In contrast to a private good, where the aggregate demand for the good is found by horizontally summing the demand curves of all the individuals, the aggregate demand for a public good is found by vertically summing the demand curves of all the individuals. Sometimes we are faced with the question of whether it is efficient to provide a public good. This depends on the cost of providing the public good, and the vertical sum of the demand for that public good.
We can think of this in terms of reservation prices. Whilst the non-rivalrous nature of a public good means that all individuals consume the same quantity, people can value the good differently at the margin. When the margin is the choice of whether to provide or not provide the good, the extent to which the individual values the good is that individual’s reservation price.
It is efficient to provide a public good when the sum of individuals’ reservation prices is greater than the cost of the good.
Consider the example of three students (A, B and C) who are flatsharing and are deciding whether or not to get a TV. Say it would cost them £150 to get a TV so they would split it three ways and each pay £50 if they got one. They are going to put the TV in the downstairs living room where it becomes a non-excludable public good, any of them can watch it and when it is on, the fact that one is watching it doesn’t ‘use it up’, the other two can be watching it at the same time, so it is non-rivalrous.
Suppose A places a value on having a TV in the house at £100. He is quite keen to get one, but not keen enough to fork out for it himself. B is considerably less keen, he values having a TV in the house at £40. C is fairly ambivalent about having one, he values it at £20. These are effectively their reservation prices.
If they are going to split the cost three ways and pay £50 each, then whilst A is keen to go ahead and get the TV, B and C aren’t, so if it comes down to a majority vote then A will be outvoted and the public good (the TV) won’t be provided, unless they can find a cheaper TV. A wants a TV but only values it at £100, he won’t pay the full amount himself.
However if you sum up the three students’ reservation prices you get £100 + £40 + £20 = £160, which is greater than the cost of the TV. So it be socially efficient to provide the TV. If everyone’s true preferences were revealed then A should pay £100 towards it, B should pay £40 towards it and C should pay £20 towards it and they will then have to decide what to do with the £10 change! But A would probably feel short-changed in that situation, not many of us would actually be willing to pay more than twice as much as the next person for something that they enjoy the same benefit from, even if we wanted it more. This is the problem of revealing preferences – individuals will have a tendency to understate the true amount they value the good and hope that other people provide it (ie they want to free ride off it). If everyone free rides, nobody provides the public good privately.
The social marginal benefit for a public good is different from the social marginal benefit for a private good. With a private good, everyone puts some value on the good at the margin, but people can consume different quantities. With a public good, everyone consumes the same quantity of the good, but individuals can value the good differently at the margin.
When a consumer consumes a private good, the benefit to society from them consuming another unit (social marginal benefit) is just the benefit to the individual. No other consumer can enjoy that unit as the good is rivalrous. So the social marginal benefit curve is just the horizontal sum of all the individuals’ demand curves.
But in the case of a public good, many users can consume the same unit of output as it is non rivalrous. So the social marginal benefit is the sum of all the private marginal benefits of each person who is consuming that unit. The social demand curve (or willingness to pay curve) for a public good is found by vertically summing the individuals’ demand curves.
Lets look at an example, firstly for a private good.
Assume there is a private good, and an economy with three consumers, A, B and C. Their respective demand functions are:
, , . The total market demand curve is the sum of all the three consumers’ individual demand curves.
Be careful not to just add the three demand functions up here, it’s not simply 50-2P+70-P+80-P = 200-4P because the consumers are not going to demand negative numbers and if you do that you will get the wrong answer. That would tell you that when P = 40, Q = 40, but that would be wrong. When P = 40, consumer A demands 0, B demands 30, C demands 40 so Q = 70. The problem with just adding up the demand function is that when P = 40, A demands 0 and not -30.
So you have to construct the demand curve by looking at what quantity each consumer would demand at each price. Eg at a price of 10, A demands 30, B demands 60, C demands 70, so overall demand is 160.
The resulting horizontal market demand looks like this:
Now consider what would happen if this was a non-excludable public good. Consider that this was say units of street lighting.
This time we look at the aggregate willingness to pay for a particular number of units of the good by vertically summing the demand curves.
So we have to think of the inverse demand functions, , , .
Say the quantity provided was 10, and as it is non rivalrous, each consumer can consume 10 units simultaneously. A would be willing to buy 10 units at a price of 20, B would be willing to buy 10 units at a price of 60 and C would be willing to buy 10 units at a price of 70, so the aggregate willingness to pay is 150.
What about if we think of a quantity of 60? A would not demand 60 units at any price. B would demand 60 units at a price of 10, and C would demand 60 units at a price of 20, so the aggregate willingness to pay is 20.
The vertical sum of demand curves looks like this.
Now consider what would happen if the market supply of this public good was horizontal at a price of P = 60.
When we vertically aggregate the demand curves to make the demand for the public good, and find the equilibrium when we have a supply curve at P = 60, we find that the social equilibrium would come at an output of 46, this is the socially efficient output for the public good. So we have an inefficiently small output of the public good.
Why is 46 the social equilibrium at a price of 60? Well remember an output of 46 of a public good means that each consumer gets to enjoy 46, it isn’t a rivalrous good which means they have to split the 46 between them, they are all enjoying 46 units of this good at the same time. Consider how much each consumer would be willing to pay for 46 units. Going back to the inverse demand functions, , , , consumer A would be willing to demand 46 units at a price of 2 per unit, B would be willing to demand 46 units at a price of 24 and C would be willing to demand 46 units at a price of 34. 2 + 24 + 34 = 60 which is the price at which the public good is being supplied.
So the most socially efficient solution would be for the three consumers to join together and for A to pay 2, B to pay 24 and C to pay 34 and that would afford the price of 60 needed to purchase 46 units of the public good which they could all enjoy.
However that is not likely to happen in practice. A gets the same benefit from the 46 units of public good as C does despite paying 17 times less! So C would think this is hardly fair. So what about a solution where they split the price of 60 three ways and each pay 20? This would suit B and C who each value the public good more than that, but A wouldn’t be interested in paying 20 as I is only willing to demand 46 units at a price of 2 per unit.
And also if all three of the consumers know how much each of them value the good, then A and B know that C values it more than anybody. So A and B could just decide not to buy any and at a price of 60, C would demand 20 units. So A and B could just free ride off the 20 that C buys, because this is a non-excludable public good. So not only would there be an inefficiently small output compared to the social optimum, but A and B can freeload off C. The implication of this is that all three consumers have an incentive to try and disguise their true preferences, ie how much they really want the good, to bluff the others into thinking that they aren’t willing to pay for it so someone else would have to.
Public goods are non-rivalrous meaning that when one consumer consumes a unit of the good, it does not prevent anybody else from consuming it and it does not use the good up in any way. An example of this would be the light provided by a street light. There is a cost of providing the street light, but the cost does not change when you supply a unit to an extra consumer, you can provide lighting for as many people as are on the street for the same price. So the marginal cost of providing to an extra consumer is 0.
Now think of this in efficiency terms. The Pareto efficient output would come in a competitive market where P = MC, so the efficient output here when MC = 0 is that P = 0. No private producer is going to sell at P = 0.
A private producer would only supply the good if non purchasers could be excluded and it could charge a price such that P > 0. Where exclusion is impossible, such as the street light example, then it is not profitable to produce.
When a public good is excludable then there is the possibility of private provision and a market existing for the good, but it will not be efficient, because again the non-rivalrous nature of a public good means MC = 0, and so if the private provider produces where P > 0, then P > MC and this will mean the output produced is less than the socially efficient amount.
Markets exist for public goods when they are excludable, but the market is likely to produce too little of the good.
The markets for software and music that is easily pirated are good ones to think about here. When exclusion is possible by pirating then it is not profitable to produce and sell software and music so no private providers would produce them. They only produce if there is some form of intellectual property rights protecting them. But given that the MC of producing another copy is very small, if the private producer is producing at a level where P > MC then the amount produced is less than the social optimum.
Two important concepts when we are thinking about classifying goods as private or public goods are the concepts of rivalry and excludability.
A good is rivalrous if one person consuming it ‘uses it up’, meaning someone else cannot consume it. If you fill your car with petrol and then use it up, nobody else can use that petrol. If you eat a sandwich nobody else can eat it. Those are rivalrous goods. However if you create a beautiful painting that people enjoy looking at, the painting is not rivalrous as it doesn’t matter how many people look at it, you aren’t ‘using it up’.
A good is excludable if you can prevent somebody from using it. If you need a ticket to go into the cinema then it’s excludable. Street lighting is not excludable though because anybody walking down the street at night benefits from it, you can’t make the light shine on some users and not on others.
These concepts allow us to classify goods into certain categories:
– Private goods are rivalrous and excludable, although sometimes the government provides publicly provided private goods (eg housing).
– Open access common property is rivalrous and non-excludable, an example of this would be fish in the ocean, it’s difficult to stop people from coming in and fishing, but when they catch fish there will be less for everybody else
– Public goods are non-rivalrous, clean air is a public good, so is national defence or street lighting. Usually you will see the definition that public goods are non-rivalrous and non-excludable, but there are some public goods like cable TV or club goods such as concerts and swimming pools that are non-rivalrous but it is feasible to exclude users.
Some public goods are impure public goods – they are not directly rivalrous, but when too many people use them, congestion becomes a problem (roads would be a good example here). The congestion means that people are effectively excluded from them when the roads are blocked.
A non-excludable public good is effectively a positive externality (or a public bad is a negative externality). If you clean up the environment then you can’t prevent other people from enjoying the cleaner environment – this is a non-excludable public good and a positive externality. Excluding anyone from consuming a public good would be inefficient.
Non-excludability causes another problem – the problem of free riding. If you can’t exclude somebody from using the good, then if one person privately provides the good, everybody else enjoys the same benefit but doesn’t have to share in the cost. So this incentivises people to not pay for provision of the public good in the hope that others will do so.
This is especially a problem in the context of revealed preferences. If you would be interested in having a public good provided, but think that somebody else is equally or maybe even more interested in it, you have an incentive to understate the extent to which you want it, so that they think you won’t pay for it and that if they want it they will have to pay for it themselves. That way you can free ride off the public good being paid for by someone else. But if everyone free rides, nobody provides the good.
Markets for public goods will only exist if non purchasers can be excluded, there are no markets for non-excludable public goods so usually if government does not provide it then nobody will. National defence is an example of a non-excludable public good. However health and education are not entirely public goods in the same way as there is an element of rivalry to them – if you are receiving some drugs in treatment for an illness then you are using them up, no other consumer can use them at the same time.