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 + nβH ≥ 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:

+∞

e^-ρt[U(ci(t),yi(t)+γz(t))
0

.  ,   4            ~     ,  4

z( t ) = Rz ( t ) — X_i ( t )

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

^yj^(t) = ryj^(t) ^{+ β}H^z(t) ^- ft[yj^(t) ⁺ Y^{z(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 ( c_i,yi+γz ) — κxi + λ ( R z—X_i )+μi ( ryi+X_i — ci )+∑ μj [ ryj + вн z — ft( yj+γz )] + ν ( вн z — X_i ),
j=i

where λ and μ_k, 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 μ_k : [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 c_i(t) = ft[y_i(t) + γz(t)] and x_i(t) = βHz(t):

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

μ_i ( 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} )⁺ Y^{z ( t} )])^{,                                          (2}8)

^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

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

μi ( t ) = ft,
μj^(t) = ^{0 for j} = i.

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