Answer to Question #307357 in Microeconomics for smarta

Question

Demonstrate how if a public good is to be provided in some fixed amount or not provided at all, then a necessary and sufficient condition for provision to be Pareto efficient is that the sum of the willingnesses to pay (the reservation prices) exceeds the cost of the public good

Answer 1

Suppose Amy and Bev live on a lake and are plagued by mosquitoes, but it is possible to control the mosquitoes by spraying regularly. The mosquito spray cannot be confined to the property of just one of the women: any spray that is used affects the entire lake shore equally. Suppose that Amy is willing to pay at most $28 for the first gallon of bug spray, $26 for the second, $24 for the third etc. and Bev is willing to pay at most $5 for the first gallon, $4 for the second, $3 for the third and so on. Suppose also that the spray costs $24 per gallon. How much of the bug spray should they buy?

To solve Amy's and Bev's problem we have to look at the marginal social value of each gallon of spray and compare it with the social marginal cost. The first gallon Amy and Bev value together at $28 + $5 = $33. The spray costs $24 per gallon. The girls value it more than it costs to obtain and thus they should purchase this unit. The second gallon has marginal social value of $26 + $4 = $30, third $24 + $3 = $27, fourth $22 + $2 = $24. Since Amy and Bev value all four units more than or equally to their marginal cost, they should buy exactly four units of bug spray, because that is the socially optimal level. Notice that this Pareto efficient amount of spray equates the marginal social value to the marginal social cost. Taking this argument further, Amy is willing to pay only $20 and Bev $1 for the fifth gallon of spray. The marginal social value yields $21 and at the cost of $24 it is inefficient to purchase another unit.