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handoutrcesp08

# handoutrcesp08 - Recursive Competitive Equilibrium Up to...

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Recursive Competitive Equilibrium Up to now, almost every problem has been formulated as an in fi nite sequence problem. Now we will focus on the equivalence of a recursive equilibrium for- mulation to the in fi nite sequence competitive equilibrium. This material draws from Harris, Ch. 4, Ch. 12 Ljundqvist and Sargent (as well as Cooley and Prescott p. 7) and Ch. 3-4 of Stokey and Lucas. The sequence problem is primitive, but it involves in fi nite sequences. Instead of working with in fi nite sequences, a recursive formulation de fi nes a fi nite number of functions (remember, however, that a function is in fi nite dimensional). 1 Stochastic Growth Environment Technology shocks: z t follow a fi nite state Markov Process π ( z t +1 ) = π ( z t +1 = z 0 | z t ) = π ( z t +1 = z 0 | z t = z ) π ( z t = z | z t 1 = b z ) · ·· The process { z t } is assumed to be stationary with bounded ergodic set Z , i.e., if z t Z then z t +1 Z with probability one. The transition probabilities π : Z × Z [0 , 1] . Productive Technology: Y t = z t F ( K t ) where F is CRS. Investment Technology: K t +1 = (1 δ ) K t + I t where K 0 is given and K t X R + where X is assumed to be closed and convex. Unit measure of ex-ante identical agents with preferences: P t =0 P z t β t π ( z t ) u ( c t ) where u is continuous, strictly increasing, strictly concave, and di ff eren- tiable. 2 Sequential CE At the beginning of each period in history z t , individual households rent capital k t X to fi rms, consume c t and invest i t .

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