1、Chapter ThirtyExchange交换交换ContentsuThe Edgeworth(艾奇沃斯艾奇沃斯)BoxuPareto improving tradeuPareto Efficient AllocationuThe Core(核心核心)uTrade in a competitive marketuWalras Law of general equilibrium(一般均一般均衡衡)uFirst Fundamental Theorem of Welfare Economics(福利经济学第一定理福利经济学第一定理)uSecond Fundamental Theorem of W
2、elfare Economics(福利经济学第二定理福利经济学第二定理)ExchangeuTwo consumers,A and B.uTheir endowments of goods 1 and 2 are uE.g.uThe total quantities available AAA (,)12 BBB (,).12and A (,)6 4 B (,).2 2and 11628AB 22426AB units of good 1units of good 2.andareExchangeuEdgeworth and Bowley devised a diagram,called an
3、Edgeworth box,to show all possible allocations of the available quantities of goods 1 and 2 between the two consumers.Starting an Edgeworth BoxStarting an Edgeworth BoxWidth=11628AB Starting an Edgeworth BoxWidth=11628AB Height=22426AB Starting an Edgeworth BoxWidth=11628AB Height=22426AB The dimens
4、ions ofthe box are thequantities availableof the goods.Feasible AllocationsuWhat allocations of the 8 units of good 1 and the 6 units of good 2 are feasible?uHow can all of the feasible allocations be depicted by the Edgeworth box diagram?uOne feasible allocation is the before-trade allocation;i.e.t
5、he endowment allocation.Width=11628AB Height=22426AB The endowmentallocation is A (,)6 4 B (,).2 2andThe Endowment AllocationWidth=11628AB Height=22426AB A (,)6 4 B (,)2 2The Endowment Allocation A (,)6 4OAOB68 B (,)2 2The Endowment Allocation A (,)6 4OAOB6846The Endowment Allocation B (,)2 2OAOB684
6、622The Endowment Allocation A (,)6 4 B (,)2 2OAOB684622TheendowmentallocationThe Endowment AllocationMore generally,The Endowment AllocationThe Endowment AllocationOAOBTheendowmentallocation 11AB 2A 22AB 1A 1B 2BOther Feasible Allocationsu denotes an allocation to consumer A.u denotes an allocation
7、to consumer B.uAn allocation is feasible if and only if(,)xxAA12(,)xxBB12xxABAB1111 xxABAB2222 .andFeasible ReallocationsOAOB 11AB xA2 22AB xA1xB1xB2Feasible ReallocationsOAOB 11AB xA2 22AB xA1xB1xB2Feasible ReallocationsuAll points in the box,including the boundary,represent feasible allocations of
8、 the combined endowments.uWhich allocations will be blocked by one or both consumers?uWhich allocations make both consumers better off?Adding Preferences to the Box 2A 1AxA2xA1OAFor consumer A.Adding Preferences to the Box 2A 1AxA2xA1More preferredFor consumer A.OAAdding Preferences to the Box 2B 1B
9、xB2xB1For consumer B.OBAdding Preferences to the BoxxB2xB1More preferredFor consumer B.OB 2B 1BAdding Preferences to the Box 2B 1BxB1xB2More preferredFor consumer B.OBAdding Preferences to the Box 2A 1AxA2xA1OAFor consumer A.Adding Preferences to the Box 2A 1AxA2xA1OA 2B 1BxB1xB2OBEdgeworths Box 2A
10、1AxA2xA1OA 2B 1BxB1xB2OBPareto-ImprovementuAn allocation of the endowment that improves the welfare of a consumer without reducing the welfare of another is a Pareto-improving allocation.uWhere are the Pareto-improving allocations?Edgeworths Box 2A 1AxA2xA1OA 2B 1BxB1xB2OBPareto-Improvements 2A 1AxA
11、2xA1OA 2B 1BxB1xB2OBThe set of Pareto-improving allocationsPareto-ImprovementsuSince each consumer can refuse to trade,the only possible outcomes from exchange are Pareto-improving allocations.uBut which particular Pareto-improving allocation will be the outcome of trade?Pareto-Improvements 2A 1AxA2
12、xA1OA 2B 1BxB1xB2OBThe set of Pareto-improving reallocationsPareto-ImprovementsPareto-ImprovementsPareto-ImprovementsTradeimproves bothAs and Bs welfares.This is a Pareto-improvementover the endowment allocation.Pareto-ImprovementsNew mutual gains-to-trade region is the set of all further Pareto-imp
13、roving reallocations.Tradeimproves bothAs and Bs welfares.This is a Pareto-improvement(帕累托改进帕累托改进)over the endowment allocation.Pareto-ImprovementsFurther trade cannot improve both A and Bs welfares.Pareto-Optimality(帕累托最优)Better forconsumer BBetter forconsumer APareto-OptimalityA is strictly better
14、 off but B is strictly worse offB is strictly betteroff but A is strictlyworse offPareto-OptimalityA is strictly better off but B is strictly worse offB is strictly betteroff but A is strictlyworse offBoth Aand B are worse offBoth A andB are worseoffPareto-OptimalityThe allocation isPareto-optimal s
15、ince theonly way one consumerswelfare can be increased is todecrease the welfare of the otherconsumer.Pareto-OptimalityThe allocation isPareto-optimal since theonly way one consumerswelfare can be increased is todecrease the welfare of the otherconsumer.The Marginal Rates of Substitution Are Equal.P
16、areto-OptimalityuWhere are all of the Pareto-optimal allocations of the endowment?Pareto-Optimality 2A 1AxA2xA1OA 2B 1BxB1xB2OBPareto-Optimality 2A 1AxA2xA1OA 2B 1BxB1xB2OBAll the allocations marked bya are Pareto-optimal.Pareto-OptimalityuThe contract curve(契约线契约线)is the set of all Pareto-optimal a
17、llocations.Pareto-Optimality 2A 1AxA2xA1OA 2B 1BxB1xB2OBAll the allocations marked bya are Pareto-optimal.The contract curvePareto-OptimalityuBut to which of the many allocations on the contract curve will consumers trade?uThat depends upon how trade is conducted.uIn perfectly competitive markets?By
18、 one-on-one bargaining?The Core(核心)2A 1AxA2xA1OA 2B 1BxB1xB2OBThe set of Pareto-improving reallocationsThe Core 2A 1AxA2xA1OA 2B 1BxB1xB2OBThe Core 2A 1AxA2xA1OA 2B 1BxB1xB2OBPareto-optimal trades blocked by BPareto-optimal trades blocked by AThe Core 2A 1AxA2xA1OA 2B 1BxB1xB2OBPareto-optimal trades
19、 not blocked by A or B are the core.The CoreuThe core is the set of all Pareto-optimal allocations that are welfare-improving for both consumers relative to their own endowments.uRational trade should achieve a core allocation.The CoreuBut which core allocation?uAgain,that depends upon the manner in
20、 which trade is conducted.Trade in Competitive MarketsuConsider trade in perfectly competitive markets.uEach consumer is a price-taker trying to maximize her own utility given p1,p2 and her own endowment.That is,.Trade in Competitive Markets 2A 1AxA2xA1OAFor consumer A.p xp xppAAAA1 12 21122 xA2*xA1
21、*Trade in Competitive MarketsuSo given p1 and p2,consumer As net demands for commodities 1 and 2 arexAA11*xAA22*.andTrade in Competitive MarketsuAnd,similarly,for consumer B Trade in Competitive Markets 2B 1BxB2xB1For consumer B.OBxB1*xB2*p xp xppBBBB1 12 21122 Trade in Competitive MarketsuSo given
22、p1 and p2,consumer Bs net demands for commodities 1 and 2 arexBB11*xBB22*.andTrade in Competitive MarketsuA general equilibrium occurs when prices p1 and p2 cause both the markets for commodities 1 and 2 to clear;i.e.xxABAB1111*xxABAB2222*.andTrade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBT
23、rade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBCan this PO allocation beachieved?Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer ATrade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer AxA2*xA1*Trade in Competitive Market
24、s 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer BxA2*xA1*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer BxA2*xA1*xB1*xB2*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBxA2*xA1*xB1*xB2*ButxxABAB1111*Trade in Competitive Markets 2A 1AxA2xA1O
25、A 2B 1BxB1xB2OBxA2*xA1*xB1*xB2*andxxABAB2222*Trade in Competitive MarketsuSo at the given prices p1 and p2 there is an excess supply of commodity 1 excess demand for commodity 2.uNeither market clears so the prices p1 and p2 do not cause a general equilibrium.Trade in Competitive Markets 2A 1AxA2xA1
26、OA 2B 1BxB1xB2OBSo this PO allocation cannot beachieved by competitive trading.Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBWhich PO allocations can beachieved by competitive trading?Trade in Competitive MarketsuSince there is an excess demand for commodity 2,p2 will rise.uSince there is
27、an excess supply of commodity 1,p1 will fall.uThe slope of the budget constraints is-p1/p2 so the budget constraints will pivot about the endowment point and become less steep.Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBWhich PO allocations can beachieved by competitive trading?Trade in
28、Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBWhich PO allocations can beachieved by competitive trading?Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBWhich PO allocations can beachieved by competitive trading?Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for c
29、onsumer ATrade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer AxA2*xA1*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer BxA2*xA1*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBBudget constraint for consumer BxA2*xA1*xB1
30、*xB2*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBxA2*xA1*xB1*xB2*SoxxABAB1111*Trade in Competitive Markets 2A 1AxA2xA1OA 2B 1BxB1xB2OBxA2*xA1*xB1*xB2*andxxABAB2222*Trade in Competitive MarketsuAt the new prices p1 and p2 both markets clear;there is a general equilibrium.uQuestion 1:Are p
31、rices p1 and p2 determined independently at equilibrium?uQuestion 2:Does the equilibrium exist?Walras Law(瓦尔拉斯法则)uWalras Law is an identity;i.e.a statement that is true for any positive prices(p1,p2),whether these are equilibrium prices or not.Walras LawuEvery consumers preferences are well-behaved
32、so,for any positive prices(p1,p2),each consumer spends all of his budget.uFor consumer A:For consumer B:p xp xppAAAA1 12 21122*p xp xppBBBB1 12 21122*Walras Lawp xp xppAAAA1 12 21122*p xp xppBBBB1 12 21122*pxxpxxppABABABBB111222111222()()()().*Summing givesWalras LawpxxpxxppABABABBB111222111222()()(
33、)().*Rearranged,pxxpxxABABABAB11111222220()().*That is,.Walras Law.0)xx(p)xx(pB2A2B*2A*22B1A1B*1A*11 This says that the summed marketvalue of excess demands is zero forany positive prices p1 and p2-this is Walras Law.Implications of Walras Law0)xx(p)xx(pB2A2B*2A*22B1A1B*1A*11 Suppose the market for
34、commodity Ais in equilibrium;that is,.0 xxB1A1B*1A*1 Thenimplies.0 xxB2A2B*2A*2 So one implication of Walras Law for a two-commodity exchange economy is that if one market is in equilibrium then the other market must also be in equilibrium.In general,if there are k markets,then only k-1 prices are d
35、etermined at equilibrium.I.e.,the general equilibrium determines only relative prices.Implications of Walras LawImplications of Walras LawWhat if,for some positive prices p1 andp2,there is an excess quantity suppliedof commodity 1?That is,.0 xxB1A1B*1A*1 0)xx(p)xx(pB2A2B*2A*22B1A1B*1A*11 Thenimplies
36、.0 xxB2A2B*2A*2 Implications of Walras LawSo a second implication of Walras Lawfor a two-commodity exchange economyis that an excess supply in one marketimplies an excess demand in the othermarket.Existence of EquilibriumuRequire that the aggregate excess demand functions be a continuous function,wh
37、ich in turn require thatEach individuals demand function be continuous,orEach consumer is small relative to the market.Trade in Competitive MarketsuTrading in competitive markets achieves a particular Pareto-optimal allocation of the endowments.uThis is an example of the First Fundamental Theorem of
38、 Welfare Economics.First Fundamental Theorem of Welfare EconomicsuGiven that consumers preferences are well-behaved,trading in perfectly competitive markets implements a Pareto-optimal allocation of the economys endowment.The Equilibrium with MonopolySee Fig.30.5 in textbook.The Equilibrium with Mon
39、opolyFigure 30.6 in book.Implicit AssumptionsuNo consumption externality(外部效应外部效应)uAgents behave competitivelyuEquilibrium existsMarket Socialism:The DebateuLange vs.HayekuSocialism with economic planningState-ownershipGovernment chooses prices to clear marketsuAdvantage of competitive marketsInform
40、ationSecond Fundamental Theorem of Welfare EconomicsuThe First Theorem is followed by a second that states that any Pareto-optimal allocation(i.e.any point on the contract curve)can be achieved by trading in competitive markets provided that endowments are first appropriately rearranged amongst the
41、consumers.uGiven that consumers preferences are well-behaved,for any Pareto-optimal allocation there are prices and an allocation of the total endowment that makes the Pareto-optimal allocation implementable by trading in competitive markets.Second Fundamental Theorem of Welfare EconomicsSecond Fund
42、amental Theorem 2A 1AxA2xA1OA 2B 1BxB1xB2OBThe contract curveSecond Fundamental Theorem 2A 1AxA2xA1OA 2B 1BxB1xB2OBA*2xB*2xA*1xB*1xSecond Fundamental Theorem 2A 1AxA2xA1OA 2B 1BxB1xB2OBA*2xB*2xA*1xB*1xImplemented by competitivetrading from the endowment .Second Fundamental Theorem 2A 1AxA2xA1OA 2B 1
43、BxB1xB2OBCan this allocation be implementedby competitive trading from?Second Fundamental Theorem 2A 1AxA2xA1OA 2B 1BxB1xB2OBCan this allocation be implementedby competitive trading from?No.Second Fundamental TheoremxA2xA1OAxB1xB2OBBut this allocation is implementedby competitive trading from q q.A1q qB2q qB1q qA2q qThe Assumption about ConvexityFigure 30.8 in book.Implications of the 2nd TheoremuMarket equilibrium is distributionally neutral.uRedistribution can be carried out for endowments.uThe market plays only the allocation role.
