Assume we have two producers. Producer 1 has an upstream factory that manufactures computers (q1). The cost of production computers has the following form c(q1), in addition c'(q1) > 0 and c''(q1) > 0. As a subproduct sheep farm that produces wool (q2). The function cost of this farm is c(q2). In addition, c'(q2) > 0 and c''(q1) > 0. However, the sheep farm also faces the damage function d1,2(q1) from the river pollution.
1. Find the first-order condition (F.O.C) from the producer 1 and 2 (∂π1 / ∂q1) and (∂π2 / ∂q2)
What is the socially optimal outcome? (Hint: Find the F.O.C of (∂(π1+π2)) / ∂q1 and (We have the same problem as in Question 1. But now we will give functional form to the cost functions and to the damage function. Assume c1(q1) = 0.1q1^2, c2(q2) = 0.5q2, and d1,2(q1) = 0.02q1^2
1. Find the first-order condition (F.O.C) from the producer 1 and 2 (∂π1 / ∂q1) and (∂π2 / ∂q2
2. Assuming p1 = 1 and p2 = 2 find q1, q2, π1, and π2.
3. What is the socially optimal outcome? (Hint: Find the F.O.C of (∂(π1+π2)) / ∂q1 and (∂(π1+π2)) / ∂q2
4. Assuming the p1 = 1 and p2 = 2 find the new q∗1, π1, and π2, using part 3. Confirm that q2 is the same as part 2.
5. If we impose a tax, such that tax = d'1,2(q1) (the derivative of the damage function), given your q1∗ from part 4, what is the value of the tax?