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ESSAYS ON THE ECONOMICS OF
LIVESTOCK DISEASE MANAGEMENT:
ON-FARM BIOSECURITY ADOPTION,

ASYMMETRIC INFORMATION IN POLICY DESIGN, AND
DECENTRALIZED BIOECONOMIC DYNAMICS

presented by
Benjamin M. Gramig

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5/08 K;IPro;/Acc8Pres/ClRC/DaleDue indd

ESSAYS ON THE ECONOMICS OF LIVESTOCK DISEASE MANAGEMENT:
ON-F ARM BIOSECURITY ADOPTION,
ASYMMETRIC INFORMATION IN POLICY DESIGN, AND
DECENTRALIZED BIOECONOMIC DYNAMICS

By

Benjamin M. Gramig

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Agricultural Economics

2008

ABSTRACT

ESSAYS ON THE ECONOMICS OF LIVESTOCK DISEASE MANAGEMENT:
ON-F ARM BIOSECURITY ADOPTION,
ASYMMETRIC INFORMATION IN POLICY DESIGN, AND
DECENTRALIZED BIOECONOMIC DYNAMICS

By
Benjamin M. Gramig
Livestock disease management involves both private and public resources and takes place
in an environment of uncertainty. An econometric procedure to estimate disease control
functions that will inform herd-level decision making is proposed and demonstrated to
shed light on the determinants of health management practice adoption. An incentive
compatible, government provided indemnity for private livestock assets culled in
response to an outbreak of contagious disease when the government is constrained by
hidden action and hidden information is characterized and compared with status quo
indemnities. A bioeconomic model with feedbacks between disease and behavioral
strategies is constructed to evaluate the nature of strategic effects between private
decision makers in a decentralized setting when government policies are a source of

externalities.

This dissertation is dedicated to my wife Melodi, without whose love, patience, and

understanding it would not have been possible.

iii

ACKNOWLEDGEMENTS

My major professor Rick Horan has been an enormous resource to me in all facets
of my doctoral education. Rick has gone out of his way to help me develop as a
researcher and has provided a solid model of professional success for which I am deeply
grateful. Chris Wolf has been constant source of information on dimensions of farm
management and has played an important role in my dissertation research as an advisor
and collaborator. This work would not have been possible without his contributions. 1
would like to thank Roy Black for being a constant source of information on most any
topic I have found to be of interest since before I came to MSU. Roy has proven to be
the voice of experience; his wealth of knowledge and insights across a wide spectrum of
applied problems is an indispensable asset in the research process. Frank Lupi has
provided expertise in choice modeling and helpful feedback at each stage of my graduate
education at MSU. Frank’s persistence always leads to higher quality outputs and I have
benefited from his many insights about research and the profession throughout my time at
MSU. I look forward to becoming someone that Rick, Chris, Roy and Frank are proud to

call a colleague and a friend.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vii
LIST OF
FIGURES ........................................................................................... viii
INTRODUCTION ................................................................................. l
ESSAY ONE
ECONOMIC ANALYSIS OF LIVESTOCK HEALTH
MANAGEMENT DECISIONS .................................................................. 5
Introduction ................................................................................ 5
Considerations for Herd-level Disease Management ................................. 7
Estimation of Disease Management Adoption Decision ............................. 13
Implementing the Two-Stage Estimation Procedure ........................ 15
First Stage: Disease Control Function Estimation ........................... 17
Second Stage: Estimation of Adoption Equations ........................... 23
Conclusions ................................................................................. 26
Appendix A ................................................................................. 30
References .................................................................................. 33
ESSAY TWO
LIVESTOCK DISEASE INDEMNITY DESIGN WHEN MORAL HAZARD IS
FOLLOWED BY ADVERSE SELECTION .................................................... 36
Introduction ................................................................................. 36
A Dynamic Model of On-Farm Decision Making ..................................... 39
Fundamental Asset Equations ................................................... 42
The Indemnity Design Problem .......................................................... 46
The Adverse Selection Problem—Reporting .................................. 48
The Moral Hazard Problem—Biosecurity Investment ....................... 49
Implications for Public Policy and Market Insurance Design ........................ 60
Appendix B ................................................................................. 63
References ................................................................................... 69
ESSAY THREE
JOINTLY-DETERMINED LIVESTOCK DISEASE DYNAMICS AND
DECENTRALIZED ECONOMIC BEHAVIOR ............................................... 72
Introduction ................................................................................. 72
Livestock Disease Dynamics ............................................................. 74
A Dynamic Model of Farmer Behavioral Choices .................................... 77
Biosecurity Dynamics and Strategic Effects ........................................... 82
Reporting Dynamics ....................................................................... 88
Numerical Simulations and Model Comparisons ..................................... 89

Discussion and Conclusion ............................................................... 93
Appendix C ................................................................................. 96
References ................................................................................. l 01

vi

LIST OF TABLES

Table 1.1 Weighted 1996 NAHMS Dairy Sample Statistics for
Bovine Leukosis Virus (BLV) ................................................................. 30

Table 1.2. Stage I Estimation Results from Fractional (flogit) and
Binary (logit) Response Models ............................................................... 31

Table 1.3. Stage II Estimation of Determinants of
BLV Reducing Management Practice Adoption ............................................. 32

Table 3.]. Baseline Modeling Parameters for the 5-1, Behavioral,
and Joint Model Simulations .................................................................... 96

vii

LIST OF FIGURES

Figure 2.1. Timing and information setting of interaction
between the government and farmers .......................................................... 65

Figure 2.2. Individual farmer decision tree: disease states
in bold and farmer choices in dashed boxes .................................................. 66

Figure 2.3. Fines f are set to ensure positive marginal incentives
to report for all prevalence levels 9 ............................................................ 67

Figure 2.4. Possible shapes for first- and second-best (SB)
indemnity payment schedules with a discontinuity at E .................................... 68

Figure 3.1. One-Dimensional Illustration of Strategic Complementarities

in Biosecurity b, When S, I, and r Are Held Fixed: (a) b=0,1 are both stable

equilibria; (b) b=0 is the only stable equilibrium; (0) b=1 is the only

stable equilibrium ................................................................................. 97

Figure 3.2. Comparison of Steady State Model Results for
Baseline Parameterization of S-I, Behavioral and Joint Model ............................. 98

Figure 3.3 Approach Paths to Steady States Exhibit Dampened
Oscillations in the Joint Model Due to F eedbacks ........................................... 99

Figure 3.4. Time Paths for Biosecurity Strategy, b(t), Under Joint Model:

(a) N0 behavioral extemalities (x=0); (b) “Small” behavioral extemalities (x=4);
(c) “Large” behavioral extemalities (x=10) .................................................. 100

viii

Introduction

The outbreak of disease in domestic livestock herds is an economic and potential
human health risk that involves both the government and individual livestock producers.
Because livestock diseases impose significant costs on society (Bennett 1992, 2003;
Bennett, Christiansen and Clifton-Hadley 1999; Bennett and ijelaar 2005; Buhr, et al.
1993; Chi, et al. 2002; National Research Council 2005), there is a need to understand
management aspects of these problems. It is useful to understand both optimal public
(centralized) management in response to a disease outbreak (Kobayashi, et al. 2007a, b;
Mahul and Durand 2000; Mahul and Gohin 1999), and the decentralized behavioral and
disease responses to common policy initiatives (Hennessy 2005, 2007; Hennessy, Roosen
and Jensen 2005).

Economic justifications for public intervention in disease control include
extemalities, public good aspects of disease problems, coordination failures, information
failures, and income distribution considerations (Ramsay, Philip and Riethmuller). The
potential of a disease outbreak to pose a large economic cost depends on many factors
including trade laws and the nature of the disease (i.e., infectiousness). For infectious
diseases of public concern, policies range from bounties for infected livestock to required
herd depopulation and farm decontamination; for diseases without international trade
implications disease management may be a purely private endeavor.

The United States Department of Agriculture’s Animal and Plant Health
Inspection Service (APHIS) provides inspection and quarantine services to prevent the
introduction of disease across national borders and also coordinates response to disease

outbreaks that originate from within the country. Border measures that protect against

incursion of disease are provided to protect the safety of the American food production
system and to prevent infection of the domestic livestock industry. Diseases not endemic
to or currently present in the US. are typically not accounted for in everyday farm-level
biosecurity measures because their risk of occurrence is viewed as exogenous to
individual farm-level decision making. However, it should be noted that private health
management measures taken with production diseases (as distinct from those of
international concern) in mind may provide social benefits in the event of outbreak of an
exotic or highly contagious disease by lowering disease transmission rates or preventing
spread altogether, as well as keeping the farm prevalence rate of tolerable endemic
diseases low overall. For these reasons, the government reasonably takes an interest in
aggregate private investments in biosecurity in a more general sense.

Livestock disease management is an inherently dynamic process that involves
feedbacks between disease ecology and private behavioral decisions. Throughout this
dissertation I focus on two common decisions that livestock managers must make: a
livestock manager must sequentially decide whether or not to invest in biosecurity as a
preventive measure and whether or not to report infection to government authorities in
the event that a disease outbreak occurs. The decision to report infection may or may not
arise depending on the uncertain disease outcome, but at some point everyone faces the
biosecurity decision. Because it is unclear in advance whether or not livestock will be
exposed to infection, private investment in biosecurity represents a definite investment of
resources in exchange for an uncertain future benefit (own herd protection).

Biosecurity may not fully protect one’s own herd, and, moreover, losses due to

increased regulatory stringency may also arise as a result of a neighbor’s herd becoming

infected — even in the absence of infection in one’s own herd. This is because all herds
within infected regions may be affected by costly regulatory actions taken by animal
health authorities to eradicate infection. Such regulatory-induced extemalities are a
common feature of livestock disease problems. For instance, all farms in the bovine
tuberculosis (bTB) infected region in Michigan’s Lower Peninsula — regardless of
infection status — incur private costs as a result of dealing with government testing,
movement restrictions and stringent testing rules for trade in live animals that go
uncompensated by the government.

It is important to note that from the government’s perspective individuals have
private information about the preventive biosecurity measures in place on their farms—
hidden actions—prior to outbreak (ex ante), and following an outbreak (ex past) they
possess private information—hidden information—about the disease status of their herd.
Once uncertainty about the occurrence of an outbreak is resolved, regardless of the ex
ante actions taken by producers, disclosure of disease status (as opposed to discovery by
another party through transmission, slaughter, or testing) is required for timely
government response to limit the spread of infectious disease and eradicate it. The length
of time between outbreak and discovery is very important in determining the duration and
severity of epidemic (UN-FAO). For these reasons, creating an incentive structure that
results in reporting of disease status is of great interest for a social planner.

This dissertation consists of three essays which each address particular elements
of livestock health management behavior and disease risks in order to fill distinct gaps in
the scientific literature and provide insights for policy design. The essays progress from

a static farm-level analysis of a single private decision, to the design of government

policy in the presence of asymmetric information when private decisions are two-
dimensional, to a jointly-determined dynamic system with extemalities and two-
dimensional private decision making.

The first essay considers the farm-level focus on production diseases and the
decision to adopt individual biosecurity practices on the basis of private benefits and
costs (even though social benefits may be provided as discussed above). An empirical
procedure is proposed to inform private herd health decision making and analyze the
determinants of biosecurity adoption behavior by accounting for preventive spillovers
(positive input extemalities) in the management of production diseases. The second
essay is concerned with the optimal design of government provided indemnities when the
regulator cannot observe preventive behavior or the infection status of individuals in the
population. The centerpiece of this essay is the use of a single policy instrument to
structure incentives so that private biosecurity and reporting actions are compatible with
policy objectives when the disease of concern is responded to by mandatory culling of
private herds. The final essay is the most complex of the three and accounts for dynamic
feedbacks between behavioral choices and disease risks. A bioeconomic model of
decentralized private decision making is constructed to examine the nature of strategic
effects associated with biosecurity and reporting behavior when government disease

eradication programs result in cross-farm extemalities.

Essay One

ECONOMIC ANALYSIS OF LIVESTOCK HEALTH MANAGEMENT
DECISIONS

Introduction

Livestock disease health management decisions are made weighing the costs of illness,
prevention and control. These decisions are informed by the research of animal
scientists, veterinarians and epidemiologists that quantifies the production effects of
disease, identifies the individual management practices that are associated with the
occurrence of specific diseases, and provides recommendations for prevention of disease.
Private economic decisions about adoption of disease management practices utilize this
information in tandem with associated market prices and the impact of the management
practices on infection level.

We examine adoption of disease prevention and management actions taken by
livestock managers for endemic production diseases.l Our research objectives are
twofold: (1) estimate disease prevalence as a function of farm practices and
characteristics and (2) analyze the determinants of health management practice adoption.2

To address our first objective we adopt the fractional logit model as a statistical method

 

' We define production diseases as those endemic to the US national herd which pose no
potential for international trade sanctions because they are neither highly contagious nor
threatening to human health. We assume that exotic or foreign animal diseases are not
taken into account in daily decision making by farm managers because the government
provides border controls to prevent entry of exotic diseases and pays indemnities in the
event that outbreak of such diseases occurs and animals are culled from private herds to
control the spread of infection and eradicate the disease.

2 The former topic is best targeted at the veterinary health sciences, while the latter is
very economic in nature. The two objectives are tackled simultaneously in a single essay,
though it is felt that the exposition may be improved by addressing them on an individual
basis.

that directly estimates herd prevalence. We find that a fractional response model
provides more useful information for disease management decision making than standard
logistic regression performed at the herd-level, which is common in veterinary
epidemiological studies. Estimated herd prevalence allows for the comparison of costs
and benefits of undertaking disease management practices when information about the
production effect of disease is known.

In the second stage, we use the results from the first stage, which provide an
estimate of the marginal effect of adopting a management practice on the herd prevalence
of disease, to create a new variable that is the estimated cost of damages from disease
suffered as a result of not adopting the management practice. This new variable becomes
an explanatory variable in an adoption equation which is estimated using the familiar
binary outcome logit model. An adoption equation is estimated in this way for each
practice found to have a negative, statistically significant association with herd
prevalence in the first stage regression. The empirical method is demonstrated utilizing
the US. Department of Agriculture’s (USDA) National Animal Health Monitoring
System (NAHMS) survey of dairy farms in 1996. Specifically, we examine the adoption
of management practices related to bovine leukosis virus (BLV).

After addressing the motivation for our two research objectives in detail and the
methods proposed to address each in the next two sections, an empirical application of
the proposed methods using herd-level data on a single disease and farm management
practices on dairy farms in the United States is presented. The article concludes with a
discussion of empirical results and possible extensions to this analysis using the full

scope of data that may be available to the researcher.

Considerations for Herd-level Disease Management

Information about the association between disease and particular management
practices often comes from veterinary epidemiological studies which refer to those
practices found to be positively associated with disease as “risk factors” for infection.
Two factors are of critical importance in evaluating the usefulness of statistical estimates
from veterinary epidemiology for economic decision making: the level at which disease
data are collected and how the disease outcome is represented. First, consider that studies
in the animal health literature present estimates based on either individual animal data or
herd-level data, which can affect the usefulness of estimates from the study for economic
analysis. Second, the disease outcome that is studied determines the form that the
dependent variable takes in the statistical model adopted. The two types of dependent
variables that may represent the disease outcome of interest are binary and fractional. A
binary disease outcome variable is the standard in the veterinary epidemiology literature

and takes the form ye {0,1} where y = ldenotes that the animal or herd is infected. A

fractional disease outcome variable, ye [0,1], falls in the unit interval, is interpreted as the
within herd prevalence of disease, and is considered only for the case of herd-level data.
Because the binary response model is the “workhorse” in the epidemiology literature we
discuss its usefulness for economic analysis first and then compare it to the fractional
response model using herd-level data.

Statistical analysis of veterinary epidemiological data to identify risk factors for
infection almost uniformly uses logistic regression and reports odds-ratios for individual
explanatory variables (management practices and demographics), which may be either

binary indicators or continuous variables (see most any general or veterinary

epidemiology text, such as Kahn and Sempos (1989) or Petrie and Watson (1999)).
Odds-ratios from binary response models indicate the odds of infection (a binary
outcome) when the covariate is present (if represented by a dummy variable) or when the
level of the covariate is increased (if represented by a continuous variable) relative to
when it is not. This interpretation is helpful for small probability events like the
occurrence of a rare disease and explains in part why management practices found to lead
to greater odds of infection are referred to by epidemiologists as “risk factors”. In a
statistical sense, when the probability of infection is very low (less than around 0.1) the
difference between relative risk and the interpretation of the odds-ratio is negligible
(Gould 2000). Odds-ratios from logistic regression are commonly reported by
epidemiologists because of their usefulness in describing associations for a variety of
sampling designs (Martin, Meek and Willeberg 1987; Selvin 1991) and because odds-
ratios for individual parameters are insensitive to the levels of other explanatory variables
(Gould 2000).

Applied econometricians tend to find marginal effects—interpreted as the
estimated increase in the probability of infection associated with a marginal change in the
explanatory variable—more intuitively palatable than odds-ratios. This is why partial
effects are normally reported when logistically distributed errors are assumed in micro-
econometric binary response models instead of the exponentiated parameter estimates
reported by epidemiologists. As we find below in our empirical example, the odds-ratio
(OR) reported by itself may suggest that undertaking a particular practice relative to some
alternative cuts in half, as given by OR=0.5, the likelihood of herd infection in a binary

response model, but the corresponding marginal effect (ME) on the probability of

infection represented by that odds-ratio may only be a several percentage point reduction,
as given by ME=-0.05. For economic purposes, the interpretation of the odds-ratio alone
is not unlike considering the statistical significance of a parameter estimate in isolation of
the practical or economic significance of the estimate. Viewed in isolation, a highly
significant odds-ratio may suggest either a large relative reduction or increase in
likelihood of infection while the associated estimate of the change in probability may be
quite small. Because of the potential for divergence between the interpretation of ORs
and MES, odds-ratios are not particularly helpful in evaluating the economic tradeoffs
between adopting competing disease management practices unless the disease in question
cannot be tolerated in the herd at any level (e. g., foot and mouth or bovine spongiform
encephalopathy). Because individual herd disease management decisions revolve around
production diseases rather than mandatory eradication programs, relying solely on odds-
ratios to evaluate resource allocation is akin to minimizing the losses from infection,
irrespective of costs.

Thus far we have only considered how results from binary response models are
reported, in general, but it is just as important to consider the nature of the dependent
variable because this affects the economic usefulness of the estimated coefficients.
Binary response models may be estimated for individual or herd-level data. For the
individual animal case, the dependent variable takes the form y,e {0,1 } where i indexes
individual animals, y,- =1 for an infected animal and 0 otherwise. The binary response

model takes the familiar form

(1) E(yi l X)= P(yi =1|X)=G(XB),

where x is a vector of explanatory variables inclusive of disease management practices,

[3 is a vector of coefficients, and 0 < G(z) <1. The function G(z) is usually a CDF and
maps the index 143 into the response probability (Wooldridge 2002). In most
epidemiological applications G(z) is the CDF for the standard logistic distribution, where
G(z) = exp(z)/[1+exp(z)]. From individual animal data the estimated probability )3,- for a
given it can be multiplied by the number of animals in the herd to get an estimated within
herd disease prevalence. A comparison of the estimated prevalence when a practice is
undertaken relative to when it is not can be used to evaluate the expected profit and

decide whether it is worthwhile to adopt the management practice.

For the case of herd-level disease data the binary dependent variable takes the
form yhe {0,1 } and is identical to (1) except for the nature of the dependent variable. The
binary response model then becomes
(2) HM IX) = P(yh =1! X) = 0045),
where h indexes the herd or farm such that yh is the herd infection status and indicates
that one or more animals on the farm is infected when yh =1. Because 52,, is the predicted
probability that a herd is infected, this estimate cannot be used to arrive at a predicted
within herd prevalence and there is no information provided by the herd-level binary
response model that allows economic decision makers to evaluate the relative cost of
different management practices.

As an alternative to the familiar binary response model already discussed,

consider that herd-level disease can be expressed as a fractional value of the form

y}{ 6 [0,1] , where h again indexes herds. The super-script f distinguishes this fractional

10

response variable from the herd-level binary response variable in (2), and the fractional

response model takes the form,

(3) E<y,{ I x) = Gar).

Fractional response variables3 are of particular interest for herd-level livestock disease
management because it is costly to draw and test blood samples from every animal on a
farm (as is required for individual animal disease data discussed in the context of binary
response models above), and for food animals this typically only occurs in a research
setting. Most practical on-farm disease diagnostic programs rely on serological tests
from a subset of the herd chosen based on the sensitivity and specificity (Kahn and
Sempos, 1989) of a particular diagnostic test in order to achieve a desired level of
statistical confidence about the estimated within herd prevalence rate. Surveys like those
conducted for the USDA’s NAHMS are designed to achieve national disease prevalence
estimates and rely on this type of sampling within private herds. Serological surveys of
wild animals are conducted based on similar principles where sampling may occur at a
higher level of aggregation than the “herd”, such as a wildlife management unit. Because

we are interested in estimating the conditional expected within herd prevalence for

 

3 Fractional response variables are frequently encountered by agricultural economists.
Examples of fractional or proportion values of interest to agricultural economists include
the share of land devoted to the production of a particular crop, the proportion of
cultivated land where a particular management practice is in use (e.g., reduced tillage,
split nitrogen application, integrated pest management), the proportion of land in a
county, township, or other jurisdiction devoted to a particular use (e. g., agricultural,
commercial, residential, protected natural area), and the disease prevalence rate in a
livestock herd or flock. A researcher studying such topics is generally interested in
inference that allows for the identification of the determinants of crop selection,
management practice or technology adoption rates, land use patterns, or rate of infection,
respectively. To this end, statistical methods that allow the researcher to estimate the
conditional expected fractional outcome of interest are desirable for empirical analysis of
public policy and individual decision making.

11

 

alternative disease management practices in order to be able to aid economic decision
making, a statistical method like fractional logit regression (Papke and Wooldridge 1996;
Wooldridge 2002) that allows the researcher to directly estimate the conditional
fractional outcome of interest with a relatively limited number of assumptions is
indispensable.4

Fractional logit regression (Papke and Wooldridge 1996) is a quasi-maximum
likelihood method (QML) based on the work of Gourieroux , Montfort and Trognon

(1984) and McCullagh and Nelder (1989). Fractional logit utilizes the Bernoulli log-

likelihood function
H f f

(4) L03) = Z {yh ln[G(xtl)]+(1-y;, >Inn —G<xr)1}
11:1

for sample size H, which is well defined for 0 < G(z) < 1. This log-likelihood function is

identical to that used in standard ML estimation of binary response index models except

that y,{ is continuous over the unit interval. Because equation (4) is a member of the

linear exponential family of distributions, the QML estimator (QMLE) Ii is consistent for

[i when (3) holds regardless of the true conditional distribution of y; (Gourieroux,

Monfort and Trognon 1984). This is a desirable characteristic of QMLEs in general

because yhf could be discrete, continuous, or some combination of the two over its range

(Papke and Wooldridge 1996).

 

4 Papke and Wooldridge (1996) and Mullahy (1998 and 2005) discuss in detail competing
models for fractional dependent variables and we do not consider this material in any
detail here except to note some of the general strengths of fractional logit over competing
models after introducing the model briefly.

12

 

Fractional logit regression improves upon alternative econometric methods for
fractional response variables by ensuring that the conditional expected outcome lies in the
unit interval, handling observations at the boundaries of the unit interval without
“arbitrary adjustment” (which is required for methods like OLS using the log-odds
transformation as discussed in Wooldridge (2002, p.662)), and directly estimating the

desired fractional response.

Estimation of Disease Management Adaption Decision

Our second research objective is to shed some light on the decision to adopt or not
adopt disease reducing practices identified by the kinds of analyses discussed in the
previous section. In this way, the estimation of the marginal effect on herd prevalence
from adopting an individual practice is the first of two stages in an econometric
procedure to investigate what influences private herd manager decisions to adopt
particular practices. Because an adoption decision is inherently binary, we consider the
choice of each practice within a binary response modeling framework where the binary
dependent variable of interest is x“, 6 {0,1 }where practice k is identified as having a
significant negative association with herd prevalence in the first stage and h indexes
private herds. The binary response model then is given by
(5) E(th I Z) = P(th =1ll)= 0(213),
where z is a vector of demographic variables that control for heterogeneity across herds
and, consistent with the conceptual framework of Mclnemey, Howe and Schepers (1992),
includes variables for the cost of output losses from avoidable infection and the cost of

undertaking practice k for the herd. The cost of output losses is a function of the

13

estimated marginal effect on within herd prevalence of adopting practice xkh from the first
stage, and the control expenditure is the estimated cost of adopting practice xkh. More
specifically, we hypothesize that when the cost of production losses from infection
associated with the marginal change in prevalence attributed to the adoption of practice k
are considered as an explanatory variable, the estimated coefficient will be positive in
sign while a negative relationship is expected between adoption and the control cost
variable in the second stage estimation.

Livestock managers may very well adopt biosecurity and disease management
practices that are not found to have a significant association with any individual disease
in a cross-sectional data set. Thus far we have only considered adoption of those
practices identified as having such an association in the first stage estimation because
without this information we cannot estimate the cost of infection associated with an
individual practice. While this information is required in order to control for the cost of
output losses avoided when investigating the determinants of adoption for individual
practices, herd demographic variables and the cost of control expenditures are still logical
explanatory variables that may be available to the researcher. We now provide an
application of the proposed two-stage econometric procedure using a cross-sectional
dataset with herd-level observations to compare the use of the fractional logit method and
standard logistic regression in the first stage and examine the findings with respect to

adoption behavior in the second stage.

14

 

Implementing the Two-Stage Estimation Procedure

Data for our empirical application come from the 1996 National Animal Health
Monitoring System (NAHMS) survey of dairy cattle conducted by the US. Department
of Agriculture’s National Agricultural Statistics Service and Animal and Plant Health
Inspection Service. Among the objectives that motivate the NAHMS survey is the
estimation of prevalence for a given disease and species at the level of the national herd.
Survey data include extensive farm-level behavioral information such as animal
inventory and operational characteristics, health management and biosecurity practices,
feeding and manure management practices, and livestock morbidity, mortality and culling
details. The details of the NAHMS survey design are enumerated elsewhere (Ott,
Johnson and Wells 2003). Statistical analysis must take the survey’s complex random
stratified sampling procedure into account for correct statistical inference when working
with NAHMS data (Dargatz and Hill 1996).5 We account for survey design effects
throughout the statistical analysis that follows.

To demonstrate our proposed two-stage estimation procedure we focus on the
production disease bovine leukosis virus, also referred to as bovine leukemia or enzootic
bovine leucosis, which has previously been studied both in veterinary epidemiology
(DiGiacomo, Darlington and Evermann 1985; DiGiacomo et al. 1986; Heald et al. 1992;
Rhodes et al. 2003) and economic decision making studies (Chi et al. 2002a; Chi et al.
2002b; Ott, Johnson and Wells 2003; Pelzer 1997; Rhodes, Pelzer and Johnson 2003).
Bovine leukosis virus (BLV) is a retrovirus that primarily affects lymphoid tissue of beef

and dairy cattle and causes malignant lymphoma and lymphosarcoma (LS), although

 

5 References on the statistical issues associated with complex survey designs are Lee and
F orthofer (2006) and Lohr (1999).

15

leukemia is not a common finding, occurring in only 2-5% of BLV infected cows (Kirk
2000). BLV is horizontally transferred within blood lymphocytes, but it is uncertain
whether or not it is transmitted vertically in utero. Economic losses to dairy farmers
associated with BLV result from reduced milk production, increased replacement costs,
and increased veterinary costs (Pelzer 1997). Because it is transmissible, the only way to
eliminate losses from a herd is to cull all infected animals and routinely test new animals
introduced to the herd to ensure the farm remains BLV free. Pelzer (1997) and Rhodes,
et al. (2003) have pointed out the important difference between the economic effect of
clinical LS and subclinical level infection (BLV seropositive status). Our data examine
BLV seropositive animals (those found to have antibodies to BLV in their blood), for
which one estimate found that “a basic BLV control program may be economically
beneficial in herds in which the prevalence of BLV infection is (greater than or equal to)
12.5%” (Rhodes, Pelzer and Johnson 2003).

We proceed by demonstrating how our two-stage econometric estimation
procedure can be used to estimate the effect of individual management practices on
livestock disease prevalence levels for a herd and how this information can be used with
the estimated cost of illness from production losses to evaluate how the economic
damages from infection associated with individual practices influence practice adoption
behavior. We utilized the Stata® statistical package (Stata Version 8.0 2003) to
implement the first stage using standard logistic regression and fractional logistic
regression. The fractional logit method (Papke and Wooldridge 1996) was implemented
using the generalized linear model command which allows the analyst to directly predict

herd prevalence and account for survey design when calculating standard errors.

16

First Stage: Disease Control Function Estimation

The first stage in the estimation procedure identified those management practice
variables that have a statistically significant effect on the herd-level dependent variable.
The 1996 NAHMS dairy dataset includes many more variables than those discussed in
this analysis. Because we were interested in a comparison of the fractional response
model and the standard binary response model used in epidemiological studies to identify
risk factors for infection, we followed a backwards stepwise procedure that is common in
the veterinary literature (e.g., Heald, et al., 1992; J ohnson-Ifearulundu and Kancene,
1998) to eliminate those management variables from our first stage regression that were
insignificant before the final estimation of the impact of individual risk factors for
infection on the herd-level disease outcome. Only the management variables that
survived stepwise elimination and regional dummies intended to capture spatial
heterogeneity were included in the first stage estimation results reported here. The
second stage estimation of the adoption equations include a cost variable which is
described in detail below and uses the results from the first stage estimation along with
farm demographic variables in the NAHMS data which are taken to be measures of
productivity. Continuous and binary variables in the 1996 NAHMS dairy survey that are
used in our two—stage estimation are reported with summary statistics in Table 1.1. The
summary statistics indicate that mean individual within herd prevalence was 40% and
that 88% (not reported in Table 1.1) of all dairy herds had at least one seropositive cow.

There was a strong statistically significant difference between positive and negative herds

17

 

only for the variables herd size, herd prevalence, and both “safe” and “unsafe” dehoming

methods (also not reported).

First stage estimation results from both the fractional response and the binary
response models are reported in Table 1.2 for comparison of the information provided by
the two models. Recall that the key difference between the two models is the nature of

the dependent variable. The fractional logit (flogit) model uses the herd prevalence
y-hf 6 [0,1] and directly estimates the fractional response associated with each
explanatory variable. The binary dependent variable yhe {0,1} in the standard logit

model uses a latent variable approach where y h = l[y;; = x8 + 8 > 0] and l[o] is an
indicator function that equals one when its argument is true; the dependent variable takes

on a value of 1 for any herd prevalence greater than zero, y h = 1[ y}: > 0] =1 , and 0

otherwise, y h =1[y;; = 0] = 0. We discuss the two models in turn because they are

fundamentally different and to contrast the nature of the information provided by the
models to decision makers.

The only regional dummy found to be significant in the flogit model was the
southeast indicator and it was associated with 16.8% higher BLV prevalence relative to
the Midwest (the control region). The herd size variable indicates that an additional
hundred animals in the herd increases the expected BLV prevalence by 0.6%, while an
additional thousand pounds of milk produced per animal per year on average is

associated with a 1% reduction in expected prevalence. Binary management practices

18

 

that survived backward stepwise elimination prior to the first stage estimation were
generally found to be significant in the flogit model.

Dehoming practices were combined into three categories based on practices
suggested to minimize the spread of infection (“safe”), practices most likely to contribute
to the spread of infection (“unsafe”), and not dehoming in order to better facilitate the
comparison of recommended and discouraged practices in subsequent regression
analysis. The use of caustic paste (to prevent the growth of horns before development) or
electric dehoming was significant and found to be associated with an 8.6% reduction in
predicted BLV prevalence compared to the discouraged practices of either saw or gouge
dehoming. Not dehoming dairy cows was not found to affect herd BLV prevalence in a
significantly different way than the use of unsafe dehoming methods. This result is
consistent with veterinary studies that have identified electric dehoming as reducing the
likelihood of BLV infection (DiGiacomo, Darlington and Evermann 1985; DiGiacomo, et
al. 1986). Saw and gouge dehoming, though not recommended practices, were observed
on 426 farms in our sample compared with 524 farms adopting a safe dehoming method.

We identified farms using “clean injection methods” as those that either use a new
needle for every animal (single-use) or sterilize needles after each use in the NAHMS
data. For the same reasons that “safe” dehoming practices are recommended, using such
injection practices is generally recommended and may be of particular importance for
BLV which is transmitted via blood lymphocytes. The flogit model finds a strong
significant relationship between clean injection practices and BLV prevalence for both
heifers (less than 24 months, never bred) and cows, but with negative and positive signs,

respectively. One possible explanation of the difference in sign is the fact that dairy

19

 

cattle normally receive vaccinations as a preventive health measure at a young age (when
designated as heifers in the survey), but injections administered later in their life (when
designated a cow in the survey) are less common and may be administered because of
other illness or because the cow was introduced to the herd from an off—farm source.
This may be a plausible explanation for the finding that clean injection practices used on
cows were associated with 16.4% higher within herd prevalence and the adoption of
clean injection practices for heifers was associated with a 13% lower estimated
prevalence than on farms where these practices were not adopted.

The predicted prevalence from the flogit model when all variables were evaluated
at their means was 39.8% (compared to the observed mean prevalence of 40% in Table
1.1) and when both BLV reducing practices were undertaken the predicted mean herd
prevalence was 23.9% (all other variables evaluated at their means) compared with
44.7% when BLV reducing practices were not undertaken.

The first stage binary response model, while modeling a related outcome, is
fundamentally different from the flogit model because of the nature of the dependent
variable already discussed. This said, the comparison of estimation results from the two
models (Table 1.2) in isolation of our underlying objective of informing economic
decision making does not make sense because these are not “competing models” in the
usual sense encountered in applied econometric research. Rather, we compare the two
models in order to point out the difference in the information provided to decision
makers.

The southeast regional dummy is found to be significantly associated with 8.5%

higher probability that one or more animals in the herd are infected in the herd-level

20

 

binary response model. The odds-ratio associated with this marginal effect indicates that
herds from the southeast are nearly seven times more likely to be BLV seropositive than
midwest herds. This finding demonstrates one point made above in the discussion about
odds-ratios for binary herd-level disease data. Viewing the odds-ratio alone finds that
herds in the southeast are multiple times more likely to be infected than those in the
midwest, but the associated marginal effect on binary herd infection status indicates that
this only translates into an eight point difference in the predicted probability that a herd is
seropositive.

As in the flogit model, herd size and average annual milk production are found to
be significant in the logit model. One hundred additional cows or an additional thousand
pounds of average annual milk production are predicted to be associated with 2.4%
higher and 0.6% lower probability that a herd is infected, respectively.

The biggest difference between the information provided to decision makers by
the two models comes from examining the variables for dehoming and injection
practices. The binary response model found only one of the four practice dummy
variables to be significant, while three were highly significant in the fractional response
model. Even though the magnitude of the estimated effect on herd prevalence of
injection practice variables found to be significant in the flogit model exceeds that of
dehoming practice dummies, these practices are not even significant in the logit model.
The odds-ratio for safe dehoming suggests that farms that adopt the practice are half as
likely to be infected relative to those who adopt unsafe methods. The binary response
model predicted that the probability a herd was infected when all variables were

evaluated at their means was 90.6% (compared to 89.1% of sampled herds being infected,

21

mentioned above). When both BLV reducing practices are undertaken (all other
variables evaluated at their means) the logit model predicts a probability of infection
equal to 90.3% compared with 92.6% when BLV reducing practices are not undertaken.
This difference in the estimated probability that one or more animals are infected
conditional on adoption does not allow a farm manager to calculate the expected change
in the cost of infection associated with the practice unless the baseline probability of
infection for the farm is known; this is information that is considered to he rarely
available to farm managers in practice.

Information of this kind provided to a decision maker is very different than the
estimated change in prevalence indicated by the flogit model (-20.8% mentioned above).
When the difference in disease prevalence indicated by the flogit model has profitability
implications, it is helpful information to have, even in the absence of information about
the baseline level of infection, in deciding which management practices to undertake.
Conversely, the information about livestock management practices provided by standard
logistic regression (adopting the practices identified in the first stage reduces the
probability that one or more animals in the herd are infected by 2.3%) provides no such
information to guide managerial decisions because the difference in probability that the
herd is infected cannot be monetized into an expected cost or benefit of undertaking the
practices for comparison with cost of undertaking the practices.

Another issue to consider is that if we have information about within herd
prevalence and not just binary herd infection status, adopting a binary response model—
even if marginal effects are reported with odds-ratios—ignores some of the information

available to the researcher which is valuable for economic decision making purposes.

22

Because of this we only use the first stage estimation results from the fractional response

model in estimating the adoption equations in stage two.

Second Stage: Estimation of Adoption Equations

In the second stage we incorporated prevalence information from the estimated
flogit model into an analysis of management practice adoption. We used an empirical
estimate from the veterinary literature based on the 1996 NAHMS dairy survey data to
construct variables that represent the cost of infection avoided (the dollar value of output
losses, not taking into account the cost of adopting the practice) by adopting BLV
reducing management practices identified in the first stage fractional logit regression.
Ott, Johnson and Wells (2003) estimated that a 1% increase in BLV prevalence costs
$1.28 per cow/year in terms of the impact on the average value of production that results
from reduced milk output, lost calves, and net replacement costs. A separate study based
on experimental data from a research herd appearing in a different journal and using a
different approach arrived at an identical estimate of the cost of subclinical BLV
infection (Rhodes, Pelzer and Johnson, 2003). Using separate estimates of the marginal
effect on herd BLV prevalence of adopting a safe dehoming method and using clean
injection practices for heifers, we constructed a variable that represented the estimated
cost of BLV avoided by adopting each individual practice for every farm in the sample.
To create a cost variable for each BLV reducing practice, the herd-specific marginal
effect from the first stage was calculated for binary management practice k according to

(Wooldridge 2002, p.458-459)

23

MEk,h =

(6) G(,BO +x1fl1+xzflz +...+xk +...+xK_1,8K_1+xK,6K +8h)-
G(flo +x1161 +X2fl2 +~-+Xk—1flk—1 +Xk+1flk+1---+XK—1flK—1+XKflK +812)

and multiplied by the product of the total number of cattle on the Operation and the $1.28
per cow/year/percentage point BLV prevalence estimate from the veterinary literature for
each practice. Two adoption equations could then be estimated in the second stage, one
for each BLV reducing practice, relating adoption of a practice to the costs avoided.
Adoption equations were estimated for each practice using standard logistic

regression because of the binary nature of practice adoption (Table 1.3). The binary

response adoption model takes the form P(xkh =1 I z) = P(x,i:h > 0) = G(z[i) and the

associated latent variable and indicator function are given by x kh = 1[x]:h = zfl + u > 0]

where z is a vector of explanatory variables for practice k in herd h.6 Explanatory
variables included the estimated economic damages from BLV that were avoided by
adopting the practice indicated by the dependent variable, farm demographic variables for
region, average annual milk production, production cost per hundred-weight, and a
constant. The variable for economic damages avoided was marginally significant in the
safe dehoming method adoption equation (p=0.074) and suggests that an additional
hundred dollars of damage avoided increases the probability of adoption by 9.8%. This

indicates that the cost of damages avoided appears to have an economically significant

 

6 In addressing the adoption of biosecurity and health management practices it is possible
that there are econometric problems that arise as result of the two stage estimation
procedure. There is an estimated regressor in the adoption equation which is known to be
a potential problem, but was unavoidable because of a lack of available information about
private costs in the data. Second, there may be simultaneity between adoption and the
conditional prevalence of disease we estimate due to correlation in the error structure of
the two equations we estimate in sequence.

24

 

impact on practice adoption in the 1996 NAHMS dairy sample. This result is consistent
with estimates from the veterinary literature that suggest mean losses from subclinical
BLV infection for a 100 cow herd with 50% prevalence is in the neighborhood of $6,400
compared to the predicted mean annual cost of a basic BLV control program of $1,765
(Rhodes, Pelzer and Johnson, 2003). The insignificance of damages in the clean injection
equation, however, goes against economic logic; especially when you consider that the
marginal effect on within herd prevalence in the disease control function is 4.4% greater
for the injection practice than for the safe dehoming method.

There are at least two potential reasons for our finding that damages do not have a
statistically significant influence on adoption of a “clean” injection practice for heifers
despite the incongruence with economic intuition. The first is that dairy farmers may not
have had information about costs of control relative to economic losses from subclinical
BLV infection. This implicitly assumes that if farmers had this information it would
have influenced their actions in an economically significant way, which is intuitively
appealing but remains an empirical question. The second is that, given farmers did have
information about the potential losses from BLV when making an adoption decision,
opportunity costs of management, labor, and capital outweighed the disease costs. This
includes other practices that are perceived to yield preventive spillovers to multiple
diseases.

Among the set of regional dummy variables, only the west and northeast regions
were found to be more than marginally significant and are positively associated with the

adoption of clean injection practices for heifers and “safe” dehoming methods,

25

respectively.7 The only other variable that significantly contributed to the adoption of
either BLV reducing practice was average annual milk production.

Goodness of fit measures for the two adoption equations favor the dehoming
equation over the clean injection practice equation. The “safe” dehoming equation
correctly predicted adoption for 62% of the sample overall compared to the best naive
prediction possible (that all farms adopt) which only resulted in 53% correctly predicted
overall. The heifer clean injection equation had a higher percentage of observations
correctly predicted overall but does no better at predicting adoption than a researcher who
naively guesses that all operations will choose not to adopt. While there was nothing
particularly remarkable about the predictive power of the latter equation, the former

seemed to do a pretty good job on the basis of this simple metric.

Conclusions

This paper presented a method for analysis (1) of herd-level livestock health data
that provides managers with information to improve resource allocation across individual

management practices and (2) the determinants of disease management practice adoption.

 

7 One explanation for the magnitude and direction associated with being from the
northeast region on adoption of a safe dehoming method is the presence in New York of
the only voluntary state BLV eradication program that the authors are aware of in the
United States. There are 268 observations in the sample from the northeastern region
states that rank among the top 20 dairy producing states (NY, VT, and PA) and 149 of
these 268 observations are from New York, the only state with a publicly funded BLV
eradication program. While we have no information to indicate if any of the sampled
farms participated in the eradication program, at the time of the survey this program
provided limited funds for serological testing, technical assistance in the form of disease
eradication and control plan development for participating operations, and among the
four baseline practices that all eradication program participants are expected to undertake
are electric dehoming and single-use sterile, disposable needles (Brunner, et al. 1997).

26

We proposed a two-stage estimation procedure designed to achieve both objectives in
sequence and demonstrated the proposed procedure in an empirical application.

When data are collected at the herd-level and the focus of analysis is a non-
epidemic disease, the information provided to livestock managers by the fractional logit
method was found to be superior to standard logistic regression commonly used in the
analysis of herd-level livestock epidemiological data. Using herd-level data on bovine
leukosis virus (BLV) from a cross-sectional survey of the top 20 dairy producing states, it
was demonstrated that the management practices identified by the two models as having
a significant negative effect on the dependent variable differ considerably and that these
differences are relevant for economic decision making. The fractional logit model
provides decision makers with information about the marginal effect of individual health
management and biosecurity practices on the estimated prevalence of disease in their
herd, while standard logistic regression provides estimates of the marginal change in the
probability that one or more animals in a herd is infected when a particular practice in
undertaken. There appears to be potentially valuable information contained in the
fractional dependent variable herd prevalence that is ignored if such data are treated as a
binary outcome in statistical analysis.

In the first stage of our estimation we consider that livestock managers must
allocate resources across individual practices associated with production and find that the
kind of information provided by the fractional logit model is particularly helpful.
Standard logistic regression remains indispensable for analysis of highly contagious
diseases, diseases with human health implications, and others that are the focus of

international trade sanctions and cannot be tolerated at any level in a national herd. The

27

 

information content of standard logistic regression results for management of production
diseases, however, is not viewed as being particularly helpful to economic decision
makers. The use of an econometric method capable of directly estimating the fractional
response of interest provides managers with information that can be used to weigh the
benefits and costs of individual practices that is not provided by binary response models
commonly used to analyze herd-level data in veterinary epidemiology.

In a second stage regression we use the information from the fractional logit
model to estimate adoption equations for practices found to be statistically and practically
significant in the first stage regression. The estimated cost of damages from BLV
infection avoided by adopting the practices identified in the first stage was calculated for
each farm in the sample, and was found to be insignificant in explaining heifer clean
injection practice adoption and marginally significant in explaining safe dehoming
practice adoption. The estimated economic damages from BLV associated with not
adopting a particular management practice might be expected to be significant from the
standpoint of economic intuition and we find this to be the case for one of the two BLV
reducing practices identified in the first stage. This estimated effect is also economically
significant for the adoption of a safe dehoming method, contributing to the probability of
practice adoption in a meaningful way.

We cannot make any statements about the second of our hypotheses abdut the
insignificance of economic damages avoided because we only consider a single disease in
our empirical application. Of particular interest is the role that preventive spillovers to
other diseases may play in the adoption of individual management practices. The

NAHMS data (in certain years and for certain species) include serological tests for

28

multiple diseases and work to quantify preventive spillovers and the role of those
spillovers in disease management and biosecurity practice adoption is underway (Grami g
and Wolf 2007).

The fractional logit method has not previously been used to analyze herd-level
livestock disease data and seems to be a promising analytical tool when compared to
standard logistic regression, which has been used almost exclusively in empirical
veterinary studies of livestock disease management. This econometric technique is likely
to be useful to agricultural economists dealing with a variety of other fractional

dependent variables.

29

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Table 1.2. Stage I Estimation Results from Fractional (flogit) and Binary (logit) Response

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Models
flogit: E(y|x), ye [0, 1] logit: E(y|x), ye { 0,1 }
Variable Coefficient Odds-Ratio
[Marginal Effecta] [Marginal Effecta]
West 0.0405 053*
[0.009] [-0.067]
0.6814“ 6.99*
somhea“ [0168“] [0.085***]
0. 1029 0.95
Nonhea” [0.024] [-0003]
Herd size 0.0275*** 1.32***
(hundreds of cows) [0006* * * ] [0.024* * *]
Average Annual Milk Production -0.042* * * 0.92 *
(thousands of pounds) [-0.010** *] [-0.006*]
Safe dehoming method -0.3647* * * 0.52* *
(caustic paste or electric dehomer) [—0.086***] [-0.056**]
. -0. 1 l 1 1 0.50
N" dehommg [-0.026] [0.076]
Clean injection methods- -0.5 829* * * l .42
Heifers [—0.130* * *] [0.026]
Clean injection methods- 0.6668* "‘ * 1 .12
Cows [0. 164* * *] [0.009]
Intercept coefficient 0.373 3 .23 7* * *
R-squared 0.0708 0.0526

 

Notes: n=980; Standard errors calculated using the delta method, account for complex

survey design

Asterisk denotes level of statistical significance: * 10%, ”5%, ***51%
a Q I O
Margrnal effects evaluated at mean for continuous varrables and 0 to 1 change for

dummies

31

Table 1.3. Stage II Estimation of Determinants of BLV Reducing Management Practice

 

 

 

 

 

 

 

 

 

 

 

 

Adoption
Safe Dehorning Method Clean Injection of
. Heifers
Variable Coefficient Coefficient
[Marginal Effecta] [Marginal Effecta]
Total damages from BLV 0_0400* -0.0660
Avoided if adopt practice ($100) [0098*13] [-0004]
. 0.3746* l.l346**
West region * * *
[0.093 ] [0.100 ]
S h . -0.2452 1.14
out east region [@059] [0095]
Northeast reglon [0'132***] [0.033]
Avg Annual Milk Production 0.1674* * a: 0.0613
(thousands of pounds) [0.041 * * * ] [0.004]
Cost per CWT 0.0069 0.0379
($ per hundred pounds milk) [0.001] [0.002]
Intercept -3.4481* ** 43191 ***
F-statistic for overall significance 1027* * * 109* * *
% Yes correctly predicted 61.6 0
% No correctly predicted 62.3 100
% Overall correctly predicted 61.9 92.1

 

Notes: n=980; Standard errors calculated using the delta method, account for complex

survey design

Asterisk denotes level of statistical significance: *10%, * *5%, * **<l %
a . . .
Marginal effects evaluated at mean for continuous varlables and 0 to 1 change for

dummies
b p-value = 0.074

32

 

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Applied Econometrics 11, no. 6(1996): 619-632.

34

 

Pelzer, K.D. “Economics of bovine leukemia virus infection.” Veterinary Clinics of
North America Food Animal Practice. Vol. l3(1997):129-141.

Petrie, A., and PB. Watson. Statistics for Veterinary and Animal Science. Oxford,
England: Blackwell Science, 1999.

Rhodes, J .K., K.D. Pelzer, and Y.J. Johnson. "Economic implications of bovine leukemia
virus infection in mid-Atlantic dairy herds." Journal of the American Veterinary
Medical Association 223, no. 3(2003): 346-3 52.

Rhodes, J.K., K.D. Pelzer, Y.J. Johnson, and E. Russek-Cohen. "Comparison of culling
rates among dairy cows grouped on the basis of serologic status for bovine
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Selvin, S. Statistical Analysis of Epidemiological Data. Vol. 17. Monographs in
Epidemiology and Biostatistics. New York City, NY: Oxford University Press,
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Stata/SE 8.0 for Windows Edition. College Station, TX, Stata Corporation, 2003.

Wooldridge, J .M. Econometric Analysis of Cross Section and Panel Data. Cambridge,
MA: MIT Press, 2002.

35

Essay Two

LIVESTOCK DISEASE INDEMNITY DESIGN WHEN MORAL HAZARD IS
FOLLOWED BY ADVERSE SELECTION

Introduction

The outbreak of disease in domestic livestock herds is an economic and potential human
health risk that involves both the government and individual livestock producers.
Economic justifications for public intervention in disease control include extemalities,
public good aspects, coordination failures, information failures, and income distribution
considerations (Ramsay, Philip and Riethmuller). The potential to pose a large economic
cost depends on many factors including trade laws and level of infectiousness. Public
policies range from bounties for infected livestock to required herd depopulation and
farm decontamination. The United States Department of Agriculture's Animal and Plant
Health Inspection Service (APHIS) provides inspection and quarantine services to
prevent the introduction of disease across national borders and also coordinates response
to disease outbreaks from within the country. Border measures that protect against
incursion of disease are provided to protect the safety of the American food production

system and to prevent infection of the domestic livestock industry.

Diseases that are highly contagious or have human health implications are often
the target of government eradication programs. When livestock is taken by the
government for public health or economic reasons, the Fifth Amendment of the US
Constitution specifies that private property taken for public use must receive just
compensation. This compensation takes the form of indemnity payments. The current

federal compensation level is defined by the Animal Health Protection Act, Subtitle E of

36

the Farm Security and Rural Investment Act of 2002 which states that compensation shall
be based on the fair market value, as determined by the Secretary of Agriculture, adjusted
for any other compensation received for that event (i.e., disaster payments or perhaps
even private market insurance). States may also offer compensation in the form of
indemnities.

This paper focuses on the structure of indemnity payments, which are currently
the primary form of compensation by the government, as the key mechanism to provide
incentives to farmers in order to achieve policy objectives. Existing indemnity payments
represent an implicit insurance policy for all livestock producers with respect to the
diseases where they are applicable, but are really more akin to ad hoc disaster payments
commonly made to farmers because of the lack of risk classification and underwriting
involved. Indeed, these payments do not have the desirable risk pooling properties
associated with insurance; there are no premiums based on the risk represented by an
insured as part of a portfolio of policies and all taxpayers fund the indemnities.
Government-provided indemnities should be designed with careful attention to the
private incentives they create and the public objectives of livestock disease risk
management. It is clear that providing adequate incentives to biosecure and report has
been an issue of concern within public agencies responsible for livestock disease
outbreak response (Ott 2006). It is our intent to address this problem in a rigorous,
systematic fashion to ensure incentive compatibility between the government and private
decision makers.

Previous economic research dealing with livestock disease (e.g., Kuchler and

Hamm; Mahul and Gohin; Bicknell, Wilen and Howitt; Horan and Wolf; Hennessy) has

37

also ignored incentive compatibility, at least in the presence of asymmetric information.8

The current paper uses a principal-agent model to examine incentive compatibility in the
presence of information asymmetry between the government and individual farmers.9
The nature of the information asymmetry is depicted in Figure 2.1. Individuals have
private information about preventive biosecurity measures they adopt on their farms prior
to outbreak (ex ante), and following outbreak (ex past) they possess private information
about the disease status of their herd. We investigate the role of incentives in individual
producer behavior that influences the occurrence, duration and magnitude of a disease
epidemic. Our focus is on farm level biosecurity choices and reporting of disease status
as the two primary areas of concern for the government. The government can set
indemnity rules in an effort to influence private incentives in order to achieve the stated
government objective that all farms invest in biosecurity and all infected farms report
infection (Ott 2006).

After setting up our model of farmer decision-making in the following two
sections, we first address the ex post problem because once uncertainty about the
occurrence of an outbreak is resolved, regardless of the ex ante actions taken by

producers, reporting of disease status (as opposed to discovery by another party through

transmission, slaughter, or testing) is required for timely government response to limit the

 

8 Prior economic research in this area has examined producer response to prices in
conjunction with a government bounty program for scrapies in the US. (Kuchler and
Hamm), optimal actions to contain Foot and Mouth Disease outbreak in France (Mahul
and Gohin), the effect of government programs to eradicate disease on prevalence level
and private control efforts in New Zealand (Bicknell, Wilen and Howitt), the dynamics of
optimally controlling infection from a disease which is transmitted between wildlife and
livestock (Horan and Wolf), and behavioral incentives when there is endemic disease in a
decentralized setting (Hennessy 2007).

9 A less formal discussion of these issues may be found in Gramig et al. (2006).

38

spread of infectious disease and eradicate it. The length of time between outbreak and
discovery is very important in determining the cost, duration, and severity of epidemic
(UN-FAO). For these reasons, an incentive structure that results in reporting of any
infection is of great interest for a social planner. Second, we investigate the design of ex
ante incentives for biosecurity investment together with ex post truthful disclosure. The
characteristics of an incentive compatibile indemnity rule are derived for the case of a
risk averse agent (the farmer) and a risk neutral principal (government agency). A
comparison of the relative size of optimal indemnities and constrained efficient ones that
are second best under information asymmetry follows. Implications of the theoretical
model results for public policy and market insurance design are considered and

conclusions are offered.

A Dynamic Model of On-Farm Decision Making

We develop a dynamic capital valuation model of the livestock enterprise fashioned after
that of Hennessy (2007), who adapts the efficiency wage model of Shapiro and Stiglitz
(1984) to the problem of livestock disease management. Our farmer decision model
departs from Hennessy (2007) by (i) introducing risk aversion on the part of a single
farmer (only briefly addressed by Shapiro and Stiglitz), and (ii) considering biosecurity
and disease reporting decisions. A diagram of the decision-making process described
below is provided in Figure 2.2. The farmer is risk averse with an instantaneous utility
function U(0)), where U’>0, U"<0. Wealth, 0), is contingent on the disease state and

farmer choices in our model.

39

The farmer will be in one of two disease states at any given point in time:
susceptible (non-infected) or infected. In the susceptible state (0:0, where 0 e [0,1] is a
random variable denoting the within-herd disease prevalence rate) farmers must choose

their biosecurity effort investment level, b. Biosecurity reduces the probability of

transitioning to the infected state, P51(b), such that 6 P51(b)/6b50. Biosecurity also

reduces the expected magnitude of a disease outbreak, should one occur. The conditional
probability density function of 0 is denoted g(0lb), such that G(0|b) is the twice
continuously differentiable conditional cumulative distribution function with
60(01b)/6b20 V b. The conditions imposed on the distribution of prevalence mean that G
satisfies first-order stochastic dominance in the sense that the cumulative density for a
given level of infection is non-decreasing (the desirable outcome) in biosecurity.lo

The farmer has a baseline profit flow when disease-free, gross of any biosecurity

investment, denoted by 7:0. An investment in biosecurity involves both initial capital

investments and the variable cost of ongoing management. In our model such
investments are, for the sake of tractability, treated as having a single variable cost w per
unit time and are incurred only in the susceptible state because once infected there is no
incentive to invest in biosecurity. The utility of wealth in the susceptible state can
therefore be expressed as

(l) U5=U(7r0—-bw).

 

'0 Because disease is a “bad”, higher outcomes of the random variable are less desirable
and so what we normally refer to as the “dominated” distribution is relatively more

attractive for our application. For b0< b1, G(0|b1)2G(0|b0) for all b, where the inequality
is strict for at least one value of b.

40

In the infected state, the farmer must decide whether or not to report infection.
Disease reporting is modeled as a mixed strategy, denoted r. Reporting results in
government testing, verification of infection, and culling of infected animals to eradicate
the disease. The farmer is compensated for any culled animals with a government
transfer denoted by 1(0). Culling results in two types of losses for the farmer—a loss of
asset value M0) associated with the livestock itself and consequential losses from
business interruption x(0). Business interruption losses may vary widely depending on
the characteristics of the individual Operation affected and possibly disease
characteristics. For instance, the presence or absence of breeding stock or having high
fixed costs associated with a specific capital asset (e.g., a dairy or egg laying operation)
could contribute to the magnitude of business interruption losses. Reporting disease
ensures that infected farms return to the susceptible state such that the transition

probability from infected to susceptible when you report is equal to one. The farmer’s
instantaneous utility when he/she reports is given by U f (no - 21(9) — 1(6) + r(6)).

Not reporting disease means that the farmer’s instantaneous expected utility is
determined by two distinct probabilistic outcomes. Government disease surveillance
activities detect non-reported infection with exogenous probability q and fail to detect
non-reported infection with probability (l-q). H Detection leads to government culling of
infected animals, compensation by government transfer 1(0) (as occurs under reporting),
and the farmer is fined an amount f for not reporting. In contrast, private culling of

infecteds when non-reported disease is not detected by the government means that

‘

I 1 Government disease surveillance is modeled as being exogenous to reflect the fact that
ongoing surveillance activities prior to reporting or discovery of outbreak are conducted
based on prior budgetary commitments independent of a given disease outbreak

41

government compensation is not forthcoming and culled animals are assumed to be sold
at salvage value 0(0). Whether infection is discovered by the government or not, all
farms that do not report incur asset value losses and associated consequential losses (as

occurs under reporting).

Government culling is followed by a certain transition to the susceptible state, as
is the case when farms report disease. However, private culling may not be as effective
and only results in transition to the susceptible state with probability h<1. This means

that the expected instantaneous utility from not reporting is given by

qUIC (no — 1(6) — 1(6) + r(6) — f) + (1 — q)U;VC (no — 2(6) — 1(6) + 0(0)). The overall
expected utility of wealth in the infected state, conditional on the current level of
infection, can therefore be expressed as

U] = rUIRbrO — 2(6) — 1(0) + r(e))+

(2) qUICOro - 4(6) — 1(6)+ r(6) —f)+
(1 — r)
(l — q)U}VC (no — 4(6) - 1(6) + 6(6))

Equations (1) and (2) are individual components of a farm’s inter-temporal decision-
making process. In the next section we incorporate state transition probabilities to derive
the system of equations that represents the full scope of the farmer’s dynamic problem.
Fundamental Asset Equations

Define V5 to be the expected lifetime utility of the decision maker in the susceptible state.

Using the continuous time discount rate p, we can define st to be the “time value” of

the livestock asset when susceptible (Hennessy, p.702). Similarly, let V] be the expected

42

lifetime utility in the infected state. This notation gives rise to the “fundamental asset
equations” (Shapiro and Stiglitz, p.436) 12,13

(3) st = Us(b)+ PSI(b)(Ee[V1] - VS),

(4) PVI = U1 (r,r(6’),f,6’) + P15 (rXVS - V1),

where

U1 (r. r(6). f, 6) = )0}? (4.1.46). 6)+ (1 — r)qu,C (4.x, r(6),f,6)+ (1 — q)U}VC (A. z. a(6)6)l
and P15 (r) = r + (l — r)[q + h(1— q)]. The time value of the susceptible livestock asset in

(3) equals the sum of the instantaneous utility in the susceptible state, Us(b), and the

expected capital loss if the disease state changes from susceptible to infected,
PSI(bXEg[V1] — VS ). Because the post-transition level of infection is unknown to farm
managers in the susceptible state, the expected capital loss associated with this state
transition is a function of the expectation of the lifetime stream of utility in the infected

state with respect to the level of disease prevalence, 0.

Similarly, the time value of the infected livestock asset in (4) equals the sum of
the expected instantaneous utility in the infected state, U} (r, r(6?),f,61), and the expected

capital gain from transitioning to the susceptible state PIS (rXV S — V1). The expectations

 

'2 Equations (3) and (4) below are provided in explicit form in Appendix B.l for the

interested reader.
'3 Equations (3) and (4) are derived following Shapiro and Stiglitz (1984, p.436).
Focusing on V5, we examine expected lifetime utility when decisions are made over small

intervals of size [0.)]: (3a) VS = U S (b): + (1 — pt)[PSI (b)tEg[V1] + (l — PS, (b)t)VS ].

Note that (l-pt) z e'p’. Equation (3) is obtained by solving (3a) for V5 and evaluating it
as t—->0. Equation (4) is derived similarly. An implicit assumption in this formulation is
that farm businesses are “infinitely lived entities”, as is assumed in Hennessy (2007,
p.702) and Shapiro and Stiglitz (1984, p.435).

43

Operator is not needed in (4) because the infected farmer is assumed to know his/her

current level Of infection.

Equations (3) and (4) may be solved as a system to get

Uslb)+PSI(b)EelVIl and

5 V b,E V = a
() S( 9[ ID p+PSI(b)

U S (b)+ PSI (b)EelV1 1]

U1 (rrr(9)»f’9)+P15(r[ p+PSI(b)

(6) V1(b,r,T(6’)rfr9sEl9[V1l)= p, 1315. (r)

Equation (5) shows the lifetime expected utility from being in the susceptible state to be

an annuity value. If there was no chance of transitioning to the infected state (i.e., P5] =
0), then lifetime utility when susceptible equals the annuity value US/p (i.e., Us is
received into perpetuity). When there is a chance of becoming infected (i.e., PSI > 0), the

annuity value changes in two ways: (i) a risk premium, P51, is added to the risk-free rate
p to yield a risk-adjusted discount rate that has the'effect of reducing the annuity value

associated with the susceptible state to U S/(PJ’PSI); (ii) we must account for the expected

annuity value that accrues in the infected state, E 9 [V1]/(p+PSI ), weighted by the

probability of transitioning to that state.
Equation (6) illustrates a similar valuation of the expected flow from the capital
asset, though conditioned on starting in the infected state and accounting for the

probability of transitioning to the susceptible state. Note that the term in brackets in

equation (6) represents V5(b,Eg [V1]), as derived in equation (5). If there were no chance

44

of transitioning to the susceptible state (i.e., P15 = 0), then lifetime utility equals the
annuity value U 1/p if U; were received in perpetuity. As with equation (5), because there
is a chance of returning to the susceptible state (i.e., P15 > 0), the discounted stream of
benefits takes this into account via the risk-adjusted discount rate (p+P15) and the

transition probability-weighted annuity value that accrues in the susceptible state, PISVS

(6,5 6 [V 1]).
Expressions (5) and (6) are not in reduced form since they both have expected

lifetime utility from infection, E 9 [V II» on the right hand side (RHS). To eliminate E9
[V1] from the RHS, take the expectation of both sides of (6) with respect to 0, isolate

E 9 [V 1] and substitute the expression back into (5) and (6). The resulting implicit

functions are

U S We + PIS (r))+ PSI(b)E6[U1]

7 V b, , = d

( ) S( r f) plp+PIS(r)+P51(b)) an
V1(b,r,r(6),f,6)=

(8) U1(r,T(9),f,9)+ P15(r{US (bXp + P1509)?“ PSI (MEBIUIIJ.

,0(P + PIS (r) + PSI (b))
p + PIS (r)

Note that the expression in square brackets in (8) is equal to (7).
The general form of these equations is similar to the comparable equations
derived in Hennessy (2007, p.702-703). However, the specific formulations are more

complicated than in Hennessy (2007, p.702). The level of complexity is greater because

45

(i) his model includes a single binary action while we account for two separate
continuous choices selected by the farmer, and (ii) we focus on the asymmetric
information problem and the design of indemnification policy which involves business
interruption and consequential losses along with government disease surveillance not

treated in the earlier work.

The Indemnity Design Problem

We use the asset value equations to solve the overarching problem of indemnity
design that achieves public objectives for private biosecurity investment and disease
reporting behavior. In practice, we normally think of indemnification in connection with
private insurance or government-subsidized risk management programs like crop and
flood insurance. Even though indemnities are required when takings occur in response to
a livestock disease outbreak, indemnities are also intended to serve a risk management
function by providing incentives to report infectious disease and invest in biosecurity that
prevents outbreak and may limit disease spread (Ott 2007). For these reasons, we
concern ourselves with the design of an indemnification scheme that achieves
government risk management objectives and address whether or not required
compensation for takings are in conflict with stated risk management objectives in the
presence of information asymmetry. Recall that our proposed payment structure pays
one amount, t(0), to an infected farmer who reports, and a lesser amount, 1(0) — f (which
may be negative), to an infected farmer who does not report and is caught. As we

describe in detail below, our proposed indemnity structure uses government transfers 1(0)

46

to address ex ante moral hazard (biosecurity actions) and fines f to address ex post
adverse selection (disease reporting)

The reporting problem is addressed by imposing a fine in response to a given
reporting strategy, r: #0 when 0=0 or when 0>0 and the farmer’s strategy is r=1 ; 1‘50
otherwise when the farmer is caught. This structure, which involves one value of f for
each action— reporting or not, given 0>0 — is intended to address the adverse selection
problem. This is analogous to how adverse selection problems, commonly encountered in
the literature (Rothschild and Stiglitz, 1976) and arising in practice, have been addressed.
We use fines to mimic a “menu of contracts” which induces the agent to reveal private
information. In the next section, we propose a method for setting the fine so that it
achieves the desired reporting behavior.

The moral hazard problem is addressed by setting the transfer, 1(0). Here the
government’s transfer to the farmer is based on 0, which is observed (verified as a result
of testing) after infection is discovered or reported. The transfer influences the farmer’s
incentives to take biosecurity actions because the likelihood of becoming infected is
influenced by b. We might therefore expect a lower marginal payment for larger
infection rates. This would mimic the risk sharing property of deductibles or co-pays
commonly used to address moral hazard problems in the principal-agent literature
(Laffont and Martimort, 2002), the crop insurance literature (Chambers, 1989), and the
broader insurance literature (Arrow, 1963; Raviv, 1979).

In the presence of both adverse selection and moral hazard, some combination of
the instruments used to address both types of information problems individually can be

expected to induce the desired biosecurity investment and reporting behaviors from the

47

agent in the current application. However, to simplify the exposition, we discuss the
different components of the indemnity rule (1(0) and f) individually. We proceed by
addressing each of the two parts of the information problem in reverse chronological
order. That is, we propose a way to solve for the indemnification scheme that will (i)
lead the farmer to report any infected animals to the government, thereby revealing
his/her hidden information, and (ii) create incentives for positive investment in
biosecurity, thereby solving the hidden action problem.

The Adverse Selection Problem—Reporting

Reporting is assumed to be socially desirable (all other things equal) because early
detection of infection limits the duration of a disease outbreak event and has been found
to be the most important factor in minimizing total economic damages from a livestock
disease epidemic (UN-FAO). The govemment’s objective is to set fines in such a way
that reporting of suspected disease always occurs. The farmer is the agent in our
Principal-Agent framework and we assume he/she chooses a reporting strategy, re [0, l ],
to maximize his/her discounted utility stream in the infected state, given by (8). This
means that the marginal incentive to report (positive, negative or zero) is given by the
sign of 6V1 (b,r,1(l9), f ,6)/ fir and the Kuhn-Tucker conditions imply the optimal
private reporting strategy.

The principal wants to set the fine so that the agent always finds reporting to be
privately optimal. The difficulty is that the marginal incentives to report will differ
depending on both b and 0, each of which are unobservable to the principal. The
principal must therefore set a single fine such that the agent finds it optimal to report

regardless of the values of b and 0. Specifically, the fine is set so that reporting occurs

48

even when the marginal incentives to report are at their minimum value, given by the

expression

[6V1(b,r,1(6), mail

(9) A(f)= mini: lim
6r

Vb,0 r—>1
Choosing f such that A(f)20 for all values of b and 0 ensures that the farmer always has a
non-negative marginal incentive to report. That is, as long as the agent, when operating
at the margin, is indifferent between increasing and decreasing his/her reporting strategy,
then the farmer will have a positive incentive to report for any value of 0>O. This
concept is depicted graphically in Figure 2.3. '4 Here b is taken as given and the dashed

curve indicates that the farmer will not have a positive incentive to report disease for all

* II!
levels of 0 when f-=0. We denote by f the fixed fine that achieves A(f )=0 and ensures

a non-negative marginal incentive to report over the range of 0.

We have introduced fines because prompt reporting yields social benefits and is
outside the scope of constitutionally required compensation for takings. We now turn to
the design of government transfers to indemnify farmers whose animals are culled by the
government, in order to provide incentives to private decision makers to make
investments in biosecurity.

The Moral Hazard Problem—Biosecurity Investment

1]:
Given that f ensures reporting, we now turn to the govemment’s problem of

designing transfer payments, 1(0), that provide incentives for biosecurity investment. As

 

'4 Figure 2.3 does not reflect the shape of any particular functional form for V1, rather it
is provided to shore up the intuition associated with the proposed method for setting fines
so that reporting occurs.

49

indicated above in relation to deductibles or co—pays used to address moral hazard
problems, some amount of risk sharing between the government and the farmer will be
necessary to solve the hidden action problem. This can be seen very easily by a quick
comparison of the instantaneous utility of wealth in the susceptible state and in the
infected state (when r = 1) for the special case in which the farmer is fully indemnified

against all losses: 1(6) = 1(6) + 1(6)). In this case, utility when infected reduces to
U (no) which is not dependent on biosecurity and is strictly greater than utility when

susceptible if there is any positive investment in biosecurity U (710 — bw). In this

situation, it is not clear why anyone would biosecure and it suggests that farmers will
need to bear some share of the risk of disease related losses in order for there to be an
incentive to invest in biosecurity. The question we turn to now is how the government
should structure indemnities to facilitate risk sharing in a constrained-efficient manner.

Assume the government takes into account the private net benefits of livestock

up
production, V3(b,l, f ), and the expected social cost of government transfers and disease

1
surveillance, K jr((9)g(6 | b)d6’ + m(q) , where m(q) represents disease surveillance

0
costs and K>O represents the constant marginal cost of diverting funds to this program,
which may include transactions costs (Alston and Hurd 1990),'5 The government’s

Objective function in setting transfers to induce biosecurity effort may be written as

 

'5 Because ours is a model of a single agent we do not account for market or social
benefits associated with private livestock production, but this model could be extended to
a multi-agent setting which would realistically consider such broader social benefits or
costs.

50

l
(9) max VS [b,l, f* ) — K [r(l9)g(6 | b)a'6 + m(q) s.t. b e arg max V5[b,l, f* j,
7(6),]? 0 b

where the use of fine f ‘ drives r to l, the agent’s unobservable choice of biosecurity
effort constrains the principal, and “argmax” denotes the set of arguments that maximize
the objective function that follows.

Denote a(b) = (p + PIS )/(p(p + PIS + PS] )) and

,6(b) = P5] / (p( p + PIS + PS] )), which are the risk-adjusted discount factors associated

with the outcomes U S and U [R . Then we can write equation (7), evaluated at r=l and

f,as

l
(10) Vs(b,l,f*) =Uslb)a(b)+fllb)[U1R(1,r(6’),9)g(9Ib)d9-
0

Equation (10) is the focus of farmer decision making in the susceptible state where the

agent optimizes to select a biosecurity investment 6 which is constrained to be b, the
government’s desired level of investment, without loss of generality by the revelation
principle (Myerson 1979; Dasgupta, Hammond and Maskin 1979). The regulator
chooses 2(0) to maximize farmer utility while taking into account the cost of indemnities,
monitoring, and response to outbreak required to implement the chosen disease risk
management policy. The agent’s first-order condition (F OC) with respect to b implies
the optimal private choice and we substitute the agent’s FOC for the constraint in

equation (9), using the notation introduced in equation (10), so that (9) can be re-written

as

51

max U3(b)a(b )+,6(b)jU,( 11(6),—6)g(6|b)d6 K ljr(6)( g(q6lb)d6+m()
(6)b O

(11) st. U§(b)a(b)+US(b)aOb( )+
[Arr/Human6(b)/t(tr(a).a)gb(9'”Iteration
0

 

The approach of substituting the farmer’s F OC in for the constraint is called the first-
order approach (FOA), e.g., Spence and Zeckhauser (1971), Ross (1973), Harris and
Raviv (1979), Holmstrom (1979), Mirrlees (1975), Rogerson (1985). The F 0A is valid
as a general solution method for (9) when the convexity of the distribution function
condition (CDF C) and the monotone likelihood ratio condition (MLRC) are satisfied

(Mirrlees 1975, Rogerson 1985). We assume both of these conditions are satisfied in

what follows. The CDFC is satisfied by 620(61 b)/ab2 2 0. The conditional pdfof
disease satisfies the MLRC if 8b (6 l b)/ g(6 | b) is non-increasing in 0 (Milgrom 1981).16

Whitt (1980) has proven that the MLRC implies F OSD, and is therefore a slightly
stronger condition than F OSD.

Because it may be hard to garner intuition from gb (6 | b)/ g(6 | b) , Milgrom

(1981) provides an alternative explanation of the relevance of the MLRC in terms of the
government’s ability to infer the agent’s hidden actions from the observation of 0. He

describes the MLRC in terms of a principal who has a prior over the agent’s choice of b,
observes the level of disease realized, and then updates her prior to calculate a posterior

on the biosecurity effort choice. Denote the posterior probability distribution of b given

 

'6 Milgrom (1981) finds the term should be non-decreasing in 0, but in his model the
action has the opposite effect on the distribution as our action b. Accordingly, the sign is
reversed here.

52

observed outcome 0 by F (bl0). Using Milgrom’s (1981) results, the nature of disease is

such that the MLRC is equivalent to 60 s a] => F(b I 61 ) 2 F(b | 60) Vb , where 00 is a

more favorable signal than 0] that the agent exerted the desired level of biosecurity

effort. '7 The importance of the assumptions about the nature of the MLRC will become
clearer when the conditions for divergence from the first-best indemnity and the
implications of the model for indemnity design are considered below.

Using the FOA, the Lagrangian for the government’s problem is

l 1
4 = Us (b)a(b)+ 6(b)]Uf (1,r(6).6)g(6 l b)d6 — K[ ]r(6)g(6 l b)d6 + m(q)
0 O

(12) U’s (b)a(b)+ Us (b)a'(b)+
l

”‘ [[fl'(b)Uf(1,r(6).6)+fl(b)Uf(1,r(6),0)§’-’-(g-l-b—)]g(l9lb)d6

0 g(6 l b)

where u is the shadow value of the constraint. The existence of the constraint, due to the
farmer’s freedom to make their own biosecurity decision, renders this a second-best
problem. In Appendix 82 we illustrate that u>0 because the government would like the
farmer to increase his investment in biosecurity given the optimal indemnity payment.
That is, the optimal indemnity here is only second-best due to the information problem; a
first-best indemnity could be used to attain greater welfare in the absence of information
asymmetry (in which case [1 would optimally vanish). Holmstrom (1979) also finds such

a result.

 

‘7 It should be noted that the interpretation of the MLRC in the current context of a
malady is different than the examples most often found in the literature because lower
realized values of the random variable prevalence represent the desirable outcome,
whereas in the typical wage contract examme higher values of the random variable output
are desirable and signal greater effort.

53

Following Holmstrom (1979), pointwise optimization with respect to 2(0) yields

the following necessary condition, which must hold for all 0

 

 

R
6Uf(l,r(6),6) 6U, (1,1(6),0)[flb ,b+() [3(1) )g_b___(6lb)]

\ _ =_
(13) W” tit-(6) K )1 61(6) (6|b)

Condition (1 3) implicitly defines 158(0), the second-best indemnity as a function of 0.

The following adjoint equation is also necessary

l

g; = Us(b)a(b )+ Us (b)a +—x ]r(6)gb (6 l b)d6
0

l
(14) [[6'(b)Uf(1,r(6),6)+6(b)Uf(1,r(6) 6) fig? '13 )]s(6lb)d6+
0

[U 1'9 (b)a(b)+ 2(119(1))6'(b)+ Us(b)a"(b)+ .
1

#t {(0}? (1,1(6),6Xfl~(b)g(6 | b)+ 26'(b)gb (g I b)+ 909%,) (9 | 01)”
.0

 

II
C

 

 

J

Using the agent’s F OC and recognizing that the final term in (14) can be written as

11(62V5(b,1,f* ) / 6‘sz , condition (14) reduces to

l
(15) x[r(6)gb(6 | b)d6 = #[62V5(b,l,f*j/6b2).
0
Condition (15) determines 11 while the constraint in (l 1) determines b.

How do second-best indemnity payments compare to first-best payments? F irst-
best payments arise when there are no constraints on the government’s problem — that is,
the regulator is neither constrained by the farmer’s first order condition nor is truthful
disclosure an issue, so that there is neither an ex post adverse selection nor an ex ante

moral hazard problem. In this case, condition (13) reduces to

54

\6Uf(1,r(6),6) __
’ 61(6) _ _ ’

 

(16) [3(1)

*
implicitly defining the first-best indemnity payment 1 (0). Comparing condition (16)

with condition (13), we find that the following must hold

(i)ZTSB(6)<z'*(6) gb(91b)+,6'(b)<

 

 

g(6lb) 6(6)
. SB .. gb(9lb) [1"(6)
(l7) (11).r (6)>r (6) g(6lb) + ,6(b) >0.

Condition (1 7) indicates that farmers receive information rents, relative to the first-best
case, under the conditions defined by (l 7ii), while the government reduces payments
below the first-best level under the conditions defined by (171') (the conditions are
described in detail below). The resulting payment level depends on the realized value of

0, which the government views as evidence of the farmer’s unobservable biosecurity

effort. Holmstrom (p.79) points out that the term gb / g is simply the derivative of the

maximum likelihood function ln[g(0|b)], when b is taken as an unknown variable, and

suggests that g ,, / g measures how strongly one is inclined to infer from 0 that the agent

did not undertake the assumed action. The second-best solution is dependent on the
distribution of 0 and its relationship to b. The deviation from perfect risk sharing implies
that the farmer (agent) must carry extra responsibility for the disease outcome, as was
discussed in terms of instantaneous utility of wealth above. We present the mathematical
conditions that facilitate the result summarized in (17) along with a description of the
intuition behind the relative magnitude of first- and second-best indemnities under the

different circumstances.

55

It follows from the definition of ,6(b) that ,6’(b)/ ,6(b) < 0. The overall sign of

gb (6 | b)/ g(6 | b)+ fl'(b)/ ,6(b) is therefore determined by both the sign and magnitude of

II: I!
the term g, / g. Define the single level of infection at which 188(0)=r (0) as 0 , such
:1: wk , . SB
that gb(6 lb)/g(6 lbj+ 6(b)/)6(b)=0. Thls means that r (6)<r*(6) for

6 > 6* , and 153(6) > r * (6) for 6 < 6*. How small is 6* ? It should be small enough

a * o
to infer that a sufficient investment in biosecurity has been made. In particular, 6 IS

smaller than the value of 0, denoted 6 , at which gb (6): 0. This point in the

distribution is of interest because for 6 < 6 the marginal benefit of biosecurity, in terms
of reducing the cumulative density of prevalence, is increasing. '3 A smaller 0 yields even

larger marginal benefits of biosecurity.
By structuring indemnities according to (l 7), the government provides very

strong incentives for a farmer to make a significant investment in biosecurity. If the

government observes 6 > 6 it will pay farmers for culled animals at a lower rate than it
would if it could observe biosecurity actions directly; this is because a relatively high

level of infection suggests a small likelihood of private biosecurity effort. Even for

observed levels of infection 6* < 6 < 6 , the level of effort inferred by the government is

still sufficiently low that the farmer cannot extract any information rents from the

government. Only if the level of observed infection falls below the critical value 0* will

 

'8 A point which might correspond with 6 on a hypothetical second-best indemnity
schedule is depicted in Figure 4, which we discuss below.

56

the government be convinced that the agent has invested in biosecurity at a high enough
level to give up information rents to the farmer.

The relationship between the prevalence level and size of indemnities is also of

great interest. To evaluate the slope of rSB(0), differentiate condition (13) with respect to

0 to get

6(ga(6lb)/g(6lb))
U ' a6

“8) "(6):”(6)+Z'(6)+i—U"i(l [sb(6lb)+6'(b)]]'

 

 

+ [1
g(91 6) 6(1))
The first two RHS terms are positive and equal the slope of the first-best indemnity r (0)

(since u=0 in the first-best case). The third term arises in the second-best case, and the

I

sign of this term depends on the value of 0. In the third RHS term, (.2117) >0 for a risk

averse agent, the numerator is negative by the MLRC, and the sign of the denominator is

gb (6 l b) + 6(b)
g(6lb) 6(b)

 

determined by the sign of y =[ Jand the relative magnitude of terms,

which for a given b depends on 0.

1]:
Several different possibilities for the shape of 158(0) relative to that of r (0) are

*
depicted in Figure 2.4. For very low levels of 0, rSB(0)> r (0). The second—best

indemnity may be initially increasing or decreasing based on the sign of equation (18).

This is because y>0 for low values of 0, making the third RHS term in (18) negative and

the sign of 1'(6) ambiguous. The negative third term means the slope of 188(0) is

*
initially less than the slope of r (0), so that these curves eventually intersect and become

57

equal at 6*. When 0> 6* , then y<0 and the denominator of the third RHS term is of

ambiguous sign. Denote 6 (with 6> 6*) to be the value of 0 such that the denominator

of the third RHS term vanishes; the slope of the second-best indemnity goes to ~00 and

there is a discontinuity in the graph of 153(0). This means the slope of 153(0) becomes

negative prior to 6. If 6 <1 , this suggests that for any observed prevalence 6* < 6 < 6 ,
the second-best indemnity should be paid at a level below the first-best level to maintain

the appropriate incentives to invest in b.'9

While it is mathematically possible for the slope of the second best indemnity

function to be positive and/or negative over the range of 0, an indemnity schedule that is

monotonically decreasing in the prevalence level like TiSB (6) would provide the

strongest incentives for biosecurity. 20 In such cases the information rents paid to the

 

19 It should be noted that for 6 > 6 the slope of the second-best indemnity is expected to
be +00 as you approach 6 from the right. This suggests that it is possible that the first-

and second-best indemnities cross again at another point 6 > 6 , but it does not seem
likely that an information reward would be optimal at high levels of 0. Moreover,

whether or not 6 falls within the unit interval is unknown.
20 For 6 < 6 , the mathematical condition required for a monotonically decreasing second-

best indemnity schedule like the one depicted by 153(6) in Figure 4 is

# 6(gbl6’ I b)/ g(9 I b))l
U’ 66

)1'(6)+x'(9)< _9» (1+6?)

 

I. This condition may be reasonable in

practice because xi’(6)+ [(6) is likely a very small positive quantity at lower values of

6. Alternatively, the condition that gives rise to the increasing segments of 1‘ng and 1393

58

c * o o
farmer would be strictly decreasing with prevalence up untll 6 = 6 . Beyond thls pOlnt,
the reduction in payments relative to the first-best case would be increasing. Biosecurity

incentives would seem to be weaker under the curves labeled 1593 and 1598 , which seem

contrary to the government’s objective.

The extent to which observed prevalence is a good signal of actual preventive bio-

securlty effort 1S clearly lmportant for implementing the 1ndemnlty schedule, 1S (0). It is

conceivable that for diseases that are extremely contagious (i.e., Foot and Mouth Disease,
Highly Pathogenic Avian Influenza, Exotic Newcastle’s Disease, or Classical Swine
Fever) an individual’s herd could become infected regardless of the extent of biosecurity
measures taken ex ante. For this reason, a “one size fits all” policy that only pays
indemnities on the basis of observed prevalence levels is likely to be problematic in
practice.

This does not preclude a disease-specific indemnification rule which would pay

*
farmers 153(0) for all but the most infectious diseases and pay them according to 1 (0)

during outbreaks of highly contagious diseases (where observed prevalence is not a good

signal of effort). Indemnifying farmers in this disease-specific manner does not diminish

i]:
reporting incentives because of the use of f ; also, if low prevalence levels are verified

when responding to an outbreak involving such pernicious diseases, this could be treated

 

#6(gb(6lb)/g(6lb))
U' 66
- U " (1+ #7)

 

 

over the relevant range of 0 is ,1'(6)+ [’(6) > for that

 

range of 0.

59

as an even stronger signal that the farmer made a significant investment in biosecurity

*
and such behavior should be rewarded by paying 153(0)> r (0). The World Organization

for Animal Health (Office International des Epizooties or OIE) maintains a list of
diseases that must be reported to the international community. The OIE’s list of
reportable diseases (formerly called “List A” diseases) could serve to determine which

diseases should be associated with such a disease-specific indemnification rule—that is,

*
diseases on the list are compensated on the basis of r (0) (with information rents if low

. . . . . . SB
prevalence lS verlfied) and all other dlsease losses are 1ndemn1fied accordlng to r (0).

An indemnification rule similar to that implemented in Belgium and the Netherlands
seems to match the suggested indemnity from our model (Horst, deVos, Tomassen, and
Stelwagen). This indemnification rule is briefly discussed below.
Implications for Public Policy and Market Insurance Design
Our model uses two distinct mechanisms to provide incentives for biosecurity and
truthful disclosure: (i) indemnities to achieve desired levels of biosecurity, and (ii)
optimal fines that induce disclosure of disease status (alternatively, these mechanisms can
be viewed as a differential indemnity schedule based on whether an infected farmer
reports or not). By using two distinct policy instruments, individually designed with each
information problem in mind, it is possible to create clear incentives for farmers to
behave in a manner that is consistent with government risk management objectives.
Status qua indemnification for livestock disease losses by USDA has paid
producers on the basis of “compensation value” equal to “fair market value assuming

disease-free status” (Ott 2006, p.72). This amount is necessarily greater than the true

60

market value of diseased animals culled by the government and is intended to create
incentives for reporting. The government has also recognized that unless farmers face
some uncompensated losses as a result of outbreak they cannot be expected to take
preventive biosecurity measures and thus does not compensate farmers for consequential
losses when issuing indemnities. Animal health authorities have relied on a single
mechanism—indemnities—to facilitate both ex ante biosecurity effort and ex post
reporting. By using a single mechanism to induce biosecurity and reporting
simultaneously, the incentives for each individual private action are not clear.

Direct comparison of the relative magnitude of status quo indemnities and the
second-best indemnities implied by our analysis is not possible, but the major difference
is the shape of the indemnity schedules implied by the different policies. Status quo

policy suggests the indemnity schedule is strictly increasing in 6(just like the first-best

up
indemnity, 2' (6), depicted in Figure 2.4), while the second-best indemnity implied by

condition (17) is strictly declining over a range of 6. It is not at all clear how the
incentives created by the status quo policy facilitate the government’s joint objectives.
An upward sloping indemnity schedule, in the absence of any penalty for not reporting,
may actually create incentives for infection when you consider that the status quo has
sought to use indemnities based on the disease-free fair market value of livestock as a
means of resolving the ex post adverse selection problem.

In an effort to induce early reporting, Belgium and the Netherlands no longer
compensate producers for dead animals and only partially compensate them for diseased
stock (Horst, deVos, Tomassen, and Stelwagen). This approach shares some important

elements with our second-best indemnities. First, while there is not an explicit fine for

61

not reporting, there is a penalty to waiting to report since dead animals fetch no payment.
This feature can help to achieve incentive compatibility with reduced or eliminated
monitoring costs. Second, the partial compensation for only diseased animals shifts some
of the risk to farmers, as do our payments. An indemnity plan that does not shift risk in
this fashion may actually create incentives for infection, which could be one problem
associated with status quo US. policy.

The discussion of incentive compatibility applies not only to public policy but
also to the development of private insurance for livestock disease protection. If private
coverage is available to farmers, the incentives provided by livestock insurance contracts
could potentially be in competition with the objectives of policy while satisfying the
individual objectives of producers (i.e., income smoothing as risk management). Careful
consideration in the design of private market coverage for livestock disease losses is
required in order to ensure that public policy and private risk management products are
jointly incentive compatible. Also, design of public policy should take into account the
role that private coverage could play in achieving public policy objectives and how
government decisions may hinder or bolster private markets for insurance. If this is not

the case then the constrained efficient result analyzed here will not be achievable.

62

Appendix B

Appendix B.l

Based on equations (1) and (2) and incorporating transition probabilities with the
expected lifetime utility notation introduced in the text, equations (3) and (4),

respectively, are written in explicit form as

(3’) st = Us (60 -bw) + Ps1(b)lE6[V1]- Vsl and

pV, = {111R (710 — /1(6)— 1(6) + r(6))+ (VS — V1)) +
(4’) ([11)? (no — 1(6) — 1(6) + r(6) — f)+ (Vs — 1(1)] +

(l — r)
(1 —q)[U}VC(rro -/1(0)-z(6)+0(9))+h(Vs 4(1)]

Appendix 82

The first order condition associated with the farmer’s problem is given by the constraint
to problem (10), or

95.: U§(b)a(b)+ Us (b)a’(b)+

60
(3.2.1) 1

. R + R T gb(9|b) =
O 6021/, (1,6616) 6(b)U,(1. (6).6)g(6lb)]e(6lb)d6 0

This is an optimal response to the indemnity payment, 2(0); however, the indemnity
payment is only second-best because the regulator is constrained by the farmer’s

response. If the regulator were not constrained, then for a first-best outcome it would

have the farmer choose addltlonal blosecurlty, such that 26% < 0. Generallze condltlon

(82-1) to be

63

(8.2-2) fl < c ,
6b

:1:
where c is a constant. c=0 for the farmer’s problem, while c=c <0 for the first-best

outcome. Using this notation, the Lagrangian can be re-written as

1
1: US(b)a(b)+,6(b)1[UR(,,)l1(6)6g()(6|b)d6—K[[r(6)g(6|b)d6+m(q)]
0 0

(82-3) Us(b)a(b)+Us(b)a'(b)+

”1]? b)U, (1 r(6) 6)+6(b)U, (1 (6) 6) -—(—;(g ‘13) (61b)d6—c

61 O O O I I 0
Clearly, — = — 12 < 0 srnce an 1ncrease 1n c when c=0 lmplles a decrease 1n welfare
c

c=0

*
as the solution moves farther away from the first-best outcome c <0. Hence, u>0.

 

64

 

.

>938 Bern 1238556 5

Fauna >ma=$ macaw

Semaoenac 98 m8 :9 coma—decry. ($593 3 so” 8 Boo:

Argue: cacao 3c 90 panace—
msa nomad—:55 Suzanna 9o
Rococo—mac em msmooooaoficaamw

_

E880 8 :5 van—3x:

 

 

Nome—82% 8330:me

 

_ _
.25 9836 mam—8 om 388
mm 83an 85 mm coma—43

wagon: 33.2w 0:; cm 90 Arman: ASSESS
A>maamv mag 90

mac/5553” @1593: mm
osmomam and so 8389
808838 mgmov

$023383

iv flan

 

3.393— Eamom
830:8 Swanson.
“—08on aoaiaocoasm.
Ba EGG Soon—=38 m2.
m3 ciao—m 2:3

 

 

Emcee N._. flamsm mag 533.30: mazmsm 0». 58830: c328: Ea mega—5.9: can @363.

65

 

Susceptible

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__________________ ‘4’
' . . l
: Make biosecurity : 4+ DR
' mvestmentchonce : ,/ l \. '\
fi/‘i (possibly no : .’ l ‘. '\
Probability ofoutbreaka : biosecurity) : -’ '. ‘. '\
. . . l | / . \ '
function of biosecurity ------------------- _,‘ ‘ .\ \.
1" ‘ ' \
Remain susceptible l‘ \, '\
\ .
. , \
. . | .
Transmon to i \-\ \_
infected state i '\ \.
. . \
l ‘ '\
‘. ‘ '\
l ‘ '\
l \ '\
\ .
Government culls -\ \-\
infected animals " -
. . . \
and pays mdgmmty ( ~\
.I. Not caught Caught by to armer, T( ) ‘. ‘.
.l by gov’t gov’t .'

 

 

 

 

 

 

 

 

 

I ./
[I /
1' / \ Gov’t culls infecteds,

. pays indemnity 1(9)-
l’nvately cull f, and transition to
mfecteds, get

I

i Privately cull infecteds,

i susceptible
salvage value :

5

get salvage value 0(9),
and transition to
susceptible

 

 

 

0(9), but remain
infected

'-°-.

Figure 2.2. Individual farmer decision tree: disease states in bold and farmer choices in
dashed boxes

66

lim[—al/L]
r—>1 5" 4

V

f= f it along solid
*

curve, where f

selected so that

A(f*)=0

       
    
 

f = 0 along dashed \\
curve, such that \
MO) is not
constrained to be
non-negative

 

0 \ / : (Glb)

 

V

Figure 2.3. Fines fare set to ensure positive marginal incentives to report for all
prevalence levels 9.

67

//

 

 

%>

Figure 2.4. Possible shapes for first- and second-best (SB) indemnity payment schedules

with a discontinuity at 6’.

68

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71

Essay Three

JOINTLY-DETERMINED LIVESTOCK DISEASE DYNAMICS AND
DECENTRALIZED ECONOMIC BEHAVIOR

Introduction

Endemic livestock diseases impose significant costs on society (Bennett 1992, 2003;
Bennett, Christiansen and Clifton-Hadley 1999; Bennett and ijelaar 2005; Buhr, et al.
1993; Chi, et al. 2002; National Research Council 2005), prompting a need to understand
management aspects of these problems. It is useful to understand both optimal public
(centralized) management in response to a disease outbreak (Kobayashi, et al. 2007a, b;
Mahul and Durand 2000; Mahul and Gohin 1999), and the decentralized behavioral and
disease responses to common policy initiatives (Hennessy 2005, 2007; Hennessy, Roosen
and Jensen 2005).

Our focus in this paper is on decentralized outcomes. The majority of prior
economic research in this area focuses on behavioral outcomes, holding disease risks
stationary (i.e., possibly a function of human choices but not reflective of the underlying
epidemiological dynamics; Hennessy 2005, 2007). Prior economic work has also
generally failed to account for cross-farm extemalities, though Hennessy (2005, 2007)
are exceptions. Analogously, the majority of prior veterinary and epidemiological
research focuses on how disease dynamics are affected by government intervention, such
as herd depopulation, without considering producer behavioral responses. In this paper,
we consider the joint-detennination of disease and behavioral dynamics across farms, in a
decentralized setting.

Livestock disease management is an inherently dynamic process that involves

72

feedbacks between disease ecology and private behavioral decisions. We consider the
impact of these feedbacks while focusing on two common decisions that livestock
managers must make: a livestock manager must sequentially decide whether or not to
invest in biosecurity as a preventive measure and whether or not to report infection to
government authorities in the event that a disease outbreak occurs. The decision to report
infection may or may not arise depending on the uncertain disease outcome, but at some
point everyone faces the biosecurity decision. Because it is unclear in advance whether
or not livestock will be exposed to infection, investment in biosecurity represents a
definite investment of resources in exchange for an uncertain future benefit (own herd
protection). But biosecurity may not fully protect one’s own herd, and, moreover, losses
due to increased regulatory stringency may also arise as a result of a neighbor’s herd
becoming infected —- even in the absence of infection in one’s own herd. This is because
all herds within infected regions may be affected by costly regulatory actions taken by
animal health authorities to eradicate infection. Such regulatory-induced extemalities are
a common feature of livestock disease problems. For instance, all farms in the bovine
tuberculosis (bTB) infected region in Michigan’s Lower Peninsula — regardless of
infection status — incur private costs as a result of dealing with government testing,
movement restrictions and stringent testing rules for trade in live animals that go
uncompensated by the government.

Our paper builds in particular on the work of Hennessy (2005, 2007), who also
accounts for cross-farm spillovers. Hennessy (2005) constructs a topological model to
analyze how disease extemalities across farms in different spatial arrangements influence

biosecurity decisions. In that model, biosecurity investments exhibit strategic

73

complementarities with respect to entry of a disease and strategic substitution with
respect to spread of an already-introduced infection. Hennessy (2007) specifies a
relationship between disease risk and biosecurity investment within a production region
such that disease risks are endogenous and biosecurity is a strategic substitute, and
derives a number of general implications about the long-run equilibrium. In both cases,
the disease risks are not based on an epidemiological model and therefore the relation
between disease risks and biosecurity is stationary. Also, group-level impacts of
government regulations are not modeled explicitly. Our model extends this previous
work in two principal ways. First, we account for the jointly determined nature of
disease and economic outcomes in a fully dynamic bioeconomic model, with endogenous
and non-stationary disease risks. Second, we characterize distinct static and inter-
temporal strategic effects that arise when government response to infection gives rise to
extemalities.

We proceed with an analytical model of disease dynamics and then integrate
economic behavior into this model. We then present simulation results to illustrate the
tradeoffs arising in a joint system, and we contrast these results with those arising from a
non-joint system. We conclude with a discussion of the general implications of this

research, identifying additional research needs in this area.

Livestock Disease Dynamics
A metapopulation disease model (Levins 1969) is adopted to model cross-farm livestock
disease dynamics. In this framework, individual farms (and not the individual animals on

each farm) are the primary unit of interest. The farms are dispersed across space, and

74

disease transmission occurs according to a simple susceptible-infected (S-I) model of
disease (Mangel (2006) provides a gentle introduction to disease models from theoretical
biology). Each farm can be in one of three states at any point in time, indexed by j:
susceptible (non-infected, j =5), infected (i=1), or empty (/'=E). As we detail below, the
rate at which farms transition between disease states depends on farmer decisions to
invest in biosecurity (in the S state) or report infection to animal health authorities (in the
I state), government disease surveillance effort (to discover non-reporters), and the rate at
which the government allows repopulation of empty farms.

Define n to be the fixed number of homogeneous farms in a region, with s farms
being susceptible, i farms being infected, and e farms being empty.l The number of
susceptible farms changes over time according to
(1) s'=ee—bfl0(s/n)i—(l—b),61(s/n)i.

The number of susceptible farms grows when empty farms are repopulated at the rate a
(the first right-hand-side (RHS) term in (1)), and s is reduced by transitions to the

infected state as represented by the remaining terms. New infections occur at different
rates based on whether or not farms invest in biosecurity. The proportion of farms that

biosecure at time t is denoted by b. For these farms, the disease transmission parameter is

,80 , and ,80 s/n represents the expected number of new infections generated by each

infected individual.2 For farms that do not biosecure, the disease transmission parameter

 

' Strictly speaking, our “production region” considers the area encompassed by a disease
surveillance zone established by a government authority to control the spread of infection
and within which all farms will be subject to inspections and possible quarantine,
depending on the disease in question.

2 This type of transmission is known as frequency-dependent because the expected
number of new infections depends on the proportion or frequency of susceptible farms

75

is ,6] > ,60: farms that biosecure face a lower risk of infection than farms that do not.

The change in the number of infected farms over time is

(2) i=bfl0(s/n)i+(1—b),81(s/n)i—ri—(l—r)qi.

The first two terms denote newly infected farms, as in (1). The last two terms represent
depopulation of infected farms. Depopulation occurs when a farm reports infection,
which is the strategy adopted by a proportion of farms r, or when the government
discovers the infection, at a rate q, on the (1 -r) proportion of infected farms that did not
initially report it.3 Finally, all transitions between disease states in (1)-(2) are balanced
by changes in the number of empty (depopulated) farms given by
(3) é=ri+(l—r)qi—£e.

The dynamic system can be rewritten in terms of proportions of farms in each
state (Hess 1990; McCallum and Dobson 2004). Define S=s/n as the proportion of
susceptible farms, I =i/n as the proportion of infected farms, and E =e/n as the proportion

of empty farms. Upon making this transformation, equations (1)-(3) can be rewritten as

(la) S = eE—bfloSI—(l—b)fllSI,

 

(McCallum, Barlow, and Hone 2001 ). The most common alternative to frequency-
dependent transmission is density-dependent transmission or what is referred to as the
“mass action model” in the disease ecology literature, where the likelihood of contact
between all farms is the same. Because it seems reasonable that all herds do not have an
equal probability of coming into contact with one another and that the probability of
contact is increasing (decreasing) in the proportion of farms that are infected
(susceptible), we model disease transmission as being a frequency-dependent process.
McCallum, Barlow and Hone (2001) and Begon, et al. (2002) discuss different ways that
pathogen transmission can be modeled and how to judge which method is appropriate for
a given application.

3 A disease model that tracks within-patch dynamics (that is, where the animal is the
primary unit) would be needed to consider a policy response that culls only individual
infected animals (“test-and-slaughter” policy) and not entire herds.

76

(2a) 1': bp051+a -b),81SI —rI-(1—r)ql,

(3a) E=rI+(1-—r)qI—6E.

Typical metapopulation models of disease transmission treat b and r in (1)-(3) as
exogenous parameters and typically compare steady state dynamics over a range of initial
values for the disease state variables and other parameters. In contrast, we take b and r to
be endogenous. Next we develop the behavioral dynamics that govern the economic
strategies b and r, which are made in response to current disease risks. In turn, the
economic choices b and r endogenously affect infection dynamics in our joint model. In

this way, we account for dynamic feedbacks between the economic and disease systems.

A Dynamic Model of Farmer Behavioral Choices
Farmers make decisions about whether to invest in biosecurity and whether to report

infection based on their current disease state. Denote individual farmer z’s (z= l ,. . .,n)

biosecurity and reporting strategies as b2 and r2, respectively. These are mixed strategies

that may be interpreted as the proportion of the time that an individual chooses each of
these binary actions. The distributions of mixed strategies in the population are denoted

as b and r, which are analytically equivalent to the proportion of individuals in the

population who choose biosecurity and reporting. The individual strategies b2 and r2 are

treated as being distinct from the strategies b and r prior to equilibrium; in equilibrium,

we will have bz=b and rz=r due to the assumption of homogeneous :farms.

The farm decision model follows that of Hennessy (2007), which is based on

Shapiro and Stiglitz (1984). In this framework, a farm in a given disease state receives an

77

expected flow of income associated with its current disease state, taking into account the
possibility that it will transition to a new disease state at some point in the future. Denote
farm z’s baseline level of profit in each period in which the farm operates (i.e.,j¢E) by It,

with baseline profits being zero during the empty state. These profits are gross of any

expected biosecurity investment costs, wbz (where w is the cost of biosecurity), private
losses from infection, 5, and regulatory costs, 621-. szk represents the rate at which farm 2

transitions from state j to state k. Both sz and szk may be influenced by farm 2’s and

others’ strategies. These terms are specified in greater detail below.

Denote sz to be the expected lifetime income (or utility) of the zth farmer who is
currently in state j. Assuming a discount rate of p, the fundamental asset equations for
susceptible, infected, and empty farms are, respectively4
(4) szs = 7r - wbz - 025 + P251 [V21 - V25]

(5) PVzI=”_5‘GZI+PZIE[VzE-Vzll

(6) szE = -GzE + PzEsles - V2131

Equation (4) is the “time value of the asset” in the susceptible state, which equals the sum

of the “instantaneous income per unit time” conditional on being susceptible, rr-wbz—ng,

and the “expected capital loss that would arise were the state to change” (Hennessy 2007,

 

4 The asset equations are derived following Shapiro and Stiglitz (1984). Focusing on the

case of j=S as an example, we take V5 and V; as given and examine expected lifetime
utility over a small time interval [0, t]:

(4a) st = [7r — wbz -st1:+<1— potPZSIerI + (I — stmrzsr
Note that (l-pt) z e_p t. Equation (4) is obtained by solving (4a) for V5 and evaluating it
as t—>0.

78

p.702) from susceptible to infected, P25[[ sz- V25]. Equations (5) and (6) have analogous

interpretations. Equations (4)-(6) can be solved simultaneously for V25 , V21 , and V2 E
as functions of the behavioral strategies, the states of the world, and model parameters.

Now consider the szk terms. P251 is the transition rate to the infected state,

which can be obtained from equation (1) as P251 = bzflol +(1— bz ) ,611 .5 Similarly, the

transition rate from the infected to the empty state is Pz IE = rz +(1— rz )q. Finally, the

transition rate from the empty state to the susceptible state is PZES = 8 .

Three types of regulatory costs may arise in various states of the world. First is a
per unit “penalty”, x, based on the number of new infections in the region,

[bflo + (1 - b) ,61 ]SI , and imposed on all operating (non-empty) farms. Hence the total

expected penalty for each farm is [b,BO +(1— b) ,61 ]SIx . This group penalty can be

thought of as the private economic cost of government sanctions that affect all operations.
For instance, all farms in the non-bTB free area in Michigan—regardless of infection
status—incur private costs as a result of dealing with government testing, movement
restrictions and stringent testing rules for trade in live animals that go uncompensated by
the government. The penalty intensifies progressively as more new farms become
infected and is gradually reduced (to a level that is effectively zero) when the number of
new infections is reduced, reflecting the fact that government responses tend to be

commensurate with the nature of the disease event. For endemic diseases that regularly

 

5 Note that while farmers are forward-looking and P251 is state-dependent, we do not
suppose that individual farms have perfect foresight about states that are beyond their
own control. Instead, farmers (naively) assume the current state of the world will persist.
This is a common assumption in decentralized models of resource use (Clark 1990).

79

occur with low frequency in the population, there may be almost no government
intervention unless the number of new infections is increasing so rapidly as to indicate
potential economic disruptions or a coming epidemic.

The second type of penalty is imposed only on infected farms that report. Farms
that self-report infection are penalized at the rate f, where f is a net cost incurred after
govemment-provided indemnification. Reported infection in our model results in herd
depopulation, resulting in an asset loss of A. The government may provide some level of
indemnification for these losses. In the United States, the status quo federal policy is to
indemnify producers based on the disease-free fair market value of animals. Specifically,
indemnities are paid to livestock owners whose herds are depopulated—regardless of
whether reporting occurred or not—to compensate for the legal taking that has occurred
under the US. Constitution and to encourage reporting (Ott 2006). Currently, all
indemnification is “. . .equal to fair market value assuming disease-free status” and this is
intended to reduce “claims by livestock owners. . .that they didn’t receive full value for
their animals” (Ott 2006, p.72). Suppose indemnities are paid as depopulation occurs in
the infected state. If they are paid at the rate A, then f represents transaction costs
associated with government intervention and business interruption.6 If indemnities are
paid at a rate less than A (or not at all), then f reflects these additional losses.

The third type of penalty is incurred by infected farms that do not report and are
discovered, at rate q. These farms are penalized at the rate g>f The transaction costs of

dealing with animal health authorities are larger under the non-reporting case, and this is

 

6 Technically, fixed costs associated with business interruption would arise in the empty
state. Without loss, we economize on notation by accounting for the discounted value of
these costs at the time of reporting in the infected state. We make similar assumptions
with respect to the cost g, defined below.

80

reflected by the larger value of g. The larger penalty under the non-reporting case
provides some incentives to report, but may not guarantee reporting if farm losses 5 and
the expected cost of being caught, (l-r)qg, are insufficiently large relative to fl Given

these types of penalties, the state-dependent regulatory costs are written as G 2 E = O , and
(7) 025 = [19,30 + (1 -b)fl1 151x
(8) 021 = [We +(1- b)fl1 151x + rzf + (1 - rz )qg -

Given the model specification, we can derive V 28 (bz , b,rz , I,S) ,

V21 (bz ,b,rz,I,S) , and V zE (bz ,b,rz , I ,S ) and use these to determine optimal levels of

biosecurity and reporting. As indicated above, a farm’s biosecurity and reporting
decisions are made at each point in time based on the farm’s current disease state. The
biosecurity decision is only made by susceptible farms because, once infected or emptied,
biosecurity investment yields no private benefits. The decision of whether or not to
report infection to animal health authorities is only made by infected farms, whereas
susceptible and empty farms face no such decision.

Consider the biosecurity decision. The marginal return to biosecurity for a

representative farm at an individual point in time is given by (3V 2 S /6bz and the optimal

biosecurity strategy b; (b,rz ,1 ,S ) is the solution to 6st / 6bz = 0. However, there is

no closed-form solution.7 Numerical evaluation of VzS (bz ,b,rz , I,S) suggests that it is

monotonic and approximately linear in bz for a wide range of feasible values of b, r2, 1, S,

 

7 If there was, then biosecurity investment dynamics would be derived from taking the
time derivative of b; (b,rz ,I,S). Without a closed form solution, it is impossible to

derive analytically the behavioral dynamics given by dbz/dt, and any dynamic analysis
based on numerical methods is complicated significantly.

81

and model parameters. We therefore develop an approximation of V zS . Using a linear
combination ofthe known endpoints V21; (b,rz , I,S) = VzS lbz =1: V25 (1,b,rz , I,S) and
VZIgB (b,rZ,I,S) = VzS lbz :0: V25 (0,b,rz,I,S), we approximate V35 at each point in

time according to I725 = b2 Vzg. + (1 — bz )Vzlng . We then use replicator dynamics to

model inter-temporal changes in behavior.
An analogous approach is used to model the reporting decision. As this decision

is made in the infected state, we approximate V2] by 172] = rz V21; + (1 — rz )Vzg/R , where

V21; (bz,b,1,S) = V21 |,Z =1: V21(bz,b,l,l,S) and

V21)R (bz,b,l,S) = V21 er =0: V21(bz,b,O,I,S).

Biosecurity Dynamics and Strategic Effects

We now focus on the biosecurity decision and introduce replicator dynamics (F udenberg
and Levine 1998; Rice 2004; Weibull 1995) to describe the change in the aggregate
frequency of biosecurity adoption, b. In a symmetric equilibrium of bz = b (Rice 2004),
b increases when the expected lifetime income from adopting the pure strategy b=l
exceeds the average expected lifetime income associated with the current distribution of

biosecurity strategies
6 B — ~
(9) Z = a[VS -VS] :> b =ab(1—b)[VSB —V§VB],

where a>0 is a speed of adjustment parameter. Equation of motion (9) indicates that

frequency of biosecurity adoption is increasing (decreasing) when the expected lifetime

82

utility from always investing in biosecurity exceeds (is less than) the expected lifetime

utility from never investing in biosecurity. From the definition of 1725 , the term

VB VNB

S — S equals the marginal value of an individual’s biosecurity strategy,

(‘3st / abz, in a symmetric equilibrium. Accordingly, from (9) the sign of b is

determined by the sign of this marginal value.

Consider the b = 0 isocline, which implicitly defines the equilibrium proportion

of farms that invest in biosecurity, conditional on the current values of other state

variables. Three values of b potentially satisfy b = 0: b=0, b=1, and
* B NB _ . . . .. . .
b (S ,I ,r) 3 VS — V S —- 0. The equ111br1um b 18 of particular importance, as

movement away from b. determines the sign of VS]? — VSNB and therefore the sign of b

in this neighborhood and also the stability of the various equilibria. Specifically, we are

interested in the sign of (XV? — V5918 )/6b in the neighborhood of b*.

Given that the term VS? — V SV B equals the marginal value of an individual’s

_VNB

biosecurity strategy, the derivative 6(VS B )/ 6b represents how an individual’ 5

marginal benefits from biosecurity, _ . . , are impacted by an increase in aggregate

biosecurity adoption, b:
621725 _ 6(81725 wbz)-
abzab ab

B NB)
a(st st _(p+8Xp+n)X82-p77512(fl1 flo)2 >0
6b Aflomfln

 

(10)

83

 

where 77 = q(1- r2) + rz > 0 and g(6/c) = —ej781,Bk + (77p + p2)(p + e + [Bk )}< 0 (for k
=O,1). The positive derivative in (10) indicates that an individual has greater incentives
to adOpt biosecurity when it is being adopted more by others in the region — a behavioral
externality. Specifically, biosecurity investments are strategic complements when x>0,

and biosecurity is neither a strategic complement nor a strategic substitute when x=0.

The strategic complement property means that VS? — VSNB > 0 for b>b¥ and

a:
VS? — VLéVB < 0 for b<b .8 The Opposite would hold if b was a strategic substitute

(though this case does not arise in our current specification). If b is neither a strategic
complement nor substitute (x=0), then the term VSB — VSNB does not depend on b and,

other things equal, there are only two potential equilibria; b=1 or b=0.
The dynamics for the case of strategic complementarity are depicted in one

dimension in Figure 3.1, holding I, S, and r fixed. Arrows denote the direction of

at:
behavioral dynamics for a given value of b relative to b (ceterus paribus). In Figure

III III
3.1a, b 6 (0,1) is unstable, while b=0 and b=1 are both locally stable. While b

represents a point of indifference between biosecuring and not (i.e., V SB — V‘SNB = 0 ), it

is also a critical point at which the marginal incentives to biosecure change (all else

at:
equal). Starting at b , if one person were to adopt just a little more biosecurity, then this

increases the marginal value of biosecurity for everyone else, triggering others to increase

 

8 The case of strategic substitutes could arise if farms undertake biosecurity actions to
prevent infections from leaving their farm. Farmers will not generally make such
investments if the primary benefits accrue to others, at least in the absence of government
programs to create incentives for this.

84

 

their investment in biosecurity. This process snowballs until everyone is an adopter. The

*
opposite occurs if, starting at b , one farmer were to adopt just a little less biosecurity—

the marginal value would decrease for everyone and aggregate investment would

Ill
diminish until no one invests. In Figures 3.1b and 3.1c, b lies outside the unit interval

and is therefore not a feasible steady state, but remains relevant for determining the
stability properties of the other equilibria; b=1 is globally unstable and b=0 is globally
stable in Figure 3.1b; b=0 is globally unstable and b=l is globally stable in Figure 3.lc.

Analogous figures could be drawn to illustrate strategic substitution by drawing

*
all arrows in opposite directions such that b in Figure 3.1a is stable—if one farmer were

to adopt just a little less (more) biosecurity, the marginal value of biosecuring increases
(decreases) for everyone and aggregate investment would increase (decrease) until the
equilibrium is re-established. This is consistent with conjectures made about the nature
of “out-of-equilibrium dynamic adjustment” in Hennessy (2007, Figure 1, p.704) where
biosecurity actions are strategic substitutes. Finally, if b is neither a strategic

complement nor substitute (x=0), then the sign of VS? — V 55V B is fixed for a given I, S,

and r, and b will increase or decrease accordingly until b=1 or b=0.

So far in our analysis of biosecurity dynamics, as in previous analyses (Hennessy
2005, 2007), we have held I, S, and r fixed as we have discussed the stability properties.
Hence the probability of becoming infected is a stationary function of b. We found that
the system always moves to a comer for the case of strategic complementarities (which
are present in our model when x>0), and also for the case of no behavioral extemalities

(x=0). The case of strategic complementarities is of particular interest because policy

85

makers seek to maintain incentives for adoption of biosecurity (and reporting). In the
context of the myopic model with I and r (and thus state transition rates) held fixed,
policies that shift b‘ to a value less than zero, so that b=1 is globally stable (Figure 3.1c),
are consistent with stated government objectives.

But I, S, and r are not fixed in the joint dynamic system. This means the
probability of becoming infected is non-stationary, and so assuming stationarity to derive
policy implications may lead to erroneous insights. It is important to consider the
interconnectedness of disease and behavioral dynamics. We focus in particular on
changes in I (and not in S, as S and I are generally inversely related), as this directly
affects the likelihood of infection and, in turn, the marginal incentives to biosecure. This

latter effect is given by

_ _ B NB
(11) a<6st/abz>=azrzs =5<st-st >>(,
a] abzal 61 < ’

 

which is too complicated to report and analytically ambiguous in sign. As I changes the

ll!
incentives to invest are altered and this affects the critical value b and thus the basins of
attraction to the equilibria b=0 and b=l. Numerical evaluation of the sign of (1 1)

suggests that (32172 S /(6b 261 ) > 0 for “low” values of I, so that an increase (decrease) in I

results in a greater (reduced) incentive to invest in biosecurity. In particular, the
incentives to invest are low when the risk of infection is low, and so it may be difficult to
encourage sufficient, sustained investments to eradicate the disease. Numerical

evaluation of (l 1) also suggests that for “high” values of I the effect of increased

infection on the marginal return to biosecurity is 6217 S /(5'bz 61) < 0 , which means that

86

an increase (decrease) in I implies there is a reduced (greater) incentive to invest in
biosecurity. The intuition behind this result is that once the prevalence of infection
reaches a certain level, the cost of direct (i.e., the group penalty) and indirect (i.e., via
transition probabilities to unfavorable economic states) behavioral extemalities becomes
such that the cost of further biosecurity investments exceeds the marginal benefits.

When we consider feedbacks between disease and behavioral dynamics the path
of economic choices over time is found to exhibit much more complex behavior than
suggested by analyses that treat I as fixed. While the usual static analysis of strategic
effects given by (10) indicates that biosecurity investments are strategic complements, the
joint dynamic system reveals that there are inter-temporal behavioral effects not
previously identified in the literature. Because of the inter-temporal relationship between
b and I and given that the marginal effect of disease prevalence on the marginal incentive
to invest in biosecurity operates in different directions based on the level of I (which is
changing over time), we find that aggregate investments in biosecurity in earlier periods
are inter-temporal substitutes for investments in later periods when prevalence is “low”,
while prior period aggregate investments in biosecurity are inter-temporal complements
to investments in later periods when prevalence is “high”.9 To avoid confusion, however,
we use the term behavioral extemalities to refer to the familiar strategic effects

represented in equation (10), and we use the term infection extemalities to refer to the

 

9 Hennessy (2005, 2007) modeled biosecurity investments as being strategic substitutes
for spreading diseases. The reason for this assumption was not clearly articulated, but it
might stem from the stationary disease risks in his models — that is, the lack of infection
dynamics. One important way (possibly the only way) a neighbor’s biosecurity actions
can substitute for one’s own is if the neighbor’s actions first prevent infection on his
farm, which in turn prevents infection spreading to one’s own farm. When infection
dynamics are not modeled explicitly, then the inter-temporal nature of these two
processes cannot be distinguished.

87

inter-temporal effects represented in equation (11).

Finally, consider the case of no strategic effects (x=0). Holding 1, S, and r fixed,

as
equation (9) suggests there is no interior steady state b . But when I, S, and r are allowed
to change, then it is possible for these values to stabilize at levels such that

VS? — V‘SNB = 0 . If and when this occurs, b = 0 and b will stabilize at whatever interior

value achieves the steady state.

Reporting Dynamics
Now consider reporting dynamics, assuming a symmetric equilibrium with rz = r. The

replicator dynamics for reporting are defined analogously to those for biosecurity and
produce an analogous result: frequency of reporting is increasing (decreasing) when the
expected lifetime utility from reporting exceeds (is less than) the expected lifetime utility

from never reporting,

(12) E=erIR —I711=> r' =;»r(1—r)[V,R -—V,NR],

where 7>O is a speed of adjustment parameter. As with the biosecurity case, the term

VR_VNR

I I equals the marginal value of an individual’s reporting strategy, 6721 /6rz ,

in a symmetric equilibrium. Hence, the sign of r is determined by the sign of this
marginal value. But in contrast to the behavioral extemalities influencing biosecurity

dynamics, there are no reporting extemalities because others’ reporting actions r do not
influence farm z’s expected asset value (i.e., 67,, /ar = 0 ).

Because there are no reporting extemalities, there is no interior equilibrium for r.

Rather, when r 6 (0,1) , reporting is either always increasing or decreasing over time

88

 

depending on the relative magnitude of asset values given by the sign of 617, /6r_,. That

is, the direction of the reporting dynamics is determined solely by the condition

VIR (b,l,I,S) — VINR (b,0,I,S) > (<)o :> r' > (<)0.

Numerical Simulations and Model Comparisons

We now develop a numerical example to simulate the model described above, which we
refer to as the joint model. We also simulate two other models to illustrate how the joint
model differs from prior work, and to highlight the importance of including feedbacks
between the disease and behavioral models. We define an S—I—only model that treats

economic choices as fixed parameters, and we define a behavior-only model that treats

infection probabilities (szk) as fixed parameters. In each model, the simulation begins at

the same initial values for all state variables. In the joint model, all behavioral and
disease variables are updated endogenously according to equations (1a)-(3a), (9), and
(12). With this structure, there are feedbacks between economic choices b and r and

disease states S, I, and E. In the S-I-only model, economic choices b and r are fixed
parameters (i.e., b = r = 0) while disease states are governed by equations of motion (la)-

(3a). In the behavior-only model, S and I are fixed parameters (i.e., I = S = E =0) while
the rate of change in biosecurity investment and reporting are governed by equations of
motion (9) and (12), respectively.

Besides shedding light on the results of traditional approaches that treat either
behavior or disease risks as fixed, comparisons across the three models for different

values of x also helps to illustrate the role of the strategic effects. In the absence of

89

behavioral extemalities (x=0), (XV: — VSNB )/ 6b = 0 so that biosecurity investments are

neither strategic complements nor strategic substitutes. Also, there are no infection
extemalities in either the behavior-only model (in which I is fixed, so as not to affect
behavior) or the S-I-only model (in which b is fixed, so as not to be influenced by
changes in 1). Infection extemalities do arise in the joint model, as changes in I affect the
marginal incentive to invest in biosecurity in each period.

Table 3.1 lists variable descriptions and the baseline parameterization used in all
numerical simulations in order to facilitate comparison of results across models given the
same starting values. The initial values for the behavioral variables in the baseline case
were selected so that farms are equally likely to invest in biosecurity (report infection)
and not invest (not report). The starting values for the disease states reflect a situation
where infection in a region has risen to a level where government chooses to intervene to
limit losses from the endemic disease.

The long run (steady state) results that emerge for each model are illustrated in
Figure 3.2. State variables are listed on the vertical axis and the results for each (in terms
of proportions of farms) are represented as bars. In the S-I-only and behavior-only
models, solid bars denote variables that are endogenous and striped bars denote variables
that are treated as fixed parameters.

In the S-I-only model, farms invest (do not invest) in biosecurity and report (do
not report) infection with constant, equal probability. The steady state is reached very
quickly, with endemic infection on about one-third of farms and less than half of farms
being susceptible. High rates of infection in the long run are due to the lack of behavioral

response to infection risks. Also note that about one-fourth of farms are depopulated in

90

the steady state as a result of the high level of infection that persists through the region.
This seems like a rather unlikely outcome when depopulation is used to respond to
discovered or reported infection for diseases that may become endemic to regions of the
US.

In the behavior-only model, the steady state consists of maximum biosecurity
investment and reporting rates (b = r =1 in Figure 3.2), due to the constant high rate of
infection that does not respond to behavioral choices. The steady state is reached even
faster than in the S-I-only model, also via a direct approach path.

Results of the joint model differ notably from both the S-I-only and behavior-only
models. The first difference arises in the steady state results because the joint model
accounts for feedbacks, which allows disease and behavioral outcomes to respond over
time. In the steady state, nearly all patches are non-infected (S=0.95), with very low
endemic infection and an equally low proportion of empty farms (I=E=0.025). This
contrasts markedly with the proportions of infected and empty farms in the behavior-only
and S-I-only models. Biosecurity (b=0.72) is at an intermediate level in the steady state:
it is less than in the behavior-only model because infection risks (a function of I) are
reduced as a result of prior biosecurity investments, reducing later incentives to invest;
biosecurity is greater than in the S-I-only model because in the joint model farmers can
respond to disease risks by increasing their level of investment. Reporting strategies
(r=1) are the same as in the behavior-only model, and greater than in the S-I-only model
so as to reduce the expected fine from getting caught in a non-reporting, infected state.

The second difference between the joint model and the other two models, due to

infection extemalities in the joint model, lies in the approach path to the steady state.

91

Specifically, the approach path for each state variable in the joint model (i) exhibits
dampened oscillations (Figures 3.3 and 3.4a), unlike the other two models, and (ii) takes
a much longer time to reach the steady state relative to the other two models, which
approach their equilibrium levels quickly without any overshooting (not pictured). The
oscillations result from the inter-temporal feedbacks between disease states and economic
behavior: the response of infection dynamics to changes in behavior and the infection
extemalities present in the behavioral dynamics (expression (11)). In particular, the sign
of (1 1) is positive in our simulation due to sustained low levels of endemic disease. This
indicates that the marginal incentives to invest (divest) in biosecurity are increasing
(decreasing) when I is increasing (decreasing). Thus, when I rises (falls) to a certain
level, b rises (falls) in the population (as in the blown-up section of Figure 3.3). For

instance, suppose b was rising in response to increased infection. In turn, these greater

biosecurity levels reduce the force of infection and there is a decline in I . If the

investments change the sign of I , then the magnitude of b will also begin to fall and
could eventually change signs. The result is a series of oscillations until a steady state is
reached. The process of overshooting the steady state greatly prolongs the period of
adjustment before the system settles down.

Now consider altering the baseline scenario by setting x>0. This group-penalty
scenario allows us to evaluate the effect of behavioral extemalities on system dynamics
in the different models. Because behaviors are fixed in the S-I-only model, there is no
change in this model relative to the baseline parameterization. We also find that
introducing behavioral extemalities has an unnoticeable effect in the behavior-only model

(not pictured); b and r both move to one very quickly. The effect of introducing

92

behavioral extemalities (for example, x=4 in Figure 3.4b) does however have significant
effects on dynamic outcomes in the joint model. In particular, by introducing strategic
complementarities within periods, the jointly-determined dynamic system no longer
reaches a steady state: overshooting persists indefinitely. Strategic complementarities
exacerbate the volatility of the system sufficiently that the system no longer settles down.
In contrast to the dampened oscillations observed in the baseline parameterization with
no extemalities (Figure 3.4a), biosecurity investment and disease states fluctuate
erratically following a path consistent with dynamic chaos over a bounded range of
values for each individual state.

Further sensitivity analysis (increasing x to 10) indicates that (i) the range over
which disease (not pictured) and biosecurity strategy states fluctuate chaotically is
increasing in x (compare Figure 3.4c to Figure 3.4b), and (ii) the strategic

complementarities are increasing in response to changes in I (i.e.,

[{de — V by B )/ 6b) / 61 > 0 ), or equivalently that the infection extemalities (or the

inter-temporal complementarities) are increasing in response to changes in b. In
particular, this second effect implies that the speeds of adjustment of both b and I are
increased, and this can be seen by the increased frequency of oscillations over the same
length of time in Figure 3.4c relative to Figure 3.4b. The overall effect on the joint
system of increased group costs x associated with government response to infection is to

increase the variance of disease and behavioral outcomes over time.

Discussion and Conclusion

There are four main implications of this research. First, when we account for feedbacks

93

in jointly-determined dynamic systems, it can yield insights not captured by myopic
models of disease or economic behavior. Such insights include the distinction between
behavioral and infection extemalities and the expectation that it may be difficult to
achieve an eradication objective without adjusting government actions over time. For
instance, the need for a shift in policy to achieve eradication has been suggested for the
bTB problem in New Zealand (Livingstone, et al. 2006), which has had a similar
experience to that in Michigan. The US experience with scrapie eradication in sheep
provides one example of a (bounty) program that had to adapt over time in order to
eventually achieve eradication (Kuchler and Hamm 2000).

Second, by jointly modeling disease and behavioral dynamics the endogeneity of
infection risk is captured. When we account for feedbacks in co-determined systems, the
influence of economic choices on infection risks is taken into account and transition rates
between economic states are non—stationary, affecting the inter-temporal tradeoffs that
arise. The notion of endogenous risk may also be instructive in thinking about
government response to widespread infection in the presence of extemalities.

Third, if economic damages from disease are assumed to be convex, then the
finding of greater variation in endemic disease levels from our simulations can be
expected to result in greater sustained average costs associated with disease in the region
affected. It is important to note that the origin of these costs in our model lies in the
structure of government disease control and indemnification policies because movement
restrictions impose costs on all farmers in a region but only infected farms that are
depopulated to control disease are compensated for their losses. This is an element

common to standing disease eradication programs as well as government response to

94

foreign animal disease outbreaks with the potential to have market-level impacts.

The final implication of this research is the critical need for data necessary to
parameterize models of this kind to analyze specific cases, as has been done in social
planner oriented models, like those developed for Foot-and-Mouth disease in the US
(Kobayashi, et al. 2007a,b) and France (Mahul and Durand 2000; Mahul and Gohin
1999). When data are available for a particular disease and geographical area, it may
even be possible to integrate disease epidemiology with decentralized strategic
interactions to better inform models of optimal allocation of public resources to respond
to an epidemic. In developing our numerical simulations we were made aware of the
general lack of empirical estimates of inter-herd disease transmission coefficients and
longitudinal data on livestock disease prevalence trends or farmer behavior necessary to
parameterize such a model. Without such data available, it will remain impossible to
evaluate the performance of joint disease ecology-economic models which is necessary in
order for such models to be of greatest use for policy making or economic decision

making purposes.

95

Appendix C

Table 3.1. Baseline Modeling Parameters for the S-I, Behavioral, and Joint Model
Simulations
Parameter or

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Description State Variable Value
Underlying disease transmission coefficient applicable when BO 0.25
invest in biosecurity
Disease transmission coefficient applicable when do not B1 3.1
invest in biosecuritL
Probability infection is discovered when don't report to Q 0.5
animal health authorities
1: 100
Instantaneous income flow at each point in time
W 10
Cost of biosecurity
6 40
Cost of infection
Per unit “penalty” imposed on all farms based on the change X 0
in overall infection
F 40
Cost of reporting infection
Cost of getting caught not having reported infection (with G 80
probability q)
p 0.05
Discount rate
Speed of adjustment parameter in the biosecurity strategy a 0.5
replicator dynamics
Speed of adjustment parameter in the reporting strategy 7 0.5
replicator dynamics
Reporting strategy on farm 2, aggregate reporting rate in the r(0) 0.5
population
Biosecurity investment strategy on farm 2, aggregate rate of b(0) 0.5
investment
[(0) 0.4
Infected proportion farms
E(0) 0.3
Empty proportion farms; Eo=r010+(1-r0)10q
8(0) 0.3
Susceptible proportion farms
8 1.0

Transition rate from empty (depopulated) to susceptible

96

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L
F"

41

O—
1
it

(a) (b)

(C)

Figure 3.1. One-Dimensional Illustration of Strategic Complementarities in Biosecurity
b, When S, I, and r Are Held Fixed: (a) b=0,l are both stable equilibria; (b) b=0 is the
only stable equilibrium; (c) b=1 is the only stable equilibrium.

97

States (solid), Fixed Parameters (striped)

 

 

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99

Proportion of Farms
1

0.8

 

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mu 1 “ I ‘l

0.8 1‘“ ‘1 :‘l M“ ‘1‘ {j " , ,3 .

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Figure 3.4. Time Paths for Biosecurity Strategy, b(t), Under Joint Model: (a) No

behavioral extemalities (x=0); (b) “Small” behavioral extemalities (x=4); (c) “Large’

behavioral extemalities (x=10)

100

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102

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