Water Quality Trading with Both Flow Pollution and

Stock Pollution in Existence

Zhiyu Wang

August 31, 2016

Abstract

Flow pollution and stock pollution commonly exist in the world. Flow pollution

causes immediate damage which does not extend to future, while stock pollution has

its damage sustain in the environment. Previous studies on permit trading either

exclusively focused on one type of pollution or neglected the difference between these

two types of pollution. Regarding that both types of pollution might coexist in a permit

trading program, this paper aims to supplement previous studies on water quality

trading in this scenario. In order to establish a water quality trading program in a

cost-effective way, it is necessary to distinguish the difference between flow pollution

and stock pollution, so that the dynamic process of stock pollution is considered in the

trading ratios. Moreover, given two separate reduction targets at flow pollution and

stock pollution, it is possible to achieve these targets together in a cost-effective way

through water quality trading, while it requires more information to derive the proper

trading ratios and the permit allocation.

Keywords: water quality trading, stock pollution, flow pollution, lake, river

1 Introduction

The success of permit trading in solving air pollution inspires people to extend its

application in solving water pollution. Usually, air pollutant either causes flow pollution

like SO2 or stock pollution like greenhouse gases, but it doesn’t cause the two types of

pollution at the same time. Permit trading as a result can focus on one type of pollution,

either flow pollution or stock pollution, in solving air pollution. However, it is not the case

for water pollution. Flow pollution and stock pollution coexist, causing damage in water

system. For example, water quality deteriorates in the Grand River and its downstream Lake

Michigan, and eutrophication problem frequently occurs in the Mississippi River and the Gulf

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of Mexico. Those types of pollution increase maintenance cost, affect economic benefit, cause

public health concern and detriment water safety and recreation. High turbidity of water,

for example, causes valves and pipes block easily, accelerates their depletion and makes

water treatment more costly. Algae bloom in Lake Erie devastated safety of tap water and

recreational activities around Toledo, Ohio.

Flow pollution and stock pollution are different in terms of damage persistence. Flow

pollution causes immediate damage to the environment but does not extend to future if no

emission is added. Stock pollution can cause short-run damage, but most importantly its

damage sustains. Pollutants, including chemicals, organisms and nutrients, generate flow

and stock pollution according to their degradation rates and locations. Within the same

area, the damage persists longer for a pollutant which degrades more slowly. Considering

locations, river pollution is usually regarded as flow pollution, because pollutants are easily

taken away by water flows1. The concentration thereby decreases quickly once emission is

ceased, and their influence to a specific area along a river is temporary (Lieb, 2004). Pollution

in a lake or a water reservoir, on the other hand, is properly regarded as stock pollution,

because turnover time is comparatively higher in a reservoir2.

Water quality trading (WQT) is a promising economic tool to control water quality.

Trading ratio is its important element which determines whether pollution can be solved

cost effectively or not. Hung and Shaw (2005) created a trading ratio system based on

physical effluent, and Farrow, Schultz, Celikkol, and Houtven (2005) developed another one

based on monetary damage. Both trading ratio systems solves, for example, high transaction

cost in WQT (Stavins, 1995) and are proved cost effective. However, they pay attentions

only to flow pollution. In fact, many studies on WQT merely focused on river pollution,

of which damage is immediate and instantaneous (e.g. Montgomery, 1972; Farrow et al.,

2005; Hung and Shaw, 2005; Mesbah, Kerachian, and Nikoo, 2009). Only a few focused on

stock pollution, after more attention shifts to nonpoint pollution (e.g. Stephenson, Norris,

and Shabman, 1998; Obropta and Rusciano, 2006; Roberts and Clark, 2008). Phosphorus

credits, for example, were created in Dillon Reservoir trading program to solve algae problem,

although low trading activities makes the program inactive currently (Stephenson, Norris,

and Shabman, 1998; Fisher-Vanden and Olmstead, 2013). Notwithstanding many think of

setting Total Maximum Daily Load (TMDL) and adjusting trading ratios in WQT to meet

a requirement of pollutant load, no one has considered the dynamics of lake pollution.

By and large, the coexistence of flow and stock pollution has been either ignored, or

1This may be not the case for all pollutants. For example, the phosphorus incorporated with sediment is

not easily taken away by water flows.2“Turnover time (T ) is the ratio of the mass (M) of a reservoir and the total rate of removal (S) from

the reservoir: T = M/S” (Houghton, Meira Filho, Callander, Harris, Kattenberg, and Maskell, 1996, qtd.

on p. 76).

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treated homogeneously in WQT. However, Hoel and Karp (2001) pointed out that these two

types of pollution require different models to analyze. Flow pollution fits a static model,

while stock pollution needs a dynamic model. Ignorance of this difference makes WQT ques-

tionable to be cost effective. There are studies on dynamic optimization of stock pollution

control. They analyzed optimal control path, while neglected how or whether it is possibly

implemented via WQT (e.g. Dechert and O’Donnell, 2006; Iwasa, Uchida, and Yokomizo,

2007; Laukkanen and Huhtala, 2008; Hediger, 2009). In all, little effort, if not none, has

been made to analyze WQT when flow pollution and stock pollution simultaneously occur.

This paper uses the trading ratio system developed by Farrow et al. in WQT. Under

this trading ratio system, environmental damage is measured in dollars. Monetary damage

coefficients are computed as the integrals of marginal damage over downstream area. The

trading ratio system contains a matrix of monetary damage coefficient ratios between trans-

action sides. Transaction can occur across river branches. Water quality information on each

location is not necessary. It is cost effective to trade according to this trading ratio system,

because every point polluter attempts to minimize its total costs through WQT, while the

trading ratios and the permit allocation limit total damage. However, there is an important

assumption: monetary environmental damage of river pollution caused by a point polluter

is linear in its emission. Without this assumption, the cost-effectiveness can hardly achieve

(Konishi, Coggins, and Wang, 2015). Therefore, this paper keeps this assumption on flow

pollution. Moreover, in terms of influence of stock pollution, Iwasa, Uchida, and Yokomizo

(2007) assume it to be a function of water pollution level, which depends on pollutant input

and limnological processes. Although a nonlinear specification of stock pollution damage

might appear more precise (e.g. Dechert and O’Donnell, 2006; Laukkanen and Huhtala,

2008), a large number of papers adopted linear functional form (e.g. Ribaudo, Osborn, and

Konyar, 1994; Hoel and Schneider, 1997; Baudry, 2000; Parry, Pizer, and Fischer, 2003;

Matsueda, Futagami, and Shibata, 2006; Dellink, Finus, and Olieman, 2008; Sbragia and

Breton, 2011; Masoudi, Santugini, and Zaccour, 2015). This paper takes linear damage

assumption on stock pollution. This linearity assumption makes pollution problem analyt-

ically tractable, and in some cases there are also scientific supports for this assumption.

For instance, Kolstad (1996) pointed out that current damage of stock pollution caused by

greenhouse gas is quite linear over a fairly broad range of current stock. Also, a significant

linear relationship may exist between total amount of certain pollutants and health outcomes

(e.g. Guha Mazumder, 2005; Ergene, Cava, Celik, Koleli, Kaya, and Karahan, 2007; Park,

Cox-Ganser, Kreiss, White, and Rao, 2008). Even if linear relationship is not obvious, a

linear specification can still be used to approximate in a certain range of stock (Matsueda,

Futagami, and Shibata, 2006). This paper extends a linear damage assumption to stock

pollution. That is, stock pollution damage is linear in pollutant load. The paper is going to

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show that even under this simple scenario, the coexistence of flow and stock pollution will

make a difference to WQT.

2 Cost Effectiveness Problem

Suppose there is a river flowing into a lake. A pollutant exists in both the river and

the lake, coming from a large area around. That area can be segregated into multiple small

areas, which is unified having 1 point polluter. These polluters are indexed by i ∈ {1, 2, ..., n}from upstream to downstream, and n is the polluter closest to the lake. Damage caused by

river pollution is immediate and instantaneous, while damage of lake pollution sustains.

The pollutant decays in the lake at a rate of γ, determined by physicochemical traits of the

pollutant and the aquatic ecosystem. Discount rate is r. Transfer coefficient from polluter

i to polluter j is τij, which tells how much pollutant from polluter i can arrive at polluter

j. It is smaller than 1 except for i = j. Transfer coefficient to the lake is τin+1. If polluter

n is at the lake, then τnn+1 = 1. The pollutant load in the lake at the time t is S(t). The

unregulated emission level at point i is e0i . The abatement effort of polluter i at the time

t is ai(t). Its abatement cost Ci(ai) is measured by the profit loss from abatement, and

is assumed twice differentiable, strictly increasing and strictly convex. Under the linearity

assumption on damage, {di}ni=1 is the marginal river damage per pound of the pollutant,

measured in monetary unit. It integrates all the river damage over downstream caused by

the emission. The instantaneous river damage caused by polluter i at the time t is Di(t),

which equals to di(ei − ai(t)). The marginal lake damage is δn+1, and δn+1S(t) stands for

the instantaneous lake damage at the time t, which depends on pollutant stock rather than

flows.

2.1 Stock Pollution Ignored

Flow pollution and stock pollution normally coexist. Albeit, they are not necessarily

both concerned by the public, especially when one pollution is more serious than the other.

This part analyzes the scenario when stock pollution is ignored. Fisher-Vanden and Olmstead

(2013) summarized active WQT programs, and found that the majority focused on flow

pollution, although a few targeted at pollution in lakes, reservoirs and Chesapeake Bay. For

those programs which only target at flow pollution, they solve the cost effectiveness problem

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below (1),

mina1,a2,...,an

n∑i=1

Ci(ai)

s.t.

n∑i=1

di(e0i − ai

)≤ DF ,

ai ≥ 0, ∀i ∈ {1, 2, ..., n},

(1)

where the instantaneous monetary damage of flow pollution cannot exceed DF . Farrow et al.

(2005) solved this problem and found that the cost to reduce additional river damage should

be the same everywhere to achieve cost effectiveness. They derived the cost effective trading

ratios in WQT which equal to the marginal cost ratios:

C′i(ai)

C′j(aj)

=didj, ∀i, j ∈ {1, 2, ..., n}. (2)

2.2 Flow Pollution Ignored

This part analyzed the scenario when flow pollution is ignored. Considering that stock

pollution is correlated across time, a constraint is imposed on the total damage caused

by stock pollution over time. For many trading programs, a long-term constraint is often

included. They attempt to solve a cost effectiveness problem below,

mina1,a2,...,an

∫ ∞0

e−rtn∑i=1

Ci(ai) dt

s.t. S = −γS +n∑i=1

τin+1

(e0i − ai

),∫ ∞

0

e−rtδn+1S dt ≤ DS,

ai ≥ 0,∀i ∈ {1, 2, ..., n}.

(3)

For parsimony, t is eliminated in all functions above. The state equation S = −γS +∑ni=1 τin+1 (e0i − ai) tells how the pollutant changes in the lake. The integrand function in

the objective is strictly convex. The feasible set of the problem is closed and bounded.

The pollutant stock is also bounded for all admissible pairs across time. By Filippov-Cesari

Existence Theorem, it assures that a solution to this problem always exists. Mangasarian

conditions also certify that the solution is unique. The Hamiltonian function H, first order

conditions and transversality condition of this problem are listed below, where λ is a Lagrange

multiplier constant over time:

H =n∑i=1

Ci(ai) + λ

(δn+1S −

DS

r

)+ µ

(−γS +

n∑i=1

τin+1

(e0i − ai

)). (4)

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C′

i(ai)− µτin+1 = 0, ∀i ∈ {1, 2, ..., n} ; (5)

µ− rµ = µγ − λδn+1; (6)

S = −γS +n∑i=1

τin+1

(e0i − ai

); (7)

limt→∞

µ(t) = 0. (8)

Solving the differential equation (6) and transversality condition (8), µ equals to λδn+1/ (r + γ),

which is constant over time. The size of λ depends on DS. The damage constraint on stock

pollution is always binding unless DS is not stringent enough. Hence, the marginal cost

ratios of the cost effective solution is:

C′i(ai)

C′j(aj)

=τin+1

τjn+1

, ∀i, j ∈ {1, 2, ..., n}. (9)

2.3 No Pollution Ignored

When both flow and stock pollution are serious, both of them tend to be considered in

the cost effectiveness problem. A constraint is imposed on the total damage of both flow

and stock pollution over time in the problem below:

mina1,a2,...,an

∫ ∞0

e−rtn∑i=1

Ci(ai)dt

s.t. S = −γS +n∑i=1

τin+1

(e0i − ai

),

∫ ∞0

e−rt

(n∑i=1

Di + δn+1S

)dt ≤ D,

ai ≥ 0, ∀i ∈ {1, 2, ..., n}.

(10)

The solution is similar to previous ones. In order to control the pollution in a cost effective

way, the marginal cost to diminish additional damage, including flow and stock pollution,

should be the same everywhere:

C′i(ai)

C′j(aj)

=

di +δn+1

r + γτin+1

dj +δn+1

r + γτjn+1

, ∀i, j ∈ {1, 2, ..., n}. (11)

3 Water Quality Trading

Trading ratios and permit allocation are two important components of WQT. Trading

ratios are introduced in WQT, because emission of polluters is not traded on a one-to-one

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basis. Trading ratios tell how much a polluter can trade with another polluter. For example,

if trading ratio tij is 2, 2 units emission from polluter i is equivalent to 1 unit emission from

polluter j in terms of environmental damage. For ease of explanation, it assumes that one

tradable permit of polluter i allows it to emit one kilogram of the pollutant. Hence, if polluter

j sells 1,000 tradable permits to polluter i, then polluter i essentially buys itself 2,000 permits

at its own site, which allows it to emit 2,000 kilograms of pollutant. Alternatively, if polluter

j buys 1,000 permits from polluter i, it can only emit 500 kilograms additional emission.

The matrix of the trading ratios is denoted as t. The trading ratios used here are more

appropriate to be called damage trading ratios, because the environmental damage they focus

on is measured in monetary unit. They are different from the ones which focus on physical

amount of the pollutant (Hung and Shaw, 2005). Damage trading ratios look at the balance

of environmental damage caused by different polluters in order to restrict the total damage,

while physical trading ratios emphasize on the balance of physical amount on the pollutant to

limit the total amount (Hung and Shaw, 2005). Because the environmental damage covers the

influence of the pollutant, which relies on demographic factors and economic development,

damage trading ratios are relatively comprehensive compared with physical trading ratios.

In the regulatory framework of WQT, river damage coefficients are assigned to polluters,

and lake damage coefficient is publicly known. The permits are distributed among polluters

allowing them to have L emission respectively, where L = (L1, L2, ..., Ln).

3.1 Equilibrium of Water Quality Trading

Once the damage trading ratios and the permit allocation are determined, polluters will

solve a cost minimization problem (12) in each period,

minrki,rsi,rji

Ci(ai)− pirsi +∑j 6=i

pjrji

s.t.(e0i − rki

)−∑j 6=i

tjirji ≤ 0,

rsi + rki = Li + ai,

rki, rsi, rji ≥ 0,∀i ∈ {1, 2, ..., n}.

(12)

where rsi is emission credit sold by polluter i3, rki is the credit kept by polluter i, rji is the

credit polluter i purchases from polluter j. The permit price is {pi}ni=1. Polluters’ abatement

is uniquely determined by t and L. An equilibrium of WQT is defined below.

An equilibrium of water quality trading is a vector of permit prices p, a vector of abate-

ment a and a sequence of trading activities {rsi, {rji}j 6=i}ni=1 given the damage trading ratios

3By abating emission, polluters create emission credits. Both credits and permits can be traded in WQT.

The words “credit” and “permit” are of the same meaning and used interchangeably in the paper.

7

t and the permit allocation L such that:

(i). Given t, L and p, a and {rsi, {rji}j 6=i}ni=1 solve the cost minimization problem for each

polluter.

(ii).∑

i 6=j rji ≤ rsj for all polluter j ∈ {1, 2, ..., n}, where the equality holds when pj > 0.

The equilibrium is not necessarily unique, because the trading activities {rsi, {rji}j 6=i}ni=1

could have multiple combinations4. Since the feasible set of problem (12) is compact, the

objective function is continuous, convex in abatement and linear in trading activities, an

equilibrium of WQT always exists. Details of the Kuhn-Tucker Conditions are in the ap-

pendix. The abatement and the marginal cost ratios in an equilibrium are:

ai = C′−1i (pi), ∀i ∈ {1, 2, ..., n} , (13)

C′i(ai)

C′j(aj)

= tij, ∀i, j ∈ {1, 2, ..., n}. (14)

The marginal cost ratios are uniquely determined by the damage trading ratios. Therefore,

it is important to figure out proper damage trading ratios.

3.2 Damage Trading Ratios

In the following, four types of damage trading ratios are going to be discussed. The

first one is derived when stock pollution is ignored. The second one is when flow pollution

is ignored. The third one is when no pollution is ignored. For these ratios, the difference

between flow pollution and stock pollution is clarified. However, this difference seems not to

be sufficiently emphasized and correctly understood. People possibly view these two types

of pollution in the same way (e.g. Stephenson, Norris, and Shabman, 1998; Obropta and

Rusciano, 2006). If lake pollution is mistakenly regarded in the same way as river pollution,

i.e. no dynamic process or correlation over time, a fourth type of ratios will reflect this

mistake.

3.2.1 Stock Pollution Ignored

Polluters trade based on the damage trading ratios t, which assure that “the increment

environmental damages from the trade are offsetting” (Farrow et al., 2005). Without loss

of generality, suppose there are 2 polluters. Polluter i buys some credits form polluter j.

These amount of credits allow polluter i additional emission ∆ei, while polluter j must have

additional abatement ∆aj to generate these credits.When stock pollution is ignored, the

4First order conditions have n2 + 2n unknowns and(n2 + 5n

)/2 non-identical equations. Therefore, the

solution {ai, rsi, {rji}j 6=i}ni=1 is not unique for n ≥ 3.

8

damage trading ratio tij satisfies equation (15), i.e. additional damage occurred in WQT is

zero:

di∆ei − dj∆aj = 0. (15)

Farrow et al. (2005) provided the corresponding ratios and the permit allocation in this

scenario. It essentially reflects that contemporary incremental environmental damages from

trading should be offsetting:

tij =didj, ∀i, j ∈ {1, 2, ..., n},n∑i=1

diLi ≤ DF .

(16)

3.2.2 Flow Pollution Ignored

When river pollution is ignored, the second type of damage trading ratios is derived.

Instead of contemporary incremental environmental damages, it assures that the continuous

incremental damage over time is offset. The additional emission ∆ei from polluter i arrives

and decays in the lake. Correspondingly, the instantaneous change of the pollutant load in

the lake at the time t is ∆S(ei, t). Similarly, the instantaneous change caused by additional

abatement from polluter j is ∆S(aj, t). The second type of the damage trading ratios tij

must satisfy: ∫ ∞0

e−rtδn+1∆S(ei, t) dt−∫ ∞0

e−rtδn+1∆S(ai, t) dt = 0. (17)

Given the state equation, the integrals above are simplified. The damage trading ratios and

the permit allocation are invariant over time, and the initial lake pollutant stock is S0:

tij =τin+1

τjn+1

∀i, j ∈ {1, 2, ..., n},n∑i=1

δn+1

r + γτin+1Li ≤

(DS −

δn+1S0

r + γ

)r.

(18)

3.2.3 No Pollution Ignored

If neither flow nor stock pollution are ignored, a third type of damage trading ratios is

derived. Similar to the discussion above, this type of ratios guarantees that the present value

of additional damage is offset, where the damage is caused by both flow and stock pollution:

di∆ei +

∫ ∞0

e−rtδn+1∆S(ei, t) dt− dj∆aj −∫ ∞0

e−rtδn+1∆S(ai, t) dt = 0. (19)

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Computing the integrals, the damage trading ratios and the permit allocation are invariant

over time:

tij =

di +δn+1

r + γτin+1

dj +δn+1

r + γτjn+1

∀i, j ∈ {1, 2, ..., n},

n∑i=1

(di +

δn+1

r + γτin+1

)Li ≤

(D − δn+1S0

r + γ

)r.

(20)

The damage trading ratios are unique for different scenarios, and the abatement vector in an

equilibrium of WQT is uniquely determined by the damage trading ratios and the amount

of permit issued. From the analysis above, it concludes that:

Conclusion 1. Suppose a water system in which a river flows into a water reservoir,

e.g. lake. A pollutant causes damages both in the river and the reservoir. The environmental

damage is measured in monetary unit and is linear in the pollutant amount. No matter

whether flow pollution, or stock pollution or no pollution is ignored, water quality trading is

still cost effective. For different scenarios, the damage trading ratios and the permit allocation

are specified in (16), (18), and (20).

3.2.4 Mistaking Stock for Flow Pollution

Although the distinctiveness of stock pollution from flow pollution has been noticed, it

is not sufficiently emphasized in WQT, and river and lake pollution are normally regarded

homogenously (e.g. Stephenson, Norris, and Shabman, 1998; Obropta and Rusciano, 2006).

The dynamic process and the continuous damage of lake pollution fail to be accounted. In

fact, current WQT programs which target at physical amount of the pollutant commonly

bear this problem. A trading ratio in WQT is frequently defined as “(a) numeric value

used to adjust water quality benefits from a particular project” (Oregon Department of

Environmental Quality (2015)) or a value “used to prevent trades from resulting in a net

increase in ambient pollution levels” (Zhang, Li, Wang, and Huang, 2015), notwithstanding

it hasn’t clarified whether pollution is measured in terms of damage or physical amount of the

pollutant. Most of WQT programs focus on physical amount of the pollutant and consider,

for example, transfer coefficients and the uncertainty between sellers and buyers. Uniform

trading ratios are also common, which neglect location of polluters (e.g. Ohio Environmental

Protection Agency, 2007; Minnesota Pollution Control Agency, 2011; Corrales, Melodie Naja,

Rivero, Miralles-Wilhelm, and Bhat, 2013; Keller, Chen, Fox, Fulda, Dorsey, Seapy, Glenday,

and Bray, 2014). These trading ratios haven’t accounted pollution damage for demographic

and economic factors, and are indifferent to the initial pollutant stock in a lake. Therefore,

WQT is hardly cost effective if stock pollution is not treated differently from flow pollution.

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In this mistake, a trading ratio is set to satisfy:

di∆ei + δn+1∆S(ei, t)− dj∆aj − δn+1∆S(aj, t) = 0. (21)

The change of the pollutant stock in the lake is only accounted for the time t. The damage

trading ratios and the permit allocation are:

tij =di + δn+1τin+1

dj + δn+1τjn+1

, ∀i, j ∈ {1, 2, ..., n},n∑i=1

(di + δn+1τin+1)Li ≤ D.

(22)

The abatement vector is unique in a cost effective solution and in an equilibrium of WQT.

Only is the trading ratios fit the cost effective result and the permit allocation is properly

determined, will water pollution be solved cost effectively via WQT. Therefore, this mistake

will lead to a loss in the cost effectiveness, unless one type of pollution is ignored in the

objective function. Yet if no pollution is ignored, it is impossible to be cost effective except

for very restrictive conditions.

Conclusion 2. Suppose a water system in which a river flows into a water reservoir,

e.g. lake. A pollutant causes damages both in the river and the reservoir. The environmental

damage is measured in monetary unit and is linear in the pollutant amount. The damage

trading ratios which haven’t accounted for the different between stock pollution and flow

pollution, are cost effective when one type of pollution is ignored in the objective function.

However, it is hardly cost effective when no pollution is ignored unless r + γ = 1.

This conclusion gives a special case when the sum of the discount rate and the decay rate

equals to 1. This says that after decay and discounting, monetary damage of lake pollution

doesn’t sustain to the next period. Interpreting it in an extreme example, the pollutant lasts

forever in the lake (i.e. γ = 0), but people are all extremely improvident (i.e. r = 1), then

the bioeconomic discount rate is 1. No one cares about continuous damage of lake pollution.

Moreover, when polluters are all far upstream, and only a trivial amount of the pollutant

can arrive in the lake, then this mistake might not affect achieving the cost effectiveness too

much.

3.3 Solving Both Pollution

Residents alongside the river are directly affected by river pollution, while residents

around the lake are directly influenced by lake pollution. An example could be the Mississippi

river and the Gulf of Mexico. Stormwater pollution in the Upper Mississippi River might

immediately influence peoples’ lives in Minnesota, Wisconsin, Iowa, Illinois and Missouri,

while not those in Texas and Louisiana who suffer from the dead zone in the Gulf of Mexico.

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However, considering heterogenous tolerance abilities to environmental damage in different

regions, the regulator might intend to set constraints on river pollution and lake pollution

respectively. An improvement in river water quality will automatically improve water quality

in the lake. Given that the limit on river pollution is DS and the limit on lake pollution is

DF , the social cost effectiveness problem is,

mina1,a2,...,an

∫ ∞0

e−rtn∑i=1

Ci(ai)dt

s.t. S = −γS +n∑i=1

τin+1

(e0i − ai

),

∫ ∞0

e−rtn∑i=1

Didt ≤ DF ,∫ ∞0

e−rtδn+1Sdt ≤ DS,

ai ≥ 0,∀i ∈ {1, 2, ..., n}.

(23)

Let λ1 be the Lagrange multiplier on river pollution and λ2 be the Lagrange multiplier

on lake pollution. Therefore, the cost effective solution is:

C′i(ai)

C′j(aj)

=

λ1di +λ2δn+1

r + γτin+1

λ1dj +λ2δn+1

r + γτjn+1

, ∀i, j ∈ {1, 2, ..., n}. (24)

When one of the pollution constraints is slack, the problem converts to a problem already

discussed in the paper. A cost effective WQT can be constructed with the trading ratios

and the permit allocation given by:

tij =

λ1di +λ2δn+1

r + γτin+1

λ1dj +λ2δn+1

r + γτjn+1

, ∀i, j ∈ {1, 2, ..., n},

n∑i=1

(λ1di +

λ2δn+1

r + γτin+1

)Li ≤

(D − λ2δn+1S0

r + γ

)r,

(25)

where D = λ1DF + λ2DS. However, in order to derive the proper values of λ1 and λ2, the

regulator has to acquire the cost information on all polluters. Otherwise, a cost effective

WQT cannot be determined.

Conclusion 3. Suppose a water system in which a river flows into a water reservoir,

e.g. lake. A pollutant causes damages both in the river and the reservoir. The environmental

damage is measured in monetary unit and is linear in the pollutant amount. Two restrictions

12

are set on river pollution and lake pollution respectively. When the cost information on all

polluters is known, water quality trading could be cost effective to solve the problem. The

damage trading ratios and the permit allocation are specified in (25).

O DF

DS

A

B

Figure 1: Combinations of Flow and Stock Pollution in WQT

For all possible WQT programs, Figure 1 shows the combinations of river and lake

pollution in their present value. The upper curve is the combinations when lake pollution

is ignored, the lower curve is the ones when river pollution is ignored. The blue area shows

all the pollution combinations under all possible WQT programs. It is also the area for all

possible combinations of flow and stock pollution. If one of the pollution constraints is slack,

by ignoring corresponding lake or river pollution, the problem can be solved efficiently in

WQT. For example, (DF , DS) is set at point A. It can be solved by a WQT in which lake

pollution is ignored and the permit allocation is set to meet the limit DF . If the pollution

constraints are at point B, under complete information on abatement cost, the value of

λ1/λ2 can be found, and thereby the corresponding WQT which is represented by the red

dashed curve can be established to solve the pollution cost effectively. If DF is set relatively

more stringent than DS, then the value λ1/λ2 tends to be higher. The corresponding cost

effective WQT approaches closer to the one which ignore lake pollution. As a result, the

pollution combinations move towards the upper curve in the figure. In general, compared

with the scenarios when there is a single target set on flow and stock pollution, it requires

cost information on all polluters in order to derive cost effective trading ratios in WQT when

there are separate targets set on flow and stock pollution respectively.

13

4 Simulation

Farrow et al. (2005) described how to compute monetary damage coefficients in their

study. They used pollutant concentrations at varied distances from emission sources to obtain

a matrix of transfer coefficients first5. The monetary damage coefficient of river pollution

integrates all the damage over the downstream area. For polluter i,

di =M∑m=0

WTP ·Hm · lm · τim ·1

Q0i

, i ∈ {1, 2, ..., n}. (28)

where WTP refers to household marginal willingness-to-pay for a small improvement on

water quality of the river. Hm is the number of households in segment m. Q0i is the stream

flow at site i, lm is a stream interval, τim is the transfer coefficient from site i to site m. M

is the total amount of segments around the river area.

The damage coefficient on lake pollution δn+1 is derived similarly. The benefit from the

lake V (W,H) depends on water quality W of the lake and households number H around the

lake. Water quality is measured in pollutant concentration C, and suppose that ∂W/∂C =

−1, and water quality is homogeneous all over the lake area. Dividing the lake area into K

segments, marginal damage of lake pollution at segment k is δkn+1. Marginal willingness-to-

pay for a small improvement in water quality of the lake is WTP , and Qn+1 is the water

capacity of the lake. Therefore, lake damage coefficient is below, where lk is the area of

5Farrow et al. (2005) provided the steps to compute transfer coefficients. First, it is to calculate concen-

trations. The pollutant concentration C (mg/L) at distance n (meters) downstream from a pollution point

i is:

Cni =eiQ

e−kθT−20 n

U , (26)

where ei (kg/day) is emission rate at site i, Q (meter3/day) is the stream flow , k (day−1) is the nominal

decay rate, θ is the sensitivity coefficient of k to a temperature T (Celsius), and U (meters/day) is stream

velocity. Second, it is to compute transfer coefficients using:

τij =CjiC0i

. (27)

These transfer coefficients can be improved by including the effect of the time which is spent to move the

pollutant from site i to site j. The modified transfer coefficient could be τij = τije−rq where q is the average

time duration of the pollutant moving from site i to site j.

14

segment k6.

δkn+1 =

∣∣∣∣∂V (W,Hk)

∂S

∣∣∣∣ =

∣∣∣∣ ∂V∂W ∂W

∂C

∂C

∂S

∣∣∣∣ = (WTP ·Hk) · 1 ·1

Qn+1

(30)

δn+1 =K∑k=1

WTP ·Hk · lk ·1

Qn+1

(31)

The river damage coefficients used here are taken from (Farrow et al., 2005), with the

damage coefficient of Clairton, Pennsylvania being normalized into 1. Their study was about

the influence of the total sewer overflow in a couple of cities in the Upper Ohio River Basin.

The total sewer overflow level without regulation is 261 kg/day. Since no detail is available,

suppose that the unregulated emission is equal everywhere, i.e. 33 kg/day. The length of

one period is one week, and the unregulated emission level is 231 kg/week. The decay rate

is set at 0.4/week, although varies under different temperatures and hydrologic conditions.

That is, 40% pollutant disappears in one period. Suppose there is a pseudo lake located in

a pseudo City X, which is at the downstream of all the other cities. The damage coefficient

of lake pollution is assumed 5 ($/kg), and the initial pollutant stock in the lake is 10 metric

tons. The discount rate is 0.5% for one period. The transfer coefficients to the lake are

assumed in an arithmetic progression order.

Table 1: Parameters in the Simulation

City di ($/kg) τin+1

Morgantown, WV 1.62 0.1

Uniontown, PA 2.21 0.2

Clairton, PA 1.00 0.3

McKeesport, PA 0.38 0.4

Greensburg, PA 3.99 0.5

Pittsburg, PA 0.36 0.6

Youngstown, OH 1.40 0.7

Steubenville, OH 0.03 0.8

City X, OH 0.00 1.0

6If δn+1(k) is a continuous function of the marginal lake pollution damage in segments, integrate δkn+1

over the lake area will get the damage coefficient of lake pollution,

δn+1 =

∫ K

k=0

δn+1(k)dt. (29)

However, in most cases this continuous and explicit function is unable to get, therefore, the coefficient is

calculated in the discrete form shown in the paper.

15

Suppose the abatement cost function is

C(ai) = αi + βiai + ξi(e0i − ai

)ln

(1− ai

e0i

), (32)

where αi is the fixed cost. Let αi = 100, βi = 1 and ξi = 1 for i ∈ {1, 2, ..., n}, i.e.

homogeneous abatement cost function everywhere. Heterogenous abatement cost function

is not a necessary condition for WQT to occur. The marginal abatement cost is strictly

positive for a nonzero abatement, and is infinitely large for full abatement.

Abatement cost goes up as the environment constraint becomes more stringent. Figure

2 shows total abatement costs for WQT under different scenarios. These curves are the

present value of total abatement costs. When reduction approaches to zero, the cost curves

all converge to the same point because polluters all barely abate. As reduction increases,

the total costs under WQT which ignores river pollution and which concerns both pollution

climb up faster. For the trading program in which lake pollution is ignored, its total cost

is always lower than the other two. Figure 3 compares the total cost under different lake

damage coefficients. TC(Lake) is the present value of the total cost when river pollution is

ignored. Similar for the meaning of TC(River) and TC(Both). The difference with TC(Both)

and TC(Lake) is much smaller than the other differences. Moreover, the differences between

TC(River) don’t alter with the changing lake damage coefficient. Nonetheless, the difference

between TC(Both) and TC(Lake) varies, especially when that coefficient is small.

16

Figure 2: Cost Comparison Under the Stock Damage Coefficient δn+1 = 5

17

Figure 3: Cost Comparison Under Stock Damage Coefficients

18

When both river and lake pollution are considered, they must be treated differently in

the cost effectiveness problem. Otherwise, the damage trading ratios will be distorted and

the equilibrium will not be cost effective. In Figure 4, the correct damage trading ratios are

called dynamic ratios, and the distorted ones are called static ratios. There is a gap between

total costs using these two types of ratios. The gap increases as the reduction goes up, yet

declines sharply after passing its peak. It is because when reduction is too high, almost

every polluter abates completely. Moreover, the gap declines when lake damage coefficient

increases. However, it is true here because the marginal damage of lake pollution is much

larger than that of river pollution. An increment in the lake damage coefficient in this sense

reduces the distortion from ignorance the difference between flow and stock pollution. If the

lake damage coefficient is close to that off river pollution, the conclusion might not hold. For

different decay rates and discount rates in Figure 5 and 6, the gap between total costs has

the same pattern, except that it shrinks with higher decay rate and higher discount rate.

19

(a) Different Stock Damage Coefficients

(b) Three Different Stock Damage Coefficients

Figure 4: Cost Comparison between Two Sets of Ratios Under Stock Damage Coefficients

20

(a) Different Decay Rates

(b) Three Different Decay Rates

Figure 5: Cost Comparison between Two Sets of Ratios Under Decay Rates

21

(a) Different Discount Rates

(b) Three Different Discount Rates

Figure 6: Cost Comparison between Two Sets of Ratios Under Discount Rates

Figure 7 is the paths of river and lake pollution damage from different WQT. DF is

22

the present value of river pollution damage, and DS is that of lake damage. In the trading

program when lake pollution is ignored, its corresponding lake pollution is the highest among

the three programs given the same level of the river pollution. As river pollution goes down,

the combinations of river and lake pollution in WQT which solves both pollution and WQT

which ignores river pollution converge to the same point. However, it is not true for WQT

which ignores lake pollution. The reason relies on the polluter at the lake, i.e. City X. Since

City X is located in the lake area, its river damage coefficient is zero. When lake pollution is

ignored, City X is in fact free from the water quality constraint due to it only affects water

quality of the lake. This figure validates the impossibility of satisfying the cost effectiveness

targets for both river community and lake community, and gives the possible area that could

be achieved via political influence.

Figure 7: Path of River and Lake Pollution Damage in WQT Programs

5 Conclusion and Further Opportunity

Previous studies on WQT haven’t sufficiently emphasized the coexistence of flow and

stock pollution as well as the difference between them. If the difference is not correctly

understood or the coexistence is simply ignored, it will increase the cost to solve water

23

pollution. Even under a simple linear damage environment, this difference could affect

achieving the cost effectiveness. When there is only one limit set on flow and stock pollution,

it doesn’t require any cost information from all polluters to derive cost effective trading ratios.

However, when there are separate limits on flow and stock pollution, complete information

on polluters’ cost functions is indispensable to construct a cost effective WQT. Difficulties

occur if damage becomes nonlinear. Konishi, Coggins, and Wang (2015) pointed out that

if flow pollution damage is nonlinear, the damage trading ratios does no longer achieve

cost effective. If stock pollution damage is nonlinear, the damage trading ratios have to

become time-dependent, and additional information on marginal cost is also required for

polluters. After all, the damage trading ratios treat linear environmental damage well.

Further research is expected to make WQT cost effective and easily applicable under less

stringent assumptions.

24

6 Appendix A. Kuhn-Tucker Conditions

For the cost minimization problem of water quality trading, the Lagrange function of

Problem (12) is

Li = Ci(ai)− pirsi +∑j

pjrji + λi

[(e0i − rki

)−∑j

tjirji

]. (33)

The Kuhn-Tucker Conditions are7

rki : C′

i(ai)− λi ≥ 0, rki ≥ 0,(C

′

i(ai)− λi)∗ rki = 0; (34)

rsi : C′

i(ai)− pi ≥ 0, rsi ≥ 0,(C

′

i(ai)− pi)∗ rsi = 0; (35)

rji : pj − λitji ≥ 0, rji ≥ 0, (pj − λitji) ∗ rji = 0; (36)

λi :(e0i − rki

)−∑j

tjirji ≤ 0, λi ≥ 0,

((e0i − rki

)−∑j

tjirji

)∗ λi = 0. (37)

If the quantity constraint Li is not binding, i.e. the first inequality in (37) is strict, then the

corresponding λi = 0. Therefore, (36) indicates that no market exists under this scenario. If

the quantity constraint is binding, there is a positive λi. Therefore, C′i(ai)− λi ≥ 0 in (34)

and C′i(ai) − pi ≥ 0 in (35) hold with equality. pj − λitji ≥ 0 in (36) must also hold with

equality due to non-arbitrage condition8.

7 Appendix B. The Permit Allocation

For each polluter in the water quality trading, it has to meet the requirement that its

actual emission should not exceed the amount of permits it possesses. That is,(e0i − rki

)−∑j 6=i

tjirji ≤ 0, ∀i ∈ {1, 2, ..., n} (38)

Since rsi + rki ≡ Li + ai, rki can be replaced with Li + ai − rsi. The above inequality can be

rearranged as follows.(e0i − ai

)≤ Li − rsi +

∑j 6=i

tjirji, ∀i ∈ {1, 2, ..., n} (39)

7Notice that the object function is strictly convex in rsi, rki, but linear in rji. Therefore, one may guess

that the solution on rsi, rki is interior, but that on rji may be corner.8Once the first inequality conditions of rki and rsi hold with equality, the first inequality condition of rji

becomes an equality, too. See the details in Konishi, Coggins, and Wang (2015).

25

Multiply both sides by the marginal damage coefficient di, aggregate over all the firms and

integrate over time, it will have∫ ∞0

n∑i=1

di(e0i − ai

)e−rtdt ≤

∫ ∞0

n∑i=1

di

(Li − rsi +

∑j 6=i

tjirji

)e−rtdt. (40)

Because the abatement vector a is constant over time, one can obtain the path of lake

pollutant load over time by the state equation S = −γS +∑n

i=1 τin+1(e0i − ai) and thereby

compute the present value of lake pollution damage.∫ ∞0

δn+1Se−rtdt =

∫ ∞0

δn+1

[(S0 −

1

γ

n∑i=1

τin+1

(e0i − ai

))e−γt +

1

γ

n∑i=1

τin+1

(e0i − ai

)]e−rtdt

=δn+1S0

r + γ+

δn+1

r(r + γ)

n∑i=1

τin+1

(e0i − ai

)≤δn+1S0

r + γ+

δn+1

r(r + γ)

n∑i=1

τin+1

(Li − rsi +

∑j 6=i

tjirji

)(41)

Combining inequalities (40) and (41), it gives∫ ∞0

n∑i=1

di(e0i − ai

)e−rtdt+

∫ ∞0

δn+1Se−rtdt ≤

∫ ∞0

n∑i=1

di

(Li − rsi +

∑j 6=i

tjirji

)e−rtdt+

δn+1S0

r + γ+

δn+1

r(r + γ)

n∑i=1

τin+1

(Li − rsi +

∑j 6=i

tjirji

)(42)

The left side of the inequality is aggregate damage of river and lake pollution over time.

Integrating the right side of the inequality and rearrange it, it gives

1

r

n∑i=1

(di +

δn+1τin+1

r + γ

)Li −

1

r

n∑i=1

(di +

δn+1τin+1

r + γ

)(rsi −

∑j 6=i

tjirji

)+δn+1S0

r + γ(43)

Because rsj =∑

i 6=j rji if permit price is nonzero, and given the trading ratios in Proposition

3.3, the middle item 1r

∑ni=1

(di + δn+1τin+1

r+γ

)(rsi −

∑j 6=i tjirji

)is zero. The initial permit

allocation indicates that the right side of the inequality is no greater than D. Therefore,

given such initial permit allocation, the aggregate damage will always be guaranteed to meet

the damage limit requirement.

26

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