Econ. 511b

" Traditional " vs. " Correct " Transversality Conditions, plus answers to problem set due 1/28/99 * 1. Where the conventional TVC's come from In a fairly wide class of growth models, the transversality conditions developed in previous lectures take on a simplified form, and under some side conditions can be converted into the transversality conditions that are usually taken to be standard. Consider an optimization problem of the following form: max C ∞ 0 ,S ∞ 0 E ∞ t=0 β t U(C t) (A1) subject to C t ≤ f(K t , K t−1 , L t , ε(t)), t = 0,. .. , ∞. (A2) The Euler equations are ∂C: D C U t = λ t (A3) ∂L: D L U t = −λ t D 3 f t (A4) ∂K: λ t D 1 f t = −βE t [λ t+1 D 2 f t+1 ]. (A5) The transversality condition is lim sup T →∞ β T E[(D C U T − λ T)dC t + (D L U T + λ T D 3 f T)dL T + λ T D 1 f T dK T ] ≤ 0. (A6) The Euler equations guarantee that the terms in dC T and dL T drop out, leaving lim sup T →∞ β T E[λ T D 1 f T dK T ] ≤ 0. A model that has an interpretation as a growth model will have D 1 f T < 0, so that increasing K t at t requires decreasing C t. If the model satisfiesˆK T ≥ 0 for every feasible choice of K's, then dK T ≥ − ¯ K T and in turn (A7) is less than lim sup T →∞ β T E[λ T D 1 f T · (− ¯ K T)]. (A8) Thus a sufficient condition for transversality to hold is lim T →∞ β T E[−λ T D 1 f T ¯ K T ] = 0. research purposes, so long as the copies contain this notice and are retained for personal use or distributed free.