STOCHASTIC

ON OPTIMAL MYOPIC PORTFOLIO POLICIES 331
Clearly, viim represents the proportion of capital Xj invested in opportunity i at
decision pointy, given that the economy is in state m at that point; thus
M.
wiym = 1 — £ » 0 (29)
for all j, all m, some n for which pjmn > 0, and all finite dt such that 6t > 0 for all
i € 5,„ and 0,- ^ 0 for at least one i. (29) is a modification of the "no-easy-moneycondition"
for the case when the lending rate equals the interest rate. 10 This condition
states that no combination of risky investment opportunities exists in any
period which provides, with probability 1, a return at least as high as the (borrowing)
rate of interest; no combination of short sales is available for which the probability
is zero that a loss will exceed the (lending) rate of interest; and no combination of
risky investments made from the proceeds of any combination of short sales can
guarantee against loss. (29) may be viewed as a condition which the prices of all
assets must satisfy in equilibrium.
(3) is now replaced by the conditional difference equations
Mjm
xi+i\ mn = J2 (/3,-y„„ — rim)zijm + rjmXj j = 1, . . . , / — 1, all m, n , (30)
i=2
and (4) becomes, for j = 1, . . . , J — 1 and all m,
Jimix,) = max Y, pim„E{ fj+1,n(xj+1\mn)} , (31)
z im n=l
where fjm(xj) is given for all m, subject to
ZiM > 0 *€5y„, (32)
and
Pr {xj+l\mn > 0} = 1
VII. OPTIMAL MYOPIC POLICIES
n = 1, . . . , Ni+i. (33)
On the basis of the finite yield assumption (28) and the "no-easy-money-condition"
(29), we obtain the following:
Theorem.—Let rym, Fim, and pimn be defined as in Section VI and let u(x) be a
monotone increasing and strictly concave function for all x > 0. Then the functions
N .., M.
t r lm ~\ i
hjn(Vjm) = 22 /"ymnfilwL J2 (flijm, — fy->iy«. + >>J | , (34)
10 Hakansson, "Optimal Investment . . ." (see n. 4 above).
408 PART IV. DYNAMIC MODELS REDUCIBLE TO STATIC MODELS