row_42 559..571 Revew of Inome and Wealh Seres 56, umber 3, Sepember 21 WELFARE EASUREET, IVOLUTARY UEPLOYET, AD HETEROGEEITY by Thomas Aronsson* Umeå Unversy Ths paper onerns welfare measuremen n an eonomy wh unon wage seng, where he equlbrum s haraerzed by unemploymen. Conrary o resuls derved n he frs bes, he urren value Hamlonan s no an exa welfare measure n an eonomy wh unemploymen. Insead, he welfare measure also depends on employmen effe, whh are aused by he dsrepany beween supply and demand n he labor marke. In addon, sne unemploymen gves rse o heerogeney, dsrbuonal effes wll also haraerze he welfare measure. 1. Inroduon The welfare eonom foundaon for soal aounng orgnaes, o a large exen, from Wezman (1976), who was able o show ha he omprehensve ne naonal produ (P) onsues an exa welfare measure n a dynam represenave-agen eonomy. The onep of omprehensve P s mean o mply an exenson of he onvenonal P, where he exenson s desgned o refle all relevan aspes of onsumpon and apal formaon for soey. To be more spef, Wezman s resul means ha, f he resoure alloaon s frs bes and he ehnology saonary, hen he urren value Hamlonan underlyng he opmal resoure alloaon wll be proporonal o he presen value of fuure uly fang he represenave onsumer. 1 The urren value Hamlonan s, n urn, nerpreable as he omprehensve P n uly erms n he sense of represenng he nsananeous uly of all urren onsumpon plus he uly value of all urren ne nvesmens. Furhermore, Wezman s resul has lear praal relevane, as seems o mply ha we an measure welfare n a dynam eonomy solely by usng nformaon ha refers o he me when he measuremen s ondued (and n some speal ases solely by observables). The purpose of hs paper s o analyze he onsequenes of nvolunary unemploymen n he onex of soal aounng. Suh a sudy s learly mo- oe: I would lke o hank Karl-Gusaf Löfgren, Tomas Sjögren, agnus Wksrön, and wo anonymous referees for helpful ommens and suggesons. A researh gran from FORAS s also graefully aknowledged. *Correspondene o: Thomas Aronsson, Deparmen of Eonoms, Umeå Unversy, SE-91 87, Umeå, Sweden (homas.aronsson@eon.umu.se). 1 In he onex of envronmenal eonoms, he omprehensve P s somemes referred o as he green P. See Harwk (199) for an early applaon of he heory of soal aounng o an eonomy, where envronmenal and/or naural resoures are mporan pars of he eonom sysem. There s also an empral leraure ha res o quanfy he green P or pars hereof; see, e.g., he leraure revew by Vnen and Harwk (1997) and he more reen applaon o Sweden by Ahlroh (23). See also Oulon (24) for a omparson beween oupu measures and a (poenal) welfare measure of P-ype. Revew of Inome and Wealh 21 Inernaonal Assoaon for Researh n Inome and Wealh Publshed by Blakwell Publshng, 96 Garsngon Road, Oxford OX4 2DQ, UK and 35 an S, alden, A, 2148, USA. 559
vaed beause of he hgh unemploymen raes n many ounres, whle he onsequenes of unemploymen for soal aounng are no ye fully undersood. I s also movaed beause earler omparable leraure on welfare measuremen n mperfe marke eonomes 2 ofen fouses on ssues oher han he labor marke, whh makes neresng o exend he heory n order o address he welfare measuremen problem under unemploymen. To my knowledge, only wo prevous heoreal sudes on soal aounng have onsdered he possbly of mperfe ompeon n he labor marke. Aronsson (1998) adds he assumpon ha he wage rae exeeds he marke learng wage rae a eah nsan o an oherwse sandard Ramsey growh model. The resuls show ha he presen value of fuure uly fang he represenave onsumer beomes proporonal o he sum of he urren value Hamlonan and he presen value of fuure hanges n employmen. The nuon s ha, f he equlbrum s haraerzed by exess supply of labor a eah nsan, fuure nreases n he employmen wll onrbue o hgher welfare and ve versa. Aronsson and Löfgren (1998) ake he analyss a sep furher by brefly addressng welfare measuremen under monopoly-unon wage formaon. The assumpon ha rade-unon wage formaon auses unemploymen makes possble o relae he presen value of fuure hanges n employmen o he wage elasy of he labor demand. However, boh hese earler sudes are based on represenaveagen models, where he unemploymen s measured n erms of work hours. In hs paper, I wll exend he sudy of soal aounng under mperfe ompeon n he labor marke o a mul-onsumer eonomy, n whh agens may be eher employed or unemployed. Suh a framework also apures ha unemploymen gves rse o heerogeney among agens; an aspe ha s no addressed by a represenave-agen model. The analyss o be arred ou n he paper s based on a dynam general equlbrum model, whh s a varan of he Ramsey growh model. Followng earler researh on welfare and fsal poly under mperfe ompeon n he labor marke (e.g. Fues and Huber, 1997; Koskela and Shöb, 22; Aronsson and Sjögren, 24), I assume ha rade-unon wage seng s he mehansm behnd he exsene of nvolunary unemploymen. However, noe ha hs assumpon s no neessarly of grea mporane per se; he man resuls wll also apply f he unemploymen s aused by mehansms oher han rade-unon wage formaon. Arguably, hs srenghens he analyss o be arred ou below. The paper onrbues o he leraure n prmarly wo ways. The frs s by showng wha he exa welfare measure looks lke n an eonomy, where some ndvduals are unemployed. In addon o he urren value Hamlonan (or a Hamlonan-lke omponen), suh a measure s lkely o onan one or several aspes of fuure hanges n he number of employed persons along he general equlbrum pah as well as dsrbuonal effes followng beause unemploymen gves rse o heerogeney; enes ha would vansh from he welfare measure n a frs bes seng. The seond s by more horoughly dsussng he dsrbuonal aspes of unemploymen n he onex of soal aounng. As I wll show below, 2 For an overvew of he leraure, see Aronsson e al. (24). See also Usher (1994) for a dsusson abou he nerpreaon of he Hamlonan. 56
he presene of unemploymen does no only mply addonal erms n he welfare measure (erms n addon o he urren value Hamlonan); also affes he way n whh we should alulae an analogue o real omprehensve P. The laer sands n sharp onras o he earler leraure on soal aounng and unemploymen referred o above, whh fouses on represenave-agen models. The oulne of he paper s as follows. In Seon 2, I presen he model and also brefly dsuss welfare measuremen n a frs bes equlbrum. Seon 3 deals wh welfare measuremen under equlbrum unemploymen and presens he man resuls along wh her empral mplaons. Fnally, Seon 4 provdes a summary and dsusson of he resuls. 2. The odel The produon sde of he eonomy s ompeve wh labor and apal beng he varable produon faors. To smplfy he noaons as muh as possble, I desrbe he produon sde as f a sngle ompeve frm produes a homogenous good. The objeve funon s wren (1) Π()= F ( (), K ()) w () () rk () () where F( ) s he produon funon, employmen (measured as he number of employed persons, see below), K he aggregae apal sok, w he wage rae (or labor nome) per employee, and r he neres rae. I assume ha he produon funon s nreasng n eah argumen and srly onave, and ha labor and apal are ehnal omplemens n produon n he sense ha he margnal produ of labor (apal) nreases wh he sze of he apal sok (employmen). The frm reas w and r as exogenous, and obeys he frs order ondons F (, K) - w = and F K(, K) - r = a eah nsan, where he subndes denoe paral dervaves. Solvng hs equaon sysem for and K gves he demands for labor and apal, respevely. For furher use, noe ha he labor demand an be wren as (2) = ( w, r) n whh w, r <. I assume ha w and r (and, onsequenly, ) are onnuously dfferenable wh respe o me. Turnng o he onsumpon sde, here are onsumers n he eonomy. Le me follow he onvenon n he leraure on soal aounng by negleng populaon growh, as well as by assumng ha he onsumers are endowed wh perfe foresgh. The onsumers share a ommon nsananeous uly funon, and eah onsumer supples one un of labor nelasally 3 a eah nsan. The nsananeous uly fang onsumer a me s gven by 3 Ths assumpon smplfes he analyss. Alhough he negle of lesure n he uly funon wll affe he exa form of he welfare measure o be derved below, s no of major mporane for how hanges n he number of employed persons, or dsrbuonal effes of hanges n he faor pres, affe he welfare measure, whh are he ssues of man onern here. 561
(3) u ()= u( () ) where s onsumpon. The funon u( ) s nreasng n s argumen and srly onave. In hs paper, I am no prmarly onerned wh a dealed desrpon of he labor marke opporunes a an ndvdual level; only ha par of he workfore may be unemployed a eah nsan. Wh hs dea n mnd, and o make he opmal onrol problem manageable, s onvenen o fnd a smple mehansm ha assgns ndvduals eher o employmen or unemploymen n fuure perods. To aheve hs bas objeve, I assume ha: () here s a unque senory rankng among ndvduals, whh s known and gven exogenously a me zero; () avalable jobs are dsrbued on he bass of senory; 4 and () senory nreases wh he me spen n employmen. Gven he sze of oal employmen, whh s deermned n he general equlbrum (see below), hese addonal assumpons make possble o assgn ndvduals o employmen or unemploymen n fuure perods. They also mply ha he senory rankng among ndvduals does no hange over me. 5 Fnally, sne ndvduals endowed wh perfe foresgh know he employmen level a eah nsan n he fuure as well as her own senory rankng, follows ha eah ndvdual, a me zero, knows hs/her employmen saus along he whole general equlbrum pah. The assumpons above enable me o formalze he opmzaon problem fang eah onsumer as f he/she solves a sandard opmal onrol problem wh he addonal resron ha he wage rae s zero durng mes of unemploymen. 6 The opmzaon problem fang ndvdual s o hoose () a eah nsan suh as o maxmze (4) U ( )= u( () ) e θ d subje o he asse aumulaon equaon and an nal apal sok (5) (6) k ()= π ()+ r() k ()+ w () () k ( )= k, where w () = w() f ndvdual s employed a me and zero oherwse. For laer use, noe ha K = kd. 4 The dea ha employmen opporunes are alloaed aordng o senory rules, and no by random draw, has been formalzed n he onex of rade-unon wage formaon by Oswald (1993). In prae, rules governng lay-offs and/or reemploymen dffer onsderably aross ounres. For nsane, Sweden apples senory rules (based on years of employmen) n ase of lay-offs, and workers who were reenly lad off wll beome reemployed f he employer needs o nrease employmen agan. 5 Ths aspe smplfes he analyss onsderably by omparson wh a more general framework, where he senory rankng may hange over me. Reall ha he dea of he paper s o apure fundamenal aspes of unemploymen for welfare measuremen a he aggreae level: no o provde onsderable deal as o who s employed and who s no (whh are of no onern here). 6 Ths means ha I dsregard unemploymen benefs. As long as unemploymen mples a loss of nome relave o beng employed, hs s no mporan for he qualave resuls derved below. 562
The aggregae profs are assumed o be shared equally among he onsumers,.e. p =P/. In addon o equaons (5) and (6), I mpose a o Ponz Game (PG) ondon. oe also ha I am dsregardng oher possble asses han physal apal. Eah ndvdual reas he wage rae, he neres rae and he prof nome as exogenous. The neessary ondons nvolve (he me ndaor has been suppressed for noaonal onvenene) (7) (8) (9) θ u ( ) e λ = λ = λ r lm λ = where l represens he osae varable assoaed wh equaon (5) and s nerpreable as he margnal uly value of apal (or he margnal uly of onsumpon) n presen value erms. Equaon (7) s he sandard frs order ondon for onsumpon, whereas equaons (8) and (9) are, respevely, he equaon of moon for he margnal uly value of apal and he ransversaly ondon. 7 Le me urn o he labor marke par of he model. As menoned n he nroduon, rade-unons are assumed o have an nfluene on wage formaon. In addon, I add hree haraerss, whh are ommon n he leraure on rade-unonzed labor markes: () All ndvduals are rade-unon members. () Trade-unons only dede (or bargan over) he wage rae. () The frm unlaerally hooses he number of persons o employ. These haraerss ypally mply ha rade-unons have some degree of marke power, and ha he equlbrum wage rae s lkely o exeed he marke learng wage rae. The employmen s assumed o be deermned by he labor demand. One spef wage formaon proess dsussed n earler leraure ha obeys hese haraerss s a monopoly rade-unon maxmzng he wage sum of s members subje o he only onsran ha he labor demand dereases wh he wage rae. However, for he analyss o be arred ou below, suffes o assume ha he wage rae s suh ha (1) = ( w, r)< a he equlbrum, mplyng ha he labor demand, measured n erms of he number of employed persons, falls shor of he labor fore. A Bref Dgresson o he Frs Bes To smplfy he analyss n laer pars of he paper, s onvenen o begn by brefly onsderng welfare measuremn n he frs bes. Ths provdes a referene 7 The ransversaly ondon s neessary, provded ha eran growh ondons are fulflled. These growh ondons serve as upper bounds for he nfluene of he sae varable on he funons nvolved. For furher deals, he reader s referred o Seersad and Sydsaeer (1987, Theorem 16 of Chaper 3). 563
ase by whh o ompare he welfare measure o be derved under mperfe ompeon n he labor marke. In he speal ase where he resoure alloaon s frs bes, we have =, and he frs order ondons look as f hey are derved from he followng presen value Hamlonan for he eonomy as a whole; (11) H* ()= [ u ( θ * () ) e + λ* () k * () ] d θ = u( * () ) e + λ* () K* () where he superndex * s used o he denoe he frs bes equlbrum. The seond lne of equaon (11) s explaned by he fa ha ndvduals have denal preferenes and nal apal soks; hey are, herefore, denal n he frs bes equlbrum assoaed wh he model se ou above. The equaon of moon for he aggregae apal sok (whh s derved by usng he ndvdual asse aumulaon equaons and he objeve funon of he frm) s gven by (12) K* ()= F( * (), K* () ) * () where * =. Dfferenang equaon (11) oally wh respe o me, and usng he frs order ondons fang he frm ogeher wh equaons (7) and (8), gves (13) dh* () = u ( * () ) θ θ e d whh means ha he only non-auonomous me dependene of he eonom sysem orgnaes from he uly dsoun faor. The ndre effes of me va onrol, sae and osae varables vansh as a onsequene of opmzaon. Sne he presen value Hamlonan approahes zero when me goes o nfny, 8 one an solve equaon (13) and ransform he soluon o urren value o oban (14) θv* ()= H *() where s V* ()= u( * () s ) e θ ( ) ds θ s he opmal value funon and H *( )= H* () e he urren value Hamlonan. Defnng he onsumer surplus, 9 s* = u(*) - u (*)* and S* = s*, he urren value Hamlonan an be rewren as (15) H *()= u( * () )+ λ *() K * ()= λ *() [ C* ()+ K * () ]+ S* () 8 Ths propery of well behaved opmal onrol problems s due o hel (1982). 9 For a more dealed analyss of he role of he onsumer surplus n soal aounng, he reader s referred o L and Löfgren (22). 564
θ n whh C* = *, whle λ *( )= λ* () e s he margnal uly value of apal measured n urren value erms a me. Equaon (14) s Wezman s welfare measure appled o he eonomy desrbed above. I means ha he urren value Hamlonan onsues an exa welfare measure n he frs bes equlbrum. oe ha he urren value Hamlonan s nerpreable as a measure of omprehensve P n uly erms; measures he uly of he urren onsumpon plus he uly value of he urren ne nvesmens. For he eonomy dsussed here, whh has a relavely smple sruure, he onsumpon onep only onans prvae goods and serves, and he ne nvesmen onep only onans physal apal; hs explans why he omprehensve P ondes wh he onvenonal P. The expresson afer he seond equaly n equaon (15) shows ha he welfare measure an also be wren as he sum of he lnearzed urren value Hamlonan and he onsumer surplus. The lnearzed urren value Hamlonan s, n urn, equal o he margnal uly of wealh mes an expresson ha wll be nerpreed as he real omprehensve P. 1 3. The Deenralzed Equlbrum wh Unemploymen Le me now reurn o he deenralzed equlbrum under mperfe ompeon n he labor marke. Wha does he relaonshp beween he opmal value funon and he sum of ndvdual Hamlonans look lke n hs ase? The sum of ndvdual presen value Hamlonans an be wren as (16) [ ( ) + () ()] H ()= u () e θ λ k d 1 ae ha he welfare measure s expressed n a uly-mer,.e. boh he lnearzed urren value Hamlonan, λ [ C + K ], and he onsumer surplus are measured n uns of uly. Alhough he hoe of mer s no mporan for he haraerzaon arred ou here, praal applaons neessae ha he uly-based welfare measure s ransformed no a money-mers welfare measure. The praal problem s ha, even f one were o negle he onsumer surplus, one anno n a meanngful way ompare C + K (referred o as real omprehensve P above) a wo dfferen pons n me, sne he margnal uly of wealh hanges over me. The key o a money-mers ransformaon of uly-based welfare measures was provded by Wezman (21) and furher elaboraed on by L and Löfgren (22). Appled o he presen model, her approah s o nrodue a pre ndex, p() suh ha ( ) p ( ) = λ ( ) λ ( ) = exp θ( )+ rsds ( ). Ths makes possble o re-sale he welfare measure suh ha C K H ( ) = λ ( ) ( ) S p ( ) + ( ) p ( ) + ( ) where l ( ) s a onsan, and he expresson whn he square brake s nerpreable as a money-merzed verson of real omprehensve P. 565
where he superndex s used o denoe he equlbrum n he mperfe marke eonomy. By analogy o equaon (11), equaon (16) an also be nerpreed as measurng he omprehensve P n uly erms for he eonomy se ou above dsouned o presen value, whh s why s neresng o onsder as a welfare measure. Le λ()= ( 1 ) λ () d denoe he average margnal uly value of apal (.e. he average margnal uly of onsumpon) a me measured n presen value erms. For furher use, noe ha (where he me ndaor has been suppressed) λ kd = λπ + r λ kd+ w λ d λd sne hose - ndvduals who are unemployed reeve no labor nome. 11 We have now ome o he poson of beng able o sudy soal aounng. By analogy o he welfare onep used n he frs bes, defne he opmal value funon, 12 and onsder Proposon 1 below: θ s V ()= u () s e ( ) d ds, ( ) Proposon 1. Whn he gven framework, and f he equlbrum s haraerzed by unemploymen, he welfare measure an be wren as (17) s s θ θv ()= H ()+ λ () s () () Π λ sd e ws () + () ẇw () s ds s + () s Π () θ λ + () sk () sder rs () λ () sds + λ sw() s θ () se ds where H ()= H () e θ s he sum of he urren value Hamlonans. Proof: See he Appendx. 11 Alhough, and l, as well as he aggregae varables w and r, ar onnuously dfferenable wh respe o me by he assumpons made earler, k s no smooh for ndvduals swhng beween employmen and unemploymen (as he wage fang he ndvdual wll jump from w o or ve versa a a swh pon). I wll assume ha hs problem s of mnor praal mporane (e.g., ha he number of swhers a eah nsan s small relave o he number of nonswhers), and proeed as f he sum of Hamlonans s onnuously dfferenable wh respe o me. An alernave would have been o approxmae he apal sok for he swhers by a smooh funon. However, as hs s of no onern for he bas mehansms va whh unemploymen affes he welfare measure, I absan from suh exensons here. 12 I s sraghforward o exend he model by assgnng dfferen dsrbuonal weghs o he employed and he unemployed. 566
Equaon (17) means ha welfare, measured by he presen value of fuure uly, s proporonal o he sum of four erms. The frs s he sum of ndvdual urren value Hamlonans, whh s nerpreable as he omprehensve P n uly erms (hs wll be furher dsussed below). One mporan aspe of Proposon 1 s o nvaldae he welfare nerpreaon of he sum of urren value Hamlonans n ase of unemploymen; nformaon referrng o me s no longer suffen for measurng welfare a ha me. Ths s undersood from he appearane of he remanng erms on he rgh hand sde of equaon (17), whh are forward lookng and, herefore, no measurable by usng enes referrng o me. These addonal erms were absen n he frs bes welfare measure gven by equaon (14). The seond and hrd erms on he rgh hand sde of he welfare measure (whh are proporonal o he hanges n w and r) represen dsrbuonal effes of he wage rae and he neres rae, respevely. These dsrbuonal effes arse beause unemploymen gves rse o heerogeney among he onsumers. Consder frs he dsrbuonal effe of a hange n he wage rae. I onsss of wo pars; he frs s he derease n prof nome from an nrease n he wage rae whh nfluenes all onsumers, and he seond s he nrease n labor nome among he employed f he wage rae nreases. Sne P/ w =-, he dsrbuonal effe a a gven pon n me (he erms whn he square brake n he frs row) an be wren as (18) Π λ + λ = λ λ w d n ( ) where λn = ( 1 ) λd s he average margnal uly value of apal n presen value erms among he employed. On average, he employed have more lfe-me resoures han he unemployed and wll, herefore, be able o onsume more a eah nsan. Ths suggess ha λn < λ, n whh ase he rgh hand sde of equaon (18) s negave. The nuon n erms of equaon (17) s ha fuure nreases (dereases) n he wage rae lead o lower (hgher) welfare. The nsananeous dsrbuonal effe of a hange n he neres rae (he erms whn he square brake n he seond row) s equal o he number of ndvduals n he eonomy mes he ovarane beween l and k; (19) Π λ + λ = ( λ ) r kd ov, k. As he margnal uly value of apal s lkely o delne wh he sze of he apal sok, one would learly expe ov(l, k) <. In hs ase, herefore, fuure nreases (dereases) n he neres rae onrbue negavely (posvely) o he welfare level. The nuon behnd equaons (18) and (19) s ha a hgher wage rae or neres rae ends o nrease he nome (earnngs and apal nome, respevely) more among ndvduals wh a low margnal uly value of apal han among ndvduals wh a hgh margnal uly value of apal. In oher 567
words, he onsumpon possbles are redsrbued oward hose wh a low margnal uly of onsumpon. The fourh erm on he rgh hand sde of equaon (17), whh s proporonal o Ṅ, measures he presen value of fuure hanges n he number of employed person. 13 The nuon s ha a margnal nrease n he number of employed persons leads o a welfare gan for he ndvdual who beomes employed a he rulng wage rae. 14 By usng he labor demand, he me dervave of an be deomposed no wo effes: (2) = w + r. w r Therefore, sne w < and r <, he employmen effe depends on how he faor pres adjus along he general equlbrum pah. If ẇ and ṙ are posve (negave) along he general equlbrum pah, hen he employmen effe onrbues o lower (hgher) welfare a me. Empral Implaons Le us now urn o he empral mplaons of equaon (17) and, n parular, how he addonal omponens assoaed wh equlbrum unemploymen affe he welfare measure. To make he nerpreaons eonomally meanngful, I wll assume hroughou hs seon ha λ < n λ and ov(l, k) < for all. The followng resul s an mmedae onsequene of Proposon 1: Corollary 1. Rsng (fallng) faor pres along he general equlbrum pah mply ha he sum of urren value Hamlonans a me overesmaes (underesmaes) welfare a me. The nuon behnd Corollary 1 s ha hgher faor pres lead o a redsrbuon from he unemployed o he employed whh redues he welfare gven he soal welfare funon assumed above as well as lead o lower employmen. In oher words, fuure nreases (dereases) n he faor pres mply redsrbuon away from (oward) hose wh a hgh margnal uly of onsumpon. oe also ha dsounng means ha he near fuure s gven a hgher wegh n he welfare measure han he dsan fuure. The laer s praally relevan for he nerpreaon of he orollary; f ẇ and ṙ are boh predomnanly posve (negave) n he near fuure, he urren value Hamlonan s lkely o overesmae (underesmae) he welfare a me. Dsrbuonal effes do no only arse beause fuure hanges n he faor pres affe he employed and he unemployed n dfferen ways (he effes 13 A smlar resul, ye wh he nerpreaon n erms of fuure hanges n he hours of work, s derved n he represenave-agen models referred o n he nroduon. 14 A smlar qualave resul apples f he unonzed labor marke s replaed by dual labor marke, where hose who do no beome employed n he unonzed seor wll fnd work n anoher (ompeve) seor, provded ha he wage rae s hgher n he unonzed seor han n he oher seor. If he wage rae were hgher n he oher seor, no ndvdual would work n he unonzed seor, n whh ase he welfare measure akes he same general form as equaon (14). 568
dsussed above). Suh effes also appear beause he dsrbuon s subopmal a presen; an aspe whh s apured by he analogue o omprehensve P. In he frs bes resoure alloaon, brefly addressed above, we saw ha he sum of lnearzed urren value Hamlonans s nerpreable as he margnal uly of onsumpon mes he real omprehensve P. We shall here derve he orrespondng expresson for he eonomy wh equlbrum unemploymen. Le ()= ( 1 ) () d and κ ()= ( 1 ) k () d denoe he average prvae onsumpon and average apal aumulaon, respevely, n he eonomy as a whole a me. The followng resul s a dre onsequene of Proposon 1: 15 Corollary 2. The sum of urren value Hamlonans an be wren as (negleng he me ndaor) (21) H = λ [ C ( 1+ ρ )+ K ( 1+ ρ )]+ S k where ρ ov λ, λ and ρk ov λ, k λκ. = ( ) ( ) = ( ) ( ) Equaon (21) s an analogue o equaon (15). The frs par s nerpreable as he lnearzed urren value Hamlonan for he eonomy as a whole (.e. sum of lnearzed urren value Hamlonans over all ndvduals) and s measured by he average margnal uly of onsumpon n urren value erms, λ, mes an analogue o real omprehensve P, whereas he seond par, S, s he onsumer surplus (measured n he same way as above). By omparson wh he frs bes, where only he sum of aggregae onsumpon and aggregae ne nvesmens onsues he omprehensve P, one an see ha he appearane of equlbrum unemploymen mples ha he dsrbuon of onsumpon and savngs (.e. he nome dsrbuon) among ndvduals a presen s also relevan, whh s he reason why he dsrbuonal haraerss, r, and r k, are par of he lnearzed urren value Hamlonan. Ths s so beause he margnal uly of onsumpon dffers aross agens, whh dd no n he frs bes referene ase dsussed above. oe ha r < aordng o equaon (7). Therefore, he wegh aahed o he aggregae onsumpon n he onex of real omprehensve P s lower here han n he frs bes. The nuon s ha he margnal welfare onrbuon of he aggregae prvae onsumpon s smaller han would have been, had he margnal uly of onsumpon been equalzed aross agens. I s no (as far as I an see) possble o sgn he dsrbuonal haraers for he ne nvesmens, r k. However, by usng λ < n λ and ov(l, k) <, one may aually show ha λ [ Cρ + K ρk]< (f evaluaed along he general equlbrum pah), whh s he expresson by whh he lnearzed urren value Hamlonan n equaon (21) dffers from s ounerpar n equaon (15). To see hs, noe ha 15 oe ha he prnples behnd he money-mers ransformaon dsussed n foonoe 1 apply here as well. See Löfgren (n press) for a reamen of exernales n he onex of money-mers welfare measures. 569
( ) = [ + ] λ + k d λ C( 1 ρ)+ K ( 1+ ρk) = λ [ Π + rk + w ]+ rov( λ, k)+ w λ n λ and K + C = Π + rk + w = F(, K). Therefore, λ [ Cρ + K ρk]= rov( λ, k)+ w[ λn λ ]< [ ] n he general equlbrum. The followng resul s also a onsequene of Proposon 1: [ ] Corollary 3. If he equlbrum s haraerzed by unemploymen, hen λ C + K overesmaes he lnearzed urren value Hamlonan. Therefore, sne he lnearzed urren value Hamlonan s nerpreable as he average margnal uly of onsumpon mes an analogue o real omprehensve P, Corollary 3 s nerpreable o mean ha C + K overesmaes he real omprehensve P. 4. Summary and Dsusson The general onluson from hs paper s ha unemploymen s mporan o onsder explly n soal aounng. Ths s so for several reasons. Frs, f he equlbrum s haraerzed by unemploymen, he urren value Hamlonan has no obvous welfare nerpreaon. Alhough hs onluson may seem obvous a a general level, he resuls show why hs dsrepany ours as well as wha he exa welfare measure looks lke. Seond, unemploymen gves rse o heerogeney whh, n urn, means ha dsrbuonal effes of hanges n he faor pres are presen n he welfare measure, and ha dsrbuonal haraerss are par of he lnearzed urren value Hamlonan. Thrd, he exa welfare measure beomes dependen on fuure hanges n he number of employed persons along he general equlbrum pah. oe ha he bas dea behnd he paper s o haraerze and nerpre he welfare measure; no o deermne he sgns or magnudes of he exra erms. One dreon of fuure researh s o apply he welfare analyss n prae n order o assess he relave welfare onrbuon ha s due o he appearane of mperfe ompeon n he labor marke. A possble approah here would be o use a numeral general equlbrum model albraed agans real world daa. Ths op s boh neresng and omplex enough o movae a sudy of s own. I leave hese and oher exensons for fuure researh. 5. Appendx Dfferenang equaon (16) oally wh respe o me and usng equaon (1), (7), (8), and (9) ogeher wh he frs order ondons of he frm gves 57
(A1) dh () = u ( () ) θ θ e d d + () () () Π λ d w () + λ () w () + () () Π λ + k d r () λ () () r () + w() λ (). By assumng ha he presen value Hamlonan approahes zero when me goes o nfny, one an derve equaon (17) by solvng equaon (A1). Referenes Ahlroh, S., Correng DP for SO 2 and O x Emssons: Implemenaon of a Theoreal odel n Prae, Revew of Inome and Wealh, 49, 426 4, 23. Aronsson, T., A oe on Soal Aounng and Unemploymen, Eonoms Leers, 59, 381 4, 1998. Aronsson, T. and K-G. Löfgren, Green Aounng n Imperfe arke Eonomes A Summary of Reen Researh, Envronmenal and Resoure Eonoms, 11, 273 87, 1998. Aronsson, T. and T. Sjögren, Effen Taxaon, Wage Barganng and Poly Coordnaon, Journal of Publ Eonoms, 88, 2711 25, 24. Aronsson, T., K-G. Löfgren, and K. Baklund, Welfare easuremen n Imperfe arke A Growh Theoreal Approah, Edward Elgar, Chelenham, 24. Fues, C. and B. Huber, Wage Barganng, Labor-Tax Progresson, and Welfare, Journal of Eonoms, 66, 127 5, 1997. Harwk, J., aural Resoures, aonal Aounng and Eonom Depreaon, Journal of Publ Eonoms, 43, 291 34, 199. Koskela, E. and R. Shöb, Opmal Faor Inome Taxaon n he Presene of Unemploymen, Journal of Publ Eonom Theory, 4, 387 44, 22. L, C-Z. and K-G. Löfgren, On he Choe of ers n Dynam Welfare Analyss: Uly versus oney easures, Umeå Eonom Sudes o. 59, 22. Löfgren, K-G., The oney ers Problem n Dynam Welfare Analyss, n T. Aronsson and K-G. Löfgren (eds), Handbook of Envronmenal Aounng, Edward Elgar, Chelenham, n press. hel, P., On he Transversaly Condons n Infne Horzon Opmal Conrol Problems, Eonomera, 5, 975 85, 1982. Oswald, A. J., Effen Conras are on he Labor Demand Curve: Theory and Fas, Labour Eonoms, 1, 85 113, 1993. Oulon,., Produvy versus Welfare; or GDP versus Wezman s DP, Revew of Inome and Wealh, 5, 329 55, 24. Seersad, A. and K. Sydsaeer, Opmal Conrol Theory wh Eonom Applaons, orh-holland, Amserdam, 1987. Usher, D., Inome and he Hamlonan, Revew of Inome and Wealh, 4, 123 41, 1994. Vnen, J. and J. Harwk, Fores Resoures and he aonal Inome Aouns: Coneps and Experene, meo, Harvard Unversy, 1997. Wezman,. L., On he Welfare Sgnfane of aonal Produ n a Dynam Eonomy, Quarerly Journal of Eonom, 9, 156 62, 1976., A Conrbuon o he Theory of Welfare Aounng, Sandnavan Journal of Eonoms, 13, 1 24, 21. 571