Public goods are good that have two special characteristics-non excludable and non-rivalable. Non-rivalable means that consume the goods of one person do not influence the consumption of others, of example traffic light, when a person use the traffic light, the quanlity of the light do not decrease, other can also use the goods under the same quanlity. Non-excludable means that every one can use the goods, nobody is exclude from using the goods, for example, national defences, no one is unprotectable when they are in the country. Merit goods are goods that are under provided because of lack of information.
Individual do not appreciate the benefit that will occur, these goods should be provided in a greater quantity. For example, museums, education there are lots of benefit to consume them. People can gets more knoeledge there and this is not only benefit for themselves but also people around and the society, the suicides may decrease and people with eider knowledge entered in works will make the process efficiency. So it is clearly that public good are goods that is under provided but this does not means that it only provided by government.
They can provided by the market mechanism but to make more available the government provides them, subsidizes them or leigislates to make consumption compulsory, and for public goods they suffer from the free rider problem, if asked whether they would pay for them, households would lie and say because once provided, they could benefit anyway. No one is willing to pay for the goods, they will not be provided in the free market. So government must provided them. Public goods must be provided by the government but merit goods not must be provided by the government. It can provided by free market mechanism.