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Group Title: Working paper - International Agricultural Trade and Policy Center. University of Florida ; WPTC 04-09
Title: Private responses to public incentives for invasive species management
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Permanent Link: http://ufdc.ufl.edu/UF00089794/00001
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Title: Private responses to public incentives for invasive species management
Series Title: Working paper - International Agricultural Trade and Policy Center. University of Florida ; WPTC 04-09
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Language: English
Creator: Ranjan, Ram
Evans, Edward
Publisher: International Agricultural Trade and Policy Center. University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2004
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Full Text

WPTC 04-09

i -ional Agricultural Trade and Policy Center

Ram Ranjan & Edward Evans

WPTC 04-09 December 2004




Institute of Food and Agricultural Sciences




The International Agricultural Trade and Policy Center (IATPC) was established in 1990
in the Institute of Food and Agriculture Sciences (IFAS) at the University of Florida
(UF). The mission of the Center is to conduct a multi-disciplinary research, education and
outreach program with a major focus on issues that influence competitiveness of specialty
crop agriculture in support of consumers, industry, resource owners and policy makers.
The Center facilitates collaborative research, education and outreach programs across
colleges of the university, with other universities and with state, national and
international organizations. The Center's objectives are to:

* Serve as the University-wide focal point for research on international trade,
domestic and foreign legal and policy issues influencing specialty crop agriculture.
* Support initiatives that enable a better understanding of state, U.S. and international
policy issues impacting the competitiveness of specialty crops locally, nationally,
and internationally.
* Serve as a nation-wide resource for research on public policy issues concerning
specialty crops.
* Disseminate research results to, and interact with, policymakers; research, business,
industry, and resource groups; and state, federal, and international agencies to
facilitate the policy debate on specialty crop issues.

Private Responses to Public Incentives for Invasive Species

Ram Ranjan (Corresponding Author)
Postdoctoral Associate
International Agricultural and Trade Policy Center
Food and Resource Economics Department
University of Florida
Email: rranjan(aifas.ufl.edu
Ph: 352 392 1881-326
Fax 352 392 9898

Edward Evans
Assistant Professor
Food and Resource Economics Department
University of Florida
Email: eaevans(ifas.ufl.edu


In this paper the impact of public policies such as subsidies and taxation on invasive
species management is explored in a Markov chain framework. Private agents react to
public incentives based upon their long term expected profits and have the option of
taking measures such as abatement, monitoring and reporting. Conditions for perverse
incentives are derived. The impact of sequencing of taxation and subsides on spread of
risks if explored. One key finding of this paper is that excessive regulation may
exacerbate the invasive species problem

Selected Paper Presented at the 25Ph West Indies Agricultural Economics Conference,
Paramaribo, Suriname Aug 20, 2004

Keywords: Invasive Species, Markov process, Perverse Incentives, Taxation and


It is generally acknowledged that the threat of invasive species cannot be

eliminated but that the risks and the potential damages can be reduced considerably

through effective management. A key ingredient to the effective management of the risks

of invasives is the degree of private participation. However inducing private

participation for controlling the invasives has become a major challenge to the policy

makers. This is so as private objectives greatly differ from the social objectives, since

private agents do not incorporate the consequences of their actions on others. Further, the

limited ability of the regulating agents to monitor private agent's actions reduces the

choice set and efficiency of such public policies. The difficulty is further compounded

when dealing with situations, as is the case with invasive species, where the

manifestations of the adverse consequences are sometimes far removed from when the

action occurred.

A growing concern for policy makers attempting to solicit private participation in

the fight against invasives is the prospect of engendering perverse incentives. Perverse

incentives provide benefits to the targeted agents from taking measures that are counter

productive towards achieving the social objectives such as risk reduction. Given the

uncertainty over biological parameters related to the pests, and behavioral parameters

related to the private agents (such as degree of monitoring and control efforts) certain

policies of the regulating agent (such as compensation for destruction of pest-infested

resources) may provide perverse incentives. Figure (1) shows the indemnity payments

for various pests by the USDA that have steeply increased over time. Concerns have

been raised over the extent of role played by perverse incentives in this increase in

payments. Another example where it has been alleged that perverse incentives could

have played a role is with the protracted time it took to control the spread of the BSE

disease in the UK. While there is no documented evidence to support such claims there it

has been suggested that because the government followed a policy of generously

compensating only those farmers whose cattle were affected by BSE or located within the

prescribed eradication region, those who were left out but faced similar loss of market

opportunities had an incentive to spread the disease amongst their cattle.

A number of studies exist that have explored the various economic dimensions of

managing the invasive species. These include Eiswerth and Johnson 2002, Eiswerth and

Van Kooten (2002), McAusland and Costello (2004), Olson, and Roy (2002), Perrings et

al. (2000), Settle and Shogren (2002), etc. However, the authors are not aware of any

study that delves into the behavioral aspects of inducing or influencing private

participation through public policies. Yet, there exists a great need to understand the

implications of such public policies both from the point of view of increased demand for

public accountability as well as ensuring that the ultimate outcomes are in keeping with

what was anticipated.

There are a number of key questions confronting the policy maker while deciding

how best to influence private behavior. Foremost is the choice between taxes and

subsidies or a combination of both. Due to political considerations and the fact that in

most cases it is difficult to assign the blame of pest infestation and spread on private

resources owners, public policies so far have mainly resorted to subsidies, thus increasing

the chances of perverse motives. In certain cases where farms have been quarantined

without any compensation being paid to the farmer, the policy maker has been taken to

court over the losses from quarantine. Second, is related to the extent of effectiveness of

such policies in terms of achieving the social objectives. Third is whether such policies

induce any further aggravation of the pest problem by providing wrong incentives to the


In this paper we explore all the three questions to a certain extent while primarily

focusing on the practicability of the policy implications that emerge from the analysis. In

order to achieve this we model the behavioral risks involved with invasive species

management in a Markov chain framework. The usefulness of this approach has already

been demonstrated in ecological modeling framework as it offers a very convenient and

transparent way of representing risk evolution through transition probability matrices.

Recently some suggestions have been made regarding the utility of Markov chain

methods in modeling ecological-economic phenomenon, specifically the invasive species

(See Perrings 2003 and 1998).

This paper is divided into two main sections; theoretical framework and

numerical simulations. In the theoretical section we briefly discuss the analytical

framework and allude to some of the major findings of our investigation. The numerical

section puts more structure on the theoretical model and derives some results that are not

easily discernible from the theoretical model.

Analytical Framework

Our analysis involves optimization of private economic benefits in presence of

taxation and indemnities. The economic benefits, however, are modeled as the long run

benefits that accrue from adoption of various levels of monitoring and prevention

activities. These long run benefits are derived as a function of the fraction of time the

resources spend in each of the various 'states' of infestation ranging from non-infestation

to high levels of infestation. Societal welfare may be related to the fraction of time the

resources spend in each 'state' both as the resource owners' revenues and also as the

impact on the 'rest-of-the-world' from the damages that these 'states' may cause in the

event of species escaping outside. Optimal monitoring and prevention efforts are first

derived under a policy of taxation and subsidies and their impact is explored with respect

to the change in societal risks from pests. The role of subsidy in providing perverse

incentives in terms of reducing the risks of detection and level of preventive efforts is

explored. Unfortunately, public policies are mostly reactive and rarely prepare in

advance for pest-infestations. In such a scenario, one key issue of concern is whether

regulatory policies can effectively dissuade a resource owner from continuing with his

production plan and risk subsequent detection rather than report an infestation at the

outset. It turns out that the absolute level of taxation can play a key role in inducing such

a behavior. Consequently, threshold level of reporting-inducing taxation is derived. This

threshold must also have an upper bound so that the resource owner is not stifled in his

entrepreneurial efforts due to a very high level of taxation.

When multiple states of infestation are observable, the regulator has a wider

choice of affecting private behavior through a mix of 'carrot and sticks' policies. Such

policies are often ad hoc and are either related to economic value of resources concerned

or exogenously specified. Under such circumstances, an important question of concern is

whether sequencing of taxation and subsidies matters in terms of affecting private

behavior. Our analysis reveals that it does matter and some of the harshest policies such

as all-out taxation may perform poorly in terms of achieving a low risk of spread of pests

as compared to those policies that combine both subsidies and taxation. Surprisingly, it

implies that over taxation can exacerbate the risk of invasive species spread. Such a

result is governed by several factors such as the differences in revenue from the resource,

risks of detection, abatement costs, etc. under various states of infestation. It highlights

the role of key parameters in influencing the dynamics of private decision-making.

Similarly, another result emphasizes the fact that an all-out subsidies program may not

increase risks as much as a mismatched policy of subsidy and taxation.

In the models below we formalize these ideas.


Consider a private resource R that yields per period economic returns to its owner

at the rate 'z(R). These resources however face the threat of invasion from a certain

species (or a class of species) to which the resources serve as a host, thus aiding in their

further growth and spread. Upon infestation, the per-period revenues from the resources

may decline and are a function of the state of infestation X. Further, the private owner is

not the only one faced with this threat of invasion. The pests threaten to spread into a

wider region outside the private property thus causing economic and ecological damages

at a much wider scale. As a consequence, there is a role for public intervention, as the

private resource owner has no incentives to consider the threats of spread beyond his


We model the dynamics related with the arrival, control, monitoring, detection

and re-growth of species (both hosts and pests) as a continuous time Markov process.

Under this framework, the arrival rate of pests is a defined as A, the control or death rate

as 5, the detection rate as d and the re-growth of host species as 0 1. The Markov chain

process is defined by the set of states and the instantaneous rates as described above. The

states of the above system are {R, X and 0}. R is the state before any invasion, X is the

invaded state and 0 is the state when the resources of the private agent are completely

destroyed due to aggressive public intervention (such as quarantine). The dynamics of

the process involving infestation, control, and resource destruction and re-growth is

shown in figure 2.


Assume that there is a regulator who is able to observe whether the private

resources have been infected or not. However, this observation requires some kind of

effort on the regulator, and detection is not possible with certainty. That is, detection is

probabilistic, with its rate given by d. Further, let's assume that the social costs of

disease infestation on this private property are high, as a consequence, the regulator is

obliged to destroy the host species (which yield economic rewards to the private owner)

on the private property. The regulator can also choose to compensate the resource owner

for the destruction of his resources for the time his resources remain destroyed and thus

do not yield any benefits2. Also, assume that the resources, once fully destroyed, can be

reestablished at the rate 0 without any additional costs3. Alternatively, resources can

grow back instantaneously after a quarantine time distributed with parameter 0. The

1 See Kulkari (1995) for a discussion of continuous time Markov chain processes.
2 There are other forms of compensating the farmer such as direct payments for the value of the destroyed
resource, uniform subsidies, etc.
3 However, the costs of resource destruction are implicitly accounted for in the amount of time spent in the
state 0.

agent takes into account the long run average revenues and costs of his actions while

deciding over level of efforts.

The steady state probabilities of the fraction of time spent in each

state, p(R), p(X) and p(0) are determined by using the fact that the rate of arrival and

exit from a state must be equal in the long run4. From figure 1, this gives us three

equations in three unknowns as:

(1) p(R)A = p,0 + px

(2) px3 + pd = pR
(3) pO = pd

However, one of the equations is redundant, and therefore we make use of the additional

fact that the sum of the fractions of time spent in each states would be one:

(4) + p, + p = 1

Solving the above system of equations we can derive the following results for the steady

state probability values:

O(d + 8)
(5) p(R) = (d
d(A + 0) + 0(3 + A)

(6) p(X) =
d(A + 0) + 0(3 + A)
(7) p(O) =
d(A + 0) + 0(3 + A)

Average expected benefits in the long run would be maximized when the agent sets his

objective function as:

4 See Kulkari (1995) for more details on the methodology.

Maximize with respect to abatement (a) and monitoring (m) with costs c(a) and c(m):

(8) {i(R)- c(m)} p(R)+ (X) c(a)} p(X),

First order conditions with respect to monitoring and abatement imply:

dp(R) dc(m) dp(x)
(9) d (R) (R) c(m)) dcnp(R) + (r(x) -c(a)) = 0
dm dm dm
dp(R) dp(x) dc(a)
(10) (r(R) c(n)) + (r(x) c(a)) p(x) c = 0
da da da

The first order conditions with respect to monitoring requires that the marginal increase

in the amount of time spent in each of the states be worth their cost in the long run. It is

intuitive that if the arrival rate of species is decreasing in preventive efforts, then the

expected amount of time spent in the invaded state x would be falling and the amount

time spent in the un-invaded state R increasing in monitoring efforts. Note that the

rewards from being in any of the states are directly proportional to the time spent in that

state. Similarly, the allocation of abatement efforts in the long run is decided by the

effectiveness of such efforts in affecting the average time spent in each of the states.

Next consider the impact of public intervention on private behavior. Upon

detection the regulator has the option of using monetary rewards or punishment. In this

simple model without multiple stages of infestation, it is hard to combine both the options

simultaneously. Therefore, for now, we assume that the regulator can either offer

rewards or taxes upon detection. Under the above assumptions we analyze the impact of

public policy on abatement and monitoring efforts. The private agent's new optimization

under taxes t, can be written as:

(11) Max: { z(R)- c(m)} p(R) + {(X-) c(a)* p(X) tp()

Note that the private owner taxes are determined by the amount of time spent in the state

with zero resources. This way of taxation makes the costs to the farmer dependent not

only upon the detection rate (which is a function of the regulator's efforts) but also upon

the biological features of the resources concerned. For instance, if 0 is high enough, the

resources would grow back faster thus inflicting less costs to the owner. Alternatively

when 0 is construed as the rate of elimination of quarantine (which is again uncertain),

the taxes t are implicitly derived as a function of 0. Modeling taxation this way allows

us to incorporate the variations in silvi-cultural aspects of the affected or threatened hosts

and is more general. Further, indemnity payments that do not cover the full costs of the

destroyed resources may also be considered as a form of taxation. The first order

conditions with respect to monitoring and abatement efforts can now be derived as:

dp(R) dc(m) dp(x) dp(X)
(12) (r(R) c(m)) p(R)+ ((X) c(a))- d = 0
dm dm dm dm

dp(R) dp(x) dc(a) dp(O)
(13) p (R (R)- c(m)) + (dr(X) c(a)) p(x) t = 0
da da da da

Note from equation (7) that p (0) is a function not only of the detection rate d and the

growth rate 0, but also of the arrival rate of species This is so as the more time the

resources spend in the infested state (x), the higher would be the rate of transformation

into the resource destruction state (0). As a consequence, the effect of increased

monitoring and abatement would be to lower the rate of arrival into the state (0). It

would be optimal to increase monitoring and abatement as compared to the previous case

of no taxation simply because now there is a cost of spending time in the state when

resources are destroyed. Therefore, taxation would lead to increased monitoring and

abatement, thus reducing the time spent in the state p(x) too. However, the degree of

effect would be governed by several factors. First, as discussed above, 0 would play a

key role in determining the cost of detection. A higher 0 would mitigate the impact of

public intervention, as the system would bounce back out of quarantine at a faster rate.

Note that an increase in d, all else remaining constant would lead to a reduction in profits

by increasing the time spent in the infested state and therefore offer additional incentives

to the resource owner towards monitoring and abatement efforts. However, if the

average revenues in the infested state do not fall substantially as compared to the un-

infested state, the level of taxation that would induce the same levels of monitoring and

abatement as before would have to rise, as the costs of spending less time in the infested

state are lower in terms of forgone revenues. Consider a special case in which the pests

are a nuisance only to the public resources surrounding the private resource, whereas the

private resources are not affected by the invasion. In such a case z(X) could be greater

than z-(R) if resources grow over time (assuming the arrival rate of invasive species are

synchronous with the growth rate of private resource). Now the incentives to reach the

state of infestation would be much higher by reducing monitoring and abatement. As a

consequence, the level of taxation needs to be increased even further.

The policy of taxation though effective may not be feasible under most situations

concerning invasive species management. This is primarily because the policy assumes

that the regulator has exclusive rights over managing the 'risk of spread' and therefore

can stop the private agent from propagating it. However, in most cases the private agent

may be the producer of the risk only to the extent that his resources act as hosts to the

invading species. Even there the regulator may not be able to pin down the responsibility

of spread entirely on the private agent as this issue is entirely different from the case of

pollution generating firms that generate pollution as an externality which is directly

linked to their production process. In fact, the production function of such firms can be

modeled as using pollution as one of the inputs. The contribution of pollution to output

in such a case can be considered as positive, whereas in the case of invasive species the

generation of species has a negative impact on the output of the resource. This is

precisely the reason why the public role in invasive species management so far has been a

policy involving all 'carrots' and no 'sticks'. However, with the recent increase in pest

infestations and their potential to affect a large section of the economy including both

consumers and producers, public policy has been increasingly viewing the risk of farm-

related pest spread as the farm owner's liability.

It is obvious that under similar circumstances involving the above analysis, a

policy of indemnities instead would lead to the opposite behavior on the part of private

agents. Both monitoring and abatement would decrease when the regulator subsidizes

the resource owner for the destruction of his resources. This is because the higher the

payment from such losses the lower would be the incentive to mitigate the loss through

spending more time in the non-infested (r). Such a policy though intuitive from the

societal welfare perspective leads to perverse incentives thus causing significant burden

to the regulators' exchequer.

Discounting and Socially Optimum taxation

So far we have ignored discounting of future profits and considered the optimal

decisions from the standpoint of the private resource owner. However, it is also of

interest to explore the role of public policies and factors such as market discount rate,

which may be beyond the private decision maker's control. In this section we derive the

average expected profits in the long run from starting in various states of the system such

as no infestation, infestation or quarantine etc. It may happen that given the costs

incurred from being in various states, the net expected benefits if one starts in the state of

infestation are lower or even negative as compared to the non-infestation state. Such

scenarios are of crucial interest to the policy maker, who can adjust the costs to the

farmers in the infested state in order to dissuade them from going ahead with their

production plan in order to reduce the social costs of pests. That is, if the resource owner

finds the discounted sum of profits form the repeated cycles of infestation detection and

quarantine and back to be negative if his current state is of the infested one, he may chose

to report the infestation rather than risk detection and quarantine. The regulator can

encourage such actions by offering rewards for voluntary disclosures and punishments

for detections.

Following the derivation of average expected discounted costs in Kulkarni (1995),

the relation between the generator matrix, per period payoffs in each state and the long

run expected profits from starting in each state can be derives as follows5:

g(r) 7z(r)
[p I-Q g() = ) (x)
g(0) 7z(0) =-h

The generator matrix Q is derived in the Appendix, g's are the long run expected benefits

from starting in each of the three states and the right hand side denotes the per period

profits in each state. Assume that the per period profits in the quarantined state are -h.

We first consider a case when the private decision maker considers the option of whether

5 See Kulkarni (1995) pgs. 306-11 for more details

or not to continue with production depending upon the state of infestation. The decision

to stop may be construed as voluntary disclosure of the pests to the regulator. For

simplicity we assume the payoffs in each period are net of the optimal abatement

decisions given the state of the art in controlling the pests. This allows us to focus upon

the role of such costs in affecting reporting decisions. The matrix of G's is derived as:

g(x) =
(0 + p)(.8(r) + A~T(x) + ;(r)p) + d(-hA + n(r)(0 + p)
p(O + p) + d( + d(-T(r)0 + h(A + p)) + ((0 + p)(8( + + ;z(x)(A + p))
p(0 + p)({ + A + p) + d({ + A + p))
SI50(r) + 02A(x) hSp hAp + 0O(r)p hp d(-O0T(r) + h(A + p))

The critical level of discount factor below which it becomes optimal for the private agent

to stop production and report infestation is derived by solving for g(7r(x)) = 0 as:

dh d7(r) (0 + 2)z(x) + V(-dh + dr(r) + 0z(x) + rz(x))2 4z(x)(-dh2 + (dO + SO)z(r) + 07r(x)
= 27(x)

Next we explore the optimal tax rate (given the private agent's discount rate) that would

make the reporting decision beneficial, and stopping optimal. This is achieved by

inducing a per period tax rate t in the infestation state X Note that in addition to the costs

of quarantine etc. incurred in the state of detection, this tax rate is charged based upon the

time spent in state X That is, the regulator punishes the private agent on the principle

that the amount of time spent in state X would have a proportional impact on the social

costs of the infestation due to its spread outside the region. This optimal threshold of

taxation using similar analysis can be derived as:

S-dh + + + p d +}0+ (r) + (OA + Op + Ap + p2 ZI(x) dhp
(6 + p)(A + p)

Any taxation beyond the above level makes the private agents profits negative from

starting in the state of infestation and continuing without reporting. The taxation

threshold for a hypothetical set of parameters is shown in figure 3.


There is a great deal of significance to knowing such a threshold level of taxation. Most

policies for invasive species control are reactive in nature and are introduced when pests

have already infested or seriously threaten to infest a neighborhood surrounding the

private resource (which is believed to be harboring them). Under such a situation, the

regulator has no knowledge of the actual scenario and would need to select his taxes such

that they are neither too low so that the private owner has no incentives to stop when he is

infested and would prefer the risk of detection, nor too high so that private enterprise is

not severely stifled. This optimal level of taxation can be derived as:

g(r) = 0 > t > g(x) = 0

where g(x) = 0 is the state when profits from continuing even in the un-infested state are


A Case with Two States of Infestation and Sequencing of Taxes and


One interesting issue of policy interest is whether the sequencing of taxation and

subsidies matters. More specifically, under what circumstances is it optimal to tax first

and subsidize later or subsidize first and tax later, or either tax or subsidize only. This

section extends the above model to a more general case involving more than one state of

infestation. In particular, we look at two states of infestation X and Y, where the state Y is

a transition from X and involves higher levels of infestation. It may be assumed that the

chances of detection are higher at higher levels of infestation. Further assume that

detection in the state X leads to total loss of resources but with an indemnity payment of /

through transition to the quarantined state (/) and detection in state Y leads to an

indemnity payment of h through transition to the quarantined state (h). The rate diagram

is shown in Figure 4.


The regulator could make one of the states less desirable than the other by reducing the

relative indemnity payment of one state with respect to the other or even making it

negative. In order to simplify things a bit, let us assume that the risk of spread to a higher

state, given by A, is exogenously determined, and the only thing the resource owner can

control is the level of his abatement efforts. Average expected profit maximization

problem of the resource owner could be stated as:

(() R) p(R) + {z-(X) c(x, a)} p(X) + (Y) c(y, a) p(Y)
(14) Max:
+ p(l)+h* p(h)

First order condition with respect to abatement effort gives

dp(R) dp(x) dc(x, a)
(R)a rz(R) + (X) (r(X)- c(x, a)) p(x) +
() da da da
dp(y) (() ) dc(y, a) dp(l) dp(h)
(r(y) c(y, a))- p(y) + I + h = 0
da da da da

In order to study the effect of sequencing of taxes and subsidies we compare the

fraction of times the system spends in each state under different policies. This would

6 The derivation of the steady state probabilities is provided in Appendix A.

give us an idea over the extent of externalities generated. For instance, the higher the

time spent in state Y, the higher may be the risk of spread into neighboring areas as

compared to the time spent in state X.

Consider a policy of equal subsidies in both the states X and Y. Three interesting

sub cases (A1-A3) may be considered:

A. 1 = h & z(y)> z(x) > p(y) > p(x)

A.2 = h & 7z(y)> T(x), d(x)< d(y) p p(y)? p(x)

A.3 = h & nc(y) < c(x), d(x) < d(y) > p(y) < p(x)

A.4 I

A. 1 may occur when the pests do minimal damage to the hosts and most of the damages

from them are to the neighboring areas outside the private resource owner. It may also

happen when state y also implies a higher level of resources, i.e. the resources grow over

time and the higher state of infestation may be possible only with a higher level of

resources. In such a case as long as the resources yield more revenues net of damages

from higher level of infestation, 7r(y) > nc(x) would imply that in the long run the resource

owner would have an incentive to keep his resources in the state y. A.2 on the other hand

is the same as A. 1 except that an additional assumption is made related to the chance of

detection being higher in the higher state of infestation. Now it is ambiguous whether the

system would still spend more time in state y as it would be determined by the parameters

of the model. Perhaps with less time spent in state y, one could make up as much

revenues as one makes with more time in state x. However, we have not made any

assumptions about the rate of regeneration back into state R. It may happen that the

regulator enforces a higher level of quarantine (for instance, number of years required to

spend as fallow land before replanting) if the detection occurs in state y. This assumption

can be easily incorporated in the differentiation of subsidies or taxes. The third sub case,

A.3 is obvious as with higher level of detection and lower level of profits in state y, it is

less profitable to spend more time in it. Finally, when indemnities are based upon the

stock of infestation, even lower revenues in state y can provide incentives to spend more

time in that state. This is shown in case A.4. There are a number of other sequences

involving various combinations of taxes and subsidies of interest and as intuition would

suggest it is hard to predict the exact outcome unless the all information is available

related to the parameters involved. In the following section we explore some specific

cases with the help of a numerical example to gain further insights.

A Numerical Example

In order to test our intuition we perform some numerical simulations using

a set of hypothetical numbers. Additional assumptions need to be undertaken regarding

the shape of the cost and revenue functions. We assume non-linearity in the costs of

abatement and effectiveness of control measures with respect to pest mortality. The

specific functional forms and parameter values are presented in Tables 1 and 2 in

Appendix B.

Figure 5 shows the value function of the resource owner under optimal policy, which is



His task is to select the optimal levels of abatement in the two infestation stages in order

to maximize his long run expected value. Next, we do some simulations by changing the

parameters of the model to see their effects on the optimal abatement efforts and the

steady state levels of the probabilities, which are of relevance to the regulator. These

simulations are shown in two sets through tables 3 and 4. We consider a case where the

revenues to the resource owner from all the three states are the same. This assumption is

made in order to highlight the impact of public policy on private managers by neglecting

the impact of lower revenues in the infested states. However, the consequences of

differential revenues should be obvious once the implications from the general case are

derived. In table 3 we also fix the arrival rate of species to be exogenous and unvarying

with the state of infestation, which is relaxed in the later set of simulations. Finally, the

rates of detection too are unvarying with the levels of infestation. First case in table 3

highlights the policy implications of a uniform taxation policy in both the states of

infestation. Notice that the fraction of time spent in state x is much higher as compared to

state y despite the revenues in the two states being similar. This in fact is true for all the

cases in table 3. This has to do with the fact that whereas the arrival rates of species are

constant amongst the states, the only way to reach state y is through x. On the other hand,

state x could be reached through both r and y. When there are subsidies in state y and

taxation in state x (case two), the fraction of time spent in state y increases marginally.

However, the fraction of time spent in state x too increases, as that is the only way to

influence higher arrival rates in state y. This detail was not readily intuitive through the

theoretical analysis above and could be of high significance for policy purposes, as it

highlights the specific linkage effect between states. In cases when the social risks posed

by state y are only marginally higher than the risks posed by state x, this policy would

backfire as the fraction of time spent in x increases despite taxation in x. Without

understanding this interlinkage, however, one may vouch for a policy of early taxation

combined with later indemnities thus producing inefficient outcomes.

Further, notice that abatement falls in state x and is zero in state. In this case

the higher risks from being in state x are compensated by the reduced levels of abatement

costs and the higher benefits from being in state y, which yields rewards through

detection in state h. In the third case, when state y is taxed and state x rewarded, time

spent in x increases and in y decreases, which should be obvious. Notice, that the time

spent in the non-infested state (r) is highest in this case and the risk of spread to outside

areas the least7. Finally, subsidies in the two states yield the highest amounts of time

spent in the infested states and the least amounts of time spent in the non-infested one.

So far, the results are very obvious. However, now allowing for more reality

we relax the assumptions of uniform levels of arrival and detection rates in the two

infested states. Besides we also assume that the amount of time spent in the detection

states (1 and h) is much higher, which is given by the lower levels of departure rate out of

these states8. The results of the new set of simulations are demonstrated in table 4 in the

Appendix. First, notice that with an increase in arrival and detection rates in state y, the

fraction of time spent in state y is uniformly higher than x all through out. Further, the

revenues are uniformly lower as the system spends more time in the detected states,

which yields much lower (10 compared to 90) revenue. Notice two striking results from

7 We have assumed that the risks of disease spread are highest in state y, which is of major policy concern
to the regulator.
8 This can be rationalized as the increase in quarantine time that the detected farm is required to spend
before resuming production.

these sets of simulations. A policy of uniform taxation in both states (case 1) does not

yield the lowest levels of risk in state y and a policy of uniform subsidies (case 4) does

not produce the highest levels of risk in the higher infested state y. Let's explore the case

of taxation first. The lowest level of risk in fact is attained when the regulator follows a

policy of subsidizing the lower infested state and taxing the higher infested state (case 3).

When y is taxed and x is subsidized, the abatement efforts in state x are lower and state y

are higher as compared to the case when both the states are taxed (case 1). In case three,

it pays to spend less time in Y by abating more. Notice that higher abatement in Y also

increase the rate of arrival into state X which is now more beneficial compare to case 1.

Lower abatement in state x serves two proposes. It increases the fraction of time spent in

state x directly and also indirectly through a higher detection rate in state 1. Note that the

later purpose is even more beneficial as it increases the time spent in state r, which yields

the most rewards. Whereas, it pays to increase abatement marginally in state y as it

increases the arrival rate back into state x and consequential bouncing back to states r via

/ while simultaneously lowering the costs of abatement efforts. Note that the reduction in

abatement efforts in state x in case 3 has a higher impact on steady state probability of

state I as compared to state y. Similarly, abatement effort in state x has a higher impact

on the steady state probability of state y than the impact of abatement effort in state y

itself. As a consequence, time spent in state / increases and that in y decreases.

Similar analysis will explain the anomaly in the case when uniform

subsidization yields lower risks in state y than a policy of taxing state x and rewarding

state y. Note that under uniform subsidies in both states (case 4), the amount of time

spent in the non-infested state is lower than that under case 2. In case 2 taxation in state 2

leads to higher abatement efforts in that state. However, in order to make up for the loss

of revenues, abatement effort in state y falls too relative to state 4. This ensures that the

time spent in state y and h are higher than before, thus leading to more societal risks.

To recoup the assumptions that make this kind of result possible: 1. Owners'

resources remain unaffected by pests and the only threats are to the outside world. 2.

Rate of further infestation increases when stocks are already infested. 3. The detection

rates are higher in the higher state of infestation. 4. The level of subsidies and taxation is

exogenously specified (which may be related to the level of resources or other factors).

5. Abatement costs are non-linear and increasing in abatement. 6. Death rates are non-

linear and increasing in abatement efforts.

When assumption 1 is relaxed it is possible that results change if there are higher

losses associated with higher levels of infestation. However, it is the difference in the

relative revenues in the three states that will determine the shape of the outcome

combined with the other parameters that played a key role above. For instance, reducing

the level of revenues in state y to 80 while keeping others constant, the same results

follow as above.


A number of results come out of the above analysis. First, the single state of

infestation model derives the condition for optimal abatement and monitoring efforts of

the private resource manager when only the long run expected rewards are considered.

Private efforts are affected not only by the biological parameters such as the arrival and

death rates of species, but also by the expected revenues in the various states. The

regulator can affect both of these factors through his policies. For instance, the length of

the quarantine would determine the time the resources will yield no or negative profits.

In order to achieve a socially optimal level of risk of spread (given by the fraction of time

spent in the higher state of infestation Y) the optimal tax rate must incorporate the above

parameters as shown in case with optimal taxation under discounting. The threshold

level of taxation is higher when the revenues in state Y are higher, and is lower when the

arrival rate of species is higher (Figure 3). The second result may seem counter intuitive,

but note that under a higher arrival rate of species the private resource owner has lower

profits. Finally, the numerical examples highlight interesting cases where sequencing of

taxation and subsidies may yield different levels of risks of spread. This is an important

issue for policy-making purposes as the regulator often has to decide about the correct

way of providing both and carrots and sticks so that minimum perverse incentives are

generated. It is even more important under situations where little information is available

over the biological parameters of the profit function of the private agents. Under such a

situation it may not be possible to design an optimal level of taxation and subsidy policy.

As a consequence, regulators resort to fixed, or lump sum payment or taxation schemes

like the ones chosen in the examples.


1. Eiswerth, M.E., and W.S. Johnson (2002): "Managing Non-Indigenous Invasive Species:

Insights from Dynamic Analysis", Environmental and Resource Economics, 23, 319-342

2. Eiswerth, M. E., and G. C. Van Kooten (2002): "Uncertainty, Economics, and the Spread

of an Invasive Plant Species". American Journal of Agricultural Economics, 84, 1317-


3. Geoffard, P.Y. and T. Philipson (1997): "Disease Eradication: Private versus Public

Vaccination", American Economic Review, 87(1): 222-230

4. Kulkarni, V.G. (1995): Modeling and Analysis of Stochastic Systems, Chapman and Hall

Publications, UK.

5. Perrings, C. (2003): "Mitigation and Adaptation Strategies in the Control of Biological

Invasions", Paper Presented at the Forth BIOECON Workshop, Venice. URL:



6. Leung, B., D.M. Lodge, D.H. Finoff, J.F. Shogren, M.A. Lewis, and G. Lamberti, (2002):

"An Ounce of Prevention or a Pound of Cure: Bioeconomic Risk Analysis of Invasive

Species", The Royal Society, 269, 2407-2413.

7. McAusland C. and C. Costello (Forthcoming 2004), "Avoiding Invasives: Trade Related

Policies for Controlling Unintentional Exotic Species Introduction", Journal of

Environmental Economics and Management.

8. Olson, L. and S. Roy (2002): "The Economics of Controlling A Stochastic Biological

Invasion", American journal ofAgricultural Economics, 84, 1311-1316

9. Perrings, C., M. Williamson and S. Dalmazzone (2000): The Economics of Biological

Invasions, Edited by Charles Perrings, Mark Williamson and Silvana Dalmazzone,

Edward Elgar Publication.

10. Perrings, C. (1998): "Resilience in the dynamics of economy-environment systems",

Environmental and Resource Economics 11(3-4): 503-520.

11. Settle, C. and J.F. Shogren, (2002): "The Economics of Invasive Species Management:

Modeling Native Exotic species Within Yellowstone Lake", American journal of

Agricultural Economics (84), 1323-38.

Appendix A

In order to solve for the long run steady state probabilities of the state we need to

make sure that the rate of exit from a state is the state is the same as the rate of entry into

it. For instance, as shown in figure 2, the rate of exit from state R is given by p(R)A,

where as the rate of entry into it is given by p(x)8 + p(l)0 + p(h) Using similar logic

we get five equations corresponding to the five states as below:

(16) p(R)A =p(x)3 + p(l)0 + p(h)O

(17) p(x)8 + p(x)A + p(x)d = p(R)A + p(y)8

(18) p(y)d + p(y)3 = p(x)A

(19) p(l) = p(x)d

(20) p(h) = p(y)d

However, one of the equations above will be redundant, and therefore, we also make use

of the fact that the sum of the fraction of times spent in each state must equal 1:

(21) p(l) + p(h) + p(x) + p(y) + P(R) = 1

Using the above equations the steady state probabilities can be solved as:

(22) p(h) =
Sd+9 d+96 + 0\ 0(d +8 +d+9 d5
d9 A dd { 2 +Aj


(d + 3)
(23) p(l)=d+ (d 8 d
+{0d 3 d+3 +1+0 + 0(d+ +5-3 d9
\\ 0-+- +1+- 0 2 JfA+d+ ----A
Sd A d 2+A

(d + 3)d
(24) p(x)=d+ 0} (d+8) d
0 --d+ d+- +1+ + A A+d+8 3 A0
d9 A d\ A2\ '5 8AJj

(25) p(y):

Ld+9 d+S 6 0} \(d+ + d+ (5 dS
+ ds A d 22 ( +A
0d~;1d Ad}{}{


61 {0(d +}{
+1+- + Od- A+d+A
d 22d

Appendix B

Table 1: Functional Forms
8(x) = a(x)^2; 3(y) = a(y)^2; c(x) = aoa(x)3; c(y) = aa(y)^3

Table 2: Base Case Parameters
7z(r) =90; 'z(x)=50; .z(y)=25; 1=10; h=10; a0=1; al=l; A =2; d=3; 0=15;

Table 3: 'z(r)= 90; 'z(x)=90; sz(y)=90; aO = 1; al = 1; = 2; d = 3; = 15;
A (x), a Revenue P (x) P (y) P (r) P (1) P (h)
L =-10, H=-10 2.1,0 83.65 .149 .099 .702 .029 .02

L=-10, H= 10 2,0 84.05 .154 .103 .692 .031 .021




(- d + 8)
8 +\


Table 4: z~(r)= 90; zr(x)= 90; .(y)= 90; a0 = 1; al= 1; A(x) = 2; A(y)= 50; d
(x)= 3;d(y)=5; = 5;
Cases Conditions a(x), a(y) Revenue P (x) P (y) P (r) P (1) P (h)

Case 1 L = -10, H = -10 6.29,1.65 71.1 .0192 0.1245 0.7203 0.0115 0.1245

Case 2 L =-10, H = 10 25.4, .9 73.9 .0177 0.1526 0.6665 0.0106 0.1526

Case 3 L = 10, H = -10 6.21, 1.74 71.3 0.0198 0.1236 0.7211 0.0119 0.1236

Case 4 L= 10, H = 10 5.34, 1 74 0.0182 0.1519 0.6670 0.0109 0.1519

Figure 1:

Trends in APHIS emergency program expenditures
Thousand $
400 00
250O 00-
200P00 -
15opoo i
sopoo -
50 0 -

DKernal burt
*CIrus canker
oAsiar LH beetle
NAIl others

1991 1993 1995 1997 1999 2001

Invasive Species Management: Trends in Emergency

Program Expenditures, USDA Briefing Room

Figure 2: Rate Diagram for the One Infestation State Case

0 R X
~ (m)


Figure 3: The threshold level of taxation under various scenarios

Sign of the numerator in g(x)

2 0C s(x)=150

-- Ba se C a se

2 0 4 06 O xe Os


-400 k=7

z(r)=150; >(x)=100;h=10; A=5;d=10; 3=2; 0= .1; p= .5;

Figure 4: Rate Diagram: Double Infestation States

8(a) 8(a)


Figure 5: The value function for the private resource owner in the base case



(/l V )

,tl \ )

2 /


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