On the Relevance of Probability Distortions in the Extended Warranties Market

Transcription

1 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 indicate 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 of Managerial Economics and Decision Sciences, y-salant@kellogg.northwestern.edu. Thanks:... 1

2 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 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 addons 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). 2 For BestBuy s presentation of selling skills, see VCorationFramework/resource?argumentRef=static&resourceRef=/files/Best Buy Vendor Selling Skills.pdf 3 See Ellison (2005) for an add-on pricing model in which firms benefit from monopoly power on the sale of the add-on due to consumers search cost. 4 Jindal (2014) studies extended warranties for washing machines using stated choices from a survey. He finds an important 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 a diminished role. For example, in the Prelec (1998) specification, the weighting function gets flatter and flatter as it crosses the 45 degree line at 1/e 0.37%. 2

3 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 transactions made by almost households between 1998 and 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 3

4 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 insignificant 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 descriptive 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. 4

5 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 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 group 5, 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 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 corresponding 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 nonwarranty 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 Attachment rates, prices, and approximate profit margins Table 1 shows the fraction of consumers who bought extended warranties (henceforth, the attachment 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 5 Income group is a number from 1 to 9 where 9 is the highest income group. We do not have additional information on income within each group. 6 We also drop the less than 1000 observations, in which the price of the good is significantly less than the price of the warranty. 5

6 Table 1: Attachment rate and price ratio by product category Attachment rate EW-Product price ratio Obs AUDIO DVS IMAGING MAJORS MOBILE MUSIC P*S*T PC HDWR TELEVISION VIDEO HDWR WIRELESS Table 2: EW information for TVs Attach rate TV price EW-TV price ratio Fail rate Margin Obs 9-16in in in in >30in 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 Consumer 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 subcategory. 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. 7 We regress the log of the product price on the log of the warranty price for each product, and estimate an average coefficient equal to with an average p-value of Failure rates come from Consumer Reports which give the likelihood that a repair has to be made within 3 to 4 years of using the product. 6

7 Table 3: Attachment rates by buyer and household characteristics All TV Characteristic Attach rate Obs Attach rate Obs Female Male Female (head of hh) Male (head of hh) Below middle income (category < 5) Above middle income (category 5) Lowest income category (category = 1) Highest income category (category = 9) Over 50 (head of hh) Under 50 (head of hh) Has child in hh No child in hh 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 R 2 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. 7

8 Table 4: Regression of extended warranty purchase on buyer and household characteristics Dependent variable: EW purchase dummy All TV Coeff SE Coeff SE Male (0.039) (0.101) Age (head) (0.0004) (0.001) Income (0.003) (0.007) Has child in hh < 10 5 (0.014) (0.038) Male Age (0.001) (0.001) Male Income (0.003) < 10 4 (0.009) Male Child (0.017) (0.045) Subcategory FE Y Y Brand FE Y Y Adjusted R No. obs (good-hh-trip) 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 accompany 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. 9 We run regressions similar Table 4 and find that none of the buyer and household characteristics in the dataset is a good predictor of this behavior. 8

9 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*** ***. (0.009) (.) (0.024) (.) Returned EW before? 0.041** ***. (0.017) (.) (0.031) (.) No. of EW bought before *** *** (.) (0.004) (.) (0.007) No. of EW returned before *** (.) (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 Y Subcat & brand FE Y Y Y Y Month & Yr FE Y Y Y Y No. obs (good-hh-trip) No. HHs 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

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

11 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 Y Subcategory FE N N Y Y Y Brand FE N N N Y Y Month FE N N N N Y Year FE N N N N Y No. obs (good-hh-trip) No. HHs 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 product category Dependent variable: EW purchase dummy OLS se Obs OLS with char se Obs FE (HH) se Television 0.31 (0.08) (0.14) (0.20) Audio 0.16 (0.05) (0.07) (0.10) Mobile 0.40 (0.19) (0.28) (.) P*S*T 0.23 (0.06) (0.08) (0.11) Imaging 0.36 (0.07) (0.12) (.) PC Hardware 0.18 (0.05) (0.08) (0.11) Music 0.16 (0.09) (0.15) (0.24) Video 0.21 (0.04) (0.07) (0.09) DVS 0.05 (0.23) (0.46) (0.28) Wireless 0.25 (0.18) (0.31) (0.25) Notes: Significance level: ***1%, **5%, *10% 11

12 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 purchase 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 ɛ < 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. 11 See footnote However, we do not need nor use the first order condition to estimate the demand side parameters. 12

13 A buyer s utility if he purchases the warranty is V EW = u(w t; r). 13 A buyer s utility if he does not purchase the warranty is V NW = 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 V NW with this lower bound in our estimation, i.e., we will have V NW = 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 making 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 experience 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 individuals 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, ɛ EW and ɛ NW, to V EW and V NW. 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: where D(t; r, ω(φ), p, σ) = Pr(ɛ NW ɛ EW Ω(t; r, ω(φ), p, σ)) = exp Ω(t; r, ω(φ), p, σ) 1 + exp Ω(t; r, ω(φ), p, σ)) Ω(t; r, ω(φ), p, σ) V EW V NW σ. 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) This assumes that there is no deductible associated with using the warranty. 14 This 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 the robustness of our estimates to this assumption in Section For example, the New York Times of August 28, 2014 writes: The company selling the warranty has the information on failure rates. You don t...that s not easy to find out. Companies aren t in the habit of telling you that their products fail 4 percent or 12 percent of the time. Failure rates are usually low. Warranty companies know that. And they know, too, that consumers tend to think the failure rate is higher. 16 The utility specification we will use in estimation imposes a specific normalization so we can identify the scale parameter σ. This scale parameter is the inverse of the marginal utility of income. 17 WTP can be uniquely obtained from choice probabilities. 13

14 Fix a product M with price p M and failure rate φ, and let ω = ω(φ). The maximum willingnessto-pay W T P (p M, 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 V EW (t, r) = V NW (p M, r, ω). The identification problem is that the same W T P 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 W T P 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 p M. 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 u x (x; r) decreases in r. Proof. Following the discussion in the main text, it suffices to prove that as we change the price p M of the main product, the slope of the iso-wtp curves dω dr do so by showing that changes monotonically. We will is strictly monotone in p M. W T P (p M,r,ω) r W T P (p M,r,ω) ω = dω dr Let Ω = u (W t; r) ωu (W p M ; r) (1 ω) u(w ; r). The WTP is defined as the price t for which Ω = 0. Thus, by the implicit function theorem, W T P r W T P ω = Ω/ r Ω/ t = u r (W t; r) ωu r (W p M ; r) (1 ω) u r (W ; r) u, and (W t; r) = u (W ; r) u (W p M; r) u > 0. (W t; r) The numerator in the first expression Ω/ r is positive because larger r implies more risk 14

15 aversion and hence a lower certainty equivalent for the lottery (ω, p M ). Thus, Ω goes up when r goes up. Observe that [ ] W T P/ r p M W T P/ ω = = [ ] ur (W t; r) ωu r (W p M ; r) (1 ω) u r (W ; r) p M u (W ; r) u (W p M ; r) { 1 ωu } r (W p M ; r) [u (W ; r) u (W p M ; r)] (u (W ; r) U (W p M ; r)) 2 [u r (W t; r) ωu r (W p M ; r) (1 ω) u r (W ; r)] U (W p M ; 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 rx } r 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. by The second order Taylor approximation of u( ) around W for some wealth deviation is given 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 absolute 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 Ω j between purchasing and not purchasing an extended warranty for product j (equation 1), we obtain that: Ω j = ω jp j t j + r 2 (ω jp 2 j t2 ). σ 18 Strictly speaking, the Arrow-Pratt coefficient of absolute risk aversion can vary with income if the utility specification is not CARA. 15

16 Let D j be the observed attachment rate for product j. Our choice model implies that 19 D j log = Ω j = ω jp j t j + r 2 (ω jp 2 j t2 ). (2) 1 D j σ The decision weight ω j acts like a (non-additive) product effect. We decompose this effect to ω j = ω(φ j ) + ξ k(j) + η j (3) where ω( ) is some unknown function of φ, ξ k(j) is a subcategory-level effect, and η j is a random shock. The parameters ξ k(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 ω j s as a function of the unknown parameters (r, σ) and the data: ω j = σω j + t j + r 2 t2 j p j + r 2 p2 j. (4) We construct moment conditions involving ω j to estimate r and σ. 20 Once we have these parameters, we calculate ω j using equation 4. Our assumption regarding the error structure in equation 3 implies the following moment condition E[ω j ω j φ j = φ j, k(j) = k(j ), p j, p j, t j, t j ] = 0 since ω j ω j = η j η j for j, j such that φ j = φ j 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 z i 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 ( ) r i = exp γ r z i + ξ r k(i), (5) 19 A complication in linking Ω j to product-level attachment rates arises because p and t vary at the product level. If we had infinite data, we could compute D j p,t = Pr(d i = 1 j(i) = j, p, t) and then invert this equation to get Ω j for each pair (p, t). However, this is not our case. The best we can do is to calculate the attachment rate for a product by aggregating over all price pairs (p, t) for this product. As for prices, we use the largest p and the smallest t in the estimation because these make the purchase of the warranty most attractive even without appealing to risk aversion and probability distortions. 20 Because of the large variation in prices across categories, we estimate separate scale parameters for each subcategory, and we allow for heteroskedasticity. Specifically we let σ k (p) = σ k p for subcategory k. 16

17 so risk aversion is a function of consumer observables z i and the product subcategory ξ r k(i). For the decision weight ω j, 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 }: ω jg = exp[ ( log(φ j )) α G ] + η jg. (6) Finally, we allow the scale of the utility function to vary with income I i : σ i = σ I i. (7) The model provides an expression for predicted attachment rates among gender G households for product j: D jg = 1 N jg {i:j(i)=j,gender{i}=g} exp ω jgp j t j + σ i 1 + exp ω jgp j t j + r i r i 2 (ω jg p 2 j t2 ) 2 (ω jg p 2 j t2 ) σ i (8) Let q jg be the corresponding attachment rate from the data. Define ˆω jg as the decision weight that equates q jg with D jg for a given set of parameters, and let ˆη jg = ˆω jg exp[ ( log(φ j )) α G ]. (9) We assume the error term η jg 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 ( η jg φ, k, p, t, z ) = 0 (10) to estimate the parameters of the model 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) where E(t; r, ω(φ), σ) is the price elasticity of demand for EWs. (11) 21 We estimate the model using a nested fixed point algorithm with inner loop tolerance of

18 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 ω j s and a local linear fit. We also include a fit based on the one-parameter Prelec (1998) function, which is close but slightly less concave than the local linear fit. ω(φ) = exp[ ( log(φ)) α ] (12) We estimate α = 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 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 weighting 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-topay 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 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 with 95% confidence interval of (0.441, 0.775) while for females, α = with 95% confidence interval of (0.443, 0.927). The difference in coefficient estimates for males and females are not statistically 22 Although we also estimated risk aversion allowing for differences in gender, age, income and having a child, we do not find any statistically significant differences in these dimensions given that the average estimate for r is close to zero. 23 The average WTPs are computed by setting the choice shocks, ɛ EW and ɛ NW, and the shock that enters the weighting function, η j, to zero. 18

19 Table 8: Average willingness-to-pay for EW on a good with value $100 Failure rate Model est Model est Standard est ω ω(φ) = φ φ 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. Constructing 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 with 95% confidence interval of (0.605, 0.802) while for the low experience group, α = 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 µφ j p j. We estimate µ = 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

20 failure rate of 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. explanatory power of α is exhausted. The estimate of standard risk aversion only becomes non-negligible once the Table 9: Parameter estimates with varying expected cost % of product price r Prelec α ω(0.05) 100% % % % % % % % % % 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 distortions 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

21 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 practice 24. 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-weighted 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 data 26 is 15.09%. Removing the bias shifts the distribution to the left and decreases the average 24 An example of this is the Minimum Advertised Price (MAP) policy, whereby retailer cannot advertise below manufacturer suggested retail prices, and the Unilateral Manufacturer s Retail Price (UMPRP) policy which imposes a fine when retailers sets a price below the one set under the UMRP. 25 Sales-weighting is at the product ID level and not the (product ID, t, p) level. 26 This 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 pricing more conservative. 21

22 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 27 Attachments rates here differ from the ones in Table 2 due to the assumption made in footnote 19. Attachment rates 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 aggregating transactions 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

23 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 firstbest fraction insured. There is overinsurance 28 with 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, q bias, is below or above the first best quantity, q F B. Welfare unambiguously decreases when we remove the bias if q F B q bias since the bias actually brings us closer to the first best quantity from below. On the other hand, if q F B < q bias, 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 q F B < q bias. 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 28 Although we view extended warranties as insurance products, in reality, these contracts do not merely involve financial transfers from the insurance company (or the retailer) to consumers. The firm has to physically prepare to service 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

24 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

25 References Baker, T. and P. Siegelman (2013), You Want Insurance with That? Using Behavioral Economics to Protect Consumers from Add-on Insurance Products, Connecticut Insurance Law Journal, 20. Barseghyan, L., F. Molinari, T. O Donoghue, and J. C. Teitelbaum (2012), The Nature of Risk Preferences: Evidence from Insurance Choices, American Economic Review, 103 (6), Cohen, A. and L. Einav (2007), Estimating Risk Preferences from Deductible Choice, American Economic Review, 97 (3): 745?88. Diamond, P. A. (1971), A Model of Price Adjustment, Journal of Economic Theory, 3, Ellison, G. (2005), A Model of Add-On Pricing, Quarterly Journal of Economics, 120, Ellison, G. and S. F. Ellison (2009), Search, Obfuscation, and Price Elasticities on the Internet, Econometrica, 77 (2), Kahneman, D. and A. Tversky (1979), Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47 (2): 263?91. Jindal, P. (2014), Risk Preferences and Demand Drivers of Extended Warranties, Marketing Science, 34 (1), Prelec, D. (1998), The Probability Weighting Function, Econometrica, 66 (3),

26 Figures Figure 1: Histogram of the Ratio of Extended Warranty Price and Product Price Figure 2: Identification: Single-crossing of willingness-to-pay Note: A 1 > A 2 26

27 Figure 3: Estimated weighting function Figure 4: Estimated weighting function: Females vs Males 27

28 Figure 5: Estimated weighting function: High vs Low Experience Figure 6: Densities and cdfs of the ratio of EW and TV price 28

29 Figure 7: Ratio of extended warranty and TV price Figure 8: Densities and cdfs of price-cost margins 29

30 Figure 9: TV extended warranty price-cost margins Figure 10: Counterfactual: Densities and cdfs of fraction insured 30

31 Figure 11: Fraction insured Figure 12: Two effects of removing the bias on Consumer Surplus 31

32 Figure 13: Consumer Surplus Figure 14: Profits 32

33 Figure 15: Comparing deadweight-loss when q F B < q bias Figure 16: Total Welfare 33