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Unformatted text preview: On Equivalence of Sequential and Recursive Representations with Private Information • Atkeson, A. and R. Lucas (1992), “On E ﬃ cient Distribution With Private Information,” Review of Economic Studies , 59, 42753. 1 Environment • Risk neutral principal (like a social planner) • Continuum of risk averse agents with preferences E " ∞ X t =0 (1 − β ) β t V ( c t ) θ t # where the momentary utility function V : R + → D ⊂ R is cts, strictly increasing, strictly concave and θ t is an i.i.d. taste shock in a f nite set Θ = { θ 1 , ..., θ n } with θ 1 > ... > θ n with f xed prob. distn μ that assigns pos. prob. to all θ values and is normalized so that E [ θ ] = 1 . • Let a shock history be denoted θ t = ( θ , ..., θ t ) ∈ Θ t +1 (the t + 1 product space of Θ ) and μ t +1 denote the prob. measure associated with histories θ t . • Let the inverse function of V be denoted by C : D → R + and let the particular value C ( x ) be the resources required to achieve utility level x . • Agents are endowed with nonstorable constant income y and an initial entitlement ω where the distn over entitlements is denoted ψ ( ω ) . For sim plicity, index an agent by ω. • Info: θ t is priv. info to agent. • Timing: at the beginning of each period, agent observes θ t , reports taste shock, makes/receives transfer, consumes. 1.1 Sequence Representation • Let z t ( θ t ) be an agent’s planned (at t = 0) report about his date t shock in history θ t . • Let z = { z t ( θ t ) } ∞ t =0 denote the agent’s reporting strategy. Let the set of all reporting strategies be denoted Z. 1 • Let a truthful reporting strategy be denoted z ∗ where in all histories z ∗ t ( θ t ) = θ t . • Let c t ( ω, z t ) denote the cons. that agent ω receives at date t for a reporting history z t and u t ( ω, z t ) = V ( c t ( ω, z t )) . • The planner assigns the sequence u = { u t ( ω, z t ) } ∞ t =0 . Let the set of all such sequences (or plans) be denoted S. • Assume u t ( · , z t ) is Borel measble and that the plan satis f es the transver sality condition: lim t →∞ β t ∞ X s =0 β s u t + s ( ω, θ t + s ) θ t + s = 0 (1) • De f ne total expected utility U : D × S × Z → D from period t = 0 on by U ( ω, u, z ) = (1 − β ) ∞ X t =0 β t Z Θ t +1 u t ¡ ω, z t ( θ t ) ¢ θ t dμ t +1 (2) Thus U ( ω, u, z ) is the total expected utility agent ω receives if the planner chooses plan u ∈ S and the agent chooses reporting strategy z ∈ Z. • Defn. An allocation is a plan u ∈ S that induces each agent to adopt the truthful reporting strategy z ∗ (4) and that delivers expected discounted utility ω to each agent ω (3)....
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas.
 Spring '07
 CORBAE

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