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Unformatted text preview: Chapter ThirtyOne Exchange Exchange ◆ Two consumers, A and B. ◆ Their endowments of goods 1 and 2 are ◆ E.g. ◆ The total quantities available ϖ ϖ ϖ A A A = ( , ) 1 2 ϖ ϖ ϖ B B B = ( , ). 1 2 and ϖ A = ( , ) 6 4 ϖ B = ( , ). 2 2 and ϖ ϖ 1 1 6 2 8 A B + = + = ϖ ϖ 2 2 4 2 6 A B + = + = units of good 1 units of good 2. and are 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. Starting an Edgeworth Box Width = ϖ ϖ 1 1 6 2 8 A B + = + = Height = ϖ ϖ 2 2 4 2 6 A B + = + = The dimensions of the box are the quantities available of the goods. 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 . Width = ϖ ϖ 1 1 6 2 8 A B + = + = Height = ϖ ϖ 2 2 4 2 6 A B + = + = The endowment allocation is ϖ A = ( , ) 6 4 ϖ B = ( , ). 2 2 and The Endowment Allocation Width = ϖ ϖ 1 1 6 2 8 A B + = + = Height = ϖ ϖ 2 2 4 2 6 A B + = + = ϖ A = ( , ) 6 4 ϖ B = ( , ) 2 2 The Endowment Allocation ϖ A = ( , ) 6 4 ϖ B = ( , ) 2 2 O A O B 6 8 4 6 2 2 The endowment allocation The Endowment Allocation More generally, … The Endowment Allocation The Endowment Allocation O A O B The endowment allocation ϖ ϖ 1 1 A B + ϖ 2 A ϖ ϖ 2 2 A B + ϖ 1 A ϖ 1 B ϖ 2 B Other Feasible Allocations ◆ denotes an allocation to consumer A. ◆ denotes an allocation to consumer B. ◆ An allocation is feasible if and only if ( , ) x x A A 1 2 ( , ) x x B B 1 2 x x A B A B 1 1 1 1 + ≤ + ϖ ϖ x x A B A B 2 2 2 2 + ≤ + ϖ ϖ . and Feasible Reallocations O A O B ϖ ϖ 1 1 A B + x A 2 ϖ ϖ 2 2 A B + x A 1 x B 1 x B 2 Feasible Reallocations O A O B ϖ ϖ 1 1 A B + x A 2 ϖ ϖ 2 2 A B + x A 1 x B 1 x B 2 Feasible Reallocations ◆ All points in the box, including the boundary, represent feasible allocations of the combined endowments. 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? Adding Preferences to the Box ϖ 2 A ϖ 1 A x A 2 x A 1 M o r e p r e f e r r e d For consumer A. O A Adding Preferences to the Box x B 2 x B 1 M o r e p r e f e r r e d For consumer B. O B ϖ 2 B ϖ 1 B Adding Preferences to the Box ϖ 2 B ϖ 1 B x B 1 x B 2 M o r e p r e f e r r e d For consumer B. O B Adding Preferences to the Box ϖ 2 A ϖ 1 A x A 2 x A 1 O A ϖ 2 B ϖ 1 B x B 1 x B 2 O B Edgeworth’s Box ϖ 2 A ϖ 1 A x A 2 x A 1 O A ϖ 2 B ϖ 1 B x B 1 x B 2 O B ParetoImprovement ◆ An allocation of the endowment that improves the welfare of a consumer without reducing the welfare of another is a Paretoimproving allocation ....
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 Spring '10
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