8 Equilibrium Analysis in a Pure Exchange Model

Size: px

Start display at page:

Download "8 Equilibrium Analysis in a Pure Exchange Model"

Shanon Harvey
5 years ago
Views:

1 8 Equilibrium Analysis in a Pur Exchang Modl So far w hav only discussd dcision thory. That is, w hav lookd at consumr choic problms on th form s.t. p 1 x 1 + p 2 x 2 m: max u (x 1 ; x 2 ) x 1 ;x 2 Th consumr choic problm gnrats dmand functions x 1 (p 1 ; p 2 ; m) and x 2 (p 1 ; p 2 ; m), which ar to b intrprtd as giving th optimal consumption bundl for any con guration of prics and incom. Howvr, th dtrmination of prics (and arguably incom as wll) is cntral in conomics and should b tratd as ndognous. Hr, th basic ida in noclassical conomics is that prics ought to b st so as to clar th markt. This ida, which gos back to Adam Smith, is intuitiv, and also surprisingly powrful. Roughly spaking, if givn a pric th dmand for a good xcds supply, thn th pric is too low and must b adjustd upwards. If instad supply xcds dmand th pric is too high and ought to b adjustd downwards to rstor quilibrium. Hnc, only prics such that supply=dmand ar stabl and w will simply postulat that quilibrium prics is st this way. Thus, w will not xplicitly modl how prics ar st or any dynamic adjustmnts (th invisibl hand or auctionr ). In conomics w mak a distinction btwn partial quilibrium and gnral quilibrium analysis. Partial quilibrium analysis is roughly about studying a markt in isolation and whil bing vry usful it has som drawbacks, in particular whn studying markts that constituts a larg shar of th conomy. Th supply/dmand graphs from 101 ar xampls of partial quilibrium and I may do som partial quilibrium xrciss latr (nd rms rst). Gnral quilibrium di rs from partial quilibrium analysis in that th conomy is studid as a closd systm, with all prics bing dtrmind simultanously. If w would insist on ralism th gnral quilibrium approach would lad to vry complx modls. Howvr, 98

2 important insights about comptitiv markts can b gaind in th simplst possibl modl with two goods and two agnts acting comptitivly in an conomy with no production. 8.1 Th 22 Modl of Pur Exchang W will study a modl whr, 2 agnts, A; B 2 goods x 1 ; x 2 ndowmnts A = A 1 ; A 2 and B = B 1 ; B 2 ar th quantitis of good on and two th agnts hav bfor any trad I will dnot consumption bundls x A = x A 1 ; x A 2 and x B = x B 1 ; x B 2 for Mr. A and Mrs. B: Th rst thing to not hr is that I hav not spci d any particular dollar incoms m A ; m B : Th rason is that what w will do is to lt agnts trad (bartr) with ach othr and if on of th agnts would hav som grn pics of papr with prsidnts on thm, thn th othr agnt wouldn t want to giv up anything h/sh cars about for thos grn pics of papr unlss thr was a consumption valu of thos grn pics of papr. It is hard to mak th cas that bills hav a consumption valu and what that mans is that this simpl (static) nvironmnt dosn t allow mony to b valud (indd, conomic thory that xplains why popl us and valu mony is a rlativly rcnt phnomnon and th thory is still vry crud and vry far from bing of much valu for th Fd). Th bottom lin of th discussion abov is that th incom of th consumr will b takn as th valu of th ndowmnt. Hnc, th rlvant maximization problm givn prics (p 1 ; p 2 ) for consumr A is max u A (x 1 ; x 2 ) x 1 ;x 2 s.t p 1 x 1 + p 2 x 2 s p 1 A 1 + p 2 A 2 99

3 and similarly for B: Now, w v spnt wks on this and th only thing to watch out for is that th incom dpnds on prics. This was actually tru for th thr applications w lookd at as wll (intrtmporal choic, uncrtainty and labor supply) but I did not strss it so much. Howvr, th rlvant dmand givn any pric p 1 ; p 2 can still b drivd from th dmand functions w drivd whn w viwd m as indpndnt from p 1 and p 2 : That is lt x A 1 (p 1 ; p 2 ; m) and x A 1 (p 1 ; p 2 ; m)b th dmand functions drivd from th standard problm with utility function u A (x 1 ; x 2 ) ; thn th dmand for good 1 and 2 from th problm abov will b givn by x A 1 (p 1 ; p 2 ; p 1 A 1 + p 2 A 2 ) and {z } valu of givn p 1 & p 2 x A 2 (p 1 ; p 2 ; p 1 A 1 + p 2 A 2 ) {z } valu of givn p 1 & p 2 D nition 1 A comptitiv (Walrasian) quilibrium in th pur xchang modl is a pric vctor (p 1; p 2) and consumption bundls x A = x A 1 ; x A 2 ; x B = x B 1 ; x B 2 satisfying: 1. [optimality] x A solvs th consumr choic problm for A givn prics p 1; p 2 x B solvs th consumr choic problm for B givn prics p 1; p 2 2. [Markt claring]. x A 1 + x B 1 = A 1 + B 1 x A 2 + x B 2 = A 2 + B 2 I will start with a graphical tratmnt, but w may not that onc w hav dmand functions x A 1 () ; x A 2 () ; x B 1 () and x B 2 () w can solv for an quilibrium by solving x A 1 p 1 ; p 2 ; p 1 A 1 + p 2 A 2 + x B 1 p 1 ; p 2 ; p 1 B 1 + p 2 B A 2 = 1 + B 1 {z } {z } aggrgat dmand for x 1 givn prics p 1 ;p 2 rsourcs of x 1 Thr is a similar condition for good 2 (which is rdundant du to Walras Law which I will discuss latr). At this point, just rcall all th graphs w did in standard dmand thory. 100

4 In almost all cass th consumption changd whn th pric changd, so if you pick a pric at random, thr is no particular rason to bliv that th quilibrium condition abov will hold. That is, only vry particular prics will b consistnt with quilibrium. 8.2 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 B 1 A 1 + B 1 x A 2 + x B 2 A 2 + B 2 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 B 1 = A 1 + B 1 and x A 2 + x B 2 = A 2 + B 2 ) 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. 101

5 A 2 + B 2 + x B 1 B 1 B? x A 2 x u x B 2 A 2 u B 2 A x A 1 A 1 A 1 + B 1 >? Figur 1: A Fasibl Allocation in th Edgworth box) Th budgt st for B consists of all (x 1 ; x 2 ) such that p 1 x 1 + p 2 x 2 p 1 B 1 + p 2 B 2 ; 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 B 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. B (i.., what B wants to buy in addition to hr ndowmnt) xcds th nt supply of Mr. A for good 1. That is: B wants to buy mor than A has to sll. Hnc 102

6 x 1 + t Budgt St For B t * t Budgt St For A t x 2 x 2?? x 1 Figur 2: Th Utility Maximization Conditions) 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 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. Both 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. 103

7 b bbb Nt Dmand Good 1, Mrs B B b bbb Nt Dmand Good 2, Mr A? b bbb u b bbb u b bbb u Nt Supply Good 2, Mrs B? 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) 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. 8.3 Slf Intrst Lads to Good Allocations in th Comptitiv Modl 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 B ar thos to th southwst of th indi rnc curv intrscting x : This mans that it is impossibl to mak on prson bttr o without making th othr prson wors o. This important fatur (which is tru undr much mor gnral circumstancs) 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 : 104

8 Q QQQ Q slop QQQ p 1 p 2 Q QQQ B Q QQQ u x Q QQQ u Q QQQ A Q? Figur 4: An Equilibrium in th Edgworth Box) In th languag of an conomist, w just argud that th quilibrium outcom is Parto cint or simply cint: 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. 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: 105

9 Q QQQ Q slop QQQ p 1 p 2 Q QQQ Bttr Bundls For B Bttr Bundls For A Q QQQ ux Q QQQ u Q QQQ B A Q? Figur 5: An Equilibrium is Parto E cint) 1. E cincy has nothing to do with distribution of rsourcs. Thr ar cint allocations whr on agnt consums vrything or almost vrything and on may disagr that such an allocation is unfair. Howvr, as long as th consumrs only car about thir own consumptions this isn t a Parto in cincy. 2. Equilibria dpnd on th initial distribution of rsourcs, th notion of cincy dos not. B A? Figur : Th contract CurvAll E cint Allocations) 10

10 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 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...). 8.4 Walras Law Svral ways to stat it, but for our purposs th only thing w ar intrstd in is th following: Claim If (p 1 ; p 2 ) is such that supply=dmand in th markt for good 1, thn supply=dmand also in th markt for good 2. 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 m B (p) = p 1 B 1 + p 2 B 2 107

11 W know (bcaus of optimization) that 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 B 1 p 1 ; p 2 ; m B (p) + p 2 x B 2 p 1 ; p 2 ; m B (p) = m B (p) = p 1 B 1 + p 2 B 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;B J=A;B 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;B 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;B 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 B 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 Gnralizations Th ons of you who ar comfortabl with th summation notation can s that thr is nothing in this that dpnds on thr bing two agnts. With n agnts indxd by i th sam rasoning givs 0 p 1 1 nx B x i 1 p 1 ; p 2 ; m i (p) 1 i i=1 A + p 2 {z } =0 by q condition for markt 1 nx i=1 x i 2 p 1 ; p 2 ; m i (p)! i 2 = 0 108

12 Now lt thr b mor than 2 goods and lt k = 1; :::; K indx th goods. Thn again th sam rasoning mans that for any p = (p 1 ; :::; p K ) w hav p 1 Thus: nx x i 1 p; m i (p) 1! nx i +p 2 x i 2 p; m i (p) 2! nx i +:::+p K x i K p; m i (p)! i 2 = 0 i=1 i=1 i=1 With 2 goods, Walras law tlls us that if w hav found prics so that on markt is in quilibrium (with utility maximizing bhavior), thn th othr markt is also in quilibrium. With K goods, Walras law tlls us that if w hav found prics so that K 1 markts ar in quilibrium, thn th K th markt is also in quilibrium. If you ar usd to counting quations and unknowns you may b confusd by this. Onc th dmand functions ar pluggd in th quilibrium conditions (for th cas with 2 goods) ar 2 quations in what sms to b 2 unknowns, p 1 and p 2 : Howvr, sinc th budgt constraints and p 1 x 1 + p 2 x 2 p 1 J 1 + p 2 J 2 tp 1 x 1 + tp 2 x 2 tp 1 J 1 + tp 2 J 2 ar quivalnt for any t > 0 w know that w can, for xampl, st p 2 = 1 (pick t = 1 p 2 ). That is, only th rlativ pric btwn th goods ar dtrmind in quilibrium. Stting p 2 = 1 is thus just a mattr of xing th unit of account so that w xprss th pric of good on in units of good 2. Hnc, w rally hav 2 quations in on unknown ( p 1 p 2 ). In gnral, such a systm would b unsolvabl. Howvr, Walras law stats that if on of th quations is solvd, thn so is th othr. That is, th two quations must hav th sam st of solutions. Hnc, th rlativ pric that clars th markt for good 1, also clars th markt for good

13 8.5 A Closd Form Exampl Considr th cas with CobbDouglas prfrncs, and lt U A (x 1 ; x 2 ) = a ln x 1 + (1 ) ln x 2 U B (x 1 ; x 2 ) = b ln x 1 + (1 b) ln x 2 ; also suppos that A = (1; 0) and B = (0; 1) : Th rlvant dmands ar thus So quilibrium rquirs that x A 1 p; m A (p) = ama (p) = a (p p 2 0) = a p 1 p 1 x B 1 p; m B (p) = bmb (p) = b (p p 2 1) = b p 2 : p 1 p 1 p 1 x A 1 p; m A (p) + x B 1 p; m B (p) = a + b p 2 p 1 = 1 = A 1 + B 1 ) p 1 p 2 Th associatd quilibrium allocation is (chck!) = b 1 a x A 1 p ; m A (p ) ; x A 2 p ; m A (p ) ; x B 1 p; m B (p ) ; x B 2 p ; m B (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. 8. 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 110

14 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 or as max u ( s) + u ( + s (1 + r)) s 1+r 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 : 111

15 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 boils down to rlation btwn th intrst rat and th discount factor which masurs how patint or impatint popl ar. 112

16 8..1 An Arbitrary Tim Horizon You should only rad this is you hav vrything ls in th cours undr complt control. No xam qustions or homwork problms will rlat dirctly to this. Suppos that th consumr livs for T priods and has prfrncs givn by TX u (c 0 ) + u (c 1 ) + 2 u (c 2 ) + 3 u (c 3 ) :::: = t u (c t ) Also, lt r t b th intrst rat on savings at tim t and s t b th savings. Maintain th assumption that th appl tr producs units pr priods. Thn, th consumption at tim t may b xprssd as t=0 c t = + s t 1 (1 + r t 1 ) s t and th optimal choic problm is max s 0;:::;s T 1 TX t u ( + s t 1 (1 + r t 1 ) s t ) t=0 This is a problm with mor than on choic variabl, but an optimal solution s 0; s 1; :::; s T 1 must b such that s t solvs th problm ovr s t givn that all othr savings ar st to th ons in th optimal solution. Hnc, th solution must b charactrizd by a rst ordr condition for ach choic variabl, so that t u 0 + s t 1 (1 + r t 1 ) s t + t+1 u 0 + s t (1 + r t ) s t+1 (1 + rt ) = 0 for vry t = 0; ::; T 1: Multiplying by 1 t w gt u 0 + s t 1 (1 + r t 1 ) s t = u 0 + s t (1 + r t ) s t+1 (1 + rt ) : By th sam rasoning as in th xampl with two priods, s 0 = s 1 =; :::; = s T 1 = 0 for appl consumption to b qual to appl production in vry priod. Hnc, th quilibrium intrst rat at tim t must satisfy u 0 () = u 0 () (1 + r t ) or, onc again r t = 1 in vry priod. Notic, that in th ral world, r t has bn around 0:04 to 0:05 ovr th last 100 yars on a yarly basis, which would imply that a around :95 is th right numbr to plug into a modl. 113

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu

Economics 201b Spring 2010 Solutions to Problem Set 3 John Zhu Economics 20b Spring 200 Solutions to Problm St 3 John Zhu. Not in th 200 vrsion of Profssor Andrson s ctur 4 Nots, th charactrization of th firm in a Robinson Cruso conomy is that it maximizs profit ovr