Question

1. There are n identical users of a common-property resource. The output of user i is denoted yi . “Pollution” is a by-product of output. Let pi = yi be the amount of pollution generated by user i’s output yi . The profit that accrues to user i—not deducting for pollution damages—when i’s output is yi is given by 2yi − y 2 i . Pollutants reduce profit. When y1 + ... + yn is total output, the loss to each user is given by the damage function (y1 + ... + yn) 2 .

(a) Derive the equilibrium profile of outputs y ∗ 1 , ..., y∗ n .

(b) Show that the equilibrium profile of outputs is inefficient.

(c) What condition must be satisfied for a profile of outputs (y1, ..., yn) to be efficient? Why?

(d) Solve for the efficient profile of outputs y ∗∗ .

(e) Are all users better-off at y ∗∗ relative to y ∗ ? If so, why don’t they coordinate at y ∗∗?

(f) Show that there is a tax t such that if each user is made to pay $t per unit of output, the equilibrium profile of outputs coincides with y ∗∗ .

Answer 1