- c) − _aq_ ∂∂ _Eq_ <u>1</u> < _E_ 20 − _E_<sup>ˆ</sup> 2<sup>(</sup> _q_<sup>)</sup> for some _q_ < _c_ (but not for _q_ = _c_ ). In this case we may or may not

have an internal solution of (11b). For instance, the fringe may be quite able to deliver profitable offsets relative to emissions reductions in the policy bloc at low levels of _q_ , but not at higher levels of _q_ . The policy bloc does not want the quota price to be too low, however, due to the environmental concern. Hence, it may be optimal to increase _q_ until _q_ ≥ _c_ , i.e., similar to a). However, we may also have an internal solution similar to b).

To sum up so far, it is optimal for the policy bloc to either ban offsets completely, and let the ′ ′ quota price be given by (13), or to have no restrictions on offsets, implying _R_ 1 ( _E_ 1) < _q_ = _R_ 2 ( _E_ 2) < _c_ . Further, the higher is _a_ and the lower is the offset potential relative to domestic emissions reductions, the more likely it is that offsets are banned.

Let us investigate case b) further. We first notice that if _a_ = 0, i.e., no updating rule, then _q_ < _c_ and _k_ = 1 is preferred over the alternative solution _q_ = _c_ and _k_ = 0. The reason is that the latter outcome is equivalent with _q_ = _c_ and _k_ = 1 (see above). But then we know from the investigation of case b) that reducing _q_ below _c_ will be beneficial. This case, i.e., without any updating rule, has already been analyzed in Strand (2012). In other words, without updating it is never optimal to put any restrictions on offset purchases (given our model assumptions), and the optimal quota price should be below the marginal damage cost. The latter conclusion is seen by replacing ′ _R_ 1 ( _E_ 1) by _q_ and setting _k_ = 1 in (11). At _q_ = _c_ the RHS is negative as the policy bloc, acting as a monopsonist, benefits from reducing _q_ due to lower costs of importing quotas.

When _a_ is marginally increased from zero, i.e., if a mild updating rule is introduced, the optimal level of _q_ will increase. This is shown in the appendix in the case of quadratic mitigation costs. The reason is that introducing updating increases the deviation between the marginal costs of abatement, given by (3b), and the marginal costs of emissions, _c_ , for a given level of _q_ . Thus, it is optimal to increase _q_ even though the policy bloc’s costs of buying offsets increase. This implies that, as long as _k_ = 1, emissions reductions in the fringe, and thus the use of offsets, will increase. That is, introducing updating will tend to increase the optimal level of offsets at low levels of _a_ . However, at some point, a further increase in _a_ will tend to reduce the optimal level of _q_ (cf. the appendix). In other words, then it would be optimal to reduce the use of offsets. The reason is, as explained above, that the policy bloc ideally would like to see a lower _q_ in the offset market in order to reduce the costs of buying offsets. This matter becomes relatively more important when _a_ is high, and hence the optimal _q_ declines. Finally, when _a_ becomes sufficiently high, it becomes optimal to stop buying offsets whatsoever, i.e., switch from _k_ = 1 to _k_ = 0.

Although the optimal quota price increases when _a_ is increased from zero, abatement in the policy bloc declines (at least, this is the case with quadratic mitigation costs, cf. the appendix).

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