Full text of DOI:10.1023/A:1008745627439

The Geneva Papers on Risk and Insurance Theory, 25: 103–130 2000
c
2001 The Geneva Association
Genetic Screening and Price Discrimination
in Insurance Markets
MICHAEL HOY
Department of Economics, University of Guelph, Ontario, Canada
PETER LAMBERT
Department of Economics, University of York, United Kingdom
Abstract
Basing insurance prices on the results of an imperfect screening test to identify risk types can reduce or increase
aggregate discrimination across insureds. We present a powerful and general new framework of analysis to examine
thisissue, drawinguponrecentworkwhichusesdecomposableinequalityindicestomeasureverticalandhorizontal
inequity in taxation. We find that, whilst improved test performance inevitably reduces vertical discrimination
(in the average prices faced by different risk types), even very accurate tests can lead to substantial horizontal
discrimination (within risk types) and enhanced overall discrimination. These conclusions are shown to be robust
to a range of different value judgements about how to aggregate individual discriminatory effects and to be
particularly relevant to the case of genetic screening.
Key words: insurance, genetic information, discrimination
1. Introduction
There is substantial debate about the fairness of insurance companies using results from
genetic screening tests to determine the price of insurance. On the one hand are those who
feel individuals should not be charged different rates for health, life or disability insurance
on the basis of unalterable and inherited genes,1 while others note that insurers should not
be required to carry substantially higher risks at what would effectively be subsidized rates.
If one adopts a standard economic view of price discrimination, it may seem clear that
prices should reflect differences in expected costs of insureds. For example, consider the
following standard definition of price discrimination:
“Discrimination may be said to occur in a market where individuals face terms of trade
that are determined by personal characteristics which do not appear directly relevant to
the transaction.”
(our italics)... Mueser, (1989, p. 856)
Since insurance prices are based on actuarial principles the following definition seems
natural:

104
MICHAEL HOY AND PETER LAMBERT
“An insurance rate structure will be considered to be unfairly discriminatory ... if, allow-
ing for practical limitations, there are premium differences that do not correspond to
expected losses and average expenses or if there are expected average cost differences
that are not reflected in premium differences.”
(our italics)... Williams (1969, pp. 211–212)
Thus, it might seem that the use of genetic testing, which provides finer information on
expected cost differences for providing insurance to different individuals, would be clearly
justified on the grounds of reducing unfair price discrimination.
We will argue in this paper that such a conclusion is not obvious if one considers carefully
how to measure the impact on discrimination from using imperfect information to classify
insureds. By adapting standard indices of inequality to measure the dispersion in price-cost
differentials and applying recent concepts from the literature on vertical and horizontal
equity within taxation,2 we show that an improvement in the quality of information used
to assign individuals to risk classes does not necessarily reduce the aggregate amount of
discrimination. As we shall show, if one presumes that the elemental measure of the degree
of discrimination resulting from charging a price different from (expected) cost should be
convex in the price/cost ratio, then a more informative signal of risk type may increase the
aggregate amount of price discrimination.
The essence of the analysis in this paper stems from the fact that, although using a more
informative (yet not perfect) signal to assign individuals to their true risk classes improves
the accuracy of the assignments, it also means that those who are misclassified face a greater
price-cost differential in insurance. Average price differences between different risk classes
better reflect actual expected cost differences when a more informative signal is used.
However, since a more informative signal leads to a greater price differential between risk
classes, those individuals who are assigned to inappropriate risk classes face a higher degree
of price discrimination than they would if a less informative signal were used. There is
some previous research that has addressed the relationship between imperfect classification
and price discrimination.3 Our contribution to this research stems from the way we separate
horizontal and vertical components of fairness and treat more explicitly the issue of how to
aggregate over individuals to obtain an overall measure of price discrimination.
2. The anatomy of price discrimination: A general model
First we present our general characterization for measuring and decomposing the extent
of price discrimination under imperfect categorization. In the following sections we will
apply this general theory to the particular case of a genetic screening test for a single disease
gene.
According to our basic concept of price discrimination, as outlined in the introduction of
the paper, the elemental measure of price discrimination faced by an individual of a given
risk type is founded on the relationship between the price that person is charged and the
expected cost of coverage the person imposes on the insurer. Let Ci be the expected cost to
the insurance company from insuring individuals of risk type i. In particular, this will reflect
the cost of standard health care for an individual and will be higher for those who possess a

GENETIC SCREENING AND PRICE DISCRIMINATION
105
particular disease gene. Let test outcomes be indexed by j and let P be the prices charged
ij
to persons whose true type is i and who receive test result j and let Nij be the number of
individuals in each of these scenarios. We will assume a competitive insurance market in
which firms charge actuarially fair prices conditional on the information available.4 Thus,
summing over individuals of each risk type i as they are assigned to each risk category
j, the following overall constraint is ensured by actuarial fairness of premiums across test
outcomes:
X X
X
Nij P =
Ni Ci
(1)
ij
i
j
i
where Ni is the number of persons of type i in the population.
Our elemental measure of price discrimination is the extent to which price differs from
expected cost to the insurer in ratio form. Thus we use the variable rij = P /Ci to capture
ij
the extent of price discrimination when an individual of type i is assigned to risk category
j and charged the relevant price. In these terms Eq. (1) can be expressed as:
(
)
X
X
Nijrij Ni Ci = 0
(2)
i
j
Under actuarially fair risk-type specific pricing, we have rij = 1, ∀i, ∀ j. This is the
condition for zero total price discrimination:
TD = 0 : rij = 1, ∀i, ∀ j
(3)
Horizontal discrimination (henceforth HD) occurs if like individuals are treated differently.
For horizontal discrimination to be zero, we require that
HD = 0 : rij independent of j, ∀i
(4)
Vertical discrimination (henceforth VD) occurs between groups if the average price charged
to members of a particular risk type i deviates from the expected cost of insuring members
of that group. For vertical discrimination to be zero, we require that
X
VD = 0 :
Nijrij = Ni , ∀i
(5)
j
Note that if (3) is true, so are (4) and (5): that is; TD = 0 implies both VD = 0 and HD = 0.
However, it is possible for one of VD = 0 or HD = 0 without TD = 0. For example,
charging everyone the same price for insurance in an environment with different risk types
implies that HD = 0, but neither VD = 0 nor TD = 0.
The variable rij is the ratio of price to expected cost to the insurer for an individual who
is of risk type i with test result j. This is a sensible value in terms of which to measure the
extent to which an individual is assessed a fair or nondiscriminatory price.5 However, it is
less clear how one ought to aggregate over individuals to determine an overall measure of

106
MICHAEL HOY AND PETER LAMBERT
price discrimination. Taking an intuition from the measurement of income inequality, one
may wish to apply increasing weight to greater differences in this ratio.6 In general terms
then, defining rij as the vector of all rij values for all individuals in the population, we can
aggregate the extent to which the P /Ci values vary (from 1) by using an inequality index
ij
I[rij] which is convex in the arguments rij.
Let rij | i represent the vector of price/cost ratios faced by individuals of a given type
but assigned (possibly) to different risk categories. So, for example, if there were five
individuals of a given risk type (say type #1), with three of them assigned to risk category
one and the remaining two assigned to risk category two, then the vector rij|i, i = 1, would
P
be the vector (r11, r11, r11, r12, r12).
j rij/Ni is the vector formed by averaging price/cost
ratios for a given risk type i over the various risk categories to which they are assigned.
This vector will then be composed of the average price-cost ratios for each risk type. Thus,
following the intuition from the inequality measurement literature, it is natural to define
total and vertical price discrimination as
TD = I[rij]
(6)
and
"
#
X
VD = I
rij/Ni
(7)
j
respectively. Horizontal discrimination locally (within type i) is defined as
HDi = I[rij | i]
(8)
(9)
and horizontal discrimination globally as
X
HD =
wi HDi
i
where wi is the proportion of the population which is of type i (wi = Ni /N where N =
P
P P
Ni =
Nij).
j
i
i
The natural inequality indices I[ ] to use in such an analysis come from the so-called gen-
eralized entropy (GE) family, because of their decomposability properties [Bourguignon,
1979; Cowell, 1980; Shorrocks, 1980, 1984]. In fact these—and monotonic transforma-
tions of them—are the only indices of relative inequality to enjoy a ‘subgroup consistency’
property, according to which overall inequality necessarily falls if it does so in a subgroup
[Shorrocks, 1984], and this will be an essential property for our analysis. All such trans-
formations satisfy the principle of transfers and scale invariance. The GE family itself,
uniquely, has the decomposability property according to which overall inequality can be
decomposed additively into between-groups inequality and a weighted sum of within-group
inequalities. The weights are independent of group income levels, or price/cost ratios in
our case, only in the case of the mean logarithmic deviation. First, however, we argue that

GENETIC SCREENING AND PRICE DISCRIMINATION
107
all of the above mentioned properties of an inequality measure are desirable for a measure
of price discrimination.
The principle of transfers requires that, for any pair of values u,v where u < v, if u (the
lower value) is reduced by some amount while v (the higher value) is increased by an equal
amount, then the index I[ ] must increase.7 In fact, any convex function I[ ] would satisfy
this property, not only the decomposable indices we are interested in. This is an intuitively
pleasing property when measuring the dispersion of price/cost ratios since it seems natural
that an individual’s perceived cost or harm from facing a higher price than is actuarially fair
would be increasing at an increasing rate in this ratio much as it is generally accepted within
inequality analysis that the cost of inequality should be a convex function of the analogous
quantities. Scale invariance requires that the relative degree of dispersion in price/cost
ratios is not changed if one multiplies the entries of the vector by any positive constant.
This implies, for example, that if the price faced by all individuals were to rise by the same
percent, the aggregate relative dispersion in price/cost ratios would remain unchanged.8
Subgroup consistency would record a higher total discrimination if, hypothetically, for one
type the degree of discrimination were increased and no change were made elsewhere.
The advantage of full decomposability is that it permits an additional subdivision of
overall discrimination into vertical and aggregate (global) horizontal components, the latter
itself a weightedsum oflocal horizontal discriminations. To see this, just note that inequality
in the vector rij used to measure discrimination is made up of the inequalities in the vectors
P
rij | i and j rij/Ni , when the groups are defined by the indexi, and that the between groups
contribution is defined by inequality in the distribution of within-group averages. For the
P
GE family, then, a decomposition of total discrimination of the form TD = VD+
i HDi
i
is achieved, where the values are aggregation weights. We state these results formally
i
below.9
Theorem 1: A continuous inequality measure I[rij] satisfies the principle of transfers,
scale invariance and decomposability, if and only if it is a member of the generalized entropy
family; i.e.,
"
#
X X
1
1
I[rij] =
for some
rij 1
2
N
i
j
∈ ( ∞, +∞), = 1, 0 (see on for other formulae for
= 1, 0). The
formulae for decomposing the measures into horizontal and vertical components is TD =
VD + HD = VD + 6i i HDi where
"
#
"
#
X
X
1
1
VD = I
rij/Ni
=
ri
1
2
Ni
j
i
"
#
X
1
1
HDi = I[rij | i] =
rij 1 , ∀i
2
Ni
j
= wi vi1
i

108
MICHAEL HOY AND PETER LAMBERT
N
where r¯ is the average price/cost ratio faced by individuals of risk type i, wi = Ni , and
i
6j rij
vi = 6 6 ri j . In the case of = 1 the functional form for I[rij] becomes
i
j
X X
1
1
I[rij] =
ln
N
rij
i
j
and aggregation for HD is performed using population weights = wi . In the case of
i
= 0 the functional form for I[rij] becomes
X X
1
I[rij] =
rij ln rij
N
i
j
and aggregation for HD is performed using the weights = vi (i.e., the average rate of
i
the price/cost ratios within each group).10
Aggregation using population weights is perhaps the most appealing on intuitive grounds.
However, by insisting on such a weighting scheme, along with the other requirements
mentioned in the theorem, we effectively adopt the logarithmic function as our particular
normative standard for comparing deviations of price/cost ratios among individuals. By
admitting weights which are not the population weights, we allow ourselves a wider range of
relative sensitivities concerning the impact of deviations of price/cost ratios on our aggregate
assessment of price discrimination. In particular, a choice of < 2 places more emphasis
on lower values of price/cost ratios while > 2 places more emphasis on higher values of
price/cost ratios.11 When measuring the overall inequality of incomes it is natural to place
more emphasis on lower incomes. However, it is perhaps more compelling in the present
context to express increasing concern with higher price/cost ratios since the higher is the
ratio rij = P /Ci , the greater is the degree of unfavourable price discrimination. We will
ij
explore this issue in Section 6.
3. An application to genetic testing
Consider the implications of genetic testing for a simple, single gene disorder. We will
model a stylized view of the relationship between genes and a disease, or risk of a person
incurring the disease, which applies in a strict sense to only a few diseases.12 However,
expanding the perspective on the relationship between genes and susceptibility to disease
to better reflect the reality of many diseases will only strengthen our argument; we discuss
this aspect later in the paper.
Let i = h, l indicate the true risk type of a person, with a person of type h (high risk)
possessing the disease gene and a person of type l (low risk) having the “normal or healthy”
gene. Let j = p, n denote the test result: p for a positive result and n for a negative result.
A person who tests positive for the gene is placed into the high risk category and charged
price Pp and a person who tests negative is placed into the low risk category and charged
price Pn. We presume that the expected cost imposed on an insurer by a person of risk

GENETIC SCREENING AND PRICE DISCRIMINATION
109
type i is Ci , with Ch > Cl, and so with actuarially fair pricing based on test results we
have Pp > Pn. Imperfect testing (i.e., the presence of false negative and false positive
test results) implies Ch > Pp > Pn > Cl. Thus, we need to consider four situations for
individuals: (i) high risk types who test positive and face price/cost ratio Pp/Ch, ii) high
risk types who test (false) negative and face price/cost ratio Pn/Ch, iii) low risk types who
test (false) positive and face price/cost ratio Pp/Cl, and iv) low risk types who test negative
and face price/cost ratio Pn/Cl, where Pp/Cl > Pn/Cl > 1 > Pp/Ch > Pn/Ch. Recall that
zero price discrimination would require P /Cj = 1 for each i and j.
i
If an individual possesses the disease gene then her probability of incurring the disease
in the insurance period is higher than it otherwise would be. For now, we presume that
the only determinant of the disease, given current knowledge, is whether the individual
possesses the gene or not. A genetic test provides imperfect information. The degree of
informativeness of the test can be summarized by the false positive and false negative rates
inherent in the test. The higher either the false positive or false negative rate, the less is the
information value of the test. The following variables describe the relevant parameters in
the population and for the genetic test:
εfn, εfp are the rates of false negatives and positives, respectively, associated with the test;
ji is the probability that a person of risk type i = h, l would receive test result j = p, n;
ij
is the probability that a person who receives test result j = p, n is of risk type
i = h, l;13
qh, ql are the proportions of the population that are of risk types h, l respectively.
It is presumed that high risk types impose higher average or expected costs on the
insurance company than do low risk types. Depending on the disease, this cost differential
could exist for one or all of the areas of health, life and disability insurance. We will assume,
for the sake of discussion, that the application concerns health insurance.
As noted earlier, we assume that the insurance industry is competitive and that insurers
are risk neutral: they price according to the actuarial costs of providing insurance.
Thus, we have the following relationships among these variables:
= 1 εfn,
= εfn,
= εfp,
= 1 εfp
(10)
ph
nh
pl
nl
and since the ij’s are the complementary probabilities of the ji’s, we can use Bayes’
theorem to derive:
qh
ph
hp
hn
lp
=
=
=
=
(11)
(12)
(13)
(14)
qh ph + ql
pl
nl
pl
nl
qh
nh
qh nh + ql
ql
pl
qh ph + ql
ql
nl
ln
qh nh + ql
In generating our pricing equations we assume all individuals purchase the same amount
of insurance regardless of whether genetic testing is used in setting prices. If individuals

110
MICHAEL HOY AND PETER LAMBERT
have better information concerning their risk type than do insurers, then one would expect
adverse selection to arise, with high risk types purchasing more insurance than low risk
types.14 For now, we assume all individuals purchase full insurance coverage regardless of
their information set and the price they face. In a pooling equilibrium, in which individ-
uals with different information sets facing the same price purchase different amounts of
insurance, the same qualitative aspects of our results would still apply. We return to this
issue in Section 6 also. Under the condition of equal insurance purchases, the actuarially
fair price of insurance with no ratemaking using results from genetic testing will be P0
where
P0 = qhCh + qlCl
(15)
If genetic testing prevails and insurers are allowed to charge prices according to test
results, then (pooled) actuarially fair pricing leads to
Pp = hpCh + lpCl
Pn = hnCh + lnCl
(16)
(17)
where Pp and Pn are the prices for those who test positive and negative respectively.
To examine the issue of vertical price discrimination we need to determine the average
price faced by individuals of each risk type, which includes both individuals who are
properly classified and those who are not. The average prices faced by individuals of risk
¯
types i = h, l ,P , are given in the two equations below:
i
¯
Ph = ph Pp + nh Pn
(18)
(19)
¯
P = pl Pp + nl Pn
l
Turning to the measurement of discrimination, for the mean logarithmic deviation (hence-
forth MLD), which is the case of = 1 in Theorem 1, total discrimination (or inequality in
price/cost ratios) is:
TD = ln(E[r]) E[ln(r)]
(20)
where r refers to a generic price/cost ratio and expectations are taken over the entire popu-
lation. Since the joint probability that a person is of risk type i (i = h, l) and assigned to
risk category j (i.e., receives test result j = p, n) is qi ji, we have
E[r] = qh phrhp + qh nhrhn + ql plrlp + ql nlrln
(21)
and
E[ln(r)] = qh ph ln(rhp) + qh nh ln(rhn) + ql pl ln(rlp) + ql nl ln(rln)
(22)

GENETIC SCREENING AND PRICE DISCRIMINATION
111
Vertical discrimination measures the extent to which the average price to cost ratio varies
between groups:
VD = ln E[r¯ ] E[ln(r¯ )]
(23)
i
i
where r¯ is the average price/cost ratio for individuals of type i. We have
i
¯
¯
Ph
P
l
E[r¯ ] = qh
+ ql
(24)
i
Ch
Cl
and
¯
¯
Ph
P
l
E[ln(r¯ )] = qh ln
+ ql ln
(25)
i
Ch
Cl
Horizontal discrimination for risk type i is the amount of discrimination for that population
subgroup; one measures it just as one measures total discrimination:
HDi = ln(E[ri ]) E[ln(ri )], i = h, l
(26)
where ri is the price faced by risk type i individuals (depending on which risk category they
are assigned to), hence:
Pp
Pn
E[ri ] =
+
,
i = h, l
(27)
pi Ci
ni Ci
and
Pp
Ci
Pn
Ci
E[ln(ri )] = pi ln
+
ni ln
,
i = h, l
(28)
Globally, horizontal discrimination is the population weighted average of horizontal
discrimination within each risk group:
HD = qhHDh + qlHDl
From Theorem 1 we know that
TD = VD + HD
(29)
(30)
We are interested to explore how all components of total discrimination (i.e., VD, HD and
its constituents HDh and HDl), as well as the total itself, are affected by changing the quality
of the screening test. We do this in the next section, where, first, we show that for any of the
family of measures that appear in Theorem 1, a higher quality test (i.e., a test with a lower
false positive and/or false negative rate) always reduces vertical discrimination. Then we

112
MICHAEL HOY AND PETER LAMBERT
examine the question of the effect of the quality of the test on horizontal discrimination,
which is more complex.
For the other measures of the GE family specified in Theorem 1, expressions correspond-
ing to (20), (23) and (26) (for = 1) generate TD, VD and the HDi , i = h, l whilst (29)
needs modified weights, and then (30) again holds. The same questions as for the MLD
can be explored in this more general context, concerning the effect of test quality changes.
4. Test quality and price discrimination
¯
¯
From Eqs. (16) through (19) it is straightforward to see that Ch > Ph > P > Cl. That is,
l
on average high risk types pay a price which is less than the expected cost they impose on
the insurer but more than do low risk types who in turn pay a price which is greater than
the price they impose on the insurer. So vertical discrimination occurs due to the fact that
¯
¯
Ph
Ch
P
Cl
l
< 1 and > 1. If a higher quality test (i.e., lower value of ε f n and/or ε f p) leads to an
¯
¯
increase in Ph and a decrease in P then vertical discrimination will be reduced for any of
l
the inequality measures with properties as described in Theorem 1.15 To show that this is
i j
indeed the case, we need to investigate how the values of the probabilities
and
and
ji
the prices Pp and Pn are affected by changes in the false positive and false negative rates
ε f n and ε f p
From Eq. (10) the following are obvious:
.
∂
∂
∂
∂
∂
∂
∂
∂
nh
ph
nl
nh
pl
nl
pl
ph
=
=
1,
=
= 1,
=
=
=
= 0
∂εfn
∂εfp
∂εfn
∂εfp
∂εfn
∂εfn
∂εfp
∂εfp
(31)
Using these results and Eqs. (11) through (14), a little algebra gives the following
i j
comparative statics results for the complementary probabilities
derivatives appear in the Appendix):
(actual values of the
hp
hp
ln
ln
hn
hn
lp
lp
∂
∂
∂
∂
∂
∂
∂
∂
,
,
,
∂εfn ∂εfp ∂εfn ∂εfp
< 0;
,
,
,
∂εfn ∂εfp ∂εfn ∂εfp
> 0
(32)
The comparative statics results for the effects of changes of the false positive and false
negative rates on the prices paid by those who test positive and negative for the disease gene
can also be derived. Again, see the Appendix for the values of these partial derivatives.
These results are required to understand the effects of changes in the accuracy of the
screening test on average prices paid by members of each risk type, but also they will be of
use later in the paper when we consider horizontal discrimination:
∂ Pp ∂ Pp
,
∂εfn ∂εfp
∂ Pn ∂ Pn
,
∂εfn ∂εfp
< 0,
> 0
(33)
These results indicate that a test with a higher rate of false negatives leads to (i) an increase
in the price paid by those who test negative, since that group will have a higher proportion

GENETIC SCREENING AND PRICE DISCRIMINATION
113
of individuals who are actually high risk types, and (ii) a decrease in the price paid by those
who do test positive since more individuals who are high risk types are in fact being placed
in the other category (i.e., the “negative” category).16 Also, a test with a higher rate of false
positives leads to (i) a decrease in the price paid by those who test positive, since that group
will have a higher proportion of individuals who are actually low risk types, and (ii) an
increase in the price paid by those who test negative since more individuals who are low
risk types are in fact being placed in the other category (i.e., the “positive” category).17
The relationships between the rates of false negatives and false positives and the average
price paid by individuals of a given risk type, however, are less clear intuitively. Consider,
for example, the impact of an increased rate of false negatives on the average price paid by
high risk types. Since Pp > Pn, the fact that more h-types are assigned to the “negative
category” and fewer to the “positive category” implies a reduction in the average price
paid by h-types. Similarly, an increased rate of false positives implies a reduction in the
average price paid by those who are assigned to the “positive category.” However, there is
also an increase in the price of insurance paid by those in the “negative category” and this,
in conjunction with the fact that the likelihood of being assigned to this category is higher,
¯
has an opposite effect on Ph.
It turns out, though, that the overall effect of an increase in the rate of false negatives
is a reduction in the average price of insurance paid by high risk types, which implies a
greater degree of price discrimination for this group. Similar results apply with respect
to the relationships between the average price paid by individuals of either risk group and
the rates of false positives and negatives. All of these results are stated in the following
theorem. Due to the amount of algebra required, they are proved in the Appendix.
Theorem 2: A more accurate screening test reduces the average price paid by members
of the low risk group and increases the average price paid by members of the high risk
group, thus reducing the overall degree of vertical price discrimination.
We now turn our attention to the relationship between the accuracy of the screening
test and the degree of horizontal discrimination. If no screening test or equivalently a
completely uninformative one is used, then all individuals are treated the same and so
there is no horizontal discrimination. Alternatively, if a perfectly informative screening
test is available and used for pricing insurance, then individuals of any risk type i are
charged the same price (i.e., their risk-type specific actuarially fair price, P = Ci ) and so
i
in this case again there is no horizontal discrimination. Hence, in between, any imperfect
screening test leads to some horizontal discrimination even if the test is almost perfectly
accurate. An improvement in the accuracy of an imperfect screening test will in some cases
increase and in other cases decrease the degree of horizontal discrimination. The effects of
increased accuracy of a screening test on the components affecting the degree of horizontal
discrimination are multifaceted and so, not surprisingly, the overall directional effect on
HD cannot be predicted unambiguously.
Consider, for example, the low risk types. Some are misclassified (i.e., receive false
positive tests) and pay the price Pp, while others are properly classified (i.e., receive true
negatives) and pay the price Pn; the range of price-cost ratios for this group is (rln, rlp) =

114
MICHAEL HOY AND PETER LAMBERT
Pp
Cl Cl
Pn
(
,
). Increasing the accuracy of the test leads to an increase in Pp and a decrease in
Pn, widening this range, implying greater inequality or price discrimination for this group.
However, this increase in accuracy also reduces the fraction of l-types who pay price Pp
and increases the fraction who pay price P and reduces the average price to cost ratio
l
¯
P
l
for this group (i.e., ) too. Thus, the net result of an increase in the accuracy of the test
can be either to incCrel ase or decrease horizontal equity for the low risk group. Similar
arguments apply to the high risk group. Our simulations in the next section confirm this
and provide us with insight on how the accuracy of a screening test is related to the degrees
of horizontal and overall discrimination. Of particular interest is the relationship between
parameter values, which should reflect the reality of genetic testing and genetic diseases,
and the effect of changing the parameter , which reflects one’s values about the degree of
horizontal discrimination for the various groups of individuals in society.
5. Simulation results for changes in test accuracy
In our simulations we consider a number of different measures of discrimination based on
the entropy family presented in Sections 2 and 3 of this paper. But first we further develop
some intuition concerning the relationship between the amount of horizontal discrimination
and the accuracy of genetic tests.
Consider the situation of low risk types facing an imperfect screening test. The degree
of horizontal discrimination for l-types is represented by the inequality in their price-cost
ratios as follows:
Pn[↓] Pp[↑]
HDl = I
,
= I (rln[↓], rlp[↑])
Cl
Cl
The arrows indicate the direction of the change in price or price-cost ratios resulting from an
increase in the degree of accuracy of a screening test. This results in increased dispersion in
prices paid by l-types who are properly classified versus those who are misclassified and so
increases horizontal discrimination. One must remember, however, that the proportion of
those l-types who are misclassified falls, while the proportion of properly classified l-types
rises, with increased test accuracy (i.e.,
falls while
rises) which can counterbalance
nl
pl
the first effect mentioned. In the limit, as the test becomes perfectly accurate
→ 0 and
pl
→ 1 and horizontal discrimination for l-types vanishes altogether.
nl
A similar analysis applies to the high risk types, with:
Pn[↓] Pp[↑]
HDh = I
,
= I (rhn[↓], rhp[↑])
Ch
Ch
As the accuracy of a test improves, the proportion of those h-types who are misclassified
falls, while the proportion properly classified rises, (i.e., falls while rises) and in
nh
ph
the limit, as the test becomes perfectly accurate
discrimination for h-types vanishes.
→ 0 and
→ 1 and so horizontal
np
ph

GENETIC SCREENING AND PRICE DISCRIMINATION
115
Overall horizontal price discrimination is
HD = qhHDh + qlHDl
By noting the relative values of the price-cost ratios for both types, we can better understand
the overall implications for horizontal price discrimination as it relates to the accuracy of
the test and the choice of measure I ( ); that is:
rhn[↓] < rhp[↑] < rln[↓] < rlp[↑]
Again, the arrows indicate the direction of changes resulting from a more accurate test.
A given Mendelian or purely genetic disease tends to afflict a very small fraction of the
population and so ql is generally much larger than qh, leading to a substantially higher
weight being placed on l-types. Moreover, in the context of a genetic testing scenario, the
difference rlp rln is likely to be large and increasing in the accuracy of the test and so
it is this difference which will tend to dominate the overall horizontal equity implications
of imperfect categorization. From a normative standpoint as well, it makes sense to place
concern more towards individuals most heavily penalized by being misclassified. These are
the low risk types who face the highest price-cost ratio. In choosing a specific member of the
entropy family of inequality measures, the larger is the value of the parameter , the greater
is the emphasis on horizontal discrimination within the l-types (i.e., the more sensitive
is the measure to variability in the higher end of the vector of price-cost ratios). Thus,
both the facts about genetic diseases and our normative concerns about misclassification
arising from imperfect categorization lead to a conclusion that, even though vertical price
discrimination is always reduced by using a more accurate genetic test for pricing insurance,
horizontal price discrimination generated by misclassification of low risk types will be a
greater concern. This is borne out by our simulation results which we now describe.
Our first simulation result, depicted in figure 1, was computed using the MLD (mean
logarithmic deviation) measure of discrimination. Recall that this is equivalent to a choice
of parameter value = 1 for the generalized entropy family of inequality measurement and
such a choice implies relatively greater emphasis on dispersion between smaller values of
price-cost ratios. Thus, this represents relatively less concern with misclassification among
l-types than among h-types. The parameter values chosen for population proportions are
qh = 0.001 and ql = 0.999, also reflecting a conservative view of the relative importance of
low-risk types in the population.18 The relative expected costs of insurance are ch = 50,000
and cl = 2,000. In the context of health insurance this would represent a situation in which
those with the genetic disease will face substantial costs for medical treatment. In the
context of disability insurance these parameter values reflect a very high probability of loss
of income. Since the range of genetic diseases generates a wide range of costs there is no
natural choice for such parameters. In a later simulation we consider a case with much
more similar costs for the two types.
To simplify the graphical analysis, we represent the accuracy of the test by a choice of
false positive and negative rates which are equal (i.e., εfp = εfn), thus allowing us to use a
two dimensional graph. A decrease in εfp, the rate of false positives, represents an increase
in the accuracy of the test. As previously noted, if the test is either completely uninformative

116
MICHAEL HOY AND PETER LAMBERT
Figure 1. Base case simulation: components of discrimination.
(εfp = 0.5) or perfectly informative (εfp = 0), there is zero horizontal price discrimination.
Consideration of increasingly accurate genetic screening tests is represented in the graphs
by movement from the value εfp = 0.5 to εfp = 0. We define εfp as that critical value
of εfp such that using any screening test that has less information value (i.e., εfp > εfp) to

GENETIC SCREENING AND PRICE DISCRIMINATION
117
categorize risks creates more total discrimination than if no categorization at all occurs (i.e.,
all consumers pay the same pooled price for insurance).
The top graph in figure 1 represents the relationship between horizontal discrimination
and the accuracy of the genetic test. As we can see, both groups face increasing horizontal
discrimination as a result of increasing the accuracy of the test over much of the range of
values for εfp. Since the relative population proportion of l-types dominates that of h-types,
not surprisingly the overall amount of horizontal discrimination, HD, virtually coincides
with HDl in this graph (but see on for some cases where this does not happen).
In the bottom graph of figure 1 we compare the size of vertical discrimination to overall
horizontal discrimination. One can see that vertical discrimination always falls as the ac-
curacy of the test improves. This conforms to the result of Theorem 2. Beginning with a
completely uninformative test, HD rises more quickly than does VD fall as one considers
tests of greater accuracy and so increasing the accuracy of testing increases total discrimi-
nation up to the point where the test is almost perfectly accurate.19 This means that unless
the screening test is almost perfect, there will be lower overall discrimination if all insureds
are charged the same price instead of using the results of screening tests for price setting.
In fact, for this case the use of any screening test with a false positive rate greater than
εfp = 0.00015 (i.e., 1.5 persons per 10,000 tested) would induce a greater amount of total
discrimination than if no categorization at all were allowed.
This result follows despite the fact that the choice of the MLD as inequality measure is
not particularly sensitive to the dispersion of price-cost ratios for low-risk types, the group
which includes those individuals most heavily discriminated against as a result of imperfect
categorization (i.e., misclassified low-risk types). In figure 2 we adopt all of the same
parameters as for the example in figure 1, except we choose parameter value = 5 for the
generalized entropy inequality index. As previously discussed, any value for this index in
excess of 2 places greater weight on dispersion of higher price-cost ratios than lower ones
and the higher is , the greater is such emphasis.20 In the top graph21 we see that the relative
contribution of horizontal discrimination among the l-types completely dominates that for
h-types even on a per capita basis and so, a fortiori, overall horizontal discrimination is
dominatedbytheHDl component. Also, therelativeimportanceofhorizontaldiscrimination
compared to vertical discrimination is greater in this case and the accuracy of testing must
be even higher before improved accuracy leads to a reduction in total discrimination.22 In
9
this case the use of any screening test with a false positive rate greater than εfp = 1.5 10
would induce a greater amount of total discrimination than if no categorization at all were
allowed. It seems very unlikely that risks could be classified so accurately that a false
positive rate on the order of only one misclassification out of a billion tested (or less) could
be met.
In the example illustrated in figure 3, one can observe the important role played by the
choice of the parameter for the generalized entropy inequality index. Here we perform
the simulations for the case of
=
3, a value which implies substantially greater em-
phasis on the dispersion of relatively smaller price-cost ratios, which in our context means
the difference in prices paid by high risk types who are properly or misproperly classified.
If one’s principal emphasis on discrimination is within the group who receive favourable
discrimination (i.e., the values rhn <rhp < 1) then, in per capita terms, HDh becomes the

118
MICHAEL HOY AND PETER LAMBERT
Figure 2. Case II simulation: components of discrimination.
more important term in the consideration of horizontal discrimination. Since h-types rep-
resent only a small fraction of the population, however, overall horizontal discrimination is
influenced substantially by HDl, as can be seen in figure 3. Overall discrimination, TD, is in
this case dominated by vertical discrimination. It is only horizontal discrimination among
h-types that is deemed very relevant according to the choice of parameter
with so few h-types in the population overall horizontal discrimination becomes relatively
=
3 and

GENETIC SCREENING AND PRICE DISCRIMINATION
119
Figure 3. Case III simulation: components of discrimination.
less important. Thus, since vertical discrimination is always less when a more informative
screening test is used for pricing insurance, we have that total discrimination is reduced
by any increase in accuracy of the screening test. We do not, however, think that this is a
relevant result from an ethical or policy perspective since it seems natural when concerned
about price discrimination to focus at least as much on those who are penalized by discrim-
ination (i.e., those facing a price greater than costs) as on those who benefit from it (i.e.,
those facing a price less than costs).

120
MICHAEL HOY AND PETER LAMBERT
Figure 4. Case IV simulation: components of discrimination.
In figure 4 we illustrate the results of simulations which are based on a case with less
difference in costs between high and low risk types. Results similar to the first two cases
are obtained. In figure 5 we illustrate a case in which the population proportions are more
balanced relative to scenarios consistent with genetic screening. In this case 25% of the
population is of the high risk type. The importance of horizontal discrimination in the two
types is similar and the contribution of vertical discrimination to total discrimination is
similar to that of overall horizontal discrimination. The result is that total discrimination is

GENETIC SCREENING AND PRICE DISCRIMINATION
121
Figure 5. Case V simulation: components of discrimination.
falling in the accuracy of the test for any initial test with accuracy εfp < 0.19 and any test with
accuracy reflected by a false positive rate εfp < εfp = 0.096 delivers a categorization scheme
with less total discrimination than if no categorization were allowed. This case illustrates
the importance of the context in which one considers the issue of price discrimination and
the degree of misclassification associated with imperfect categorization. In a case, unlike

122
MICHAEL HOY AND PETER LAMBERT
genetic diseases, in which the relative proportion of high risk types in the population is not
so small, using imperfect screening to assign individuals to risk categories can more easily
be justified on the grounds of reducing overall price discrimination.
6. Generalizations
In our formal analysis in Sections 3 and 4 we have taken a simplistic view of the relationship
between genes and health risks. Although a few genetic diseases are quite well represented
by the straightforward mapping between geno-type and risk-type,23 that we have employed,
most relationships between genes and health risks are more complex. For the majority of
genetic diseases, the propensity to sustain the disease depends on a combination of several
genetic determinants as well as environmental influences.24 These cases are referred to as
multifactorial genetic diseases. Recognizing this aspect of genetic diseases strengthens
the conclusion that basing insurance prices on genetic screening tests is likely to increase,
rather than reduce, the extent of price discrimination.
In the case of multifactorial genetic diseases, knowing whether an individual possesses
a given gene does not provide a precise assessment of risk type and so the link between a
given, identified gene and its associated disease(s) may be quite weak. This implies that
even if a particular gene of interest is always correctly identified by the screening test, there
will be substantial misclassification of individuals to risk categories. Not only must one
know that a given gene is “defective” in order to correctly ascertain an individuals future
health risks, one must also know the particular mutation for that gene, the status of other
related genes, and also the environmental factors involved with that disease for the particular
individual. The last of these factors would include life-style choices, not all of which are
likely to be perfectly observable to the insurer.
Not only is accurate classification of risks by genetic screening tests unlikely in the con-
text of multifactorial genetic diseases, the use of only two risk classes does not properly
reflect the true degree of complexity. As our analysis in Section 2 of this paper readily
demonstrates, however, one can introduce as many risk factors and classification groups
as one likes without changing the nature of the analysis. The importance of being able
to do so is even more evident when one considers that it is not possible to treat in a
dichotomous fashion even those diseases which appear to be entirely dependent on a sin-
gle gene and no other genetic or environmental factors. The disease cystic fibrosis, for
example, has been identified with a single gene. However, as of 1997, more than six
hundred different mutations of the gene responsible for this disease had been discovered
[see Casey, 1997]. The severity of the disease depends on which of the mutations is re-
sponsible and testing for these is imperfect. Other illustrative examples are represented
by the so-called breast cancer genes (BRCA-1, 2), which are contributors to the multi-
factorial genetic diseases of breast cancer, ovarian cancer, and others. BRCA-1, 2 are
very long genes with many possible different variations/mutations creating many differ-
ent scenarios concerning penetrance of the diseases and, hence, relative health risks.25
Such complexities represent the standard, rather than the exception, and so scientific or
technological accuracy in relating genes to health risks is likely to remain less than per-
fect.

GENETIC SCREENING AND PRICE DISCRIMINATION
123
Besides generalizing the context of genetic diseases, we can also apply our analysis to
study insurance pricing environments which are more complex. In our analysis we base the
price of insurance, as determined by test results (i.e., positive or negative), on the implicit
assumption that both risk types purchase the same amount of insurance. Thus, recalling
Eqs. (16) and (17), the unit price for those receiving test result j = p, n, is
Pj = hjCh + ljCl
As noted earlier, many models of the insurance market take account of the fact that, in
the presence of asymmetric information, high risk types will typically purchase greater
amounts of insurance than will low risk types when the two types are faced with the same
price of insurance. Thus, the actual price of insurance will be a demand-weighted average
of the risk-type specific actuarial costs. If we let Lij represent the demand for insurance by
¯
an individual of risk type i with test result j, and let L j be the average amount of insurance
purchased by individuals with test result j, then the pooled actuarially fair price of insurance
would be26
hj Lhj
¯
L j
lj Llj
¯
L j
Pj =
Ch +
Cl
The qualitative nature of our results concerning price discrimination and the accuracy of
the screening test would persist under these conditions. In fact, the extent of horizontal
price discrimination faced by misclassified low risk types, which we saw in the previous
section tends to dominate the overall degree of price discrimination in the context of genetic
screening, is even greater given this consideration, while discrimination for high risk types
falls as before, but this has a negligible effect on overall horizontal discrimination due to
¯
the insignificant weight attached to HDh. This follows because, with Lhp > L p > Llp, the
extent to which low risk types who are misclassified face unfavourable discrimination is
exacerbated by adverse selection considerations.
Under conditions of asymmetric information, adverse selection may also lead to a sep-
arating equilibrium as described in Rothschild and Stiglitz [1976]. In this case high risk
types purchase full coverage insurance at their actuarially fair price while low risk types
purchase less than full coverage at their actuarially fair price. The contracts are self-
selecting in that the high risk types do not purchase the contract designed for low risk types
despite the lower per unit price because of the fact that high risk types prefer a higher
level of coverage. In this scenario, the presence of unidentifiable high risk types creates
an externality for low risk types but this is not reflected in a higher than actuarially fair
price for the low risks but rather a rationing of the amount of insurance that they end up
with relative to what they would purchase if there were no high risk typesin the market.
Thus, this scenario is not amenable to an analysis of price discrimination. This outcome,
however, requires that there be a sufficiently large fraction of high risks in the population
that low risk types prefer the separating contract, which offers them a relatively low level
of coverage, to a pooling equilibrium, which offers higher coverage but at an actuarially
unfair (higher) price. In the context of genetic information, the fraction of high risk types
in the population is very small and so a separating equilibrium is not plausible for the

124
MICHAEL HOY AND PETER LAMBERT
case of no categorical information allowed and so our analysis of price discrimination is
appropriate.27
Thus, natural generalizations in regards to the reality of both genetic diseases and insur-
ance market pricing reinforces our conclusions that improved accuracy of genetic screening
is likely to lead to an increase in the overall amount of price discrimination.
7. Conclusions
In this paper we have exploited the relationship between decomposable inequality measures
and the concepts of horizontal and vertical equity in order to provide a powerful and general
framework within which to address the issue of price discrimination arising from the use
of imperfect categorization in insurance pricing. We developed a particular application to
determine whether using results from screening tests for geno-type to risk-rate insurance
premiums is more or less discriminatory than charging all heterogeneous individuals the
same average price. We conclude that the use of such information leads to more, rather than
less, aggregate discrimination. The reasons for this conclusion revolve around the realities
associated with genetic diseases as well as compelling normative concerns.
AnysingleMendeliandiseaseaffectsonlyasmallfractionofthepopulationbutmanysuch
diseases have extremely detrimental health implications. Even very accurate genetic tests,
which lead to a small fraction of individuals being misclassified, can generate substantial
horizontal inequity within either risk class due to the significant differences in the costs of
insurance provision. Although vertical discrimination is always reduced by the introduction
of a more accurate screening test, the fact that most individuals are of the low risk type
means that use of population weights by risk type de-emphasizes the importance of vertical
discrimination relative to horizontal discrimination.
Price discrimination persists when individuals face different price-cost ratios for a prod-
uct. From a normative perspective, it is compelling to place increased concern on the
dispersion in price-cost ratios between individuals for whom the levels of those ratios are
higher. The reason for doing so is that those with the highest price-cost ratios resulting
from imperfect categorization are those who are most heavily discriminated against. Using
simulations we found that, even with a conservative value judgement regarding the relative
importance of price-cost ratios across different groups, increased accuracy of a screening
test leads to an overall increase in price discrimination up to very high degrees of accuracy.
This conclusion follows because, even though increased accuracy in screening reduces
the number of individuals who are misclassified, at the same time improved test accuracy
increases the dispersion in price-cost ratios between individuals properly classified and
misclassified.
Two further considerations reinforce our conclusions. Firstly, the fact that most genetic
diseases are multifactoral weakens the potential to use genetic testing to predict health risks
with perfect accuracy. Thus, the limiting case of perfect testing, in which discrimination
wouldbeeliminated, isunlikelyformostgeneticdiseases. Second, adverseselectionconsid-
erations exacerbate the importance of penalizing those low risk types who are misclassified
and it is this aspect of imperfect testing which naturally arises as the most prominent one
in our analysis and conclusions.

GENETIC SCREENING AND PRICE DISCRIMINATION
125
The consideration of asymmetric versus symmetric information also raises an interesting
question about the interpretation of the measures of price discrimination. In the case of
asymmetric information, each individual knows her risk type and so under any scheme of
imperfect categorization each person knows with certainty the extent of price discrimina-
tion she faces. In the case of symmetric information we presume that individuals know
that within the population there are high risk types (i.e., unlucky genomes) and low risk
types (i.e., lucky genomes) and also know the population parameters that apply (e.g., the
proportions of risk types, the rate of false positives and negatives of the screening test),
but no individual knows who are the high and low risk types.28 Thus, in an environment
of asymmetric information the weights used to aggregate discrimination reflect relative
frequencies of the various groups of properly and improperly classified individuals, while
under symmetric information the weights reflect the probabilities that people hold regarding
their classification status. For our application to genetic screening, it is the latter case that
is relevant. In general, intermediate cases are also possible; that is, individuals may have
better information than insurers but not perfect information about their true risk type, in
which case a mixture of the above two arguments would be needed for the interpretation of
the weights.
Finally, we emphasize that considerations of discrimination from using, or not using,
available imperfect information to categorize risks in an insurance market is not the only
relevant issue when deciding on whether or not insurers should be allowed to use such
information.29 It is also important to address efficiency and distributional implications.
However, concern over equity in the context of discriminatory pricing will undoubtedly
remain a legal and regulatory concern for policy-makers and so having a firm foundation
for measurement of discrimination is and will remain important.
8. Appendix
8.1. Values of Partial Derivatives
hp
∂
qhql
[qh ph + ql
qhql
pl
=
=
=
=
=
=
< 0
< 0
> 0
> 0
> 0
> 0
2
∂εfn
]
pl
ph
hp
∂
∂
∂
2
∂εfp
[qh
+ ql
]
ph
pl
hn
qhql
nl
2
∂εfn
[qh nh + ql
]
nl
hn
qhql
nh
2
2
2
∂εfp
[qh nh + ql
]
]
]
nl
pl
pl
lp
∂
qhql
pl
∂εfn
lp
[qh ph + ql
∂
qhql
ph
∂εfp
[qh ph + ql

126
MICHAEL HOY AND PETER LAMBERT
ln
∂
qhql
[qh nh + ql
qhql
nl
=
< 0
< 0
2
2
∂εfn
ln
]
nl
∂
nh
=
=
=
=
=
∂εfp
∂ Pp
∂εfn
∂ Pp
∂εfp
∂ Pn
∂εfn
∂ Pn
∂εfp
[qh nh + ql
]
]
]
]
]
nl
pl
pl
pl
nl
qhql
pl
2 (Cl Ch) < 0
2 (Cl Ch) < 0
2 (Ch Cl) > 0
2 (Ch Cl) > 0
[qh ph + ql
qhql
ph
[qh ph + ql
qhql
nl
[qh nh + ql
qhql
nh
[qh nh + ql
8.2. Proof of Theorem 2
By a more accurate screening test we mean one in which either the rate of false negatives or
positives (or both) is lower. We show that a higher rate of either false negatives or positives
implies a higher average price for low risk types and a lower average price for higher risk
¯
¯
Ph
Ch
P
Cl
l
types. This implies an increase in the ratio and a decrease in the ratio
which means
an unambiguous increase in the overall degree of price discrimination.
Recalling Eqs. (18) and (19), and using the results from part 1 of the Appendix, we have
∂ P
∂
∂ Pp
pl ∂εfn
∂ Pn
∂
∂ Pn
l
=
pl Pp +
+
∂εfn
nl Pn +
∂εfn
∂εfn
nl ∂εfn
∂ Pp
=
=
+
pl ∂εfn
nl ∂εfn
pl
plqhql
nlqhql
nl
2 (Ch Cl) +
2 (Ch Cl)
[qh ph + ql
]
[qh nh + ql
]
nl
2
pl
pl
2
nl
= (Ch Cl)qhql
2
2
[qh nh + ql
]
[qh ph + ql
]
nl
pl
This expression will be positive provided the term in the brackets{ } is positive; i.e., provided
nl
pl
>
[qh nh + ql
]
[qh ph + ql
]
pl
nl
or
nl[qh ph + ql pl] > pl[qh nh + ql
]
nl

GENETIC SCREENING AND PRICE DISCRIMINATION
or
127
and
>
nl ph
pl nh
and this is so since for any informative test εfp, εfn
<
12 , which implies
>
nl
nh
>
pl. Using similar steps, we can show that
ph
2
2
¯
∂ Ph
ph
nh
= (Ch Cl)qhql
2
2
∂εfp
[qh nh + ql
]
[qh ph + ql
]
pl
nl
which is negative provided the term in brackets { } is negative, which, after a few algebraic
steps, is equivalent to requiring that
>
nl ph
nh pl
1
2
which we saw above is so for any informative test (εfp, εfn < ).
Thus, we have established the following two results:
¯
∂ P
l
> 0
(A1)
(A2)
∂εfn
¯
∂ Ph
< 0
∂ε f p
Due to our assumption of pooled actuarially fair pricing we have that
¯
¯
qh Ph + ql P = qhCh + qlCl
l
in which the right side is independent of εfn and εfp (as are qh and ql). Thus, we have the
following two results.
¯
¯
∂ Ph
∂ P
l
qh
qh
+ ql
+ ql
= 0
= 0
∂εfn
∂εfn
¯
¯
∂ P
l
∂εfp
∂ Ph
∂εfp
which, in conjunction with Eqs. (A1) and (A2) imply
¯
∂ P
l
> 0
(A3)
(A4)
¯
∂εfp
¯
∂ Ph
< 0
∂εfn
Thus, an increase in either or both of the rates of false positives and negatives increases P
l
¯
¯
Ph
Ch
P
l
¯
and decreases Ph, increasing the dispersion in the values
<
, proving Theorem 2.
Cl

128
MICHAEL HOY AND PETER LAMBERT
Acknowledgments
We thank the Canadian Genome Analysis and Technology programme (#G0-12776) and
the Social Sciences and Humanities Research Council of Canada for financial support. The
first author also thanks the CNRS of France for financial support and GREQAM, Universite´
d’Aix Marseille, where he was a visitor during much of the writing of this paper, for their
hospitality. Mattias Polborn and the editor provided very insightful comments which have
led to substantial improvements in the paper. All views, and any errors or omissions, are
our own responsibility.
Notes
1. For a breadth of views on this controversial issue, see Hook [1992], Lapham [1996], Lowden [1992], Murray
[1992], Pokorski [1995], van Leeuwen and Hertogh [1992] and the background statement on genetic testing
and insurance by the ASHG Ad Hoc Committee on Genetic Testing/Insurance Issues [1995].
2. In particular, see Lambert and Ramos [1997].
3. For example, see Schmalensee [1984] and Tryfos [1987].
4. We also implicitly assume no administrative costs. However, loading factors could be included in the pricing
equations without loss of generality.
5. One could use the difference P
Ci as an individual measure of price discrimination. However, this
ij
would imply, for example, that an individual charged $10 more than the cost of providing him with either an
automobile or a litre of milk would feel equally aggrieved or discriminated against in each case, which seems
unlikely. Such an approach is used by Schmalensee [1984].
6. That is, the degree of discrimination created by a person facing a price which exceeds expected cost by 20%
should be deemed worse by a factor somewhat greater than two than the degree of discrimination created by
a person who faces a price which exceeds expected cost by 10%.
7. For example, the pair (r1, r2) = (1.3, 1.7) should be deemed overall less discriminatory than the pair (r1, r2) =
(1.2, 1.8).
8. Thus, forexample, chargingeveryindividualanequalmultiplicativeloadingwouldleaveoveralldiscrimination
unchanged. Alternatively, one could remove any loading factors from the price and consider discrimination
in terms of the relationship between the pure premium and the actuarial cost of the indemnities.
9. For proofs see Bourguignon [1979], Cowell [1980], and Shorrocks [1980, 1984]. Since prices are actuarially
fair the overall mean of the rij’s is 1; otherwise, one should divide rij by its mean in the formulae in Theorem
1.
10. If = 1, the inequality measure is often referred to as the mean logarithmic deviation (MLD) while if = 0,
it is often referred to as the Theil entropy measure. In both cases, the method of computing HDi , HD, and VD
are the same as for the general case, just making the substitution of the appropriate function. The case = 2
provides an index which is ordinally equivalent to the coefficient of variation as the measure of inequality.
11. For more discussion on this and other properties of this family of inequality measures, see Jenkins [1991].
See Kuga [1980] for a link to the popular Gini index of inequality and Lambert and Aronson [1993] for a
discussion on decomposing the Gini index.
12. Huntington’s Chorea is a disease which fits this stylized view very well. If an individual possesses the
“Huntington’s gene” then it is effectively certain that person will eventually succumb to the disease. Many
diseases, however, are multifactoral in nature in that the risk of incurring the disease depends on many
environmental factors and possibly other genes as well.
13. Note that
= Pr( j | i), the probability that a person receives test result j given that she is of risk type i,
ji
i j
while
j.
= Pr(i | j), the probability that an individual is of risk type i given that she has received test result
14. In such circumstances differential insurance purchases by risk type are predicted under models with both
linear and nonlinear pricing. For examples of the former, see Villeneuve [1996] and Hoy and Polborn [2000],
while for examples of the latter, see Rothschild and Stiglitz [1976].

GENETIC SCREENING AND PRICE DISCRIMINATION
129
¯
¯
15. That is, if a lower value of ε f n and/or ε f p leads to an increase in Ph and a decrease in P then the dispersion
l
¯
¯
,
l ) falls unambiguously.
c
l
in the values of ( Pc
P
h
h
∂ P
p
16. This latter result,
< 0, in fact requires that there exist low risk types among those who test positive (i.e.,
∂ε
f n
> 0, see the Appendix) since otherwise an increased false negative rate would just mean fewer h-types
pl
in the group testing positive but not a smaller fraction of h-types in the group that tests positive, and then Pp
would not change.
17. As in the previous footnote, a caveat is required here:
∂ P
f p
n
< 0 requires that there are high risk types among
∂ε
those who test negative (
> 0).
nh
18. As noted in Strachan and Reid [1997], Mendelian diseases typically affect less than 1 person in 1000.
19. Computation indicates that HD is increasing in the accuracy of the test up to a false negative/positive rate of
0.00340 while TD is increasing in accuracy up to a false negative/positive rate of 0.00385.
20. We performed the simulations for the cases of = 2 and = 3 and obtained very similar results.
21. The functions are relatively flat for values of ε f p greater than 0.006 and so only that part of each function
near the origin is graphed.
22. Computation indicates that TD is increasing in the accuracy of the test up to a false negative/positive rate of
0.000271 and essentially the same result applies to HD.
23. Familial retinoblastoma is such an example with a single, short gene being the indicator.
24. See Strachan and Read [1997, chapter 3] for a full discussion.
25. The BRCA-1 gene spans almost 100,000 bases of the genome and encodes a protein of 1,863 amino acids.
As of 1996, 235 known sequence variations had been identified. The BRCA-2 gene is more complex and
less well studied but is even longer and is known to contain a large range of sequence variations as well. [See
Science, 1996, October issue.]
26. In Villeneuve [1996], for example, this formula is referred to as the average clientele risk. Other formulae
would be relevant if nonlinear pricing persists but the essence of the argument remains.
27. The semi-separating, semi-pooling equilibrium possibility analyzed, for example, by Wilson [1977] involves
cross-subsidization between high and low risk types and so the analysis developed in this paper could be
applied with minor modifications to such cases.
28. Later in life, however, one may determine whether or not one actually was a high or low risk type and so
know the extent of discrimination that one faced in the insurance market.
29. Such considerations are addressed, for example, in Crocker and Snow [1986], Doherty and Thistle [1996],
Ligon and Thistle [1996], Doherty and Posey [1998], Hoy [1982, 1984], Hoy and Polborn [1998], Tabarrock
[1994].
References
BOURGUIGNON, F. [1979]: “Decomposable Income Inequality Measures,” Econometrica, 47, 901–920.
CASEY, D. [1997]: “What Can the New Gene Tests Tell Us?,” The Judges Journal, American Bar Association,
Summer, 36(3).
COWELL, F.A. [1980]: “On the Structure of Additive Inequality Measures,” Review of Economic Studies, 47,
521–531.
COWELL, F.A. [2000]: “Measurement of Inequality,” in Handbook of Income Distribution, A.B. Atkinson and
F. Bourguignon (Eds.)., Elsevier Science, North Holland, Amsterdam, New York and Oxford, 1, 87–166.
CROCKER, K.J. and SNOW, A. [1986]: “The Efficiency Effects of Categorical Discrimination in the Insurance
Industry,” Journal of Political Economy, 94, 321–344.
DAVIES, J.B. and HOY, M. [1995]: “Making Inequality Comparisons When Lorenz Curves Intersect,” American
Economic Review, 85, 980–986.
DOHERTY, N.A. and POSEY, L. [1998]: “On the Value of a Checkup: Adverse Selection, Moral Hazard and the
Value of Information,” Journal of Risk and Insurance, 65(2), 189–212.
DOHERTY, N.A. and THISTLE, P.D. [1996]: “Adverse Selection with Endogenous Information in Insurance
Markets,” Journal of Public Economics, 63, 83–102.
HOOK, E.B. [1992]: “Genetic Distinctions Are Not Necessarily Examples of Genetic Discrimination,” American
Journal of Human Genetics, 51, 897–898.

130
MICHAEL HOY AND PETER LAMBERT
HOY, M. [1982]: “Categorizing Risks in the Insurance Industry,” Quarterly Journal of Economics, 97, 321–336.
HOY, M. [1984]: “The Impact of Imperfectly Categorizing Risks on Income Inequality and Social Welfare,”
Canadian Journal of Economics, 17, 557–568.
HOY, M. [1989]: “The Value of Screening Mechanisms Under alternative Insurance Possibilities,” Journal of
Public Economics, 39, 177–206.
HOY, M. and POLBORN, M. [2000]: “The Value of Genetic Information in the Life Insurance Market,” Journal
of Public Economics.
JENKINS, S. [1991]: “The Measurement of Income Inequality,” in Economic Inequality and Poverty, International
Perspectives, Lars Osberg (Ed.), M.E. Sharpe, Armonk, New York, 3–38.
KUGA, K. [1980]: “Gini Index and the Generalized Entropy Class: Further Results and a Vindication,” Economic
Studies Quarterly, 31, 217–228.
LAPHAM, E.V., KOZMA, C. and WEISS, J.D. [1996]: “Genetic Discrimination: Perspectives of Consumers,”
Science, 274, 621–624.
LAMBERT, P.J. and RAMOS, X. [1997]: “Vertical Redistribution and Horizontal Inequity,” International Tax
and Public Finance, 4, 25–37.
LAMBERT, P.J. and ARONSON J.R. [1993]: “Inequality Decomposition Analysis and the Gini Coefficient
Revisited,” Economic Journal, 103, 1221–1227.
LIGON, J.A. and THISTLE, P.D. [1996]: “Consumer Risk Perceptions and Information in Insurance Markets
with Adverse Selection,” The Geneva Papers on Risk and Insurance Theory, 21, 191–200.
LOWDEN, J.A. [1992]: “Genetic Discrimination and Insurance Underwriting,” American Journal of Human
Genetics, 51, 901–903.
MUESER, P. [1989]: “Price Discrimination,” in The New Palgrave, Eatwell, J., Milgrate, M. and Newman, P.
(Eds.), W.W. Norton, New York.
MURRAY, T.H. [1992]: “Genetics and the Moral Mission of Health Insurance,” Hastings Center Report,
November–December, 12–17.
POKORSKI, R.J. [1995]: “Genetic Information and Life Insurance,” Nature, 376, 13–14.
ROTHSCHILD, M. and STIGLITZ, J. [1976]: “Equilibrium in Competitive Insurance Markets: An Essay on the
Economics of Imperfect Information,” Quarterly Journal of Economics, 90, 629–650.
SHORROCKS, A.F. [1980]: “The Class of Additively Decomposable Inequality Measures,” Econometrica, 48,
613–625.
SHORROCKS, A.F. [1984]: “Inequality Decomposition by Population Subgroups,” Econometrica, 52, 1369–
1385.
Science [1996]: October, 274.
SCHMALENSEE, R. [1984]: “Imperfect Information and the Equitability of Competitive Prices,” Quarterly
Journal of Economics, 99, 441–460.
STRACHAN, T. and READ, A.P. [1996]: Human Molecular Genetics. Oxford, UK: BIOS Scientific Publishers
Limited.
TABARROK, A. [1994]: “Genetic Testing: An Economic and Contractarian Analysis,” Journal of Health Eco-
nomics, 13, 75–91.
TRYFOS, P. [1987]: “The Equity of Classification Systems in Automobile Insurance,” The Journal of Risk and
Insurance, 54(3), 569–581.
VAN LEEUWEN, E. and HERTOGH, C. [1992]: “The Right to Genetic Information: Some Reflections on Dutch
Developments,” Journal of Medical Philosophy, 17, 381–393.
VILLENEUVE, B. [1996]: Mandatory Insurance and Intensity of Adverse Selection, CREST/INSEE working
paper.
WILLIAMS, A.C.J. [1969]: “Price Discrimination in Property and Liability Insurance,” in Insurance, Government
and Social Policy, H.S. Denenberg and S.L. Kimball (Eds.), Richard D. Irwin Inc., University of Pennsylvania.
WILSON, C. [1977]: “A Model of Insurance Markets with Incomplete Information,” Journal of Economic Theory,
16, 167–207.