Question

Suppose marginal benefit from a hectare of for a public park (assume it is a pure public good) for two groups of consumers (A and B) is given by: MBa = 10 − Q and MBb = (8 – Q)/2 where Q is the number of hectares of the park. To simplify our analysis, assume that there are only 1 consumer of each type. The marginal cost to provide the park is a constant $5.

a) What is the socially efficient number of hectares for the park?

b) Assume that the consumers each makes a voluntary contribution to a fund which will be used to build the park. The size of the park depends on the amount of money collected. How many hectares will be built in the end? Assume both consumers know the marginal cost and marginal benefit function of each type.

Answer 1

Suppose marginal benefit from a hectare of for a public park (assume it is a pure public good) for two groups of consumers (A and B) is given by: MBa = 10 − Q and MBb = (8 – Q)/2 where Q is the number of hectares of the park. To simplify our analysis, assume that there are only 1 consumer of each type. The marginal cost to provide the park is a constant $5.

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Since there is only 1 consumer of each type, society consists of two consumers - A and B.

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a) What is the socially efficient number of hectares for the park?

This is the point where MC = MB

Here, MB = MBa + MBb

MB = 10 - Q + (8 - Q)/2

MB = 10 - Q + 4 - Q/2

MB = 14 - 3Q/2

and MC = 5

Thus, 14 - 3Q/2 = 5

3Q/2 = 9

Q = 6 hectares

Thus, the socially efficient number of hectares is 6.

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b) Assume that the consumers each makes a voluntary contribution to a fund which will be used to build the park. The size of the park depends on the amount of money collected. How many hectares will be built in the end? Assume both consumers know the marginal cost and marginal benefit function of each type.

Both consumers will equate their marginal benefits.

MBa = 10 − Q = 5

Thus, Qa = 5 hectares

MBb = (8 – Q)/2 = 5

8 - Q = 10

Thus, Qb = (-2) hectares

From the information, Consumer A values the park, while Consumer B doesn't

Consumer B will not pay for the park.

In the end, only 5 hectares will be built.