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Chapter 16
STOCK POLLUTION PROBLEMS
STOCK POLLUTION PROBLEMS
Extending the resource depletion model to
incorporate pollution damage
Figure 16.1 The structure of the aggregate stock pollution model
Fossil fuel
stock
Pollution flows
Fossil fuel
extraction
Production
Capital
stock
Clean-up or
defensive
expenditure
Consumption
Environmental
pressure
Pollution
stocks
Welfare
Utility
The aggregate stock pollution
model: notation
• E = index of environmental pressures or degradation
• higher E means worse environmental quality
• A = The concentration level of pollution
• M = pollution emissions
• Γ = Extraction and processing costs
• V = clean up expenditure
• F = reduction in pollution stock from cleaning up
• α = constant decay or assimilation rate of the pollution stock
• Other variables (U, C, R, M) defined as before
Ut = U(Ct, Et)
[UC >0, UE<0]
(16.1)
Et = E(Rt, At)
[ER >0, EA>0]
(16.2)
→
Ut = U(Ct, E(Rt, At)
Qt = Q (Kt, Rt, E(Rt, At)
(16.3)
(16.4)
Γt= Γ(Rt)
Ft=F(Vt)
[FV > 0]
(16.6)
  M(R )  A - F(Vt)
A
t
t
t
S t  R t
 t  Q(Kt , R t , E t )  Ct  Γ(R t )  Vt
K
 C ΓV
which uses the accounting identity: Q = K
(16.7)
Pollution stock accumulation
•
•
Equation 16.5 specifies that emissions depend upon the amount of resource
use, R.
By integration of equation 16.5 we obtain
 t
At 
 M R   A  d


0
•
So for a pollutant which is not infinitely long-lived (α > 0) the pollution stock
at time t will be the sum of all previous pollution emissions less the sum of all
previous pollution decay, while for a perfectly persistent pollutant (α = 0) A
grows without bounds as long as M is positive.
The optimisation problem:
Select values for control variables Ct, Rt and Vt, for t = 0, ..., ∞, so
as to maximise
t 
W

U(C t , E(R t , A t ))e ρt dt
t 0
subject to the constraints
S  R t
 t  M(R t )  αAt
A
- F(Vt)
 t  Q(Kt , R t , E(R t , A t )  Ct  Γ(R t )  Vt
K
The Hamiltonian:
(
H = U Ct, E(Rt, At)
)
( )
+
Pt –Rt
+
ωt Q[Kt, Rt, E(Rt, At)] – Ct – Г(Rt) – Vt
+
t M(Rt) – αAt – F(Vt)
(
(
)
)
First Order Conditions from the Hamiltonian:
First Order Conditions from the Hamiltonian can be rewritten as:
Interpreting the FOC
UC,t = ωt
(16.8a)
• A static efficiency condition: in each use to which a resource is put, the
marginal value of the services from it should be equal to the marginal
value of that resource stock in situ.
• In each period, the marginal utility of consumption UC,t must be equal
to the shadow price of capital ωt (prices are measured in units of utility
here).
• The marginal net benefit of using one unit of output for consumption is
equal to its marginal net benefit when it is added to the capital stock.
P  P
• 16.8d is a Hotelling dynamic efficiency condition for the
resource net price
• Provided that the utility discount rate is positive, the
resource net price must always grow at a positive rate.
• Note that the ambient pollution level does not affect the
growth rate of the resource net price.
Equation 16.8f
•
Equation 16.8f is a dynamic efficiency condition which describes how the
shadow price of pollution, , must move along an efficient path.
•
As this condition is not central to our analysis, and because obtaining an
intuitive understanding of it is difficult, we shall consider it no further.
    QK 

( 16.8e )
• A dynamic efficiency condition, which requires that each
asset or resource earns a rate of return at all points in time
that is equal to the social rate of discount.
• Dividing both sides of 16.8e by ω, we obtain
t

 QK ,t t  
t
( 16.8e )
i.e. the return to physical capital (its capital appreciation
plus its marginal productivity) must equal the social
discount rate.
t   t FV ,t
•
•
•
•
•
•
( 16.8c )
A static efficiency condition: efficiency in clean-up expenditure
ω is the shadow price of capital; the amount of utility lost when a unit of
output is diverted from consumption or investment to clean-up expenditure.
 is the optimal value of one unit of ambient pollution (a negative quantity, as
pollution is harmful)
FV is the amount of pollution stock clean-up from an additional unit of cleanup expenditure.
So the right-hand side, –FV, is the utility value gained from pollution cleanup
when one unit of output is used for clean-up expenditure. This must be set
equal to the value of utility lost by reducing consumption (or investment) by
one unit.
Equivalently, optimal pollution clean-up expenditure will be the level at which
the marginal costs and the marginal benefits of clean-up are equal.
Equation 16.8b
First, rearrange Equation 16.8(b) to read as:
Gross Price(t) =tQR,t = Pt + tR,t - UE,tER,t - tQE,tER,t-tMR,t
or, suppressing time subscripts:
Gross Price =QR= P+R-UEER-QEER-MR
That is:
Gross Price = Net Price + Extraction cost + 3 Pigovian taxes
applied to the stock and flow pollution externalities
The following slides show
the steps building up to
Figure 16.2, then Figure
16.2 itself.
Again, time subscripts on
variables have been
suppressed for simplicity.
Units of utility
P
Net Price

Time, t
Units of utility
P+R
Marginal extraction cost
P
Net Price

Time, t
Units of utility
P+R-UEER
Utility flow damage
P+R
P

Time, t
Units of utility
P+R-UEER-QEER
P+R-UEER
Pollution flow damage
P+R
P

Time, t
Figure 16.2 Optimal time paths for the variables of the pollution model
Units of utility
Gross Price =QR= P+R-UEER-QEER-MR
P+R-UEER-QEER
Stock damage
P+R-UEER
P+R
P

Time, t
Units of utility
Figure 16.3 Optimal ‘three part’ pollution taxes
Gross price
Note: All elements
are in general time
dependent
Stock damage tax
Pollution flow damage tax
Utility damage tax
Net Price +
Extraction cost
Net
price, Pt

Time, t
utility units
utility units
MC
MC (Social)
MB
MB = QR
MC
(Private)
QR
Optimal
tax
M*
M
M*
Notes: MC = MC (social) = P + R – UE ER - QE ER - MR
MC (private) = P + R
Optimal tax = – UE ER - QE ER - MR
MB = QR
Figure 16.4 Optimal taxes and the wedge between private and social costs.
M
A complication: variable decay
of the pollution stock
•
•
Throughout this chapter, we have assumed that the proportionate rate of natural decay of
the pollution stock, α, is constant.
That is, although a larger amount of decay will take place the greater is the size of the
pollution stock, the proportion that naturally decays is unaffected by the pollution stock
size (or by anything else).
•
This assumption is very commonly employed in environmental economics analysis.
•
However, this assumption is usually made for convenience and analytical simplicity.
•
Whether it is reasonable or not depends on the problem under study.
•
Often it will not be reasonable, because the rate of decay changes over time, or depends
on the size of the pollution stock.
Of particular importance are the existence of threshold effects, irreversibilities, and time
lags in flows between various environmental media (as with greenhouse gases).
•
A complication: variable decay
of the pollution stock (2)
•
Another way of thinking about this issue is in terms of carrying capacities (or
assimilative capacities, as they are sometimes called) of environmental media.
•
In the case of water pollution, for example, we can think of a water system as having
some pollution-assimilative capacity. This enables the system to carry and to
continuously transform some proportion of these pollutants into harmless forms through
physical or biological process.
•
Our model has in effect assumed unlimited carrying capacities: no matter how large the
load on the system, more can always be transformed at a constant proportionate rate.
•
Whether this assumption is plausible is, in the last resort, an empirical question.
However, there are good reasons to believe that it is not plausible for many types of
pollution.
Where the assumption of a constant decay rate is grossly invalid, it will be necessary to
respecify the pollutant stock–flow relationship in an appropriate way.
•
An application of a stock pollution model to
the problem of global climate change
•
•
•
•
•
William Nordhaus – with various collaborators – has been developing a suite of
integrated economic–scientific models to examine policy responses to global climate
change.
While being different from the model we have developed here in the way it is structured,
in its degree of richness and complexity, and in many of its specifics, Nordhaus’s DICE
(Dynamic Integrated model of Climate and the Economy) model shares much of the
essence of the model developed above.
It is an optimal growth model, augmented by the presence of a rather sophisticated
science component that models emissions paths into greenhouse gas atmospheric
concentrations, and then on into implied time paths for global mean temperature
changes.
Using the latest implementation of this model, DICE-2007, Nordhaus is able to estimate
a global efficient emissions reduction schedule over time, and to suggest particular rates
of carbon tax that would bring about an efficient outcome.
The DICE model, along with some other models of global climate change and the body
of work that has emanated from the IPCC, were looked at earlier in Chapter 9.
The DICE-2007 model of global warming
The Structure of DICE
•
DICE-2007 employs an optimal economic growth modelling framework,
integrated with a science component which comprises a carbon cycle and a
system of climate equations.
•
The carbon cycle is based on a three-reservoir model calibrated to existing
carbon-cycle models and historical data.
•
The three reservoirs are the atmosphere, the upper oceans and the biosphere,
and the deep oceans.
•
Carbon flows in both directions between adjacent reservoirs. The mixing
between the deep oceans and other reservoirs is extremely slow.
The DICE-2007 model of global warming
The Structure of DICE (2)
•
The climate equations include an equation for radiative forcing and two equations for
the climate system.
•
The radiative-forcing equation calculates the impact of the accumulation of GHGs on
the radiation balance of the globe.
•
The climate equations calculate the mean surface temperature of the globe and the
average temperature of the deep oceans for each time-step. These equations draw upon
and are calibrated to large-scale general circulation models of the atmosphere and ocean
systems.
•
In DICE-2007 model, the only GHG that is subject to controls is industrial CO 2. This
approach is taken as CO2 is the major contributor to global warming, and because other
GHGs are likely to be controlled in different ways. Other GHGs (CO2 emissions from
land-use changes, other well-mixed GHGs, and aerosols) are included as exogenous
trends in radiative forcing.
The DICE-2007 model of global warming
The Structure of DICE (3)
•
CO2 emissions depend on:
1.
2.
3.
total output
a time-varying emissions-output ratio
an emissions-control rate.
•
The emissions-output ratio is estimated for individual regions, and then aggregated to
the global ratio.
•
The emissions-control rate is determined by the climate-change policy being examined.
•
The emissions abatement cost function is parameterized by a log-linear function
calibrated to recent studies of the cost of emissions reductions.
•
Projected damages from the economic impacts of climate change rely on estimates from
earlier syntheses of the damages, with updates in light of more recent information.
–
The basic assumption is that the damages from gradual and small climate changes are modest, but the damages
rise nonlinearly with the extent of climate change.
The Baseline Case Using DICE
•
Using consensus estimates of values for the key driving variables (such as
growth of labour force, capital stock, technical change, emissions-to-output
ratios, and so on) and assuming that no significant additional emissions
reductions are imposed, one can use DICE-2007 to generate predictions about
GDP, climate change, and the impacts of climate change over some suitable
future span of time.
•
These model inputs and outputs define what is called the baseline case.
(Although Nordhaus does not use this term, such a baseline simulation is also
known as a “Business As Usual” case.)
Some of the principal characteristics of the
baseline case from a DICE-2007 simulation are:
•
Rapid continued increase in CO2 emissions from 7.4 billion tons of carbon per year in
2005 to 19 billion tons per year in 2100.
•
A rapid increase in atmospheric concentrations of CO2 from 280 parts per million (ppm)
in preindustrial times to 380 ppm in 2005 and to 685 ppm in 2100.
•
Measured mean global surface temperature in 2005 increased by 0.7°C relative to 1900
levels and is projected to increase by 3.1°C in 2100 relative to 1900, and by 5.3°C in
2200 relative to 1900.
•
The climate changes associated with these temperature changes are estimated to increase
damages by around 2.5 percent of global output in 2100 and by close to 8 percent of
global output in 2200.
•
The damages are likely to be most heavily concentrated in low-income and tropical
regions such as tropical Africa and India.
Optimisation and Policy
Analysis Using DICE
•
One can run DICE-2007 in full optimising mode.
•
This results in an idealised policy that can be called the “optimal” economic response,
one that establishes a time profile of emissions control which maximises the global
economic welfare function.
•
Such a policy entails GHG emissions being reduced (below baseline levels) in a way
that is efficient across industries, countries, and time.
•
To obtain such an efficient policy outcome the marginal costs of reducing CO 2 and other
GHGs should be equalized in each sector and country.
•
Furthermore, in every year the marginal abatement cost should be equal to the marginal
benefit in lower future damages from climate change.
Optimisation and Policy
Analysis Using DICE
•
This optimal time profile of emissions control is of interest in itself.
•
An important output of the optimal policy response run of DICE-2007 is an
estimate of the “optimal carbon price,” or “optimal carbon tax.”
•
This is the price on carbon emissions that balances the incremental costs of
reducing carbon emissions with the incremental benefits of reducing climate
damages.
•
However, the optimal time profile of emissions also serves as a benchmark
against which various alternative policy responses to global warming can be
assessed.
Major Results from DICE
Policy Analyses
A. Efficient Policy Response
•
Efficient emissions reductions follow a “policy ramp” in which policies involve modest rates of
emissions reductions in the near term, followed by sharp reductions in the medium and long terms.
•
For the efficient climate-change policy, the net present-value global benefit of the optimal policy is
$3 trillion relative to no controls. This total involves $2 trillion of abatement costs and $5 trillion of
reduced climatic damages.
•
The economically optimal carbon price or carbon tax would be $27 per metric ton in 2005 in 2005
prices. For carbon dioxide (smaller by a factor of 3.67) the optimal tax is $7.40 per ton of CO 2. The
optimal carbon price would rise steadily over time, at a rate between 2 and 3 percent per year in real
terms, to reflect the rising damages from climate change.
•
The upper limit on the carbon price is determined by the price at which all uses of fossil fuels can be
economically replaced by other technologies. This is cost of the backstop technology. DICE
estimates this to have an upper limit of around $1,000 per ton of carbon over the next half century or
so, falling thereafter at an unknown rate.
B. DICE: Alternative Policy Responses
•
For most of the climatic-limits cases examined (such as adding a constraint that limits the
atmospheric concentration of CO2 to two times its preindustrial level, or adding a constraint that
limits the global temperature increase to 2.5°C), DICE finds that the net present value of global
economic welfare is close to that of the optimal case.
•
Moreover, the near-term carbon taxes that would apply to the climatic limits, except for the most
stringent cases, are close to that of the economic optimum.
•
All existing programmes fail to meet the efficiency criteria.
•
DICE modeling results point to the importance of near-universal participation in programs to reduce
greenhouse gases.
•
A low-cost and environmentally benign substitute for fossil fuels would be highly beneficial. A lowcost zero-carbon technology would have a net value of around $17 trillion in present value because it
would allow the globe to avoid most of the damages from climate change. No such technology
presently exists, none of the mooted options is currently feasible, but the net benefits of zero-carbon
substitutes are so high as to warrant very intensive research.
Steady-state outcomes: climate example
•
UNFCCC: ‘The ultimate objective of this convention...is to achieve...stabilization of GHG
concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference
with the climate system’.
•
There is no consensus as to what this level is, nor perhaps could there be, given the judgement that
inevitably must surround the word ‘dangerous’.
•
A widely held opinion among natural and physical scientists is that the target should be set at twice
the pre-industrial concentration (i.e. at 560 ppm of CO2 or 1190 GtC in the atmosphere). Many
climate-change research teams have employed this value for one of the scenarios they have
investigated. Nordhaus’s DICE model simulations now suggest that imposing such a doubling as a
constraint on our emissions behaviour is close to being economically optimal.
•
More recently, a growing number of climate scientists are warning that if temperatures were to rise
more than 2°C above 1900 mean levels, a ‘tipping point’ would be passed beyond which positive
feedback effects would generate runaway warming, possibly leading to catastrophic outcomes. Many
climate change modellers are now also looking at what kinds of emissions control programmes
would satisfy the constraint that mean global temperature increases do not exceed that particular
level. not necessarily the lowest cost one).
Steady-state outcomes: more generally
•
Is the notion of a steady state useful, or even meaningful, in the context of the modelling
framework we have been examining in Section 16,1, in which a long-lived stock
pollutant is emitted as the result of combustion of a fossil fuel that is a non-renewable
resource for which there is a finite total stock?
•
Intuition suggests that it may not be. Looking back at the various 4-quadrant diagrams in
Chapter 15, it is evident that nothing resembling a steady state is attained until the nonrenewable resource is fully depleted, or until that resource ceases to be used for some
reason other than simple exhaustion of the resource.
•
Furthermore, our analysis in Chapter 14, demonstrated that if there is no substitute for
such a non-renewable resource and that resource is essential in production, then a steady
state will not be reached in any finite span of time.
•
Thus, for the pollution model we have just been studying, the notion of a steady state is
logically inconsistent and so not meaningful.
Substitution possibilities
•
•
It seems rather implausible, though, to imagine that fossil fuels (or indeed any
other non-renewable resource) have zero substitution possibilities.
So if we wish to think about policy over indefinitely long spans of time, it is
necessary to consider substitutes for non-renewables. Two alternatives suggest
themselves.
1. First, there is the category of backstop technologies.
2. A second possibility is that economic activity might increasingly switch to
using renewable resources in place of non-renewables.
•
That suggests that one might generalise the model specified in Section 16.1 to
version in which the production function is of the form Q = Q(R1, R2, K, E)
where R1 and R2 denote non-renewable and renewable natural resources
respectively.
16.4 A model of waste accumulation and
disposal
• Deals with efficient emissions targets for stock pollutants
where it is not necessary to take account of aggregate level
resource stock-flow constraints.
• Ignoring such constraints appropriate where pollution
derives from the extraction and use of some resource on a
scale sufficiently small that a resource stock constraint is
not binding.
• We call these ‘local’ models of stock pollution.
• Typically more disaggregated than the models previously
examined in this chapter.
Local models of waste accumulation and disposal
• Examples:
• pollution associated with the use of nitrates in agricultural chemicals
• discharges of toxic substances and radioactive substances
• various forms of groundwater and marine water contamination.
• These models best thought of as partial equilibrium cost–
benefit analysis in a dynamic modelling framework.
• Variables now being measured in consumption units rather
than in utility units, so appropriate rate of discount is r
rather than ρ.
• We pay particular attention to the economically efficient
steady-state outcome.
Nitrate pollution example
• There are benefits in terms of increased agricultural yields from using
nitrate-based fertilisers.
• However, the application of nitrate fertilisers cannot be perfectly
controlled, in the sense that only the precise amount taken up by the
growing biomass is applied.
• Surplus-to-growth applications of fertiliser will eventually contaminate
groundwater resources, or will via water runoff processes be
transported into other local water systems, which will themselves be
contaminated.
Nitrate pollution example
• M denotes quantity of fertiliser ‘emissions’
• B(M) denotes the total benefits of fertiliser use
• D denotes economic value of damages, expressed in terms of lost
consumption-equivalent possibilities.
• As we dealing with stock pollutants, the damage caused by excess
fertiliser applications depends on the concentration rate of the pollutant
in relevant environmental receptors. So D = D(A).
• Stock-flow relationship between M and A: dA/dt = Mt – αAt where α
is the decay rate of the pollutant.
• Seek to maximise the net present value of benefits less damages over
some suitably defined interval of time.
16.4.1The steady state
In a steady state, all variables are constant over time
• So dµ/dt in equation 16.12 will be zero.
• Time subscripts redundant and no longer necessary.
The two first-order conditions become
• dB/dM = -µ
(16.13)
• dD/dA = -(R + α)µ
(16.14)
In steady state the pollution-stock differential equation collapses to
• M = αA
(16.15)
16.4.1The steady state
Combining equations 16.13, 16.14 and 16.15 we can obtain:
dD
dB
 dA
dM r  
•
(16.17)
An efficient solution requires that the present value of net benefit of a
marginal unit of pollution equals the present value of the loss in future
net benefit that arises from the marginal unit of pollution.
• Intuition given on next slide.
Intuition
dD
dB
 dA
dM r  
• The term on the LHS is the increase in current net benefit that arises
when the rate of emissions is allowed to rise by one unit. This marginal
benefit takes place in the current period only.
• The term on the RHS is the present value of the loss in future net
benefit that arises when the output of the pollutant is allowed to rise by
one unit.
• dD/dA lasts for ever; it is a form of perpetual annuity.
• To obtain the present value of an annuity, we divide its annual flow,
dD/dA, by a relevant discount rate
• This discount rate is the sum of r and α: the greater is the rate of
decay, the larger is the ‘effective’ discount rate applied to the annuity
and so the smaller is its present value.
Special cases
dD dB  r 

1  
dM dM   
(16.20)
This is a transformed version of (16.17).
Four special cases
•
•
•
•
Case A:
Case B:
Case C:
Case D:
r = 0, α > 0
r > 0, α = 0
r > 0, α > 0
r = 0, α = 0
Case A
r = 0, α > 0
dD dB  r 
dD dB

1





dM dM   
dM dM
(16.21)
• With no discounting , an efficient steady-state rate of emissions for a
stock pollutant requires that the contribution to benefits from a
marginal unit of pollution flow be equal to the contribution to damage
from a marginal unit of pollution flow.
Case C: r > 0, α > 0
dD dB  r 

1   unchanged
dM dM   
• The marginal equality we noted in Case A (the contribution to benefits
from a marginal unit of pollution flow be equal to the contribution to
damage from a marginal unit of pollution flow) remains true but in an
amended form (to reflect the presence of discounting at a positive
rate).
• Discounting, therefore, increases the steady-state level of emissions.
• This is because a larger value of r reduces the present value of the
future damages that are associated with the pollutant stock.
Case B: r > 0, α = 0 and Case D: r = 0, α = 0
dD dB  r 

1   undefined
dM dM   
•
•
•
•
•
•
•
Given that α = 0, cases B and D are each one in which the pollutant is perfectly
persistent – the pollutant does not decay to a harmless form.
No positive and finite steady-state level of emissions can be efficient.
The only possible steady-state level of emissions is zero.
If emissions were positive, the stock would increase without bound, and so
stock-pollution damage would rise to infinity.
The steady-state equilibrium solution for any value of r when α = 0, therefore,
gives zero emissions.
The pollution stock level in that steady state will be whatever level A had risen
to by the time the steady state was first achieved. Pollution damage continues
indefinitely, but no additional damage is being caused in any period.
Any activity generating perfectly persistent pollutants that lead to any positive
level of damage cannot be carried on indefinitely.
b
 0
A
a
A*  18
c
 0
M
d
M*  9
Figure 16.5 Steady state solution and dynamics of the waste
accumulation and disposal model.