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

Lemma 1 Let κ < к be given and assume that condition (10) is satisfied. Then it holds that
β > γq and R — nβ < ρ.

Proof: Using (9) it is easy to see that the (10) is equivalent to β > γq. To prove the second
statement of the lemma, we distinguish two cases. If ρ<R, then it follows from the assumptions
κ < K and ρ > r that (ρ — r)γκ/(κ — κ) > 0 > (ρ — R)/n. On the other hand, if ρ ≥ R, then
we obtain from (10) that (ρ — r)γκ/(κ — κ) > ρ — R +(n — 1)γq ≥ ρ — R ≥ (ρ — R)/n. In both
cases it therefore follows that (ρ — r)γκ/ (κ — κ) > (ρ — R)/n. Using (9) again, it is easy to see
that the latter condition is equivalent to R — nβ < ρ. <ι

Proof of Theorem 1: Suppose that all players j = i use the equilibrium strategies φj. In
this situation, player i faces the optimal control problem P defined as follows:

where R = R — (n — 1)β. The Hamiltonian function of this problem is

H = U(ci, yi + γz) — KXi + λ(Rz — Xi) + μ(ryi + x_i — ci) + V μj [ryj + βz — ¢(Vj + Yz)],
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 P as well as the following optimality
conditions are satisfied by c_i(t) = q[y_i(t) + γz(t)] and χ_i(t) = βz(t):

HCi = ^{u ( c}i⁽ t ) /^[ yi⁽ t ) ⁺ Yz⁽ t )]) ^— μi⁽ t ) = ^0,

Hxi = ^μi ⁽ t ) ^{— λ (} t ) ^{— κ} = ^0,

λ( t ) = ρλ ( t ) — Hz = ( ρ — R) λ ( t ) — γw ( ci ( t ) / [ yi ( t ) + γz ( t )]) — ( β — γq)∑ μj ( t ), (13)
j=i

^μ i⁽ t ) = ρμi⁽ t ) ^— Hyi = ⁽ ρ — r ) μi⁽ t )

^μ j⁽ t ) = ρμj⁽ t ) ^— Hyj = ⁽ ρ — r + q) μj⁽ t )

We define the costate trajectories by λ(t) = K — κ, μ_i(t) = к, and μj (t) = 0 for all j = i.
Note that all costate trajectories are constant and non-negative. Condition (11) holds because
of u'(q) = к and condition (12) follows immediately from the definitions of λ(t) and μ_i(t).
Substituting c_i(t)/[y_i(t) + γz(t)] = q and the definitions of the costate trajectories into (13) we

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