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

The following parameter restriction will be crucial for the main result of this section.

n 1A1 - (n 1 - 1) A2 > n 1 ρ 1 - (nɪ - 1)ρ2 + (n - 1)γ2q2 - R,

n2A2 - (n2 - 1) A1 > n2P2 - (n2 - 1)P 1 + (n - 1)Y1«1 - R.

Let us denote the type of player j ∈ {1, 2,... ,n} by £ ( j ). We are now ready to characterize the
interior equilibrium of the game with heterogeneous players.

Theorem 2 Assume that the parameter values satisfy κ_t < κ_t, £ ∈ {1, 2, }, as well as condition
(22). The strategy profile ( φ 1, φ 2,..., φn ) defined by φ^c_i (y, z ) = eg ( _i )( y_i + γ_i ( _i ) z ) and φχ (y, z ) =
/3_t(_i)z forms a Markov-perfect Nash equilibrium.

The proof of this theorem is very similar to that of theorem 1, which is why we do not present
it here.⁵

Theorem 2 can be used to study the influence of heterogeneity on the equilibrium outcome.
As an illustration, we discuss the effect of differences in appropriation costs on the equilibrium
growth rate of the public asset. Suppose that κ ₁ = ^ + ε∕n ₁ and κ₂ = ^ - ε∕n₂, where K and
ε are real numbers such that n₂(K - κ₂) < ε < n ₁(κ₁ - ^). The latter condition ensures that
κ_t < κ_t holds for each £ ∈ {1, 2, }, as required by theorem 2. An increase in ∣ε∣ corresponds to
a mean-preserving spread in the distribution of extraction costs across players. Defining
we have the following result

Lemma 5 Suppose that γ1 and γ2 are strictly positive. The equilibrium growth rate of the
common property asset, g = R - n1β^g1 - n2β^g2, is a strictly concave function of ε which attains
its unique maximum at

*   n 1 n2[(1 - W)^ + W«1 - «2]

ε =------------------------------.

n1 + n2W

Proof: From the definitions of β1 and β2 we have

n₁B₁ + n₂B₂     R + n₁ (A₁ - ρ₁) + n₂(A₂ - ρ₂)

n-1                   n-1

It is therefore sufficient to prove that f (ε)=n1 (A1 - ρ1) + n2(A2 - ρ2) is strictly convex with
respect to ε and that it attains its minimum at ε = ε^*. Using the definitions of A₁ and A₂ we
obtain

_f, () =   ⁽P1 ^{- r} 1) Y1 ^g1 _   ⁽P2 ^{- r}2) Y2 ^κ2

^ε    («1 - K - ε∕n 1)²   (K2 - ^ + ε∕n2)²

⁵The proof is available from the authors upon request.

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