Proof of Proposition 2. Consider y0 e (0,ζ] and the optimal policy generated by the maximal selection
from X(y). (Under Caa(a,y) + Cay(a,y) ≥ 0, this policy function is non-decreasing in y.) It is sufficient to
show that the optimal path generated by this policy is strictly decreasing over time. Suppose not. Then
there exists an optimal program (yt,xt,at)0~ where y1 = f(x0) ≥ y0. This, in turn, implies that x1 ≥x0. Since,
y1 e [y0,f(y0)] ⊂ (0,f(ζ)), it follows that x1<y1 = f(x0). Therefore, using Lemma 4, we have
Ca(y0 -Xo, Уо) ≥Dx(xo) + δ[Ca(f(x0)-x1,f(x0))+Cy(f(x0)-x1,f(x0))]fs(x0)
and using Caa(a,y) + Cay(a,y) ≥ 0 and f(x0) ≥ y0 we have
Ca(f(Xθ) - Xo, f(Xo)) ≥Dχ(Xo) + δ[Ca(f(X0)-X1,f(X0))+Cy(f(X0)-X1,f(X0))]fχ(X0
which violates the inequality in the antecedent of the proposition. ■
Proof of Proposition 3. Consider the set of optimal paths generated by the maximal selection from X(y).
It is sufficient to show that complete eradication occurs on every path generated by this selection. From
Lemma 2, we know that this is an non-decreasing function and therefore, every optimal path generated
by this selection is weakly monotone and hence convergent (they are all bounded). Suppose to the
contrary that there is an optimal path generated by the maXimal selection from X(y) which is bounded
away from zero. Then, it must converge to a strictly positive optimal steady state y* = f(X*). Note that
y0 e (0,K) implies that every optimal program is bounded above by K so that x* and y* lie in [0,K]. If
x* e (0,K) then equation (3.1) implies Ca(f(x*)-x*,f(x*)) = Dx(x*) +
δ[Ca(f(x*)-x*,f(x*))+Cy(f(x*)-x*,f(x*))]ft(x*) which contradicts the inequality in the proposition. Also,
y*= K = f(K) is not an optimal steady state as the inequality in the proposition implies Ca(0,K)) < Dx(K)
+ δ[Ca(0,K))+Cy(0,K)]fχ(K) = Dx(K) + δCa(0,K))fχ(K) which implies C>K)) < [Dχ(K)Z(1-δfX(K))].
The latter can be used to show that a program where the control is infinitesimal but greater than zero in
period 0 and equal to zero every period thereafter dominates a program where at=0 in every period.
26
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