Demand for Insurance: Which Theory Fits Best? Some VERY preliminary - PowerPoint PPT Presentation

Demand for Insurance: Which Theory Fits Best? Some VERY preliminary experimental results from Peru Jean ¡Paul ¡Petraud ¡ Steve ¡Boucher ¡ Michael ¡Carter ¡ UC ¡Davis ¡ UC ¡Davis ¡ UC ¡Davis ¡ ¡ I4 ¡Technical ¡Mee;ng ¡ Hotel ¡Capo ¡D’Africa, ¡Rome ¡ June ¡14, ¡2012 ¡

Goals Today 2 ¨ Theory ¤ Consider a specific empirical context (Pisco, Peru); ¤ Develop two alternative contracts: A) Linear, B) Lump Sum; ¤ Compare predictions of insurance demand under: n Expected Utility Theory; n Cumulative Prospect Theory. ¤ Highlight preference parameter spaces such that theories generate different demand predictions. ¤ Preference parameters: Risk aversion, Probability weighting, Loss aversion. ¨ Empirical Approach ¤ Experimental insurance games with Pisco cotton farmers ¤ Part I: Elicit farmer-specific values of preference parameters ¤ Part II: Elicit farmers’ choice across contracts (Linear vs. Lump Sum vs. None) ¨ Descriptive evaluation of theories: Which theory seems to be most consistent with elicited parameters?

Linear vs. Lump Sum Contracts 3 ¨ Income under No Insurance : ¤ Y N = Apq A : Area (ha); p : Output price ($/qq); q : yield (qq/ha) ¤ Compare Linear vs Lump Sum contracts with identical: A) Strikepoint; B) Premium and C) Expected ¨ Indemnity payment (i.e., same Expected Income) ¨ Income under Linear Insurance : ¤ Y L = Ap[(T – q) – π ] if q ≤ T ¤ Y L = Ap(q – π ) if q > T ¤ T: strikepoint (qq/ha); π : premium (qq/insured ha) ¨ Income under Lump Sum Insurance : ¤ Y S = Ap(q + s – π ) if q ≤ T ¤ Y S = Ap(q – π ) if q > T ¤ s: Lump sum indemnity (qq/insured ha) ¨ Parameterize for Pisco ¤ A = 5 ha; p = 100 S./qq; ¤ T = 32 qq/ha; π = 620 S./ha; s = 1,060 S./ha

Linear vs. Lump Sum Contracts 4 45 Income 40 35 30 25 No Insurance Linear Contract 20 Lump Sum Contract 15 10 5 0 0 10 20 30 40 50 60 70 80 90 T = 32 Yield (qq/ha)

Discrete Version 5 ¨ Discrete yield distribution with 5 possible outcomes: ¤ Start with empirical distribution of average yield in Pisco; ¤ Collapse all density above mean into 1 outcome with 55% prob; ¤ Collapse density below mean into 5 outcomes with smaller probabilities; ¨ End up with:

Linear vs. Lump Sum Contracts 6 45 60 55% ¡ Income Probability 15% ¡ 10% 0 0 24 ¡ 60 ¡ 8 ¡ 32 ¡ 43 ¡= ¡E(yield) ¡ 16 Yield (qq/ha)

Linear vs. Lump Sum Contracts 7 45 60 Income 55% ¡ (000 S.) 30 26.9 Probability 18.2 16 14.2 12.9 12 15% ¡ 10.2 8 10% 6.2 4 0 0 24 ¡ 43 ¡= ¡E(yield) ¡ 60 ¡ 32 ¡ 8 ¡ 16 Yield (qq/ha)

¨ How do we choose between Red vs. Green vs. Blue stars? ¨ Need to see how insurance effects PMF of income. 8 45 60 Income 55% ¡ (000 S.) 30 26.9 Probability 18.2 16 14.2 12.9 12 15% ¡ 10.2 8 10% 6.2 4 0 0 24 ¡ 43 ¡= ¡E(yield) ¡ 60 ¡ 32 ¡ 8 ¡ 16 Yield (qq/ha)

PMF’s of income under different contracts 9 60 50 40 Prob. 30 None 20 10 0 0 40 4 8 ¡ 12 ¡ 16 ¡ 30 ¡ Income

PMF’s of income under different contracts 10 60 50 40 None Prob. 30 Linear 20 10 0 0 40 12 ¡ 12.9 ¡ 16 ¡ 4 8 ¡ 26.9 ¡ 30 ¡ Income

PMF’s of income under different contracts 11 60 50 40 None Prob. 30 Linear Lump Sum 20 10 0 0 40 6.2 ¡ 8 ¡ 10.2 ¡ 12 ¡ 12.9 ¡14.2 ¡ 16 ¡ 18.2 ¡ 4 26.9 ¡ 30 ¡ Income

PMF’s of income under different contracts 12 60 50 40 Linear Prob. 30 Lump Sum 20 10 0 0 40 10.2 ¡ 12 ¡ 12.9 ¡14.2 ¡ 16 ¡ 4 6.2 ¡ 8 ¡ 18.2 ¡ 26.9 ¡ 30 ¡ Income

Contract choice under EUT versus CPT 13 ¨ What matters under EUT? ¤ Degree of risk aversion n γ : Coefficient of Relative Risk Aversion ¨ What matters under CPT? ¤ Degree of risk aversion ¤ Subjective probabilities n Decision weights assigned to each outcome may differ from objective probabilities n α : Coefficient from probability weighting function ¤ Reference point and reflection n Do I treat “gains” systematically differently than “losses” n R : Reference point above which lie gains, below which lie losses. ¤ Loss aversion n Degree of asymmetry of valuation of losses versus gains n λ : Coefficient of loss aversion

Contract Choice under EUT 14 ¨ u(Y) = Y 1- γ ¤ Constant Relative Risk Aversion ¤ γ is coefficient of relative risk aversion ¤ γ > 0 à risk averse; γ < 0 à risk loving ¨ Linear contract gives greater risk reduction than lump sum contract. ¨ Risk averse farmers will: ¤ Never prefer lump sum to linear; ¤ Buy linear if they are sufficiently risk averse ( γ > γ *), such that risk premium > insurance premium. ¨ Risk neutral & risk loving farmers will: ¤ Always prefer no-insurance n Highest variance; n Loading à Highest E(Y)

Expected Utility Theory 15 70 60 50 40 None Prob., EU Linear Lump Sum 30 20 10 0 0 40 10.2 ¡ 12 ¡ 12.9 ¡14.2 ¡ 16 ¡ 4 6.2 ¡ 8 ¡ 18.2 ¡ 26.9 ¡ 30 ¡ Income

EUT Departure 1: Subjective Probability Weights 16 ¨ People tend to: ¤ Overweight small probabilities; ¤ Underweight larger probabilities. ¨ Probability weighting function from Prelec (1998): ¤ w(p) = exp(-(-ln(p) α ) ¨ Cumulative Prospect Theory (Kahneman & Tversky, 1992) transform w(p) into decision weights that: ¤ Sum to 1; ¤ Maintain monotonicity

Impact of Prob. Weighting on Insurance Demand 17 70 ¨ In each option, relatively bad outcomes are lower 60 prob.; 50 ¨ Thus expected utility falls for ALL options as α à 0 40 None Prob. Linear 30 Lump Sum ¨ Linear becomes relatively more attractive because it truncates lowest outcomes 20 10 0 0 40 4 6.2 ¡ 8 ¡ 10.2 ¡ 12 ¡ 12.9 ¡14.2 ¡ 16 ¡ 18.2 ¡ 26.9 ¡ 30 ¡ Income

Impact of Probability Weighting: Summary 18 Linear ¨ γ * is CRRA such that indifferent between Linear & No contracts; γ *( α ) ¨ ∂γ */ ∂α > 0 ¤ As α falls from 1 to 0, None n Linear becomes relatively more attractive n So marginally less risk averse people prefer Linear ¤ As α increases above 1 n Overweight high prob events; n Linear becomes less attractive; n Eventually prefer Lump Sum (area C). D γ *(1) ¨ Demand Flip-floppers? E ¤ E: None (EUT) à Linear (CPT) ¤ D: Linear (EUT) à None (CPT) ¤ C: Linear (EUT) à Lump Sum (CPT)

Departure #2: Reflection & Reference Point ¨ u(Y) = (Y-R) 1- γ if Y > R 60 EU(R = 16) ¨ u(Y) = -((R-Y) 1- γ ) if Y > R 50 40 ¨ Utility function “reflected” around 30 reference point, R . 20 10 Losses Gains ¡ ¨ Risk averse behavior over “gains” 0 0 10 20 30 40 50 16 ¡ -10 ¨ Risk loving behavior over “losses” -20 ¨ How does Reflection affect insurance -30 demand? -40 ¤ Depends where R is… -50 ¤ (Wouter’s Proposition 5 )

Low R à Insurance evaluated over “gains” 20 70 EU(R=2) 60 50 40 30 None Prob., Linear EU Lump Sum 20 10 0 8 ¡ 2 ¡ 12 ¡ 30 ¡ 4 16 ¡ 0 40 Income -10 -20

High R à Insurance evaluated over “losses” 21 EU(R=2) 60 Prob., None EU Linear 40 Lump Sum EU(R = 32 ) 20 0 12 ¡ 16 ¡ 30 ¡ 4 8 ¡ 32 ¡ 0 40 Income -20 -40 -60

Intermediate R à Insurance evaluated over “gains” & “losses” 22 60 Prob., EU None 40 Linear Lump Sum 20 0 32 ¡ 16 ¡ 4 8 ¡ 12 ¡ 30 ¡ 0 40 Income -20 -40 -60

Impact of Reference Point: Summary 23 ¨ As R increases: ¤ Relatively more insured 60 outcomes evaluated over Prob., EU losses; None 40 Linear Lump Sum ¤ Lump sum becomes relatively more attractive than linear; 20 ¤ Eventually no-insurance dominates 0 8 ¡ 4 32 ¡ 12 ¡ 16 ¡ 0 40 Income -20 ¨ In intermediate range (insured outcomes over both losses & -40 gains), any ranking can obtain; -60

Departure #3: Loss Aversion ( λ ) 60 ¨ u(Y) = (Y-R) 1- γ if Y > R ¨ u(Y) = -( λ (R-Y) 1- γ ) if Y > R 40 20 ¨ λ introduces asymmetry in magnitude of loss and gain of given size; 0 0 5 10 15 20 25 30 35 40 45 Income ¡ EU( λ =1) -20 ¨ λ > 1 à Loss hurts more than a gain of equal size gain. 16 ¡ -40 ¨ How does λ affect insurance demand? -60 ¤ It depends on R ( Wouter’s Proposition 6 ☺ ) EU( λ =2) -80 -100

R < 12.9 = Apq(T- π ) 80 ¨ Impact of ↑λ on EU: ¤ No effect under LC; 60 ¤ Falls under LS; 40 ¤ Falls more under NC. 20 ¨ Impact of ↑λ on demand: Prob. 0 ¤ Can flip from LS à LC or NC à 12.9 ¡ 0 40 Income LC if LS initially preferred. -20 ¤ No impact if LC initially preferred. None -40 Linear Lump Sum -60 -80