obtain 0 = (ρ — R)(κ — κ) — γw(q). This equation holds because of the definition of R, equation
(9), K = u,(K), and (6). Analogously, by substituting ci(t)/[yi(t) + γz(t)] = K and μi(t) = κ
into (14), one obtains 0 = (ρ — r)K — w (q). This equation holds because of K = u'(K) and (6).
Condition (IS) holds trivially because of μi (t) = 0 for all t.
To verify the transversality condition (IG), let us define g = R — nβ. Lemma 1 implies that
g < ρ. From the state equation z(t) = Rz(t) — xi(t) and from xi(t) = βz(t) it follows that
z(t) = z0e9t. Consequently, we have limt→+∞ e-ρtλ(t)z(t) = (K — κ)z0 limt→+∞ e(9-ρ)t = 0. From
xi(t) = βz(t), z(t) = z0e9t, and the state equation yi(t) = ryi(t) + xi(t) — ci(t) we obtain
Уi (t) ≤ ryi (t) + βz0 e9t. It follows therefore that there exists a positive constant Y such that
yi(t) ≤ Yemax{r,9}t. This implies that limt→+∞ e-ρtμi(t)yi(t) ≤ κY limt→+∞ emax{r-ρ,9-ρ}t = 0,
whereby the last equation follows from g < ρ and r < ρ. Finally, limt→+∞ e-ρtμj (t)yj (t) = 0
holds trivially for all j = i because μj(t) = 0. The transversality condition (IG) is therefore
satisfied.
It remains to be shown that the feasibility conditions are not violated. More specifically, we
have to show that the application of the strategy φi implies that z(t), yi(t), xi(t), and ci(t)
remain non-negative for all t (independently of the initial conditions z0 and yi0). We have
already proved that z (t) = z0e9t which shows that, starting from any initial state z0 > 0, the
asset stock z (t) remains strictly positive for all t. Now note that lemma 1 implies that β > γ(
and therefore β > 0. It follows that xi(t) = βz(t) ≥ 0. Application of φi implies furthermore
that ci(t) = K[yi(t) + γz(t)] and, hence, yi(t) = (β — γ()z(t) + (r — K)yi(t). This shows that
yi(t)∣yi(t)=0 = (β — γ()z(t) > 0, whereby the inequality follows from z(t) > 0 and β > γ( (see
lemma 1). Therefore, yi (t) cannot become negative if it starts at a non-negative initial value.
Because ci (t) = (K[yi (t) + γz(t)], it follows also that ci (t) remains non-negative. This completes
the proof of the theorem. <ι
The equilibrium described in the above theorem corresponds to the interior equilibrium discussed
in the papers by Tornell and Velasco [11] and Tornell and Lane [10]. It is an equilibrium in
which players are indifferent about how much to extract from the common property resource.
In the proof of theorem 1 this can be seen from the fact that the condition Hxi = 0 holds.
Theorem 1 has two crucial assumptions, namely, κ<κK and condition (10). Note that the latter
is satisfied if R — ρ is sufficiently large or if γ > 0 and κ is sufficiently close to (but smaller than)
κK. Note furthermore that, for γ = 0, condition (10) reduces to the simple inequality R>ρ.
A further remark on theorem 1 concerns the relative size of R and r. In Tornell and Velasco [11]
and Tornell and Lane [10] it is assumed that R>rholds. If γ = 0, this is exactly what
condition (10) implies. In the present model, however, the assumption R>ris not necessary.
As a matter of fact, let us consider the case where R and r are chosen such that R<r<ρ
holds and where γ is positive. As has been mentioned before, condition (10) will be satisfied
in this situation provided that κ is sufficiently close to κK. Thus, the interior Markov-perfect
Nash equilibrium described in theorem 1 exists also in situations where the total return on the
common property asset, R, is strictly smaller than the return on the private asset, r .
In Tornell and Velasco [11] and Tornell and Lane [10] it has been shown that, in the interior
equilibrium corresponding to the equilibrium from theorem 1, the private rate of return on the