Seminar Exercise 3

ECN 4910

Exercise 1

• a) For the problem to make sense a > d.

• Social welfare is maximised by solving the following problem:

maxxx (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)

Graphic Solution• Make the area under a-bx minus the area

under d-gx as large as possiblea

d

a/b(a – d)/(b + g)

Gain from regulation

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.

Graphic Solution

a

d

(a – d)/(b + g) x*

Net surplus always positive

Loss to polluter

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

Graphic Solution with taxes

• The total tax is not enough to compensate the pollutera

d

(a – d)/(b + g) x*

Net surplus always positive

Loss to polluter+

Graphic Solution with Subsidy

• The total tax is not enough to compensate the pollutera

d

(a – d)/(b + g) x*

Net surplus always positive

Loss to polluter from subsidy

Net gain to polluter

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.

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!

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

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.

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.

Properties of λ(y)

• limy→0 λ(y) = ∞

• λ(y) = 0 → y ≥ y=p3/(27r2w) (Why?)

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