9/24/11

Finding demand curves for public goods, and social demand curves


This post goes over a public goods question, where the individual demand curves need to summed up in order to get a legitimate social demand curve.  The reason we need a social demand curve is because everyone benefits from the good in question, so if each consumer only buys until they are satisfied, then they are ignoring the potential benefits to others in the market.

Four towns share a common water source. By buying open land along the watershed (area from which the water flows) the towns can preserve its purity from sewage, road runoff, and such. The land demand schedule for each town based on water treatment costs saved can be expressed as

P = $34,000-10Qd

where Qd is acres purchased and P is the price the town would be willing to pay.

a. If the cost of land is $30,000 per acre, how much land will be purchased if each town operates independently? How much if they form a joint commission for land purchases? Graph the different possibilities. (If the economic theory is not clear, imagine representatives of the four towns sitting around a table, discussing the costs and benefits of purchasing different amounts of land.)

b. Which is the socially efficient solution and why? How would the answers change if the price of land was $36,000 per acre?
c. Discuss this in terms of the demand for clean water. Is clean water a public good in this case? Can water generally be considered a public good?

The first step to answering a. is to plug in the price of land into the demand function:
$30,000 = $34,000 – 10*Qd
If we add 10Qd to both sides, subtract 30,000 from both sides, and then divide by 10, we will get:
Qd = 400

If each town operates independently then they will each buy 400 acres of land, because that is what their individual demand curves would suggest. 

If the representatives get together, they will realize that when one town buys an acre of land, the other three benefit from that purchase.  In order to internalize this positive externality, we need to account for it.  To that to this problem, we will need to add together the 4 different demand functions, and come up with a new one (this process is similar to the way the demand curve for a public good is created).

So our new demand function will be P = $136,000 – 40Qd, and the associated graph will look like:

In the above graph, the D curve represents each town’s individual demand curve, and it shows that the intersection between the given price and the individual demand curve occurs at Q = 400.  If they each operate independently then a total of 1,600 acres get purchased.  However when we construct the social demand line, we see that the new intersection occurs at a Q of 2650, which is much higher than the 1600 with everyone operating independently.

We can solve for Qd with the new demand function.  To do this we have to add 40Qd to both sides, subtract 30,000 from both sides, and then divide by 40, and we get:
Qd = 2650

This joint venture amount is much larger than the individual amount because the towns now understand that when they purchase more, they are benefiting their neighbors as well.  Under this scenario, each town would purchase 662.5 acres instead of 400.

For part b we need to consider the benefits to society, so the social demand curve is the correct choice.  The reason we need to use the social demand curve is because we have externalities occurring in the market that need to be accounted for.  You could also use the definition of a public good, which states that the individual demand curves for the public good need to summed up vertically instead of horizontally as they are for normal (non-public) goods.  

Now if the price increased to $36,000 per acre, no land would be bought if each town operated independently.  This is because the optimal amount of land being bought would be negative, and since this is not possible, we default to zero. 

However, if we sum up the demand curves, and account for the externality then a positive amount would be bought, see the graph below:

With a price of $36,000, the optimal amount of land bought jointly would be:
$36,000 = $136,000 – 40Qd
Add 40Qd to both sides, subtract 36,000 from both sides, and divide by 40 to get:
Qd = 2,500
So a total of 2,500 acres would be purchased, meaning 625 acres would be bought by each town.

For the last question, water is definitely a public good.  But water can’t always be considered a public good, for example bottled water is consumed by someone then disposed of.  I suppose dirty water would be a public good, but clean fresh water is definitely rival (only one person can use it) but not necessarily excludible is it is a lake or aquifer.  Also, recreation on lakes and the ocean would be considered a public good.  It really depends on how exactly you look at it.

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