微观经济理论英文版课件：ch12 GENERAL EQUILIBRIUM AND WELFARE

1、Chapter 12GENERAL EQUILIBRIUM AND WELFARECopyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.1Perfectly CompetitivePrice SystemWe will assume that all markets are perfectly competitivethere is some large number of homogeneous goods in the economyboth consumption good

2、s and factors of productioneach good has an equilibrium pricethere are no transaction or transportation costsindividuals and firms have perfect information2Law of One PriceA homogeneous good trades at the same price no matter who buys it or who sells itif one good traded at two different prices, dem

3、anders would rush to buy the good where it was cheaper and firms would try to sell their output where the price was higherthese actions would tend to equalize the price of the good3Assumptions of Perfect CompetitionThere are a large number of people buying any one goodeach person takes all prices as

4、 given and seeks to maximize utility given his budget constraintThere are a large number of firms producing each goodeach firm takes all prices as given and attempts to maximize profits4General EquilibriumAssume that there are only two goods, x and yAll individuals are assumed to have identical pref

5、erencesrepresented by an indifference mapThe production possibility curve can be used to show how outputs and inputs are related5Edgeworth Box DiagramConstruction of the production possibility curve for x and y starts with the assumption that the amounts of k and l are fixedAn Edgeworth box shows ev

6、ery possible way the existing k and l might be used to produce x and yany point in the box represents a fully employed allocation of the available resources to x and y6Edgeworth Box DiagramOxOyTotal LaborTotal CapitalACapital for xCapital for yLabor for yLabor for xCapitalin yproductionCapitalin xpr

7、oductionLabor in y productionLabor in x production7Edgeworth Box DiagramMany of the allocations in the Edgeworth box are technically inefficientit is possible to produce more x and more y by shifting capital and labor aroundWe will assume that competitive markets will not exhibit inefficient input c

8、hoicesWe want to find the efficient allocationsthey illustrate the actual production outcomes8Edgeworth Box DiagramWe will use isoquant maps for the two goodsthe isoquant map for good x uses Ox as the originthe isoquant map for good y uses Oy as the originThe efficient allocations will occur where t

9、he isoquants are tangent to one another9Edgeworth Box DiagramOxOyTotal LaborTotal Capitalx2x1y1y2APoint A is inefficient because, by moving along y1, we can increasex from x1 to x2 while holding y constant10Edgeworth Box DiagramOxOyTotal LaborTotal Capitalx2x1y1y2AWe could also increase y from y1 to

10、 y2 while holding x constantby moving along x111Edgeworth Box DiagramOxOyTotal LaborTotal CapitalAt each efficient point, the RTS (of k for l) is equal in bothx and y productionx2x1x4x3y1y2y3y4p4p3p2p112Production Possibility FrontierThe locus of efficient points shows the maximum output of y that c

11、an be produced for any level of xwe can use this information to construct a production possibility frontiershows the alternative outputs of x and y that can be produced with the fixed capital and labor inputs that are employed efficiently13Production Possibility FrontierQuantity of xQuantity of yp4p

12、3p2p1y1y2y3y4x1x2x3x4OxOyEach efficient point of productionbecomes a point on the productionpossibility frontierThe negative of the slope ofthe production possibilityfrontier is the rate of producttransformation (RPT)14Rate of Product TransformationThe rate of product transformation (RPT) between tw

13、o outputs is the negative of the slope of the production possibility frontier15Rate of Product TransformationThe rate of product transformation shows how x can be technically traded for y while continuing to keep the available productive inputs efficiently employed16Shape of the Production Possibili

14、ty FrontierThe production possibility frontier shown earlier exhibited an increasing RPTthis concave shape will characterize most production situationsRPT is equal to the ratio of MCx to MCy17Shape of the Production Possibility FrontierSuppose that the costs of any output combination are C(x,y)along

15、 the production possibility frontier, C(x,y) is constantWe can write the total differential of the cost function as18Shape of the Production Possibility FrontierRewriting, we getThe RPT is a measure of the relative marginal costs of the two goods19Shape of the Production Possibility FrontierAs produ

16、ction of x rises and production of y falls, the ratio of MCx to MCy risesthis occurs if both goods are produced under diminishing returnsincreasing the production of x raises MCx, while reducing the production of y lowers MCythis could also occur if some inputs were more suited for x production than

17、 for y production20Shape of the Production Possibility FrontierBut we have assumed that inputs are homogeneousWe need an explanation that allows homogeneous inputs and constant returns to scaleThe production possibility frontier will be concave if goods x and y use inputs in different proportions21O

18、pportunity CostThe production possibility frontier demonstrates that there are many possible efficient combinations of two goodsProducing more of one good necessitates lowering the production of the other goodthis is what economists mean by opportunity cost22Opportunity CostThe opportunity cost of o

19、ne more unit of x is the reduction in y that this entailsThus, the opportunity cost is best measured as the RPT (of x for y) at the prevailing point on the production possibility frontierthis opportunity cost rises as more x is produced23Concavity of the Production Possibility FrontierSuppose that t

20、he production of x and y depends only on labor and the production functions areIf labor supply is fixed at 100, thenlx + ly = 100The production possibility frontier isx2 + y2 = 100for x,y 024Concavity of the Production Possibility FrontierThe RPT can be calculated by taking the total differential:Th

21、e slope of the production possibility frontier increases as x output increasesthe frontier is concave25Determination ofEquilibrium PricesWe can use the production possibility frontier along with a set of indifference curves to show how equilibrium prices are determinedthe indifference curves represe

22、nt individuals preferences for the two goods26Determination ofEquilibrium PricesQuantity of xQuantity of yU1U2U3y1x1Output will be x1, y1If the prices of x and y are px and py, societys budget constraint is CCCIndividuals will demand x1, y1x1y127Determination ofEquilibrium PricesQuantity of xQuantit

23、y of yy1x1U1U2U3CCThe price of x will rise and the price of y will fallx1y1There is excess demand for x and excess supply of y excesssupplyexcess demand28Determination ofEquilibrium PricesQuantity of xQuantity of yy1x1U1U2U3CCx1y1The equilibrium output willbe x1* and y1*y1*x1*The equilibrium prices

24、will be px* and py*C*C*29Comparative Statics AnalysisThe equilibrium price ratio will tend to persist until either preferences or production technologies changeIf preferences were to shift toward good x, px /py would rise and more x and less y would be producedwe would move in a clockwise direction

25、along the production possibility frontier30Comparative Statics AnalysisTechnical progress in the production of good x will shift the production possibility curve outwardthis will lower the relative price of xmore x will be consumedif x is a normal goodthe effect on y is ambiguous31Technical Progress

26、 in the Production of xQuantity of xQuantity of yU1U2U3x1*The relative price of x will fallMore x will be consumedx2*Technical progress in the production of x will shift the production possibility curve out32General Equilibrium PricingSuppose that the production possibility frontier can be represent

27、ed byx 2 + y 2 = 100Suppose also that the communitys preferences can be represented byU(x,y) = x0.5y0.533General Equilibrium PricingProfit-maximizing firms will equate RPT and the ratio of px /pyUtility maximization requires that34General Equilibrium PricingEquilibrium requires that firms and indivi

28、duals face the same price ratio orx* = y*35The Corn Laws DebateHigh tariffs on grain imports were imposed by the British government after the Napoleonic warsEconomists debated the effects of these “corn laws” between 1829 and 1845what effect would the elimination of these tariffs have on factor pric

29、es?36The Corn Laws DebateQuantity of Grain (x)Quantity of manufactured goods (y)U1U2x0If the corn laws completely prevented trade, output would be x0 and y0y0The equilibrium prices will bepx* and py*37The Corn Laws DebateQuantity of Grain (x)Quantity of manufactured goods (y)x0U1U2y0Removal of the c

30、orn laws will change the prices to px and pyOutput will be x1 and y1x1y1y1x1Individuals will demand x1 and y138The Corn Laws DebateQuantity of Grain (x)Quantity of manufactured goods (y)y1x0U1U2x1y0 x1y1Grain imports will be x1 x1imports of grainThese imports will be financed bythe export of manufac

31、tured goodsequal to y1 y1exportsofgoods39The Corn Laws DebateWe can use an Edgeworth box diagram to see the effects of tariff reduction on the use of labor and capitalIf the corn laws were repealed, there would be an increase in the production of manufactured goods and a decline in the production of

32、 grain40The Corn Laws DebateOxOyTotal LaborTotal CapitalA repeal of the corn laws would result in a movement from p3 to p1 where more y and less x is producedx2x1x4x3y1y2y3y4p4p3p2p141The Corn Laws DebateIf we assume that grain production is relatively capital intensive, the movement from p3 to p1 c

33、auses the ratio of k to l to rise in both industriesthe relative price of capital will fallthe relative price of labor will riseThe repeal of the corn laws will be harmful to capital owners and helpful to laborers42Political Support forTrade PoliciesTrade policies may affect the relative incomes of

34、various factors of productionIn the United States, exports tend to be intensive in their use of skilled labor whereas imports tend to be intensive in their use of unskilled laborfree trade policies will result in rising relative wages for skilled workers and in falling relative wages for unskilled w

35、orkers43Existence of General Equilibrium PricesBeginning with 19th century investigations by Leon Walras, economists have examined whether there exists a set of prices that equilibrates all markets simultaneouslyif this set of prices exists, how can it be found?44Existence of General Equilibrium Pri

36、cesSuppose that there are n goods in fixed supply in this economylet Si (i =1,n) be the total supply of good i availablelet pi (i =1,n) be the price of good iThe total demand for good i depends on all pricesDi (p1,pn) for i =1,n45Existence of General Equilibrium PricesWe will write this demand funct

37、ion as dependent on the whole set of prices (P)Di (P)Walras problem: Does there exist an equilibrium set of prices such thatDi (P*) = Si for all values of i ?46Excess Demand FunctionsThe excess demand function for any good i at any set of prices (P) is defined to beEDi (P) = Di (P) SiThis means that

38、 the equilibrium condition can be rewritten asEDi (P*) = Di (P*) Si = 047Excess Demand FunctionsDemand functions are homogeneous of degree zerothis implies that we can only establish equilibrium relative prices in a Walrasian-type modelWalras also assumed that demand functions are continuoussmall ch

39、anges in price lead to small changes in quantity demanded48Walras LawA final observation that Walras made was that the n excess demand equations are not independent of one anotherWalras law shows that the total value of excess demand is zero at any set of prices49Walras LawWalras law holds for any s

40、et of prices (not just equilibrium prices)There can be neither excess demand for all goods together nor excess supply50Walras Proof of the Existence of Equilibrium PricesThe market equilibrium conditions provide (n-1) independent equations in (n-1) unknown relative pricescan we solve the system for

41、an equilibrium condition?the equations are not necessarily linearall prices must be nonnegativeTo attack these difficulties, Walras set up a complicated proof51Walras Proof of the Existence of Equilibrium PricesStart with an arbitrary set of pricesHolding the other n-1 prices constant, find the equi

42、librium price for good 1 (p1)Holding p1 and the other n-2 prices constant, solve for the equilibrium price of good 2 (p2)in changing p2 from its initial position to p2, the price calculated for good 1 does not need to remain an equilibrium price52Walras Proof of the Existence of Equilibrium PricesUs

43、ing the provisional prices p1 and p2, solve for p3proceed in this way until an entire set of provisional relative prices has been foundIn the 2nd iteration of Walras proof, p2,pn are held constant while a new equilibrium price is calculated for good 1proceed in this way until an entire new set of pr

44、ices is found53Walras Proof of the Existence of Equilibrium PricesThe importance of Walras proof is its ability to demonstrate the simultaneous nature of the problem of finding equilibrium pricesBecause it is cumbersome, it is not generally used todayMore recent work uses some relatively simple tool

45、s from advanced mathematics54Brouwers Fixed-Point TheoremAny continuous mapping F(X) of a closed, bounded, convex set into itself has at least one fixed point (X*) such that F(X*) = X*55Brouwers Fixed-Point Theoremxf (X)11045Any continuous function must cross the 45 lineSuppose that f(X) is a contin

46、uous function defined on the interval 0,1 and that f(X) takes on the values also on the interval 0,1This point of crossing is a “fixed point” because f maps this point (X*) into itselfX*f (X*)56Brouwers Fixed-Point TheoremA mapping is a rule that associates the points in one set with points in anoth

47、er setlet X be a point for which a mapping (F) is definedthe mapping associates X with some point Y = F(X)if a mapping is defined over a subset of n-dimensional space (S), and if every point in S is associated (by the rule F) with some other point in S, the mapping is said to map S into itself57Brou

48、wers Fixed-Point TheoremA mapping is continuous if points that are “close” to each other are mapped into other points that are “close” to each otherThe Brouwer fixed-point theorem considers mappings defined on certain kinds of setsclosed (they contain their boundaries)bounded (none of their dimensio

49、ns is infinitely large)convex (they have no “holes” in them)58Proof of the Existence of Equilibrium PricesBecause only relative prices matter, it is convenient to assume that prices have been defined so that the sum of all prices is equal to 1Thus, for any arbitrary set of prices (p1,pn), we can use

50、 normalized prices of the form59Proof of the Existence of Equilibrium PricesThese new prices will retain their original relative values and will sum to 1These new prices will sum to 160Proof of the Existence of Equilibrium PricesWe will assume that the feasible set of prices (S) is composed of all n

51、onnegative numbers that sum to 1S is the set to which we will apply Brouwers theoremS is closed, bounded, and convexwe will need to define a continuous mapping of S into itself61Free GoodsEquilibrium does not really require that excess demand be zero for every marketGoods may exist for which the mar

52、kets are in equilibrium where supply exceeds demand (negative excess demand)it is necessary for the prices of these goods to be equal to zero“free goods”62Free GoodsThe equilibrium conditions are EDi (P*) = 0 for pi* 0EDi (P*) 0 for pi* = 0Note that this set of equilibrium prices continues to obey W

53、alras law63Mapping the Set of Prices Into ItselfIn order to achieve equilibrium, prices of goods in excess demand should be raised, whereas those in excess supply should have their prices lowered64Mapping the Set of Prices Into ItselfWe define the mapping F(P) for any normalized set of prices (P), s

54、uch that the ith component of F(P) is given byF i(P) = pi + EDi (P)The mapping performs the necessary task of appropriately raising or lowering prices65Mapping the Set of Prices Into ItselfTwo problems exist with this mappingFirst, nothing ensures that the prices will be nonnegativethe mapping must

55、be redefined to beF i(P) = Max pi + EDi (P),0the new prices defined by the mapping must be positive or zero66Mapping the Set of Prices Into ItselfSecond, the recalculated prices are not necessarily normalizedthey will not sum to 1it will be simple to normalize such thatwe will assume that this norma

56、lization has been done67Application of Brouwers TheoremThus, F satisfies the conditions of the Brouwer fixed-point theoremit is a continuous mapping of the set S into itselfThere exists a point (P*) that is mapped into itselfFor this point,pi* = Max pi* + EDi (P*),0 for all i68Application of Brouwer

57、s TheoremThis says that P* is an equilibrium set of pricesfor pi* 0,pi* = pi* + EDi (P*)EDi (P*) = 0For pi* = 0,pi* + EDi (P*) 0EDi (P*) 069A General Equilibrium with Three GoodsThe economy of Oz is composed only of three precious metals: (1) silver, (2) gold, and (3) platinumthere are 10 (thousand)

58、 ounces of each metal availableThe demands for gold and platinum are70A General Equilibrium with Three GoodsEquilibrium in the gold and platinum markets requires that demand equal supply in both markets simultaneously71A General Equilibrium with Three GoodsThis system of simultaneous equations can b

59、e solved asp2/p1 = 2p3/p1 = 3In equilibrium:gold will have a price twice that of silverplatinum will have a price three times that of silverthe price of platinum will be 1.5 times that of gold72A General Equilibrium with Three GoodsBecause Walras law must hold, we knowp1ED1 = p2ED2 p3ED3Substituting

60、 the excess demand functions for gold and silver and substituting, we get73Smiths Invisible Hand HypothesisAdam Smith believed that the competitive market system provided a powerful “invisible hand” that ensured resources would find their way to where they were most valuedReliance on the economic se