# Efficiency Criteria to provide a public good?

Suppose there are two consumers in the whole economy. If the cost of a public good is 100, and the maximum willingness to pay for a public good for consumers 1 and 2 is 80 and 25, would it still be efficient to provide public good?

Here is my confusion:

• Because total willingness is greater than the cost, the good should be efficient to provide.

• If I think of max willingness to pay as benefit derived, then total benefit > total cost, so it would be inefficient because they need to be equal.

• Can you think of a way to distribute the cost of producing the public good among the two consumers such that both are better off with the good being provided? May 22 at 10:05
• @MichaelGreinecker Lindahl pricing? May 22 at 12:58
• Not that complicated. You just need to find two positive numbers $c_1$ and $c_2$ such that $c_1+c_2=100$, $c_1<80$, $c_2<25$. Then, if consumer $1$ pays $c_1$ and consumer $2$ plays $c_2$, then the public good is funded (since $c_1+c_2=100$) and both consumers get more benefit from the public good than the cost they have to bear (since $c_1<80$ and $c_2<25$). May 22 at 13:34
• @MichaelGreinecker thanks. If you post this as an answer I will mark this resolved. May 22 at 15:14

A situation in which the public good is not provided is inefficient; it is possible to make both consumers better off. To see this, let $$c_1$$ and $$c_2$$ be to numbers such that $$c_1+c_2=100$$, $$c_1<80$$ and $$c_2<25$$. Such numbers are necessarily positve, for $$c_1=100-c_2>100-25$$ and $$c_2=100-c_1>100-80$$.
Now, provide the public good, let consumer $$1$$ pay $$c_1$$, and let consumer $$2$$ play $$c_2$$. Then the public good can be funded, since $$c_1+c_2=100$$. Consumer $$1$$ is better off than before, because $$80-c_1>0$$; their personal benefit exceeds their personal cost. Consumer $$2$$ is better off than before, because $$25-c_2>0$$; their personal benefit exceeds their personal cost too.