The construction of public libraries is typically financed in two ways.
The first way in which the construction of public libraries is financed is through the use of public funds: that is, money from the government. The government allocates some money, on an annual basis, for the construction of public libraries. That is money you (as an elected community leader) can use to finance the construction of such libraries. Admittedly, the amounts of money going into such programs has been going down over the years: as the government keeps on succumbing to pressure for it to reduce spending on ‘social programs.’ The unfortunate thing is that the people who put pressure on the government to reduce such social spending are stingy folks who imagine that all social programs involve ‘giving cash handouts to lazy people’ (as the pressure groups, mainly made of conservatives, put it). In the process, even good programs, like that of constructing public libraries, end up suffering.
The second way in which the construction of public libraries is financed is through the use of private funds. These can be in the form of donations, raised by members of the community where the library is to be built. The private funds can also be in the form of grants, from local corporations: those being funds that the corporations give out as part of their corporate social responsibility.