General equilibrium pure exchange economy

Transcription

1 Mchael Bar ECON05 General eqlbrm re echange econom Descrton of the econom No rodcton Agents Consmers: { } wth references descrbed b { } Goods: { } Intal endowment: where s the endowment of consmer and denotes the amonts of good and that consmer s endowed wth { } Notce that s the total amont of good n the econom and s the total amont of good n the econom Allocaton: a collecton of consmton bndles that secf how mch each agent consmes of each good That s s an allocaton whch secfes that consmer consmes nts of good and nts of good and smlarl for consmer Feasble allocaton: the allocaton s feasble f and That s the allocaton s feasble f the total consmton of s eqal to the total endowment of and same for Ths comletes the descrton of a re echange It s called re echange snce we gnore the rodcton at ths ont and concentrate onl on how the goods that are alread rodced can be allocated The roblems that we are gong to address wth ths model are Among al feasble allocatons n the econom whch ones are n some sense good? In other words can we fnd a crteron that everbod wll agree that a good allocaton shold ossess? One sch crteron s Pareto effcenc to be dscssed later We need to fnd a mechansm for allocatng the goods n the econom One sch mechansm s the comettve market We wold lke to know weather comettve eqlbrm allocatons are effcent The answer to ths qeston s es nder some weak assmtons

2 Edgeworth bo A sefl grahcal tool for descrbng feasble allocatons for econom wth agents and goods s the Edgeworth bo Eamle: Sose that the ntal endowment s ; Ths means that the total amont of good avalable n the econom s and the total amont of avalable s The Edgeworth bo combnes the as of the two consmers sch that when we allocate more to consmer there s less avalable for consmer Fgre shows the Edgeworth bo for ths econom All the allocatons nsde the bo are feasble However the references do not deend on the set of feasble allocatons I mght refer Mercedes to BMW bt none of them s feasble for me Fgre The ntal endowment The net fgre shows the ndfference crves of both consmers In ths eamle the references are conve and ncreasng n both goods To dstngsh between the two agents agent s red and agent s ble Fgre The ntal endowment

3 Notce that t s ossble to make both consmers better of b movng them to allocaton nsde the lens area formed b the ndfference crves Ths sggests that f agents consme ther ntal endowment than ths allocaton s not effcent there s a wast of resorces Common sense sas that f there s a ossblt to make both consmers better off than wh not do t? Now we want to be more recse abot wastefl or neffcent allocaton Pareto Otmal allocatons Defnton: Pareto Otmal or Effcent allocaton A feasble allocaton s PO f there s no other feasble allocaton ~ ~ ~ ~ ~ sch that ~ ~ for all { } and ~ ~ for some { } In words an allocaton s PO f t s feasble and there s no other feasble allocaton that all consmers weakl refer and at least one strctl refers to In other words t s mossble to make one agent better off wthot makng the other worse off Fgre shows one sch allocaton Fgre Pareto Otmal allocaton We wold lke to fnd the set of all Pareto Otmal allocatons snce now we have a crteron that dstngshes between wastefl allocatons and effcent allocatons An allocaton that we consder we mght want to check f t s Pareto Otmal or not The above fgre sggests that the onts of tangenc between the ndfference crves of both agents are PO Ths s not alwas tre however If references are not conve then t s not tre Moreover even n the case of strctl conve references the set of Pareto Otmal allocatons ma contan other onts besdes the onts of tangences of ndfference crves Pareto Vlfredo 75-90

4 Fndng the set of PO allocatons The set of PO allocatons s the solton of the followng roblem ma s t The roblem sas that we want to mamze the tlt of agent nder the gven tlt of agent Ths s eqvalent to sang that t s mossble to make agent better off wthot makng agent worse off the defnton of PO The last two constrants are the feasblt constrants After all the defnton of Pareto Otmalt corresonds onl to feasble allocatons To solve ths roblem we sbsttte the feasblt constrants nto the frst constrant The roblem s s t ma Denote the total amonts of and n the econom b The roblem now s s t ma Lagrangan ] [ L λ FONC 0 L λ the mns sgn follows from the chan rle 0 L λ denote the artal dervatves of wth resect to and : The left hand sde s the sloe of agent ndfference crves MRS and the rght hand sde s the sloe of agent ndfference crves MRS Ths f we have well behaved references we can fnd the set of PO allocatons sng MRS MRS

5 5 Eamle Sose that the ntal endowment s ; The references are reresented b the followng tlt fnctons: Notce that the er ndees are not owers The corresond to the nmber of the agent agent and agent Fnd the set of all PO allocatons Solton MRS MRS the set of Pareto Otmal allocatons Fgre The set of Pareto Otmal allocatons conssts of all onts of tangenc between the agents ndfference crves Notce that gvng all the goods n the econom to one erson s effcent n the Pareto sense However most eole wll sa that t s nfar We never sa that all that effcenc s all that matters Bt f we have two alternatves to acheve the same socal goal and one alternatve s not effcent whle the other s most eole wll advse n favor of the effcent alternatve After all f t s ossble to make some eole better off wthot makng the rest worse off then wh not do t? The set of Pareto Otmal allocatons

6 Comettve Eqlbrm Walrasan Eqlbrm Notce that fgre dects a staton n whch the two agents can get better off b tradng wth each other An allocaton that s nsde the lens shae area makes both consmers better off For eamle the allocaton 5 makes both better off verf that Recall that the ntal endowment s ; One eamle of trade between the agents s that agent gves one nt of good to agent and the latter gves one nt of good n retrn Relatve rces In ths econom there s no mone so eole trade one good for another echange barter Nevertheless we can talk abot rces n ths econom We can sa for eamle that the rce of one nt of good s one nt of good Ths rce s called the relatve rce of good In realt we observe rces n terms of nts of mone bt we alwas can fnd the relatve rces Sose that the rce of tomatoes s $ er ond and the rce of ales $5 er ond The relatve rce of tomatoes n terms of ales s each nt of tomatoes s worth nts of ales Smlarl the relatve rce of ales n terms of tomatoes s 05 each nt of ales s worth half nt of tomatoes The bdgets Sose that the rces are n terms of dollars Then the bdgets wold look lke ths Agent Agent As sal the bdget has the eendtres on the left hand sde and ncome on the rght hand sde Show that ths bdget s homogeneos of degree zero n rces Now snce there s no mone n the econom we need to se relatve rces As a conventon we eress the rces of goods n terms of nts of good Dvde both bdgets b to get Agent Agent The rce of n terms of s srrsngl Agent Agent and the rce of n terms of s not Leon Walras 90

7 Defnton: Comettve eqlbrm A comettve eqlbrm conssts of the rces rce rato and allocaton sch that Gven the allocaton s the best bndle nder ther bdget constrant The markets are cleared That s the qanttes of and demanded b both consmers s eqal to the qanttes sled b both consmers The frst art sas that gven the rces the allocaton s n the consmer s demand snce the demand gves s the best choce at an rce vector The second art of the defnton reqres that demand sl n all markets and ths s wh we call t a general eqlbrm Solvng for general eqlbrm Ste : fnd the demand of each consmer Ste : eqate demand sl n one 5 of the markets to fnd the eqlbrm rces Ste : lg the rces n each consmer s demand to fnd the eqlbrm allocaton Eamle Sose that the ntal endowment s ; The references are reresented b the followng tlt fnctons: Fnd the general eqlbrm n ths econom Solton Ste : fnd the demand of each consmer We are famlar wth the demand that reslts form Cobb-Doglass references We know that when the eonents of the tlt fncton are eqal the consmer sends half of hs ncome on each good Ths the demand s Consmer : Consmer : It s mortant to sa that we never roved that the general eqlbrm ests Ths sse s wa beond the scoe of or corse bt I wll menton that nder some week condtons t ests Now o see wh I dd not want o to forget how to derve the demand 5 Wlaras law mles that f there are n markets and n n- are n eqlbrm then the last market s also n eqlbrm Later I rove ths reslt Now o see wh I dd not want o to forget how to derve the demand 7

8 Ste : eqate demand sl n the market for ths s the eqlbrm rce rato Ste : lg the rces n each consmer s demand to fnd the eqlbrm allocaton Consmer : Consmer : Alwas check that the eqlbrm allocaton s feasble Indeed and Now llstrate grahcall the eqlbrm Fgre Notce that the comettve eqlbrm allocaton s Pareto Otmal Ths s one of the most mortant reslts n welfare economcs and called The frst Fndamental theorem of Welfare Economcs It bascall means that s agents trade comettvel each actng n hs own self nterest mamzes hs tlt then the resltng allocaton s effcent Ths theorem s a formalzaton of Adam Smth s nvsble hand argment and t eresses or confdence n the market econom The ntal endowment Comettve eqlbrm allocaton