On the Relevance of Probability Distortions in the Extended

Warranties Market∗

[Preliminary and incomplete]

Jose Miguel Abito Yuval Salant

October 14, 2015

Abstract

We use panel data on extended warranty purchases from a large electronics retailer to study

what drives the retailer’s very high profit on these warranties. Households’ purchase behavior

over time, differences between in-store and online purchase behavior, and warranty returns in-

dicate that product misperception is a much more important driver than households’ preference

such as risk aversion or innate biases in decision-making. We postulate that the misperception

is about the rate in which the insured product fails, and proceed to estimate a structural model

in which a profit-maximizing retailer sells warranties to a population of risk-averse consumers

who may misperceive failure rates. Our estimates indicate that more than 80% of the retailer’s

profit on warranties is due to overweighting of insured products’ failure rates, and correcting

households’ misperception substantially increases consumer welfare.

∗Abito: University of Pennsylvania, Wharton School, Business Economics and Public Policy,abito@wharton.upenn.edu. Salant: Northwestern University, Kellogg School of Management, Department ofManagerial Economics and Decision Sciences, y-salant@kellogg.northwestern.edu. Thanks: ...

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1 Introduction

The market for extended warranties has been highly profitable since the early 2000s. According to

analysts’ estimates, extended warranties accounted for almost half of BestBuy’s operating income

in 2003, and profit margins on warranties ranged from 50% to 60%.1 BestBuy gradually reduced

the transparency of reporting on its extended warranty business since 2001. Around the same time,

concerns about the high profitibility in the extended warranties market led to a UK Competition

Commission investigation of their main consumer electronics retailers. The Commission attributed

the high profit margins to a “complex monopoly situation,” the solution of which likely falls outside

the scope of standard competition policy (Baker and Siegelman, 2013).

Retailers indeed benefit from significant market power when selling warranties. This is because

warranties are an add-on offered to consBarsegumers after they finalize their decision to purchase the

insured product. For example, BestBuy sales people are trained to offer warranties and other add-

ons to buyers only after they finalize their decision to purchase the product,2 and online retailers

offer warranties to buyers only during checkout. At this stage, it may be costly for consumers

to revisit their decision to purchase the main product, thus giving the warranty seller significant

pricing power.3

But pricing power cannot explain the strong demand for extended warranties. In our dataset

from a big US consumer electronics retailer on extended warranty purchases between 1998 and

2004, about 27% of the consumers purchase an extended warranty for a TV despite the warranty’s

high price (about 22% of the insured product price) and the low likelihood that a TV fails (about

7% within three to four years of its purchase). To put things in context, this implies that about

one out of every four consumers who purchase a median-priced TV is willing to pay $110 or more

to insure himself against a loss of at most $500 with 7% probability.

A very high level of risk aversion in the form of diminishing marginal utility for wealth is required

to explain this behavior. On the other hand, Prospect theory (Kahneman and Tversky, 1979)

proposes that choice under uncertainty is governed by probability weighting and loss aversion, and

the recent empirical work of Barseghyan et al (2012) establishes the important role of probability

weighting in driving home and auto insurance choice. In line with prospect theory, Barseghyan

et al (2012) show that overweighting of small claim probabilities and insensitivity to probability

changes explains a sizable part of consumers’ willingness to pay for insurance.4

In this paper, we first provide evidence that consumers in our dataset likely misperceive the

1“The Warranty Windfall”, Business Week (Dec 19, 2004).2For BestBuy’s presentation of selling skills, seehttps://www.extendingthereach.com/wps/PA VCorationFramework/resource?argumentRef=static&resourceRef=/files/Best Buy Vendor Selling Skills.pdf

3See Ellison (2005) for an add-on pricing model in which firms benefit from monopoly power on the sale of theadd-on due to consumers’ search cost.

4Jindal (2014) studies extended warranties for washing machines using stated choices from a survey. He finds animportant role for loss aversion in this context. Since washing machines tend to have a higher failure rate (25-29%for the first four years of service) compared to TVs, it is not surprising that probability distortions have adiminished role. For example, in the Prelec (1998) specification, the weighting function gets flatter and flatter as itcrosses the 45 degree line at 1/e ≈ 0.37%.

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value of the warranty at the point of sale. Building on the findings of Barseghyan et al (2012) on

the importance of probability weighting in insurance choices, we postulate that the misperception

is about the rate in which the insured product fails. We proceed to estimate a structural model

in which a monopolistic profit-maximizing retailer sells warranties to a population of risk-averse

consumers who may misperceive failure rates. Our parameter estimates indicate that misperception

of failure rates is economically relevant: more than 80% of our retailer’s profit on warranties is due

to the overweighting of failure rates.

Our panel data comes from a large US consumer electronics retailer. It documents about

45000 transactions made by almost 20000 households between 1998 and 2004. Almost 30% of the

transactions involve the purchase of an extended warranty.

Households’ warranty purchases exhibit several patterns. First, as mentioned above, demand for

warranties is very strong despite high prices and low failure rates. Second, controlling for household

characteristics, the likelihood of purchasing an extended warranty drops by 3 percentage points for

every warranty the household bought in the past. This implies a 10% or larger decrease in the

likelihood of buying warranties. Third, there is an even larger drop of 8 percentage points in the

likelihood of buying warranty for any warranty the household returned in the past. Returns data

also indicates that conditional on returning a warranty, about 33% of the returns do not involve

the return of the insured product. Fourth, the shopping environment affects warranty purchases:

controlling for household, product subcategory and brand fixed effects, the likelihood of buying a

warranty in-store is around 17 percentage points higher than online.

Taken together, we interpret these patterns as suggestive that buyers misperceive the value of

the warranty in the store, and learn about it over time and with experience. The returns data is

especially interesting in this context. It seems unlikely that consumers return warranties because

they are unsatisfied with the “warranty experience”. This is because they are unlikely to use the

warranty in a 30-day return window due to the very low failure rates of the insured products. It

is also unlikely that consumers find more attractive warranty offers because stand alone extended

warranty providers for electronics were not easy to find in the early 2000s. Such returns are

perhaps related to consumers learning after purchase but within a relatively short period of time

that insuring against product failure is less attractive than they initially thought.

Motivated by this reduced-form evidence, we postulate a choice model in which risk-averse

consumers who may misperceive the value of the warranty make purchase decisions. There are two

possible sources of value misperception in the context of warranties: the rate in which the insured

product fails and the cost of repair. These may be perceived as higher than they actually are, thus

increasing the attractiveness of purchasing a warranty. As we are unable to separately estimate

both sources, we focus on failure rate mispercpetion putting a natural upper bound on the cost of

return in the form of the price of the insured product.

We construct demand for warranties from choice behavior, and separately identify and estimate

the consumer’s degree of (standard) risk aversion and failure rate distortions. Our identification

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strategy relies on how a household’s maximum willingness-to-pay for a warranty varies across

products that have different repair costs but the same failure rates. We argue that a single-crossing

condition of the willingness-to-pay function is sufficient to separately identify standard risk aversion

and probability distortions in this context. We provide examples of utility functions that satisfy

the single-crossing property, including the one that we will use for estimation.

On the seller side, we assume that our retailer is a profit-maximizing monopolist. Monopoly

power is an immediate consequence of consumers incurring search costs to visit a store. Ellison

(2005) motivates this assumption in his add-on pricing model by arguing that while product prices

are advertised and easily accessible to consumers, those of warranties are not and buyers have to

incur a cost to actually figure them out. This indeed seems to have been the case in the extended

warranty market in the early 2000s. While TV prices were advertised, warranty prices were often

revealed to consumers after finalizing their TV purchase decision. We assume that the retailer’s

cost is a linear function of the expected repair cost and estimate its slope.

Our first main finding is that failure rate distortions play an important role in driving the

purchase behavior of households. There is a substantial overweighting of failure rates below 15%.

For example, a 5% failure rate is seen as if a product has a failure rate of 12% when an average

household is evaluating the warranty. Standard risk aversion, on the other hand, plays an insignif-

icant role in buyers’ decision making: the average willingness-to-pay of buyers with our estimated

risk aversion parameter is close to actuarially fair rates, consistent with behavior of a risk neutral

consumer. When we estimate the model under the assumption that consumers perceive failure

rates correctly, an extreme degree of risk aversion is required to estimate the data.

Our second main finding is about the significance of failure rate distortions in the seller’s profit

and consumer welfare. We perform counterfactual experiments to assess the impact of probability

distortions on extended warranty prices, price-cost margins, and welfare. We find that the ratio

of extended warranty price to the main product price declines from 17% to 16%, with price-cost

margins going down from 31% to 24% when we remove the bias. Removing the bias drastically

reduces the fraction insured from 39% to 7%. When the bias is removed, profits of the retailer

falls from $265 million to $44 million, or a decrease of about 83%. In terms of consumer welfare,

removing the bias leads to an increase of $217 million, roughly the same amount as the decrease in

profits. This change reflects a more than twofold improvement in consumer welfare.

The paper is organized as follows. The next section introduces the data and provides descrip-

tive analysis of extended warranty purchase. Section 3 presents the model and our identification

strategy. Section 4 discusses estimation presents the results. The last section contains the welfare

analysis.

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2 Data Characteristics

We use the INFORMS Society of Marketing Science (ISMS) Durables Dataset 1, which is a panel

data of household durable goods transactions from a major U.S. electronics retailer. The full sample

contains about 170,000 transactions made by almost 20,000 households across the retailer’s 1,176

outlets and its online store. Prices across outlets and the online store are essentially identical.

Transactions took place between December 1998 and November 2004.

There are four main types of transactions in the data. About 117,000 transactions involve the

purchase of a specific product other than an extended warranty. About 15,000 transactions involve

the purchase of an extended warranty. About 5000 transactions involve the return of a product

other than an extended warranty and about 1000 transactions involve the return of an extended

warranty. For each transaction, we observe a unique product ID (similar to an SKU), the price

of the product, the brand, and the category and subcategory of the product. A shopping trip is

defined as a collection of transactions made by a given household at a given store in a given date

and time. We observe unique household and store IDs. For each household and shopping trip, we

observe the buyer’s gender, the age and gender of the head of the household, income group5, and

whether there are children in the household.

There are two data issues that we have to deal with. First, the data only tells us the product

subcategory (e.g. 9-16 inch TVs) for which the warranty is for. We restrict our sample to shopping

trips in which there is a clear one-to-one mapping between the extended warranty and the corre-

sponding product. For example, we drop shopping trips involving a purchase of two 9-16 inch TVs

but only one extended warranty purchased for this subcategory. We lose about 2,000 observations

for this reason.

Second, if a household did not purchase an extended warranty for a given product, we do not

observe the warranty’s price. To identify the warranty price in such cases, we match the non-

warranty transaction with a corresponding warranty transaction for the same product ID from the

closest transaction date. After dropping transactions for which we cannot find a corresponding

warranty transaction, we end up with a sample of about 45,000 observations.6

2.1 Attachment rates, prices, and approximate profit margins

Table 1 shows the fraction of consumers who bought extended warranties (henceforth, the attach-

ment rate) and the extended warranty-to-product price ratio for each product category. Attachment

rates range from about 20% for items such as VCRs (VIDEO HDWR), music CDs and video games

(MUSIC), to as high as about 40% for items like car stereos and speakers (MOBILE). Warranties

are priced on average at about 24% of the price of the insured product, and the standard deviation

5Income group is a number from 1 to 9 where 9 is the highest income group. We do not have additional informationon income within each group.

6We also drop the less than 1000 observations, in which the price of the good is significantly less than the price ofthe warranty.

5

Table 1: Attachment rate and price ratio by product categoryAttachment rate EW-Product price ratio Obs

AUDIO 0.281 0.232 6450DVS 0.295 0.207 1439IMAGING 0.377 0.199 3001MAJORS 0.356 0.197 864MOBILE 0.398 0.249 5176MUSIC 0.208 0.169 1189P*S*T 0.245 0.237 3765PC HDWR 0.258 0.274 8773TELEVISION 0.311 0.217 6307VIDEO HDWR 0.206 0.240 5828WIRELESS 0.245 0.317 1485

0.287 0.239 44277

Table 2: EW information for TVs

Attach rate TV price EW-TV price ratio Fail rate Margin Obs

9-16in 0.149 122.99 0.284 0.072 0.729 42219-20in 0.176 173.97 0.240 0.065 0.710 106725in 0.270 245.33 0.220 0.069 0.643 52227in 0.299 354.06 0.197 0.059 0.681 1477>30in 0.348 812.53 0.219 0.076 0.619 1229

0.268 400.07 0.223 0.067 0.672 4717

Notes: Fail rates are from Consumer Reports. Margin = (EW price - fail rate × TV price)/EW price.

of the warranty-to-product price ratio is about 11% (see Figure 1 for the distribution of ratios).

There is no significant correlation at the product level between variations in the product price and

variations in the warranty price.7

Our structural analysis focuses on TVs due to availability of published failure rates from Con-

sumer Reports. Table 2 provides attachment rates, TV prices, extended warranty-to-product price

ratios, published failure rates, and approximate price-cost margins, broken down by TV subcat-

egory.8 Attachment rates range from about 15% to 35%, with larger attachment rates for more

expensive categories. The price ratio for TVs is about 22% on average with a standard deviation

of about 8% (see Figure 1 for the distribution of ratios.)

To provide a rough upper bound on the expected marginal cost of servicing a TV warranty,

we multiply the failure rates from Consumer Reports by the price of the product. This estimate

implies a lower bound on the price-cost margin of 62% to 73% for different subcategories, which is

close to what is cited in the popular press. Note that we expect the seller in our dataset to have

lower margins due to revenue sharing with warranty providers and commissions to sales people.

7We regress the log of the product price on the log of the warranty price for each product, and estimate an averagecoefficient equal to 0.046 with an average p-value of 0.26.

8Failure rates come from Consumer Reports which give the likelihood that a repair has to be made within 3 to 4years of using the product.

6

Table 3: Attachment rates by buyer and household characteristics

All TV

Characteristic Attach rate Obs Attach rate Obs

Female 0.305 13976 0.286 1534Male 0.280 26228 0.261 2768

Female (head of hh) 0.308 12412 0.285 1380Male (head of hh) 0.280 24760 0.260 2620

Below middle income (category < 5) 0.321 10404 0.310 1170Above middle income (category ≥ 5) 0.276 33900 0.253 3547Lowest income category (category = 1) 0.343 2656 0.340 300Highest income category (category = 9) 0.253 6452 0.233 660

Over 50 (head of hh) 0.293 23259 0.282 2717Under 50 (head of hh) 0.280 20882 0.250 1975

Has child in hh 0.282 13940 0.248 1279No child in hh 0.296 6234 0.323 779

2.2 Buyers’ characteristics

Tables 3 and 4 examine the relationship between attachment rates and buyers’ characteristics for

all product categories and for TV purchases. In Table 3, attachment rates are broken down by

buyer’s gender, gender and age of the head of the household, whether income is above or below

the median income category in the data, and whether there is a child in the household. Differences

in attachment rates along these dimensions are small, except for income and having a child. For

example, attachment rate for TVs among females is almost 29% in comparison to 26% among

males. In contrast, the attachment rate of households above the middle income category 25% in

comparison to 31% among those below the middle income category. Moreover, we see a decrease

of 11 percentage points (from 34% to 23%) in the attachment rate of the highest income level

households relative to the lowest one. Finally, although having a child seems to decrease the

likelihood of purchasing a warranty by 7 percentage points. For income, the size of the difference

in attachment rates goes down significantly once we introduce controls in the regression analysis,

while the difference for having a child goes away.

Table 4 presents the results of regressing an extended warranty purchase dummy on buyers’

and households’ characteristics and their interactions with gender. The regressions include brand

and subcategory fixed effects to account for average differences in purchasing behavior across these

dimensions. Consistent with most of the raw means in table 3, the only characteristic that is

statistically and economically significant is income when including all product categories. Moreover,

adjusted R2 are extremely low, despite including subcategory and brand fixed effects. All in all, the

two tables indicate that the above buyers’ and households’ characteristics (except perhaps income)

are not strong predictors of warranty purchases.

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Table 4: Regression of extended warranty purchase on buyer and household characteristics

Dependent variable: EW purchase dummy

All TVCoeff SE Coeff SE

Male -0.064 (0.039) -0.069 (0.101)Age (head) 0.001∗ (0.0004) 0.001 (0.001)Income -0.014∗∗∗ (0.003) -0.013∗ (0.007)Has child in hh < 10−5 (0.014) -0.017 (0.038)Male × Age 0.001 (0.001) 0.001 (0.001)Male × Income 0.002 (0.003) < 10−4 (0.009)Male × Child 0.004 (0.017) -0.006 (0.045)Subcategory FE Y YBrand FE Y Y

Adjusted R2 0.08 0.06No. obs (good-hh-trip) 19375 1973

Notes: Standard errors in parentheses are clustered at shopping trip level.

Significance level: ***1%, **5%, *10%

2.3 Warranty returns

Our data contains 1239 warranty return transactions. About 67% of these are returns that ac-

company the insured product return. These returns are made due to the add-on feature of the

warranty – it has no value if the insured product is returned. The more interesting returns are

the 33% warranty returns that are made without returning the main product.9 The reason for

this type of warranty returns seems to be different than the usual reasons for product returns. It

seems unlikely that consumers return the warranty because they are unsatisfied with the “warranty

experience”. This is because they are unlikely to use the warranty within a 30 day return window

due to the very low failure rates of the insured products. It is also unlikely that consumers found

more attractive warranty offers because stand alone extended warranty providers were not easy

to find in the early 2000s. Such returns are perhaps related to consumers learning ex-post that

insuring against product failure is less attractive than they initially thought.

2.4 Warranty purchases over time

The data tracks households over time, so we can examine how past purchases and returns of

warranties influence future purchases of warranties. Table 5 contains the results of regressing

warranty purchase for a given product on whether an extended warranty was purchased and whether

an extended warranty was returned in the past for any other product. We also include a dummy,

which is equal to 1 if the transaction was made in the store as opposed to online. Our preferred

specification includes household, subcategory, brand, month and year fixed effects, and uses the

number of past extended warranty purchases and returns as regressors.

9We run regressions similar Table 4 and find that none of the buyer and household characteristics in the dataset is agood predictor of this behavior.

8

Table 5: Regression of current EW purchase behavior on past EW purchases and returns

Dependent variable: Buy EW today?

I II III IV

Bought EW before? 0.257*** . -0.152*** .(0.009) (.) (0.024) (.)

Returned EW before? 0.041** . -0.197*** .(0.017) (.) (0.031) (.)

No. of EW bought before . 0.071*** . -0.033***(.) (0.004) (.) (0.007)

No. of EW returned before . -0.015 . -0.082***(.) (0.012) (.) (0.023)

In-store? 0.154*** 0.169*** 0.181*** 0.173***(0.016) (0.015) (0.039) (0.038)

Household FE N N Y YSubcat & brand FE Y Y Y YMonth & Yr FE Y Y Y YNo. obs 14878 14878 14878 14878(good-hh-trip)No. HHs 6321 6321 6321 6321

Notes: Standard errors in parentheses are clustered at shopping trip level. Dependent

variable refers to a given product while “Bought” and “Returned” dummy regressors

refer to buying and returning an EW for any product at some point in the past.

Model III and IV use the number of extended warranties bought and returned on any

other product before as regressors. Significance level: ***1%, **5%, *10%

9

The results provide evidence of learning.10 Experience with extended warranties in the past is

associated with a decrease in the likelihood of buying a warranty in the present by 15 percentage

points, which is more than half of the average attachment rate across products (28.7%). When

experience is measured in terms of the number of extended warranties bought in the past, buying

an additional extended warranty in the past is associated with a decrease of 3 percentage points in

the likelihood of buying an extended warranty today.

The effect of past returns on the likelihood of buying an extended warranty is even more

pronounced. Returning an extended warranty in the past is associated with a 20% decrease in the

likelihood of purchasing another warranty, and each returned warranty is associated with a decrease

of 8% percentage points in the likelihood of buying another warranty.

2.5 In-store versus online transactions

About 1% of the transactions in the data were made online. The attachment rate for these trans-

actions is about 4% relative to the in-store and overall attachment rates of about 29%. To examine

what drives this sevenfold difference and its robustness, we explore various regressions in Table 6.

The first model does not include any controls so it gives numbers that are very similar to the raw

attachment rates. The other models turn on various fixed effects. Subcategory and brand fixed

effects allow us to soak up any differences in mean purchasing behavior induced by the nature of

the product. We also include household, month and year fixed effects as further controls.

We see a drop of the effect of in-store purchases as we include more fixed effects. Including

just a household fixed effects reduces the effect by about 5 percentage points. The reduction in

the effect is much larger when including product-related fixed effects. Including all of the fixed

effects lead to a reduction in the effect from 25 percentage points to 17. That is, the likelihood of

purchasing an extended warranty jumps from 12% to 29% when being in the store.

As an additional robustness check, table 7 contains the results of regressing the extended war-

ranty purchase dummy on shopping mode but broken down by product category. The first two

columns come from a simple OLS regression without additional controls. The middle two columns

include the household characteristics in the data as controls. Finally the last two columns include

a household fixed effect. There is significant variation in the effect of in-store purchases across

the product categories but overall, the effect remains large. The effect survives even if we include

household characteristics. Although we lose statistical significance once we include household fixed

effects due to a small number of observations, the magnitudes are roughly the same across the

different regression models.

10When household fixed effects are not included, a past purchase of a warranty has a positive effect on the likelihoodof buying one today, contrary to learning. This reflects the classic problem of disentangling unobserved persistentheterogeneity and state dependence.

10

Table 6: Regression of extended warranty purchase on shopping mode

Dependent variable: EW purchase dummy

I II III IV V

In-store? 0.247*** 0.200*** 0.180*** 0.175*** 0.166***(0.022) (0.027) (0.027) (0.027) (0.027)

Household FE N Y Y Y YSubcategory FE N N Y Y YBrand FE N N N Y YMonth FE N N N N YYear FE N N N N YNo. obs 44304 44304 44304 44304 44304(good-hh-trip)No. HHs 17158 17158 17158 17158 17158

Notes: Standard errors in parentheses are clustered at shopping trip level.

Significance level: ***1%, **5%, *10%

Table 7: Regression of extended warranty purchase on shopping mode broken down by productcategory

Dependent variable: EW purchase dummy

OLS se Obs OLS with char se Obs FE (HH) se

Television 0.31∗∗∗ (0.08) 6307 0.32∗∗ (0.14) 2360 0.22 (0.20)Audio 0.16∗∗∗ (0.05) 6450 0.12∗ (0.07) 2517 0.07 (0.10)Mobile 0.40∗∗ (0.19) 5176 0.37 (0.28) 1883 . (.)P*S*T 0.23∗∗∗ (0.06) 3765 0.23∗∗∗ (0.08) 1519 0.19∗ (0.11)Imaging 0.36∗∗∗ (0.07) 3001 0.39∗∗∗ (0.12) 1197 . (.)PC Hardware 0.18∗∗∗ (0.05) 8773 0.09 (0.08) 3471 0.03 (0.11)Music 0.16∗ (0.09) 1189 0.20 (0.15) 469 0.30 (0.24)Video 0.21∗∗∗ (0.04) 5828 0.22∗∗∗ (0.07) 2151 0.17∗ (0.09)DVS 0.05 (0.23) 1439 0.34 (0.46) 586 0.50∗ (0.28)Wireless 0.25 (0.18) 1485 0.27 (0.31) 483 0.57∗∗ (0.25)

Notes:

Significance level: ***1%, **5%, *10%

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3 Model and Identification

We postulate an add-on pricing model a-la Ellison (2005) and Ellison and Ellison (2009). There

are several sellers of a main product M and an extended warranty EW for the product M . Each

seller sets a price p for the main product that is observable to buyers, and a price t for the extended

warranty that is not observable to buyers.

The assumption that the product price is observable and the warranty price is not seems to be

case in practice. For example, BestBuy advertises products’ prices whereas it trains its sales people

to offer warranties and other add-ons to buyers only after they finalize their decision to purchase

the product.11

Buyers decide which seller to visit after observing the main product prices and forming rational

expectations about warranty prices. Consumers’ utility is additive in the warranty component, and

they wish to buy at most one unit of the main product and of the warranty.

The assumption that buyers form rational expectations about warranty prices is not necessary

for our empirical analysis. One could alternatively assume that buyers do not anticipate the pur-

chase of the warranty before visiting the seller, and decide which seller to visit based on the prices

of the main product. In this alternative specification, buyers form rational expectations about

warranty prices of other sellers after visiting the first seller and being offered the warranty.

Buyers visit the seller of their choice at a cost of s and learn the price of the warranty. The

cost s corresponds to the hassle or time involved in visiting a store and going through the purchase

process. Buyers then decide whether to buy the main product, the main product plus the warranty,

or visit another store at a cost of s, where they will face the same decision.

Relevant equilibrium properties. There are two properties of any pure strategy sequential

equilibrium of the above game that we will use in our empirical analysis. The first is that the

price of the warranty set by any seller is the monopoly price relative to the residual demand for

warranties among buyers of the main product. Otherwise, as in Diamond (1971), the seller can

raise the price of the warranty by some ε < s and buyers will not switch to another seller. We use

the first order condition of this monopoly pricing program to estimate the seller’s cost of providing

warranties.12

The second property is that buyers visit only one seller and always buy the main product in

equilibrium. This is because buyers incur a cost of visiting a seller. Thus, if they anticipate they

will not buy the main product, they will not visit the store. We therefore focus below on buyers’

decision to buy the warranty conditional on already purchasing the main product.

Warranty purchase decisions. Let W denote the buyer’s wealth after buying the main

product, t the price of the warranty, and u(·, r) the buyer’s concave utility over wealth levels that

is parameterized by r, the buyer’s degree of risk aversion around W .

11See footnote 2.12However, we do not need nor use the first order condition to estimate the demand side parameters.

12

A buyer’s utility if he purchases the warranty is VEW = u(W − t; r).13 A buyer’s utility if he

does not purchase the warranty is VNW = ω(φ)E(u(W − X; r)) + (1 − ω(φ))u(W ; r) where ω(φ)

is the buyer’s perception of the objective failure probability φ, which increases in φ, and X is the

random cost of repair. Clearly, X is weakly smaller than the price of the main product p because

the buyer can always buy a new product instead of fixing the existing one. Thus, the buyer’s utility

if he does not purchase the warranty is bounded below by ω(φ)u(W −p; r)+(1−ω(φ))u(W ; r). We

will identify VNW with this lower bound in our estimation, i.e., we will have VNW = ω(φ)u(W −p; r) + (1− ω(φ))u(W ; r).14

The non-standard component in the buyer’s utility is the probability distortion function ω(·)that reflects how the buyer assesses objective failure probabilities and how he uses them in mak-

ing decisions. There are at least two reasons for probability distortions. First, estimating failure

probabilities is not straightforward. This is because buyers usually have limited personal experi-

ence about failures of durable goods, and credible information about failure rates is not readily

available. The common view is that this leads to over-estimation of failure probabilities.15 Second,

even if individuals estimate failure probabilities correctly, Prospect Theory proposes that individ-

uals incorporate these probabilities in decision making by using decision weights. In particular,

individuals tend to put too much weight on low probability events, such as the failure probability

of a durable good.

Demand for warranties. To construct the demand for warranties, we add individual choice

shocks, εEW and εNW , to VEW and VNW . Assuming these shocks are iid Type I Extreme Value

with scale parameter σ and normalizing the buyer population to 1, we can derive the demand for

warranties:

D(t; r, ω(φ), p, σ) = Pr(εNW − εEW ≤ Ω(t; r, ω(φ), p, σ))

=exp Ω(t; r, ω(φ), p, σ)

1 + exp Ω(t; r, ω(φ), p, σ))

where

Ω(t; r, ω(φ), p, σ) ≡ VEW − VNWσ

.16 (1)

Identification of risk aversion and probability distortion. We focus on the identification

of risk aversion and probability distortions from maximum willingness to pay (WTP).17

13This assumes that there is no deductible associated with using the warranty.14This implies that we likely underestimate ω and r because using a higher repair cost makes the purchase of the

warranty more attractive even without appealing to risk aversion and probability distortion. We discuss therobustness of our estimates to this assumption in Section 5.

15For example, the New York Times of August 28, 2014 writes: “The company selling the warranty has theinformation on failure rates. You don’t....That’s not easy to find out. Companies aren’t in the habit of telling youthat their products fail 4 percent or 12 percent of the time. Failure rates are usually low. Warranty companies knowthat. And they know, too, that consumers tend to think the failure rate is higher.”

16The utility specification we will use in estimation imposes a specific normalization so we can identify the scaleparameter σ. This scale parameter is the inverse of the marginal utility of income.

17WTP can be uniquely obtained from choice probabilities.

13

Fix a product M with price pM and failure rate φ, and let ω = ω(φ). The maximum willingness-

to-pay WTP (pM , r, ω) of buyers with risk aversion r and the distorted probability ω for a warranty

to product M is the price t that solves VEW (t, r) = VNW (pM , r, ω). The identification problem is

that the same WTP can be explained by a continuum of pairs (r, ω(r)) where r is the degree of

risk aversion and ω(r) is the distorted probability as a function of r. This is because an increase

in r can be undone by a decrease in ω. We exploit variation across products in order to uniquely

identify the pair (r, ω).

We use variation in prices for two products with the same failure rate φ for unique identification.

Because failure rates are the same, the same pair (r, ω) should explain the different WTP for

warranties to these two products. The pair (r, ω) can be identified uniquely if the two iso-WTP

“curves”, i.e., the two continuums of pairs (r, ω(r)) that explain the different WTPs, cross each

other exactly once.

Figure 2 provides graphical intuition. It sketches in solid black the iso-WTP curve for the

product M with price pM . Without additional variation, we cannot uniquely identify risk aversion

and probability distortion. If, however, we also have data on the WTP for the product M ′ with

the same failure probability but a different price and the corresponding iso-WTP curve in dashed

red intersects the curve for product M exactly once, then the pair (r, ω) can be uniquely identified.

Proposition 1 presents a condition on a family of instantaneous utility functions that guarantees

such “single-crossing” and hence unique identification.

Proposition 1 Let u(·, r)r be a family of utility functions parametrized by the degree of risk

aversion r such that larger r is associated with more aversion to risk. The pair (r, ω) is uniquely

identified if the marginal utility of wealth ux(x; r) decreases in r.

Proof. Following the discussion in the main text, it suffices to prove that as we change the

price pM of the main product, the slope of the iso-WTP curves dωdr changes monotonically. We will

do so by showing that

−∂WTP (pM ,r,ω)

∂r∂WTP (pM ,r,ω)

∂ω

=dω

dr

is strictly monotone in pM .

Let Ω = u (W − t; r) − ωu (W − pM ; r) − (1− ω)u(W ; r). The WTP is defined as the price t

for which Ω = 0. Thus, by the implicit function theorem,

∂WTP

∂r=

∂Ω/∂r

∂Ω/∂t=ur (W − t; r)− ωur (W − pM ; r)− (1− ω)ur (W ; r)

u′ (W − t; r), and

∂WTP

∂ω=

u (W ; r)− u (W − pM ; r)

u′ (W − t; r)> 0.

The numerator in the first expression ∂Ω/∂r is positive because larger r implies more risk

14

aversion and hence a lower certainty equivalent for the lottery (ω,−pM ). Thus, Ω goes up when r

goes up.Observe that

∂

∂pM

[∂WTP/∂r

∂WTP/∂ω

]=

∂

∂pM

[ur (W − t; r)− ωur (W − pM ; r)− (1− ω)ur (W ; r)

u (W ; r)− u (W − pM ; r)

]

=1

(u (W ; r)− U (W − pM ; r))2

ωu′

r (W − pM ; r) [u (W ; r)− u (W − pM ; r)]

− [ur (W − t; r)− ωur (W − pM ; r)− (1− ω)ur (W ; r)] U′ (W − pM ; r)

Let us examine the second term first. As indicated above the term in parenthesis is positive. The

derivative of u with respect to W is positive, and hence the entire expression with the minus before

is negative. Monotonicity is thus guaranteed if the cross derivative of u with respect to (W, r) is

negative.

The family of CARA utility functions −e−rxr satisfies the condition of the proposition for

sufficiently large wealth levels. This is because the cross derivative with respect to x and r, e−rx−rxe−rx, is negative for x > 1

r . It is also straightforward to verify that the utility specification we

use in estimation satisfies the condition of the proposition.

4 Estimation

We begin by describing how we estimate risk aversion and the probability weighting function when

buyers are homogeneous. We then discuss estimation when risk aversion and probability weighting

can depend on consumer characteristics. Finally, we describe how we estimate the seller’s cost.

Following Cohen and Einav (2007) and Barseghyan et. al. (2012), we use a second order Taylor

approximation of buyers’ utility function u(·) in estimating the model. The main benefit of using

this specification is that it does not require data on wealth.

The second order Taylor approximation of u(·) around W for some wealth deviation ∆ is given

by

u(W + ∆) ≈ u(W ) + u′(W )∆ +u′′(W )

2∆2.

Dividing by u′(W ) and letting r = −u′′(W )/u′(W ) denote the Arrow-Pratt coefficient of abso-

lute risk aversion,18 we obtain that

u(W + ∆)

u′(W )≈ u(W )

u′(W )+ ∆− r

2∆2.

Using this specification to evaluate the utility difference Ωj between purchasing and not purchasing

an extended warranty for product j (equation 1), we obtain that:

Ωj =ωjpj − tj + r

2(ωjp2j − t2)

σ.

18Strictly speaking, the Arrow-Pratt coefficient of absolute risk aversion can vary with income if the utilityspecification is not CARA.

15

Let Dj be the observed attachment rate for product j. Our choice model implies that19

logDj

1−Dj= Ωj =

ωjpj − tj + r2(ωjp

2j − t2)

σ. (2)

The decision weight ωj acts like a (non-additive) product effect. We decompose this effect to

ωj = ω(φj) + ξk(j) + ηj (3)

where ω(·) is some unknown function of φ, ξk(j) is a subcategory-level effect, and ηj is a random

shock. The parameters ξk(j) allow decision weights to vary between subcategories, thus capturing

the possibility that consumers may apply different decision weights for TVs of different sizes,

different projection technology, etc.

Using equation 2, we can express the ωj ’s as a function of the unknown parameters (r, σ) and

the data:

ωj =σΩj + tj + r

2 t2j

pj + r2p

2j

. (4)

We construct moment conditions involving ωj to estimate r and σ.20 Once we have these parame-

ters, we calculate ωj using equation 4. Our assumption regarding the error structure in equation 3

implies the following moment condition

E[ωj − ωj′ |φj = φj′ , k(j) = k(j′), pj , pj′ , tj , tj′ ] = 0

since ωj −ωj′ = ηj − ηj′ for j, j′ such that φj = φj′ and k(j) = k(j′). Thus, as long as failure rates

for any two products j and j′ belonging in the same subcategory are the same, decision weights

associated with extended warranties for these products will be equal, on average.

4.1 Estimation with heterogeneous buyers

We extend the previous model to allow for risk aversion, the probability weighting function, and the

scale parameter to be functions of consumer observables. Denote by zi the observed characteristics

of household i which includes gender, income, age and whether there is a child in the household.

We parameterize risk aversion by

ri = exp(γrzi + ξrk(i)

), (5)

19A complication in linking Ωj to product-level attachment rates arises because p and t vary at the product level. Ifwe had infinite data, we could compute Dj|p,t = Pr(di = 1|j(i) = j, p, t) and then invert this equation to get Ωj foreach pair (p, t). However, this is not our case. The best we can do is to calculate the attachment rate for a productby aggregating over all price pairs (p, t) for this product. As for prices, we use the largest p and the smallest t in theestimation because these make the purchase of the warranty most attractive even without appealing to risk aversionand probability distortions.

20Because of the large variation in prices across categories, we estimate separate scale parameters for eachsubcategory, and we allow for heteroskedasticity. Specifically we let σk(p) = σk

√p for subcategory k.

16

so risk aversion is a function of consumer observables zi and the product subcategory ξrk(i). For

the decision weight ωj , we assume that it is the sum of a one-parameter Prelec (1998) weighting

function and a mean zero error term η, allowing the parameter α to vary by gender G ∈ M,F:

ωjG = exp[−(− log(φj))αG ] + ηjG. (6)

Finally, we allow the scale of the utility function to vary with income Ii:

σi = σ√Ii. (7)

The model provides an expression for predicted attachment rates among gender G households

for product j:

DjG =1

NjG

∑i:j(i)=j,genderi=G

expωjGpj−tj+

ri2(ωjGp

2j−t2)

σi

1 + expωjGpj−tj+

ri2(ωjGp

2j−t2)

σi

(8)

Let qjG be the corresponding attachment rate from the data. Define ωjG as the decision weight

that equates qjG with DjG for a given set of parameters, and let

ηjG = ωjG − exp[−(− log(φj))αG ]. (9)

We assume the error term ηjG is orthogonal to published failure rates, product subcategory, product

price, price of the extended warranty, and consumer characteristics and so we can use the moment

condition

E(ηjG|φ, k, p, t, z

)= 0 (10)

to estimate the parameters of the model.21

4.2 Seller’s cost

We also estimate is the marginal cost of the seller, c(p, φ). We assume that c(p, φ) = µφp where

µ absorbs factors such as revenue sharing with the warranty provider, sales commission, etc. We

estimate this parameter from the first order condition of the seller’s profit maximization problem:

p− c(φ)

p=

1

|E(p; r, ω(φ), σ)|(11)

where E(t; r, ω(φ), σ) is the price elasticity of demand for EWs.

21We estimate the model using a nested fixed point algorithm with inner loop tolerance of 10−8.

17

5 Results

5.1 Probability weighting and risk aversion

Figure 3 plots our estimate of the probability weighting function ω(·). We include a scatter plot

of the estimated product effects ωj ’s and a local linear fit. We also include a fit based on the

one-parameter Prelec (1998) function,

ω(φ) = exp[−(− log(φ))α] (12)

which is close but slightly less concave than the local linear fit. We estimate α = 0.685 with

bootstrapped 95% and 90% confidence intervals of (0.659, 1.023) and (0.668, 0.980), respectively.

Our estimates are in line with Prospect Theory. First, there is substantial overweighting of

small probabilities. For example, a product with a 5% failure rate is perceived as a product with

a 12% failure rate. Second, the degree of overweighting declines as failure rates increase.

We estimate the risk aversion parameter r to be practically zero (≈ 10−6 with a 95% confidence

interval that has width of less than 10−6).22 We also estimate a risk aversion parameter under what

we refer to as the standard model in which ω(φ) = φ. We estimate this parameter to be equal to

0.036 with 95% confidence interval of (0.026, 0.046) .

To interpret our estimates, Table 8 presents the average willingness-to-pay (WTP)23 for an

extended warranty for a product worth $100 under various failure rates. Columns 2 and 3 present

the WTP using the estimated risk aversion parameter from the full model (using the Prelec weight-

ing function). In column 2, we compute the WTP with our estimated weighting function, and in

column 3, we impose ω(φ) = φ. Column 4 uses the estimated risk aversion parameter from the

standard model in which ω(φ) = φ.

Columns 2, 3, and 4 illustrate that in the full model, the contribution of probability distortions

is much more significant than that of standard risk aversion. As column 3 shows, willingness-to-

pay when there are no probability distortions is equal to the actuarially fair rate. In contrast, the

risk aversion parameter in Column 4 that comes from the standard model (i.e. without probability

distortions) requires a very high degree of aversion to risk in order to fit the data well. For example,

a consumer with this risk aversion parameter will only accept a 50-50 gamble with a loss of $10 if

the gain is at least $17.

When we allow different estimates of the Prelec parameter by gender, we find that males tend

to overweight probabilities slightly more than females. The estimated α for males is 0.608 with

95% confidence interval of (0.441, 0.775) while for females, α = 0.685 with 95% confidence interval

of (0.443, 0.927). The difference in coefficient estimates for males and females are not statistically

22Although we also estimated risk aversion allowing for differences in gender, age, income and having a child, we do notfind any statistically significant differences in these dimensions given that the average estimate for r is close to zero.

23The average WTPs are computed by setting the choice shocks, εEW and εNW , and the shock that enters theweighting function, ηj , to zero.

18

Table 8: Average willingness-to-pay for EW on a good with value $100

Failure rate Model est Model est Standard estω ω(φ) = φ φ

0.01 6.686 1.000 2.6730.02 8.783 2.000 5.1300.03 10.384 3.000 7.4150.04 11.742 4.000 9.5610.05 12.950 5.000 11.5900.06 14.056 6.000 13.5190.07 15.086 7.000 15.3620.08 16.058 8.000 17.1290.09 16.984 9.000 18.8300.10 17.872 10.000 20.470

significant. Nevertheless, the difference in coefficient estimates is economically significant: females

view a 5% failure rate as 12% while males view it as 14%. Figure 4 plots the weighting function

for females and males.

Finally, we investigate the effect of experience on our structural parameter estimates. Construct-

ing a measure of experience turns out to be tricky, especially if one does not control for unobserved

consumer characteristics as in the reduced form regressions in Table 5. As an approximate measue

of experience, we take a household’s number of trips in the data and divide it by the difference

between the very last transaction date (for all households) and this household’s first transaction

date. A higher value for this measure reflects higher experience. Intuitively, holding a household’s

first transaction date constant, the more trips in the store, the more familiar the consumer is with

sales practices, etc. Similarly, holding the same number of trips constant, a shorter horizon upon

which these trips were made makes exposure to sales practices more salient. While this measure of

experience is not perfect, we do find strong reduced form support for its relevance.

We estimate our full structural model separately for the set of households who are above and

below the experience measure. The estimated α for the high experience group is 0.703 with 95%

confidence interval of (0.605, 0.802) while for the low experience group, α = 0.637 with 95%

confidence interval of (0.550, 0.723). The difference in Prelec weighting parameter estimates for

these two groups are not statistically, significant, at least at the 5% level. Nevertheless, in terms of

economic significance, higher experienced households weight a 5% failure rate as 11% versus lower

experienced households as 13%. Figure 5 graphs the two weighting functions.

5.2 Retailer’s cost

Expected marginal cost for an extended warranty for product j is µφjpj . We estimate µ = 1.477

with 95% confidence interval of (0.163, 2.397). This implies a back-of-the-envelope seller’s profit

margin of 45%, because an extended warranty is priced at about 22% of the price of the good, and

the marginal cost of selling and servicing the warranty is the product of µ = 1.477, the average

19

failure rate of 0.067 and the price of the good. Estimates from the popular press indicate that

BestBuy transfers about 40% of the price of the warranty to the company that handles service,

suggesting that BestBuy’s cost of selling the warranty, mostly in the form of sales commission, is

about 15%.

5.3 Robustness against different expected loss of the consumer

Our benchmark model uses the upper bound loss p to compute the value of not buying the extended

warranty. In Table 9, we present the estimates of probability weighting and risk aversion when

we allow the expected loss to vary as a fraction of p. Clearly, by using the upper bound, we

underestimate the extent of the bias since the estimate of the Prelec parameter α decreases as we

decrease the loss. The estimate of standard risk aversion only becomes non-negligible once the

explanatory power of α is exhausted.

Table 9: Parameter estimates with varying expected cost

% of product price r Prelec α ω(0.05)

100% ≈ 10−6 0.652 0.12990% ≈ 10−6 0.597 0.14680% ≈ 10−6 0.532 0.16670% ≈ 10−6 0.455 0.19260% ≈ 10−6 0.367 0.22450% ≈ 10−6 0.268 0.26140% ≈ 10−6 0.178 0.29730% ≈ 10−6 0.119 0.32020% ≈ 10−6 0.079 0.33610% ≈ 10−6 0.049 0.348

5.4 Discussion

Our estimation indicates that there is a substantial distortion of probabilities in the extended

warranties market. The reduced form evidence about returns, learning and online versus offline all

indicate that the bias is not a preference parameter but rather misperception that is corrected with

time. This is the first message of the paper: probability distortions rather than risk aversion explain

consumer behavior in the extended warranties market and the distortion is a mistake in decision

making rather than a preference parameter. This observation motivates our welfare analysis in the

next section in the sense that removing the bias enhances consumer surplus.

6 Counterfactuals

We are interested in quantifying the profit and consumer surplus implications of probability distor-

tions in the extended warranty market. We thus focus on a counterfactual exercise in which we fix

the strategic environment, and study how optimal prices and quantities change when consumers

20

do not exhibit the bias. Since we find evidence that the bias is triggered in part by the store

environment or can be alleviated with learning and experience, this exercise gives us quantitative

insight on the effectiveness of consumer protection policies, and informational campaigns.

In order to make progress on analyzing the effects of the bias on profits and consumer surplus,

we assume that the price of the main product will not change when the bias is removed and

consumers no longer are willing to pay as much as before for extended warranties. We have two

supportive evidence for this assumption. First is institutional. The market for TVs and most

consumer electronics have two interesting features: there is little differentiation across retailers

selling the same TV, and manufacturers have tight controls on pricing and marketing practices of

retailers. Even if retailers would like to decrease the price of TVs to attract extended warranty

sales, say, below the wholesale price, the manufacturers have policies that indirectly discourage

such practice24. Second, we compared TV prices from BestBuy and Target in 2003 and found that

prices are the same on average across these two retailers. BestBuy offers extended warranties in

2003 but Target does not (they only started in October 2006). To the extent that the two retailers

face roughly the same TV wholesale price, if selling extended warranties affect the main product

price, we would expect TV prices from BestBuy to be significantly lower. Therefore we do find

empirical support for our assumption.

We compare two settings. In the first, we use the estimated weighting function and estimated

risk aversion, and in the second we turn off the bias, i.e. set ω(φ) = φ but keep the same risk

aversion parameter. In each setting, we construct demand based on these estimates, on the failure

rate φ, and on the product price p, then derive optimal prices. This gives us two distributions

of optimal prices: one for biased consumers and another for unbiased consumers. In calculating

optimal prices, we use the risk aversion parameter estimated from the full model and also the

one-parameter Prelec (1998) function (i.e. equation 12 with estimated α = 0.685) as our weighting

function whenever relevant (biased consumers). These numbers depend on the failure rates φ and

product price p. We compute these numbers for each value of the failure rate observed in the data.

Finally, for the product price, we take the (conditional) mean price for each observed failure rate.

In what follows, distributions and averages are sales-weighted25.

6.1 Prices and profit margins

Figure 6 plots the density and cdf of the extended warranty price-to-product ratio with and without

the bias. Our model (with bias) predicts an average ratio of 15.06%, while the average ratio in the

data26 is 15.09%. Removing the bias shifts the distribution to the left and decreases the average

24An example of this is the Minimum Advertised Price (MAP) policy, whereby retailer cannot advertise belowmanufacturer suggested retail prices, and the Unilateral Manufacturer’s Retail Price (UMPRP) policy whichimposes a fine when retailers sets a price below the one set under the UMRP.

25Sales-weighting is at the product ID level and not the (product ID, t, p) level.26This number differs from the one reported in Table 2 since we used the minimum extended warranty price and

maximum product price observed for each product in estimation. See footnote 19. Our assumption makes pricingmore conservative.

21

ratio to 13.17%.

Figure 7 plots the ratios as a function of failure rate. Ratios generally increase as failure rate

increases because marginal cost is increasing in failure rates. The ratio with the bias ranges from

about 12.52% to 24.21% and without the bias, from 9.85% to 23.36%. The gap between the ratios

with and without the bias tends to decrease as the failure rate increases due to the concavity of

the weighting function in the range of observed failure rates.

Figure 8 presents the effect of the bias on price-cost margins. With the bias, average price-cost

margin is 37.37%. The average price-cost margin in the data (i.e. computed using our estimate

of µ but with observed prices) is 37.07%. Similar to the extended warranty-to-product price ratio,

removing the bias shifts the distribution to the left and decreases the mean price-cost margin to

28.95%.

The left panel of figure 9 plots price-cost margins with and without the bias as a function of

failure rates. The right panel shows the correspoonding percent reduction in price-cost margins

from removing the bias. With the bias, price-cost margins range from 15.84% to 56.05%, while

without the bias, the range is from 12.80% to 47.83%. The percent reduction in price-cost margins

peaks at about a failure rate of 5%. At this failure rate, removing the bias reduces price-cost

margins by about 45%.

6.2 Quantity

The effect of the bias on the fraction of insured individuals is even more profound. Figure 10 plots

the density and cdf of the fraction insured with and without the bias. With the bias, the average

fraction insured is 37.84%, while the average fraction insured in the data is 30.20%.27 Without

the bias, the average fraction insured decreases to 11.21%. This reflects an 70% reduction in the

average fraction of consumers who buy the extended warranty.

6.3 Welfare

When consumers overweight failure probabilities, demand for extended warranties goes up. Our

reduced-form results provide evidence that this “excess” demand for extended warranties are driven

by mistakes rather than an innate bias on decision-making (which will be embedded in a consumer’s

preferences). Hence, the increase in consumers’ willingness-to-pay for the warranty due to the bias

does not reflect a true increase in consumer surplus. From the point of view of welfare, the first

best level of insurance is characterized by the intersection of the demand curve without the bias,

with the retailer’s marginal cost t = µφA. To get a realistic dollar equivalent measure for consumer

27Attachments rates here differ from the ones in Table 2 due to the assumption made in footnote 19. Attachmentrates in Table 2 are at the (product ID, t, p). However, attachment rates used in estimation (and counterfactual)are only sales-weighted at the product ID level. Specifically, we only take a simple average when aggregatingtransactions with different (t, p) at the product ID level, keeping the number of product ID level sales.Sales-weighting is then based on the aggregate number of sales for the product ID.

22

surplus, profits and total welfare, we assume that there are 30 million potential buyers of TV

extended warranties which is in the order of magnitude quoted in the popular press.

Figure 11 compares the fraction of insured individuals with and without the bias to the first-

best fraction insured. There is overinsurance28with the bias relative to the first-best. On the other

hand, there is underinsurance without the bias relative to the first best, which is a consequence of

monopoly pricing.

Consumer surplus increases when the bias is removed. There are two channels for this increase.

First, holding the extended warranty price constant, removing the bias shifts the demand curve

to the left and reduces the fraction insured. Consumers who now forgo buying the warranty are

exactly those who pay more than their unbiased willingness-to-pay, hence increasing consumer

surplus. We refer to this as the ripoff effect. Second, since extended warranty prices go down

without the bias, additional consumers would now like to buy the warranty, increasing the fraction

insured and consumer surplus. We refer to this as the price effect. Figure 12 illustrates these two

effects.

Figure 13 plots consumer surplus as a function of failure rate for the first best, and with and

without the bias. In the first best, consumer surplus ranges from $43 million to $166 million.

Without the bias, consumer surplus ranges from $16 million to $60 million and with the bias, it

ranges from -$116 million to -$34 million. The average consumer surplus with the bias is -$176

million, while without the bias, average consumer surplus is $41. The ripoff effect accounts for 92%

of the gap between consumer surplus with and without the bias.

Figure 14 plots profits as a function of failure rates. Profits are zero in the first best while

profits range from $16 million to $64 million when there is no bias. When there is bias, profits

range from $135 million to $1.06 billion. Average profits fall from $265 million with the bias, to

$44 million without the bias, a decrease of about 83%. Most of the profits when consumers are

biased comes from the surplus extracted from consumers who would not have bought the warranty

otherwise, i.e. the ripoff effect.

We now turn to the effect of removing the bias on total welfare. The effect depends on whether

the quantity insured with the bias, qbias, is below or above the first best quantity, qFB. Welfare

unambiguously decreases when we remove the bias if qFB ≥ qbias since the bias actually brings

us closer to the first best quantity from below. On the other hand, if qFB < qbias, the effect

of removing the bias is ambiguous. In this case, one needs to compare the deadweight-loss from

overinsurance with the deadweight-loss from underinsurance. Figure 15 illustrates the comparison

of deadweight-losses for qFB < qbias.

Figure 16 plots total welfare as a function of failure rates. Depending on the failure rate, welfare

is sometimes higher with the bias than without. However, for failure rates where welfare is higher

28Although we view extended warranties as insurance products, in reality, these contracts do not merely involvefinancial transfers from the insurance company (or the retailer) to consumers. The firm has to physically prepare toservice a potential claim by maintaining a service center, whose size depends on the number of warranties sold,holding inventories of parts, employing customer service agents, etc.

23

without the bias, the difference is much larger. Nevertheless, average welfare do decrease by $3

million when we remove the bias.

To summarize our welfare analysis, although we do find that policies that eliminatying the bias

may slightly reduce welfare due to the large decrease in profits, there is overwhelming reason to

adopt such policies as the impact of the bias on consumer welfare is sunstantial.

24

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25

Figures

Figure 1: Histogram of the Ratio of ExtendedWarranty Price and Product Price

Figure 2: Identification: Single-crossing ofwillingness-to-pay

Note: A1 > A2

26

Figure 3: Estimated weighting function

Figure 4: Estimated weighting function: Fe-males vs Males

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Figure 5: Estimated weighting function: Highvs Low Experience

Figure 6: Densities and cdfs of the ratio of EW and TV price

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Figure 7: Ratio of extended warranty and TV price

Figure 8: Densities and cdfs of price-cost margins

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Figure 9: TV extended warranty price-cost margins

Figure 10: Counterfactual: Densities and cdfs of fraction in-sured

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Figure 11: Fraction insured

Figure 12: Two effects of removing the bias on ConsumerSurplus

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Figure 13: Consumer Surplus

Figure 14: Profits

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Figure 15: Comparing deadweight-loss when qFB < qbias

Figure 16: Total Welfare

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