STOCHASTIC

292 REVIEW OF ECONOMIC STUDIES
In addition to (1.4) we have
Combining (1.4) and (1.5) we obtain
where
y = f+ o, = r"+6ar
B =
°,
Equations (1.5) and (1.6) can be represented as in Figure 3
1 *
U V U 2 U3IU/O.6)
FIGURE 3
(1.6a)
• •
...(1.5)
...(1.6)
Ul>U2>U3> t/4 indicate utility indifference curves. The point A is first determined
and then the degree of lending/borrowing w is selected.
If we introduce a proportional tax, (1.5) will take the form
and (1.6) will take the form
(7y = w(i-t)ar,
y = r*(l-l)+Z^lay.
...(1.5a)
...(1.6a)
These two curves are drawn with dotted lines in Figure 3. Thus we see that (1.6)
moves down to the position (1.6a) and (1.5) swings to position (1.5a). The effects of the
first factor is indicated by the difference between A and A 1 and the second effect by the
difference B—B l . Increased w means greater risk-taking. Whether w increases or not
depends upon the size of these two effects. It seems that borrowing, i.e. operating to the
right of T implies a move towards greater risk-taking since the effects there are much
greater on the difference between (1.5a) and (1.5).
A somewhat different approach to the investment problem under risk is given in
Roy [17]. He assumes that the investor makes investments in order to minimize the upper
316 PART III. STATIC PORTFOLIO SELECTION MODELS