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# Class Note for EconS 500 with Professor Gibson at WSU 03

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This 3 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 19 views.

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Date Created: 02/06/15
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 k0 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 ck39 1 6k s wKzfKk cZO 0Sl 1k3921 6k rK aBKa 1 K1 MK 1 a6K K1 6K1 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 l 5k WKzlrKzk 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

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