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
More intriguing information
1. A Review of Kuhnian and Lakatosian “Explanations” in Economics2. BILL 187 - THE AGRICULTURAL EMPLOYEES PROTECTION ACT: A SPECIAL REPORT
3. The name is absent
4. Unilateral Actions the Case of International Environmental Problems
5. The name is absent
6. The name is absent
7. Graphical Data Representation in Bankruptcy Analysis
8. The name is absent
9. Confusion and Reinforcement Learning in Experimental Public Goods Games
10. Restructuring of industrial economies in countries in transition: Experience of Ukraine