and
2(P1 - r 1)Y1 Ki + 2(P2 — r2)Y2K2
ni(Ki - K - ε∕ni)3 n2(K2 - ^ + ε∕n2)3.
Obviously, we have f''(ε) > 0 for all ε ∈ (n2(K - κ2),n 1(κ1 - ^)) and f'(ε) = 0 if and only if
ε = ε* This completes the proof of the lemma. <1
As a special case, suppose that the cost parameters are the only source of heterogeneity. For-
mally, let us assume that the conditions γ 1 = γ2, r 1 = r2, K1 = K2, and ρ 1 = ρ2 hold. In this
case, W = 1 and ε* = 0. We therefore conclude that, in a model in which the appropriation
cost is the only source of heterogeneity, a mean-preserving spread in the distribution of this
cost across players leads to a reduction of the equilibrium growth rate of the public asset. On
the other hand, if players differ also in other characteristics, then it holds that ε* = 0 and that
there exists a non-zero degree of cost heterogeneity that maximizes the net growth rate of the
public asset.
5 The pessimistic equilibrium
In this section we assume that the players have to obey the additional constraint
xi(t) ≤ βHz(t)
(23)
for all t and all i, where βH is a given positive constant. In other words, each player’s extraction
intensity must not exceed the exogenously given upper bound βH . Such a constraint may
interpreted, for example, as a certain form of capital control. If βH ≥ β and the conditions of
theorem 1 are satisfied, then the strategy profile described in that theorem remains to qualify as
a Markov-perfect Nash equilibrium. However, there will also exists another equilibrium along
which all agents extract the common property asset at maximum speed. This is proved in the
following theorem.
Theorem 3 Consider the differential game specified by (1)-(5) and (23). Let κ < κ be given
and assume that the inequalities βH > γq and
ρ κ ≤ ρ - R +( n - 1) 3h
(24)
K - K
are satisfied. The strategy profile (φ 1, φ2,..., φn) defined by φc(y, z) = q(yi + γz) and φχ(y, z) =
3Hz forms a symmetric Markov-perfect Nash equilibrium.
Before we prove this theorem we state the following auxiliary result.
Lemma 6 Let κ < K be given and assume that (24) holds. Then it follows that βH > (R -ρ)/n.
14