STOCHASTIC

Assume that the changes in cash balance are a Bernoulli random walk with equal probability
of unit positive and negative steps. The average cost of the cash balance over a time T is
K{T) Nd(T) £,*,*,
CT = C„ — \-Cd — 1- ch — ,
where X, is the cash balance at time t and NU(T), Na(T) are the number of transfers up and
down to z, respectively, in time T.
(a) Show that
CT ~* (cjfto) + (CilMh) + cbE„X w.p. 1 as T-+ oo,
where na and nh are the expected times between hits at 0 and h, respectively, and n is the
limiting distribution of the Markov chain {X,}.
(b) Show that C(z, h) = limr_ » CT is given by
(c) Neglecting the integer restrictions on z and h, show that the z* and h* which minimize
C are related:
h* - z* = kz*, where cuk 3 = 3cdk + 2cA.
(d) For cu = cd, show that
v 1/3
h* = 3z* z* =(£)'
(e) Show that k is relatively insensitive to the ratio cjcd.
20. Referring to Exercise 19, suppose that the cash balance policy is of the form (0, u\ d, h):
When the balance falls to 0, return it to H; when the balance reaches h, return it to d; otherwise,
do nothing (0 g u g rfg A).Assume the cash balance follows a Bernoulli random walk
with equal probability of unit steps up and down. Assume there is a unit holding cost ch
per unit time, a fixed cost c,, and a unit cost T for transfers to or from the cash balance.
(a) Show that the long-run average cost of the cash balance policy is
^, , .x c h h 2 + d 2 -u 2 + hd c + ux ct + (h-d)x
C(«, d,h) = — • — + ——-—- +
3 h+d-u uQi+d-u) Qi-d)(h+d-u)'
Let «*, d*, h* minimize C(u, d, h).
(b) Show that u* = d* if r = 0.
(c) Show that «* < d* if x > 0.
(d) Show that u* = l,d* = h*-\ if c,= 0 and x > 0.
21. Consider an investor whose utility for lifetime consumption is «(c0,ci,...,cr). For
each /', suppose the elasticity of marginal utility of ct is constant and equal to yt— 1.
(a) Show that for all i
ta, + b,cV if y, * 0,
"(Co.Ci CT) .
\at + b, logd if yt = 0,
where at and bt are independent of c( (but may be functions of cs fory ^ ;')•
Note that the additive utility functions of Samuelson and Hakansson are of this form, with
y, constant over i, and bt independent of all the c,. Assume instead that either yt =£ 0 and
a( is independent of the cs, for all / and i", or y, = 0 and b, is independent of the Cj, for
ally and /.
694 PART V DYNAMIC MODELS