A TURNPIKE THEOREM FOR A FAMILY OF FUNCTIONS* - Ivie

satis…es that for anyu2 U;uis ¹ strictly concave at(k u ;k u ); where U ¹ is the
closure of the normalized family
¾
U =
½u:D! < j u= u0
ju 0 X j ; u 0 2U 0 :
1
4. Conclusions
The papers by Nishimura, Sorger and Yano [11] and Nishimura and Yano [12],
make the point that the incompatibility between ergodic chaos and patience is
not as strong as would appear from the standard Turnpike Theorem. Our result
clari…es the extent to which an incompatibility exists. In fact, roughly speaking,
interiority, smoothness (C 4 ¡felicity functions), and, fundamentally, the uniform
strict concavity of the family is enough to have uniform convergence to the steady
state for any member of the family, hence the phenomena shown in [11] and [12]
is ruled out. An interesting question here is to see if it is possible to improve this
Uniform Turnpike Theorem assuming onlyC 2 ¡felicity functions and/or relaxing
the uniform strict concavity of the family (assumption A6). We think that this
is not the case, but this can only be proved by means of an example, which we
leave for further research.
Another question to investigate is to determine if the restriction to models
with interior solutions is a su¢cient condition to rule out the existence of a family
of optimal growth problems displaying ergodic chaos for any value of the discount
factor. We have already pointed out in footnote 1 that, to our knowledge, the
existence of a family of optimal growth problems with interior solutions displaying
ergodic chaos for any value of the discount factor is still an open question.
A more precise knowledge of the relation between the discount factor and the
extent of chaos that optimal paths can display is a problem that we are currently
investigating. We recall that in [11] the authors show in precise formulation that,
in spite of having ergodic chaos for any ± 2 (0;1); the chaos disappears as the
discount factor tends to one, and that a general result of a similar ‡avor can
be found in [9], where it is shown that topological entropy tends to zero as the
discount factor tends to one, in a family of strictly concave felicity functions. Also,
very recently in [5], as a consequence of our Theorem 2, it is proven in precise
formulation that, for a givenC 1 ¡family of felicity functions as in Remark 6, if it
displays some kind of chaos for any value of the discount factor, the chaos has to
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