj E R. This is a feasible reaction as O=CAw,+c Awj=c Ad,-CAri. As A(wj+rj)=O VjE R, AG=O. Each iE T maintains the same consumption of X, so that there is no change in any marginal rate of substitution. Also there is no change in the one-sided derivatives of the budget constraint. Thus, in this new position all agents are maximising utility and we have again constructed the new (unique) Nash Equilibrium. Q.E.D. We now consider
‘large’ redistributions.
Theorem 4. If a redistribution of income occurs between the members then G cannot fall. Jf G rises, then the contributors after the redistribution a proper subset of 7:
of T form
Proof. Suppose, in contrast to the statement of the theorem, AG ~0. Then there must exist at least one individual iE T such that Axi>O. For this i, MRS& > MRS&, where the superscripts refer to the pre (1) and post (2) redistribution equilibria. As dG < 0, (1 + D’)&, s( 1 + 0’);“. Thus, MRSzx > for utility maximisation in case 1. This (I+ D’)iL, using the conditions contradicts the condition for utility maximisation in case 2. Hence, AG 2 0.
S. Vicary, Transfers
and the weakest-link
387
Now suppose AG >O, and that all previous donors continue to donate. There must exist at least one individual jE T who now has a lower consumption of the private good. Thus, MRS-&< MRS’,‘,. As AC>0 we have which, with the conditions for utility maximisation in (1 +D’);“5(1 +D’);,, case 1, yields MRS& <( 1 + D’)tn, contradicting the condition for utility maximisation by j in case 2. As it is obvious that with AG>O no member of Q.E.D. S or R will start to contribute, the theorem is proved. Theorem 4 produces results parallel to those in BBV, and represents an extension of Theorem 3 in that we do not restrict ourselves to ‘small’ redistributions. We may now consider what happens when redistribution raises the aggregate income of the members of set 7: As we have dealt with redistributions within 7: we can now focus on bilateral transfers involving sets R and S. It is easy to see that a redistribution from an individual in R to an individual in T has no effect on G. Furthermore, a redistribution from someone in set S to someone in set T cannot cause G to fall. This can be seen by supposing the ‘loser’ remains in set S and by pursuing the reasoning in the first part of the proof of Theorem 4 to set 7: Should the loser join R, then there will be no effect on G for that part of the redistribution which gives him an income before transfers of less than +, the borderline income for this individual (allowing for any change in G up to this point). Hence, for redistributions involving members of set T for which the aggregate wealth of the members of T rises G cannot fall, this result following from combining the results we have derived up to now. However, some complications arise from considering set S. Obviously ‘small’ redistributions within S which do not change the membership of S do not change G. But if such a redistribution causes someone to move into T the quantity of G will rise (by definition members of T make positive donations). If a loser joins R, then G may fa1k9 the presumption that this will be so if the redistribution is large enough. Similarly, a redistribution from a member of R to a member of S may cause G to fall, this being the presumption for a large enough enough change. Consequently, we are not able to say that AG 20 for a general redistribution in which zieT Aw,>O. 4. Identical tastes It will now be useful to make
the assumption
of identical
tastes
between
‘Initially we have (1+ D’),,, 2 MRS,,8( 1 + D’),,, V in 7: The consequence of bringing an individual in S down to the border with R is to raise the value of (1 + D’),,,. A further fall in this individual’s income will cause (1+ D’),,, to rise by an equal amount. If the initial equilibrium was not at a kink then, as fixing G causes MRScx to fall, dG=O is not possible, and, pursuing familiar reasoning, neither is dG>O. However, if the initial equilibrium is at a kink the rise in (1 +D’),, may not be enough to reverse the inequality with MRS’,,. For a large enough transfer this will occur, however, due to the effect on MRSL,, iE 7:
388
S. Vicary,
Transfers
and the weakest-link
individuals. This will enable us to gain some intuition about the pattern of transfers we might observe in an economy with a weakest-link public good, as well as helping us to analyse the consequence of greater equality for the provision of G. Given our previous work there will be a unique equilibrium level of G determined by the Nash process. This will in turn give us particular values and contributors will be maximising utility for (1 SD’),, and (1 +D’),,, when (1 + D’hus MRS,,s( 1 +D’),,. Let w’ and w” be the levels of income such that in the absence of donations MRS,, equals (1 +D’),, and i is definitely (not) a donor if wi > w’ (1 +D’)I.H, respectively. An individual (wi < ~3”). An individual i with w” < wil w’ may or may not become a donor, but no donor will have a post-transfer income of less than w”. As a certain quantity of G is voluntarily provided and as G is a normal good, it follows that all individuals with income below some level w* will receive donations in final equilibrium. Also, unlike the conventional case, individuals in set T do not necessarily end up with the same utility level unless w’=w”. We now examine the impact of equalising income redistributions G. Obviously, equalising redistributions within sets R and S have no effect on G. In the former case redistribution is anyway made redundant by the transfers from set 7: The case of equalising redistributions within set T is a special case of Theorem 3 and there is likewise no effect on G. If we consider a redistribution from a member of set T to a member of set S, then dG 50. In the light of the last paragraph we need only consider ‘small’ changes in initial income which do not change either individual’s status. Suppose dG>O. In the new equilibrium (2) there must be an individual iE T for whom xi has fallen so that MRSgx> MRS$. As AC>0 which implies (1 + LY);, > MRS& which we have (l+D’)~H~(l+D’)~H, contradicts the condition for utility maximisation in the conjectured new equilibrium. We cannot, however, eliminate the possibility that AC=0 as initially the economy may be at a kink in the G(D) function. For large enough redistributions, however, AC < 0. For the case of equalising redistributions between R and S we have the following. For ‘small’ (no change of status) redistributions, the effect of increasing the income of some i E R is to cause a local upward shift in the budget constraint facing the members of T around the initial level of G. This is illustrated in fig. 3. Given utility maximisation in the original position, if will rise. It follows from familiar AC ~0, then for some jg T MRS’,, reasoning that this is not possible and that AGZO. Again, the possibility that initially we were at a kink means we cannot eliminate AG=O. The case of equalising income between sets R and T follows immediately from what has just been said. Where there is no change in status there is the familiar neutrality result. When the ie T joins the set S we have AC 20, and where je R joins S we have AGSO.
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and the weakest-link
389
X
, Fig. 3. The impact of increased donations on a donor’s budget constraint. After the redistribution to jE R, the output at which resources begin to be transferred to j rises from M to N.
Table Income
redistribution
1 and the impact
on
G.
XT
S
R
T
AC20
AGjO
AC=?
S
AC20
AC=?
AC20
R
AC=0
AC50
AC50
It may be useful to summarise our results by means of table 1. It is evident from this that there is no general presumption about equalising redistributions on the provision of G, particularly when we focus on redistributions from the very wealthy. On the other hand, redistributions from the middle income range to the poor may incrase the provision of G. It should, however,
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be noted that the weak inequalities in the table generally result from an initial equilibrium at a kink, and that for sufficiently large redistributions the strong inequality usually holds.”
5. Analysis of disasters Our final analytical section deals with a question posed by Hirshleifer. We consider how co-operative behaviour responds to a disaster. We interpret ‘disaster’ here as a widespread fall in income in the community. Co-operative behaviour here can mean two things: total provision of the public good G; or the total amount of transfers made. Two cases will capture the main likely consequences of a general disaster. (a) Suppose dwi
S. Vicary, Transfers and the weakest-link
391
Fig. 4. The change in the post-disaster equilibrium of a donor. The fall in the income of recipients causes a downward shift in the budget constraint of a donor. The slope of the budget constraint to the right of B does not change, and given the definition of R, the slope to the left of B cannot become gentler. This must induce a fall in G for a sulliciently large disaster.
indifference curve at point B must be steeper than at point A, all individual members of T would wish to increase donations. Using familiar reasoning, dG
392
S. Vicary, Transfers and the weakest-link
Fig. 5. The impact of a disaster on donations. If all donors maintain consumption of the private good, then movement is from B to A. At B, however, all donors wish to increase donations. B cannot be an equilibrium.
better way of understanding some adversity, and at other times as well.
aspects
of the
social
alliance
during
6. Conclusion In this concluding section we examine some of the predictions that the model seems to suggest once we accept that co-operative behaviour (not necessarily at a time of adversity) is in some way captured by the weakestlink idea. As it happens there is a general consistency with observations made by a number of writers, notably Tullock (1986). Firstly, transfers in the model are predominantly from the very wealthy to the very poor, and will be greater the more unequally income is distributed. This accords at least with the casual observation that people do not give money (for redistributional purposes) to those who have an income level similar to their own. As a consequence of this, the middle income groups are not greatly involved in
S. Vicary, Transfers and the weakest-link
393
voluntary transfers, although they benefit from the transferring that does takes place. One awkward piece of evidence is that charitable donations as a proportion of income do not seem to increase with income as the analysis of section 4 seems to suggest [see, for example, Collard (1978)]. However, many charities are not redistributive in the sense meant here, and often can be thought of as providing a conventional public good. Also, the existence of government policy on income distribution needs to be taken into account. Nevertheless a disaggregated study of charities may represent one way of testing the model. Secondly, it will be noted from our analysis that transferring is an expensive way to raise G. If donors could, relatively cheaply, force recipients to increase expenditure on G, then they will do so. Hence, legislation to fix certain levels of expenditure, or quantity restrictions on the way donations may be spent, are likely to be observed. Thus, this model provides one explanation as to why so much redistribution is in kind, and it does so without any paternalistic assumptions. It should be noted, however, that in our model it is not necessarily cheaper to provide recipients with G (assuming this is possible), because recipients could substitute x for G in their expenditure. To achieve a substantial fall in costs donors must directly interfere with the consumption quantity decisions of recipients. In the context of some of the examples listed at the beginning of the paper, this may not necessarily be a problem, however. If an actor is experiencing difficulty with his part, then the other players may consider time well spent in helping him to improve, and there are various ways in which compensation can be made for a poor individual performance on the night. It is possible, though, that there may be some moral hazard problem. Thirdly, when we consider the reasons for transferring it is obvious that transfers only take place to the extent that recipients provide extra G for donors. If we suppose the number of public goods thus provided varies inversely with the ‘distance’ (geographic, class or whatever) between the individuals involved we have a possible explanation as to why the bulk of the redistribution we observe is within rather than between communities (Tullock puts a lot of emphasis on this point). Fourthly, we may note that donations are determined by a process directly comparable to the private allocation of a standard public good. This means that there is underprovision of G after the transfer [( 1 + D’)Rr, > 11, even from the point of view of donors. Thus, the model provides an explanation of the fact that governments are involved in redistribution. Naturally, this will be limited if donors maintain control of the political machine. None of this denies the existence of other factors in income redistribution. Nevertheless the extent to which the predicted pattern of redistribution corresponds to observations is intriguing, and may by itself make the weakest-link model worthy of further scrutiny.
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References Agnelli, S., 1986, Street children (Weidenfeld). Bergstrom, T., L. Blume and H. Varian, 1986, On the private provision of public goods, Journal of Public Economics 29, 25549. Bliss, C. and B. Nalebuff, 1984, Dragon-slaying and ballroom dancing: The private supply of a public good, Journal of Public Economics 25. 1-l 2. Collard, D., 1978, Altruism and Economy (Martin Robertson). Cornes, R., Dyke maintenance, dragon-slaying and other stories: Some neglected types of public goods, Unpublished paper, Australian National University. Cornes. R. and T. Sandier, 1985, The theory of externalities, public goods and club goods (Cambridge University Press, Cambridge). Harrison, G.W. and J. Hirshleifer, 1989, An experimental evaluation of weakest-link/best-shot models of public goods, Journal of Political Economy 97, 201-225. Hirshleifer, J., 1983, From weakest-link to best-shot: The voluntary provision of public goods, Public Choice 41, 37 I -386. Hirshleifer, J., 1985, From weakest-link to best-shot: Correction, Public Choice 56, 221-223. Hirshleifer, J., 1987, Economic behaviour in adversity (Harvester Press). Myerson, R.B., 1978, Refinements of the Nash equilibrium concept, International Journal of Game Theory 7, 73-80. Olson, M., 1965, The logic of collective action (Harvard University, Cambridge). Palfrey, T.R. and H. Rosenthal, 1984, Participation and the provision of discrete public goods: A strategic analysis, Journal of Public Economics 24, 171-193. Samuelson, P.A., 1955, A diagrammatic exposition of a theory of public expenditure, Review of Economics and Statistics 37, 35&356. Selten, R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4, 25555. Sen, A.K., 1967, Isolation, assurance and the social rate of discount, Quarterly Journal of Economics 81, 112-124. Sugden, R., 1984, Reciprocity: The supply of public goods through voluntary contributions, Economic Journal 94, 772-787. Tullock, G., 1986, The economics of wealth and poverty (Harvester Press)