1General Equilibrium/ExchangeECON 370: Microeconomic TheorySummer 2004 – Rice UniversityStanley GilbertEcon 370 - Exchange 2General vs. Partial Equilibrium• Partial equilibrium– Analysis of a single market– Assumes price, quantity interactions with all other markets small enough to be ignored• General equilibrium– Considers all interactions among markets– Simultaneously obtains prices in all markets– Consistent relationships among all markets for all variables (factors, incomes, etc.)Econ 370 - Exchange 3General Equilibrium Analysis• Start with theory of pure exchange – Assume fixed set of consumer goods (ignore how these goods are produced)– Assign goods to individuals: endowments– Analyze exchange of endowments• Will consider “production” of these goods later in general equilibrium context• Will analyze 2-consumer, 2-good modelEcon 370 - Exchange 4Exchange: Introduction• Two consumers, A and B• Endowments of goods 1 and 2 are• ωA= (ω1A, ω2A) for Consumer A• ωB= (ω1B, ω2B) for Consumer B– E.g., ωA= (6, 4) and ωB= (2, 2)• Total quantities available are– ω1A+ ω1B= 6 + 2 = 8 units of good 1– ω2A+ ω2B= 4 + 2 = 6 units of good 22Econ 370 - Exchange 5Exchange: The Edgeworth Box• Theory of Exchange– Modeled using Edgeworth box– Shows all possible allocations of fixed quantities of goods 1 and 2 between consumers A and B– Start with initial allocation = endowments– After trade, end up with final allocation– Traded quantities must be consistentEcon 370 - Exchange 6Starting an Edgeworth BoxWidth = 826B1A1=+=+ωωHeight =624B2A2=+=+ωωThe dimensions ofthe box are thequantities availableof the goodsEcon 370 - Exchange 7ωA= (,)64OAOB6846Edgeworth Box: The Endowment22ωB= (,)22TheendowmentallocationEEcon 370 - Exchange 8Other Feasible Allocations•(x1A, x2A) denotes an allocation to consumer A•(x1B, x2B) denotes an allocation to consumer B• Feasible allocation satisfies – x1A+ x1B≤ ω1A+ ω1Band– x2A+ x2B≤ ω2A+ ω2B• So, all points in Edgeworth box (including those on boundary) are Feasible3Econ 370 - Exchange 9OAOBω2A+ ω2Bω1A+ ω1BFeasible Reallocations: Diagramx2Ax1Ax2Bx1BEEcon 370 - Exchange 10Adding Preferences - Consumer AMore preferredFor consumer AxA2xA1OAω2Aω1AEEcon 370 - Exchange 11Adding Preferences - Consumer BxB2xB1More preferredFor consumer BOBω2Bω1BEEcon 370 - Exchange 12Adding Preferences – Consumer BB2ωB1ωB1xMore preferredOBB2xEFor consumer B4Econ 370 - Exchange 13Edgeworth BoxA2ωA1ωA2xA1xOAω2BB1ωB1xOBB2xEcon 370 - Exchange 14Pareto-Improving Reallocations• Pareto-improving reallocation– Re-allocation of endowment improving welfare of one consumer w/o reducing welfare of the other• Only Pareto-improving reallocations will occur through voluntary trade • Can be easily identified on Edgeworth box diagramEcon 370 - Exchange 15Pareto-Improving TradesPareto Improving TradesA2ωA1ωA2xA1xOAω2BB1ωB1xOBB2xEcon 370 - Exchange 16Pareto-Optimality: DiagramOnly way one consumer’swelfare can be increased is todecrease that of the otherPareto-optimalAllocation where convexindifference curves are tangent5Econ 370 - Exchange 17Pareto-Optimality: DiagramBoth Aand B areworse offBoth A and B are worse offA is strictly better off but B is strictly worse offB is strictly betteroff but A is strictlyworse offEcon 370 - Exchange 18Contract CurveA2ωA1ωB2ωB1ωA2xA1xOAB1xOBB2x“Contract Curve” is the set of all Pareto-optimal allocations Contract curveEcon 370 - Exchange 19The Core: Definition• Core - set of all allocations that are:– Pareto optimal allocations, and– Welfare-improving for both consumers, given original endowments• Rational trade should achieve allocation in core• Which core allocation? – Depends on allocation mechanism: Competition, Monopoly, negotiation, etc.Econ 370 - Exchange 20Pareto-Improving Allocations“Core”A2ωA1ωB2ωB1ωA2xA1xOAB1xOBB2xCore6Econ 370 - Exchange 21Trade in Competitive Markets• Suppose final allocation is determined by a competitive price mechanism– Consumers are price takers– Endowment determines location of budget line (analogous to income constraint)– Endowment may not be optimal Consumption bundle– Consumers trade to reach optimum• Note: not realistic in two person case– Two groups– True for many individualsEcon 370 - Exchange 22Trade in Competitive MarketsA2ωA1ωA2xA1xOAFor consumer AAAAAppxpxp22112211ωω+=+A*2xA*1xOptimumEndowmentBudget LineEcon 370 - Exchange 23Trade in Competitive Markets• So, given p1, p2, consumer A wants to– Reduce consumption of good 1– Increase consumption of good 2– (Relative to endowments of 1 and 2) • Thus, net demands for commodities 1, 2 are– x1*A– ω1A,– x2*A– ω2AEcon 370 - Exchange 24Trade in Competitive MarketsB2ωB1ωB2xB1xFor consumer BOBB*1xB*2xBBBBppxpxp22112211ωω+=+OptimumEndowmentBudget Constraint7Econ 370 - Exchange 25Trade in Competitive Markets• So, given p1, p2, consumer B wants to– Increase consumption of good 1– Decrease consumption of good 2– (Relative to endowments of 1 and 2) • Thus, net demands for commodities 1, 2 are– x1*B– ω1B,– x2*B– ω2BEcon 370 - Exchange 26General equilibrium• General Equilibrium occurs for (p1, p2) such that both commodity markets clear:– x1*A+ x1*B= ω1A+ ω1B– x2*A+ x2*B= ω2A+ ω2B• Requires (p1, p2) tangent to both IC’s at same point, so Supply = Demand for both goods• Recall price line is budget line for both A, B, and endowment point is same for A, BEcon 370 - Exchange 27Trade in Competitive MarketsOAOBA2ωA1ωA2xA1xB2ωB1ωB1xB2xEcon 370 - Exchange 28Trade in Competitive Markets• At prices (p1, p2) both markets clear• There is a general equilibrium• Trading in competitive markets achieves a specificPareto optimal reallocation of endowments• Pareto optimality in exchange also referred to as efficiency in consumption8Econ 370 - Exchange 29Efficiency in Consumption• Efficiency in consumption requiresBB1B2A1A2AMRSdxdxdxdxMRS ===• Or mutually beneficial trades exist• (MRS’s reflect relative valuations of x1)Econ 370 - Exchange 30Efficiency in Consumption• Efficiency in consumption achieved with competitive markets• All consumers face the same p1/ p2• So, all have the same MRS• So, efficiency in consumption achieved• Or, Pareto optimal exchangeEcon 370 - Exchange 31Walras’ Law: Introduction• Walras’ Law is an identity• Identity– a