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

Appendix (available upon request)

This appendix contains the proof of theorem 2. We start with the following auxiliary result.

Lemma 7 Assume that the parameter values satisfy κ_i < κ_i, I ∈ {1, 2, }, as well as
condition (22). Then it holds that β_i > γ_iq_i and R — n ₁ ∕3₁ — n₂β₂ < min{ρ ₁ ,ρ₂}.

Proof: Using the definitions of β ₁ and β_i it is straightforward to verify that the two
inequalities in (22) are equivalent to β₁ > γ ₁ q₁ and β₂ > γ₂q₂, respectively. Furthermore,
because of the first inequality in (22), γ1 ≥ 0, and A1 ≥ 0 we have

n 2 ρ 2 — ( n 2 — 1) ρ 1 — R  < n 2 A 2 - ( n 2 - 1) A1 - ( n - 1) Y1 11

≤ n₂A₂ — (n₂ — 1)A₁

= n₂A₂ + n₁A₁ — (n — 1)A₁

≤ n2A2 + n1A1.

Using the definitions of β₁ and /3_i it is easy to verify that this condition is equivalent to
R — n ₁ ∕3ι — n₂β₂ < ρ ₁. The inequality R — n ₁ β₁ — n₂β₂ < ρ₂ can be proved in analogous
way by starting from the second inequality in (22) instead of the first one. This completes
the proof of the lemma. <ι

Proof of Theorem 2: Without loss of generality we may assume that player i is of type
1. If all players j = i use their equilibrium strategies φj , then it follows that player i faces
the following optimal control problem P1:

+∞

e^-ρ1t[U1 (ci(t),yi(t)+γ1z(t))
0

z( t ) = R z ( t ) — x_i ( t )

yi ( t ) = Г1 yi ( t ) + X_i ( t ) — Ci ( t )

^yj^(t) = ri(j)yj^(t) ^{+ β}i(j)^z(t) ^— ⅛(j)[Uj^(t) ⁺ Yi(j)^{z(t)] for j} = ⁱ
xi(t) ≥ 0 , Ci(t) ≥ 0 , y(t) ≥ 0 , z(t) ≥ 0,

where R1 = R — (n1 — 1)β1 — n2β2. The Hamiltonian function of this problem is

H = U1(Ci, Ui + γ 1 z) — κ 1 Xi + λ(R1 z — x_i) + μi(r 1 yi + x_i — c_i)
⁺ ∑ ^μj [ ri ( j ) ^yj ^{+ β}i ( j )^{z — q}i ( j )^{( y}j ^{+ γ}i ( j )^{z )]},
j=i

where λ and μ_k, k = 1, 2 ,...,n, are the costate variables corresponding to z and y_k,
respectively. Note that the Hamiltonian is jointly concave in (z, y,Ci,Xi). The theorem
is therefore proved if there exist costate trajectories λ : [0, + ∞) → [0, + ∞) and μ_k :
[0, +∞) → [0, +∞), k =1, 2,...,n, such that the feasibility conditions of problem P1 as

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