Dynamics of social ties and local public good provision

JOURNALOF

PUBLIC ELSEVIER

Journal of Public Economics 64 (1997) 323-341

ECONOMICS

Dynamics of social ties and local public good provision Frans van Dijk*, Frans van Winden CREED, Facul~.' of Economics and Econometrics, University of Amsterdam, lOI8 WB Amsterdam The Netherlands

Received ! December 1993; revised I June 1996; accepted ! June 1996

Abstract A model is presented in which social ties between individuals and private contributions to a local public good are interrelated. Ties are formalized by means of utility interde~i~deiice, aild dei~id oa ihc |-~i~ioryof social interaction, in this case the joint provision of the public good. The resulting dynamic model generates equilibrium values of the intensity of ties and the private provision level. The impact of public provision on these variables is analyzed. Our results are very different from those obtained with the standard model, where individuals are only interested in the utility from own consumption. Kevwords: Local public goods; Social ties; Voluntary provision JEL classification: Ai3; D64: H41; H70

1. Introduction In many economic issues social ties between individuals are important. This has been argued by social-psychologists and sociologists over the years and, more recently, by a number of economists, among which is Becker (1974). Positive social ties provide individuals with additional resources in times of need, help to control externalities, form a basis for a cooperative outcome o f a prisoner's dilemma, and ensure fair business dealings (Coleman, 1990; Granovetter, 1985). Negative ties may bring about the reverse. Where these ties are relevant, the *Corresponding author. 0047-2727/97/$17.00 O 1997 Elsevier Science S.A. All rights reserved Pll S0047-2727(96)01620-9

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factors determining their formation become of interest. In this article we investigate the development and impact of such ties in the context o f the private provision o f public goods in small communities where they are expected to play a significant role'. Studies of the private provision of a public good usually assume that individuals determine their contributions non-cooperatively in a one-shot game. Contributions of other individuals are taken as given, and only the utility derived from own consumption is considered. We refer to this set-up as the standard model (for a rigorous treatment, see Bergstrom e t a l . (1986)). Equilibrium contributions fall below the social optimum in this model, and this shortfall increases with the number of players. Disregarding comer solutions, the total availability of the public good is invariant to the income distribution as well as public provision. When the interaction is repeated a known finite number o f times, the outcome is the same as that of the one-shot game. With an infinite or uncertain timespan any outcome can be sustained (see, e.g., Fudenberg and Tirole (1991)). However, if individuals are myopic, the results of the standard model are still valid. We will use this model as a bench-mark. Many variants of the standard model and alternative models have been developed (e.g., Sugden (1984); Comes and Sandier (1985); Andreoni (1989)), but also these models do not explicitly allow for social ties. Interestingly, experimental results show that 'group identity' affects contributions, indicating that there is more to the provision of public goods than is currently captured by economic models (see Dawes and Thaler (1988)). Our model focuses on the affective component of social ties, its development, and its impact on private provision. Ties are formalized by means of weighted interdependent utility functions z. Weights attached to the utility of other individuals can be positive, zero or negative. An important difference with the existing literature is that these weights are not constant over time, but depend on the history o f interaction between the individuals. The formation o f ties is determined by the contributions of individuals, resulting in a simultaneous, dynamic model of social ties and public good contributions. We also investigate the impact of public provision and taxation. It should be emphasized immediately, however, that we do not claim to fully capture the very complex social process that ~s at stake here. The incorporation of the weights in the model, and the use made o f psychological notions and findings is still rather ad hoc. In the absence of a fundamental theory of interpersonal relations that duly takes account of the ~For convenience,we refer to these goods as local public goods, in contrast with the .public finance literature, where the term generally refers to public goods provided by municipalities. -'In the economic literature utility interdependence is frequently assumed (ad hoc) to tackle phenomena that are hard to explain by pure individualism, such as behavior within families (including intergenerational transfers), voluntary redistribution, charity and crime (e.g., Barro (1974); Becker (1981); Hochman and Nitzan (1985)). In these models utility interdependence is used, but not explained. An exception is Guttman et al. (1992) where a uniform level of altruism is determined by education.

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affective component (sentiments and emotions), this is unavoidable. Hopefully, the analysis presented below will motivate such fundamental research. Our main results are in stark contrast with those of the standard model. F i r s t , in many cases private provision is higher. It may even approach the provision level that follows from the maximization of a social welfare function. A lower provision level can also occur, however. The outcome is determined by the combined effect of the spread of incomes and preferences, and the so-called tolerance level of individuals, which refers to the appreciation by an individual of the contributions of others. When any contribution is valued positively, and incomes are equal, a high provision level is reached. If people have a negative appreciation of any contribution smaller than their own, high provision levels cannot be attained. When in addition incomes differ the provision level falls below that of the standard model. Thus, in our model private provision is not invariant to the income distribution. S e c o n d , public provision leads to a decrease of total provision, by impeding the development of social ties. Again invariance is not found. Total provisian reaches a minimum when, in the two-person case presented in detail, at least one of the private contributions has just been reduced to nil. It is only, when public provision is further raised that total provision starts to increase. Furthermore, as the development of ties takes time, a reduction in public provision is not immediately taken over by private initiative, if at all. Consequently, public programs are more difficult to end than to initiate. The organization of the article is as follows. Social ties are formalized in Section 2. In addition, the impact of ties on public good provision is briefly examined. Section 3 goes into the dynamics of social ties. In Section 4 the complete model of private provision in the absence of government intervention is analyzed, while public provision is added in Section 5. Section 6 concludes.

2. Social ties and their impact on public good provision Among the variables used by sociologists and social psychologists to characterize social ties--emotional intensity, reciprocal services, mutual confidence and others--the affective component is considered to be the key element of a tie; see Granovetter (1973) and the review of the empirical literature on personal attachments by Baumeister and Leary (1995). This enables a one-dimensional conceptualization as a first approach, and it provides a natural way of defining positive and negative ties, as well as the intensity of ties (which may be different for the subjects involved; see Weiiman (1988)). A social tie between two individuals i a n d j is assumed to consist of i's (j's) sentiments about j (i), where sentiments or feelings are defined as the extent to which i (j) cares about j ' s (i's) welfare and derives satisfaction from it. This is formalized by using the standard concept of utility interdependence. A tie between two individuals i and j is described by the pair (%, aii) in the following expression:

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F. van Dijk, F. van Winden I Journal of Public Economics 64 (1997) 323-341 V, = UhU~h~ (h, k = i, j, h ~ k)

(1)

where Vh stands for the total utility o f individual h, and Uh for the utility h derives from his own consumption.. If individuals maintain ties with other persons as well, their weighted utility enters Eq. ( l ) in the same way. Alteruativ¢ly, one could substitute total utility Vk for Uk, as in Bernheim and Stark (1988). In the case o f two individuals, that will be focused on below, results are not affected by the choice o f specification. The domain of ah~ is assumed to be restricted to the interval ( - l , l), which implies that people neither 'love' nor "hate' others more than they love themselves. This seems to be a reasonable restriction for which substantial experimental evidence exists (e.g., Sawyer (1966); Liebrand (1984)). We now turn to the impact o f social ties on public good provision. To bring out ~he interesting aspects o f the problem, it suffices to focus on a stylized local community with two inhabitants. The results can easily be generalized to larger communities (see Section 6). Let Uh=Uh(Xh,g) ( h = i , j ) , where x denotes the consumption of a private good, and g the consumption of a public good. Preferences are assumed to be strictly convex, and the prices of both goods are taken to be one. With Yh and gh, respectively, denoting h's income and contribution to the public good, yh----xh+gh and g=g~+g~. Individuals take each other's contribution to the public good as given. When both ahk (h, k = i, j, h # k ) are zero, the standard model is obtained. The first-order condition for the maximization of Eq. (1) is given by: •

3

Og + ahk Og Uk-- Oxh.

(2)

Eq. (2) presuppose' ,hat the left-hand side is positive, which is not necessarily so. When non-positive, gh is zero and x h equals Yh" This case is covered by the requirement that contributions are non-negative. It follows for h's reaction function g h r that gh=max{ghr(yh, Yk, gk, ahk), 0}. In general, OghrlOt~h~>O. Assuming that x and g are normal goods, O < 0 , as the direction of change resulting from an increase in Yk (X~, given g~) is not determined. It can be shown that the preferences represented by the extended utility functions are strictly convex, as long as the individuals derive positive utility from the public good (the problem is trivial if this is not the case). As a result the structure of the problem is the same as that of the standard model, and a unique equilibrium exists (see Bergstrom et al. (1986)). Assuming an interior solution, ~The resulting total (or extended) utility functions are formally identical to the individual social welfare functions used by Arrow (1981). However,the underlyingconceptsare completelydifferent.In Arrow's study a social welfare function reflects an individual's 'social conscience' concerning the distribution of income (a normative problem), whereas total utility Vh representsthe actual concern for and satisfaction derived from the welfare of specific others.

F. van Dijk, F. van Winden I Journal of Public Economics 64 (1997) 323-341 gh = gh(Yh" Yk" ahk, Otkh) (h, k = i, j, h ~ k)

327

(3)

It follows from the properties of the reaction functions that OghlOahk>O, Ogkl OahkO. Thus, the impact of an increase in ahk is unambiguous:. Three special cases with regard to the a ' s are of interest. First, the standard model is obtained when a 0 = aji=0. A second interesting case is % = a j : - - > l , where V~= Vj, which means that i and j maximize the same function under separate budget constraints. Contributions reach a maximum (gmaX), and it can be easily shown that this leads to the same result as the maximization of a social welfare function S=Vy~, or S=U~Uj, under an aggregate budget constraint and the constraint that the consumption of the private good by an individual does not exceed her or his income. Consequently, full cooperation is achieved. A third special case holds when a o = a~:-->-I. In this case i and j become maximally uncooperative and g reaches a minimum (groin). When preferences and incomes of i a n d j are equal, groin is zero. Otherwise, gl or gj is zero. The standard model is an intermediate case, where the private provision level falls short of the maximum, but is above the potential minimum. We conclude that the sign and intensity of social ties among the inhabitants of a community influence private provision. The development of these ties is discussed in the next section. In concluding this section we consider the consequences of government intervention, assuming that public provision is financed through (local) taxation. The budget constraint of individual h (h = h j ) now reads Yh =Xh + gh + 7"hgg, where gg denotes the quantity of the public good provided by the government, and 7"h is h's tax share. Furthermore, g = g i + g j + g g . In case of an interior solution, the equilibrium is given by: gh

"~

gh(Yh, Yk, ~lhk, ~tkh)

--

'~hgg (h, k = i, j, h ~ k).

(4)

This is the familiar invariance result obtained by Warr (1982) for the standard case. For given social ties, public provision crowds out private contributions one to one.

3. Formation of social ties Evidently, a uniform level of sentiments towards other individuals does not exist. Sentiments not only differ between individuals, they also vary over time. Whether some base leve! of affection or concern occurs among individuals who have never interacted before, is open to debate, and we abstract from it here. Regarding the dynamics of social ties, we take into account that sentiments 4Due to the secondary,undeterminedcross-effectof Yhon & in the reaction function,the signs of the derivatives with respect to Yhaxe not as clear-cut. In general, OgJ~yh>O. OgklOy~O, and this will be assumed throughout.

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develop through prolonged interaction (Baumeister and Leary, 1995). Furthermore, as argued by Coleman (1990), ties are subject to decay over time, when not actively maintained. As to the development of ties we use the hypothesis that positively valued interaction is likely to generate positive sentiments, whereas the reverse holds for negatively valued interaction. This correspondence between interpersonal experiences and sentiments has been argued and substantiated by many social scientists over the years (see Homans (1950); Simon (1952); Fararo (1989); Frijda (1986)). Whether in our case the interaction will be valued positively or negatively depends on the contributions. Two aspects are likely to play an important role. First, as any contribution by j is helpful in producing the public good which is positively valued by i, this is expected to lead to a positive valuation by i of the interaction (with similar reasoning applying to j). Second, it stands to reason that the valuation of the ;,~teraction will be positively (negatively) affected if the contribution by the other exceeds (falls short of) i's or j ' s own contribution. This comparison of contributions may be elicited by a notion of fairness. 5 Weighing the one aspect against the other leads to the following criterion for positive (negative) sentiments: Ghk--gk --sh'gh > ( < ) 0 (h, k = i, j, h # k ) , where the weight ~, (O-
(5)

The change of sentiments depends on the weighted relative contributions (Ghk) and the state of the sentiments (ahk). A diminishing marginal (positive) effect of Ghk is assumed. The effect of ahk is twofold. First, ahk is taken to influence the impact of Ghk such that sentiments remain within their boundaries ( - 1 , 1). Second, sentiments decay over time. Consequently, fh is taken to be an increasing, S-shaped, function of Ghk (the 'impulse'), such that ~---> 1 - %k ( - 1 - ot~,k) when Ghk--->~ (--~c). When Ghk =0, fh = 0 if ties do not decay over time, whereasfh < 0 if decay occurs and ahk#0. The functionft , is continuous and twice differentiable. In the numerical example of the appendix (Appendix A) a function exhibiting these properties is presented. Fig. 1 gives the phase-diagram of Eq. (5) for e~ >0. For clarification, suppose that ties would not decay over time. In that case the stationary points, where dah~ldt=fh=O, are represented by the broken line b. When the impulse Ghk is zero, stationarity is obtained for any ahk. For all positive values of Ghk, trhk grows to its upperbound 1, and for all negative values to its lower bound - 1. For any ahk # 0 to qualify as a stationary point the countereffect of the decay of the tie requires Ghk # 0 . This is shown by the fat line in the figure. ~The relevance of fairness has been shown particularly in the context of ultimatum-bargaining games (Thaler, 1988). Its role in public good experiments is still subject of research (Ledyard, 1993). See also Rabin {1993) on fairness in general.

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329

It

i/

L/. -1

Fig. 1. Phase-diagramof Eq. 15) (~h>O) (a)f~=O with attrition; (b)fh =0 without attrition.

In that case ahk only approaches its boundary 1( - 1) for infinite positive(negative) values of the impulse Ghk. Thus, in the presence of attrition there exists an upward sloping differentiable function through the origin ( f * ) describing the stationary points of Eq. (5): Othk = f~h (Ght).

(6)

4. Dynamics of social ties a n d private provision 4.1. M o d e l a n d e q u i l i b r i u m o u t c o m e s

Combining the results of Section 2 Section 3, we can now examine the development of the social tie (%, aj~) and the corresponding contributions (gi, gj)Following Coleman (1990) we assume that social ties are generally the unconscious byproduct of social interaction. Individuals are supposed to be myopic, which implies that people do not take into account the potential impact of their contributions on their own feelings and the feelings of others towards them.6 The decision structure can then be modeled as a sequence of contribution decisions ~Strategic behavior is discussed in van Dijk and van Winden (1995).

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represented by Eq. (3), which are only connected by the development of the tie over time. Finally, we ignore the possibility that social ties are affected by mortality or migration (see Section 6). Using Eqs. (3) and (5) the development of the social tie is described by the following differential equations:

dahkldt =fh(Ghk(Othk, akh), ahk) (h, k = i, j, h # k)

(7)

where the income variables have been deleted, for convenience. The stationary points of these equations, determined by fh = 0, can be mapped in (%, %)-space. Neglecting extreme situations where g~ or gj is zero, the curves representing these stationary points are upward sloping. Using Eq. (6), the slopes are given by:

[ dab, ~ l\ dctkh/hl =

OOh, OCt,h / Of~ { Og_._._k_k Ogh "~ > O.

As regards the position of the curves in the (aij, %i)-plane, the situation where individuals are identical is considered first. Fig. 2 gives the phase-diagram for the general case where O < e i = ~ = e < 1. For negative values of ahk, the decay of the tie over time causes an increase of ahk. For stationarity (fh = 0), this increase needs to be offset by a negative value of Ghk, and h's contribution has to be larger than k's. This implies that Otkh would have to be more negative than Othk. For ahk=0, stationarity requires that Ghk =0, and thus gk =ehgh, which again implies that akh would have to be smaller than ahk, and negative. When aJ,k becomes positive, Ghk must also be positive to compensate the decline of sentiments due to attrition. Consequently, akh increases and switches sign. Given the properties of fh, the

ap

!

iV-.

/

J

,,

ag

a

~...,

. . . . . .

,_

. . . . . . . . .

,

14 Fig. 2. Phase diagram of Eq. (7) (identical individuals) i: f, =0, j: f~=0.

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marginal impact of Ghk then decreases with the increases of ahk and Ghk. But the rate of decline of ahk due to decay does not similarly decrease. Thus, as ahk becomes more and more positive, akh has to increase at an ever faster rate, and at some point a~h would have to become necessarily larger than ahk. Since individuals are assumed to be identical, the curves for i and j are symmetric, and intersect where a~j = air Differences between i andj may exist with regard to income, [-.references for the private and public goods, the tolerance level, and the speed at which ties develop and decay. For expositional reasons, only differences in income or preferences are considered. With regard to income we assume that yi ~/~ and 71ig<7/J~ for all xi=x j and g, where 7/h denotes the elasticity of utility of h with regard to z (Z=Xh, g and h=i, j). This implies that (OU~lOg)l (OU~lx,)<(OUJOg)l(OUJOxj). In both cases f~=0 and ~ = 0 are no longer symmetric. Symmetry requires G o for (a o, % ) = (a, b) to be equal to Gj, for (a 0, aj~)=(b, a), whereas in the cases considered Go>Gjr Both curves '=hilt to the right, and at the point of intersection a 0 must be larger than aji. When incomes or preferences differ sufficiently, f~=0 shifts to such an extent that a o is positive for the whole domain of a#, whilefj = 0 may shift so far that a~ is negative for all a 0. The following proposition presents the equilibrium outcomes. e

¢

Proposition 1. (a) A unique social-tie equilibrium, denoted by (a o, aj~), exists; (b ) with equal preferences and incomes, the tie is symmetric and positive in equilibrium (ot~j= oeje = t~e >0); (c) with differing preferences or income levels, the e tie is asymmetric in equilibrium (in the examined case oteo >aj~), and the sentiment ¢ with the lower intensity may be negative (ctji >-<0): (d) the equilibrium is locally stable. Parts (a) and (b) follow directly from the geometry of Fig. 2. Note that the functions cannot intersect in the negative orthant, since f~=0 then requires that gi>gj (G0<0) and f j = 0 that gi--O. Consequently, fh cannot be stationary for negative values of ahk. Both curves intersect once in the positive oRhant, irrespective of differences in preferences or income. The equilibrium a ' s are larger

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than in the general case. When 6 = 1, Ghk =gk--gh" With equal preferences and incomes, both f~=O and f/=O pass through the origin. Hence, (0,0) is the equilibrium. With differing preferences or incomes, both curves shift to the right. In that case, the equilibrium a ' s are of opposite sign, but equal in absolute value (since G 0 = - G i , ) . Moreover, they are smaller than in the general case.

4.2. Provision levels: comparison with the standard model As a benchmark we use the standard model, with % = % =0. Comparing the benchmark provision level (gb) with the equilibrium levels generated by our model (g~), the following proposition holds for the general case, where 0 < 6 < 1: Proposition 2. (a) With equal preferences and incomes: g~>gb; (b) with different preferences and equal incomes: ge>gb, (C) with different incomes and equal preferences: ge >- gb, because the equilibrium a ' s are always positive irrespective of differences in preferences or income. Now, consider e = 1. When preferences and incomes, and, therefore, contributions are equal, the a ' s are zero in equilibrium, and ge =gb. Allowiiig for differences in preferences or income gives in this case: (1) with different preferences regarding x and g, but equal income: g ¢ > g b and (II) with differences in income, but equal preferences: g¢
F. van Dijk, F. van Winden I Journal of Public Economics 64 (1997) 323-341 5. P r i v a t e p r o v i s i o n a n d public

333

provision

We now examine the impact o f public provision. Treating public provision (gg) as exogenous and using Eq. (4), the development o f Cehk is described by:

dcthk l dt = fh ( gk ( othk, Otkh) -- sh gh ( Othk, Otkh) -- ('rl, -- eh en )g x, a ~ )

(g)

(h, k = i , j, h ¢ k ) 7. Assuming that the tax burden is equally split 0"h=0.5), s and concentrating again on the general case where 0 < e < l , the impact o f public provision on the position o f the phase-lines in Fig. 2 is readily seen. Publ/c provision causes a decline o f Ghk if both private contributions are positive: g~(ahk, O L k h ) - - i ~ g h ( ~ g h k , akh)--O.5(l--e)gg'<&(ahk, O l k h ) - - 6 g h ( G l h k , a~h). When gg is increased from zero, f~ = 0 shifts to the left, and fj = 0 to the right. With differing preferences or incomes the phase-diagram is again asymmetrical. The following two propositions present the equilibrium results for the social tie and the to~d provision level o f the public good. Proposition 3. If g~ increases from zero, then: (a) with equal preferences and income, the equilibrium ot's decline uniformly until they become zero; (b) with ¢ differing preferences or incomes, the equilibrium ot"s decline, and the lower (aji) becomes negative, or, when already negative, becomes more negative; when g~ increases further and the lower contribution (gi ) becomes zero, both equilibrium ¢ a" s go to zero, i.e., the positive one (a~j) decreases further, while the negative one (a j~i) increases. Proposition 4. If gg increases from zero, then: (a) the total provision level o f the public good ge (which includes gg) decreases until, in case of identical individuals, both contributions are zero or~ else, the lower contribution ( gi ) is zero; when g~ increases further, g¢ increases; (b) when incomes differ, total provision starts to increase at a lower value of gg than in case o f identi~.zal individuals (keeping total income constant); with equidistant differences in preferences generally the same result is found 9.

7Note that we focus on observable effort, i.e.. the voluntary contributions to the public good. We assume that contributing to the public good by paying taxes does not generate affective ties. Political processes seem to constitute a weaker mechanism for the formation of social ties than v o l ~ provision. In case of voting in elections the link between decisions and contributions is much less direct. Also, the secrecy typical for voting results in limited information about the willingness of individuals to contribute. Slf ¢,~0.5 the tax rate can be split in an average tax rate and an equally sized redistributive tax and subsidy rate. In that case the analysis of the effect of differences in income of Section 4.2 app]ies. ~Equidistant means that the marginal rate of substitution dx/dg for i is (! - 3') and forj (1 + ~,) times that rate in case of uniform preferences. As to the generality of the result, an exception occurs in the extreme case where e is close to ! and tie attrition approaches zero for both individuals (see Appendix C).

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Part (a) of proposition 3 follows directly from the discussed inward shift off~ = 0 andfj =0. As to part (b), it is easily seen that increasing gg in a case of differences in preferences or incomes causes one equilibrium a to become negative. In the case where i attaches lower importance to the public good, or has a lower income than j, i's contribution becomes zero before j's. In that event the impulse on aj~ becomes negative, since Gj~ = -e[gj(%, % ~ ) - 0 . 5 g J < 0 . It follows that %i must be negative f o r f j = 0 . The intersection o f f ~ = 0 and f j = 0 occurs at a negative % and a positive a 0. A further increase of g~ results in a decline of the negative e impulse, which causes %~ to become less negative. Eventually both equilibrium a ' s are reduced to zero. Part (a) of proposition 4 follows from the decline of the equilibrium a's. The increase of g~ is overcompensated by the decrease of the contributions of i and j. Total provision starts to increase only when at least one of the voluntary contributions has been reduced to zero. Part (b) reflects that with differences in income the lower contribution is always, and with differences in preferences in nearly all cases, smaller in equilibrium than the contribution in case of equal preferences and incomes. The proof of proposition 4 is straightforward, but tedious (see Appendix C). The numerical example of the appendix (Appendix A) illustrates the results. Together, the propositions lead to the following conclusions. Public provision impedes the development of positive and negative sentiments, where the latter occurs if the provision level is sufficiently high. People become more neutral/ indifferent towards each other. For identical individuals, ties and, therefore, the total provision level decline uniformly as public provision increases. When preferences or incomes differ, the impact is more complex. At low levels of public provision, sentiments become even (more) negative, and again total provision decreases. Thus, in general, increasing public provision from zero first causes total provision to decline. This constitutes a major departure from the standard model. R is only when at least one of the contributions is reduced to zero that total provision starts to increase. The larger the differences between individuals the lower the level of public provision at which this reversal occurs, and the smaller the decline of total provision. The consequences of decreasing public provision, to cut back government expenditure for instance, can also be examined. Suppose that for a long time public provision completely crowded out private contributions, and is now totally abolished. At first, total provision declines sharply to the level gi(0,0)+gj(0,0). When tolerance is low and large differences in income exist, private provision subsequently declines further. If tolerance is sufficiently high, private provision starts to increase and reaches a substantial level, but only after some time. The community would face a difficult transition period. Because the electorate may not accept the temporary decline of provision, the government might feel urged to maintain expenditures. Also note that, from a long term, social welfare perspective limited expenditure reductions only make sense as long as the reduced public provision level remains above the level that would be privately attained in the

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absence of public provision. Otherwise, total provision is reduced below the level that can be privately achieved. The model suggests that there is a clear choice to he made: either rely on private or on public provision.

6. Concluding remarks Our analysis points at conditions under which individuals succeed in providing local public goods without government intervention) ° They will fail to do so in :ases of low tolerance with respect to the contributions of others, in particular when combined with an unequal income distribution. In that case there is a rationale for government intervention. Moreover, crowding out private contributions would also forestall negative sentiments. From a wider perspective, this is important in itself as such sentiments may cause social tensions that endanger public order and result in costs to society. Another rationale for government intervention suggested by the model concerns geographical mobility. In contrast with the standard model, migration affects the provision level of the public good in our model, because it disrupts existing social ties and necessitates the build-up of new ties. van Dijk (1997) shows that generally the average intensity of social ties decreases and that the provision level gets closer to the prediction of the standard model. Community size could provide a third rationale for government intervention. In the standard model the provision level, relative to the level that is found maximizing a social welfare function, becomes rapidly inconsequential when the number of inhabitants increases. Extending our model to larger communities, it can he shown that the relative decline of the provision level proceeds much more gradually as long as the inhabitants are informed about the contributions of a substantial number of their fellow-inhabitants. If this condition is not met, the results approach those of the standard model. In other instances private provision can reach a high level, and public provision is unnecessary. When it does not crowd out private provision completely, public provision is counterproductive, as it reduces the total provision level. At any rate, it hampers the development of positive social ties that would develop through the private provision process, and may thereby induce further public intervention. A case in point is voluntary assistance to people in need. Following our model, there is no general willingness to lend such support; only individuals who have positive sentiments towards a person in need would be willing to help. When the development of such sentiments is blocked, the need for costly welfare and social security arrangements increases. While staying close to standard economic theory, our model underscores the H'lt may also shed a new light on the occurrence of cooperation in public good experiments. The model suggeststhat account should be taken of the possibilitythat the motivationof subjectschanges under the social interaction that takes place within the experiment.

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i m p o r t a n t role p l a y e d b y social ties in the p r o v i s i o o o f public goods. Ignoring the effects o f p u b l i c p o l i c y on these ties, m a y , on the one hand, unwittingly result in the d e c l i n e o f social c o n d i t i o n s that stimulate i n d i v i d u a l s to take care o f their o w n needs, and. on the other hand, in social tensions and insufficient p u b l i c amenities. In the first case, g o v e r n m e n t intervention is too strong, in the s e c o n d too weak. in both c a s e s p u b l i c p o l i c y is not attuned to the social e n v i r o n m e n t .

Acknowledgments W e thank G a r y Becker, F r e d e r i k Carlsen, Eric Drissen, H e i n z Hollander, S h m u e l Nitzan, T h o m a s R o m e r and, in particular, an a n o n y m o u s referee for s t i m u l a t i n g c o m m e n t s o n e a r l i e r versions o f this paper. W e are also grateful to the Netherlands" O r g a n i z a t i o n for Scientific R e s e a r c h for financial support.

Appendix A

Numerical example T o illustrate the social-ties m e c h a n i s m , T a b l e 1 presents s o m e numerical e x a m p l e s . T h e (own c o n s u m p t i o n ) utility functions are specified as: U h = xhahg ~-[3h ( h = i , j ) . F o r Eq. (5) a d i s c r e t e - t i m e version o f the f o l l o w i n g specification is used:

dothk

e "h~g~-~'g'~ - (1 - 8hahk)l(l + 8hOthk)

dt

e"'qgk-~g"' + (1 - 6hahk)l(l + 6hOthk)

-- ahk.

Table i Provision of a public good and social ties y, =y,= I00

y, =70, y,= 130

g~

c

gh

g¢

a~)

ate,

g"

g~

vt~)

ot~,

0 0 0 20 50 67 80

0 0.5 1 0.5 0.5 0.5 0.5

67 67 67 67 67 67 67

95 89 67 85 76 67 80

0.81 0.59 0 0.49 0.23 0 0

0.81 0.59 0 0.49 0.23 0 0

67 67 67 70 78 82 85

92 78 61 74 66 75 83

0.89 0.76 0.36 0.69 0.47 0.28 0. !0

0.58 0.01 -0.36 - 0. i 0 -0.26 -0.15 - 0.05

Note: flh =0.5, 8h =0.8 and trh =0.015 for h=i, j. The minimum provision level is reached at g = 5 0 for yi=70, y,= 130, and at g~=67 for y, =y,= 100. In case of the maximization of a social welfare function S=V,V, for a,,=ot,,= l, or S = U U , the optimum is !00.

F. van Dijk, F. van Winden I Journal of Public Economics 64 (1997) 323-341

337

The parameter o-h reflects the speed with which ties develop, while ~ is 1 minus the linear rate of decay over time. Table 1 gives the social-ties equilibria and the corresponding provision levels for equal as well as unequal incomes. In the first three rows public provision is zero, and the influence of different values of ~ is examined. In the lower part of the table results are given for different levels of public provision. Results are compared with those of the standard model.

Appendix B

Proofs of parts (b) and (c) of proposition 2 The conclusions as to e = 1 remain to be proven: when preferences or incomes b b differ such that i has a lower demand for the public good. then: g~ g~, gj gj, and g~ < g b . In the benchmark case a 0 and o~ are both zero. Change of the a ' s is always symmetrical. Thus, it must be proven that: agi/~ot- ~gj/~a >-<0 for a 0 = o~ = 0 and for a~j=S and o ~ = - 8 , Og~. It is assumed that an interior solution exists. When oto=aj~=0, in equilibrium individuals i and j choose points denoted by (l) and (2) in the (x, g)-plane, such that: [(SU~/Og)/(~U~/~x~)]~,=[(OU~/~g)/(SUj/~x~)]~2 ~. We first consider differences in preferences. Of course, g is equal for both individuals and x, >x~. By setting a0 = o~ = a , where o~ is marginally larger than zero, we can examine the change of the slopes of i and j ' s indifference curves through the points (1) and (2), when a~ and ~i increase from zero. For positive a the slopes are with regard to, respectively, i and j in absolute value:

s~ = [(aUJag)/(aujax~)l,,, + al(auJag)/Ujl,~,/I(aUJax~)/U,l,,,, st

=

[(ouj/og)l(OUj/%)l,2,

+ a{(avi/og)/U,l,,,/l(auj/Oxj)!Ujl,,,.

For a wide range of utility functions (e.g., CES functions with elasticity of substitution <-- - 1, and thus also a Cobb-Douglas function): O[(aUilOg)lUi]iOxi <-0. This implies: [(aUklOg)lUkl,~<--[(aUklOg)lUkl,z~ for k=i, j. And thus: si -> s,(l) = [(aU,/Og)/(aU,/Ox,)l,,, + a[(aUj/~g)/Ujl, ,,/{(OUJOx,)/U,],,,.

sj <-sj(X) = [(auj/og)/(avj/oxj)],2,

+ a[(ov,/a~)/V,l,2,/{(ovj/Oxj)/uj],2,.

As defined in Section 4.1, with differences in preferences: (tgU~h3g)lU~<(a~l ag)lUj for all x i =x~ and g. Thus: si(1) > (l + a)[(aU, la~)/(au,/axi)l,,,

F. van Dijk. F. van Winden I Journal of Public Economics 64 (1997) 323-341

338

and st(2) < ( 1 + a)[(oUJOg)l(OUJaxj)L2 ,. As the r.h.s, of both inequalities are equal, s , ( l ) > s j ( 2 ) and thus also s , > s t. Consequently, the slope dx/dg of i's new indifference curve through ( l ) is steeper than the slope of j ' s new indifference curve through (2). As also i's effective income y, + g t is larger than j ' s effective income Yt +g,' it must necessarily hold that ag ir l a % > Ogt r/ T,. And, using the results of Section 2, Ogi 10% > Ogt I ~ , as it can be shown that the difference between Ogjr/ag, and Og,'/Ogj works in the same direction. For a 0 = 8 and a~, = - & with 8 marginally different from zero, we can repeat this analysis. This is left to the reader (see, however, below for the case of differences in income). With differences in incomes ( y i < y j ) , the indifference curves of i and j are identical, when a~j = a~, = 0 and also when % = ~, = tx. In the case a 0 = ~ = 0, in equilibrium x, =xj and thus y, + g j =Yt + g r This implies that at a = 0 0 g , r/Oa = OgtrlOt~. When % = 8 and % ~ : : - & i and j choose points in the (x. g)-plane t t denoted again as ( l ) and (2), such that: s, = s t , where: t

s, = [ ( a u , / a g ) / ( o u , / o x , ) ] , s'j = [ ( a u j / o ~ ) / ( a u ~ / a x t ) ] , ~

,, + 8 l ( a U J O g ) / U , ] , ~ . , / [ ( a u i / a w , ) / u , ] , , - a[(au,/ae,)/u,],,

,/[(aujawt)/ujl=2

,,. ,.

Now, in equilibrium x, < x t. It may also be noted that: [(JU,/Jg)/(0U, lax~)]~=~< [(oUJOg)/(oUJOxt)]c..rlncreasing % and a~, marginally with e, the slopes of the indifference curves become: t

s, = s, + ~ [ ( a u j a g ) / u t ] , 2 , / [ ( a u , / a x , ) / u , ] , ,

,,

st = s'j + ~ [ ( a v , / a g ) / U , ] , , , / [ ( a U J a x j ) / u t ] , 2

,.

As x, < x t and (direct) preferences are equal, [( a U j ag)/ Ujl,2,/[( a u , / ax,)/ u,], , , <_ [( a U j ag)l Ut], , ,/ [( a u , / ax, )/ U,l,

,,

= [(au,/ag)/U,],,,l[(au,/ax,)/u,l,,,,

[(au,/0g)/U,],,,/l(0UJ0x,)/~],:, = [(aut/ag)/utl,~,/[(aut/oxt)/Utl,2

l(0u,/og)/U,l,~.,/[(ouj/oxj)/~l,2,

z ,.

And thus: & < s t, as the extreme r.h.s, for i is smaller than that tbrj. Consequently,

d:qldg rotates less than dxJdg, and, as x, < x i implies that also $~ + g l
F. van Dijk. F. van Winden I Journal o f Public Economics 64 ( 1 9 9 7 ) 3 2 3 - 3 4 1

339

Appendix C Proof of proposition 4 Part (a): we define i's equilibrium private contribution in the absence of public provision (g, =0) as: giC =&(a~, a/i) ; e and with public provision as: gi e* =&(a~j¢*, a~*)-0.5gg>0. We first examine &~*>0 and gi~*>0. As follows from e~.J / e ¢ proposition 3(a), a~* e aj~(unequal). And & e-. g~ - j . e The inequality regarding ot~ implies: ( 1 8)gj¢(equal)g,¢(unequal)-egj~(unequal). We will first examine the extreme cases with regard to e. When e = 0 , the inequalities reduce to: &~(equal)< gj~(unequat) and gie(equal)>gf(unequal), which implies that g~¢(unequal) is reduced to zero before &~(equal) when increasing public provision. This result holds for differences in income as well as preferences. When s = 1, both inequalities reduce to: gj¢(unequal)>&~(unequal), which does not allow direct conclusions. We know, however, that: g~(equal)=gb(equal), and in case of differences in income: gb(equal)=g~(unequal) and from the proof in Appendix B: g¢(unequal)g¢(unequal). And, as

340

F. van Dijk, F. ran Winden I Journal o f Public Economics 64 (1997) 3 2 3 - 3 4 1

c > e e < g, (unequal) g, (unequal) and o f course gj~(equal)=g~(equal), gi (unequal) g~e(equal). This implies that in case o f differences in income gf(unequal) is again reduced to zero before g~e(equal) when increasing public provision. This result now obviously holds also for the general case O < e < I. In case o f equidistant preferences as defined earlier (still for e = 1), it can be easily shown that g b(equal)>g b(uneqv,al). As g~(equal)=gb(equal) and from the the proof in Appendix B ge(unequal)>gb(unequal), we can not reach immediate conclusions as to g~(equal) >
-in all situations with regard to the allocation over x and g, i's equilibrium contribution in case o f different preferences is equal or close to his equilibrium contribution in case of equal preferences; -when both are not equal, g,~(unequal) may be larger or smaller than g~¢(equal), depending on the allocation over x and g. To summarize the results for differing preferences, only when e is close to 1 and the rate of decay is close to zero, i's equilibrium contribution in case of different preferences may be (somewhat) larger than (or equal to) his equilibrium contribution in case of equal preferences, in all other cases g~¢(unequal) is smaller than g~(equal), and, consequently, gf(unequal) is again reduced to zero before g~¢(equai), when public provision is increased.

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