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 Front Cover
 Center information
 Abstract
 Introduction
 Model of producer behavior with...
 Results
 Conclusion
 Endnotes
 Reference
 Figures






Group Title: Working paper - International Agricultural Trade and Policy Center. University of Florida ; WPTC 04-07
Title: Model of producer incentives for livestock disease management
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Table of Contents
    Front Cover
        Page i
    Center information
        Page ii
    Abstract
        Page 1
    Introduction
        Page 2
        Page 3
    Model of producer behavior with endogenous risk of detection
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Results
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
    Conclusion
        Page 21
        Page 22
    Endnotes
        Page 23
    Reference
        Page 24
        Page 25
    Figures
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
Full Text

WPTC 04-07


i -ional Agricultural Trade and Policy Center



A MODEL OF PRODUCER INCENTIVES FOR LIVESTOCK
DISEASE MANAGEMENT
By
Ram Ranjan & Ruben N. Lubowski

WPTC 04-07 December 2004


WORKING PAPER SERIES


i~fr


UNIVERSITY OF
FLORIDA


Institute of Food and Agricultural Sciences


"j_









INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER


THE INTERNATIONAL AGRICULTURAL TRADE AND POLICY CENTER
(IATPC)

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.











A Model of Producer Incentives for Livestock

Disease Management

RAM RANJAN*
&
RUBEN N. LUBOWSKI


ABSTRACT

We examine the management of livestock diseases from the producers' perspective,
incorporating information and incentive asymmetries between producers and regulators.
Using a dynamic model, we examine responses to different policy options including
indemnity payments, subsidies to report at-risk animals, monitoring, and regulatory
approaches to decreasing infection risks when perverse incentives and multiple policies
interact. This conceptual analysis illustrates the importance of designing efficient
combinations of regulatory and incentive-based policies.




Key Words: livestock disease; asymmetric information; reporting; indemnities; risk
management.


JEL Codes: C61; D82; Q12; Q18; Q28.


*Ranjan is a post-doctoral researcher at the University of Florida. Please send correspondence to Ram
Ranjan, Food and Resource Economics Department, University of Florida, G 097, McCarty Hall B, P.O.
Box 110240, Gainesville, FL 32611-0240; Phone: (352) 392 1881 Ext. 326, Fax: (352) 392 9898; Email:
RRanjaniifas.ufl.edu. Lubowski is an Economist at the US Department of Agriculture, Economic
Research Service. The authors thank Michael Livingston, Erik O'Donoghue, and Stephen Ott for helpful
comments. All views expressed here are the authors' alone and do not necessarily correspond to those of
the U.S. Department of Agriculture.









1. Introduction


Governments are under pressure to manage the threat of livestock diseases
because of public health concerns and the negative impacts on livestock producers.
Traditional policies for addressing livestock diseases include testing and monitoring
activities, conducted by the government, and regulations imposed on livestock producers
and processors. Such policies might have limited success, however, if producers do not
cooperate with the government. Payments for reporting sick animals, indemnity
payments for livestock destroyed for disease control, and other incentive-based policies
could encourage producers to aid in disease detection. By creating a suboptimal mix of
incentives, however, regulators could fail to reduce, and even exacerbate disease
outbreaks. If indemnities are too high, producers could find it beneficial to submit low-
grade (cheap) cattle for testing or increase the probability of disease outbreaks. With
insufficient indemnity payments, producers may slaughter too many animals to avoid
future losses; similarly, regulatory policies that ban the use of sick animals may promote
early slaughter to avoid detection.
Designing policies to address animal diseases requires understanding the
incentives faced by livestock owners. In this paper, we develop a model to examine
livestock disease management when both the government and producers can affect
disease risks. Economic studies of livestock diseases have focused on the effects of
health concerns on prices (Piggott and Marsh 2004; Lloyd et al. 2001) and on estimating
potential economic impacts (Matthews and Buzby 2001; Matthews and Perry 2003).
Studies of livestock owner behavior and livestock populations focus chiefly on
explaining cyclical patterns in cattle stocks (e.g. Aadland 2004). Bicknell, Wilen, and
Howitt (1999) examined cattle owners' incentives to control bovine tuberculosis. In
their model, producers select marketing levels, private testing, and eradication of wild
animal vectors based upon prices, biological parameters, indemnity payments, the cost
and efficacy of testing, and government monitoring of slaughterhouse activity. When
government monitoring is 100% effective and producers have no private information
about disease infection, they show that government policies reduce aggregated disease
outbreaks, as well as private incentives to control disease.









One particular livestock disease that needs immediate public involvement at
various levels is the Bovine Spongiform Encephalopathy (BSE) commonly known as the
mad cow disease. Recent outbreak in the US has caused widespread concerns about the
impact on the beef industry from international trade restrictions. Besides, there are
significant risks to the disease passing on to the humans in the form of BSE-CJD. There
is limited scientific evidence currently available over the real cause of disease eruption.
The disease is caused by a malformation in the healthy proteins, prions, in cows.
Conflicting opinion exists as to the real cause of this malformation. Recent experiments
have claimed the cause of this malformation in prions to be spontaneous and infectious
(Legname et al. 2004). Consequently, it is not possible to incorporate the
epidemiological aspect of such a cattle disease in detail at this stage. However, it is still
possible to look at the producer behavior when the risks of such disease spread are
unknown or partially known.
Using a stochastic dynamic model, we examine the incentives of livestock
producers to take private actions that can increase or decrease the potential for disease
detection. We incorporate the asymmetry of information between producers and
government regulators. Producers maximize expected economic benefits from livestock
sales and government incentive payments in pre and post disease-detection scenarios.
Livestock producers make decisions over harvest, reporting, and other activities that aid
government monitoring efforts. They consider expected prices before and after disease
detection, government incentive payments, and subjective probabilities of disease
detection that can be influenced by federal monitoring and by producers themselves via
reporting and other activities such disposal methods of sick animals. We use this model
to examine producers' responses to a range of policy options including indemnity
payments for destroyed livestock, subsidies to voluntarily aid in disease detection,
government monitoring, and regulations that raise the cost of maintaining livestock. We
also consider measures that reduce demand losses upon disease detection, such as
improving animal tracking and identification systems.
The analysis characterizes the complex incentives produced by multiple related
policies. Increased monitoring by the government, regulations to reduce disease
transmission, payments to producers for reporting sick animals, indemnity payments for









destroyed livestock, and policies to identify diseased animals may all potentially increase
the stock of disease. However, perverse incentives may be mitigated in some cases
through changes in payments for reporting, but whether payments should be raised or
lowered may vary depending on the level of monitoring and other variables. We
highlight the significance of designing the right combination of regulatory and incentive-
based policies.


2. Model of Producer Behavior with Endogenous Risk of Detection
The livestock producer maximizes the expected economic benefits from livestock
sales and from government payments before and after the disease is detected by the
government. Upon detection, the presence of the disease becomes public knowledge, and
prices fall due to lowered demand domestically and/or internationally. The model
includes three state variables and two control variables. The first state variable is ct, the
stock of livestock at time t.


The second state variable is qt, the stock of the disease in the population. We
model the stock of disease directly rather than the stock of infected animals, as in
Bicknell, Wilen and Howitt (1999). This formulation is general enough to include
diseases that continue to spread infection after the death of the host animal.
The probability that the presence of the disease in the population is detected by
the government is treated in the form of a third state variable, allowing us to endogenize
the risk faced by producers. We model this endogenous risk using a survivor function,
following previous work that examines behavior given risks from an environmental
catastrophe (e.g. Clark and Reed 1994; Gjerde, Grepperud and Kvemdokk 1999).
In each time period, the livestock producer faces a certain instantaneous

probability of disease detection denoted as A(t). For tractability, we specify a Poisson

distribution, so the probability of detection in any interval dt

is A(t)dt where A(t) = A(t)ds. A Poisson distribution is often used to represent counts of
0
events across time and involves the assumptions that the probability of observing an
event is approximately proportional to the size of a time interval; that there is virtually no









probability of two events occurring within the same interval; that the process that
determines the probability does not change over time; and that the probabilities are
independent across intervals. While these assumptions will affect the exact results, the
model is simply intended to illustrate potential producer behavior given endogenous risks
of detection.
If T is a stochastic variable that represents the time of disease detection in the
entire population of infected animals, the cumulative probability density function

associated with detection is F(t) = Pr(T < t) and F(t) = -e (t) given the Poisson

specification. The survivor function represents the probability of the livestock producers
continuing to market cattle without disease detection up to each time period t and is given

by S(t) = Pr(T> t) = 1- F(t). Under the Poisson specification, S(t) = e t)and the

probability of detection in a particular period t is A(t)e 2). This equals the probability of

detection at time t given survival up to time t without detection.
In our model, the livestock producer affects ct as well as the stock disease and the
risk of detection by choosing two variables at each point in time: ht, the level of livestock
harvested (and marketed), and dr, the level of reporting. This reporting embodies the idea
that livestock owners have certain private information regarding the likelihood that their
animals are infected, which is not available to the government regulator. Thus, producers
have the choice of reporting such information to the government or taking private actions
that increase the chances that the disease is detected (if it is actually present in the
population). For example, there might be reporting activities that entail some private
costs and which will thus perhaps not be undertaken unless the producer receives an
incentive.
Given this framework, the producer's problem is to maximize the present
discounted value of an infinite stream of livestock harvests, net of carrying costs, plus
government payments. In our base model, we consider only government payments
associated with reporting activities. The producer's problem is thus to choose levels of
ht and dt in each period to maximize:

J = croh- c(t)f +zd +Av(t) e (t)e rt (1)
0 1









where ;, is the price per unit of livestock prior to disease detection; h is the amount of

livestock harvested (and sold); f is the cost of carrying (feeding) a unit of livestock; z is

the reward (penalty) faced by a producer for each unit d of reporting activity; v(t) denotes

the value function in the post-detection scenario as of time t; and r is the instantaneous
discount rate. The producer's choice in (1) is subject to the state equations for livestock,
disease, and risk evolution, (2), (3), and (4) below.
Dropping the time notation for simplicity, livestock increase simply as a function
of the existing stock, c, times a fixed growth rate, p, and decline with the level of

harvest, h, and with the stock of disease, q, where u denotes the extent to which the stock
of disease contributes to livestock mortality:

c = pc- h -uq (2)

The contagious disease stock, q, increases with c and an exogenous component :

q = cq- 0 (3)

Disease evolves in proportion to its existing level times the size of the livestock
population net of spontaneous introduction or remission of the disease. Negative
(positive) 0 implies an increase (decay) over time due to exogenous effects. The
multiplicative term captures the element of contagion so that the greater the stock of
disease and the greater the size of the population, the greater the amount of infection over
each time period. This reflects the case of a contagious disease which is directly
transmitted across living animals. The formulation also applies to cases where
transmission occurs through other pathways, such as feed contaminated with tissues from
infected animals.
The change in the probability (risk) of disease detection is modeled as an additive
function of the stock of disease, the amount of reporting, and a function that depends on
the level of government monitoring activity m:

A = aq+ ad -e m (4)

The probability of disease detection depends positively on the disease stock, which will
affect the probability of detection given some base level of surveillance activity.1 Higher
levels of government monitoring m also increase the chances of detection. However,









monitoring and base-level surveillance alone may not be effective in detecting the
disease. Producers can also directly affect the degree of detection through reporting
actions d, which include measures that a producer can take to affect the detection
probability given private information or behavior that is not observable by the monitoring
agency. This formulation for the probability of detection highlights the importance of
private participation in disease control. For simplicity, the marginal impacts ofd or m on
the detection probability are assumed independent of q. This may be realistic if
producers or the monitoring agency can target testing or reporting in a manner that does
not depend on the disease stock.
For the base case, we specify a simple post-detection scenario, in which the price
of livestock declines but the government is able to eradicate the disease completely and
prevent its future introduction.2 Thus in the post-detection scenario the stock of livestock
grows as:

c = pc- h (5)

so livestock growth depends only on the growth rate and level of harvest, with no death
from disease. A more realistic representation would be a scenario where the exogenous
risks of disease evolution remains positive and the price of beef recovers over time.
However, the results from the above formulation can be generalized in a straightforward
fashion to incorporate this and the implications of more complex and realistic scenarios
are discussed in section 3.3. In the base-case post-detection scenario, the livestock owner
realizes returns from livestock sales at the reduced price ofi,1 <;ro. The producer's

objective is then simply to choose harvest levels to maximize the infinite stream of net
returns from livestock sales starting at detection time T:

v(T) = Max (;h c(t)f)e tt (6)
h
T
subject to (5).

Restricting attention to the steady-state level of livestock (c=0), producers

receive an infinite stream of net benefits equal to c(t)(;rp- f)and the value function

can be rewritten as:









v(T)= c(t) -' er (7)
r
As discussed below, we focus on behavior in the steady state even though there is no
guarantee that this equilibrium exists or will actually be reached. As discussed by Clark
and Reed (1994), we assume that the steady state solution provides a useful guide for the
direction in which the system is headed. This will be true if the system converges rapidly
towards the steady state behavior, even if the equilibrium is never actually attained.
Substituting equation (6) into equation (1) and using the result in (7), the
producer's optimization problem can be solved using Pontryagin's maximum principle.
The current value Hamiltonian is written as:

H = 7,/(t) -c(t)f + zd(t) + A v(t) e'(t) + 11 c+ l, q+1, A (8)

where 1, 12, /3 are, respectively, the shadow prices with respect to livestock c, disease q

and the hazard rate A. Substituting (2), (3), and (4), the first order necessary condition
for an optimum with respect to the harvest level h is:

= e -(t)-1 = 0 (9)
Oh
The first order condition with respect to reporting is:
8H
= ze +(t) +av(t)et +13al = 0 (10)
8d
Further, the rate of evolution of shadow prices is given by:

OH rl, -(- -fe +A( -f )e-'e +1p+q +rl (11)
8c at r
8H a /
2 a -- +rl a- = v(t)ae- -l/u+12c(t)+l3a +r2l (12)
8q 2 t

13 -+rl3- roh(t)-fc(t)+d(t)+Av(t) e t)+r (13)

These necessary conditions will also be sufficient conditions for maximization of the

Hamiltonian if it is jointly concave in both the state and control variables (Mangasarian's









theorem).3 In this paper, we assume that the conditions for sufficiency are satisfied (see

Kamien and Schwartz, 1981 for further details).


The steady state requires =0, 12 0, 13 0. Transforming le into present
value shadow prices /, we obtain:

f- (pr- f)ert- Nr- mq+ rn = 0 (14)
r
-v(t)ao +11u -u2C -c-3a0 + r/2 =0 (15)
( *
.,h(t) fc(t) + zd(t) + A v(t) + r3 = 0 (16)

ac 0aq A
Further, =, = 0, and =0 imply:
at at at
h = pc uq (17)

q=-- (18)
c
In the steady state, harvest equals the growth in the stock of livestock net of death from
disease. In the steady-state, disease growth from contagion equals cqwhich is perfectly

offset by exogenous decay .4 The risk of disease detection represented by the hazard
function, A, remains constant as the impact of reporting behavior, monitoring, and
disease levels are balanced as follows:

emao
d e a q (19)
a, a,

Equations (14)-(19) and first order conditions given by (9) and (10) comprise eight
equations in eight unknowns, namely: c, q, d, h, /,, p2, p3, and A .


3. Results
In this section, we examine how the steady-state levels of the state variables
change with changes in the model parameters.5 We emphasize the impacts of policy
parameters on the livestock stock, which is inversely proportional to the stock of disease
in the steady state as shown in (18). We first describe comparative static results and then
illustrate the system's dynamics using numerical simulations.











3.1 Comparative Static Analysis
Using (7) and (14)-(19), we derive an implicit function for the steady state level

of ,3, the rate of change of the shadow price of detection probability A, in terms of the

model parameters:

G= = c2(,p-f -a1Q(p-f)e -r)-c(-Z)(r -e m) (.+ za0)= 0 (20)
a, a,
This equation is quadratic in the steady state level of the livestock stock. We
illustrate the shape of this function given purely hypothetical values for the model

parameters. Figure 1 shows an example of how /u varies with the stock of livestock for

particular values that were selected for producing steady state livestock and disease
levels, as discussed further in section 3.3 on the numerical simulations.6 Because the
parameter values are purely hypothetical, the livestock stock numbers are just an
indicator of the total herd size and do not correspond to any particular physical units,
such as number of cows. The U-shape of the function 1 is a function of the parameters
selected and is simply intended to illustrate some possible results of the model. In the
present case, rewards from extra livestock beyond a certain threshold exceed their impact
on risk from increased growth of disease. Given other values, for example if the growth
rate in disease is highly susceptible to livestock stock or if livestock mortality highly

sensitive to stock of disease, the shadow price of A could follow an inverted U-shaped
curve with respect to livestock levels.
While the chosen values are hypothetical, the figure illustrates how the function
depends on the stock level. The figure indicates that at high levels of livestock, the
shadow price of increased risk of detection is positive while at lower (positive) levels of

livestock, it is actually negative, with ~3 =0 at a level of c about equal to 11. Given our

specification of the post-detection value function, A is a "bad" from a producer's
perspective. However, if rewards in the post-detection scenario exceed the pre-detection

scenario, A would be a "good" and it would pay to increase the probability of disease









detection. Factors that may lead to an increase in the post-detection reward may be
higher prices of livestock (or greater market share) for some producers or indemnity
payments from the government. Given our specification, all else equal, the shadow price

of A should be negative as greater risks of detection imply lower expected profits. A
positive shadow price indicates that higher levels of risks enable higher steady state
levels of livestock sales or of reporting, which increase expected profits.
To examine possible policy impacts, we conduct a comparative static analysis of
the steady-state stock of livestock with respect to key exogenous parameters in the model.
We use the implicit function theorem to obtain partial of the stock of livestock with
respect to the different parameters, with particular emphasis on those which the
government can directly influence. Understanding the impacts of key variable on the
stock of livestock helps in understanding the impacts on the steady-state level of the stock
of disease. As indicated in (18), in the steady state, livestock and disease stock are
inversely related in proportion to the exogenous disease decay parameterO. Depending
on this exogenous rate of disease evolution, livestock and disease must remain in a fixed
proportion in order to maintain the steady state level of detection risk. As the steady state
stock of livestock rises, the steady state stock of disease falls, and vice versa.
We first consider the change in the steady state livestock with respect to the pre-
detection price (;0):

OG
dSc &., uO c2p
S(21)
as OG z
o 2c{op f -a1 (1p- f)e rt} (r-e ')
Oc a1

This equation reflects the tradeoff in terms of balancing risk of detection and increased
mortality due to disease from a marginal increase in livestock versus the increased
benefits from that unit of livestock in terms of current and future harvests. The

denominator (which is the same in all of the partial of c) equals the change in /u, with

respect to the stock of livestock in the steady state. As such, it is the partial derivative of
the instantaneous expected benefits with respect to a marginal change in livestock in the
steady state. The sign of this term varies depending on whether the benefits of an









additional unit of livestock in the pre-detection world exceed the costs of reducing
reporting to compensate for the added risk from the additional livestock.
In the pre-detection scenario, the benefits from an additional unit of livestock are
its contribution to the profits from harvest and additional stock growth minus the
additional carrying costs f and the foregone benefits from the post-detection scenario.

These effects comprise the terms 2c {rp f a (rp f)e rt in the denominator. The

bracketed terms are magnified by the level of livestock. This dependence on the
livestock level arises because both livestock growth and disease growth depends upon the
livestock stock. Livestock growth (before harvest and mortality) equals the stock times
the constant growth rate (cp) in (2) and (5). The dependence of disease on the livestock

stock arises from the biological feature of disease contagion embodied in the term cq in
equation (3). This term indicates that the chances of disease spread increase in
proportion to the size of the population. Thus, for a given level of disease, contagion is
greater for a higher animal population. As a result, when the steady-state livestock level
is high (and disease levels are thus low), the added risk produced on the margin by an
extra unit of disease is greater than when the livestock population is lower (and disease
higher). As a result, the cost-benefit tradeoff in terms of added risk is relatively more
favorable to livestock versus reporting when the livestock stock is higher. This
relationship provides an essential feature of the comparative static results discussed
further below.
An additional unit of livestock raises the growth rate of the disease, which in turn
increases the risk of detection. Thus, reporting must be lowered as disease rises to

maintain a steady state level of risk. The term-(r-e e") captures the cost of marginal
a,

livestock unit on forgone benefits from reporting. As the effectiveness (a1) of reporting
increases, reporting levels need to be reduced by less for the same reduction in risk.
Thus, as a, increases, fewer benefits from reporting need to be foregone for each
additional unit of livestock. Also, the level of monitoring augments the effect of the

discount rate r as the term e decreases with m. The effect of the monitoring level is to
increase the importance of reporting benefits. When monitoring is lowered, the term









(r-e ") may actually turn negative, reversing the impact on livestock stock of the

exogenous variable, if the denominator was negative. This dependence on the monitoring
level is discussed further below.
The net effect on the sign of the partial in (21) depends on the sum of the two
denominator terms, as well as the sign of the numerator. As long as the benefit of an
added unit of livestock exceeds the opportunity cost in terms of foregone reporting, the
denominator will be positive. For a given value of all exogenous variables, there will be
a threshold level of livestock above which the sign of the partial will change. The switch
in the signs of equation (21) depending on the livestock level is depicted in figure 2. For
values that produce a negative numerator, there is a level of stock above which the partial
is negative and, below which, it is positive. This change in sign depending on the
livestock level is a feature of all the partial for the livestock stock in the steady state.
The denominator captures the benefits of livestock versus reporting. The
numerator reflects the tradeoff between added livestock and a higher level of risk in the
steady state. While higher levels of prices in the pre-detection period increase current
benefits from livestock, higher levels of livestock add to risk by producing disease and
higher risk levels increase the chances of detection and of transition into the post-
detection scenario where producers will face a new level of prices. This tradeoff is

captured in the numerator as the numerator is the marginal change in /3, the shadow

price of A, resulting from the marginal change in the exogenous variable, the pre-
detection price. The numerator is also the partial derivative of the instantaneous
expected benefits resulting from the change in risk with respect to a marginal change in
the exogenous variable.
The numerator of (21) shows that the cost of the increased risk resulting from the
higher stock of livestock after an increase in prices is mitigated by the death rate of
livestock from the disease and the exogenous rate of disease decay, given uO, the first
term in the numerator. The increment in risk is augmented by the growth rate of

livestock as given inc2p, the second term in the numerator. If the parameters and steady

state levels are such that this second numerator term outweighs the first, then marginal









livestock adds so much risk that the optimal response is actually to reduce livestock as
prices rise to maintain current livestock benefits in the pre-detection state.
We now consider the impacts on the steady-state livestock stock of livestock
carrying costs These costs will potentially be affected by government policies such as
regulations banning certain types of feed, which presumably increase the costs of
maintaining a unit of livestock. The change in steady state livestock with respect tofis:

a-c c2(1 ale (22)
L'f 2c{, p-f -af(r p-f)e-} z (r-e-m)
al
This partial is illustrated in figure 3. This shows the change in sign depending on the
level of stock, but in the opposite direction than (21), with a positive partial at high
livestock levels and a negative partial at low livestock levels. The reverse direction
makes sense because the impact off is to decrease the value of livestock stock while
higher prices serve to increase the value of this stock in terms of potential harvests. The
instantaneous benefits from a change in risk resulting from higher decrease in a, and
the discount rate. This is because the value of pre-detection livestock (which is now
costlier to maintain) increases in these variables relative to the alternatives of reporting
and post-detection profits. As a result, as a, and r increase, the costs of decreasing
livestock stock (and increasing reporting) in response to greater are greater.
The partial with respect to the post-detection price-level is:
dc c21pe rt
S-c pe (23)
dc 2c{ op- f-a (;Tp- f)e t} (r -e-)
a,
As long as the value of additional livestock sales exceeds the foregone benefits from
reporting (the denominator is positive), an increase in post-detection prices will increase
livestock. This is because now there are greater benefits from adding risk through
increased livestock given that post-detection profits are greater. The benefit of adding
risk through more livestock will be higher for higher levels of a, because this decreases

the foregone benefits from reporting from higher livestock. Post-detection profits
generate greater instantaneous benefits from a change in risk when the growth rate is









higher and the discount rate is lower, as these raise the post-detection livestock and the
value of future livestock harvests, respectively.
We now consider the effect of maintaining the livestock stock with respect to
reporting subsidies (z):

C (r -e -m)+a
c = a, a, (24)
aZ 2c{7.p- f -a1(.1p-f)e rt}- (r-e m)
a,

Figure 4 shows the shift in this partial as stock of livestock increases. For the particular
values of the exogenous variables, higher rewards for reporting imply the denominator is
negative at low levels of livestock. Thus, when the stock of livestock is low (and rewards
for reporting are sufficiently high), the denominator is negative and an increase in
rewards would further reduce the relative benefits of livestock versus reporting, further
lowering the steady-state level livestock. On the other hand beyond a threshold level of
livestock (about c=5 in the example), the partial becomes positive and rewards would
have a positive impact on livestock.
The numerator indicates that the instantaneous benefits from a change in risk in
response to z increases in a, as reporting rewards can be obtained for less added risk, and

increases in c, a0 and 0 as less livestock stock needs to be foregone to offset the added

risk from reporting. The level of monitoring enters in both the numerator and
denominator to adjust the discount rate for the change in the risk of detection. Both the
magnitude and direction of the impact of rewards on livestock could depend critically on
the monitoring level as this can potentially switch the sign of both the numerator and the

denominator if e > r. In order for this to happen, monitoring must fall below some
critical level m*. Consider the implications of a monitoring level below m* combined
with a high level of reporting rewards. Earlier we saw that the response to increasing
reporting payments was to lower livestock at high levels of reward. However, the
incentives are reversed for monitoring below m*. This highlights the role of designing
the optimal mix of public policies in order to reach the desired objectives.
Equation (25) indicates the relationship between livestock stock and monitoring:









Cz
-e
Sc a
C= a,1 (25)
am 2c{zp- f -a(p- f)e rt z (r-e m)
a,

An interesting feature of this equation is that the impact of monitoring will vary based on
its level. Under high monitoring, chances of detection are higher, thus making increases
in livestock more costly in terms of foregone current reporting benefits. This implies that
higher monitoring will lower the livestock stock. This incentive is augmented at high
levels of reporting rewards, adjusted for the contribution of reporting to risk in the

term-. These reporting rewards will also be less important at higher levels of livestock
a,

because, simplifying further, c drops out except for the last term in the denominator

which becomes (r e ).
ca1

Under low monitoring, the sign of the denominator could switch from negative to
positive. Thus, when the benefits from reporting are relatively high, greater monitoring
can switch the tradeoff towards increasing livestock and away from reporting. At low
enough levels of monitoring risk, producers are willing to raise livestock despite high
reporting rewards. The monitoring level in this equation serves to augment the market
rate of discount by increasing the risks of detection.
The comparative static relationships described above illustrate the risk
management tradeoffs that govern producer's responses to different possible policies.
The results suggest potentially perverse policy outcomes given the public health
implications of livestock diseases. Policies that increase benefits from livestock (such as
subsidies to beef or dairy industries or other livestock producers), that increase costs of
carrying livestock (such as regulations on feed), that reduce post-detection livestock
losses (through improved tracking and surveillance), that pay producers for reporting, or
that increase monitoring can each lead to either increases or decreases in the livestock
stock with opposite implications for the level of the disease. Both the magnitude and
direction on the steady-state disease stock will depend on the value of all of the
exogenous parameters, as well as the steady-state level of the livestock stock itself. For

example, equation (21) shows that for the case of r > e m policies that increase livestock









prices will increase stock and reduce steady-state level of disease only when the livestock
level (and growth rate) is relatively low compared to the degree of lethality and decay of
the disease. This effect could be reversed at low levels of stock.
The comparative static results underscore the importance of selecting an efficient
mix of incentive-based policies and monitoring. The importance of sufficient monitoring
is evident from the incentives induced by a scheme in which payments are increased but
monitoring is low. These will be the reverse of the incentives induced by higher
payments and high monitoring. Policies that lead producers to increase disease stock
might increase the livestock sector's profits, but may not necessarily increase the overall
public good. Understanding the risk calculus of producers is thus essential for making
policy adjustments and developing an efficient portfolio of government interventions.


3.2 Alternative Scenarios for Post-Detection Values
Our analysis illustrates certain elements of optimizing behavior under simple
assumptions about the nature of disease spread and livestock dynamics. Several
additional complexities are worth considering. One case is that of high sensitivity of
import demand to an outbreak of the disease. In this case, the fall in world prices after
detection may be related to the extent of the disease in the environment. This is reflected
in the following post-detection value function:

v(T,q, c,r,;(q))= (Q1 -lq)h -c(t)f)e rtt (26)
t
where 1 is the parameter measuring the impact of the disease stock on prices upon
detection. In this case, producers will have additional incentives to reduce the disease.
Similarly, if indemnities i are provided in case of disease detection and indemnities are
based upon the level of q, then the post-detection value function becomes:

v(T, q,c,r, ', p)= (lh c(t)f)e t + i(q)e (27)
t
Given (27), producers would face greater incentives to report but also would potentially
face perverse incentives to raise the level of q in order to increase detection risk and thus
obtain indemnities. Both the cases represented in (26) and (27) could be present, with the









effect of indemnities and of price losses correspondingly decreasing and increasing the
costs (incentives to avoid) disease detection.
Another potentially relevant scenario is one in which the disease cannot be
eliminated completely and recurs after detection. For simplicity, consider a case where
the second detection leads to complete destruction of the cattle stock and thus a total
shutdown of the industry. The value function after the second detection is:

v(T, q, c,r, p) =0 (28)

For the period between the first and the second detections, the current value Hamiltonian
is:


H = ,lh(t)- c(t)f + zd(t)e- t + 11c + 1,q + 1,1


(29)


The owner's objective is to maximize the sum of the discounted value of his returns from
cattle and reporting rewards net of the costs of carrying the stock. In contrast to equation

(8), A in this equation serves only as an additional discounting term because the
livestock owner receives no benefits in the post-detection scenario. Falling prices after
the first detection lower the optimal steady state level of cattle as shown by equation
(23).7 Assuming that the amount of cattle stock eradication to reach the steady state is
trivial, we can derive the steady state level of cattle as a solution to the equation below:

Tlu8 +z a0
S- rc a = 0 (30)
f + 7al (r p)


Equation (30) is a quadratic form whose roots are given by:

r 2a
c = --+r2+4
2 f + r, (r- p)

The value function after the first detection is now:


a m
v= {p1rc- cf+ z(e aq)}e-t
a1 a, c


(31)


(32)


where the first term in the integral is the steady-state benefits from cattle harvest, and the
third term is the benefits from steady-state reporting activities. Using these equations, the
Hamiltonian is:









H = [oh(t)- c(t)f + zd(t)+ I vye- + 1,c + 12q + 131 (33)

This Hamiltonian differs from (8) in that there is now a constant reward to be had after
the first detection. The steady state value of cattle satisfies this implicit function:
(a0 + v)
p .,q + zq (ao )
a
c2- -= 0 (34)
f+po(r- r)

The roots of this equation are:

I (+ v)
pouq + zq (a + )
c = r2 + 4 a1 (35)
2 f f+po(r- r)

It is interesting to compare the steady state values of cattle before first and second
detections. Note that the steady state value of cattle given by equation (31) after the first
detection would be higher than the steady state value before the first detection (35) as
long as the effect of lower profits zo in equation (35) dominates the extra term (V) within

the roots, which could be negative. Intuitively, as the number of detections increase and
the value from cattle falls as additional detections occur, it pays to incorporate the impact
of current actions on future losses in advance. This fact is confirmed after solving for the
steady state levels of cattle using the same set of parameters as before, as the value
function (V) after the first detection turns out to be negative. Solving for the steady state
levels of cattle using the same set of parameters as before, we find that the steady state
level of cattle equals about 3.2 units after the first detection. The cattle stock does not
converge to a steady state for the period before first detection, implying that the risks
from cattle increase at a much faster rate than could be compensated for by the
exogenous parameters. In contrast, the steady state level of cattle in the single detection
scenario is equal to 13, as presented in figure 1. This confirms the intuition that multiple
detection scenarios imply a lower optimal cattle stock. This suggests that expectations
about continuing government efforts-and how they will affect future livestock profits-
will be important in shaping livestock owners' risk mitigation decisions.
So far we have focused on comparisons of state variables. However, it is also
important to examine the dynamics involved with the non-linear nature of disease
evolution. We explore these issues in the next section.











3.3 Numerical Analysis of the Dynamics


Given the non-linear nature of the state equations, examining the time path of the
key policy variables such as livestock, disease, and reporting may provide some insights
into potential policy effects. In this section, we briefly explore the role of some key
parameters on the system dynamics. We use numerical simulations to examine the
impacts of key parameters on the time path of the state variables. Figure 5 shows the
time path of the stock of livestock under various situations. The base case reflects a set of
hypothetical parameter values that were selected for producing steady state livestock and
disease levels, as shown in figures 5 and 6 respectively.
While the stock of livestock falls to a steady state level in the base case, when the
growth rate of disease--affected by the stocks of both livestock and disease---is lowered
exogenously, the stock of livestock falls. This parameter, termed a3, lowers the disease
transmission mechanism in equation (3) and could reflect regulatory measures, such as
restrictions on livestock feed, that could reduce disease spread8. A lower impact of the
stock of disease and livestock to the growth rate of disease would allow for a larger stock
of livestock. The stock of livestock falls initially along an optimal path up to a certain
point and then increases beyond it. This ability to raise the stock of livestock at later
stages is made possible after the exogenous rate of decay of disease has a higher
(negative) impact on the growth rate of disease as compared to the much lower (but
positive) impact from the combined effect of increased livestock but decreased disease.
Stock of livestock is also lower when the disease-induced death of livestock, given by the
parameter u, is higher. The state stock of livestock, however, later rises above the base
case even though the death rate is higher. This is again made possible by the high
reduction in the stock of livestock in the beginning stages, which has a significant
lowering impact on the growth rate of disease in the later stages, thus allowing for a
higher stock of livestock. This reveals the complex nature of disease dynamics that can
arise.
Finally, we examine the impact of discounting. Initially, a higher discount rate
lowers the stock of cow livestock but eventually the steady state level of livestock is









higher than the base case. While this may seem counter-intuitive, a larger stock of
livestock adds to the death rate and risk of disease detection, as well as providing
revenues from livestock sales. The negative impacts are reduced once the stock of
livestock is significantly lowered in the initial stages. As a return, later stages allow for a
higher stock of livestock. Although the livestock stock is higher in the later periods, the
disease is growing at a lower rate compared to the base case.
Figure 6 depicts the time path of disease under similar conditions. In the base
case, disease increases at the same time as the stock of livestock. When the growth rate
of disease (parameter a3) is lower, disease actually falls in the later stages as the
exogenous impact of the decay parameter takes over. This fall in the disease is driving
the counter intuitive results above. Disease falls under the case of higher disease-induced
death of livestock as livestock levels are reduced. Finally, the impact of discounting is to
stabilize the stock of disease above the base case levels.
Figure 7 illustrates the impact on reporting behavior under similar scenarios.
There is no reporting in the base case. When a3 is lower, stockowners avail of the
benefits of reporting rewards by considering the costs and benefits of increased detection
risks. Higher disease-induced death rate increase also allows for reporting due to a
reduction in the growth of disease due to the reduction in livestock stock. All the
reporting actions take place at a later stage when the discounted value of the costs of
reporting in terms of livestock sales is lower.
This examination of the dynamic aspects of the model reveals that the time path
of disease may follow highly counter-intuitive patterns. These responses would be
difficult to explain without understanding the underlying patterns of private optimizing
behavior.


4. Conclusion
This paper examined the behavioral aspects of livestock disease management
from the livestock owner's perspective. We developed a stochastic, dynamic model of
livestock levels and disease for a representative producer who can take private actions to
increase the government's chances of disease detection. In this model, the producer









maximizes expected revenues from the optimal management of livestock sales and any
behavior that increases the chances of disease detection.
Several insights arise from the comparative statics and the numerical dynamic
analysis. The comparative statics indicated that it is critical for the regulator to use the
efficient mix of available options, lest they should lead to perverse incentives. The
dynamic analysis further revealed complex interactions of the biological and economic
processes that may lead to counter-intuitive behavior on the part of the private stock
owners faced with various sources of risk. Next steps in this work will focus on
determining the existence of steady-state equilibria under different modeling
assumptions.
Future research would benefit from a better understanding of the biological
processes and their relationship to the potential economic and policy responses.
Additional insights could potentially be gained from modeling the variation in individual
producer behavior and the relationship to the livestock industry at the national level.
Operations of different sizes and types could also respond differently to prices, costs,
disease, and government policies. The level and nature of disease in the national herd or
in different subpopulations might also affect the risk calculus of individual producers
given different levels of contagion as well as market segmentation and traceability.
Realistic estimates for key parameters would also enable comparisons of producer
responses to different policies in the context of actual economic and biological scenarios.









Endnotes


1 Equation (4) only applies to positive and nonzero levels of disease.
2 For simplicity, we assume that the number of cattle that need to be eradicated for
disease eradication are minor and do not to affect the livestock owner's incentives.
3 This would require that the 5x5 Hessian matrix comprising the second order partial
derivatives of the three state and two control variables is negative semi-definite. In order
to establish negative semi-definiteness, it must be shown that all the principal minors
have discriminants that alternate in sign, with the first one being negative.

4 In examining the steady state solution, we assume the existence of a steady state in
which monitoring and an exogenous decay of the disease stock lead to constant A and q.

5 Note that for diseases that do not experience any exogenous decay, there may not be a
steady state. However, a steady state analysis is only a comparison of relative values and
it may still be possible to redefine variables in order to study their steady state behavior.

6 The steady-state cattle and disease levels are shown in figures 5 and 6, respectively, for
a0 = 0.2, a, = 0.7, f = 0.4, m = 10, r = 0.5, t = 10, u = 0.6, z = 30, o = 5, r = 1, p = 0.1, 0 = 0.3.
7 The owner will raise the cattle stock as compared to the steady state if the post-detection
prices actually increase or if the indemnities paid by the government ex-post are
sufficiently high. To model these cases, the value from cattle after the first detection
would have to be broken into two parts. The first part would equal the stream of benefits
from cattle until the cattle stock reaches its steady state value and the second part would
equal the stream of benefits at the steady state value.

SWe redefine equation (3) as q = a3(cq)- .









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Figure 1:
Rate of Change of Shadow Price of Risk with respect to Steady-State Cattle Stock


j113
10



5-

Cows

5 10 15 2C


-5



-10

Parameters: ao =.2, a1 =.7, f =.4,m = 10,r =.05,t = 10,u =.6,z =30, /o = 5, 7,p = 1,p =.1,0 = .3
Results for Solutions with negative cattle stock are omitted.











ac
Figure 2: Change in Steady-State Cattle Stock with Respect to Current Prices ( )
C*,7,


Parameters: ao =.2, a = .7,f = .4,m = 10,r =.05, t =10,u =.6,z =30, /o = 5, l1 =, p =.1, 0 = .3
Solutions with negative cattle stock are omitted.











Qc
Figure 3: Change in Steady-State Cattle Stock with Respect to Carrying Costs (-)
Of



100









25



Catf6 stock
^ 5 10





-50


-75



Parameters: ao =.2, a = .7, f =.4, m = 10, r =.05, = 10,u =.6, z =30, To =5, 1 =1,p =.1, 0 =.3
Solutions with negative cattle stock are omitted.













Figure 4: Change in Steady-State Cattle Stock with Respect to Reporting Rewards
(c

8z

















S Catt stock (c)
10 15 20

-2



-4


-6



Parameters: ao =.2, a, =.7,f =.4, m =10, r =.05,t =10, u =.6,z =30, To =5, l = ,p =.1,0 =.3
Solutions with negative cattle stock are omitted.












Figure 5: Evolution of Cattle Stock (c) under Alternative Parameter Values


0 45


1 21 41 61


Parameters: m 100,r = .05,r
po = 2.5,p, = .95,a3 = 1.3,qo


S.05,ao
.005,co


.2,a = .01,u = .01,z = .005,q = .003,f = .681,
3,1 .01.












Figure 6: Evolution of Disease Stock (q) under Alternative Parameter Values


11 21 31 41 51 61 71


Parameters: m = 100,r = .05,r = .05,ao
po = 2.5,p, = .95,a3 = 1.3,qo = .005,co


.2,a, = .01,u
3,1 o .01.


.Ol,z


.005,q .003,f = .681,












Figure 7: Evolution of Reporting Actions (d) under Alternative Parameter Values


21 41
Time


Parameters: m = 100,r = .05,r = .05,ao
po = 2.5,p, = .95,a3 = 1.3,qo = .005,co


.2,a = .01,u
3,1 o .01.


.01,z = .005,q = .003,f = .681,




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