Gen e r a l E q uilib riu mJonathan Levin∗November 2006“From the time of Adam Smith’s Wealth of Nations in 1776, one re-curren t theme of economic analysis has been the remarkable degree ofcoherence among the vast n umbers of individual and seemingly sepa-rate decisions about the buying and selling of commodities. In ev ery-day, normal experience, there is something of a balance between theamoun ts of goods and services that some individuals w an t to supplyand the amounts that other, differerent individuals wan t to sell [sic].Would-be buy ers ordinarily count correctly on being able to carry outtheir in tentions, and w ould-be sellers do not ordinarily find themselvesproducing great amoun ts of goods that they cannot sell. This expe-rience of balance is ind eed so widesp read that it raises no in tellectualdisquiet am on g la ym en ; they take it so m uch for gran ted that they arenot disposed to understand the mechanism by which it occurs.”Kenneth Arro w (1973)1 IntroductionGenera l equilibrium analysis addresses precisely how these “vast numbers of indi-vidual and seemingly separate decisions” referred to b y Arrow aggregate in a waythat coordinates productiv e effort, balances supply and demand, and leads to anefficient allocation of goods and services in the economy. The answer economistsha v e pro vided, beginning with Adam Smith and continuing through to Jevons and∗Various sections of these notes draw heavily on lecture notes written by Felix Kubler; someof the other sections draw on Mas-Colell, Whinston and Green.1Wa lras is that it is the price system plays the crucial coordin atin g and equilibratin grole: the fact the ev eryone in the economy faces the same prices is what generatesthe comm on information needed to coordinate disparate individual decisions.You doubtless are fam ilia r w ith the standard tr eatmen t of equilibrium in asingle ma rket. Pric e play s the role of equilibrating deman d and supply so that allbuy ers who want to buy at the going price can, and do, and similarly all sellerswho wan t to sell at the going price also can and do, with no excess or shor tageson either side. The extension from this partial equilibriu m in a single market togeneral equilibrium reflectstheideathatitmaynotbelegitimatetospeakofequilibrium with respect to a single commodit y when supply and demand in thatmarket depend on the prices of other g oods. O n this view, a coherent theory ofthe price system and the coordination o f economic activity has to con sider thesim ultaneous general equilibrium of all mark ets in the econom y. This of courseraises the questions of (i) whether suc h a general equilibrium exists; and (ii) whatare its properties.A recurring them e in genera l equilibriu m analysis, and economic theor y moregenerally, has been the idea that the competitive price mec hanism leads to out-comes that are efficient in a way that outcomes under other systems such as plannedeconom ies are not. The relevant notion of efficiency was form alized and tied tocompetitiv e equilibrium b y Vilfredo P are to (1909) and Abram Bergson (1938).This line of inquiry culminates in the Welfare Theorems of Arro w (1951) and De-breu (1951). These theorem s state that there is in essence an equivalence betweenPareto efficient outcomes and competitive price equilibria.Our goal in the next few lectures is to do some small justice to the main ideasof general equilibrium. We ’ll start with the basic concepts and definitions, thew elfare theorems, and the efficiency properties of equilibrium. We’ll then providea proof that a general equilibrium exists under certain conditions. From there, we’llinv e stigate a few important ideas about general equilibrium: whether equilibrium isuniqu e, how prices might adjust to their equilibrium lev els and whether these levelsare stable, and the exten t to which equilibria can be cha racterize d and changes inexogenous preferences or endow m ents will have predictable consequen ces. Finallyw e’ll discuss ho w one can incorporate production in to the model and then time2and uncertainty, leading to a brief discussion of financial markets.2TheWalrasianModelWe’re going to focus initially on a pure exchange economy. An exchange ec on om y isan economy without production. There are a finite n u mber of agents and a finitenumber of comm odities. Eac h agent is endo wed with a bundle of comm odities.Shortly the world will end and ev eryone will consume their commodities, but beforethishappenstherewillbeanopportunityfortradeatsomesetprices. Wewantto kno w whether there exist prices such that when every o ne tries to trade theirdesired amounts at these prices, demand will just equal supply, and also what theresulting outcom e will look like – whethe r it will be efficien t in a w ell-definedsense and how it will depend on preferences and endo w m ents.2.1 The ModelConsid er an economy with I agents i ∈ I = {1, ..., I} and L commodities l ∈L = {1, ..., L}. A bun dle of comm odities is a v ector x ∈ RL+.Eachagenti has anendowm en t ei∈ RL+and a utility function ui: RL+→ R. These endo wments andutilities are the prim itives of the exchange economy, so we write E =((ui,ei)i∈I).Agents are assumed to tak e as giv en the market prices for the goods. We w on ’tha v e m uc h to say about where these prices come from, although we’ll say a bitlater on. The vector of market prices is p ∈ RL+; all prices are nonnegative.Each agen t c h ooses consum pt io n to maximize her utilit y given her budget con-straint. Therefore, agent i solv es:maxx∈RL+ui(x) s.t. p · x ≤ p · ei.The bu dget constraint is slightly different th an in standa rd price th eory. Recallthat the familiar budget constraint is p · x ≤ w,wherew is the consu m er ’s initialw ealth. Here the consumer’s “w ealth” is p · ei, the amount she could get if she sold3her entire endowm en t. We can write the budget set asBi(p)={x : p · x ≤ p · ei}.We’ll occasionally use this notation below .2.2 Wal ra sia n Equ ilib riu mWe now define a W a lrasian equilibrium for the exchange economy. A Walrasianequilibrium is a vector of prices, and a consumption bundle for eac h agent, suchthat (i) ev ery agent’s consumption maxim izes her utility given prices, and (ii)markets clear: the total demand for eac h commodity just equals the aggregateendow men t.Definition 1 A Wa lrasia n equilibrium for the economy E is a vector (p, (xi)i∈I)such that:1. Agents are