with respect to the number of players n. Furthermore, the aggregate extraction intensity nβ is
strictly decreasing with respect to n.
Proof: The first part of this lemma follows immediately from lemma 2 and RR = R — ( n — 1) β.
The second part follows from the facts that R is independent of n (see lemma 2) and that β is
strictly decreasing with respect to n (this lemma), and from nβ = R — RR + β. <ι
The results of this lemma are basically the same as in Tornell and Velasco [11] and Tornell and
Lane [10]. Finally, we investigate monotonicity properties of the equilibrium growth rate of the
public asset. This growth rate is given by g = R — nβ. The following results hold.
Lemma 4 The equilibrium growth rate of the common property asset, g, is decreasing with
respect to the appropriation cost κ, strictly decreasing with respect to the weight γ and the raw
rate of return R, and strictly increasing with respect to the number of players n.
Proof: From the definition of g and from (9) it follows that
n ∣^ ( ρ — r ) γκR∖ R
ρ ρ - 7∙
n — 1 к — κ n — 1
From this expression it is easy to see that the first three monotonicity properties stated in the
lemma hold true. The last one follows from g = R — nβ and lemma 3. <ι
Tornell and Lane [10] call the fact that g decreases with respect to R the voracity effect, which
they define as “a more-than-proportional increase in discretionary redistribution in response to
an increase in the raw rate of return in the efficient sector” ([10, p. 34]). In our model, the
discretionary redistribution corresponds to the term nβ. Our model allows also another inter-
esting observation that follows from the monotonicity of g with respect to κ. More specifically,
if one interprets κ as the cost of money laundering, then it follows that by reducing the cost of
money laundering, a government can increase the net growth rate of the public asset.
Given that both consumption and wealth are arguments of the utility function, it is natural to
ask whether the elasticity of substitution between consumption and wealth has any effect on
the equilibrium growth rate or the intensity of extraction. In order to address this question we
assume that the utility function is of the CES type
U ( c,A ) = ( cε + Aε )1 /ε,
where ε is a real number smaller than 1 and different from 0. The elasticity of substitution is
then given by σ = 1 /(1 — ε). The corresponding functions u and w are given by u(q) = (1 + qε)1 /ε
and w(q) = (1 + qε)(1 ')/ε, respectively. It has to be mentioned that the CES function does
not satisfy the Inada conditions in assumption U. Nor does it satisfy the condition u(0) = 0
when ε is positive. However, we have used these conditions only to ascertain the existence of
a solution to equation (6). Here, we shall show directly that (6) has a unique solution such
that the rest of our analysis remains valid. Indeed, it is straightforward to see that equation (6)
11