pricing under noisy signaling

ORI GINAL RESEARCH

Pricing under noisy signaling

David Feldman • Charles Trzcinka • Russell S. Winer

� Springer Science+Business Media New York 2014

Abstract We provide rationale, conditions, and insights for ‘‘customized’’ pricing in

markets, that is, for equilibria where different buyers pay different prices for similar

products. We use a Spence/Riley signaling model enhanced by a signaling methodology

under random relations between costs and attributes, developed by Feldman (Math Soc Sci

48:93–101, 2004) and Feldman and Winer (Math Soc Sci 48:81–91, 2004). Examples

include markets for new cars, retail, human capital, trades where transaction costs are

negotiable, and transactions where sellers affect buyers’ costs by offering different levels

of service or support for the same products and prices. These encompass a large fraction of

all assets, prices, and transactions. Our results help explain the different levels of seg-

mentation and product/service differentiation that we observe in markets and the efficiency

of these equilibria. We note that we can demonstrate the results within competitive sellers’

markets. Financial markets examples include dividend, initial public offerings, market

microstructure and capital structure signaling, and share class distinctions in mutual funds.

Keywords Pricing � Signaling � Asymmetric information � Dividends � Initial

public offerings � Capital structure

JEL Classification D82 � D49 � G12 � G35 � G32 � M30

D. Feldman (&)Banking and Finance, The Australian School of Business, University of New South Wales, UNSWSydney, NSW 2052, Australiae-mail: [email protected]

C. TrzcinkaKelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47405-1701,USAe-mail: [email protected]

R. S. WinerStern School of Business, New York University, 44 West 4th Street, New York, NY 10012, USAe-mail: [email protected]

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Rev Quant Finan AccDOI 10.1007/s11156-014-0442-8

1 Introduction

We provide rationale, conditions, and insights for ‘‘customized’’ pricing in markets, that is,

for equilibria where different buyers pay different prices for similar products. We use a

Spence/Riley signaling model enhanced by a signaling methodology under random rela-

tions between costs and attributes, developed by Feldman (2004) (henceforth, F) and

Feldman and Winer (2004) (henceforth, FW). Examples include markets for new cars,

retail, human capital, trades where transaction prices and/or costs are negotiable, and

transactions where sellers affect buyers’ costs by offering different levels of service or

support for the same products and prices. These encompass a large fraction, perhaps the

majority, of all assets, prices, and transactions. Our results help explain the different levels

of segmentation and product/service differentiation that we observe in markets, and the

efficiency of these equilibria. We note that we can demonstrate the results within com-

petitive sellers’ markets. Financial markets examples include dividend, initial public

offerings (IPO), market microstructure and capital structure signaling, and share class

distinctions in mutual funds.

Only sometimes are prices fully set ex ante by sellers, with buyers acting as price takers.

This is the case, for example, in some catalogue purchases, internet e-trades, and retail

transactions. Alternatively, prices are set during buyer–seller interactions. This is the case,

for example, in the US new car market, the market for human capital, and some retail

markets. It is also the case when we consider as part of the transaction the quality of

service, product support level, customer support level, and other factors that affect buyers’

or sellers’ costs.

We call prices noisy if they are functions of factors undetermined by assets’ ‘‘funda-

mental’’ values and include errors. While many real-world prices are noisy, much of

financial economics research focuses on non-noisy pricing.

In the partial equilibrium (henceforth ‘‘equilibrium’’) that we model, sellers’ values are

functions of buyers’ attributes that sellers cannot directly observe. Particularly, buyers’

propensity to purchase affects sellers’ selling costs. Consequently, transaction prices are

functions of interactions between buyers and sellers. Buyers signal to sellers their unob-

servable propensity to purchase by manifesting their level of information about the relevant

product/service and the selling process. Before signaling, buyers draw their propensities to

purchase at random. Then, conditional on realized propensities, they draw their signaling

costs, their per-unit cost of acquiring information. The competitive sellers, facing higher

expenses when selling to lower attribute/propensity-to-purchase buyers, set their markups

as monotonic functions of buyers’ propensities perceived from buyers’ signaled infor-

mation levels. Then, buyers choose to signal the information level that minimizes their

total costs, that is, the sum of their product/service purchase costs and signaling (infor-

mation acquisition) costs.

Under the seminal Spence/Riley [Spence (1973), Riley (1975)] signaling equilibria,

costs and attributes are one to one. In Spence’s labor market model, for example, a higher/

lower productivity necessarily implies a lower/higher cost of acquiring education and vice

versa. In the real world, however, it is nearly impossible to find examples of such deter-

minism. In reality, a certain productivity level is associated with several costs and vice

versa. Thus, in contrast to Spence’s classical structure, we assume that each level of

propensity might be associated with all levels of costs; thus, each level of cost is associated

with all levels of propensities (as in F and FW). Therefore, the best separation that one

could achieve is by costs, which does not imply separation by propensities but, rather,

induces a probability distribution function over propensities. Specifically, each cost level

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induces a (conditional) distribution over propensities. Because selling costs are not fully

determined by assets’ fundamental values and because propensities to purchase (attributes)

do not fully determine information acquisition (signaling) costs, we have pricing under a

noisy signaling equilibrium.

We achieve pricing under a noisy signaling equilibrium if there is a markup schedule

such that (1) lower cost buyers choose higher information levels (separation), (2) buyers

minimize their total costs (buyers’ rationality), and (3) sellers, on average, cover their

selling costs by the markups they set (sellers’ rationality).

In our market, costs can take a continuum of values, and propensities can take either

discrete values or a continuum of values. Following F, a necessary and sufficient condition

for separating,1 by information acquisition costs, noisy signaling equilibrium, under dis-

crete propensities is the ordering of the cost-density functions induced by realized pro-

pensities by the monotone likelihood ratio property2 (MLRP). This condition also implies

the ordering by cost elasticity of these cost-density functions. Under a continuum of

propensities, following FW, the necessary and sufficient conditions become the ranking of

the cost-probability distributions with respect to the original probability measure and with

respect to a probability measure modified by the ‘‘propensity-to-purchase payoff function’’

by the generalized monotone likelihood ratio property3 (GMLRP), which, in turn, implies

the ranking by cost elasticities of these distributions.

An advantage of the analysis in discrete attributes is its use of simpler analytics, which

might make it easier for some readers to follow. The analysis under a continuum of

attributes, however, is more general. It allows a more parsimonious representation and

facilitates an intuitive economic interpretation. In particular, a change of probability

measure captures a major insight into our noisy signaling equilibrium, as explained below.

We already know from the Spence/Riley model that a one-to-one relationship between

costs and attributes (the most degenerate relationship possible) might allow a separating

signaling equilibrium. On the other hand, it is intuitively straightforward to infer that if

costs are independent of attributes (the most degenerate relationship possible), there will be

no signaling equilibrium (involving these costs). The problem, then, is to identify condi-

tions on the distributions of costs and attributes that are both stronger than independence

and weaker than a deterministic relations, and that allow a separating signaling equilib-

rium. We not only identify such conditions here, but we also show that these conditions

depend on a factor outside the probability distribution functions of costs and attributes.

These equilibrium conditions depend also on the effect that attributes have on the relevant

(sellers’) ‘‘reward function.’’

When changes in attribute levels induce bigger/smaller changes in sellers’ rewards,

there are higher/lower incentives for separating/pooling equilibria. Thus, in addition to the

costs/attribute distribution properties, the effects of attributes on rewards become part of

the equilibrium conditions.

1 Throughout the paper, we use the term ‘‘separating’’ to mean fully separating. This paper’s results,however, hold for partially separating equilibria as well. When the appropriate boundary conditions hold, wehave full separation.2 See Lehmann (1959) for definitions and use of the MLRP in probability and statistics, and Milgrom (1981)for its role in information economics.3 The GMLRP, defined by FW, can order probability distributions induced by probability distributions. TheMLRP orders probability distributions induced by realizations only.


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Under a continuum of attributes, the effects of attributes on rewards parsimoniously

enter the equilibrium conditions by a modification of the original probability measure of

costs, conditional on attributes.

We examine the market for new cars, where the analysis provides the rationale and

conditions for equilibria where different buyers step into car dealerships and come out with

the same products (car/options combinations) after having paid different prices for them.

We assume that buyers differ by their, unobservable to sellers, propensities to purchase and

by their per-unit search costs of information about cars and the car selling process. Car

sellers incur selling, showroom, labor, and test drive costs (henceforth, selling costs) that

they cover by assigning markups to the cars they sell. Sellers assign (different) markups to

(different) buyers, depending on the level of information4 about cars that buyers demon-

strate/signal. Sellers assume that, on average, buyers who signal a high level of information

have a higher propensity to purchase and, thus, induce lower selling costs. To cover their

expenses, sellers assign high markups to low-propensity, high-selling-cost buyers; and to

compete over the high-propensity, low-selling-cost buyers, they assign low ones. We also

assume that the relations between buyers’ per-unit information acquisition (signaling) costs

and their propensity to purchase (attribute) is not one to one. We assume the most general

case: each level of buyers’ (signaling) cost, and thus level of information, might be

associated with each level of propensity to purchase. In our pricing under noisy signaling

equilibrium, buyers are fully separated by their signaling/information search costs but each

level of cost is associated with all possible levels of propensity to purchase. Thus, there is

no separation by propensity to purchase, and the equilibrium is a noisy separating signaling

equilibrium. Of course, a higher level of information should, on average, induce lower

selling costs, and indeed this is a necessary condition for the equilibrium. But to prevent

‘‘mimicking,’’ a stronger, point-by-point monotone ordering condition on the cost-density

functions should hold.

It is interesting to note that this paper’s results provide, for example, a rationale for car-

buying strategies suggested by CitiMatters, a publication of CitiBank. The May 1998,

article ‘‘Smart Car-Buying Strategies to Help You Make the Best Showroom Deal,’’

emphasizes the advantage in preparing oneself before starting to negotiate a car purchase.

First, the article advises, one should choose the desired car including its options; and,

second, one should collect all relevant information about the car ahead of time. In our

model, selecting the car and options before entering the dealership saves selling time, and

the high level of car information signals a high propensity to purchase. Observing both, the

dealer presumes that the buyer is likely to purchase while inducing low selling costs and

will offer a lower markup.

The market for new cars example represents all markets where prices are not ex ante

fixed. The bigger, more liquid, and more competitive markets are, the more likely it is that

prices would be ex ante fixed. On the other hand, the higher the uncertainty regarding the

selling costs, the more likely it is that prices would be customized and negotiated.

Our analysis is relevant to equilibria in markets for goods and services that are not fully

segmented. We define a fully segmented market as one where buyers a priori choose a

4 We interpret ‘‘level of information’’ as one dimensional measure of both quality and quantity ofinformation.

D. Feldman et al.

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specific product/service at a pre-specified price. In markets that are not fully segmented,

sellers set ‘‘customized’’ prices as a function of a signal conveyed by the buyer. In partially

segmented markets, there is ‘‘customization’’ of prices within segments.

Our analysis leads to insights regarding market segmentation and product/service dif-

ferentiation. As markets grow and become more global and liquid, various products and

services that were customized become segmented or differentiated. Consider two exam-

ples. First, for several years now, one can buy new cars through the internet, paying lower

markups and inducing very low selling costs. We argue, however, that the important

property regarding the selling costs is their variance rather than their level. Under selling

costs with no variance, any level of selling costs could be ex ante deterministically priced.

The product/service plus the selling service and selling costs combination are fully spec-

ified ex ante, so we say that this product/service is fully segmented or differentiated.

For a financial market example consider the retail brokerage service. Not too long ago,

there were only full-service brokerage houses. In these full-service houses, fees are cus-

tomized/negotiated on the one hand, and the level of service is not fully specified on the

other. For example, different customers use different amounts and levels of brokers’

advice. Today, the market for brokerage services is partially segmented. Side by side, there

are full-service brokerage houses, limited-service/discount brokerage houses with lower

fees and fewer services, and no-service outlets, where one can buy securities for fixed pre-

specified fees and receive no service. Thus, customers may choose to buy a fully seg-

mented service at the no-service outlets (including online), a somewhat customized service

at discount brokerage houses, and a fully customized service at the full-service brokerage

houses. We discuss additional examples of product/service market structure, in which we

gain insights into the level of product/service differentiation.

Additional financial examples include the following: equilibrium dividend and IPO

(non-noisy) signaling models that have not been strongly supported by data, and the

creations and offerings of mutual fund share classes. The latter are actually screening

equilibria as price discrimination of different mutual fund buyers is prohibited by regu-

lation. Within a market microstructure context, private information is signaled by

increasing transaction sizes which, in turn, induce higher bid-ask spreads. Finally, we

discuss the classical capital structure signaling. We suggest that dividend, IPO, and even

capital structure noisy signaling models might be supported better by data and that the ban

on price discrimination in mutual fund shares has induced a socially inferior equilibrium

(because it allows only partial and not segmentation). This is because the number of share

classes must be low, it is difficult for investors to identify their optimal share class choice,

and brokers’ incentives are structured to further obfuscate this choice.

Related signaling equilibria are in Lundtofte (2013), where investors filter noisy public

signals to learn about the economy’s growth rates, and in Feldman and Bar Niv (Bur-

novski) (2004), where executives signal conformity/home bias by consistently deciding to

litigate international business contracts at their home courts, forgoing the benefits of

diversifying litigation forums.

The presentation of our results in terms of the ratio of the probability distribution

functions of the cost under the original and a modified probability measure has several

benefits. It not only allows for a concise presentation but also has an intuitive appeal: we

change the measure by multiplying the original measure of each propensity by the reali-

zation of its ‘‘full information’’ reward/markup. If the ratio of the densities is monotonic,

buyers will not ‘‘cheat,’’ or mimic other buyers with lower costs. Their optimal policy will

be to signal ‘‘truthfully.’’


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For expositional convenience, we examine the case in which only one signal is available

regarding the unobservable propensity. All the model’s results, however, hold for the case

of multiple signals.

The paper is organized as follows. Section 2 describes the model and the conditions for

the equilibria, Sect. 3 discusses additional applications and examples, and Sect. 4 sum-

marizes and concludes.

2 Pricing under noisy signaling: buyer–seller interaction

We adapt the signaling methodology developed in F and FW within Spence’s labor market

model to identify pricing under a noisy signaling separating equilibrium. We do that for

both discrete and a continuum of propensities to purchase. In the next subsection, we

identify necessary conditions and three equivalent necessary and sufficient conditions for

noisy signaling equilibria under continuum of costs and discrete propensities. In the fol-

lowing subsection, we do it under a continuum of propensities.

2.1 Discrete propensities

In a market with asymmetric information, buyers and competitive sellers trade a single

good/service. First, buyers draw a realization of an attribute, propensity to purchase, from a

real valued, positive, finite mean, binary random variable h, with possible realizations hi,

hi [ 0, i = l, h, where, without loss of generality, hl \ hh.5,6 A fraction of the population,

qi, 0 \ qi \ 1, i = l, h, ql ? qh = 1, is of attribute hi.

Then buyers draw a per-unit, real valued (positive) signaling cost c, from an interval,

c 2 ½c; �c�, 0\c\�c, according to a twice continuously differentiable probability distribution

conditional on their attribute type hi, fi(c), i = l, h.7 Buyers observe their attributes and

costs realizations (their types), but these are unobservable to others.8

Identical sellers face a fixed wholesale price for the good. Because the level of this

wholesale price is not relevant to our analysis, for simplicity, we assume it is zero. Sellers,

however, incur various selling costs (real estate, labor, technology, communication,

showroom, demonstration, etc.). To cover their selling costs, sellers markup the wholesale

price of the good. Because selling costs are inversely related to buyers’ attributes/pro-

pensities to purchase, the competitive sellers wish to discriminate among buyers. When

sellers can observe buyers’ propensities, to match their selling costs they choose low

markups for high-propensity buyers and high markups for low-propensity buyers. In the

absence of relevant signals, however, sellers are forced to assign each buyer the ‘‘pooling’’

markup, i.e., the average markup over the population’s propensities. However, when sellers

can use signals to better differentiate between the different buyers’ types, they try to attract

the high-propensity/low-selling-costs buyers by lowering their markups. Correspondingly,

sellers try to protect themselves from low-propensity/high- selling-costs buyers by

increasing their markups. To stay competitive, other sellers also have to use signals while

setting markups, and the ‘‘pooling’’ equilibrium is then replaced by a separating signaling

5 The case where hl = hh is economically redundant, thus excluded.6 For simplicity and brevity, we assume a binary attribute. Our results hold for multiple possible attributevalues.7 We choose the simpler notation f(c) over more elaborate ones, such as fcjhðcjhiÞ, and even f ðcjhiÞ:8 Our analysis stands if buyers observe their costs but not their attributes.

D. Feldman et al.

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one. In such an equilibrium, sellers draw inferences from signals conveyed by buyers and

set markups as a function of the perceived attributes/propensities of the buyers.

Sellers, here, cannot separate buyers by the realizations of their unobservable (to sell-

ers9) propensities to purchase h; thus, they separate buyers by their signaling costs, their

costs of acquiring information. Sellers conjecture that buyers use a strictly monotonic and

twice continuously differentiable signaling rule b(�) to map costs c into positive10 signals of

information level s, s 2 ½s; �s�, 0\s\�s. If a buyer sends a signal s, a seller uses the inverse

signaling function b-1(�) to infer the buyer’s cost. Therefore, the seller faces the posterior

distribution of propensity to purchase p½hijb�1ðsÞ�; i = l, h. Let Mh, Mh 2 ½Mhh;Mhl� be a

positive competitive markup for a propensity level h. Because lower propensity buyers

require more of the sellers’ time and resources per unit sold, Mh is strictly decreasing in h(implying Mhh

\Mhl). Then, sellers announce a markup schedule, that is, a function that

maps the information-level signals s to markups M(s). Thus, the competitive markup

M(s) that sellers assign to buyers who signal s, is

MðsÞ ¼ Mhlp hljb�1ðsÞ� �

þMhhp hhjb�1ðsÞ� �

; ð1Þ

which is the average markup, weighted by the conditional (posterior) distribution of

productivity levels given realized signals. Clearly, the markup rule is rational only if it

induces buyers with costs c to signal their ‘‘true type,’’ s = b(c).

Let Cðs; cÞ be the total costs of buyers who signal s and have signaling costs c. Then11

Cðs; cÞ ¼ MðsÞ þ sc ¼ MðsÞ þ sb�1ðsÞ: ð2ÞThe first term on the right-hand side of Eq. (2) is the markup charged to a buyer who

signals information level s, and the second term is this buyer’s total signaling cost. Buyers

choose signal level s to minimize their overall costs Cðs; cÞ, which are the sum of their

signaling costs and the markup. In pricing under a noisy signaling equilibrium, buyers

signal truthfully, i.e., they optimally choose to signal s ¼ s, and in this case

b�1ðsÞ ¼ b�1ðsÞ ¼ c:The information structure is as follows. All probability distribution functions and the

markup schedule that sellers announce are common knowledge. Buyers know their own

propensity-to-purchase type and realized cost of acquiring information, but these are

unobservable to others. Alternatively, without other changes to the model, buyers might

know their realized costs but not their propensity types. The latter, obviously, is less

economically appealing.

In a noisy (differentiable) signaling equilibrium (1) rational buyers choose signals that

minimize their costs, that is, their markups and signaling costs (buyers’ rationality), (2)

rational sellers offer buyers markups that cover their, inferred, selling costs to these buyers

(sellers’ rationality), and (3) buyers with lower costs choose to acquire higher information

levels (monotone signaling or separation).

By construction, following the markup function, Eq. (1), guarantees sellers’ rationality.

Buyers’ choice of cost-minimizing signals [defined in Eq. (2)] guarantees buyers

9 As said earlier, buyers’ attributes could be unobservable to buyers as well.10 Without loss of generality, for simplicity, we assume positive signals.11 To simplify the exposition and notation, we assume a linear signaling cost structure where the signalingcost is sc. Our results hold for a general signaling cost function C(s, c) under appropriate assumptions aboutthe function’s partial derivatives.


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rationality. Finally, buyers’ minimizing costs by ‘‘truth telling,’’ that is, choosing signals

that reveal their ‘‘true types’’ (which are costs, here) guarantees monotonicity and

separation.

To identify a noisy signaling equilibrium, we now need to identify a signaling rule b(�)that, given the markup structure, costs structure, and probability distribution functions,

minimizes buyers’ costs when buyers choose truth-revealing signals.

We start by writing the posterior distribution of attributes that a cost c induces. By

Bayes rule,

p½hijc� ¼fi½c�qi

fl½c�ql þ fh½c�qh

; i ¼ l; h: ð3Þ

Under the signaling rule, b(c) = s, sellers observing a signal s, infer an underlying cost

c; b�1ðsÞ ¼ c: We can, thus, use this property and write the posterior distribution of

attributes as a function of observable signals, rather than as functions of unobservable

costs. Equation (3), then, becomes,

p hijb�1ðsÞ� �

¼fi b�1ðsÞ� �

qi

fl b�1ðsÞ� �

ql þ fh b�1ðsÞ� �

qh

; i ¼ l; h: ð4Þ

We can now use the posterior distribution functions of attributes given signals in the

optimal markup schedule. Substituting Eq. (4) into Eq. (1), we get

MðsÞ ¼Mhl


ql þMhhfh b�1ðsÞ� �

qh



qh

ð5Þ

Rearranging Eq. (5) yields,

MðsÞ ¼ Mhhþ ðMhl

�MhhÞ


ql



qh

ð6Þ

In equilibrium, buyers choose signal levels s to minimize their costs, specified in Eq.

(2), setting oC=os ¼ 0; which, in turn, implies M0ðsÞ þ c ¼ 0, or

M0ðsÞ þ b�1ðsÞ ¼ 0: ð7ÞSubstituting Eq. (6) into Eq. (7), we have

qlqhðMhl�Mhh

Þfl b�1ðsÞ� �

f 0h b�1ðsÞ� �

� f 0l b�1ðsÞ� �

fh b�1ðsÞ� �

qlfl b�1ðsÞ� �

þ qhfh b�1ðsÞ� �� 2

b�10ðsÞ þ b�1ðsÞ ¼ 0: ð8Þ

Requesting that Eq. (8) is satisfied at s ¼ s, and because b�1ðsÞ ¼ c, we have

b½b�1ðsÞ� ¼ bðcÞ ¼ s. Taking the derivatives of both sides with respect to s yields

b0 b�1ðsÞ� �

b�10ðsÞ ¼ 1; ð9Þ

or

b0ðcÞb�10ðsÞ ¼ 1; ð10Þ

or

b�10ðsÞ ¼ 1=b0ðcÞ: ð11ÞSubstituting Eq. (11) into Eq. (8), again requesting s ¼ s and using b-1(s) = c, yields

D. Feldman et al.

123

qlqhðMhl�Mhh

Þ f0l ðcÞfhðcÞ � flðcÞf 0l ðcÞ

qlflðcÞ þ qhfhðcÞ½ �21

b0ðcÞ þ c ¼ 0; ð12Þ

or

b0ðcÞ ¼ qlqhðMhl�Mhh

Þc

� �flðcÞfhðcÞ

qlflðcÞ þ qhfhðcÞ½ �2f 0hðcÞfhðcÞ

� f 0l ðcÞflðcÞ

� : ð13Þ

Equation (13) is an ordinary differential equation with a unique solution that satisfies the

boundary condition bðcÞ ¼ �s. Because the first and second multiplicands of the right side of

Eq. (13) are positive for all cost levels (either by definition or construction), we have

monotone signaling or separation; that is, b0(c) \ 0, 8c, if and only if the third multiplicand

of Eq. (13) is negative for all cost levels, or

f 0hðcÞfhðcÞ

� f 0l ðcÞflðcÞ

\0; 8c; ð14Þ

which is (a version of) the (strict) MLRP.12 A simple rearrangement of Eq. (14) shows that the

condition is equivalent to monotonicity in the cost elasticities; see the proposition below.

Finally, we confirm that the first-order conditions of the total cost function, Eq. (2),

indeed identify a minimum. We do that by examining the second-order conditions, spe-

cifically, o2Cos2

s¼s

[ 0. For brevity, we omit the second-order analysis here and refer the

reader to a sufficiently similar one in F, pages 99–101.

We have now identified a differentiable separating/monotone signaling rule b(�) that is

both truth telling with respect to buyers’ types and costs minimizing with respect to the

signaling costs, and where sellers assign the rational competitive markups. We have, thus,

proved the following proposition.

Proposition 1 There exists a differentiable noisy signaling equilibrium under discrete

propensities, in the buyers-sellers market described above, if and only if

1. For all costs, the probability density functions of per-unit signaling costs conditional

on attributes obey the strict MLRP, specifically,

f 0l ðcÞflðcÞ

[f 0hðcÞfhðcÞ

; 8c; ð15Þ

and, equivalently,13 if and only if,

2. For all costs, the elasticities of cost probability density functions, conditional on lower

attributes’ realizations are lower than those conditional on higher attributes’

realizations, specifically,

oflðcÞoc

flðcÞc

[ofhðcÞ

ocfhðcÞ

c

; 8c; ð16Þ

and, or, equivalently,14 if and only if

12 See Lehmann (1959) for the definition of MLRP, and see Milgrom (1981) for an examination of theinformational role of the MLRP.13 After multiplying both sides of inequality (15) by c.14 See Milgrom (1981, Proposition 2).


123

3 8cl; ch, cl; ch 2 ½c; c�, cl \ ch, p ðhjc ¼ clÞ dominates p ðhjc ¼ chÞ by first-order

stochastic dominance.

The above necessary and sufficient conditions imply necessary, only, conditions when

aggregating the above point-by-point (of costs) conditions, over the costs probability

distribution function. The aggregation results are in the following corollary.15

Corollary 1 A necessary condition for a noisy signaling equilibrium, under discrete

propensities, is the ranking of the probability distribution functions of per-unit costs

conditional on lower propensities and those conditional on higher propensities by first-

order stochastic dominance.

We will now characterize the noisy signaling equilibrium under a continuum of propensities.

For the reader’s benefit, we will choose a different method of identifying the equilibrium.

2.2 Continuum of propensities

For brevity, we will highlight only the market aspects that are different under a continuum

of propensities. Buyers, first, draw a realization of the attribute, propensity to purchase,

from a real valued, positive, finite mean, random variable h distributed according to a

probability measure lh with a corresponding twice differentiable density function, f(h).

Then, conditional on the realization of the propensity, buyers draw a (positive) per-unit

signaling cost c, c 2 ½c; c�, 0\c\c, from a positive, twice continuously differentiable

probability distribution function,16 f ðcjhÞ.Sellers, unable to separate buyers by the realizations of their unobservable (to sellers)

propensities to purchase h, conjecture that buyers use a strictly monotonic and twice

continuously differentiable signaling rule b(�) to map costs c into signals of information

level s, s 2 ½s; �s�. If buyers send signals s, sellers use the inverse signaling function b-1(�) to

infer buyers’ costs. Therefore, sellers face the posterior distribution of propensities to

purchase f ½hjb�1ðsÞ�. Let Mh be a positive competitive markup for a propensity level h.

Because lower propensity buyers require more of the sellers’ time and resources per unit

sold, Mh is strictly decreasing in h. Then, sellers announce a markup schedule, that is, a

function that maps the information-level signals s to markups M(s). Thus, the competitive

markup M(s) that sellers assign to buyers who signal s, is

MðsÞ ¼Z

Mhf hjb�1ðsÞ� �

dh: ð17Þ

Let C(s, c) be the total costs of buyers who signal s and have signaling costs c. Then17

Cðs; cÞ ¼ MðsÞ þ sc: ð18ÞThe first term on the right-hand side of Eq. (18) is the markup charged to a buyer who

signals information level s, and the second term is this buyer’s total signaling cost. Buyers

choose signal level s to minimize their overall costs, the sum of their signaling costs and

15 See, F and, for example, Lehmann (1959, p. 74).16 For notational brevity, we suppress the subscripts of the probability distribution functions. The arguments

will define the functions, i.e., f ðcjhÞ,fcjhðcjhÞ, etc.17 As before, in the previous subsection, to simplify exposition and notation, we assume a linear signalingcost structure where the signaling cost is sc. Our results hold for a general signaling cost function C(s,c) under appropriate assumptions about the function’s partial derivatives.

D. Feldman et al.

123

the markup C(s, c). In pricing under a noisy signaling equilibrium, buyers signal truthfully,

i.e., b-1(s) = c.

While in the previous subsection we identified the equilibrium following optimal pol-

icies of buyers and sellers, we identify the equilibrium here using implications of equi-

librium conditions. Let the equilibrium markup for a given cost level be N(c). Then,

pricing under a noisy signaling equilibrium is a differentiable equilibrium with mapping of

per-unit acquisition costs to the level-of-information signals c ? b(c) = s, and signals to

markups s ? M(s), such that

(C.1) Separation by cost. Signals are a strictly monotonic function of costs,18 that is

b0(c) \ 0, 8c.

(C.2) Buyers’ rationality. Buyers minimize purchase and information acquisition/

signaling costs,19 that is [first- and second-order conditions of Eq. (18)]

M0½bðcÞ� ¼ �c; M00½bðcÞ�[ 0; 8c:

(C.3) Sellers’ rationality. The markup rule M(�) assigns the competitive markup N(�),that is M[b(c)] = N(c).

We will now use these conditions to identify the equilibrium. Denote the cumulative

distribution function of propensities conditional on costs FðhjcÞ. Thus, N(c), the com-

petitive markup equals the expected markup conditional on the cost level c, or

NðcÞ ¼Z

MhdFðhjcÞ; 8c; ð19Þ

and is a function of both the ‘‘propensity payoff function’’ Mh and the probability distri-

bution functions of propensities conditional on costs FðhjcÞ.Differentiating Condition (C.3) and substituting Condition (C.2) yields

b0ðcÞ ¼ �N0ðcÞc

; 8c: ð20Þ

Therefore, to satisfy Condition (C.1), N0(c) must be positive. From Eq. (19), this impliesZ

MhdFhðhÞ[Z

MhdFlðhÞ; 8cl; ch; cl\ch; ð21Þ

where Fi(h) , Fðhjc ¼ ciÞ; i = l, h.

Differentiating N(c), defined in Eq. (19), yields

N0ðcÞ ¼ o

oc

ZMhdFðhjcÞ

� ; ð22Þ

and because f(c) is positive, by Bayes rule

N0ðcÞ ¼ o

oc

RMhf ðcjhÞdlh

f ðcÞ

� : ð23Þ

18 To simplify the exposition, we assume mathematical conditions for separation by cost that are morerestricting than necessary. For example, we can allow b0(c) = 0 for some c.19 These are the necessary and sufficient conditions for buyers’ cost minimization under the cost structurespecified in Eq. (18). Different cost structures would induce corresponding conditions.


123

We define a new probability measure l�h; l�h,

MhlhRMhdlh

; and denote densities under lh* as

f*(�). In particular, we denote f �ðcÞ ¼R

f ½cjh�dl�h:Rewriting Eq. (23) after multiplying and dividing the integrand by

RMhdlh;

N0ðcÞ ¼ o

oc

ZMhdlh

Rf ðcjhÞ MhdlhR

Mhdlh

f ðcÞ

2

64

3

75 ¼o

oc

ZMhdlh

Rf ðcjhÞdl�h

f ðcÞ

� ; ð24Þ

and if E is the expectation operator under lh, E½gð�Þ�,R

gð�Þdlh; we have

N0ðcÞ ¼ E½Mh�o

oc

f �ðcÞf ðcÞ

� : ð25Þ

Performing the differentiation on the right-hand side of Eq. (25) yields

N0ðcÞ ¼ E½Mh�f �ðcÞf ðcÞ

f �0ðcÞf �ðcÞ �

f 0ðcÞf ðcÞ

� ; 8c; ð26Þ

and substituting Eq. (26) into Eq. (20), we get

b0ðcÞ ¼ �E½Mh�c

f �ðcÞf ðcÞ

f �0ðcÞf �ðcÞ �

f 0ðcÞf ðcÞ

� ; 8c: ð27Þ

Equation (27) is a first-order ordinary differential equation with a unique solution for a

given boundary condition, bðcÞ ¼ �s; for example.

Equation (27) reveals that separation, or Condition (C.1), holds if and only if the third

multiplicand of the right-hand side is positive, that is, if and only iff �0ðcÞf �ðcÞ [ f 0ðcÞ

f ðcÞ ; 8c:We have proven the following proposition under a continuum of propensities, corre-

sponding to the previous results in Proposition 1 under discrete propensities.

Proposition 2 There exists a differentiable noisy signaling equilibrium under a contin-

uum of propensities, in the buyers-sellers market described above, if and only if.

1. For all costs, the probability density functions of per-unit signaling costs with respect

to the original probability measure lhand with respect to the modified probability

measure lh*, obey the strict GMLRP, specifically

f �0ðcÞf �ðcÞ [

f 0ðcÞf ðcÞ ; 8c; ð28Þ

and, equivalently,20 if and only if.

2. For all costs, the costs, elasticities of the probability density functions of the per-unit

signaling costs with respect to the original probability measure lhare lower than those

with respect to the modified probability measure lh*, specifically,

of �ðcÞoðcÞf �ðcÞ

c

[of ðcÞoðcÞf ðcÞ

c

; 8c: ð29Þ

20 After multiplying both sides of inequality (15) by c.

D. Feldman et al.

123

The above necessary and sufficient conditions imply necessary, only, conditions when

aggregating the above point-by-point (of costs) conditions, over the costs distribution. The

aggregation results in the following corollary.21

Corollary 2 A necessary condition for a noisy signaling equilibrium, under a continuum

of propensities, is the ranking of the probability distribution functions of per unit signaling

costs with respect to the modified probability measure lh* and with respect to the original

probability measure lh by first-order stochastic dominance.

In the following subsection, we explain the economic rationale that induces the (ana-

lytical) change of measure in the equilibrium condition.

2.3 The economic rationale of the equilibrium conditions

2.3.1 Economic rationale of the necessary conditions

Consider two extreme cases, one where costs and propensities to purchase are uncorrelated

and the other where costs and propensities are one to one, that is, deterministically related.

When costs and propensities are uncorrelated, there are no signaling equilibria, of course,

as costs convey no information on propensities. In the other extreme case, as Spence

showed, when the conditional probability distributions of propensities and costs degenerate

to a single realization, separating signaling equilibria exist. In this paper we study ‘‘in

between’’ cases where each propensity realization might be associated with every cost

realization, and vice versa.

When a certain cost might be associated with any propensity, full separation by pro-

pensities is impossible. Thus, intuitively, a necessary condition for equilibrium is that, at

least on average, higher propensities are associated with lower costs. Analytically, we find

that this condition becomes first-order stochastic dominance.

2.3.2 Economic rationale of the necessary and sufficient conditions

Furthermore, first-order stochastic dominance is an aggregate measure (an integral over the

whole domain of the probability distribution function). It could be possible to reverse the

relative changes of conditional distributions locally, over a finite interval, without changing

the aggregate result, thus without triggering a violation of the condition. However, such

reversal invites mimicking, of ‘‘high types’’ by ‘‘low types,’’ over this interval, destroying

the separation of types necessary for the signaling equilibrium. Thus, the necessary and

sufficient condition for a separating signaling equilibrium requires a point-by-point ful-

fillment of conditions on functions of the conditional probability density functions. Ana-

lytically, this becomes the MRLP, GMLRP, or, equivalently, the point-by-point elasticities

rankings.

2.3.3 Economic rationale of the change of measure

Finally, we explain the economic role of the change of measure as part of the equilibrium

conditions. The above discussion analyzed the economic implications of properties of the

joint distribution functions of propensities and costs. Consider now the impact of buyers’

unobservable propensities to purchase on the sellers’ rewards. In particular, consider,

21 See, F and, for example, Lehmann (1959, p. 74).


123

again, two extreme cases, one case where sellers’ rewards are highly sensitive to (even

minute) changes in propensities and the other where propensity changes bear little impact

on sellers’ rewards. Because incentives for having separating signaling equilibria in the

former cases are higher than in the latter, equilibrium conditions are affected by sellers’

rewards’ sensitivities to propensities. This effect on the equilibrium conditions occurs

through a change, by the sensitivity of sellers’ rewards to propensities to purchase, in the

original probability measure of the population’s joint probability distribution functions of

propensities and costs. This change of measure under continuum of propensities is, we

believe, an elegant parsimonious expression of the economic equilibrium.

A different way to describe the intuition behind the change of measure is as follows.

Again, consider two extreme cases—one where only very coarse separation is feasible but

sellers’ payoff functions are highly sensitive to separation, and the other where fine sep-

aration is possible but sellers’ payoff functions are hardly sensitive to separation. Thus, in

addition to the probability measure of costs induced by propensities, which determines the

ability to separate propensities given costs, another probability measure plays a crucial

role, the probability measure of costs induced by propensities modified by the sensitivity of

the sellers’ propensities payoff function.

3 Applications

3.1 The market for goods and services: new cars

In the US market for new cars, there is pricing under a noisy signaling equilibrium. That is,

transaction prices of goods and services are often determined by the interaction between

buyers and sellers. Except for a small market share of internet sales, where prices are set ex

ante and reflect minimal or no selling costs, car purchase prices are negotiated between

buyers and salespeople in car dealerships. Thus, if several individuals step into a dealership

and each end up buying the same car/options combination, chances are that they each pay a

different price. We will now examine this market referring to the results in the previous

section.

Assume that new car dealers incur selling costs. To cover these costs, dealers mark up

the prices of the cars that they sell. To align dealers and salespeople incentives, we

assume that salespeople’s commission fees are proportional to dealers’ markups. Buyers

differ in their propensity to purchase cars h distributed according to a probability

measure lh with a corresponding twice differentiable density function f(h). A high-

propensity buyer requires relatively fewer dealership resources per car sold, and a low-

propensity buyer requires relatively more dealership resources. Resources include

salespeople’s time and test drives.

We assume that buyers are randomly endowed with a positive per-unit cost of obtaining

information about cars and car-selling procedures c, c [ 0, c 2 ½c; c�; drawn from a

positive, twice continuously differentiable probability distribution function f ðcjhÞ: This

per-unit cost of information is related to several factors that are also related to propensities

to purchase. First, on average, buyers with higher levels of education might be more

efficient in absorbing and processing information from the environment. Second, on

average, buyers from higher socioeconomic strata might have easier access to sources of

information, greater exposure to buyer–seller interactions, and more consumption expe-

rience with new cars.

D. Feldman et al.

123

The timing sequence in this market is as follows. Buyers are randomly endowed with

propensity attributes h. They then draw information-acquisition costs c from distributions

conditional on the realization of their propensity attributes. Sellers announce a markup

schedule M(s), a function of the buyers’ signal s, signaling information about cars. Sellers

design the markup schedule as a function of the distributions of propensities to purchase

and information acquisition costs, inducing different buyers to acquire different levels of

information about cars [see Eq. (17)]. Buyers observe M(s) and, knowing their costs,

choose a signal that minimizes the sum of their signaling costs and assigned markups [see

Eq. (18)]. Buyers determine the optimal mapping from costs to signals given the sellers’

markup schedule to be b(c) = s (see Proposition 1). Then, buyers acquire their optimal

information level about cars (their signal), enter car dealerships, and, by interacting with

salespeople, signal their information level. They demonstrate, for example, their knowl-

edge about dealers’ costs, car specifications, and available car options both for the sellers’

cars and for competing models. A buyer with an extremely high level of information, for

example, would enter a dealership and offer to purchase a specified car/options combi-

nation at a specified price instantaneously. If the dealer agrees, the dealer sells a car on the

spot, incurring no selling costs.

Salespeople offer a price to buyers as a function of the signaled information level. For

example, a buyer who signals s1 is offered a markup M(s1). Buyers with high information

levels impress sellers that ‘‘they know what they want’’ and are, thus, better ready to make

a quick purchase decision if ‘‘their terms are met.’’ Moreover, sellers perceive the better

informed buyers as less likely to agree to pay higher markups.

Propositions 1 and 2 specify the conditions on the distributions of buyers’ propensities to

purchase, buyers’ information-acquisition costs, and the effect of propensities to purchase on

sellers’ costs, all of which induce noisy signaling equilibria in the new car market. While we

chose to describe a new car market where buyers have a continuum of propensities to

purchase, we can repeat the analysis for the case of buyers with discrete propensities. We do

not present this analysis, for brevity. As we see a noisy signaling equilibrium in the US new

car market, it seems that these distributional conditions hold in the real world.

While our model is a single-period one and we do not model sequential interactions, we

may regard the process of negotiating and receiving price quotes from one or several

sellers as part of the information acquisition process, eventually leading to lower markups.

If dealers neglect to use the markup schedule associated with a signaling equilibrium,

they will lose sales by assigning higher markups to some high-propensity buyers and will

not compensate themselves sufficiently for selling costs of some low-propensity buyers to

whom they assigned lower markups. Thus, on one hand, dealers will be careful while

increasing markups because of the risk of losing customers; on the other hand, dealers will

be careful to charge sufficiently high markups in order to cover their selling costs. The

dealership, then, will have to adopt pricing under noisy signaling or be driven out of the

market. For the same reason, dealers will not be able to uniformly assign the ‘‘pooling’’

markup, that is, the average of the markups weighted by the population proportions of the

buyer propensities.

Thus, pricing under a noisy signaling equilibrium, as described above, might exist in

this market. In this noisy equilibrium, there will be some high-propensity buyers (with high

per-unit cost of information) who will be assigned higher markups than those of some low-

propensity buyers (with low per-unit information costs).


123

3.2 Product/service market structure: allocational efficiency

We now look at the extent to which prices are predetermined in product/service markets.

We define a fully segmented product/service market as one where buyers choose among a

priori selected and priced products or services. In fully segmented markets, buyers are

unable to affect (in the short run) the price of products or services. Transaction prices and

products are not ‘‘customized,’’ i.e., are not functions of buyers’ attributes, actions, or

interaction with sellers. Examples of such environments include supermarkets, department

stores, and most internet purchases. In a non-segmented market, the product/service sold

and/or its price are functions of an interaction between buyers and sellers. Such markets

include the markets for new cars, human capital, and all markets where assets and prices

are accompanied by different levels of service or support.

A partially segmented product/service market is one where there are both a priori

differentiation and ‘‘customization’’ of the products or services. The market for brokerage

services is an example of a partially segmented one. Side by side, there are full-service

brokerage firms, discount brokerage firms, and no-service electronic brokerage firms. A

buyer might opt for a fully segmented e-trade, for a somewhat customized discount bro-

kerage, or for a fully customized full-service brokerage.

The previous analysis suggests that pricing under a noisy signaling equilibrium in the

spirit of the one modeled here occurs in product/service markets that are not fully seg-

mented and where buyers have attributes that are relevant but not directly observable to

sellers. It seems, however, that sellers are better off under segmentation for two reasons.

First, without segmentation, pricing errors are bound to occur. High prices are offered to

high-propensity buyers, thus potentially alienating them; and low prices are offered to low-

propensity buyers, thus forgoing revenues and incurring unduly high costs. Second, seg-

mentation increases the allocational efficiency of resources. Buyers of different propen-

sities require different types of resources; this, in turn, requires sellers to have all these

resources available. Under full segmentation, there is a perfect match between the demand

and supply of these resources. Sellers have to provide only the resources needed to service

the particular buyers’ type, and buyers will self-select themselves. Without segmentation,

sellers must make all resources available to all buyers. This reduces the efficiency of their

fixed costs allocation and some of the variable costs investments. For example, highly

informed buyers do not require company research reports offered by full-service brokers;

less informed buyers may make good use of the new computer-based information services

now available in automobile showrooms.

However, not all markets are fully segmented. If the market size is not large enough, the

number of buyers in a certain propensity group might not support the fixed costs. In

addition, regulatory conditions, such as dealer-manufacturer arrangements, may prohibit

the free choice of selling methods.

3.3 Allocating selling resources: retail markets

In some markets, for example, because of regulations or legal mandates, sellers are con-

strained in their ability to set prices as functions of buyers’ propensities and signals. In this

case, sellers are unable to freely price their selling resources costs. Instead, they might opt

to control the amount of selling resources allocated to buyers, for example, modifying the

level of time and attention they allocate to buyers as a function of the buyers’ attributes and

signals. As an example, in the retail market for clothes, the attribute is the propensity to

purchase, and signals are the buyers’ manners and clothing. Buyers vary in their signaling

D. Feldman et al.

123

costs in terms of their means and disposition to be well-dressed. In equilibrium, buyers

who wish to receive a higher level of service might dress better when shopping for clothes.

3.4 Financial signaling—the classic case: capital structure

Following two seminal contributions to finance, the appearance of signaling mechanisms in

financial modeling became imminent. These contributions are, first, Modigliani–Miller

paradigms within a world of symmetric information and no agency problems [Modigliani

and Miller (1958)] and, second, Ross’s (1973) introduction of agency problems. Indeed,

Ross (1977) and Leland and Pyle (1977) introduced financial signaling equilibria.

Asymmetric information, for example, between corporate insiders and outsiders results in

debt levels and, consequently, level of debt interest payments, that signal, with respect to

the more refined information set of informed corporate insiders, the quality of a firm’s

future investment opportunities. Future investment opportunities, in turn, define expected,

then, realized future revenue cash flows.

Needless to say that in the real world, we should not expect to see non-noisy signaling

equilibria where corporate debt interest payments and corporate investment opportunities

are one to one. However, the noisy signaling model that we developed here, where cor-

porate debt interest payments noisily signal corporate investment opportunities and the

resulting revenue cash flows, might not be rejected by data. We outline below a framework

for empirical studies of such noisy signaling equilibria, following the empirical procedure

in Feldman et al. (2014).

Using the example above, one can first identify, using nonparametric methods, the

empirical joint probability distribution function of levels of corporate debt interest pay-

ments and levels of corporate revenue cash flows. Then, one would use this distribution to

construct a hierarchy of distributions of revenue cash flows conditional on ever-growing

levels of debt interest payments. If this hierarchical structure of conditional probability

distribution functions is ordered by first-order stochastic dominance, the necessary con-

ditions for a noisy signaling equilibrium cannot be rejected.

To refute the necessary and sufficient conditions, one would have to further construct a

hierarchy of functions of the derivatives of the probability density functions of corporate

revenue cash flows, normalized by the values of the probability density functions, con-

ditional on ever-growing levels of debt interest payments. When one presents these

functions as a hierarchy of graphs, if this hierarchy demonstrates monotone ordering—that

is, the graphs of this hierarchy lie on top of one another, not crossing each other—we

cannot reject the necessary and sufficient conditions for a signaling equilibrium, the

GMLRP.

3.5 Dividend signaling

Following the non-noisy corporate dividend signaling models of Bhattacharya (1979), John

and Williams (1985), and Miller and Rock (1985), empirical evidence supporting the

models has been weak. Downes and Heinkel (1982) found no evidence in favor of dividend

signaling firm value; DeAngelo et al. (1996) found no evidence of dividend signaling

future earnings; and Bernhardt and Robertson (2005) rejected conditions for dividend

signaling. It might well be the case that these models’ lack of better empirical support

stems from their being a degenerate special case of the noisy signaling equilibria we have

presented here.


123

3.6 IPO signaling

Models of IPO underpricing as a non-noisy signal, such as those of Allen and Faulhaber

(1989) and Welch (1989), received mixed empirical support. Narasimhan et al. (1993) find

more attractive alternatives to Welch (1989), and Michaely and Shaw (1994), in examining

several models of IPO underpricing, find in favor of adverse selection models and no

evidence in favor of signaling models. Again, it would be interesting to find out how noisy

signaling models would fare empirically where their degenerate special case the non-noisy

ones did not fare well.

3.7 Mutual fund share classes

Finally, we note that, unlike dealers in the new car market, sellers of mutual fund shares are

constrained sell to all buyers at the same price. They were not able to customize buyers’

prices in a signaling equilibrium and not happily resorting to a pooling equilibrium;

screening equilibria arose. Mutual fund shares were divided into classes, each with a

different fee structure. While some funds offer seven classes, the mode is three. With the

reduced pricing flexibility that such screening equilibrium offers, buyers are only partially

not fully segmented, and it seems that sellers have opted for other techniques. It is argued

by O’Neal (1999, 2003) that not only do sellers complicate and obfuscate the fee structure,

they also incentivize brokers to sell suboptimal classes to buyers. It seems, therefore, that

disallowing the noisy signaling equilibrium reduces societal welfare.

3.8 Market microstructure signaling

A central question in financial market microstructure studies is the association between

transaction sizes and spreads. If transaction sizes are increasing in private information,

market makers, in order to protect themselves, set effective bid-ask spreads to be

increasing in transaction sizes. Indeed, Feldman et al. (2014) find that when eliminating

transactions of zero effective spreads (which are not executed through market makers) and

also eliminating very large transactions (which are generally executed ‘‘upstairs,’’ not as

‘‘regular’’ size transactions), all partitions that split transaction sizes to higher sizes and

lower ones induce distributions of spreads, conditional on transaction sizes that are con-

sistent with the necessary and necessary and sufficient conditions for noisy signaling

equilibria.

4 Summary and conclusion

In many real-world situations, transaction prices are determined by interactions between

buyers and sellers. We apply the Spence/Riley signaling model enhanced by Feldman

(2004) and Feldman and Winer (2004) models for separating signaling equilibria under

random relations between costs and attributes to model such equilibria. We call these

equilibria pricing under noisy signaling equilibria. In these equilibria, buyers signal their

unobservable propensities to purchase by the information levels they have collected about

the relevant product/service and the selling process. Competitive sellers’ average selling

costs are decreasing functions of buyers’ information; thus, they set markups to be a

decreasing function of the signaled information. Buyers search for information up to the

level that minimizes their total costs: the markups they pay plus their search costs. Because

D. Feldman et al.

123

the relations between propensities and per-unit information search costs are not one to one,

the equilibrium separation by information search costs does not imply separation by

propensities. Thus, some buyers with low information search costs have lower propensities

than other buyers with higher costs. Because buyers with lower (higher) costs acquire more

(less) information and pay lower (higher) markups, this asymmetric information equilib-

rium allows pricing errors and is noisy.

Clearly, the existence of such equilibria depends on the relations between the proba-

bility distribution functions of buyers’ propensities to purchase and their search costs, and

on the effect of their propensities on sellers’ selling costs. We model buyers’ propensities

both as discrete and as continuous. Under discrete propensities, we show that a necessary

and sufficient condition for equilibrium is the ordering of buyers’ cost-density functions

conditional on propensities by the MLRP. Under a continuum of propensities, the neces-

sary and sufficient condition for pricing under a noisy signaling equilibrium is the ranking

of the cost-density functions under the original probability measure and those with respect

to the same probability measure modified by the ‘‘propensity payoff function’’ by the

GMLRP. Necessary conditions are the ordering of these probability distribution functions

by first-order stochastic dominance and also the ordering by first-order stochastic domi-

nance of the distribution of propensities conditional on costs.

We show that sellers who deviate from pricing under noisy signaling will be driven out

of our market and that buyers who do not ‘‘signal truthfully’’ incur extra costs. We gain

insights into markets’ product/service segmentation. While full segmentation—that is, a

situation where prices of all products, services, and product/service combinations are ex

ante fixed—prevents pricing errors because sellers screen buyers who self-select, ‘‘smaller

markets’’ might not justify the fixed costs of full segmentation. We also show how the US

market for new cars and retail shopping are examples of our model.

In the financial markets, we identify both segmented and signaling equilibria in bro-

kerage fees, discuss the potential inferiority of forced (only partially segmented) screening

equilibria in mutual fund share classes, signaling equilibria in corporate dividends, IPOs,

capital structure, and market microstructure. We hypothesize that as markets grow and

become more global and transaction costs decline, there will be less pricing under noisy

signaling equilibria and more fully segmented, more efficient markets.

Acknowledgments We thank Linda Pesante, participants of The Annual Conference on Pacific BasinFinance, Economics, Accounting, and Management, The Econometric Society European Meetings, TheAustralasian Finance and Banking Conference, and seminar participants at The University of New SouthWales, The University of Melbourne, and The University of Western Australia.

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