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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.

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,

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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

in question - in this case, finite stocks of fossil fuels – is properly built into the modelling

framework.

1 The term “ A model of waste accumulation and disposal” is borrowed from the title of Plourde’s (1972)

seminal paper.

3

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

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

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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

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

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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:

(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

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.

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16.1.4 Defensive or clean-up expenditure

We now introduce an additional control variable (or instrument) - expenditure on cleaning-

up 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

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

(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:

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Select values for the control variables Ct , Rt and Vt for t = 0,..., so as to maximise

subject to the constraints

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

pressures, and so reduces production. The overall effect of R on Q is, therefore, ambiguous

and cannot be determined a priori.

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16.1.6 The optimal solution to the model

The current-valued Hamiltonian is

Ignoring time subscripts, the necessary conditions for a social welfare maximum are2:

2 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.

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These can be rewritten as:

(16.8a)

(16.8b)

(16.8c)

(16.8d)

(16.8e)

(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:

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

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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:

We can read this as saying that:

Gross price = net price + extraction cost + value of flow damage operating on utility + value

of flow damage operating on production + value of stock damage

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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.

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

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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, . 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

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.

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.

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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 clean-

up from an additional unit of defensive expenditure.

Putting these pieces together, we can deduce the meaning of Equation 16.8c. The right-

hand 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

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.

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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

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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. Two-

way 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

AT, UP and LO denote the atmosphere, upper oceans/biosphere, and lower (deep) oceans

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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.

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 See Equations 2.13a -2.13c in Nordhaus and Boyer (1999), page 2-24, electronic manuscript version.

5 To verify this, just consider the role played by the final term in the first of the three equations.

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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

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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

the economically-efficient level of emissions) can be achieved either (a) by setting a

19

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

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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 question typically suggested massive reductions in emissions. It soon became clear that

21

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 non-

renewable 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

to cease using them. In each of these cases, production will switch from the non-renewable

22

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.

Insert Box 16.3 near here

Caption: The local stock pollution model.

23

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

subject to

(16.9)

A0 = A(0), a non-negative constant

Mt 0

Optimisation conditions

The current-valued Hamiltonian for this problem is

(16.10)

The necessary first order conditions for a maximum (assuming an interior solution) include:

(16.11)

(16.12)

24

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

(16.13)

(16.14)

Also, in a steady state the pollution stock differential equation

collapses to

(16.15)

Equation 16.14 can also be written as

(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:

(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

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.

25

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:

(16.18)

and so

(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

This allows us to write Equation 16.19 as

or

(16.20)

26

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

(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

(16.22)

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

6 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.

27

Equation 16.20 remains unchanged here.

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 steady-

state 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.

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

28

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:

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:

(16.23)

Then substituting Equation 16.23 into Equation 16.12 we have:

(16.24)

Finally, substituting Equation 16.11 into Equation 16.24 yields the second differential

equation we require 7

7 Notice that D and B are functions of M or A.

29

(16.25)

The differential 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

(16.9b)

(16.25b)

In steady state, variables are unchanging through time, so dA/dt =0 and dM/dt =0.

Imposing these values, and solving the two resulting equations yields M* = 9 and A* = 18

(as we found previously). This steady state solution is shown in the ‘phase plane’ diagram,

Figure 16.5. The intersection of the two lines labelled =0 and =0 (which are here 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

initial values {A0, M0 }. The two lines =0 and =0 (known as isoclines) divide the space

into four quadrants. Above the line =0, A > 2M , decay exceeds emissions flows, and so A

is falling. Conversely below the line =0, A < 2M , decay is less than emissions flows, and

so A is rising. These movements are shown by the downward facing directional arrows in

30

the two quadrants labelled a and b, and by upward facing directional arrows in the two

quadrants labelled c and d.

Above the line =0, 0.6M > 14.4 – 0.5A, and so from Equation 16.25b we see that M is

rising. Below the line =0, 0.6M < 14.4 – 0.5A, and so M is falling. These movements are

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

over time. As this trajectory crosses the =0 isocline into quadrant c, A will continue to

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 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

31

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 from Equation 16.9 and

from Equation 16.17.

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.

32

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.

33

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.

34

Discussion questions

1. In what principal ways do stock pollution models differ from models of flow pollutants?

35

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*

36

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

1

2

Table 16.2 A comparison between the RICE model and the dynamic pollution model of Section

16.1.

Component Model in Section

16.1

RICE-99

Objective function RICE has a similar objective function.

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)

variables

Ct , Rt and Vt for t = 0,..., RICE does not deal with defensive expenditure as such.

Resource stock

constraint

RICE recognises the finiteness of fossil fuel stocks. A carbon

supply curve describes the availability of carbon fuels at rising

marginal costs. As stocks are increasingly depleted, price rises

along a Hotelling-type path over time.

Pollution stock-flow

relationship

RICE does not deal with defensive expenditure as such, nor

does it assume a constant decay parameter (see Box 16.1).

Production

function

RICE does not include E in function, but labour (= population) is

specified as an input. Production function is constant returns to scale,

3

Cobb Douglas form. Population growth is exogenous. Exogenous

technological change has two forms, economy-wide and energy-saving.

Capital

accumulation

1

2

3