handoutfernandezphelan - Hidden Info with Persistent Shocks...

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Hidden Info with Persistent Shocks A. Fernandez and C. Phelan (2000), “A Recursive Formulation for Re- peated Agency with History Dependence,” Journal of Economic Theory , 91, 223-47. 1 Environment Realizations of the nonstorable endowment take 2 values h t H = { h H ,h L } where h H >h L which follows a f rst order markov where π ( h t 1 )denotes the prob that h t = h H given h t 1 . The initial seed is h 1 assumed to be public. The conclusion considers what happens in the general N state Markov process (which necessitates expanding the state space to N 1threats) . Let Π ( h t + j | h t ) denote the probability of future history { h t +1 , ..., h t + j } given h t . Endowment realizations are iid across agents. All agents are also endowed with an initial entitlement to future utility w 0 . A planner can transfer resources across time at rate q (0 , 1) . Agents are risk averse with preferences U ( c )= E " X t =0 β t U ( c t ) # where U : B R with B =[ b , b ] is cts, strictly increasing, strictly concave. The implied values for feasible momentary utility is D d , d ]wh e r e d = U ( b )and d = U ( b ) . Info: h t is priv. info to agent (except initial seed). Timing: at the beginning of each period, agent observes h t , reports en- dowment shock, makes/receives transfer, consumes. 1.1 Sequence Representation The only real decision for agents to make is about their reports. Let e h denote a reporting strategy given by the sequence { e h t ( h t ) } t =0 map- ping histories h t =( h 0 , ..., h t ) H t into a report of the current endowment e h t . 1
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De f ne a transfer system τ as a sequence of functions { τ t } t =0 such that τ t : H t R and τ t ( h t ) ≥− h t for all t and h t . Notational note: the dependence of τ t ( h t ) on the initial state ( h 1 ,w 0 )isle ftimp l ic it(thatis , should really write τ t ( h t ; h 1 0 ) . The constraint does not rule out the possibility that c t <b . This can be taken care of many ways, but they simply assume that the agent cannot claim a higher than actual endowment. See footnote 3 of F-P. If we let an agent’s current utility b u t after report contingent transfer be denoted b u t = U ( h t + τ t ( e h t ( h t ))) and de f ne C : D B by C ( b u t )= U 1 ( b u t ) . Since C is the inverse function of U, it is a uniquely de f ned strictly in- creasing and convex function.
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handoutfernandezphelan - Hidden Info with Persistent Shocks...

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