FROM LOCAL TO GLOBAL PUBLIC GOODS: HOW SHOULD EXTERNALITIES BE REPRESENTED? Rosella Levaggi. Discussion Paper n. 0903

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1 Dipartimento di Scienze Economiche Università degli Studi di Brescia Via San Faustino 74/B Brescia Italy Tel: /840/848, Fax: FROM LOCAL TO GLOBAL PUBLIC GOODS: HOW SHOULD EXTERNALITIES BE REPRESENTED? By Rosella Levaggi Discussion Paper n These Discussion Papers often represent preliminary or incomplete work, circulated to encourage discussion and comments. Citation and use of such a paper should take account of its provisional character. A revised version may be available directly from the author(s). Any opinions expressed here are those of the author(s) and not those of the Dipartimento di Scienze Economiche, Università degli Studi di Brescia. Research disseminated by the Department may include views on policy, but the Department itself takes no institutional policy position.

2 From local to global public goods: how should externalities be represented? Rosella Levaggi Dipartimento di Scienze Economiche Via San Faustino 74b BRESCIA (ITALY) March 11, 2009 Abstract The literature on public goods provision has experienced a signi cant increase since Samuelson s (1954) paper. The common goal is to make the model more suitable to describe a more general class of non-rival goods. However, there does not seem to be a consensus in the literature on the form of the function to be used to describe the externality created by the public good. In this note we try to show the hypotheses underlying the functions used. The di erent models proposed are optimal when used in the right framework and this should be kept in mind when choosing the function in relation to the type of public good to be studied. Keywords: public goods, externalities, optimal conditions J.E.L. H4,H7 1

3 1 Introduction Public goods are non-rival, non-excludable commodities: once a unit has been produced, it increases the utility of all the individuals in the same way. In other words, each unit of public good produces a positive externality for the rest of the community. The extent of the community to which the good spreads its bene ts and the externality produced are the distinguishing characteristics of the di erent types of public goods studied by the literature 1. Since Samuleson s (1954) seminal paper, an increasing number of contributions have been proposed to study public goods provision but less and less consensus exists on the functional form to represent them. In public goods literature it is possible to distinguish between two basic views: a) the global public goods approach and b) the local public goods with spillovers approach. In this note we explore the di erences in the assumptions behind these two approaches and suggest when they should be applied. 2 Modelling utility for public goods All the literature agrees that public goods generate an externality, that the latter is anonymous, additive, and that it produces utility at a marginal decreasing rate. What the literature does not agree on is whether the distribution across jurisdictions is important. For simplicity, l et s assume that N individuals live in two regions, A, and B. Their utility function depends on the consumption of a private good x and a public good z. 2.1 Global public goods This literature assumes that the externality generated by the public good is anonymous and additive across individuals and across areas. The total quantity of public good produced enters the utility function of each individual, no matter where they are located. The utility function for a representative individual i living in local authority A can be written as: U(x i ; kz A + (1 k) z B ) where x i is the quantity of private good demanded, z A is the quantity of public good provided in A; and z B is the quantity provided in B. For k = 0 we have a local public good; for k = 1=2 a pure public good and for k < 1=2 a local public good with spillovers. For k = 1=2, the idea behind this function is that the public good spreads its bene ts across jurisdictions in the same way: what matters is the total quantity produced, not how it is distributed. For a pure public good and heterogeneous production costs, such function implies a corner solution: the good should be produced in the most e cient jurisdiction and the other should contribute to its nance, independently of 1 See Buchanan, Cornes and Sandler etc. for a review of the di erent types. 2

4 income distribution. This result has important policy implications at supernational level on the direction of the ow of resources and on the form of the subsidy to use to correct for the externality. Since Warr s neutrality theorem (1982,1983) the literature has concentrated on the study of the e ects of matching grants in a context of information asymmetry (Buchholz and Konrad, 1995; Huber and Runkel, 2006). 2.2 Utility in variety Another stream of literature, mostly concentrated on decentralisation vs scal federalism 2, assumes a di erent functional form for the utility of public goods. The external e ects are anonymous and additive, but only within the same local authority. The shape of the utility function can be written as: U(y i ; (1 k) f (z A ) + kf (z B )) where k is a spillover parameter. As before, for k = 0 we have a local public good; for k = 1=2 a pure public good and for k < 1=2 a local public good with spillover. The idea behind this formulation is that public goods have marginal decreasing utility at community level, hence the utility of an additional unit of good depends on where it is produced. Another way to interpret this assumption is to think about basic needs. The rst units of the public good produced respond to a basic need (for example having drinking water), subsequent units do not produce the same level of utility (having completely puri ed drinking water). In this case, more utility may be gained by a more even distribution of the public good. Wildasin (1997) has extended the use of this function to represent impure public goods: U(y k ; f (z k ) + k P f (z i )) In this case, the utility derived from the impure public good is anonymous across households, but its distribution at this lower level matters. Also in this case the good has a marginal decreasing utility, but since its bene t is mainly concentrated at individual level, it is at this level that the distribution matters. The use of the second type of utility function aims at reducing the di erences in the level of the public good produced across jurisdictions. In this case corner solutions are never optimal. The use of this utility function opens an interesting debate as regards coordination problems (Petretto, 2000, Levaggi and Menoncin, 2008b; Wildasin and Ogawa, 2008), asymmetry of information (Levaggi, 2008) bailout and the soft budget constraint (Wildasin, 2004, Crivelli and Staal, 2006, Levaggi and Menoncin, 2008a). The two approaches concide if the good modelled is a local public good. For the other cases, the di erence in the functional form produces interesting and important di erences in the results and the policy implications. 2 Besley and Coate (2003), Wildasin (1997). 3

5 3 Discussion To show the di erences in the two approaches it is possible to work out a simple example using the following lin-log functions: for global public goods and U i = y i + log(z i + kz j ) U i = y i + k log z i + (1 k) log z j for utility in variety. The number of individuals in each jurisdiction is normalised to one and the public good is nanced using taxation; its form and distribution among the two local authorities is unin uential given the assumption of linear utility in income. Let s also assume that jurisdiction A is more productive, i.e. p A < p B : The rst best allocation for a benevolent central planner is the quantity that maximises total welfare, de ned as the sum of the utility for both jurisdictions. For global public goods we have two solutions according to the value of k: k be written as: q pj p i p j+pi. In this case there is an internal solution that can z i = z j = For k = 0, the solution becomes: pj 2(pj+pi)k(1: k) p jp i (p j+p i) 2 k(1 k) pi 2(pj+pi)k(1 k) p jp i (p j+p i) 2 k(1 k) z r = 1 p r r = i; j k > q pj p i p j+p i. In this case we will have a corner solution given by: z i = 2 p i z j = 0 For the utility in variety case, the optimal solution is equal to: z r = 1 p r r = i; j and it is independent from k. The total quantity is the same for the local public good case and it will then be di erent using the two approaches. In this case, due to the assumption of a lin-log utility, the total quantity for the global public good case is higher but this result is not general: it simply depends on the type of utility function used. 4

6 Even in a very simple framework with a single decision-maker and full information, the choice of the utility function has important implications from a policy point of view. In the rst case, given the global public goods characteristic, production can be concentrated in a single jurisdiction and the problem will be to make the more productive provide the quantity at low cost and the others pay their part. In the long run competition on the production side may lead to improved e ciency. In the second case, utility in variety implies imperfect substitution. Production will have to be located in each jurisdiction and the incentive to cost minimisation may be less e ective. 4 A "superior" model? The natural question arising from this discussion is which functional form should be used to represent public goods? The answer is that there is no good or right model: the two approaches re ect di erent ways of producing externalities and they should be applied to the right framework. For negative external e ects such as pollutants and CO2 emission in the environment, the global public good approach is more reasonable since it is the concentration of the pollutant in the air that produces the public bad, no matter its source. In this respect papers such those proposed by Buckolz and Konrad (1995), Wildsin and Ogawa (2008) are used in the right context. Health care, health promotion and education relates more to the second type of literature, perhaps the right context being the one proposed by Wildasin (1997). Some of the literature on global public goods uses these services as examples for the application of their framework and this does not seem to be correct 3. Peace, global knowledge and environment protection are somehow more controversial issues. However, given the importance of the way such goods are modelled, the literature should pay more attention to this aspect. 3 Sandler and Arce (2002) make a very interesting categorisation of health care goods, but they do not formalise the utility function. 5

7 A Optimal conditions for global public goods The welfare function for the community can be written as: W = y i + y j T + log((1 k)z i + kz j ) + log((1 k)z j + kz i ) which has to be maximised under the following constraints: T = p i z i + p j z j z i 0 z j 0 The rst constraint can be substituted in the utility function. The second one is always veri ed given that p i < p j by assumption. To take account of the third constraint we can use a Khun Tucker maximisation approach. The problem can be written as: W = y a + y b p a z a p b z b + log((1 k)z a + kz b ) + log((1 k)z b + kz a The F.O.C can be written i : = j : p az az b +2p az akz b p az 2 a k 2pazak2 z b +p az 2 a k2 p akz 2 b +pak2 z 2 b +z b 2kz b +2z ak+2k 2 z b 2z ak 2 ( z a+z ak kz b )(z b kz b +z ak) p b z az b +2p b z akz b p b z 2 a k 2p bz ak 2 z b +p b z 2 a k2 p b kz 2 b +p bk 2 z 2 b +2kz b 2k 2 z b +2z ak 2 +z a 2z ak ( z a+z ak kz b )(z b kz b +z j 0; z j 0; z j = 0 q For k 1 pj p 2 1 i p i = 0; z j 0; z j = 0 and the solution qcan bewritten as: For k > 1 pj p 2 1 i p i = 0; z j = 0; z j < 0 and the solution is equal to: z a = 2 p a z b = 0 B Utility in variety In this case welfare can be written as: W = y i + y j p i z i p j z j + (1 k) log z i + k log z j + (1 k) log z j + k log z i ) The F.O.C can be a = paza 1 b = p bz b 1 z b which gives the solution in the text. 6

8 References [1] Besley, T. and S.Coate (2003) Centralized versus decentralised provision of local public goods: a political economy approach, Journal of Public Economics, 87,261 [2] Buchholz W., Konrad K. (1995), Strategic transfers and private provision of public goods, Journal of Public Economics, 57, [3] Crivelli, E. and K. Staal (2006) Size and Soft Budget Constraint, CESifo Working Paper Series No Available at SSRN: [4] Huber, B. and Runkel, M. (2006). Optimal Design of Intergovernment Grants Under Asymmetric Information, International Tax and Public Finance, 13, [5] Levaggi, R. and M. Menoncin (2008a) mimeo [6] Levaggi, R, and M. Menoncin (2008b) Fiscal Federalism, Patient Mobility and the Soft Budget Constraint,Politica Economica, forthcoming [7] Samuelson, P.A. (1954). The Pure Theory of Public Expenditure, Review of Economics and Statistics 36 (4): doi: / [8] Warr, P.G. (1983). The private provision of a public good is independent of the distribution of income, Economic Letters 13, [9] Wildasin D.E. Ogawa, H. (2008) Think locally, Act locally: spillovers, spillbacks and E cient Decentralized Policy Making, American Economic Review, forthcoming [10] Wildasin, DE (2004) The institutions of federalism: towards an analytical framework, National Tax Journal, LVII, [11] Wildasin, D.E. (1997) Externalities and bailouts, Hard and Soft budget constraints in Intergovernmental Fiscal Relations, mimeo 7

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