The name is absent

decreasing in y. In such cases all of the conditions for controlling an invasion will be more likely to hold,
the larger the invasion.

In the following let (y_t,x_t,a_t)₀~ be an optimal program from y₀. The next result characterizes the
intertemporal tradeoffs between marginal costs and damages along an optimal program.

Lemma 4. a. If at > 0 then C_a(¾yt) ≤ Dx(x) + δ[C_a(at+1,yt+1)+Cy(at₊1,yt₊1)]fx(xt).

b. If x > 0 and at+1 > 0 then C_a(¾yt) ≥ Dx(xt) + δ[Ca(at+1,yt+1)+Cy(at₊1,yt₊1)]f_s(x_t).

c. If 0 < x_t < y_t and a_t+1 > 0 then

Ca(at,yt) = Dx(Xt) + δ[Ca(at+1,yt+1)+Cy(at+1,yt+1)]fx(xt).                                                   (3.1)

Since the value function in nonconvex models may not be differentiable, Lemma 4 cannot be obtained by
applying standard envelope theorem arguments such as those of Benveniste and Scheinkman [1979].

Majumdar and Mitra [1982] use variational methods to obtain the Euler equation in a nonconvex growth
model. Our version of Lemma 4 generalizes the envelope theorem of Benveniste and Scheinkman by
using an alternative approach based on the principle of optimality and the fact that Dini derivatives of V
exist everywhere.

Corollary to Lemma 4c. If 0 < x_t < y_t and 0 < x_t+1 < f(x_t) for all t then

∞                                          i -1

C.(∙t.yt) = Dx(xt) + Σδⁱ [Dx(xt+i) + C_y(a,+_i,y,+i)]∏f_x(xt+j).
i=1                                         j=0

This has a simple interpretation when the costs of control are independent of the size of the
invasion. For an interior policy the optimal control equates the marginal costs of control with the
discounted sum of marginal damages over time multiplied by the compounded marginal growth of the
invasion. This is a simple cost-benefit criterion which balances the cost of removing a unit of the

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