1. Chapter Thirty Exchange Tausch 交换

2. Hu Jingbei’s declaration to copyright May 17, 2007 u This ppt-file may come from an American source which is not known to me at the moment. It based on Varian’s Intermediate Microeconomics, 5th ed. In many places of this file, I made changes for my lectures with Varian’s 6th ed. I will be grateful to persons who may show me the original sources of this file.

3. Contents u The Edgeworth Box u Pareto improving trade u Pareto Efficient Allocation u The Core ( 核 ) u Trade in a competitive market u Walras’ Law u First Fundamental Theorem of Welfare Economics ( 福利经济学第一定理 ) u Second Fundamental Theorem of Welfare Economics ( 福利经济学第二定理 )

4. Exchange u Two consumers, A and B. u Their endowments of goods 1 and 2 are u E.g. u The total quantities available and units of good 1 units of good 2. and are

5. Exchange u The English economist Edgeworth devised a diagram, called an Edgeworth box (Edgeworth Diagramm), to show all possible allocations of the available quantities of goods 1 and 2 between the two consumers.

6. Starting an Edgeworth Box Width = Height = The dimensions of the box are the quantities available of the goods.

7. Feasible Allocations u What allocations of the 8 units of good 1 and the 6 units of good 2 are feasible? u How can all of the feasible allocations be depicted by the Edgeworth box diagram? u One feasible allocation is the before- trade allocation; i.e. the endowment allocation (urspaengliche Allokation).

8. Width = Height = The endowment allocation is and The Endowment Allocation

9. Width = Height = The Endowment Allocation

10. OAOA OBOB 6 8

11. OAOA OBOB 6 8 4 6

12. OAOA OBOB 6 8 4 6 2 2

13. OAOA OBOB 6 8 4 6 2 2 The endowment allocation The Endowment Allocation

14. More generally, … The Endowment Allocation

15. OAOA OBOB The endowment allocation

16. Other Feasible Allocations u denotes an allocation to consumer A. u denotes an allocation to consumer B. u An allocation is feasible if and only if and

17. Feasible Reallocations OAOA OBOB

18. OAOA OBOB

19. u All points in the box, including the boundary, represent feasible allocations of the combined endowments. u Which allocations will be blocked by one or both consumers? u Which allocations make both consumers better off?

20. Adding Preferences to the Box OAOA For consumer A.

21. Adding Preferences to the Box More preferred For consumer A. OAOA

22. Adding Preferences to the Box For consumer B. OBOB

23. Adding Preferences to the Box More preferred For consumer B. OBOB

24. Adding Preferences to the Box More preferred For consumer B. OBOB

25. Adding Preferences to the Box OAOA For consumer A.

26. Adding Preferences to the Box OAOA OBOB

27. Edgeworth’s Box OAOA OBOB

28. Pareto-Improvement u An allocation of the endowment that improves the welfare of at least a consumer without reducing the welfare of another is a Pareto- improving allocation (Allokation der Pareto-Verbesserung) u Where are the Pareto-improving allocations?

29. Edgeworth’s Box OAOA OBOB

30. Pareto-Improvements OAOA OBOB The set of Pareto- improving allocations

31. Pareto-Improvements u Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations (Allokationen der Pareto- Verbessrung). u But which particular Pareto- improving allocation will be the outcome of trade?

32. Pareto-Improvements OAOA OBOB The set of Pareto- improving reallocations

33. Pareto-Improvements

35. Trade improves both A’s and B’s welfares. This is a Pareto-improvement over the endowment allocation.

36. Pareto-Improvements New mutual gains-to-trade region is the set of all further Pareto- improving reallocations. Trade improves both A’s and B’s welfares. This is a Pareto-improvement ( 帕累托改进 )over the endowment allocation.

37. Pareto-Improvements Further trade cannot improve both A and B’s welfares.

38. Pareto-Optimality (Pareto- Optimum, 帕累托最优 ) Better for consumer B Better for consumer A

39. Pareto-Optimality A is strictly better off but B is strictly worse off B is strictly better off but A is strictly worse off

40. Pareto-Optimality A is strictly better off but B is strictly worse off B is strictly better off but A is strictly worse off Both A and B are worse off

41. Pareto-Optimality The allocation is Pareto-optimal since the only way one consumer’s welfare can be increased is to decrease the welfare of the other consumer.

42. Pareto-Optimality The allocation is Pareto-optimal since the only way one consumer’s welfare can be increased is to decrease the welfare of the other consumer. At the Pareto-optimality both of the Marginal Rates of Substitution must be equal.

43. Pareto-Optimality u Where are all of the Pareto-optimal allocations of the endowment?

44. Pareto-Optimality u Condition of the Pareto-Optimality: u MRS A = MRS B u To explain the condition, we can assume one consumer’s utility constant and see how to maximize another’s utility. With both the budget constraints, a Lagrange method can be used to show the condition.

45. Pareto-Optimality OAOA OBOB

46. OAOA OBOB All the allocations marked by a are Pareto-optimal.

47. Pareto-Optimality u The contract curve ( 契约线 ) is the set of all Pareto-optimal allocations.

48. Pareto-Optimality OAOA OBOB All the allocations marked by a are Pareto-optimal. The contract curve

49. Pareto-Optimality u But to which of the many allocations on the contract curve will consumers trade? u That depends upon how trade is conducted. u In perfectly competitive markets? By one-on-one bargaining?

50. The Core (Kore, 核 ) OAOA OBOB The set of Pareto- improving reallocations

51. The Core OAOA OBOB

52. OAOA OBOB Pareto-optimal trades blocked by B Pareto-optimal trades blocked by A

53. The Core OAOA OBOB Pareto-optimal trades not blocked by A or B are the core.

54. The Core u The core is the set of all Pareto- optimal allocations that are welfare- improving for both consumers relative to their own endowments. u Rational trade should achieve a core allocation.

55. The Core u But which core allocation? u Again, that depends upon the manner in which trade is conducted.

56. Trade in Competitive Markets u Consider trade in perfectly competitive markets. u Each consumer is a price-taker trying to maximize her own utility given p 1, p 2 and her own endowment. That is,...

57. Trade in Competitive Markets OAOA consumer A’s budget line:

58. Trade in Competitive Markets u So given p 1 and p 2, consumer A’s net demands for commodities 1 and 2 are and

59. Trade in Competitive Markets u And, similarly, for consumer B …

60. Trade in Competitive Markets consumer B’s budget line: OBOB

61. Trade in Competitive Markets u So given p 1 and p 2, consumer B’s net demands for commodities 1 and 2 are and

62. Trade in Competitive Markets u A general equilibrium (Allgemeines Gleichgewicht, 一般均衡 ) occurs when prices p 1 and p 2 cause both the markets for commodities 1 and 2 to clear; i.e. and

63. Trade in Competitive Markets OAOA OBOB

64. OAOA OBOB Can this PO allocation be achieved?

65. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

66. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

67. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

68. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

69. Trade in Competitive Markets OAOA OBOB But

70. Trade in Competitive Markets OAOA OBOB and

71. Trade in Competitive Markets u So at the given prices p 1 and p 2 there is an – excess supply of commodity 1 – excess demand for commodity 2. u Neither market clears so the prices p 1 and p 2 do not cause a general equilibrium.

72. Trade in Competitive Markets OAOA OBOB So this PO allocation cannot be achieved by the given prices.

73. Trade in Competitive Markets OAOA OBOB How can PO allocations be achieved by competitive trading?

74. Trade in Competitive Markets u Since there is an excess demand for commodity 2, p 2 will rise. u Since there is an excess supply of commodity 1, p 1 will fall. u The slope of the budget constraints is - p 1 /p 2 so the budget constraints will pivot about the endowment point and become less steep.

75. Trade in Competitive Markets OAOA OBOB Which PO allocations can be achieved by competitive trading?

76. Trade in Competitive Markets OAOA OBOB Which PO allocations can be achieved by competitive trading?

77. Trade in Competitive Markets OAOA OBOB Which PO allocations can be achieved by competitive trading?

78. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

79. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

80. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

81. Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

82. Trade in Competitive Markets OAOA OBOB So

83. Trade in Competitive Markets OAOA OBOB and

84. Trade in Competitive Markets u At the new prices p 1 and p 2 both markets clear; there is a general equilibrium. u Question 1: Are prices p 1 and p 2 determined independently at equilibrium? u Question 2: Does the equilibrium exist?

85. Walras’ Law Walras’sches Gesetz 瓦尔拉斯法则 u Walras’ Law is an identity; i.e. a statement that is true for any positive prices (p 1,p 2 ), whether these are equilibrium prices or not.

86. Terms of demand u w: goods in initial endowment allocation u x: goods in the final allocation u x: gross demand (Bruttonachfrage) for the u good x u (x – w): net demand (Nettonachfrage) for u the good x, excess demand u (Ueberschussnachfrage) u (x A + x B ): aggregate demand (aggregierte u Nachfrage)

87. Walras’ Law u Every consumer’s preferences are well-behaved so, for any positive prices (p 1,p 2 ), each consumer spends all of his budget. u For consumer A (budget constraint): For consumer B:

88. Walras’ Law Summing gives

89. Walras’ Law Rearranged, The first parenthesis can be seen as the excess demand for good 1 and the second one as the excess demand for good 2.

90. Walras’ Law This says that the summed market value of excess demands is zero for any positive prices p 1 and p 2 -- this is Walras’ Law.

91. Walras’ Law u We often name z the excess demand function and, because x is a demand function with p, we get u It stands for any p(p 1, p 2 ) > 0. It is the Walras’ law.

92. Implications of Walras’ Law Suppose the market for commodity A is in equilibrium; that is, Then implies

93.  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 determined at equilibrium.  I.e., the general equilibrium determines only relative prices. Implications of Walras’ Law

94. What if, for some positive prices p 1 and p 2, there is an excess quantity supplied of commodity 1? That is, Then implies

95. Implications of Walras’ Law So a second implication of Walras’ Law for a two-commodity exchange economy is that an excess supply in one market implies an excess demand in the other market. It also means that not any p(p 1, p 2 ) > 0 can ensure z 1 = 0 and z 2 = 0 at the same time, even if any p(p 1, p 2 ) > 0 can ensure p 1 z 1 +p 2 z 2 =0

96. Implications of Walras’ Law u Because the equilibrium prices are prices which lead demand for and supply of every good to be equal at the same time, the Walras’ law does not mean there must be at least a set of prices p E (p E 1, p E 2 ) > 0 which could result in z 1 = 0 and z 2 = 0 simultaneously. u Therefore: We must find the equilibrium prices p E (p E 1, p E 2 ) > 0 if it/they would exist.

97. Relative Prices u Walras’ law means there are only k-1 prices that can be determined independently in an economy with k goods. That means also, we can only get p i for good i that u p i = p i (p j ) i=1,2,…,k, i≠j u Only after p j is determined, can we get all p i. u p j is called a numeraire price. Good j is the numeraire.

98. Relative prices u All prices except the numeraire price are prices relative to the numeraire price. Making it as 1, all prices are relative to the numeraire. u That is: All prices are relative prices. There are no absolute prices. u Changing p j from being 1 to being 2, all other prices will be double. But the exchange relations between each pair of all kinds of goods remain the same. u In the economy with paper money, governments can multiply the numeraire price. We refer it to change in price level or in absolute price level.

99. Solve for General Equilibrium u Individual optimization problem u X i j =X i j (P 1, P 2, M j ), i=1, 2; j=A, B u M j =M j (P 1, P 2,  j 1,  j 2 ) u X i j =X i j (P 1, P 2 ) u Market clearing for market 1: X 1 A (P 1,P 2 )+ X 1 B (P 1,P 2 ) =  1 A +  1 B u Can solve for equilibrium P 1 /P 2 ? u Can solve for demand at equilibrium?

100. Existence of Equilibrium u Require that the aggregate excess demand functions be a continuous function, which in turn require that –Each individual’s demand function be continuous, or –Each consumer is small relative to the market. The second requirement is just one of conditions for perfectly competitive makerts. Therefore: Competitive markets will necessarily reach equilibriums.

101. Existence of Equilibrium u Example: u The endowment allocation u Apple Orange u (good 1) (good 2) u A w a 1 w a 2 u B w b 1 w b 2 u (The whole example will come…)

102. Trade in Competitive Markets u Trading in competitive markets achieves a particular Pareto-optimal allocation of the endowments. u This is an example of the First Fundamental Theorem of Welfare Economics.

103. First Fundamental Theorem of Welfare Economics u Given that consumers’ preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy’s endowment. u It means that competitive equilibrium will necessarily be Pareto efficient.

104. The Equilibrium with Monopoly Figure 30.5 in textbook. It was taken off to reduce the size of this ppt.

105. The Equilibrium with Monopoly Figure 30.6 in textbook. It was taken off to reduce the size of this ppt.

106. Implicit Assumptions u No consumption externality ( 外部效应 ) u Agents behave competitively u Equilibrium exists u This theorem gives a general mechanism—the competitive market— to ensure Pareto efficiency or Pareto optimality.

107. Market Socialism: The Debate u Lange vs. Hayek u Socialism with economic planning –State-ownership –Government chooses prices to clear markets u Advantage of competitive markets –Information

108. Second Fundamental Theorem of Welfare Economics u The 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 consumers.

109. u Given 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. u That means any Pareto efficient or optimal allocation is the competitive equilibrium allocation. Second Fundamental Theorem of Welfare Economics

110. Second Fundamental Theorem OAOA OBOB The contract curve

111. Second Fundamental Theorem OAOA OBOB

112. OAOA OBOB Implemented by competitive trading from the endowment .

113. Second Fundamental Theorem OAOA OBOB Can this allocation be implemented by competitive trading from  ?

114. Second Fundamental Theorem OAOA OBOB Can this allocation be implemented by competitive trading from  ? No.

115. Second Fundamental Theorem OAOA OBOB But this allocation is implemented by competitive trading from .

116. The Assumption about Convexity

117. Implications of the 2 nd Theorem u Market equilibrium is distributionally neutral. u Redistribution can be carried out for endowments. u The market plays only the allocation role.