ABSTRACT
We can address this problem of determining an appropriate functional form for the adaptation value of equity by seeking a second solution of the auxiliary equation that takes the form Y (η) = u(η)X (η), where u(η) is a twice-differentiable function of η. It then follows that
Y ′(η) = u′(η)X (η)+ u(η)X ′(η)
and
Y ′′(η) = u′′(η)X (η)+ 2u′(η)X ′(η)+ u(η)X ′′(η)
When we substitute these latter two expressions into the auxiliary equation, we obtain
2 ζ 2η
d2Y
dη2 + (r −α)ηdY
dη − rY (η) = 1
2 ζ 2η[u′′(η)X (η)+ 2u′(η)X ′(η)]
+ (r −α)ηu′(η)X (η)
+ u(η) [ 1
2 ζ 2ηX ′′(η)+ (r −α)ηX ′(η)− rX (η)
] = 0
Note that the final term on the right-hand side of the above expression is merely the auxiliary equation multiplied by u(η) and, from §11-9, it will have to be equal to zero. We are then left with
2 ζ 2η[u′′(η)X (η)+ 2u′(η)X ′(η)]+ (r −α)ηu′(η)X (η) = 0
We can separate this latter equation into terms involving the function u(η) and terms involving the function X (η) by dividing all terms in the equation by 12ζ
2ηu′(η)X (η). Doing so, we have
u′′(η) u′(η)
= − [ 2X ′(η) X (η)
+ 2(r −α) ζ 2
] Now, if we integrate across both sides of this differential equation, we find
log[u′(η)] = −2log[X (η)]+ 2(α − r) ζ 2
η+ c1
where c1 is a constant of integration.However, it can be shown that c1 is a redundant parameter in determining the adaptation value of equity (as with c1 in §7-12) and, given this, we set it to zero. We can then apply the exponential operator to both sides of the above equation and thereby show
u′(η) = exp
[ 2(α − r)η
ζ 2
] X 2(η)
Finally, if we integrate across both sides of this expression (again ignoring the redundant constant of integration), we find
u(η) = ∞∫
exp
[ 2(α − r)y
ζ 2
] X 2(y)
dy
This in turn means that a second solution of the auxiliary equation will be
Y (η) = u(η)X (η) = X (η) ∞∫
exp [
] X 2(y)
dy
Now, for this second solution of the auxiliary equation, it can be shown that
lim η→0
Y (η) = 1 a0
where a0 is the parameter associated with the power series expansion for X (η) as summarized in §11-9 above. Likewise, it can also be shown that
lim η→∞Y (η) = 0
Hence, Y (η) has the important property that it asymptotically declines towards zero as the recursion value of equity η grows in magnitude. This in turn will mean that when the firm is highly profitable (i.e. η is large) the adaptation value of equity Y (η) will be relatively small and it is highly unlikely the firm will exercise the option it possesses to change its investment opportunity set in the foreseeable future. If, however, the firm is in a loss-making situation (i.e η is small), the adaptation value of equity Y (η) will be relatively large and there is a much greater likelihood the firm will exercise its option to change its investment opportunity set. Moreover, from §11-9 above and from §9-8, we know that the market value of a firm’s equity is the sum of its recursion value and its adaptation value:
P(η) = η+Y (η) Substituting the expression for Y (η) into the above expression for P(η) shows that the market value of the firm’s equity will be
P(η) = η+X (η) ∞∫
exp [
] X 2(y)
dy
where X (η) is the formal solution of the auxiliary equation determined in §11-9 above. Moreover, the fact that our earlier analysis shows that Y (η) has a limiting value of 1/a0 as the recursion value of equity η approaches its lower limit of zero will mean
lim η→0
P(η) = lim η→0
⎧⎨⎩η+X (η) ∞∫
exp [
] X 2(y)
dy
⎫⎬⎭= 1a0 = P(0)
where P(0) represents the adaptation value of the firm’s equity should its recursion value fall away to nothing. This in turn shows that a0 = 1/P(0) will be the parameter associated with the formal solution X (η) of the auxiliary equation determined in §11-9 above.
