STOCHASTIC

where
and
k"
1 f (Mt-Ui) 2
4k {Oi 1 +Mi 2 + CT2 2 + Hz 2 ~2/iifi2
UiG2 2 +M2 (/*2 2 + i2 2 )lM2, show that (dl) holds if w e [0, M>_]. For w > w+, show that
where
U(w) = Mzlw — k^w 2 ],
,„, ,a2 2 + /i2 2
(g) In (f) show that (el) holds for w e [w_, w+].
(h) Discuss the results above for the special cases ft2 = r, a2 = 0 (p2 is money) and
Mi = r, a > 0,
Consider the derived utility function t/(vf) for the class u such that — u'(£)/u"(() = // + !{,
for constants A and /*. According to Exercise ME-5, this class of utility function propagates
unchanged, up to a positive linear transformation, if the maximization problem has an
interior solution.
(a) For «((?) = — e' ai , show that x* = w, i.e., the maximum does not occur at an interior
point. Thus
[e-"»-e-
Note that the function U(w) is not equivalent to u(w).
(b) For «(£) = (a+() c , show that
U(w) =
1 1
(c+l)G?-a)w
(a 4- w(0+ l)) c
Note that U(w) is not equivalent to u(w) except when a = 0.
(c) For H(£) = log(£+a), show that
'
(a+w(a+l)) c+1
1 [« + 08+l)w]log(a + G8+l)w)'
£/(w) = -1 + •
(fi-a)w !-[o + (a+l)H']log(a-l-(a+l)M')
Show that [/ is equivalent to u (in the sense of a positive linear transformation) if and
only if a = 0.
414 PART IV DYNAMIC MODELS REDUCIBLE TO STATIC MODELS
(fl)