trades the marginal increase in expected payoff from an increase in appropriation against the
marginal loss in the discounted value of the future stream of payoffs which results from a
reduction of productive effort. If the payoff gain from an increase in ai is larger than the
payoff loss implied by the decrease in li for all levels of ai , then agent i will rationally devote
all resources to appropriation. In contrast, the agent chooses ai =0in cases where the
discounted marginal gain from productive investment exceeds the instantaneous marginal gain
from aggressive behavior for all levels of ai .
2.2 Equilibrium
We can then make use of (8) to characterize subgame perfect equilibria of the differential game.
Since we have started our analysis assuming identical agents, a natural focus is on symmetric
equilibria. The symmetry assumption allows us to drop the subscript i in the subsequent
discussion, and we will suppress this index unless strictly necessary for expositional clarity.
Let us first analyze interior solutions of ai . Differentiation of the interior first-order condi-
tion in (8) gives
V 00 (Z)=
n
X
j=1
d2p lJ∣^ 7j⅛
∂ai∂aj j ∂ai
r(n
1)
n2a2
a0 (Z)Z +
r(n- 1)
n2a
(9)
At an interior solution of a (Z) we may apply the envelope theorem to characterize a0 (Z).
Using the symmetry assumption, we obtain
a0 (Z)=
+ + n2a ZZ) [(1 - a (Z)) n - (р + 2δ) Z]
n n a (Z )
(10)
r (n ~ 1) Z (a - δZ)
a2a (Z)2 V ,
We will employ phase-plane methods to characterize the qualitative solution of the nonlin-
ear differential equation (10) and the associated MP strategies. For this purpose we have to
identify the steady state locus where Zi = 0, called C1 in the following. Let us denote by C2
the loci where a0 (Z) goes to plus infinity, and by C3 the loci where a0 (Z) equals zero in the