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 κi < κi, I {1, 2, }, as well as
condition (22). Then it holds that
βi > γiqi and R — n 1 ∕31 — n2β2 < min 1 2}.

Proof: Using the definitions of β 1 and βi 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 /3i 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:

maximize

subject to


+

— κ1xi(t)] dt


e-ρ1t[U1 (ci(t),yi(t)+γ1z(t))
0

z( t ) = R z ( t ) — xi ( t )

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

yj(t) = ri(j)yj(t) + βi(j)z(t) (j)[Uj(t) + Yi(j)z(t)] for j = i
x
i(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 — xi) + μi(r 1 yi + xi — ci)
+
μj [ ri ( j ) yj + βi ( j )z qi ( j )( yj + γi ( j )z )],
j=i

where λ and μk, k = 1, 2 ,...,n, are the costate variables corresponding to z and yk,
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 μk :
[0
, +) [0, +), k =1, 2,...,n, such that the feasibility conditions of problem P1 as

19



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