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# Macroeconomic Theory I EconS 500

WSU

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This 9 page Class Notes was uploaded by Maurine Kuhic on Thursday September 17, 2015. The Class Notes belongs to EconS 500 at Washington State University taught by Mark Gibson in Fall. Since its upload, it has received 93 views. For similar materials see /class/205981/econs-500-washington-state-university in Economic Sciences at Washington State University.

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Date Created: 09/17/15

Mark J Gibson Recursive Competitive Equilibrium To de ne equilibrium recursively we need to specify the state of the economy In a recursive equilibrium all aggregates are functions of the state of the economy In addition we need to specify the state of the individual In a recursive equilibrium all household decisions are functions of the state of the individual and the state of the economy In solving their dynamic programming problems households take the evolution of the aggregate capital stock as given but choose the evolution of their individual capital stocks In equilibrium they are consistent with one another Homogeneous households no uncertainty There is measure one of identical households each endowed with one unit of labor in each period and 0 units of initial capital Each household s utility function is 20 7logq1710g1 The aggregate resource constraint is C K 1 52 KfL 39 1 Here the state of the economy is the aggregate capital stock K and the state of an individual household is its capital stock k A recursive competitive equilibrium is vkK ckK 2kK QUC IO 6K 00 1600 rAK MK such that 8kK lkK 39kK solve vkK max ylogc 1 ylogl l 0k391 391lt st c k39 1 6k s vvKl rKk cZO 0531 k3921 6k rK adKa 1 K1 MK 1 a6K K1 K1 39K 1 6K 6K K1 8KK CK 2KK K I 39KK 1 39K Homogeneous households uncertainty Now suppose that there is uncertainty about the evolution of total factor productivity The resource constraint is Ct KH1 l 6Kt z Kt sz z Here 2 evolves according to a Markov process with transition function Qz dz39 The state of the economy is now K 2 The state of an individual household remains k A recursive competitive equilibrium is 3kK z ckK z lkKz h39kK z Kz Kz 1639Kz rKz wKz such that 8kKz 2kKz I 39kKz solve 3kK z maX ylogc l ylog1 l J3k39I 39K z z39Qz 61239 st ck39 1 5k S WKzl rKzk cZO 0531 k3921 6k KK z az Ka 1 Kz1 MK 2 1 az 9K K Zr 6K z1 39K z 1 6K 26K K Zr 5KKz Kz 2KK z K z I 39KK z 1 z Heterogeneous households uncertainty There are i l2 household types The measure oftype i is Li Type i is endowed with one unit of labor in each period and 1 units of initial capital The state of the economy is Kz K1K2KIZ where K is capital per household oftype i A shock Z with transition function Qzz39 affects TFP as above and also affects the labor efficiencies of households One unit of labor provides 7712 efficiency units of labor for households of type i Arrow securities are used to share risk across households A security bz39 is a promise of delivery of one unit of capital conditional on next period s shock being 239 If these securities were not available then there would be incomplete markets A recursive competitive equilibrium is rKZ WKZ QKZz39 51kKz AclkKz Mum 12kK 2239 GYM XKz 1Kz Kz K39K z z39K139K z z39K K 2239 such that 61kKz Juarez 21kKz b2kKzz39 solve QkK z maX ylogc 1 ylogl r J39Q 1 6k x bz K39K z z39 z39Qz 51239 st 0 xI K z z39bdz39 S viK 37712 rK zk 020 x20 OSESI Z M81K1Kz Kz ZIMJEIK1K2AK2 Z Lmlz21KlKz Kz leLllbKlKZZl 0 Kl39Kzz39 1 61 K1Kz 13 K1 Kzz39 KKz 2 AK 6K z2K z 26KK z K zk KK z a26KKz 391 Kzl39 MK 2 1 azt9KK z K 2y 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 818118 8282822suchthat 0 Given e Ajay consumer 139 139 12 chooses 8 8 to solve max 20 it log 8 DO A O A st 220 p28 S 220 ptw 620 o ajafswjwf if 2 gt0 t0l2 Characterization of equilibrium using calculus The KuhnTucker theorem says that 8 8 8 solves the consumer s maximization x 0 1 2 problem if and only if there exists a Lagrange multiplier 2 0 such that 1 AA A l A l llptSO 0 1fc gt0 cl ELEM Z0 c 20 0 if 1 gt0 l A A For any t I 012 11m 00 1mp11es that c gt 0 wh1ch1mp11es that A gt 0 It gt0 c also implies that t gt 0 t 01 2 Consequently of71f72 518115 8 512 522 is an equilibrium if and only if there exist Lagrange multipliers 21 ii i gt 0 such that o fif72i12t012 ct ZZO zc ZZO zwp i12 0 95 wtlwft012 Pareto ef ciency An allocation 8 8118i 52512522 is Pareto efficient if it is feasible quot1 quot2 1 2 c cZ S w wZ t0l2 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 8118i 52512522 is Pareto efficient if and only ifthere exist numbers 021022 0 Z 0 and not both 0 such that 853113 53812822 solves A no 2 1 A no 2 2 max aIZH l log 0 a2 220 z log 0 st c th S w21wtzt0l2 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 c 39 502812522 solves the social planner s quot1 0 1 pm 712 7139 ZOsuchthat problem if and only if there eXists a Lagrange multipliers 710 711 A l A A 01 71392S0 01fcjgt0 11A C 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 A1 A2 0 90 wtlwft012 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 e lf72 83511 8 53512 is an equilibrium Then the A1 A1 A1 A2 A2 2 allocation c0clcz c0 01 c is Pareto ef cient 2 Proof Since 11301131 farm 53511 8 5381252 is an equilibrium we know that there exist Lagrange multipliers 21 xi J gt 0 such that 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 is a Pareto efficient allocation In other words we are A1 A1 A2 A2 A2 quot1 quot2 cl 02 c0 c1 c2 and l 1 that satisfy certain properties A1 A1 A1 A2 A2 A2 then coclcz 0001 cz A A A A1 glVen p09p19p29quotquot co and we want to construct 021 022 and 73quot 721 73392 that together with 528282satisf certain other ro erties To rove the theorem we set 0 1 2 y p p p A1 A1 A1 coclc2 Equilibrium with transfers An Arrow Debreu equilibrium with transfers is a sequence of prices e 71 z an quot1 quot2 quot2 quot2 A A c0 c1 c2 and transfers VIZ such that A1 A1 allocatlon c0clcz A1 0 Given 0j71f72 consumer 139 139 12 chooses 53013 to solve 0 2 max Zko l logc st 220 pic S 220 ptw tl c 2 0 AZSw1w2 if in 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 ii z l2r012 1 A cl 0 220 pic 220p2wt1 139 12 0 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 t1 t2 AAA A1A1A1A2A2A2AA suchthat p0p1p2 coclcz coc1c2t1t2 1s an equilibrium Proof Since 8 511 8 03 812 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 Apt and Lagrange multipliers xlvxl J gt 0 such that l A A it T A p c 2 no A 1 no A I A 220 pic 220 p w 1 A1 A2 1 2 ct 02 wt wt then oj71f72 538125 533123 t1t2 is an equilibrium with transfers In other words we are given 58i5 03812522 021022 and 0733733 that satisfy certain properties and we want to construct e 71 rm fpfz and 2122 that together with 8 8i 8 5 512 822 satisfy certain other properties To prove the theorem we set A no A I no A 1 t1 Zzoplcl Zz0 plwl39 I

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