Insecure Property Rights and Growth: The Roles of Appropriation Costs, Wealth Effects, and Heterogeneity

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)

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 β_H . 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

K - K

are satisfied. The strategy profile (φ ₁, φ₂,..., φ_n) defined by φc(y, z) = q(y_i + γz) and φχ(y, z) =
3_Hz 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.

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