microlower.jpg © 2010 W. W. Norton & Company, Inc. microtitle.jpg microedition.jpg varianname.jpg 31 Exchange microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Exchange uTwo consumers, A and B. uTheir endowments of goods 1 and 2 are uE.g. uThe total quantities available and and units of good 1 units of good 2. and are microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Exchange uEdgeworth 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Starting an Edgeworth Box microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Starting an Edgeworth Box Width = microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Starting an Edgeworth Box Width = Height = microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Starting an Edgeworth Box Width = Height = The dimensions of the box are the quantities available of the goods. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Allocations uWhat 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Allocations uWhat 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. the endowment allocation. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Width = Height = The endowment allocation is and The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Width = Height = The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› OA OB 6 8 The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› OA OB 6 8 4 6 The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› OA OB 6 8 4 6 2 2 The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› OA OB 6 8 4 6 2 2 The endowment allocation The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› More generally, … The Endowment Allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Endowment Allocation OA OB The endowment allocation microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Other Feasible Allocations u denotes an allocation to consumer A. u denotes an allocation to consumer B. uAn allocation is feasible if and only if and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Reallocations OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Reallocations OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Reallocations uAll points in the box, including the boundary, represent feasible allocations of the combined endowments. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Feasible Reallocations uAll points in the box, including the boundary, represent feasible allocations of the combined endowments. uWhich allocations will be blocked by one or both consumers? uWhich allocations make both consumers better off? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box OA For consumer A. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box For consumer A. OA microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box For consumer B. OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box For consumer B. OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box For consumer B. OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box OA For consumer A. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Adding Preferences to the Box OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Edgeworth’s Box OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvement uAn 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Edgeworth’s Box OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements OA OB The set of Pareto- improving allocations microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements uSince 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements OA OB The set of Pareto- improving reallocations microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements Trade improves both A’s and B’s welfares. This is a Pareto-improvement over the endowment allocation. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Improvements Further trade cannot improve both A and B’s welfares. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality Better for consumer B Better for consumer A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality A is strictly better off but B is strictly worse off microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality A is strictly better off but B is strictly worse off B is strictly better off but A is strictly worse off microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 Both A and B are worse off microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality uWhere are all of the Pareto-optimal allocations of the endowment? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality OA OB All the allocations marked by a are Pareto-optimal. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality uThe contract curve is the set of all Pareto-optimal allocations. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality OA OB All the allocations marked by a are Pareto-optimal. The contract curve microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Pareto-Optimality uBut 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 one-on-one bargaining? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core OA OB The set of Pareto- improving reallocations microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core OA OB Pareto-optimal trades blocked by B Pareto-optimal trades blocked by A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core OA OB Pareto-optimal trades not blocked by A or B microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core OA OB Pareto-optimal trades not blocked by A or B are the core. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core uThe 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› The Core uBut which core allocation? uAgain, that depends upon the manner in which trade is conducted. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uConsider 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, ... microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA For consumer A. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uSo given p1 and p2, consumer A’s net demands for commodities 1 and 2 are and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uAnd, similarly, for consumer B … microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets For consumer B. OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uSo given p1 and p2, consumer B’s net demands for commodities 1 and 2 are and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uA general equilibrium occurs when prices p1 and p2 cause both the markets for commodities 1 and 2 to clear; i.e. and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Can this PO allocation be achieved? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer B microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer B microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB But microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uSo 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB So this PO allocation cannot be achieved by competitive trading. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Which PO allocations can be achieved by competitive trading? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uSince there is an excess demand for commodity 2, p2 will rise. uSince there is 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Which PO allocations can be achieved by competitive trading? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Which PO allocations can be achieved by competitive trading? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Which PO allocations can be achieved by competitive trading? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer A microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer B microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB Budget constraint for consumer B microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB So microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets OA OB and microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Trade in Competitive Markets uAt the new prices p1 and p2 both markets clear; there is a general equilibrium. uTrading in competitive markets achieves a particular Pareto-optimal allocation of the endowments. uThis is an example of the First Fundamental Theorem of Welfare Economics. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› First Fundamental Theorem of Welfare Economics uGiven that consumers’ preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy’s endowment. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem of Welfare Economics uThe 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem of Welfare Economics 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB The contract curve microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB Implemented by competitive trading from the endowment w. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB Can this allocation be implemented by competitive trading from w? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB Can this allocation be implemented by competitive trading from w? No. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Second Fundamental Theorem OA OB But this allocation is implemented by competitive trading from q. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Walras’ Law 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Walras’ Law uEvery consumer’s preferences are well-behaved so, for any positive prices (p1,p2), each consumer spends all of his budget. uFor consumer A: For consumer B: microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Walras’ Law Summing gives microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Walras’ Law Rearranged, That is, ... microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Walras’ Law This says that the summed market value of excess demands is zero for any positive prices p1 and p2 -- this is Walras’ Law. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Implications of Walras’ Law Suppose the market for commodity A is in equilibrium; that is, Then implies microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Implications of Walras’ Law What if, for some positive prices p1 and p2, there is an excess quantity supplied of commodity 1? That is, Then implies microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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.