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working paper
department
of economics
THE INFORMATIVENESS OF PRICES:
SEARCH WITH T.KAJINING AND
INFLATION UNCERTAINTY
Roland Benabou
Robert Gertner
No.
555
April 1990
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE INFORMATIVENESS OF PRICES;
SEARCH WITH LEARNING AND
INFLATION UNCERTAINTY
Roland Benabou
Robert Gertner
No.
555
April 1990
THE INFORMATIVENESS OF
PRICES:
SEARCH WITH LEARNING AND
INFLATION UNCERTAINTY
Roland Benabou
M.I.T.
Robert Gertner
University of Chicago
April 1990
This
We
is
a
much
would
revised
like to
and expanded version of "Inflation and Market Power
in a
Duopoly Model wiih Search."
thank Olivier Blanchard, Dennis Carlton, Peter Diamond, Robert Lucas, Kevin Murphy,
David Scharfstein, Jean Tirole, and Robert Vishny for beneficial discussions.
THE INFORMATTV^ENESS OF PRICES:
SEARCH \M1 H LEARNING AND
INFLATION UNCERTAINTY'
Roland Benabou
M.I.T.
Robert Gertner
University of Chicago
ABSTRACT
Sources of aggregate cost uncertainty, such as real shocks or unanticipated
reduce the information content of prices by making
aggregate price variations.
difficult to
it
may
in
fact lead
them
equilibrium, and prices will reflect increased competition.
a model of search with learning,
in
whether
is
become
To examine
consumers' search
inflation variabilit}' increases or
thereby market efficiency,
to
for instance by
better informed in
this issue,
we develop
which consumers search optimally from an unknown price
distribution, while firms price optimally given
decisive factor in
separate relative and
But when agents can acquire information,
searching, such increased noise
inflation,
rules.
We
show
that the
reduces the incentive to search, and
the size of informational costs.
I.
Introduction
Consider the problem of a consumer
who
She must estimate how much of
station.
is
it
suppliers, such as an oil shock or high inflation
much
of
it
is
opposite case
deemed more
it
manufacturer
is
relevant,
may pay to do
it is
good
behavior
in turn
deal, or
will
working
when
on new
factors which are specific to this particular
a truly
to a
its
common
factor affecting
all
other
way through the economy; and how
for this particular seller.
If
the
first
not worth looking for a better deal elsewhere; in the
so. Similarly,
offering large rebates
due
demand shock
reflects a specific supply or
explanation
observes an unexpectedly high price at a gas
this
cars,
brand
whether the whole industry
(e.g.
is
consumer observes
that an automobile
she must resolve whether this reflects
excess inventories) and
perhaps having a
sale.
make
the offer
Again, her search
vary with the inferences she draws from observed prices. These inferences are
based on her knowledge of the relative variability of idiosyncratic and aggregate
shocks, inflation being one of the latter kind.
This problem of search with learning from prices
to us important
on several counts.
First,
is
the main focus of the paper;
it
seems
from a microeconomic point of view, the inferences
which buyers draw from prices underlie their demand functions, and should therefore play an
important role
in
determining
technological innovations, etc.
implications for
what
is
how markets
react
to
oil
shocks,
weather variations,
Secondly, from a macroeconomic point of view,
it
has
often thought to be an important source of welfare losses from
unanticipated inflation: a deterioration
through increased relative price
in
variability.
the information content of prices, operating
^
According to
this view, stochastic inflation
Several studies documenting the correlation between the variance of unanticipated
and relative price variability (Vining and Elwertowski (1976), Parks (1978), Fischer
(1981)) seem to lend support to this idea. Hercowitz (1982), on the other hand, finds that
aggregate real shocks, not monetar)' shocks, explain relative price dispersion in the United
States. As the average rate and the variabilit}' of inflation are also correlated (e.g. Taylor
(1981). Pagan, Hall and Trivedi (1983)), high inflation is often seen as indirectly responsible
for any informational costs of unanticipated inflation. The empirical literature on these
issues is surveyed extensively in Fischer (1981) and Cukierman (1983).
1.
inflation
constitutes a source of aggregate noise in
While
decisions.
this
market
well understood in
is
signals, leading to inefficient allocation
models where the information structure
is
exogenous (Lucas (1973), Barro (1976), Cukierman (1979)), what happens when agents can
decide to acquire additional information, by searching or otherwise, has not been explored.
Finally,
another common, but not yet formalized idea,
is
that sellers can "hide" behind
aggregate or inflationary noise to charge higher real prices, taking advantage of consumers'
reduced information to increase their markups. 2
In this
paper we attempt to
cost uncertainty, arising
of
efficiency
from
allocations
ex'plore these issues
real shocks or
market
a
in
by analyzing the effects of aggregate
from unexpected general
endogenous
with
inflation,
on the
and
information-gathering
price-setting.3
The
literature
traditionally
on the
effects of inflation
on the information content of prices has
focused on prices' role as signals for producers rather than consumers.
Misperceptions by producers concerning aggregate and relative price movements (Lucas
(1973), Barro (1976),
movements
relationship
first
in
Cukierman
(1979), Hercowitz (1981)) or
(Cukierman (1982)) generate
relative prices
between the variance of inflation
type of misperceptions
availability of
macroeconomic
is
in
make
transitory
inefficiencies as well a as
the price level and that of relative prices.
price or
monetary
statistics
should solve the problem. This
models too
literally,
rather than as a
the point that inflationary' noise interacts with the functioning of
2. For instance, whether or not oil companies and gasoline retailers engaged in such
behavior following the shutting down of the Alaskan pipeline after the "Exxon Valdez"
spill has been the subject of intense debate.
3.
For a related
(1988), (1989).
The
often dismissed as implausible on the grounds that the
criticism takes the informational structure of these
simplifying device to
permanent and
analysis of the effects of anticipated, or trend inflation, see
Benabou
oil
the price system.
A
more
and constructive restatement of
valid,
some
pointing out that agents can indeed acquire information, but at
example of this
this criticism consists in
cost; the clearest
search. This in turn has two important consequences:
is
(i)
Sellers generally have
{}{)
Information
is
market power.
in
is
.
endogenous: information acquisition and pricing strategies
are determined jointly
Our paper
.
equilibrium, and both
in
depend on
inflation.
the "misperceptions" or "learning" tradition but emphasizes the
informational problems of consumers as well as producers, and embodies the above remarks,
on which we now elaborate.
First,
informational costs realistically imply that sellers have market power.
growing more sophisticated
literature has
in
its
treatment of signal-extraction, the "misperceptions"
maintained the assumption of perfect competition, although
with that of imperfect price information.
asymmetry between informational
In
it
modern economies, no market
is
more
likely that
both
t}'pes
it
seems
at
odds
Indeed, their coexistence relies on a drastic
costs within a
is
While
really cut off
market (zero) and across markets
from the others
of costs are of the
in
(infinite).
terms of information, and
same order of magnitude.
A
realistic
information-based model should therefore probably recognize the presence of some (possibly
small)
monopoly power
.
As
a result, inflationary uncertainty
these are not Walrasian to start with; the issue then
will
be worsened or
Secondly,
may
alter relative prices, but
becomes whether
preexisting distortions
alleviated.
in the
absence of
fully
adequate price
statistics,
buyers and sellers have an
incentive to seek information, for instance by searching (within a market) or just by checking
other prices (across markets).
therefore
embody
For instance, we
An
equilibrium analysis of inflationary misperceptions should
the effects of the inflation process on this incentive to acquire information.
shall see that a deterioration in the quality of price signals
due
to increased
inflationary noise can lead agents to acquire
informed
more
in equilibrium. Naturally, the costs of
information, making
them
actually better
acquiring information (e.g. search costs) will
play a determinant role.
To
take account of these points,
we build
a model with a stochastic structure somewhat
Lucas (1973) and Barro (1976), but where information can be acquired
similar to those of
endogenously and the market structure
is
own
very different. Duopolistic firms observe their
production costs, then set prices. Since costs are correlated, buyers learn from one firm's price
something about the price of the other. Given these inferences and their knowledge of firms'
strategies, they decide
whether or not to search. Conversely, when setting their prices firms
take account of buyers' search rule, as well as of their
cost
and
own inferences about their competitor's
price.
This requires addressing the
To
Bayesian learning.
represent the
first
more general problem of search market equilibrium
our knowledge
this
paper, and independent
models which combine optimal adaptive search
Not
Rosenfield and Shapiro (1981)) with equilibrium pricing.
quite complex,
and
this
is
why we have
general equilibrium framework.
microeconomic
identify
to focus attention
The
in
inflation
industry-wide cost shock.
inter-industry costs,
work by Dana (1990)
(e.g.
Rothschild (1973),
surprisingly, the
a single market, as
question then arises of
setting, the idea that inflation
an increase
on
with
how
to
problem
opposed
capture,
is
to a
in
reduces the informativeness of prices.
a
We
uncertainty with an increase in the variance of an
In effect,
we
take as given the fact that inflation affects
and examine how such shocks
affect intra-industry pricing behavior
and
market performance.
Of
course, this
should affect the
is
only a partial and crude representation of inflation. First, inflation
demand
timing: realistically,
side as well as the supply side.
But
this
when consumers see prices change unexpectedly,
is
essentially
an issue of
their resources
have not
yet
been
fully
and unambiguously affected by the inflationary shock
change would not be unexpected, and there would be no
Similarly,
full
when
demand which consumers
our model has no money. However,
one takes
if
no substantial problem with
is
they had, the price
difficulty in assessing relative prices).
firms discover unexpected changes in costs, they have not yet ex'perienced the
increase in nominal
there
(if
will
eventually address to them. Secondly,
nominal
as given that inflation affects
calling the
numeraire good money.
Since
all
costs,
price
changes are unex-pected by agents given their information (any inflationary trend has already
been factored
out), the dollar prices
and may enter
prices,
Thus
in spite
their utility
and
which they observe are their best assessments of real
profit calculations without implying
of the model's obvious limitations,
we feel
that
any money
what we learn from
illusion.
it
about
the effects of real, aggregate cost shocks remains relevant for genuine, money-driven inflation.
The reader who does
not share these convictions can maintain a purely "real" reading of the
paper; indeed the relationship between the stochastic structure of supply shocks and market
efficiency
is
of interest independently of any possible link to inflation.
A well-known
discussion of the relationship
macroeconomic shocks
is
contained
in
Okun
repeat purchase behavior of consumers
shocks.
raises
The reasoning
its
price.
He argues informally that search and the
may reduce
the adjustment of prices to
that a firm's previous customers will only search
Lowering price can only
therefore have high
justification for
is
(1981).
between consumer search behavior and
demand
elasticities.
attract
if
demand
the original firm
consumers who are good shoppers and
The argument, although
consumers' search behavior. In particular,
if
intuitively plausible, lacks a
inflation
is
expected, a nominal
price increase at the inflation rate should not induce search.
In our model,
where search and
pricing decisions are optimal given the stochastic
structure of costs, the variability of the joint cost shock will affect consumers' signal-extraction
problem, hence their search
rules.
This
in
turn will determine the elasticity of
demand
faced
by each firm, hence
between aggregate
its
pricing strategy
and ultimately
cost or inflation uncertainty,
created by this mechanism
is
the
The
social welfare.
relationship
monopoly power and market
efficiency
main focus of the paper.
We identify two major effects of an increase in the variance of inflationary shocks. We
refer to the
first
one as the correlation
an increase
high,
while
if
in
effect. It
operates so that
if
search costs are sufficiently
power and
the variability of inflation works to increase market
real prices,
search costs are sufficiently low, an increase in the variability of inflation works to
decrease market power and real prices.
Intuitively, the
argument
the following.
is
As
the
variance of the joint cost shock increases, the correlation between firms' costs increases. Since
optimal prices depend positively upon cost, prices also
Bayesian consumers to put more weight on the
when forming
their posterior belief
the observed price
is
first
become more
correlated. This leads
observed price and
less
on
their prior,
about the price distribution at the second firm. Thus,
if
high, greater variance of the joint cost shock implies a higher conditional
expectation of the price at the second firm. This reduces the value of search for a given price
at the first store, as
the observed price
it
is
becomes
less likely that its
high price
is
really a
bad
deal. Conversely,
if
low, greater variance of the joint cost shock implies a lower conditional
expectation of the price at the second firm. This increases the value of search, as even this low
price
is
price
is
it
less likely to
low,
is
with
good
deal.
Now
if
search costs are low, buyers' reservation
effect inflation variability tends to decrease
making the market more competitive. Conversely
if
search costs are high,
buyers' reservation price, and inflation variability will tend to increase
it
firms'
An
costs,
This
really
and therefore through the correlation
further, thereby
so
be a
it
even more, and
market power.
increase in the variance of the joint cost shock affects not only the correlation of
but also their unconditional variance, and generally the conditional variance as well.
in turn will
tend to increase the conditional variance of prices, and thereby the option
value of search, as
we
well known. This effect, to which
is
to raise the value of search
and lower
firms'
refer as the variance effect, tends
market power, as the variance of the inflationary
shock increases.
We identify several
other channels through which aggregate cost shocks affect market
performance, but the correlation and variance effects stand out as both more important and
general.
They formalize and
fill
the gaps in the intuition one generally has about the impact
of inflation uncertainty on buyers' incentives to search.
The main
conclusion of the paper
is
that
when
is
it
recognized that informational
imperfections generate imperfect competition and endogenous information gathering, the a
from variable
priori case for information-related welfare losses
many
Indeed, inflation has
effects (which
tend to improve market efficiency.
crucially
depends on how
costly
anticipated inflation and search,
whether through
efficiency cost
its
trend or
its
it is
We
we
show
whether
Benabou (1989) comes
still
In analyzing the interaction of
to quite similar conclusions.
seems
to
The
Thus,
have no single identifiable
which could be expected to swamp the benefits across
inflationary regimes.
much weaker.
inflation lowers or raises welfare
to acquire information.
variance, inflation
is
on the price system, several of which
identify)
that
inflation
all
market structures and
strong aversion to inflation which people almost universally
proclaim remains hard to reconcile with the consequences of the sophisticated behavior and
information utilization with which economic models
The paper proceeds
as follows: Section
the construction of equilibrium strategies.
II
The
endow them.
describes the
effects of
changes
discussed in Section IV, both through analytic examples and a
V concludes.
model and Section
III
contains
in inflation variability
are
number of simulations. Section
The Model
II.
In this section
and the next we present
model of a duopolistic search market
a
equilibrium with Bayesian learning. Buyers' search decisions depend on the inferences they
make from observed
prices, taking into
account firms' strategies. Conversely, firms' pricing
inferences about their competitors' prices,
decisions incorporate their ovkH
knowledge of buyers' inference and search
1.
Market and information structure
costs
[c~
,
and
Ci
c*]'x[c'
product of a
observes
i
Cy
.
,
C2
c*]
own
0<c"<c*<-*-'». For
cost c
^
,
instance, each firm's cost c
cost c
its rival's
j
,
although c
,
We denote by f
on firm
I's.
(
c
2
I
c
1
and / ( c 2
)
We assume that
and C2 almost everywhere, and that
I
/ (C2
c
,
1
sum
y, (real cost).
or
Firm
provides information about
is
common
the distribution and density of firm 2's
)
c
support
could be the
,
and a private cost shock
(e.g. inflation)
but not
joint distribution with
4 Buyers do not observe cost realizations but the joint distribution of costs
cost, conditional
]
,
identical firms, with constant marginal
which are drawn from a symmetric
common cost shock
its
knowledge.
c
,
their
rules.
There are two
.
and
1
) is
continuously differentiable in both
costs are positively correlated in the following sense:
dF(C2 |c,)
dCi
d
<
dCx
4.
Suppose for instance
C2 even though
that c, = Q + y
we assume
Vcj
,
(2)
.
Ll-F(C2|Ci)J c,-c,-0
that firm
1
^
,
i=
\
,2; then c
does not know
,
1
contains information about
but only c
1
.
This assumption
is
not essential but fits well with the idea of aggregate uncertainty. Firms are themselves
buyers of inputs, and do not have perfect information on whether high or low materials and
labor prices are specific to their own suppliers or economy-wide. In a richer model, of
course, firms could also decide to become informed at some cost.
Condition (1)
signals a higher c
2
is
•
just first-order stochastic
To
interpret
it,
would then say that the hazard rate
decreasing in Cj
monotone
C2
'
> C2
.
cost)
But
is
need only hold
(2)
will
it
that such will
Cg
as Cj tends to c
There
turn to buyers.
D(p)
quasi-concave
2.
,
at ever}'
(
c
c
,
1
2 )
It
•
to c'
j
,
is
/(C2 Ci) has the usual
if
I
from below, so
,
that a higher c
one moves from c"
as
be the case
=
S{p)
is
a
continuum of
increases in c
it is
in fact
for
1
any
quite weak. ^
its
in
-S'(p)is
p for all
c
in
c,letn^(c) denote
measure
identical consumers, with
= j'°pD{r)dr be the surplus each of them derives from buying
her
demand
function in the absence of search.
standard assumption that a firm's profit per customer
with cost
were required
/(C2'|ci)//(c2lci)
i.e., if
means
ensure that a firm's equilibrium expected profit function (conditional on
normalized to one. Let
p, where
that
(2) also
quasi-concave.
We now
at
shown
easily
first
for finding the true
likelihood ratio property,
Condition (2)
own
It is
.
suppose
dominance;
c
[
"
its
Search and learning from prices
c
.
*
].
Since
maximum
Initially,
.
11 (
p
,
c
)
n(p,c)
is
=
We make the
(p-c)D(p)
is
strictly
the profit function of a monopolist
value, achieved at
Pmic).
half the buyers observe firm one's price
and half
firm two's price, at no cost. Given the observed price, they must decide whether to purchase
or to search and find out the other firm's price.
consumer
to
buy
at the
recalled costlessly
communication
5.
Searching entails a cost
but allows the
cheapest of the two prices. The assumption that the
means
that a
is
a
pure informational
cost, rather
first
offer can
be
than a transportation or
cost.
Given the
first
Note also
that (1) tends to
observed price, say
lower probability that c 2
is
in
make
[c"
,
Ci
p
1
,
a
(2) hold,
], i.e.
consumer must
and that
if
first infer
an increase
/2(C2|C))<0
the extent to which
in c
for C2 < Ci
,
1
leads to a
then (2) holds.
10
it
reflects a firm-specific
From
shock or a joint shock.
about the other firm's price, with distribution G
price support
[
p" p*
,
p2
finding out
=
1^(P,)
]
•
(
p2 p
and density
)
i
I
'[-5(P2)-S(p,)]g(p2|p,)dp2
search.
otherwise she
;
distribution, a higher
found, and this tends to increase Ii^'Cp
p
1
)
As
.
distribution
is
1
G(p2 p
news" about the distribution
U'' (
I
)
1
;
)
p
\'
>
p
\
I
effect
is
Pi)
accepted.
)
To
(
P
1
)
effects
more
away
at
p
,
worth
(3)
.
on the expected return from
likely that a better deal
firms' prices are correlated, a high
p
1
can be
is
"bad
of the other firm's price, and this tends to reduce
and on consumer preferences
from above.
where a price
make assumptions on
not too strong, so that U^(Pi)
=
it is
p
is
1
(1981)) this
rejected but a higher price
preser\'e the reservation price property, Rothschild (1973)
impose the condition that the return
'^''
it
right
(1970), Rothschild (1973), Rosenfield and Shapiro
Rosenfield and Shapiro (1979)
G^(P2
on the
well-known from the literature on optimal search from an exogenous unknov^n
(DeGroot
is
if
buy
two
makes
1
but
learning effect can result in search strategies
p
that
)
1
'D(p2)G(p2|p,)dp2,
r
=
will just
(3) that observing a high price has
For a given
I
p'
larger than the search cost a
Note from
p2 p
(
the expected benefit from search,
if
p'
is
g
consumer will decide
Finally, given these beliefs, the
before buying
r
inference she then forms posteriors
this
to
is
(
D(p)
the distribution of prices (equivalently
=
1
)
which ensure that the learning
monotone. Alternatively, Rosenfield and Shapiro
an additional search never cross the horizontal
an equilibrium model, however, C
In
and
(
p2 p
I
1
)
line
incorporates firms'
optimal pricing rule, therefore no such assumptions can be made. Moreover, the pricing rule
itself
depends on buyers' inferences, making the problem quite complicated.
We shall nonetheless follow similar lines of reasoning, but with respect to the exogenous
distribution of costs
f (cj
I
c
1
)
.
We
mainly focus on equilibria where buyers' search rules
.
11
have the reservation price property (reservation price
equilibria),
and show that a pure
strategy reservation price equilibrium exists provided that either:
i)
Firm's costs are not too correlated
ii)
Buyers' search cost
The
will
first
assumption
W
ensure that
The model
(
p
1
)
is
will
f2
is
not too large
falls
W
(
p
,
) is
below a once
it
monotonic. " Alternatively, the second
has risen above.
on Reinganum (1979). The two essential differences are that we
also builds
analyze a duopoly while she assumes a continuum of firms, and
she has independent
costs.
),
relatively small.
ensure that
never
(
we have correlated
These two features allow respectively
costs while
for search to take place in
We expand on these remarks in Section
equilibrium, and for consumers to learn from prices.
III.
3.
Technical conditions
Since they offer
First, in
intuition,
little
some readers might want
f (c
that this implies
Turning now
6.
to
go
technical assumptions.
directly to Section
to the
|
c)
>
demand
>
,
side,
for
0.
all
Wc£[c',c'],
c>
(4)
c'
we assume
that
-Z)'(p) = S"(p)
Consider a parameterized family of conditional distributions F
derivatives
F\
=
f^ and
Then
(i)
7.
We define /
holds for
X
f2
limits
in >^,
and such
(C2
I
that for
c
>^
is
1
=
)
bounded on
with partial
,
f2
-
not too large.
c
(
which are continuous
"
|
c
" )
and /(c'|c')aslim/(c|c) and
c->c'
These
III.
addition to (1) and (2), the distribution of costs must satisfy:'
/(c|c)
Note
more
This paragraph gathers additional,
.
1
1
m/
(
c
|
c)
,
respectively.
c->c'
can be positive even with /(c"|c,) = Oor/(c*|Ci) =
,
Vcie(s",c*).
•
12
[c\c^]
strict
.
i-e.
A
-
sup {- Z)'(p) c-<p<c'><c». This may
quasi-concavity of n
(
•
,
c
)
c->0-
This also holds in the limit at
on the compact
Finally, the
implies that:
P(p.c)-—V^
continuity
require
I
p^(c)
set A' =
>
,
Vp.
0.
since
c<p<p^ic).
<0
U pp^p^^c) ,c)
.
Therefore, by uniform
{(c,p)|c"<c<c*,c<p<p^(c)>:
0<m = min{p(p,c),(c,p)e/v}<max{p(p,c),(c,p)G/^} = M<oo.
(5)
We shall assume:
n„(c-)
>
S(c-)-S(c
)
+
AV.
..2
.
A^|^-J (c -C-)
This basically requires that the monopoly profit functions
Fl
(p
.
,
c) be neither too
(6)
flat
nor too spiked, that monopoly profits for the most efficient firm be sufficiently large, and that
the range of possible costs (c' ,c^) not be too wide. Condition (6) will ensure the existence
of a solution to the differential equation defining the optimal price strategy of firms with
relatively high costs.
:
14
We
intuitive conditions.
exists,
do not have conditions
exists
when
Nor do we
or a very general existence theorem.
parameter values there
for
more than one
the mixed strategy equilibrium
rule out the possibility that for
some
of the four equilibrium types, or even
some
other, less intuitive type.°
The
first
proposition shows that
all
equilibria share
charge their monopoly price.
Proposition
c£[c'
1
In
an
,
any equilibrium of the game, there
,c' + e] sets price equal to
p„(c)
intuitive feature:
low cost firms
.
exists
an e >
such that a firm with cost
.
Proof: See appendix.
The
intuition
is
quite simple.
Since search costs are
observe a price sufficiently close to the lowest price charged
the lowest price charged in equilibrium
deviate and raise
argument
if
its
is
less
strictly positive,
consumers who
equilibrium
not search.
in
will
than Pmi^^') the firm charging
price without losing any customer, thereby increasing
the lowest price
is
above
pm(
c
" ) is
its
this price
profits.
even simpler. Given Proposition
If
can
The
1, it will
be
useful to define:
V.
Sc)^j\s(ip^Cc^)-S(ip„ic))]fic^\c)dc^
8.
For instance, one can not even exclude
decreases with
its
here because c
and p2
Pic)
.
is
i
cost c
i
=
j[Dip^(ic,))p^'ic,)Fic,\c)dc,.
where a
pCci)
The usual revealed preference argument fails
demand function through its correlation with C2
equilibria
firm's price
over some range.
affects firm I's ex^pected
We shall, however, restrict attention throughout the paper to equilibria where
non-decreasing.
(7)
13
III.
1.
Equilibrium
General properties
.
We
between firms and buyers, with a consistency
(see
Fudemberg and
restriction
on
beliefs off the equilibrium
Tirole [1989]): a consumer's observation of firm
directly affects her beliefs
about firms Ts cost
c
.
i
Thus,
not to search, she must use these arbitrary beliefs about c
p
if
there are no restrictions on the consumer's beliefs about c
;
i
i
is
i
I's
p~
if
a
consumer observes
played with positive probability, she must
to
form
firm's having a price less than
p
"
Moreover, these
each firm
will
.
This
beliefs
will
be
than the lowest
put zero probability on the other
.
We identify four different types of equilibria to the model.
high,
only
about c 2 which
beliefs
at firm 1 a price less
still
i
off the equilibrium path,
about C2 and the equilibrium strategy must be used to form beliefs about P2
important below; for example,
p
price
path
but in determining whether or
are consistent with the joint distribution of costs and Bayes' rule.
price
game
look for a symmetric, perfect Bayesian equilibrium of the
be able to charge
its
monopoly
If
search costs are sufficiently
price without triggering
search costs are not quite large enough to support
this
any search.
outcome, there may
still
If
be an
equilibrium without search, where firms with higher costs bunch at consumers' reservation
price to prevent search.
Reinganum model.
These two
equilibria are qualitatively similar to those of the
A new type of equilibrium
arises for
lower search costs: a pure strategy,
reservation price equilibrium with buyers searching at higher prices and firms'
decreasing to zero as their costs increase toward the
maximum
c*
.
The
fourth possible type
of equilibrium involves mixed strategies. High cost firms charge prices which
indifferent
price
between buying and searching, and the
makes
We
this pricing rule
fraction of
markups
make consumers
consumers which search
at
any
optimal for firms.
prove the existence of the
first,
second or third types of equilibria, under quite
15
l/„(c)
is
the value of search
W(p) when
p
observing a price
=
Pm(c),
with cost below c charge their monopoly price, and no firm with cost above
than
Pmic)
•
We now move
1
.
When
2:
can cover
all
cost realizations,
firm charges
its
and consumers
o for
all
monopoly
price
p^(c), and consumers never
c
in
[
c
"
,
c
*
]
,
attract
is
quite intuitive.
search.
more than
.-
.
.
No
firm can earn
half of the customers, since
V^
is
optimal given the definition of
3.
No-search equilibrium with bunching
the equilibrium of Proposition
2,
not search,
there exists an equilibrium in which each
Proof: See appendix.
This
will still
first store.
K„(c) <
If
...
search costs are large enough, the range of monopolistic
independently of the price observed at the
Proposition
c charges less
.
Monopolistic equilibrium
pricing of Proposition
firms
to a characterization of equilibrium, starting with the case of
large search costs.
2.
if all
more per customer by
..
-
•
'.
deviating, nor can
it
none of them search. Conversely, not searching
•
there
.
When
may
search costs are not large enough to support
still
be an equilibrium
in
which no consumers
search. Define c' as the smallest solution to:
I/^(c')-
f
D(p„(c2))p„-(c2)f-(c2lc')dC2 = o.
(8)
16
and
n' "
let
charging
less
p
'
p
,
(c*)
if all
A consumer
.
than p* (so that p' reveals c'
p
Firms with
c<
c' are
firm with cost just above c
up
i
*
.
to
We
).
p
charge
°
between search and purchasing
p^ic) and no firm with
c>c'
cost
charges
but reject higher ones.9
charges
it
at a store
on reservation price equilibria where
shall focus
able to charge their
still
If
indifferent
c<c'
firms with cost
consumers accept prices
is
its
monopoly
monopoly price,
will
it
Consider, however, a
price.
induce search; rather than
accept the resulting first-order loss in customers (they search and find a lower price with a
probability of at least
f (c
and only a second-order
is
an equilibrium
in
|
c)
effect
which
ajl
on
p'
prefers charging
), it
profits
per customer. In
firms with cost above
which causes no
,
Proposition
3:
If
search costs o are such that
which consumers have reservation price p*
c<
c'
and p(c) = p*
for
p' >
and
we show that if p
fact,
c* charge
consumers do not search but prices are constrained by the
loss of
p'
*
customers
> c*
,
there
In this equilibrium,
.
possibility of search.
c* > c'
,
there exists an equilibrium in
firms' pricing rule
is:
p(c) = p^ic)
for
c*<c<c*.
Proof: See appendix.
To
sustain this equilibrium,
search, in response to any price
p
we can
i
>
p
*
;
simply assign beliefs which
for instance, c
such a price then earns zero profits, while
9.
It
it
i
= c'
.
Any
make
it
profitable to
firm which deviates to
could earn positive profits by playing
its
can be shown that any reservation price equilibrium with p(c) non-decreasing
must have the same form
as those
we examine
< p* necessarily
(with
p
'
simply replaced by
p
<
p
*
);
on very implausible out-of-equilibrium beliefs.
p
Since the basic features of the equilibrium and the spirit of the results remain unchanged,
we do not think it worthwhile to go into the complexities of equilibrium refinement, and
moreover those with
simply concentrate on the
more
rest
natural reservation price equilibrium
where
p
=
p\
17
equilibrium strategy. Therefore such deviations will not occur, and this justifies the imposed
beliefs.
Each
firm chooses instead the price
and prevents search.
Finally,
p<
p' which maximizes
its
profit
by definition of c', accepting offers below p'
per customer
optimal for
is
consumers.
The
and 3 are analogous
equilibria in Propositions 2
model, except that consumers' reservation price p'
to those of the
Reinganum (1979)
depends here on the learning which
occurs in equilibrium.
4.
Reservation price equilibrium with search
more
difficult case, in
p"
cost exceeds
which consumers' reservation price p"
can then not avoid search, unless
contained a continuum of firms (as
because
its
Smaller search costs lead to a
.
in
all
still
expect positive profits
a firm
find a lower price.
firms in the market (more generally a finite number),
induces search, but
when
than c*.
makes negative
it
Reinganum's model), such
consumers who searched would
less
is
it is
one's
new but much
A firm whose
If
the market
would have
to stay out,
profits.
However, with only two
possible to charge a price which
rival,
who follows the same strategy,
has an even higher cost, hence an even higher price.
We
now
characterize a pure strategy reservation price equilibrium in which search
actually takes place.
realizations,
it is
Firms' pricing strategy
p' = p^{c'). Afirm with higher
all
all
consumers who
visit this
firm
We now derive
Pfic).
If
consumers
on Figure
similar to that of Proposition 3: a firm v-ith cost
and a firm with cost between c' and some c^ >
so
illustrated
is
if its
rival
first
firm
i
For low
cost
below c' charges Pm(c)
charges consumers' reservation price
c'
cost realization c
1.
,
however, charges a price
Pf{c)>p',
search.
charges a price
has a higher price, and to none
p
,
if its
which induces search,
rival
it
will sell to
has a lower price. Therefore
18
EQUILIBRIUM PRICING STRATEGY WITH SEARCH
p(c)
Pf(c)
P'
PmiC)
Figure
1
.
.
19
firm
probability of making sales
i's
on firm
i's
true cost
c,
,
and
its
Y(p,.c,)
where
p(
•
) is
differentiable
=
is
the probability that firm
j
has a higher price, conditional
expected profits are:
n(p.,c,)-[l-/^ro6[p(c,)<p,|c,))].
Assume
the equilibrium pricing strateg)'.
on [c' ,c'], with p(c')> p'
for
this will
;
now that p (
be
(9)
) is
increasing and
We
verified below.
can then
rewrite:
Prob[picp<p,\c,]
and
=
F[p;\p,)\c,]
differentiate (9) with respect to p, to obtain the first-order condition
which p(c) must
solve:10
•
n(P(c).c)
p^Cc^^ ^ ^ i-f(cic) np(p(c).c)
^^^'^^
.
_
.
This differential equation must be satisfied along the price path
[c\c^] which
lead to search.
the highest possible cost, c
to find a lower price.
would lower
its
sales at a price
does not
Instead,
*
.
The boundar)'
Such a firm makes no
Therefore,
price by a
condition
little bit
it
price
c
we
sales since
consumers search and are sure
(
'
,
c
;
otherwise the firm
positive expected profits, since
its rival's
satisfy Lipschitz conditions at
the region of costs
found by considering a firm with
must be the case that p{c') = c'
and make
above cost whenever
is
in
' )
,1 ^
was
higher.
it
would make
Because p(c') = c^
is
(10)
so standard theorems are not applicable.
construct the solution as the fixed-point of a contraction mapping; this
condition (6)
,
is
where
needed.
Since charging
p>
p' leads consumers
become
informed, it is not surprising
that this pricing rule is quite similar to the optimal bidding rule in an auction with correlated
values (Milgrom and Weber [19S2]). The difference, and source of difficult}', is that
n ( p c ) cannot be expressed as a function U (p- c)
10,
,
11.
Nor
at points
(Pm(c),
c)
to
fully
.
20
Lemma
The
1:
differential equation (10)
has a unique solution
Moreover,
p
f
'( c
)>
Pf(c)
,
for
on
all
c
in [c"
,c^)
c*]
,
with terminal condition p^^*)^^,-
»
c< p f(c) <
satisfying
,
(^c'
Prrt(.c)
,
with equality only at
c*
.
.
Proof: See appendix.
While Pf{c) solves the
characterize optimal prices for
we show that
first-order condition (10),
c>
c'
equilibrium profits are
First,
.
level
it
really
using condition (2) on the distribution of costs,
quasiconcave.
strictly
Lemma 2: Assume that buyers have reservation price p'
some
remains to prove that
it
=
p^(c')
If
firms with cost above
c''>c* charge Pf(c), while firms with cost c<c'' charge
min(p^(c),p'),
p^
then a firm's profits from charging any price
p'
.
are:
w(p,c)^ n(p,c).[i-F(p;'(p)ic)].
They are
strictly
quasiconcave and maximized over
pe[p^,c*]
at
(II)
p
= p/r(c).
Proof: See appendix.
The
final step in characterizing firms' strategies
separating those which prefer to charge
The
first
'
to find the threshold cost
those which prefer to charge
p f(c)
strategy prevents search but the second yields greater profit per consumer,
come back. At
c ^ a firm
is
l-^-F(c'\c')
The
all
p from
is
indifferent
between the two, so c ^
•n(p*,c^)
=
visit
it
first
(1/2)
and to
all
p
'
.
> p'
they
if
defined by:
[l-F{c'\c')]-n(pF{c'),c').
left-hand side represents profits from charging
consumers who
is
c^
(12)
A firm which does this sells to
consumers who
visit
the other firm
first
and
.
21
observe a price above
p
Given the symmetry of the equilibrium pricing strategy,
*
1/2 times the probability that the other firm has cost
The
right-hand represents profits from charging
customers
who
visit
find a lower price, which
Lemma
c'
3:
Assume
£ic' ,c^]
,
^-[l
p ^(c'). A firm which
does
and
also sells to customers
It
is
also -[
1
-
/^ (
c
""
I
which
is
search, find a higher price
it first,
-[l-fCc'^lc^)] customers.
and
above c'
c
^
)]
that buyers have reservation price
,
- Fie'
just
c')]
\
this sells to all
the other firm
visit
is
which given symmetry,
return,
who
this
is
first,
search,
exists a
unique
customers.
p' = p ^^c')
There
.
such that the following strategies are mutual best responses for firms:
(i)
Force
(ii)
For
c e [c*
(iii)
For
c e
p(c)
[c",c"].
=
p„(c);
c^)
,
p(c) = p';
[c\c"]
,
p(c) = pp(ic).
,
Moreover, c^ and p' are continuous and non-decreasing
in c"
.
Proof: See appendix.
This result concludes the characterization of firms' strategies:
strategy equilibrium
c'
.
P f(')
'
where consumers have reservation price
and c'
only remains
to
,
which are the unique solutions to
verify
that,
given
firms'
characterized by the reservation price p*
is
where
,
strategies,
i.e.
that
p' = P mi^' )
first
Next,
p^<p'\
if
p
1
=
p
then Ii''(Pi
'
,
)
= ^'^(Ci
consumers only
'
it is
,
(8), (10),
and
there exists a pure
uniquely defined by
(12), respectively.
optimal buyer search
U'(Pi)>a
issues associated v.ith equilibrium learning will
Consider
p
if
if
and only if p,
is
>p'
It
indeed
.
This
be most important.
)
< Km(c') = a
since by definition
,
c
infer that c
,
6
[
'
,
c
^ )
,
so,
22
l^(Pl)=
D(p„.(C2))P^'(C2)f(C2|C°<C,<C^)dC2
1
< [' D(p„(C2))p^-(C2)/'(C2lC,=C*)G!C2 = K„(C°) = 0.
and they still do not want to search. For out-of-equilibrium prices
they lead to the belief that c
P
i
= c
°
(4)
p^), assume that
Dip,)Gip,\p,)dp,-[Sip')-Sip,)]Gip'\p,)
J^
I
l/„(c')-[5(p')-S(p,)]F(c'|c')>a.
and
weight on c
,
r p
,
\
due to
^
so that:
,
I^(P>) = /_'-D(p2)G(p2lp,)dp2
D{P2)C{P2 Pi)dp2==
=
p e(p°
i
c'
> c'
.
More
generally, any out-of-equilibrium beliefs
being closer to c ' than to c ^
Finally, buyers will search at all
strategies, this
is
equivalent to
K(c
i
)
will
> a
for
if
all c
,
and only
>
c
^
if
I-/ (
p
sufficient
pie(p*,p^).12
lead to search at prices
pi> p'
which put
,
)
> o
;
given firms'
where:
l/(ci)- f' D(p^(c2))p„'(c2)/^(c2|c,)dc2-[S(p*)-5(pO]/^(c^|c,)
•
1
^(Pf(C2))Pf'(C2)^(C2lCi)dC2.
(13)
c
12.
The fact that consumers' beliefs do not remain monotonic in the observed price as it
moves off the equilibrium path is admittedly unappealing. But one has to choose between
such monotonicity and the reservation price property. Indeed, due to bunching, consumers
who observe their reservation price p strictly do not want to search; if observing p + e
did not lead them to infer a lower c
hence a lower Cj and p 2 they would not want to
'
i
search, a contradition.
,
'
,
23
Using the definition
(8) of c", the condition for search
[5(p')-S(p^)]f(c^|c,)
'D(p,(c2))p/(c2)f(c2|c,)dc2
r
+
becomes:
[' Z)(p^(c2))p^'(C2)[/^(C2|c')-f(C2lc,)]a!c2,
The
P2
p
G
*
[
.
hand
left
p
1
]
;
side
denote
C] rather than c*
,
it
the
is
Wc,>c'
expected return, conditional on
as ^{c^ ,g)
The
.
right
hand side
rule,
x(c
i
.
,
(14)
.
from finding
,
the "bad news" from inferring
P2<p";
about the likelihood of finding a price
For buyers to have a reservation price
is
c
>
denote
o) must not be too large.
x(Ci,a).
as
it
From
(14)
we
see that such will be the case in two intuitive cases, which extend the insights of Rosenfield and
Shapiro [1981] and Rothschild [1973] to an equilibrium context. The
small, so that c'
is
close to c'
,
making the
one
is
when
a
is
small; the following proposition
x
integral in
first
formalizes this intuition.
Proposition
4: If
search costs are relatively low, there exists a pure-strategy, reservation-price
equilibrium characterized by
p' = p„{c').
have reservation price
c 6
[
c
*
c
,
^
)
,
and
critical cost levels
p f{c)
if
cg[c^c*]
c' and c^
Firms charge
,
with
,
with c' <c'
p,n(c)
if
<c^ <c'
c£[c'
.c']
.
Buyers
,
p'
if
is
when
a
c<p;r(c)<Pm(c).
Proof: See appendix.
The second case
firm's cost
^2(^2 Ci)
I
in
which a reservation price rule
will
be optimal for buyers
does not reveal too much information about that of
is
small, so that the
two distribution functions
in
x
its
competitor,
i.e.,
when
are close to one another.
:
24
Proposition
5:
If firms' costs
are not too correlated,
i.e., if,
F2'^niax{F2(c2 Cj) |c"<C2^Ci <c*}
I
is
not too large, there exists a pure-strategy reservation price equilibrium. For a above
level
ct'
,
whether
it
involves
V mCc*}
is
no search and corresponds
larger or smaller than a
.
to that of Proposition 2 or 3,
For a < o °
,
it
some
depending on
involves search and corresponds
to the equilibrium in Proposition 4.
Proof:
5.
See appendix.
Mixed
strategy equilibrium
hold, there
may be no
.
If
the assumptions of low search costs or low correlation do not
reservation price equilibrium, as one should expect.
Indeed,
simulations suggest that there exists an intermediate range of search costs in which none of the
three types of equilibria discussed above exists (see Section IV).
briefly to a fourth type of equilibrium,
We
must therefore turn
which involves mixed strategies by consumers and
generally does not have the reservation price property.
In such
p(ic) =
Pm(c)
indifferent
an equilibrium, pricing
,
for c<ic'.
From
between searching and
c'
not,
low levels of costs remains unchanged,
at
the pricing rule
to c*
and they
will
randomize
i.e.,
Pr(c) makes consumers
this decision.
Thus
for
all
c, e [c' ,c^]
f
•^
[S(p„(c,))-S(p,(c,))]/(c2|c,)dc2+
r
-^
c'
'[S(p,(c2))-S(p,(c,))]/(c,|c,)dc2
Differentiating this expression with respect to Cj yields
Pr(c2) ,forc2<Ci
;
= o.
(15)
c'
this allows the function
p
/?
(
)
to
Pr'(c^)
as a function of
all
be constructed, moving up from the
:
25
initial
condition
p>p'
p
rc'\ = p'
.
The
must then be such that
fraction
it
makes the
it
that in this type of equilibrium
because not
all its
would then make zero
price equilibria.
It is
pii{c')>
optimal; for
all
p>p
c'
if
it
.
The
raised
highest-cost firm
its
price they
all
>
makes
(16)
positive
would search, and
profits.
also
is
somewhat less appealing intuitively than the previous reservation
much more
difficult to
must show the existence of solutions p^(c)
which are extremely complicated.
We
construct for a general specification: one
to (15)
and
ouy^
(p)
g
[0,
1 ]
to (16), both of
have not established general conditions under which
equilibrium exists, but report below on simulations using simple functional forms which
indicate that
do.
search at any price
Pr{c)
t«,(p^(C2))/(C2|C,)dC2
consumers search; yet
Such an equilibrium
this
pricing rule
who
.
Pg{c^)£argmax{ nCp.c,) 1-CU,(P)+|
/::
profits
of consumers
-
Cj e (c', c*]
Note
uo^(p)
it
does
exists in the
intermediate range where none of the other three equilibria
'
26
The
IV.
EfTects of Inflation Uncertainty
We are now ready to analyze
changes
in
how
equilibrium pricing and search rules are affected by
We
the underlying distribution of costs.
are interested in particular in
increase in inflation uncertainty impacts firms' market power, the
and consumer surplus.
Again, what
view
detailed
its
it
interactions with price information
we demonstrate
correlation effect
c
'
,
is
also the cost at
and the variance
effect.
and market
efficiency,
rise to similar issues
This
and
she
will
which firms are
first
is
power
in the
model;
this
specification (costs are the
which we report
in
is
we
done by analyzing the comparative
partial but generally
.
in c' is
the
statics of
demand
p^^c")
,
and
sum of uniform private and joint cost shocks and demand is linear),
the second part of this section.
which are equal up
prices are again equal
However, the
call
good indicator of monopoly
to c'
3,
it is
clear that an increase in c' to c'
and greater above. The
effects of
on the search equilibrium of Proposition 4 are more complicated. Since
,
of
confirmed by a variety of simulations using an alternative
In the no-search equilibrium of Proposition
results in prices
some
share
constrained by search from charging their monopoly
As explained below, c' provide a
p'
we have
effects.
characterizes consumer's reservation price p' =
price.
'
speaking
subscribe to our interpretation can thus
the two most important intuitions, regarding what
Recall that c'
iso-elastic.
c^
strictly
using a particularly convenient specification where costs are jointly log-linear and
is
in
is
search, profits
as arising from real shocks (raw materials, weather). Hopefully, once
our conviction that nominal inflation gives
First
label inflation uncertainty
The reader who does not
aggregate cost uncertainty.
safely
we
amount of
how an
effect
ambiguous.
it
an increase
in c'
implies an increase
from c' to c' and greater from c* to the lower of c" and
on c^ of a change
We know that if the
in
the cost distribution which causes an increase
differential equation (10)
which determines prices
;
27
Pp(c) above
c"
from c^to
prices
p'
since
is
less
than
Thus by no means do the comparative
show
.^^
Pf{c)
(10) will generally be affected in a complicated
simulations
(Lemma 3).
This will imply lower
But of course the
differential equation
c ^ remains unchanged, c * will also be greater
statics
manner by changes
of c'
tell
Nonetheless our
the entire story.
do provide robust intuitions about the
that they
in the inflation process.
full
equilibrium effects of
inflationary uncertainty in this model.
Case
1
:
Log-Normal
We
costs,
Costs, Iso-Elastic
begin the analysis by making the following distributional assumptions about firms'
now denoted
C,
=
,
i
(i)
C, = er,
;
(ii)
log
(iii)
log r, ^ Yi
(jv)
Yi
The
Demand
^ G
.
Y2
•
1 ,2:
c.^logC,
is
distributed normally with
is
distributed normally with
log-costs c
1
(1 - 9^){'-'q + v^)
and C2
.
mean
and variance
lie
,
on
where
c
p -
We shall examine how
is
,
c and variance fy
therefore normal, with
^-^
i/g
uncertaint)', affects the conditional distribution
,
is
mean
makes
it
+
- p
( 1
)
c
F(c, Icj) and consumers' return
inflation
to search.
does not
it
< + °° This
to prove the existence of an equilibrium with search, largely because we
this
difficult
1
which measures aggregate cost or
however, that a firm earns greater profits at this lower price since
induce search, while it could have charged p f(c) and permitted search.
Note that
p c
the correlation coefficient of the two
13. Note,
14.
;
3nd 9 are independent. 14
distribution of c 2 conditional
and variance
mean
example does not
satisfy the
assumption of the model that
can not use Pfic') = c' as a terminal condition for (10). However,
this
c
*
.
formulation
provides the clearest way to analyze the effects of inflation uncertainty on c'
closed-form ex^pressions can be obtained.
,
because
28
Assume that demand is iso-elastic, D{P)
priceis:
P;„(c) - log /'^(c) = log(;^) +
n
p^iog
M(P)
+ c;
Ti-1
The unconditional
The
distribution of
variance s^
moment
.
To
distribution of
Pm(c2)
.
V[>
1
.
so that the log of the monopoly
Define:
PP^(1-P)P:
Pm(c)
is
s'-(l-p')(fye^i^v):
mean p and
normal, with
conditional on Ci = c
,
,
,
is
consider p and s^, rather than
at the
5(p^(c))
monopoly
=
Therefore the return to search
variance Ug
normal with mean |-i(Pm(c))
i^e
and v^, as the parameters of interest.
1
1
We can
^P„(c)'-'^ = ^exp[(l-ri)p^(c)].
in the
region of monopoly prices
is:
•{exp[(l -ri)P2]-exp[(l - Ti)p„(c)]}exp
(P2-^)'
dp-.
2s^
with
|j.
= |iCPm(c))
.
Rewriting
in
terms of the distribution
and density ^ of a standard
4>
normal:
i^.(c)
-exp
^(Ti-l)[2p
+
2p(p„(c)-p)-s'(Ti-l)]
exp[-(Ti- l)p„(c)]- *
(l-p)(p^(c)-p)+S^(Tl-l)
4>
(l-p)(p„(c)-p)
(18)
Tl-1
Hence:
cp
•
and
price as:
•p„(o
l^.(c) =
(1^)'
better demonstrate the two effects of inflation uncertainty, let us for the
consumer surplus
write
=
c
= P"^
(p„(c)-p)-exp-^(ri-l)[2p + 2p(p^(c)-p)-s'(Tl-l)]
^
*
(i-p)(p„(c)-p)-^s2(Tn-i)
,
29
=
ds
\\-9){Pm{c)-p)^s\^-\)
+ s(r|-
(l-p)(p„(c)-p) + s^(ri-l)
1
)4>
<J)
exp--(n-l)[2p^2p(p„(c)-p)-s^(Ti-l)]
So
finally:
sg^
—
sgnip- p^{c))
=
;
dp
>
sgnic-c)
=
(19)
(20)
.
ds
The
first result
(19) shows
what we
call
the distribution of C2 conditional on 0^ =
the unconditional
mean
c
,
this conditional distribution
p
,
the
effect of
first
to lower
it
at c
uncertainty
>
is
increasing in p
in
v^
is
if
mean
of
a weighted average of the observation c and
- p
1
c>
c
,
.
respectively. Therefore, the
and decreasing
if
c
< c
.
By
mean
of
increasing
therefore to raise the value of search at c < c
,
and
Xhjs correlation effect captures the idea that inflation or aggregate cost
makes people search
things are just as
is
with weights p and
an increase
c .15
the correlation effect. Recall that the
less
when
bad elsewhere. However,
they see a high price because they think that
,
it
makes them search more when they see
also
a
low price because they think that there are other, possibly better bargains to be found.
,
The second
return to the
15.
first
result (20)
shows what we
really
determines search
but the conditional distribution of surplus
involves the equilibrium pricing rule
)
.
the variance effect. Given that buyers can
store costlesslv, an increase in the variance s
Of course, what
S(p
call
p{c)
-S (
is
^
of the conditional distribution
not the conditional distribution of cost c 2
p2)
;
see (3).
The
difference between the two
as well as the convexit)' of the surplus function
This discussion, and the one which follows, are meant to give the main qualitative
intuitions.
30
increases the option value of search.
But note that an increase
in
Vq
,
the unconditional
variance of the joint cost shock, does lead to such an increase in the conditional variance:
vl
ds^
ye + ^Y.
thereby raising the value of search and reducing the market power of firms.
We now examine how these two effects impact consumers' reservation price and firms'
pricing.
We focus on the case where search matters, i.e. where c* = inf{c|K„(c) = o><oo.
Then K„'(c')>0
,
so (19) and (20) imply:
sgn—-
=
sgn[p^ic )-p]
dp
—
sgn(c -c)
=
(21)
dc'
<
0.
(22)
ds
If
exceeds
c'
c
,
monopoly markups over
opposite direction.
With a
relatively costly.
this
By
the correlation effect
tends to increase
it
further, resulting in
a wider range of costs; the variance effect, however,
definition, such a configuration with
relatively
case both the variance and the correlation effects reduce
when
c' > c occurs
low cost of search, on the other hand, c'
it
works
is
less
in
the
search
is
than c;
in
even more, and the market
becomes more competitive.
While
log-normal case
this
is
variance effects seem quite robust.
of any
common
shock to firms'
very specific, the intuitions behind the correlation and
One can
costs, as
think in general of the variability of inflation, or
having two effects on the conditional distribution of
by making costs more correlated,
costs (and through equilibrium pricing, of surplus). First,
shifts
it is
F(c2
I
c
1
)
and
its
conditional mean, up for high c
a source of additional uncertainty,
it
i
,
down
for
low c
j
;
it
secondly, since
causes a mean-preserving spread in the shifted
31
The
distribution.
generality of these intuitions
is
also supported by the simulations, using a
very different specification, reported below.
In addition to these two central, information-related effects, inflationary uncertainty has
other consequences in our model.
First,
even
in the
absence of search, an increase
variance of prices can be beneficial to consumers, because consumer surplus
price. This
of income
that
it
is
is
is
clearly a result of our partial equilibrium
constant; 1"
we
positively affected because they
framework, where the marginal
in the simulations).
depend
in
part on
Similarly, equilibrium profits
monopoly
profits,
is
raise
more robust and
each
interesting.
firm's elasticit)' of
Even though an increase
demand, an increase
consumers and purchases toward the firm with the lower
profitable, this tends to increase expected profits
cost.
and efficiency
in cost.
The
do with information, since
and consumers always bought at the cheapest
in
it
price.
the value of
shifts
Since these firms are
more
same
time.
would remain even
16. Also, this effect has little to
third
search activity
in
at the
was
costless
utility'
that firms face both ex-ante and ex-post uncertainty, because they set their
additional effect
may
in
may be
which are convex
price after learning their cost, but before learning that of their competitor.
search
convex
thus tend to de-emphasize this effect, but one should be aware
present (for instance
Note however
is
in the
if
search
;
32
Case 2
Uniform
:
Let us
Costs. Linear
now assume
and that demand
is
Demand
that costs are the
sum
of a joint cost shock and a private cost shock,
linear:
(i)
c,
=
c
(ii)
9
is
uniformly distributed on
(iii)
Y
is
uniformly distributed on [-6,6];
(iv)
Yi
(v)
D{p)
We assume that
represents the
i
,
+ e + Y,;
Y2
a<
•
3rid
[
-a
,
a
]
9 are independent;
A-p.
=
6 which reduces the
,
volatilit}'
of inflation.
The
number of cases
to analyze but
not essential;
is
unconditional density of cost, /z(c)
is
the
sum
a
of
two uniform distributions and has the familiar trapeze shape:
h(c) =
^
^
h(c) =
The
4ab
^
h(c) =
^
c-c+a+6
—
26
if
^
c-
if
^
c
c+a+6~c
on
from a price observation causes a consumer
c
to
,
<
c<
is
also the
update
a- b
c->-
a-b <c<c- a +
c-a + 6<c<c + a +
sum
his beliefs
is
Q~U[-a
6] the posterior of 9
is
Q~U[-a,a]
6,c + a + 6] the posterior of 9
is
Q~U[ci- c-
e[c-a-6,c + a-6]
if
Ci
e[c +
if
Ci
G[c-a
a-6,c-a +
+
1
about the joint cost shock 9
the posterior of 9
Ci
6.
of two uniforms; inferring c
as follows:
if
b
-
-
if
4a6
distribution of Cg conditional
+
a-b
,Ci - c + b]
;
b ,a]
.
;
33
Note
in the
,
if
c
the intermediate region there
falls in
,
lowest region, the conditional expectation of c 2
variability
c
that
a
while
,
and increasing
if
c
a
in
falls in
1
log-normal case.
The same
always widens as
a
is
is
no learning. However,
if
,
works here
effect
c
than c and decreasing
less
the highest region, the conditional expectation of c
Thus the correlation
.
is
in a
way
2 is
falls
i
in the
above
similar to the
true of the variance effect, since the support of 6
,
given c
1
,
increases.
We now look at a number of simulations of this example, in order to get some feeling for
the relative size of the various effects of an increase in aggregate uncertaint}'.
simulations,
Z?(p)=15-p,
c
dispersion of the joint cost shock
costs are given in Tables
1. 2.
and
'
to a c' near c,
and high search
first
6,
to vary.
3.
first at
We
5 = 3.
and
allow search costs
The results for low,
respectively.
well below the unconditional
costs lead to a c
Looking
a
=
We
c
= 6
above
costs lead to a c' well
we
all
and the
intermediate, and high search
define these terms so that low search
mean of
the effects of search costs,
a
In
,
intermediate search costs lead
c.
see that as they increase, the equilibrium
involves reservation price strategies and search, then mixed strategies, then monopolistic
pricing plus bunching at
p
support the intuitive way
in
'
,
and
finally
unconstrained monopolistic pricing. These results
which we associated each
t\'pe
of equilibrium to a different range
of search costs.
Next
we
turn to our main subject of interest: the effects of inflation uncertainty on
monopoly power and on the components of welfare
For low search
costs,
which are reported
the variance and correlation effects
make
equilibrium pricing strategies with increases
gains in
consumer
surplus.
in
in
in
Table
search
equilibrium.
1
.
c
'
is
more
decreasing
valuable.
in
a
,
because both
This reduction in
the variance of the joint cost shock leads to
34
LOW SEARCH COSTS
g
=
0.1:
*
cs
ps
n
CS
Welfare
E(c)
E(P)
3.6413
~
-
18.863
16.227
35.090
6.0000
9.3034
3
3.5719
9.3189
9.3789
18.827
16.425
35.252
5.9919
9.2688
1.00
3
3.0871
8.9957
9.3204
18.814
17.779
36.593
5.8663
9.0151
1.50
3
2.5647
8.6442
9.2282
18.834
19.270
38.104
5.6789
8.6989
2.00
3
2.0438
8.2318
9.0783
18.899
20.692
39.591
5.4329
8.3351
2.50
3
1.5240
7.7904
8.8921
18.995
21.966
40.961
5.1407
7.9309
cs
ps
n
CS
Welfare
E(c)
E(P)
a
Type
0.10
2
0.50
=
c
0.5:
*
a
Type
0.10
2
4.4747
—
~
19.791
14.248
34.039
6.0000
9.6466
0.50
2
4.4435
-
—
19.691
14.430
34.121
6.0000
9.6315
1.00
3
4.3420
9.6651
9.7756
19.687
14.697
34.384
5.9860
9.5816
1.50
3
3.9877
9.4448
9.7844
19.745
15.550
35.295
5.9004
9.4066
2.00
3
3.4317
9.0605
9.6659
19.793
16.923
36.716
5.7316
9.0955
2.50
3
2.8796
8.6373
9.5031
19.880
18.281
38.161
5.5029
8.7325
c
Table
1
NOTES:
In all simulations
Type
D(p) =
=
15-p, c
6,
b
=
3.
refers to the equilibrium pricing strategy in the following way:
1= p(c)-p„(c)
2= p(c)- p„{c)
Vc
.
Nosearch.
p(c)- p' Voc'. Nosearch.
3=p(c)-p„(c) Vc<c'. p(c)-p' yc£[c'.c'). p(c)-pf(c) Vc>c'. Search above p"
4=p(c)-p„(c) Vc <c' p(c)- p,(c) Vc>c'. Consumerssearch with a mixed slraleg)' above Max {c',
'Vc<c'
.
.
.
n and CS
are, respectively,
E(p) and E(c)
c- a* b>
expected industry profits and consumer surplus per customer (net of search costs).
are, respectively, the
expected price that a customer pays and the expected cost
at the store
she purchases from.
.
35
INTERMEDIATE SEARCH COSTS
a
=15:
*
cs
ps
n
CS
Welfare
E(c)
E(P)
--
-
20.790
11.476
32.266
6.0000
10.2448
6.5098
-
-
20.798
11.510
32.308
6.0000
10.2381
2
6.4592
-
-
20.820
11.782
32.602
6.0000
10.2171
1.50
2
6.3740
--
--
20.854
11.801
32.655
6.0000
10.1814
2.00
4
6.2524
-
--
21.358
11.581
32.939
5.9482
10.2333
2.50
4
6.0917
--
~
21.584
11.573
33.157
5.9431
10.2504
CS
p^
n
CS
Welfare
E(c)
E(p)
--
--
20.854
]].170
32.024
6.0000
10.319
-
~
20.886
11.200
32.086
6.0000
10.313
20.916
11.297
32.213
6.0000
10.294
a
Type
0.10
2
6.5259
0.50
2
1.00
=
c
3.0:
a
Type
0.10
2
6.9156
0.50
2
6.9004
1.00
2
6.85 28
--
1.50
2
6.7727
--
-
20.964
11.461
32.425
6.0000
10.262
2.00
4
6.6590
-
-
21.396
11.165
32.561
5.9645
10.342
2.50
4
6.5462
-
—
21.588
11.074
32.662
5.9695
10.389
c
Table 2
NOTES:
In all simulalions
"Ape
D(p) =
=
15-p, c
6.
b
=
3.
refers to the equilibrium pricing stralcg\- in the following way:
1=
p(,c)- p„(,c)
Vc
.
Noscarch.
2=p(c)-p„(c) Vc<c'. p{z)- p' 'Vc>c'. Nosearch.
3=p(c)-p„(c) Vc<c', p(c)-p' Vce[c'.c'). p(c)-pf(c) Wc>c' Search above p"
4=p(c)-p„(c) Vc <c' p(c)- p,(c) ydc' Consumerssearch with a mixed sirategT,' above Max{c'. c-a*
.
.
.
.
n and CSare.
respectively, expected industrj' profits
E(p) and E(c)
are, respectively, the expected price that a
and consumer surplus per customer (net of search
customer pays and the expected cost
b}
costs).
at the store
she purchases from.
.
.
36
HIGH SEARCH COSTS
g
=
4.0:
*
CS
ps
n
CS
Welfare
E(c)
E(P)
7.6430
—
~
20.966
10.766
31.732
6.0000
10.423
2
7.6289
~
—
20.980
10.792
31.772
6.0000
10.418
1.00
2
7.5849
~
~
21.024
10.873
31.897
6.0000
10.403
1.50
2
7.5435
—
~
21.100
10.996
32.096
6.0000
10.380
2.00
2
8.7607
~
~
21.312
10.777
32.089
6.0000
10.461
2.50
2
10.448
~
—
21.520
10.767
32.287
6.0000
10.497
CS
ps
n
CS
Welfare
E(c)
E(P)
a
Type
0.10
2
0.50
=
c
5.0:
*
a
Type
0.10
2
8.3300
-
-
20.986
10.561
31.547
6.0000
10.481
0.50
2
8.3165
—
—
21.014
10.584
31.598
6.0000
10.477
1.00
2
9.1327
—
—
21.082
10.554
31.636
6.0000
10.496
1.50
1
~
~
21.188
10.594
31.782
6.0000
10.500
2.00
1
2.50
1
c
~
~
-
21.333
10.667
32.000
6.0000
10.500
—
-
21.520
10.760
32.280
6.0000
10.500
Table 3
NOTES:
In all simulations
Type
D(p) = 15-p, c =
6.
b
=
3.
refers to the equilibrium pricing strategy in the following way:
l=p(c)-p„(c) Vc. Nosearch.
2=p(c)-p„(c) Vc<c'. p(c)- p' '</c>c'. Nosearch.
3=p(c)-p„(c) Vc<c', p(c}- p' Vcc[c'.c'). p(c)-Pf(c) 'Vc>c'. Search above p'
4=p(c)-p„(c) 'Vc<c' p{c)- p,(_c) Vc>c'. Consumerssearch with a mixed strategy above Max{c'. c-a*
.
n and CS
are, resf)ectively, expected industry profits
and consumer surplus per customer (net of search
E(p) and E(c) are, respectively, the ex-pected price that a customer pays and the expected cost
b>
costs).
at the store
she purchases from.
.
37
Another interesting aspect of these simulations
increases in
a
two reasons
for this.
consumers
.
a
In fact, for high levels of
a
First as
is
that profits
do not decrease much with
profits are actually increasing in
a
increases, search increases. This raises the likelihood that
better matching of consumers to low-cost firms actually tends to increase profits.
and the
is
offset
prices
two columns of Table
1,
by the decrease
is
The second
in cost.
expectation w^th the unconditional variability
Table 2 reports simulation
the correlation effect affects
These simulations
because
in
some ranges
Table
is
,
ver}' little or
c'
is
dominant. In
profits are
a
is
not at
is
when
this range,
convex
in cost,
all.
The variance
is
decreasing, yet profits increase and
when
c"
is
a
is
WTien c'
.
1.
all
is
made
c
.
reduce
not the entire story,
consumer surplus decreases.
.
In
all
equilibria, there
cost specification.
less
When
then becomes
than profits increase.
The
of the correlation effect.
the simulations
is
that social welfare
increasing or almost constant in the variance of inflation. In those cases
adversely affected, producers are
near
Similarly, the total
sufficiently large; the correlation effect
Indeed, one noteworthy feature of
in
small relative to the previous
demand, uniform
some of the harm
is
effect alone acts to
significantly greater than c
consumer surplus decreases, but by
increased variance effect mitigates
and thus increase
.
also indicate that the determination of c"
'
the decrease in price
,
very limited compared to Table
partially a result of the linear
increases
a
levels of
on consumer surplus and profits
3 reports the results
no search. This
= 4
c
At higher
results for intermediate values of a
increases, but this reduction
effect of inflation variability
case.
it
see
factor which offsets the deleterious effecfeof lower
monopoly
the fact that
One can
which report the price paid by the average consumer
cost incurred for the average customer.
on producers
c ' as a
There are
purchase at a low-cost firm, which has higher profits per customer. Thus, the
will
this in the last
.
is
always
where consumer are
better off by at least an equal magnitude.
38
The
is
lesson to take
away from these simulations, however,
is
not that stochastic inflation
welfare-improving. This result clearly depends a great deal on the exact specification of the
model. Rather, we interpret them as saying that one needs to know a great deal more,
particular about the size of informational costs, before
inflation
effects are at play,
and one cannot say a
into equilibrium price setting, several
priori
what
be without simultaneously making a statement about market
that
it is
that increases in stochastic
reduce the informativeness of prices and thereby decrease welfare. In an economy
where endogenous information gathering feeds back
complex
one can say
in
their net
structure.
impact on welfare
will
The simulations show
conceivable that the benefits of increases in the variance of joint cost shocks outweigh
the losses.
39
V. Conclusion
Aggregate cost uncertainty, whether due to
inflation,
input prices or to stochastic
reduces the information content of prices by making
and aggregate price
in
common
variations. In this
in a single
The
difficuh to separate relative
paper we have explored how
an environment where agents can act to enhance
how the stochastic structure
it
this
mechanism operates
their information via search.
of shocks, consumer search, and non-competitive pricing interact
product market.
v
results indicate that the a priori case for significant welfare losses
associated with reduced informativeness of prices
endogenous information acquisition and
agents to seek
and prices
more
will
We studied
from
inflation
becomes much weaker when one allows for
Indeed, inflationar}' noise can lead
price-setting.
information, so that in equilibrium they will in fact be better informed,
The
reflect increased competition.
whether
decisive factor in
inflation
uncertainty improves or deteriorates market efficiency will be the size of informational costs.
Another contribution of
this
paper
to the theoretical literature
analysis will
be useful
search behavior
rules of
develops an
consumers.
We hope that this
between pricing and
in general.
to
be done before we can
information-related welfare losses from stochastic inflation.
market structure and information technology,
incentive to search and on pricing
It
it
price distribution,
for attaining a better understanding of the relation
There remains of course much
structure.
that
unknown
equilibrium model in which consumers search optimally from an
and firms price optimally given the learning and search
is
would be useful
may ver)' well
in
fully assess
the possibility of
Our model has
a very specific
which the effect of noisy inflation on the
offset the
reduced informativeness of the cost
to extend the analysis to other
market structures and information
technologies, so as to ascertain the robustness of our results. In particular,
making the model
40
dynamic by incorporating repeat purchases seems quite desirable, although quite
Finally,
one should
also introduce general
inflationary interpretation
more
precise.
equilibrium effects, in order to
difficult.
make
the
41
A ppendix:
Proof of Proposition
p' > p^(,c')
price. This
is
.
If
Let the lowest price charged in a given equilibrium of the game be p"
1:
firm
i
has cost c" then
i,
since
it
consumers who
all
will not search since there
is
profitable.
Assume p' < p^(^c~)
arbitrarily small positive
.
Thus p' < p^(_c')
No consumer will wish
sure to find the lowest price possible, the gain
of profits, a firm with cost c ' earns
many customers by
is
any other
as at
zero probability of finding
is
i's
customer are maximized
profits per
S(p")-o
to search
if
5-S(p"
+
p
he observes
insufficient to cover the search costs.
more per customer
at
deviating. Therefore, the deviation
than at
p"
profitable and
p"
p
is
'
+ e
,
+
"
+ e,),
e
where
since even
,
number,
will
charge
profits per customer.
Proof of Proposition
first
its
monopoly
price, since
it
is
of buyers
is
he were
will get
= p^(_c').
is
a small
Q.E.D.
2:
Km(c,) <
equilibrium, the
an
Given the quasi-concavity
Furthermore, the firm
.
is
maximizes the number of customers as well as the
A consumer's search decision for any price played in equilibrium
store and not search. Given the equilibrium pricing strategy
to search
5
if
Since no consumers search at prices sufficiently close to p" a firm with cost c" + e where e
positive
at
.
Let us implicitly define e, by
number.
assume
go to the other store and decide to search will purchase from
has the lowest price charged in equilibrium. Since firm
p„(c"), the deviation
as
first
First,
.
many customers
charges p„(c"),it gets at least as
if it
because a consumer who observes P;n(c")
a lower price. In addition,
firm
Proofs of Propositions
max
I'^Cc) < o
Thus,
.
it
maximum number of customers
not sensitive to price, (except
if
if
is
to purchase
from the
at the first store, the
he observes p,
value
does not pay to search. Given that no consumer searches in
the firm can attract
is
half of all customers. Since the
a firm charges a price greater than
number
Pm(c') the number of customers
could be lower) a firm charges a price which maximizes profit per customer which
is
p„ (c)
.
This
is
the optimal
price to charge independent of off-the-equilibrium-path behaxior of consumers. Q.E.D.
Proof of Proposition
price exceeds
p'
consumer knows
.
is
3:
In this equilibrium, consumers' search rule
By the
definition of c
that c^ £[c' .c'].
',
no consumer wishes
By the
definition of c
'
he
is
to search
if
and only
to search at prices
is
indifferent
if
the
below p'
.
first
observed
At p'
,
between searching and not
all
if
a
he
42
p° and knows
observes
Given the positive correlation
knewc, =
search
if
c°.
c
is
in this equilibrium,
observing p' only reveals that c, > c°
consumer's beliefs of Cj are at least as great as
in costs this implies that the
he observes p°
must be such that
sufficient to
he
proposed equilibrium. This shows that consumer search decisions are optimal
in the
it
all
below p^{c').
prices are
If
consumer observes a price above p'
a
pays for him to search. Believing that c
,
= c
',
or more generally that
If
any firm deviates to a price greater than p°
since they search and find a lower price. Thus, so long as a firm charges a price
the consumers. Thus, the optimal price
of profits) the optimal price
Proof of Lemma
c
is
,
his
,
close to
ensure that he does wish to search.
Pricing rules are clearly optimal.
1:
is
p
'.
is
p„(c) unless
it
gets zero consumers,
no greater than
p
°
,
it
gets half
exceeds p'in which case (given the quasi-concavity
it
Q.E.D.
Since the differential equation:
not applicable. Instead,
p f{c)
will
^^-'^
rT7(^-n,(p(c),c)
=
(p(c),c)^
satisfy Lipschitz conditions at
n(p(c).c)
/(c|c)
,,
^^^),
does not
if
This combined with the (weak) monotonicity of the pricing rule implies that he does not wish to
on the equilibrium path, where
beliefs
However,
that c, = c'-
(c' ,c') nor at any
{p„(^c),c)
,
standard theorems are
be constructed as the fixed point of a contraction mapping. Lxt
Co be
the
spaceof continuous functions on [0,c*],and:
C
{p(-)GColp(c)e[c,p„(c)]
-
Denote by v(c) = ^^tt^ the hazard
rate entering (Al).
p(-
)
Vc}.
solves (A. 1)
if
(/1.
and only
if
/(c)-n(p(c),c)
2)
the function
(.'1.3)
obeys the differential equation:
/'(c)
=
W^{p{c),c)p'{c)-D{p{c))
with terminal condition /(c') = n(c',c*) = 0. Integrating
/(c)
This integral
is
=
Z)(p(x))e
J'
convergent at c", since
/(c)
p{x^>x
<
{'
''
(A 4)
""'"'dx
v{c)I{c)-D{p{c))
=
back-wards:
-
J(p(-).c).
implies:
D{x)dx
=
{AA)
S(c)-S(c*).
(^.5)
43
We have transformed
the differential equation (A,l) into an equiv-alent integral equation:
By assumption n(-,c)
is
fixed point formulation (clearly /
Pic)
endowed with the sup norm:
I
iA.6)
[0,n„(c)]
)
sup |p(c)|
pi -
(
) e [
c
,
n „ ( c ) ])
into
[c.p„,{c)].
is:
n(-.c)-'(J(p(-).c)).
=
mapping T:p(-)-* T p{-
the
Vc.
quasi-concave, hence can be ins-erted from
strictly
Hence, with obvious notation, the
We now show that
n(p(c),c).
-
J(pi-).c)
.
where T p{c)
Let p(-
.
e
)
is
(A.7)
the
r.h.s.
of (A.7),
is
C; byconsiruction, 7p(c)
a contraction
e [c,
p„(c)]
on C
Vc.
,
ctlO.c")
Moreover,
it is
ne[0.n„(c)].
Tp{-)
£
C
.
n(-,c)"'(n)
easily verified that
J{pi-).c)- I{c)
Since
Consider
X>y
denote
,
n(- .c)"'(A')
=
y-
.
n(- ,c)''(>)
^
<
^
definition of
m and
.X>Z>y.
.\f
•
or z - n(-
.
n„(c).x-
=
is
(Z)
for all
then continuous, hence
We have:
unbounded. For
is
all .V, )' e
(y!.8)
[0,
n„(c)] .with
Weclaim:
^
<
.
x->
p„(c).
>--
= .V.
<
<
Pm(.c)-z
y<2< x<p„(c).
x-y
m p„(c)-x
, ^ ^,
(A-9)
\
z>
y such that
.
^- 11^(2. c)
.
But, by (4), the
,
Finally, replace (A.10) in the
M
.
Nexn, apply the
first
inequality in (A.9) with
y-=.v:
n„(c)-A-
[p^ic)-x]'>^^
n„(c)-A'
or
p^ic)-x>^^^
(/1.
10)
second part of (A.9), to obtain:
.,
.,
n(-.c) '(A-)-n(-.c) '(y) =
Therefore,
Tp
and
:
Inequality (A.9) then follows from
=
.
x-y
c)"'
m
J-
.
c.
c']
.
|n(-,c)-'(J(p(-).c)-n(-,c)-'(J(q(-).c))|.
x-y
M p„(c)-y
—
Indeed there exists Z
clearly continuous in
l/np(n(-,c)''.c). which
c)"' has derivative
x-
jointly continuous in (c. fl), for all c e [c'
now (p.q)£ C>^C and any c£[c' .c']
|7p(c)-7q(c)!
Note that n(-
is
is
sfJi
X-y
x--y<^
m vn^(c)-A'
(^.ii)
.
44
./; v(y)rty
i:
\Tp(c)-Tq(c)\<^-^-
[D(p(.v))-Z)(g(x-))]dx
Vn„(c)-J(p(-).c)
'c'
,,.
e
,
v(y)dy
-J
'
|p(x)-g(.v)|d>r
<^.Ar^^
m
,
Vn^(c)-J(p(-).c)
<lp-q|Thus
T
will
be a contraction
AVM
m
(^.12)
Vn„(c)-J(p(-).c)
for all c e
if,
[
,
c
*
]
.
-
n„(c)-J(p(-).c)>M
ic'-c-)
Indeed.uniformcontinuitywill then imply that (A.13) holds with
so that
I
7p- 7g
|<
(J
I
p- g
.
I
p(c)>c.
Because,
G
n„(c)-S(c)
is
remains to show that p^'(c)>
while /(c
The
I
c) >
by assumption
case of the
denominator
in
(Al) go
left
(4),
,
for
some pe(0,
1)
,
,
=
n„(c)-s(c) + s(c-),
increasing, (A.13) then holds by assumption (6). This concludes
Pf{c)
.
Wc^[c',c"\.
hence the
derivative
D(x)dx
z)(p(x))dx>n„(c)-J^
''
the proof of the existence and uniqueness of
It
replaced by M/%j\i
But note that we have,
n„(c)-j(p(-).c) = n„(c)-j^
since
M
(A. 13)
result,
Pf'{c')
to zero as c goes to c*
.
is
For
For ce [c',c*)
T\{pp{c),c)>
,
by (A.5)
by (A.1)
more complicated because both
all e
>
the numerator and
,
/(c--€|c--e)n(p,(c--e),c--e)-[l-F(c--e|c--e)]n^(p,(c--£).c--e)p/(c--e)
=
0.
(/1.
14)
But,
f(c*-e|c--e)-f(cn(Pf(c*).c-)-n(p,(c--e).c*-e)
Since
f(c*|c')=l and n(p,(c*),
|c-) = [/(c-
|
c*
)
+
f ^Cc*
c*
)](-e) + o(e)
.
=
[n^(p,(c*-e),c'-e)p,-(c--€)-^n,(p,(c*-e).c'-e)]e
=
[n^(c-|c-)p/(c--e)+n,(c-,c')]e
+ o(e)
+ o(e).
c*) = 0, (A.14) becomes:
{/(c- |c-)[-n^(c-lc*)p/(c--e)+D(c-)]-[/(c-|c-)+f2(cor
|
|
c-)]n^(c*
.
c'
)p,
'(c* -
e)>e = o(e)
.
45
p/(c*-e)[2/(c* |c")+f2(c' |c*)]n^(c-,c')=/(c* c-)D(c-)+
|
p/(c*-e)
which means that the limit Pf'ic')- lim
exists and, since
o(
1
)
OpCc' (c')- Z)(c')
c-O
^I
/(c-|c-)
^'
^^
2f(c-\c-)*F,{c-\c-)
2-
This proves the result. Q.E.D.
Proof of
Lemma
firm has
Pz> p
if
and only
if
2:
Consumers observe p ,> p'
Given
i
Thus
it
^ p{p.c) =
p
C2<
c' and
p fic^)
is
Y „p p
(
^^
(
c
)
.
c
)
<
,
,
.
.
The
the other
this
occurs
first-order
which p^(c)is the only
ct
c')
But by the implicit function theorem,
.
.
the second-order condition becomes Yp;.(p/^(c),c)>0.
Y,(p.c)
^ {p .c)
if
implying that equilibrium profits (for
-Vpe(Pf(c).c)
Pf ic)=
^'
H'^/p,(c).c)
Given that p^' >
only
increasing in Cj for Cz ^ c'
precisely the differential equation (A.1), to
is
only remains to show that
are strictly quasiconcave in
it
come back
given that they search, they will
has probability /^(p}'(Pi)|c); hence the form of
Pf(c2)>p, and
condition for maximization
solution.
PriCi) < p'
that
:
But,
-D(p)[i-f(p;'(p)ic)]-n(p.c)f2(p;'(p)|c).
=
Ypc(PA(c).c) = -D-(p,(c))[l-/^(c|c)]*D(p,(c))-:^^^7^-n(p.c)-:^^^^-np(p(c).c)f2(clc).
Pf
The
first
two terms are
positive.
Pf
(.c;
Replacing Pr'(c) from (A.1), the sum of the
(.c;
last
two terms has the sign
of:
-np(p,(c),c)[/,(cic)(l-f(c|c))*f,(cic)/(c|c)]>0.
from assumption
Proof of
Lemma
(4)
and the
We
3:
first
np(p,(c).c)>0.
fact that
find c'
Q.E.D.
the point of indifference between
.
p' and
Pf(,c) (when
c'<c'). by
examining:
5(c)-n(p",c)
We first show that
if
5(c) =
.
then 6'(c) <
n(p'.c)
Then:
l-^f(clc)
=
.
n(p,(c).c)[l-f(c|c)].
Indeed, 5(c) =
ifandonly
(/1.
15)
if,
-!-^^^^i^n(p,(c),c)<n(p,(c).c).
l-;f(c|c)
(^.i6)
'
46
5'(c)--D(p
^(P.(c))[l-f(c|c)]-n/p,(c).c)[l-f(c|c)]p/(c)
l--f(clc)
)
-n(p*.c)-n(p,(c).c) [f(c\c)^F,(c\c)]
-Dip
l--f(c|c)
)
-T^^iP .C)/(C|C)
2
Pf(c)-cJ
n(p,(c).c)-in(p".c) f^Cdc).
where we used both (A.1) and (A.16).
Now
p° < Pf(c) since n (
on
that the
first
,
•
c
)
increasing
is
6(c*) = jn(p',c*) =
If
(i)
therefore, also (by
j(p'-c')D(p')
p
price for c > c'
that
c<c°
it is
.
c
*
6
,
2) to
is its
*
)
Q.E.D.
.
This
and also
in
that
turn implies
'
(
c
)
<
is
c
°
.
has the sign of p' - c'
,
so
.
,
p'
all
,
Thus two cases can
.
Thus p' maximizes
while
arise:
p
to
'
Pf{c) and,
profits.
preferred price
6(c)<0so
those which do not induce search. By
the firm would rather charge
p'<p„(c)
p^Cc)
among
those which induce search. Thus Pf(c)
.
Note
Moreover, for c>c'
'
.
c°) > n(p;r(c'), c"),
firms with cost in [c° ,c"] prefer charging
is its
among
no search below
increases in c
n(p',
has a unique zero c^£{c',c'). and a firm with cost above c^ prefers
)
c>c'
For ce[c' .c')
continuous in
>
preferred price
since there
""
and p f{c)
negative,
always negative, we have showed that 6
is
so
,
any price above p"
6(-
since
;
(
clearly will prefer charging
Therefore, c
c"
.
>
p'<c*,
charging Pf(c) to p'
Lemma 2, Pf(c)
c
Lemma
If
(ii)
p
is
l-^f(c°|c') -n(p,(c°).c')[l-f(c"|c-)]>0
p' = p„{c')> Pf{c')
because
.
which contains
,
term
can have at most one zero. Moreover,
)
•
6(c°) = n(p'.c")
p
p„{c)]
last
.
Therefore, 5(
with
[c,
term above is also negative. Since the second term
whenever 6(c) =
any p <
(A.16) and (2) imply that the
.
is
the globally optimal
p' than Pfic)
and n(-,c) increases on
[c,
p^(c)]
,
.
and also than
Finally, a firm
that the uniqueness of the solution c'^to 5(c) =
Censures
p'<p„(c)
p
and so does p' = Pf(c')
,
since
so
n(p°.c) and 6(c)
p,(
)
•
is
increase in
increasing and does not
'
or c
'
.
depend on
,
.
47
Proof of Proposition
c] ,cl
a
for small
^
make
.
when a
(14) holds
!''„()
if
not monotonic on
is
because
V„'(c")>0.
interval,
i.e.,
Wc£[c'.c']
for a <
from
a"
K„(c)
>
K^
>
V^(c)>V^
.
of [c'
all
,
c'
(
•
.
c
'
on o
explicit,
by denoting them as
'
if
and only
decreases to c"
p\<c'
if
if
c>c".
and only
.
In particular,
J is
defined by (see
) is
the limit:
strictly
o"
Let
c>
if
Lemma
3,
K^C
)
to
is
some (maximal)
'
and
cl
.
r„(c")
-
c" e[c'
=
F^
this
such that:
.c^]
for all
a<a",
namely c^- max{ce[c'.c"]lI'„(c)<o}
to the cost
The
Then
.
c
'
,
and not above. Clearly,
we shall assume that c'c<Cn(c')
.
c^ £ (^c' .c']
bounded away from zero on
there exists a unique
differential
,
equation (10) and
i.e,
its
as
o
decreases
p"- p„(^cl)<
solution
.
Pf{)
c'
.
do
Lemma 3):
i-|^(c:ic:)
n(p,(cn.c:)[i-f(c:ic:)]
ensures that a unique solution c\£{c'„.c')
p„,(c") so by
in
closely the determination of c
monotonic up
it is
.
K„(c")-0.
Since
.
n(p;.c:)
otherwise
]
monopoly pricing can be sustained up
to zero, c
(Recall that
we examine more
Moreover, l'„(c)>Oin [c',c*]. so
not depend on a, while c
n
etc.,
.
D(p„(c3))p„'(c2)f(c2|c')dc,-o.
Let us turn next to the determination of c^
po-
c'
c'
3lc^e[c".c"]Vce[c'c"], I„,(c)>o
Thus
small enough,
is
f'
•^
to
,
Recall that
V^Cc')Even
the dependence of c'
etc.
,
To show that
c
We shall
4:
c^ decreases to a limit
el's. c~
exists).
.
n(p„(c).c')= W{pf{c').c'). which is
As o decreases
In fact c
J
to zero,
p\
decreases
remains bounded away from c",
impossible since c' <
Pf{c')< p^{c~) and
quasiconcave.
We are now ready to examine (14) for low values of
o
.
First, for all
c^t c\:
X(c,.o)< [' Z)(p„(c2))p„-(c2)f(c2|c')dc,-o
(,-1.17)
c"
SO x(ci.o) goes to zero (uniformly in
cjuith
o
.
On
the other
hand we show
§(o)-min{^(o,c,).c,e[c:.c-]}>0
for
o low enough.
First note, that for all
o<a"and c,>cJ>Co:
that:
(/1.
18)
48
I^(0.c,)-^(0.c,)|<|5(p:)-5(p;)|*|S(p;)-5(pJ)|+
|5(p:)-S(p;)|+2 1S(p:)-S(p')|.
=
SO ^(a.c^) converges to ^(0 c
,
if
|(0)>
,
i.e.,
,
uniformly in c
)
g[co.c*]
forall c,
D(p,(c2))p/(Cj)dc2
/
,
as
o goes
(^.19)
to zero. Therefore, by continuity (A. 18) will hold
:
^(0.c,)-[S(p;)-S(p,(c^))]f(cS|c,)+J^'D(p,(C2))p/(c2)f(c2|c,)dc2>0. (A. 20)
Since pS =
P/^(Co)>Po''Pm(c').
c,>Co.
requires
The
the
first
second
term can only be zero
term
in
(A.20)
f (co c,)
if
1
then
can
continuous and non-negative,
is
D(P/:(ci))>£)(c*)>0 by Lemma
p,'(ci)>0, and by assumption
;
for c
2
near c
1,
is
,
be
only
D(.PF(c2))Pf'ic2)F(c2\ci), which
=
which by assumption
zero
identically zero
on
Proof of Proposition
We consider conditional distributions
5:
this
,
]
But
.
cannot be
f (cj
I
Q.E.D.
c,) satisfying assumptions (1),(2), and
(4),
for which:
Fj " sup{f 2(^2 c,)
c'<C2^c,<c*}
.
I
is
c
co
and (A.20) must therefore hold. Thus (A18) holds, which, with (A17), implies that for alow
,
I
enough (below some o'e(0,o"]), y(C|)-o-=^(o,C|)-x(o.C|)>0,Vc,£[c°,c*].
and
[
f(c, |c,)>0 ,so
(4),
integrand
the
if
(4)
small.
More
precisely,
we
consider a family of distributions indexed by
continuously. For instance, ifc, = c + e + Yi .i'
(i)
This defines c
3; if
p^{c°)<
(ii)
(iii)
l^
',
m(
c
1
)
c'
,
we can
l^ (
c
F(c
1
'
)
)
,
p^(c')>c'
if
construct
monopoly
< a < 6
This will prove the theorem.
,
,
it
can be shown that
/"g
^
—
.
pricing at
p
"
p^(c)
.
is
increasing.
the equilibrium corresponds to either Proposition 2 or Proposition
p f{-) and
define c' £{c' ,c']
the equilibrium return to search at
>
fj
low enough:
the returns to search under
.
uniquely;
is
affects
1.2, where G is uniformly distributed on[-a,a]andY,,Y2
independently drawn from a uniform distribution on [-6,6]. with
We show that for distributions for which F 2
some parameter which
so that searching above
p
^
is
.
Then
for
F 2 low enough, we show:
p=p,:(c).c>c'. is
optimal.
increasing;
:
49
As before with a
c'f
.
c'f
,
,
we
shall
make
the
dependence of c'
c'
.
on
e'c.,
.
F
explicit
by denoting them as
etc. Since,
K„-(c,)-Z)(p„(c,))p„-(c,)f(c, |c,)+
Dip^(c,))p„-(c2)F,(c^\c,)dc^.
r
c'
(i)
will hold
if
for
c
all
,€( c ',
c
]
_ ^Z)(p„(c,))p„-(c,)^(c,|c,)
F,<
(yl.21)
We can impose (A,21) directly because the r.h.s. is a continuous, positive function of c
away from
zero. Indeed, the
/(c"|c')>0
limit
r.h.s.
at c, - c'
.
of (A.21)
is
strictly positive for c
D(Pr,ic'))
similarly the term in brackets
For c^t
c'f
>
Alternatively, using the convexity of
^ ,i>(P^(c-))
^2^
where
,
is
c
"
,
.
and therefore bounded
and apphing L'Hopital's
,
S(p)
(A.21)
.
is
rule,
it
has
implied by,
Pm'(C|)f (c, |c,)
min
iA.22)
Pm(c^)- p^ic-)
c,c:c-.c-]L
bounded away from zero on [c"
.
c
"
]
,
due to
(3)
and L'Hopital's
rule.
we have:
,
K'(c,)
=
r 'D(p„,(c,))p„'(c,)f2(c,|c,)dc3
D(p,(c,))p/(c,)f(c,lc,)+
c'
-[S(p;)-S(p=)]f3(c'Jc,)* [
'
Dip,ic,))p/{c,)F,(c,\c,)dc^
>Z?(p,(c,))p/(c,)f(c,|c,)-f,[S(p„(c-))-S(p„(c;))-S(p;)-S(p^)-S(pn-S(p,(c,))]
>Z)(p,(c,))p/(c,)f(c,|c,)-f2[S(p„(c-))-S(c,)].
But,
.,
/(c,ic,)
,
l-f(c,
|c,)
n(p,(c,).c,)
/(c,ic,)
n„(p,(c,).c,)
l-f(c,|c,)
/(c,|c,)
n(p„(c,).c,)
Dipfic,))
n„(c,)
l-f(c,|c,) D(p,(c,))
SO;
l"(c,)>
Thus,
(ii)
will
hold
/(Ci
,
I
Ci)f (C|
'
I
'
,
Ci)
"
-
n„(C|)-f,[5(p )-S(c,)].
if:
F 2- min
ce c,.c
n„(c-)
/(c|c)f(c|c)
l-f(clc)
.
S(p„(c-))-5(c-)
(^.23)
.
50
By assumption
if
we
(4),
the
but both sides of (A.23) involve the distribution F. Nonetheless,
r.h.s. is strictly positive,
assumptions
to zero, so
(1), (2)
does the
and
(4),
and such that
F*"
of (A.23) whereas the
l.h.s.
Finally,
.
Therefore, (A.23) holds for
by (14),
(iii)
will
hold
;
in the r.h.s.
satisfy
then as K goes
converges to (by L'Hopital's rule):
f\c\c)F'(c\c) ^^
>0,
\-F\c\c)
cclCc.c
> c~
\ e[0,X] which
,
minimum
.
c'f°
,
depend continuously on \ with F° -
j'"
.
min
since
F'"(c2\c,)
consider a one parameter family of conditional distributions
F''
with X small enough.
if:
l^(cn = [S(p;)-S(p^)]F(c'Jc=)> [ '/)(p„(c,))p„-(c,)[f(c,|c;)-f(c2|c=)]dc2.
for
which
it
suffices that:
F,<
r^^-^
Again,
this
condition involves
as to find c }
small
This
X the
is
F on both
sides;
Sip-)-S(p'f)
moreover,
Nonetheless, given a family of distribution
l.h.s.
of (A.24)
v.ill
be small, while the
because the equality (12) defining
Cf = p' = p^f; but c = Pf(.c)
excluded by focusing (as
we
is
c^
only possible at
r.h.s.
it
requires that the function P/r(
f
will
''
(
c
2
1
have) on the case where
c'
.
c
,
)
Thus p^o
)
be computed, so
with the properties described above, for
be close to the
always requires
c=
(/1. 24)
.
c}-Cf
Pf>
= p):o
finite
p'f.
value corresponding to f"°
unless
n{p' .Cf)
would require p^o =
c'
/^D(p„(c2))p„'(c^)f°(c2lc)dc2>o.
=
,
i.e.,
which can be
Q.E.D.
51
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