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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 t i i t t c β = where 0 1 i β < < , 1,2 i = . Endowments: 0 1 2 ( , , ,...) i i i w w w where 0 i t w > , 1,2 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 0 1 2 ˆ ˆ ˆ , , , p p p and an allocation 1 1 1 0 1 2 ˆ ˆ ˆ , , ,... c c c ; 2 2 2 0 1 2 ˆ ˆ ˆ , , ,... c c c such that Given 0 1 2 ˆ ˆ ˆ , , , p p p , consumer i , 1,2 i = , chooses 0 1 2 ˆ ˆ ˆ , , ,... i i i c c c to solve 0 max log t i i t t c β = 0 0 ˆ ˆ s.t. i i t t t t t t p c p w = = 0 i t c . 1 2 1 2 ˆ ˆ t t t t c c w w + + , = if ˆ 0 t p > , 0,1,2, t = . Characterization of equilibrium using calculus: The Kuhn-Tucker theorem says that 0 1 2 ˆ ˆ ˆ , , ,... i i i c c c 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 i i t i t p c β λ , 0 = if ˆ 0 i t c >
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0 0 ˆ ˆ 0 i i t t t t t t p w p c = = , 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 λ > . It also implies that ˆ 0 t p > , 0,1,2, t = . Consequently, 0 1 2 ˆ ˆ ˆ , , , p p p ; 1 1 1 0 1 2 ˆ ˆ ˆ , , ,... c c c ; 2 2 2 0 1 2 ˆ ˆ ˆ , , ,... c c c is an equilibrium if and only if there exist Lagrange multipliers 1 2 ˆ ˆ , λ λ , ˆ 0 i λ > , such that 1 ˆ ˆ ˆ t i i t i t p c β λ = , 1,2 i = , 0,1,2, t = 0 0 ˆ ˆ i i t t t t t t p c p w = = = , 1,2 i = 1 2 1 2 ˆ ˆ t t t t c c w w + = + , 0,1,2, t = . Pareto efficiency: An allocation 1 1 1 0 1 2 ˆ ˆ ˆ , , ,... c c c ; 2 2 2 0 1 2 ˆ ˆ ˆ , , ,... c c c is Pareto efficient if it is feasible, 1 2 1 2 ˆ ˆ t t t t c c w w + + , 0,1,2, t = ., and there exists no other allocation, 1 1 1 0 1 2 , , ,... c c c ; 2 2 2 0 1 2 , , ,... c c c that is also feasible and is such that 0 0 ˆ log log t i t i i t i t t t c c β β = = > , some i , 1,2 i =
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