File: Thirdch16.doc
Date: 18 July 2002
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EDITOR: Figures for this chapter are located in the following separate files:
Chapter16pictures.doc (Figure 16.1)
Picturesforchapter16.ppt (Figures 16.2, 16.3, 16.5)
Chapter 16
Stock pollution problems
Look before you leap.
Proverb, source unknown; most
likely source is Aesop's fables
Learning Objectives
In this chapter you will

Investigate two models of optimal emissions which are suitable for the analysis of
persistent (long lasting) pollutants. Each of these models is a variant of the optimal
growth model framework that we have addressed before at several places in the text.

The first model you will study is an ‘aggregate stock pollution model’ which is
appropriate for dealing with pollution problems where the researcher considers it
appropriate to link emissions flows to the processes of resource extraction and use.
This will enable you to see how optimal pollution targets can be obtained from
generalised versions of the resource depletion models we investigated in Chapters 14
and 15.
1

The second - a ‘model of waste accumulation and disposal’ - provides a framework
that is suitable for analysing stock pollution problems of a local, or less pervasive,
type, such as the accumulation of lead in water systems or contamination of water
systems by effluent discharges.

We stress, more strongly than has been the case hitherto, the dynamics of pollution
generation and pollution regulation processes, using phase plane analysis for this
purpose.
Introduction
Our analysis of pollution targets in Chapter 6 recognised that some residuals are durable.
Their emissions accumulate, impose loads upon environmental systems which persist
through time, and can result in harmful impacts. Processes of this form were called stock
pollution problems. In this chapter, we revisit our previous analysis of pollution targets (in
Chapter 6). Two modelling frameworks will be examined. We refer to these as an ‘aggregate
stock pollution model’ and a ‘model of waste accumulation and disposal’.1 The first is
appropriate for dealing with pollution problems at a highly aggregated level, and where it is
necessary to place pollution problems explicitly in the context of the material basis of the
economy, by linking residual flows to the processes of resource extraction and use. In doing
so, it will be possible to generate pollution targets from the resource depletion models we
investigated in Chapters 14 and 15.
This approach is appropriate for dealing with economy wide or global stock pollution
problems arising from the use of fossil fuels. Climate change modelling falls into this
category, and several of the illustrations we use in the chapter refer to that example. Most
climate change models are highly aggregated using, for example, an aggregate “fossil fuels”
resource as an input into production. And they require that the material basis of the pollution
1
The term “ A model of waste accumulation and disposal” is borrowed from the title of Plourde’s (1972)
seminal paper.
2
in question - in this case, finite stocks of fossil fuels – is properly built into the modelling
framework.
The second framework – the waste accumulation and disposal model – is appropriate for
analysing stock pollution problems of a local, or less pervasive, type. Examples of such
problems include the accumulation of lead, mercury and other heavy metals in water
systems, the accumulation of particulates in air, the build up of chemicals from pesticides
and fertilisers in soils, and contamination of inland and coastal water systems by effluent
discharges. In these cases, resource use is of a sufficiently small scale (in the problem being
considered), so that limits on resource stocks do not become binding constraints. Hence, the
researcher can focus on the dynamics of the pollution problem but need not explicitly build
into the model a component which links pollutant emissions to the resources from which
they are derived.
For both modelling frameworks, though, we shall take the analysis of previous chapters
further by giving a more complete account of the dynamics of the pollution processes, the
properties of their steady states (if they exist), and the implications for pollution control
targets and instruments.
16.1 An aggregate dynamic model of pollution
Pollution problems come in many forms. Yet many have one thing in common; they are
associated with the use of fossil fuels. In this section, we present a simple and highly
aggregated stock pollution model. To fix ideas, it will be useful to think of this as a global
climate change model, although that is by no means the only context in which the model
could be used.
16.1.1. Basic structure of the model
The model developed in this section is a simple, aggregate stock pollution model. It can be
thought of as an optimal growth model – of the type covered in Chapter 14 – but including
3
some additional components, one of which models the way in which pollution flows are
related to the extraction and use of a composite non-renewable resource. We employ here
equivalent notation to that used in Chapter 14 and, wherever appropriate, adopt equivalent
functional forms. Being an optimal growth model, we look for its ‘solution’ by using
dynamic optimisation techniques. Specifically, we are trying to find the characteristics of an
emissions path for the pollutant that will maximise a suitably defined objective function.
We suppose that the production process utilises two inputs: capital and a non-renewable
environmental resource. Obtaining that non-renewable resource involves extraction and
processing costs. There is a is a fixed (and known) total stock of the non-renewable resource.
From now on we shall refer to this resource as ‘fossil fuels’. Use of fossil fuels involves two
kinds of trade-offs. First, there is an intertemporal trade-off: given that the total stock is
fixed, using fossil fuels today means that less will be available tomorrow. So different paths
of fossil fuel extraction can affect the welfare of different generations. Second, using fossil
fuels leads to more production (which is welfare enhancing) but also generates more
pollution (which is welfare detracting). The principal concern of Chapters 14 and 15 was
with the intertemporal trade-off. Here we are interested in both of these trade offs.
Insert Figure 16.1 near here
Caption: The structure of the aggregate stock pollution model
The pollution model used is an extension of that developed in Chapter 14. Its structure –
elements and key relationships - is illustrated in Figure 16.1. We retain the assumption that
extracting the resource is costly, but simplify the earlier analysis by having those costs
dependent on the rate of extraction but not on the size of the remaining stock. Pollution is
generated from the use of the fossil fuel resource.
16.1.2 Pollution damages
There are various ways in which pollution damages can be incorporated into a resource
depletion model. Two of these are commonly used in environmental economics:
 damages operate through the utility function
4
 damages operate through the production function.
In order to handle these kinds of effects in a fairly general way, we use the symbol E to
denote an index of environmental pressures. These environmental pressures have a negative
effect upon utility. To capture these effects, we write the utility function as
U = U(C, E)
(16.1)
in which, by assumption, UC > 0 and UE < 0. The index of environmental pressures E
depends on the rate of fossil fuel use (R) and on the accumulated stock of pollutant in the
relevant environmental medium (A). So we have
E = E(R, A)
(16.2)
Higher rates of fossil fuel use and higher ambient pollution levels each increase
environmental pressures, so that ER > 0 and EA > 0. Substituting Equation 16.2 into Equation
16.1 we obtain
U = U(C, E(R, A)
(16.3)
This deals with the case where damages operate through the utility function. But many forms
of damage operate through production functions. For example, greenhouse gas-induced
climate change might reduce crop yields, or tree growth may be damaged by sulphur dioxide
emissions. A production function that incorporates damages of this kind is
Q = Q(R, K, E(R, A)
(16.4)
Obtaining the non-renewable resource involves extraction and processing costs, , which
depend on the quantity of the resource used, hence we have
 = (R).
16.1.3 The resource stock-flow relationship
The utility and production functions both depend on A, the ambient level of pollution. The
way in which A changes over time is modelled in the same way as in Chapter 6. That is:
  M(R )  A
A
(16.5)
which assumes that a constant proportion  of the ambient pollutant stock decays at each
point in time. Note that Equation 16.5 specifies that emissions depend upon the amount of
resource use, R. By integration of Equation 16.5 we obtain
5
t
A t =  (M(R  ) - A )d
0
so for a pollutant which is not infinitely long-lived ( > 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 ( = 0) A grows without bounds as long as M is
positive.
16.1.4 Defensive or clean-up expenditure
We now introduce an additional control variable (or instrument) - expenditure on cleaningup pollution. Expenditure on V is an alternative use of output to investment expenditure,
consumption, or resource extraction and processing costs, and so must satisfy the identity
 CV
QK
In the model clean-up activity operates as additional to natural decay of the pollution stock.
For example rivers may be treated to reduce biological oxygen demand or air may be filtered
to remove particles. The level of such activity will be measured by expenditure on it, V. We
shall refer to V as ‘defensive expenditure'. This is a term which is widely used in the
literature but in an ambiguous way. Sometimes it refers to expenditure on coping with, or
ameliorating the effects of, an existing level of pollution. Thus, for example, in some
contexts the term would be used to cover expenditure by individuals on personal air filters,
'gas masks', for wear while walking the streets of a city with an air pollution problem. As we
use the term here, it would in that context refer to expenditure on an activity intended to
reduce the level of air pollution in the city.
The consequences of defensive expenditure on the pollutant stock is described by the
equation:
F = F(V)
(16.6)
in which FV > 0. The term F, therefore, describes the reduction in the pollution stock brought
about by some level of defensive expenditure V. Incorporating this in the differential
equation for the pollutant stock gives
6
  M(R )  A  F(V)
A
(16.7)
which says that the pollution stock is increased by emissions arising from resource use and is
decreased by natural decay and by defensive expenditure.
16.1.5. The optimisation problem
The dynamic optimisation problem can now be stated as:
7
Select values for the control variables Ct , Rt and Vt for t = 0,..., so as to maximise
t 
W
 UC ,E(R ,A )e
t
t
t
t
dt
t 0
subject to the constraints
S t  R t
  M(R )  A  F(V )
A
t
t
t
t
  Q(K ,R ,E(R ,A ))  C  (R )  V
K
t
t
t
t
t
t
t
t
Insert Table 16.1 immediately below the above boxed set of equations (table located at foot
of this document).
Caption: Key variables and prices in the model.
As shown in Table 16.1, there are three state variables in this problem: St, the resource
stock at time t; At, the level of ambient pollution stock at time t; and Kt, the capital stock at
time t. Associated with each state variable is a shadow price, P (for the resource stock), 
(for the capital stock) and  (for the ambient pollution stock). Be careful to note that,
because we are maximising a utility-based social welfare function, the discount rate being
used here is a utility discount rate (not a consumption discount rate) and the shadow prices
are denominated in units of utility (not in units of consumption). This should be taken into
account when comparing the shadow price of the ambient pollution stock in this chapter ()
with the shadow price  used in Chapter 6 (which was measured in consumption units).
In the production function specified by Equation 16.4 we assume that QE < 0 (and also, as
before, ER > 0 and EA > 0). The rate of extraction of environmental resources thus has a
direct and an indirect effect upon production. The direct effect is that using more resources
increases Q. The indirect effect is that using more resources increases environmental
8
pressures, and so reduces production. The overall effect of R on Q is, therefore, ambiguous
and cannot be determined a priori.
16.1.6 The optimal solution to the model
The current-valued Hamiltonian is
H  UC t ,E(R t ,A t )   Pt  R t   t Q[K t ,R t ,E(R t ,A t )]  C t  (R t )  Vt 
  t M(R t )  A t  F(Vt ) 
Ignoring time subscripts, the necessary conditions for a social welfare maximum are2:
H
 UC    0
C
H
 U E E R  P  Q R  Q E E R  R  M R  0
R
H
   FV  0
V
H
P  
 P  P  P
S
 

H
    Q K 
   
K
H
  
         U E E A  Q E E A
A
These can be rewritten as:
UC  
2
(16.8a)
We will leave you to verify that these first-order conditions are correct, using the method of the Maximum
Principle explained in Appendix 14.1.
9
P  U E E R  QR  QE E R  R  M R
(16.8b)
  FV
(16.8c)
P  P
(16.8d)
    QK 

(16.8e)
      U E E A  Q E E A
(16.8f)
16.1.7 Interpreting the solution
Three of these first-order conditions for an optimal solution - Equations 16.8a, 16.8d and
16.8e - have interpretations essentially the same as those we offered in Chapter 14. No
further discussion of them is warranted here, except to note that Equation 16.8d is a
Hotelling dynamic efficiency condition for the resource net price, which can be written as:
P

P
Provided that the utility discount rate is positive, this implies that 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.
Three conditions appear that we have not seen before, Equations 16.8b, 16.8c and 16.8f.
The last of these 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. However, some important interpretations can be drawn from Equations 16.8b
and 16.8c. We now turn to these.
16.1.7.1 The static efficiency condition for the resource net price
Equation 16.8b gives the shadow net price of the environmental resource. It shows that the
net price of the environmental resource equals the value of the marginal net product of the
10
environmental resource (that is, QR, the value of the marginal product less R, the value
of the extraction costs) minus three kinds of damage cost:
 UEER, the loss of utility arising from the impact of a marginal unit of resource use on
environmental pressures;
 QEER, the loss of production arising from the impact of a marginal unit of resource use
on environmental pressures;
 MR the value of the damage arising indirectly from resource extraction and use. This
corresponds to what we have called previously stock-damage pollution damage. This
‘indirect’ damage cost arises because a marginal increase in resource extraction and use
results in pollution emissions and then an increase in the ambient pollution level, A. To
convert this into value terms, we need to multiply this by a price per unit of ambient
pollution.
Note that we have stated that these three forms of damage cost must be subtracted from
the marginal net product of the environmental resource, even though they are each preceded
by an addition symbol in Equation 16.8b. This can be verified by noting that UE and QE are
each negative, as is the shadow price is t, given that ambient pollution is a ‘bad’ rather than
a ‘good’ and so will have a negative price.
In a competitive market economy, none of these pollution damage costs will be
internalised - they are not paid by whoever it is that generates them. This has implications
for efficient and optimal pollution policy. A pollution control agency could set a tax rate per
unit of resource extracted equal to the value of marginal pollution damages, UEER, + QEER
+ MR.
The nature of the required tax is shown more clearly in Figure 16.2 and 16.3. To interpret
these diagrams, it will be convenient to rearrange Equation 16.8b to:
QR  P  R  U E E R  QE E R  M R
We can read this as saying that:
11
Gross price = net price + extraction cost + value of flow damage operating on utility + value
of flow damage operating on production + value of stock damage
Insert Figure 16.2 near here
Caption: Optimal time paths for the variables of the pollution model.
Insert Figure 16.3 near here (directly below Figure 16.2 if possible)
Caption: Optimal ‘three part’ pollution taxes.
Figure 16.2 can be interpreted in the following way. In a perfectly functioning market
economy with no market failure, in which all costs and benefits are fully and correctly
incorporated in market prices, the gross (or market) price of the resource would follow a
path through time indicated by the uppermost curve in the diagram (and denoted by QR).
We can distinguish several different cost components of this socially optimal gross price:
1. the net price of the resource (the rent that must be paid to the resource owner to persuade
him or her to extract the resource);
2. the marginal cost of extracting the resource;
3. the marginal pollution damage cost. This consists of three different types of damage:
 pollution flow damage operating through the utility function;
 pollution flow damage operating through the production function;
 pollution stock damage (which in our model can work through both production and utility
functions).
However, in a competitive market economy where damage costs are not internalised and
so do not enter firms’ cost calculations), the market price will not include the pollution
damage components, and so would not be equal to the gross price just described. The market
price would only include two components: the net price (or resource royalty) and the
marginal extraction cost. It would then be given by the curve drawn second from the bottom
in Figure 16.2.
12
But now suppose that government were to introduce a socially optimal tax in order to
bring market prices into line with the socially optimal gross price. It is now easy to see what
such a tax would consist of. The tax should be set at a rate equal in value (per unit of
resource) to the sum of the three forms of damage cost, thereby internalising the damages
arising from resource use. We could regard this tax as a single pollution tax, or we might
think of it as a three-part tax (one on utility flow damages, one on production flow damages
and one on stock damages). Such an interpretation is shown in Figure 16.3. The three-part
tax has the advantage that it shows clearly what the government has to calculate in order to
arrive at a socially-optimal tax rate.
Figure 16.4 shows this interpretation of the optimal tax rate in terms of a ‘wedge’ between
the private and the social marginal cost. As you can see from the notes that accompany the
diagram, the private marginal cost is given by P + R. The optimal tax is set equal to the
marginal value of the three damage costs. When imposed on firms, the wedge between
private and social marginal costs is closed. Be careful to note, however, that Figure 16.4 can
only be true at one point in time. We know that all the components of costs change over
time, and so the functions shown in the diagram will be shifting as well.
Insert Figure 16.4 near here (a new picture)
Optimal taxes and the wedge between private and social costs.
16.1.7.2 Efficiency in defensive expenditure
The necessary conditions for a solution of our pollution problem include one equation,
Equation 16.8c, that concerns defensive expenditure,   FV . To understand this
condition, let us recall the meanings of its terms. First, the variable  is the shadow price of
capital; it is the amount of utility lost when one unit of output is diverted from consumption
(or investment in capital) to be used for defensive expenditure. Be careful to note that these
values are being measured at the optimal solution. That is, it is the amount of utility lost
when output is diverted to pollution clean-up when consumption and clean-up are already at
13
their socially optimal levels.3 You can imagine that finding out what these values are going
to be is a very difficult task indeed; this is a matter we shall return to shortly.
Second,  is the optimal value of one unit of ambient pollution; remember that this is a
negative quantity, as pollution is harmful. Third, FV is the amount of pollution stock cleanup from an additional unit of defensive expenditure.
Putting these pieces together, we can deduce the meaning of Equation 16.8c. The righthand side, -FV , is the utility value gained from pollution clean-up when one unit of output
is used for defensive expenditure. This must be set equal to the value of utility lost by
reducing consumption (or investment) by one unit. Put in another way, the optimal amount
of pollution clean-up expenditure will be the level at which the marginal costs and the
marginal benefits of clean-up are equal.
16.2 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, , is constant. 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 is 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). For an example of threshold effects and irreversibilities, consider river
3
The reason why we can use the value lost by diverting expenditure from either consumption or investment
follows from this point: at the social optimum, the value of an incremental unit of consumption will be identical
to the value of an incremental unit of investment. They will not be equal away from such an optimum.
14
pollution. At some threshold level of biological oxygen demand (BOD) on a river, the decay
rate of a pollutant may collapse to zero. An irreversibility exists if the decay rate of the
pollutant in the environmental medium remains below its previous levels even when the
pollutant stock falls below the threshold level. An irreversibility implies some hysteresis in
the environmental system: the history of pollutant flows matters, and reversing pollution
pressures does not bring one back to the status quo ex ante.
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 is grossly invalid, it will be necessary to respecify the pollutant stock-flow
relationship in an appropriate way. Box 16.1 illustrates how one important climate change
model – the RICE-99 model of Nordhaus and Boyer (1999) - deals with variable decay rates
of atmospheric greenhouse gases. Suggestions for further reading at the end of this chapter
point you to some literature that explores models with variable pollution decay rates.
Insert Box 16.1 near here
Caption: Decay rates for greenhouse gases in the RICE-99 model
15
Box 16.1 Decay rates for greenhouse gases in the RICE-99 model.
In early versions of Nordhaus’ climate change models, the GHG emissions-concentrations
relationship adopted an extended form of Equation 16.7, although without the F(V) term
being present. Estimates of the decay rate parameters were obtained from historical data on
M and A. Notice that in Equation 16.7 (and other similar expressions) a clear distinction is
drawn between stocks and flows: A is a stock of accumulated pollutants, measured in units
of mass at some point in time; M is a flow of pollutant emissions, measured in units of mass
per period of time.
A problem with this approach when it comes to climate change modelling is that a constant
decay rate parameter implies that the deep oceans are an infinite sink for carbon, which does
not appear to be consistent with either theory or evidence.
RICE-99 uses a structural approach to the model GHG decay rates, based on current thinking
about the carbon cycle. There are three reservoirs for carbon in the model: (i) the
atmosphere; (ii) the upper oceans and biosphere; and (iii) the deep oceans. Carbon
emissions, labelled ET in the equations below, enter the atmosphere reservoir. There are
exchanges of carbon mass (the various M terms on the right-hand sides of the equations
below) between the three reservoirs). Some carbon flows from the atmosphere to the upper
oceans/biosphere, and some flows in the opposite direction. These flows need not be equal
(and indeed would only be equal in a steady state equilibrium of the system). A two way
flow relationship also exists between the upper oceans/biosphere and the deep oceans. Twoway mixing also takes place between the upper oceans/biosphere and the deep oceans but is
very slow. There is, though no direct interchange of carbon mass between the atmosphere
and the deep oceans, although the structure of the model implies that some carbon emissions
are indirectly taken up by deep oceans. The model equations also have the property that the
deep oceans provides a finite – rather than an infinite – sink for carbon in the long term. This
has important implications for climate change modelling.
Carbon flows within and between the sinks are given by the following three equations, in
which M denotes carbon mass (not emissions, which are notated as ET), and the subscripts
16
AT, UP and LO denote the atmosphere, upper oceans/biosphere, and lower (deep) oceans
respectively.4 Note that in this three box carbon cycle model, not only are all the terms
denoted by the letter M stocks (being measured in mass units) but so too are ‘emissions’.
The term emissions is, therefore, being used by Nordhaus and Boyer here in a different way
to that in the rest of this chapter where they are treated as a flow variable.
M AT ( t )  10  ET ( t )  11M AT ( t  1)  12 M AT ( t  1)   21M UP ( t  1)
M UP ( t )   22 M UP ( t  1)  12 M AT ( t  1)   21M UP ( t  1)  32 M LO ( t  1)   23 M UP ( t  1)
M LO ( t )  33 M LO ( t  1)  32 M LO ( t  1)   23 M UP ( t  1)
It is evident by inspection of this system of dynamic equations that it does not specify a
constant decay rate of atmospheric GHG’s (whether that is measured in proportionate or
absolute terms).5 Chapter 3 of Nordhaus and Boyer (1999) explains how these equations are
parameterised from existing carbon cycle models.
End of Box 16.1
4
5
See Equations 2.13a -2.13c in Nordhaus and Boyer (1999), page 2-24, electronic manuscript version.
To verify this, just consider the role played by the final term in the first of the three equations.
17
Box 16.2 Nordhaus: DICE and RICE models of Global Warming
During the last fifteen years, Nordhaus – with various collaborators - has been developing a
suite of integrated economic-scientific models of global warming. The most recent version,
the RICE-99 model, is described in Nordhaus and Boyer (1999). This model is publicly
accessible; Nordhaus’ personal web site (at
http://www.econ.yale.edu/~nordhaus/homepage/homepage.htm) contains links to the full
electronic version of the book, together with versions of the model programmed in GAMS
and in EXCEL. The latter is relatively easy to use.
RICE (Regional Integrated model of Climate and the Economy) employs an optimal growth
modelling framework, augmented by the addition of an environmental sector. As in all
optimal growth models, choices essentially concern a trade-off between consumption today
and in the future. GHG emissions control reduces current consumption but increases future
consumption. Optimisation is used to manage this trade-off to maximise social welfare. In
this respect, RICE is broadly similar in structure to the optimal growth model we developed
in Section 16.1. Table 16.2 shows the similarity between RICE and the model you examined
earlier, and notes several of the particular characteristics of the RICE model.
Insert Table 16.2 near here (table located near foot of this document)
Caption: A comparison between the RICE model and the dynamic pollution
model of Section 16.1.
Not only is RICE an operational, empirically-parameterised model, but also it is far more
richly developed than the simple, stylised model we developed earlier. As its name employs
RICE is regionally disaggregated. The world is composed of 13 large sovereign countries or
groups of countries. Each “country” selects values of the control variables - consumption;
investment in tangible capital, and climate investment (GHG reduction) to maximises a
utilitarian intertemporal social welfare function subject to relevant economic and
technological constraints. Utility is discounted at a positive pure rate of time preference,
which is assumed to decline over time (due to decreasing impatience) from 3% per year in
18
1995 to 1.8% in 2200. Global social welfare is a population-weighted sum of individual
country per capita social welfares.
RICE consists of two main sectors: a (relatively conventional) economic sector, and a
geophysical (which embodies climate change modelling). In RICE’s economic sector, each
country is endowed with initial stocks of capital, labour and region-specific technology.
Capital and labour, together with a composite “carbon energy” resource, are inputs in the
each economy’s production function. The carbon energy resource is in finite supply, and
becomes available at a rising marginal cost. Using the carbon energy resource generates CO2
emissions as a joint product. The production function is calibrated against data on energy
use, energy prices and energy-use elasticities. This generates an empirically-based CO2
marginal abatement cost function.
The geophysical component of RICE consists of simplified versions of current best-practice
climate science. It contains a 3-box carbon cycle (see Box 16.1); a radiative forcing
equation; climate change equations; and climate damage relationships. The global impact is
derived by aggregation of regional impact estimates. The latter include market, non-market
and potential catastrophic impacts.
Policy analysis
To undertake policy analysis, RICE can be used to simulate the effects of policy makers
imposing a carbon tax or issuing tradable emissions permits, under a variety of assumptions
about whether tax rates are equal, and whether emissions trading is allowed, between
alternative configurations of blocs of countries. A country specific carbon-tax rate can also
be interpreted as the price of a carbon emission permit in that country. A uniform rate of
carbon tax over all countries is equivalent to a system of marketable permits in which
trading is allowed between all countries. RICE also allows permit trading to take place
within blocs, so that the carbon tax (permit price) is equalised within a bloc. Setting the
carbon tax at zero yields the Reference or Baseline case. With these various configurations
of policy instruments, RICE can be used to analyse the relative costs and benefits of a wide
variety of possible global warming policies. In particular, a Pareto optimal policy (inducing
19
the economically-efficient level of emissions) can be achieved either (a) by setting a
uniform carbon tax in every country equal to the global environmental shadow price of
carbon – that being the present value of all future consumption reductions in all regions of
one unit of carbon emissions today or (b) by distributing to each country permits equal to the
quantity of emissions they would produce if the Pareto optimal tax were imposed.
Major results
1. The Reference case is used to simulate the consequences of climate change where no
action is taken by policy makers to reduce global warming. Nordhaus finds that
impacts differ sharply between regions. Russia and some high-income countries
(principally Canada) will benefit slightly from a modest global warming. Low
income countries – particularly Africa and India – appear to be quite vulnerable to
climate change. For example, regional impacts of a 2.5 degrees C global warming
ranges from a net benefit of 0.7% f output for Russia to damage of 5% of output for
India.
2. RICE can be used to compare the relative efficiency of different approaches to
climate-change policy. A path that limits CO2 concentrations to no more than a
doubling of pre-industrial levels is close to the “optimal” or efficient policy. Current
approaches – such as that in the Kyoto Protocol – are highly inefficient with
abatement costs approximately ten times their benefits in avoided damages.
3. Investigate role of carbon taxes. (As a measure of the stringency of global warming
policy). Optimal carbon price in range $5 to $10 per ton of carbon. Kyoto policy
targets yield carbon taxes close to $100 per ton. These fail a cost-benefit test because
they impose excessive near-term abatement.
See also Table 10.9 in Chapter 10.
End of Box 16.2
20
16.3 Steady state outcomes
In some of the previous chapters, we have examined steady state outcomes – equilibria in
which the levels of variables of interest are unchanging through time. Is the notion of a
steady state useful, or even meaningful, in the context of the modelling framework we have
been examining? In the rest of this section, we show that it is not a meaningful concept if the
optimising model used above is used to think about policy over indefinitely long spans of
time. In that case, some attention must be paid to renewable resources too, and a steady state
can only make sense where those resources are brought into consideration.
However, one may object in principle to the use of optimal growth models for policy
analysis when major pollution problems are the object of concern, perhaps arguing that
policy should be constrained by some form of precautionary principle (see Chapter 6). For
example, in thinking about the greenhouse effect, using the precautionary principle suggests
that the policy maker should try and identify what kinds of states are acceptable in terms of
avoiding risks of catastrophic climate change. Such states might be defined in terms of
maximum allowable global mean temperature levels, or perhaps (less directly) in terms of
maximum allowable GHG concentration rates. Much of the current discussion about the
greenhouse effect is couched in this kind of framework – particularly about what GHG
concentration rates are acceptable.
Article 2 of the United Nations Framework Convention on Climate Change (UNFCCC)
states that “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. When global warming first began to attract the attention of policy analysts,
many scientists implicitly took stabilisation of atmospheric concentrations at the then current
levels as the appropriate target, and posed the question of what level of GHG emissions
reduction would be required to achieve this. Not surprisingly, the answer given to this
21
question typically suggested massive reductions in emissions. It soon became clear that
stabilisation at current concentration rates was politically infeasible, and probably
economically indefensible. A widely held opinion among natural and physical scientists
today is that this should 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. As we mentioned
in Box 16.2, Nordhaus’ RICE model simulations suggest that this is close to an optimal
target.
16.3.1 Is a steady state meaningful in our current model?
For the pollution model we have just been studying, however, the notion of a steady state is
logically inconsistent and so not meaningful. No constant positive quantity of a nonrenewable resource can be extracted indefinitely, given the limited stock size. The only
constant amount that could be used indefinitely is zero. That case is of no interest in our
model as it stands. For if R were zero, production would be using produced capital as the
sole input. That is at odds with the laws of thermodynamics; unless the capital itself were
consumed in the process of production, it implies that physical output could be produced
without using any physical inputs, which is clearly impossible.
It is evident that production cannot rely forever only on the use of non-renewable
resources. At some point in time – it will be necessary to make use of renewable resources as
productive inputs. This points us to a way in which the model investigated above should be
extended if it is to be useful for very long term analysis. And as we shall see, it also leads to
the notion of a steady state being a meaningful and relevant concept.
To fix ideas, let us return to the example of using fossil fuels as a non-renewable resource
input. Fossil fuels cannot be used for ever. Eventually one of three things must happen.
Stocks will become completely exhausted; or the price of fossil fuels will rise so high as to
make them uneconomic (in which case the economically relevant stock becomes zero); or
the pollution consequences of using fossil fuels will become intolerable, and we are forced
22
to cease using them. In each of these cases, production will switch from the non-renewable
to a renewable resource input. If a steady state is ever attained, it would be one in which the
renewable resource is used at a constant rate through time.
That suggests that we should generalise the model specified above so that 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. The index of environmental pressures, E, may
depend on R2 as well as R1. We leave it to you as an exercise to investigate how the model
we have been examining could be generalised in this way, and what its steady state would
be. A possible answer is provided in the file Enlarged model.doc in the Additional
Materials for Chapter 16.
16.4 A model of waste accumulation and disposal
In this section, we investigate efficient emissions targets for stock pollutants where it is not
necessary to take account of resource constraints in the way we did in Section 2. Ignoring
such constraints may be appropriate where pollution derives from the extraction and use of
some resource on a scale sufficiently small that resource stock constraints are not binding.
We might call these ‘local’ modals of stock pollution. They are typically much less highly
aggregated than those previously studied. Examples include models of pollution associated
with the use of nitrates in agricultural chemicals, with discharges of toxic substances and
radioactive substances, and with various forms of groundwater and marine water
contamination.
The models we are looking at here are best thought of as examples of partial equilibrium
cost-benefit analysis, albeit in a dynamic modelling framework. Because variables are now
being measured in monetary (or consumption unit) terms, rather than in utility units, the
appropriate rate of discount is now r rather than . We shall pay particular attention to the
economically-efficient steady state model outcome. Box 16.3 lays out the problem we shall
be considering and the first order conditions for its solution.
23
Insert Box 16.3 near here
Caption: The local stock pollution model.
24
Box 16.3 The local stock pollution model
The problem
The objective is to choose a sequence of pollutant emission flows, Mt, t = 0 to t = , to
maximise
t 
 B(M t )  D(A t ) e
 rt
dt
t 0
subject to
dA t
= M t - A t
dt
(16.9)
A0 = A(0), a non-negative constant
Mt  0
Optimisation conditions
The current-valued Hamiltonian for this problem is
H t  B(M t )  D(A t )   t (M t  A t )
(16.10)
The necessary first order conditions for a maximum (assuming an interior solution) include:
H t
0 
M t
dB t
 t  0
dM t
(16.11)
H t
d
dD
 r t 
 r t 
 t
dt
A t
dA t
(16.12)
25
The steady state
In a steady state, all variables are constant over time and so d/dt is zero. Time subscripts are
no longer necessary. The two first order conditions become
dB
 
dM
(16.13)
dD
 (r  )
dA
(16.14)
Also, in a steady state the pollution stock differential equation
dA t
= M t - A t
dt
collapses to
M  A
(16.15)
Equation 16.14 can also be written as
dD
  = dA
r
(16.16)
The variable  is the shadow price of one unit of pollutant emissions. It is equal to the
marginal social value of a unit of emissions at a social net benefits maximum. As pollution
is a bad, not a good, the shadow price, , will be negative (and so - will be positive).
The conditions 16.13 and 16.16 say that two things have to be equal to - at a net benefit
maximum. Therefore those two things must be equal to one another. Combining those
conditions we obtain:
dD
dB
 dA
dM r  
(16.17)
Equation 16.17 is one example of a familiar marginal condition for efficiency: in this case,
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
26
marginal unit of pollution. However, it is quite tricky to get this interpretation from
Equation 16.17, so we shall take you through it in steps.
The term on the left-hand side of Equation 16.17 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. In contrast, the right-hand side of Equation 16.17 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. Note that dD/dA itself lasts forever; it is a form of perpetual
annuity (although an annuity with a negative effect on utility). To obtain the present value of
an annuity, we divide its annual flow, dD/dA, by the relevant discount rate, which in this
case is r. The reason why we also divide the annuity by  is because of the ongoing decay
process of the pollutant. If the pollutant stock were allowed to rise, then the amount of decay
in steady state will also rise by a proportion  of that increment in the stock size. This
reduces the magnitude of the damage. Note that  acts in an equivalent way to the discount
rate. 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.
For the purpose of looking at some special cases of Equation 16.17, it will be convenient to
rearrange that expression as follows:
dD dB
dB


r
dA dM
dM
(16.18)
and so
dD 1 dB dB r


dA  dM dM 
(16.19)
Given that in steady-state A = (1/)M, then from the damage function D = D(A), and using
the chain-rule of differentiation, we can write
dD dD dA dD 1



dM dA dM dA 
This allows us to write Equation 16.19 as
dD dB dB r


dM dM dM 
27
or
dD dB 
r

1  
dM dM   
(16.20)
If we knew the values of the parameters  and r, and the functions dB/dA and dD/dM (or
dD/DA, from which dD/dM could be derived for any given value of ), Equation 16.20
could be solved for the numerical steady state solution value of M, M*. Then from the
relationship A = (1/)M the steady state solution for A is obtained, A*.
Four special cases of Equation 16.20 can be obtained, depending on whether r = 0 or r > 0,
and on whether  = 0 or  > 0. These were laid out in Table 6.4 in Chapter 6. We briefly
summarise here our earlier conclusions.
Case A: r = 0,  > 0
Given that  > 0, the pollutant is imperfectly persistent and eventually decays to a harmless
form. With r = 0, no discounting of costs and benefits is being undertaken. Equation 16.20
collapses to: 6
dD dB

dM dM
(16.21)
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. We can also write this expression as
dD 1 dB

dA  dM
6
(16.22)
We can arrive at this result another way. Recall that NB(M) = B(M) - D(A). Maximisation of net benefits
requires that the following first-order condition is satisfied: dNB/dM = dB/dM - dD/dM = 0. Differentiating
(using the chain rule in the damage function) and then rearranging we obtain dB/dM = (1/)(dD/dA) = dD/dM.
28
which says that the contribution to damage of a marginal unit of emissions flow should be
set equal to the damage caused by an additional unit of ambient pollutant stock divided by
.
Case C: r > 0,  > 0
Equation 16.20 remains unchanged here.
dD dB 
r

1  
dM dM   
The marginal equality we noted in Case A remains true but in an amended form (to reflect
the presence of discounting at a positive rate). Discounting, therefore, increases the steadystate level of emissions. Intuitively, the reason it does so is because a larger value of r
reduces the present value of the future damages that are associated with the pollutant stock.
In effect, higher weighting is given to present benefits relative to future costs the larger is r.
However, the shadow price of one unit of the pollutant emissions becomes larger as r
increases.
Cases B ( r > 0,  = 0 ) and D ( r = 0,  = 0 )
Given that  = 0, case 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 will increase without bound, and so stock-pollution
damage will rise to infinity. The steady-state equilibrium solution for any value of r when 
= 0, therefore, gives zero pollution. 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, say time T.
Pollution damage continues indefinitely, but no additional damage is being caused in any
period.
29
This is a very strong result - any activity generating perfectly persistent pollutants that lead
to any positive level of damage cannot be carried on indefinitely. At some finite time in the
future, a technology switch is required so that the pollutant is not emitted. If that is not
possible, the activity itself must cease. Note that even though a perfectly persistent pollutant
has a zero natural decay rate, policy makers may be able to find some technique by which the
pollutant may be artificially reduced. This is known as clean-up expenditure. We examine
this possibility in the following section.
Dynamics
The previous sub-section outlined the nature of the steady state solution to the local stock
pollution model. However, without some form of policy intervention, it is very unlikely that
variables will actually be at their optimal steady state levels. How could the policy maker
“control” the economy to move it from some arbitrary initial position to its optimal steady
state?
To answer this question, we need to carry out some analysis of the dynamic of the model
solution. Our interest is with the dynamics of the state variable (At) and the instrument or
control variable (Mt) in our problem. Specifically, we are looking for two differential
equations of the form:
dA
 f ( A, M )
dt
dM
 g ( A, M )
dt
We already have the first of these – it is given by Equation 16.9, the pollution stock-flow
relationship. To obtain the second of this pair of differential equations we proceed as
follows. First, take the time derivative of Equation 16.11 yielding:
 d2 B  dM
d

=  
2  dt
dt
d
M


(16.23)
30
Then substituting Equation 16.23 into Equation 16.12 we have:
(r  ) 
 d2 B  dM
dD

=  
2  dt
dA
d
M


(16.24)
Finally, substituting Equation 16.11 into Equation 16.24 yields the second differential
equation we require 7
dM
=
dt
dB
dD
)
dM dA
d2 B
d M2
(r  )(
The differential
(16.25)
equations 16.25 and 16.9 will provide the necessary information from which
the efficient time paths of {Mt, At}can be obtained. In the absence of particular functions,
the solutions can only be qualitative. However, if we select particular functions and
parameter values, then a quantitative solution can be obtained. In the example which
follows, we choose the functions and parameter values used earlier in the text, in Box 6.7 of
Chapter 6. There we had  = 0.5, r = 0.1, D = A2 and B = 96M - 2M2, and so
dB/dM = 96 - 4M, dD/dA = 2A and dD/dM = 8M (in steady state).
It will be convenient to obtain the steady state solution before finding the dynamic adjustment
path. Inserting the function and parameter values given in the previous paragraph into the
differential equations 16.9 and 16.25 gives
dA
= M  0.5A
dt
(16.9b)
dM (0.6)(96  4M )  2A
=
 0.6M  0.5A  14.4
dt
4
(16.25b)
 =0 and dM/dt  M
 =0.
In steady state, variables are unchanging through time, so dA/dt  A
Imposing these values, and solving the two resulting equations yields M* = 9 and A* = 18
7
Notice that D and B are functions of M or A.
31
(as we found previously). This steady state solution is shown in the ‘phase plane’ diagram,
 =0 and M
 =0 (which are here A =
Figure 16.5. The intersection of the two lines labelled A
2M and A = (-0.6/0.5)M + 28.8 from 16.9a and 16.9b) gives M* = 9 and A* = 18.
Next, we establish in which direction A and M will move over time from any pair of
 =0 and M
 =0 (known as isoclines) divide the space
initial values {A0, M0 }. The two lines A
 =0, A > 2M , decay exceeds emissions flows, and so A
into four quadrants. Above the line A
 =0, A < 2M , decay is less than emissions flows, and
is falling. Conversely below the line A
so A is rising. These movements are shown by the downward facing directional arrows in
the two quadrants labelled a and b, and by upward facing directional arrows in the two
quadrants labelled c and d.
 =0, 0.6M > 14.4 – 0.5A, and so from Equation 16.25b we see that M is
Above the line M
 =0, 0.6M < 14.4 – 0.5A, and so M is falling. These movements are
rising. Below the line M
shown by the leftward facing directional arrows in the two quadrants labelled a and d, and
by rightward facing directional arrows in the two quadrants labelled b and c.
Taking these results together we obtain the pairs of direction indicators for movements in
A and M for each of the four quadrants when the system is not in steady state. The curved
and arrowed lines illustrate four paths that the variables would take from particular initial
values. Thus, for example, if the initial values in quadrant d with M = 15 and A = 2, the
differential equations which determine A and M would at first cause M to fall and A to rise
 =0 isocline into quadrant c, A will continue to
over time. As this trajectory crosses the M
rise but now M will also rise too. Left alone, the system would not reach the steady state
optimal solution, diverging ever further from it as time passes.
Inspection of the other three trajectories shows that these also fail to attain the steady state
optimum, and eventually diverge ever further from it. Indeed, there are only two paths which
do lead to that optimum. These are shown by the dotted lines whose arrows point towards
32
the bliss point, together known as the stable arm of the problem. For any dynamic process
with a saddle point equilibrium such as this, the only way of reaching the optimum is for the
policy maker to control M so as to reach the stable arm, and then to adjust M accordingly
along the stable arm until the bliss point is reached.
Insert Figure 16.5 near here.
Caption: Steady state solution and dynamics of the waste accumulation and disposal model.
Source: Located as image at end of this document
From all of this we have the following conclusion. If the initial level of pollution stock lies
to the left of the stable arm, emissions should be increased until they reach the level
indicated by the stable arm (for that level of pollution stock). The pollution stock will then
rise (fall) if A0 were less (more) than A* , and the policy maker would need to increase
(decrease) emissions to stay on the stable path until the bliss point were reached.
There are several instruments by means of which the environmental protection agency
could control emissions in this way. For example, it could issue quantity regulations (by
issue of licenses; it could use a marketable permit system; or it could use an emissions tax or
abatement subsidy. Note that the regulator will need to keep in mind both the steady state
solution which it wishes to be ultimately achieved, and the transition path to it. For the latter
purpose, regulation will typically change in severity over time if an optimal approach to the
equilibrium is to be achieved.
In the steady-state, the terminal condition (transversality condition) will be satisfied. At =T
=A* and Mt =T =M*. Here Mt = MT = AT so that dA/dt = 0, and the pollution stock remains
at the steady-state level.
The terminal conditions for pollution emissions are M T  A T from Equation 16.9 and
(r   )
dB(M T ) dD(A T )

0
dM T
dA T
from Equation 16.17.
33
If the reader would like to see in more detail how these properties can be discovered using a
computer software package, we suggest you examine the Maple file Stock pollution 1.mws)
This file is set up to generate the picture reproduced here as Figure 16.5. For a much more
extensive account of the techniques of dynamic analysis using phase plane diagrams, see the
file Phase.doc. Both of these are available in the Additional Materials for Chapter 16.
34
Summary
In this chapter you will

Investigate two models of optimal emissions which are suitable for the analysis of
persistent (long lasting) pollutants. Each of these models is a variant of the optimal
growth model framework that we have addressed before at several places in the text.

The first model you will study is an ‘aggregate stock pollution model’ which is
appropriate for dealing with pollution problems where the researcher considers it
appropriate to link emissions flows to the processes of resource extraction and use.
This will enable you to see how optimal pollution targets can be obtained from
generalised versions of the resource depletion models we investigated in Chapters 14
and 15.

The second - a ‘model of waste accumulation and disposal’ - provides a framework
that is suitable for analysing stock pollution problems of a local, or less pervasive,
type, such as the accumulation of lead in water systems or contamination of water
systems by effluent discharges.

We stress, more strongly than has been the case hitherto, the dynamics of pollution
generation and pollution regulation processes, using phase plane analysis for this
purpose.
35
Further reading
Baumol and Oates (1988) is a classic source in this area, although the analysis is formal and
quite difficult. Other useful treatments which complement the discussion in this chapter are
Dasgupta (1982, Chapter 8) and Smith (1972) which gives a very interesting mathematical
presentation of the theory. Several excellent articles can be found in the edited volume by
Bromley (1995).
The original references for stock pollution are Plourde (1972) and Forster (1975). Conrad
and Olson (1992) apply this body of theory to one case, Aldicarb on Long Island. One of the
first studies about the difficulties in designing optimal taxes (and still an excellent read) is
Rose-Ackerman (1973). Pezzey (1996) surveys the economic literature on assimilative
capacity, and an application can be found in Tahvonen (1995). Forster (1975) analyses a
model of stock pollution in which the decay rate is variable.
Some journals provide regular applications of the economic theory of pollution. Of particular
interest are the Journal of Environmental Economics and Management, Ambio,
Environmental and Resource Economics, Land Economics, Ecological Modelling, Marine
Pollution Bulletin, Ecological Economics and Natural Resources Journal.
36
Discussion questions
1. In what principal ways do stock pollution models differ from models of flow pollutants?
37
Problems
.
1. Using Equation 11.18, deduce the effect of an increase in  for a given value of r, all other
things being equal, on:
(a) M*
(b) A*
38
Table 16.1 Key variables and prices in the model
Variables (t = 0, …,)
Instrument (control) variables:
Ct
Rt
Vt
State variables:
Co-state variables (Shadow
Prices) (t = 0, …,)
St
Pt
Kt
t
At
t
39
1
Table 16.2 A comparison between the RICE model and the dynamic pollution model of Section 16.1.
Component
Model in Section
RICE-99
16.1
Objective function
t 
W
 UC ,E(R ,A )e
t
t
t
t
t 0
RICE has a similar objective function.
dt
Differences and specifics:
Discrete time model; no environmental degradation index term (E) enters
utility function; RICE has a global objective function which aggregates over
regions. Specifically, objective function is a discounted sum of population
weighted sum of utility of per capita consumption. Logarithmic form of utility
function embodies assumption of diminishing valuation of consumption as
consumption rises. Utility discount rate falls over time.
Control (instrument)
Ct , Rt and Vt for t = 0,...,
RICE does not deal with defensive expenditure as such.
S t  R t
RICE recognises the finiteness of fossil fuel stocks. A carbon
variables
Resource stock
supply curve describes the availability of carbon fuels at rising
constraint
marginal costs. As stocks are increasingly depleted, price rises
along a Hotelling-type path over time.
Pollution stock-flow
  M(R )  A  F(V )
A
t
t
t
t
it assume a constant decay parameter (see Box 16.1).
relationship
Production
RICE does not deal with defensive expenditure as such, nor does
Q t  Q(K t ,R t ,E(R t ,A t )
RICE does not include E in function, but labour (= population) is
specified as an input. Production function is constant returns to scale,
function
Cobb Douglas form. Population growth is exogenous. Exogenous
2
technological change has two forms, economy-wide and energy-saving.
Capital
accumulation
  Q  C  ( R )  V
K
t
t
t
t
t
3
1
2
3