STOCHASTIC

Then
* ( 8xj \ V -
s u = \-r-\ -ax, ) pk D
[Hint: Totally differentiate the conditions in (£>).]
(d) Conrider the effect of a change in the total expected return to the investor without
any change in risk. Then E(w) = p'x— x and var(w) = x"Lx, where x is a tax if x £ 0 or a
subsidy if x gj 0. Show that the Kuhn-Tucker conditions are
H
Xl
Xn
-A
Differentiate both sides by x to show that
dx,
BE(w)
Pi =const
(e) Utilize (d) to show that
S, ,,j -*, -(w-
-(1 + «T)A,
-(\+*x)p„
— w0
- a EiUi/>*£*.
g^ 1
Show how S0- may be interpreted as the effect of the change in p:, on Xj provided that the
investor is compensated for the change in p, so as to enable him to enjoy the same expected
wealth with the same risk—analogous to the Slutsky equation in consumer theory.
We may say that assets i and,/ (i^j) are substitutes or complements if S,j is negative or
positive, respectively.
(f) Show that S,j = Sj, so that the substitution effect is symmetric.
(g) Show that Sit > 0. Hence the rise in expected return on an asset will increase its
own demand—if the wealth effect is neglected.
(h) Show that LJSU = 0. Hence T.J*ISIJ 0
1=1 1=1
and £ £ Su < 0 if lgm