obtain a general solution to (10):
a(Z)=
(n — 1) rZ (n — δZ) δ (ρ + δ)
(14)
ρ+ δ
n [1 — r (n — 1)] (n — δZ) ~ + (n — 1) r (ρ + δ) c1
where c1 represents an arbitrary constant of integration and may take a positive, zero or
negative value. When c1 =0, (14) simplifies to
aL (Z)=
r(n— 1)(ρ+δ)
Z Z Tvi Z.
[1 — r(n — 1)]n
(15)
It is easy to confirm by the well-established guessing method for a value function that strat-
egy aL stands for the linear strategy (see Appendix B). Moreover, the left branch of the
linear strategy aL to the left of the steady state line C1 starts from the origin, and then
reaches point S on the steady state line C1 , while its right branch starts from any initial
value Z0 >ZS [if we do not here take into account the resource constraint (1)], then reaching
point S also. Moreover, it can be verified by substitution that the linear strategy aL also goes
through the singular point E .
The a4- (a3-) family of strategies represents the solution curve of (14) coupled with c1 > 0
when n<δZ (n>δZ), while the a1- (a2-) family of strategies represents the solution curve
of (14) coupled with c1 < 0 when n<δZ (n > δZ). Moreover, the left-branch of the a4-
family of strategies also starts from the origin, while its right-branch starts from any initial
value Z0 < ZE, both of which reach the same point on line C1 .Theleftbranchofthea1-family
of strategies starts from the origin and reaches a point on the steady state line C1 , while its
right branch starts from point (n∕δ, 0) and reaches the same point on line C1; therefore, those
strategies never hit the horizontal axis except for the origin and point (n∕δ, 0). On the other
hand, when the a2- and a3-families of strategies start from any initial value Z0 >ZE , the a2-
family of strategies approaches point (n∕δ, 0), while the a3-family of strategies goes to plus
infinity, as illustrated in Figs.1 and 2.
Nevertheless, not all integral curves in Figs.1 and 2 are qualified as MP equilibrium strate-
gies. There are three additional requirements which have to be met. The first prerequisite is
that strategies should not violate the resource constraint (1). This implies that a (Z) should
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