In public economics there are two vital economic terms known as public and private goods. Here I will examine the public goods and the crucial characteristics that a public good is required to have, to be a public good as well as the issues and problems that it presents in the society when it comes to determining public policy for such goods.
Public goods are services provided by the government whose benefits extend to a group of individuals or the society as a whole. There are two different types of public goods. These include pure public goods and impure public goods. Several member of the society can consume pure public goods without diminishing in value to any individual by using particular services. The street signs are one of such examples, as they will not wear out even if large numbers of people are watching it. It would be extremely difficult to limit its use to one person or preventing other from looking at it. These goods are therefore non-profitable to be provided privately. The interest in these types of public goods can be traced back to classical economics. It is where "David Hume and Adam Smith agreed that government intervention is needed to supply goods and services characterized by collective benefits"  . If left to individuals or organisations, these goods would not be provided adequately.
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There is also issue of externalities connected to the supply of public goods due to its non-excludability factor. For example pollution is a "public bad". It is where actions of people harm or benefit other people. For instance Victims of negative externalities can't practically charge polluters a fee for suffering, and beneficiaries of positive externalities can't practically be charged for their enjoyment due to spill over and R&D. Public goods have two main characteristics that distinguish it from a private good. Pure public goods are non-rival in consumption and non-excludable. Pure public goods, consumption of which is non-rival if a person can increase its consumption without reducing the quantity available for other's consumption, when they are provided to the individual's. As a principle stated by "Sameulsion in 1954 one individual's consumption does not reduce the other's consumption "It implies that marginal cost of extending the service to additional individual user is zero"  . For example a non-rival good is terrestrial TV, Radio station and fireworks. Many people can consume those goods without paying for them, as marginal cost of extending the service is zero, as one-person's consumption does not diminish the use by other people.
The second characteristic is non-excludability. The "non-excludability of goods is where no one can prevent other person from the consumption of the good's because it might be expensive or impossible"  . For Example NHS service and civil defence in United Kingdom is considered to be a public service being non- excludable. Because when government spends money on civil defence it brings protection and safety to the society or individual regardless of their contribution or non-contributions to the country. As they are non-excludable therefore entry fee cannot be charged. These are provided free and to exclude individuals would be very costly and impossible to stop once it has been provided.
David Hume pointed out that there is a "free-rider problem associated with public goods, even before the time of Adam Smith's writings. Each citizen who can enjoy the benefit of a public good has an incentive to try to lay the whole burden of provision on others, whenever the exclusion of non-payers is very costly or impossible." 
The provision of pure public goods by so-called public sector is itself subject to government failures as externalities and public goods are often linked due to non-excludability factor that is a prominent feature of public goods. Pure public goods are seen as an extreme form of externality and have problems of efficiency in allocation of resources. Efficiency is defined by (Samuleson 1954), requires marginal rate of benefit to equalise to marginal cost of providing the good. "Simuleson argues that public goods cause market failure in the form of underproduction and over financing"  caused by negative, positive externalities and the free rider problem. An example of a positive externality is that of a person who produces a utility and it affects other person's utility. If I don't take this into account and stop doing the activity, my satisfaction will be maximized but it has failed to take into its affect on the other person. It would result in an under- production from paretian point of view.
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Free riding refers to a problem where under production of goods or services provided by the public sector lead to Pareto inefficiency or where there is an excessive use of the public goods. It is relevant in the case where non-excludability and non-rivalry in consumption of public good by the consumer will establish the consumption of the goods without contributing towards its production. This leads to underproduction of public goods and gives rise to free riding. Potential users may wait for the good to be supplied and then consume the good for free. For example NHS is non-rivalry in consumption and non-excludable as non-payers can take a free ride and enjoy the benefits of consumption (free riders). This creates market failure as smaller amount of money is being regenerated into the health service. Thus there is less money for the government to invest into the health services. This results in hospitals having out dated equipment, decrease in hospital beds and cuts in hospital jobs.
To solve the problems of such free riders we can look at following three issues as identified by McMillian: The first issue is that consumer may not be willing to pay for the good such as NHS. For example to solve this issue government can collect taxes for the provision of public goods and the problem might disappear by further restricting the deals with free rider problem  .
The second issue; what is the optimal level of production that requires information on individual demand for public good. People can give false information about their taste as consumer might be hoping to bring about a non-Pareto optimal equilibrium that gives them higher utility. That is caused by lack of information as observed by Samuleson. It causes particular problems in the case of Non-rival good consumption aspect of public goods that leads to "Demand Revelation" problems, as consumer might not reveal their true marginal rate of substitution. Each person's marginal rate of substitution is different which is required to establish welfare maximization. This can lead to excessive supply of funds and inefficiently allocation of the goods as most public goods and services are financed through a process of taxation, having no choice. Optimal levels of expenditures are difficult to be established. The provision of public goods can easily be over-financed or under-financed.
Demand revelation problem in the absence of perfect information. This can be solved by relaying on potential competition in the market in a way to minimise adverse impact of collusion or coalition. "[Tideman and Tullock (1976)] pointed out that Demand revelation is a superior for making choices. Principle person is given a choice of accepting a decision made without his participation or changing to whatever he wants by paying for it and having it done otherwise" 
Furthermore government could simply make an estimate of the optimum output distribution of tax and permit them to do better. The "advantage of this approach is that it would provide the market with information that is otherwise not available to government."  They can also use mechanism that has been designed to induce true preferences revelation under specific condition to attain efficient provision of public goods. For example "Clark (1971) demand revelation mechanism is one such mechanism"  .
The third issue of free riding concern is as to what happens if the number of people grows larger in the economy. The issue can lead to under provision of services and goods. For example, problems like congestion costs arise, due to increase in number of users thus reducing the benefits to each individual. For example, the NHS, where the demand for healthcare is greater than supply; and in the absence of the price mechanism, demand is rationed through waiting lists. That waiting list reduces the benefits of individuals due to overcrowding as fewer resources like beds or doctors are available.
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The Coasian solution could also be used to solve free rider problem, where theorem, suggests the idea that a legal system would be unnecessary for the regulation of externalities in a world without transaction. It is where beneficiaries come to an agreement that satisfies both parties, thus reaching a Pareto effect allocation of resources. For example a farmer's cattle are consuming the protective plants at the edge of another farmer field, which reduces his output. This is not necessary for the government to step in and force parties to act properly. It is in their own interest to come to an agreement, as costs of implementation could be higher for both parties. Therefore, they will be able to find an agreement that is Pareto efficient. Since long transaction costs will be too high.
Societies have dealt with government provision of public goods and services in different ways. That could include where a representative could be elected and represent people preferences. Voting is a common way to decide the allocation of problems. Whether people vote on public issues or elected representatives decide the public goods and their provision. Under majority voting, maximum spending will be based on median voter that might not be an efficient level. The main issue of public good is that an individual is paying for someone else's consumption and sometime people don't pay enough.
Provision of public goods will face economic difficulties. As non-rivalry forces will pose challenges in trying to define the optional level of supply. Given an increase in the number of consumers will enhance the level of aggregate well-being. Secondly the non-exclusive nature of a public good is the source of undersupply, since agents tend to hide their preferences. Efficient provision needs to take into account the costs of design and promotion of efficient collective action. Reaching optimum supply levels require not only financial resources, but also the incentives to get agents involved and the institutional responses to ensure effective fulfillment .The government can correct market failure and provide the socially optimal levels of public goods by financing the provision of public goods with tax revenue. Public financing of public goods may be the only option in cases where the public good is non-excludable.
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Charles K.Rowley, Friedrick Schneider, (2004), The Encyclopaedia of Public Choice Volume 2, P457, P459
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Richard Cornes and Todd Sandler, (1996), The Theory of Externalities, Public Goods and Club Goods, 2nd Edition, P8-9
Stephen Shmanske Public Goods, (1991), Mixed Goods and Monopolistic Competition, P3, P31