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A behavioral model of the adoption of protective activities
Journal of Economic Behavior and Organization 6 (1985) 1-15. North-Holland
A BEHAVIORAL MODEL OF THE ADOPTION OF PROTECTIVE ACTIVITIES* Howard KUNREUTHER University of Pennsylvania, Philadelphia,PA 19104, USA
Warren SANDERSON State University of New York, Stony Brook, NY 11790, USA
Rudolf VETSCHERA University of Vienna, 1010 Vienna, Austria Received November 1983, final version received May 1984 Considerable empirical evidence suggests that individuals are unwilling to protect themselves against low probability-high loss events even if the costs of protection are subsidized. This behavior has been difficult to rationalize using the traditional expected utility model. This paper proposes a model of adoption of protective activities which emphasizes the importance of interpersonal communication and past experience. The time path of adoption and an equilibrium rate is characterized. Properties of the model shed light on the reasons for low usage of seat belts and limited purchase of federally subsidized flood insurance.
1. Introduction
The study of empirical data on the adoption of protective activities reveals a puzzling phenomenon. People are often willing to protect themselves against risks which have a moderate frequency of occurrence even though the potential losses are small. On the other hand, they are often reluctant to protect themselves against low probability events with high losses. The unwillingness of individuals to obtain protection against low probability events is illustrated by two examples which have attracted public attention in the United States. One involves the use of seat belts, the other insurance against floods. Only between 10 and 20 percent of automobile passengers utilize seat belts although engineering studies reveal that these safety devices are highly *We would like James Vaupel for anonymous referee SES81-12512, and Austria.
to thank Brian Arthur, Lawrence Berger, Mark Pauly, Philip Parker and helpful discussions and comments on earlier drafts of this paper. An also provided useful comments. This research is supported by NSF Grant by the International Institute of Applied Systems Analysis, Laxenburg,
0167-2681/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
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H. Kunreuther et al., A behavioral model of the adoption of protective activities
effective in reducing the probability of death or serious injury from automobile accidents. Consumers have shown a noticeable lack of interest in purchasing cars with an automatic seat belt even though the annual costs of this feature are as low as $24.50 [Arnould and Grabowski (1981)1. In flood prone areas, homeowners have had an opportunity to purchase highly subsidized flood insurance since 1968, yet relatively few had done so before the U.S. government required coverage as a condition for subsidized disaster relief and Federally funded mortgages [Anderson (1974),1. People do have an interest in insuring themselves against moderate probability-low loss events. One Pennsylvania state insurance commissioner found this out the hard way when he instituted a mandatory $100 deductible for automobile collision policies which would have saved consumers millions of dollars per year. His action proved so unpopular that it eventually had to be rescinded in favor of the previously existing $50 deductible [Cummins et al. (1974)-1. These examples are at odds with predictions from the standard economic approach toward risk, expected utility theory. According to this theory a risk averse person increases demand for a protective activity if the probability of the event decreases and the negative outcome proportionately increases so that the expected loss remains constant. Recently, Paul Schoemaker (1982) summarized data from laboratory and field experiments which suggest that individuals often violate the axioms upon which the expected utility model is grounded and make choices which are inconsistent with those predicted by the theory. This paper proposes an alternative model of behavior toward protective activities. It differs from the individualistic model in emphasizing the importance of interpersonal influence and past experience on the adoption of protective activities. These factors are introduced as a way of characterizing a dynamic equilibrium. The model is similar in spirit to one developed by Mark Satterthwaite (1979), who looks at the importance of friends and neighbors in providing information. It also relates the adoption of protective activities to the proportion of individuals who have recently experienced a disaster. The paper is organized as follows: The next section reviews the relevant literature indicating the importance of interpersonal influence and past experience on the adoption of protective activities. Section 3 develops a process model which shows how these two factors can characterize an equilibrium adoption rate. This model can explain observed aggregate protective behavior over time which cannot be rationalized with the traditional expected utility model. Section 4 contains a numerical example which specifies plausible parameters that generate values consistent with the observed adoption rates on seat belt usage and flood insurance purchase. The concluding section summarizes the findings and links them to policy considerations.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
3
2. The importance of social context
2.1. Interpersonal influence The role of interpersonal influence on decisions about protective activities can be illustrated by two studies. A survey that National Analysts (1971) conducted for the Department of Transportation, revealed that a significant number of individuals began to wear their seat belts on a permanent basis after they were asked by others to buckle up. In a similar vein, face-to-face interviews with over 3,000 homeowners in flood and earthquake prone areas [Kunreuther et al. (1978)] indicated that one critical factor influencing purchase of insurance was knowledge of someone else who had bought coverage, or discussions with an insured individual. Individuals had imperfect information on the price, loss and probability of a flood and presumably used personal contacts as sources for those data. The literature on the diffusion of innovations also provides a useful perspective. The pioneering studies of Griliches (1957), Mansfield (1963), and Nelson and Winter (1982) have pointed out the importance of learning from others and imitation as an element of the adoption process. 1 Cohen and Axelrod (1984) have indicated that preferences for a product may change as a result of imitation. With respect to consumer behavior the literature in marketing discusses the adoption process for new products in a similar fashion. Bass (1969) successfully tested a diffusion model of the purchase of consumer durables in which there are two classes of people: innovators and imitators. Innovators are not influenced in their initial purchase by the number of people who have already bought the product while imitators are. This approach generates logistic diffusion paths for a number of different consumer goods which are consistent with the observed data. Easingwood, Mahajan and Miller (1983), and Gatignon and Robertson (1985) summarize more recent studies that build on the Bass model. For many the adoption process occurs only after there has been personal communication with peers. Whyte (1954) shows that air conditioners were purchased on one side of the street but not on the other because of the conversations which took place in an alley between the mothers who were caring for their children. Farmers and doctors frequently became aware of a new product through impersonal sources such as the mass media, but discussed its performance with a colleague or neighbor before adopting it themselves [Katz (1961)].
2.2. Influence of experience Protective activities differ from consumption goods because they are contingent claims. Rewards are received only when a particular state of 1See Rogers (1983) for a summary of a large number of empirical studies of the innovation process which supports this view.
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H. Kunreuther et al., A behavioral model of the adoption of protective activities
nature occurs (e.g., an accident). For some protective activities the return is visible (e.g., a payment from an insurance company). For others the return is less noticeable, such as a fatality avoided by wearing a seat belt. People learn about the magnitudes, probabilities and consequences of losses from personal experience and the experiences of others. Tversky and Kahneman (1973), have provided evidence that individuals exhibit an availability bias, whereby the subjective probability assigned to an event is influenced by the ease with which one can recall an event. A recent accident or disaster is thus likely to have a disproportionate impact on the estimate of the probability compared to one in the distant past and thus may generate demand for protective activities. Cohen and Axelrod (1984) would attribute this interest in protection to the element of surprise that a negative event generates if the world is too complex for an individual to develop a correctly specified model of the environment. Evidence from natural disasters illustrates the importance of a recent flood or earthquake in triggering interest in insurance. Three communities in northern New Jersey sampled in the flood insurance field survey reported in Kunreuther et al. (1978), suffered flood damage in 1973. Virtually the entire year's sale of flood insurance policies occurred in the two months following the flood. Similarly, the purchase of earthquake insurance increases sharply after a quake, but decreases steadily thereafter, as the memories become less vivid [Steinbrugge et al. (1969),1. The trend of flood insurance coverage in Pennsylvania also suggests that if there is no disaster for a sufficient period of time, people will cancel their policies. For example, from 1977 to 1982 the number of flood insurance policies in the state dropped approximately 50 percent, from 127,000 to 64,000, because of relatively minor flooding in the state during this period [-Kury (1982),1. People appear to view such protective activities as an investment rather than a contingent claim. If they do not collect on their policies for a prolonged period they tend to let them lapse. Hence the importance of exogenous events as personal reminders of the value of insurance coverage. People also use the experiences of their friends, relatives, neighbors, and aquaintances to judge the probabilities and impacts of various misfortunes. In addition to the low costs of obtaining such information [see Porath (1980)1, peers provide certain types of qualitative data which cannot easily be obtained from other sources [Pauly and Satterthwaite (1981),1. The experience of a neighbor who wore a seat belt during an accident has tangible meaning where dry highway statistics do not. An individual may be able to obtain more insight into the terms of his insurance policy after speaking with a friend or neighbor who recently settled a claim, than discussing this subject with an insurance agent. In addition, one can trust the judgments of friends since they have no monetary incentives, as sellers do, to promote certain activities.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
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3. A wocess oriented madd In this section, we present and analyze a formal model which highlights the importance of social context and past experience in the adoption of protective activities. 3.1. The formal model The model of choice is grounded on a set of simplifying assumptions. All consumers are assumed to face the same objective risk, but their perceived expected loss due to a particular event differs depending upon their past experience, as well as upon the information they have collected from publications and the media. Only one homogeneous form of protection is considered. We assume that individuals learn about the costs and benefits of a protective activity from friends and neighbors before making direct inquiries to sellers. This assumption is similar to the one used by Satterthwaite (1979) in developing a model for the adoption of a reputation good such as a physician's services where information on quality was obtained through a series of inquiries from personal acquaintances. In the case of protective activities some individuals may inadvertently learn about the product from others rather than actively searching for information. At each point of time people are continuing or discontinuing the use of a protective activity depending upon its price, p, the subjective probability of the event and the estimated loss. The latter two factors are captured in the parameter, L, which represents the expected loss. Behavior is assumed to be different for individuals who recently have suffered a loss and those who have not. Hence the overall adoption and discontinuation rates will depend on the fraction Ht of the population who have experienced a loss during the most recent time interval At. The magnitude of these variables, p, L a n d / / , , influence the parameters characterizing the adoption and discontinuation process. We assume a fixed population of relevance to the problem and specify the fraction of that population, At, who protect themselves against a given risk at time t. This fraction determines the probability that a person who has not adopted a protective activity will meet someone who is currently protected [i.e., At ( 1 - A t ) ] . The variable is assumed to change over time according to the following differential equation: A(t) = ac, A,(1 - A,) + ao,(1 - A,) - 3c, A, (1 - A,) - 5o, A,
(1)
where A(t) = the increase (or decrease) in A at instant t, A, =the fraction of the (fixed) population protecting themselves against a given contingency at time t,
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H. Kunreuther et al., A behavioral model of the adoption of protective activities
~ct =the fraction of interpersonal contacts between adopter and nonadopters, which result in a non-adopter adopting protective behavior at time t, ~ot =the fraction of non-adopters who adopt protective behavior for all other reasons at time t, ~ct =the fraction of interpersonal contacts between adopter and nonadopters, which result in an adopter discontinuing protection at time t, and ~ot =the fraction of adopters who discontinue protective behavior for all other reasons at time t. We have already argued in section 2 that actual experiences with hazards provide an important stimulation to people to protect themselves against these hazards. Social interaction is affected in two ways by these experiences. First, the diffusion of information about the protective activity increases when /7, increases. People who experience an accident or disaster communicate more about the importance, availability and convenience of certain forms of protection as well as about their subjective feelings toward protective activities. Second, these recent experiences raise their estimate of the probability of a future disaster so that they are more likely to favor protection at any given price and loss estimate. Hence, we expect communication between protected and unprotected individuals to result more frequently in the unprotected person deciding to adopt protection and less frequently in the protected individual discontinuing it when one or both of them have recently experienced the loss. We capture this effect in our model by specifying that ~ct is positively related to /7t and that ~Sct is negatively related to/7tA recent loss experience increases the chance that people who have adopted a protective activity will continue their use of it, and that others initiate protection. People who have recently experienced flooding are more likely to renew their coverage, than those who have been spared this event. Seat belt users who have been in an automobile accident are more likely to buckle up than accident-free drivers. We postulate, therefore, that recent experiences with accidents, other things being held constant, increases the adoption parameter ~ot and decreases the discontinuation parameter ~ot. A relatively low price will induce consumers to buy the product and continue its use over time even if they don't suffer a loss in the future. A high price is less likely to convince individuals to purchase the product initially. If they do purchase it, they are less likely to continue protection in the future. Hence, we assume that the magnitudes of the adoption parameters (~ot and ~ct) vary inversely with p while the discontinuation parameters (6or and 6ct) vary directly with p. The reverse holds for the expected loss, L. As estimated expected losses increase, a larger percentage of the population will adopt and maintain protection for given levels of/7, and p.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
7
3.2. Comparative statics and dynamics The equilibrium proportion of people protecting themselves against a given risk is attained when the numbers of new adopters and discontinuers are identical, A(t)=0. Given positive values of the parameters ~c,, ~o,, 6o,, and 6c, there exists a unique and stable equilibrium proportion of adopters which lies in the (0, 1) interval. Let Ae represent this equilibrium proportion. Appendix A shows that this equilibrium value is - (OCo,+ 6o, -- =c, + act) + x/(=o, + 6ot - ~c, + act)= + 4O(o,(Occ,- 6c,)
Ae
(2)
2(ot¢,--6c,),
where ~c, =/=6or If ~c, =6c~ there is no net effect of interpersonal communication. In this case eq. (1) implies that Ae=otoJ(O~ot+6o,). The following three statements are implications of this model.
Statement 1. Holding the expected loss and the proportion of the population recently experiencing the risk constant, the equilibrium proportion of the population protecting themselves decreases as the price of protection increases. Statement 2. Holding the price of protection and the proportion of the population recently experiencing the risk constant, the equilibrium proportion of the population protecting themselves increases as the expected loss increases. Statement 3. Holding the price of protection and the expected loss constant, the equilibrium proportion of the population protecting themselves increases as the proportion of the population recently experiencing the risk increases. Proofs of these three statements appear in appendix B. Statements 1 and 2 are intuitively clear and are consistent with predictions from an expected utility model. Statement 3, however, is quite different from anything derived from expected utility but is consistent with observed behavior. Protection against frequent risks with lower losses is more common than against infrequent but more dangerous risks; hence the demand for low deductible insurance and the reluctance to wear seat belts. Our formulation of the problem also sheds light on the case of disasters like floods and earthquakes. Unlike automobile accidents which occur roughly to a constant proportion of drivers each year, disasters tend to affect large proportions of certain population groups in a given year and then no one for a long period of time. In this case, the level of protection immediately
8
H. Kunreuther et al., A behavioral model of the adoption of protective activities
jumps after the disaster as people react to the experience of the loss. The level of protection may even continue to increase for a while as people learn about the losses of their friends and neighbors. As the disaster recedes in time, fit becomes zero and the proportion protecting themselves falls asymptotically toward A*, the equilibrium level of protection associated with no one in the population having had recent disaster experience. With the coming of the next disaster H t increases and the entire process repeats itself. This verbal description of the time path of adoption of protective activities in the case of disasters can be formalized in two statements whose proofs are contained in appendix B. 4. A disaster with its attendant change in 11 t from zero to a strictly positive number brings about an increase in the proportion protecting themselves. Statement
Following the disaster, when f i t reverts back to zero, the proportion protecting themselves falls and asympotically approaches A* from above. S t a t e m e n t 5.
To summarize, our model produces five formal implications, three of which pertain to behavior toward accidents and two of which relate to disasters. We show with regard to accidents that the equilibrium level of risk Coverage is negatively related to its price, positively related to the expected loss, and positively related to the fraction of the group who have recently experienced a loss. Immediately after a disaster the coverage level rises and subsequently falls asymptotically to the level of coverage which would obtain if no one in the group had had any recent risk experience.
4. Numerical examples Two numerical examples, one concerning seat belts and one concerning flood insurance, illustrate the model. Arnould and Grabowski (1981), reported a seat belt usage rate of about 14 percent in 1978. This figure is, of course, consistent with an infinite number of values of the four adoption and discontinuation parameters of our model. What is important here, however, is to determine if, in fact, there exist plausible parameter sets which generate the 14 percent figure. Table 1 contains two parameter sets, those in columns (1) and (2) which produce the requisite 14 percent usage rate with reasonable underlying assumptions. In column (1) we have assumed that the net effect of interpersonal communication is favorable to seat belt use (ac,-gct =0.03) and the impact of external factors such as past experience favoring adoption is
H. Kunreuther et al., A behavioral model of the adoption of protective activities
9
Table 1 (1)
(2)
(3)
act aot 6c, ~5ot
0.1 t 0.02 0.08 0.15
0.01 0.01 0.07 0.01
0.14 0.05 0.05 0.12
Ae
0.14
0.14
0.42
relatively small (~ot =0.02). To generate the observed usage rate requires an external discontinuation rate of 6¢t = 0.15. In column (2) we have considered a situation where both the external adoption and discontinuation rates are close to zero. Under this scenario the observed usage rate of 14 percent can only be obtained if there is a net negative impact of interpersonal communication (~¢,-6ct=-0.06). This might be a realistic vector of parameters for societies in which high prestige is attached to driving skills and seat belt usage might be seen as a lack of confidence in one's driving. Column (3) describes the dramatic increase in overall adoption which would occur if people tend to react more favorably to information provided to them. The adoption parameters ~ct and ~ot are increased by 0.03 from column (1), while the discontinuation parameters are decreased by the same amount. Although the first two columns in table 1 produce the same (static) equilibrium level of adoption, the underlying dynamics of the process are quite different. To note this difference, one can determine the net adoption (NA) and net discontinuation (ND) rates in equilibrium, which are computed as
NA=o~eAe(1-Ae)+O~o(1--Ae)
and
ND=feAe(1-Ae)+foA e.
Of course, in an equilibrium situation, we have NA=ND. For columns (1) and (2) in table 1, the rates are, respectively, NA=ND=O.03 and N A = ND=0.0098. For the parameters in column (1), about 3 percent of the population newly adopt (or discontinue) seat belt usage in each period; for those in column (2) that figure is less than 1 percent. This corresponds to 27 percent of seat belt users in the first case and 7 percent in the second case. If we had statistics on the change in behavior regarding seat belt usage, we would be able to identify which of the two parameter sets more accurately characterizes reality. We can also determine the impact of a change in any of the adoption or discontinuation parameters on the equilibrium level of protection. W h e n / i t = 0, eq. (1) can be rewritten as
10
H. Kunreuther et al., A behavioral model of the adoption of protective activities
3or
%t
where
~)ct = (Xct -- (~ct"
In order to maintain a given equilibrium, Ae, changes in either of the exogenous parameters, 0tot or tSot, require a larger offsetting change in Yc,, the net interpersonal contact effect. Suppose Ae =0.14, then for a 0.01 increase in % , yet must decrease by 0.07 (i.e., 0.01/0.14) to maintain this equilibrium level. If 5or increases by 0.01, then ~'~tmust increase by 0.0116 (i.e., 0.01/0.86). The second example shows the behavior of the model where an exogenous event like a flood suddenly induces a large part of the population to adopt protection. We compare two kinds of such events, a more frequent but less severe one with a rarer, but more severe flood. The parameters depicted in table 2 refer to the case where //t = 0 and, in this example, apply to both types of floods. Table 2 ctct ~o, t$ct 3°,
0.01 0.01 0.08 0.05
Ae
0.08
For the severe flood, we chose an interval of 70 periods and an adoption level after the disaster of 75 percent. For the small flood, the interval between disasters is 30 periods and only 50 percent of the population adopt protective activities immediately following it. The adoption path for both scenarios is depicted in fig. 1. This figure shows that in most years a higher portion of the population protects themselves against the smaller flood than against the large one.
5. Conclusions
We argue in this paper for a new approach to the adoption of protective measures against risk. Expected utility theory takes the isolated individual as the unit of analysis. The individual, given information on probabilities and losses, decides on whether to insure himself against various risks. Aggregating across these individuals yields the proportion of the group who protect themselves. Our approach, on the other hand, is dynamic and treats people in their social context. It is based on the idea that the group protecting themselves from a given risk is always in flux, with some people entering it and others leaving it, based, in part, on their own experiences and in part on
H. Kunreuther et al., A behavioral model of the adoption of protective activities
t
11
Relatively Large and Infrequent Flood
0.7 .4~
0.6 Proportions of Population with Insurance Coverage
Relatively Small and I~ Frequent Flood
0.5
0.4
0.3
0.2
0.1
I
10
I
20
I
30
I
40
I
50
1
60
I
70 Time
Fig. 1. Time paths of protection for two types of flood.
the experience of others. We define the equilibrium rate of protection in this dynamic model as the value where the adoption and discontinuation rates balance. The model enables one to analyze the deviations from equilibrium and the dynamic path of adjustment. In our view, it is natural that adoption of protection increases after a catastrophe and falls back to an equilibrium level over time. The adoption model and expected-utility theory yield qualitatively identical implications with respect to the effects of changes in the price of coverage and the expected loss on the equilibrium level of protection. They differ with respect to their implications of a change in the probability of the loss-producing event. The adoption model suggests that, other things being equal, the protection rate will be positively associated with the probability of the event for a fixed expected loss while the expected utility model predicts the reverse. The adoption model is only suggestive about the role of the public sector with respect to information provision and regulation. If the private market
12
H. Kunreuther et al., A behavioral model of the adoption of protective activities
fails some governmental action may be deemed appropriate. For example, the public provision of information may improve markets for protective activities but only if the data arc sufficiently salient to decision makers to influence adoption. Financial incentives may also be useful in inducing more people to adopt protection. However, suppose the initial fraction of the group adopting protection is small, and the event protected against is relatively infrequent. Then government policies aimed at subsidizing the activity (as with flood insurance) and the provision of information (as in seat belt usage) are likely to have little impact. Direct governmental regulation of protectivc activities, as a last resort, may therefore be appropriate in those instances. These might include requirements coupled with appropriate penalties such as fines if one does not wear a scat belt (as in France) or nonpayment of medical payments on an insurance claim if the injured person had not worn a seat belt (as in Austria). Measures such as these may explain why thc use of scat belts in other countries is somewhat higher than in the United States. The adoption modcl implies that there is no magic formula which guarantees a socially optimal solution to the problem of protection against risk. Proper public policy toward low probability events is an inherently complicated puzzle, and simple-minded solutions to it may do more harm than good.
Appendix A: Stability and uniqueness of the equilibrium level Proof that A e is a unique and stable equilibrium level. A e
An equilibrium level
is defined by the condition that Ae =0,
or, equivalently, that
(A.I)
occtAe(1 - A e ) + o~ot(1 - A e ) - 6 c t A e ( 1
ae
- A e ) - t ~ o t A e =0.
(A.2)
-(OCot+ 3ot--°c~t+ 6ct)-+x/(~ot+ 6ot'~z~t+ 6~02+ 4~tot(°c~t"-3~t) 2(cto, - 6ot) (h.3)
where ~c, ¢3¢r. Let us denote the equilibrium value computed using the positive root as Ate+) and the equilibrium value using the negative root as Ate-). It can easily be shown that for parameters ~o,, t~o,, ~c, and 3~t in the (0, 1) interval: 0 < Ate+) < 1,
A~-) < 0
(A.4)
for
0to,--3c, > 0, and
H. Kunreuther et al., A behavioral model of the adoption of protective activities
13
A~-) > 1 for 0t¢t--6~, < 0. Therefore A(~+) is the unique equilibrium level in the (0, 1) interval. This completes the proof of uniqueness. To show that A~+) is also a stable solution, we have to show that for At > A(e+), A, <0,
and
for At0.
(A.5)
We rewrite the basic eq. (1) as in product form as At = -- 2(0tot- tic,)'(At- A~+)) "(A,- A~-)).
(A.6)
This product enables us to determine the sign of At for all values of A, depending on the sign of (otct-~ct). Using the inequalities in (A.4), the inequalities in (A.5) now follow directly. Appendix B: Proof of Statements 1 to 5
It follows from eq. (2) for the equilibrium rate of protection that 0A~
aA~<0
d~ot > 0 ,
dgot
and
cOA~ 0(~o _ 6¢,) > 0 .
(B.1,2, 3)
It is also intuitively appealing that the equilibrium level of adoption rises as people become more sensitive to information favoring adoption, or the net effect of communication works in favor of adoption. Conversely, the level will drop as people are more sensitive to information against adoption. Our assumptions on the effects of//t, P a n d / , may be summarized as
O~ct OH, > O,
aOt°t > 0, aH---~
(B.4) 06 ~t
--
OH,
< O,
OOtct ~<0, @
06or
--<
0//,
O.
aOtot --<0, 0p
(B.5) 06¢t > 0, @
06o, > 0.
@
14
H. Kunreuther et al., A behavioral model of the adoption of protective activities OOtct
O~°t > O,
~L > 0 ,
OL (B.6)
06°,
060,
--<0, 0L
--<0. 0L
Statements 1-3 now can conveniently be re-formulated as follows:
Statement 1
OA~ OA~ O~ot OA~ 060,
~=
dp
.
.
.
.
OOto, Op + 06o, dp >0
<0
<0
+
>0
OA~
c9(ot~,~6~,) >0
0(0~¢t--0et )
"
Op
<0.
(B.7)
<0
Statement 2
cgAe OA~ 0~o, OAe 060, OL =0 >0
OA~
O(%-g¢,)
o,'0L + o o," OL >0
<0
<0
>0. >0
>0
OAe
0(~et--~ct)
(B.8)
Statement 3 t3Ae
OAe Ofio, OA~ O~ot
?E
-5-E,=o o,ort, + o o,or1, + >0
>0
<0
<0
>0
>o.
(B.9)
>0
Statement 4 is an immediate consequence of the fact that a disaster causes a discontinuous increase in/-/t and that aAe/OIIt > O. Statement 5 follows from (B.9), the stability property shown in appendix A and the fact after the disaster At increases to a level higher than the equilibrium level associated with/-/t = 0.
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H. Kunreuther et aL, A behavioral model of the adoption of protective activities
15
Easingwood, C., V. Mahajan and E. Muller, 1983, A non uniform influence innovation diffusion model of new product acceptance, Marketing Science 2, 273-295. Gatignon, H. and T. Robcrtson, 1985, A propositional inventory for new diffusion research, Journal of Consumer Research, June. Griliches, Z., 1957, Hybrid corn: An exploration in the economics of technological change, Econometrica 25, 501-522. Katz, E., 1961, The social itinerary of technical change: Two studies on the diffusion of innovation, Human Organization 20, 70-82. Kunreuthcr, H., R. Ginsberg, L. Miller, P. Sagi, P. Slovic, B. Borkan and N. Katz, 1978, Disaster insurance protection: Public policy lessons (Wiley, New York). Kury, F., 1982, 10 years after Agnes: The lessons still unlearned, The Philadelphia Inquirer, June 21. Mansfield, E., 1963, The speed of response of firms to new technologies, Quarterly Journal of Economics 77, 290-311. National Analysts, Inc., 1971, Motivating factors in the use of restraint systems, Final Report Contract FH-11-7610, prepared for the U.S. Department of Transportation (Philadelphia, PA). Nelson, R. and S. Winter, 1982, An evolutionary theory of economic change (Harvard University Press, Cambridge, MA). Pauly, M. and M. Sattcrthwaite, 1981, The pricing of primary care physicians' services: A test of the role of consumer information, Bell Journal of Economics 12, 488-506. Porath, Y., 1980, The F-connection: Families, friends and firms and the organization of exchange, Population and Development Review 6, 1-30. Rogers, E., 1983, Diffusion of innovations (Free Press, New York). Satterthwaite, M., 1979, Consumer information: Equilibrium industry price and the number of sellers, Bell Journal of Economics 10, 483-502. Schoemaker, P., 1982, The expected utility model: Its variants, purposes, evidence and limitations, Journal of Economic Literature 20, 529-563. Steinbruggc, K., F. McClure and A. Snow, 1969, Studies in scismicity and earthquake damage statistics, U.S. Department of Commerce report, app. A, COM-71-00053 (Washington, DC). Tversky, A. and D. Kahneman, 1973, Availability: A heuristic for judging frequency and probability, Cognitive Psyc_hology5, 207-232. Whyte, W., 1954, The web of word of mouth, Fortune.
A BEHAVIORAL MODEL OF THE ADOPTION OF PROTECTIVE ACTIVITIES* Howard KUNREUTHER University of Pennsylvania, Philadelphia,PA 19104, USA
Warren SANDERSON State University of New York, Stony Brook, NY 11790, USA
Rudolf VETSCHERA University of Vienna, 1010 Vienna, Austria Received November 1983, final version received May 1984 Considerable empirical evidence suggests that individuals are unwilling to protect themselves against low probability-high loss events even if the costs of protection are subsidized. This behavior has been difficult to rationalize using the traditional expected utility model. This paper proposes a model of adoption of protective activities which emphasizes the importance of interpersonal communication and past experience. The time path of adoption and an equilibrium rate is characterized. Properties of the model shed light on the reasons for low usage of seat belts and limited purchase of federally subsidized flood insurance.
1. Introduction
The study of empirical data on the adoption of protective activities reveals a puzzling phenomenon. People are often willing to protect themselves against risks which have a moderate frequency of occurrence even though the potential losses are small. On the other hand, they are often reluctant to protect themselves against low probability events with high losses. The unwillingness of individuals to obtain protection against low probability events is illustrated by two examples which have attracted public attention in the United States. One involves the use of seat belts, the other insurance against floods. Only between 10 and 20 percent of automobile passengers utilize seat belts although engineering studies reveal that these safety devices are highly *We would like James Vaupel for anonymous referee SES81-12512, and Austria.
to thank Brian Arthur, Lawrence Berger, Mark Pauly, Philip Parker and helpful discussions and comments on earlier drafts of this paper. An also provided useful comments. This research is supported by NSF Grant by the International Institute of Applied Systems Analysis, Laxenburg,
0167-2681/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
2
H. Kunreuther et al., A behavioral model of the adoption of protective activities
effective in reducing the probability of death or serious injury from automobile accidents. Consumers have shown a noticeable lack of interest in purchasing cars with an automatic seat belt even though the annual costs of this feature are as low as $24.50 [Arnould and Grabowski (1981)1. In flood prone areas, homeowners have had an opportunity to purchase highly subsidized flood insurance since 1968, yet relatively few had done so before the U.S. government required coverage as a condition for subsidized disaster relief and Federally funded mortgages [Anderson (1974),1. People do have an interest in insuring themselves against moderate probability-low loss events. One Pennsylvania state insurance commissioner found this out the hard way when he instituted a mandatory $100 deductible for automobile collision policies which would have saved consumers millions of dollars per year. His action proved so unpopular that it eventually had to be rescinded in favor of the previously existing $50 deductible [Cummins et al. (1974)-1. These examples are at odds with predictions from the standard economic approach toward risk, expected utility theory. According to this theory a risk averse person increases demand for a protective activity if the probability of the event decreases and the negative outcome proportionately increases so that the expected loss remains constant. Recently, Paul Schoemaker (1982) summarized data from laboratory and field experiments which suggest that individuals often violate the axioms upon which the expected utility model is grounded and make choices which are inconsistent with those predicted by the theory. This paper proposes an alternative model of behavior toward protective activities. It differs from the individualistic model in emphasizing the importance of interpersonal influence and past experience on the adoption of protective activities. These factors are introduced as a way of characterizing a dynamic equilibrium. The model is similar in spirit to one developed by Mark Satterthwaite (1979), who looks at the importance of friends and neighbors in providing information. It also relates the adoption of protective activities to the proportion of individuals who have recently experienced a disaster. The paper is organized as follows: The next section reviews the relevant literature indicating the importance of interpersonal influence and past experience on the adoption of protective activities. Section 3 develops a process model which shows how these two factors can characterize an equilibrium adoption rate. This model can explain observed aggregate protective behavior over time which cannot be rationalized with the traditional expected utility model. Section 4 contains a numerical example which specifies plausible parameters that generate values consistent with the observed adoption rates on seat belt usage and flood insurance purchase. The concluding section summarizes the findings and links them to policy considerations.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
3
2. The importance of social context
2.1. Interpersonal influence The role of interpersonal influence on decisions about protective activities can be illustrated by two studies. A survey that National Analysts (1971) conducted for the Department of Transportation, revealed that a significant number of individuals began to wear their seat belts on a permanent basis after they were asked by others to buckle up. In a similar vein, face-to-face interviews with over 3,000 homeowners in flood and earthquake prone areas [Kunreuther et al. (1978)] indicated that one critical factor influencing purchase of insurance was knowledge of someone else who had bought coverage, or discussions with an insured individual. Individuals had imperfect information on the price, loss and probability of a flood and presumably used personal contacts as sources for those data. The literature on the diffusion of innovations also provides a useful perspective. The pioneering studies of Griliches (1957), Mansfield (1963), and Nelson and Winter (1982) have pointed out the importance of learning from others and imitation as an element of the adoption process. 1 Cohen and Axelrod (1984) have indicated that preferences for a product may change as a result of imitation. With respect to consumer behavior the literature in marketing discusses the adoption process for new products in a similar fashion. Bass (1969) successfully tested a diffusion model of the purchase of consumer durables in which there are two classes of people: innovators and imitators. Innovators are not influenced in their initial purchase by the number of people who have already bought the product while imitators are. This approach generates logistic diffusion paths for a number of different consumer goods which are consistent with the observed data. Easingwood, Mahajan and Miller (1983), and Gatignon and Robertson (1985) summarize more recent studies that build on the Bass model. For many the adoption process occurs only after there has been personal communication with peers. Whyte (1954) shows that air conditioners were purchased on one side of the street but not on the other because of the conversations which took place in an alley between the mothers who were caring for their children. Farmers and doctors frequently became aware of a new product through impersonal sources such as the mass media, but discussed its performance with a colleague or neighbor before adopting it themselves [Katz (1961)].
2.2. Influence of experience Protective activities differ from consumption goods because they are contingent claims. Rewards are received only when a particular state of 1See Rogers (1983) for a summary of a large number of empirical studies of the innovation process which supports this view.
4
H. Kunreuther et al., A behavioral model of the adoption of protective activities
nature occurs (e.g., an accident). For some protective activities the return is visible (e.g., a payment from an insurance company). For others the return is less noticeable, such as a fatality avoided by wearing a seat belt. People learn about the magnitudes, probabilities and consequences of losses from personal experience and the experiences of others. Tversky and Kahneman (1973), have provided evidence that individuals exhibit an availability bias, whereby the subjective probability assigned to an event is influenced by the ease with which one can recall an event. A recent accident or disaster is thus likely to have a disproportionate impact on the estimate of the probability compared to one in the distant past and thus may generate demand for protective activities. Cohen and Axelrod (1984) would attribute this interest in protection to the element of surprise that a negative event generates if the world is too complex for an individual to develop a correctly specified model of the environment. Evidence from natural disasters illustrates the importance of a recent flood or earthquake in triggering interest in insurance. Three communities in northern New Jersey sampled in the flood insurance field survey reported in Kunreuther et al. (1978), suffered flood damage in 1973. Virtually the entire year's sale of flood insurance policies occurred in the two months following the flood. Similarly, the purchase of earthquake insurance increases sharply after a quake, but decreases steadily thereafter, as the memories become less vivid [Steinbrugge et al. (1969),1. The trend of flood insurance coverage in Pennsylvania also suggests that if there is no disaster for a sufficient period of time, people will cancel their policies. For example, from 1977 to 1982 the number of flood insurance policies in the state dropped approximately 50 percent, from 127,000 to 64,000, because of relatively minor flooding in the state during this period [-Kury (1982),1. People appear to view such protective activities as an investment rather than a contingent claim. If they do not collect on their policies for a prolonged period they tend to let them lapse. Hence the importance of exogenous events as personal reminders of the value of insurance coverage. People also use the experiences of their friends, relatives, neighbors, and aquaintances to judge the probabilities and impacts of various misfortunes. In addition to the low costs of obtaining such information [see Porath (1980)1, peers provide certain types of qualitative data which cannot easily be obtained from other sources [Pauly and Satterthwaite (1981),1. The experience of a neighbor who wore a seat belt during an accident has tangible meaning where dry highway statistics do not. An individual may be able to obtain more insight into the terms of his insurance policy after speaking with a friend or neighbor who recently settled a claim, than discussing this subject with an insurance agent. In addition, one can trust the judgments of friends since they have no monetary incentives, as sellers do, to promote certain activities.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
5
3. A wocess oriented madd In this section, we present and analyze a formal model which highlights the importance of social context and past experience in the adoption of protective activities. 3.1. The formal model The model of choice is grounded on a set of simplifying assumptions. All consumers are assumed to face the same objective risk, but their perceived expected loss due to a particular event differs depending upon their past experience, as well as upon the information they have collected from publications and the media. Only one homogeneous form of protection is considered. We assume that individuals learn about the costs and benefits of a protective activity from friends and neighbors before making direct inquiries to sellers. This assumption is similar to the one used by Satterthwaite (1979) in developing a model for the adoption of a reputation good such as a physician's services where information on quality was obtained through a series of inquiries from personal acquaintances. In the case of protective activities some individuals may inadvertently learn about the product from others rather than actively searching for information. At each point of time people are continuing or discontinuing the use of a protective activity depending upon its price, p, the subjective probability of the event and the estimated loss. The latter two factors are captured in the parameter, L, which represents the expected loss. Behavior is assumed to be different for individuals who recently have suffered a loss and those who have not. Hence the overall adoption and discontinuation rates will depend on the fraction Ht of the population who have experienced a loss during the most recent time interval At. The magnitude of these variables, p, L a n d / / , , influence the parameters characterizing the adoption and discontinuation process. We assume a fixed population of relevance to the problem and specify the fraction of that population, At, who protect themselves against a given risk at time t. This fraction determines the probability that a person who has not adopted a protective activity will meet someone who is currently protected [i.e., At ( 1 - A t ) ] . The variable is assumed to change over time according to the following differential equation: A(t) = ac, A,(1 - A,) + ao,(1 - A,) - 3c, A, (1 - A,) - 5o, A,
(1)
where A(t) = the increase (or decrease) in A at instant t, A, =the fraction of the (fixed) population protecting themselves against a given contingency at time t,
6
H. Kunreuther et al., A behavioral model of the adoption of protective activities
~ct =the fraction of interpersonal contacts between adopter and nonadopters, which result in a non-adopter adopting protective behavior at time t, ~ot =the fraction of non-adopters who adopt protective behavior for all other reasons at time t, ~ct =the fraction of interpersonal contacts between adopter and nonadopters, which result in an adopter discontinuing protection at time t, and ~ot =the fraction of adopters who discontinue protective behavior for all other reasons at time t. We have already argued in section 2 that actual experiences with hazards provide an important stimulation to people to protect themselves against these hazards. Social interaction is affected in two ways by these experiences. First, the diffusion of information about the protective activity increases when /7, increases. People who experience an accident or disaster communicate more about the importance, availability and convenience of certain forms of protection as well as about their subjective feelings toward protective activities. Second, these recent experiences raise their estimate of the probability of a future disaster so that they are more likely to favor protection at any given price and loss estimate. Hence, we expect communication between protected and unprotected individuals to result more frequently in the unprotected person deciding to adopt protection and less frequently in the protected individual discontinuing it when one or both of them have recently experienced the loss. We capture this effect in our model by specifying that ~ct is positively related to /7t and that ~Sct is negatively related to/7tA recent loss experience increases the chance that people who have adopted a protective activity will continue their use of it, and that others initiate protection. People who have recently experienced flooding are more likely to renew their coverage, than those who have been spared this event. Seat belt users who have been in an automobile accident are more likely to buckle up than accident-free drivers. We postulate, therefore, that recent experiences with accidents, other things being held constant, increases the adoption parameter ~ot and decreases the discontinuation parameter ~ot. A relatively low price will induce consumers to buy the product and continue its use over time even if they don't suffer a loss in the future. A high price is less likely to convince individuals to purchase the product initially. If they do purchase it, they are less likely to continue protection in the future. Hence, we assume that the magnitudes of the adoption parameters (~ot and ~ct) vary inversely with p while the discontinuation parameters (6or and 6ct) vary directly with p. The reverse holds for the expected loss, L. As estimated expected losses increase, a larger percentage of the population will adopt and maintain protection for given levels of/7, and p.
H. Kunreuther et al., A behavioral model of the adoption of protective activities
7
3.2. Comparative statics and dynamics The equilibrium proportion of people protecting themselves against a given risk is attained when the numbers of new adopters and discontinuers are identical, A(t)=0. Given positive values of the parameters ~c,, ~o,, 6o,, and 6c, there exists a unique and stable equilibrium proportion of adopters which lies in the (0, 1) interval. Let Ae represent this equilibrium proportion. Appendix A shows that this equilibrium value is - (OCo,+ 6o, -- =c, + act) + x/(=o, + 6ot - ~c, + act)= + 4O(o,(Occ,- 6c,)
Ae
(2)
2(ot¢,--6c,),
where ~c, =/=6or If ~c, =6c~ there is no net effect of interpersonal communication. In this case eq. (1) implies that Ae=otoJ(O~ot+6o,). The following three statements are implications of this model.
Statement 1. Holding the expected loss and the proportion of the population recently experiencing the risk constant, the equilibrium proportion of the population protecting themselves decreases as the price of protection increases. Statement 2. Holding the price of protection and the proportion of the population recently experiencing the risk constant, the equilibrium proportion of the population protecting themselves increases as the expected loss increases. Statement 3. Holding the price of protection and the expected loss constant, the equilibrium proportion of the population protecting themselves increases as the proportion of the population recently experiencing the risk increases. Proofs of these three statements appear in appendix B. Statements 1 and 2 are intuitively clear and are consistent with predictions from an expected utility model. Statement 3, however, is quite different from anything derived from expected utility but is consistent with observed behavior. Protection against frequent risks with lower losses is more common than against infrequent but more dangerous risks; hence the demand for low deductible insurance and the reluctance to wear seat belts. Our formulation of the problem also sheds light on the case of disasters like floods and earthquakes. Unlike automobile accidents which occur roughly to a constant proportion of drivers each year, disasters tend to affect large proportions of certain population groups in a given year and then no one for a long period of time. In this case, the level of protection immediately
8
H. Kunreuther et al., A behavioral model of the adoption of protective activities
jumps after the disaster as people react to the experience of the loss. The level of protection may even continue to increase for a while as people learn about the losses of their friends and neighbors. As the disaster recedes in time, fit becomes zero and the proportion protecting themselves falls asymptotically toward A*, the equilibrium level of protection associated with no one in the population having had recent disaster experience. With the coming of the next disaster H t increases and the entire process repeats itself. This verbal description of the time path of adoption of protective activities in the case of disasters can be formalized in two statements whose proofs are contained in appendix B. 4. A disaster with its attendant change in 11 t from zero to a strictly positive number brings about an increase in the proportion protecting themselves. Statement
Following the disaster, when f i t reverts back to zero, the proportion protecting themselves falls and asympotically approaches A* from above. S t a t e m e n t 5.
To summarize, our model produces five formal implications, three of which pertain to behavior toward accidents and two of which relate to disasters. We show with regard to accidents that the equilibrium level of risk Coverage is negatively related to its price, positively related to the expected loss, and positively related to the fraction of the group who have recently experienced a loss. Immediately after a disaster the coverage level rises and subsequently falls asymptotically to the level of coverage which would obtain if no one in the group had had any recent risk experience.
4. Numerical examples Two numerical examples, one concerning seat belts and one concerning flood insurance, illustrate the model. Arnould and Grabowski (1981), reported a seat belt usage rate of about 14 percent in 1978. This figure is, of course, consistent with an infinite number of values of the four adoption and discontinuation parameters of our model. What is important here, however, is to determine if, in fact, there exist plausible parameter sets which generate the 14 percent figure. Table 1 contains two parameter sets, those in columns (1) and (2) which produce the requisite 14 percent usage rate with reasonable underlying assumptions. In column (1) we have assumed that the net effect of interpersonal communication is favorable to seat belt use (ac,-gct =0.03) and the impact of external factors such as past experience favoring adoption is
H. Kunreuther et al., A behavioral model of the adoption of protective activities
9
Table 1 (1)
(2)
(3)
act aot 6c, ~5ot
0.1 t 0.02 0.08 0.15
0.01 0.01 0.07 0.01
0.14 0.05 0.05 0.12
Ae
0.14
0.14
0.42
relatively small (~ot =0.02). To generate the observed usage rate requires an external discontinuation rate of 6¢t = 0.15. In column (2) we have considered a situation where both the external adoption and discontinuation rates are close to zero. Under this scenario the observed usage rate of 14 percent can only be obtained if there is a net negative impact of interpersonal communication (~¢,-6ct=-0.06). This might be a realistic vector of parameters for societies in which high prestige is attached to driving skills and seat belt usage might be seen as a lack of confidence in one's driving. Column (3) describes the dramatic increase in overall adoption which would occur if people tend to react more favorably to information provided to them. The adoption parameters ~ct and ~ot are increased by 0.03 from column (1), while the discontinuation parameters are decreased by the same amount. Although the first two columns in table 1 produce the same (static) equilibrium level of adoption, the underlying dynamics of the process are quite different. To note this difference, one can determine the net adoption (NA) and net discontinuation (ND) rates in equilibrium, which are computed as
NA=o~eAe(1-Ae)+O~o(1--Ae)
and
ND=feAe(1-Ae)+foA e.
Of course, in an equilibrium situation, we have NA=ND. For columns (1) and (2) in table 1, the rates are, respectively, NA=ND=O.03 and N A = ND=0.0098. For the parameters in column (1), about 3 percent of the population newly adopt (or discontinue) seat belt usage in each period; for those in column (2) that figure is less than 1 percent. This corresponds to 27 percent of seat belt users in the first case and 7 percent in the second case. If we had statistics on the change in behavior regarding seat belt usage, we would be able to identify which of the two parameter sets more accurately characterizes reality. We can also determine the impact of a change in any of the adoption or discontinuation parameters on the equilibrium level of protection. W h e n / i t = 0, eq. (1) can be rewritten as
10
H. Kunreuther et al., A behavioral model of the adoption of protective activities
3or
%t
where
~)ct = (Xct -- (~ct"
In order to maintain a given equilibrium, Ae, changes in either of the exogenous parameters, 0tot or tSot, require a larger offsetting change in Yc,, the net interpersonal contact effect. Suppose Ae =0.14, then for a 0.01 increase in % , yet must decrease by 0.07 (i.e., 0.01/0.14) to maintain this equilibrium level. If 5or increases by 0.01, then ~'~tmust increase by 0.0116 (i.e., 0.01/0.86). The second example shows the behavior of the model where an exogenous event like a flood suddenly induces a large part of the population to adopt protection. We compare two kinds of such events, a more frequent but less severe one with a rarer, but more severe flood. The parameters depicted in table 2 refer to the case where //t = 0 and, in this example, apply to both types of floods. Table 2 ctct ~o, t$ct 3°,
0.01 0.01 0.08 0.05
Ae
0.08
For the severe flood, we chose an interval of 70 periods and an adoption level after the disaster of 75 percent. For the small flood, the interval between disasters is 30 periods and only 50 percent of the population adopt protective activities immediately following it. The adoption path for both scenarios is depicted in fig. 1. This figure shows that in most years a higher portion of the population protects themselves against the smaller flood than against the large one.
5. Conclusions
We argue in this paper for a new approach to the adoption of protective measures against risk. Expected utility theory takes the isolated individual as the unit of analysis. The individual, given information on probabilities and losses, decides on whether to insure himself against various risks. Aggregating across these individuals yields the proportion of the group who protect themselves. Our approach, on the other hand, is dynamic and treats people in their social context. It is based on the idea that the group protecting themselves from a given risk is always in flux, with some people entering it and others leaving it, based, in part, on their own experiences and in part on
H. Kunreuther et al., A behavioral model of the adoption of protective activities
t
11
Relatively Large and Infrequent Flood
0.7 .4~
0.6 Proportions of Population with Insurance Coverage
Relatively Small and I~ Frequent Flood
0.5
0.4
0.3
0.2
0.1
I
10
I
20
I
30
I
40
I
50
1
60
I
70 Time
Fig. 1. Time paths of protection for two types of flood.
the experience of others. We define the equilibrium rate of protection in this dynamic model as the value where the adoption and discontinuation rates balance. The model enables one to analyze the deviations from equilibrium and the dynamic path of adjustment. In our view, it is natural that adoption of protection increases after a catastrophe and falls back to an equilibrium level over time. The adoption model and expected-utility theory yield qualitatively identical implications with respect to the effects of changes in the price of coverage and the expected loss on the equilibrium level of protection. They differ with respect to their implications of a change in the probability of the loss-producing event. The adoption model suggests that, other things being equal, the protection rate will be positively associated with the probability of the event for a fixed expected loss while the expected utility model predicts the reverse. The adoption model is only suggestive about the role of the public sector with respect to information provision and regulation. If the private market
12
H. Kunreuther et al., A behavioral model of the adoption of protective activities
fails some governmental action may be deemed appropriate. For example, the public provision of information may improve markets for protective activities but only if the data arc sufficiently salient to decision makers to influence adoption. Financial incentives may also be useful in inducing more people to adopt protection. However, suppose the initial fraction of the group adopting protection is small, and the event protected against is relatively infrequent. Then government policies aimed at subsidizing the activity (as with flood insurance) and the provision of information (as in seat belt usage) are likely to have little impact. Direct governmental regulation of protectivc activities, as a last resort, may therefore be appropriate in those instances. These might include requirements coupled with appropriate penalties such as fines if one does not wear a scat belt (as in France) or nonpayment of medical payments on an insurance claim if the injured person had not worn a seat belt (as in Austria). Measures such as these may explain why thc use of scat belts in other countries is somewhat higher than in the United States. The adoption modcl implies that there is no magic formula which guarantees a socially optimal solution to the problem of protection against risk. Proper public policy toward low probability events is an inherently complicated puzzle, and simple-minded solutions to it may do more harm than good.
Appendix A: Stability and uniqueness of the equilibrium level Proof that A e is a unique and stable equilibrium level. A e
An equilibrium level
is defined by the condition that Ae =0,
or, equivalently, that
(A.I)
occtAe(1 - A e ) + o~ot(1 - A e ) - 6 c t A e ( 1
ae
- A e ) - t ~ o t A e =0.
(A.2)
-(OCot+ 3ot--°c~t+ 6ct)-+x/(~ot+ 6ot'~z~t+ 6~02+ 4~tot(°c~t"-3~t) 2(cto, - 6ot) (h.3)
where ~c, ¢3¢r. Let us denote the equilibrium value computed using the positive root as Ate+) and the equilibrium value using the negative root as Ate-). It can easily be shown that for parameters ~o,, t~o,, ~c, and 3~t in the (0, 1) interval: 0 < Ate+) < 1,
A~-) < 0
(A.4)
for
0to,--3c, > 0, and
H. Kunreuther et al., A behavioral model of the adoption of protective activities
13
A~-) > 1 for 0t¢t--6~, < 0. Therefore A(~+) is the unique equilibrium level in the (0, 1) interval. This completes the proof of uniqueness. To show that A~+) is also a stable solution, we have to show that for At > A(e+), A, <0,
and
for At
(A.5)
We rewrite the basic eq. (1) as in product form as At = -- 2(0tot- tic,)'(At- A~+)) "(A,- A~-)).
(A.6)
This product enables us to determine the sign of At for all values of A, depending on the sign of (otct-~ct). Using the inequalities in (A.4), the inequalities in (A.5) now follow directly. Appendix B: Proof of Statements 1 to 5
It follows from eq. (2) for the equilibrium rate of protection that 0A~
aA~<0
d~ot > 0 ,
dgot
and
cOA~ 0(~o _ 6¢,) > 0 .
(B.1,2, 3)
It is also intuitively appealing that the equilibrium level of adoption rises as people become more sensitive to information favoring adoption, or the net effect of communication works in favor of adoption. Conversely, the level will drop as people are more sensitive to information against adoption. Our assumptions on the effects of//t, P a n d / , may be summarized as
O~ct OH, > O,
aOt°t > 0, aH---~
(B.4) 06 ~t
--
OH,
< O,
OOtct ~<0, @
06or
--<
0//,
O.
aOtot --<0, 0p
(B.5) 06¢t > 0, @
06o, > 0.
@
14
H. Kunreuther et al., A behavioral model of the adoption of protective activities OOtct
O~°t > O,
~L > 0 ,
OL (B.6)
06°,
060,
--<0, 0L
--<0. 0L
Statements 1-3 now can conveniently be re-formulated as follows:
Statement 1
OA~ OA~ O~ot OA~ 060,
~=
dp
.
.
.
.
OOto, Op + 06o, dp >0
<0
<0
+
>0
OA~
c9(ot~,~6~,) >0
0(0~¢t--0et )
"
Op
<0.
(B.7)
<0
Statement 2
cgAe OA~ 0~o, OAe 060, OL =0 >0
OA~
O(%-g¢,)
o,'0L + o o," OL >0
<0
<0
>0. >0
>0
OAe
0(~et--~ct)
(B.8)
Statement 3 t3Ae
OAe Ofio, OA~ O~ot
?E
-5-E,=o o,ort, + o o,or1, + >0
>0
<0
<0
>0
>o.
(B.9)
>0
Statement 4 is an immediate consequence of the fact that a disaster causes a discontinuous increase in/-/t and that aAe/OIIt > O. Statement 5 follows from (B.9), the stability property shown in appendix A and the fact after the disaster At increases to a level higher than the equilibrium level associated with/-/t = 0.
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H. Kunreuther et aL, A behavioral model of the adoption of protective activities
15
Easingwood, C., V. Mahajan and E. Muller, 1983, A non uniform influence innovation diffusion model of new product acceptance, Marketing Science 2, 273-295. Gatignon, H. and T. Robcrtson, 1985, A propositional inventory for new diffusion research, Journal of Consumer Research, June. Griliches, Z., 1957, Hybrid corn: An exploration in the economics of technological change, Econometrica 25, 501-522. Katz, E., 1961, The social itinerary of technical change: Two studies on the diffusion of innovation, Human Organization 20, 70-82. Kunreuthcr, H., R. Ginsberg, L. Miller, P. Sagi, P. Slovic, B. Borkan and N. Katz, 1978, Disaster insurance protection: Public policy lessons (Wiley, New York). Kury, F., 1982, 10 years after Agnes: The lessons still unlearned, The Philadelphia Inquirer, June 21. Mansfield, E., 1963, The speed of response of firms to new technologies, Quarterly Journal of Economics 77, 290-311. National Analysts, Inc., 1971, Motivating factors in the use of restraint systems, Final Report Contract FH-11-7610, prepared for the U.S. Department of Transportation (Philadelphia, PA). Nelson, R. and S. Winter, 1982, An evolutionary theory of economic change (Harvard University Press, Cambridge, MA). Pauly, M. and M. Sattcrthwaite, 1981, The pricing of primary care physicians' services: A test of the role of consumer information, Bell Journal of Economics 12, 488-506. Porath, Y., 1980, The F-connection: Families, friends and firms and the organization of exchange, Population and Development Review 6, 1-30. Rogers, E., 1983, Diffusion of innovations (Free Press, New York). Satterthwaite, M., 1979, Consumer information: Equilibrium industry price and the number of sellers, Bell Journal of Economics 10, 483-502. Schoemaker, P., 1982, The expected utility model: Its variants, purposes, evidence and limitations, Journal of Economic Literature 20, 529-563. Steinbruggc, K., F. McClure and A. Snow, 1969, Studies in scismicity and earthquake damage statistics, U.S. Department of Commerce report, app. A, COM-71-00053 (Washington, DC). Tversky, A. and D. Kahneman, 1973, Availability: A heuristic for judging frequency and probability, Cognitive Psyc_hology5, 207-232. Whyte, W., 1954, The web of word of mouth, Fortune.