CONSUMERS HETEROGENEITY, PUBLICNESS OF GOODS AND THE SIZE OF PUBLIC SECTOR

DEPARTENT OF ECONOICS UNIVERSITY OF CYPRUS CONSUERS HETEROGENEITY, PUBLICNESS OF GOODS AND THE SIZE OF PUBLIC SECTOR Chrstos Blanakos Dscusson Paper 18-01 P.O. Box 0537, 1678 Ncosa, CYPRUS Tel.: +357-893700, Fax: +357-89508 Web ste: http://www.econ.ucy.ac.cy

Consumers Heterogenety, Publcness of Goods and the Sze of Publc Sector Chrstos Blanakos Abstract 01 Ths artcle studes the relatonshp between the level of consumers nequalty (or heterogenety) and the sze of government for the case of an mpure publc good. It s shown that the sze of redstrbuton (represented by the level of subsdy provded to the frm) ncreases wth the publcness of the good but may decrease wth the level of consumers nequalty. Under the assumpton of Nash barganng between consumers and producers wth respect to the level of subsdy, t s also shown that the actual sze of government wll be neffcently small f the level of nequalty s relatvely low but can be neffcently large (mplyng that the good wll be overproduced n equlbrum) f the level of nequalty s relatvely low and the publcness of the good s hgh enough. Keywords: Impure publc good, perfect competton and monopoly, subsdy, Inequalty, Nash Barganng. JEL Classfcatons: C71, H11, H1, H41. 1. Introducton It s well known that the Frst Welfare Theorem fals.e. the compettve equlbrum s not Pareto effcent n the presence of a publc good. Ths knd of neffcency s due to the free-rder problem assocated wth the presence of a postve consumpton externalty. If, at the same tme, the market of the good under queston s monopolstc, then the neffcency problem s accentuated and the output quantty s further dstorted below the optmum f the monopolst cannot mplement perfect prce dscrmnaton. In such cases, a benevolent socal planner who has complete nformaton about consumers demand and frms cost functons can restore effcency by mplementng an approprately desgned quantty-based or prce-based nterventon mechansm (such as a subsdy scheme). 1 Of course, a good may be nether purely publc (nondepletable and nonexcludable) nor purely prvate. The concept of an mpure publc good was frst examned n studes focusng on publcly provded goods that are subject to crowdng effects (Bergstrom and Goodman, 197; Borcherdng and Deacon, 197). These studes use a crowdng functon Department of Economcs, Unversty of Cyprus, P.O. Box 0537, Ncosa 1678, Cyprus. E-mal: blanakos.chrstos@ucy.ac.cy. Telephone: + 35789369. 1 See as-colell et al (1995, Ch. 11.C) for more detals on the prvate provson of publc goods. 1

to take nto account the fact that the amount of good (or the servce level ) captured by each ndvdual s negatvely related to the number of consumers. For the case of a prvately provded good (wth whch ths paper s also concerned), the mpure publc good model has been based on the characterstcs approach (Cornes and Sandler, 1984; 1994). In ths framework, each consumer purchases some quantty of the good under queston but derves utlty from the good s characterstcs rather than from the good tself. In partcular, t s assumed that each unt of the good bought by the consumer jontly generates β unts of a prvate characterstc and γ unts of a publc characterstc (where β,γ>0) over whch the ndvdual utlty functon s defned. Ths paper studes the case of an mpure publc good by abstractng both from crowdng effects and from the ndrect characterstcs approach. In order to take ntermedate cases of publcness nto account, we assume that each ndvdual captures (consumes) the quantty bought by herself n the market plus a proporton of the total quantty bought by other consumers. Ths proporton represents the degree of publcness for the good under queston and can vary between zero (mplyng that the good s purely prvate) and one (mplyng that the good s purely publc). There s clear evdence that the degree of government nterventon n both perfectly and mperfectly compettve markets can often be ether neffcently hgh or neffcently low. Therefore, some goods wll be underproduced (relatve to the frst-best level of output) and other goods wll be overproduced n equlbrum. Ths knd of government falure can be attrbuted to the based objectve functon of a non-benevolent government servng the nterests of some partcular socal group to the detrment of others. For nstance, the hgh levels of subsdes provded to producers of agrcultural products n USA or n EU countres have often been attrbuted to governmental decsons favorng large farmers nterests at the expense of consumers-taxpayers. However, the man branch of the lterature has followed the medan voter approach to account for the actual degree of government nterventon n the market. Accordng to ths approach, the sze of redstrbuton schemes (mplemented through taxes and subsdes) smply reflects the medan voter s preferences or ncome n poltcal regmes where decsons are made by majorty votng. In ths context, economc models usng the medan voter approach (e.g. Romer, 1975; Roberts, 1977; eltzer and Rchard, 1981; Alesna and Rodrk, 1994; Persson and Tabelln, 1994) typcally reach the concluson that the sze of redstrbuton

(whch can be nterpreted as the sze of publc sector) ncreases wth the level of ncome nequalty between consumers. Ths s due to the fact that any ncrease n nequalty also ncreases the degree of government nterventon preferred by the medan voter. However, emprcal evdence on the relatonshp between nequalty and redstrbuton remans ambguous and does not fully support ths theoretcal predcton, thus perhaps questonng the valdty of the medan voter approach tself. 3 In lght of these controverses, more recent studes (e.g. Ackert et al, 007; Durante and Putterman, 009; Grosser and Reuben 010; Hocht et al, 01) reman wthn the medan voter framework but assume that consumers-voters have other-regardng preferences (reflectng, for example, ther nequalty averson or farness consderatons) n order to explan the actual sze of redstrbuton and, n partcular, the relatonshp between ncome nequalty and the sze of government. Ths paper studes the relatonshp between the sze of government, the level of consumers heterogenety or nequalty and the publcness of the good under queston by abstractng both from exogenous assumptons about the government s objectve functon and from the largely unrealstc medan voter noncooperatve approach. In partcular, t s assumed here that the level of subsdy (representng the government sze) s determned by an mplct arbtrator who dvdes the gans from trade accordng to the barganng power of dfferent socal groups. ore specfcally, we adopt an axomatc barganng approach n the form of the Nash soluton (Nash, 1950), accordng to whch the fnal polcy selecton s the result of a reasonable socal compromse between the nterests of consumers and producers. 4 The model studes the degree of government nterventon n the market for a good of varyng publcness under the assumpton that consumers have heterogeneous preferences. To the extent that people wth a hgher ncome are more wllng to pay and thus have a For an excepton to the rule, see Katsm and outos (006) who show that the relatonshp between nequalty and redstrbuton can be non-postve even n a medan voter context when the government uses tax revenues to fnance the provson of a publc good also produced by the prvate sector. For other studes based on the medan voter approach, see Lee and Roemer (1999) and Benabou (000). 3 Some emprcal studes (e.g. eltzer and Rchard (1983), Easterly and Rebelo (1993), Alesna and Rodrk (1994) and lanovc (000)) support the hypothess of a postve relatonshp between nequalty and redstrbuton. However, other studes (e.g. Clarke (1995), Lndert (1996), Perott (1996) and Rodrguez (1999)) do not confrm ths theoretcal predcton. 4 Ths approach seems to be a rather more realstc descrpton of decson-makng n modern nsttutonalzed democraces than the drectly democratc medan voter approach. 3

more ntense preference for the good under queston, the degree of preference heterogenety can also be nterpreted as the level of ncome nequalty between consumers 5. The man fndngs of the paper can be summarzed as follows: Frst, the optmal sze of government s larger n a monopolstc market than n a compettve market. Ths s a hardly surprsng result gven the double source of neffcency that dstorts the monopolstc market and calls for a relatvely heaver government nterventon to mplement the Pareto optmal allocaton n equlbrum. Second, both the optmal and the actual sze of government strctly ncrease wth the publcness of the good and weakly decrease wth the degree of consumers heterogenety (nequalty). Ths means that a hgher level of ncome nequalty or preference heterogenety may mply a smaller government sze, contrastng the usual predcton of medan voter models whch fnd a postve relatonshp between nequalty and redstrbuton. Thrd, the actual level of subsdy (.e. the sze of government) can be neffcently hgh when the publcness of the good s hgh enough and the level of consumers nequalty s relatvely low. That s, a low enough level of nequalty may mply the overproducton of quas-publc goods due to the neffcently hgh degree of government nterventon n the market. Ths result contrasts the standard predcton of equlbrum underproducton for publc goods (or, more generally, for goods subject to postve consumpton externaltes). On the other hand, f the publcness of the good s relatvely low or the level of nequalty s relatvely hgh, then the level of subsdy wll be neffcently low (.e. the government sze wll be too small) and the good wll be underproduced n equlbrum. The rest of the paper s organzed as follows: Secton ntroduces the basc setup of the model. Secton 3 characterzes the compettve and monopolstc equlbrum for a good of varyng publcness as a functon of the subsdy scheme mplemented by the government. Secton 4 computes the Pareto optmal allocaton and uses t as a benchmark to determne the optmal correctve subsdy that restores effcency both n a compettve and n a monopolstc market. Secton 5 characterzes the actual level of subsdy n a monopolstc market under the assumpton of Nash barganng between consumers and producers and evaluates the actual sze of government from a welfare pont of vew. Secton 6 concludes the paper and dscusses ts possble extensons. 5 The next secton provdes a justfcaton of ths nterpretaton n more detal. 4

. The odel Consder a settng wth n consumers and one good (labeled as good A) of varyng publcness n addton to m prvate goods. We assume a constant-returns-to-scale technology for the sngle proft-maxmzng frm producng good A, mplyng the followng cost functon: cq ( ) = cq, where c > 0 and q s the produced quantty of good A. Defne a beneft functon V () =1,,n, where: V (0) = 0, V () > 0 and V () < 0. over the level of good A captured by each consumer Imagne that a market exsts for good A and each consumer chooses the quantty (A ) of the good bought n the market by takng as gven ts prce (p). In order to capture the varyng publcness of the good (rather than assume that the good has a purely prvate property and a purely publc property, as does the characterstcs approach), we assume that each ndvdual consumes the quantty of the good bought by herself plus a proporton δ [0,1] of the total quantty bought by other consumers: A + δ A j j Accordng to ths formulaton, the publcness of good A ncreases wth the value of parameter δ, whch represents the degree of nondepletablty and/or nonexcludablty of the good. The corner values δ=0 and δ=1 correspond to the extreme cases of a purely prvate and a purely publc good, respectvely. We assume n= for smplcty. The redstrbuton scheme s effectuated through a subsdy s [0, c] provded to the frm per unt of produced output. The cost of the subsdy s pad by consumers n the form of lump-sum taxes j T1, T (where T s the tax pad by consumer ) and the total tax burden s equally shared between consumers 1 and : T 1 =T =T, where T1+ T = T = sq or T = sq/. We also assume m=1 and lesure (l) s the purely prvate good used as the numerare commodty. Each consumer =1, has a dfferent ncome ( ) and her preferences are represented by a utlty functon whch s separable n l and A: φ = u( l ) + V( A + δ A ), where u > 0 and u < 0 (1) 5

The budget constrant faced by consumer wll be bndng at the soluton of her utlty maxmzaton problem and has the followng form: l + pa = T () We can substtute () nto (1) to get: φ = u ( T pa) + V( A+ δ A) (3) j Then, we can use a frst-order Taylor expanson to approxmate u() as: 6 u ( T pa) u ( ) ( pa+ T) u ( ) (4) Rewrte (3) by use of (4) as: φ = u ( ) ( pa+ T) u ( ) + V( A+ δ A) Of course, ths utlty functon represents the same preferences as the followng one: 1 U = V( A + δ Aj) pa T u ( ) We conclude that consumer s utlty functon can be wrtten as: j U ( A, A ) = θ V( A + δ A ) pa T (5) j j whereθ 1/ u ( ) > 0 s the nverse of the margnal utlty of ncome. Accordng to ths formulaton, f u() s concave then a hgher level of ncome () s assocated wth a lower u ( ) and therefore wth a hgher value of θ. We assume 1, mplyng θ θ. Snce the parameter θ can also be nterpreted as the ntensty of consumer s 1 preferences for good A, we conclude that the wealther consumer also has a stronger preference for the good under queston. In the same sense, the degree of preference heterogenety (captured by the ratoθ 1 / θ ) can also be nterpreted as the level of ncome nequalty between consumers (see also Trole, 1989, Ch. 3.3). In order to get reduced form solutons, we use the specfc form of beneft functon V( x) = x for the rest of the paper. 3. Compettve and onopolstc Equlbrum At a compettve equlbrum, each consumer chooses the quantty A so as to maxmze her utlty functon (takng as gven the prce p and the quantty A j purchased by the other consumer) and, therefore, solves the followng problem: 6 The general formula s f ( x) f( x0) ( x x0) f ( x ) +. In ths case, 0 f = u x= pa T x0 =, and. 6

max U ( A, A ) = θ V ( A + δ A ) pa T { A } j j st.. A 0 The soluton of ths problem mples the followng best-response functons for each consumer : θ A( Aj) = max δ A,0 j p (6) In order to derve the set of ndvdual demand functons, we mpose the requrement that each consumer s choce of purchased quantty must be her best response to the quantty purchased by the other consumer (as n a Nash equlbrum): A = A( A ) (7) j We solve the system of equatons (6) and (7) to get consumers ndvdual demand functons A( p), A ( p) and the aggregate demand functon D(p): 1 ( A 1 ( p), A ( p), D( p ))= θ δθ θ δθ θ + θ (1 δ) p (1 δ) p (1 + δ) p 1 1 1,,, f δ θ (8) where θ θ / θ (0, 1) 1 between consumers and D( p) = A ( p). θ θ 0,,, f θ δ 1 p p s a measure of preference heterogenety or ncome nequalty n = 1 Note that consumer 1 purchases a zero amount of good A for hgh enough values of δ. In other words, f the good s smlar enough to a pure publc good then the free-rder problem takes the usual extreme form and only the consumer who derves the largest margnal beneft from the good purchases a postve quantty n equlbrum. The compettve frm chooses the suppled quantty of the good so as to maxmze profts and, therefore, solves the followng problem: max π = ( p+ s c) q { q} st.. q 0 The soluton yelds the supply functon of the compettve frm: 7

0, f p< c s q( p ) = 0, f p = c s (9), f p > c s At a compettve equlbrum, aggregate demand equals aggregate supply: D( p) = q( p) (10) The system of equatons (8), (9) and (10) mples the compettve equlbrum prce and allocaton summarzed below along wth equlbrum profts, utltes and surpluses (gven the subsdy s). Case 1. For δ θ, the compettve equlbrum s: θ + θ θ δθ θ δθ ( p*, q*, A, A ) c s,,, (1 + δ)( c s) (1 δ )( c s) (1 δ )( c s) * * 1 1 1 1 = * * (1 δ ) θ1 + δθ sq* (1 δ ) θ + δθ1 sq* ( U1, U) =, (1 δ )( c s) (1 δ )( c s) (1+ δ)( θ1 + θ) (1+ δ)( θ1 + θ) ( CS*, PS*, TS*) = sq*, 0, sq* (1 + δ)( c s) (1 + δ)( c s) (11a) Case. For θ δ 1, the compettve equlbrum s: θ θ ( p*, q*, A, A ) c s,, 0, ( c s) ( c s) * * 1 = * * θθ 1 δ sq* θ sq* ( U1, U) =, c s c s θ ( θ δ + θ ) θ ( θ δ + θ ) CS PS TS = sq sq c s c s 1 1 ( *, *, *) *, 0, * (11b) n * where CS* = U s the consumer surplus, PS*=π* s the producer surplus (whch = 1 concdes wth the frm s proft) and TS* s the total surplus n compettve equlbrum. On the other hand, f the market s monopolstc then the frm chooses the prce of the good so as to maxmze ts profts gven the consumers behavor descrbed by the aggregate demand functon and, therefore, solves the followng problem: max π = ( p+ s c) q { p} st.. q= D( p) 8

The soluton of ths problem yelds the monopolstc prce (p ), whch can then be substtuted nto ndvdual demand functons to compute the equlbrum allocaton summarzed below along wth profts, utltes and surpluses (gven the subsdy s): Case 1. For δ θ, the monopolstc equlbrum s: θ + θ θ δθ θ δθ ( p, q, A, A ) ( c s),,, 4(1 + δ)( c s) 4(1 δ )( c s) 4(1 δ )( c s) 1 1 1 1 = (1 δ ) θ1 + δθ sq (1 δ ) θ + δθ1 sq ( U1, U ) =, (1 δ )( c s) (1 δ )( c s) (1a) (1+ δ)( θ + θ ) θ + θ (3 + 4 δ)( θ + θ ) ( CS, PS, TS ) = sq,, sq (1 + δ)( c s) 4(1 + δ)( c s) 4(1 + δ)( c s) 1 1 1 Case. For θ δ 1, the monopolstc equlbrum s: θ θ ( p, q, A, A ) ( c s),, 0, 4( c s) 4( c s) 1 = θθ 1 δ sq θ sq ( U1, U ) =, c s ( c s) θ ( θ δ + θ ) θ θ (4θ δ + 3 θ ) ( CS, PS, TS ) = sq,, sq ( c s) 4( c s) 4( c s) 1 1 (1b) 4. The Frst-Best Allocaton and the Optmal Sze of Government The Pareto optmal allocaton maxmzes aggregate surplus subject to the feasblty constrants and thus solves the followng problem: n max TS = θv ( A + δ Aj) cq qa1 A = 1 j {,, } n st.. A q = 1 qa, 0, = 1,..., n We solve ths problem to get the frst-best allocaton summarzed below. Case 1. For δ θ, the Pareto effcent allocaton s : P P P (1 + δ)( θ1 + θ) (1 + δ)( θ1 δθ) (1 + δ)( θ δθ1 ) ( q, A1, A ) =,, c (1 δ) c (1 δ) c (13a) 9

Case. For θ δ 1, the Pareto effcent allocaton s : P P P ( θ1 δ + θ) ( θ1 δ + θ) ( q, A1, A ) =, 0, c c (13b) The set of subsdes S = ( s*, s ) mplementng the frst-best allocaton n a compettve and n a monopolstc market, respectvely, s characterzed by the followng condton 7 : q*( s*) = q ( s ) = q P (14) Equaton (14) can be solved to yeld the set of optmal correctve subsdes representng the optmal sze of government n a compettve and n a monopolstc market: (*, s s ) = δ 1+ δ c, c 1+ δ (1 + δ) δθ 1+ δθ c, c 1+ δθ (1+ δθ ), f δ θ, f θ δ 1 (15) Proposton 1 mmedately follows from (15). Proposton 1. (a) The optmal sze of government s always larger n a monopolstc market than n a compettve market: s > s* for all δ, θ [0,1]. (b) The optmal sze of government strctly ncreases wth the publcness of the good and weakly decreases wth the degree of consumers heterogenety (nequalty) both n a compettve and n a monopolstc market: s*/ δ > 0, s / δ > 0 for δ [0,1] s*/ θ > 0, s / θ > 0 for θ [0, δ ) s*/ θ = 0, s / θ = 0 for θ ( δ,1] Fgures 1 and graphcally depct s* and s as a functon of δ and θ, respectvely. 7 Equvalently, s* and s are the solutons to the problem of a benevolent socal planner who chooses the level of subsdy so as to maxmze the aggregate surplus assocated wth the compettve and monopolstc equlbrum found above: s* = arg max TS *, where TS* and TS are gven n (11) and (1), respectvely. s = arg max TS 10

s*, s s (δ) s*(δ) 0 θ 1 δ Fgure 1. The optmal correctve subsdy as a functon of δ s*, s s (θ) s*(θ) 0 δ 1 θ Fgure. The optmal correctve subsdy as a functon of θ Part (a) of Proposton 1 s hardly surprsng: The degree of government nterventon requred to mplement the effcent allocaton n a monopolstc market s hgher than n a compettve market, snce the former suffers both from the free-rder problem (as does the compettve market) and from the standard output dstorton assocated wth a nondscrmnatng monopolst. As for part (b), t s also farly ntutve that the level of 11

correctve subsdy ncreases wth the publcness of the good, snce the free-rder problem s accentuated as the good becomes more smlar to a publc good. The most nterestng fndng s the negatve relatonshp between consumers heterogenety (nequalty) and the optmal sze of government forδ θ. Ths result s worth explanng n more detal. For the case of a monopolstc market, the optmal correctve subsdy (s ) must satsfy the frst-order condton 8 : TS ( s ( θ1, θ), θ1, θ) s 0 We dfferentate both sdes of the above dentty wth respect to θ and get: TS s TS + 0 s θ s θ, or: s TS = θ / s θ s TS / where all second-order partal dervatves are evaluated at s=s. Snce the denomnator of the expresson n the rght hand of (16) s negatve due to the second-order condton for maxmzaton, we conclude that: s TS sgn = sgn = θ s θ / s s Consder the case θ δ frst (.e. the case where the level of nequalty s relatvely hgh). Then, consumer 1 purchases nothng n equlbrum and we know that: TS θ ( θ δ 1 + ) = θ s θ + θ ( c s) 4( c s) 4( c s) where the frst term s the pre-tax consumer surplus, the second term s the cost of subsdy (.e. the tax pad by consumers) and the thrd term s the frm s proft. Then, t s straghtforward to see that: (16) (17) TS θ ( θ δ 1 + θ ) θ ( c+ s) = + θ 3 s ( c s) 4( c s) 4( c s) and: TS θ δ s 0 > 0 (18) (17) / s = s = > s θ1 ( c s ) θ1 In ths case, an ncrease n consumer 1 s ncome or taste parameter θ 1 (mplyng a hgher value of θ and therefore a lower degree of nequalty) ncreases the margnal pre-tax consumer surplus assocated wth a hgher value of s (due to a hgher valuaton of the quantty captured by consumer 1) and leaves both margnal profts and the margnal cost 8 The analyss here follows Varan (199, p. 491-495) and s qualtatvely the same for the case of a compettve market. 1

of subsdy unaffected (because the equlbrum prce and output do not depend on θ 1 n ths case), mplyng that the socal planner optmally chooses a hgher level of the correctve subsdy s. Smlarly, we can fnd: TS 1 c + s s [ 3 ] 0 < 0 (17) / s = s = θ 1 δ + θ θ < s θ( c s ) c s θ In ths case, an ncrease n consumer s ncome or taste parameter θ (mplyng a lower value of θ and therefore a hgher degree of nequalty) ncreases the margnal pre-tax consumer surplus and the frm s margnal proft assocated wth a hgher value of s but, at the same tme, ncreases even more the margnal cost of subsdy, mplyng that the socal planner optmally chooses a lower level of the correctve subsdy s. (19) From (18) and (19), we conclude that both an ncrease n θ 1 and a decrease n θ (.e. any ncrease n θ, whch means a lower level of nequalty between consumers) mples a hgher optmal level of subsdy ( s / θ > 0). In sum, an ncrease n the level of equalty calls for a bgger government to mplement the effcent resource allocaton n equlbrum 9. 5. onopolstc Equlbrum wth Nash Barganng Ths secton characterzes and evaluates the actual level of subsdy chosen n a monopolstc market under the assumpton of Nash barganng between consumers and producers. For ths purpose, we defne a barganng problem ( uu, ) where the set u = ( CS ( s), PS ( s)) represents the payoff allocatons that can be settled on f there s cooperaton between consumers and producers wth respect to the level of subsdy s [0, c] and the threat pont ( u = CS (0), PS (0)) u s the outcome that occurs f there s a breakdown of cooperaton. In other words, f barganng between consumers and producers collapses then the government does not ntervene n the market at all and agents receve the set of payoffs correspondng to a monopolstc equlbrum wthout any subsdy scheme (s=0). The symmetrc Nash barganng soluton (s N ) maxmzes the 9 For θ δ, a smlar analyss shows that TS ( s )/ s θ = 0, mplyng s / θ = 0. As a result, any change n the level of nequalty leaves the optmal level of subsdy unaffected n ths case ( s / θ = 0). 13

product of agents payoff dfferences from the threat pont and solves the followng problem: max W = [ CS ( s) CS (0)] [ PS ( s) PS (0)] {} s st.. 0 s c where CS (s) and PS (s) are gven n (1). The Nash soluton s summarzed below. N s = + 8δ 5+ 8δ c + 8 5+ 8 δθ c δθ, f δ θ, f θ δ 1 (0) The results from the comparatve statcs analyss concernng the effect of publcness (δ) or nequalty (θ) on the actual sze of government (s N ) are qualtatvely the same as descrbed N N n Proposton 1 for the benchmark case: s / δ > 0, s / θ 0. We proceed to evaluate the Nash soluton (whch s our predcton for the actual sze of government n a monopolstc market) by comparng t to the optmal level of subsdy (s ) found n (15). If s N > s, the actual government sze s neffcently large and the good s overproduced (relatve to the frst-best) n equlbrum. In order to examne more closely the possblty of overproducton, we remnd frst that the socally optmal level of subsdy (s ) satsfes the frst-order condton: TS s / s = s CS PS = 0, or: / s = s / s s = s s = (1) Then, overproducton ( s NB > s ) wll be the case f and only f: W CS ( ( ) (0)) ( ( ) (0)) PS / s = s = / s = s PS s PS + / s = s CS s CS > 0 s s s (1) PS / ( ( ) (0)) ( ( ) (0)) s = s CS s CS PS s PS > 0 s Snce the dervatve n the last expresson s postve, we conclude: NB s > s CS ( s ) CS (0) > PS ( s ) PS (0) () In other words, overproducton wll be the case f and only f consumers gans from the mplementaton of the frst-best level of subsdy (relatve to the case where there s no subsdy at all) exceed producers respectve gans. Note that a decrease n consumers 14

threat pont or an ncrease n producers threat pont makes overproducton more lkely. Therefore, the possblty of overproducton depends on our reasonable assumpton that consumers receve the low enough payoff CS (0) and producers receve the hgh enough payoff PS (0) f cooperaton breaks down. Intutvely, producers always beneft from a hgher level of subsdy.e. the level of subsdy that maxmzes the producer surplus s P s = c > s C s < s. On the other hand, the level of subsdy that maxmzes consumer surplus s. Snce consumers outsde opton s low enough and producers outsde opton s hgh enough, consumers are wllng to accept (and producers are able to mpose) a level of subsdy that s hgher than s C and, ndeed, mght be even hgher than s as a Nash soluton nstead of endng up wth ther low reservaton payoff. Defne the functon: f( δθ, 1, θ) CS ( s ) CS (0) PS ( s ) PS (0) Then, from () we see that s NB > s f and only f 1 f ( δ, θ, θ ) > 0. Smple calculatons yeld: f ( δ, θ1, θ ) = (4δ 1)( θ1 + θ) 4(1 + δ )c 4δθ θ 4c 1, f δ θ, f θ δ 1 The above expresson mmedately mples the results stated n Proposton, whch s the man fndng of the paper. (3) Proposton. (a) For low values of θ (.e. f the level of nequalty s hgh enough), the equlbrum level of subsdy s neffcently low and the good s underproduced n equlbrum for any value of publcness (δ): If θ < 1/4, then s N N < s (.e. ( ) q s < q P ) for all δ [0,1]. (see Fgure 3) (b) For ntermedate or hgh values of θ (.e. f the level of nequalty s relatvely low), the equlbrum level of subsdy s neffcently low (the good s underproduced) when the publcness of the good s low but the equlbrum level of subsdy s neffcently hgh (the good s overproduced) when the publcness of the good s hgh enough: If 1/4 θ 1/, then: s s N N q s < q P ) for δ < 1/4θ N < s ( ( ) q s > q P ) for 1/4θ < δ 1 (see Fgure 4) N > s ( ( ) In ths case, the monopolstc equlbrum s effcent for δ=1/4θ. 15

If 1/ θ 1, then: s s N N N P < s ( q ( s ) q N > s ( ( ) < ) for δ < 1/ q s > q P ) for 1/ < δ 1 (see Fgure 5) In ths case, the monopolstc equlbrum s effcent for δ=1/. s Μ, s N s (δ) s N (δ) 0 θ 1 Fgure 3. θ < 1/4 δ s Μ, s N s N (δ) s (δ) 0 θ 1/4θ 1 δ Fgure 4. 1/4 θ 1/ 16

s Μ, s N s N (δ) s (δ) 0 1/ θ 1 δ Fgure 5. 1/ < θ 1 In sum, both the actual and the optmal government sze decrease wth the level of nequalty and ncrease wth the publcness of the good. If the level of nequalty s relatvely hgh, then the government sze wll be neffcently low and the good s underproduced n equlbrum. But f the level of nequalty s relatvely low and the publcness of the good s hgh enough, then the government sze wll be neffcently large and the good s overproduced n equlbrum. That s, quas-publc goods tend to be oversubsdzed when consumers ncomes are relatvely equal (.e. when preferences are relatvely homogeneous). 6. Concluson ost medan voter models predct a postve relatonshp between the sze of government redstrbuton and the level of consumers-voters nequalty. However, ths theoretcal predcton s not fully supported by emprcal evdence. Ths paper has departed from the medan voter approach n order to study the relatonshp between the level of nequalty and the degree of government nterventon n the market of a good whch s subject to postve consumpton externaltes. In ths context, t has been shown that the government sze (represented by the level of subsdy provded to the frm) ncreases wth the 17

publcness of the good but may decrease wth the level of consumers nequalty. In contrast to the medan voter approach, we have assumed that the actual sze of redstrbuton s the result of a compromse between consumers and producers who bargan over the level of subsdy. Then, we have shown that the actual level of subsdy (as gven by the Nash soluton of ths barganng problem) can be ether neffcently low or hgh dependng on the publcness of the good and on the level of consumers nequalty. In partcular, f the level of nequalty s suffcently hgh then the government sze wll be neffcently small and the good wll be underproduced n equlbrum. But f the level of nequalty s relatvely low and the publcness of the good s hgh enough, then the actual level of subsdy wll be neffcently hgh and the good s overproduced n equlbrum. These results may shed some lght on cases of potentally oversubsdzed or undersubsdzed goods observed n real economc lfe. A potental extenson of our analyss mght be to examne the optmal government sze after takng nto account the possblty of entry of new frms n the market. In ths framework, t mght also be possble to ntroduce nformatonal asymmetres n order to examne the scope and welfare mplcatons of a lmt prcng polcy exercsed by the subsdzed ncumbent. These ssues are left for future research. References - Ackert, Lucy, Jorge artnez-vazquez and ark Rder (007), Socal Preferences and Tax Polcy Desgn: Some Expermental Evdence, Economc Inqury, 45, 3, 487-501. - Alesna, Alberto and Dan Rodrk (1994), Dstrbutve poltcs and economc growth, Quarterly Journal of Economcs, 109, 465-490. - Benabou, Roland (000), Unequal socetes: Income dstrbuton and the socal contract, Amercan Economc Revew, 90, 96-19. - Bergstrom, Theodore and Robert Goodman (197), Prvate Demand for Publc Goods, Amercan Economc Revew, 63, 3, 80-96. 18

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