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Edgeworth Box: Feasible Allocations, Pareto Improvements, and Pareto-Optimality

Explore the concept of feasible allocations, Pareto improvements, and Pareto-optimality using the Edgeworth box diagram. Discover how trade can improve both consumers' welfare and identify the set of Pareto-optimal allocations.

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Edgeworth Box: Feasible Allocations, Pareto Improvements, and Pareto-Optimality

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  1. Chapter Twenty-Nine Exchange

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

  3. Exchange • 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.

  4. Starting an Edgeworth Box

  5. Starting an Edgeworth Box Width =

  6. Starting an Edgeworth Box Height = Width =

  7. Starting an Edgeworth Box Height = The dimensions ofthe box are thequantities availableof the goods. Width =

  8. Feasible Allocations • What allocations of the 8 units of good 1 and the 6 units of good 2 are feasible? • How can all of the feasible allocations be depicted by the Edgeworth box diagram?

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

  10. The Endowment Allocation The endowmentallocation is Height = and Width =

  11. The Endowment Allocation Height = Width =

  12. The Endowment Allocation OB 6 OA 8

  13. The Endowment Allocation OB 6 4 OA 6 8

  14. The Endowment Allocation 2 OB 2 6 4 OA 6 8

  15. The Endowment Allocation 2 OB 2 6 Theendowmentallocation 4 OA 6 8

  16. The Endowment Allocation More generally, …

  17. The Endowment Allocation OB Theendowmentallocation OA

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

  19. Feasible Reallocations OB OA

  20. Feasible Reallocations OB OA

  21. Feasible Reallocations • All points in the box, including the boundary, represent feasible allocations of the combined endowments.

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

  23. Adding Preferences to the Box For consumer A. OA

  24. Adding Preferences to the Box For consumer A. More preferred OA

  25. Adding Preferences to the Box For consumer B. OB

  26. Adding Preferences to the Box For consumer B. More preferred OB

  27. Adding Preferences to the Box For consumer B. OB More preferred

  28. Adding Preferences to the Box For consumer A. OA

  29. Adding Preferences to the Box OB OA

  30. Edgeworth’s Box OB OA

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

  32. Edgeworth’s Box OB OA

  33. Pareto-Improvements OB OA The set of Pareto-improving allocations

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

  35. Pareto-Improvements OB OA The set of Pareto-improving reallocations

  36. Pareto-Improvements

  37. Pareto-Improvements

  38. Pareto-Improvements Tradeimproves bothA’s and B’s welfares.This is a Pareto-improvementover the endowment allocation.

  39. Pareto-Improvements New mutual gains-to-trade region is the set of all further Pareto- improving reallocations. Tradeimproves bothA’s and B’s welfares.This is a Pareto-improvementover the endowment allocation.

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

  41. Pareto-Optimality Better forconsumer A Better forconsumer B

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

  43. Pareto-Optimality A is strictly better off but B is strictly worse off B is strictly betteroff but A is strictlyworse off

  44. Pareto-Optimality Both A andB are worseoff A is strictly better off but B is strictly worse off B is strictly betteroff but A is strictlyworse off

  45. Pareto-Optimality Both A andB are worseoff A is strictly better off but B is strictly worse off B is strictly betteroff but A is strictlyworse off Both Aand B are worse off

  46. Pareto-Optimality The allocation isPareto-optimal since theonly way one consumer’s welfare can be increased is todecrease the welfare of the otherconsumer.

  47. Pareto-Optimality An allocation where convexindifference curves are “only just back-to-back” is Pareto-optimal. The allocation isPareto-optimal since theonly way one consumer’s welfare can be increased is todecrease the welfare of the otherconsumer.

  48. Pareto-Optimality • Where are all of the Pareto-optimal allocations of the endowment?

  49. Pareto-Optimality OB OA

  50. Pareto-Optimality All the allocations marked bya are Pareto-optimal. OB OA

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