Home > Micro concepts, Public Goods > The demand for public goods

## The demand for public goods

October 22, 2011

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:
$Q_A = 50 - 2P$, $Q_B = 70 - P$, $Q_C = 80 - P$. 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, $P = 25-0.5Q_A$, $P = 70 - Q_B$, $P = 80 - Q_C$.

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, $P = 25-0.5Q_A$, $P = 70 - Q_B$, $P = 80 - Q_C$, 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.