Can radiation-contaminated food be marketed?∗
Makoto SAITO†

Masataka SUZUKI

November 2011

Abstract
This paper presents a simple theoretical model to explain consistently heterogeneous patterns in consumers’ valuation on radiation-contaminated milk by explicitly
incorporating a strong preference for zero radiation risks. In particular, it establishes
a rigorous condition under which contaminated milk is still traded at discount prices
even when contamination levels are relatively high. Using an internet-based questionnaire survey consisting of 7,600 respondents, we empirically explore whether theabove condition holds. According to estimation results, as milk contains more radiation, a contaminated milk market disappears quickly among those who originallyperceive their own cancer risks to be rather low. Conversely, contaminated milk
is still traded at discount prices among those who are regarded as having alreadycarried considerable cancer risks.

∗

The authors would like to thank Tao Gu, Kyosuke Haseyama, and Teruyuki Sakaue for excellent
research assistance, and acknowledge Makoto Nirei, and seminar participants at Hitotsubashi University
for their helpful comments. The authors are grateful for a grant-in-aid from the Ministry of Educationand Science, Japan.
†
Correspondence to: Makoto SAITO, Faculty of Economics, Hitotsubashi University, 2-1, Naka,
Kunitachi, Tokyo, 186-8601, Japan, E-mail: makoto@econ.hit-u.ac.jp, phone: +81-42–580-8807, fax:
+81-42-580-8882.

1

1

Motivation

Many kinds of food were contaminated by radiation which was released from the explosion at the Fukushima No. 1 nuclear plant in March 2011. Some of such radiationcontaminated food immediately lost market liquidity even if they satisﬁed safety standards. Many researchers, including economists, interpreted such phenomena as heavily
driven by groundless and unscientiﬁc rumors, and as a consequence of irrational excessive
reaction to slight radiation possibility.
However, a particular consumer may be acutely sensitive to a possibility that foodis contaminated by slight radiation. Given such a strong aversion to low-level radiation
risks, his/her extreme reluctance to purchase slightly contaminated food can be interpreted as not irrational adverse reaction, but rational proper response. Thus, it may
not be interesting for us to ask ourselves whether a particular consumer’s reaction toradiation-contaminated food is irrational or rational.
A more appealing question may be whether consumers’ attitudes toward radiation
risks are heterogeneous enough to sustain market liquidity for contaminated food. If
there emerges a market in which contaminated food is discounted to the extent thatfood is contaminated, by risk-generous consumers, then shrinking demand among keenly
risk-averse consumers is replaced to some extent by steady demand from relatively riskgenerous consumers. We call such a discount transaction a ‘secondary’ market as opposed
to a ‘primary’ market in which food, considered as harmless, is traded without any
discount for radiation risks.
We usually interpret taking radiation risks by eating radiation-contaminated food astaking additional cancer risks. According to the prospect theory, those who originallyperceive their own cancer risks to be rather low tend to overreact to additional cancer
risks, while those who are regarded as having already carried considerable cancer risks
are likely to be insensitive to a tiny increase in cancer risks. In other words, the former(latter) type of consumers demonstrates a strong (weak) preference for zero cancer risks.
In this paper, we present a simple theoretical model to explain consistently heterogeneous patterns in consumers’ valuation on radiation-contaminated food by explicitly
incorporating varying degrees of preferences for zero radiation risks. For this purpose,
we construct as a structural model, a discrete/continuous choice model from a set of sim2

ple assumptions of idiosyncratic preference shocks. From this simple theoretical model,
we can analytically derive, given a contamination level, how many consumers purchaseradiation-contaminated milk at a primary market, how many discount it at a secondary
market, and how many reveal a refusal to buy it. In addition, this model yields ananalytical form of a density function for prices quoted by those who discount contaminated milk. Consequently, we can deﬁne statistical likelihood functions rigorously, andestimate this model by the maximum likelihood estimator.
As reasonably expected, both primary and secondary markets shrink rather quickly
among those who reveal a strong preference for zero radiation risks. Conversely, asecondary market is expected to be active among those whose preference for zero risks
is fairly weak. In the above model, we theoretically explore when active secondary
markets are substituted for disappearing primary markets, and when even secondary
markets disappear rather quickly. If the former is a case, then a market mechanism
works to some extent to sustain liquidity for contaminated food markets.
Taking radiation-contaminated milk for example, we conducted an internet-basedquestionnaire survey consisting of 7,600 consumers who were living in the Tokyo metropolitan area in August 2011. We provided each respondent with a brief description about
how safety standards are set oﬃcially for radiation-contaminated milk. That is, following the instruction issued by the International Commission on Radiological Protection(ICRP), the Japanese government set 200 Becquerel (Bq) of cesium per kilogram ofmilk as the upper limit, thereby reducing radiation-driven cancer risks to extremely low
levels.
We then asked them how they responded to milk contaminated at a level of 10 Bq
per kilogram, 50 Bq/kg, 100 Bq/kg, and 200 Bq/kg (corresponding to the upper limit).
Each responded by either purchasing it at a normal price (set at 200 yen per one literpack), discounting it below 200 yen, or refusing to purchase it at any price. We also
conducted a survey on the characteristics of the respondents, including their perception
about own cancer risks.
According to our estimation results, there are consumers whose degree of zero risk
preferences is on either side of the critical value below which a secondary market partially substitutes for a primary market. More concretely, those who originally perceive
their own cancer risks to be rather low are unlikely to purchase contaminated milk ateven heavily discount prices. Conversely, those who are regarded as having alreadycarried considerable cancer risks, including heavy smokers and regular drinkers, are rel3

atively generous to radiation-contaminated milk. Thus, there is still a possibility that asecondary market works for such risk-generous consumers.
(to be completed)2

A simple theoretical model

2.1

A basic setup

2.1.1

A case with/without a thresholdIn this section, we construct as a structural form, a discrete/continuous choice model toexplain heterogeneous patterns in consumers’ valuation about radiation-contaminatedmilk. We below formalize how the valuation of contaminated milk decreases with acontamination level.
Suppose that one liter pack of milk is usually sold at 200 yen, and that it is evaluatedat vi by consumer i when it is possibly contaminated by radiation. As mentioned in the
introduction, the upper limit of a contamination level was set at 200 Bq of cesium per
kilogram for milk by the government. We thus consider several cases in which milk is
contaminated at D Bq per kilogram, where 0 < D ≤ 200.
Suppose that consumer i evaluates one liter pack of radiation-contaminated milkaccording to the following equation:vi = −pi di + 200(1 − pi ),(1)

where pi represents a subjective evaluation about the probability that cancer risks are
realized, and di denotes a subjective assessment of a damage which would result fromthe realization of cancer risks. As equation (1) implies, the valuation of contaminatedmilk decreases with pi . We here deﬁne p̂i as the physical upper limit of pi where 0 =−p̂i di + 200(1 − p̂i ).
We consider cases with/without a threshold point. In a case without any threshold,vi = 200 if pi = 0, 0 < vi < 200 if p̂i > pi > 0, and vi = 0 if pi = p̂i . As depicted
in Figure 1-1, the ﬁrst (second, or third) case is called a case where a radiation risk is
considered as harmless (tolerable, or intolerable).
In a case with a threshold, consumer i lowers the admissible upper limit of pi fromp̂i to pi . We here deﬁne v i as the lower limit of vi where v i = −pi di + 200(1 − pi ). As
4

shown in Figure 1-2, the valuation of contaminated milk (vi ) jumps from v i to zero oncea radiation risk reaches pi in terms of probability.
Here, we interpret p̂pi as a risk-adjusted weight for the probability that an intolerable
icancer risk is realized. Once v i is available, we obtain the above risk-adjusted weight asfollows:
p̂i
200
=
.
pi
200 − v i(2)

In the estimation procedure, we indeed estimate v i .
2.1.2

Introducing idiosyncratic shocks

We below model the above individual valuation of radiation-contaminated milk by introducing a set of idiosyncratic shocks. Because deﬁnite opinions about the potential
impact of radiation contamination may not be formed among consumers, we here formulate idiosyncratic shocks such that there emerge large degrees of heterogeneity in risk
attitudes even among observationally equivalent consumers.
As described above, when a radiation risk is regarded as harmless, a consumer purchases one liter pack of contaminated milk at a normal price or 200 yen. That is,vi = 200. On the other hand, a consumer does not purchase contaminated milk at any
price when a radiation risk is regarded as intolerable. Consequently, vi = 0. As an intermediate case where a radiation risk is tolerable, a consumer purchases contaminatedmilk, but discounts it. Accordingly, 0 < vi < 200 in a case without any threshold, andv i < vi < 200 in a case with a threshold.
The above three cases are modeled as follows. The valuation vi revealed by consumer
i is characterized by two random variables, x and y, as idiosyncratic shocks. Here, x
is uniformly distributed between Di (≤ 0) and Di (≥ 200), while given x = X, y is
uniformly distributed between X and Di . That is,[]x ∼ U Di , Di ,[]y|x=X ∼ U X, Di ,

(3)
(4)

where U [., .] denotes the uniform distribution operator.
Figure 2 depicts a valuation function where vi is decreasing with D in a situation
where y is drawn from the uniform distribution given x = X. In Case 1, the consumer’svaluation (vi ) is equal to 200 yen when the contamination level (D) is lower than X (see5Figure 3-1). In Case 2, vi is between v i (≥ 0) and 200 yen when X ≤ D ≤ y (see Figure3-2). Here, the consumer’s valuation (vi ) is assumed to decrease linearly with D as longas D is below y, but v jumps from v i to zero at D = y in the presence of a threshold v i .
In Case 3, vi is equal to zero when D > y (see Figure 3-3).
Di , Di , and v i jointly parameterize the consumer’s attitude toward radiation risks.
More concretely, lower Di implies stronger preference for zero risks, when Di is given at
D (> 200). As discussed before, on the other hand, larger v i indicates a lower admissible
upper limit of the probability that a radiation risk is realized.

2.2

Computing the unconditional probability of the three cases

We below compute the unconditional probability that each of the three cases takes
place. From now on, we assume Di to be set at D (> 200) for all consumers, thereby
interpreting the value of Di relative to D as the degree of a preference for zero radiation
risks.
In Case 1, vi = 200 when D < X. Thus, how Case 1 is likely to occur corresponds
to the probability that x is between D and D.
∫ D

1
dx
D D − DiD−D=.
D − DiPr (D ≤ x) =(5)

In Case 2, v i < vi < 200 when D is between x and y. The conditional probability
that X < D < y given x = X is computed as follows:
∫ D

1
dy
D D−X
D−D
=
.
D−X

Pr ( D < y| x = X) =(6)

Thus, the unconditional probability for Case 2 is computed by integrating equation(6) over Di < x < D:
∫ D(

)
D−D
1
dx
Pr (x < D < y) =D − x D − Di
Di
)]
)
(
D−D [ (=
ln D − Di − ln D − D .
D − Di

6(7)

In Case 3, vi = 0 when y ≤ D. The conditional probability of Case 3 given x = X iscomputed as follows:
Pr ( y ≤ D| x = X) = 1 − Pr ( D < y| x = X)
=

D−X
.
D−X

(8)

By integrating equation (8) over Di < x < D, the unconditional probability of Case
3 is computed as below:
)
D−x
1
dx
D − x D − Di
Di
)
(
)]
D − DiD−D [ (=−
ln D − Di − ln D − D .
D − Di D − Di

Pr (y ≤ D) =

∫ D(

(9)

From equations (5), (7) and (9), we have Pr (D ≤ x)+Pr (x < D < y)+Pr (y ≤ D) =
1.

2.3
2.3.1

Derivation of the conditional density function of v for Case 2
The conditional density function

In Case 2, consumers purchase radiation-contaminated milk, but discount it when acontamination level is tolerable. More concretely, 0 ≤ v i < vi < 200 when X < D < ygiven x = X. As described before, the valuation function is linear with a discontinuous
valuation point where vi jumps from v i to zero at D = y. Note that v i = 0 in a casewithout any threshold.
Thus, the function of vi for Case 2 is formulated as follows:vi ( y| x = X) = 200 −200 − v i
(D − X) .
y−X

(10)

Using a change of random variable technique, we below derive the conditional density
function for vi given X < D < y from equations (4) and (10).
(200 − v i ) (D − X).
φ ( vi (y)| X < D < y) = ()
D − D (200 − vi )2

(11)

{
)}
(
i
Given vi , the lower limit of x is Dl (v) ≡ max Di , D − 200−v
D−
D. Then, byvi −viintegrating over x, the conditional density function for vi given x = X < D < y is7derived as follows:()
1(200 − v i ) (D − x)φ ( vi (y)| x < D < y) =dx)(
2
D − DiDl (v) D − D (200 − vi )(200 − v i )2=()
2 {D − Dl (v)}
2 D − D (D − Di ) (200 − vi ) (200−vi )(D−Di )2 , v i ≤ vi < vbi
2(D−D)(200−vi )= (200−vi )(D−D)2 , vbi ≤ vi < 200,
∫ D

(12)

2(D−Di )(vi −v i )where:
vbi = v i + (200 − v i )
2.3.2

D−D
.
D − Di(13)

Some statistics of the conditional distribution of v

As the appendix proves, the above-derived conditional density function of v yields theD−Didentical median and mode at vbi = v i + (200 − v i ) D−D.
i

On the other hand, the conditional expectation of v is calculated as follows:∫ 200
vi · φ ( vi (y)| x < D < y) dv

E [ vi | x < D < y] =
vi

(
)
∫ 200
v (200 − v i ) D − D
vi (200 − v i ) (D − Di )
dv
+
(
)
2
2 dv
2 (D − Di ) (vi − v i )v i 2 D − D (200 − vi )
v
bi
[
(
)
(
)]
200 − v i D − Di
D − Di200 + v i
D−D
D−D
+
ln
−
ln
.
=
2
2
D − DiD−D
D − DiD − Di
=

∫ vbi

(14)

We below present some properties of the above statistics. As reasonably expected, aconsumer discounts contaminated milk heavily as a contamination level (D) increases.
Thus, the mode (identical to the median) and the conditional average is expected to be
decreasing in D.
The mode (median) of vi , equal to vbi , indeed decreases as D increases.
∂b
vi
=−
∂D

(

200 − v i
D − Di

8

)
≤ 0.

(15)

As shown below, the conditional expectation of v is also decreasing in D:
]
(
)
(
)
D − DiD − DiD − DiD−D
D − Di
)
+
+(
(
)2 ln
2 ln
D − DiD − DiD − D (D − Di )(D − Di )
D−D
[
]
(
)
(
)
200 − v i
D − DiD−D
D − DiDi − DD − Di
)
≤
+
+(
(
)2
2
2
D − DiD − DiD − D (D − Di )(D − Di )D−D∂E [ vi | x < D < y]200 − v i
=
∂D
2

[

= 0,

(16)

where we use the inequality ln x ≤ x − 1 in the second line of equation (16).
A consumer with lower Di relative to D is more sensitive to radiation risks, anddiscounts contaminated milk more heavily. It is easy to show that both the mode (orthe median, vb) and the conditional expectation (E [ v| x < D < y]) is increasing in both
Di and v i .

2.4

A possibility that secondary markets substitute for primary markets

In this subsection, we explore whether secondary markets (Case 2) may substitute forprimary markets (Case 1) as contamination becomes serious, thereby preventing markets for contaminated milk from disappearing quickly (Case 3). That is, we examine
whether consumers trade contaminated milk actively at discount prices, even when acontamination level increases.
As shown below, the unconditional probability that Case 1 emerges is monotonicallydecreasing in a contamination level (D).
∂ Pr (vi = 200)
1
< 0.
=−
∂D
D − Di(17)

On the other hand, the unconditional probability that Case 3 emerges is monotonically increasing in D.
∂ Pr (vi = 0)
1
=
ln
∂D
D − Di(D − Di
D−D

)
> 0.

(18)

Consequently, how the probability that Case 2 behaves as contamination levels increase depends on whether a decrease in the probability of Case 1 is dominated by anincrease in the probability of Case 3.
From equation (7), the partial derivative of Pr (0 < vi < 200) with respect to D is

9calculated as follows:∂ Pr (0 < vi < 200)
1=∂D
D − Di(D−D
1 + lnD − Di

)
.

(19)

)
(
)(
1
From equation (19), if D ≤ Di + D − Di 1 − exp(1), then ∂ Pr(0<v<200)
≥ 0.
∂D
−D

1
Hence, if D−Di ≤ 1 − exp(1)≈ 0.632, ∂ Pr(0<v<200)
can be positive when D is positive.1∂D
i

In this case, a secondary market substitutes to some extent for a primary market, before
both markets shrink.
−D1≈ 0.632, then ∂ Pr(0<v<200)is always negOn the other hand, if D−Di > 1 − exp(1)∂Diative. In this case, an increase in contamination levels necessarily dampens demandin a secondary market where radiation-contaminated milk is discounted continuously
according to the contamination level.
−D

As the above argument suggests, κi ≡ D−Di can be interpreted as a key parameteriabout the degree of zero risk preferences. As discussed before, the absolute value of Di
represents how strong a preference for zero radiation risks is. Thus, as a preference forzero radiation risks is weaker, |Di | is smaller, and κi is lower. Thus, to the extent thata preference for zero radiation risks is weak, a secondary market can be substituted fora primary market.
Our empirical interest lies in how high κi is for a consumer with particular characteristics. We want to infer from the estimated value of κi how milk markets are robust(or fragile) with respect to radiation contamination.

1
1 − exp(1)≈ 0.632 has the following interesting interpretation. Choose the population size q where arare catastrophic event with occurrence probability 1/q hits averagely on one person in the population.
Then, the probability that at least one person is hit by this rare event is computed as
1

1 − lim (1 − 1/q)q = 1 −q→∞

1.
exp (1)

The fact that at least one person living in a society suﬀers from cancer due to radiation may trigger an
extremely adverse impact on a market transaction of contaminated food. In this sense, we may claimthat a market of contaminated food breaks down with probability of 0.632, when a substantial portionof consumers in a society are extremely attentive to the realization of such infrequent events. On theother hand, κi represents the probability that x is negative for consumer i. Thus, κi is interpreted as
the probability that consumer i has a strong preference for zero risks. Therefore, our proposition canbe interpreted as follows: if a consumer turns out to be keenly averse to such a rare catastrophic eventwith probability of 63.2%, then a secondary market of contaminated food indeed fails to work eﬀectively
with probability of 63.2%.

10

2.4.1

Some numerical examples−D

As discussed above, κi (= D−Di ) plays a key role in determining how the probability ofi1Case 2 behaves. If κi is greater than 1 − exp(1)(≈ 0.632), then primary and secondary

markets shrink simultaneously as D increases.
Figures 4-1 through 4-3 depict how the unconditional probability of each of the threecases changes as contamination levels change. Figure 4-1 (4-2, 4-3) assumes that κi isequal to 0.038 (0.167, 0.688). Since κi < 0.632 in the ﬁrst two ﬁgures, the probability ofCase 2 is increasing in D unless D is fairly high. In these cases, secondary markets are
somewhat robust with respect to radiation contamination. In the last ﬁgure, however,κi is greater than 0.632, and the probability of Case 2 is monotonically decreasing in
D. That is, primary and secondary markets shrink simultaneously in response to moreradiation-contaminated milk.
The above key parameter κi also plays a signiﬁcant role in determining the shapeof the density functions for Case 2. As demonstrated in Figures 5-1 (κi = 0.4) and
5-2 (κi = 0.25), the mode (equivalently the median) of the density function shifts more
downward in response to an increase in D when κi is large.

3

A statistical model

In this section, we present a statistical model by introducing a linear speciﬁcation into
the two systematic parts (Di and v i ) in the discrete/continuous choice model of theprevious section. We thus deﬁne logarithmic likelihood functions depending on how aconsumer evaluate one liter pack of milk contaminated at D. With such preparation,
we estimate a set of structural parameters by the maximum likelihood estimation.
One potential complication here is that we may observe the individual valuation ofmilk not at a single, but multiple contamination levels. In the questionnaire survey
conducted in this study, we indeed asked each respondent about his/her valuation ofmilk contaminated at four levels, that is, 10 Bq, 50 Bq, 100 Bq, and 200 Bq.
In this section, we thus propose a statistical model ﬁrst for a case with a single
contamination level, and then move to a case with multiple contamination levels.

11

3.1

A case where the individual valuation is observed at a single contamination level()
In our theoretical model, there are three consumer-speciﬁc parameters Θ ≡ D, Di , v ithat determine how an individual consumer responds to radiation risks. We expect that
Di and v i are systemically correlated with individual characteristics such as gender, age,income level, and so on. Given such systematic parts, the random variables x and y canbe interpreted as idiosyncratic preference shocks.
We thus assume the following linear speciﬁcation for these systemic parts:
Di = zi β + constD ,(20)v i = zi γ + constv ,(21)

where zi is a 1 × K vector which represents individual characteristics, and β and γ are
respectively K × 1 coeﬃcient vectors.
It is in principle possible to formulate Di = zi α + constD as well. But, we treat
D as a constant parameter to avoid potential identiﬁcation problems in estimating Di
together with Di and v i .
Given the individual valuation (vi ) of milk contaminated at D, we derive from equation (5), (7), (9), and (12), the following logarithmic likelihood functions:
 [
(
)
(
) ]
D−Di

D−D

ln
−
{ln
D
−
D
−
ln
D
−
D
} ,

i


[ D−Di D−Di
]


(
)
(
)

(200−v i )(D−Di )ln 2 D−D (200−v )2 {ln D − Di − ln D − D } ,
i
i)
[ (
]
ln L ( v| Θ, D) =
2
(
)
(
)
(200−v
)
D−D

)
i (

ln 2 D−D (D−D )(v −v )2 {ln D − Di − ln D − D } ,


(
i

i)
i
i

 [
]


ln D−D ,D−Dvi = 0v i ≤ vi < vb
vb ≤ vi < 200vi = 200.

i(22)
From equations (20) through (22), we can estimate coeﬃcients (β, γ) by the maximum likelihood estimation. In this case, D cannot be pinned down without any information about a slope of linear valuation through two or more observations included inCase 2. Thus, D must be predetermined.

12

3.2

A case where the individual valuation is observed at multiple contamination levels

In this subsection, we consider a case where the individual valuation of contaminatedmilk is observed at not a single level D, but multiple levels Dj where j = 1, 2, 3, and 4.
With more observations of the individual valuation, we can pin down more speciﬁcallythe range of two random variables x and y, thereby narrowing the range of integrationin deriving distribution (density) functions.
More concretely, we narrow the range of these random variables as follows. If Case
1 is observed for Dj , then the lower limit of x is not Di , but Dj . If Case 2 is observedfor Dj , then the upper limit of x is not D, but Dj , and the lower limit of y is not Di ,but Dj . If Case 3 is observed for Dj , then the upper limit of y is Dj .
When there are two or more observations in Case 2, the range of the valuation vi isnarrowed as well. Taking a case with two observations at both D2 and D3 included inCase 2 for example, as shown in Figure 6-1, the upper limit of x reduces to D2 whena threshold is absent (v i = 0). Consequently, the upper limit of vi at D3 becomes low
relative to 200 yen. As depicted in Figure 6-2, on the other hand, the lower limit of yincreases to D3 . As a result, the lower limit of vi at D2 becomes high relative to zeroyen. The above argument suggests that D can be estimated from the information about
the distribution of discount prices, which are observed in Case 2.
In the presence of a threshold (v i > 0), the upper limit of vi is relaxed to someextent. As shown in Figure 6-1, it increases from (a) to (b) at D3 . On the other hand,
the lower limit of vi is restricted more with v i > 0. As shown in Figure 6-2, it increases
from (a) to (b) at D2 .
Taking the above aspects into consideration, we below derive a set of logarithmic
likelihood functions in a case where the individual valuation of contaminated milk isobserved at multiple contamination levels Dj for j = 1, 2, 3, 4. In our questionnaire()
survey, D1 , D2 , D3 , D4 = (10, 50, 100, 200). For each Dj , the valuation of radiationcontaminated milk by consumer i is denoted by vij .

()
To simplify the notation, we deﬁne as follows Case ≡ k 1 , k 2 , k 3 , k 4 where k j =1, 2, 3. Each k j represents one of the three cases when a contamination level is Dj . For
example, if the valuation of contaminated milk by consumer i is observed as vi1 = vi2 =
200, v i < vi3 < 200, and vi4 = 0, then Case = (1, 1, 2, 3).
By construction, vij is non-increasing in Dj . Consequently, there are ﬁfteen combina-

13

tions in Case:( 1 2 3 4)
k , k , k , k = (1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 1, 3), (1, 1, 2, 2), (1, 1, 2, 3),(1, 1, 3, 3), (1, 2, 2, 2), (1, 2, 2, 3), (1, 2, 3, 3), (1, 3, 3, 3), (2, 2, 2, 2), (2, 2, 2, 3), (2, 2, 3, 3),(2, 3, 3, 3), and (3, 3, 3, 3).
Then, the unconditional probability of each combination is derived as follows:
Pr (case)

(
)
4
i −D

,
Pr D4 ≤ x = D


D
−D
i
i

(
)

( 3
)

Di −D 4D i −D 3
4

Pr
D
≤
x
<
D
<
y
=
ln
,

4

D i −D i [
D −D


( 3
)
( 4i
) (
) ( Di −D3 )]

1
4
3
4

Pr
D
≤
x
≤
y
≤
D
=
D
−
D
−
D
−
D
ln D −D4 ,iDi −D i
i

(
)

(
)
4
2

D i −D
Di −D
2
3
4


Pr
D
≤
x
<
D
<
D
<
y
=
ln
,
3

D −D
D −D


( 2
) D4i −Di3 ( Dii −D2 )


3
4

=
Pr
D
≤
x
<
D
<
y
≤
D
ln
,

D
D i −D 3
[(i −Di())() ( Di −D2 )]


1
3
2
3
2
3

D
−
D
−
D
−
D
ln D −D3 ,
Pr
D
≤
x
≤
y
≤
D
=
i

Di −D i

i
(
)

( 1
)

4
1

D i −D
Di −D
2
4


 Pr D ≤ x < D < D < y = Di −Di ln Di −D(2 ,

)
(
)
1
4
−D 3
i −D
ln D
Pr D1 ≤ x < D2 < D3 < y ≤ D4 = D
,
=
2
D
−D
D
−D
i ( i

)i

( 1
)
1
3
2

−D
i −D

ln D
,
Pr D ≤ x < D2 < y ≤ D3 = DDD i −D 2
i −D i

[

(
)
(
)
(
) ( i −D1 )]1Pr D1 ≤ x ≤ y ≤ D2 = D −D
D2 − D1 − Di − D2 ln D,
D i −D 2
() Dii −D4i ( Di −Di )


1
4

Pr x < D < D < y = D −D ln D −D1 ,


i
i
i
(
)


 Pr (x < D1 < D3 < y ≤ D4 ) = D4 −D3 ln Di −Di ,


1

D −D
D −D


(
) D3i −Di2 ( Dii −Di )

1
2
3

ln
Pr
x
<
D
<
D
<
y
≤
D
=
,


Di −D
D

(i
)i −D1()
2
1

Di −D i−Dln,
Pr x < D1 < y ≤ D2 = D
D i −D i ( D i −D 1)


1
(
)
1

D
−D
D
−D
 Pr y ≤ D1 =
i
i −D
−D
ln D i−D1i ,
D −D
D −D
i

i

i

i

Case = (1, 1, 1, 1)
Case = (1, 1, 1, 2)Case = (1, 1, 1, 3)
Case = (1, 1, 2, 2)Case = (1, 1, 2, 3)
Case = (1, 1, 3, 3)Case = (1, 2, 2, 2)Case = (1, 2, 2, 3)
Case = (1, 2, 3, 3)Case = (1, 3, 3, 3)Case = (2, 2, 2, 2)Case = (2, 2, 2, 3)
Case = (2, 2, 3, 3)Case = (2, 3, 3, 3)Case = (3, 3, 3, 3) .

i

(23)

Next, we derive the conditional density function for vij when k j = 2. With obser[)
vations at multiple levels, the range of x and y is in general restricted to Dx , Dx and(]
Dy , Dy , where Di ≤ Dx ≤ Dx ≤ Dy ≤ Dy ≤ Di . Given vij , the possible range of x is
[ ( )
( ))
further restricted to Dl vij , Dh vij , where:
}
{
j (
( )
)
200
−
v
i
,
Dl vij= max Dx , Dj − j
Dy − Djvi − v i}{( ))200 − vij (
j
j
j
Dh vi
.
= min Dx , D − j
Dy − Dvi − v i

(24)
(25)

Using equation (11) and considering narrower integration ranges, the conditional density

14

function for vij is computed as follows:
(
φ

¯

(
) (
)
(200 − v i ) Dj − x
1
dx
(26)
)2
(
)(
Dx − Dx
Dl (vij )Dy − Dy 200 − vij[{( )}2 {( )}2 ]
200 − v ijjj
D − Dl vi− D − Dh vij
.
)2
)(
)(
(
2 Dx − Dx Dy − Dy 200 − vij∫ Dh ( v j ))¯
vij ¯ Case

i

=

=

When Case 2 is observed at multiple contamination levels, we use the conditional
density for vij at the highest Dj to compute the corresponding likelihood function. In
the estimation procedure, we thus use the following logarithmic likelihood function for()
vi = vi1 , vi2 , vi3 , vi4 :()
ln L vi | Θ, D1 , D2 , D3 , D4 =ln Pr (Case) ,( ¯)ln Pr (Case) + ln φ v j ∗ ¯¯ Case ,iif k j ̸= 2 f or j = 1, 2, 3, 4
otherwise,
(27)

where

3.3

}
{
j ∗ = max j| v i < vij < 200 .

(28)

On measurement errors

In applying our statistical model with idiosyncratic shocks to the survey data for an
estimation purpose, we may face a serious measurement error problem. We have so
far assumed that when Case 2 is observed, the actual valuation by consumer i (vi ) is
identical to the true valuation (vi∗ ). As discussed above, however, with two or more
observations included in Case 2, the range of vi∗ is restricted further from either theabove or the below. In addition, the range of vi∗ is bounded from the below by thepresence of a threshold (v i ). Consequently, the actual valuation vi may be outside thetheoretically consistent range of the true valuation vi∗ .
The ﬁrst type of measurement errors: To deal with these issues, we introduce
measurement errors in two ways; one is rather crude, and the other is relatively sophisticated. Both types of measurement errors are applied to a case where the discounted
valuation at the highest contamination level is adopted. In this case, as mentioned
before, the range of vi∗ is bounded from the above because the upper bound of x isdetermined jointly by D and the lowest contamination level among those included inCase 2, and it is bounded from the below due to the presence of a threshold v i (≥ 0).
15

For the ﬁrst type of measurement errors, it is assumed that not only D, but alsov i is exogenously given; that is, v i is set at v for all consumers. We then treat theobserved valuation as the true valuation when vi is between the upper limit of vi∗ andits lower limit. However, if vi is greater than the upper limit of vi∗ , then vi is resetat max(vi∗ ), and if vi is smaller than the lower limit of vi∗ , then vi is reset at v. It isfurther assumed that the overvaluation (vi > max(vi∗ )) takes place with probability π,
and that the undervaluation (vi < v) occurs with probability π. Consequently, vi isobserved without any measurement error with probability 1 − π − π. The above way totreat measurement errors is ad hoc, but operational in that the magnitude of errors is
independent of the true valuation vi∗ .
The second type of measurement errors: As the second method, we specify a
simple density function for measurement errors (g1 (vi | vi∗ )) by two linear functions.

g1 (vi | vi∗ ) =By construction,

∫ 200
0


∗

 200−v∗ 2i vi ,if vi ≤ vi∗ ,otherwise.

100vi

vi∗
(200 − vi ),100(200−vi∗ )2

g1 (v | vi∗ )dv = 1 and

∫ 200
0

(29)

[vg1 (v | vi∗ )] dv = vi∗ hold. Hence,

the above formulation of measurement errors does not yield any systematic bias. See
Figure 7-1 for a shape of the above density function.
An alternative, and even simpler density function for measurement errors (g2 (vi | vi∗ ))is formulated by two uniform distributions.

g2 (vi | vi∗ ) =By construction,

∫ 200
0


∗

 200−v∗i ,if vi ≤ vi∗ ,otherwise.

100vi

vi∗
100(200−vi∗ ) ,g2 (v | vi∗ )dv = 1 and

∫ 200
0

(30)

[vg2 (v | vi∗ )] dv = vi∗ hold again.

Hence, the above formulation of measurement errors does not yield any systematic bias
either. See Figure 7-2 for a shape of the above density function.
Given the above type of measurement errors, the logarithmic likelihood is deﬁned asfollows:∫ max v∗ [iln Pr (Case) + lnvi]∗
φ ( v| Case) gk (vij | v) dv,(31)

where k is 1 or 2. It is possible to compute analytically the integral in equation (31) inboth cases (k = 1 and 2).

16

4

Questionnaire survey results

4.1

The way in which an internet-based questionnaire survey was conducted

To investigate the consumers’ response to radiation-contaminated milk, we charged theSurvey Research Center (hereafter, SRC), a Tokyo-based private research institute, toconduct an internet-based questionnaire survey in the second half of August, 2011. The
SRC had a large-scale panel of those who were currently living in the Tokyo metropolitanarea, and provided us with the sample consisting of 760 male and 760 female respondents
for each age group of twenties, thirties, forties, ﬁfties, and sixty or older. Thus, the total
sample size amounts to 7,600.
As mentioned in the introduction, following the safety measure recommended by
the ICRP, the Japanese government set the upper limit of cesium contained in milk at
200 Bq/kg. Given this safety standard, the annual radiation exposure through drinking
contaminated milk every day amounts to at most 5 millisievert per year, and a life-time
probability of the incidence of cancer increases by only 0.025%.
After providing a brief description about the oﬃcial safety standard of radiation
contamination, we asked each respondent which kind of preference he/she revealed in
response to one liter pack of milk which was supposed to be contaminated at the levelof 10 Bq/kg, 50 Bq/kg, 100 Bq/kg, or 200 Bq/kg, when he/she was assumed to purchase such slightly radiation-contaminated milk for his/her own drinking purpose. Each
respondent chose either to purchase it at a normal price (assumed to be 200 yen), todiscount it below 200 yen, or to refuse to purchase it at any positive price.2 A respondent who chose to discount it was further asked to write down a discount price up to
one digit between 0 yen and 200 yen; for example, a quoted discount price may be 121
yen or 98 yen.
Besides preferences for radiation-contaminated milk, we inquired about some of respondents’ characteristics. With respect to annual income classes, 726 respondents (9.6%of the entire sample) belonged to (i) less than two million yen, 2,685 (35.3%) to (ii) between two and ﬁve million yen, 3,042 (40.0%) to (iii) between ﬁve and ten million yen,and 1,147 (15.1%) to (iv) ten million yen or more.
2

A respondent who lived with children was given a set of the same questions in a case where he/shepurchased radiation-contaminated milk for his/her children. However, the corresponding responses are
not used in this study.

17

In terms of household structure, 4,700 respondents out of 7,600 (61.8%) had a
spouse. 4,571 respondents (60.1%) did not live with any children, while 1,508 respondents (19.8%) lived with one child, 1,231 (16.2%) with two children, 246 (3.2%) withthree children, and 44 (0.6%) with four or more children. The age of the youngest childwas between zero and two years old for 506 respondents (16.7%), between three andﬁve for 289 (9.5%), between six and ten for 368 (12.1%), between eleven and ﬁfteen for
409 (13.5%), between sixteen and twenty for 454 (15.0%), and twenty or older for 1,003(33.1%).
About a preference for luxury items, 1,546 respondent (20.3%) smoked regularly,
while 1,871 (24.6%) drank frequently. 363 respondents (4.8%) purchased organic vegetables regularly, 3,631 (47.8%) sometimes, and 3,606 (47.4%) never. In terms of other
health-related issues, 3,335 respondents (43.9%) participated in cancer insurance, and
4,071 (53.6%) had a checkup at least once a year.
As a somewhat delicate question, we asked each respondent about the possibility
that he/she would suﬀer from fatal cancer in the course of lifetime. 634 respondents(8.3%) thought that they were unlikely to suﬀer from fatal cancer. On the other hand,
1,392 respondents (18.3%) considered the probability to be below the national average(around thirty percent per life-time), 2,798 (36.8%) approximately equal to it, and 1,235(16.3%) above it. 1,514 respondents (19.9%) answered that they could not judge about
the probability of cancer incidence, while 27 (0.4%) did not answer at all.
As to a preference for normal milk, which may have a substantial impact on thevaluation of radiation-contaminated milk, 2,594 respondents (34.1%) drank milk almost
every day, 3,496 (46.0%) sometimes, and 1,510 (19.9%) never.

4.2

The frequency of the three cases and the distribution of quoted
discount prices

As reported in Table 1, respondents are more and more averse toward contaminatedmilk as the contamination level is higher. The share of Case 1 where a respondentpurchases contaminated milk without any discounting decreases as the contaminationlevel increases from 10 Bq/kg to 200 Bq/kg; it is 15.6% for 10 Bq/kg, 11.7% for 50
Bq/kg, 8.8% for 100 Bq/kg, and 5.6% for 200 Bq/kg. The proportion of Case 2 wherea respondent discounts contaminated milk also decreases with the contamination level;
it is 38.6% for 10 Bq/kg, 28.1% for 50 Bq/kg, 19.9% for 100 Bq/kg, and 13.2% for 200

18

Bq/kg. On the other hand, the share of Case 3 where a respondent never purchases
contaminated milk at any positive price increases with the contamination level; it is
45.8% for 10 Bq/kg, 60.1% for 50 Bq/kg, 71.2% for 100 Bq/kg, and 81.2% for 200
Bq/kg.
We observe all possible ﬁfteen patterns in the combination of the three cases which
emerges as a contamination level increases from 10 Bq/kg to 50 Bq/kg, 100 Bq/kg,
and 200 Bq/kg. Among 7,204 respondents whose valuation of contaminated milk is notincreasing in contamination levels, 382 (5.3%) evaluate contaminated milk at 200 yen
for the four levels (Case = (1, 1, 1, 1)), while 3,353 (46.5%) ﬁnd zero valuation for all
levels (Case = (3, 3, 3, 3)).
Out of 7,204 respondents, 449 (6.2%) skip Case 2 as a contamination level increases,or belong to Case = (1, 1, 1, 3), (1, 1, 3, 3), or (1, 3, 3, 3), while 1,185 (16.4%) include
Case 2 only once, or belong to Case = (1, 1, 1, 2), (1, 1, 2, 3), (1, 2, 3, 3), or (2, 3, 3, 3). On
the other hand, 1,142 (15.9%) individuals include Case 2 two or three times, or belong
to Case = (1, 1, 2, 2), (1, 2, 2, 2), (1, 2, 2, 3), (2, 2, 2, 3), or (2, 2, 3, 3). 693 respondents(9.6%) quote discount prices for the four levels (Case = (2, 2, 2, 2)).
As mentioned above, those who were interested in purchasing contaminated milk at
discount prices were asked to write down a discount price up to one digit between zero
yen and 200 yen. As shown in Table 2, the average quoted price decreases with thecontamination level; it is 134.9 yen for 10 Bq/kg, 130.8 yen for 50 Bq/kg, 125.4 yen for
100 Bq/kg, and 119.9 for 200 Bq/kg.
As shown in Figure 8-1, a respondent tended to quote a discount price at 10-yen
intervals for a higher range, and at 50-yen intervals for a lower range. Accordingly, themedian of quoted prices is a round number; it is 150 yen for 10 Bq/kg, 150 yen for 50
Bq/kg, 140 yen for 100 Bq/kg, and 120 yen for 200 Bq/kg.
Figure 8-2 depicts the histogram of the quoted discount prices at 50-yen intervals. As
demonstrated by this ﬁgure, the overall frequency reduces, and the mode shift downward
as the contamination level increases.

4.3

Determinants of the respondents’ original perception of their own
cancer risks

In our questionnaire survey, each respondent was inquired about the risk perceptionof cancer incidence. Here, we construct a dummy variable which takes one for those19

who considered their own cancer risks to be lower than the national average (about
thirty percent per life-time), and otherwise zero. Then, using this dummy variable
as a dependent variable, we conduct the logit estimation. The summary statistics of
explanatory variables are documented in Table 3, while the estimation result is reported
in Table 4.
According to Table 4, the tendency for respondents to consider cancer risks to berather low is noticeable among those above the age of sixty, non-smokers, those witha habit of eating organic vegetables and having a regular checkup, and non-holders ofcancer insurance. If older respondents might have thought that they had survived cancer
risks, the above estimation result may be subject to a sample selection bias. In the next
section, we use as an explanatory variable the predicted cancer risk perception which isbased on the logit estimation. The average of the predicted value is 26.8%.

5Estimation results

5.1

A preliminary examination of the distribution of quoted discountprices in Case 2

Before conducting estimation procedures, we examine with extreme care how quoted
discount prices are distributed in Case 2. First of all, we drop the observations in which
the valuation of radiation-contaminated milk increases with the contamination levels.
Consequently, the sample size reduces from 7,600 to 7,204.
As discussed before, when the discounted valuation at the highest level is adopted
among multiple observations included in Case 2, the range of the true valuation (vi∗ ) is
restricted from the above, depending on the value of D. In the presence of a threshold(v i ), its range is further bounded from both the below and the above.
Let us ﬁrst examine the case where a threshold is absent (v i = 0), and D is set at 500
Bq/kg.3 In Case = (k 1 , k 2 , k 3 , k 4 ) = (1, 1, 2, 2) (Pattern 4), (1, 2, 2, 2) (Pattern 7),(1, 2, 2, 3) (Pattern 8), (2, 2, 2, 2) (Pattern 11), (2, 2, 2, 3) (Pattern 12), and (2, 2, 3, 3)(Pattern 13), among not a few respondents, the reported valuation vi is beyond thetheoretically consistent upper limit of vi∗ . More precisely, 924 out of 1,835 respondents
included in the above six patterns violates the theoretical upper limit of vi∗ . That is,
the number of the observations with violation is 20 out of 45 in Pattern 4, 13 out of 32
3

As of November, 2011, there had been cases where the contamination level was reported to outside
the established standard (200 Bq/kg), but the reported level had never exceeded 500 Bq/kg.

20

in Pattern 7, 15 out of 44 in Pattern 8, 317 out of 693 in Pattern 11, 238 out of 457 inPattern 12, and 321 out of 564 in Pattern 13.
In the presence of a threshold (v i > 0), two additional eﬀects are generated. First,
the valuation of a discount price is bounded not only from the above, but also fromthe below at v i in the above patterns. In addition to the six patterns, the lower boundarises in the following four patterns: Case = (k 1 , k 2 , k 3 , k 4 ) = (1, 1, 1, 2) (Pattern 2),(1, 1, 2, 3) (Pattern 5), (1, 2, 3, 3) (Pattern 9), and (2, 3, 3, 3) (Pattern 14). Obviously,
the number of the observations which violate the lower bound of vi∗ increases with v i .
Second, the theoretical upper bound of vi∗ is relaxed due to a larger threshold v i . Accordingly, the number of the observations which violate the upper bound of vi∗ decreases
with v i .
To see a change in the number of the observations with violation by an increase in a
threshold, we set D at 500, and raise v as a common value for a threshold from zero to
190 at 10-yen intervals. The number of the observations which violate the lower boundincreases from zero (v = 0) to 3,015 out of 3,020 (v = 190). On the other hand, thenumber of the observations which violate the upper bound decreases from 924 out of
1,835 (v = 0) to 3 (v = 190). Consequently, the total number of the observations withviolation is smallest at v = 90.
We can observe more details about the above phenomena by Figures 9-1 and 9-2.
The six histograms in Figure 9-1 depict the distribution of quoted discount prices forthe patterns in which two or more observations are included in Case 2 (Patterns 4, 7, 8,
11, 12, and 13). In each histogram, two black (yellow or red) vertical lines represent thetheoretical range of vi given v = 0 (50 or 100). The four histograms in Figure 9-2, on
the other hand, depict the distribution of quoited prices for the patterns in which only
one observation is included in Case 2 (Pattern 2, 5, 9, and 14).
Given a large number of the observations with violation, the treatment of measurement errors is quite important in estimation procedures.

5.2

Estimation results under the ﬁrst type of measurement errors

Under the ﬁrst type of measurement errors, once the observed valuation violates the theoretical range of the true valuation, it is replaced by the theoretical upper (lower) limit.
A major drawback of this adjustment method is that neither D nor v i can be estimated,and both need to be predetermined. Given such a restrictive nature, estimation results

21under the ﬁrst type of measurement errors have to be treated as tentative ones.
Table 5 reports the estimation results for Di in the cases with and without a threshold(v i ).4 In the absence of a threshold, it is assumed that D = 500 and v i = 0 for all
consumers. As reported in the ﬁrst panel of Table 5, most individual characteristics
have signiﬁcant impacts on the determinants of Di in the case without any threshold.
Recall that lower Di implies stronger preferences for zero radiation risks.
A preference for zero radiation risks is strong among younger females with infants,
while it is weak among older males without any child. An interesting observation isthat a preference for zero risks is strong among high income individuals. Probably due
to more restrictive budget constraints, individuals with more children tend to have aweaker preference for zero risks.
As to cancer-risk-related variables, a preference for zero risks is weaker among regular
smokers and/or drinkers, those who are not interested in eating organic vegetables or
having a regular health checkup. Restating these results, a preference for zero radiation
risks is weak among those who are regarded as having already carried considerable cancer
risks.
When a set of the above cancer-risk-related variables are replaced by the predictionof a cancer risk perception based on the logit estimation (Table 4), we ﬁnd that apreference for zero risks is strong among those who perceive their own cancer risks tobe small. That is, those who originally perceive their own cancer risks to be rather low
are unlikely to purchase contaminated milk at even heavily discount prices. Note thatthose who never drink evaluate contaminated milk substantially downward.
The second panel of Table 5 reports the estimation results for the case with a threshold (v i > 0). With D = 500, we assume v i = 90 for all consumers because the numberof the observations which violate the theoretical range of the true valuation is smallest
at v i = 90 ∀i.
The overall estimation results do not diﬀer substantially between the cases with/withouta threshold. But, the signiﬁcance level of estimated coeﬃcients is lower in the case witha threshold.

4

It is assumed that the overvaluation, exact valuation, and undervaluation take place with equal
probability.

22

5.3

Estimation results under the second type of measurement errors

Under the second type of measurement errors, we can estimate not only a systematicpart of Di , but also a systematic part of v i , and a parameter D, which is assumed to becommon among respondents. Accordingly, not only the degree of zero risk preferences−D

which is measured by κi = D−Di , but also the risk-adjusted weight which is representedi200
or equation (2), can be inferred from such estimation results. Note thatby pp̂i = 200−viilower Di implies stronger zero risk preferences, while higher v i (a threshold) indicateslarger risk-adjusted weights.
Table 6 reports estimation results in a case where a combination of two linear functions is adopted as a density function of measurement errors. As shown in the estimation
about Di , the estimated eﬀects of individual characteristics on zero risk preferences are
similar to those of Table 5. A preference for zero radiation risks is strong among younger
females with infants, while it is weak among older males without any child. However,neither income class nor the number of children has any signiﬁcant eﬀect on a preference
for zero risks. As to cancer-risk-related variables, a preference for zero risks is weaker
among regular smokers and/or drinkers. In addition, those who never drink evaluate
contaminated milk downward.
In terms of the estimation of v i , females, the old, the rich, and those with infants,
but fewer children tend to apply larger risk-adjusted weights, thereby having higher
threshold points in valuation of radiation-contaminated milk. The estimated parameterof D, signiﬁcant at 260.8, is above 210 Bq/kg of cesium which was detected on March
20th, 2011.
Table 6 reports another speciﬁcation where a set of the above cancer-risk-related
variables are replaced by the prediction of a cancer risk perception based on the logit
estimation (Table 4).5 As to a systematic part of v i , we also drop age dummies because
older respondents have both larger risk-adjusted weights and lower perception of cancer
risks. We ﬁnd that those who perceive their own cancer risks to be small tend to havea fairly strong preference for zero risks as well as a pretty large risk-adjusted weight.
More concretely, the estimated coeﬃcient on the logit-ﬁtted-value is −1003.8 in Di and41.0 in v i .

5Rigorously, when we include the logit-ﬁtted-value as an explanatory variable in a nonlinear system
in order to treat endogeneity, its coeﬃcient may not be consistent. Here, we consider an endogeneity
problem in a somewhat ad hoc way at the expense of improper treatment of nonlinearity.

23

5.4

On a possibility of secondary markets−D

As discussed in Section 2, the degree of zero risk preferences κi , deﬁned as D−Di , servesi

as a key parameter in determining whether secondary markets are robust with respect toradiation contamination. That is, if κi < 0.632, then a secondary market substitutes tosome extent for a primary market among those whose characteristics are similar to that ofconsumer i. Conversely, if κi > 0.632, then both primary and secondary markets shrink
simultaneously among those resemble to consumer i in terms of individual characteristics.
In addition, we can infer from estimated v i , the risk-adjusted weight which is rep200
resented by p̂pi = 200−v. This risk-adjusted weight implies the extent that a subjectiveiiprobability of cancer occurrence is augmented by extreme aversion to radiation risks.
Obviously, the higher v i is, the larger the risk-adjusted weight is.
Table 7 reports the value of κi is computed together with a 95% conﬁdence interval
based on the estimation results reported by Table 5. Given that v i cannot be estimatedin this case, we do not have any inference about the risk-adjusted weight. According tothe ﬁrst panel of Table 7, the computed value of κi for the case without any thresholdis on either side of 0.632, depending on individual characteristics. For example, a point
estimate of κi is 0.686 with a 95% conﬁdence interval between 0.653 and 0.718 for afemale regular drinker of milk, of the age of twenties, with one infant, an annual income
between two and ﬁve million yen, uninterested in smoking or drinking, but interested ineating organic vegetables and having a regular health checkup.
On the other hand, a point estimate of κi is 0.288 with a 95% conﬁdence interval
between 0.202 and 0.374 for a male regular drinker of milk, sixty years old or older, notliving with any child, an annual income between two and ﬁve million yen, interested insmoking and drinking, but uninterested in eating organic vegetables or having a regular
health checkup. As shown in the second panel of Table 7, the pattern observed in thecases with thresholds is similar to that of those without any threshold, but the overall
valuation of κi increases in any case.
Table 8-1 reports the computed values of both κi and p̂pi for the case where canceririsk-related variables are used as explanatory variables, while Table 8-2 briefs thosevalues for the case where they are replaced by the logit-ﬁtted-value. According to theﬁrst panel of Table 8-1, most point estimates of κi are above 0.632. The computed valueof κi is barely below 0.632 for a male regular drinker of milk, sixty years old or older,
not living with any child, and interested in smoking and drinking. As shown in the24second panel of Table 8-1, the computed value of the risk-adjusted weight does not diﬀer
substantially among respondents. For example, it is around two for young males andabout three for old females.
As reported in Table 8-2, on the other hand, the degree of zero risk preferences aswell as the risk-adjusted weight diﬀer substantially among respondents, depending onwhether he/she perceives his/her own cancer risk to be rather low. For example, an oldmale who perceives his cancer risk to be high carries κi = 0.466 much lower than 0.632,
and applies risk-adjusted weights around two times as large as his subjective probability.
On the other hand, a young female with an infant who perceives her cancer risk to be
low has κi = 0.895 much higher than 0.632, and applies risk-adjusted weights more thanﬁve times as large as her subjective probability.
The overall results suggest that if the individual characteristics are conditioned byrespondents’ perception about cancer risks, then how a preference for zero risks is strong,and how large the risk-adjusted weight is depend critically on whether a respondent perceives his/her cancer risk to be low. Concretely, those who perceive their own cancer risk
to be rather low reveal strong preferences for zero risks in deciding whether they purchase radiation-contaminated milk, and demonstrate extreme aversion toward radiation
risks in determining how much contaminated milk is discounted.

6Conclusion

This paper presents a simple theoretical model to explain consistently heterogeneous
patterns in consumers’ valuation on radiation-contaminated milk by explicitly incorporating a strong preference for zero radiation risks. In particular, it establishes a rigorous
condition under which contaminated milk is still traded at discount prices even whencontamination levels are relatively high. Using an internet-based questionnaire survey
consisting of 7,600 respondents, we empirically explore whether the above condition
holds. According to estimation results, as milk contains more radiation, a contaminatedmilk market disappears quickly among those who originally perceive their own cancer
risks to be rather low. Conversely, contaminated milk is still traded at discount prices
among those who are regarded as having already carried considerable cancer risks.
(to be completed)25

Appendix 1: Conditional distribution for v in Case 2
In this appendix, we prove two propositions about the conditional distribution for v inCase 2.
Proposition 1 vb in equation (13) is the median of the conditional distribution for v
given v i ≤ v < 200.
Proof. From equation (12), the conditional density function φ ( v (y)| x < D < y) is
continuous for v ∈ [v i , 200). And, the conditional probability of the event ‘v i ≤ v < vb’
given v i ≤ v < 200 is calculated as follows:

Pr ( v i ≤ v < vb| v i ≤ v < 200) =

∫ vb

=

(200 − v i ) (D − Di )()
2 dv
v i 2 D − D (200 − v)
]vb[
(200 − v i ) (D − Di )1()
200 − v v
2 D−D

i

=

1
.
2

(32)

Because φ ( v (y)| x < D < y) is continuous for v ∈ [v i , 200), equation (32) means that
Pr ( vb ≤ v < 200| v i ≤ v < 200) = 1 − Pr ( v i ≤ v < vb| v i ≤ v < 200) = 21 . Therefore, vb is
the median of the conditional distribution for v given v i ≤ v < 200.
Proposition 2 vb in equation (13) is the mode of the conditional distribution for v given
v i ≤ v < 200.
Proof. From equation (12), the derivative of φ ( v (y)| x < D < y) with respect to v is
calculated as follows: (200−vi )(D−Di )∂φ ( v (y)| x < D < y)  2(D−D)(200−v)3 ≥ 0, v i ≤ v < vb=∂v
− (200−vi )(D−D)3 ≤ 0, vb ≤ v < 200.
2(D−D )(v−v )
i

(33)

i

Further, φ ( v (y)| x < D < y) is continuous at v = vb. Therefore, φ ( v (y)| x < D < y)takes the largest value at v = vb, and vb is the mode of the conditional distribution.

26

References[1] Kahneman, Daniel, and Amos Tversky, 1979, “Prospect theory: An analysis of decision under uncertainty,” Econometrica 47:2, 263–291.
(to be completed)27

iTable 1: The share of the respondents classified according to thethree cases
Case 1: apurchase
without any
discounting
1,189(15.6%)
892(11.7%)
670(8.8%)
428(5.6%)

10 Bq
50 Bq
100 Bq200 BqCase 2: apurchase withdiscounting

Case 3: nopurchase at any
price

2,934
(38.6%)
2,137(28.1%)
1,516(19.9%)
1,000(13.2%)

3,477(45.8%)
4,571(60.1%)
5,414(71.2%)
6,172(81.2%)

total
7,600
7,600
7,600
7,600

Table 2: Distribution and statistics of discount prices quoted in Case 2
The contaminationlevel (Bq)

total

10 Bq

2,934

50 Bq

2,137

100 Bq1,516

200 Bq

1,000

quoted discount prices (yen)

～19

20～39

40～59

60～79

80～99

20

33

99

11

72

mean

mediankurtosis

231

134.9

150

4.15

162

130.8

150

3.60

104

125.4

140

3.10

58

119.9

120

2.79

100～119 120～139 140～159 160～179 180～199
595

323

1,010

540

500

218

670

364

370

145

437

231

263

145

(0.7%)(1.1%)(3.4%)(0.4%)(2.5%)(20.3%) (11.0%) (34.4%) (18.4%)(7.9%)(0.9%)(1.2%)(4.2%)(0.5%)(3.6%)(23.4%) (10.2%) (31.4%) (17.0%)(7.6%)(1.4%)(2.1%)(5.9%)(1.5%)(4.2%)(24.4%)(9.6%)(28.8%) (15.2%)(6.9%)(2.7%)(2.1%)(7.7%)(1.9%)(5.7%)(25.9%)(7.4%)(26.3%) (14.5%)(5.8%)

20

21

27

26

32

21

90

90

77

11

23

19

76

63

57

259

74

ii

Table 3: Descriptive statistics of explanatory
variables
Mean
male dummy
age dummy(twenties)(thirties)(fourties)(fifties)
income classspouse dummy
the number of children
the age of the youngest childwithout any childyounger than 3 years old
between 3 and 10
between 11 and 15
between 16 and 20
over 20
smoker dummy
drinker dummyno habit of eating organic
vegetables
participation in cancer insurance
regular health checkup
no habit of drinking milk
the predicted cancer risks basedon the logit estimation

0.5
0.2
0.2
0.2
0.2
2.607
0.618
1.643
0.601
0.067
0.086
0.054
0.060
0.132
0.203
0.246
0.474
0.439
0.536
0.199
0.268

Standard
Deviation

0.855
0.904

iii

Table 4: Estimation result of the logit model for respondents’ perception of cancer risks

explanatory variables
male dummy
age dummy (twenties)(thirties)(fourties)(fifties)
smoker dummy
drinker dummyno habit of eating organic
vegetables
participation in cancer insurance
regular health checkup
constant
Number of observations
Wald chi-squared
P-value of chi-square test
Pseudo R squared
Log likelihood

A dummy variable of those who perceive cancer
risks to be lower than the national average
Coefficient
Marginal effect
0.135
**
0.026
**
(0.055)(0.011)-0.467***-0.085***(0.086)(0.014)-0.481***-0.087***(0.084)(0.014)-0.364***-0.067***(0.082)(0.014)-0.068-0.013(0.079)(0.015)-0.314***-0.058***(0.071)(0.012)-0.039-0.008(0.064)(0.012)-0.137**-0.026**(0.054)(0.011)-0.365***-0.070***(0.056)(0.011)0.046
0.009(0.055)(0.011)-0.545***(0.075)7573
125.5
0.00000.0143-4335.5

Note 1: A dependent dummy variable takes one for a respondent who perceive life-time cancer risk to be lower
than the national average.
Note 2: *, **, and *** implies the significance level at 10%, 5% and 1% respectively.
Note 3: Among 7600 respondents, 27 did not answer the question concerning the perception of cancer risks.

iv

Table 5: Maximum likelihood estimation results for D with D = 500
under the first type of measurement errors
v=0

v = 90Specification 1

male dummy
age dummy (twenties)(thirties)(fourties)(fifties)
income classspouse dummy
the number of children
youngest child age dummy(between 0 and 2)(between 3 and 10)(between 11 and 15)(between 16 and 20)(over 20)
smoker dummy
drinker dummyno habit of eating organic
vegetables
participation in cancer
insurance
regular health checkup
no habit of drinking milk
The predicted cancer risks
based on the logit estimation
constant
Number of observations
Wald chi-squared
P-value of chi-square test
Log likelihood268.55(29.39)-160.32(43.45)-171.86(44.90)-160.48(45.22)-105.09(39.68)-71.73(15.67)-28.36(32.82)
68.69(33.78)-321.21(97.81)-238.56(87.57)-194.12(93.95)-159.58(83.90)-153.34(66.41)
54.03(29.40)68.44(28.10)70.77(25.94)-17.82(26.37)-13.73(25.99)-635.01(55.98)-492.52(51.10)7204
420.1
0.0000-41720.4

Specification 2

***
***
***
***
***
***

**
***
***
**
*
**
*

309.87(27.34)-180.64(47.78)-203.62(51.60)-189.02(50.30)-96.12(39.89)-79.02(15.07)-41.15(32.98)74.24(28.55)-327.43(90.66)-249.19(81.59)-205.49(88.61)-164.73(78.99)-179.61(60.52)

Specification 1

***
***
***
***
**
***

***
***
***
**
**
***

**
***

***

***

-626.43(56.00)-496.89(272.76)-292.00(102.23)7204
441.6
0.0000-41728.5

***
*
***

Specification 2

202.67(39.30)-272.36(65.90)-321.45(52.35)-283.18(66.39)-202.08(49.12)-28.14(23.83)-22.21(44.68)38.69(43.57)-238.61(121.95)-116.61(114.27)-159.67(128.95)-87.62(114.95)-97.48(73.18)71.41(44.65)102.95(42.43)45.34(37.82)-22.88(37.66)-9.36(38.35)-791.24(81.05)***-748.26(72.37)7204
232.0
0.0000-33711.5

***

***
***
***
***

*

248.14(37.89)-305.29(75.01)-368.66(67.71)-318.76(76.43)-198.49(57.07)-37.66(23.14)-21.62(46.63)46.91(44.60)-262.45(124.01)-130.20(114.34)-173.90(82.11)-99.96(115.19)-135.56(89.32)***-776.34(81.16)-534.53(425.57)-538.97(159.30)7204
210.7
0.0000-33717.4

***

***
***
***
***

**

**

**

***

***

Note 1: *, **, and *** implies the significance level at 10%, 5% and 1% respectively.
Note 2: The observations in which respondents who raise vi as the contamination level increases are dropped.
Consequently, the number of observations reduces to 7204.

v

Table 6: Maximum likelihood estimation results for D , v and D
under the second type of measurement errors (linear density functions)
Specification 2

Specification 1
lower D
male dummy
age dummy (twenties)(thirties)(fourties)(fifties)
income classspouse dummy
the number of children
youngest child age dummy(between 0 and 2)(between 3 and 10)(between 11 and 15)(between 16 and 20)(over 20)
smoker dummy
drinker dummyno habit of eating organic
vegetables
participation in cancer
insurance
regular health checkup
The predicted cancer risks
based on the logit estimationno habit of drinking milk
constant
Number of observations
Wald chi-squared
P-value of chi-square test
Log Likelihood

lower v

361.24 ***(47.08)-124.55*(69.68)-207.50 ***(72.99)-171.42 **(72.12)-135.68 **(64.22)-24.76(25.46)-55.06(54.03)52.03(55.33)-392.62 **-10.11(1.28)-28.89(2.19)-30.69(2.07)-25.91(2.36)-17.11(1.44)
1.79(0.72)-1.69(1.60)-4.24(1.21)
22.98(155.25)-207.55(138.22)-192.77(150.21)-237.26(141.89)-192.70(109.01)
81.59(46.56)90.36(44.16)48.76(41.18)(2.24)
16.48(4.02)
21.81(4.02)
15.31(3.09)
10.18(2.15)-0.48(1.43)
1.39(1.34)-1.29(1.17)

*
*
*
**

10.58

1.52

(42.08)-0.08(42.24)(1.12)-0.13(1.11)-681.61 ***(85.76)-913.90 ***(85.87)7204
215.6
0.0000-29465.6-1.26(1.82)
134.28(2.31)

upper D***

lower D
412.81(45.26)-249.92(75.33)-341.00(83.29)-285.03(78.72)-171.09(64.15)-23.29(24.50)-80.77(53.48)53.40(55.43)-394.43

***
***
***
***
**

***
***

(157.40)-189.61(138.85)-176.25(149.15)-232.93(141.36)-225.10(108.71)

***
***
***
***

-1003.82***

260.84(2.02)***

lower v******

-14.01(1.09)

upper D

***

***
***
***

**

*
**

0.67
(0.65)
11.38(1.38)-3.06(1.21)
8.84(2.18)
0.70(3.35)
12.41(4.21)
9.17(2.94)
19.75(2.03)**

40.98(450.00)-671.62 ***(85.49)-513.25 ***(166.03)7204
218.3
0.0000-29507.9(10.09)-1.90(1.64)
100.98(3.03)

***
**
***

***
***
***

***

***

260.95(2.04)***

Note 1: *, **, and *** implies the significance level at 10%, 5% and 1% respectively.
Note 2: The observations in which respondents who raise vi as the contamination level increases are dropped.
Consequently, the number of observations reduces to 7204.

vi

Table 7: Estimation of κ i for a consumer with particular characteristics based on the estimation results of Table 5
Panel 1: v = 0
genderage

income
level

male
male
male
male
femalefemalefemale

20-29
30-39
40-49
over 60
20-29
40-49
over 600-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

number ofage of themarriagechildren youngest child
no
yes
yes
yes
yes
yes
yes

0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20smoking

drinking

no
no
no
yes
no
no
no

yes
no
no
yes
no
no
no

habit ofparticipation
eatingin cancer
organic
insurance
vegetables
no
no
yes
yes
yes
yes
no
no
yes
no
yes
yes
no
no

havingregularhealthcheckup
no
yes
yes
no
yes
yes
yes

habit ofparticipation
eatingin cancer
organic
insurance
vegetables
no
no
yes
yes
yes
yes
no
no
yes
no
yes
yes
no
no

havingregularhealthcheckup
no
yes
yes
no
yes
yes
yes

habit ofdrinking milk

κi

(9 5 % in te rval)

yes
yes
yes
yes
yes
yes
yes

0 .3 8 8
0 .6 4 9
0 .5 7 7
0 .2 8 8
0 .6 8 6
0 .6 5 5
0 .5 2 6

0 .3 3 0
0 .6 0 9
0 .5 2 7
0 .2 0 2
0 .6 5 3
0 .6 2 3
0 .4 6 4

habit ofdrinking milk

κi

(9 5 % in te rval)

0 .5 8 3
0 .7 0 7
0 .6 6 2
0 .4 4 7
0 .7 2 4
0 .7 0 2
0 .6 0 7

0 .5 3 8
0 .6 7 0
0 .6 1 6
0 .3 7 4
0 .6 8 9
0 .6 6 8
0 .5 4 6

0 .4 4 6
0 .6 8 8
0 .6 2 6
0 .3 7 4
0 .7 1 8
0 .6 8 7
0 .5 8 8Panel 2: v = 90

genderage

income
level

male
male
male
male
femalefemalefemale

20-29
30-39
40-49
over 60
20-29
40-49
over 600-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

number ofage of themarriagechildren youngest child
no
yes
yes
yes
yes
yes
yes

0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20smoking

drinking

no
no
no
yes
no
no
no

yes
no
no
yes
no
no
no

Note 1: 95% confidence intervals for estimated κ i are computed by the delta method.

yes
yes
yes
yes
yes
yes
yes

0 .6 2 7
0 .7 4 4
0 .7 0 7
0 .5 2 0
0 .7 5 8
0 .7 3 7
0 .6 6 8

vii

Table 8-1: Estimation of κ i and risk-adjusted weights for a consumer with particular characteristics based on the estimation results of Table 6

Panel 1: Estimation of κ igenderage

income
level

male
male
male
male
femalefemalefemale

20-29
30-39
40-49
over 60
20-29
40-49
over 600-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

number ofage of themarriagechildren youngest child
no
yes
yes
yes
yes
yes
yes

0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20smoking

drinking

no
no
no
yes
no
no
no

yes
no
no
yes
no
no
no

habit ofparticipation
eatingin cancer
organic
insurance
vegetables
no
no
yes
yes
yes
yes
no
no
yes
no
yes
yes
no
no

havingregularhealthcheckup
no
yes
yes
no
yes
yes
no

habit ofparticipation
eatingin cancer
organic
insurance
vegetables
no
no
yes
yes
yes
yes
no
no
yes
no
yes
yes
no
no

havingregularhealthcheckup
no
yes
yes
no
yes
yes
no

habit ofdrinking milk
yes
yes
yes
yes
yes
yes
yes

κi

(9 5 % in te rval)

0 .6 8 3
0 .8 2 4
0 .7 8 9
0 .6 2 6
0 .8 5 0
0 .8 3 7
0 .7 9 4

0 .6 3 5
0 .7 9 3
0 .7 5 0
0 .5 5 3
0 .8 2 7
0 .8 1 3
0 .7 5 7

0 .7 3 2
0 .8 5 5
0 .8 2 8
0 .6 9 9
0 .8 7 4
0 .8 6 0
0 .8 3 2

Panel 2: Estimation of risk-adjusted weights
genderage

income
level

male
male
male
male
femalefemalefemale

20-29
30-39
40-49
over 60
20-29
40-49
over 600-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

number ofage of themarriagechildren youngest child
no
yes
yes
yes
yes
yes
yes

0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20smoking

drinking

no
no
no
yes
no
no
no

yes
no
no
yes
no
no
no

riskhabit ofadju ste ddrinking milk
we igh t
yes
yes
yes
yes
yes
yes
yes

1 .9 5
2 .4 2
2 .2 3
2 .6 92 .7 02 .5 1
2 .9 6(9 5 % in te rval)
1 .8 72 .3 0
2 .0 82 .5 4
2 .5 3
2 .3 2
2 .7 4

2 .0 3
2 .5 4
2 .3 82 .8 5
2 .8 82 .7 03 .1 8Note 1: 95% confidence intervals are computed for estimated κ i by the delta method, and for estimated risk-adjusted weights by one million times simulation.

viii

Table 8-2: Estimation of κ i and risk-adjusted weights for a consumer with particular characteristics based on the estimation results of Table 6

Panel 1: Estimation of κ igenderage

incomelevel

marriage

male
male
male
male
femalefemalefemale

20-29
30-39
40-49
over 60
20-29
40-49
over 600-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

no
yes
yes
yes
yes
yes
yes

number ofage of thechildren youngest child
0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20

perception
habit ofof cancer
drinking milk
risk
high
low
low
high
low
low
high

yes
yes
yes
yes
yes
yes
yes

κi

(9 5 % in te rval)

0 .5 8 9
0 .8 8 1
0 .8 6 5
0 .4 6 6
0 .8 9 5
0 .8 8 8
0 .7 3 0

0 .4 2 9
0 .8 4 2
0 .8 1 5
0 .1 4 5
0 .8 6 5
0 .8 5 4
0 .6 3 4

0 .7 4 9
0 .9 2 0
0 .9 1 4
0 .7 8 7
0 .9 2 5
0 .9 2 2
0 .8 2 6

Panel 2: Estimation of risk-adjusted weights
genderincomelevel

marriage

male
male
male
male
femalefemalefemale0-200
500-1000
500-1000
200-500
200-500
500-1000
200-500

no
yes
yes
yes
yes
yes
yes

number ofage of thechildren youngest child
0
1
2
0
1
2
3

0-2
16-20
0-2
16-20
over 20

perception
riskhabit ofof cancer
adju ste ddrinking milkrisk
we igh t
high
low
low
high
low
low
high

yes
yes
yes
yes
yes
yes
yes

1 .7 8
3 .8 8
3 .6 6
2 .0 05 .3 1
4 .9 6
2 .6 5(9 5 % in te rval)
1 .7 02 .6 1
2 .6 6
1 .8 82 .6 93 .0 92 .3 8

1 .8 7
5 .1 4
4 .6 62 .1 1
7 .9 2
6 .8 4
2 .9 1

Note 1: 95% confidence intervals are computed for estimated κ i by the delta method, and for estimated risk-adjusted weights by one million times simulation.

ix

Figure 1-1: A case without any thresholdvi (yen)
200 yen200 Bq0

contamination
harmlesstolerable

intolerable

level (Bq)

Figure 1-2: A case with a thresholdvi (yen)
200 yenthresholdvi

200 Bq0

contamination
harmlesstolerable

intolerable

level (Bq)xFigure 2: A pattern in a consumer’s valuation of radiation-contaminated milkvi (yen)

200

vi
Bq

Di

0

X

y

D

200

Di

Figure 3-1: Case 1, vi = 200vi (yen)

200

vi
Bq

Di

0

D

X

y

200

Di

xi

Figure 3-2: Case 2, 0 < vi < 200vi (yen)

200

vi
Bq

Di

0

X

y

D

200

Di

Figure 3-3: Case 3, vi = 0
vi (yen)

200

vi
Bq

Di

0

X

y D 200

Di

xii
Figure 4-1: Numerical examples

Unconditional Probability(upper D = 250, lower D = -10, kappa = 0.038)
120.0%
100.0%
80.0%
60.0%
40.0%
20.0%

v=200

0<v<200

200.0

190.0

180.0

170.0

160.0

150.0

140.0

130.0

120.0

110.0

90.0

100.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

0.0

10.0

0.0%

v=0

Figure 4-2:

Unconditional Probability(upper D = 250, lower D = -50, kappa = 0.167)

170.0

180.0

190.0

200.0

180.0

190.0

200.0

160.0

150.0

140.0

130.0

120.0

110.0

100.0

0<v<200

170.0

v=200

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

0.0

10.0

90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%

v=0

Figure 4-3:

Unconditional Probability(upper D = 250, lower D = -550, kappa = 0.688)

v=200

0<v<200

v=0

160.0

150.0

140.0

130.0

120.0

110.0

100.0

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

0.0

10.0

90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%

xiii
Figure 5-1:

Conditional Density(upper D = 300, lower D = -200, kappa = 0.4, lower v = 0)0.01800
0.01600
0.01400
0.01200
0.01000
0.00800
0.00600
0.00400
0.00200
0.00000
0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
10

50

100

200

Figure 5-2:

Conditional Density(upper D = 3000, lower D = -100, kappa = 0.25, lower v = 0)0.01400
0.01200
0.01000
0.00800
0.00600
0.00400
0.00200
0.00000
0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
10

50

100

200

xiv

Figure 6-1: Lower upper limit of the valuation (vi) with two observations included in Case 2
vi (yen)

200

(b)

(a)

vi

Bq

Di

0

200

D3

D2

D

Figure 6-2: Higher lower limit of the valuation (vi) with two observations included in Case 2
vi (yen)

200

(b)

(a)

Di

0

D2

vi
D3

Bq
200

D

xv

Figure 7-1: A shape of a density function of measurement errors (linear functions)density (g(vi|vi*))

200 − vi*
100vi*vi*
100 ( 200 − vi* )observed
0

truevaluation (vi*)

valuation (vi)200

Figure 7-2: A shape of a density function of measurement errors (uniform distributions)density (g(vi|vi*))

200 − vi*
200vi*vi*
200 ( 200 − vi* )observed
0

truevaluation (vi*)

valuation (vi)
200

xvi

0

0

Frequency
500
1000

Frequency
500
1000

Figure 8-1: Histograms of the distribution of quoted discount prices at 10-yen intervals

50

100
10 Bq

150

200

0

50

100
100 Bq

150

200

0

50

100
50 Bq

150

200

0

50

100
200 Bq

150

200

0

0

Frequency
500
1000

Frequency
500
1000

0

0

0

Frequency
500
1000

Frequency
500
1000

1500

1500

Figure 8-2: Histograms of the distribution of quoted discount prices at 50-yen intervals

0

50

100

150

0

50

10 Bq

100

150

100

150

0

0

Frequency
500
1000

Frequency
500
1000

1500

1500

50 Bq

0

50

100
100 Bq150

050
200 Bq

xvii

50
100
150
pattern=4, 200Bq

200

0

50
100
150
pattern=11, 200Bq

200

20
15
Frequency
10
5
0

50
100
150
pattern=7, 200Bq

200

0

50
100
150
pattern=12, 100Bq

200

50
100
150
pattern=8, 100Bq

200

0

50
100
150
pattern=13, 50Bq

200

200

0

Frequency
100
150
50
0

0

0

50

50

Frequency
100
150

Frequency
100
150

200

200

0

0

0

0

5

5

Frequency
10

Frequency
10

15

15

20

20

Figure 9-1: Histograms of the distribution of quoted discount prices at the highest contaminationlevel in Pattern 4, 7, 8, 11, 12, and 13

30
25
Frequency
10
15
20
5
0

0

5

Frequency
10
15
20

25

30

Figure 9-2: Histograms of the distribution of quoted discount prices in Pattern 2, 5, 9, and 14

50

100
pattern=2, 200Bq

150

200

0

50

100
pattern=9, 50Bq

150

200

0

50

100
pattern=5, 100Bq

150

200

0

50

100
pattern=14, 10Bq

150

200

0

0

5

50

Frequency
101520

Frequency
100 150 200

25

250

30

300

0