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## Public Goods, The Private Sector And Policy Intervention Notes

This is an extract of our Public Goods, The Private Sector And Policy Intervention document, which we sell as part of our Microeconomics Notes collection written by the top tier of University Of Cambridge students.

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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. Private provision of the public good means that we are trying to maximise A's utility, taking B's choice of gB as given. This gives rise to the maximisation problem in the box. The result is, that A chooses his contribution to the public goods such that his marginal rate of substitution equals the marginal rate of transformation between the private and public good.

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
[?]
+
[?]
=0
[?]g A [?]x A [?]g A [?]G

- -p

[?] u A [?]u A
+
=0
[?]x A [?] G

? [?]u A ??
- ? [?]G ?

[?]u A
[?]x A

=p -

MU AG
= p - MRS = MRT MU Ax

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:

1

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