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Public Goods, The Private Sector And Policy Intervention Notes

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Microeconomics, Paper 2, Part IIA Supervision 04

2nd Dec 2009

"If a public good such as a lighthouse is provided by the private sector, there is clearly no case for
policy intervention." Discuss. Public goods have the property that they are non­excludable and non­rivalrous. This means that no
consumer can be excluded from consuming it and that the consumption of the public good by an
additional consumer does not reduce the quantity available for other consumers. Because of these
properties, the private provision of public goods has positive externalities, as not only the provider, but
also other individuals benefit from the provision of the public good. However, this generally leads to
underprovision of the public good compared with a socially efficient level, as a private provider does
not take these positive spillover effects into account.
We can show this inefficiency by considering the results of the private provision of a public good: We
have individuals A and B, whose utility functions depends positively on a private good x, and a public
good G. Furthermore, individual A/B possesses e(A)/e(B) units of the private good as an endowment
and the production of one unit of G requires to give up p units of the private good. gA and gB denote
the contribution of A and B respectively to the public good. Then, we have the budget constraint for individual A

x A + pg A = e A Consumption by A of the public good is given by

G = gA + gB because the public good, once provided,
can be consumed by A and B equally. Thus,
A's consumption is the sum of the provision
of A and B.

max u A ( x A , G )

s.t. x A + pg A = e A

max u A (e A − pg A , g A + g B )

∂x A ∂u A ∂G ∂u A


∂g A ∂x A ∂g A ∂G

Private provision of the public good means
that we are trying to maximise A's utility,
→ −p A + A = 0 taking B's choice of gB as given. This gives
∂x A ∂ G rise to the maximisation problem in the box.
 ∂u A 
The result is, that A chooses his contribution

→  ∂G 
=p →
to the public goods such that his marginal
∂u A rate of substitution equals the marginal rate
∂x A of transformation between the private and
public good.

= p → MRS = MRT x MU A

However, this is not the pareto efficient amount. To find out the pareto efficient amount of public good
provision, we maximize A's utility subject to a constant utility of B. The results are shown in the
following box:


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