STOCHASTIC

(v) UHEV-E^CV-Af- 1 )^ w, V&0, w>0,0^g,^G< oo}.
(c) Interpret the conomic meaning of these constraint sets.
*(d) When is the result in (a) valid for the constraint set in part (v) of (b) ?
(e) Suppose VN = £ MX") rt N and X" e A". Show that the following modified value
function and constraint set are permissible for the model under consideration:
?N = £ Vr,", where
X" e {%»| X," = /.(A,"), X" e A"}.
(f) Show how the value function in (e) permits certain forms of taxation to be included
in the model.
Some of the constraint sets in (b) are intended to include borrowing, in which case it is not
appropriate to assume that all rt > a > 0. Hence we will suppose that each VN satisfies
0 < VN < p < oo and Breiman's condition (A), namely that
E(?I\RN\=E(^L RN.\ S J^I .
\sK.\ ) \vm Jsz^
That is, given the wealth levels, SN _ i and S£ _ x are a known function of the past outcomes in
periods l,...,N—l, namely RN _ t. The key idea in the proof is to show that
for then E(SNISm \ RN-1) is a supermartingale; nartingale; that is, is
But then
4
E(^L R^^^zl for all JV.
(£Mfe)'
Hence E(S»ISN.) £ 1 for all N. [Note: This is a stronger statement than Elim(SNISN*) g 1
since it holds for all N.]
(g) Show that lim„^ „o(l + x/(n -1))" = e* if |jc| < oo.
(h) Show that AK is sufficiently regular for there to exist a strategy X N (e) such that
VN = X K (e)r N =:(l-e)VN* + eVN for any 8 £ 0. [Hint: Let A w (s) = (\ - e) X"'+ eX".]
Since X N ' is the unique expected log maximizer, it follows that for e S 0,
(i) Show that this last line is equivalent to
E log [(1 - e) Vs* + eVN~] - log VK* :£ 0.
/ i v„\
£log 1 +
l " i
I 1 + 77—T7T) ^-l°g 1
(j) Show that the assumptions imply that VNIVN. is bounded. Let n •• 1/E and x = VNIVN..
Use (g) to show that
ik *-) s •
Hence by the supermartingale theorem, lim SKjSNt exists almost surely and
£(lim«SyS*.)gl.
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PART V DYNAMIC MODELS