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# Class Note for EconS 500 with Professor Gibson at WSU

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Washington State University taught by a professor in Fall. Since its upload, it has received 11 views.

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Date Created: 02/06/15

EQUILIBRIUM AND PARETO EFFICIENCY Environment Pure exchange economy with two in nitely lived consumers and one good per period Utility 20 it logc where 0 lt E lt1 139 12 Endowments wgwiw where w gt0 il2 t0l2 Market structure With an ArrowDebreu 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 e l 72 and an allocation quot1 quot1 quot1 quot2 quot2 quot2 c clc2 c c czsuchthat 8 8 to solve 0 Given 0f71f72consumer 139 il2chooses 80 1 2 max 20 it log 0 DO A O A st 220 pic S 220 ptw c220 o ajafswjwf if 2 gt0 t0l2 Characterization of equilibrium using calculus The KuhnTucker theorem says that 8 5 8 2 solves the consumer s maximization problem if and only if there exists a Lagrange multiplier 2 0 such that ELEM Z0 c 20 0 if 1 gt0 gt0 l A A For any t I 012 11m 00 1mp11es that c gt 0 wh1ch1mp11es that A gt 0 It c also implies that t gt0 t0l2 Consequently of71f72 clclc 8 512 522 is an equilibrium if and only if there exist Lagrange multipliers 21 ii i gt 0 such that o fif72i12t012 ct 2027 220 w l i1 2 0 85 wtlwft012 Pareto ef ciency An allocation 8 8118i 52512522 is Pareto efficient if it is feasible ajafswjwft012 and there exists no other allocation 51511521 502512522 that is also feasible and is such that 20 1 log gt 20 it log j some 139 139 12 and 20 f mg 2 20 f 1oga all 139 139 12 Alternatively An allocation 8 811 81 82512 52 is Pareto efficient if and only if there exist 239 2 A A1 A1 A1 A2A2A2 04 20 and not both 0 suchthat coclc2 c0clcz solves numbers a1 a2 A no 2 1 A no 2 2 max alz l logct otZZZH 2 logct st 021c22 w21w22 t0l2 c220 Note It is easy to show that if an allocation solves the above social planner s problem it satis es the rst de nition of Pareto ef ciency It is a little more dif cult to show that if an allocation satis es the rst de nition of Pareto ef ciency there exist welfare weights 011 612 such that the allocation solves the social planner s problem Characterization of Pareto ef ciency using calculus The KuhnTucker theorem says that 51 51 51 39 52 52 52 solves the social planner s 0 1 pm 0 1 2 problem if and only if there eXists a Lagrange multipliers A0 711 712 7339 Z 0 such that A l A A 01 71392S0 01fcjgt0 l w21w22 52152220 0 if 7gt0 l A A For any t 1 012 11m 00 implies that C gt 0 wh1ch1mp11es that 7139 gt 0 gt0 6 Consequently 51 811 8 52 512 522 is a Pareto ef cient allocation if and only if there exist Lagrange multipliers 710 711 712 7339 gt 0 such that o emf 72 112 1012 2 o 521522wtlw22 t0l2 Note Since 64 gt 0 for at least one 1 1 12 we know that for that consumer 1 5 gt 0 for all t t 01 2 and consequently that 7339 gt 0 If one of the welfare weights 0 equals 0 then 8 0 We can imagine the rst order conditions for that consumer 1 as being satis ed in the limit or we can simply ignore them In what follows we avoid the case where one of the welfare weights equals 0 First welfare theorem Suppose that 0j71f72 853113 53812822 is an equilibrium Then the A1 A2 A2 A2 A1 A1 allocation c0clcz c0 c1 c2 1s Pareto ef cient Proof Since 11301131 farm 53511 8 5381252 is an equilibrium we know that there exist Lagrange multipliers 21 xi J gt 0 such that h 2 N3 A A1 A2 1 2 CC WW We also know that if there eXist welfare weights 021 022 0 gt 0 and Lagrange multipliers 73quot 1 5392 7339 gt 0 such that A2 2 ct 02 w W then 535115 53512522 is a Pareto efficient allocation In other words we are AAA A1A1A1A2A2A2 glVen POAPIAPZAu c c c c c c 0 1 2 0 1 2 and 212 that satisfy certain properties A A A A A A1 and we want to construct a1 a2 and no 72391 72392 that together With CO 01 2 5 512 522 satisfy certain other properties To prove the theorem we set A1 A1 Equilibrium with transfers An Arrow Debreu equilibrium with transfers is a sequence of prices e 71 z an A1A1A1A2A2A2 AA allocatlon 0001 2 coclc2 and transfers VIZ such that 0 Given 0j71f72 consumer 139 139 12 chooses 5351 5 to solve max 20 it log 0 st 220 pic S 220 ptw tl c 2 0 o cj fSw1w2 if f7 gt0 t012 Characterization of equilibrium with transfers using calculus Once again we use the KuhnTucker theorem to show that e l 72 58118 82 512 522 f1 fl is an equilibrium with transfers if and only if there exist Lagrange multipliers 2112 2 gt 0 such that f i niLLr012 cl 0 Zkopt 2220p2wt1 i12 o 521522wtlw t012 Second welfare theorem Suppose that 533125 502812822 is a Pareto ef cient allocation where each consumer receives strictly positive consumption Then there exist prices e l 72 and transfers f t quot1 quot1 quot1 quot2 quot2 quot2 A A 1 2 t suchthat p0p1p2 coclcz coc1czt1 2 1s an equilibrium Proof Since 8 511 8 5 812 is a Pareto efficient allocation equilibrium we know that there exist welfare weights 021 022 0 Z 0 and Lagrange multipliers 73quot 1 5392 7339 gt 0 such that Since 5 gt 0 we know that 0 gt 0 139 l 2 We also know that ifthere exist prices e l 72 transfers 15 and Lagrange multipliers xlvxl J gt 0 such that IE Q2 New g gt no A 1 no A I A 220 pic 220 p w 1 A1 A2 1 2 C2Ct W2Wt then 0f71f72 535115 533123 t1t2 is an equilibrium withtransfers In other words we are given 58i5 83812522 021022 and 0733733 that satisfy certain properties and we want to construct e 71 rm I t 39 and 2122 that together 1 2 with 8 8i 8 5 512 822 satisfy certain other properties To prove the theorem we set

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