Imperfect Product Testing and Market Size

Article (PDF Available)inInternational Economic Review 35(1):61-86 · February 1994with18 Reads
DOI: 10.2307/2527090 · Source: RePEc
Abstract
The authors consider an imperfect test of product quality and ask how it interacts with adverse selection to affect market size. Although one might expect adverse selection to be mitigated, there are scenarios where it is exacerbated. Also, two counterintuitive comparative static results emerge. First, a small increase in the test cost can increase the equilibrium expected profits earned by sellers of higher quality units and so expand the market. Second, the equilibrium expected profits earned by sellers with lower quality units can be increased by a small improvement in the accuracy of an imperfect test. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Full-text (PDF)

Available from: Charles F Mason, Nov 17, 2015
INTERNATIONAL
ECONOMIC REVIEW
Vol. 35, No. 1,
February 1994
IMPERFECT PRODUCT
TESTING AND
MARKET SIZE*
BY
CHARLES F. MASON AND
FREDERIC P.
STERBENZ1
We consider an
imperfect test of product
quality,
and ask how
it
interacts
with adverse
selection to affect
market size. Although
one might expect
adverse
selection
to be mitigated, there are scenarios where it is exacerbated.
Also,
two
counter-intuitive comparative static results
emerge.
First,
a
small
increase in the
test cost can increase
the equilibrium
expected profits earned
by sellers of
higher quality units,
and
so expand the market.
Second,
the
equilibrium expected
profits earned by
sellers
with lower
quality
units
can be
increased by
a
small
improvement
in
the accuracy of an imperfect test.
1.
INTRODUCTION
For several
years,
it has
been
understood
that
uncertainty
about product
quality
may
limit the size
of
the market. As
Akerlof (1970) showed
in
his seminal
paper,
market
price will
reflect average quality;
but when sellers
are better informed than
buyers
regarding
product quality, any seller who values his wares
greater
than
market
price
will exit.
Thus, the market
may degenerate to
lower quality
items. In
this context
of
adverse selection, there
is
a
natural
question: Is the market's
tendency to degenerate to lower
quality
items reduced
if
sellers are able to
provide
believable information about the
quality
of their
products
to consumers?
Information
regarding
a
product's
quality
could be
provided
by
the
commodity's
distributor
at
the point of
sale,
as with the
vacuum cleaner salesman
demonstrating
his
product.
Alternatively,
it
might be
provided by
an
impartial
intermediary's
appraisal
of the
-good,
as with
the Underwriter Laboratories' seal of
approval,
a
prominent
critic's review
of
a
movie,
or the
awards
given by
judges
at
a
prestigious
wine
tasting.
In
the
context of labor
markets, potential
employees might
willingly
take
a
certification
test,
as
with
accountants,
financial
analysts,
or real estate
salesmen.
An
important common
aspect
of
these
examples
is that a
test of limited
reliability
is
applied to the
product. Our paper
investigates the impact of
such tests on
the
nature
of
equilibria
in
markets
where
asymmetric
information can
be
partially
mitigated by
a
test of limited
reliability. In
particular,
we
are
interested
in
the effect
of
testing
on
the
incentives for sellers of
higher quality
units to
participate
in the
market and in
the
comparative static effects of
changes
in
test cost and test
accuracy
on sellers' incentives.
*
Manuscript
received October 1991;
revision received October
1992.
1
An
earlier version
of this paper was
presented
at
the American
Economic Association
meetings
in
Dallas. We thank Ted
Bergstrom, Mark
Machina, Owen Phillips,
Steve Raymar, Mike
Riordan,
Todd
Sander, John
Tschirhart, and seminar
participants at Carnegie Mellon
University for helpful
comments.
The constructive comments of
two
anonymous referees lead
to
a
significant improvement
in
the
manuscript. The usual
disclaimer applies.
61
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
62 CHARLES F. MASON AND FREDERIC
P. STERBENZ
Although
one
might anticipate
that
such information
would
necessarily
encour-
age sellers
to participate
in
the
market,
and so
expand
the
market,
our
analysis
indicates
that this need not
be
the case:
testing may
induce some sellers
of
good
units to
exit, shrinking
the
market. This can
result
because the increment in
expected
profits
a
good
seller can
hope
to realize
by having
his
product
tested is
influenced
by
the test
accuracy
and the number of bad units
that are
tested. When
this increment in
expected profits
is insufficient to cover
the
test
cost,
the
testing
equilibrium
will
contain
fewer
good
units
than
some no-information
equilibrium.
Sellers
of
good
units who remain in the market take
a
test which
yields
a smaller
increment
in their
expected profits
than
the test cost
precisely
because failure
to
take
the
test would mark their wares as lower
quality,
and
yield
an even
larger
reduction
in profits. This result is similar
to
Akerlofs
(1976)
"rat race"
phenom-
enon,
wherein
imperfect
information about
quality
leads to incentives for lower
quality
items
to
masquerade
as
higher
quality items,
and
higher
quality
items must
consequently
work "harder" than is
socially optimal
to
distinguish
themselves.
Our
comparative
statics
analysis produces
two results
in the
context of
imperfect
tests, both of which
fail
to hold in the context of
perfectly
discriminating
tests
and
both of
which
are somewhat counterintuitive. With
respect
to
changes
in
test
cost,
we show that sellers
with
good
units can
benefit
from
more
costly
tests. Even
though the
increased
cost of the
test
raises
the
cost of
marketing good units,
it
also
deters some sellers with bad units from
testing
their
products.
This
screening
effect
raises the
price
a
passed
unit
fetches;
it
is possible that this
increase in price is large
enough to more than
offset the
increase in test cost, and so increases the
expected
return from
testing
a
good
unit.
With
respect
to test
accuracy,
we
show that the
expected
profits
earned from
testing
can be
positively
affected
by
increases
in test
accuracy,
for all sellers. An increase in
test accuracy
unambiguously raises the
expected
profit
from
testing
all
good units which were
marketed
in
the original
equilibrium, leading to
an
increase
in
the
volume
of
good units
which
are
tested and
marketed.
This pushes up the price of
passed units;
if
the supply
curve of good units
is
sufficiently elastic, the increase in
test accuracy can
engender a large enough
increase
in
the
quantity
of
good units
as to
raise the expected
profits
from
testing
a bad
unit.
Because our model allows for an
imperfect
test and
does not
require
sellers to
test their
products,
the
act of testing
indirectly delivers
information to the market.
This indirect
information resembles
Spence's (1973, 1974)
notion of
a
signal,
and
forms
the
basis for the
investigations
of
advertising by
Nelson
(1974),
Kihlstrom
and Riordan
(1984),
and
Milgrom
and
Roberts
(1986).
However,
our model
also
includes
information
directly
relevant
to determining
product quality,
whereas
these
papers focus on the role of the
respective signals
in
providing
information
to
the market.
One
consequence
of this
aspect of our model
is that
our
equilibria
are
commonly
pooling; by contrast, only
separating equilibria
are
possible
with
perfect
tests.
Many
authors have
considered
the
role of
testing
in markets
where more than one
quality
exists,
and
sellers have
better information
than
buyers.
Examples include
Stiglitz's
(1975) study
of
screening,
Leland's
(1979) study
of
minimum
quality
standards,
and
studies
by
Guasch and Weiss
(1980, 1982)
and Burdett and
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
IMPERFECT
PRODUCT TESTING
63
Mortensen (1981) of testing in
labor markets. Other papers that
discuss testing are
Heinkel (1981), Jovanovic (1982),
Matthews and Postlewaite
(1985), and Titman
and Trueman (1986). The crucial
distinctions between our
paper and the extant
literature are that we allow
for information from an imperfect
test, and that we
allow sellers to remove their
products from the market, i.e.
we do not assume
perfectly inelastic supply.2
Hence, our model allows
us
to determine
the impact of
an
imperfect test,
as
opposed
to a perfect test, on both the
incentive to
test and to
sell
a
product.
As
we remarked
above,
this leads to rather different conclusions
regarding
the
comparative static
effects of changes in test cost
and
test accuracy.
In
many respects our model
is most similar to the models in
Heinkel (1981) and
Jovanovic (1982), although
there are
important
differences between our work
and
theirs.
Heinkel considers
the
use of
an
imperfect
test
of product
quality
in
a
market
where
sellers control
both
quantity
and
quality. However,
his
analysis focuses on
the use of this test
to impose
an ex
post penalty
on sellers
who
provide relatively
lower quality products. Jovanovic
is
concerned
with the incentives for sellers to
reveal privately held information.
His sellers
know the signal which will be
disclosed,
and so do not have to
pay
to obtain
it;
disclosure
is
costly. By contrast,
our
sellers
do not
know the
signal to
be disclosed
(i.e.,
the test
results);
disclosure
is
costless, although obtaining
the signal
is
costly. Since
the decision to pay the test
cost serves a
screening function
in
our model, our equilibria
are generally
different
from his.3
2. THE MODEL
The market we consider
has
two
important
characteristics:
the
product
in
question
can
be
of more than one quality,
and sellers have better information
than
buyers regarding quality.
Our
model captures
the former
aspect
by considering
two
possible qualities, "good"
and
"bad." The
asymmetry
of information
is
modeled
by allowing
sellers
perfect
knowledge of
their units' true
quality,
while
buyers
are
ignorant of quality.
This is essentially the framework employed
by
Akerlof
(1970).
Buyers
would
be
willing
to pay XG for
a
good
unit
and XB
for
a
bad
unit.
2
See Stiglitz (1975,
p. 286). Leland (1979)
does allow sellers
to exit the market, although
he does not
allow them the option
of not testing and
his screening process is
perfect. Burdett and
Mortensen's (1981)
models
in
Sections
4a and 4b address incentives
for workers
to
test for
ability,
and
to reveal this
information. While
their models are similar
in many respects
to ours, their test perfectly
reveals true
quality, while we
allow for an imperfect
test. Guasch and Weiss
(1980, 1982) consider testing
in
labor
markets where the
test is perfectly discriminating
but workers
perceive the results
as random, because
they do not know
their true quality. In contrast,
our sellers do
know their products' true
quality and the
test may be random.
This allows us the
flexibility to model sellers'
reservation prices
as asymmetric, so
that the optimal choice
for some of our
sellers is to exit the
market rather than test.
For an additional
discussion on the incentives to test and
disclose product quality
when the seller is imperfectly
informed,
see Matthews and
Postlewaite (1985).
3
In one sense
the disclosed information
in Jovanovic's model
is imperfect because
it is a noisy signal
of true quality. However,
in another sense
the disclosed information
is perfect because
disclosure entails
fully revealing
the seller's knowledge
to
buyers. We note
that one can draw
an
analogy
between
Jovanovic's equilibria
with costless revelation
and equilibria
which would emerge from
our model if
the
test were
costless.
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
64 CHARLES
F.
MASON
AND FREDERIC P. STERBENZ
Presumably, higher quality units are worth more, so
that
XG
>
XB.
We
assume
that these numbers are exogenously given and are known by
all
sellers.
We
will write
Yi(j)
as the
reservation price
for
the
seller of
the
jth type
i
unit,
where i
=
G if the unit is good and i
=
B
if the unit is bad.
We
assume
that
the
seller
valuations
Yi(-)
are
exogenously
determined. The seller's valuation could be the
salvage value of
the unit. In the
context of
labor
markets,
Yi(j)
could be the
wage
the jth type
i
worker could earn by being self-employed,
with
Xi
measuring
the
value of his marginal product to a potential employer.
To
simplify
the
analysis
which
follows,
we assume that there is
a
continuum of
buyers and
sellers.
The assumption of
a
large
number of
buyers
is
made to
insure
that the
product
trades at
the consumers' reservation
price,
as in Akerlof
(1970).
For
expositional purposes,
we use
"g"
to
index
good
units
and
"b"
to index bad
units,
and we
order sellers
in
increasing
terms
of
their reservation
prices.
Thus,
both
YG(g)
and
YB(b)
are
upward sloping.
In a market with
no signals of quality, buyers
can
only
draw
inferences from
sellers'
actions.
In
particular,
all units
appear
identical to
buyers
and
are
sold at the
same price; buyers base their reservation price
on their
perception of
average
quality
in the
market.4 Sellers choose between marketing
and
not marketing their
wares; they enter only
if
market price exceeds
their
reservation price.
Buyers form
an
expectation of
the
probability
that a unit
brought to market
is
good,
0. Based on this
belief, buyers
form
a
reservation price, RP(0):
(1) RP(0)
=
0
*
XG
+
[1
-
0] *
XB
=
XB
+
0
*
(XG
-
XB)-
We note that the assumed
ordering
of
units
implies
that
the
identity
of the
marginal
good (respectively, bad),
unit measures
the volume of good (respectively, bad)
units in
any equilibrium.
A
rational expectations equilibrium
is
a collection of units in the market, and
an
associated
price,
such
that
(i)
no seller would
unilaterally add
or
remove
a unit from
the market,
and
(ii) the buyers' expectation equals the true fraction of good units
in
the market.
As in
many
models with
imperfect information, our
model
may
have
multiple equilibria.
Since
RP(O)
=
XB,
if
XB
<
YG(g)
for
all
g, one equilibrium
is a
"lemons"
market, containing only
bad units. If an
equilibrium exists wherein
some
good
units
are
marketed,
then this
equilibrium
contains b* bad units and g*
good units,
where
(2)
YG(g*)
=
RP(0);
(3)
YB(b*)
=
RP(0);
and
(4)
0 =
g*l(g*
+
b*).
4
We adopt the
interpretation
of rational
expectations used
previously
in
the literature (Chan
and
Leland
1982,
Cooper and Ross 1984, and
Grossman
and
Shapiro 1984).
In
this
interpretation,
buyers form
an
expectation
of some observable
market statistic; a
rational expectations
equilibrium
occurs
when this
expectation is
confirmed. Note that this
implies
competitive behavior, since each
supplier
ignores the
implication of
its actions on the
equilibrium quality
distribution.
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
IMPERFECT PRODUCT TESTING
65
We
also describe
an
alternative approach to determining
an
equilibrium.
For
any
candidate value g for the equilibrium number of good units, one can use equations
(1), (3), and (4) to determine b*, as a function of g. This yields a potential
equilibrium price, RP*(g), as a function of the proposed number of good units.
This potential equilibrium price is determined by inserting b*(g) into equation (4)
to give 0 in terms of g, and then combining
with
equation (1) to get
(1') RP*(g)
=
XB
+
g(XG
-
XB)![g
+
b*(g)].
An equilibrium is then fully described by g*:
YG(g*)
=
RP*(g*),
and b*
-
b*(9
*)
S
We
define for
purposes
of reference the
"largest
sales"
equilibrium
as that
equilibrium
-with
the most good
units marketed. Because
YG(')
is
upward-sloping,
this
equilibrium necessarily
has
the
highest price among
no information
equilibria;
it
follows that it also has
the
largest total volume of
units as well.
Hence,
each seller
is
at least as well off
in
this equilibrium as
in
any other no-information equilibrium,
so that the largest sales equilibrium Pareto dominates
all
other no-information
equilibria.
When
comparing
the
no-information
market to the market
with
testing,
we shall generally use the largest sales equilibrium as the point of reference.
3. THE MARKET WITH TESTING
Now suppose that any seller can employ an independent intermediary to test
his
product. To ensure the impartiality of this intermediary,
we rule
out
side
payments.
One
might
think of the test
as
determining
the
presence
or absence of some
characteristic
which is
correlated
with
quality.
The
higher
is this
correlation,
the
more accurate
is
the test. Also, given that
the
characteristic's presence
is
exactly
determined, there
is
no point
in
repeating
the test:
any
unit that has failed once
will
always
fail.
We
assume that the
results of the test cannot
be
falsely reported.
That
is,
neither
a failed
unit
nor an
untested
unit
can
be
represented
as a
passed
unit.
One
might
imagine
the
intermediary publishing
the
results of the test
(subject
to the seller's
approval),
as
with the
accolades at
a wine
judging
or
the successful
acquisition
of
the Underwriter Laboratories'
seal
of
approval.
In
this
scenario,
the
seller can
(privately)
observe the test result
and then
arrange
to
have the information
credibly
revealed.
We
do allow sellers to conceal the test results, so
that a failed unit can
be
represented
as an
untested
unit.
In
our,
testing model,
hfi
denotes the
probability
that
a
type
i
unit
will
pass
the
test,
i
=
G or
B,
and A
denotes
the
test cost.
As the test is informative,
MOB
<
COG.
We restrict our attention
to
tests
with 0
<
A
<
XG
-
XB.
The
upper
bound
on test
cost
is
natural since
the
largest possible gain
in
price
which
can result from
taking
the test is
XG
-
XB;
the lower
bound allows
testing
to
have a
signaling
effect.
Upon observing
the available
information, buyers
form
a
prediction
that
a
given
5
We note for later reference that RP*( g) is increasing in
g. For fixed b*(g),
it
is clear from equation
(1) that this holds; if b* increases with g, it follows from equation (3) that RP* must also have increased.
Therefore, larger values of g must yield larger values of RP*.
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
66 CHARLES F. MASON AND FREDERIC
P.
STERBENZ
YG(
)G)
exi
YG(
)
test\
Good
Bad~~~~~~~~~~~a
FIGURE
1
THE
DECISION TREE
unit is
good. Because
there are
three
possible signals
buyers
could receive,
they
form
three
predictions. The
subjective
probability that
a
unit
is good
is
Pp,
if it
is
sold as passed,
Pf if it
is sold as
failed, and
P~n
if
it is
sold as untested.
Given
these
buyers'
expectations,
reservation
prices
are
(5a)
RPP = XB + PP
* (XG XB),
(5b)
RPf = XB
+ Pf *
(XG-XB)
(5c)
RP~n
=
XB
+
Pu
(XG
-XB).
A decision
tree showing
the
sellers' options
in our
model is depicted
in Figure
1.
Each
seller has three
options available to
him.
Under the
first option, the
seller
chooses
to exit
the market,
realizing zero
profits. Under the
second
option,
the
seller
chooses not to
test his
unit, and
receives the price
untested
units
command,
RPn
This yields
the seller
with the jth
type i
unit profits of RP
n
- Y1 (j),
where
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
IMPERFECT
PRODUCT TESTING 67
i
=
G or B.
Under the third option, the seller
pays the
test
cost,
has his unit
tested,
privately
observes
the
test results, and then
decides
his
course of action.
In
this
event,
the seller
receives some expected gross
payment,
wi(j);
his ex ante
expected
profits
are
Wi (j)
= io
(i)
-
Y
(i)-
-
A
We define the "value"
of
the
jth type
i
unit as
the
maximum of
these three
expected
profits,
and
denote
it
by
Vi(j):
(6)
Vi(j)
=
max
{O,
RP1,
-
Yj(j),
Wi(j)}.
The
expected
profits from testing
a
unit
are
determined
by
the reward
received
if
the unit
passes,
and the seller's best
option
if the
unit fails.
As in the
no-information
market,
we assume
that
products trade at buyers'
reservation
prices. Thus, the
contribution to expected
profits from a passed type i unit is
qiRPp,
i
=
G or
B.
Should
the
jth type
i unit
fail
the
test, which occurs with
probability
1
-
hi,
its seller
has three
suboptions. First, he can sell the unit as
failed, whence it earns
RPf. Second, he can
conceal the test result, and sell it at the
untested price.
In
this
case,
the
failed unit
would earn
RPun
Third, he
can
withdraw
the
unit
from the market, and
thereby
earn
Yj(j).
Combining these
observations
produces
the
relation
(7)
CO(j)
=
iRP
+ (
1
-
[max
{RPf,
RP1dn, Yi(i)}],
so that the
expected
profits from testing
the
jth type
i unit are
(8)
Wi(j)
=
qji
*
[RPP
-
Yi(j)]
+
(1
-
qi)
* [max
{RPf
-
Yj(j),
RPM,,
-
Yj
(j),
O}]
-
A.
Let
gT
gf,
and
gfun
be
the volumes of
good
units
that
are
tested,
that
would
be
left on the market after
failing
the
test,
and that are marketed but not tested. Let bT
bf, and bun be the
similarly
defined
volumes of
bad units. An
equilibrium
is a
six-tuple of volumes
(g
T
gf,
gun,
b
T, be,
b
an)
and a
triple
of
prices
(RPP,
RPf,
RPUn)
such
that
(i)
the
six-tuple
of volumes confirms
the
expectations
implicit
in
the
triple
of
prices,
and
(ii) given
the
triple
of
prices,
sellers' best
responses
yield
the
six-tuple
of volumes.
In
equilibrium,
the
probability
that
a
passed
unit is
good
is
(9a)
PFp
=
q,~gT,/[qGg
T+
qNBb
T].
Because
failed units
may
be sold
as
either
failed
or
untested,
the
equilibrium
expectations
for these two
classes
are
more
subtle than for
passed
units. All failed
units that are
marketed could
be
sold as
such,
in
which case
the
probability
that a
unit
sold
as failed is
good
is
(9b)
Pf
=
(1
-
qfG)gf/[(1
-
qfG)gf
+
(1
-B)b]
and the
probability
that
an untested unit is
good
is
given by equation (4),
using gufn
in
place
of
g*
and b
uf
in
place of
b*.
Alternatively,
all failed units
could
be
sold
as
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
68
CHARLES F. MASON AND FREDERIC
P. STERBENZ
untested. In this case, no units are sold as failed, so that Pf cannot be determined
by Bayes' rule;
the
only requirement
is that
Pf
?
P,,
(so
that
placing
a failed
unit
in the untested market is optimal).
The
probability
that a unit sold
as untested
is
good is then
(9c)
P,~,,
=
I[(1
-
)gf
+
gun]/[(1 -
G)gf
+
gUf
+
(1 -
P)b
+
b
u].
Finally,
it is
possible
that some failed
units
could
be
placed
in
each of
the
failed
and
untested markets. It can
be
shown
that this
is
qualitatively
identical to
the
market
where all failed units are
sold
as
untested,
and so we
shall
focus on
equilibria
where
all
failed
units that are not withdrawn
are
placed
in the same
market,
i.e. either
all
are sold as failed or
all
are sold
as untested.6
Two observations
on consumers'
equilibrium expectations
are
germane. First,
these
expectations
are consistent
with
Bayes'
law.
Consider,
for
example,
the true
equilibrium fraction of
tested units which
are
good:
6T
=
gT/(gT
+ b
T).
Combined
with the
probabilities
of
good
and bad units
passing, 'PG
and
qiB,
Bayes'
law gives the posterior probability
that
a
passed
unit
is
good
as
Pp
=
rGOTII0GOT
+
rB(l
-OTA;
multiplying both numerator
and
denominator by g
T +
b
T
then
produces
the
right
side of equation (9a).
Similar
arguments verify
the
Bayesian
nature
of the formulae
in
equations (9b)
and
(9C).7 Second,
our
assumption
that
qPG
>
VPB
implies
RPP
strictly exceeds RPf
unless
gT
>
0
=
b
T,
or gT
=
0;
in
either case
we
could have
Pp
=
Pf.
Similarly,
RPp
will
generally
exceed
RP,
6
In such a
scenario,
we must have
Pf
=
Pun.
Imagine
a
perturbation
that moved all failed units from
the failed market to the untested market. Using equations (9b) and (9c), we infer that this would leave
Puns
and so the untested
price, unchanged.
Hence,
any equilibrium
wherein sellers with failed units are
indifferent between
selling
them as failed and
selling
them as untested must
yield
the same
price
for failed
units as the
regime
where all marketed failed units are
placed
in the untested market.
7
Equation (9b)
is
produced by noting
that
the
proportion
of units that would be
placed
in the
failed
market that are
good
is
Of
=
gfl(gf
+
bf), using Bayes'
law to
derive
the
posterior probability
of
drawing
a
good
unit from the set of units sold as failed as
Pf
=
(1
-
IJG)Ofl[(l
-
kG)Of
+ (1
-
B)(
-
Of)],
and then
multiplying
both numerator and denominator
by gf
+
bf
.
To
get equation (9c),
note that of those
units that
would
be
placed
in the untested market
the
proportion
of units that
are
good
and have failed the
test is
gf/(gf
+
gun
+
bf
+
bun);
the
proportion
of units that are
good
and have not taken the test is
gunl/(gf
+
gufn
+
bf
+
btn);
the
probability
that a candidate tested
good (respectively, bad)
unit will
fail,
and so
wind
up
in the untested
market,
is
1
-
DIG (respectively;
1 -
I'B);
and
the
probability
that an
untested
unit,
whether
good
or
bad,
will wind
up
in
the
untested market
is 1. Then
use
Bayes'
law
to
derive the
posterior probability
of
drawing
a
good
unit from the set of units sold as untested as
Pun
{[(1
-
JG)gf
+
gun2]I(gf
+
gun
+
bf
+
b
un)}I
{[(1
-
p)gf
+
gun
+
(1
-
B)bf
+
bn]I(gf
+
gun
+
bf
+
bun)}
and
clear the fraction.
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
IMPERFECT PRODUCT TESTING
69
For
purposes of comparison, we
discuss first perfect tests. Such
a
test has
qIG
=
1
and
'OB
=
0. It is clear that no bad
unit can gain from taking this
test, so that
only
good
units are
tested.
Since only good
units are tested
(and
passed), only good
units
would pay
for this test, and so taking
the test perfectly signals a
unit as good. It
follows that
RPP=
XG, and that the
expected profits from testing
the
gth good
unit
are
XG
-
A
-
YG(g)- Consequently,
the marginal good unit
is
go,
where YG(go)
= XG -
AA. With
Y'G(*)
>
0,
those good
units
g
<
go
will
prefer
testing
to
exiting.
In
addition, all good units will
generally prefer testing to not
testing, so that all
marketed
good units are tested.
Finally, no bad units that are
marketed are tested,
so that
RPU,
=
XB.
Will
this
test increase the quantity
of good
units in the market?
So long
as
some
good
units
are
marketed
in
the
no-information equilibrium, for
sufficiently expen-
sive tests
the answer is no. Let
9LS
represent the number of
good units
in
the
largest
sales no-information
equilibrium, and RPLS the
associated price. With
9LS
>
0,
it
follows from
equations
(1) through (4)
that
YG(gLS)
>
XB,
and so there
exist values
of
A
for which XB
<
XG
-
A
<
YG(gLS).
But the
quantity of good
units in the
testing market must be
go,
as defined above.
Consequently, for perfect
tests
with
costs
in the
range XG
-
XB
>
A
>
XG
-
YG(gLS) there will be fewer
good
units in
the
testing equilibrium
than
in
the
largest
sales
no-information
equilibrium. With
the test cost less
than the gain in price, XG
-
XB,
the
unraveling
result
of Matthews and
Postlewaite
(1985) implies
that sellers
of
good
units are
induced to
test in any equilibrium.
However,
if
the difference
between the new
equilibrium
price
and the
old equilibrium
price, XG
-
RPLS,
is
less than the
cost
of the
test,
then
some
good
units are induced to
exit, thereby
causing
the
testing
equilibrium
to contain fewer good units than the
largest
sales
no-information
equilibrium.8
We
observe that an
increase
in
the cost of a
perfect
test
necessarily
lowers the
expected
profits from testing good
units,
and
so lowers
the value of
all
good
units
which are
tested.
In
turn,
this
reduces
the
quantity
of
good
units
in
the
market.
Although
this
point
seems rather
trivial,
we
make
it
to emphasize
a
distinction
between
perfect
and
imperfect
tests. As we shall
show
below,
increasing
the
cost
of
an
imperfect
test
can
serve
to
raise the
expected profits
from
testing good units,
thereby
raising
the value of all marketed
good
units and
increasing
the
quantity
of
good
units
in
the
market.
A
second distinction between
testing equilibria
with
perfect
tests and
testing
equilibria
with
imperfect
tests concerns the effect of a
slight
change
in
test
accuracy,
In the
context of
a
perfect
test,
a
slight
reduction
in
test
accuracy
cannot
have
a deleterious
effect on bad units. What
is
also true
in
the context of
perfect
tests
is
that
a small increase in
MPB (above
zero)
will
not
change
the
expected profits
from
testing
for
any unit, so long
as
qIG
remains
equal to unity.
In
particular,
such
a
reduction
in
test
accuracy
need
not lower the
value
of
any
good
units.9
By
8
Even if the test is not perfect, when consumers
expect that all good units in the market will be tested
the existence of a test forces good units to abandon the
untested portion of the market.
Thus,
a similar
result
can
hold in
the context of imperfect tests.
9 This statement is based on the presumption that
consumers expect all failed units to be bad. (Since
G
=
1,
clearly
no
failed unit can be
good.)
Based on this
belief,
the
expected
profits
from
testing
a bad
This content downloaded from 129.72.96.98 on Tue, 17 Nov 2015 17:29:32 UTC
All use subject to JSTOR Terms and Conditions
70 CHARLES
F.
MASON AND FREDERIC
P.
STERBENZ
contrast,
in
the context of
testing equilibria
with
imperfect
tests a
reduction
in
test
accuracy
will
lower
the
value of
all
good
units
in the
market.
This
can induce
sufficiently many
good
units
to be withdrawn from the
market as
to
yield
a
decrease
in the
expected profits from
testing,
and
consequently
the
value
of,
all
bad units.
We now
turn
our
attention to
imperfect
tests,
imposing 'PG
<
1
and
NPB
>
0.
In
light of the
observation
above that the
price passed
units fetch
is
at least as
large
as
the
prices paid to units sold as failed or
untested,
our restriction that
FPG
>,
FPB,
and
equation (8),
we
infer that
for
all
good
units
g
and bad units
b,
(10)
WG(g)
-
(RPU.
-
YG(g))
?
WB(b)
-
(RPU.
-
YB(b))
That is, the option
to
test is at least as
appealing to
sellers
of
good
units as
it
is
to
sellers
of
bad
units,
relative
to
choosing
not to
test.
Indeed,
this
inequality
is
strict
unless
RPp
=
XG
=
RPf.
Correspondingly,
in
equilibrium
sellers of
good
units
cannot
be indifferent between
testing
and
not
testing
unless
RPp
=
XG.
In
such
an
event, no bad units would be
tested,
so
that either
RPf
=
XG
also
(in
which case
all
sellers must be indifferent between
testing
and
not
testing),
or
RPf c
RPu,
(in
which case all sellers
of
bad units
prefer not
to
test).
To
analyze
the
first of these classes
of
separating
equilibrium,
we
recall
the
definition
of
go: YG(g0)
=
XG
-
A.
We
then define an
"expenditure
signal
equilibrium" by
(a)
go
=
g
T
+
gun good
units
are
marketed,
with
g
T
gf
>
0
and
gun
>
0; (b)
bT
b
=
0
<
bU; (c)
RPP
=
RPf
=
XG;
and
(d)
RPun
=
RP*(gun) = XG
-
A. Notice that in
such
an
equilibrium
the
payment
of
the
test
cost
perfectly signals
that the
product
is
good;
the
test
results
provide
no
useful
information.
This class is similar
to
the
separating
equilibria
in
Nelson
(1974),
Kihlstrom
and
Riordan
(1984),
and
Milgrom
and Roberts
(1986),
wherein an
expenditure
on
advertising
can
perfectly signal
a
product's
quality.
Existence of
such
an
equilibrium
is tied
to
the
no-information reservation
price
when
go
good
units are
marketed,
the value of a
good
unit,
and the test
cost.
LEMMA
1. A
necessary
and
sufficient
condition
for
the
existence
of
an
expen-
diture
signal equilibrium is
RP*(go)
>
XG
-
A.
PROOF. Assume an
expenditure
signal equilibrium
exists.
Since
RP* is
increas-
ing
in
g (compare footnote
4),
and
go
>
gfun,
it
follows that
RP*(go)
>
RP*(gun)
=
XG
-
A. For
sufficiency, notice
that
RP*(go)
>
XG
-
A
implies
the
existence
of
a value
for g
<
go
such that
RP*(g)
=
XG
- A.
If
gun
=
o
gT
=go
-
go
b
T =
bf
=
0,
and
YB(bun)
=
XG
-
A,
then
the
proportion