The Complexity of General Equilibrium

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1 Prof. Ja Bhattachara Eco --Sprg 200 Welfare Propertes of Market Outcomes Last tme, we covered equlbrum oe market partal equlbrum. We foud that uder perfect competto, the equlbrum prce ad quatt mamzed the sum of producer ad cosumer surplus. I a echage ecoom, we foud that trade at the equlbrum market prce led automatcall to Pareto optmal outcomes (o the cotract curve). The et two lectures wll ask whether these ce welfare propertes hold geeral equlbrum uder perfect competto. Lecture 7 Eco --Sprg 200 The Complet of Geeral Equlbrum I partal equlbrum aalss, there s a clea ad sharp theoretcal dstcto betwee producers ad cosumers. I geeral equlbrum, oe perso ma partcpate smultaeousl ma dfferet markets. Sometmes as a cosumer Sometmes as a producer There are bllos of people the ecoom. Outcomes oe market affect outcomes coutless other markets. What s eeded s some mechasm to cocsel cove to everoe formato about everoe else s eeds ad about the dffcult fulfllg them. Lecture 7 Eco --Sprg The Role of Market Prces The role of prces s to cove sgals to ever partcpat about relatve scarct all markets smultaeousl. We wll see that uder compettve codtos, market prces succctl cove all ecessar formato about a umagabl complcated realt so that all markets are equlbrum smultaeousl. Lecture 7 Eco --Sprg Compettve Equlbrum Cosumers, takg prces as gve, choose cosumpto goods ad suppl puts (captal, labor) to frms to mamze utlt. Producers, takg prces as gve, bu puts from cosumers ad make cosumpto goods to mamze profts. Cosumers ow frms ad collect a profts based o how ma shares the ow. I compettve equlbrum, all put ad output markets clear ad frms make zero profts. Lecture 7 Eco --Sprg Cotget Commodtes ad Futures Cotget commodtes are goods that are delvered depedg upo the state of the world. Fre surace pas f there s a fre. Bets agast the Lakers must be pad whe the w. Futures are commodtes that are delvered at some future tme for a prce pad toda. Cattle futures Orga futures Ecoomsts have costructed proofs that show that geeral equlbrum ests eve whe cotget ad futures markets are allowed. Lecture 7 Eco --Sprg Pla for the Rest of the Lecture Rather tha showg the most geeral forms of the proof of geeral equlbrum (whch requres advaced math), I wll cosder the estece of geeral equlbrum two models: Walras echage ecoom wth ma markets but o producto. A ecoom wth two producto goods ad two factors of producto. Lecture 7 Eco --Sprg Lecture 7

2 Prof. Ja Bhattachara Eco --Sprg 200 Walras Echage Ecoom goods ( ) fed suppl, S S. Each good has a assocated prce, P P. There are K people ad each take prces as gve. Icome s the value of each perso s holdgs at market prces Perso k s come s I k = P Sk Perso k s budget costrat s: P = I Market Demad Walras Ecoom Each perso pcks hs optmal cosumpto budle to mamze utlt Ths leads to dvdual demad fuctos for each dvdual whch are a fucto of all market prces. D k (P P ), D 2k (P P ), D k (P P ) The sum of dvdual demad fuctos elds market demad fuctos: D (P P ), D 2 (P P ), D (P P ) k k Lecture 7 Eco --Sprg Lecture 7 Eco --Sprg Equlbrum Walras Ecoom ( ) Equlbrum s a set of prces P = P, P2,... P such that all of the markets clear: D ( P ) = S ecess demad equatos: ED ( P ) = D ( P ) S = 0 There are equatos wth ukows Ths meas there wll automatcall be a soluto, rght? No! Sce the equatos characterzg the equlbrum are o-lear, there s o guaratee that there wll be a solutos. Lecture 7 Eco --Sprg Walras Law The total value of demad must equal the total value of suppl the ecoom. Ths s true eve whe o-equlbrum prces hold. Walras Law follows drectl from summg the dvdual budget costrats. Each perso s budget costrat s: Dk ( P) = P Sk Summg all the budget costrats elds: Total value of demad P ( P) = PS = Total value of suppl = PD Lecture 7 Eco --Sprg Zero Degree Homogeet of Demad Now there are + equatos P, ad ol ukows. If ths were a lear sstem, ths would mea there are ftel ma solutos. Ths happes because demad s homogeous of degree zero. Suppose we have foud the equlbrum. Doublg all prces would also double come We have see that doublg prces ad come at the same tme does ot chage demad. Lecture 7 Eco --Sprg 200 Ol Relatve Prces Are Idetfed The sstem of demad equatos detfes ol - relatve prces, ot all absolute prces. Ths meas we ca pck a - of the equlbrum ecess demad equatos, whch, prcple should be able to detf all - relatve prces. We re back to where we were--o guaratee of a soluto, sce the sstem s o-lear. Lecture 7 Eco --Sprg Lecture 7 2

3 Prof. Ja Bhattachara Eco --Sprg 200 Mathematcal Dgresso-- Brouwer s Fed Pot Theorem A cotuous mappg F() of a closed, bouded, cove set to tself has at least oe fed pot ( ) such that F( )= Mappg: A rule assocatg pots a set wth aother set of pots. Closed: The set cotas ts edge. Cove: If the set cotas two pots, t also cotas all pots o the le coectg the two pots. Bouded: The set s dmesos are fte. Eample for a Uvarate Mappg Ituto: Cosder a cotuous fucto f() wth doma ad rage [0,]. A cotuous fucto s a cotuous mappg. The set = [0,] s closed, bouded, ad cove. A fed pot of f() s a pot such that f( )=. O a graph of f(), t s fed pots are o the 45 o le. Lecture 7 Eco --Sprg Lecture 7 Eco --Sprg Eample: Brouwer s Fed Pot Theorem f() 45 o le f( ) Lecture 7 Eco --Sprg Back to Walras Ecoom The et step to fdg a equlbrum s to ormalze the prces so that the add to oe. For the ew prce set, dvde each prce b the sum of all the prces. P j = Pj j P Redefg prces ths wa wll ot chage demad because of zero degree homogeet. Lecture 7 Eco --Sprg Applg Brouwer s FP Theorem After ormalzg prces, the prce set s closed, cove, ad bouded. Use the ecess demad fuctos to defe a mappg o the ormalzed prce set. F ( P) = P + ED ( P) To be rgorous, we eed to worr about what happes to demad whe some prce s zero. Demad could be less tha suppl for a good wth zero prce. We wll gore ths case here. Lecture 7 Eco --Sprg A Equlbrum Ests B Brouwer s FP Theorem, a fed pot must est. F ( P ) = P + ED ( P ) = P At the fed pot P, all of the ecess demad fuctos equal zero. ED ( P ) = D ( P ) S = 0 All markets clear, so a compettve equlbrum must est. Lecture 7 Eco --Sprg Lecture 7 3

4 Prof. Ja Bhattachara Eco --Sprg 200 Two-Good Two-Iput Ecoom Now, a ecoom that cludes producto. Two tpes of frms, each producg oe of two goods ad, whch are sold to cosumers. The frms bu puts K ad L from cosumers to produce the outputs usg producto techologes =F(K, L) ad =G(K,L). Cosumers ow the frms--the collect a profts made b the frms. Lecture 7 Eco --Sprg Cosumer ad Producer Goals Cosumers ( = ) mamze utlt subject to ther budget costrat. Icome from labor ad captal: r L + w K Icome from profts: α π + β π Producers mamze profts π ad π. Proft shares sum to oe: α = β = Labor ad captal costrats: L = L K = K Lecture 7 Eco --Sprg Demostratg Equlbrum The formal approach to demostratg the estece of compettve equlbrum s smlar to the approach the Walras echage ecoom. Two factor demad equatos; two factor prces w & r. Two product demad equatos; two good prces P ad P. Profts equal zero compettve equlbrum. Addg all partcpats budgets show Walras Law, so ol relatve prces ca be detfed. Istead, equlbrum wll be demostrated graphcall. Lecture 7 Eco --Sprg Producto Possblt Froter ( L ) = G K, The PPF represets the dfferet combatos of ad that ca be produced the ecoom f K ad L are ot wasted. Lecture 7 Eco --Sprg ( K L ) = F, Rate of Product Trasformato The rate of product trasformato s defed as (- tmes) the slope alog the producto possblt froter. d MC RPT ( for ) = = d MC RPT represets how much ca be traded for whle keepg puts K ad L productvel emploed. RPT s equal to the rato of margal costs of producto. At ever pot o the PPF, K ad L are effcetl emploed B defto, ths meas that costs are mmzed. Sce puts are fed suppl, mmum costs of producg ad wll be costat. Lecture 7 Eco --Sprg Wh s the PPF Cocave? Dmshg returs. Icreasg output of rases ts margal cost. Icreasg output of rases ts margal cost. Specalzed puts Some puts ma be better suted to the producto of oe good, rather tha aother Icreasg the producto of oe good evetuall requres usg puts that are poorl suted for that good s producto. Dfferg factor test ad ma requre K ad L dfferet proportos. The, eve uder costat returs to scale ad o-specalzed puts, makg more of or ma requre the use of relatvel more of the less tesvel requred factor, rasg margal costs. Lecture 7 Eco --Sprg Lecture 7 4

5 Prof. Ja Bhattachara Eco --Sprg 200 PPF, Cosumer Utlt ad Prces S D S Lecture 7 Eco --Sprg D At these prces, there s ecess demad for ad ecess suppl of. P should crease ad P should decrease. U(, ) Slope = P /P Equlbrum Two Good Ecoom Lecture 7 Eco --Sprg U(, ) Slope = P /P Propertes of Equlbrum P MC MU = RTS = = P MC MU = MRS At P ad P, both product markets clear. Profts ad utlt are mamzed. Labor ad captal are effcetl used sce fal cosumpto s o the PPF. Profts equal zero. (Need to dscuss put markets to show ths). Lecture 7 Eco --Sprg Lecture 7 5

The Firm. The Firm. Maximizing Profits. Decisions. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert

The Firm. The Firm. Maximizing Profits. Decisions. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert The Frm The Frm ECON 370: Mcroecoomc Theory Summer 004 Rce Uversty Staley Glbert A Frm s a mechasm for covertg labor, captal ad raw materals to desrable goods A frm s owed by cosumers ad operated for the