Hierarchical Modeling and Hierarchical Modeling and the Economics of Risk in the Economics of Risk in

IPMIPMPaul D. MitchellPaul D. Mitchell

Agricultural and Applied Agricultural and Applied EconomicsEconomics

University of Wisconsin-MadisonUniversity of Wisconsin-MadisonEntomology ColloquiumEntomology Colloquium

March 28, 2008March 28, 2008

Goal TodayGoal Today

Explain hierarchical modeling for Explain hierarchical modeling for economic analysis of pest-crop economic analysis of pest-crop systems and IPMsystems and IPM

Overview how economists represent Overview how economists represent and compare risky systems and and compare risky systems and identify the preferred systemidentify the preferred system

Hierarchical ModelingHierarchical Modeling

Purpose: To model the natural variability Purpose: To model the natural variability of the insect-crop system analyzing itof the insect-crop system analyzing it

Insect-crop system is highly variableInsect-crop system is highly variable Pest population density, crop injury/damage Pest population density, crop injury/damage

for given pest density, crop loss for given for given pest density, crop loss for given injury/damage, control for given pest densityinjury/damage, control for given pest density

Missing a key component of system if Missing a key component of system if economic analysis ignores this variabilityeconomic analysis ignores this variability

Hierarchical ModelingHierarchical Modeling Treat variables as random and use linked Treat variables as random and use linked

conditional probability distributionsconditional probability distributions pdf of a variable has parameters that depend pdf of a variable has parameters that depend

(are conditional) on variables from another (are conditional) on variables from another pdf, which has parameters that are pdf, which has parameters that are conditional on variables from another pdf, … conditional on variables from another pdf, … … … … …

Final result: pdf for economic value that has Final result: pdf for economic value that has much/most of the system’s natural variabilitymuch/most of the system’s natural variability

Key is having right the data to estimate the Key is having right the data to estimate the conditional probability distributionsconditional probability distributions

Stylized ExampleStylized Example

Pest Density

Crop Damage

% Yield LossPest-Free Yield

Pest Control Used?

Crop Price

Net Returns

Corn Rootworm in CornCorn Rootworm in CornPest Density

Adults BTD

Crop DamageNode Injury Scale

Yield Loss% loss

Pest-Free Yield

Pest Controlsoil insecticide, seed

treatment, RW Bt corn

Crop Price

Net Returns

Pest Density: Pest Density: Adult beetles/trap/dayAdult beetles/trap/day

Unconditional beta distributionUnconditional beta distribution Eileen Cullen’s data from the Soybean Eileen Cullen’s data from the Soybean

Trapping Network in southern WI, 2003-2007Trapping Network in southern WI, 2003-2007

YearYear N obsN obs MeanMean St DevSt Dev CVCV

20032003 3434 2.692.69 2.092.09 77.6%77.6%

20042004 2828 2.622.62 2.602.60 99.1%99.1%

20052005 3131 2.422.42 1.961.96 81.1%81.1%

20062006 1919 2.682.68 1.661.66 62.2%62.2%

20072007 1010 1.101.10 0.530.53 47.9%47.9%

AllAll 122122 2.472.47 2.062.06 83.5%83.5%

Pest Density: Pest Density: Adult beetles/trap/dayAdult beetles/trap/day

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

BTD

Cou

nt

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2 4 6 8 10 12

BTD

pro

bab

ility

den

sity

ParmParm EstimatEstimatee

ErrorError t-statt-stat P-valueP-value

meanmean 2.482.48 0.180.18 13.6413.64 0.0000.000

st devst dev 2.002.00 0.150.15 13.6713.67 0.0000.000

maxmax 12.6412.64 4.764.76 2.662.66 0.0080.008

Crop Damage: Node Injury Crop Damage: Node Injury ScaleScale

Oleson et al. 2005. Oleson et al. 2005. JEE 98(1):1-8JEE 98(1):1-8

Quantifies feeding Quantifies feeding damage on corn damage on corn roots by corn roots by corn rootworm larvaerootworm larvae

http://www.ent.iastate.edu/pest/http://www.ent.iastate.edu/pest/rootworm/nodeinjury/rootworm/nodeinjury/nodeinjury.htmlnodeinjury.html

NINISS

DescriptionDescription

0.00.0 No feeding damage No feeding damage (lowest score)(lowest score)

1.01.0 One node (circle of roots) One node (circle of roots) or equivalent of an entire or equivalent of an entire node eaten back to within node eaten back to within approximately 1½ inches approximately 1½ inches of stalk (soil line on 7of stalk (soil line on 7thth node)node)

2.02.0 Two complete nodes Two complete nodes eateneaten

3.03.0 Three or more nodes Three or more nodes eaten (highest score)eaten (highest score)

Crop Damage: Node Injury Crop Damage: Node Injury ScaleScale

Untreated NIS (NISUntreated NIS (NIS00) conditional on ) conditional on previous summer’s BTDprevious summer’s BTD

Eileen Cullen’s data from the Soybean Eileen Cullen’s data from the Soybean Trapping Network (2003-2007)Trapping Network (2003-2007)

Conditional beta pdf estimatedConditional beta pdf estimated Min 0 and max 3 set, then MLE conditional Min 0 and max 3 set, then MLE conditional

mean = Amean = A00 + A + A11BTD and constant st devBTD and constant st dev AA00 = base level of damage even if observe = base level of damage even if observe

BTD = 0BTD = 0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10

BTD

Unt

reat

ed N

IS

2004

2005

2006

2007

Fit

Untreated NIS Conditional on Untreated NIS Conditional on BTDBTD

CorrelationCorrelation

2004: 0.1022004: 0.102

2005: 0.4962005: 0.496

2006: 0.4402006: 0.440

2007: 0.2412007: 0.241

All: 0.292All: 0.292

Mean = 0.393 + 0.0388BTD St. Dev. 0.530Mean = 0.393 + 0.0388BTD St. Dev. 0.530

Untreated NIS Conditional on Untreated NIS Conditional on BTDBTD

0

3

6

9

12

15

0 0.5 1 1.5 2 2.5 3

Untreated NIS

prob

abili

ty d

ensi

ty

BTD = 3

BTD = 6

BTD = 9

MeanMean0.510.510.630.630.740.74

Untreated NIS: Estimation Untreated NIS: Estimation ResultsResults

ParmParm EstimateEstimate ErrorError t-statistict-statistic P-valueP-value

InterceptIntercept 0.3930.393 0.0660.066 5.9625.962 0.0000.000

SlopeSlope 0.0390.039 0.0160.016 2.4252.425 0.0150.015

St DevSt Dev 0.5300.530 0.0520.052 10.10610.106 0.0000.000

LogLLogL -12.7248-12.7248 Mean = mMean = m00 + m + m11*BTD*BTD

SlopeSlope 0.4400.440 0.0580.058 7.5477.547 0.0000.000

ExponentExponent 0.1810.181 0.0720.072 2.5162.516 0.0120.012

St DevSt Dev 0.5330.533 0.0520.052 10.19310.193 0.0000.000

LogLLogL -12.4867-12.4867 Mean = mMean = m11*BTD*BTDmeme

Untreated NISUntreated NISLinear vs Cobb-Douglas MeanLinear vs Cobb-Douglas Mean

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10

BTD

Un

trea

ted

NIS

NIS

linear

Cobb-Dgls

NIS conditional on Untreated NIS conditional on Untreated NIS and control methodNIS and control method

Treated (NISTreated (NIStt): beta density with conditional ): beta density with conditional mean = Bmean = B11NISNIS00, conditional st. dev. = S, conditional st. dev. = S11NISNIS00, , min = 0, max = NISmin = 0, max = NIS00

Parameter BParameter B11 varies by control method varies by control method No intercept and expect BNo intercept and expect B11 < 1 and S < 1 and S11 < 1 < 1 Drive NISDrive NIStt to zero as NIS to zero as NIS00 goes to zero goes to zero

University rootworm insecticide trial data University rootworm insecticide trial data from 2002-2006 (U of IL, IA State, UW, etc.)from 2002-2006 (U of IL, IA State, UW, etc.) NIS for side-by-side plots, treated and untreated NIS for side-by-side plots, treated and untreated

control, with trap crop planted previous yearcontrol, with trap crop planted previous year

Treated NIS vs Untreated Treated NIS vs Untreated NISNIS

N obsN obs 1010 2121 2727 2121 1818

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Untreated NIS

Tre

ated

NIS

Bt

Force

Aztec

Fortress

Lorsban

NIS conditional on Untreated NIS conditional on Untreated NIS and control methodNIS and control method

ParameterParameter est.est. errorerror t-t-statstat

p-p-val.val.

Bt (MON863 Cry 3Bb1)Bt (MON863 Cry 3Bb1) 0.080.0822

0.010.0144

5.735.73 0.000.0000

Force (tefluthrin)Force (tefluthrin) 0.170.1799

0.010.0177

10.410.422

0.000.0000

Aztec (tebupiriphos & Aztec (tebupiriphos & cyfluthrincyfluthrin

0.150.1566

0.010.0155

10.610.600

0.000.0000

Fortress (chlorethoxyfos)Fortress (chlorethoxyfos) 0.150.1511

0.010.0188

8.508.50 0.000.0000

Lorsban (chlorpyrifos)Lorsban (chlorpyrifos) 0.250.2522

0.020.0233

11.011.044

0.000.0000

SS11 0.090.0988

0.000.0088

12.212.233

0.000.0000

NIS conditional on Untreated NIS conditional on Untreated NIS and control methodNIS and control method

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 1.0 2.0 3.0

Untreated NIS

Mea

n Tr

eate

d N

IS Bt

Force

Aztec

Fortress

Lorsban

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5 3

Untreated NIS

Trea

ted

NIS

Bt

Force

Aztec

Fortress

Lorsban

Fit

Effect of Untreated NISEffect of Untreated NIS

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3

Lorsban Treated NIS

Pro

bab

ilit

y D

ensi

ty

NIS 1

NIS 2

NIS 3

Effect of Control Method (NISEffect of Control Method (NIS00 = = 1.5)1.5)

0

2

4

6

8

10

12

14

16

18

20

0 0.5 1 1.5

Treated NIS

Pro

babi

lty

Den

sity

Bt

Aztec

Lorsban

Proportional Yield Loss Proportional Yield Loss conditional on NIS differenceconditional on NIS difference

No control method gives complete control No control method gives complete control (treated NIS (treated NIS ≠≠ 0), so how determine the 0), so how determine the yield effect of rootworm control?yield effect of rootworm control?

Proportional yield loss conditional on the Proportional yield loss conditional on the observed difference in the NISobserved difference in the NIS

Need NISNeed NISaa, NIS, NISbb, Yield, Yieldaa, and Yield, and Yieldbb from from plots in same locationplots in same location

University rootworm insecticide trial data University rootworm insecticide trial data plus JEE publications (Oleson et al. 2005)plus JEE publications (Oleson et al. 2005)

Multiple pairings per location: if n Multiple pairings per location: if n treatments at a location, then have n!/[2!treatments at a location, then have n!/[2!(n-2)!] pairings(n-2)!] pairings

Proportional Yield Loss Proportional Yield Loss conditional on NIS differenceconditional on NIS difference

If NISIf NISaa > NIS > NISbb, then , then NIS = NISNIS = NISaa – NIS – NISbb and loss is and loss is = (Y = (Ybb – Y – Yaa)/Y)/Ybb

If NISIf NISaa < NIS < NISbb, then , then NIS = NISNIS = NISbb – NIS – NISaa and loss is and loss is = (Y = (Yaa – Y – Ybb)/Y)/Yaa

NIS always > 0, but NIS always > 0, but not always > 0 not always > 0 Conditional pdf: Normal distribution Conditional pdf: Normal distribution

with with = = 11NIS and NIS and = = 00 + + 11NISNIS If If NIS = 0, then NIS = 0, then = 0 and “noise” = 0 and “noise” 00

causes the observed yield differencecauses the observed yield difference

Proportional Yield Loss Proportional Yield Loss conditional on NIS differenceconditional on NIS difference

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

NIS difference

Pro

port

iona

l Yie

ld L

oss

loss

fit

00 0.1480.148

11 --0.0190.019

11 0.1300.130

Hierarchical Model Hierarchical Model SummarySummary

1)1) Draw BTDDraw BTD

2)2) Use BTD to calculate conditional mean Use BTD to calculate conditional mean of NISof NIS00 and then draw NIS and then draw NIS00

3)3) Use NISUse NIS00 to calculate conditional mean to calculate conditional mean and st dev of NISand st dev of NIStrttrt and then draw NIS and then draw NIStrttrt

4)4) Calculate Calculate NIS and conditional mean NIS and conditional mean and st dev of loss, then draw lossand st dev of loss, then draw loss

5)5) Draw yield and calculate net returnsDraw yield and calculate net returns

Net Return to ControlNet Return to Control

Returns without controlReturns without control RR00 = PY(1 – = PY(1 – 00) – C) – C

Returns with controlReturns with control RRtrttrt = PY(1 – = PY(1 – trttrt) – C – C) – C – Ctrttrt

Net Benefit of ControlNet Benefit of Control RRtrttrt– R– R00 = PY( = PY(00 – – trttrt) – C) – Ctrttrt

Hierarchical Model ProblemsHierarchical Model Problems

If specify a data-based multi-step If specify a data-based multi-step hierarchical model, often no closed form hierarchical model, often no closed form expression exists for the pdf of some/all expression exists for the pdf of some/all the intermediate and final variablesthe intermediate and final variables

What is pdf of NISWhat is pdf of NIS00 ~ beta with a mean ~ beta with a mean that is a linear function of BTD ~ beta?that is a linear function of BTD ~ beta?

What is the pdf of net benefit to control?What is the pdf of net benefit to control? Must use Monte Carlo methods to analyzeMust use Monte Carlo methods to analyze

Monte Carlo MethodsMonte Carlo Methods

Use computer programs to draw Use computer programs to draw random variables in a linked fashionrandom variables in a linked fashion

Draw sufficiently large number to get Draw sufficiently large number to get convergence to “true” pdfconvergence to “true” pdf

Excel with 1,000 draws to illustrate Excel with 1,000 draws to illustrate for today’s presentationfor today’s presentation

Economics of Risk in IPMEconomics of Risk in IPM

Current simulations assume blanket Current simulations assume blanket treatment (always use control)treatment (always use control)

IPM: make decision to use control a IPM: make decision to use control a function of drawn BTDfunction of drawn BTD If BTD If BTD ≥≥ T, returns = R T, returns = Rtrttrt

If BTD < T, returns = RIf BTD < T, returns = R00

3 returns to compare: R3 returns to compare: R00, R, Rtrttrt, and R, and Ripmipm

How do you compare systems when How do you compare systems when returns or net benefit is random?returns or net benefit is random?

Economics of Risk in IPMEconomics of Risk in IPM

This problem is what economists call This problem is what economists call “risk”“risk” When decisions have random outcomesWhen decisions have random outcomes

Economics (& other fields) have methods Economics (& other fields) have methods and criteria for comparing random and criteria for comparing random systems to decide which is “best”systems to decide which is “best”

Quick review today using IPM in cabbageQuick review today using IPM in cabbage Mitchell and Hutchison (2008) “Decision Mitchell and Hutchison (2008) “Decision

making and economic risk in IPM”making and economic risk in IPM”

Representing Risky SystemsRepresenting Risky Systems Three elements neededThree elements needed

Actions: Apply insecticide, Adopt IPMActions: Apply insecticide, Adopt IPM Events/States of Nature with Events/States of Nature with

Probabilities: Rainy, Low pest pressureProbabilities: Rainy, Low pest pressure Outcomes: Insecticide washed off, IPM Outcomes: Insecticide washed off, IPM

prevents unneeded applicationsprevents unneeded applications Represent with decision trees, payoff Represent with decision trees, payoff

matrices or statistical functionsmatrices or statistical functions All are equivalent representationsAll are equivalent representations

Cabbage IPM Case StudyCabbage IPM Case Study

Cabbage looper (Cabbage looper (Trichoplusia niTrichoplusia ni) in ) in cabbage (cabbage (Brassica oleraceaBrassica oleracea) ) Mitchell and Hutchison (2008) Book Mitchell and Hutchison (2008) Book

Chapter Chapter Based on Minnesota cabbage IPM Based on Minnesota cabbage IPM

1998-2001 case study 1998-2001 case study Hutchison et al. 2006ab; Burkness & Hutchison et al. 2006ab; Burkness &

Hutchison 2008Hutchison 2008

Payoff MatrixPayoff Matrix

Event or State of NatureEvent or State of Nature ------ Action to choose ------------ Action to choose ------

Pest intensity (Pest intensity (T. niT. ni) ) (# sprays)(# sprays) ProbabilityProbability

Biologically-Biologically-Based IPMBased IPM

Conventional Conventional SystemSystem

Low (1-2 sprays)Low (1-2 sprays) 0.700.70 17951795 285285

Moderate (3-4 sprays)Moderate (3-4 sprays) 0.200.20 775775 -366-366

High (> 5 sprays)High (> 5 sprays) 0.100.10 13811381 871871

Subjective probabilities based on expert opinion of cabbage IPM Subjective probabilities based on expert opinion of cabbage IPM specialistspecialist

Decision TreeDecision Tree Action Event Probability Outcome (Pest Management System) (Pest Intensity) (Subjective) (Net Returns $US /ha) Low 0.70 1795

□ Moderate 0.20 775

IPM High 0.10 1381

□ Low 0.70 285

Conventional □ Moderate 0.20 -366

High 0.10 871

Statistical FunctionsStatistical Functions(pdf and cdf change with (pdf and cdf change with

action)action)

0.0

0.2

0.4

0.6

0.8

-500 0 500 1000 1500 2000

Net Returns ($/ha)

Pro

bab

ility

Den

sity

IPM

Conventional

0.0

0.2

0.4

0.6

0.8

1.0

-500 0 500 1000 1500 2000

Net Returns ($/ha)

Cu

mu

lati

ve P

rob

abili

ty

IPM

Conventional

Measuring Risky SystemsMeasuring Risky Systems

Central Tendency: Mean, Median, ModeCentral Tendency: Mean, Median, Mode But what about variability?But what about variability?

Variability/Spread/DispersionVariability/Spread/Dispersion Variance/Standard DeviationVariance/Standard Deviation Coefficient of Variation: Coefficient of Variation: Return-Risk Ratio (Sharpe Ratio): Return-Risk Ratio (Sharpe Ratio):

Asymmetry or “Downside Risk”Asymmetry or “Downside Risk” SkewnessSkewness Returns for critical probabilities (Value at Risk Returns for critical probabilities (Value at Risk

VaR)VaR) Probabilities of key events (Break-Even Probabilities of key events (Break-Even

Probability)Probability)

Value at Risk and Value at Risk and Probability of Critical OutcomeProbability of Critical Outcome

F(z)

1.0

0.0 Outcome z

Pr(z ≥ zt)

zt

F(z)

1.0

0.0 Outcome z

VaR(c)

c

ProbabilitiesProbabilities

Subjective: what the decision maker Subjective: what the decision maker thinks based on beliefs and thinks based on beliefs and experienceexperience Often based on expert opinionOften based on expert opinion

Objective: based on collected dataObjective: based on collected data Cabbage case studyCabbage case study

24 observations (6 fields x 4 years)24 observations (6 fields x 4 years) Smoother plots than in previous figuresSmoother plots than in previous figures

Cabbage Case StudyCabbage Case StudyObjective ProbabilitiesObjective Probabilities

01

23

45

67

89

-4000 -3000 -2000 -1000 0 1000 2000

Net Returns ($/ha)

Ob

serv

ato

ns

IPM

Conventional

0.0

0.2

0.4

0.6

0.8

1.0

-4000 -3000 -2000 -1000 0 1000 2000

Net Returns ($/ha)

Cu

mu

lativ

e P

rob

ab

ility

IPM

Conventional

Measures of RiskMeasures of RiskCabbage IPM Case StudyCabbage IPM Case Study

Subjective ProbabilitiesSubjective Probabilities Objective ProbabilitiesObjective Probabilities

MeasureMeasure IPMIPM ConventionalConventional IPMIPM ConventionalConventional

MeanMean 15501550 213213 11771177 186186

MedianMedian 17951795 285285 15601560 771771

ModeMode 17951795 285285 18961896 1107 & 12721107 & 1272

Standard DeviationStandard Deviation 406406 338338 970970 17091709

Coefficient of VariationCoefficient of Variation 0.260.26 1.581.58 0.820.82 9.209.20

Return-Risk RatioReturn-Risk Ratio 3.813.81 0.630.63 1.211.21 0.110.11

Break Even ProbabilityBreak Even Probability 1.01.0 0.80.8 0.880.88 0.710.71

Probability of $1000/haProbability of $1000/ha 0.80.8 0.00.0 0.750.75 0.500.50

Comparing Risky SystemsComparing Risky Systems

How to choose between risky How to choose between risky systems? systems? Need a criterion to combine with one of Need a criterion to combine with one of

these representations of risk to make these representations of risk to make decisiondecision

Many criteria exist, quickly overview Many criteria exist, quickly overview some commonly used in ag & pest some commonly used in ag & pest management management

Decision-Making CriteriaDecision-Making CriteriaRisk NeutralRisk Neutral

Choose system to maximize mean Choose system to maximize mean returnsreturns ““Risk neutral” because indifferent to Risk neutral” because indifferent to

variabilityvariability

= $10/ac with = $10/ac with = $5/ac = $5/ac

= $10/ac with = $10/ac with = $50/ac = $50/ac These equivalent with this criterion, but These equivalent with this criterion, but

most would choose the first system (“risk most would choose the first system (“risk averse”)averse”)

Decision-Making CriteriaDecision-Making CriteriaSafety FirstSafety First

Minimize probability returns fall Minimize probability returns fall below a critical levelbelow a critical level Places no value on meanPlaces no value on mean

Maximize mean returns, subject to Maximize mean returns, subject to ensuring a specific probability that a ensuring a specific probability that a certain minimum return is achievedcertain minimum return is achieved Max Max s.t. Pr(Returns > $100/ha) s.t. Pr(Returns > $100/ha) ≥≥ 5% 5%

Useful for limited income farmers Useful for limited income farmers such as in developing nationssuch as in developing nations

Decision-Making CriteriaDecision-Making CriteriaTradeoff Mean and VariabilityTradeoff Mean and Variability

Most people willing to tradeoff Most people willing to tradeoff between mean and variabilitybetween mean and variability Buy insurance because indemnities Buy insurance because indemnities

reduce variability of returns, though reduce variability of returns, though premiums reduce mean returnspremiums reduce mean returns

Preference (or Utility) function to Preference (or Utility) function to formalize this trading off between formalize this trading off between risk and returnsrisk and returns

Certainty Equivalent and Certainty Equivalent and Risk PremiumRisk Premium

Certainty EquivalentCertainty Equivalent: non-random return : non-random return that makes person indifferent between that makes person indifferent between taking risky return and this certain taking risky return and this certain returnreturn Certain return that is equivalent for a Certain return that is equivalent for a

person to the risky returnperson to the risky return Risk Premium: Mean Return minus the Risk Premium: Mean Return minus the

Certainty EquivalentCertainty Equivalent How much the person discounts the risk How much the person discounts the risk

return because of the riskreturn because of the risk

Risk Preferences/Utility Risk Preferences/Utility FunctionsFunctions

Takes each outcome and converts it Takes each outcome and converts it into benefit measure after accounting into benefit measure after accounting for riskfor risk

Mean-Variance: U(R) = Mean-Variance: U(R) = RR – – RR22

Mean-St Dev: U(R) = Mean-St Dev: U(R) = RR – – R R Mean-St Mean-St U(R) is the certainty equivalent U(R) is the certainty equivalent or or 22 is the risk premium is the risk premium is the personal parameter determining is the personal parameter determining

their tradeoff between risk and returnstheir tradeoff between risk and returns

Risk Preferences/Utility Risk Preferences/Utility FunctionsFunctions

Constant Absolute Risk AversionConstant Absolute Risk Aversion U(R) = 1 – exp(–U(R) = 1 – exp(–R)R) = coefficient of absolute risk aversion= coefficient of absolute risk aversion

Constant Relative Risk AversionConstant Relative Risk Aversion U(R) = RU(R) = R1 – 1 – ( ( < 1), U(R) = –R < 1), U(R) = –R1 – 1 – ( ( > 1), > 1),

and U(R) = ln(R) (and U(R) = ln(R) ( =1) =1) = coefficient of relative risk aversion= coefficient of relative risk aversion

Others exist that I’m not mentioningOthers exist that I’m not mentioning

Cabbage IPM Case StudyCabbage IPM Case StudyCertainty Equivalents ($/ac)Certainty Equivalents ($/ac)

Subjective Subjective ProbabilitiesProbabilities

Objective Objective ProbabilitiesProbabilities

CriterionCriterion IPMIPM ConventionConventionalal

IPMIPM ConventionConventionalal

Mean-Variance Mean-Variance (( = 0.0005) = 0.0005) 14671467 156156 706706 -1274-1274

CARACARA (( = 0.0001) = 0.0001) 15411541 208208

11211277 2828

CRRACRRA (( = 1.2) = 1.2) 14611461 ** ** **

*CRRA preferences not defined for negative returns *CRRA preferences not defined for negative returns outcomesoutcomes

Decision-Making CriteriaDecision-Making CriteriaStochastic DominanceStochastic Dominance

Assume little structure for utility function Assume little structure for utility function and still rank risky systems in some cases and still rank risky systems in some cases using cdfusing cdf

First Oder Stochastic DominanceFirst Oder Stochastic Dominance If FIf Faa(R) (R) ≤≤ F Fbb(R) for all R, then R(R) for all R, then Raa preferred to preferred to

RRbb if U’ > 0 (prefer more to less) if U’ > 0 (prefer more to less)

Second Order Stochastic DominanceSecond Order Stochastic Dominance RRaa preferred to R preferred to Rbb if U’’ < 0 (risk averse) and if U’’ < 0 (risk averse) and

min

*

min max( ) ( ) 0, *R

b a

R

F R F R dR R R R

IPM FOSD’s Conventional pest IPM FOSD’s Conventional pest control control when using subjective when using subjective

probabilitiesprobabilities

0.0

0.2

0.4

0.6

0.8

1.0

-500 0 500 1000 1500 2000

Net Returns ($/ha)

Cum

ulat

ive

Prob

abili

ty

IPM

Conventional

FFipmipm(R) ≤ F(R) ≤ Fconvconv(R)(R)

IPM SOSD’s Conventional pest IPM SOSD’s Conventional pest control when using objective control when using objective

probabilitiesprobabilities

0.0

0.2

0.4

0.6

0.8

1.0

-4000 -3000 -2000 -1000 0 1000 2000

Net Returns ($/ha)

Cu

mu

lativ

e P

rob

abili

ty

IPM

Conventional

min

*

min max( ) ( ) 0, *R

conv ipm

R

F R F R dR R R R

SummarySummary

Illustrated Hierarchical Modeling as Illustrated Hierarchical Modeling as way to include natural variability in way to include natural variability in the analysis of pest-crop systemsthe analysis of pest-crop systems Example: western corn rootworm in cornExample: western corn rootworm in corn Still need to add in IPM component and Still need to add in IPM component and

apply decision making criteria to derive apply decision making criteria to derive optimal IPM thresholds and estimate the optimal IPM thresholds and estimate the value of IPMvalue of IPM

SummarySummary

Overviewed how economists Overviewed how economists represent and compare risky systems represent and compare risky systems Illustrated with cabbage IPM case study Illustrated with cabbage IPM case study

of cabbage looperof cabbage looper Representing risky systemsRepresenting risky systems Measures of risk Measures of risk Decision-making criteria to choose Decision-making criteria to choose

systemsystem