Why would a narrow elite ever extend the vote to the masses? Perhaps the hand of the elite is forced by the threat of revolution. To convince the masses that the elite is committed to giving them surplus, the elite extend the franchise. This is argument of Acemoglu and Robinson.
Lizzeri and Persico have a quite different argument which has particular resonance for Britain’s Age of Reform in the nineteenth century. Suppose only a fraction of the population can vote. Two parties, the Whigs and the Tories compete for their vote. The parties can either offer a public good or a transfer with revenue generated via taxation. When the enfranchised group is a small elite, there is an incentive to tax the entire population and then target transfers to swing voters in the elite. That way a party can give them as much as they would get with pubic good provision and get into power. The mass of the elite that is not targeted gets no transfer.
When the franchise is extended pork barrel politics is not as powerful as the taxable endowment is not large enough to offer the now larger majority enough to compensate them for zero pubic good production. Each political party can at least get a 50% chance of getting elected by offering public goods. Hence, an extension of the franchise leads to less pork barrel politics and more public good production. Some members of the elite are indifferent to this change and others – those who were not receiving transfers when the franchise was small – strictly prefer it. Hence, extension of the franchise Pareto-dominates a small franchise. The franchise can be extended even when there is no threat of revolution by the disenfranchised masses.
The Pareto-domination property does not obtain in general (when voters are ideological and pubic good production is not zero-one) but the majority of the elite prefers extension of the franchise. In nineteenth century Britain, members of the elite clamoured for the extension of the franchise. There was less pork barrel transfer and more public good production after the franchise was extended.