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

public asset has to coincide with the rate of return on the private asset. To see whether this
holds true in our model, let us compute the private rate of return on the common property asset
in the equilibrium from theorem 1. It is given by RR = R — (n — 1)/ which, because of (9), is
equal to

R = ρ — ρ ^κ                                    (17)

κ3 — κ

Tornell and Velasco [11] and Tornell and Lane [10] consider a situation where none of the two
asset stocks is an argument of the utility function. This implies of course that both asset stocks
are treated equally which corresponds in our model to the case γ = 1. Furthermore, Tornell and
Velasco [11] and Tornell and Lane [10] do not consider any appropriation costs which would be
reflected in our model byκ = 0. Substituting γ = 1 andκ = 0 into (17) it follows indeed that
R³ = r. Thus, if we choose γ andκ such as to make our model as similar as possible to that of
Tornell and Velasco [11] and Tornell and Lane [10], we obtain exactly their result. In general,
however, R³ can be smaller or larger than r, as can be easily shown by inspection of (17). As a
matter of fact, R³ = r holds if and only ifκ =(1— γ)κ3, R³ > r holds if and only ifκ<(1 — γ)κ3,
and R³ < r holds if and only ifκ>(1 — γ)κ3. These results involve the monetary returns r and
R³ . One could define ‘full’ rates of return for the two types of assets by the left-hand sides of
equation (7) and (8), respectively. In that case, both rates of return would obviously be equal
to the time-preference ρ such that return equalization across assets would hold.

Let us now discuss the monotonicity properties of R³ and β³ with respect to the model parameters.

Lemma 2 The private rate of return on the public asset, R³ from equation (17), is decreasing
with respect to the appropriation cost κ, strictly decreasing with respect to the weight γ, and it
is independent of the number of players n.

Proof: The lemma follows immediately from (17), ρ> r, and γ ≥ 0. <ι

The monotonicity with respect to κ can be explained as follows. The interior equilibrium is an
equilibrium, in which agents extract the resource more quickly than they consume it. Positive
appropriation costs form an impediment to this voracious behavior, because these costs are
incurred before the resource is consumed and the agents have a strict time-preference. The
agents will only be voracious if the rate of return on the private asset is sufficiently high relative
to the return of the public asset. Since the rate of return on the private asset is a fixed constant
r, this implies that, for high appropriation costs, the rate of return on the common property
asset, R³ , must be very low. The fact that R³ is strictly decreasing with respect to γ is also easy
to interpret. A higher value of γ means that agents attach less weight to their private asset
holdings. Therefore, they have a weaker incentive to be voracious and, hence, the rate of return
on the private asset has to be high relative to the rate of return on the public asset in order for
voracious behavior to qualify as an equilibrium. This, in turn, means that the private rate of
return on the common property asset must be low if γ is high. The fact that R³ is independent
of n follows immediately from the no-arbitrage condition (8).

Lemma 3 The equilibrium extraction intensity, β³ from equation (9), is increasing with respect
to the appropriation cost κ, strictly increasing with respect to the weight γ, and strictly decreasing

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