Seminar Exercise 3

1. Seminar Exercise 3 ECN 4910

2. Exercise 1 • a) For the problem to make sense a > d. • Social welfare is maximised by solving the following problem: maxxx (a – by)-(d + gy)dy = maxx (a – d) x –½(b + g)x2  foc: (a – d) – (b + g)x = 0  x = (a – d)/(b + g)

3. Graphic Solution • Make the area under a-bx minus the area under d-gx as large as possible a Gain from regulation d (a – d)/(b + g) a/b

4. b) Coase theorem • Important point. A set of property rights must be defined. Some point on the line [0, a/b] defines how much can be polluted. • Let this point be x*. • Then for every x* a net surplus is generated by moving from x* to (a – d)/(b + g). The gross gain to the winner is always larger than the gross loss to the loser so the winner can compensate the loser.

5. Graphic Solution a Net surplus always positive Loss to polluter d x* (a – d)/(b + g)

6. c) Pigouvian taxes/Subsidies • The basic idea of this exercise is that the total compensation should be generated by taxes or subsidies. The polluter pays a tax or receives a subsidy. The total of this sum is used as compensation for participation

7. Graphic Solution with taxes • The total tax is not enough to compensate the polluter a Net surplus always positive + Loss to polluter d x* (a – d)/(b + g)

8. Graphic Solution with Subsidy • The total tax is not enough to compensate the polluter a Net surplus always positive Loss to polluter from subsidy d Net gain to polluter x* (a – d)/(b + g)

9. Exercise 2 • First problem. Standard theory of the firm. Solve: Maxx,y (px⅓y⅓ – wx – ry) Solution: x = p3/(27rw2) , y=p3/(27r2w) • These are the unconstrained levels. If y is required to be constrained at some level where y > p3/(27r2w). The marginal benefit of y is obviously zero. Inthe following it assumed that y is constrained below this level.

10. The effect of constraining y • Two approaches. • Direct approach. Solve Maxx (px⅓y⅓ – wx – ry) Find Solution: x* = Insert into px⅓y⅓ – wx – ry and take the derivative with respect to y. • Awful math!

11. Now recall the envelope theorem • Let x be a vector and y be a parameter • Consider the problem Maxx (F(x,a)) subject to G(x,a) ≤ 0. • Optimal solution may be written x*(a) • The derivative dF(x*(a),a)/da = ∂L/∂a where L is the Lagrangian. • THIS STUFF IS IMPORTANT. YOU CAN READ THE MATH IN ESSENTIAL MATHEMATICS FOR ECONOMISTS, K SYDSÆTER, section 14.2

12. The shadow price approach • Solve Maxx,Y (px⅓Y⅓ – wx – rY) s.t. Y ≤ y • Form Lagrangian: • L= (px⅓Y⅓ – wx – rY) – λ(Y – y) • F.o.c: ∂L/∂x = ⅓px-⅔Y⅓ – w = 0 ∂L/∂Y = ⅓px⅓Y-⅔ – r – λ = 0 As y is now at or below the unconstrained level, the constraint is binding. Therefore y = Y and λ ≥ 0.

13. The math is still pretty bad • Solution Y = y. x = and, tada... • λ = λ(y) = • By the envelope theorem the marginal benefit of y is given by λ.

14. Properties of λ(y) • limy→0λ(y) = ∞ • λ(y) = 0 → y ≥ y=p3/(27r2w) (Why?)

15. Total benefit • Again: Two approaches • Direct approach. We already have x* = • Benefit from y is found by inserting x* into px⅓Y⅓ – wx – rY • Show-offs may find it by integrating λ(Y) from 0 to y