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Public Goods

Part 1. Definitions

Economics 313

Public Goods

Slide 2

 We have seen that if externalities are present the market equilibrium may not be efficient

 Another source of market failure that is quite similar to externalities is the presence of public goods

 We have assumed until now that all goods are private goods

Private Goods v Public Goods

Slide 3

 Private goods are  excludable: If you don’t pay you won’t get the good.  rival: if you consume a certain amount of the good there is

less to consume for others.

 Public goods are  non-excludable: If you don’t pay you can still get the good.  non-rival: your consumption of the good does not diminish

the amount available for others.

Mixed Goods

Slide 4

 People can be excluded, but consumption is non-rival: (example: cable TV)

 Nobody is excluded but as more people use it becomes rival: (example: highways)

 Lots of examples where goods “sort of” fit the definition of public goods (impure public goods)

 Note that just because a good is publicly provided it doesn’t mean it is necessarily a public good

Concept Check  The Internet is A. A private good B. A public good C. A mixed good D. All of the above E. None of the above

Concept Check  A taco is a A. private good B. public good C. mixed good D. All of the above E. None of the above

Public Goods  From the defining characteristics we can see that

public goods are a special type of nondepletable multilateral externality-producing good

 Where there are public goods the first welfare theorem will fail, and we cannot be sure that our equilibrium will be efficient

 It does not necessarily mean our equilibrium will be inefficient - just that it is not guaranteed to be efficient (more on this later).

-- end of part 1 --

Public Goods

Part 2. Streetlight Example

Economics 313

Public Goods: Example  A simple example: streetlights can be thought of as a

Public Good (PG)  Non-excludable: Once streetlights are installed, we can’t exclude some

consumers from consuming their services

 Non-rival: If I consume the services, there is no less light left over for others to consume

 Suppose there are currently no streetlights, there are just 2 residents, and streetlights cost $35 each

Public Good: Example  Suppose that Consumer A is willing to pay:  $50 for the first streetlight  $30 for the second streetlight  $10 for the third streetlight  Nothing for the fourth

 And that Consumer B is willing to pay:  $40 for the first streetlight  $10 for the second streetlight  Nothing for the third

Public Good: Example  How many streetlights will be installed in the street, if A &

B don’t coordinate on streetlight purchases?

 Note that A & B’s decisions here are inter-related: what A wants to do depends on what B has done (or what A thinks B will do), and vice versa

 Suppose (arbitrarily) in this example, that A makes decisions first.  A will to buy the first streetlight because its worth $50 to her and only

costs $35.

 A won’t buy a second one, as its only worth $30 to her.

Public Good: Example  Given that A has purchased a streetlight, what will B

do?

 There is already one installed, the second is worth just $10 to him, so he won’t buy any  He may not buy any, but he gets to consume 1

 Note how non-rivalry and non-excludability is key here

Public Good: Example  Now suppose (also arbitrarily) that instead B makes

purchase decisions first.

 He will also choose to buy the first streetlight because its worth $40 to him and only costs $35.

 Given that B has purchased a streetlight, what will A do?

 There is already one installed, the second is worth just $30 to A, so she won’t buy any.

Public Good: Example  In equilibrium then, one streetlight is installed no

matter who makes the first decision.

 Indeterminacy (in this ex) as who provides the public good, but not on the level of provision

 Is this level of PG provision efficient? A. Yes B. No

Public Good: Example  Can we make at least one individual better off without

making the other one worse off?  If so, then the equilibrium can’t have been efficient.

 Suppose A and B coordinate on the purchase of a second streetlight.  If A contributes $27 and B contributes $8 they raise enough money to

purchase the second light ($35).

 A paid $27 for something she valued at $30 and B paid $8 for some they valued at 10 ⇒ They are both better off.

 An actual PI! ⇒ the original allocation (of one streetlight) can’t have been efficient.

Public Good: Example  What is the source of the inefficiency?

 A looks only at their individual willingness to pay (or benefit from consumption) and compares it to the price.  A fails to account for the benefit that flows to B.

 B looks only at their individual willingness to pay (or benefit from consumption) and compares it to the price.

 Consequently, A & B do not provide enough of the public good.

Public Good: Example  Note that under-provision (inefficient provision) of the

public good will not always occur.  Suppose in our example the price of streetlights is $41.

 Recall:  A is willing to pay $50 for the first streetlight, $30 for

the second, $10 for the third, and nothing for the fourth.

 B is willing to pay $40 for the first streetlight, $10 for the second, and nothing for the third.

Public Good: Example  If price is $41, then in equilibrium only one streetlight

is installed and that is efficient

 Because the other party would not be willing to pay for a second if they coordinated

 Failure of the 1st welfare theorem means that the equilibrium is not guaranteed to be efficient, not that it is necessarily inefficient

Concept Check  Private provision of a public good A. is necessarily inefficient B. is necessarily efficient C. may not be efficient D. would be included under the first welfare theorem E. would be the best thing for everyone

-- end of part 2 --

Public Goods

Part 3. Efficient Provision

Economics 313

Public Goods Generally  Recall, that when each individual is maximizing their utility,

Individual willingness to pay = MRS and individuals are consuming on their budget line

 Equilibrium provision of the PG will therefore be where consumer marginal rates of substitution are equal to the budget trade off with other goods

 That is, the equilibrium in competitive markets looks very similar to every other consumer choice problem we have seen so far. But in this case, the equilibrium isn’t efficient

Public Goods Generally  Very important difference: A only pays for units of x that she

provides, but she gets to consume units that B provides also.

 That is, 𝑥𝑥𝐴𝐴 enters A’s budget constraint, while 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 enters A’s utility function (and hence her MRS). Similarly, 𝑥𝑥𝐵𝐵 enters B’s budget constraint, while 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 enters B’s utility function (and hence his MRS)

 This implies individuals will not fully account for how their provision of a public good impacts others. Also, they don’t need to pay for the benefits of other’s expenditures, and can “free ride” on other peoples’ contributions.

Public Good: Example  With public goods (when it is non-rival and non-

excludable) aggregate willingness to pay is what is relevant for social efficiency, not just individual willingness to pay

 Aggregate willingness to pay = 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵

 So need to compare 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 to the price ratio ( 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

) in order to determining whether the socially efficient

amount of the public good is being supplied

Public Good: Example  𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 > 𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦 ⇒consumers are willing to pay

more for a good in aggregate than it costs society to produce it → efficiency requires increasing the amount of x

 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 < 𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦 ⇒consumers were willing to pay less in aggregate than it cost society to produce it → efficiency requires decreasing the amount of x

 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦 ⇒aggregate willingness to pay for the last unit purchased just equals the cost → efficient

Public Good: Example  If preferences are strictly convex, then in competitive

equilibrium:  𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 =

𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

 The equilibrium level of PG provision, consumers are in aggregate willing to pay more for an extra unit of the PG than that it would cost.

 They should buy more, but don’t → under-provision (inefficiency)

⇒ 𝑴𝑴𝑴𝑴𝑺𝑺𝑨𝑨 + 𝑴𝑴𝑴𝑴𝑺𝑺𝑩𝑩 = 𝟐𝟐 𝒑𝒑𝒙𝒙 𝒑𝒑𝒚𝒚

> 𝒑𝒑𝒚𝒚 𝒑𝒑𝒚𝒚

Optimality Conditions Generally

Slide 28

 Private goods (X and Y):

𝑀𝑀𝑀𝑀𝑆𝑆𝑌𝑌 𝑓𝑓𝑓𝑓𝑓𝑓 𝑋𝑋 𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝑌𝑌 𝑓𝑓𝑓𝑓𝑓𝑓 𝑋𝑋

𝐵𝐵 = 𝑀𝑀𝐶𝐶𝑥𝑥 𝑀𝑀𝐶𝐶𝑌𝑌

 One private (X), one public good (G):

𝑀𝑀𝑀𝑀𝑆𝑆𝑋𝑋 𝑓𝑓𝑓𝑓𝑓𝑓 𝐺𝐺 𝐴𝐴 + 𝑀𝑀𝑀𝑀𝑆𝑆𝑋𝑋 𝑓𝑓𝑓𝑓𝑓𝑓 𝐺𝐺

𝐵𝐵 = 𝑀𝑀𝐶𝐶𝐺𝐺 𝑀𝑀𝐶𝐶𝑥𝑥

 The condition for the efficient provision of a public good is called the Samuelson condition after Paul Samuelson.

-- end of part 3 --

Public Goods

Part 4. Strategic Interactions

Economics 313

Private vs Public Goods  In our discussion of private goods we were able to show

that a particular social institution – the competitive market – was capable of achieving a Pareto efficient allocation of private goods

 A major assumption of this analysis was that individuals consumption did not affect other peoples utility – thus everyone optimizing with respect to their own consumption was sufficient to achieve optimality

 This is not the case for public goods since individuals utilities are linked

Interdependent Utilities & Game Theory  When utilities are interdependent, my choices will affect

the choices of others.

 To predict how agents will behave in these sorts of circumstances (a game) we need a model of equilibrium behavior.

 Nash Equilibrium (NE) is when all players are simultaneously playing a best response to the choices of other players.

 I am assuming you learned this concept 203, but will review

An Example of NE with Public Goods  Assume we have two room mates who want to get a TV.

Suppose they both have $500 in wealth and each value the TV at $100. Assume the cost of the TV is $150. Since the sum of the two evaluations of the good are greater than $150, it would be Pareto efficient for them to purchase the TV.

 Assume that there is no way for one of the roommates to exclude the other from watching the TV and each roommate will decide independently whether to buy the TV or not.

An Example of NE with Public Goods  Label one of the roommates “player A” and the other

“player B”.

 If Player A buys the TV, he gets net benefits of -$50 and Player B gets to watch TV for free and gets net benefits of $100.

 We can show these payoffs in a game matrix and find the Nash equilibrium outcome (where each player is best responding to the actions of the other player).

An Example of NE with Public Goods

-50, -50 -50, 100

100, -50 0, 0

Player B

Buy Don’t Buy

Player A

Buy

Don’t Buy

*Player A’s payoffs are first, then Player B

NE: mutual best responses: Don’t Buy, Don’t Buy

-- end of part 4 --

Public Goods

Part 5. Mechanisms

Economics 313

Ways to Deliver Public Goods?  Individuals may naturally form associations to provide

public goods

 Social norms about donations to public goods may also evolve

 Complete government provision funded through taxation

 Will majority voting result in efficient public good provision?

Will Voting Deliver Efficient Provision?  For the ease of the argument, assume we have 𝑛𝑛 voters

where 𝑛𝑛 is an odd number. Let’s say also consumers are voting on the amount of expenditure on a public good.

 Each consumer has a most preferred level of expenditure and their valuations of other levels of expenditure depend on how close to their ideal it is.

 It is perfectly possible that if consumers have vote on three levels of expenditure, a majority might prefer A to B and B to C, but a majority may also prefer C to A!

Will Voting Deliver Efficient Provision?  This intransitivity can result in “cycles” around different

policy choices and the order of how policies are introduced will matter for what is selected

 For example, if you vote on A vs. B and then A vs C, C will be the outcome. However if you vote on C vs A, then C vs B, B will be the outcome

 What are the conditions on preferences that would avoid this sort of cycling?

Will Voting Deliver Efficient Provision?  It turns out if voters have “single peaked preferences” this

sort of situation can be avoided.

Net Utility

Expenditure

Net Utility

Expenditure

“Multiple-Peaked”“Single-Peaked”

Will Voting Deliver Efficient Provision?  Assuming single-peaked preferences for everyone, what

would be the result of a majority vote on public goods provision?

 Solution: Median level of expenditure – that expenditure where half of the population wants to spend more and the other half wants to spend less

 Intuition: if more than one-half wanted more expenditure on the public good, they would vote for more, so the only possible equilibrium voting outcome is when the votes for increasing and decreasing expenditure are balanced.

Will Voting Deliver Efficient Provision?  Is the median level the efficient level? In general, no.

 The median outcome just means that half the population wants more and the other half wants less: it doesn’t say anything about how much more they want the good. Since efficiency also takes into account values, voting will not in general lead to an efficient outcomes.

 Economist have proposed various schemes to estimate demands for PGs including the “Vickery-Clarke-Groves” mechanism (see Econ 325 & 452)

What I expect you to know  Know the technical requirements for what a public good

is and be able to give examples

 Solve basic public goods problems like these examples and those in your problem set

 Suggest methods to insure efficient provision and discuss whether majority voting will deliver efficiency

 Next week we will go through some more sophisticated examples of public goods problem and solving them.

-- end --