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Chapter Twenty-Nine Exchange

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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

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Exchange u Edgeworth and Bowley devised a diagram, called an Edgeworth box, to show all possible allocations of the available quantities of goods 1 and 2 between the two consumers.

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Starting an Edgeworth Box

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Width =

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Starting an Edgeworth Box Width = Height =

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

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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?

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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.

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

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Width = Height = The Endowment Allocation

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OAOA OBOB 6 8

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OAOA OBOB 6 8 4 6

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OAOA OBOB 6 8 4 6 2 2

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

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More generally, … The Endowment Allocation

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OAOA OBOB The endowment allocation

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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

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Feasible Reallocations OAOA OBOB

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OAOA OBOB

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u All points in the box, including the boundary, represent feasible allocations of the combined endowments.

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Feasible Reallocations 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?

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Adding Preferences to the Box OAOA For consumer A.

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

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Adding Preferences to the Box For consumer B. OBOB

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

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

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Adding Preferences to the Box OAOA For consumer A.

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Adding Preferences to the Box OAOA OBOB

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Edgeworth’s Box OAOA OBOB

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Pareto-Improvement u An allocation of the endowment that improves the welfare of a consumer without reducing the welfare of another is a Pareto-improving allocation. u Where are the Pareto-improving allocations?

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Edgeworth’s Box OAOA OBOB

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

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

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

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Pareto-Improvements

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

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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 over the endowment allocation.

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

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Pareto-Optimality Better for consumer B Better for consumer A

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

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

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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

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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

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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.

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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. An allocation where convex indifference curves are “only just back-to-back” is Pareto-optimal.

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

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Pareto-Optimality OAOA OBOB

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

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Pareto-Optimality u The contract curve is the set of all Pareto-optimal allocations.

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

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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?

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The Core OAOA OBOB The set of Pareto- improving reallocations

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The Core OAOA OBOB

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

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The Core OAOA OBOB Pareto-optimal trades not blocked by A or B

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

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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.

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

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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,...

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Trade in Competitive Markets OAOA For consumer A.

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

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

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Trade in Competitive Markets For consumer B. OBOB

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

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Trade in Competitive Markets u A general equilibrium occurs when prices p 1 and p 2 cause both the markets for commodities 1 and 2 to clear; i.e. and

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Trade in Competitive Markets OAOA OBOB

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OAOA OBOB Can this PO allocation be achieved?

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

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Trade in Competitive Markets OAOA OBOB But

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Trade in Competitive Markets OAOA OBOB and

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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.

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Trade in Competitive Markets OAOA OBOB So this PO allocation cannot be achieved by competitive trading.

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

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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.

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

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

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

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer A

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

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Trade in Competitive Markets OAOA OBOB Budget constraint for consumer B

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Trade in Competitive Markets OAOA OBOB So

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Trade in Competitive Markets OAOA OBOB and

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Trade in Competitive Markets u At the new prices p 1 and p 2 both markets clear; there is a general equilibrium. 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.

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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.

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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.

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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. Second Fundamental Theorem of Welfare Economics

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Second Fundamental Theorem OAOA OBOB The contract curve

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Second Fundamental Theorem OAOA OBOB

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OAOA OBOB Implemented by competitive trading from the endowment .

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

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

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

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Walras’ Law 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.

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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: For consumer B:

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Walras’ Law Summing gives

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Walras’ Law Rearranged, That is,...

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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.

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

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Implications of Walras’ Law 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.

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Implications of Walras’ Law What if, for some positive prices p 1 and p 2, there is an excess quantity supplied of commodity 1? That is, Then implies

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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.

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