AD sequentyial Pareto lecture - EQUILIBRIUM AND PARETO...

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EQUILIBRIUM AND PARETO EFFICIENCY Environment: Pure exchange economy with two infinitely lived consumers and one good per period. Utility: 0 log ti it t c β = where 0 1 i << , 1, 2 i = . Endowments: 012 ( , , ,...) iii www where 0 i t w > , i = , 0,1,2,. .. t = . Market structure: With an Arrow-Debreu markets structure, futures markets for goods are open in period 0. Consumers trade futures contracts among themselves. Equilibrium: An Arrow-Debreu equilibrium is a sequence of prices ˆˆˆ ,,, ppp and an allocation 111 , , ,... ccc ; 222 01 2 . . . such that Given , consumer i , i = , chooses , , ,... to solve 0 max log t c = 00 ˆˆ s.t. ii tt t t pc pw ∞∞ == 0 i t c . 12 1 2 t t cc ww +≤+ , = if ˆ 0 t p > , 0,1,2, t = . Characterization of equilibrium using calculus: The Kuhn-Tucker theorem says that , , ,... solves the consumer’s maximization problem if and only if there exists a Lagrange multiplier ˆ 0 i λ such that 1 ˆ ˆ 0 ˆ t t i t p c βλ , 0 = if ˆ 0 i t c >
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00 ˆˆ 0 ii tt t t pw pc ∞∞ == ∑∑ , 0 = if ˆ 0 i λ > . For any t , 0,1,2, t = , 0 1 lim t i c c β = ∞ implies that ˆ 0 i t c > , which implies that ˆ 0 i > . It also implies that ˆ 0 t p > , 0,1,2, t = . Consequently, 012 ˆˆˆ ,,, ppp ; 111 , , ,... ccc ; 222 01 2 . . . is an equilibrium if and only if there exist Lagrange multipliers 12 , λλ , ˆ 0 i > , such that 1 ˆ ˆ ˆ t t i t p c = , 1, 2 i = , 0,1,2, t = t t = , i = 1 2 t t cc ww +=+ , 0,1,2, t = . Pareto efficiency: An allocation , , ,... ; . . . is Pareto efficient if it is feasible, 1 2 t t +≤+ , 0,1,2, t = ., and there exists no other allocation, , , ,... ; . . . that is also feasible and is such that ˆ log log ti it cc ββ > , some i , i = , and ˆ log log , all i , 1,2 i = . Alternatively, An allocation , , ,... ; . . . is Pareto efficient if and only if there exist numbers , αα , ˆ 0 i α and not both 0, such that , , ,... ; . . . solves 11 2 2 max log log αβ + 1 2 s.t t t + ≤+ , 0,1,2, t =
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This note was uploaded on 01/29/2012 for the course ECONOMICS 101 taught by Professor Tikk during the Spring '11 term at University of Toronto- Toronto.

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AD sequentyial Pareto lecture - EQUILIBRIUM AND PARETO...

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