Public goods are defined as products where, for any given output, consumption by additional consumers does not reduce the quantity consumed by existing consumers. There are very few absolutely public goods, but common examples include law, parks, street-lighting, defence etc. As there is no marginal cost in producing the public goods, it is generally argued that they must be provided free of charge, because otherwise the people who benefit less than the cost of using the public good, will not use it. That will lead to a loss of welfare. Also the goods are mostly non-excludable, that means that if once provided everybody can use them, which when charged will lead to "free-riding". So these goods will not be provided by free markets as there is no way to charge for the usage, the solution is, that state must provide these goods and finance them from taxes collected from everybody.
Merit goods on the other hand are products generally not distributed by means of the price system, but based on merit or need, because people although having perfect knowledge would buy the wrong amount of them. These goods can be supplied by free market, but not on the right quantity. Merit goods are, for example, education and to some extent the health-care. They are provided by state as "good for you".
Health and education are merit goods, that can be provided by free market, but have major drawbacks when only provided privately.
Main problem is that people will not buy the sufficient amount of them even if they can afford it. People would not go into expensive insurance schemes as they think nothing will happen to them, but if something does happen, they can't afford to pay. This will lead to an uneven distribution of income as rich will be more educated and thus can potentially earn more in the future.
The merit goods when provided privately are usually affordable only to the very rich. As they carry external benefits, government thinks everybody should have them. For example if one is vaccinated it benefits the others as it decreases the risk of epidemic diseases, whereas good education (or investment to human capital) will increase the economic growth and the overall well-being(the supply-side of labour) later on. This view is best explained by social marginal benefits(SMB) and personal marginal benefits (PMB). In this case SMB are bigger than PMB, in free market loss of welfare occurs as shown in the diagram below:
Q0= socially efficient level of usage
Q1= free market equilibrium
ABC-loss of welfare due to underproduction
The provision of health-care (and to some extent education) is also very expensive and requires investment (fixed costs are high), so there is a big scope for economies of scale. It is argued that it is not morally permissible to let a private monopoly control such vital services, so a state provision would be the cheapest. There is also no evidence that bigger classes reduce the quality of education (but they are a lot cheaper to maintain). But there have been several hardships and inconvenience caused due to hospitals concentrating to one location, because the journey times increase. Still UK has one of the best records in efficiency measured in terms of the expenditure per person compared to other developed OECD countries. The life expectancy is the same in UK as in other OECD countries, but treatment of some diseases (e.g. breast cancer) is not improving very fast, and the life expectancy is not growing fast, too.
State can also take account of the long-term demographic trends (e.g. in education build more universities when the number of A-level candidates increases).
Uniform standards are quite important for education as it makes life easier for personnel managers to recruit people later on, as they can compare people better. State has introduced National Curriculum for that purpose.
That is why at present 10% of government expenditure goes to education and people have to pay 10% of their income to social security payments (only from their first tax band). State has set up a National Health Service to deal with health.
Still it would not be beneficial to have only state sector. State cannot provide the most up to date and very expensive service to some people, if the effects are not very big, because then everybody would have to pay for it. Same thing applies to education where the best possible service would be too expensive. That is why an independent sector of 7-8% of the total is established for people who can afford to pay.
This includes public schools and private health-insurance schemes like BUPA. They can provide best possible outcome without charging less wealthy people. Private hospitals for example can afford to have short waiting lists and can keep the patients in care for longer.
The state sector is generally less efficient than private sector as it has got no profit motives. Conservative government has solved this dilemma by introducing competition to health-care (GP can choose to which hospital they send their patients, the creation of internal markets) and to education (universities are funded based on performance, schools are encouraged to transfer from local authority control to state control or to "opt out". Then they will be funded based on the number of students, so better schools attracting more students will receive more funding). This competition is not very efficient at the moment, because the knowledge in education market is poor and the government is unlikely to allow schools to leave the market. It also carries disadvantages as GP-s refuse to accept elderly people whose treatment is more expensive than average etc.
The bureaucracy costs are generally high in state sectors. The creation of competitive internal markets involved recruitment of extra managers etc.
Education and health are very expensive, so if government needs to cut public expenditure, it is likely to affect the quality of education and health.
There are also general disadvantages for high government intervention. If the government builds schools it can crowd out the private investment in the local area etc.
The NHS started out in 1940 based on the Beveridge report as a completely free service for life, but subscription charges were soon introduced. The people might think that they were fooled. On the other hand private providers are more likely to change their policies or go to bankruptcy altogether.