微观经济学讲义19

19 EXCHANGEPartial equilibrium analysis : how demand and supply were affected by the price of the particular good we were examining.General equilibrium analysis : how demand and supply conditions interact in several markets to determine the prices of many goods.The structure of the model: Competitive market.Two goods and two consumers. In two stages.The Edgeworth BoxTwo people: A and B. Two goods: 1 and 2.A’s consumption bundle: X A = (),,21A A x x B’s consumption bundle: X B = (21,B B x x ).An allocation: (X A , X B ).The initial endowment allocation, (21,A A ωω) and 21,(B B ωω).A feasible allocation (X A , XB ):1111B A B A x x ωω+=+ 2222B A B A x x ωω+=+Make a rectangle: Height: w 2A + w 2B Width: w 1A + w 1BDepict A’s and B’s indifference curves ----Edgeworth boxX B 1 O BX A 2 W X B 2O A X A 1TradeAny trade from endowment can be in the box.X B1O BMX A2W X B2O A X A1There are some trades make A and B both better off: M. Equilibrium: the state no body would like change.----Pareto efficientallocation.O BO AContract curve: the set of all Pareto efficient points. MRS12A = MRS12BMarket TradeMarket price: (p1, p2)Consumer A’s problem:Max u(x A1, x A2 )s. t. p1x A1 + p2x A2 p1w A1 + p2w A2The solution:[x A1(p1, p2), x A2(p1, p2)]The net demand or “excess demand” of A:e A1 = x A1–w A1e A2 = x A2–w A2The net demand or “excess demand” of B:e B1 = x B1–w B1e B2 = x B2–w B2e B 1 O BX A 2 W X B 2O A e A 1A market equilibrium : e A 1 + eB 1 = 0 e A 2 + e A 2 = 0We have 6 equations for 6 variables: MRS 12A = MRS 12B = p 1/p 2 p 1x A 1 + p 2x A 2 = p 1w A 1 + p 2w A 2 p 1x B 1 + p 2x B 2 = p 1w B 1 + p 2w B 2 x A 1 + x B 1 = w A 1 + w B 1 x A 2 + x B 2 = w A 2 + w B 2The aggregate excess demands),(),(),(211211211p p e p p e p p z B A +=),(),(),(212212212p p e p p e p p z B A +=an equilibrium ),(*2*1p p : 0),(*2*11=p p z0),(*2*12=p p zWalras’ LawBy consumer’s budget 222121222111),(),(A A A A p p p p x p p p x p ωω+≡+or[][]0),(),(2212212111≡-+-A A A A p p x p p p x p ωωThe value of agent A’s net demand is zero (for any (p 1, p 2)).0),(),(21222111≡+p p e p p p e p A AThe similar equation for agent B:0),(),(21222111≡+p p e p p p e p B Badd up [][]0),(),(),(),(2122122112111≡+++p p e p p e p p p e p p e p B e A B Aor.0),(),(21222111≡+p p z p p p z pWalras’ law : the value of aggregate excess demand is identically zero for all possible choices of prices. ----only 5 equations are independent.Relative PricesWalras’ law implies that there are only k – 1 independent equations in a k-good general equilibrium model.On the other hand, the prices of all goods are relative in equilibrium.Equilibrium and EfficiencyIs the market equilibrium Pareto efficient?If not, there is some other feasible allocation ),,,(2121B B A A y y y y such that1111B A B A y y ωω+=+ 2222B A B A y y ωω+=+and),(),(2121A A A A A x x y y).,(),(2121B B B B B x x y yBy hypothesis of market equilibrium:22112211A A A A p p y p y p ωω+>+22112211B B B B p p y p y p ωω+>+Now add them).()()()(222111222111B A B A B A B A p p y y p y y p ωωωω+++>+++Substitute from equations (19.1) and (19.2) to get)()()()(222111222111B A B A B A B A p p p p ωωωωωωωω+++>+++First Theorem of Welfare Economics: all market equilibria are Pareto efficient.Efficiency and EquilibriumGiven a Pareto efficient allocation, can we find prices such that it is a market equilibrium?O BO ASecond Theorem of Welfare Economics: if all agents have convex preferences, then there will always be a set of prices such that Pareto efficient allocation is a market equilibrium for an appropriate assignment of endowments.20 PRODUCTIONThe Robinson Crusoe EconomyRobinson Crusoe economy: economic model with one consumer, one firm, and two goods.Robinson Crusoe plays a dual role: both a consumer and a producer.Two goods: coconuts and leisure.Preferences for coconuts and leisurecLproduction function：using time to collect coconutscLRobinson’s choice: consuming leisure, or gathering coconuts.The most preferred combination: the slope of the indifference curve must equal the slope of the production function.Crusoe, Inc.Robinson divides himself into two roles: producer and consumer.The firm: how much labor to hire and how many coconuts to produce.Consumer: collect income as worker and profits as shareholder; purchas e the firm’s output.set the price of coconuts at one dollar apiece.The FirmThe firm’s profitπ = C – wL The isoprofit lines:L C ωπ+=.Tangency condition: the marginal product of labor must equalw .cπLThe vertical intercept of the isoprofit line measures the maximal level of profits:*π.It’s the stock dividend to sole shareholder, Robinson.From the viewpoint of Robinson, the consumer.Robinson receives his dividend of *π dollars. He contemplates how much he wants to work and consume.cπLThe budget lineL C ωπ+=.He consumes at exactly the same point as he would have if he had made all the decisions at once.Production and the First Welfare TheoremIf all firms act as competitive profit maximizers, then a competitive equilibrium will be Pareto efficient.Production and the Second Welfare TheoremAny Pareto efficient allocation can be achieved through the use of competitive markets.Production PossibilitiesThe two-good case: Robinson can devote his time to gathering coconuts or to fishing.CoconutsC*O F* FishProduction possibilities set: the set of output goods that is feasible. Production possibilities frontier: the boundary of production possibilities set.ExampleConstant returns to scale: the production functions will be linear functions of labor.Fish: 10 L fCoconuts: 20 L cSuppose that Robinson decides to work 10 hours a day. Then the production possibilities setF = 10L f C = 20L c L c + L f = 10.The production possibilities frontier .102010=+C F Marginal rate of transformation: The slope of this production possibilities set.Pareto EfficiencyWhich choices from the production possibilities set will be Pareto efficient choices?Pareto efficient allocation: MRS is equal to the MRT.Suppose, MRS = 1; MRT = 2?Good 2X 2UVO X 1 Good 1Castaways, Inc.Economy contains tow individuals, Robinson and Friday. There are four goods:Two factors of production (Robinson’s labor and Friday’s labor) andTwo output goods (coconuts and fish).Robinson and Friday are both shareholders of the firm, Castaways, Inc.Castaways, Inc.:profit maximization problemf f c c f c L L F C F w L w F p C p cf --+,,,m axsubject to the technological constraints.The profits of the firmπ= p c C + p f F – L *where ***f f c c L w L w L += is the labor costs of production.Take the profits as constant, rearranging the equationcf cp F p p L C -+=*π.----the isoprofit linesCoconutsO Fishthe slope of isoprofit lines －p f /p cVertical intercept: ( +L*)/p c.Firm wants to maximize its profits: the isoprofit line must be tangent to the production possibilities setMRT= －p f /p c.Robinson and Friday as ConsumersRobinson and Friday receive wages and dividendsEach person chooses the best bundle of goods that he can afford at the prices p f and p c.Decentralized Resource AllocationEach individual and each firm only has to worry about its own maximization problem.The only facts that need to be communicated among the firms and the consumers are the prices of the goods.。