JKH: I'll have a go at that (no doubt Nick will add more insight too).
The multiplier effect depends on the extent to which Ricardian equivalence holds. If it holds perfectly - i.e. people expect all increase in income to be eaten up by future tax increases - then the multiplier effect should in theory be zero.
However, the truth of Ricardian equivalence also depends on the multiplier effect! Full Ricardian equivalence is based on the idea that national income will not be affected by a stimulus. If instead the stimulus does increase income, then future tax revenues will be increased automatically, partly offsetting the marginal tax rises that are expected. And does the stimulus increase income? Well, that depends on the multiplier effect!
Although I haven't worked out the mathematics, I expect there would be at least two stable equilibria: one where Ricardian equivalence is complete and self-fulfilling, and another where it is not complete and also self-fulfilling.
My choice between them is to throw in the fact that at least some people do not rationally optimise their lifetime predicted income, and therefore a stimulus will have at least some effect for those people. This is enough to give a non-zero multiplier, which in turn means Ricardian equivalence in its traditional sense will not hold, and thus the rational expectation of the investor should be for increasing income.