Keeping Up with the Joneses: Who Loses Out? School of Economics & Finance Discussion Papers. David Ulph

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1 Shool of Eoomis & Fiae Olie Disussio Paper Series iss X ifo: Shool of Eoomis & Fiae Disussio Papers Keepig Up with the Joeses: Who Loses Out? David Ulph Shool of Eoomis ad Fiae Disussio Paper No Sep 2014 JEL Classifiatio: D110; I31; J22 Keywords: Veble Effets; osumer behaviour; Nash equilibrium; wages ad well- beig

2 Keepig Up with the Joeses: Who Loses Out? David Ulph 1 Abstrat This paper ivestigates how well-beig varies with idividual wage rates whe idividuals are about relative osumptio ad so there are Veble effets Keepig up with the Joeses leadig idividuals to over-work. I the ase where idividuals ompare themselves with their peers those with the same wage-rate - it is show that Keepig up with the Joeses leads some idividuals to work who otherwise would have hose ot to. Moreover for these idividuals well-beig is a dereasig futio of the wage rate - otrary to stadard theory. So those who are worst-off i soiety are o loger those o the lowest wage. Keywords: Veble Effets; osumer behaviour; Nash equilibrium; wages ad wellbeig JEL Codes: D110; I31; J22 September Professor of Eoomis, Uiversity of St Adrews ad Diretor, Sottish Istitute for Researh i Eoomis (SIRE). Shool of Eoomi & Fiae, Uiversity of St Adrews, St Adrews KY16 9AL, Sotlad. du1@st-adrews.a.uk. Tel: +44 (0)

3 Keepig Up with the Joeses: Who Loses? Itrodutio Datig bak to Veble (1924), there is a extesive literature o ospiuous osumptio whereby idividuals lose esteem if their osumptio of some good(s) whih sigal their status is below the average of the referee/peer group ad gai esteem if their osumptio exeeds the average. It is reogised that this a lead to a rat rae i whih idividuals over-osume, with a osequet eed to fud this extra osumptio by either workig harder or savig less ( Frak (1985), Shor (1998)). This over-osumptio is referred to as the Veble Effet 2 or the Keepig up with the Joeses Effet 3. This paper develops some further impliatios for behaviour ad well-beig whe people are oered about their osumptio relative to their peers take to be those with a similar wage rate. It is show that the Keepig up with the Joeses Effet a lead people to work who would otherwise have hose ot to, ad that, for suh idividuals wellbeig will be a stritly dereasig futio of their wage rate. Thus those who are least well off i soiety are ot those with the lowest wage. 1. The Model Idividuals are edowed solely with 1 uit of time that a be spet o work or leisure. There is a tax/beefit system whereby everyoe reeives a tax-free uiversal beefit, σ > 0 ad all eared iome is taxed at the rate τ, 0< τ < 1. Idividuals differ i their produtivity whih is refleted i their et wage rate ω 0. A idividual with et wage ω who speds a fratio l, 0 l 1of time o leisure will ed up with osumptio = ω(1 l ) + σ. Idividual well-beig is a ombiatio of well-offess, y, ad happiess, h, as give by the futio: 1 w h θ θ = y, 0 θ 1. (1) Here: (i) Well-offess, y, is aptured by a utility futio y= u, l (2) ( ) satisfyig the stadard assumptios e.g. oavity. (ii) Happiess measures idividuals pereptios of how well their life is goig i ompariso to their peers those with the same et wage-rate, ω. It is assumed that this depeds o a idividual s osumptio relative to the average osumptio > 0 of their peers, ad that happiess is give by: 2 The Veble effet has also bee ivoked to help explai the Easterli Paradox - Easterli (2001). 3 This has led to argumets for either taxig suh ospiuous osumptio or ireasig the rate of iome tax see Boski ad Sheshiski (1978) - to orret the osumptio exterality. 1

4 h = = (3) The two reasos for adoptig this futioal form for happiess are: a) Happiess is thereby bouded betwee 0 ad 1, refletig the way happiess is traditioally measured o some fiite sale. b) Labour supply deisios deped o the average osumptio of others. If, istead, happiess depeds solely o the, give (1), the average osumptio of others would exert a egative exterality o idividual wellbeig but would ot affet behaviour thereby missig a ruial feature of the Keepig up with the Joeses effet 4. The parameter θ determies how muh idividual well-beig depeds o relative osumptio 5. So if θ = 0 we have the ovetioal eoomists story about wellbeig, ad there will be o Keepig up with the Joeses Effet. If 0< θ 1 the the Keepig up with the Joeses Effet is preset, ad is ireasig i θ. Combiig (1) (3) well-beig a be writte as: (,, ; ) θ = (, ) 1 θ θ wl ul, (4) + 2. Idividual Labour Supply ad Well-Beig Cosider a idividual with et wage rate ω. The idividual takes as give > 0 - the average osumptio of those with the same et wage rate - ad hooses labour supply (effort) e = 1 l to maximise well-beig, Let σ + ω (,1,, ) e σ + ω θ ( σ + ω,1 ) 1 w e e u e e σ + ωe+ ( ωσ,, ; argmax ( σ ω,1, ; e= f w + e e (6) 0 e 1 be the well-beig-maximisig labour supply deisio, ad ( ωσ,, ; ( σ ω,1, ; v = MAX w + e e (7) 0 e 1 the assoiated idiret well-beig futio. The f.o.. for maximisatio is θ 1 1 [ ωu u ω l ] + 0, 1 θ σ + ωe σ + ωe+ u e 0, (8) θ θ (5) 4 This is true of the formulatio adopted by Boski ad Sheshiski (1978). 5 This formulatio is osistet with that adopted by Boski ad Sheshiski (1978). 2

5 where the iequalities hold with omplemetary slakess. From (8) there is a reservatio et wage rate ul ( σ,1) ωσ (,, = (9) θ u ( σ,1) u ( σ,1) θ σ + σ at or below whih labour supply is zero ad above whih it is positive. This reservatio wage rate is: a stritly ireasig futio of ueared iome, σ; a stritly dereasig futio of average osumptio, ; a stritly dereasig futio of the weight, θ, give to happiess. Whe θ = 0, the reservatio wage is just the ovetioal margial rate of substitutio betwee osumptio ad leisure at zero hours of work. The fat that it is dereasig i both ad θ meas that the Keepig up with the Joeses Effet is iduig people to work who would ot otherwise have doe so. Sie, oditioig o ad θ, the labour supply deisio is a ovetioal utilitymaximisig deisio, it follows that, whe idividual labour supply is positive, it is a stritly dereasig futio of ueared iome, while the effet of a irease i the (et) wage rate is ambiguous, though the ompesated labour supply respose is positive. From (8) it follows that whe labour-supply is positive it is a stritly ireasig futio of - the Keepig up with the Joeses Effet ad, osistet with this, is also a ireasig futio of θ. I summary we have the followig omparative stati laboursupply preditios i the ase where labour supply is positive: i.e. ω > ω( σ,, f f f f f f f < 0; > 0; = e. > 0; > 0; > 0 6. (10) σ ω < ω ω σ θ Turig to the idiret well-beig futio, this agai will satisfy the stadard oditios, iludig Roy s idetity, so: ( ωσ v > 0; v = e. v = f,,,. v > 0, (11) σ ω σ σ So, oditioig o average osumptio,, for idividuals who work, well-beig is a stritly ireasig futio of the et wage rate. From (5) ad (7) the evelope theorem implies that v < 0. (12) Thus idividuals are worse off the greater is the average osumptio of others. 6 The supersript deotes the ompesated labour supply futio. 3

6 3. Nash Equilibrium Labour Supply ad Well-beig So far we have examied labour supply ad well-beig for ay arbitrary level of average osumptio of the peer group - those with the same et wage rate. To omplete the aalysis we eed to determie this average level of osumptio. Sie everyoe maximises well-beig takig as give the deisios of everyoe else as refleted i the average osumptio of the group, the relevat equilibrium oept is o-ooperative Nash. Sie everyoe i the omparator group is idetial, i the Nash equilibrium everyoe eds up with the same level of labour supply ad osumptio. This ommo osumptio is therefore the average osumptio of eah group, whih implies that for everyoe h = 1/2 3.1 Labour Supply From (6) the Nash equilibrium level of labour supply a be haraterised as the impliit solutio to the equatio: ( ωσσ,, ω, e= f + e. (13) To esure that there is a uique well-defied Nash equilibrium assume that: f ω ω < 1. (14) f ωσθ, ;. Deote the Nash equilibrium labour supply futio by ( ) Note that it follows from (8) that the reservatio wage is ow give by: ω ( σ, = u ( σ ) u l 4 ( σ,1) ( σ,1) θ u,1 +. 2(1 σ, (15) whih is a stritly ireasig futio of σ ad a stritly dereasig futio of θ with ω 0 as θ 1. The fat that the reservatio wage falls with θ is a maifestatio of the Keepig up with the Joeses Effet sie idividuals are beig idued to work who otherwise have hose ot to. From (13) it follows that, whe Nash labour supply is positive: f. f f + e f + ω = ; = σ, (16) ω f 1 ω. σ 1 ω. so Nash labour supply resposes to ireases i the wage rate ad ueared iome differ from the idividual labour supply respose i two ways:

7 (i) (ii) Ireases i the wage rate ad i ueared iome raise the value of peer osumptio whih idues additioal work effort; There is a multiplier effet at work whereby hages i labour supply idue hages i peer osumptio whih geerates further hages i labour supply. The sig of both of these terms is idetermiate. However, from (16) it follows that = e. = ω > 0 (17) ω ω σ 1 ω so the Slutsky-Hiks deompositio still applies to the Nash labour supply futio, ad the ompesated Nash labour supply respose is positive ad is just the idividual ompesated respose saled up by the multiplier effet. Now, from (8), the Nash labour supply a be haraterised through the oditio: u θ u ω 1. l +, e 0. (18) 2(1 u u So, whe labour supply is positive, the, i the traditioal ase where happiess does ot θ = 0 the margial rate of substitutio betwee leisure ad affet well-beig ( ) osumptio equals the (et) wage. However whe happiess does affet well-beig, θ > 0, the margial rate of substitutio is greater tha the wage rate multiplied by a ( ) fator that (a) depeds o the ratio of average to margial utility of osumptio, ad (b) is ireasig i the weight idividuals plae o happiess. This additioal term aptures the distortio i Nash equilibrium labour supply idued by the Keepig up with the Joeses Effet. It is this distortio that leads idividuals to supply too muh labour sie it ireases the attrativeess of work Well-beig By substitutig the Nash equilibrium level of effort bak ito the well-beig futio give i (4) we obtai the Nash idiret well-beig futio: where v θ 1 θ = 2 % (19) ( ωσθ,, ) v ( ωσθ,, ) 1 7 Ideed it follows from (15) ad (18) that i the extreme ase where 1 θ = the ω ( σ,1) = 0 ad f ω, σ,1 1, so everybody speds their etire time i work. ( ) 5

8 ( ωσθ,, ) σ+ ω ( ωσθ,, ),1 ( ωσθ,, ) % (20) v u f f is the Nash idiret well-offess futio. To uderstad what happes to well-beig all we eed to uderstad is what happes to well-offess. If ω ω ( σ, labour supply is zero ad v% v% v% v u u ( ωσθ,, ) = ( σ,1) = = 0; = ( σ,1) % ω θ σ (21) so Roy s idetity holds: v v = e. ω σ (22) If ω ω ( σ, we get: > labour supply is positive, the, by differetiatig (20) ad usig (18) u v% θ = u e ω. ; ω ω 2(1 u (23) u v% θ = u 1 ω. ; σ σ 2(1 θ ) u (24) I the traditioal ase where idividuals plae o weight o happiess ( 0) θ = the (24) ad (23) just redue to their ovetioal forms. I partiular Roy s idetity (22) holds. However if θ > 0 a margial hage i the wage or beefit idues a additioal effet o well-beig that is positive (resp. egative) if the hage auses labour supply to fall (resp. rise) ad so redue (resp. irease) the distortio o labour supply. I ertai irumstaes a irease i the wage rate ould atually make people worse off as the distortio-itesifyig effet domiates the diret beefit from a higher et wage. Propositio 1 If θ > 0 well-beig is a stritly dereasig futio of the wage rate for ω ω σ, θ e 0i.e. for some of those who are beig those idividuals for whom ( ) idued to work oly beause of their desire to Keep up with the Joeses. Proof: If e 0 the first term o the RHS of (23) is approximately zero. Moreover from the Slutsky-Hiks equatio, (17), > 0so the oly effet of the higher wage is ω ω to itesify the distortio ad so make people worse off. Corollary 1.1 The idividuals with the lowest level of well-offess ad hee wellbeig are o loger those with the lowest level of ability. 6

9 4. Example If the well-offess futio is Cobb-Douglas, straightforward to hek that 1 u (,) α l = l α, 0< α < 1. It is 0, ω ω ( σ, θ (1 ασ ) α + ω (1 α) σ ω ( σ, = ; f ( ω, σ, = 2(1 θ ) ad θ, ω ω ( σ, α + θ 2(1 θ ) 1+ ω 2(1 θ ) α σ, ω ω ( σ, α θ 1 α v% ( ωσθ,, ) = α + (1 α) 2(1 θ ) (25) (1 α ).( ω+ σ). ω, ω ω ( σ, θ 1+ 2(1 θ ) v% From (29) it follows that > 0 σ ad that, for ω > ω ( σ, v% ω (1 ασ ) v% = α < 0 ω, ω σ, θ < ω < ω σ,0 v% ω ω σ ( ) ( ) (26) Thus well-offess ad hee well-beig are stritly dereasig i the wage rate for preisely the group of idividuals that are beig idued to work purely beause of the keepig up with the Joeses effet. This is illustrated i Figure 1 i the Appedix. 5. Colusio Whe we situate osumers i a soial otext ad their osumptio may deped o that of others, the may of the stadard preditios of the ovetioal theory of osumer behaviour may be overtured. Most strikigly those who are worst off i soiety are o loger those o the lowest wage. The worst off will be people with a suffiietly high wage that they are idued ito work beause of the Keepig up with the Joeses Effet. This has impliatios for the uderstadig of poverty ad iequality ad the desig of tax/beefit systems that warrat further ivestigatio. 7

10 Appedix Figure 1 v% % v ( ωσ,,0) α σ % v ( ωσθ,, ) 0 ωσθ (, ) ωσ,0 ( ) ω 8

11 Referees Boski, M., Sheshiski, E., Optimal Redistributive Taxatio whe Idividual Welfare Depeds upo Relative Iome. Quarterly Joural of Eoomis. 92, Easterli, R., Iome ad Happiess: Toward a Uified Theory, The Eoomi Joural, 111, Frak, R., Choosig the Right Pod: Huma Behavior ad the Quest for Status. Oxford Uiversity Press, New York. Shor, J., The Overspet Ameria. Basi Books, New York. Veble, T., The Theory of the Leisure Class: A Eoomi Study of Istitutios. George, Alle ad Uwi, Lodo 9