# 6. Public goods (+ public bads) (2016)

Apunte InglésUniversidad | Universidad Autónoma de Barcelona (UAB) |

Grado | Economía - 3º curso |

Asignatura | Public Sector Economics |

Año del apunte | 2016 |

Páginas | 6 |

Fecha de subida | 18/04/2016 |

Descargas | 7 |

Subido por | alex_izquierdo |

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6. PUBLIC GOODS (+ PUBLIC BADS)
“Private goods”
xR + xF = X ––> “Rival Consumption” (Rivalry) + “Excludability”.

Pareto-Efficient Condition
MVR(xR) = MVF(xF) = MC (xR + xF) –> MRSRx,y = MRSFx,y = MRTx,y
–> Institutional Framework –> Px
“Public goods”
Non-rival consumption (non-rivalry) + non-excludability.

Pareto-Efficient Condition
“National Defense”
MVR (Y*) + MVF (Y*) = MC (Y*)
MRSRxR,y = MRSFxF,y = MRTx,y
max UR (xR , Y)
xR,xF,Y
s.t. UF (xF,Y)=UF
F (XR + XF, Y) = 0
There are infinitely ways to achieve the
point we want. Robinson could pay 3 and Friday
6, 2 and 7, 4 and 5, etc. But, with the particular
distribution which every one pay an amount equal
to their marginal valuation of the good at point Y*,
they will agree providing Y*. F.e. 3,6.

MVR (Y*) + MVF (Y*) = MC (Y*)
f (Y) = MC (Y)
Lindahl Prices
P*R = MVR(Y*)
P*F = MVF (Y*)
UNANIMITY
P*R - ε = 3 - ε
P*F + ε = 6 ε
YR = Y F = Y
The marginal valuation for the good could be very different. But, with Lindahl prices, we can
assure that the provision of the public good will conduct to a Pareto-improvement: both will prefer Y*
than 0. Lindahl prices provide a reasonable benchmark in the provision of the public good.

Lindahl Prices
Assume that MC is constant.

PF = αc
PR = (1-α)c
P*R = α*c
P*F = (1-α*)c
Lindahl prices
This only works if there is a truthful revelation of preferences. But individuals prefer to not reveal
it, so they can save their money. This is known as the “free ride” problem. This is not incentive
compatible to reveal your preferences. So, market mechanism and unanimity will be inefficient.

Will the provision of public goods throw majority voting be consistent with an efficient provision
of public goods?
- Can we assure existence of a majority notion? Yes? Ok. No? Bad luck. Under certain
circumstances, we can assure exist a majority notion, but not always.

- If yes, will it be Pareto-efficient? No in general,
but…
Example:
We assume three groups of people with
the following preferences.

- I “Bored boring” (muermos): A > B > C
- II “Middle of the way”: B > A > C
- II “Exalted”: C > B > A
With individuals voting their preference
action, no majority will be achieved. Let’s consider
the “Robin-round vote” (todos contra todos).

Alternative B is preferred to any other.

A
B
B
C
I
II
I
III
III
II
Median Voter Theorem (D. Black)
If:
- Unidimensional issue.

- Odd number of individuals (voters).

- Preferences are “single peaked” (there are at least and no more than one peak).

Then, there will always exist a majority motion, and it will coincide with the maximum preference
of the median voter.

When will not exist a majority?
- I “Bored boring” (muermos): A > C > B
- II “Middle of the way”: B > A > C
A
B
I
II
III
- II “Exalted”: C > B > A
Cycling. No majority motion.

B
C
C
A
II
I
III
I
III
II
- Three individuals (A, B, C).

- One public good (or bad) to be elucidated.

- Key assumption: the fiscal system is exogenous (ta, tb, tc). Individual preferences concerning the
public good are single-peaked. (ta + tb + tc = MC constant).

Theorem (Bowen)
If:
- Each individual pays the same
“tax price” for the public good. (MC=c; ti=c/3).

- There is a symmetric distribution if individual
preferences (median=mean).

1/HMC = tmedian = MVmedian = MVmean = 1/H ∑ MV
hypothesis
median voter theorem
hypothesis
definition
∑MV = MC
Then, the majority rule will lead to the Paretoefficient level of Y.

If kurtosis is not 0 (the distribution is symmetric), the Bowen theorem will not work, so the
majority voting will not be Pareto-efficient.

With two public goods (n public goods) can we
ensure existence of a majority motion? No.

- Separate votes: You choose one input a forget the
other in one vote, and the other way round in
another.

–> (Y1, _); (_, Y2): This conduct to a no cycling result.

The amount of Y1 will be the maximum preference
of the medium voter of input Y1, and the same for Y2.

–> (Y1/Y2): First, individuals choose the preferred
ratio between Y1 and Y2. Here, B is the medium voter,
so his line will be the chosen.

Then, they choose the amount they prefer. B will choose point B. A and C will choose the more
closed point to A and C, so the medium voter amount of every input separately will be the amount
of Y1 and Y2 decided.

Theorem (Plott)
Under certain conditions (stringent), cycling will not be the outcome. This condition say that if the
maximum preference of one of individuals lay on the political bilateral contract curve between the
other two, there will be no cycling.

Collective consumption of private goods
For example education or health services. They are rival, so they are private goods, but and
consume collectively, and in the same amount.

∑Hh=1 Zh = Z
Z1 = … = Zh = ZH => Zmine· H = Z
MV1 (X1) = … = MVH (XH) = MC (X1 + … + XH) (Private)
MV1 (Y) + … + MVH (Y) = MC (Y) => ∑MVh(Y) = MC (Y) (Public)
(Private goods collectively consumed) =>
∑ MVh (Zmine) = H· MC (Zmine)
=
1/H ∑ MVh (Zmine) = MC (Zmine)
Is N/2 optimal?
K/N is the optimal. N/2 is inclusive in K/N? The optimal inclusivity will be the one that minimize
the cost in the voting procedure. If unanimity is required, the cost of the voting procedure could be
infinite. So, we have to decide what is the optimal inclusivity point.

We have to take into account the
decision-making (CD-M) costs and the
inefficiency costs (CI).

Buchan + Tullock (The Calculus of Consent)
–> “Public Choice” –> an economic theory
of politics.

min C (N) = CDM (N) + CI (N)
K* => MCDM (K*) = - MCI (K*)
Social choice
“a (mathematical) economic
theory of voting results.”
Public choice (Buchanan + Tullock)
“an economic theory of Politics.”
Individuals are not political eunuchs.

...