Welfare Economics: Lecture 12

Transcription

1 Welfare Economics: Lecture 12 Ram Singh Course 001 October 20, 2014 Ram Singh: (DSE) Welfare Economics October 20, / 16

2 Fair Vs Efficient Question 1 What is a fair allocation? 2 Is a fair allocation also an efficient allocation? 3 Is a fair allocation Pareto efficient? 4 Does a competitive market lead to fair allocations? Ram Singh: (DSE) Welfare Economics October 20, / 16

3 Fairness: Two Definitions Consider a N M pure exchange economy. Let, (e 1,..., e N ) be the vector of initial endowments. Definition Allocation x = (x 1,..., x n ) is Fair if it is equal division of endowments. That is, if [ ] ( i, j N) x i = x j = N i=1 ei N. Definition Allocation x = (x 1,..., x n ) is Fair if it is Non-envious/envy-free. That is, if ( i, j N)[x i R i x j ], i.e., ( i, j N)[u i (x i ) u i (x j )]. Are these definitions equivalent to each other? Ram Singh: (DSE) Welfare Economics October 20, / 16

4 Pareto Optimal Allocations I Theorem An allocation ˆx = (ˆx 1,..., ˆx I ) is PO iff: For some utility levels Ū2,..., ŪI, ˆx = (ˆx 1,..., ˆx I ) is a solution of the following s.t. x i 0, and See MWG, Section 16.F max {u 1 (x 1 )} x 1,...,x I u i (x i ) Ūi, for all i > 1 I x i = I e i i=1 i=1 Ram Singh: (DSE) Welfare Economics October 20, / 16

5 Fair Vs Pareto Efficient Question 1 Is an Equal division allocation Non-envious? 2 Is a Non-envious allocation also an Equal division allocation? 3 Is an Equal division allocation Pareto Efficient? 4 Is a Non-envious allocation Pareto Efficient? Ram Singh: (DSE) Welfare Economics October 20, / 16

6 Fair but Not Efficient Consider a 3 3 exchange economy. Let u 1 = 3x 1 + 2y 1 + z 1 u 2 = 2x 2 + y 2 + 3z 2 u 3 = x 1 + 3y 1 + 2z 1 e 1 = e 2 = e 3 = (1, 1, 1) Consider an allocation y = (y 1, y 2, y 3 ), where y 1 = (3, 2 3, 0), y2 = (0, 0, 2) and y 3 = (0, 7 3, 1) Ram Singh: (DSE) Welfare Economics October 20, / 16

7 Can Fair be Efficient? I Consider a 2 2 pure exchange economy: Let The goods are; x and y. u 1 (.) = x α.y 1 α and u 2 (.) = x β.y 1 β, and α = β = 1 2 the total initial endowment vector is ( x, ȳ) >> (0, 0). x 1 and x 2 denote the amounts of good x allocated to individuals 1 and 2, resp. y 1 and y 2 denote the amounts of good y allocated to individuals 1 and 2, resp. Ram Singh: (DSE) Welfare Economics October 20, / 16

8 Can Fair be Efficient? II Clearly MRS 1 = y 1 x 1 MRS 2 = y 2 x 2 = ȳ y 1 x x 1 So the set of PO allocations is solution to y 1 x 1 = y 2 x 2 = ȳ y 1 x x 1 But, ( y1 = ȳ y ) ( 1 y1 = ȳ ) x 1 x x 1 x 1 x Ram Singh: (DSE) Welfare Economics October 20, / 16

9 Can Fair be Efficient? III Question Is fair (equal) division of endowments a P.O allocation? Consider a 2 2 exchange economy: The goods are; x and y. u 1 (x 1, y 1 ) = x 1.y 1 and u 2 (x 2, y 2 ) = x 2.y 2 the total initial endowment vector is ( x, ȳ) >> (0, 0). So the set of PO allocations is solution to MRS 1 = MRS 2, i.e., u(.) x 1 u(.) y 1 = u(.) x 2 u(.) y 2 Ram Singh: (DSE) Welfare Economics October 20, / 16

10 Can Fair be Efficient? IV Under equal division allocation, i.e., when x 1 = x 2 and y 1 = y 2, we have u(.) x 1 u(.) y 1 = u(.) x 2 u(.) y 2 Ram Singh: (DSE) Welfare Economics October 20, / 16

11 Fairness under Markets Question Is competitive equilibrium allocation fair? Proposition When preferences are strongly monotonic and initial allocation is Equal, the competitive equilibrium is fair (non-envious) Suppose, e 1 = e 2 =... = e n, and (x 1,..., x n ) is a Walrasian equilibrium allocation. Suppose, ( i, j {1,..., n})[u i (x i ) < u i (x j )]. Then, (x 1,..., x n ) not a WEA. So, the following holds. ( i, j {1,..., n})[u i (x i ) u i (x j )]. Ram Singh: (DSE) Welfare Economics October 20, / 16

12 Effect of Endowments I Question Does an increase in endowment make the person better off? Answer is Yes, for well behaved utilities. Consider a 2 2 pure exchange economy: The goods are; x and y. u 1 (.) = x α.y 1 α and u 2 (.) = x β.y 1 β, α, β (0, 1) and α β the initial endowments are e 1 (.) = ( x 1, ȳ 1 ) >> (0, 0) and e 2 (.) = ( x 2, ȳ 2 ) >> (0, 0). Ram Singh: (DSE) Welfare Economics October 20, / 16

13 Effect of Endowments II Question Find out the competitive equilibrium prices and allocations. How do the competitive equilibrium allocations change with initial endowments For any given price vector p = (p 1, p 2 ) >> (0, 0), you can check that demands are: x 1 (p) = α(p 1 x 1 + p 2 ȳ 1 ) p 1 y 1 (p) = (1 α)(p 1 x 1 + p 2 ȳ 1 ) p 2 x 2 (p) = β(p 1 x 2 + p 2 ȳ 2 ) p 1 y 2 (p) = (1 β)(p 1 x 2 + p 2 ȳ 2 ) p 2 Ram Singh: (DSE) Welfare Economics October 20, / 16

14 Effect of Endowments III Market Clearance for 1st good implies: x 1 (p) + x 2 (p) = x 1 + x 2 α(p 1 x 1 + p 2 ȳ 1 ) + β(p 1 x 2 + p 2 ȳ 2 ) p 1 p 1 = x 1 + x 2 αp 1 x 1 + αp 2 ȳ 1 + βp 1 x 2 + βp 2 ȳ 2 = p 1 x 1 + p 1 x 2 Re-arranging the last equality, we get p 2 (αȳ 1 + βȳ 2 ) = [(1 α) x 1 + (1 β) x 2 ]p 1, i.e., p1 p2 = (αȳ 1 + βȳ 2 ) (1 α) x 1 + (1 β) x 2 (1) Ram Singh: (DSE) Welfare Economics October 20, / 16

15 Effect of Endowments IV Let p2 = 1. In view of (1), the equilibrium demands can be written as ( ) x1 (p (1 α) x 1 + (1 β) x 2 ) = α x 1 + ȳ 1 (αȳ 1 + βȳ 2 ) ( ) y1 (p (αȳ 1 + βȳ 2 ) ) = (1 α) x 1 + ȳ 1 (1 α) x 1 + (1 β) x 2 ( ) x2 (p (1 α) x 1 + (1 β) x 2 ) = β x 2 + ȳ 2 (αȳ 1 + βȳ 2 ) ( ) y2 (p (αȳ 1 + βȳ 2 ) ) = (1 β) x 2 + ȳ 2 (1 α) x 1 + (1 β) x 2 From (1) ( p 1 p ) 2 = (αȳ 1 + βȳ 2 )(1 α) x 1 [(1 α) x 1 + (1 β) x 2 ] 2 < 0. Ram Singh: (DSE) Welfare Economics October 20, / 16

16 Effect of Endowments V Now, you can verify the following: x 1 (1 α) = α + ȳ 1 x 1 [(αȳ 1 + βȳ 2 )] > 0 y 1 = (1 α) (αȳ 1 + βȳ 2 )(1 β) x 2 x 1 [(1 α) x 1 + (1 β) x 2 ] 2 > 0 x 2 (1 α) = βȳ 2 x 1 (αȳ 1 + βȳ 2 ) > 0 y2 (1 α)(αȳ 1 + βȳ 2 ) = x 2 (1 β) x 1 [(1 α) x 1 + (1 β) x 2 ] 2 < 0 Ram Singh: (DSE) Welfare Economics October 20, / 16