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



Proof: Because of (24), γ ≥ 0, ρ > r, κ < κ, and βH > 0 it holds that

ρ - R + H ≥ вн +  '     ' > 0.

κ — κ

This proves the lemma. <ι

Proof of Theorem 3: Suppose that all players j = i use the equilibrium strategies φj .In
this situation, player
i faces the following optimal control problem Q:

+

- κxi(t)] dt


maximize

subject to


e-ρt[U(ci(t),yi(t)+γz(t))
0

.  ,   4            ~     ,  4

z( t ) = Rz ( t ) — Xi ( t )

Уi ( t ) = ryi ( t ) + Xi ( t ) — Ci ( t )

yj(t) = ryj(t) + βHz(t) - ft[yj(t) + Yz(t)] for j = i
0 ≤ xi(t) ≤ βHz(t) , ci(t) 0 , y(t) 0 , z(t) 0,

where R = R — (n — 1)βH. The Lagrangian function of this problem is

L = U ( ci,yi+γz ) — κxi + λ ( R z—Xi )+μi ( ryi+Xi — ci )+∑ μj [ ryj + вн z — ft( yj+γz )] + ν ( вн z — Xi ),
j=i

where λ and μk, k = 1, 2,... ,n, have the same interpretation as in the proof of theorem 1 and
where
ν is the Lagrange multiplier for (23). Note that the Lagrangian 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, as well as a multiplier ν : [0, + )
[0, +) such that the feasibility conditions of problem Q as well as the following optimality
conditions are satisfied by
ci(t) = ft[yi(t) + γz(t)] and xi(t) = βHz(t):

u' ( Ci ( t ) / [ yi ( t ) + γz ( t )]) — μi ( t ) = 0,                                                          (25)

μi ( t ) — λ ( t ) — κ — ν ( t ) = 0,                                                             (26)

λ(t) = (ρ — R)λ(t) — γw(Ci(t)/[yi(t) + γz(t)]) (вн — γq)μj(t) — внν(t),   (27)

j=i

μ i( t ) = ( ρ r ) μ( t )— w ( ci( t ) /[ yi( t )+ Yz ( t )]),                                          (28)

μ j( t ) = ( ρ r+ ft) μj( t )                                                                 (29)

ν ( t )[вн z ( t ) — Xi ( t )] = 0,                                                                    (30)

lim e-ρt λ (t) z (t) + μk (t) yk (t)   0.                                             (31)

t+

k=1

We define the costate trajectories by

λ(t) =


γ ( ρ — r ) κ + вн ( ft — κ )
ρ — R + пвн

μi ( t ) = ft,
μ
j(t) = 0 for j = i.

15



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