Appendix A: Derivation on the HJB equation
In this appendix we show how to derive (10) in the text. Assuming an interior solution, we
solve (8) for each agent to get the optimal strategy ai = ai (Z). By substituting this optimal
strategy into (6), the HJB equation (6) associated with agent i is transformed into
ρVi(Z)=pi(a1(Z),...,an(Z))Z+Vi0(Z)
n
(1-aj(Z))-δZ
j=1
(A.1)
By differentiating (A.1) with respect to Z and applying the envelope theorem to the resulting
expression, we obtain
nn
ρVi (Z) = ∑dpiaj (Z) Z + p,(.) + V (Z) £(1
a
- aj (Z)) - δZ
j=1 j j=1
n
+Vi0(Z) - a0j(Z)-δ .
(A.2)
j=1
Substituting (8) and (9) into Vi0 (Z) and Vi00 (Z) in (A.2), respectively, and exploiting symmetry
yields
∂2pi
∂ai2
"=(■ - <Z - Z
a0i(Z)+(n - 1)
∂2pi
∂ak∂ai
a0 (Z) + p(.)+
a0k(Z) Z[n (1 - a(Z)) -δZ]
+dpi [n (1 - a(Z)) - δZ] - (δ + ρ)∂piZ, k = i.
ai ai
(A.3)
Since the assumption of symmetry further allows us to make use of the following simple
expressions:
pi
∂2pi
∂ aO2
1 ∂pi r (n - 1) ∂pi r
n , ∂ai n2a , ∂ak an2 ,
n (r - 1) - 2r ∂2pi r2 (-n +2)
n3a2 , ∂ak ∂ai n3a2
r(n- 1)
(A.4)
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