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

∂U(c,A)/∂A which follows from A = y + γz. Note that the wealth-adjusted modified golden
rule is not about the steady-state holdings of capital as such, rather it is about the ‘holdings’
of capital/consumption ratio. It is in fact a no-arbitrage condition.

The no-arbitrage condition (7) involves only the private asset with return r. It is reasonable to
conjecture that an analogous no-arbitrage condition must hold for the common property asset.
Furthermore, because γ is the weight given to the common property asset and κ is the cost of
extracting it, it is intuitively clear that this second condition will depend on the parameters γ
and κ. To derive it, let us assume that all agents, except agent i, use the appropriation rate
xj (t) = βz(t), where β is a non-negative number. The private return on the common property
asset for agent i is then obviously given by RR = R — (n — 1) β. Since consumption of the resource

is not possible without prior extraction, the marginal utility of consumption from the common
property asset is given by [∂U(c, A)/∂c] — κ = u' (q) — κ. Finally, the marginal utility of an
additional unit of the common property resource must be [∂U(c,A)/∂z] = γ [∂U(c, A)/∂A] =
γw(q). In analogy to equation (7), the no-arbitrage condition for the common property asset

The formal derivation of this condition follows exactly the same logic as that of equation (6) or
(7). Suppose that the consumption/wealth ratio is constant and equal to q. The cost of reducing
appropriation and consumption at time t by an infinitesimally small amount dx is [u'(q) — κ]dx.
The gain from doing so is that the common property asset holdings at time t increase by dx.
This implies that there are e^R(τ-t)dx additional units of the public asset available at time τ ≥ t.
The additional discounted utility that can be derived from these additional asset holdings is
given by γw(q)dx/(ρ — Rl). Equation (8) therefore says that, at q = q, the marginal cost of a
reduction of appropriation equals its marginal gain.

Let us denote the marginal utility of consumption at q = t by t, that is, κ = u'(q). Furthermore,
provided that κ < t, let us define the number β by

β=ɪ R—ρ +^{( ρ—r} )^γκ
n — 1           κt — κ

Using (6) and R^t = R — (n — 1)β it is easy to see that β = β^t is the only value of β that satisfies
the second no-arbitrage condition (8). We are now ready to state the main result of the present
section.

Theorem 1 Let κ<κt be given and assume that the condition
is satisfied. The strategy profile (φ1,φ2,...,φn) defined by φ_i^c(y,z)=qt(yi+γz) and φ_i^x(y, z) = β^tz
forms a symmetric Markov-perfect Nash equilibrium.

Before we present the proof of the theorem, it will be useful to derive the following auxiliary
result.

<<   5   6   7   8   9   10   11   >>

More intriguing information