Question

QA = 10 – P

QB = 18 – 3P

QC = 14 – .5P

If the marginal cost is $4 a unit, how many units should be bought if the good is a public good? How much would each person be willing to pay?

Do consumers of public goods have the same incentives to reveal their true valuations of Public goods as they do of Private goods? Why or why not?

Answer 1

The purpose of this problem is to help you understand the difference in market

demand for purely private and purely public good. For each of the following situations suppose

that private marginal benefits are given by

MP B

= 10

−

Q

, where

Q

is the quantity of a

good, keep in mind that

MP B

measures the maximum price a person will be willing to pay

for an additional unit of the good.

a) Suppose the good is purely private, for example, apples. On a diagram plot individual

demand for apples and construct market demand for apples if there are two consumers

in the market.

Market demand for private good is obtained by adding quantities demanded

at each price.

You can construct market demand by points: at price

P

= 10

none of the consumers is

willing to purchase any quantity, then total quantity demanded is

Q

= 0

, this is the first

point on your diagram (0,10); at

P

= 9

each person will demand 1 apple and the total

quantity demanded in the market is

2

·

1 = 2

, the second point is (2; 9) and so on.

You can also use algebra: from the MPB you can find

individual quantity demanded

as a function of price

Q

1

= 10

−

P

is demand by the first person and

Q

2

= 10

−

P

is

quantity purchased by the second person at each price;

market demand is the sum

of individual quantities

Q

M

=

Q

1

+

Q

2

, substitute the expressions for the individual

demands (plug 10-P for

Q

1

and

Q

2

) which will give you

Q

M

= 10

−

P

+ 10

−

P

, simplify

to

Q

M

= 20

−

2

P

and single out price as a function of

Q

M

. Both methods give you

exactly the came market demand curve:

P

= 10

−

.

5

Q

M

where

Q

M

is the total quantity

demanded at each price. Notice that the market demand curve has the same vertical

intercept as individual demands, has half of the slope and twice the horizontal intercept.

b) Suppose the good is purely public, for example a street lights installed in the neighbor-

hood. On a diagram plot private marginal benefits and construct market demand for

the public good if there are two consumers.

Pure public good is

nonexcludable

and

nonrival

in consumption. Nonexcludable

means that if one of the consumers purchases one unit of the good, the other consumer

will be able to consume that unit as well and there is no way to preclude the second

consumer from enjoying the benefits. Nonrival means that the fact that the second per-

son is consuming the good does not diminish the benefits to the first person. When you

construct the market demand for public good use the notion of ‘maximum amount all

consumers will be willing to pay in order to purchase an additional unit’.

You can construct the market demand curve by points: for example, for the first unit

Q

= 1

of the good each of the consumers is willing to pay P=9 dollars, it means that the

maximum that both will be willing to pay in order to ‘install’ the first unit of the public

good is 18, so one of the points on the market demand for the public good is (1, 18); for

the second unit one person will pay up to 8 dollars, max. that two will pay is 16, the

second point is (2, 16) etc.

You can also notice that

when you construct market demand for public good

1

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you are summing up the individual marginal benefits for each unit

,

therefore

D

M

= 2

·

MP B

= 2(10

−

Q

) = 20

−

2

Q

. The market demand curve has

vertical intercept twice higher compared to the individual demand, as twice the slope and

the same horizontal intercept.

c) Compare the two diagrams. What will happen to the shape of market demand curves

if we add more consumers in each case?

The most important thing to notice about diagrams is the following. When we aggre-

gate the individual demands for a private good, the resulting market demand curve is

a horizontal summation of individual demands. When we find market demand for the

public good we use a vertical summation of the individual demands. I think that in this

case ‘demand for public good’ might seem like a misnomer, because the curve literally

tells you ‘the maximum amount that consumers would collectively pay for an additional

unit’, and therefore rather represents the marginal social benefits, as we will see from

the next problem it has little to do with the quantities that will be privately purchased in

the market if consumers do not share the costs of purchasing each unit.

In this simplified case with identical consumers adding more people will look as follows.

In case of private good the vertical intercept will always be at 10, but as we add more

consumers the horizontal intercept will move proportionately further from the origin,

therefore the slope will always be equal to the slope of individual demands divided by

the number of consumers. In case of public good adding more people to the society will

move vertical intercept proportionately higher while leaving the horizontal intercept in

the same spot, this means that the slope will be getting steeper