TARIFF AGREEMENTS AND NON-RENEWABLE RESOURCE ... - Ivie

so that the dynamics of the accumulated extractions is given by
Then, we have that
ẋ = n 2 (a + β A + β M − (c − α A − α M )x) .
dẋ
dx < 0 → dẋ
dx = −n 2 (c − α A − α M )=− n (c − y) < 0,
2
so that the stability condition requires that c − y > 0. This condition is
satisfied by the lowest root of (55) yielding
δ = c − y = 2 3cnr + r
2 0.5 − r
> 0. (57)
3n
With this result α A and α M can be obtained directly from (47) and (48)
Next, we calculate z using (54)
α A = nδ2
4r , α M = nδ2
2r . (58)
z = −
3anδ
4r +3nδ < 0,
and then β A and β M from (49) and (50)
β A = −
anδ
4r +3nδ < 0, β M = −
2anδ < 0. (59)
4r +3nδ
By substitution in (56) we obtain the linear Makov-perfect Nash equilibrium
strategies for the tariff and the price (23) and (24). Finally, using (59)
in (51) and (52) we obtain
µ A =
2a 2 nr
(4r +3nδ) 2 , µ M =
and by substitution the value functions (28).
B Proof of Lemma 1
4a 2 nr
(4r +3nδ) 2 ,
Let us suppose that θ S (0) ≤ θ N (0). 10 Then using (26) and (42) for t =0we
obtain after obvious simplifications that 36cr +16nγ 2 ≤ 27nδ 2 . In Appendix
10 Superscript N stands for the MPNE and S for the MPSE.
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