1 Walrasian Equilibria and Market E ciency

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1 1 Walrasian Equilibria and Markt E cincy Rading in Txtbook: Chaptr 3 in Stiglitz. 1.1 Motivation Whn thinking about th rol of govrnmnt w hav to considr a numbr of rathr fundamntal qustions, such as: What should a govrnmnt do? Why ar som activitis undrtakn in th privat sctor and othrs in th public sctor, What ar th pros and cons of having th govrnmnt intrvn in a particular markt? Assuming that th govrnmnt should intrvn in a particular markt, how can w dtrmin what a good intrvntion would b? As is pointd out in th vry bginning of th txtbook, on way to start thinking about ths vry basic qustions is to look at som facts from th ral world. Th rst two chaptrs in th book ar intndd to giv you a bit of background, and I think you should rad ths chaptrs. Howvr, for now I think that it is su cint that w agr on th following: 1. Govrnmnt spnding accounts for a big chunk of total conomic activity. In sum, fdral and local govrnmnt spnding about 30% of GDP in th US, and mor than this in almost all othr dvlopd countris. 2. Crtain goods (military dfns, watr and swag, national parks, highway construction tc.) ar almost xclusivly providd by govrnmnts. 3. Othr goods (ducation, mail dlivry, policing) ar providd by both th public and th privat sctor. 4. Th govrnmnt also provids a lgal systm. 1

2 5. Evn for sctors lik stl, autos, tomato growing, sugar tc. th govrnmnt intrvns by providing xplicit or implicit subsidis, tari protction, and othr rgulatory masurs that in unc th markt outcoms.. Th govrnmnt also rdistributs incom. Howvr, whil th public sctor is important, w rly mainly on th privat sctor for th production and distribution of most goods and srvics. Indd, most conomists would argu that a dcntralizd (=privat) systm has many virtus, and that on should b carful with govrnmnt intrvntion in markts that work. This viw is of cours partially groundd in mpirical obsrvations (that is, comparing th Sovit Union and its satllits with th US and its Europan allis). Howvr, anothr rason for th blif that on should not mss with markt is that thr ar appaling thortical argumnts for this viw: th logic of th invisibl hand pionrd by Adam Smith is th cornrston of th most important modl in conomics th comptitiv (or Walrasian or noclassicall) quilibrium modl. My viw is that, bfor vn thinking about why govrnmnt intrvntion can in som cass b justi d, it is crucial to undrstand this modl, which tlls us that pric taking bhavior has crtain advantags that sm hard for th public sctor to rplicat. Th comptitiv/walrasian/noclassical modl is what for a quit long tim has bn th bnchmark modl in conomics. Th cntral assumptions ar: 1. Pric taking bhavior; individual agnts (consumr and rms) bliv that thir own actions hav no in unc on prics. Hnc, a consumr taks th quilibrium prics as givn and picks th bst consumption plan givn this quilibrium pric. Similarly, rms tak input and output prics as givn and choos th production plan that maximizs pro ts. 2. Markt Claring; in quilibrium, prics ar st so that supply=dmand on all markts. Notic that thr is a bit of magic involvd. Thr is no xplicit mchanism in th modl for how prics ar formd. Intuitivly, on would think that if th dmand for a good should 2

3 xcd th supply, thn th pric ought to b adjustd upwards, whras if thr is xcss supply, th dmand should fall. Hnc, it appars that only prics whr th supply quals dmand ar stabl, which is th loos justi cation of th markt claring assumption. Our intuition says that this is n for markts whr participants ar small, but it is still an opn qustion how accurat this intuition is. A nal rmark bfor looking at th formal analysis is that thr is a distinction btwn partial quilibrium and gnral quilibrium analysis. Th supplydmand graphs from Econ 101 ar prim xampls of partial quilibrium analysis. Ths graphs can b instructiv and usful, but hr w sk to illustrat why comptitiv quilibria ar conomically cint, and for this purpos it is mor instructiv to considr a gnral quilibrium stup, which mans that all prics and quantitis ar dtrmind simultanously as a closd systm. 1.2 Th 22 Modl of Pur Exchang To bgin with, w ignor production and considr th simplst nontrivial gnral quilibrium modl possibl. Suppos thr ar; 2 goods labld 1 and 2:Quantitis dnotd by x 1 ; x 2 2 agnts, A; with prfrncs givn by utility functions u A (x 1 ; x 2 ) and u (x 1 ; x 2 ) Agnts liv a onpriod lif. Thy wak up in th morning with ndowmnts A = A 1 ; A 2 and = 1 ; 2, which ar quantitis th two goods that th agnts hav bfor any trad I will dnot consumption bundls x A = x A 1 ; x A 2 and x = x 1 ; x 2 for Mr. A and Mrs. : Th Consumr Choic Problm Th rst thing to not hr is that w hav not spci d any particular dollar incoms m A ; m : Instad, w will lt trads b bartr trads whr on agnt givs th othr goods 3

4 in rturn for othr goods. Indd, thr is no room for intrinsically uslss pics of papr in this or any othr noclassical quilibrium modl. That is, as long as agnts don t driv any plasur from mony (say as wallpapr) nobody would accpt mony unlss mony was xplicitly backd by th right to purchas goods with it. Hnc, th incom of th consumr will b takn as th valu of th ndowmnt. Th rlvant maximization problm for consumr A is thus s.t p 1 x 1 + p 2 x 2 p 1 A 1 + p 2 A 2 max u A (x 1 ; x 2 ) (1) x 1 ;x 2 and similarly for (Just rplac A with ). For comparison, th problm (1) is rally just th gnric utility maximization problm ovr appls and bananas from Econ 101, xcpt that th incom is ndognously dtrmins as th valu of th ndowmnt (p 1 A 1 + p 2 A 2 ) instad of bing an xognous paramtr Comptitiv Equilibria Th concpt of a comptitiv quilibrium is on of th most important in conomics: D nition 1 A comptitiv (Walrasian) quilibrium in th 2 2 pur xchang modl is a pric vctor p = (p 1; p 2) and consumption bundls x A = x A 1 ; x A 2 ; x = x 1 ; x 2 satisfying: 1. Th bundl consumd by ach agnt is th bst a ordabl bundl givn pric vctor p : That is x A solvs th consumr choic problm (1) and x solvs th symmtric consumr choic problm for agnt givn prics (p 1 ; p 2 ) = (p 1; p 2) : 2. Markts clar (fasibility). x A 1 + x 1 = A x A 2 + x 2 = A

5 1.3 Graphical Tratmnt In latr discussions it will b usful to distinguish btwn th parts in th d nition of quilibrium that has to do with fasibility from th part that has to do with optimizing bhavior. D nition 2 An allocation (a list of consumption bundls for ach agnt) is fasibl if x A 1 + x 1 A x A 2 + x 2 A It is rathr clar that in quilibrium ( that is if w add optimal bhavior as wll) all rsourcs must b usd maning that th mor intrsting fasibl allocations ar thos whr th rsourc constraints hold with quality. Graphically any fasibl allocation that uss all rsourcs (x A 1 + x 1 = A and x A 2 + x 2 = A ) can b convnintly dscribd as a point in a box as in gur 1. In th gur, th lngth of ach sid is th total rsourcs of ach good which immdiatly mans that if w pick any point di rnt from in th box total consumption of ach good will b qual to th total rsourcs. Now, optimal bhavior is dtrmind xactly as bfor. Givn a pric vctor (p 1 ; p 2 ) w hav that: Th budgt st for A consists of all (x 1 ; x 2 ) such that p 1 x 1 + p 2 x 2 p 1 A 1 + p 2 A 2 ; which ar just all points blow a lin with slop p 1 p 2 that gos through th ndowmnt point (not that whn w look at it from th point of viw of A th ndowmnt is locatd at ( A 1 ; A 2 ) from th rlvant origin in th southwst cornr. Th budgt st for consists of all (x 1 ; x 2 ) such that p 1 x 1 + p 2 x 2 p p 2 2 ; 5

6 A x 1 1? x A 2 x u x 2 A 2 u 2 A x A 1 A 1 A >? Figur 1: A Fasibl Allocation in th Edgworth box) which ar just all points abov a lin with slop p 1 p 2 that gos through th ndowmnt point. That is, from th point of viw of th origin is in th northast cornr. This is illustratd in gur 2. Obsrv that thr is absolutly no rason that th budgt st must b in th st of fasibl allocation. In th pictur this is indicatd by th budgt lins continuing across th dgs in th box (but only for positiv consumptions). Th optimality rquirmnt is thn as usual graphically dpictd as a tangncy btwn th highst achivabl indi rnc curv and th budgt lin. Now, w can simply put th two picturs togthr in th box for som arbitrary prics (p 1 ; p 2 ) as in Figur 3. Th way th pictur is drawn w hav that th nt dmand for good on of Mrs. (i.., what wants to buy in addition to hr ndowmnt) xcds th nt supply of Mr. A for good 1. That is: wants to buy mor than A has to sll. Hnc thr is xcss dmand for good 1 : at th givn prics th consumrs want to consum mor than is availabl in th markt of good 1, so th markt is not in quilibrium in Figur 3. Th mirror imag of this xcss dmand for good 1 is xcss supply for good 2, but this is

7 x 1 + t udgt St For t * t udgt St For A t x 2 x 2?? x 1 Figur 2: Th Utility Maximization Conditions) automatic givn that w hav xcss dmand for good 1 as will b discussd latr. So, how will an quilibrium look lik in th box? 1. Allocation must b fasibl) graphically this mans that both agnts choos sam point in th Edgworth box. 2. oth agnts must choos th bst bundl givn th prics) th quilibrium must b such that both agnts hav a tangncy btwn pric lin and indi rnc curv at quilibrium allocation. An quilibrium can thus b dpictd as in Figur 4 as a budgt lin that gos through th ndowmnt which is such that both agnts hav a tangncy with th pric lin at th sam point. 7

8 b bbb Nt Dmand Good 1, Mrs b bbb Nt Dmand Good 2, Mr A? b bbb u b bbb u b bbb u Nt Supply Good 2, Mrs? slop b bbb p 1 p 2 A Nt Supply Good 1, Mr A b bb b? Figur 3: Exampl of Prics NOT Consistnt with Equilibrium) 1.4 Grd is Good: Slf Intrst Lads to E cint Allocations Som xamination of this pictur rvals a rathr rmarkabl proprty of comptitiv (Walrasian) quilibria. Givn th quilibrium allocation x all bundls that ar bttr for A ar thos to th northast of th indi rnc curv intrscting x : Similarly, th bundls that ar bttr for ar thos to th southwst of th indi rnc curv intrscting x : This mans THAT IT IS IMPOSSILE TO MAKE ONE PERSON ETTER OFF WITHOUT MAKING THE OTHER AGENT WORSE OFF Tru undr much mor gnral circumstancs (mor consumrs, rms, goods, a tim dimnsion, uncrtainty...) This important fatur is mphasizd in Figur 5 whr th only di rnc from Figur 4 is that I v takn away all indi rnc curvs not going through x : An conomist would say that th quilibrium outcom is Parto cint: 8

9 Q slop QQQ p 1 p 2 u x u A Q? Figur 4: An Equilibrium in th Edgworth ox) D nition 3 An allocation is Parto cint if it is fasibl an if thr is no othr fasibl allocation that maks both agnts bttr o. Parto cincy is th concpt of cincy in conomics. Indd, conomists usually just rfr to it as cincy and it is thn commonly undrstood that Clarly, allocations that ar not Parto cint ar undsirabl. Thn, thr is a way to mak all agnts in th conomy bttr o and if vryon is happir thn that is clarly a bttr us of th rsourcs. Not that thr is an in nit numbr of Parto optimal allocations vn in th simply 2 2 pur xchang modl. To s this not that for any point such that thr is a tangncy btwn th indi rnc curvs of th agnts it is impossibl to incras th happinss of on agnt without making th othr lss happy. On can thus trac out th st of Parto optimal allocations in th Edgworth box as th st of tangncis as in Figur. Th curv that conncts all th Parto optima is somtims calld th contract curv. Important to not is: 1. E cincy has nothing to do with distribution of rsourcs. 9

10 Q slop QQQ p 1 p 2 ttr undls For ttr undls For A ux u Q QQQ A Q? Figur 5: An Equilibrium is Parto E cint) 2. Equilibria dpnd on th initial distribution of rsourcs, th notion of cincy dos not. 3. Dspit potntial issus about fairnss th rsult that comptitiv quilibria ar cint may b thought of as a grd is good typ of rsult. Indd it is th basic rason for why conomists ar oftn vry scptical towards markt intrvntions. Laving th markt alon (undr th comptitiv assumptions which ar loosly basd A? Figur : Th contract CurvAll E cint Allocations) 10

11 on idas of many rms and many consumrs) w hav rasons to bliv that th markt outcom is at last approximatly cint. Mssing with th markt w may hlp som individuals or groups, but, as w ll s with mor concrt xampls of intrvntionist policis, cincy is typically lost. 4. Latr in th cours w will analyz and discuss rasons for why th markt may not produc Parto cint outcoms. In spit of th sming gnrality of th rsult that quilibria ar cint (w hav only considrd th simplst xchang modl, but it holds also whn w hav arbitrary numbrs of goods and/or agnts and production by rms...) thr ar lots of rasons why th markt could produc in cint quilibrium outcoms (public goods, xtrnalitis, informational issus, monopoly powr...). 1.5 Walras Law Th graphs ar instructiv, but somtims it is hlpful to b abl to actually comput an quilibrium. W not that givn prics (p 1 ; p 2 ) ; th aggrgat dmand is x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 2 for good 1 x A 2 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 2 p 1 ; p 2 ; p p 2 2 for good 2 Whr x A 1 () ; x A 2 () ; x 1 () and x 2 () ar th rgular dmand functions you considrd in th rst half of th smstr. Hnc, w can solv for an quilibrium by solving x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 A 2 = {z } {z } aggrgat dmand for good 1 givn prics p 1 ;p 2 x A 2 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 2 p 1 ; p 2 ; p p 2 A 2 = {z } {z } aggrgat dmand for good 2 givn prics p 1 ;p 2 rsourcs of x 1 rsourcs of x 1 ; for (p 1 ; p 2 ). At a rst glanc, this looks promising. Two quations in two unknowns. ut s.t p 1 x 1 + p 2 x 2 p 1 J 1 + p 2 J 2 max x 1 ;x 2 u J (x 1 ; x 2 ) (2) 11

12 and s.t p 1 p 2 x 1 + x 2 p 1 p 2 J 1 + p 2 J 2 max u J (x 1 ; x 2 ) (3) x 1 ;x 2 ar quivalnt problms. Hnc, w may normaliz, for xampl by stting p 2 = 1 which givs th systm x A 1 p 1 ; 1; p 1 A 1 + A 2 + x 1 p 1 ; 1; p = A x A 2 p 1 ; 1; p 1 A 1 + A 2 + x 2 p 1 ; 1; p = A That is, w gt two quilibrium conditions and a singl unknown. Luckily, it turns out that th two quilibrium conditions ar quivalnt. This is oftn rfrrd to as Walras law (although somtims th trm Walras law is usd for th fact that th valu of xcss dmand is zro, which is th proprty that is usd to prov th claim; Proposition 1 Suppos that (p 1 ; p 2 ) clars th markt for good 1, that is x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x 1 p 1 ; p 2 ; p p 2 2 = A : Thn, th markt for good 2 clars as wll. This coms dirctly from th fact that th budgt constraint holds with quality for vry agnt for any prics. For simplicity of notation, lt m A (p) = p 1 A 1 + p 2 A 2 W know (bcaus of optimization) that m (p) = p p 2 2 p 1 x A 1 p 1 ; p 2 ; m A (p) + p 2 x A 2 p 1 ; p 2 ; m A (p) = m A (p) = p 1 A 1 + p 2 A 2 p 1 x 1 p 1 ; p 2 ; m (p) + p 2 x 2 p 1 ; p 2 ; m (p) = m (p) = p p 2 2 Summing w gt (writ out sums if you don t lik P signs) p 1 X x J 1 p 1 ; p 2 ; m J (p)! X J 1 + p 2 x J 2 p 1 ; p 2 ; m J (p)! J 2 = 0 J=A; J=A; 12

13 Sinc p 1 > 0 and p 2 > 0 it follows that if X x J 1 p 1 ; p 2 ; m J (p) J 1 = 0 (markt for good 1 clars) J=A; thn th quality abov guarants that X x J 2 p 1 ; p 2 ; m J (p) J 2 = 0 (markt for good 2 clars) J=A; Th conomics bhind ths summations ar actually straightforward. W bgin by obsrving that agnts will us thir full budgts, which mans that th valu of th optimal dmand givn any pric quals th valu of th ndowmnt for both agnts. Summing ovr th agnts, th valu of th optimal dmand for A+th valu for th optimal dmand for must qual th val of th sum of th ndowmnts. This mans, rgardlss of whthr th pric is an quilibrium pric or not, that th valu of th xcss dmand/supply for good 1+th valu of th xcss dmand/supply for good 2 must b idntical to zro, rgardlss of whthr th prics clar th markt or not. 1. Exampl 1: Calculating a Comptitiv Equilibrium Explicitly in th 22 Modl Assum that th agnts hav CobbDouglas prfrncs, U A (x 1 ; x 2 ) = a ln x 1 + (1 ) ln x 2 U (x 1 ; x 2 ) = b ln x 1 + (1 b) ln x 2 ; and that th ndowmnts ar A = (1; 0) and = (0; 1) : In words, agnt A is a sllr of good 1 and a buyr of good 2 and agnt is th othr way around. Th rlvant dmands can thrfor b calculatd to b, x A 1 p; m A (p) = ama (p) = a (p p 2 0) = a p 1 p 1 x 1 p; m (p) = bm (p) = b (p p 2 1) = b p 2 : p 1 p 1 p 1 13

14 So quilibrium rquirs that x A 1 p; m A (p) + x 1 p; m (p) = a + b p 2 p 1 = 1 = A ) p 1 p 2 = b 1 a Plugging th rlativ pric back into th dmand xprssions abov w thn hav that th quilibrium allocation is x A 1 p ; m A (p ) ; x A 2 p ; m A (p ) ; x 1 p; m (p ) ; x 2 p ; m (p ) ha; b; 1 a; 1 bi Notic that th mor th othr agnt liks th good that th agnt has in hr ndowmnt, th bttr o th agnt is, simply r cting that incrasd dmand drivs up th pric, which is good for th sllr of th good. 1.7 Exampl 2: A Rprsntativ Agnt Exampl A common way to writ down gnral quilibrium modl that ar simpl is to assum that all agnts in th conomy ar idntical clons of ach othr. This typ of modls ar usd a lot in macroconomics (bcaus it allows th choic problm for th individual to b quit rich) and ar calld rprsntativ agnt modls. Considr a world populatd with lots of agnts consuming only appls. Each agnt livs for two priods and has an appl tr that producs units of appls in vry priod. Thr is a comptitiv markt for borrowing and saving and r dnots th intrst rat. All agnts hav idntical prfrncs givn by u (c 1 ) + u (c 2 ) : Th choic problm for an individual is thus to dcid how much to borrow or sav. As w hav sn bfor, w may writ this ithr as max u ( s) + u ( + s (1 + r)) s 1+r 14

15 or as max u (c 1 ) + u (c 2 ) c 1 ;c 2 s.t. c r c r : For our purposs, th rst xprssion is simplr. Th rst ordr condition is simply u 0 ( s) + u 0 ( + s (1 + r)) (1 + r) = 0 Now, assuming that th appls ar nonstorabl w not that: 1. In quilibrium, r must b such that s = 0: If not, rsourcs will not balanc. That is, if s < 0 thn all agnts borrow, so total appl consumption in th rst priod xcds total appl production. Symmtrically, if s > 0 all agnts sav, so total appl consumption in th scond priod xcds what is availabl. 2. Hnc, s = 0 must solv th optimization problm, and thrfor satisfy th rst ordr condition. W conclud that u 0 () + u 0 () (1 + r ) = 0 or r = 1 : This is vry vry simpl, but it is a usful thory of how th quilibrium intrst rat is dtrmind. It simply says that th quilibrium intrst rat must b dtrmind so that popl ar happy to consum what is availabl in vry priod, which boild down to rlation btwn th intrst rat and th discount factor which masurs how patint or impatint popl ar. For futur rfrnc, w obsrv that this worldviw is not so asy to rconcil with on whr popl sav to littl for thir rtirmnt (which is a common claim by thos in favor of compulsiv savings programs such as social scurity). 15

16 1.8 Exchang E cincy: Summary So far w hav dwlt mainly on what is Stiglitz labls xchang cincy. Th basic point is that, if th agnts act as pric takrs and th pric somhow is st to clar th markt, thn th outcom will b Parto cint. In trms of th Edgworth box, it is mor or lss obvious that both agnts must hav indi rnc curvs that ar tangnt to th (common) budgt lin. In th conomics lingo this is usually xprssd by saying that both agnts must hav marginal rats of substitutions that ar qual to th rlativ pric ratio. This is n, but it is important to rcall that th marginal rat of substitution (or th slop of an indi rnc curv) is just a fancy nam for th rat that an agnt is willing to trad on (small unit of a good) to anothr. Th logic is thrfor simply that th intrnal trms of trad must b qualizd across agnts. To undrstand this intuitivly, it is bst to think about th consquncs if th slops/mrs/intrnal rats of trads would di r. Thn, on agnt would b willing to trad, say, on banana for an appl. Th othr agnt howvr would (for th slops to di r) ithr b willing to giv up mor than a banana for an appl or mor than an appl for a banana. In ithr cas w hav an in cincy, as th two agnts could nd a trad that lads to a Parto improvmnt. 1.9 Production E cincy An advantag with bing carful about xchang cincy is that th principls from a pur xchang conomy carry ovr to a modl with production without too much work. Suppos that good 1 and good 2 ar both producd from two (for asy graphical rprsntation) inputs, and call ths inputs land (L) and labor (N). A pric taking pro t maximizing rm in sctor 1 will thn sk to maximiz its pro t by solving subj to y 1 f 1 (L; N) max p 1y 1 w L L w N N y 1 ;L;N whr f 1 () is th production function for good 1. Th ky obsrvation for production cincy is that rgardlss of which quantity y 1 is part of th solution to this pro t maximiza 1

17 tion problm, th pro t maximizing quantity must b producd in th chapst way possibl. Hnc, if (y 1; L ; N ) is a solution to th pro t maximization problm, thn (L ; N ) is a solution to subj. to y 1 f 1 (L; N) max L;N p 1y 1 w L L w N N or, sinc max x h (x) = min x h (x) and p 1 y 1 is a constant to th problm min w LL + w N N L;N subj. to f 1 (L; N) L C w L b bbb 3 b b bbb bbb Highr costs b b b bbb bbb bbb b b b bbb slop bbb bbb w N wl b b b C w N b bb N Figur 7: Constant Cost Curvs (aka Isocosts) W can draw this prtty much lik w illustrat a utility maximization problm in an indi rnc curv graph. That is, rst considr combinations of inputs (L and N) such that costs ar th sam. That is C = w L L + w N N, L = C w L w N w L N This is prtty intuitiv. It just says that costs ar kpt constant along a straight lin with slop givn by th rlativ factor pric. For xampl, say that w L = 2 and w N = 4; so that a 17

18 unit of labor is twic as xpnsiv as a unit of land. Thn, what th linar rlationship says is simply that to kp costs constant w nd to rduc th input of land with 2 units if w hir an additional unit of labor. Nxt, considr combinations L; N such that output is constant. That is, (L; N) such that y 1 = f 1 (L; N) : If th production function is linar, this will again just rsults in straight lins. Howvr, this would b rathr nasty (almost always this would lad to cornr solutions ) and txtbooks hav a tndncy to shy away from anything inconvnint. Instad, it is usually assumd that th curvs associatd to th quation y 1 = f 1 (L; N) hav th sam nic convx shap as th gnric txtbook indi rnc curvs. W thn obsrv that th lowst costs to produc any particular output y1 must occur at th lin that touchs th givn isoquant (=curv of L and N that produc xactly y1 units of good 1) which is closst to th origin of th graph. It is thn graphically apparnt that th solution must occur whr thr is a tangncy btwn th isoquant and th isocost sinc othrwis it is possibl to mov towards lowr cost lvls and still produc th sam output. L C w L L 3 b b bbb bbb Highr costs b b bbb bbb sb b bbb bbb y = f(n; L) b b bbb b N N C w N Figur 8: Th Cost Minimization Problm This maks prfct sns: Th slop of th curv d nd by th condition y 1 = f 1 (L; N) (aka th marginal rat 18

19 of tchnical substitution) tlls us how w can rduc th us of land if w incras th labor input by on small unit and kp output constant That is, th slop tlls us at which rat factors can b substitutd. Th rlativ factor pric w N wl markt. is th rat at which factors can b xchangd in th For cincy (and pro t maximization) in production th rat at which factors can b substitutd must qual th rlativ factor pric (othrwis w can produc mor at a givn cost). At this point w can combin cost minimization in th two industris to gt a graph which is virtually idntical to th on illustrating th quilibrium in an xchang conomy. Q slop QQQ w N wl Q QQ Industry2 u L,N Industry1 Q Q? Figur 9: Comptitiv Factor Markts and E cincy in Edgworth ox) 1.10 Product Mix E cincy Exchang cincy is purly about prfrncs (and ndowmnts) and production cincy is purly about tchnology (production or cost functions). Th nal cincy critrion 19

20 combins consumr prfrncs and tchnological constraints. Tracing th Parto frontir for productiv cincy w can gnrat a production possibilitis frontir lik th on illustratd in Fig 3. in book. Product mix cincy thn rquirs that goods ar producd in such a way that th slop of all indi rnc curvs coincid with th slop of th production possibilitis frontir. 20

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