well as the following optimality conditions are satisfied by ci (t) = 31[yi (t) + γ 1 z(t)] and
xi ( t ) = ∕31 z ( t ):
u'1(ci(t)/[yi(t) + γ 1 z(t)]) - μi(t) = 0, (32)
μi(t) - λ(t) - κ 1 = 0, (33)
ʌ(t) = (ρ 1 - R1)λ(t) - γ 1 w 1(ci(t)/[yi(t) + γ 1z(t)]) - ^μj(t)[βt(j) - γt(j)3(j)], (34)
j=i
μi(t) = (P 1 - r 1)μi(t) - w 1(ci(t)/[yi(t)+ γ 1 z(t)]), (35)
/3j(t) = (P 1 - rt(j) + q(j) )μj(t) (36)
lim e-ρ11 λ(t)z(t) + £ μk (t)yk (t) ≤ 0. (37)
t→+∞
k=1
We define the costate trajectories by λ(t) = K1 - κ 1, μi(t) = κ1, and μj (t) = 0 for all
j = i. Note that all costate trajectories are constant and non-negative. Condition (32)
holds because of u'1(31) = K1 and condition (33) follows immediately from the definitions
of λ(t) and μi(t). Substituting ci(t)/[yi(t) + γ 1 z(t)] = 31 and the definitions of the costate
trajectories into (34) we obtain 0 = (P1 - RK1)(κK1 - κ1) - γ1w1 (qK1). This equation holds
because of the definition of R1, equation (21), 31 = u'1(31), and (20). Analogously, by
substituting ci(t)/[yi(t) + γ 1 z(t)] = 31 and μi(t) = 31 into (35), one obtains 0 = (ρ 1 -
r 1)κ1 - w 1(31). This equation holds because K1 = u'1(31) and because of (20). Condition
(36) holds trivially because of μj (t) = 0 for all t.
To verify the transversality condition (37), let us define g = R - n1β31 - n2β32. Lemma 7
implies that g < ρ 1. From the state equation z(t) = R3 z (t) - xi (t) and from xi (t) = /1 z (t^)
it follows that z(t) = z0egt. Consequently, we have limt→+∞ e-ρ1tλ(t)z(t) = (κ31 -
κ1)z0 limt→+∞ e(g-ρ1)t = 0. From xi(t) = /1z(t), z(t) = z0egt, and the state equation
yi (t) = r 1 yi (t) + xi (t) - ci (t) we obtain yi (t) ≤ r 1 yi (t) + /1 z0 egt. It follows therefore
that there exists a positive constant Y such that yi(t) ≤ Y emax{r1,g}t. This implies that
limt→+∞ e-ρ 1 tμi(t)yi(t) ≤ K1 Y limt→+∞ emax'' 1 -ρ 1 ,g-ρ 1 }t = 0, whereby the last equation fol-
lows from g < ρ 1 and r 1 < ρ 1. Finally, limt→+∞ e-1 γj> (t)yi (t) = 0 holds trivially for all
j = i because μj (t) = 0. The transversality condition (37) is therefore satisfied.
It remains to be shown that the feasibility conditions are not violated. More specifically, we
have to show that the application of the strategy φi implies that z(t), yi(t), xi(t), and ci(t)
remain non-negative for all t (independently of the initial conditions z0 and yi0). We have
already proved that z(t) = z0egt which shows that, starting from any initial state z0 > 0,
the asset stock z(t) remains strictly positive for all t. Now note that lemma 7 implies that
/31 > γ1q31 and therefore /31 > 0. It follows that xi(t) = /1z(t) ≥ 0. Application of φi implies
furthermore that ci(t) = 31 [yi(t) + γ 1 z(t)] and, hence, yi(t) = [/31 - γ 1 31]z(t) + [r 1 - 31]yi(t).
This shows that yi(t)∖yi,(t)=0 = [β 1 - γ 1 31]z(t) > 0, whereby the inequality follows from
z(t) > 0 and /31 > γ1q31 (see lemma 7). Therefore, yi(t) cannot become negative if it starts
at a non-negative initial value. Because ci(t) = q31 [yi(t)+γ1z(t)], it follows also that ci(t)
remains non-negative. This completes the proof of the theorem. <1
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