A policy decision needs to be taken. Do you let technocrats conduct a cost-benefit analysis or do you let people vote on it? Our first intuition would tell us that a referendum is best when people's opinions are widely dispersed, and a cost-benefit analysis is best when not.
All wrong, claim Martin Osborne and Matthew Turner. The reason is that referenda only reveal ordinal information about preferences, whereas cost-benefit analysis also reveals cardinal information: If a majority is only slightly bothered by a policy, but a minority sees huge benefits, social welfare would indicate such a policy should be implemented. A referendum would not allow it, whereas a cost-benefit analysis would. But referenda have the advantage to attract well-informed voters, as uninformed ones typically abstain. A cost-benefit analysis would also sample uninformed ones, which is bad.
Thus: use referenda when there is poor information and there is little dispersion in preferences, whereas use cost-benefit analyses when there is good information and large dispersion in preferences.