we substitute those expressions into (A.3) yielding
0 = nan21 [-r - r (n - 1)] Za0(Z) + 1+
(32 ) [r (—n + 2) + n (r — 1) — 2r] Z [n (1 — a(Z)) — δZ] a0(Z)
+ r (n - 1) [n (i — a(z )) — δz ] — (δ + ρ) r n —1 z. (A.5)
n2a n2a
Further rearranging (A.5) gives rise to (10) in the text.
Appendix B: Linear Strategy
We will show below that (15) represents a linear strategy. Under symmetry, rewrite the HJB
equation (6) as follows:
ρV (Z)= max [p (a1, a2,..., an) Z + V0(Z) {n (1 — a) — δZ}] . (B.1)
ai∈[0,1]
Suppose that the value function is linear, that is, V (Z)=A+BZ,whereA and B are unknown
constants. Substitute this hypothetical value function into the above HJB equation to get
ρ [A + BZ] = max
— Z + B {n (1 — a) — δZ}
(B.2)
Substituting further the (interior) first-order condition (8) , that is, a = r (n — 1) Z/Bn2 into
a in (B.2), we obtain
pA + pBZ = 1Z + B
n
n1
(n —1) z\ — δZ∣ .
Bn2 J ʃ
(B.3)
Comparing the coefficient of Z and the constant in both sides of (B3) yields
pA — Bn = 0 and pB---1-------+ + Bδ = 0.
nn
Solving the above simultaneous system of equations in terms of A and B to yield
23
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