# VCG - A Public Good Problem N individuals 1 N Individuals...

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A Public Good Problem N individuals 1 ,...,N Individuals have to decide whether or not to build a bridge i.e. on the level of a public good y where y takes one of two values 0 (“Do Not Build”) or 1 (“Build”) Each individual i has a net valuation (i.e. beneﬁt - her share of the cost) of the bridge θ i where θ i could be either positive or negative x i : private good (money) received by i . Utility of i is θ i + x i if y = 1 and x i if y = 0, i.e. U ( y,x i ; θ i ) = i + x i 1

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Eﬃciency What is the eﬃcient level of the public good? Suppose there are two individuals 1 and 2 with θ 1 = 6 and θ 2 = - 4. Suppose y = 0. This is not eﬃcient because the bridge could be built and 1 could pay 5 units of money to 2 making them both better oﬀ. Eﬃciency requires N X i =1 θ i 0 y = 1 and N X i =1 θ i < 0 y = 0 SAMUELSON-LINDAHL CONDITION 2
θ i is likely to be the private infor- mation of individual i . But knowledge of these numbers is vital for a planner whose goal is ef- ﬁciency. The θ i ’s must therefore be elicited But will individuals reveal their private infor- mation truthfully? Suppose that individuals announced their net valuations and it was decided to build the bridge only if their sum was non-negative. Then those with positive (negative) net valuations would announce unboundedly large positive (negative) net valuations. Clearly unsatisfactory. How do we proceed?

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## This note was uploaded on 03/03/2011 for the course MARKETING 101 taught by Professor Singh during the Spring '11 term at Management Development Institute.

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VCG - A Public Good Problem N individuals 1 N Individuals...

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