STOCHASTIC

24 THE REVIEW OF ECONOMICS AND STATISTICS
of this latter assumption, it follows that risk
classes of securities should be scaled in terms of
variances of returns rather than standard deviations
(with the level of covariances reflected in
the parameters of the linear function). The
complexities involved when indifference contours
are scaled on covariances or standard deviations
are indicated below.
The conclusion that the indifference contour
between xt and the variance a? is linear in the
general case when all covariances cr,y are held
constant is established in the appendix, note II,
by totally differentiating the equilibrium conditions
(12) [or the equivalent set (220) restricted
to the m! stocks held in the portfolio]. But all
pairs of values of Xi and a< 2 along the linear
indifference coutour which holds A,° fixed at
some given level also rigorously imply that the
proportionate mix of all other stocks in the portfolio
is also unchanged. Consequently, we may
proceed to derive other properties of this indifference
contour by examining a simple "two
security" portfolio. (The s' h security is renumbered
"1," and "all other" securities are called
the second security.) If we then solve the
equilibrium conditions 33 (12) in this two-stock
case and hold K = hi°/hi° constant, we have
(24) K = hi'/h^" = constant = (*i 0 if ] p I < Km/n2 < I u _1 |.
On the other hand, from (25c) we have
142 PART II. QUALITATIVE ECONOMIC RESULTS