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

Because of βH > (R — ρ)/n (see lemma 6) and κ < κ it follows that all costate trajectories are
constant and non-negative. The Lagrange multiplier ν is defined by

( κ — κ )[ ρ — R +( n — 1) вн ] — γ ( ρ — r ) κ
ρ — R + nβH

Because of lemma 6) and (24) we have ν(t) ≥ 0. Using these definitions and (6), it is straight-
forward to verify (25)-(29). Condition (30) holds because of xi (t)=βHz(t).

Defining gH by gH = R — nβH , it follows from lemma 6) that gH <ρ. The transversality
condition (31) can now be verified in exactly the same way as in the proof of theorem 1.
Finally, we have to show that the application of the strategy φi implies that z(t), yi (t), xi (t),
and c_i (t) remain non-negative for all t. Because β_H >γqn has been assumed, this property can
be verified in essentially the same way as in the proof of theorem 1. This completes the proof
of the theorem. <ι

If the assumptions of theorem 1 and βH ≥ βⁿ hold, then the condition βH >γqn required in
theorem 3 is satisfied. This follows trivially from lemma 1. Moreover, condition (24) is true
provided that βH is sufficiently large. The assumptions of theorem 3 are therefore satisfied
whenever those of theorem 1 hold and when βH is sufficiently large. It follows that in this case
the pessimistic equilibrium from theorem 3 and the interior equilibrium from theorem 1 coexist.
The equilibrium stated in theorem 3 does not exist if extraction intensities are not bounded
above, that is, if the constraint (23) is missing. If one would introduce a similar lower bound,
that is a constraint of the form xi (t) ≥ βLz(t), then it would be possible to derive also an
optimistic equilibrium, in which all players extract with intensity βL . The structure of the
equilibrium strategies in this case would be much more complicated because of the co-existence
of the constraints xi(t) ≥ βLz(t) and yi (t) ≥ 0. In particular, we believe that the strategies
of player i in any Markov-perfect Nash equilibrium would have to depend non-trivially on all
private asset stocks and not only on the own private asset holdings yi.⁶ We therefore leave the
characterization of the optimistic equilibrium for a separate paper.

6 Concluding remarks

In this paper, we have shown that new insights can be gained by adding appropriation costs and
wealth effects to the model developed by Tornell and Velasco [11] and Tornell and Lane [10].
A number of testable implications have been derived. For example, countries where powerful
groups have equal appropriation costs have higher growth rates than countries where powerful
groups have unequal costs. Furthermore, an increase in the cost of money laundering reduces
growth.

Our model can be extended in several directions. One may suppose that agents care about both
relative wealth and absolute wealth. This would make the status-seeking motive for the agents

⁶Tornell and Velasco [11] characterize optimistic equilibria in a model without non-negativity constraints on
private asset holdings.

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