# Search and Competition with Flexible Investigations

Vasudha Jain University of Texas at Austin Mark Whitmeyer Hausdorff Center for
Mathematics & Institute for Microeconomics, University of Bonn
Email<EMAIL_ADDRESS>
We thank Francesc Dilmé, Joseph Whitmeyer, Thomas Wiseman, and a seminar
audience at the University of Bonn for helpful comments and feedback. The
second author’s work was funded by the DFG under Germany’s Excellence
Strategy-GZ 2047/1, Projekt-ID 390685813.

###### Abstract

We modify the standard model of price competition with horizontally
differentiated products, imperfect information, and search frictions by
allowing consumers to flexibly acquire information about a product’s match
value during their visits. We characterize a consumer’s optimal search and
information acquisition protocol and analyze the pricing game between firms.
Notably, we establish that in search markets there are fundamental differences
between search frictions and information frictions, which affect market
prices, profits, and consumer welfare in markedly different ways. Although
higher search costs beget higher prices (and profits for firms), higher
information acquisition costs lead to lower prices and may benefit consumers.
We discuss implications of our findings for policies concerning disclosure
rules and hidden fees.

Keywords: Sequential Search, Optimal Stopping, Information Acquisition,
Information Frictions, Rational Inattention, Hold-up Problem
JEL Classifications: C72; D82; D83

##  1 Introduction

There is a large and rich literature on sequential consumer search. Many
papers in this area share a similar common structure: there are collections of
possibly heterogeneous firms and consumers, the latter of which wish to
purchase products from the firms. These consumers (or at least some subset of
them) visit firms in sequence in order to discover important information about
the products including, e.g., their qualities, if the products are vertically
differentiated; match values, if there is horizontal differentiation; and/or
prices. The essential decisions made by consumers are whom to visit (if search
is directed), when to stop, and whether to purchase upon stopping (and from
whom).

Despite the considerable control they possess regarding their search behavior,
consumers in these models are surprising passive in a different sense. Namely,
when a consumer visits a firm she is unable to affect the information her
visit reveals–there is an equivalence between visiting a firm and acquiring
information about its product(s). She observes (at least some of) the
pertinent information (prices, qualities, and/or match values) or the draw of
some signal correlated with these values, but may not control this information
directly.

In many settings this is unrealistic. Consider, for instance, a prospective
homeowner looking for a house. She visits homes on the market in sequence; but
rather than absorb information about a house effortlessly upon visiting it,
she must acquire this information actively. Not only does the consumer face a
sequential search problem as she moves from house to house, but within each
visit, she has a dynamic information acquisition problem. She chooses which
rooms to visit and how long to linger in each; what questions to ask the real-
estate agent; how thoroughly to inspect the plumbing, lighting, or paint
condition; and how many dark corners to scrutinize for mold.111In fact there
are a number of guides online that advise consumers how to behave optimally
during such visits; one urges its readers, “Once inside a home, try
everything. Follow common courtesy but don’t be shy.”
(https://www.redfin.com/home-buying-guide/what-to-look-for)

This phenomenon is not limited merely to the housing market. Indeed, it is
difficult to think of many markets in which such information acquisition does
not take place. A prospective car buyer chooses how many features of a car to
inspect when at a dealer, a consumer shopping for groceries chooses how much
of each product’s label to peruse, a teenager looking for a prom dress
scrutinizes candidates in the many mirrors of a dress store’s changing room,
and a music enthusiast looking for new headphones plays various songs to test
out different facets of a device’s sound quality and feel.

Importantly, this information acquisition process is dynamic. If a prospective
homeowner sees that a house’s roof is caving in, she stops her tour right away
and moves on to the next property. Likewise, a young businesswoman may not
test drive a mini-van in garish pink–she has already moved on to the sleek
sedan in dark grey. Information is costly and so consumers investigate until
they have “seen enough,” at which point they decide whether to stop or move on
to the next firm or product.

Our goal in this paper is to explore the effect of this flexible information
acquisition on consumer and firm behavior in an otherwise standard sequential
search market. There are a large number of consumers with unit demand for a
horizontally differentiated product and a large mass of ex ante
indistinguishable firms that sell the product. It costs a consumer $c>0$ to
visit a firm; and when at a firm she acquires information about the product’s
idiosyncratic fit by sampling various “attributes”–which inform her
valuation–in sequence, at a cost of $\gamma>0$ per attribute.

The firms compete by setting prices, which–following much of the
literature–can only be discovered by the consumer upon visiting a firm. In our
main analysis we assume that when visiting a firm, a consumer observes the
firm’s price before acquiring information (before she starts sampling
attributes at the firm). In many circumstances this is realistic: most
retailers of consumer goods post prices that are non-negotiable and which
consumers note before inspecting the items. Due to the symmetry of the model
we focus on symmetric equilibria.

Within this setting, we tackle a number of questions. First, it is a standard
result that in search models with hidden prices, an increase in the search
cost leads to an increase in prices. Does this hold when there are information
frictions as well? What about the effect of information frictions on prices?
That is, do greater information frictions lead to higher prices? Do laws
mandating maximal disclosure always benefit consumers? Do they always hurt
firms?

Importantly, a consumer’s optimal information acquisition protocol is shaped
by the prices set by firms. Consequently, this gives her considerable power to
react to a price deviation by a firm–even though she can not observe a price
change before her visit, such a deviation will affect her optimal information
acquisition protocol at that firm. This is realistic–consumers, upon seeing an
unexpectedly high price, may just decide that the product is too expensive to
consider at all.

We find that (explicit) search costs continue to have the same effect as in
models without flexible learning. That is, prices and firm profits increase in
(explicit) search costs, whereas consumer welfare decreases. On the other
hand, the effect of information frictions is unexpectedly ambiguous. Prices
decrease in the level of information frictions, and so consumer welfare may
rise as a result. In fact, for certain regions of the parameter universe, an
increase in information frictions leads to a Pareto increase in welfare. The
expected duration of search is also non-monotone in the amount of information
frictions.

One notable implication of our results is that a firm’s market power is
generated entirely by the search costs (think switching costs) and not by the
information (learning) costs. To elaborate, it is only the existence of this
search cost that prevents a consumer from departing the firm at the first sign
of bad news. The information frictions, in turn, generate a downward sloping
demand curve for firms, which becomes steeper (ceteris paribus) as information
frictions increase. Thus, an increase in information frictions results in a
decrease in the equilibrium price, and this decrease may be so great as to
outweigh a consumer’s direct welfare loss as a result of increased frictions.
Firms are unaffected by the change in information frictions, since their
profits rely on the market power granted to them by the search cost.

Later on we explore a natural modification of the model: we stipulate that a
consumer does not observe a firm’s price until after she has acquired
information. This leads to an “informational hold-up problem,” and in the
unique equilibrium, firms mix over prices. Thus, our model generates price
dispersion even when prices are not posted. Surprisingly, the qualitative
results from the main setting continue to hold: explicit search costs benefit
firms at the expense of consumers but information frictions may be Pareto
improving.

We also compare the results from the two timing specifications. Consumers
strictly prefer that prices be observable before learning. Firms, on the other
hand, may prefer that prices be hidden, but only if the expected quality of
the product and the level of information frictions are sufficiently high.
Otherwise, firms prefer that prices be observable as well. The comparisons in
this section allow us to speak to the recent debate about hidden fees, drip
pricing, and other varieties of hidden prices. Our results suggest that laws
that mandate transparent prices do indeed benefit consumers, since they allow
them to learn efficiently. Moreover, firms may benefit from such laws as well.

We finish this section by discussing the relevant papers in the literature.
Section 2 lays out the model and establishes some preliminary results. Section
3 characterizes a consumer’s optimal search and information acquisition
protocol, Section 4 briefly describes a monopolist’s pricing behavior, and
Section 5 explores the equilibrium in the pricing game between the firms. In
Section 6 we modify the game so that prices are now unobservable to the
consumer before she acquires information and we look at the pricing game
between the firms. Section 7 compares welfare in the two timing regimes and
Section 8 concludes. All proofs are left to the appendix,222Due to the
algebra-heavy nature of some of the comparative statics derivations, we also
have a Supplementary Appendix that provides Mathematica code to corroborate
our calculations. unless otherwise noted.

###  1.1 Related Work

The paper closest to this one is Branco, Sun, and Villas-Boas (2012),333We
discuss their model in detail in the next section, when we describe our setup.
Henceforth, we refer to them as BSV. who look at the monopolist’s problem when
consumers flexibly acquire information after observing the monopolist’s price.
As in our search setting, they find that information frictions may Pareto-
improve welfare. The portion of our paper in which prices are not observable
before consumers acquire information is closely related to Ravid, Roesler, and
Szentes (2020), who explore a bilateral-trade setting in which a consumer
acquires costly information about a monopolist’s product before observing the
seller’s price. Their focus is on the limiting case as information frictions
vanish, and they establish that in the limit, the equilibrium converges to the
Pareto-worst free learning equilibrium.444A consumer’s learning problem in our
setting is not quite the same as that in Ravid, Roesler, and Szentes (2020).
In their model the consumer faces a static problem: she chooses a signal about
the value of a product, about which she has some prior distribution. As shown
by Morris and Strack (2017), when the prior is binary there is an equivalence
between the static information acquisition problem, in which a decision-maker
chooses a distribution over posteriors subject to a posterior-separable cost,
and a dynamic Wald stopping problem. Beyond the binary-prior setting, this
equivalence vanishes. Moreover, our problem is not a Wald problem–a consumer
directly observes various components that affect her expected value for the
product.

Other papers that look at a monopolist selling to consumers who may acquire
information flexibly include Branco, Sun, and Villas-Boas (2016), Pease
(2018), and Lang (2019). In the latter two papers, a consumer observes a
monopolist’s posted price before observing a diffusion process correlated with
the product’s quality at a flow cost. Lang (2019) assumes that the consumer’s
prior valuation for the product is distributed normally and finds that in
regions of the parameters with active information acquisition, increased
information frictions improve the monopolist’s profit if and only if the ex
ante expected value for the product is sufficiently high. Pease (2018) assumes
a binary value for the product, and also finds that the level of information
frictions has an ambiguous effect on firm profit. Branco, Sun, and Villas-Boas
(2016) modify the setting of BSV by allowing the seller to choose how much
information to provide about the product. They find that the seller may want
to induce information overload and provide more information than is optimal
for the consumer.

Ke, Shen, and Villas-Boas (2016) alter BSV by looking at the information
acquisition problem of a consumer inspecting the attributes of multiple
products in continuous time. At an instant, the consumer may shift her
attention from one product to another and acquires information at each in the
same fashion as BSV (and our paper)–by observing Brownian motion subject to a
constant flow cost. Given the difficulty of the consumer’s problem (it is a
PDE problem with ambiguous boundary conditions), the focus of their paper is
on the optimal information acquisition protocol of a consumer when there are
just two products. They also provide numerical analysis of the pricing problem
faced by a multi-product monopolist.

To our knowledge, the only other work to explore the dual effects of
information and search frictions in a sequential search model is Guo (2021).
In his model, a consumer searches sequentially and when at a firm draws her
value from a family of rotation ordered distributions. The primary focus of
his paper is the case in which a consumer chooses from a collection of uniform
distributions. There, he finds that the explicit search cost and the cost of
informativeness (in the rotation order) of a consumer’s posterior value
distribution operate in similar ways: equilibrium price and profit increase in
both types of friction but may drop discontinuously for high frictions (of
either variety). The reverse happens for consumer welfare. Our setup is
different in that our consumers tackle a sequential information acquisition
problem–one that is equivalent to a static problem in which they choose any
distribution over posterior values whose mean is the prior, subject to a cost.
This allows us to illustrate how search and information frictions operate in
fundamentally different ways and have dramatically different effects on
equilibrium behavior and outcomes.

When a consumer does not observe a firm’s price before acquiring information
about its product, she faces an informational hold-up problem. Similar to
Ravid, Roesler, and Szentes (2020), Roesler and Szentes (2017) investigate the
consumer-optimal signal in a bilateral trade environment, which is
subsequently extended by Yang (2019), who allows the seller to have a more
general production technology. In the same setting, Condorelli and Szentes
(2020) allow the buyer to choose her distribution over values for the product.
Dogan and Hu (2018), in turn, explore the buyer-optimal signal in the random
sequential search setting of Wolinsky (1986). In each paper, some form of
truncated Pareto distribution is optimal (which is intuitive, since it is
those distributions over values/posteriors that generate unit-elastic demand).

The literature that explores more standard versions of the hold-up problem in
bilateral trade is also relevant. Bar-Isaac, Caruana, and Cuñat (2012) look at
the scenario where a seller can privately invest in the quality along one
dimension of a multi-characteristic good, after which consumers may (at a
cost) acquire information about one dimension of the product. The cost of
information affects the firm’s investment via its influence on the consumers’
behavior and may lower consumer welfare and the firm’s profit. In contrast,
there is no investment in our model and so the cost of information affects
firms’ behavior only through its influence on their prices.

More recently, Dilmé (2019) looks at a scenario in which a seller can
privately invest in the quality of its product before a buyer makes a take-it-
or-leave-it offer. As we find in the hidden prices portion of this paper, and
following similar logic, Dilmé shows that the seller randomizes over
investment levels. He also finds that modifying the information structure of
the game–so that the buyer observes the mean of the seller’s investment
policy–aids the seller, who increases his investment level and his profit at
the expense of the buyer. Along these lines, Lau (2008), Hermalin and Katz
(2009), and Nguyen and Tan (2019) all look at various facets of information
design in the hold-up problem. Rao (2021) embeds a similar hold-up problem
into a labor search model, in which workers invest in human capital. With
hidden investment, Rao shows that the equilibrium exhibits both wage and skill
dispersion akin to our information and price dispersion.

Naturally, our paper is also related to the broader collection of works that
look at the effects of consumer rational inattention on market behavior. Of
particular pertinence is Matějka and McKay (2012), who explore a model in
which a consumer must incur a cost to evaluate the offers of firms within a
market. Notably, they show that prices are increasing (and, hence, consumer
welfare decreasing) in the size of the information frictions. Also related is
Liu and Dukes (2016), who look at a simultaneous search model, in which a
consumer must evaluate products in an oligopolistic market subject to an
evaluation cost. ex ante she chooses how many firms about which to acquire
information and to which depth. Crucially, the dynamic aspect of our
problem–in which a consumer’s optimal behavior (both regarding her search
behavior and her information acquisition protocol) is determined each
period–allows a consumer to react to price deviations by firms. They find that
an increase in information costs may benefit consumers, though at the expense
of total welfare. Other studies involve incentives for firms to “stay under
consumers’ radar” (De Clippel, Eliaz, and Rozen (2014)) or “get on consumers’
radar” (Bordalo, Gennaioli, and Shleifer (2016)), which forces are absent from
our paper.

##  2 The Model

Our model extends the framework of BSV to a random search setting. There is a
continuum (unit mass) of selling firms and a continuum (unit mass) of
consumers. Each firm’s product has $T$ different attributes that are ex ante
unknown. The value of each attribute to a consumer is a random variable. For
simplicity, these random variables are i.i.d. and have mean $0$. More
specifically, the value of attribute $j$ to a consumer is the random variable
$X_{j}$, which takes values $-z$ and $z$ with equal probability, where $z>0$.

If firm $i$ sets price $p_{i}$ then a consumer’s realized utility from product
$i$ is $\mu-p_{i}+\sum_{j=1}^{T}X_{j}$, where $\mu\in\mathbb{R}$ is the
consumer’s prior value for the product. We interpret $\mu$ to be a consumer’s
baseline value for the products for sale, and assume that it is the same for
each firm. Given this, it is the idiosyncratic attributes of a firm’s product
that set it apart from its competitors. For example, suppose the consumer is
shopping for a bicycle. Given her habits and needs she has some underlying
valuation $\mu$ for a generic bike. Then, once she arrives at a bike store,
she may test out the various attributes of a bike, each of which shapes her
value for that specific bicycle further–she likes its color but not the shape
of its seat, she finds that the handlebars are angled too far forward but
likes their grip, and so on. Crucially, these various attributes do not affect
her valuation for other bikes.

Following BSV, we assume that the number of attributes, $T$, is infinity. As
they note, this has the reasonable interpretation that the set of attributes
can never be “fully searched.” Naturally, this assumption also helps with the
problem’s tractability.

We make a distinction between search and information acquisition. Each period
a consumer decides whether to stop and purchase from the current firm, select
her outside option, or continue her search, in which case she is randomly
matched with another firm in the market. When a consumer visits a firm, she
acquires information flexibly. As is routine in the literature, a consumer’s
first period visit is free, but any subsequent search imposes on her some cost
$c>0$ per period.

When a consumer visits a firm, she acquires information flexibly. She does
this by examining attributes one-by-one, in sequence, at a cost of $\gamma>0$
per attribute. Examining attribute $j$ reveals the realization of random
variable $X_{j}$, and after examining $t$ attributes, a consumer’s expected
valuation from purchasing firm $i$’s product (so, net of that firm’s price) is

$u_{i}(t)=v_{i}+\sum_{j=1}^{t}x_{j}+\sum_{j=t+1}^{T}\mathbb{E}\left[X_{j}|t\right]=v_{i}+\sum_{j=1}^{t}x_{j}\text{
,}$

where $v_{i}\coloneqq\mu-p_{i}$; and since, by assumption,
$\mathbb{E}\left[X_{j}|t\right]=0$ for all $j,t$.

We continue to follow BSV by simplifying the information acquisition problem
faced by a consumer at firm $i$. We assume that the change in utility from
each attribute, $z$, becomes infinitesimal as $T$ grows infinitely large. This
captures the fact that the importance of any one attribute is negligible
compared to the value of the product on the whole. In the limit, the
$u_{i}(t)$ process becomes a Brownian motion–cropping the argument, $du=\sigma
dW$, where $W$ is a standardized Brownian motion–with standard deviation
$\sigma$, where $z=\sigma\sqrt{dt}$.555As noted by Lang (2019), there is a
subtle technical issue to this approach in that the “actual value for the
product $\lim_{t\to\infty}\int_{0}^{t}W_{s}ds$ is ill-defined.” However, we
argue that this modeling choice is akin to the use of the improper uniform
distribution in the global games literature, and which greatly helps us with
simplicity at the cost of slight impropriety. Furthermore, for most of this
paper, we use a convenient reformulation of the consumer’s stopping problem as
a static information acquisition problem, and we urge any objectors to the
dynamic set-up to view it exclusively as the static problem instead. When a
consumer is at a firm, each instant she decides whether to keep observing
attributes or to stop doing so. Once she stops acquiring information at a
firm, she either purchases from the current firm, selects her outside option,
or continues her search. Throughout, we assume that
$c<\sigma^{2}/\left(4\gamma\right)$. This assumption–that search costs are not
too high–is a necessary condition for the existence of equilibria with active
search. For simplicity, we normalize the value of the consumer’s outside
option to $0$.

The timing of the game is summarized in Figure 1 below.

Firms simultaneously (and privately) set prices. Consumer is randomly matched
with a firm whose price she then observes. Consumer decides when to stop
costly acquisition of information about that firm’s attributes. Consumer pays
$c$ to be randomly matched with another firm, and the same process is
repeated. Consumer buys from the current firm or selects her outside option
and payoffs are realized. Continues searchStops search Figure 1: Timing

###  2.1 Simplifying the Consumer’s Information Acquisition Problem

Next, we observe that the consumer’s information acquisition problem can be
reformulated from a dynamic optimal stopping problem to a much simpler static
problem. Evidently, at a firm, a consumer’s information acquisition strategy
is a stopping time $\tau$, and any such $\tau$ generates a distribution over
(expected values) $u$, which we denote $F_{\tau}$. That is,

$F_{\tau}\left(u\right)\coloneqq\mathbb{P}\left(U_{\tau}\leq u\right)\text{
.}$

We denote by $\mathcal{M}\left(\mu\right)$ the set of (Borel) probability
measures on $\mathbb{R}$ with mean $\mu$ and define the ex ante cost of
distribution $Q\in\mathcal{M}\left(\mu\right)$ over values to be the minimal
cost at which the consumer can generate $Q$:

$C\left(Q\right)\coloneqq\inf_{\tau:F_{\tau}=Q}\mathbb{E}\left[\gamma\tau\right]\text{
.}$

Then, the results of, e.g., Ankirchner, Hobson, Strack, et al. (2015) imply
the following remark:666This remark is a trivial modification of the main
result of Root (1969) (Theorem 2.1). See also Georgiadis and Szentes (2020)
for a recent use of this result in an economic setting. This is an offshoot of
the Skorokhod embedding problem in mathematics.

######  Remark 2.1.

The ex ante cost of distribution $Q\in\mathcal{M}\left(\mu\right)$ over
values, $C:\mathcal{M}\left(\mu\right)\to\mathbb{R}^{+}$, is

$C\left(Q\right)=\kappa\int_{-\infty}^{\infty}\left(x-\mu\right)^{2}dQ\left(x\right),\quad\text{where}\quad\kappa\coloneqq\gamma/\sigma^{2}\text{
.}$

The precise formula follows via Ito’s lemma: for value $u_{\tau}$,

$c\left(u_{\tau}\right)=c\left(\mu\right)+\int_{0}^{\tau}\sigma
c^{\prime}\left(u_{t}\right)dW_{t}+\frac{1}{2}\int_{0}^{\tau}\sigma^{2}c^{\prime\prime}\left(u_{t}\right)dt\text{
,}$

where $c\left(u\right)\coloneqq\kappa\left(u-\mu\right)^{2}$. Because $\tau$
is a.s. finite, taking expectations, the first term on the right side of the
equal sign is $0$, and the middle term, being a martingale, vanishes.

Accordingly, instead of solving a consumer’s dynamic information acquisition
problem at each firm, we may instead solve the following static problem. Let
$g\left(u\right)$ be a consumer’s payoff from stopping her information
acquisition at value $u$. She solves

$\max_{Q\in\mathcal{M}\left(\mu\right)}\left\\{\mathbb{E}_{Q}\left[g(u)\right]-C\left(Q\right)\right\\}=\max_{Q\in\mathcal{M}\left(\mu\right)}\left\\{\int_{-\infty}^{\infty}g(u)dQ(u)-\kappa\int_{-\infty}^{\infty}\left(u-\mu\right)^{2}dQ\left(u\right)\right\\}\text{
.}$

##  3 A Consumer’s Behavior

Our first step in characterizing a consumer’s optimal search and information
acquisition protocol is to determine how she optimally acquires information at
a seller $i$ charging price $p_{i}$ when she has some outside option of $a\geq
0$ “in hand.”777$a$ is the maximum of her maximal continuation value and her
explicit outside option (which is given exogenously). Her payoff from
obtaining value $u$ is

$W\left(u\right)=\begin{cases}a-\kappa\left(u-\mu\right)^{2},\quad&u-p_{i}\leq
a\\\ u-p_{i}-\kappa\left(u-\mu\right)^{2},\quad&u-p_{i}\geq a\\\
\end{cases}\text{ .}$

Her optimal learning is given by the concavification888This approach is
famously used by Aumann, Maschler, and Stearns (1995) to characterize the
limiting value of repeated games with one-sided incomplete information.
Kamenica and Gentzkow (2011) provide a recent promulgation of this technique.
of this function and is pinned down by the two posterior beliefs $u_{L}$ and
$u_{H}$. Explicitly,

$u_{L}=a+p_{i}-\frac{1}{4\kappa},\quad\text{and}\quad
u_{H}=a+p_{i}+\frac{1}{4\kappa},\quad\text{where}\quad\mathbb{P}\left(u_{H}\right)=\frac{1}{2}+2\kappa\left(\mu-
p_{i}-a\right)$

An example of $W$ and its concavification are depicted in Figure 2.

Figure 2: $W$ (in red) and its concavification (in blue) for $a=1/4$, and
$v_{i}\coloneqq\mu-p_{i}=\kappa=1/2$.

A consumer, given prior value $\mu$, price $p_{i}$, with $a\geq 0$ in hand,
and behaving optimally, obtains the following payoff:

$W^{*}\left(\mu,p_{i};a\right)=\begin{cases}a,\quad&\mu\leq
a+p_{i}-\frac{1}{4\kappa}\\\ \frac{1}{16\kappa}+\frac{\mu-
p_{i}+a}{2}+\kappa\left(\mu-
p_{i}-a\right)^{2},\quad&a+p_{i}-\frac{1}{4\kappa}\leq\mu\leq
a+p_{i}+\frac{1}{4\kappa}\\\ \mu-
p_{i},\quad&a+p_{i}+\frac{1}{4\kappa}\leq\mu\\\ \end{cases}\text{ .}$

If the prior value, $\mu$, is in an interior region, a consumer learns and
either becomes very optimistic about firm $i$’s product, in which case she
purchases it; or becomes pessimistic about firm $i$’s product and leaves
without purchase. Equivalently, for an interior $\mu$, a consumer observes
process $W$ until its value hits either $u_{L}$ or $u_{H}$ at which point she
stops acquiring information and either purchases the product or continues her
search elsewhere. Outside of the interior region, a consumer does not learn
and either purchases from firm $i$ immediately or pursues other options.
Predictably, as $\kappa$ rises, learning occurs over a smaller range of prior
values. Furthermore, for higher values of $a+p_{i}$, an immediate purchase
from firm $i$ is optimal for a smaller range of priors, while pursuing other
options without learning about firm $i$ is optimal for a larger range of
priors.

###  3.1 A Consumer’s Search Protocol

Because of our focus on symmetric equilibria, a consumer faces a stationary
environment. It is easy to see that a consumer’s optimal search protocol can
be calculated recursively. Given this, suppose that the parameters are such
that there exists active search and information acquisition (we will determine
these precise values shortly). As a result, we define $\Phi$ to be a
consumer’s payoff under her optimal search and information acquisition
protocol. Given that we have assumed there to be active search and information
acquisition, the consumer’s optimal information acquisition protocol is
stipulated by Expression 3, in which we set $a=\Phi-c$. Moreover, using
Expression 3 (and again setting $a=\Phi-c$) we can calculate $\Phi$:

$\Phi=\frac{1}{16\kappa}+\frac{\mu-p+\Phi-c}{2}+\kappa\left(\Phi-c-\left(\mu-p\right)\right)^{2}\text{
,}$

where we cast aside the subscript $i$ due to symmetry. Rearranging this
equation, we obtain

$\Phi_{l}=\mu-p+c+\frac{1}{4\kappa}-\sqrt{\frac{c}{\kappa}}\text{ ,}$

where the subscript denotes “low price,” introduced so as to distinguish this
value from the maximal payoffs in other parameter regions.

Now, let us tackle our (implicit) parametric assumptions. The first is that
$\Phi_{l}-c\geq 0$, since the consumer must (weakly) prefer continuing her
search over stopping to take her outside option. This holds if and only if

$\mu-p\geq\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\text{ .}$

The second is that the learning is feasible; namely, that

$\Phi_{l}-c-\frac{1}{4\kappa}\leq\mu-p\leq\Phi_{l}-c+\frac{1}{4\kappa}\text{
.}$

Since $c>0$, the left-hand inequality must always hold, but the right-hand one
is satisfied if and only if our assumption $c\leq 1/\left(4\kappa\right)$
holds.

If Inequality 3.1 does not hold, a consumer’s optimal learning is again given
by Expression 3, in which $a=0$; and we find that this protocol is optimal
provided
$-1/\left(4\kappa\right)\leq\mu-p\leq\sqrt{c/\kappa}-1/\left(4\kappa\right)$.
The consumer’s payoff is

$\Phi_{m}=\frac{1}{16\kappa}+\frac{\mu-p}{2}+\kappa\left(\mu-p\right)^{2}\text{
,}$

where the subscript stands for “medium price.” If
$\mu-p\leq-1/\left(4\kappa\right)$, consumers neither search nor acquire
information and there are no transactions. A consumer’s payoff is merely $0$.

Compiling these observations, we have

######  Proposition 3.1.

If $\mu-p\leq-1/\left(4\kappa\right)$, there is no trade, and the consumer’s
payoff is $0$. If
$-1/\left(4\kappa\right)\leq\mu-p\leq\sqrt{c/\kappa}-1/\left(4\kappa\right)$,
a consumer learns at one firm and obtains $\Phi_{m}$; and if
$\mu-p\geq\sqrt{c/\kappa}-1/\left(4\kappa\right)$, a consumer both learns and
searches and obtains $\Phi_{l}$.

##  4 A Monopolist’s Problem

With a view towards our pricing equilibrium, let us briefly characterize a
monopolist’s optimal pricing policy in this environment. This is precisely the
problem studied in BSV, so this section’s results merely repeat theirs (with
the trivial generalization of allowing the the consumer to have an outside
option of $a\geq 0$).

The monopolist’s profit as a function of its price $p_{m}$ is

$\Pi_{M}\left(p\right)=\begin{cases}0,\quad&p\geq\mu+\frac{1}{4\kappa}-a\\\
p\left(\frac{1}{2}+2\kappa\left(\mu-
p-a\right)\right),\quad&\mu-\frac{1}{4\kappa}-a\leq
p\leq\mu+\frac{1}{4\kappa}-a\\\ p,\quad&p\leq\mu-\frac{1}{4\kappa}-a\\\
\end{cases}\text{ ,}$

and so its optimal price is

$p_{M}=\begin{cases}\mu-\frac{1}{4\kappa}-a,\quad&\mu\geq\frac{3}{4\kappa}+a\\\
\frac{1}{8\kappa}+\frac{\mu-a}{2},\quad&\frac{3}{4\kappa}+a\geq\mu\geq
a-\frac{1}{4\kappa}\end{cases}\text{ .}$

The maximal profit, $\Pi_{M}^{*}$, and corresponding consumer payoff,
$\Phi_{M}^{*}$, are

$\Pi_{M}^{*}=\begin{cases}\mu-\frac{1}{4\kappa}-a,\quad&\mu\geq\frac{3}{4\kappa}+a\\\
\frac{\left(1+4\kappa\left(\mu-a\right)\right)^{2}}{32\kappa},\quad&\frac{3}{4\kappa}+a\geq\mu\geq
a-\frac{1}{4\kappa}\\\ 0,\quad&\mu\leq a-\frac{1}{4\kappa}\end{cases},\text{
and
}\quad\Phi_{M}^{*}=\begin{cases}\frac{1}{4\kappa}+a,\quad&\mu\geq\frac{3}{4\kappa}+a\\\
\frac{\Pi_{M}^{*}}{2}+a,&\frac{3}{4\kappa}+a\geq\mu\\\ \end{cases}\text{ .}$

These findings are quite intuitive: if the expected quality (net of the
outside option), $\mu-a$, is sufficiently high; or, equivalently, information
frictions are sufficiently high, the monopolist prices so as to dissuade
learning. If $\mu-a$ is in a middle region, or $\kappa$ is sufficiently low,
the monopolist wants to encourage learning. In fact, as $\kappa$ becomes
increasingly small, the monopolist prices ever higher, to take advantage of
the consumers who learn that they “absolutely must have the product.”

##  5 The Firms’ Pricing Game

Having pinned down consumers’ behavior, we may now describe the equilibrium
pricing decisions of the firms. Following much of the literature, we assume
that each firm’s price is hidden and can only be discovered by a consumer when
she visits the firm. To determine her optimal search and information
acquisition protocol, the consumer conjectures a price, $\tilde{p}$ set by
each firm, and at equilibrium these conjectures must be correct, i.e.,
$\tilde{p}=p$.

For tractability, we focus on symmetric, pure strategy equilibria with random
search.999Henceforth, we drop the modifier symmetric. As is standard, we
require that if the consumer observes a deviation at a firm, her beliefs about
the other firms’ prices are unchanged.101010This is a standard assumption in
the literature, though one that may not be realistic in situations with
vertical relations (see, e.g., Janssen and Shelegia (2020)). However, these
are absent from our model, so we claim that our handling of off-path beliefs
is appropriate.

We begin by searching for a low price equilibrium, in which consumers search
actively. A firm’s profit is

$\Pi\left(p\right)=\begin{cases}0,\quad&\tilde{p}+\sqrt{\frac{c}{\kappa}}\leq
p\\\ 2\kappa
p\left(\tilde{p}-p+\sqrt{\frac{c}{\kappa}}\right),\quad&\tilde{p}+\frac{1}{2\kappa}-\sqrt{\frac{c}{\kappa}}\leq
p\leq\tilde{p}+\sqrt{\frac{c}{\kappa}}\\\
p,\quad&p\leq\tilde{p}+\frac{1}{2\kappa}-\sqrt{\frac{c}{\kappa}}\end{cases}\text{
.}$

Taking the first order condition and imposing symmetry, we obtain the
equilibrium price $p_{l}=\sqrt{c/\kappa}$ and substituting this into $\Phi$,
we see that the consumer prefers to continue her search over her outside
option ($0$) if and only $\mu\geq 2\sqrt{c/\kappa}-1/\left(4\kappa\right)$. A
consumer’s payoff, a firm’s profit, and a consumer’s probability of stopping
at the current firm are, respectively,

$\Phi_{l}^{*}=\mu+\frac{1}{4\kappa}+c-2\sqrt{\frac{c}{\kappa}},\quad\Pi_{l}^{*}=2c,\quad\text{and}\quad\mathbb{P}\left(u_{H}\right)=\frac{1}{2}+2\kappa\left(\mu-
p-a\right)=2\sqrt{c\kappa}\text{ .}$

Next, it is easy to see that the medium price equilibrium–in which there is no
active search–is given by the interior monopoly solution when consumers have
an outside option of $0$. Namely, $p_{m}=1/\left(8\kappa\right)+\mu/2$, and
the necessary and sufficient condition for this equilibrium to exist (that the
consumer prefers to take her outside option over continuing her search and
that trade occurs) is
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)\geq\mu\geq-1/\left(4\kappa\right)$. A
consumer’s payoff and a firm’s profit are, respectively,

$\Phi_{m}^{*}=\frac{\left(1+4\kappa\mu\right)^{2}}{64\kappa},\quad\text{and}\quad\Pi_{m}^{*}=\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}\text{
.}$

Compiling these observations,

######  Proposition 5.1.

If $\mu\leq-1/\left(4\kappa\right)$, no equilibria with trade exist. If
$-1/\left(4\kappa\right)\leq\mu\leq 2\sqrt{c/\kappa}-1/\left(4\kappa\right)$,
the unique equilibrium is one in which consumers visit only one firm and learn
there (but do not search actively). The equilibrium price is $p_{m}$. If
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)\leq\mu$, the unique equilibrium is
one in which consumers both search and acquire information. The equilibrium
price is $p_{l}$.

One noteworthy aspect of this result is that there exist no equilibria in
which consumers neither search nor acquire information. This stands in sharp
contrast to the monopolist scenario, in which the monopolist may wish to set a
price so low that the consumer buys without learning. This cannot happen in
the search market, since lower prices increase the consumer’s outside option
(her continuation value), thereby encouraging both learning and search.

###  5.1 Comparative Statics

At last, we can tackle some of the questions raised in the introduction.
First, we observe the unsurprising result that an increase in the prior
expected quality of the good, $\mu$, improves welfare in a Pareto sense.
Naturally, prices are also increasing in $\mu$.

######  Proposition 5.2.

Increasing $\mu$ is Pareto improving, possibly strictly. If $\mu$ is
sufficiently high (so there is active search), the market price and a firm’s
payoff are unaffected by $\mu$, but a consumer’s payoff is strictly increasing
in $\mu$. If $\mu$ is moderate (so there is no active search), prices, profits
and consumer payoffs are all strictly increasing in $\mu$.

Second, we find that an increase in the explicit search cost, $c$, harms
consumers but benefits firms.

######  Proposition 5.3.

If $c$ is sufficiently small (so there is active search), the market price and
a firm’s payoff are strictly increasing and a consumer’s payoff is strictly
decreasing in $c$. Otherwise (when there is no active search), the market
price, a firm’s payoff and a consumer’s payoff are all unchanging in $c$.

It is easy to see that firms require explicit search costs in order to make
profits. Indeed, observe that the lower posterior belief $u_{L}$ equals
$\mu-\sqrt{c/\kappa}$, which is decreasing in $c$ and goes to $\mu$ as $c$
vanishes. As costs vanish, the willingness of the consumer to depart for a
different firm increases, eroding the rents that each firm can extract.

The effect of a change in $\kappa$ is more subtle.

######  Proposition 5.4.

Prices are strictly decreasing in $\kappa$. If $\kappa$ is sufficiently small,
welfare is Pareto decreasing in $\kappa$. If $\mu$ is sufficiently large (or
$c$ is sufficiently small) and $\kappa$ is sufficiently large, welfare is
Pareto increasing in $\kappa$, possibly strictly (if the parameters are such
that there is no active search).

This result is somewhat surprising: it is possible that an increase in
information frictions can be Pareto improving. The effect of these frictions
on the price seems natural: notice that when the information friction $\kappa$
is higher, $\mathbb{P}(u_{H})$ is more responsive to changes in $p$, holding
the continuation value fixed–a firm’s “demand curve” is steeper. Consequently,
at equilibrium, the solution of the optimization problem of a firm entails
lowering the price slightly as a reaction to an increase in $\kappa$.
Moreover, as it turns out, this balance is perfect: in regions of the
parameter space with active search, a firm’s profit is unaffected by changes
in $\kappa$ since, as noted above, the rents it can extract are limited by the
explicit search cost.

In contrast, a change in the information acquisition cost affects a consumer’s
welfare in two ways. Ceteris Paribus, an increase in $\kappa$ hurts consumers,
as information is more difficult to obtain. However, it also results in a
decrease in the market price, which is to a consumer’s benefit. The overall
effect of $\kappa$ on consumer welfare; therefore, is determined by which of
these two forces dominates. When $\kappa$ is low (and therefore prices are
high) the effect of an increase in $\kappa$ on the price is too small to
outweigh the negative direct affect on learning. Conversely, a high $\kappa$
flips the relationship.

It is easy to see that if there is active search, the expected search duration
is $1/\left(2\sqrt{c\kappa}\right)$, which is predictably decreasing in $c$
and $\kappa$. If there is no active search, the expected search duration is
just $1$.

(a) Case 1: $\mu>0$ and $c\leq\mu/4$.

(b) Case 2: $\mu>0$ and $\mu/3\geq c\geq\mu/4$.

(c) Case 3: $\mu>0$ and $c\geq\mu/3$.

Figure 3: Consumer welfare (solid red), profit (solid blue), price (dotted
green), and expected search duration (dashed purple) as functions of $\kappa$.

######  Proposition 5.5.

The expected duration of search as a function of $\mu$ is increasing in $\mu$:
it is a step function with one discontinuity at
$\mu=2\sqrt{c/\kappa}-1/\left(4\kappa\right)$, where it jumps up from $1$ to
$1/\left(2\sqrt{c\kappa}\right)$. The expected duration of search is
decreasing in $c$: it is strictly decreasing in $c$ for $c$ sufficiently small
(in the active search region) and constant thereafter (in the region without
active search). The expected duration of search may be non-monotone in
$\kappa$. In particular, for a region of the parameter space, there is active
search for $\kappa$ sufficiently low or sufficiently high, but no active
search in an intermediate region.

Figure 3 illustrates the results of Propositions 5.4 and 5.5. In the proof of
the former (in Appendix A.3) we find that the parameter space (for $\mu>0$)
can be divided into three cases, each of which is depicted in the figure. The
portions of the domain in which $\Pi$ is flat are those regions of $\kappa$
(for the fixed $\kappa$ and $\mu$) that facilitate active search. Note that
the graph exhibits considerable non-monotonicity in $\kappa$: over different
intervals of $\kappa$ profit and consumer welfare may both be increasing or
decreasing. Moreover, the equilibrium may involve active search for small and
large $\kappa$, but not for moderate $\kappa$. Accordingly, the expected
search duration may have discrete jumps both up and down.

###  5.2 Obfuscation

By now, a sizeable collection of papers on obfuscation in search markets has
emerged. These papers, both theoretical111111See, e.g., Wolinsky (1986), Gu
and Wenzel (2014), Hämäläinen (2018), and Petrikaitė (2018). and
empirical121212See, e.g., Ellison and Ellison (2009) and Richards, Klein,
Bonnet, and Bouamra-Mechemache (2019). In addition, Ellison (2016) discusses
obfuscation in her survey of price search. explore how firms may strategically
increase consumers’ search costs to allow for higher mark-ups. Instead of
firms obfuscating by increasing search costs (the typical type of obfuscation
explored in the literature), a different form of obfuscation suggests itself
in our paper. Namely, we may allow each firm to not only set a price but also
choose a level of information friction $\kappa$ from some interval of possible
frictions $\left[\underaccent{\bar}{\kappa},\bar{\kappa}\right]$, where
$\underaccent{\bar}{\kappa}>0$ and $\bar{\kappa}\leq\infty$.

How might firms be able to choose $\kappa$? Recall that in a consumer’s
dynamic problem, she observes in sequence each attribute, which is a symmetric
(binary) random variable. By adding noise to this random variable, a firm
would decrease the variance of the corresponding Brownian process, thereby
increasing $\kappa$. Firms could also increase the explicit cost of inspecting
each attribute, $\gamma$, which also raises $\kappa$.

A pair $(\kappa,p)$ is an equilibrium provided it is the solution to a firm’s
monopoly problem with $a$ determined by the equilibrium; and so looking at the
monopolist’s profit (given in Expression 4), we see that the equilibrium
$\kappa$ must be a corner solution. Viz.,

######  Remark 5.6.

If firms may choose both $p$ and $\kappa$, the equilibrium
$\kappa^{*}\in\\{\underaccent{\bar}{\kappa},\bar{\kappa}\\}$.

It is straightforward, though rather tedious to characterize the explicit
equilibria of this game, but we believe the main insight we discover regarding
obfuscation is contained in the remark and so do not do so. As the remark
states, when firms may strategically obfuscate the product value, by hindering
the observability of a product’s attributes, they either muddle this process
as much as possible or as minimally as possible. An interior level of
obfuscation never constitutes an equilibrium.

##  6 Hidden Prices

So far, we have assumed that when a consumer visits a firm, she observes the
price for the item before acquiring information. This seems like reasonable
assumption for many sorts of consumer goods like, e.g., shoes, bicycles,
perfumes, headphones, foodstuffs. However, there are other markets in which it
may be more natural to assume that a consumer acquires information about a
product before seeing its price. One way this may manifest is when firms
include hidden fees or charges that can only be observed later on in the
purchase process.

In this section, we analyze the situation in which a consumer may not observe
a firm’s price until after she has acquired information at the
firm.131313Importantly, each firm sets its price in advance and may not
observe consumers’ behavior beforehand. As we will shortly see, this changes
the equilibrium behavior significantly: there is now an informational hold-up
problem, which any equilibrium must accommodate.

As in our main specification, we begin by stating the outcome when there is a
single (monopolist) firm.

###  6.1 Monopoly With Hidden Prices

As we did in the observable prices case, we stipulate that a consumer has an
outside option of $a\geq 0$. In contrast to the observable prices scenario, we
find that the existence of a pure strategy pricing equilibrium for the
monopolist is incompatible with trade.

######  Lemma 6.1.

There are equilibria in which the monopolist chooses a deterministic price if
and only if $\mu\leq a$. In any such pure-strategy equilibrium, there is no
trade.

If the monopolist mixes over prices, it must be indifferent over all prices in
the support of her mixture. That is, given a consumer’s distribution over
posteriors, $F$, the monopolist’s profit is given by

$\hat{\Pi}\left(p\right)=p\left(1-F\left(p+a\right)\right)\text{ ,}$

which equals some constant $\lambda>0$ for all $p$ in the support of her mixed
strategy.

######  Theorem 6.2.

If $\mu>a$, there is an essentially unique141414That is, any equilibrium must
be such that the empirical distribution over valuations is $F$. One way this
could be obtained is if each consumer randomized and chose distribution $F$
herself. Because there is a unit mass of consumers, one might worry about
whether one could construct a continuum of independent individual random
variables that yields the desired empirical distribution over values by the
Exact Law of Large Numbers. Sun (2006) allows for precisely such a
construction to be done rigorously. Alternatively, we could assume that there
is a random variable $Y\sim H$ with support $[0,1]$, which is a consumer’s
label. By, e.g., Winkler (1988) there is a distribution $H$ such that a
consumer with label $y$ (deterministically) chooses a binary distribution over
values with support $\left\\{u_{L}\left(y\right),u_{H}\left(y\right)\right\\}$
such that the resulting compound distribution is precisely $F$. Yet another
approach would be to assume that there is a single representative consumer,
who herself mixes over values according to $F$. equilibrium in the monopolist
scenario. The empirical distribution over consumer valuations is

$F\left(x\right)\coloneqq
1-\frac{\underaccent{\bar}{x}_{M}}{x},\quad\text{on}\quad\left[\underaccent{\bar}{x}_{M},\bar{x}_{M}\right]\text{
,}$

and the monopolist chooses a uniform distribution over prices

$G\left(p\right)\coloneqq
2\kappa\left(p-\underaccent{\bar}{p}_{M}\right),\quad\text{on}\quad\left[\underaccent{\bar}{p}_{M},\bar{p}_{M}\right]\text{
.}$

$\underaccent{\bar}{p}_{M}$ solves

$\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}\right\\}}=\frac{\mu-a}{\underaccent{\bar}{p}_{M}}-1\text{
,}$

and $\bar{p}_{M}=\underaccent{\bar}{p}_{M}+1/\left(2\kappa\right)$, and
$\underaccent{\bar}{x}_{M}=\underaccent{\bar}{p}_{M}+a$. The equilibrium
vector of payoffs is

$\hat{\Pi}_{M}^{*}=\underaccent{\bar}{p}_{M},\quad\text{and}\quad\hat{\Phi}_{M}^{*}=\kappa\left(\mu-\underaccent{\bar}{x}_{M}\right)^{2}+a\text{
.}$

Unsurprisingly, a truncated Pareto distribution ($F$) over valuations emerges,
which yields unit elastic demand for the monopolist, which is, therefore,
willing to mix over prices. The monopolist chooses a distribution over prices
that yields a payoff for consumers (net of information acquisition costs) that
is an affine function of the realized posterior values. They, in turn, are
therefore willing to choose any distribution with support on
$\left[\underaccent{\bar}{x}_{M},\bar{x}_{M}\right]$. Indeed, as the
concavification approach suggests, if a consumer’s payoff were strictly convex
in her value, a binary distribution would be uniquely optimal. Conversely, if
her payoff were strictly concave in her value, she would not acquire any
information.

####  6.1.1 Monopoly Comparative Statics

Just as how the equilibrium price when prices are observable increases in
$\mu$ and $\kappa$, so does the lower bound for its distribution over prices
when prices are hidden.

######  Lemma 6.3.

The lower bound for the monopoly price, $\underaccent{\bar}{p}_{M}$, is
strictly increasing in $\kappa$ and $\mu$.

However, because the monopolist’s profit is precisely this lower bound,
$\underaccent{\bar}{p}_{M}$, information frictions are unambiguously good for
the monopolist when prices are hidden. In the limit, as $\kappa$ explodes,
$\underaccent{\bar}{p}_{M}$ and $\bar{p}_{M}$ converge to $\mu-a$ and so the
monopolist leaves consumers with zero rents. In contrast to the observable
prices case–when consumer welfare is initially decreasing in $\kappa$, then
increasing, then decreasing again (after entering the region in which the
monopolist does not induce learning)–when prices are hidden, consumer welfare
is initially increasing in $\kappa$ then decreasing. In the limit, as $\kappa$
explodes, both profit and consumer welfare are the same regardless of when
consumers may observe the price.

######  Lemma 6.4.

The monopolist’s profit is strictly increasing in $\kappa$ and $\mu$. Consumer
welfare is strictly increasing in $\mu$ and is (strictly) increasing in
$\kappa$ if and only if $\underaccent{\bar}{p}_{M}\kappa\lesssim.337$, i.e.,
if and only if $\kappa$ is sufficiently low.

###  6.2 The Monopolistic Competition Equilibrium

If there exists an equilibrium with learning and active search, the
equilibrium is precisely that given in the monopolist scenario, where we
substitute a consumer’s continuation value, $\hat{\Phi}-c$, in for $a$. Some
simple algebra yields the following:

######  Theorem 6.5.

If $\mu>0$, there is an essentially unique equilibrium in the monopolistic
competition scenario. The equilibrium involves active search if and only if
$\mu-\sqrt{c/\kappa}>\underaccent{\bar}{p}$, where the price
$\underaccent{\bar}{p}$ solves

$\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}=\frac{1}{\underaccent{\bar}{p}}\sqrt{\frac{c}{\kappa}}\text{
.}$

The empirical distribution over consumer valuations is

$F\left(x\right)\coloneqq
1-\frac{\mu-\sqrt{\frac{c}{\kappa}}}{x},\quad\text{on}\quad\left[\mu-\sqrt{\frac{c}{\kappa}},\mu-\sqrt{\frac{c}{\kappa}}+\frac{1}{2\kappa}\right]\text{
,}$

and the firms choose uniform distributions over prices

$G\left(p\right)\coloneqq
2\kappa\left(p-\underaccent{\bar}{p}\right),\quad\text{on}\quad\left[\underaccent{\bar}{p},\bar{p}\right]\text{
.}$

The equilibrium vector of payoffs is

$\hat{\Pi}=\underaccent{\bar}{p},\quad\text{and}\quad\hat{\Phi}=\mu-\underaccent{\bar}{p}+c-\sqrt{\frac{c}{\kappa}}\text{
.}$

If $\mu-\sqrt{c/\kappa}\leq\underaccent{\bar}{p}$, the unique equilibrium is
given in Theorem 6.2, with $a=0$.

Because $\underaccent{\bar}{p}$ is only given implicitly, the condition that
demarcates the active search equilibrium is difficult to parse (since it
itself is in terms of the equilibrium object $\underaccent{\bar}{p}$). As a
result, it is helpful to supplement the theorem with the following
proposition.

######  Proposition 6.6.

For any search cost $c$, there is a $\tilde{\mu}\left(c\right)>0$ such that if
$\mu<\tilde{\mu}\left(c\right)$ the equilibrium does not involve active search
irrespective of $\kappa$. If $\mu>\tilde{\mu}\left(c\right)$, there exist
$\underaccent{\bar}{\kappa}\left(c,\mu\right)>0$ and
$\bar{\kappa}\left(c,\mu\right)<1/\left(4c\right)$ (with
$\bar{\kappa}>\underaccent{\bar}{\kappa}$) such that the equilibrium involves
active search if and only if
$\kappa\in\left[\underaccent{\bar}{\kappa},\bar{\kappa}\right]$.

Thus, we see that either $\mu$ is so low that there is no active search,
regardless of $\kappa$; or it is sufficiently high that there is active search
if and only if $\kappa$ lies in an intermediate region.

There are a number of interesting implications of this equilibrium.
Strikingly, despite the homogeneity of the market, and even though search is
not directed, the unique equilibrium exhibits price dispersion. Furthermore,
there may also be significant diversity of consumer behavior. Namely, the
empirical distribution over consumer valuations may be driven by varying
levels of learning by different consumers: some learn a lot at each firm while
others learn very little. The unique equilibrium; therefore, justifies a broad
spectrum of different behavior by consumers, even though they are identical.

####  6.2.1 Monopolistic Competition Comparative Statics

######  Proposition 6.7.

The lower bound for the price, $\underaccent{\bar}{p}$, is strictly increasing
in $c$. In the active-search region (if it exists), $\underaccent{\bar}{p}$ is
independent of $\mu$ and is (strictly) decreasing in $\kappa$ if and only if
$h\left(c,\kappa\right)\coloneqq\ln{\left(\sqrt{c\kappa}\right)}+2-2\sqrt{c\kappa}\leq(<)\
0$. Outside of the active-search region, $\underaccent{\bar}{p}$ is strictly
increasing in $\kappa$ and $\mu$.

A simple calculation (or a glance at its graph) reveals that $h$ is strictly
increasing in $\kappa$. Moreover, $\lim_{\kappa\searrow 0}h=-\infty$ and
$h\left(1/\left(4\kappa\right)\right)=\ln\left(1/2\right)+1>0$, so $h$ has a
unique root at some $\tilde{\kappa}\in\left(0,1/\left(4c\right)\right)$.

Because a firm’s profit is merely the lower bound for the support of the price
distribution, the following result is evident.

######  Corollary 6.8.

A firm’s profit is strictly increasing in $c$. In the active-search region, a
firm’s profit is independent of $\mu$ and is (strictly) decreasing in $\kappa$
if and only if $h\left(c,\kappa\right)\leq(<)\ 0$. Outside of the active-
search region, profits are strictly increasing in $\kappa$ and $\mu$.

A direct calculation reveals that a consumer’s welfare is strictly decreasing
in $c$. On the other hand, note that

$\hat{\Phi}^{\prime}\left(\kappa\right)=-\underaccent{\bar}{p}^{\prime}\left(\kappa\right)+\frac{1}{2\kappa}\sqrt{\frac{c}{\kappa}}\text{
,}$

and so if the price is decreasing in $\kappa$, a consumer’s welfare is
increasing in $\kappa$. Naturally, even if the price is increasing in
$\kappa$, a consumer’s welfare may still be increasing in $\kappa$, provided
the price increase is not too large. We have the following lemma.

######  Lemma 6.9.

A consumer’s payoff, $\hat{\Phi}$, is strictly decreasing in $c$. In the
active-search region, $\hat{\Phi}$ is independent of $\mu$ and is (strictly)
increasing in $\kappa$ if and only if $\kappa\ (<)\leq\kappa^{\prime}$, where
$\kappa^{\prime}\in\left(\tilde{\kappa},1/\left(4c\right)\right)$. Outside of
the active search region, a consumer’s payoff is (strictly) increasing in
$\kappa$ if and only if $\kappa$ is sufficiently low.

##  7 Who (if Anyone) Benefits From Hidden Prices?

One interpretation of the “hidden prices” version of the model is as a market
in which firms can introduce hidden fees late in the purchase. This issue has
come to the forefront of recent policy debates; and in the United States of
America alone, a number of bills have been proposed, or rules imposed,
recently to restrict drip pricing and other forms of hidden (additional)
prices.151515In their background section, Santana, Dallas, and Morwitz (2020)
include a nice discussion of such recent rules and proposed legislation.
Additional papers with similar analyses of the effect of hidden fees and
prices on consumer shopping behavior include Blake, Moshary, Sweeney, and
Tadelis (2018) and Bradley and Feldman (2020). The modal explanation for the
adverse effects of hidden fees is behavioral: and in particular price (and
tax) salience is often cited. Viz., consumers do not fully perceive the extra
price. In contrast, hidden prices affect behavior in our model
strategically–our consumers are fully rational and understand the
ramifications of the hold-up problem in which they find themselves. Our paper;
therefore, can provide another perspective on this debate, since we may
compare the various market outcomes (profits, consumer welfare, and prices)
when prices are observable before learning versus when they are not.

A priori one might suppose that hidden prices are to consumers’ detriment,
since it affects their learning adversely–a consumer understands that learning
that she loves the product may allow the firm to charge a high price to
exploit this enthusiasm. Furthermore, it also removes the ability of consumers
to react to price changes and thereby discipline firms. As we shortly
discover, this prediction is correct: consumers prefer observable fees, both
when there is a search market as well as when there is a single monopolistic
seller of the product. It is less clear which information regime firms prefer.
On one hand, firms might have more impunity to set higher prices since
consumers have less flexibility to react. However, consumers are strategic and
so may not learn as much as the firms would like when prices are hidden.

We begin our analysis with the monopoly case.

###  7.1 Hidden Versus Observable Prices With Monopoly

For simplicity, we assume that the consumer’s outside option is $0$. Recall
that with observable prices, the payoffs for the monopolist and a consumer
are, respectively,

$\Pi_{M}^{*}=\begin{cases}\mu-\frac{1}{4\kappa},\quad&\mu\geq\frac{3}{4\kappa}\\\
\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa},\quad&\frac{3}{4\kappa}\geq\mu\geq-\frac{1}{4\kappa}\\\
0,\quad&\mu\leq-\frac{1}{4\kappa}\end{cases},\quad\text{and}\quad\Phi_{M}^{*}=\begin{cases}\frac{1}{4\kappa},\quad&\mu\geq\frac{3}{4\kappa}\\\
\frac{\Pi_{M}^{*}}{2},&\frac{3}{4\kappa}\geq\mu\\\ \end{cases}\text{ .}$

In contrast, with hidden prices, the payoffs are

$\hat{\Pi}_{M}=\begin{cases}\underaccent{\bar}{p}_{M},\quad&\mu>0\\\
0,\quad&\mu\leq
0\end{cases},\quad\text{and}\quad\hat{\Phi}_{M}=\begin{cases}\kappa\left(\mu-\underaccent{\bar}{p}_{M}\right)^{2},\quad&\mu>0\\\
0,\quad&\mu\leq 0\end{cases}\text{ ,}$

where $\underaccent{\bar}{p}_{M}$ is given implicitly by

$\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}\right\\}}=\frac{\mu}{\underaccent{\bar}{p}_{M}}-1\text{
.}$

Obviously, if the prior is too low ($\mu\leq-1/\left(4\kappa\right)$) there is
no trade in either case. On the other hand, if
$-1/\left(4\kappa\right)<\mu\leq 0$, there is trade if and only if prices are
observed before learning. Thus, in that region of the parameter space, both
monopolist and consumers are strictly better off when prices are not hidden.

If the prior is high ($\mu\geq 3/\left(4\kappa\right)$), the consumer is
better off with observable prices if and only if

$\frac{1}{4\kappa}\geq\kappa\left(\mu-\underaccent{\bar}{p}_{M}\right)^{2}\quad\Leftrightarrow\quad\underaccent{\bar}{p}_{M}\geq\mu-\frac{1}{2\kappa}\text{
.}$

As we show in Appendix A.11, this always holds. The monopolist is better off
with observable prices if and only if
$\underaccent{\bar}{p}_{M}\leq\mu-1/\left(4\kappa\right)$. This must hold in
this region of the parameter space as well.

For intermediate priors ($3/\left(4\kappa\right)\geq\mu>0$), a consumer is
better off with observable prices if and only if

$\frac{\left(1+4\kappa\mu\right)^{2}}{64\kappa}\geq\kappa\left(\mu-\underaccent{\bar}{p}_{M}\right)^{2}\quad\Leftrightarrow\quad\underaccent{\bar}{p}_{M}\geq\frac{\mu}{2}-\frac{1}{8\kappa}\text{
,}$

which is always true. Finally, the monopolist is better off with observable
prices if and only if

$\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}\geq\underaccent{\bar}{p}_{M}\text{
,}$

which is also always true.

The following proposition sums up our stark conclusion:

######  Proposition 7.1.

Both monopolist and consumers prefer prices to be observable before learning.

Figure 4 depicts the monopolist’s profits with observable prices and hidden
prices as well as its price when prices are observable and the lower bound of
its distribution over prices when prices are hidden. In some sense, this
result is unsurprising. By allowing the monopolist the power to commit to its
price, consumers have more incentive to learn, since they know that they will
not be exploited.

Figure 4: Monopolist profit with observable prices (solid blue), monopolist
price with observable prices (dotted green), and monopolist profit and price
lower bound with hidden prices (solid orange) as functions of $\kappa$.

###  7.2 Hidden Versus Observable Prices With Monopolistic Competition

As we stated above, consumers prefer to observe prices before learning in the
search market with multiple firms. The intuition is analogous to that when
there is a single seller: they no longer have to worry about being exploited
and can therefore learn optimally as a result.

######  Proposition 7.2.

Consumer welfare when prices are observed before learning is strictly higher
than when prices are observed after learning.

Surprising, the result from the monopoly scenario does not always carry over
for firms. That is, it is possible that firms may actually prefer hidden
prices. This occurs when $\mu$ and $\kappa$ are sufficiently high (in
comparison to $c$). The explanation for this finding is as follows: if the
parameters are such that there is active search when prices are hidden, there
must be active search when prices are observed. Moreover, recall that when
there is active search, a firm’s profit is independent of both the prior
quality and the level of information frictions and is determined only by $c$,
which is quite low by assumption. In contrast, when prices are hidden (and
there is active search), $c$ must be so low (or $\mu$ and $\kappa$ so high)
that the lower bound of the equilibrium price, $\underaccent{\bar}{p}$ is
greater than $2c$, the profit in the observable prices set-up. Of course,
$\underaccent{\bar}{p}$ is precisely a firm’s profit when prices are hidden,
and so its profits are, therefore, higher.

When the parameters are such that consumers do not actively search in either
observation regime, we are back in the monopolist setting and so observable
prices are preferred by the firms. The third and final possibility is that the
equilibrium involves active search if and only if prices are observable. This
is a mix of the other two cases: if $\mu$ and $\kappa$ are sufficiently high,
firms prefer hidden prices, but not if they are too low.

(a) Case 1: $\mu>0$ and $c\leq\mu/4$.

(b) Case 2: $\mu>0$ and $\mu/3\geq c\geq\mu/4$.

(c) Case 3: $\mu>0$ and $c\geq\mu/3$.

Figure 5: Profit with observable prices (solid blue), price with observable
prices (dotted green), and profit and price lower bound with hidden prices
(solid orange) as functions of $\kappa$.

######  Proposition 7.3.

Firms prefer that consumers observe prices before learning if and only if
$\mu$ is sufficiently low.

Figure 5 depicts firms’ profits with observable prices and hidden prices as
well as their prices when prices are observable and the lower bound of the
market distribution over prices when prices are hidden.

##  8 Discussion

With the exception of Guo (2021), the existing literature that explores
sequential consumer search does not allow for flexible information acquisition
by consumers. Similarly, the existing rational inattention literature does not
tackle the dynamic problem of consumers searching and flexibly acquiring
information in sequence. We formulate and solve a model that incorporates
these aspects, which allows us to identify and highlight the vital differences
between search and information frictions.

Despite the ability of consumers to adapt their learning to the prices they
observe, higher search frictions benefit firms and lead to greater market
prices. Increased search costs make consumers more reluctant to leave firms,
making them less responsive to price increases, which firms can exploit. In
contrast, higher information frictions make consumers more responsive to price
increases, thereby decreasing market prices and potentially benefiting
consumers. We find that these relationships hold (for the most part) even when
consumers cannot observe prices before acquiring information: firms benefit at
the expense of consumers as search costs increase, whereas an increase in
information frictions may improve the lot of consumers.

Our analysis; therefore, suggests that mandated decreases in information
frictions (mandatory disclosure rules or transparency requirements, say) may
be to the market’s detriment. Moreover, our comparison of cases in which a
firm’s price is observed before or after information acquisition allows us to
provide input on policies banning hidden fees. We find that consumers are
always better off when they can observe prices before learning but that firms
may be hurt by the transparency, in spite of the hold-up problem that ensues
when prices are hidden.

## References

  * Ankirchner et al. (2015) Stefan Ankirchner, David Hobson, Philipp Strack, et al.  Finite, integrable and bounded time embeddings for diffusions.  _Bernoulli_ , 21(2):1067–1088, 2015.
  * Aumann et al. (1995) Robert J Aumann, Michael Maschler, and Richard E Stearns.  _Repeated games with incomplete information_.  MIT press, 1995.
  * Bar-Isaac et al. (2012) Heski Bar-Isaac, Guillermo Caruana, and Vicente Cuñat.  Information gathering externalities for a multi-attribute good.  _The Journal of Industrial Economics_ , 60(1):162–185, 2012.
  * Blake et al. (2018) Thomas Blake, Sarah Moshary, Kane Sweeney, and Steven Tadelis.  Price salience and product choice.  _Mimeo_ , 2018.
  * Bordalo et al. (2016) Pedro Bordalo, Nicola Gennaioli, and Andrei Shleifer.  Competition for attention.  _The Review of Economic Studies_ , 83(2):481–513, 2016.
  * Bradley and Feldman (2020) Sebastien Bradley and Naomi E Feldman.  Hidden baggage: Behavioral responses to changes in airline ticket tax disclosure.  _American Economic Journal: Economic Policy_ , 12(4):58–87, 2020.
  * Branco et al. (2012) Fernando Branco, Monic Sun, and J Miguel Villas-Boas.  Optimal search for product information.  _Management Science_ , 58(11):2037–2056, 2012\.
  * Branco et al. (2016) Fernando Branco, Monic Sun, and J Miguel Villas-Boas.  Too much information? information provision and search costs.  _Marketing Science_ , 35(4):605–618, 2016.
  * Condorelli and Szentes (2020) Daniele Condorelli and Balazs Szentes.  Information design in the holdup problem.  _Journal of Political Economy_ , 128(2):681–709, February 2020.
  * De Clippel et al. (2014) Geoffroy De Clippel, Kfir Eliaz, and Kareen Rozen.  Competing for consumer inattention.  _Journal of Political Economy_ , 122(6):1203–1234, 2014.
  * Dilmé (2019) Francesc Dilmé.  Pre-trade private investments.  _Games and Economic Behavior_ , 117:98–119, 2019.
  * Dogan and Hu (2018) Mustafa Dogan and Ju Hu.  Consumer search and optimal information.  _Mimeo_ , August 2018.
  * Ellison and Ellison (2009) Glenn Ellison and Sara Fisher Ellison.  Search, obfuscation, and price elasticities on the internet.  _Econometrica_ , 77(2):427–452, 2009.
  * Ellison (2016) Sara Fisher Ellison.  Price search and obfuscation: an overview of the theory and empirics.  _Handbook on the Economics of Retailing and Distribution_ , 2016.
  * Georgiadis and Szentes (2020) George Georgiadis and Balazs Szentes.  Optimal monitoring design.  _Econometrica_ , 88(5):2075–2107, 2020.
  * Gu and Wenzel (2014) Yiquan Gu and Tobias Wenzel.  Strategic obfuscation and consumer protection policy.  _The Journal of Industrial Economics_ , 62(4):632–660, 2014.
  * Guo (2021) Liang Guo.  Endogenous evaluation and sequential search.  _Marketing Science_ , 2021.
  * Hämäläinen (2018) Saara Hämäläinen.  Competitive search obfuscation.  _Journal of Economic Dynamics and Control_ , 97:38–63, 2018\.
  * Hermalin and Katz (2009) Benjamin E Hermalin and Michael L Katz.  Information and the hold-up problem.  _The RAND Journal of Economics_ , 40(3):405–423, 2009.
  * Janssen and Shelegia (2020) Maarten Janssen and Sandro Shelegia.  Beliefs and consumer search in a vertical industry.  _Journal of the European Economic Association_ , 18(5):2359–2393, 2020.
  * Kamenica and Gentzkow (2011) Emir Kamenica and Matthew Gentzkow.  Bayesian persuasion.  _The American Economic Review_ , 101(6):2590–2615, 2011.
  * Ke et al. (2016) T Tony Ke, Zuo-Jun Max Shen, and J Miguel Villas-Boas.  Search for information on multiple products.  _Management Science_ , 62(12):3576–3603, 2016\.
  * Lang (2019) Ruitian Lang.  Try before you buy: A theory of dynamic information acquisition.  _Journal of Economic Theory_ , 183:1057–1093, 2019.
  * Lau (2008) Stephanie Lau.  Information and bargaining in the hold-up problem.  _The RAND Journal of Economics_ , 39(1):266–282, 2008.
  * Liu and Dukes (2016) Lin Liu and Anthony Dukes.  Consumer search with limited product evaluation.  _Journal of Economics & Management Strategy_, 25(1):32–55, 2016.
  * Matějka and McKay (2012) Filip Matějka and Alisdair McKay.  Simple market equilibria with rationally inattentive consumers.  _American Economic Review_ , 102(3):24–29, 2012\.
  * Morris and Strack (2017) Stephen Morris and Philipp Strack.  The wald problem and the relation of sequential sampling and ex-ante information costs.  _Mimeo_ , 2017.
  * Nguyen and Tan (2019) Anh Nguyen and Teck Yong Tan.  Information control in the hold-up problem.  _The RAND Journal of Economics_ , 50(4):768–786, 2019.
  * Pease (2018) Marilyn Pease.  Shopping for information: Consumer learning with optimal pricing and product design.  _Mimeo_ , 2018.
  * Petrikaitė (2018) Vaiva Petrikaitė.  Consumer obfuscation by a multiproduct firm.  _The RAND Journal of Economics_ , 49(1):206–223, 2018.
  * Rao (2021) Neel Rao.  Search equilibrium with unobservable investment.  _Mimeo_ , 2021.
  * Ravid et al. (2020) Doron Ravid, Anne-Katrin Roesler, and Balázs Szentes.  Learning before trading: on the inefficiency of ignoring free information.  _Mimeo_ , 2020.
  * Richards et al. (2019) Timothy J Richards, Gordon J Klein, Celine Bonnet, and Zohra Bouamra-Mechemache.  Strategic obfuscation and retail pricing.  _Review of Industrial Organization_ , pages 1–31, 2019.
  * Roesler and Szentes (2017) Anne-Katrin Roesler and Balazs Szentes.  Buyer-optimal learning and monopoly pricing.  _American Economic Review_ , 107(7), 7 2017.
  * Root (1969) David H Root.  The existence of certain stopping times on brownian motion.  _The Annals of Mathematical Statistics_ , 40(2):715–718, 1969.
  * Santana et al. (2020) Shelle Santana, Steven K Dallas, and Vicki G Morwitz.  Consumer reactions to drip pricing.  _Marketing Science_ , 39(1):188–210, 2020.
  * Sun (2006) Yeneng Sun.  The exact law of large numbers via fubini extension and characterization of insurable risks.  _Journal of Economic Theory_ , 126(1):31–69, 2006\.
  * Winkler (1988) Gerhard Winkler.  Extreme points of moment sets.  _Mathematics of Operations Research_ , 13(4):581–587, 1988.
  * Wolinsky (1986) Asher Wolinsky.  True Monopolistic Competition as a Result of Imperfect Information.  _The Quarterly Journal of Economics_ , 101(3):493–511, 08 1986.
  * Yang (2019) Kai Hao Yang.  Buyer-optimal information with nonlinear technology.  _Mimeo_ , June 2019.

##  Appendix A Omitted Proofs and Derivations

###  A.1 Proposition 5.2 Proof

###### Proof.

By direct computation, if
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)>\mu\geq-1/\left(4\kappa\right)$, the
market price, a firm’s payoff and a consumer’s payoff are strictly increasing
in $\mu$. Analogously, if $\mu\geq 2\sqrt{c/\kappa}-1/\left(4\kappa\right)$,
the market price and a firm’s payoff are unaffected by $\mu$, but a consumer’s
payoff is strictly increasing in $\mu$. ∎

###  A.2 Proposition 5.3 Proof

###### Proof.

Recall that there is active search and information acquisition in equilibrium
provided

$\mu\geq 2\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\text{ .}$

Rearranging, we get a number of cases.

Case 1: If $\mu\geq 3/\left(4\kappa\right)$, Inequality A.2 holds (since
$0<c<1/\left(4\kappa\right)$).

Case 2: If $-1/\left(4\kappa\right)\leq\mu<3/\left(4\kappa\right)$, Inequality
A.2 holds if and only if

$0<c<\frac{\left(1+4\kappa\mu\right)^{2}}{64\kappa}\text{ .}$

Case 3: If $\mu\leq-1/\left(4\kappa\right)$, Inequality A.2 does not hold.

In the active-search region, a simple calculation reveals that $\Phi^{*}_{l}$
is strictly decreasing in $c$. In this region, $\Pi^{*}_{l}$ and $p_{l}$ are
strictly increasing in $c$. Trivially, these equilibrium objects are
independent of $c$ in the region without active search. ∎

###  A.3 Proposition 5.4 Proof

###### Proof.

In the active-search region,

$\frac{\partial{\Phi_{l}^{*}}}{\partial{\kappa}}=-\frac{1}{4\kappa^{2}}+\frac{1}{\kappa}\sqrt{\frac{c}{\kappa}}\geq\
(>)\ 0\quad\Leftrightarrow\quad\kappa\geq\ (>)\ \frac{1}{16c}\text{ .}$

In this region, $\Pi_{l}^{*}$ is unchanging in $\kappa$, and $p_{l}$ is
strictly decreasing in $\kappa$. In the region without active search,

$\frac{\partial{\Phi_{m}^{*}}}{\partial{\kappa}}=\frac{\mu^{2}}{4}-\frac{1}{64\kappa^{2}}\geq\
(>)\ 0\quad\Leftrightarrow\quad\kappa\geq\ (>)\
\left|\frac{1}{4\mu}\right|\text{ .}$

Since $\Pi_{m}^{*}=2\Phi_{m}^{*}$, the effect of a change of $\kappa$ on a
firm’s profit is identical to its effect on consumer welfare in this (no-
search) region. $p_{m}$ is strictly decreasing in $\kappa$.

Rearranging Equation A.2, we observe that there are four cases to exhaust:

Case 1: $\mu>0$ and $0<c\leq\mu/4$. Here, $\kappa<1/\left(4c\right)$ implies
inequality A.2. Consequently,

$\operatorname{sgn}{\Phi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}-1,\quad&0<\kappa<\frac{1}{16c}\\\
0,\quad&\kappa=\frac{1}{16c}\\\
1,\quad&\frac{1}{16c}<\kappa<\frac{1}{4c}\end{cases},\quad\text{and}\quad\operatorname{sgn}{\Pi^{*^{\prime}}\left(\kappa\right)}=0\text{
.}$

Case 2: $\mu>0$ and $\mu/4<c<\mu/3$. Here, A.2 holds either if

$0<\kappa\leq\frac{8c-\mu-4\sqrt{\left(4c-\mu\right)c}}{4\mu^{2}}\eqqcolon\underaccent{\bar}{\kappa},\quad\text{or}\quad\bar{\kappa}\coloneqq\frac{8c-\mu+4\sqrt{\left(4c-\mu\right)c}}{4\mu^{2}}\leq\kappa<\frac{1}{4c}\text{
.}$

Consequently,

$\operatorname{sgn}{\Phi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}-1,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
-1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4\mu}\\\
0,\quad&\kappa=\frac{1}{4\mu}\\\ 1,\quad&\frac{1}{4\mu}<\kappa<\bar{\kappa}\\\
1,\quad&\bar{\kappa}\leq\kappa<\frac{1}{4c}\end{cases},\quad\text{and}\quad\operatorname{sgn}{\Pi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}0,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
-1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4\mu}\\\
0,\quad&\kappa=\frac{1}{4\mu}\\\ 1,\quad&\frac{1}{4\mu}<\kappa<\bar{\kappa}\\\
0,\quad&\bar{\kappa}\leq\kappa<\frac{1}{4c}\end{cases}\text{ .}$

Case 3: $\mu\geq 0$ and $c\geq\mu/3$. Here, A.2 holds if
$0<\kappa\leq\underaccent{\bar}{\kappa}$. Accordingly,

$\operatorname{sgn}{\Phi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}-1,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
-1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4\mu}\\\
0,\quad&\kappa=\frac{1}{4\mu}\\\
1,\quad&\frac{1}{4\mu}<\kappa<\frac{1}{4c}\end{cases},\quad\text{and}\quad\operatorname{sgn}{\Pi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}0,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
-1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4\mu}\\\
0,\quad&\kappa=\frac{1}{4\mu}\\\
1,\quad&\frac{1}{4\mu}<\kappa<\frac{1}{4c}\end{cases}\text{ .}$

Case 4: $\mu<0$. Again, A.2 holds if $0<\kappa\leq\underaccent{\bar}{\kappa}$.
Accordingly,

$\operatorname{sgn}{\Phi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}-1,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4c}\end{cases},\quad\text{and}\quad\operatorname{sgn}{\Pi^{*^{\prime}}\left(\kappa\right)}=\begin{cases}0,\quad&0<\kappa<\underaccent{\bar}{\kappa}\\\
1,\quad&\underaccent{\bar}{\kappa}\leq\kappa<\frac{1}{4c}\end{cases}\text{ .}$

∎

###  A.4 Lemma 6.1 Proof

###### Proof.

Let us begin by establishing that if $\mu>a$ there exist no pure-strategy
equilibria. For convenience, we assume that there is a single representative
consumer. Suppose for the sake of contradiction that there is an equilibrium
in which the seller sets a deterministic $p$. Recall that if
$\mu\in\left(a+p-1/\left(4\kappa\right),a+p+1/\left(4\kappa\right)\right)$,
the consumer’s learning has support
$\left\\{a+p-1/\left(4\kappa\right),a+p+1/\left(4\kappa\right)\right\\}$.
Otherwise, the consumer does not learn and either buys with certainty (if
$\mu$ is sufficiently high) or takes her outside option with certainty. That
leaves us with three cases.

Suppose first that $p\geq\mu-a+1/\left(4\kappa\right)$, so that the consumer
does not learn and takes her outside option (on path). The seller’s profit is
$0$. Evidently, it can deviate by charging some price $\tilde{p}\in(0,\mu-a)$,
which yields it profit $\tilde{p}>0$.

Second, suppose that
$p\in\left(\mu-a-1/\left(4\kappa\right),\mu-a+1/\left(4\kappa\right)\right)$.
Conditional on the consumer purchasing, the seller’s profit is $p$. Suppose it
deviates to a price $\tilde{p}\coloneqq p+1/\left(8\kappa\right)$. Evidently,

$u_{H}-\tilde{p}=a+p+\frac{1}{4\kappa}-\left(p+\frac{1}{8\kappa}\right)=a+\frac{1}{8\kappa}>a\text{
,}$

so the consumer still purchases with the same probability at a strictly higher
price, yielding a strictly higher profit for the seller.

Third, suppose that $p\leq\mu-a-1/\left(4\kappa\right)$, so that the consumer
does not learn and purchases from the seller. Evidently, the firm’s profit is
$p$. However, it can deviate to some $\tilde{p}=\mu-a-\epsilon$, which is
strictly higher than $p$ and $0$ provided
$\epsilon<\min\left\\{1/\left(4\kappa\right),\mu-a\right\\}$.

We have established the necessity of $\mu\leq a$ for the existence of a pure-
strategy equilibrium. Now let us deduce the no-trade result.

Let $\mu\leq a$, and suppose for the sake of contradiction that there exists a
pure-strategy equilibrium in which trade occurs. Consequently, either
$p\in\left(\mu-a-1/\left(4\kappa\right),\mu-a+1/\left(4\kappa\right)\right)$
or $p\leq\mu-a-1/\left(4\kappa\right)$. The latter is clearly impossible,
since the seller is making negative profits. In the first case, we must have
$p\leq\mu-a+1/\left(4\kappa\right)$. Clearly, if
$\mu-a+1/\left(4\kappa\right)\leq 0$, there are no pure-strategy equilibria
with trade. Thus, let us assume $\mu-a+1/\left(4\kappa\right)>0$. If there is
trade (with positive probability), then $0\leq
p<\mu-a+1/\left(4\kappa\right)$. But then the firm can raise its price
slightly, thereby increasing its profit.

Finally, let us construct pure strategy equilibria in which there is no trade.
Evidently, we must have $p\geq\mu-a+1/\left(4\kappa\right)$. The consumer will
not learn and so (on-path) the consumer will take her outside option. On path,
the firm gets $0$ and any deviation would yield it at most $0$ since $\mu\leq
a$. ∎

###  A.5 Theorem 6.2 Proof

###### Proof.

Again, for convenience, we assume that there is a single representative
consumer. From Lemma 6.1, we know that there exist no pure strategy equilibria
in this region. As noted in the main part of the text, given the consumer’s
equilibrium distribution over values, $F$, the monopolist’s profit is

$\hat{\Pi}\left(p\right)=p\left(1-F\left(p+a\right)\right)\text{ ,}$

which must equal some constant $\lambda\geq 0$ for all $p$ in the support of
its mixed strategy since it is mixing. Denote the support of its mixed
strategy by $\left[\underaccent{\bar}{p},\bar{p}\right]$, where
$\underaccent{\bar}{p}\geq 0$. Given the monopolist’s mixed strategy over
prices $G$, the consumer’s distribution ($F$) must solve

$\max_{F}\int_{-\infty}^{\infty}\left\\{\int_{\underaccent{\bar}{p}}^{x-a}\left(x-p\right)dG\left(p\right)+\int_{x-a}^{\bar{p}}adG\left(p\right)-\kappa\left(x-\mu\right)^{2}\right\\}dF\left(x\right)\text{
.}$

It is also helpful to write down the consumer’s payoff as a function of the
realized value:

$V\left(x\right)=\begin{cases}a-\kappa\left(x-\mu\right)^{2},\quad&x<a+\underaccent{\bar}{p}\\\
a\left(1-G\left(x-a\right)\right)+\int_{\underaccent{\bar}{p}}^{x-a}\left(x-p\right)dG\left(p\right)-\kappa\left(x-\mu\right)^{2},\quad&a+\underaccent{\bar}{p}\leq
x\leq a+\bar{p}\\\
x-\mathbb{E}_{G}\left[p\right]-\kappa\left(x-\mu\right)^{2},\quad&a+\bar{p}\leq
x\\\ \end{cases}\text{ .}$

Concavifying $V$ yields the consumer’s optimal distribution(s). Our first pair
of claims ensures that the upper and lower bounds of the support of any
equilibrium distributions over values and prices differ from each other
(respectively) only by the value of the outside option.

######  Claim A.1.

In any equilibrium, the upper bound of the support of the consumer’s
distribution, $\bar{x}$, satisfies $\bar{x}=a+\bar{p}$.

###### Proof.

It is easy to see that we cannot have $\bar{x}<a+\bar{p}$. Indeed, the
monopolist’s profit from any price $p>\bar{x}-a$ is $0$, whereas it can always
obtain a strictly positive profit by setting price
$p^{\prime}\in\left(0,\mu-a\right)$. Can we have $\bar{x}>\bar{p}+a$? In this
case the consumer’s optimal distribution above $\bar{p}+a$ must consist of a
single mass point, since information is costly. However, then we have
$F\left(\bar{p}+a\right)=F\left(\bar{p}+a+\eta\right)$ for all (sufficiently
small) strictly positive $\eta$, contradicting the optimality of the
monopolist’s distribution over prices. ∎

######  Claim A.2.

In any equilibrium, the lower bound of the support of the consumer’s
distribution, $\underaccent{\bar}{x}$, satisfies
$\underaccent{\bar}{x}=a+\underaccent{\bar}{p}$.

###### Proof.

That $\underaccent{\bar}{x}\leq a+\underaccent{\bar}{p}$ is also
straightforward. Otherwise, the monopolist obtains a strictly higher profit
from $\underaccent{\bar}{p}+\epsilon$ than from $\underaccent{\bar}{p}$, a
violation. Can we have $\underaccent{\bar}{x}<\underaccent{\bar}{p}+a$? In
this case the consumer’s optimal distribution below $\bar{p}+a$ must consist
of a single mass point, since information is costly. Moreover, $F$ must have
no support on $\left(\underaccent{\bar}{x},\tilde{x}\right)$, where
$\tilde{x}>\underaccent{\bar}{p}+a$ since $V$ is strictly concave on
$\left(-\infty,\underaccent{\bar}{p}+a\right)$. However, this means that the
monopolist strictly prefers price $\underaccent{\bar}{p}+\eta$ to
$\underaccent{\bar}{p}$ since demand is locally perfectly inelastic in a
neighborhood of $\underaccent{\bar}{p}$. ∎

We now establish that the consumer’s distribution over values has full
support.

######  Claim A.3.

In any equilibrium the distributions chosen by the consumer and the monopolist
have full support on $\left[\underaccent{\bar}{x},\bar{x}\right]$ and
$\left[\underaccent{\bar}{p},\bar{p}\right]$, respectively.

###### Proof.

Suppose that there is a gap in the support of the consumer’s distribution,
i.e., $F\left(x_{1}\right)=F\left(x_{2}\right)$ for some $x_{2}>x_{1}$. Then,
$\hat{\Pi}\left(p\right)<\hat{\Pi}\left(p_{2}\right)$ for all
$p\in\left(p_{1},p_{2}\right)$, where $p_{i}=x_{i}-a$. Accordingly, the firm
does not charge any price $p$ in this interval, i.e.,
$G\left(p_{1}\right)=G^{-}\left(p_{2}\right)$.161616We define
$G^{-}\left(p\right)\coloneqq\sup_{w<p}G\left(w\right)$. Then,

$V\left(x_{1}\right)=a\left(1-G\left(p_{1}\right)\right)+\int_{\underaccent{\bar}{p}}^{p_{1}}\left(x_{1}-p\right)dG\left(p\right)-\kappa\left(x_{1}-\mu\right)^{2}\text{
,}$

and

$\begin{split}V\left(x_{2}\right)&=a\left(1-G\left(p_{2}\right)\right)+\int_{\underaccent{\bar}{p}}^{p_{2}}\left(x_{2}-p\right)dG\left(p\right)-\kappa\left(x_{2}-\mu\right)^{2}\\\
&=a\left(1-G\left(p_{1}\right)\right)+\int_{\underaccent{\bar}{p}}^{p_{1}}\left(x_{2}-p\right)dG\left(p\right)-\kappa\left(x_{2}-\mu\right)^{2}\\\
\end{split}{.}$

By the strict convexity of the cost function, $F$ cannot be optimal for the
consumer, who could strictly benefit by having value
$x^{\prime}\coloneqq\lambda x_{1}+\left(1-\lambda\right)x_{2}$ for any
$\lambda\in\left(0,1\right)$ in the support of $F$, rather than $x_{1}$ and
$x_{2}$.

The analysis of the previous paragraph also implies that the monopolist’s
distribution can have no gaps in its support. ∎

The next claim establishes that the monopolist’s distribution over prices may
not have mass points and neither does the consumer’s distribution over values
except possibly at the upper bound of its support.

######  Claim A.4.

In any equilibrium the distribution chosen by the monopolist has no mass
points and the distribution chosen by the consumer has no mass points except
possibly at $\bar{x}$.

###### Proof.

Suppose for the sake of contradiction that $F$ has a mass point at some value
$\hat{x}<\bar{x}$ and let $\hat{p}=\hat{x}+a$. Evidently,

$\hat{\Pi}\left(\hat{p}-\epsilon\right)=\left(\hat{p}-\epsilon\right)\left(1-F\left(\hat{x}-\epsilon\right)\right)>\left(\hat{p}+\epsilon\right)\left(1-F\left(\hat{x}+\epsilon\right)\right)=\hat{\Pi}\left(\hat{p}+\epsilon\right)\text{
,}$

for sufficiently small $\epsilon>0$. This implies that the support of $G$ has
a gap, contradicting our previous claim.

Because the consumer’s distribution over values has full support, the
consumer’s payoff as a function of value $x$, $V\left(x\right)$, must be
affine. The distribution over prices cannot have a mass point, since otherwise
$V$ would have a discrete change in its slope induced by the mass point. ∎

At last we may derive the unique equilibrium. Setting
$\hat{\Pi}\left(p\right)=\lambda$, we obtain

$F\left(x\right)=1-\frac{\lambda}{x-a}\text{ .}$

Since $F$ has no mass points on $\left[\underaccent{\bar}{x},\bar{x}\right)$,
$\lambda=\underaccent{\bar}{x}-a$. Moreover, since
$\mathbb{E}_{F}\left[x\right]=\mu$, we have

$1+\ln{\left\\{\frac{\bar{x}-a}{\underaccent{\bar}{x}-a}\right\\}}=\frac{\mu-a}{\underaccent{\bar}{x}-a}\text{
.}$

Because $V$ is affine,

$a\left(1-G\left(x-a\right)\right)+\int_{\underaccent{\bar}{x}-a}^{x-a}\left(x-p\right)dG\left(p\right)-\kappa\left(x-\mu\right)^{2}=\alpha
x+\beta\text{ ,}\ \forall\
x\in\left[\underaccent{\bar}{x},\bar{x}\right]\text{ ,}$

where $\alpha,\beta\in\mathbb{R}$. We may differentiate both sides and do some
algebra, which yields

$G\left(p\right)=2\kappa\left(a+p-\mu\right)+\alpha\text{ ,}$

and substituting this back in to Equation A.5, we obtain

$\alpha=2\kappa\left(\mu-\underaccent{\bar}{x}\right)\text{ ,}$

and

$\beta=\kappa\left(\underaccent{\bar}{x}^{2}-\mu^{2}\right)+a\text{ .}$

Since $G\left(\bar{p}\right)=1$,

$\alpha+2\kappa\left(\bar{p}+a-\mu\right)=1\text{ .}$

Thus, combining Equations A.5 and A.5, we get

$\bar{p}=\underaccent{\bar}{p}+\frac{1}{2\kappa}\text{ .}$

Substituting in for $\bar{p}$ in Equation A.5 via Equation A.5 and using the
fact that $\underaccent{\bar}{x}=a+\underaccent{\bar}{p}$, we have

$\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}=\frac{\mu-a}{\underaccent{\bar}{p}}-1\text{
.}$

The firm’s profit is clearly $\underaccent{\bar}{p}$ since if it charges that
price it is purchased from with certainty, and the consumer’s payoff is just
$V\left(\mu\right)$. ∎

###  A.6 Lemma 6.3 Proof

###### Proof.

Let us define the function
$\iota\left(\kappa,\underaccent{\bar}{p},\mu\right)$ as

$\iota\left(\underaccent{\bar}{p}_{M},\kappa,\mu\right)\coloneqq\underaccent{\bar}{p}_{M}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}\right\\}}+\underaccent{\bar}{p}_{M}-\mu+a\text{
.}$

Evidently, $\iota$ is twice continuously differentiable in each of its
arguments on $\mathbb{R}_{++}^{3}$.

######  Claim A.5.

$\iota$ has a strictly positive root if and only if $\mu>a$. This root,
$\underaccent{\bar}{p}_{M}$, is the unique positive root and lies in the
interval $\underaccent{\bar}{p}_{M}\in\left(0,\mu-a\right)$.

###### Proof.

Directly,

$\iota^{\prime}\left(\underaccent{\bar}{p}_{M}\right)=-\frac{1}{2\kappa\underaccent{\bar}{p}_{M}+1}+\ln\left\\{\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}+1\right\\}+1,\quad\text{and}\quad\iota^{\prime\prime}\left(\underaccent{\bar}{p}_{M}\right)=-\frac{1}{\underaccent{\bar}{p}_{M}\left(2\kappa\underaccent{\bar}{p}_{M}+1\right)^{2}}\text{
.}$

Evidently, $\iota^{\prime\prime}\left(\underaccent{\bar}{p}_{M}\right)<0$ for
all $\underaccent{\bar}{p}_{M}>0$. We claim that $\iota$ is strictly
increasing in $\underaccent{\bar}{p}_{M}$ for $\underaccent{\bar}{p}_{M}>0$.
For all such $\underaccent{\bar}{p}_{M}$,
$\iota^{\prime}\left(\underaccent{\bar}{p}_{M}\right)$ is strictly decreasing
in $\underaccent{\bar}{p}_{M}$. This, plus the fact that
$\lim_{\underaccent{\bar}{p}_{M}\to\infty}\iota^{\prime}\left(\underaccent{\bar}{p}_{M}\right)=1$,
implies that $\iota$ is strictly increasing in $\underaccent{\bar}{p}_{M}$ for
all $\underaccent{\bar}{p}_{M}>0$.

Then, we see that as $\underaccent{\bar}{p}_{M}\searrow 0$, $\iota\searrow
a-\mu$, which is strictly negative if and only if $\mu>a$; and as
$\underaccent{\bar}{p}_{M}\nearrow\mu-a$,
$\iota\nearrow\left(\mu-a\right)\ln\left\\{1+1/\left(2\kappa\left(\mu-a\right)\right)\right\\}$,
which is strictly positive if and only if $\mu>a$. Thus, if $\mu>a$,
$\iota\left(\underaccent{\bar}{p}_{M}\right)$ has a unique positive root at
some $\underaccent{\bar}{p}_{M}\in\left(0,\mu-a\right)$. ∎

Directly, $\iota^{\prime}\left(\mu\right)=-1$ and

$\iota^{\prime}\left(\kappa\right)=-\frac{\underaccent{\bar}{p}_{M}}{2\underaccent{\bar}{p}_{M}\kappa^{2}+\kappa}<0\text{
;}$

and so by the implicit function theorem,
$\underaccent{\bar}{p}_{M}^{\prime}\left(\mu\right)>0$ and
$\underaccent{\bar}{p}_{M}^{\prime}\left(\kappa\right)>0$. ∎

###  A.7 Lemma 6.4 Proof

###### Proof.

Recall that the monopolist’s profit is just the lower bound for the price,
$\underaccent{\bar}{p}_{M}$, and so its profit is strictly increasing in
$\kappa$ and $\mu$.

The consumer’s payoff is
$\hat{\Phi}=\kappa\left(\mu-a-\underaccent{\bar}{p}_{M}\right)^{2}+a$.
Accordingly,

$\hat{\Phi}^{\prime}\left(\mu\right)=2\kappa\left(\mu-a-\underaccent{\bar}{p}_{M}\right)\left(1-\underaccent{\bar}{p}_{M}^{\prime}\left(\mu\right)\right)\text{
.}$

Then,

$\underaccent{\bar}{p}_{M}^{\prime}\left(\mu\right)=\frac{1}{-\frac{1}{2\kappa\underaccent{\bar}{p}_{M}+1}+\ln\left\\{\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}+1\right\\}+1}=\frac{z+1}{\left(z+1\right)\ln\left\\{\frac{z+1}{z}\right\\}+z}\text{
,}$

where $z\coloneqq 2\underaccent{\bar}{p}_{M}\kappa$. Moreover, by the
definition of $\underaccent{\bar}{p}_{M}$

$2\kappa\left(\mu-a-\underaccent{\bar}{p}_{M}\right)=z\ln{\left\\{\frac{z+1}{z}\right\\}}\text{
.}$

Consequently, $\hat{\Phi}^{\prime}\left(\mu\right)$ reduces to

$\hat{\Phi}^{\prime}\left(\mu\right)=z\ln{\left\\{\frac{z+1}{z}\right\\}}\left(1-\frac{z+1}{\left(z+1\right)\ln\left\\{\frac{z+1}{z}\right\\}+z}\right)\text{
,}$

which is strictly positive for all $z>0$.

Similarly,

$\hat{\Phi}^{\prime}\left(\kappa\right)=\left(\mu-a-\underaccent{\bar}{p}_{M}\right)^{2}-2\kappa\left(\mu-a-\underaccent{\bar}{p}_{M}\right)\underaccent{\bar}{p}_{M}^{\prime}\left(\kappa\right)\text{
,}$

which is positive if and only if

$\kappa\left(\mu-a-\underaccent{\bar}{p}_{M}\right)-2\kappa^{2}\underaccent{\bar}{p}_{M}^{\prime}\left(\kappa\right)\geq
0\text{ ,}$

since $\mu-a>\underaccent{\bar}{p}_{M}$. Then,

$\underaccent{\bar}{p}_{M}^{\prime}\left(\kappa\right)=\frac{\frac{\underaccent{\bar}{p}_{M}}{2\underaccent{\bar}{p}_{M}\kappa^{2}+\kappa}}{-\frac{1}{2\kappa\underaccent{\bar}{p}_{M}+1}+\ln\left\\{\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}+1\right\\}+1}=\left(\frac{\underaccent{\bar}{p}_{M}}{\kappa}\right)\frac{1}{\left(z+1\right)\ln\left\\{\frac{z+1}{z}\right\\}+z}\text{
,}$

where again $z\coloneqq 2\underaccent{\bar}{p}_{M}\kappa$. Inequality A.7 is,
therefore, equivalent to

$\frac{1}{2}\ln{\left\\{\frac{z+1}{z}\right\\}}-\frac{1}{\left(z+1\right)\ln\left\\{\frac{z+1}{z}\right\\}+z}\geq
0\text{ ,}$

which is equivalent to requiring that
$\kappa\underaccent{\bar}{p}_{M}\lesssim.337$. ∎

###  A.8 Proposition 6.6 Proof

###### Proof.

First, we show that $\underaccent{\bar}{p}_{M}$ is strictly concave in
$\kappa$ and $\underaccent{\bar}{p}$ is strictly convex in $\kappa$.

######  Claim A.6.

$\underaccent{\bar}{p}_{M}$ is strictly concave in $\kappa$.

###### Proof.

By the implicit function theorem (and dropping the subscript $M$),

$\underaccent{\bar}{p}^{\prime\prime}\left(\kappa\right)=\frac{-\iota_{\underaccent{\bar}{p}}^{2}\iota_{\kappa\kappa}+2\iota_{\underaccent{\bar}{p}}\iota_{\kappa}\iota_{\kappa\underaccent{\bar}{p}}-\iota_{\kappa}^{2}\iota_{\underaccent{\bar}{p}\underaccent{\bar}{p}}}{\iota_{\underaccent{\bar}{p}}^{3}}\text{
.}$

Since $\iota_{\underaccent{\bar}{p}}>0$ this is strictly negative if and only
if

$-\iota_{\underaccent{\bar}{p}}^{2}\iota_{\kappa\kappa}+2\iota_{\underaccent{\bar}{p}}\iota_{\kappa}\iota_{\kappa\underaccent{\bar}{p}}-\iota_{\kappa}^{2}\iota_{\underaccent{\bar}{p}\underaccent{\bar}{p}}<0\text{
.}$

Let us go through each term one by one. First,

$-\iota_{\underaccent{\bar}{p}}^{2}\iota_{\kappa\kappa}=-\left(\frac{z}{z+1}+\ln\left\\{\frac{1}{z}+1\right\\}\right)^{2}\frac{\underaccent{\bar}{p}\left(2z+1\right)}{\kappa^{2}\left(z+1\right)^{2}}\text{
,}$

where $z\coloneqq 2\kappa\underaccent{\bar}{p}$. Second,

$2\iota_{\underaccent{\bar}{p}}\iota_{\kappa}\iota_{\kappa\underaccent{\bar}{p}}=2\left(\frac{z}{z+1}+\ln\left\\{\frac{1}{z}+1\right\\}\right)\frac{\underaccent{\bar}{p}}{\kappa^{2}}\left(\frac{1}{\left(z+1\right)^{3}}\right)\text{
.}$

Third,

$-\iota_{\kappa}^{2}\iota_{\underaccent{\bar}{p}\underaccent{\bar}{p}}=\frac{\underaccent{\bar}{p}}{\kappa^{2}}\left(\frac{1}{\left(z+1\right)^{4}}\right)$

Summing these together (and dividing out by positive terms), Inequality A.11
holds if and only if

$2\ln\left(\frac{1}{z}+1\right)\left(1-2z\right)+\frac{2z+1}{z+1}\left(1-z\right)-\left(\ln\left(\frac{1}{z}+1\right)\right)^{2}\left(2z+1\right)<0\text{
,}$

which holds for all strictly positive $z$. ∎

######  Claim A.7.

$\underaccent{\bar}{p}$ is strictly convex in $\kappa$.

###### Proof.

Let us define the function
$\varphi\left(\kappa,\underaccent{\bar}{p},c\right)$ as

$\varphi\left(\kappa,\underaccent{\bar}{p},c\right)\coloneqq\underaccent{\bar}{p}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}-\sqrt{\frac{c}{\kappa}}\text{
.}$

Evidently, $\varphi$ is twice continuously differentiable in each of its
variables on $\mathbb{R}_{++}^{3}$.

We shall break protocol slightly in this proof and use results that are
derived (not much) later on in the paper (that $\varphi$ is strictly
increasing in $\underaccent{\bar}{p}$ for $\underaccent{\bar}{p}>0$, which we
establish in Section A.9). Mirroring the proof of the previous claim, we need
to show that

$-\varphi_{\underaccent{\bar}{p}}^{2}\varphi_{\kappa\kappa}+2\varphi_{\underaccent{\bar}{p}}\varphi_{\kappa}\varphi_{\kappa\underaccent{\bar}{p}}-\varphi_{\kappa}^{2}\varphi_{\underaccent{\bar}{p}\underaccent{\bar}{p}}>0\text{
.}$

Again, let us go through each term one by one. First,

$\begin{split}-\varphi_{\underaccent{\bar}{p}}^{2}\varphi_{\kappa\kappa}&=-\left(\ln{\left\\{1+\frac{1}{x}\right\\}}-\frac{1}{1+x}\right)^{2}\left(\frac{x\left(2x+1\right)}{2\left(1+x\right)^{2}\kappa^{3}}-\frac{3\sqrt{c\kappa}}{4\kappa^{3}}\right)\\\
&=-\left(\ln{\left\\{1+\frac{1}{x}\right\\}}-\frac{1}{1+x}\right)^{2}\left(\frac{x\left(2x+1\right)}{2\left(1+x\right)^{2}\kappa^{3}}-\frac{3x\ln{\left\\{1+\frac{1}{x}\right\\}}}{8\kappa^{3}}\right)\end{split}\text{
,}$

where $x\coloneqq 2\kappa\underaccent{\bar}{p}$ and since we must have

$\sqrt{c\kappa}=\frac{x}{2}\ln{\left\\{1+\frac{1}{x}\right\\}}\text{ .}$

Second,

$\begin{split}2\varphi_{\underaccent{\bar}{p}}\varphi_{\kappa}\varphi_{\kappa\underaccent{\bar}{p}}&=-2\left(\ln{\left\\{1+\frac{1}{x}\right\\}}-\frac{1}{1+x}\right)\left(\frac{\sqrt{c\kappa}}{2\kappa^{2}}-\frac{x}{2\left(x+1\right)\kappa^{2}}\right)\left(\frac{1}{\kappa\left(x+1\right)^{2}}\right)\\\
&=-2\left(\ln{\left\\{1+\frac{1}{x}\right\\}}-\frac{1}{1+x}\right)\left(\frac{x\ln{\left\\{1+\frac{1}{x}\right\\}}}{4\kappa^{2}}-\frac{x}{2\left(x+1\right)\kappa^{2}}\right)\left(\frac{1}{\kappa\left(x+1\right)^{2}}\right)\\\
\end{split}\text{.}$

Third,

$\begin{split}-\varphi_{\kappa}^{2}\varphi_{\underaccent{\bar}{p}\underaccent{\bar}{p}}&=\left(\frac{x\ln{\left\\{1+\frac{1}{x}\right\\}}}{4\kappa^{2}}-\frac{x}{2\left(x+1\right)\kappa^{2}}\right)^{2}\left(\frac{1}{\underaccent{\bar}{p}}\right)\frac{1}{\left(1+x\right)^{2}}\end{split}\text{.}$

Summing these together (and dividing out by positive terms), Inequality A.8
holds if and only if

$3\ln^{3}\left(\frac{1}{x}+1\right)\left(x+1\right)^{3}+\ln^{2}\left(\frac{1}{x}+1\right)\left(x+1\right)\left(-14x-13\right)+19\ln\left(\frac{1}{x}+1\right)\left(x+1\right)-8>0\text{
,}$

which holds for all strictly positive $x$. ∎

Next, we establish that $\underaccent{\bar}{p}_{M}$ ranges from $0$ to some
finite positive number and that $\underaccent{\bar}{p}$ is U-shaped in
$\kappa$ and explodes at $\kappa=0$ and $\kappa=1/\left(4c\right)$.

######  Claim A.8.

$\infty>\lim_{\kappa\nearrow 1/\left(4c\right)}\underaccent{\bar}{p}_{M}>0$
and $\lim_{\kappa\searrow 0}\underaccent{\bar}{p}_{M}=0$.

###### Proof.

The first statement in the claim follows immediately from the analysis in
Section A.6. Suppose for the sake of contradiction that
$\underaccent{\bar}{p}_{M}>L$ for some constant $L>0$ for all $\kappa>0$.
Thus,

$\lim_{\kappa\searrow
0}\underaccent{\bar}{p}_{M}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}_{M}}\right\\}}+\underaccent{\bar}{p}_{M}-\mu\geq\lim_{\kappa\searrow
0}L\ln{\left\\{1+\frac{1}{2\kappa L}\right\\}}+L-\mu=\infty\text{ .}$

Since $\iota$ is continuous and strictly decreasing in $\kappa$, this means
that there is some $\hat{\kappa}>0$ such that for all $0<\kappa<\hat{\kappa}$,
$\iota\left(\kappa\right)>0$, a contradiction. ∎

######  Claim A.9.

$\lim_{\kappa\nearrow
1/\left(4c\right)}\underaccent{\bar}{p}=\lim_{\kappa\searrow
0}\underaccent{\bar}{p}=\infty$.

###### Proof.

As is shown in Section A.9, $\underaccent{\bar}{p}$ is strictly increasing in
$\kappa$ for all $\kappa\in\left(\tilde{\kappa},1/\left(4c\right)\right)$,
where $\tilde{\kappa}\in\left(0,1/\left(4c\right)\right)$. Suppose for the
sake of contradiction that $\underaccent{\bar}{p}$ is bounded above by
$K<\infty$ for all $\kappa\in\left(\tilde{\kappa},1/\left(4c\right)\right)$.
We have

$\lim_{\kappa\nearrow\frac{1}{4c}}\underaccent{\bar}{p}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}-\sqrt{\frac{c}{\kappa}}\leq\lim_{\kappa\nearrow\frac{1}{4c}}K\ln{\left\\{1+\frac{1}{2\kappa
K}\right\\}}-\sqrt{\frac{c}{\kappa}}=K\ln{\left\\{1+\frac{2c}{K}\right\\}}-2c<0\text{
,}$

a contradiction.

Similarly, $\underaccent{\bar}{p}$ is strictly decreasing in $\kappa$ for all
$\kappa\in\left(0,\tilde{\kappa}\right)$. Suppose for the sake of
contradiction that $\underaccent{\bar}{p}$ is bounded above by $M<\infty$ for
all $\kappa\in\left(0,\tilde{\kappa}\right)$. We have

$\lim_{\kappa\searrow
0}\frac{\underaccent{\bar}{p}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}}{\sqrt{\frac{c}{\kappa}}}\leq\lim_{\kappa\searrow
0}\frac{M\ln{\left\\{1+\frac{1}{2\kappa
M}\right\\}}}{\sqrt{\frac{c}{\kappa}}}=0<1\text{ ,}$

a contradiction. ∎

Next, (setting $a=0$)

$\iota\left(\mu-\sqrt{\frac{c}{\kappa}}\right)=\left(\mu-\sqrt{\frac{c}{\kappa}}\right)\ln{\left\\{1+\frac{1}{2\kappa\left(\mu-\sqrt{\frac{c}{\kappa}}\right)}\right\\}}-\sqrt{\frac{c}{\kappa}}\eqqcolon\rho\left(\mu\right)\text{
,}$

which is equal to $0$ for $\kappa>0$ if and only if

$x\left(\mu\right)\ln{\left(1+\frac{1}{x\left(\mu\right)}\right)}=2\sqrt{c\kappa}\text{
,}$

where $x\left(\mu\right)\coloneqq 2\kappa\mu-2\sqrt{c\kappa}$. Then, it is
easy to see that the left-hand side of this equation is strictly increasing
and strictly concave in $\mu$ and converges to $1$ as $\mu$ goes to $\infty$.
On the other hand, the right-hand side is a constant (for $c$ and $\kappa$
fixed) and is strictly less than $1$, since $\kappa<1/\left(4c\right)$. Thus,
for all (allowed) $\kappa$ and $c$, there exists a $\mu$ such that the
equilibrium involves active search. Furthermore, if $\mu\leq\
(<)\sqrt{c/\kappa}$, $\underaccent{\bar}{p}_{M}$ is obviously (strictly)
greater than $\mu-\sqrt{c/\kappa}$.

Finally, since $\underaccent{\bar}{p}_{M}$ is continuous and strictly
increasing in $\mu$ for $\mu>0$ and ranges from $0$ to some positive value,
$\underaccent{\bar}{p}$ is U-shaped and explodes at its endpoints, and
$\underaccent{\bar}{p}_{M}$ is strictly concave in $\kappa$ and
$\underaccent{\bar}{p}$ is strictly convex in $\kappa$, the result follows. ∎

###  A.9 Proposition 6.7 Proof

###### Proof.

Let us begin with the following claim.

######  Claim A.10.

$\varphi\left(\underaccent{\bar}{p}\right)$ has a real root at some
$\underaccent{\bar}{p}>0$ if and only if $c<1/\left(4\kappa\right)$. This root
is unique.

###### Proof.

First, we argue that $\varphi$ cannot have a root at some
$\underaccent{\bar}{p}<0$. Directly,

$\varphi^{\prime}\left(\underaccent{\bar}{p}\right)=\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}-\frac{1}{1+2\kappa\underaccent{\bar}{p}},\quad\text{and}\quad\varphi^{\prime\prime}\left(\underaccent{\bar}{p}\right)=-\frac{1}{\underaccent{\bar}{p}\left(2\kappa\underaccent{\bar}{p}+1\right)^{2}}\text{
.}$

Evidently, $\varphi^{\prime\prime}\left(\underaccent{\bar}{p}\right)>0$ for
all $\underaccent{\bar}{p}<0$. Consequently, since
$\varphi^{\prime}\left(\underaccent{\bar}{p}\right)$ is strictly increasing in
$\underaccent{\bar}{p}$ for all
$\underaccent{\bar}{p}\in\left(-\infty,0\right)$ and
$\lim_{\underaccent{\bar}{p}\to-\infty}\varphi^{\prime}\left(\underaccent{\bar}{p}\right)=0$,
$\varphi$ is strictly increasing in $\underaccent{\bar}{p}$ for all
$\underaccent{\bar}{p}\in\left(-\infty,0\right)$. Since
$\lim_{\underaccent{\bar}{p}\nearrow
0}\iota\left(\underaccent{\bar}{p}_{M}\right)=-\sqrt{c/\kappa}<0$, we conclude
that there is no root.

Second, $\lim_{\underaccent{\bar}{p}\searrow
0}\varphi\left(\underaccent{\bar}{p}\right)=-\sqrt{c/\kappa}$, thereby
eliminating $0$ as a candidate root.

Third, we establish that $\varphi$ is strictly increasing in
$\underaccent{\bar}{p}$ for strictly positive $\underaccent{\bar}{p}$. Since
$\varphi^{\prime}\left(\underaccent{\bar}{p}\right)$ is strictly decreasing in
$\underaccent{\bar}{p}$ and
$\lim_{\underaccent{\bar}{p}\to\infty}\varphi^{\prime}\left(\underaccent{\bar}{p}\right)=0$,
$\varphi$ is strictly increasing in $\underaccent{\bar}{p}$ for all
$\underaccent{\bar}{p}$.

Fourth, as $\underaccent{\bar}{p}\searrow 0$,
$\varphi\searrow-\sqrt{c/\kappa}<0$; and as
$\underaccent{\bar}{p}\nearrow+\infty$, $\varphi\nearrow
1/\left(2\kappa\right)-\sqrt{c/\kappa}$. Thus,
$\varphi\left(\underaccent{\bar}{p}\right)$ has a positive real root if and
only if $c<1/\left(4\kappa\right)$. It is unique. ∎

Evidently, $\varphi^{\prime}\left(c\right)<0$. Then, by the implicit function
theorem,

$\underaccent{\bar}{p}^{\prime}\left(c\right)=-\frac{\varphi^{\prime}\left(c\right)}{\varphi^{\prime}\left(\underaccent{\bar}{p}\right)}>0\text{,}$

where we used the result from Claim A.10 that
$\varphi^{\prime}\left(\underaccent{\bar}{p}\right)>0$.

Next,

$\varphi^{\prime}\left(\kappa\right)=\frac{\sqrt{c\kappa}}{2\kappa^{2}}-\frac{1}{2\left(\frac{1}{2\underaccent{\bar}{p}\kappa}+1\right)\kappa^{2}}\text{
,}$

which is positive if and only if

$\underaccent{\bar}{p}\leq\frac{\sqrt{c}}{2\sqrt{\kappa}-2\kappa\sqrt{c}}\text{
.}$

Then,

$\begin{split}\varphi\left(\frac{\sqrt{c}}{2\sqrt{\kappa}-2\kappa\sqrt{c}}\right)\geq
0\quad&\Leftrightarrow\quad\frac{\sqrt{c}}{2\sqrt{\kappa}-2\kappa\sqrt{c}}\ln{\left\\{1+\frac{1}{2\kappa\frac{\sqrt{c}}{2\sqrt{\kappa}-2\kappa\sqrt{c}}}\right\\}}-\sqrt{\frac{c}{\kappa}}\geq
0\\\
&\Leftrightarrow\quad-\ln{\left\\{\sqrt{c\kappa}\right\\}}-2+2\sqrt{c\kappa}\geq
0\end{split}\text{ .}$

By the implicit function theorem, $\underaccent{\bar}{p}$ is decreasing in
$\kappa$ if and only if $\varphi^{\prime}\left(\kappa\right)\geq 0$, if and
only if
$h\left(c,\kappa\right)\coloneqq\ln{\left\\{\sqrt{c\kappa}\right\\}}+2-2\sqrt{c\kappa}\leq
0$. Note that $h$ has a unique root at
$\tilde{\kappa}\in\left(0,1/\left(4c\right)\right)$. ∎

###  A.10 Lemma 6.9 Proof

###### Proof.

Let us begin with the result pertaining to the explicit search cost $c$.
Directly,

$\hat{\Phi}^{\prime}\left(c\right)=1-\underaccent{\bar}{p}^{\prime}\left(c\right)-\frac{1}{2\sqrt{c\kappa}}<0\text{
,}$

since $\underaccent{\bar}{p}^{\prime}\left(c\right)>0$ and
$c<1/\left(4\kappa\right)$. Next,

$\hat{\Phi}^{\prime}\left(\kappa\right)=-\underaccent{\bar}{p}^{\prime}\left(\kappa\right)+\frac{1}{2\kappa}\sqrt{\frac{c}{\kappa}}\geq
0\quad\Leftrightarrow\quad\frac{\varphi^{\prime}\left(\kappa\right)}{\varphi^{\prime}\left(\underaccent{\bar}{p}\right)}\geq-\frac{1}{2\kappa}\sqrt{\frac{c}{\kappa}}\text{
.}$

Substituting in for $\varphi^{\prime}\left(\kappa\right)$ and
$\varphi^{\prime}\left(\underaccent{\bar}{p}\right)$, this holds if and only
if

$\frac{\frac{\sqrt{c\kappa}}{2\kappa^{2}}-\frac{1}{2\left(\frac{1}{2\underaccent{\bar}{p}\kappa}+1\right)\kappa^{2}}}{\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}-\frac{1}{1+2\kappa\underaccent{\bar}{p}}}\geq-\frac{1}{2\kappa}\sqrt{\frac{c}{\kappa}}\quad\Leftrightarrow\quad
2\underaccent{\bar}{p}\kappa\left(\sqrt{c\kappa}-1\right)+\sqrt{c\kappa}\ln\left(1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right)\left(1+2\underaccent{\bar}{p}\kappa\right)\geq
0\text{ .}$

Enacting a change of variable with $x\coloneqq\kappa\underaccent{\bar}{p}$ and
substituting in using $\varphi=0$, the latter inequality becomes

$w\left(x\right)\coloneqq\left(1+\frac{1}{2x}\right)\ln{\left\\{1+\frac{1}{2x}\right\\}}+1-\frac{1}{x\ln{\left\\{1+\frac{1}{2x}\right\\}}}\geq
0\text{ .}$

Note that $x$ is strictly increasing in $\kappa$ for all
$\kappa\in\left(\tilde{\kappa},1/\left(4c\right)\right)$. Define
$\tilde{x}\coloneqq\tilde{\kappa}\underaccent{\bar}{p}\left(\tilde{\kappa}\right)$
and observe that by construction, $w\left(\tilde{x}\right)>0$. Moreover, $w$
is strictly decreasing in $x$ and $\lim_{x\to\infty}w\left(x\right)=-1$.
Appealing to the intermediate value theorem, we deduce the result. ∎

###  A.11 Proposition 7.1 Proof

###### Proof.

Recall

$\iota\left(\underaccent{\bar}{p},\kappa,\mu\right)\coloneqq\underaccent{\bar}{p}\ln{\left\\{1+\frac{1}{2\kappa\underaccent{\bar}{p}}\right\\}}+\underaccent{\bar}{p}-\mu\text{
,}$

(where we have set $a=0$), which is strictly increasing in
$\underaccent{\bar}{p}$.

First, let $\mu\geq 3/\left(4\kappa\right)$. We have

$\iota\left(p\right)<\iota\left(\mu-\frac{1}{2\kappa}\right)=\left(\mu-\frac{1}{2\kappa}\right)\ln{\left\\{1+\frac{1}{2\kappa\left(\mu-\frac{1}{2\kappa}\right)}\right\\}}+\left(\mu-\frac{1}{2\kappa}\right)-\mu\text{
,}$

for all $p<\mu-1/\left(2\kappa\right)$. The right-hand side of this expression
is weakly positive if and only if

$\left(z-1\right)\ln{\left\\{\frac{z}{z-1}\right\\}}-1\geq 0\text{ , }$

where $z\coloneqq 2\kappa\mu$, which never holds. Thus,
$\underaccent{\bar}{p}\geq\mu-1/\left(2\kappa\right)$. We also have

$\iota\left(p\right)>\iota\left(\mu-\frac{1}{4\kappa}\right)=\left(\mu-\frac{1}{4\kappa}\right)\ln{\left\\{1+\frac{1}{2\kappa\left(\mu-\frac{1}{4\kappa}\right)}\right\\}}+\left(\mu-\frac{1}{4\kappa}\right)-\mu\text{
,}$

for all $p>\mu-1/\left(4\kappa\right)$. The right-hand side of this expression
is weakly positive if and only if

$\left(z-1\right)\ln{\left\\{\frac{z+1}{z-1}\right\\}}-1\geq 0\text{ , }$

where $z\coloneqq 4\kappa\mu$. Since $\mu\geq 3/\left(4\kappa\right)$, $z\geq
3$ and so Inequality A.11 holds. Consequently,
$\underaccent{\bar}{p}\leq\mu-1/\left(4\kappa\right)$.

Second, let $3/\left(4\kappa\right)\geq\mu>0$. We have

$\iota\left(p\right)<\iota\left(\frac{\mu}{2}-\frac{1}{8\kappa}\right)=\left(\frac{\mu}{2}-\frac{1}{8\kappa}\right)\ln{\left\\{1+\frac{1}{2\kappa\left(\frac{\mu}{2}-\frac{1}{8\kappa}\right)}\right\\}}+\left(\frac{\mu}{2}-\frac{1}{8\kappa}\right)-\mu\text{
,}$

for all $p<\mu/2-1/\left(8\kappa\right)$. The right-hand side of this
expression is weakly positive if and only if

$\left(z-1\right)\ln{\left\\{\frac{z+3}{z-1}\right\\}}-1-z\geq 0\text{ , }$

where $z\coloneqq 4\kappa\mu$, which never holds. Thus,
$\underaccent{\bar}{p}\geq\mu/2-1/\left(8\kappa\right)$. Finally, we also have

$\iota\left(p\right)>\iota\left(\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}\right)=\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}\ln{\left\\{1+\frac{1}{2\kappa\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}}\right\\}}+\frac{\left(1+4\kappa\mu\right)^{2}}{32\kappa}-\mu\text{
,}$

for all $p>\left(1+4\kappa\mu\right)^{2}/\left(32\kappa\right)$. The right-
hand side of this expression is weakly positive if and only if

$\left(1+z\right)^{2}\ln{\left\\{\frac{(1+z)^{2}+16}{(1+z)^{2}}\right\\}}+\left(1+z\right)^{2}-8z\geq
0\text{ , }$

where $z\coloneqq 4\kappa\mu$. This clearly holds. Consequently,
$\underaccent{\bar}{p}<\left(1+4\kappa\mu\right)^{2}/\left(32\kappa\right)$. ∎

###  A.12 Proposition 7.2 Proof

###### Proof.

First, we establish an auxiliary result.

######  Claim A.11.

If $\mu\geq\underaccent{\bar}{p}+\sqrt{c/\kappa}$, then
$\underaccent{\bar}{p}>\sqrt{c/\kappa}-1/\left(4\kappa\right)$.

###### Proof.

If $c\leq 1/\left(16\kappa\right)$, the result is immediate, so let
$c>1/\left(16\kappa\right)$. Since $\varphi$ is strictly increasing in
$\underaccent{\bar}{p}$ for all $\underaccent{\bar}{p}>0$, we have

$\varphi\left(\underaccent{\bar}{p}\right)\leq\varphi\left(\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\right)=\left(\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\right)\ln{\left\\{\frac{4\sqrt{c\kappa}+1}{4\sqrt{c\kappa}-1}\right\\}}-\sqrt{\frac{c}{\kappa}}\text{
,}$

for all $\underaccent{\bar}{p}\leq\mu-\sqrt{c/\kappa}$. The right hand side of
this expression is weakly positive if and only if

$f\left(x\right)\coloneqq\left(x-1\right)\ln{\left\\{\frac{x+1}{x-1}\right\\}}-x\geq
0\text{ ,}$

where $x\coloneqq 4\sqrt{c\kappa}$. Note that $x$ takes values in the interval
$\left(1,2\right)$, and it is easy to see that $f<0$ for all such $x$.
Therefore, $\underaccent{\bar}{p}>\sqrt{c/\kappa}-1/\left(4\kappa\right)$. ∎

Second, we establish the result for high $\mu$.

######  Claim A.12.

If $\mu\geq\underaccent{\bar}{p}+\sqrt{c/\kappa}$, then $\Phi_{l}>\hat{\Phi}$.

###### Proof.

$\Phi_{l}>\hat{\Phi}\quad\Leftrightarrow\quad\mu+\frac{1}{4\kappa}+c-2\sqrt{\frac{c}{\kappa}}>\mu-\underaccent{\bar}{p}+c-\sqrt{\frac{c}{\kappa}}\quad\Leftrightarrow\quad\underaccent{\bar}{p}>\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\text{
,}$

which follows from Claim A.11. ∎

Third, we establish the result for moderate $\mu$.

######  Claim A.13.

If $\underaccent{\bar}{p}+\sqrt{c/\kappa}>\mu\geq
2\sqrt{c/\kappa}-1/\left(4\kappa\right)$, then $\Phi_{l}>\hat{\Phi}_{M}$.

###### Proof.

Using the conditions of the claim,

$\Phi=\mu+\frac{1}{4\kappa}+c-2\sqrt{\frac{c}{\kappa}}\geq
c>\kappa\left(\mu-\underaccent{\bar}{p}\right)^{2}=\hat{\Phi}_{M}\text{ .}$

∎

Finally, if $\underaccent{\bar}{p}+\sqrt{c/\kappa}>\mu$ and
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)>\mu$, the result is derived in the
proof of Proposition 7.1. ∎

###  A.13 Proposition 7.3 Proof

###### Proof.

First, we establish the result for high $\mu$.

######  Claim A.14.

If $\mu\geq\underaccent{\bar}{p}+\sqrt{c/\kappa}$, then
$\underaccent{\bar}{p}>2c$.

###### Proof.

By Claim A.11,if $\mu\geq\underaccent{\bar}{p}+\sqrt{c/\kappa}$, then
$\underaccent{\bar}{p}>\sqrt{c/\kappa}-1/\left(4\kappa\right)$, i.e., if there
is active search when prices are hidden, there must be active search when
prices are observable. Since $\varphi$ is strictly increasing in
$\underaccent{\bar}{p}$ for all $\underaccent{\bar}{p}>0$, we have

$\varphi\left(\underaccent{\bar}{p}\right)\leq\varphi\left(2c\right)=2c\ln{\left\\{\frac{4\kappa
c+1}{4\kappa c}\right\\}}-\sqrt{\frac{c}{\kappa}}\text{ ,}$

for all $\underaccent{\bar}{p}\leq 2c$. The right hand side of this expression
is negative if and only if

$x\ln{\left\\{\frac{x+1}{x}\right\\}}-\sqrt{x}<0\text{ ,}$

where $x\coloneqq 4\kappa c$. This clearly holds for all
$x\in\left(0,1\right)$. ∎

Thus, if the mean is sufficiently high so that there is active search when
prices are hidden, firms prefer that prices be hidden. Second, we establish
that for medium $\mu$, firms prefer hidden before learning if and only if
$\mu$ is sufficiently high.

######  Lemma A.15.

Let $\mu>0$. There exist $\kappa$, $\mu$ and $c$ that generate
$\underaccent{\bar}{p}+\sqrt{c/\kappa}>\mu\geq
2\sqrt{c/\kappa}-1/\left(4\kappa\right)$, such that
$\underaccent{\bar}{p}>2c$, $\underaccent{\bar}{p}=2c$, or
$\underaccent{\bar}{p}<2c$.

###### Proof.

Note, that the condition for this result is that there is active search if and
only if prices are observable. Since $\iota$ is strictly decreasing in $\mu$
it suffices to show that there exist values of the parameters such that
$\underaccent{\bar}{p}=2c$, since at the two endpoints
($\underaccent{\bar}{p}+\sqrt{c/\kappa}$ and
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)$) firms strictly prefer active or
hidden search, respectively. Directly, we have

$\iota\left(2c\right)=2c\ln{\left\\{\frac{4\kappa c+1}{4\kappa
c}\right\\}}+2c-\mu\text{ .}$

This must equal $0$ at equilibrium, i.e.,

$\mu=2c\ln{\left\\{\frac{4\kappa c+1}{4\kappa c}\right\\}}+2c\text{ .}$

Consequently, we must have

$2c+\sqrt{\frac{c}{\kappa}}>2c\ln{\left\\{\frac{4\kappa c+1}{4\kappa
c}\right\\}}+2c\geq 2\sqrt{\frac{c}{\kappa}}-\frac{1}{4\kappa}\text{ ,}$

which holds if and only if

$\sqrt{x}>x\ln\left\\{\frac{x+1}{x}\right\\}\geq 2\sqrt{x}-x-\frac{1}{2}\text{
,}$

where $x\coloneqq 4\kappa c$. This holds for all $x\in\left(0,1\right)$. ∎

Third, if $\underaccent{\bar}{p}+\sqrt{c/\kappa}>\mu$ and
$2\sqrt{c/\kappa}-1/\left(4\kappa\right)>\mu$, then the proof of Proposition
7.1 implies that
$\underaccent{\bar}{p}<\left(1+4\kappa\mu\right)^{2}/\left(32\kappa\right)$.
That is, if there is no active search for either timing regime, firms prefer
that prices be observable.

Finally, if $0\geq\mu>-1/\left(4\kappa\right)$, the firms obviously prefer
that prices are not hidden since otherwise there is no trade. ∎