General Equilibrium with Imperfect Competition

Transcription

1 General Equlbrum wth Imperfect mpettn 99 General Equlbrum wth Imperfect mpettn In rder t examne prce-settng behavr, we need a mdel n whch frms make a nntrval chce abut prces erfect cmpettn cannt supprt ths because frms all must set the marketequlbrum prce we cannt d the mcrfundatns f prce-settng and stcky prces wthut a macr general-equlbrum mdel f mperfect cmpettn Mdel s lke shrt-run f mnplstc cmpettn mdel n Ecn 20 (but n fxed cst and n entry t drve lng-run prft t zer) Many frms n ecnmy (nfntely many) Each frm prduces a varant f the prduct that s dstnct frm all ther frms an thnk f lcatn varatn: each frm s lcated alng the nterval [0, ] and they are dense n that lne Each frm s prduct s a clse substtute fr thse f ther frms, but nt a perfect substtute The elastcty f each frm s demand s < < Hgher means clser t perfect cmpettn lygply rductn functn: L N dmnshng returns (nne necessary) N captal r ther fxed nputs hse unts s that ne unt f utput = prduct f ne unt f labr nput Utlty functn: U L > t make the margnal dsutlty f wrk ncreasng n wrk effrt Nte the addtvty f the utlty functn rss partals are zer des nt affect MU L and L des nt affect MU Ths makes analyss much smpler by allwng us t analyze ptmal cnsumptn and ptmal labr effrt separately N nvestment, gvernment, r fregn sectr t all utput s cnsumed: = Behavr f sngle husehld n slatn uppse that the ecnmy cnssted f ust ne husehld cnsumng nly ts ne varety f the cnsumptn gd

2 00 General Equlbrum wth Imperfect mpettn We rule ths ut n the mdel by assumng that peple desre a varety f varants f cnsumptn gds We use t here because t s a useful benchmark t assess the effcency f the market ecnmy Husehld s nly chce s hw much t wrk L, whch determnes ts utput and cnsumptn at L as well Gven the prductn functn and the absence f nn-cnsumptn uses fr utput, = L, s U L L U L L 0 Optmal L and = = as well Behavr f husehld wth nfnte varety f gds We nw must thnk carefully abut what we mean by There are many gds and they are nt dentcal, s there s n sngle that shuld unambguusly be n the utlty functn A tractable apprach t ths prblem s the cnstruct an aggregate ndex f the husehld s cnsumptn f all f the gds (a knd f average) A smple arthmetc average r sum wuld be a lgcal chce, but desn t turn ut t have cnvenent prpertes We use the Dxt-tgltz aggregate ndex t calculate an average f cnsumptn acrss all f the gds Wth fnte number f gds N the Dxt-tgltz ndex wuld be N Nte RT prperty f ndex: dublng all dubles We want the number f gds t apprach Only as N des each frm becme a vanshngly small part f the verall market Instead f ndexng frms/gds by =, 2,, N we ndex them by [0, ] Wth a cntnuum f gds, the Dxt-tgltz ndex becmes 0 (Nte mssn f d n Rmer s equatn (6.39)) d We analyze husehld decsn n tw steps:

3 General Equlbrum wth Imperfect mpettn 0 Gven ttal nmnal expendtures, hw des the husehld decde whch f the prducts t buy? In ther wrds, what s the demand fr gven the verall level f (and ) chsen? What determnes the verall level f cnsumptn and spendng and? There s n savng here, s these wll be determned by the amunt f ncme earned, whch depends n labr suppled Bascally, = = L, s ths wll depend n labr supply Demand fr takng as gven Lagrangan: L d 0 d 0 Frst expressn s bectve functn t be maxmzed s the Lagrange multpler, a varable whse value s t be determned The secnd parenthetcal expressn s the cnstrant 0 d, that ttal nmnal spendng s the sum f spendng n all gds Ths s expressed n a dfference frm that equals zer when the cnstrant s satsfed Maxmzng L wth respect t the chce varables and s equvalent t maxmzng the utlty functn subect t the cnstrant L 0 d 0 assures that the cnstrant s satsfed L 0 d 0, [0,] 0 d A 0 d Because s nt yet knwn, we must slve the mdel further t determne A and thus the cnsumer s demand functn fr

4 02 General Equlbrum wth Imperfect mpettn L luggng A frm FO 0 nt the FO 0 A d 0 0 A d A. d L 0, r 0 d : Fllwng the analyss f Rmer s equatn (6.46), we can plug ths value f A nt the cnsumptn equatn t get aggregatn equatn fr t get 0 d d 0 d d d d d, and then substtute ths nt the d 0 d. We defne t be the prce ndex defned by the denmnatr f ths expressn, whch s natural because t means that =. Ths prce ndex s nt unlke the I n that t averages prces acrss gds, but t des s n a pwer metrc rather than lnearly. Gven ths slutn fr A, upply f labr nsumer demand fr prduct s average cnsumer demand acrss prducts tmes the relatve prce f prduct taken t the pwer s the prce elastcty f demand fr prduct Agan we take advantage f the separablty f the utlty functn t analyze the decsn f hw much t wrk n slatn frm the cnsumptn decsn

5 General Equlbrum wth Imperfect mpettn 03 Fr the husehld, cnsumptn s equal t ts ncme (there s n savng) f WL R, wth R beng ts share f frms prfts. rfts were zer n ur grwth and RB mdels because frms were perfectly cmpettve. There s n captal ncme here because there s n captal. Husehld chses labr effrt t maxmze utlty: W L 0 W L Frms behavr Nte that ths wll crrespnd t ptmal (ndvdual) labr supply f L = nly f W/ = Fllwng ur aggregate ndex fr, let utput, averaged acrss gds nce = (and thus = ), the demand fr gd s 0 be an ndex f aggregate R WL W Frm s real prft s L There are three chce varables f the frm here: /,, and L The three varables are cnstraned by The prductn functn = L and The demand equatn ubsttutng fr and L yelds nly chce varable s / R W Dfferentatng wth respect t / and settng equal t zer: R / W 0 / Dvdng by,, n whch the

6 04 General Equlbrum wth Imperfect mpettn W r W. Ths s a key equatn shwng the mnply frm s prcng plcy The relatve prce f frm s a markup n margnal cst W/ The markup rat s /( ) > It s a standard result frm mnply thery that the markup rat depends n the elastcty f demand n ths way As the ecnmy becmes mre cmpettve, and the markup rat erfectly cmpettve frms prce at margnal cst wth n markup (and earn zer prfts) As frm abve, the frm s ptmal prce Equlbrum n the aggregate ecnmy uppse that all frms and husehlds behave accrdng t the equatns derved abve and that markets clear W W L L Only ne pece s mssng t determne the equlbrum levels f all the varables f the mdel: aggregate demand What s?, but what are and? We need a thery f aggregate demand t cuple wth ur supply mdel Quantty thery: MV s smpler than I/LM framewrk, but generates a dwnward-slpng AD curve We make t even smpler by defnng the unts f the mney stck n such a way that V =, s M. We can thnk f M as the mney stck wth a velcty cnstraned t ne Or we can thnk f M as a general varable reflectng the level f aggregate demand

7 General Equlbrum wth Imperfect mpettn 05 In ether case, we take t t be exgenus and subect t shcks and t manpulatn by mnetary (and perhaps fscal) plcy M s ur thery f aggregate demand luggng ths n, W L W, r luggng ths nt the prcng equatn, M r All frms n the mdel are symmetrc (they all have the same ), s they must all chse the same prce If all are the same, then the aggregatn f the prces = and Thus, the equlbrum level f utput n the mdel s M M Equlbrum prce s rpertes f equlbrum n the mdel Equlbrum s neffcent We shwed that husehlds that nternalze all aspects f prductn and cnsumptn wuld chse L The decentralzed equlbrum has < Mnples prduce t lttle due t cntrved scarcty Ths leads t the cnclusn that bms can be gd and recessns partcularly bad If equlbrum s effcent, then devatns bth abve and belw the natural level f utput are bad Mney s neutral hange n M leads t prprtnal change n and W wth n change n In rder t ntrduce a rle fr AD-based mnetary f fscal stablzatn plcy we must add prce stckness; mperfect cmpettn alne s nt enugh There s a pstve AD externalty f each frm s prductn If ne frm were t prduce mre (sacrfcng prft), t wuld rase a lttle bt, whch wuld add t the demand fr ther frms and make everyne better ff