A Note on the Provision of a Public Service of Different Quality
aa r X i v : . [ ec on . GN ] M a r A Note on the Provision of a Public Serviceof Different Quality ∗ Monica Anna Giovanniello † Simone Tonin ‡ February 2020
Abstract
We study how the quality dimension affects the social optimum in a model of spatialdifferentiation where two facilities provide a public service. If quality enters linearly inthe individuals’ utility function, a symmetric configuration, in which both facilities havethe same quality and serve groups of individuals of the same size, does not maximize thesocial welfare. This is a surprising result as all individuals are symmetrically identicalhaving the same quality valuation. We also show that a symmetric configuration offacilities may maximize the social welfare if the individuals’ marginal utility of qualityis decreasing.
Keywords : Public goods; spatial differentiation; quality; social welfare.
JEL classification : D60, H41.
We study the problem of providing a public service in different locations. We build amodel where individuals are uniformly distributed along the Hotelling line and the serviceis provided by a configuration of two facilities. We introduce a new dimension by assumingthat each facility is not only characterized by its location but also by its quality. Indi-viduals have homogeneous quality valuation and they bear quadratic transportation costto consume the service. In this framework, we study the configuration of facilities thatmaximizes the social welfare function. A trade-off arises between the location and qualityof the facilities and the size of the groups of individuals served by them.We first consider the case in which quality enters linearly in the individuals’ utility func-tion. We show that, when the quality valuation is high enough, the optimal configurationis such that one facility has the highest quality and serves all the individuals. Differently,when the quality valuation is low enough, it is optimal to have a facility with the highestquality serving a large group of individuals and another one with the lowest quality servinga smaller group of individuals. It is surprising that the symmetric configuration, two facil-ities having the same quality and serving groups of individuals of the same size, is neveroptimal despite individuals have homogeneous quality valuation.We continue our analysis by considering an extension in which individuals’ marginalutility of quality is decreasing. In this case, we show that the symmetric configuration offacilities is a social optimum when individuals’ quality valuation is low enough. The intu-ition behind our results is the following. First, when individuals’ marginal utility of quality ∗ We would like to thank Mauro Bambi, Daniel Cardona, and Leslie Reinhorn for their useful comments. † Departament d’Economia Aplicada, Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain,[email protected] ‡ Dipartimento di Scienze Economiche e Statistiche, Universit`a degli Studi di Udine, 33100 Udine, Italy,[email protected]
1s constant and the quality valuation is low enough, there is an individual who is indifferentbetween consuming the service in a distant high quality facility or in a closer one of lowquality, i.e. higher quality may compensate for higher transportation costs. Differently,when individuals’ marginal utility of quality is decreasing and the quality valuation is lowenough, the symmetric configuration of facilities becomes a social optimum as quality doesnot compensate for transportation costs.To the best of our knowledge this is the first paper that analyse the provision of a publicservice where both location and quality are considered. Starting from Tiebout (1956) thereis a large literature on local public goods where their provision is constrained by a division ofthe space into jurisdictions. This literature mainly focuses on jurisdictions’ optimal number,size, and composition (see Rubinfeld (1987) and Scotchmer (2002) for further references).Cremer et al. (1985) proposed an approach based on spatial competition theory to studiesthe optimal number and locations of facilities producing the public good. Differently fromtheir contribution, we analyse the optimal provision of a public good characterized by bothquality and location, but we astray from organizational and financing issues.The rest of the paper is organized as follows. Section 2 describes the mathematicalmodel. Section 3 is devoted to the study of the social optimum when individuals haveconstant marginal utility of quality. Section 4 considers a case in which individuals havedecreasing marginal utility of quality. Section 5 concludes.
We consider a spatial model described by a uniform distribution of individuals over theinterval I = [0 ,
1] and by two facilities providing a public service without congestion.The two facilities are located in a and b , with 0 ≤ a ≤ b ≤
1, and their quality are q a and q b respectively. The amount of resources available to finance the public service isexogenously given and it results in the following constraint q a + q b = 1. Individuals are assumed to have the same quality valuation for the public service but todiffer for their locations. Each individual consumes the public service in only one location x ∈ { a, b } and he has to bear the corresponding transportation cost. For simplicity, we alsoassume that all individuals consume the public service only once. We can then define theutility function of an individual located in i ∈ I consuming the public service in x ∈ { a, b } as u ( i, x ) = θq x − ( i − x ) , (1)with θ > The social welfare function is then given by Z i θq x ( i ) − ( i − x ( i )) di, (2)where x ( i ) ∈ arg max x ∈{ a,b } θq x − ( i − x ) is the optimal facility that maximizes the individual i ’s utility function. Let ( a, b, q ), with 0 ≤ a ≤ b ≤ ≤ q ≤
1, be a configuration of facilities thatspecifies their locations, a and b , and their qualities, q a = q and q b = 1 − q . A social Neven and Thisse (1989) introduced vertically differentiation in an Hotelling model with private goods. See Gabszewicz and Thisse (1992) for a survey on spatial competition theory. Given the individual’s utility in (1), considering the case q a + q b ≤ We astray from considering the tax to finance the public service because subtracting a lump-sum taxfrom the utility of all individuals would not change our analysis. a and the other defined over the interval of individuals consumingthe public service in the facility located in b . This allows us to use the first order necessaryconditions for a maximum to find the social optimum.The function ˆ j ( a, b, q ) associates to any configuration of facilities the unique indifferentindividual as followsˆ j ( a, b, q ) = j <
0) or ( a = b and q < ) , a − b − θ (2 q − a − b ) if j ∈ I, j >
1) or ( a = b and q > ) , with j = a − b − θ (2 q − a − b ) being the unique solution of the equation u ( i, a ) = u ( i, b ) with a = b .When j does not belong to the interval I , ˆ j ( a, b, q ) is equal to 0 or 1 depending of j beingless than 0 or greater than 1. Note also that j is not defined at configurations of facilitieswhere a = b while, in such cases, ˆ j ( a, b, q ) is equal to 0 or 1 as the public service is consumedby all individuals in the facility with the highest quality. The only configuration of facilitiesfor which ˆ j ( a, b, q ) is not defined is (cid:0) a, b, (cid:1) with a = b . However, as the following remarkpoints out, this is not a problem for our analysis. Remark.
Any configuration of facilities (cid:0) a, b, (cid:1) with a = b is not a social optimum becauseit is always possible to find a configuration of facilities (cid:0) a ′ , b ′ , (cid:1) with a ′ < b ′ that gives ahigher social welfare.Given the function ˆ j ( a, b, q ), the social welfare function can be written as W ( a, b, q ) = Z ˆ j ( a,b,q )0 θq − ( i − a ) di + Z j ( a,b,q ) θ (1 − q ) − ( i − b ) di, (3)We now state our main result which identifies the social optima for different values ofthe quality valuation θ . Theorem 1.
The social optima configurations of facilities are– for θ ∈ (cid:0) , (cid:1) : (cid:0) + θ, + θ, (cid:1) and (cid:0) − θ, − θ, (cid:1) ;– for θ ∈ (cid:2) , ∞ (cid:1) : (cid:0) , b, (cid:1) with b ∈ (cid:2) , (cid:3) and (cid:0) a, , (cid:1) with a ∈ (cid:2) , (cid:3) .The key insight on which is based this result is that the marginal utility of quality isconstant while the marginal transportation cost is increasing. Consequently, for θ ∈ (cid:2) , ∞ (cid:1) at the social optimum there is one facility with quality 1 serving all individuals while theother one has quality 0 and serves nobody. This configuration is optimal because for allindividuals the gains derived from consuming in the facility with the highest quality morethan compensate for the higher transportation costs borne to reach it. Note also that theindividual who is located in the same point of the facility with quality 0 prefers to consumefrom the highest quality facility. For θ ∈ (cid:0) , (cid:1) at the social optima there are again afacility with quality 1 and a facility with quality 0. But now individuals are segregatedin two different groups. A large group of individuals consume the public service from thehighest quality facility while the other smaller group is served from the facility with quality0. Heuristically speaking, for a small group of individuals quality does not compensate forthe transportation costs and such group is then served by the nearer facility with quality0. 3 roof. The social welfare maximization problem ismax a,b,q W ( a, b, q ) , subject to − a ≤ i ) a − b ≤ ii ) b ≤ iii ) − q ≤ iv ) q ≤ v ) (4)To prove the theorem, we first find the configurations of facilities ( a ∗ h , b ∗ h , q ∗ h ) that satisfy thefirst order necessary conditions for a maximum and are candidate to be the social optimum.We then compare the social welfare associated to each configuration of facilities found andwe finally identify the social optimum.We first consider the first order necessary conditions for a maximum associated to triples( a, b, q ) such that ˆ j ( a, b, q ) ∈ (0 , j ( a, b, q ) ∈ (0 , θ (1 − q ) − b + b − − ( a − b − θ (2 q − a − b ) . (5)By the Lemma in the Appendix, when we consider function (5) to derive the first ordernecessary conditions for a maximum, we just focus on the cases in which only constraints( iv ) or ( v ) may be binding. We then have three cases to analyze. First, neither constraints( iv ) nor ( v ) are binding. Then, the first order necessary conditions for a maximum withrespect to a , b , and q are respectively − (cid:0) a − b − θ (2 q − (cid:1) (cid:0) a − ab + b + θ (2 q − (cid:1) a − b ) = 0 , (6) − (( a − a − ( b − b − θ (2 q − (cid:0) a + a (2 − b ) + b (3 b − − θ (2 q − (cid:1) a − b ) = 0 , (7) θ (cid:18) θ (1 − q ) a − b + a + b − (cid:19) = 0 . The unique solution of the system of equations above is the configuration of facilities( a ∗ , b ∗ , q ∗ ) = (cid:18) , , (cid:19) . Second, only constraint ( iv ) is binding. Then, the first order necessary conditions for amaximum with respect to a and b are still (6) and (7). While the first order necessarycondition for a maximum with respect to q becomes θ (cid:18) θ (1 − q ) a − b + a + b − (cid:19) + λ iv = 0 , with λ iv being the Lagrangian multiplier associated to the constraint ( iv ). Then, theconfiguration of facilities ( a ∗ , b ∗ , q ∗ ) = (cid:18) − θ, − θ, (cid:19) In the paper we have solved all the first order necessary conditions for a maximum by using a computeralgebra system. θ ∈ (0 , ) is the unique solution of first order necessary conditions for a maximum.Third, only constraint ( v ) is binding. By following, mutatis mutandis , the same stepsabove, it is possible to verify that the configuration of facilities( a ∗ , b ∗ , q ∗ ) = (cid:18)
14 + θ,
34 + θ, (cid:19) for θ ∈ (0 , ) is the unique solution of first order necessary conditions for a maximum.We next consider the first order necessary conditions for a maximum associated to triples( a, b, q ) such that ˆ j ( a, b, q ) = 0. By solving the integrals of the social welfare function (3),when ˆ j ( a, b, q ) = 0, we obtain θ (1 − q ) − b + b − . It is straightforward to verify that the configurations of facilities( a ∗ , b ∗ , q ∗ ) = (cid:18) a, , (cid:19) , such that a ∈ [0 , ], are solutions of the first order necessary conditions for a maximum.We finally consider the first order necessary conditions for a maximum associated totriples ( a, b, q ) such that ˆ j ( a, b, q ) = 1. By following, mutatis mutandis , the same stepsabove, it is straightforward to verify that the configurations of facilities( a ∗ , b ∗ , q ∗ ) = (cid:18) , b, (cid:19) , such that b ∈ [ , W ( a ∗ , b ∗ , q ∗ ) = 24 θ − ,W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) = 48 θ + 24 θ − ,W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) = 48 θ − . For θ ∈ (0 , ), the configurations of facilities ( a ∗ h , b ∗ h , q ∗ h ) for h = 1 , , , , W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) > W ( a ∗ , b ∗ , q ∗ )and that W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) > W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) . We can now conclude that, for θ ∈ (0 , ), the social optima are ( a ∗ , b ∗ , q ∗ ) and ( a ∗ , b ∗ , q ∗ ).For θ ∈ [ , ∞ ), the configurations of facilities ( a ∗ h , b ∗ h , q ∗ h ) for h = 1 , , W ( a ∗ , b ∗ , q ∗ ) = W ( a ∗ , b ∗ , q ∗ ) > W ( a ∗ , b ∗ , q ∗ ) , the social optimal are ( a ∗ , b ∗ , q ∗ ) and ( a ∗ , b ∗ , q ∗ ), for θ ∈ [ , ∞ ).5 The symmetric configuration of facilities
In this section we study an example in which the symmetric configuration of facilities (cid:0) , , (cid:1) is a social optimum. The reason why it is worth investigating such configurationlies in its characteristics: all individuals consume the public service from a facility of thesame quality, q a = q b , and both facilities serve groups of individuals of the same size. Thiscase can then be considered as the most egalitarian provision of the public service.We now consider the same framework described in Section 2 with the exception thatindividuals have decreasing marginal utility of quality. Specifically, we assume the utilityfunction of an individual located in i ∈ I consuming the public service in x ∈ { a, b } is˜ u ( i, x ) = θ √ q x − ( i − x ) . (8)The function ˜ j ( a, b, q ) that associates to any configuration of facilities the unique indif-ferent individual becomes˜ j ( a, b, q ) = j <
0) or ( a = b and q < ) , a − b − θ ( √ q −√ − q )2( a − b ) if j ∈ I, j >
1) or ( a = b and q > ) . abusing of notation let j = a − b − θ ( √ q −√ − q )2( a − b ) be the unique solution of the equation ˜ u ( i, a ) =˜ u ( i, b ), with a = b . As above, the only configuration of facilities for which ˜ j ( a, b, q ) is notdefined is (cid:0) a, b, (cid:1) with a = b . However, the Remark holds also in this case.Given the function ˜ j ( a, b, q ), the social welfare function can be written as˜ W ( a, b, q ) = Z ˜ j ( a,b,q )0 θ √ q − ( i − a ) di + Z j ( a,b,q ) θ p − q − ( i − b ) di. (9)We now state our result on the symmetric configuration of facilities. Theorem 2.
The social optimum is the symmetric configuration of facilities (cid:0) , , (cid:1) for θ ∈ (cid:16) , √ (cid:17) .The key insight on which is based this result is that the marginal utility of qualityis decreasing. In other words, for θ ∈ (0 , √ ), a high quality facility does not longercompensate for the higher transportation cost borne to reach it. Therefore, the symmetricconfiguration of facilities emerges as a social optimum. Proof.
The proof of Theorem 2 follows closely the one above. The social welfare maxi-mization problem is still represented by (4) where, instead of having W ( a, b, q ), we have˜ W ( a, b, q ) defined in (9). We begin by considering the first order necessary conditions for amaximum associated to triples ( a, b, q ) such that ˜ j ( a, b, q ) ∈ (0 , j ( a, b, q ) ∈ (0 , θ p − q − b + b − − (cid:0) a − b − θ (cid:0) √ q − √ − q (cid:1)(cid:1) a − b ) . (10)The Lemma holds also when individuals have the utility function in (8) and then at a socialoptimum ( a ∗ , b ∗ , q ∗ ) with ˜ j ( a ∗ , b ∗ , q ∗ ) ∈ (0 ,
1) constraints ( i ), ( ii ), and ( iii ) are not binding.Therefore, we focus on the cases in which only constraints ( iv ) or ( v ) may be binding. First,neither constraints ( iv ) and ( v ) are binding. Then, the first order necessary conditions fora maximum with respect to a , b , and q are respectively6 (cid:0) a − b − θ (cid:0) √ q − √ − q (cid:1)(cid:1) (cid:0) a − ab + b + θ (cid:0) √ q − √ − q (cid:1)(cid:1) a − b ) = 0 , − (cid:0) ( a − a − ( b − b − θ (cid:0) √ q − √ − q (cid:1)(cid:1) (cid:0) a + a (2 − b ) + b (3 b − − θ (cid:0) √ q − √ − q (cid:1)(cid:1) a − b ) = 0 , θ (cid:16) √ q + √ − q (cid:17) (cid:0) a − b − θ (cid:0) √ q − √ − q (cid:1)(cid:1) a − b − √ − q = 0 . When θ ∈ (cid:16) , √ (cid:17) the unique solution of the system of equation above is the configurationof facilities ( a ∗ , b ∗ , q ∗ ) = (cid:18) , , (cid:19) . In the case in which either constrain ( iv ) or constrain ( v ) is binding, there are no solutionsof the first order necessary conditions for a maximum.We now consider the first order necessary conditions for a maximum associated to triples( a, b, q ) such that ˜ j ( a, b, q ) = 0. By solving the integrals of the social welfare function (9)we obtain ˜ W ( a, b, q ) = θ p − q − b + b − . It is possible to verify that the configurations of facilities( a ∗ , a ∗ , q ∗ ) = (cid:18) a, , (cid:19) , such that a ∈ [0 , ], are solutions of the first order necessary conditions for a maximum.We finally consider the first order necessary conditions for a maximum associated totriples ( a, b, q ) such that ˜ j ( a, b, q ) = 1. By following, mutatis mutandis , the same stepsabove, it is possible to show that the configurations of facilities( a ∗ , b ∗ , q ∗ ) = (cid:18) , b, (cid:19) , such that b ∈ [ , W ( a ∗ , b ∗ , q ∗ ) = 24 √ θ − W ( a ∗ , b ∗ , q ∗ ) = ˜ W ( a ∗ , b ∗ , q ∗ ) = 48 θ − . Comparing the above levels of social welfare for each facility configuration, it is immediateto see that ˜ W ( a ∗ , b ∗ , q ∗ ) > ˜ W ( a ∗ , b ∗ , q ∗ ) = ˜ W ( a ∗ , b ∗ , q ∗ ) , for θ ∈ (cid:16) , √ (cid:17) . But, then, the social optimum is ( a ∗ , b ∗ , q ∗ ). For θ ∈ [ √ , ∞ ) there are also other solutions of the first order necessary conditions for a maximumwhich are different from the symmetric configuration of facilities. Conclusion
Our framework allows us to study the provision of a public service horizontally (location)and vertically (quality) differentiated. Theorem 1 shows that the symmetric configurationof facilities is never a social optimum even if individuals have an identical and low enoughquality valuation. This result differs from models where the public service is only horizon-tally differentiated in which, at the optimum, each facility serves a group of individuals ofthe same size for any number of facilities (see Cremer et al., 1985). Theorem 2 highlightsthe role of decreasing marginal utility of quality by showing that the symmetric configura-tion of facilities may arise as a social optimum when it is not possible to compensate highertransportation costs by providing a public service of higher quality. Our results point tomany possible extensions that are worthy of further study. It would be very interesting toendogenize the optimal number of facilities and to formalize the taxation scheme requiredto finance the provision of the public service.
Appendix
To prove our main results we need the following lemma showing which constraints arenot binding at the social optima with ˆ j ( a ∗ , b ∗ , q ∗ ) ∈ (0 , Lemma.
Let ( a ∗ , b ∗ , q ∗ ) be a social optimum. If ˆ j ( a ∗ , b ∗ , q ∗ ) ∈ (0 , i ), ( ii ), and ( iii ) are not binding. Proof.
Let ( a ∗ , b ∗ , q ∗ ) be the social optimum and assume that ˆ j ( a ∗ , b ∗ , q ∗ ) ∈ (0 , i ) is binding. Consider a location a ′ such that a ′ > a ∗ = 0. First, note thatˆ j ( a ′ , b ∗ , q ∗ ) > ˆ j ( a ∗ , b ∗ , q ∗ ) as ∂ ˆ j ( a,b,q ) ∂a >
0. Then, it is straightforward to verify that Z j ( a ′ ,b ∗ ,q ∗ )0 u ( i, a ′ ) di > Z j ( a ∗ ,b ∗ ,q ∗ )0 u ( i, a ∗ ) di + Z j ( a ′ ,b ∗ ,q ∗ ) j ( a ∗ ,b ∗ ,q ∗ ) u ( i, b ∗ ) di. As the individuals i ∈ [ j ( a ′ , b ∗ , q ∗ ) , b ∗ , obtainthe same utility under both configurations of facilities, it follows that W ( a ′ , b ∗ , q ∗ ) >W ( a ∗ , b ∗ , q ∗ ), a contradiction. Hence, constraints ( i ) is not binding. By following, mutatismutandis , the same steps, it is easy to show that constraint ( iii ) is not binding. Finally,constraint ( ii ) is not binding as ˆ j ( a ∗ , b ∗ , q ∗ ) ∈ (0 , References [1] Cremer, H., De Kerchove, A.M. and Thisse, J.F., 1985. An economic theory of public facilitiesin space.
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