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On neutrality with multiple private and public goods
Mathematical Social Sciences 76 (2015) 103–106
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
On neutrality with multiple private and public goods Marta Faias a,b,∗ , Emma Moreno-García c , Myrna Wooders d a
Universidade Nova de Lisboa, FCT, Portugal
b
CMA, Portugal
c
Universidad de Salamanca, Spain
d
Vanderbilt University, United States
highlights • We consider an economy with multiple private goods and multiple public goods. • We obtain an analogue of the neutrality result of Warr (1983) and Bergstrom et al. (1986). • We provide an algorithm showing how endowment redistributions can be ‘‘neutralized’’ by changes in the amounts contributed to each public good.
article
info
Article history: Received 7 April 2014 Received in revised form 5 February 2015 Accepted 11 April 2015 Available online 30 April 2015
abstract We obtain an analogue of the neutrality result of Warr (1983) and Bergstrom et al. (1986) for economies with both multiple private and public goods. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In a classic paper, Warr (1983) demonstrated that, in an economy with one private good and one public good, small redistributions of endowments of the private good among contributors to public good provision will leave the equilibrium total provision of public good provided unchanged.1 This has become known as ‘‘Warr’s neutrality result’’. Bergstrom et al. (1986) provide an elegant formulation of the model and obtain a neutrality result without recourse to first order conditions; they rely on properties resulting from optimization by individual agents. Consider an equilibrium for an economy and a perturbation of endowments with the property that, after the perturbation, every consumer can afford his equilibrium private good allocation. Provided that the perturbation does not change the total amount of endowment of the economy an equilibrium for the economy generates an equilibrium for the perturbed economy in which all consumers have the same private goods allocation and the total public good
contribution is unchanged.2 The assumptions of one-private good, one public good underpins their model and results. Indeed, the authors write ‘‘whether there are less restrictive assumptions that give rise to the same neutrality result is an open question’’. We obtain an analogue of the neutrality result of Warr (1983) and BBV for economies with both multiple private and public goods. Since BBV’s celebrated paper there has been a number of insightful papers addressing neutrality issues. To treat multiple private goods, Villanacci and Zenginobuz (2006a) introduce the concept of a private provision equilibrium. This is an analogue of Walrasian equilibrium with private provision of a public good; prices for private goods are Walrasian and individual contributions to public good provision have the property that, in equilibrium, no consumer can benefit by changing his provision. The importance of the one-private-good assumption of Warr and BBV is highlighted by the work of Villanacci and Zenginobuz (2006b, 2007, 2012), which addresses related issues considering multiple private commodities and one public good. They obtain, under a strictly concave production technology assumption, nonneutrality results within several scenarios. To be more precise,
∗
Corresponding author at: Universidade Nova de Lisboa, FCT, Portugal. E-mail addresses: [email protected] (M. Faias), [email protected] (E. Moreno-García), [email protected] (M. Wooders). URL: http://www.myrnawooders.com (M. Wooders). 1 See also Kemp (1984). http://dx.doi.org/10.1016/j.mathsocsci.2015.04.005 0165-4896/© 2015 Elsevier B.V. All rights reserved.
2 There are numerous precursors to the BBV model and results; see their paper for references. Many other authors have studied existence of equilibrium and Warr’s neutrality result in a variety of contexts; see, for example, Kemp (1984), Itaya et al. (2002), Cornes and Itaya (2010), Silvestre (2012), Allouch (2015) and others.
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M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
Villanacci and Zenginobuz (2006b) show a non-neutrality result (in terms of utilities) even when all households are strict contributors to the public good. To obtain further non-neutrality results, Villanacci and Zenginobuz (2007) consider redistributions of one of the multiple private goods, which is treated as numeraire, among contributors and non-contributors, to the production of the single public good. The work of Villanacci and Zenginobuz again raises the BBV question: If there are multiple private goods and even multiple public goods under what conditions, if any, will neutrality hold? Is it possible to obtain an analogue of the neutrality result of Warr and BBV without the assumptions of one-private-good, one-public good? The research of Villanacci and Zenginobuz may suggest the answer to the BBV question is completely negative. In this paper we show that, with both multiple private and public goods, if endowments of private good are redistributed in such a way that each consumer can afford his initial equilibrium allocation of private goods at the initial equilibrium prices, then there exists an equilibrium for the post-redistribution economy that satisfies neutrality. We obtain our results in a model allowing multiple private and public goods and with a modification of the equilibrium concept of Villanacci and Zenginobuz (2006a) to allow multiple public goods. The sufficient conditions for our neutrality result rely on both the consumers that contribute to all the public goods and the wealth (value of endowments and profits) that each of them has in the initial equilibrium. Thus, the set of redistributions that allows us to obtain the same equilibrium depends on the initial equilibrium prices and, therefore, differ for each initial situation when there is multiplicity of equilibria. From our Theorem it follows that non-neutrality can only occur when (i) redistributions involve at least one non-contributor for some public good; or (ii) the value of the new endowments at the initial equilibrium prices does not allow consumption of the initial bundle for at least one consumer. In addition, we introduce a lemma with a constructive proof that provides an algorithm showing how endowment redistributions can be ‘‘neutralized’’ by changes in the amounts contributed to each public good. 2. The model
L +K Given a price system (p, q) ∈ R+ and profits Πk for each firm k, consumers choose private goods consumption and voluntary contributions to public good provision. Each consumer takes as given the contributions of the other consumers to public goods. That is, given a vector (gj , j ∈ N , j ̸= i) of voluntary contributions, each consumer i solves the problem:
Ui (x, g−i + ϱ)
max
(x,ϱ)∈RL+ ×RK+
such that
p · x + q · ϱ ≤ p · ei +
where g−i =
K
δik Πk ,
k=1 j̸=i
gj .
Definition. A private provision equilibrium for the economy E is a price system (p, q), a vector of inputs y = (yk ∈ RL+ ; k = 1, . . . , K ) for firms, a private commodities allocation x = (xi ∈ RL+ ; i = 1, . . . , N ) and an assignment of voluntary contributions (g k ∈ R+ ; k = 1, . . . , K ) = g such that,
N
i=1
gi =
(i) (xi , gi ) solves the optimization problem of consumer i for every i ∈ N . (ii) yk maximizes firm k′ profit, for every k. (iii)
N
i=1
xi +
K
k=1
yk ≤
N
i=1
ei .
(iv) g ≤ Fk (yk ) for every public good k. k
We will typically denote a private provision equilibrium by a list (p∗ , q∗ , x∗ , g ∗ , y∗ ). 4. Neutrality Let us consider the economy E described in Section 2. A redisN tribution of endowments is any allocation eˆ such that i=1 ei =
ˆ i . Let E (ˆe) denote the economy that coincides with E except i =1 e for the endowment which is given by eˆ , a redistribution of e. N
Lemma 4.1. Let (p, q, x, g ) be a vector of prices, allocations, and contributions and let Πk be profits for each firm k, such that p · xi + q · K gi = p·ei +δik k=1 Πk for every consumer i. Consider a redistribution
We consider an economy E with a finite number L of private goods and a finite number K of public goods. There is a set N of N consumers who individually consume private goods and collectively consume public goods. Each consumer i ∈ N = {1, . . . , N } is characterized by her endowment of private goods ei ∈ RL++ and by her preference relation over commodity space
eˆ of endowments such that p · xi ≤ p · eˆ i + δik k=1 Πk , for every i. Define 1ei by eˆ i = ei + 1ei . Then there exists a vector of voluntary contributions gˆ such that q · (ˆgi − gi ) = p · 1ei for every consumer i N N N ˆi . and i=1 (ˆgi − gi ) = 0, that is, i=1 gi = i=1 g
RL++K . Her preferences are represented by a continuous, concave +K and monotone-increasing utility function Ui : RL++ → R+ . Define N e= e . i=1 i There are K firms that produce public goods. A firm k ∈ {1, . . . , K } is characterized by a production function Fk : RL+ → R+ that converts private goods into public good k. We assume that each Fk is continuous and concave. Each consumer i ∈ N owns N a share δik ≥ 0 of the firm k’s profit and i=1 δik = 1 for each k.
Note that after the redistribution, each consumer can afford her initial equilibrium bundle of private goods. Our neutrality result requires more than this. Define a contributing consumer as a consumer whose contribution to every public good is positive in the initial equilibrium. We now restrict redistributions of endowments to redistributions among contributing consumers (that is, eˆ i = ei for all non-contributing consumers). This is an important assumption for our result below. (see Footnote 4).
3. Private provision equilibrium A price system is a vector (p, q) ∈ RL++K , where p = (pℓ , ℓ = 1, . . . , L) denotes the vector of prices for the L private commodities and q = (qk , k = 1, . . . , K ) denotes the vector of prices for the K public goods. Given a price system (p, q) ∈ RL++K , each firm k, k = 1, . . . , K , chooses the vector of inputs in RL+ that maximizes its profits Πk (y) = qk · Fk (y) − p · y.
K
Theorem 4.1 (Neutrality). Let (p∗ , q∗ , x∗ , g ∗ , y∗ ) be a private provision equilibrium for the economy E and let Πk∗ denote the equilibrium profits of firm k. Let eˆ be a redistribution of endowments such K that p∗ · x∗i ≤ p∗ · eˆ i + k=1 δik Πk∗ for every consumer i and eˆ i = ei for all non-contributing consumers. Then there exists a vector of voluntary contributions to public goods (ˆgi , i = 1 . . . , N ) such that (p∗ , q∗ , x∗ , gˆ , y∗ ) is aprivate provision equilibrium for the economy N N E (ˆe) and i=1 gˆi = i=1 gi∗ .
M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
Now, with the formal statement of our Theorem in hand, we remark, as noted in the introduction, that our result shows that non-neutrality requires either: (i) redistributions involve at least one non-contributor for some public good; or (ii) the value of the post-redistribution endowments at the original equilibrium prices renders her initial bundle unaffordable for at least one consumer. Consider the initial equilibrium (p∗ , q∗ , x∗ , g ∗ , y∗ ). One candidate for an equilibrium after a redistribution satisfying the condiN tions of the lemma is given by (p∗ , q∗ , x∗ , gˆ , y∗ ), where i=1 gˆi =
N
gi∗ , that is, each agent is allocated the same amount of private goods as in the initial equilibrium and, although contributors to public goods provision may make different contributions to public good provision, the total amounts of public goods provided remains unchanged. Clearly, each consumer is optimizing at his new equilibrium allocation; his private goods allocation is unchanged and also his marginal utilities for public goods remain unchanged. Moreover, by the conditions on the re-distribution, he can afford his original allocation of private goods and the total amount of public goods is the same. Thus, since marginal rates of substitution are unchanged, the same prices hold in the new equilibrium. In the one-private-good case, this is essentially the same as the BBV neutrality result. The multiple-private-goods case is more subtle and requires proof. Note that we have not imposed sufficient conditions to ensure uniqueness of equilibrium.3 If, however, there are multiple equilibria for the initial economy then our result applies to each equilibrium of the original economy; each equilibrium of the economy determines a set of redistributions of endowments such that the initial equilibrium prices are also post-redistribution equilibrium prices. In other words, for each equilibrium price in the original economy there is a set of redistributions of endowments that allows the same equilibrium price in the economy after redistributions. Our lemma constructs individual contributions to each public good in a manner ensuring that the total amounts of public goods provided are unchanged. To further relate our work to the literature, if we consider the particular scenario of one public good, strictly concave production functions, and redistributions involving only a numeraire private good then, following the approach by Villanacci and Zenginobuz (2006b, 2007), their non-neutrality results would hold. However, if redistributions involving a numeraire private good are only among contributors to the unique public good, we obtain the neutrality result of Theorem 5 of Villanacci and Zenginobuz (2007). Finally, for simplicity we have assumed that there is no production of private goods. If we had allowed such production and defined redistributions taking profits into account in budget constraints, our Theorem would, in essence, remain unchanged. We would simply have to introduce production plans for private goods and note that, when the post-redistribution prices are the same as the initial equilibrium prices, inputs and outputs of private goods are unchanged in the equilibrium of the perturbed economy. i=1
5. Conclusions For our neutrality result to be meaningful, an existence theorem is required. Existence of Nash equilibrium for a voluntary contributions game with one private good is demonstrated in BBV. For the private provision equilibrium with multiple private goods and one public good existence was established in Villanacci and
3 We conjecture that taking each consumer’s public goods as a vector of private goods (as in Foley, 1970, for example), if conditions are satisfied for uniqueness of equilibrium (as in Balasko, 1975), then the equilibrium of the initial economy will be unique, as will the equilibrium of the perturbed economy. The validity of this conjecture is beyond the scope of our paper.
105
Zenginobuz (2006a). Florenzano (2009) also provides an existence result and, as a by product of their main results obtained through a strategic approach, Faias et al. (2014), also demonstrate existence in the case of multiple private and public goods and production functions with constant returns to scale. We note here, as discussed in Faias et al. (2014), a privateprovision public goods economy can be modelled as a strategic market game, as in the research initiated by Shapley and Shubik (1977) for private goods exchange economies. Whether there are some conditions on redistributions that would enable a neutrality result for total public good provision in such a market game approach is an open question. Acknowledgements This work is partially supported by Research Grants ECO201238860-C02-01 (Ministerio de Economía y Competitividad) and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2014 (CMA). The third author is indebted to the Douglas Grey Fund for Research in Economics at Vanderbilt University for financial support for this collaboration. We thank an associate editor and two anonymous referees for comments and suggestions that have helped to improve this work. Appendix. Proofs Proof of Lemma 4.1. We will define three different sets of consumers. To which set a consumer i belongs depends on whether p1ei is less than, greater than, or equal to zero. For each set, we develop an algorithm to assign new levels of public good contributions to each agent in the set. These assignments are made in such N ˆi equals a way as to ensure that the total of contributions i=1 g the total amounts of public goods given by the initial allocation N i =1 g i . Set A. Let A denote the set of consumers i for whom p1ei < 0. Thus, A consists of those consumers for whom the values of endowments have decreased. Given i ∈ A, denote by k(i) the lowest index number on public goods for which the change in the absolute value of endowment |p1ei | is less than the total value of her contributions summed over all the public goods with lower index numbers; that is, k(i) = k h h min{k : |p1ei | ≤ γik := h=1 q gi }. For every agent i ∈ A let us define 1gi as follows:
k −g i k−1 1gik = p · 1ei + γi k q 0
if k < k(i) if k = k(i) otherwise.
That is, for the kth public good, k < k(i), the ith consumer’s new assignment of public good contribution is equal to zero. For k = k(i) the ith consumer’s contribution is equal, in value, to the difference between the change in the value of her endowment and the amount that she initially spent on the public goods indexed 1, . . . , k(i), and, for k > k(i) the ith consumer’s new assignment of public good contribution is equal to her initial assignment. For each i ∈ A, define gˆi = gi + 1gi . Set B. Now let B denote the set of consumers i for whom p · 1ei > 0; that is, B is the set of consumers for whom the value of endowment increases under the redistribution. To construct variations of the public goods contributions for consumers in B by induction let us write B = {b1 , . . . , bn }. k k For each public good k, let ηk (b1 ) = i∈A q 1gi (the sum, over the members of A, of the values of the changes in public good k
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M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
k defined above) and let ρ1k = h=1 ηh (b1 ) denote the absolute value of the sum of these changes. Now, for the first consumer in the set B, b1 , let k(b1 ) = min{k : p · 1eb1 ≤ ρ1k }, the lowest index number on public goods for which the change in the value of endowment (p1eb1 ) is (weakly) less than the amounts by which the values of the public goods contributions of consumers in A of good k have decreased.4 For k(b1 ) = 1, define
1gbk1 =
1gbk1
1gbk1
0
if k < k(b1 ) if k = k(b1 ) otherwise.
k
h η ( b ) . i h =1
For every i ∈ B let k(bi ) = min{k : p · 1ebi ≤ ρik }. The modification 1gbki is defined as follows. If k(bi ) = 1, define
p · 1ebi
if k = 1 q1 0 otherwise.
Otherwise, define
0
if k < k(bi ) if k = k(bi ) otherwise.
Finally, 1gi = 0 if p · ei = p · eˆ i . N N ˆi . By construction q · 1gi = p · 1ei and i=1 gi = i =1 g
Proof of Theorem 4.1. For every consumer i the bundle (xi , G∗ ), ∗
max (x,G)∈RL++K
such that
N
i=1
gi∗ , solves the following individual problem:
Ui (x, G) ∗ p∗ · x + q∗ · G ≤ p∗ · ei + q∗ · g− i + ∗ G ≥ g− i
∗ where g− i =
j̸=i
δik Πk∗
Define ηk (bi ) = ηk (bi−1 ) + qk 1gbki−1 . Let ρik =
k η (bi ) qk k 1gbi = p · 1ebi − ρik−1 qk
K
G ≥ gˆ−i
Suppose that |B| > 1.
where G∗ =
p∗ · x + q∗ · G ≤ p∗ · eˆ i + q∗ · gˆ−i +
k=1
is defined as follows:
k η (b1 ) qk = p · 1eb1 − ρ1k−1 qk
1gbki =
Ui (x, G)
if k = 1
q1 0 otherwise.
For k(b1 ) > 1,
max (x,G)∈RL++K
such that
p · 1eb1
By the previous lemma we can take gˆi such that q∗ · gˆi = N p · eˆ i − p∗ · x∗i for every i and i=1 gˆi = G∗ . ∗ ∗ Since given prices (p , q ), each y∗k is a solution for the firm k problem, it remains to show that, for every i, the bundle (x∗i , G∗ ) is a solution for the following problem: ∗
K
δik Πk∗
k=1
gj∗ .
4 1) since q · 1gi = p · 1ei for all i ∈ A (that is, i∈A q · 1gi = There is such a k( i∈A p · 1ei ) and i ∈ A p · 1e i = bi ∈B p · 1ebi . Then, i∈A q · 1gi ≥ p · 1ebi for all bi ∈ B.
ˆj . where gˆ−i = j̸=i g ∗ Note that p∗ · eˆ i + q∗ · gˆ−i = p∗ · ei + q∗ · g− i. Let us write eˆ i = ei + 1ei . If p∗ · 1ei < 0 the budget set for consumer i is smaller and (x∗i , G∗ ) belongs to it. Consider the case p∗ · 1ei > 0 and assume that there is a bundle (x, G) which is possible for agent i after redistribution of endowments and is preferred to (x∗i , G∗ ). Then, for every λ sufficiently close to 1, λ(x∗i , G∗ ) + (1 − λ)(x, G) is affordable5 for agent i before the redistribution of endowments and by convexity of preferences this bundle is also preferred to (x∗i , G∗ ), which is a contradiction. References Allouch, N., 2015. On the private provision of public goods on networks. J. Econom. Theory 157, 527–552. Balasko, Y., 1975. Some results on uniqueness and on stability of equilibrium in general equilibrium theory. J. Math. Econom. 2, 95–118. Bergstrom, T., Blume, L., Varian, H., 1986. On the private provision of public goods. J. Public Econ. 29, 25–49. Cornes, R., Itaya, J., 2010. On the private provision of two or more public goods. J. Public Econ. Theory 12 (2), 363–385. Faias, M., Moreno-García, E., Wooders, M., 2014. A strategic market game approach for the private provision of public goods. J. Dyn. Games 1 (2), 283–298. Florenzano, M., 2009. Walras–Lindahl–Wicksell: What equilibrium concept for public goods provision? I—The convex case. CES Working Papers 2009.09. Documents de Travail du Centre d’Economie de la Sorbonne. Foley, D., 1970. Lindahl’s solution and the core of an economy with public goods. Econometrica 38, 66–72. Itaya, J., de Meza, D., Myles, G., 2002. Income distribution, taxation, and the private provision of public goods. J. Public Econ. Theory 4 (3), 273–297. Kemp, M.C., 1984. A note of the theory of international transfers. Econom. Lett. 14, 259–262. Shapley, L.S., Shubik, M., 1977. Trade using one commodity as a means of payment. J. Polit. Econ. 85, 937–968. Silvestre, J., 2012. All but one free ride when wealth effects are small. SERIEs 3, 201–207. Villanacci, A., Zenginobuz, U., 2006a. Existence and regularity of equilibria in a general equilibrium model with private provision of a public good. J. Math. Econom. 41, 617–636. Villanacci, A., Zenginobuz, U., 2006b. Pareto improving interventions in a general equilibrium model with private provision of public goods. Rev. Econ. Des. 10, 249–271. Villanacci, A., Zenginobuz, Ü, 2007. On the neutrality of redistribution in a general equilibrium model with public goods. J. Public Econ. Theory 9 (2), 183–200. Villanacci, A., Zenginobuz, Ü, 2012. Subscription equilibrium with production: Nonneutrality and constrained suboptimality. J. Econom. Theory 147, 407–425. Warr, P.G., 1983. The private provision of a public good Is independent of the distribution of income. Econom. Lett. 13, 207–211.
5 Note that λG∗ + (1 − λ)G ≥ g ∗ for λ close enough to 1, provided that the −i redistribution of endowments is among contributors.
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
On neutrality with multiple private and public goods Marta Faias a,b,∗ , Emma Moreno-García c , Myrna Wooders d a
Universidade Nova de Lisboa, FCT, Portugal
b
CMA, Portugal
c
Universidad de Salamanca, Spain
d
Vanderbilt University, United States
highlights • We consider an economy with multiple private goods and multiple public goods. • We obtain an analogue of the neutrality result of Warr (1983) and Bergstrom et al. (1986). • We provide an algorithm showing how endowment redistributions can be ‘‘neutralized’’ by changes in the amounts contributed to each public good.
article
info
Article history: Received 7 April 2014 Received in revised form 5 February 2015 Accepted 11 April 2015 Available online 30 April 2015
abstract We obtain an analogue of the neutrality result of Warr (1983) and Bergstrom et al. (1986) for economies with both multiple private and public goods. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In a classic paper, Warr (1983) demonstrated that, in an economy with one private good and one public good, small redistributions of endowments of the private good among contributors to public good provision will leave the equilibrium total provision of public good provided unchanged.1 This has become known as ‘‘Warr’s neutrality result’’. Bergstrom et al. (1986) provide an elegant formulation of the model and obtain a neutrality result without recourse to first order conditions; they rely on properties resulting from optimization by individual agents. Consider an equilibrium for an economy and a perturbation of endowments with the property that, after the perturbation, every consumer can afford his equilibrium private good allocation. Provided that the perturbation does not change the total amount of endowment of the economy an equilibrium for the economy generates an equilibrium for the perturbed economy in which all consumers have the same private goods allocation and the total public good
contribution is unchanged.2 The assumptions of one-private good, one public good underpins their model and results. Indeed, the authors write ‘‘whether there are less restrictive assumptions that give rise to the same neutrality result is an open question’’. We obtain an analogue of the neutrality result of Warr (1983) and BBV for economies with both multiple private and public goods. Since BBV’s celebrated paper there has been a number of insightful papers addressing neutrality issues. To treat multiple private goods, Villanacci and Zenginobuz (2006a) introduce the concept of a private provision equilibrium. This is an analogue of Walrasian equilibrium with private provision of a public good; prices for private goods are Walrasian and individual contributions to public good provision have the property that, in equilibrium, no consumer can benefit by changing his provision. The importance of the one-private-good assumption of Warr and BBV is highlighted by the work of Villanacci and Zenginobuz (2006b, 2007, 2012), which addresses related issues considering multiple private commodities and one public good. They obtain, under a strictly concave production technology assumption, nonneutrality results within several scenarios. To be more precise,
∗
Corresponding author at: Universidade Nova de Lisboa, FCT, Portugal. E-mail addresses: [email protected] (M. Faias), [email protected] (E. Moreno-García), [email protected] (M. Wooders). URL: http://www.myrnawooders.com (M. Wooders). 1 See also Kemp (1984). http://dx.doi.org/10.1016/j.mathsocsci.2015.04.005 0165-4896/© 2015 Elsevier B.V. All rights reserved.
2 There are numerous precursors to the BBV model and results; see their paper for references. Many other authors have studied existence of equilibrium and Warr’s neutrality result in a variety of contexts; see, for example, Kemp (1984), Itaya et al. (2002), Cornes and Itaya (2010), Silvestre (2012), Allouch (2015) and others.
104
M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
Villanacci and Zenginobuz (2006b) show a non-neutrality result (in terms of utilities) even when all households are strict contributors to the public good. To obtain further non-neutrality results, Villanacci and Zenginobuz (2007) consider redistributions of one of the multiple private goods, which is treated as numeraire, among contributors and non-contributors, to the production of the single public good. The work of Villanacci and Zenginobuz again raises the BBV question: If there are multiple private goods and even multiple public goods under what conditions, if any, will neutrality hold? Is it possible to obtain an analogue of the neutrality result of Warr and BBV without the assumptions of one-private-good, one-public good? The research of Villanacci and Zenginobuz may suggest the answer to the BBV question is completely negative. In this paper we show that, with both multiple private and public goods, if endowments of private good are redistributed in such a way that each consumer can afford his initial equilibrium allocation of private goods at the initial equilibrium prices, then there exists an equilibrium for the post-redistribution economy that satisfies neutrality. We obtain our results in a model allowing multiple private and public goods and with a modification of the equilibrium concept of Villanacci and Zenginobuz (2006a) to allow multiple public goods. The sufficient conditions for our neutrality result rely on both the consumers that contribute to all the public goods and the wealth (value of endowments and profits) that each of them has in the initial equilibrium. Thus, the set of redistributions that allows us to obtain the same equilibrium depends on the initial equilibrium prices and, therefore, differ for each initial situation when there is multiplicity of equilibria. From our Theorem it follows that non-neutrality can only occur when (i) redistributions involve at least one non-contributor for some public good; or (ii) the value of the new endowments at the initial equilibrium prices does not allow consumption of the initial bundle for at least one consumer. In addition, we introduce a lemma with a constructive proof that provides an algorithm showing how endowment redistributions can be ‘‘neutralized’’ by changes in the amounts contributed to each public good. 2. The model
L +K Given a price system (p, q) ∈ R+ and profits Πk for each firm k, consumers choose private goods consumption and voluntary contributions to public good provision. Each consumer takes as given the contributions of the other consumers to public goods. That is, given a vector (gj , j ∈ N , j ̸= i) of voluntary contributions, each consumer i solves the problem:
Ui (x, g−i + ϱ)
max
(x,ϱ)∈RL+ ×RK+
such that
p · x + q · ϱ ≤ p · ei +
where g−i =
K
δik Πk ,
k=1 j̸=i
gj .
Definition. A private provision equilibrium for the economy E is a price system (p, q), a vector of inputs y = (yk ∈ RL+ ; k = 1, . . . , K ) for firms, a private commodities allocation x = (xi ∈ RL+ ; i = 1, . . . , N ) and an assignment of voluntary contributions (g k ∈ R+ ; k = 1, . . . , K ) = g such that,
N
i=1
gi =
(i) (xi , gi ) solves the optimization problem of consumer i for every i ∈ N . (ii) yk maximizes firm k′ profit, for every k. (iii)
N
i=1
xi +
K
k=1
yk ≤
N
i=1
ei .
(iv) g ≤ Fk (yk ) for every public good k. k
We will typically denote a private provision equilibrium by a list (p∗ , q∗ , x∗ , g ∗ , y∗ ). 4. Neutrality Let us consider the economy E described in Section 2. A redisN tribution of endowments is any allocation eˆ such that i=1 ei =
ˆ i . Let E (ˆe) denote the economy that coincides with E except i =1 e for the endowment which is given by eˆ , a redistribution of e. N
Lemma 4.1. Let (p, q, x, g ) be a vector of prices, allocations, and contributions and let Πk be profits for each firm k, such that p · xi + q · K gi = p·ei +δik k=1 Πk for every consumer i. Consider a redistribution
We consider an economy E with a finite number L of private goods and a finite number K of public goods. There is a set N of N consumers who individually consume private goods and collectively consume public goods. Each consumer i ∈ N = {1, . . . , N } is characterized by her endowment of private goods ei ∈ RL++ and by her preference relation over commodity space
eˆ of endowments such that p · xi ≤ p · eˆ i + δik k=1 Πk , for every i. Define 1ei by eˆ i = ei + 1ei . Then there exists a vector of voluntary contributions gˆ such that q · (ˆgi − gi ) = p · 1ei for every consumer i N N N ˆi . and i=1 (ˆgi − gi ) = 0, that is, i=1 gi = i=1 g
RL++K . Her preferences are represented by a continuous, concave +K and monotone-increasing utility function Ui : RL++ → R+ . Define N e= e . i=1 i There are K firms that produce public goods. A firm k ∈ {1, . . . , K } is characterized by a production function Fk : RL+ → R+ that converts private goods into public good k. We assume that each Fk is continuous and concave. Each consumer i ∈ N owns N a share δik ≥ 0 of the firm k’s profit and i=1 δik = 1 for each k.
Note that after the redistribution, each consumer can afford her initial equilibrium bundle of private goods. Our neutrality result requires more than this. Define a contributing consumer as a consumer whose contribution to every public good is positive in the initial equilibrium. We now restrict redistributions of endowments to redistributions among contributing consumers (that is, eˆ i = ei for all non-contributing consumers). This is an important assumption for our result below. (see Footnote 4).
3. Private provision equilibrium A price system is a vector (p, q) ∈ RL++K , where p = (pℓ , ℓ = 1, . . . , L) denotes the vector of prices for the L private commodities and q = (qk , k = 1, . . . , K ) denotes the vector of prices for the K public goods. Given a price system (p, q) ∈ RL++K , each firm k, k = 1, . . . , K , chooses the vector of inputs in RL+ that maximizes its profits Πk (y) = qk · Fk (y) − p · y.
K
Theorem 4.1 (Neutrality). Let (p∗ , q∗ , x∗ , g ∗ , y∗ ) be a private provision equilibrium for the economy E and let Πk∗ denote the equilibrium profits of firm k. Let eˆ be a redistribution of endowments such K that p∗ · x∗i ≤ p∗ · eˆ i + k=1 δik Πk∗ for every consumer i and eˆ i = ei for all non-contributing consumers. Then there exists a vector of voluntary contributions to public goods (ˆgi , i = 1 . . . , N ) such that (p∗ , q∗ , x∗ , gˆ , y∗ ) is aprivate provision equilibrium for the economy N N E (ˆe) and i=1 gˆi = i=1 gi∗ .
M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
Now, with the formal statement of our Theorem in hand, we remark, as noted in the introduction, that our result shows that non-neutrality requires either: (i) redistributions involve at least one non-contributor for some public good; or (ii) the value of the post-redistribution endowments at the original equilibrium prices renders her initial bundle unaffordable for at least one consumer. Consider the initial equilibrium (p∗ , q∗ , x∗ , g ∗ , y∗ ). One candidate for an equilibrium after a redistribution satisfying the condiN tions of the lemma is given by (p∗ , q∗ , x∗ , gˆ , y∗ ), where i=1 gˆi =
N
gi∗ , that is, each agent is allocated the same amount of private goods as in the initial equilibrium and, although contributors to public goods provision may make different contributions to public good provision, the total amounts of public goods provided remains unchanged. Clearly, each consumer is optimizing at his new equilibrium allocation; his private goods allocation is unchanged and also his marginal utilities for public goods remain unchanged. Moreover, by the conditions on the re-distribution, he can afford his original allocation of private goods and the total amount of public goods is the same. Thus, since marginal rates of substitution are unchanged, the same prices hold in the new equilibrium. In the one-private-good case, this is essentially the same as the BBV neutrality result. The multiple-private-goods case is more subtle and requires proof. Note that we have not imposed sufficient conditions to ensure uniqueness of equilibrium.3 If, however, there are multiple equilibria for the initial economy then our result applies to each equilibrium of the original economy; each equilibrium of the economy determines a set of redistributions of endowments such that the initial equilibrium prices are also post-redistribution equilibrium prices. In other words, for each equilibrium price in the original economy there is a set of redistributions of endowments that allows the same equilibrium price in the economy after redistributions. Our lemma constructs individual contributions to each public good in a manner ensuring that the total amounts of public goods provided are unchanged. To further relate our work to the literature, if we consider the particular scenario of one public good, strictly concave production functions, and redistributions involving only a numeraire private good then, following the approach by Villanacci and Zenginobuz (2006b, 2007), their non-neutrality results would hold. However, if redistributions involving a numeraire private good are only among contributors to the unique public good, we obtain the neutrality result of Theorem 5 of Villanacci and Zenginobuz (2007). Finally, for simplicity we have assumed that there is no production of private goods. If we had allowed such production and defined redistributions taking profits into account in budget constraints, our Theorem would, in essence, remain unchanged. We would simply have to introduce production plans for private goods and note that, when the post-redistribution prices are the same as the initial equilibrium prices, inputs and outputs of private goods are unchanged in the equilibrium of the perturbed economy. i=1
5. Conclusions For our neutrality result to be meaningful, an existence theorem is required. Existence of Nash equilibrium for a voluntary contributions game with one private good is demonstrated in BBV. For the private provision equilibrium with multiple private goods and one public good existence was established in Villanacci and
3 We conjecture that taking each consumer’s public goods as a vector of private goods (as in Foley, 1970, for example), if conditions are satisfied for uniqueness of equilibrium (as in Balasko, 1975), then the equilibrium of the initial economy will be unique, as will the equilibrium of the perturbed economy. The validity of this conjecture is beyond the scope of our paper.
105
Zenginobuz (2006a). Florenzano (2009) also provides an existence result and, as a by product of their main results obtained through a strategic approach, Faias et al. (2014), also demonstrate existence in the case of multiple private and public goods and production functions with constant returns to scale. We note here, as discussed in Faias et al. (2014), a privateprovision public goods economy can be modelled as a strategic market game, as in the research initiated by Shapley and Shubik (1977) for private goods exchange economies. Whether there are some conditions on redistributions that would enable a neutrality result for total public good provision in such a market game approach is an open question. Acknowledgements This work is partially supported by Research Grants ECO201238860-C02-01 (Ministerio de Economía y Competitividad) and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2014 (CMA). The third author is indebted to the Douglas Grey Fund for Research in Economics at Vanderbilt University for financial support for this collaboration. We thank an associate editor and two anonymous referees for comments and suggestions that have helped to improve this work. Appendix. Proofs Proof of Lemma 4.1. We will define three different sets of consumers. To which set a consumer i belongs depends on whether p1ei is less than, greater than, or equal to zero. For each set, we develop an algorithm to assign new levels of public good contributions to each agent in the set. These assignments are made in such N ˆi equals a way as to ensure that the total of contributions i=1 g the total amounts of public goods given by the initial allocation N i =1 g i . Set A. Let A denote the set of consumers i for whom p1ei < 0. Thus, A consists of those consumers for whom the values of endowments have decreased. Given i ∈ A, denote by k(i) the lowest index number on public goods for which the change in the absolute value of endowment |p1ei | is less than the total value of her contributions summed over all the public goods with lower index numbers; that is, k(i) = k h h min{k : |p1ei | ≤ γik := h=1 q gi }. For every agent i ∈ A let us define 1gi as follows:
k −g i k−1 1gik = p · 1ei + γi k q 0
if k < k(i) if k = k(i) otherwise.
That is, for the kth public good, k < k(i), the ith consumer’s new assignment of public good contribution is equal to zero. For k = k(i) the ith consumer’s contribution is equal, in value, to the difference between the change in the value of her endowment and the amount that she initially spent on the public goods indexed 1, . . . , k(i), and, for k > k(i) the ith consumer’s new assignment of public good contribution is equal to her initial assignment. For each i ∈ A, define gˆi = gi + 1gi . Set B. Now let B denote the set of consumers i for whom p · 1ei > 0; that is, B is the set of consumers for whom the value of endowment increases under the redistribution. To construct variations of the public goods contributions for consumers in B by induction let us write B = {b1 , . . . , bn }. k k For each public good k, let ηk (b1 ) = i∈A q 1gi (the sum, over the members of A, of the values of the changes in public good k
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M. Faias et al. / Mathematical Social Sciences 76 (2015) 103–106
k defined above) and let ρ1k = h=1 ηh (b1 ) denote the absolute value of the sum of these changes. Now, for the first consumer in the set B, b1 , let k(b1 ) = min{k : p · 1eb1 ≤ ρ1k }, the lowest index number on public goods for which the change in the value of endowment (p1eb1 ) is (weakly) less than the amounts by which the values of the public goods contributions of consumers in A of good k have decreased.4 For k(b1 ) = 1, define
1gbk1 =
1gbk1
1gbk1
0
if k < k(b1 ) if k = k(b1 ) otherwise.
k
h η ( b ) . i h =1
For every i ∈ B let k(bi ) = min{k : p · 1ebi ≤ ρik }. The modification 1gbki is defined as follows. If k(bi ) = 1, define
p · 1ebi
if k = 1 q1 0 otherwise.
Otherwise, define
0
if k < k(bi ) if k = k(bi ) otherwise.
Finally, 1gi = 0 if p · ei = p · eˆ i . N N ˆi . By construction q · 1gi = p · 1ei and i=1 gi = i =1 g
Proof of Theorem 4.1. For every consumer i the bundle (xi , G∗ ), ∗
max (x,G)∈RL++K
such that
N
i=1
gi∗ , solves the following individual problem:
Ui (x, G) ∗ p∗ · x + q∗ · G ≤ p∗ · ei + q∗ · g− i + ∗ G ≥ g− i
∗ where g− i =
j̸=i
δik Πk∗
Define ηk (bi ) = ηk (bi−1 ) + qk 1gbki−1 . Let ρik =
k η (bi ) qk k 1gbi = p · 1ebi − ρik−1 qk
K
G ≥ gˆ−i
Suppose that |B| > 1.
where G∗ =
p∗ · x + q∗ · G ≤ p∗ · eˆ i + q∗ · gˆ−i +
k=1
is defined as follows:
k η (b1 ) qk = p · 1eb1 − ρ1k−1 qk
1gbki =
Ui (x, G)
if k = 1
q1 0 otherwise.
For k(b1 ) > 1,
max (x,G)∈RL++K
such that
p · 1eb1
By the previous lemma we can take gˆi such that q∗ · gˆi = N p · eˆ i − p∗ · x∗i for every i and i=1 gˆi = G∗ . ∗ ∗ Since given prices (p , q ), each y∗k is a solution for the firm k problem, it remains to show that, for every i, the bundle (x∗i , G∗ ) is a solution for the following problem: ∗
K
δik Πk∗
k=1
gj∗ .
4 1) since q · 1gi = p · 1ei for all i ∈ A (that is, i∈A q · 1gi = There is such a k( i∈A p · 1ei ) and i ∈ A p · 1e i = bi ∈B p · 1ebi . Then, i∈A q · 1gi ≥ p · 1ebi for all bi ∈ B.
ˆj . where gˆ−i = j̸=i g ∗ Note that p∗ · eˆ i + q∗ · gˆ−i = p∗ · ei + q∗ · g− i. Let us write eˆ i = ei + 1ei . If p∗ · 1ei < 0 the budget set for consumer i is smaller and (x∗i , G∗ ) belongs to it. Consider the case p∗ · 1ei > 0 and assume that there is a bundle (x, G) which is possible for agent i after redistribution of endowments and is preferred to (x∗i , G∗ ). Then, for every λ sufficiently close to 1, λ(x∗i , G∗ ) + (1 − λ)(x, G) is affordable5 for agent i before the redistribution of endowments and by convexity of preferences this bundle is also preferred to (x∗i , G∗ ), which is a contradiction. References Allouch, N., 2015. On the private provision of public goods on networks. J. Econom. Theory 157, 527–552. Balasko, Y., 1975. Some results on uniqueness and on stability of equilibrium in general equilibrium theory. J. Math. Econom. 2, 95–118. Bergstrom, T., Blume, L., Varian, H., 1986. On the private provision of public goods. J. Public Econ. 29, 25–49. Cornes, R., Itaya, J., 2010. On the private provision of two or more public goods. J. Public Econ. Theory 12 (2), 363–385. Faias, M., Moreno-García, E., Wooders, M., 2014. A strategic market game approach for the private provision of public goods. J. Dyn. Games 1 (2), 283–298. Florenzano, M., 2009. Walras–Lindahl–Wicksell: What equilibrium concept for public goods provision? I—The convex case. CES Working Papers 2009.09. Documents de Travail du Centre d’Economie de la Sorbonne. Foley, D., 1970. Lindahl’s solution and the core of an economy with public goods. Econometrica 38, 66–72. Itaya, J., de Meza, D., Myles, G., 2002. Income distribution, taxation, and the private provision of public goods. J. Public Econ. Theory 4 (3), 273–297. Kemp, M.C., 1984. A note of the theory of international transfers. Econom. Lett. 14, 259–262. Shapley, L.S., Shubik, M., 1977. Trade using one commodity as a means of payment. J. Polit. Econ. 85, 937–968. Silvestre, J., 2012. All but one free ride when wealth effects are small. SERIEs 3, 201–207. Villanacci, A., Zenginobuz, U., 2006a. Existence and regularity of equilibria in a general equilibrium model with private provision of a public good. J. Math. Econom. 41, 617–636. Villanacci, A., Zenginobuz, U., 2006b. Pareto improving interventions in a general equilibrium model with private provision of public goods. Rev. Econ. Des. 10, 249–271. Villanacci, A., Zenginobuz, Ü, 2007. On the neutrality of redistribution in a general equilibrium model with public goods. J. Public Econ. Theory 9 (2), 183–200. Villanacci, A., Zenginobuz, Ü, 2012. Subscription equilibrium with production: Nonneutrality and constrained suboptimality. J. Econom. Theory 147, 407–425. Warr, P.G., 1983. The private provision of a public good Is independent of the distribution of income. Econom. Lett. 13, 207–211.
5 Note that λG∗ + (1 − λ)G ≥ g ∗ for λ close enough to 1, provided that the −i redistribution of endowments is among contributors.