Stock pollution 1

ECON 4910 Spring 2007 Environmental Economics Lecture 9: Stock pollutionPerman et al. Chapter 16

Lecturer: Finn R. Førsund

Stock pollution 2

Effects of pollution Two ways

Environmental effects of pollution within a time period as function of pollutants discharged during the same period

Environmental effects within a time periods a function of accumulated amounts of pollutants from earlier periods

The time dimension of accumulation effects Real time: accumulation over short time periods, e.g.

accumulation of organic waste over a few hours in a river, important when ecosystems are highly vulnerable to extreme values in real time, threshold values for when “bad things” happen, e.g. day variation in oxygen in rivers

Discrete time with longer time periods, day, week , month , year: total load is the determining the pollution effect, not variation within the chosen time period, e.g. death of fish when snow melts in spring with accumulated acidity.

Stock pollution 3

Waste accumulation model

Stock of pollutants at time t from previous emissions eo ,e1……. et

Environmental impact from the stock of pollutants At and not from the current emission et

Damage function( ), 0t t t tD D A D

0

t

t tA e dt

Stock pollution 4

Accumulation as entropy The long-run situation: the Haavelmo

predictament

The only solution is to stop accumulating with At < Ao

A more general situation with depreciation of stocks due to natural processes: decay of materials, decomposition due to bacteria, sunlight, carbon sinks, chemical reactions Generation of pollution balanced against decay

( )limo

tA A

D A

Stock pollution 5

Waste accumulation model with decay Decay due to natural processes in Nature

α ”radioactive” decay coefficient Decay balancing current emissions

Critical loads: current emission corresponding to the decay of a stock that does not yield significant damages in the ecosystem

, 1 0t t tA e A

0t t tA e A

Stock pollution 6

A two-period problem Most simple model of waste dynamics without

decay damage function Dt(.) in accumulated waste benefit function in current emissions

Social planning problem faced in period 1

β discount factor: 0 < β < 1 Must know the emission in period 2 to decide on

emission in period 1 implying that the whole path of emissions must be decided in period 1

1 1 1 1 2 2 2 1 2{ ( ) ( ) ( ( ) ( ))}Max B e D e B e D e e

( ), 0, 0t t t t tB B e B B

Stock pollution 7

A two-period problem, cont. First-order conditions

Period 1

Period 2 (as decided in period 1)

Assumption: no restriction on emission in period 2 Solving simultaneously for e1 and e2.

1 1 1 1 2 1 2( ) ( ) ( )B e D e D e e

2 2 2 1 2( ) ( )B e D e e

Stock pollution 8

Illustration of the two-period case for period 1

B1’

βD2’

D1’

D1’+βD2’

e1

B’,D’

e1*(e2)

Infinite horizon model for waste accumulation in continuous time The planning problem with decay of

accumulated waste

Stock pollution 9

0

[ ( ) ( )]

0,

0

t

rte t t

t t t

o

t

Max B e D A e dt

subject to

A e A

A

e

The mathematical method

The Hamiltonian plays same role as the Lagrange function Current value Hamiltonian: variables are not

discounted to the time of the planning decisions; time zero, but refer to time t

The Hamiltionan consists of the objective function and a constraint expressing the change in the stock variable over time

Stock pollution 10

( ) ( ) ( )t t t t tH B e D A e A

Rules for using the Hamiltonian to get first-order conditions The static first-order condition for the control

(flow) variable

The dyamic first-order condition for the state (stock) variable

Stock pollution 11

( ) 0t tt

HB e

e

( )t t t t tt

Hr D A r

A

Interpreting the first-order conditions Rearranging the static condition

The shadow price on the stock-accumulation equation is negative: more waste reduces the objective function

Balance between marginal benefit of emission at time t and damage created by this emission from t to infinity expressed by the shadow price at t

Stock pollution 12

( )t tB e

Interpreting the first-order conditions, cont. Interpreting the dynamic first-order condition

Shadow price decrease (increase) if marginal damage is less (greater) than the rental value of the shadow price

The sum (α+r) can be interpreted as a ”gross rate of discount”:

future damages are discouted with r, and the decay coefficient also reduces the damage by reducing the stock with a fixed rate

Stock pollution 13

( ) ( )t t tD A r

Interpreting the first-order conditions, cont. Combining the static and the dynamic

conditions introducing marginal benefit to facilitate interpretation

The shadow price increases (decreases) when marginal damage of the stock is higher (lower) than the “gross interest” on the marginal benefit of emission.

Stock pollution 14

( ) ( )( )t t tD A B e r

Steady state

In steady state the stock of waste and the shadow price on the stock are constant

Inserting the static condition into the dynamic condition and setting the change in the shadow price equal to zero (dropping index t)

Marginal damage is set equal to the gross interest rate on the stock shadow price

Stock pollution 15

0 ( ) ( )( ) ( ) ( )

( ) ( )

D A B e r D A r

D A r

Steady state, cont. From the growth equation for waste

The flow of emission is equal to the amount of decay of the accumulated stock taking place

Interpretation of the shadow price on the stock of waste in steady state

Shadow price equal to present value of the marginal damage,using gross rate of interest

Stock pollution 16

0 , /e A e A A e

( )D A

r

Stock PollutionStock pollution 17

Steady state, cont.

The present value of damages shall be equal to the marginal benefit in steady state

Two equations to determine the variables A and e in steady state, eliminating A

( )( )

( )

D AB e

r

( ) ( / ) ( / )( ) ( )

( ) ( )

D A D e D eB e B e

r r r

Stock PollutionStock pollution 18

Studying steady state using a phase diagram Variables:, the stock A, and the control

variable (the instrument) e. The differential equations governing the

development of these variables

Growth equation for the stock of waste

( , ) , ( , )t t t t

dA def A e g A e

dt dt

t t tA e A

Stock PollutionStock pollution 19

Phase diagram, cont.

Second equation: start with differentiating w.r.t time the static first order condition

Inserting into the dynamic condition yields

0t tde dBdt dt

( )tt

deB D rdt

Stock PollutionStock pollution 20

Phase diagram, cont.

Substituting for the shadow price from the static first order condition

( ) ( )

( )( ) ( )

( )

t

t t t

t

deB D r D B rdt

de B e r D A

dt B e

Stock PollutionStock pollution 21

Phase diagram, cont.

Solving

From the steady state solution

The curve for no change in A is a straight line through the origin with a positive slope of α

, 0t tA e

/A e

Stock PollutionStock pollution 22

Phase diagram, cont.

Finding the location of curve where

Solving for A

A falling curve in (A, e) space with convex damage function and concave benefit function

( )0 ( )( ) ( )t

B r De B e r D A

B

1[ ( )( )]A D B e r

Stock PollutionStock pollution 23

The phase diagram

1/α

A

Ao

eo

b

d

ca

e

de/dt = 0

dA/dt = 0

e*

A*