Strategic Price Complexity in Retail Financial Markets
Bruce Ian Carlin∗
***MARKET PAPER***
Abstract
There is mounting empirical evidence to suggest that the law of one price is violated in retail financial markets: there is significant price dispersion even when products are homogeneous. Also,
despite the large number of firms in the market, prices remain above marginal cost and may even
rise as more firms enter. In a non-cooperative oligopoly pricing model, I show that these anomalies
arise when firms add complexity to their price structures. Complexity preserves market power
and corporate profits by bounding the financial literacy of consumers. As consumers find it more
difficult to find the best deal, more of them optimally choose to remain uninformed about industry
prices, which ultimately leads to price dispersion and failure of competition. Professional advice
(i.e. an advice channel) removes this advantage, unless the firms increase aggregate complexity, decrease price dispersion across the industry, or institute incentive contracts with the advice channel.
Because retail markets are extremely large, such practices have important welfare implications.
∗
Fuqua School of Business, Duke University, bruce.carlin@duke.edu I thank Jim Anton, Tom Beale, Gary
Biglaiser, Michael Brandt, Joe Cima, Jim Friedman, Simon Gervais, Dan Graham, Joe Harrington, Pete Kyle,
Tracy Lewis, Pino Lopomo, Leslie Marx, Rich Mathews, Christine Parlour, Uday Rajan, Adriano Rampini, David
Robinson, Stephen Ross, Ariel Rubinstein, Dale Stahl, Curt Taylor, S. Viswanathan, Bob Winkler, Huseyin Yildirim
and seminar participants in the 2006 Duke Economics and Finance seminars.
1
1
Introduction
Price formation in retail financial markets deviates from the predictions of standard price theory in
several important ways. The law of one price is violated: significant price dispersion is present when
goods and services are homogeneous. Despite the large number of firms in each market, prices do
not converge to marginal cost. Even when new firms enter the industry, prices often do not decrease
and may in fact rise. These pricing irregularities have been documented empirically in the markets
for credit cards (Ausubel 1991), conventional fixed-rate mortgages (Baye and Morgan 2001), S&P
Index funds (Hortacsu and Syverson 2004), life annuities (Mitchell, Poterba, Warshawsky, and
Brown 1999), and term life insurance (Brown and Goolsbee 2002).
What is responsible for this seeming departure from classic microeconomics? The answer that
I explore builds on the fifty year-old observation by Scitovsky (1950) that ignorance is a source
of oligopoly power. Producers of retail financial products create ignorance by making their prices
more complex, thereby gaining market power and the ability to preserve industry profits.1 Many
of the households who purchase retail financial products do not understand clearly what they are
buying and how much they are paying for these goods.2 The complexity that firms add directly
affects consumer literacy about prices and can generate the irregularities noted above.3
In this paper, I consider the following important questions: Given the “bounded” financial
literacy of households, how do firms compete on price and optimally add complexity to their
price structures? How does complexity affect price formation? What is the effect of firm entry
on complexity and pricing? Do professional financial advisors (i.e., an advice channel) alleviate
complexity or make things more complicated? What are the policy implications of this analysis?4
1
While this strategy remains relatively unexplored in financial markets, it has been documented in other industries
(Ellison and Ellison 2004).
2
There is now a large empirical literature which documents this fact. Evidence from the mutual fund industry
includes Capon, Fitzsimons, and Prince 1996; Sirri and Tufano 1998; Alexander, Jones, and Nigro 1998; Jain and Wu
2000; Barber, Odean, and Zheng 2003; Jones and Smythe 2003; Wilcox 2003, and Choi, Laibson, and Madrian 2006.
Evidence from the 401k and retirement investment literature includes Agnew and Szykman 2004, Choi, Laibson,
and Madrian (2004, 2005, 2005a, 2005b) and Calvet, Campbell, and Sodini (2006). Evidence from the marketing
literature includes Morwitz, Greenleaf, and Johnson 1998; Iyegar and Lepper 2000; Chernev 2003 and Gourville and
Soman 2005.
3
Admittedly, complexity may not be solely responsible for the pricing artifacts in each of these markets. Other
frictions induced by differentiation and switching costs may also play a role (Calem and Mester 1995). However,
differentiation and switching costs do not completely account for the failure of competition in these markets (Hortacsu
and Syverson 2004). According to Ausubel (1991), there are not enough dimensions on which to differentiate credit
cards to account for the absence of perfect competition with 4,000 lenders.
4
The importance of these issues was outlined in the 2006 Presidential Address to the American Finance Association
2
I address these questions using a three-stage pricing complexity game in which homogeneous
firms produce an identical financial product and compete on price for market share. In the first
period, firms simultaneously choose their prices (mutual fund fees, interest rates, etc.) and the
complexity of their price structures. That is, they decide how much to partition their prices into
direct and indirect fees, whether to hide some charges in the fine print, or whether to generate
new classes of fees not used by other firms. In the second period, consumers with heterogeneous
search costs choose whether to gain perfect information about the price structure in the industry or
whether to remain uninformed. The search costs in this model could represent the price of financial
education or the time it takes to evaluate each firm’s price structure. Those who pay this cost are
called the informed and those who do not are the uninformed. The decision to become informed is
endogenous in the model and is affected by both the prices and complexity in the market. In the
third period, informed consumers purchase from the low-priced firm, whereas uninformed consumers
choose randomly from all of the firms.
In equilibrium, price dispersion arises because the firms compete strategically for market share
from both types of consumers. The firm with the actual low-price captures the entire share of
the informed. All of the firms, however, receive some demand from the uninformed. The firms
never charge marginal cost because they gain positive expected profits from sales to uninformed
consumers. Also, it is impossible to have a “one-price” equilibrium in which all firms charge the
same prices for their products. If they did so, one firm could undercut their competitors by a small
amount and gain the market share from the informed consumers. So, equilibrium prices are strictly
higher than marginal cost and there is always dispersion in prices.
The aggregate complexity of the industry is determined through strategic interaction between
the firms. In equilibrium, the firms choose complexity as a monotonic function of the price that they
charge. The low-price firms choose a complexity level that is lower than that for high-price firms.
Since the low-price firms want consumers to know that they have the cheapest prices, they want
the industry to be as clear as possible. Industry clarity allows them to “undercut” their rivals and
gain market share. In contrast, high-price firms desire more complexity. The more “difficult” the
industry, the more uninformed consumers there will be and the more market share the high-price
by John Campbell (Campbell 2006). Given the size of retail markets, frictions imposed by firms in the market have
significant economic impact. In 2005, $2.8 trillion of home mortgages were initiated (Freddie Mac, June 2006 Economic
Outlook). Also, 53.7 million American households owned mutual fund shares, accounting for over $8 trillion in funds
(Investment Company Institute 2005).
3
firms will receive. Industry confusion is the way high-price firms gain market share.
After deriving the equilibrium of the game, I consider several implications of the model. I first
evaluate the effect of entry of new firms on the distribution of prices and complexity. I find that
increased competition always leads to higher industry complexity. When more firms compete for
market share, the probability that they receive demand from the informed consumers decreases. To
maximize expected profits, the firms tend to increase complexity in order to optimize the revenues
they receive from uninformed consumers. Increased complexity and higher cognitive load makes
it harder for consumers to become informed. If a larger fraction of consumers remain uninformed
when more firms are present, then prices rise. Hortacsu and Syverson (2004) have previously shown
that entry into the S&P index fund industry in 1996-1999 was associated with a rightward shift in
the distribution of prices.
The second major implication that I consider is the effect of an advice channel. In the mutual
fund industry, the advice channel is a system of brokers who offer investment advice to household
investors. In the credit card, life insurance, and mortgage industries, price search engines assume
the role of advisor. Search engines “advise” consumers about the range of prices (interest rates
and fees) available in the industry and detect the low-price product. To study the pricing effects
induced by an advisor, I extend the original three-stage model to a four-stage model. The firms
establish the industry as before, but in the second stage, the industry experts (those with negligible
search costs) form an advice channel and market their information services to consumers. They
charge consumers an advisory fee and incur an education cost of providing this service. In the third
stage, consumers decide whether to pay their own search costs, pay for advice from the channel, or
remain uninformed. In the fourth stage, consumers make their purchases based on the information
that they have.
In this version of the model, if the education costs to provide information are less than the
benefits to gaining information, perfect competition arises because the advice channel markets
their services to consumers with high search costs. All consumers become informed and price
dispersion disappears. According to the model, the firms can prevent this outcome in three ways:
they can increase the complexity in the industry (driving up the costs of educating consumers),
they can decrease the price dispersion in the market (lowering the benefit of perfect information),
or they can sign contracts with the advice channel that align incentives to preserve industry profits.
4
All three of these strategies have been documented empirically (Brown and Goolsbee 2002; Ellison
and Ellison 2004), though they remain relatively unexplored in financial markets.
Finally, I consider the policy implications of the analysis. The oligopoly power induced by complexity may be limited in three general ways. First, if the firms were forced to disclose prices clearly,
the industry would be more competitive. Requiring firms to disclose prices, but not restricting the
form in which these disclosures are made, will not improve consumer surplus. The second way to
increase consumer surplus is to restrict the types of contracts that firms may have with the advice
channel. By limiting the incentives that firms may offer the advice channel, this would remove an
important mechanism whereby the firms can protect industry profits. Finally, increasing financial
literacy by making financial education more efficient (lowering the cost of education in the model)
should also push the industry toward clarity and lower prices.
The model in this paper has several empirical implications. Since complexity is an important
source of value for firms, changes in complexity should be positively correlated with firm profitability. The following quote provides a recent example of how Wall Street values the effect of
complexity on firm profitability (New York Times, July 13, 2006):
“...Dell Inc. said Thursday it plans to gradually reduce its use of mail-in rebates and
its often confusing array of other promotional offers. Ro Parra, senior vice president
of Dell’s Home and Small Business Group, acknowledged the various programs had
become increasingly complex and cumbersome for customers trying to get the best deals
on computers and other electronics....The move, however, did not impress Wall Street.
Dell shares fell 3 percent on the Nasdaq Stock Market.”
This quote suggests that as prices become less opaque, producer surplus decreases leading to a
lower stock valuation. It also suggests that financial firms are aware that complexity may be used
as a strategic tool when they price their own products.
The model also predicts that as competitive pressures rise in an industry, firms will respond by
adding more complexity to their prices. This implies that as firms enter the market and industry
concentration falls, the complexity should increase.5 The model also suggests that the presence of
an advice channel should cause industry complexity to rise and price dispersion to fall. Note that
5
One way to test this prediction would be a time-series that correlates industry concentration (i.e., HerfindahlHirschman index) to the level of price complexity present in public disclosures (i.e., mutual fund prospectus or a
credit card disclosure).
5
this does not imply that prices will fall, rather they should be more concentrated when an advisor
is present.
Besides modeling pricing behavior in retail financial markets, another contribution of this paper
is to extend the literature on oligopoly pricing with consumer search.6 Many existing models address
pricing strategies, but do not provide insight into the role of information in these markets. Prices
are dispersed in equilibrium, but firms are unable to affect the search efficiency of their consumers.7
It seems natural to examine all of the strategic alternatives that are available to firms, that is to
model how firms “hide” information about prices when consumers search. The “hide and seek”
pricing game that I model is new to this literature, providing insight into the role of information
in achieving industry price dispersion.
The rest of the paper is organized as follows. In Section 2, I set up and solve the three-stage
pricing complexity game. I characterize the strategic behavior of the firms and prove existence of
a symmetric mixed strategy equilibrium of the game. In Section 3, I analyze the effect of entry on
industry prices and complexity. In Section 4, I analyze the four-stage game with an advice channel.
In Section 5, I discuss the policy implications of the paper. Section 6 concludes. The Appendix
contains all of the proofs.
2
Strategic Pricing and Complexity
2.1
The Market
Consider an economy in which n firms, indexed by j ∈ N = {1, ..., n}, produce a homogeneous retail
financial product. The product may be used by households to finance the purchase of consumption
goods (for example, a credit card) or as an investment vehicle to maximize lifetime utility (for
example, an index fund). The firms face zero marginal costs and have no capacity constraints.
None of the firms have an informational or skill advantage compared to the others. They have
6
This literature includes Diamond 1971; Salop and Stiglitz 1977; Weitzman 1979; Stiglitz 1979; Reinganum 1979;
Braverman 1980; Varian 1980; Braverman and Dixit 1981; Salop and Stiglitz 1982; Burdett and Judd 1983; Carlson
and McAfee 1983; Rob 1985; Perloff and Salop 1985; Stiglitz 1987; Stahl 1989; Robert and Stahl 1993; Stahl 1996;
Anderson and de Palma 2003; Gabaix and Laibson 2005; Gabaix, Laibson, and Li 2005; Spiegler 2005.
7
One exception is the model by Robert and Stahl (1993). In their model, firms choose prices and whether to
advertise (inform) their price to a proportion of consumers, given that consumers undertake sequential search. In
this paper, I consider the flip-side of the problem. That is, I evaluate the extent to which consumers are uninformed
in the first place.
6
equal access to consumer marketing data. In the case of providing investment services, they make
identical portfolio decisions based on market conditions and achieve the same expected return
and return variance. The only difference between the firms is the price that they charge and the
complexity that they add to their price structures.8
In the market, there is a unit mass of consumers M who each have unit demand for the retail
good. Every consumer i is risk-neutral and maximizes the expected payoff from their purchase.
Their utility is given by
Ui = v − pi − δi ci
where v is the fundamental value of the product, pi is the price that they pay, ci is the cost of learning
about prices, and δi ∈ {0, 1} represents consumer i’s binary decision whether to become informed.
The value v is commonly known among the consumers and can be considered the monopoly price
for the good. Because consumers receive v when they purchase the good from any firm, maximizing
utility in this market is equivalent to minimizing the sum of price and effort (search costs).
A consumer who chooses to learn about industry prices through costly search (ci ) will purchase the good at the lowest price available, pmin = minj {pj }nj=1 . Alternatively, if they remain
uninformed, they purchase the good randomly from the firms with probability n1 , resulting in an
P
expected price p = n1 nj=1 pj . The decision to become informed is an “all or nothing” choice, that
is, consumers may either become fully informed or may remain completely uninformed. For tech-
nical simplicity, I do not include other adaptive decision-making procedures (i.e., rules of thumb or
heuristics) that would allow consumers to narrow the field of choices and make a partially-informed
decision.9 The model could be generalized to include such intermediate levels of knowledge, at the
expense of clarity.
All or nothing search has been used previously by Salop and Stiglitz (1977) and Braverman
(1980).10 Searching in this fashion is costly when prices are complex. If it is difficult for consumers
to find the important price components (hidden fees and add-ons) and computation is necessary,
8
Restricting the firms to produce an undifferentiated good is not for technical convenience. Rather, if complexity
in fee structures causes failure of competition and price dispersion in the market, adding heterogeneity in the product
attributes will only make price dispersion more likely.
9
Making choices randomly ( n1 ) is the most severe form of adaptive decision making. Camerer, Ho, and Chong
(2004) refer to this type of consumer as a “step 0” type. For a review of adaptive decision-making, see Payne,
Bettman, and Johnson (1993).
10
Shugan (1980) describes all or nothing search as a “lowest-price tournament” in which n − 1 sequential price
comparisons are required to determine an overall winner. For a full review of search technology, see Stahl (1989).
7
t=1
Each firm chooses
Price pj and Complexity kj
t=2
Consumers choose whether to be informed.
They pay ci if ci ≤ B.
t=3
Consumers make purchase
Figure 1: Pricing Complexity Game. At t = 1, n financial services firms establish the industry
structure. They each choose prices pj for their products and the complexity kj of their prices. At
t = 2, consumers calculate the expected benefit B of perfect information. They choose whether
to pay a cost ci to fully observe prices as long as ci ≤ B. Otherwise, if they remain uninformed,
they purchase their products from the firms randomly. Finally, at t = 3, informed
consumers pay
P
pmin = minj {pj }nj=1 and uninformed consumers pay an expected price p = n1 nj=1 pj .
it takes more effort to compare prices and find the cheapest price. Also, every firm’s complexity
choice affects the cost of the entire analysis because it is mandatory to compare every firm’s price
to all of the others.
The pricing complexity game is a three-stage game (Figure 1), which proceeds as follows. In the
first period (t = 1), the firms simultaneously set prices for the product and decide how complicated
to make their price structure. That is, each firm chooses pj ∈ [0, v] and kj ∈ [k, k] where pj is the
actual (total) price that firm j charges and kj is a measure of how difficult it is to sift through the
price on all dimensions.11 I define Σj = [0, v] × [k, k] to be the strategy space for firm j and σj ∈ Σj
to be firm j’s (mixed) strategy over prices and complexity. In any Nash equilibrium, the strategies
of the firms are given by the vector σ ∗ = [σ1∗ , . . . , σn∗ ].
Once the first period is over, consumers decide whether to become informed about pricing in
the industry. For simplicity, I assume at this point that consumers are endowed with specific
statistics about industry prices before making search decisions. Explicitly, they observe the mean
p and minimum pmin prices in the industry, but they do not know which firm offered the actual
low price. This simplification seems compelling because in reality consumers often know important
statistics about industry prices, but do not know where to get the best deal. Also, this assumption
is consistent with traditional models in the search literature. For example, in Salop and Stiglitz
−
→
→
(1977), consumers are given a vector of prices −
p and a vector of locations l , but they do not know
11
The firms may choose any value for kj in [k, k] without paying a cost for doing so. The goal is to generate
strategic choices for kj that are independent of an imposed exogenous cost function. The model could be extended
easily to handle costly choices of kj .
8
which location offers which price.12
Before choosing whether to become informed, consumers also observe the aggregate industry
complexity, which I denote by K. However, they do not observe the vector of complexity choices
k∗ = [k1∗ , . . . , kn∗ ] without spending effort analyzing all of the price structures in the industry. In
this model, complexity is considered to be an “experience” good since consumers do not know how
complicated the process is for each firm until they commit ci to analyze prices. K is a multivariate
map
K : [k, k]n → R+
such that K ∈ C 2 and
∂K
∂kj
> 0. That is, as any one firm makes their price more difficult to evaluate,
it makes the whole industry more costly to analyze. A high level of K means that the price structure
in the industry as a whole is cumbersome and opaque, whereas a low level of K means that the
industry price structure is lucid. I define Kmin = K(k, . . . , k) as the lower bound on K.
At t = 2, investors are allowed to pay a personal cost ci to gain perfect information about which
firm has the lowest price. The cost ci could represent the price of financial education or the time it
takes to evaluate every firm’s price structure. These search costs are distributed in the consumer
population according to the distribution function G(c|K) ∈ C 2 such that c ∈ [0, c] and G(0|K) > 0.
The distribution function G(·) can be considered a gauge of the population’s cognitive ability to
analyze a financial product. For example, if
G1 (c|K) ≥ G2 (c|K)
holds for all c, then population 1 is more financially literate than population 2. The key is that the
distribution of search costs drives the population’s ability to make informed purchasing decisions.
I do make the weak assumption that G(0|K) > 0, so that there will always be a small group
of industry “professionals” who have negligible search costs. So, no matter how complicated the
industry is, there will always be some informed consumers.13
The distribution of search costs also depends on the level of industry complexity K. For two
12
According to Salop and Stiglitz (1977), they “move only one step away from perfect information” in order to
analyze how consumers search for the best alternative.
13
The mass point in the distribution at c = 0 can be made arbitrarily small, making the assumption as weak as
one wants. However, having at least an atom at c = 0 implies that there will always be some consumers who are
informed. In Section 4, these are the consumers who can form an advice channel in the industry.
9
levels of complexity K′ and K′′ such that K′ > K′′ , then
G(c|K′ ) ≤ G(c|K′′ )
for all c. Increasing the aggregate complexity (K) raises the costs that consumers need to pay to
become informed. This is an important channel in which the firms’ strategic interaction affects the
consumer types that they face in the market.
Consumers will rationally pay ci as long as it is less than the expected benefit to clearly observing
the prices from all n firms. I denote this benefit by B, which is simply
B = p − pmin .
If the consumer remains uninformed and chooses randomly from the n firms, the expected price
that they will pay is p. If they become informed, they will purchase the product from the low-price
firm and pay pmin . For a given B, G(B|K) is the fraction of the consumer population that observes
prices perfectly.
In the third stage of the model, consumers make their purchases. The firm with the low price
receives demand from the entire mass of informed consumers as well as their share ( n1 ) of the
uninformed consumers. The other firms receive only
2.2
1
n
of the uninformed consumers’ demand.
Firm Behavior
In this section, I consider the firms’ problem of creating an industry price structure. I pose the
optimization problem faced by the firms and prove general existence of a Nash equilibrium for the
game. I show that the equilibrium involves a positive price mark-up over marginal cost and that a
symmetric equilibrium in pure strategies cannot exist. That is, there will always be price dispersion
in this industry. In order to study the effect of certain market conditions on the distribution of
industry prices, I derive a symmetric mixed strategy equilibrium for the game.
Define J ∗ to be the set of firms who quote the lowest price. Let nj ∗ be the number of firms in
J ∗ , so that the nj ∗ firms in J ∗ split the demand from the informed consumers equally. Each firm
10
j chooses pj and kj to maximize its expected profit, that is, they solve
max
p j Qj ,
pj ∈[0,v]
kj ∈[k,k]
where the expected demand Qj is calculated as
Qj =
G(B|K) 1{j∈J ∗ } 1 − G(B|K)
+
.
nj ∗
n
Qj is composed of two parts. The first expression represents the demand from the fraction G(B|K)
of informed consumers. Firm j receives
1
nj ∗
of this demand if they are one of the nj ∗ firms that
quote the lowest price in the industry. The second expression represents the expected demand from
[1 − G(B|K)] uninformed consumers.
The following proposition establishes the general existence of a Nash equilibrium in this pricing
game and characterizes some of its properties.
Proposition 1. (General Existence) A Nash equilibrium of the pricing complexity game exists and
has the following properties:
1. (Absence of the Bertrand Paradox) There does not exist an equilibrium in pure strategies in
which all firms choose a price p = 0. If c > v, then choosing p = 0 is a strictly dominated
strategy.
2. (Price Dispersion) There does not exist a symmetric equilibrium in pure strategies.
3. (No ties for lowest price) There does not exist a pure strategy equilibrium in which nj ∗ ≥ 2.
4. When n = 2, no asymmetric equilibrium in pure strategies exists.
According to Proposition 1, prices are always above marginal cost (Property 1). Consider,
without loss of generality, firm 1’s payoff in the game and suppose that all other firms choose
p−1 = 0. If p1 = 0, then the benefit to becoming informed is B = 0 and the expected profit is
π1 (0) = 0. For any p1 > 0, the expected benefit is positive (B =
11
p1
n ),
so that as long as B < c,
there will be a positive measure of uninformed consumers that will purchase randomly from all
of the firms. Therefore, if firm 1 chooses p1 ∈ (0, nc), π1 (p1 ) > 0, which rules out a Bertrand
equilibrium.14 It follows that if the upper bound on search costs c is greater than the value of the
financial product v, then there will always be uninformed consumers in the market. In this case,
setting p = 0 is a strictly dominated strategy.
Proposition 1 also implies that a one-price equilibrium is impossible (Property 2). This means
that, in equilibrium, there will be positive price dispersion between the firms in the industry.
Consider, again without loss of generality, that all firms except firm 1 choose a pure pricing strategy
such that p−1 = p. Then,
π1 (p) =
p
.
n
Since for ǫ > 0,
"
#
[1 − G(B(p − ǫ)|K)]
p
> ,
π1 (p − ǫ) = (p − ǫ) G B(p − ǫ)|K +
n
n
then firm 1 has an incentive to undercut its competitors. The last inequality results from the fact
that G(·|K) is continuous and there is a market share (mass of informed consumers) that firm 1 no
longer has to share. In the same way, it follows that two or more firms cannot have the lowest price
in the industry in a pure strategy equilibrium (Property 3). If that were so, a profitable epsilon
deviation would be available that would break the tie. Therefore, we know that there is always a
non-degenerate distribution of prices in the market.
Price dispersion will arise if the firms play a mixed strategy in equilibrium or if an asymmetric
equilibrium in pure strategies exists. By Property 4, this latter case is impossible with only two
firms (and may be for n ≥ 3, but this remains unknown). In order to evaluate how the distribution
of prices changes in response to other market conditions (in particular the effect of entry and the
presence of an advice channel), it is useful to analyze the strategies that firms will choose in a
mixed strategy equilibrium. I now derive a symmetric mixed strategy equilibrium for the game.15
14
In the model, the lower limit for kj is k. If k > 0, then there will always exist some complexity in each price
structure. Proposition 1 also holds when k = 0. In this case, the if all firms choose p = 0 and k = 0, there is a
profitable deviation for firm 1 to choose p1 > 0 and k1 > 0 so that K > 0 and E[π1 ] > 0. A proof ruling out a “double
Bertrand” equilibrium is straightforward to construct.
15
This type of mixed-strategy is present in other models of competitive pricing with consumer search (Varian 1980;
Rosenthal 1980; Stahl 1989; Robert and Stahl 1993; Stahl 1996).
12
Define Πj pj , kj |σ−j as the expected profit for firm j when it chooses pj and kj , given the
symmetric mixed strategies of the other firms σ−j . From firm j’s point of view, the prices of the
other firms are random variables (e
p1 , ..e
pj−1 , pej+1 , ...e
pn ), which implies that G(B|K) is a random
variable. So, when firm j computes their optimal mixing function and expected profit, they must
consider the conditional expectation of the fraction of informed consumers, given the mixing distributions of the other firms. As such, let us define Γ(pj , kj , σ−j ) to be the conditional expectation
of G(B|K) given a choice of kj and pj for firm j and the strategies of the other firms. That is,
Γ(pj , kj , σ−j ) ≡ Eσ−j [G(B|K)|pj , kj ].16
Therefore, each firm maximizes their expected profit, which is expressed as
h
1 − Γi
Π(pj , kj |σ−j ) = pj Γ[1 − F (pj )]n−1 +
.
n
The following proposition establishes the existence of a symmetric mixed strategy equilibrium and
characterizes the strategic choices of the firms.
Proposition 2. (Symmetric Mixed Strategy Equilibrium) In the pricing complexity game, there
exists a symmetric equilibrium σ ∗ = {F ∗ (p), k∗ (p)} in which firms randomize over prices in [p, v]
according to F ∗ (p) and choose complexity as a deterministic function of p according to k∗ (p). The
lower bound p is computed as
p=
v[1 − ΓH ]
,
(n − 1)ΓL + 1
(1)
where ΓH = Γ(v, k, σ−j ), and ΓL = Γ(p, k, σ−j ).
The function F ∗ (p) solves
"
v[1 − ΓH ] − p[1 − Γ])
F (p) = 1 −
npΓ
#
1
n−1
(2)
16
For clarity, I will use Γ in the rest of the paper to represent Γ(pj , kj , σ−j ), unless it is necessary to specify its
arguments.
13
and k∗ (p) is given by
where

 k
k∗ (p) =
 k
p̂ = F
−1
1−
if
p < p̂
if
p > p̂,
h1i
1
n−1
n
(3)
.
F ∗ (p) is continuous and strictly increasing in p, ∀p ∈ [p, v].
According to Proposition 2, the firms randomize over prices p ∈ [p, v] according to a distribution
function F ∗ (p) and choose kj using a pure strategy given their draw from F ∗ (p). It is not optimal
for any firm j to mix over the entire strategy space Σj , that is, k∗ (·) is a deterministic function of
p. The firms choose kj in a binary fashion according to the p that they draw from F ∗ (p).17 Highpriced draws (above a threshold level p̂) will choose maximal complexity and low-priced draws will
choose minimal complexity. This means that if a firm has a low price, they want consumers to be
informed about it. The low-price firms will choose k so that their contribution to the aggregate
complexity K is minimized. In this case, they desire the fraction of informed consumers G(B|K)
to be the highest it can be. In contrast, if a firm has a high price (above p̂), they want consumers
to be poorly informed. The high-price firms will choose k to maximize their contribution to K in
order to minimize G(B|K).
By inspection,
p=
v[1 − ΓH ]
(n − 1)ΓL + 1
is monotonically decreasing in both ΓH and ΓL . That is, as the expected fraction of informed
consumers increases, the lower bound on the support of their mixing distribution decreases. As
ΓH and ΓL approach one, p → 0 and as ΓH and ΓL approach zero, p → v. This means that if
consumers are less literate about financial decisions, the bound on the lowest price possible rises.
As the fraction of uninformed consumers reaches its upper limit, prices will approach the monopoly
price. The relationship between p and n, however, is more complicated and is addressed in Section 3.
17
Since the equilibrium distribution F ∗ (p) is strictly increasing and continuous, the event that a consumer draws
p = p̂ from F ∗ (p) is of measure zero. Therefore, I do not consider this outcome in the analysis.
14
The expression in (2) is implicit in F (p), since both ΓH and Γ depend on F (p). It is not
possible to express the general solution to (2) explicitly, but a solution (fixed point) is guaranteed
by Schauder’s fixed point theorem (see proof in Appendix). It is possible to use the expression in
(2) and the lower bound p in (1) in specific cases to analyze the amount of price dispersion that
is expected in the market. The examples in Section 3 highlight this. Also, since k∗ (p) depends on
F ∗ (p), market conditions such as entry and an advice channel will affect price dispersion as well as
complexity through their effects on F ∗ (p). The effect of these market conditions are the subject of
Sections 3 and 4.
3
The Effect of Entry on Industry Complexity and Prices
When the number of firms in the industry increases, how does the distribution of prices change?
What effect does entry have on complexity (K)? In what follows, I show that as more firms
compete in the industry, the probability that each firm adds complexity rises, which leads to higher
aggregate complexity. This occurs because the firms use complexity as a way to preserve oligopoly
power. Interestingly, as the level of complexity rises, the prices in the industry may rise as well.
This is counter-intuitive as classic theory tells us that more competition leads to lower prices and
convergence to marginal cost. In contrast, this model predicts that the use of complexity may cause
failure of price competition and rising prices following entry.18
Proposition 2 shows that some firms will choose a baseline level of complexity (k) and some will
choose to add more complexity than is required (k). The probability of choosing kj = k for each
firm is
1 − Fn (p̂) =
h1i
1
n−1
n
.
(4)
The subscript n is added to indicate that Fn (p̂) depends on the number of firms, not a partial derivative with respect to n. Since the right-hand side of (4) is increasing in n, each firm’s probability of
choosing high complexity is monotonically increasing in n.
To evaluate the effect of n on the expected amount of industry complexity E[K], it is helpful to
e
define the function K(m)
as the increase in aggregate complexity over Kmin when m firms choose
e
k and the rest choose k. Clearly, if all firms choose low complexity, K = Kmin and K(0)
= 0.
18
Rosenthal (1980) and Stiglitz (1987) also find rising prices with increased industry competition.
15
Also, since
∂K
∂kj
e
> 0, then K(m)
is monotonically increasing in m, that is,
e
dK(m)
dm
> 0. Given this,
the expected aggregate complexity may be computed as Kmin plus the expectation of a binomial
distribution
E[K] = Kmin +
n X
n
m=0
h 1 1 im h
1 1 in−m
n−1
n−1
e
K(m)
1−
.
m
n
n
The following proposition characterizes the effect of entry on industry complexity.
Proposition 3. (The Effect of Entry) In the pricing complexity game, the probability that a firm
chooses high complexity (k) is monotonically increasing in n. In the limit as n → ∞, all firms
choose k. The expected aggregate industry complexity E[K] is increasing in n.
Proposition 3 implies that firms will use complexity as a way to protect market power and
oligopoly profits as the number of competitors increases. This is because their chances of winning
the business of the informed consumers decreases as the number of competing firms increases. By
adding more complexity, each firm increases the fraction of consumers who are uninformed, which
increases their profits in the case that they do not win demand from the informed consumers.
To see how n affects the expected aggregate industry complexity E[K], consider the following
example. Suppose that the aggregate complexity is the average of the choices of all n firms, that
P
e
= mα
is, K = n1 j∈N kj . Let α = k − k. Then, Kmin = k and K(m)
n . The expected aggregate
complexity is calculated as
E[K] = k +
n 1 i h
X
m
n mα h 1 n−1
m=0
m
n
n
1−
1
n
1
n−1
in−m
.
Since the second term is an expectation of a binomial distribution with probability
h 1
n
1
n−1
i
,
h 1 1 i
n−1
E[K] = k + α
.
n
It is straightforward to show that
∂E[K]
∂n
> 0, that is, entry leads to higher expected industry
complexity.
To gain some intuition about how the added industry complexity affects the distribution of
16
0.9
0.8
0.7
0.6
F (p) 0.5
0.4
0.3
α=1
α=2
α=4
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
p
0.6
0.7
0.8
0.9
Figure 2: The distribution of prices as a function of industry complexity. The parameter α is
defined such that α = k − k. For a given k, as α increases, F (p) shifts to the right, indicating
P rising
prices. Plots are given for α = 1, α = 2, and α = 4. The other parameters are K = n1 j∈N kj ,
v = 1, and n = 15.
prices, consider the example in Figure 2.19
Suppose that K =
1
n
P
j∈N
kj and let v = 1 and
n = 15. Again, let α = k − k and set k = 14 . The figure plots the distribution of prices for α = 1,
α = 2, and α = 4. As the upper bound of complexity rises from α = 1 to α = 4, the distributions
(and p) shift to the right. In fact,
F (p|α = 4) ≤ F (p|α = 2) ≤ F (p|α = 1).
for all p, which means that the distribution of prices with a higher bound on complexity first-order
stochastically dominates that with a lower bound. A similar plot may be generated by allowing k
to vary as well. For higher levels of k, the distribution of prices also shifts to the right. Clearly, in
this example if a policy-maker was able to bound α, then prices would become more favorable to
consumers. I explore these policy implications in Section 5.
Now, let us consider an example in which we fix the bounds on complexity and evaluate the
distribution of prices when n varies. Consider the plots in Figure 3. Using the same algorithm
19
The plot in Figure 2 was generated by discretizing F (p) over a finite set of prices {0, 0.1, . . . , 1} and running a
fixed point numerical algorithm. In less than ten iterations, convergence was reached. The process was repeated for
finer partitions of prices without any difference in the rate of convergence.
17
F (p)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
F (p)
0.5
0.4
0.4
0.3
0.3
n = 12
0.2
0.1
0
0.2
n=8
0
n = 10
0.2
n = 50
0.1
n=4
0.4
p
n = 100
0.6
0.8
0
1
0
0.2
0.4
p
0.6
0.8
1
Figure 3: The distribution of prices as a function of the number of firms in the industry. In the
first panel, plots are given for n = 4, n = 8, and n = 12. As n increases for small n, more weight
is placed in the tails of the distribution. However,as n increases for large n, Fn+1 (p) stochastically
dominates Fn (p). This is observed in the second panel where plots are given for n = 10, n = 50,
and n = 100. The other parameters are v = 1, k = 34 , and k = 14 .
as in Figure 2, the left-hand panel plots Fn (p) for n = 4, n = 8, and n = 12, and the right-hand
P
panel plots Fn (p) for n = 10, n = 50, and n = 100. The other parameters are K = n1 j∈N kj ,
v = 1, k =
3
4,
and k =
1
4.
When n is low, entry induces the firms to add more density to the
tails of the distribution. As competition grows initially, firms add density to the low prices to
increase their chances of winning market share from the informed consumers. However, because
competition lowers their overall chances of winning, they also add density to the higher prices to
gain more surplus from the uninformed consumers when they lose. Therefore, price dispersion is
higher on average as n grows initially from four firms.
Once n becomes large, the probability that each firm receives demand from the informed consumers becomes very small compared to the gain in surplus (from raising prices) available from the
uninformed consumers. Therefore, once n becomes large the effect of entry is for the distribution
of prices to shift to the right. That is, firms will transfer more density to the right tail of the
distribution. This trend is evident in the right-hand panel and was documented in the S&P index
fund industry in 1996-1999 by Hortacsu and Syverson (2004).
The preceding example shows that entry may not only lead to higher complexity, but also
to higher prices. It would also be nice to know conditions under which we are guaranteed that
Fn+1 (p) first-order stochastically dominates Fn (p). For stochastic dominance to occur with entry,
18
it is necessary that pn+1 ≥ pn and sufficient that Γn+1 ≤ Γn . If both of these conditions are
present, then prices will increase as more firms enter the market. Two factors may cause Γn
to decrease: cognitive overload and rising complexity. Increased cognitive load is captured in
this model by comparing the underlying distribution of search costs when n + 1 and n firms are
present. If Gn+1 (c|·) ≤ Gn (c|·) for all c, then some consumers will reach cognitive overload when
considering an extra option in the market and fewer will choose to become informed. Likewise, if
the underlying distribution changes such that G(B|Kn+1 ) ≤ G(B|Kn ), then rising complexity will
lead to Γn+1 ≤ Γn . Both of these effects may coincide simultaneously.
It is impossible, though, to prove a stronger stochastic dominance result based on the primitives
in the model because the expected benefit to information with n firms Bn is not monotonic in Fn (p).
That is, Fn+1 (p) < Fn (p) does not imply that Bn+1 > Bn . If Fn+1 (p) < Fn (p), then the mean rises
with n (pn+1 > pn ) and so does the expected minimum price (En+1 [pmin ] > En [pmin ]). However,
Fn+1 (p) < Fn (p) does not imply any monotonic relationship between the differences between these
two statistics.
4
The Effect of an Advice Channel
In Section 2, I consider a model in which firms directly market their goods and services to consumers. In this section, I determine how prices and complexity change when advisors can help
consumers select which financial instruments to purchase. I show that if the advice channel markets their expertise to consumers, they target consumers with high search costs, which causes all
consumers to become informed. In this case, price dispersion disappears and the market is perfectly
competitive. However, the firms may prevent this outcome in three ways: they can raise industry
complexity, decrease the benefit to becoming informed, or offer the advice channel incentives to
share in industry profits. Rising industry complexity causes the cost of education to rise, making
it less profitable for the advice channel to market its information services. Decreasing the expected
benefit of information involves making industry prices more concentrated (decreasing price dispersion). Concentration in prices is not equivalent to decreasing prices, however, as it is possible
for prices to rise, while still preventing the advice channel from enforcing competition. Signing
incentive contracts makes it more profitable for the advice channel to hold back information from
19
Purchase Channel by Households
Retirement Plan
Advice Channel
Direct Sale
Discount Broker/Supermarket
Share
48%
37%
10%
5%
Share of Mutual Fund Assets
Advice Channel
Retirement Plan
Institutional
Direct Sale
Supermarket
Percentage
55%
16%
13%
12%
5%
Table 1: Primary Distribution Channel Used by Households. Left-hand column lists the percentage
of households that purchase mutual funds though each channel. The right-hand column lists the
fraction of mutual fund assets purchased through each channel. Source: Investment Company
Institute July 2003.
consumers and preserve industry profits.
In the mutual fund industry, the advice channel is a system of brokers who have expertise about
the industry, but do not manage portfolios or trade directly. They offer advice and sell information
to investors. They distribute investment vehicles for the funds and resell investment products. In
the United States, the advice channel accounts for 55% of the assets under management in the
industry and 37% of households purchase mutual fund shares through the advice channel. Only
10% of mutual fund sales occur via a direct sale between mutual fund companies and individual
investors (Table 1).
In the mortgage, life insurance, and credit card industries, price search engines play the role of
advisor. Websites established by intermediaries allow consumers to search for the best rates. Even
though there is no “human” interaction at the website, the consumer depends on the search engine
in the same way that they depend on a financial advisor. Therefore, in the following analysis I
treat the objective functions (and resultant behavior) of the search engine and the financial advisor
as isomorphic.
Consider the four-period extension of the pricing complexity game in Figure 4. In the first
stage, the firms choose pj and kj as in Section 2. In the second stage, market participants who
have negligible search costs (the “professionals”) form an advice channel and sell their information
to fellow consumers for a price w.20 The professionals pay an advisory cost A(K) to educate each
20
I assume that the advice channel has monopoly power so that they earn positive profits from selling their
information to consumers. Monopoly power may arise if the advisor is a sole entity or if multiple advisors implicitly
collude over time to avoid the Bertrand paradox and maintain monopoly levels of w (Abreu 1988). If the advice
channel is competitive, then the results from the extended game are identical to those in Section 2.
20
t=1
Firms choose
prices pj and
complexity kj
t=2
Advice channel
sells its information for w
t=3
Consumers pay ci , w,
or remain uninformed
t=4
Consumers make
purchase decision
Figure 4: Pricing Complexity Game with an Advice Channel. At t = 1, the firms choose pj and
kj . At t = 2, the advice channel offers retail information services to the consumers at price w. At
t = 3, consumers choose whether to pay a cost ci to fully observe prices, pay w for advice, or choose
randomly from the firms. At t = 4, consumers purchase, given the information they possess.
consumer that they serve. This cost could be interpreted as the time-cost to educate others or the
cost of developing learning materials. I assume that A(Kmin ) = 0 and that A(·) is non-decreasing
in K. In the third stage, consumers decide whether to pay a personal cost ci , pay the price w for
advice, or remain uninformed. In the last period, consumers make their purchase decisions based
on what they know.
As in Section 2, we can start in period three and work backwards. At t = 3, consumers
calculate the expected benefit of perfect information B and then decide whether to pay for perfect
information. When w > B, no consumer pays the advice channel for information and the advice
channel makes zero profits. If ci ≤ w ≤ B, then consumer i becomes informed independent of the
advice channel. If w ≤ ci and w ≤ B, then they rely on the advice channel to become informed.
So, as long as w ≤ B, then 1 − G(w|K) of the consumers will purchase services from the advice
channel and G(w|K) will pay their own personal costs ci to gain perfect information.
In period two the advice channel solves
max πA = [w − A(K)][1 − G(w|K)]
w≤B
(5)
If A(K) < B, it is profitable for the advice channel to offer information services. In this case, since
the objective function is continuous and the action space [A(K), B] is compact, there exists a w∗
that maximizes the profits. All consumers become informed and firms with higher prices receive
zero demand. As long as A(K) < B , it is a weakly dominant strategy for all firms to charge
marginal cost. Price dispersion disappears and the market is perfectly competitive.
If A(K) > B, then price dispersion and industry profits persists because it is no longer profitable
21
for the advice channel to provide information services. There are two potential ways in which the
industry may achieve A(K) > B and prevent perfect competition. First, in aggregate the industry
complexity K can increase so that the cost of educating consumers becomes exceedingly high.
Second, the level of price dispersion may decrease, thereby lowering B, given a particular level of
A(K). It is important to note that a decrease in price dispersion is not equivalent to a drop in
prices. In fact, it is possible for prices to rise but become more concentrated (lowering B), which
would preserve industry profits and avoid erosion due to the presence of an advice channel.
Both of these strategies have been documented empirically, though they remain relatively unexplored in financial markets. Ellison and Ellison (2004) show that firms in the market for computer
memory and central processing units (CPU’s) increase complexity when there are price search engines in the market. By increasing K, the firms in this market add search frictions, which makes it
harder for the search engine to assist consumers search (rising A(K)). The second way to preserve
profits (changing B) has been studied in the market for life insurance. Brown and Goolsbee (2002)
show that price search engines cause price dispersion to decrease and prices to drop by 8 − 15%.
It is important to note, however, that price dispersion in this market did not disappear altogether
and that it is still not perfectly competitive.
To derive a symmetric mixed strategy equilibrium for the extended game, define
Ψ(pj , kj , σ−j ) ≡ Pr(A(K) > B|pj , kj , σ−j ).
The function Ψ(pj , kj , σ−j ) is the conditional probability that the costs of educating consumers
exceeds the benefits of perfect information, given firm j’s choices of pj and kj , and given the
symmetric mixed strategies of the other firms.21 Because the firms cannot coordinate when they
choose their prices and complexity, it is possible ex post that A(K) < B. In this case, the low-price
firm gets all of the industry demand. If in fact A(K) > B, then all firms receive positive profits
because some consumers remain uninformed. Therefore, each firm maximizes the expected profit
n h
o
1 − Γi
Π(pj , kj |σ−j ) = pj Ψ Γ(1 − F (pj ))n−1 +
+ (1 − Ψ)[1 − F (pj )]n−1 .
n
The following proposition establishes a symmetric mixed strategy equilibrium of the extended game
21
Again, for ease of notation, I will denote Ψ(pj , kj , σ−j ) as Ψ, unless it is necessary to specify its arguments.
22
and characterizes the strategic choices of the firms.
Proposition 4. (Symmetric Mixed Strategy Equilibrium with Advice Channel) There exists a sym∗ (p)} such that k ∗ (p) = k ∗ (p) and firms randomize
metric mixed strategy equilibrium σ ∗ = {FA∗ (p), kA
A
over prices in [pA , v] according to FA∗ (p) which solves
"
(vΨH (1 − ΓH ) − pΨ(1 − Γ))
FA (p) = 1 −
np[1 − Ψ(1 − Γ)]
#
1
n−1
.
(6)
The lower bound pA is computed as
pA =
vΨH [1 − ΓH ]
,
n − (n − 1)Ψ(1 − ΓL )
(7)
and the other parameters are ΓH = Γ(v, k, σ−j ), ΓL = Γ(pA , k, σ−j ), ΨH = Ψ(v, k, σ−j ), and
ΨL = Ψ(pA , k, σ−j ).
The structure of the equilibrium in Proposition 4 is similar to that when an advice channel is
not present (Proposition 2). The expression in (6) is implicit in FA (p) since both Ψ and Γ depend
on FA (p). As in Section 2, it is impossible to write the general solution for FA∗ (p) explicitly, but
a solution is guaranteed by Schauder’s fixed point theorem. Note that the function in (6) and the
lower bound in (7) embed the result in Proposition 2. If Ψ = 1, we get the same implicit function
of F (p) and lower bound p as in (1) and (2).
To gain some intuition about how education costs affect prices, consider two cost functions A1 (·)
and A2 (·), such that A1 (·) > A2 (·) for all values of K. Since
∂Ψ
∂A(K)
> 0, then Ψ(A1 (K)) > Ψ(A2 (K)).
The probability that price dispersion and industry profitability persists ex post increases when the
cost of education rises. According to (7), the lower bound of the support pA is monotonically
increasing in A(K). So, as A(K) → ∞, pA → p and FA∗ (p) → F ∗ (p). In this case, the distribution
of industry prices is always non-degenerate. In contrast, as A(K) → 0, then pA → 0 and FA∗ (p) → 1,
which means that as the costs of providing information to uninformed consumers approaches zero,
the industry becomes competitive and prices converge to marginal cost.
The binary complexity decision rule in Proposition 4 is identical to that in Proposition 2 and the
cutoff price is the same (p̂A = p̂). This implies that the expected level of industry complexity E[K]
23
does not change when an advisor is present and an increase in A(K) does not occur in equilibrium
on average. The firms only use the strategy of lowering B to protect industry profits. Because
complexity is costless, this is a suboptimal outcome for the firms and likely arises because of the
non-cooperative nature of the game. If the firms could coordinate their choices, increasing aggregate
complexity would indeed be an optimal strategy.
There are other ways in which the firms in the industry may preserve price dispersion and
oligopoly profits. One way is for the firms to align themselves with the advice channel and sign
incentive contracts that are mutually profitable. With these types of incentives, advisors may find
that their commission payments from the industry exceed the profits that they can generate from
selling information to consumers. In this case, they will no longer provide complete information
to consumers. Also in this case, as we shall see, if the marginal costs to providing education
are sufficiently high, then the firms will also optimally add more complexity to the industry. By
contracting with the advisors, the firms are able to coordinate their complexity choices and increase
the aggregate complexity when it is advantageous to do so.
Incentive agreements are quite common in retail markets. In the mutual fund industry, frontend and back-end load fees are paid to the advice channel when shares in funds are purchased.
Commissions are usually paid as a percentage of the assets invested. Mutual funds also pay the
advice channel incentives in the form of marketing dollar funds to reimburse the channel for pushing
their funds. In the mortgage, life insurance, and credit card industries, payments to the search
engine occur in the form of sales commissions. These commissions may be a flat “referral fee” or
may be a percentage of the sale. The details of the agreement are contractually set before the retail
products are made available for sale though the channel.
To model this, consider the extended game again with one change. At t = 1, the firms not
only choose pj and kj , but also make a take-it-or-leave-it offer (contract) to the advice channel
to share in the expected industry profits. The contract space that I consider allows the firms in
∗ to resell their products, where
aggregate to pay the advice channel a monetary transfer T = πA
∗ = [w∗ − A(K)][1 − G(w∗ |K)] from (5). In doing so, the channel now has an incentive not to
πA
disclose private information about the industry price structure. I assume for technical convenience
that each firm pays an equal share of this transfer.
Admittedly, this contract space is narrow, since the ex post transfer must make the advice
24
channel at least as well-off as if they sold their information to consumers. The firms offer a statedependent contract that pays the advice channel a monetary transfer based on the independent
draws that each firm receives from F ∗ (p).22 In this way, I avoid any coordination problems in which
the firms ex post do not satisfy the advice channel’s incentive compatibility constraint
∗ 23
T ≥ πA
.
To derive a symmetric mixed strategy equilibrium when incentive contracts are allowed, define
∗
Ω(pj , kj , σ−j ) ≡ Eσ−j [πA
|pj , kj ]
and
λG =
1
∂G(w ∗ |K)
∂kj
.
− G(w∗ |K)
The function Ω(pj , kj , σ−j ) is the conditional expectation of the transfer required to meet the advice
channel’s incentive compatibility constraint, given firm j’s choices of pj and kj , and given the
symmetric mixed strategies of the other firms.24 The hazard rate function λG is the instantaneous
decrease in the fraction of consumers who become informed independent of the advice channel,
given the fraction of consumers who depend on the advice channel for information. As such, λG is
a measure of the sensitivity of the marginal consumer to higher complexity. It always holds that
λH ≤ 0.
Making the channel’s incentive compatibility constraint bind, the firms in the industry now
maximize the expected profit
n
1 − Γo Ω
− .
Πj (pj , kj |σ−j ) = pj Γ[1 − F (pj )]n−1 +
n
n
(8)
The next proposition establishes a symmetric mixed strategy equilibrium when contracting between
22
Another approach, which would be more consistent with classic contract theory, would be to allow the firms to
offer the advice channel a fraction γ of their profits. However, this type of model is analytically intractable and loses
much of the economic intuition that the monetary transfer model allows. Focusing on a narrow contract space also
allows me to avoid problems like coordination and free-riding, in order to capture the effect of the surplus transfer
on the complexity and pricing in the industry.
23
∗
For technical simplicity, I do not consider contracts in which the firms offer the advice channel less than πA
in
exchange for revealing only partial information about the industry.
24
Again, for ease of notation, I will denote Ω(pj , kj , σ−j ) as Ω, unless it is necessary to specify its arguments.
25
the firms and the advice channel is possible.
Proposition 5. (Equilibrium with Advice Channel and Contracts) Suppose that
∂Ω
∂k
< 0 for all
∗ (p), k ∗ (p)} such
k ∈ [k, k]. Then, there exists a symmetric mixed strategy equilibrium σ ∗ = {FAC
AC
∗ (p) which solves
that firms randomize over prices in [pAC , v] according to FAC
"
v[1 − ΓH ] − p[1 − Γ] + (Ω − ΩH )
FAC (p) = 1 −
npΓ
∗ (p) is given by
and kAC
where

 k
∗
kAC
(p) =
 k
if
p < p̂AC
if
p > p̂AC ,
#
1
n−1
(9)
(10)
FAC (p̂AC ) < F (p̂).
The lower bound pAC is computed as
pAC =
v[1 − ΓH ] − (ΩH − ΩL )
,
(n − 1)ΓL + 1
(11)
and the other parameters are ΓH = Γ(v, k, σ−j ), ΓL = Γ(pAC , k, σ−j ), ΩH = Ω(v, k, σ−j ), and
ΩL = Ω(pAC , k, σ−j ).
Suppose that A(K) = A, a constant. Then, there exists an equilibrium of the same form, except
that
FAC (p̂AC ) > F (p̂).
The equilibrium in Proposition 5 shares the same structure as is Propositions 2 and 4. The
distribution of prices is non-degenerate and perfect competition does not arise because the firms
and advice channel share the common goal to preserve industry profits.
Adding complexity to the industry has two effects. It increases the revenue that the advice
channel receives when they market their information to consumers. It also increases the education
costs incurred when they provide these services. Both of these income effects change the monetary
transfer that is required to satisfy the advisor’s incentive compatibility constraint. As long as
26
∂Ω
∂k
<0
for all k ∈ [k, k], increases in complexity cause education costs to rise faster than the revenue that
is generated from information services. In this case, the firms will add high complexity to their
pricing schedules with higher probability. A sufficient condition which ensures this is
∂A(K)
∂w∗
>
− [w∗ − A(K)]λG ,
∂k
∂k
(12)
for all k ∈ [k, k]. That is, if the marginal cost of educating consumers is sufficiently high, then firms
will have an incentive to increase complexity in the industry. The density FAC (p̂AC ) is less than
the density F (p̂) derived in Proposition 2, and the expected industry complexity E[K] rises.
If the firms cannot affect the cost of education (A(K) is constant), then they will choose high
complexity with lower probability. That is, 1 − FAC (p̂AC ) < 1 − F (p̂). This occurs because
high complexity increases the profits to the advice channel when they market their information
services, which increases the monetary transfer needed to meet the advisor’s incentive compatibility
constraint. When higher complexity does not make education more expensive, it destroys value for
the firms because they need to provide higher incentives to the advice channel.
∗
are not monotonic in the primitive parameters of the
In Proposition 5, since pAC and FAC
model, it is not possible to predict whether prices will increase or decrease when contracts are
present. However, it is clear (also from Proposition 4) that the actions of an advice channel affect
complexity and prices in the market and may preserve producer surplus. From a social welfare
standpoint, these relationships have important welfare implications, which are analyzed next.
5
Policy Considerations
In this section, I consider the welfare effects of the pricing strategies outlined in the paper. I start
by considering the effects of complexity on consumer and producer surplus. Then, I propose some
of the policies that might provide relief to consumers.
Welfare Analysis
Without an advice channel present, consumer surplus (CS) is computed for the unit mass of
financial consumers as the aggregate value of the retail financial product (V ) minus the price paid
27
minus the sum of search costs. So, CS is calculated as
CS = V − G(B|K)E[pmin ] − [1 − G(B|K)]p −
Z
B
c dG(c|K).
(13)
0
The second and third terms in (13) aggregate the expected revenue that is collected from consumers
and the last term is the aggregate search cost spent by consumers to learn about industry prices.
In the absence of an advice channel, total welfare in the market is the sum of producer and
consumer surplus
W = CS + P S.
Since the marginal cost of production is zero, producer surplus is given by
P S = G(B|K)E[pmin ] + [1 − G(B|K)]p.
Therefore, the welfare loss in the market due to complexity is given by
∆W = −
Z
B
c dG(c|K).
0
By inspection, the welfare loss is non-monotonic in K. That is, with K = 0, no search costs are paid
because the industry is perfectly clear and prices converge to marginal cost. Everyone is informed in
this case, so perfect competition arises. When K is maximized, the fraction of informed consumers
is minimized. In this case, as fewer consumers pay search costs, the welfare loss due to complexity
decreases. If K rises high enough such that the fraction of uninformed consumers approaches one,
the welfare loss to search converges to zero. Therefore, from a social planner’s standpoint, the
absence of complexity or an infinite amount of complexity both achieve social efficiency. However,
with zero complexity, consumers maximize their surplus because perfect competition arises and with
infinite complexity, producers maximize their surplus because prices approach monopoly levels.
In the presence of an advice channel and no contract with the firms, W reaches first-best when
the cost of education is negligible, that is, when A(K) = 0. In this case, there is no price dispersion,
consumers do not pay search costs, and CS = V . When contracts are possible and education
is costly, however, consumer welfare decreases because price dispersion persists, inducing some
consumers to pay search costs. Therefore, the contracts that firms sign with the advice channel
28
and the cost of education have significant effects on consumer welfare.
Restoring Consumer Surplus
The model in this paper is a partial equilibrium analysis. What is left out is that consumers and
producers use their rents to fund their lifetime consumption. If aggregate surplus decreases because
of strategic complexity, this has general effects on the economy. In what follows, I focus on policies
to restore consumer surplus.
What should be evident is that there are three policies that can maximize both consumer
surplus and overall welfare. First, the government could limit the complexity of the industry. Since
∆W = 0 when K = 0, limiting complexity would drive down welfare losses. The policy implication
is that requiring disclosure is not enough. To limit welfare losses, the government needs to require
clear disclosure. The second way to affect welfare is to limit contracting between producers and
the advice channel. Doing so limits search costs and maximizes both total welfare and consumer
surplus. As already noted, perfect competition may arise in the absence of contracts. Finally, the
third way to restore consumer surplus is for the government to make education more efficient. This
may require making financial education mandatory in school systems or designing a mechanism (for
example, a web tool) for consumers to better learn about the industry. Lowering the marginal cost
of educating consumers would help limit complexity and would make the industry more competitive.
6
Conclusion
Purchasing a retail financial product requires effort. Because prices in the market are complex,
consumers must pay a cost (time or money) to compare prices in the market. Some consumers
gain sufficient expertise and get the best deal. Those with high search costs forego value, and often
make purchases without knowing exactly what they are getting or how much they are paying. In
fact, they may also be unaware that they are indeed over-paying.
In this paper, I develop a model of pricing complexity in which firms compete on price for
market share and strategically add complexity to preserve market power in the face of competitive
pressures. The resulting equilibrium matches empirical observation: price dispersion persists even
when goods are homogeneous, prices do not converge to marginal cost despite a large number of
29
firms, and entry of firms into the industry may cause prices to rise.
The analysis in the paper has important social implications, given the large size of retail financial markets. I outline three policies that could restore consumer surplus. First, clear disclosure
of prices would limit complexity and loss of consumer surplus. Second, limiting the incentive contracts between firms and financial advisors would make the industry more competitive. Finally,
improving the delivery of financial education and increasing financial literacy would restore surplus
to households who purchase retail products.
There are several possible extensions to this paper, which are the subjects of future research.
First, the model assumes that consumers have unit demand for the retail product and all participate
in the market. One extension would be to include a more sophisticated demand function, in which
some consumers might not participate if complexity were too high. In such a model, firms would
have to make a trade-off between adding complexity and losing consumers to non-participation.
The welfare losses might be higher if participation in the market is sufficiently low.
A second extension would be to consider the multi-market effect of complexity. Consumers are
endowed with limited resources to devote to all of their financial decisions. They cannot become
experts in all markets and may choose to become differentially knowledgable about the various
markets in which they participate. Complexity in one market may make it more difficult to become
informed in other markets. This externality may lead to higher general complexity across the
economy and an over-emphasis on financial innovation. Importantly, it is possible that such an
externality might cause a state of “consumer apathy” to develop. In this case, consumers would
lose maximal surplus because they become overloaded with information that they are unable to
distill. One possible result is that they make purchases randomly across all financial markets.
A third extension would evaluate a more competitive advice channel. In the paper, I assume
that the advice channel has monopoly power. This could be relaxed to allow for an analysis of how
bargaining power between the parties affects equilibrium complexity. If heterogeneity were added
between advisors, it would be interesting to evaluate what would happen if consumers had varied
abilities to choose their advisor.
In the paper, I assume that the good that is produced is homogeneous, so that if price dispersion
occurs in this case, then it would be even more likely to occur when products are heterogeneous.
Another extension might include differentiated goods where the value v is also a strategic variable
30
chosen by the firms. In such a model, consumers would have random utility over both the value
and the price of the good. This type of analysis may predict what type of products will be present
in the market, given the consumer literacy present.
The model that I present in this paper is a one-shot game. Consumers do not learn over time
and cannot switch products once they realize what they have purchased. Another extension of this
model would be to embed the pricing complexity game in a dynamic game in which consumers
could observe at a later date (at a cost) the price that they are paying for their goods after they
make their purchase. At this later time, they would have the choice to either stay with this good or
purchase the good from another firm. This type of model would allow for an analysis of switching
costs in the industry and for evaluating how literacy and complexity evolve over time.
Finally, the last extension would be to apply the analysis in this paper to security design in
general. At some level, even institutional investors have limits to their expertise and struggle to
value certain aspects of securities, such as embedded options, collars, or default protection. It is
possible for experts to construct securities with sufficient complexity to gain an advantage because
buyers of the product may not have sufficient expertise to properly value the security. This type
of strategic complexity in security design is implied in Harris and Piwowar (2006).
I believe that this paper presents a plausible argument for considering complexity as an important determinant of price formation in retail financial markets. Given the large number of potential
extensions of this analysis, this paper hopefully represents a first step toward greater understanding
of the effect of complexity on security design, asset prices, and market structure.
31
References
Abreu, D. (1988). On the theory of infinitely repeated games with discounting. Econometrica 56 (2),
383–396.
Agnew, J. and L. Szykman (2004). Asset allocation and information overload: The influence of information
display, asset choice, and investor experience. Working paper, Boston College.
Alexander, G., J. Jones, and P. Nigro (1998). Mutual fund shareholders: Characteristics, investor knowledge, and sources of information. Financial Services Review 7, 301–316.
Anderson, S. and A. de Palma (2003). Price dispersion. University of Virginia, unpublished.
Ausubel, L. (1991). The failure of competition in the credit card market. American Economic Review 81,
50–81.
Barber, B., T. Odean, and L. Zheng (2003). Out of sight, out of mind: The effects of expenses on mutual
fund flows. Forthcoming, Journal of Business.
Baye, M. and J. Morgan (2001). Information gatekeepers on the internet and the competitiveness of
homogeneous product markets. American Economic Review 91, 454–474.
Braverman, A. (1980). Consumer search and alternative market equilibria. Review of Economic Studies 47,
487–502.
Braverman, A. and A. Dixit (1981). Consumer search and market equilibria: A note. Review of Economic
Studies 48, 657–658.
Brown, J. and A. Goolsbee (2002). Does the internet make markets more competitive? evidence from the
life insurance industry. Journal of Political Economy 110, 481–507.
Burdett, K. and K. Judd (1983). Equilibrium price dispersion. Econometrica 51, 955–970.
Calem, P. and L. Mester (1995). Consumer behavior and the stickiness of credit-card interest rates.
American Economic Review 85, 1327–1336.
32
Calvet, L., J. Campbell, and P. Sodini (2006). Down or out: Assessing the welfare costs of household
investment mistakes. Harvard, unpublished.
Camerer, C., T. Ho, and J. Chong (2004). A cognitive hierarchy model of games. Quarterly Journal of
Economics 119, 861–898.
Campbell, J. (2006). Household finance. Presidential Address to the American Finance Association, forthcoming Journal of Finance.
Capon, N., G. Fitzsimons, and R. Prince (1996). An individual level analysis of the mutual fund investment
decision. Journal of Financial Services Research 10, 59–82.
Carlson, J. and P. McAfee (1983). Discrete equilibrium price dispersion. Journal of Political Economy 91,
480–493.
Chernev, A. (2003). When more is less and less is more: The role of ideal point availability and assortment
in consumer choice. Journal of Consumer Research 30, 170–183.
Choi, J., D. Laibson, and B. Madrian (2006). Why does the law of one price fail? an experiment on index
mutual funds. mimeo Harvard, Yale, and University of Pennsylvania.
Davis, M. (1966). A First Course in Functional Analysis. London: Gordon and Breach.
Diamond, P. (1971). A model of price adjustment. Journal of Economic Theory 3, 156–168.
Ellison, G. and S. Ellison (2004). Search, obfuscation, and price elasticities on the internet. MIT, unpublished.
Gabaix, X. and D. Laibson (2005). Shrouded attributes, consumer myopia, and information suppression
in competitive markets. Harvard University and MIT, unpublished.
Gabaix, X., D. Laibson, and H. Li (2005). Extreme value theory and the effects of competition on profits.
Harvard University and MIT, unpublished.
Gourville, J. and D. Soman (2005). Overchoice and assortment type: When and why variety backfires.
Marketing Science 24, 382–395.
33
Harris, L. and M. Piwowar (2006). Municipal bond liquidity. University of Southern California, unpublished.
Hortacsu, A. and C. Syverson (2004). Product differentiation, search costs, and competition in the mutual
fund industry: A case study of S&P index funds. Quarterly Journal of Economics 119 (2), 457–488.
Istratescu, V. (1981). Fixed Point Theory. Dordrecht, Holland: D. Reide Publishing Company.
Iyegar, S. and M. Lepper (2000). When choice is demotivating: Can one desire too much of a good thing?
Journal of Personality and Social Psychology 79, 995–1006.
Jain, P. and J. Wu (2000). Truth in mutual fund advertising: Evidence on future performance and fund
flows. Journal of Finance 55, 937–958.
Jones, M. and T. Smythe (2003). The information content of mutual fund print advertising. Journal of
Consumer Affairs 37, 22–41.
Mitchell, O., J. Poterba, M. Warshawsky, and J. Brown (1999). New evidence on the money’s worth of
individual annuities. American Economic Review 89, 1299–1318.
Morwitz, V., E. Greenleaf, and E. Johnson (1998). Divide and prosper: Consumers’ reactions to partitioned
prices. Journal of Marketing Research 35, 453–463.
Payne, J., J. Bettman, and E. Johnson (1993). The Adaptive Decision Maker. New York: Cambridge
University Press.
Perloff, J. and S. Salop (1985). Equilibrium with product differentiation. Review of Economic Studies 52,
107–120.
Reinganum, J. (1979). A simple model of equilibrium price dispersion. Journal of Political Economy 87,
851–858.
Rob, R. (1985). Equilibrium price distributions. Review of Economic Studies 52, 487–504.
Robert, J. and D. Stahl (1993). Informative price advertising in a sequential search model. Econometrica 61, 657–686.
34
Rosenthal, R. (1980). A model in which an increase in the number of sellers leads to a higher price.
Econometrica 48, 1575–1579.
Salop, S. and J. Stiglitz (1977). Bargains and ripoffs: A model of monopolistically competitive price
dispersion. Review of Economic Studies 44, 493–510.
Salop, S. and J. Stiglitz (1982). The theory of sales: A simple model of equilibrium price dispersion with
identical agents. American Economic Review 72, 1121–1130.
Scitovsky, T. (1950). Ignorance as a source of oligopoly power. American Economic Review 40, 48–53.
Shugan, S. (1980). The cost of thinking. Journal of Consumer Research 7, 99–111.
Sirri, E. and P. Tufano (1998). Costly search and mutual fund flows. Journal of Finance 53, 1589–1622.
Spiegler, R. (2005). Competition over agents with boundedly rational expectations. forthcoming Theoretical Economics.
Stahl, D. (1989). Oligopolistic pricing with sequential consumer search. American Economic Review 79,
700–712.
Stahl, D. (1996). Oligopolistic pricing with heterogeneous consumer search. International Journal of Industrial Organization 14, 243–268.
Stiglitz, J. (1979). Equilibrium in product markets with imperfect information. American Economic Review 69, 339–345.
Stiglitz, J. (1987). Competition and the number of firms in a market: Are duopolies more competitive
than atomistic markets. Journal of Political Economy 95, 1041–1061.
Varian, H. (1980). A model of sales. American Economic Review 70, 651–659.
Weitzman, M. (1979). Optimal search for the best alternative. Econometrica 47, 641–654.
Wilcox, R. (2003). Bargain hunting or star gazing? investors’ preferences for stock mutual funds. Journal
of Business 76, 645–663.
35
Appendix
PROOF OF PROPOSITION 1: First I prove existence and then prove properties 1 − 4.
Let Σ = ×j∈N Σj . Define the best response correspondence
R:Σ→Σ
with component functions
Rj (σ) ≡ {σj′ |σj′ ∈argmax [πj (σj , σ−j )]}.
σj ∈Σj
Since Σ is convex and compact and since π is continuous, R(·) is non-empty. Convexity of R is
proved by contradiction. Suppose that R is not convex. Then, there exists two mixed strategies
σ ′ , σ ′′ ∈ R, and α ∈ (0, 1) such that σ ′′′ = ασ ′ + (1 − α)σ ′′ 6∈ R. However,
πj [ασj′ + (1 − α)σj′′ , σ−j ] = απj [σj′ , σ−j ] + (1 − α)πj [σj′′ , σ−j ],
so σ ′′′ ∈ R. Therefore, R is convex. R has a closed graph since π is continuous.25 By Kakutani’s
fixed-point theorem, there exists a Nash equilibrium of the game.
Properties 1 − 4:
1. Suppose that p−j = 0, for all firms other than j. If pj is zero, then B = 0 and πj (0) = 0. Firm
j can profitably deviate by choosing a price pj ∈ (0, nc) and earn positive expected profits,
which rules out a Bertrand equilibrium. If c > v, then it is always true that
1−G(B|K)
n
> 0,
so that πj (pj ) > 0 whenever pj > 0. Therefore, when c > v, setting pj = 0 is a strictly
dominated strategy.
25
Alternatively, the upper hemi-continuity of R can be proven by contradiction. Suppose that R is not upper
hemi-continuous. Then, there exists a sequence (σ k , σ̃ k ) → (σ, σ̃) such that σ̃ k ∈ R(σ k ) and σ̃ 6∈ R(σ). Then for some
j, σ̃j 6∈ R(σj ). Therefore, ∃ǫ > 0 and a σj′ such that πj [σj′ , σ−j ] > πj [σ̃j , σ−j ] + 3ǫ. Given convergence of (σ k , σ̃ k )
and π continuous,
k
k
πj [σj′ , σ−j
] > πj [σj′ , σ−j ] − ǫ > πj [σ̃j , σ−j ] + 2ǫ > πj [σ̃jk , σ−j
] + ǫ,
which contradicts σ̃ k ∈ R(σ k ). So, R is upper hemi-continuous.
36
2. Suppose that σ−j (p, ·) = p̂ > 0. Then,
πj (p̂) =
Since
p̂
.
n
"
#
[1 − G(B(p̂ − ǫ)|K)]
> πj (p̂),
πj (p̂ − ǫ) = (p̂ − ǫ) G B(p̂ − ǫ)|K +
n
then no symmetric equilibrium in pure strategies can exist.
3. Follows in the same fashion as Property 2.
4. Suppose that it is possible to have an asymmetric equilibrium in pure strategies and, without
loss of generality, that p2 > p1 . Firm 2 solves
max
p2 ∈[0,v]
k2 ∈[k,k]
and Firm 1 solves
max
p1 ∈[0,v]
k1 ∈[k,k]
p 2 Q2 =
p2 [1 − G(B|K)]
,
n
h
[1 − G(B|K)] i
p1 Q1 = p1 G(B|K) +
.
n
For p2 > p1 to be an equilibrium, it must be that
p 2 Q2 >
p1
2
and p1 Q1 >
p2
.
2
These two conditions mean that firm 2 has no incentive to lower their price and undercut
firm 1 by a small amount and that firm 1 has no incentive to raise their price just below firm
2’s price. For these conditions to simultaneously hold it must be that
Q1 Q2 >
1
,
4
which is impossible. By contradiction, there cannot exist an asymmetric equilibrium in pure
strategies when n = 2. 37
PROOF OF PROPOSITION 2:
Derivation of k∗ (p): The expected profit to firm j, given that all of the other firms use the
strategy {F (p), k(p)}, is
h
1 − Γi
Π(pj , kj |σ−j ) = pj Γ[1 − F (pj )]n−1 +
.
n
(A1)
First order conditions with respect to kj and symmetry (kj = k) imply that
∂Π
p ∂Γ
∂Γ
1
n−1 ∂Γ
n−1
= p[1 − F (p)]
−
=p
[1 − F (p)]
−
.
∂k
∂k
n ∂k
∂k
n
Since
∂Γ
∂k
< 0, if
[1 − F (p)]n−1 ≥
we have
(A2)
∂Π
∂k
1
,
n
< 0 and obtain the corner solution k = k. This occurs when p < p̂, where the threshold
level p̂ is
p̂ = F
−1
1−
h1i
1
n−1
n
.
When p > p̂, that is when
[1 − F (p)]n−1 <
we have
∂Π
∂k
1
,
n
> 0 and obtain the other corner solution k = k.
Properties of F (p): I prove the continuity of F (p) as follows. Suppose that there did exist a
countable number of mass points in the distribution of F (p). Then, we can find a mass point p∗
and an ǫ > 0 such that f (p∗ ) = a > 0 and f (p − ǫ) = 0. Now consider a deviation by firm j to
choose F̂ (p) such that f (p∗ ) = 0 and f (p − ǫ) = a. Since E[Πj (p, k)] using F (p) is strictly less
than using F̂ (p), this would be a profitable deviation. Therefore, in equilibrium, no mass points
can exist.
I prove strict monotonicity (Increasing) of F (p) as follows. Suppose there exists an interval
[pa , pb ] within [p, v] such that F (pb ) − F (pa ) = 0. Then, for any p̂ such that pa < p̂ < pb ,
[1 − F (p̂)]n−1 = [1 − F (pa )]n−1 . Since p̂[1 − F (p̂)]n−1 > pa [1 − F (pa )]n−1 and p̂[1 − (1 − F (p̂))n−1 ] >
pa [1 − (1 − F (pa ))n−1 ], then there exists a profitable deviation. Thus, F (pb ) − F (pa ) 6= 0 for any
interval [pa , pb ] within [p, v].
38
Existence of F ∗ (p): Since in any mixed strategy equilibrium, each firm must be indifferent between
choosing any p ∈ [p, v], it must be that Π(pj , kj |σ−j ) = π, a constant, for all p. We can use (A1)
to write
π=
h
v[1 − ΓH ]
1 − ΓL i
= p ΓL +
.
n
n
Solving for p yields (1). Since Π(pj , kj |σ−j ) = π for all p ∈ [p, v], we can write (A1) as
h
1 − Γi
π = pj Γ[1 − F (pj )]n−1 +
.
n
Solving for F (p) and dropping subscripts yields
F (p) = 1 −
Since π =
v(1−ΓH )
,
n
then
h π − p (1 − Γ) i 1
n−1
n
.
pΓ
"
v[1 − ΓH ] − p[1 − Γ])
F (p) = 1 −
npΓ
#
1
n−1
(A3)
.
Equation A3 is implicit in F (p), since ΓH and Γ depend on F (p). The following lemma establishes
that a solution (fixed point) to Equation A3 exists, which completes the proof.
LEMMA A.1. Define F C as the space of continuous functions that map [p, v] into [0, 1]. Define
the operator T : F C → F C as
"
v[1 − ΓH ] − p[1 − Γ])
(T F )(p) = 1 −
npΓ
#
1
n−1
.
T has a fixed point in F C .
Proof. Define X = C([p, v], R) as the space of continuous functions mapping [p, v] into R. X is
a Banach space under the norm ||F ||0 = supp∈[p,v] |F (p)| (Davis (1966), Theorem 2.2, page 43).
Define F C ⊆ F C to be the space of continuous functions that map [p, v] into [0, 1] such that
∀F ∈ F C , |F ′ | < M for some M > 0.
F C is compact and convex, which I prove as follows. For any ǫ > 0, define δ =
39
ǫ
M.
For all
F ∈ F C and for any two points p, q ∈ [p, v] such that |p − q| < δ,
|F (p) − F (q)| < ||F ′ (p)|| |p − q| < M δ = ǫ.
Therefore, F C is equicontinuous. Since, ||F ||0 ≤ 1, F C is bounded. By the Arzela-Ascoli Theorem,
F C is compact.
Next, consider two functions F, G ∈ F C . For any α ∈ [0, 1], define
H = αF + (1 − α)G.
Without loss of generality consider that α < 12 . Since F and G are continuous, for every ǫ > 0,
there exists a δF > 0 and a δG > 0 such that for all p1 , p2 ∈ [p, v]
||p2 − p1 || < δF ⇒ ||F (p2 ) − F (p1 )|| < αǫ.
and
||p2 − p1 || < δG ⇒ ||G(p2 ) − G(p1 )|| < αǫ.
Let δ = αδF + (1 − α)δG . Then,
||p2 − p1 || < δ ⇒ ||F (p2 ) − F (p1 )|| + ||G(p2 ) − G(p1 )|| < 2αǫ < ǫ.
Therefore, H is continuous.
Since the first derivative of F and G is bounded, α|F ′ | < αM and (1 − α)|G′ | < (1 − α)M , which
implies that |H ′ | < M . Finally, since R(F ) ∈ [0, 1] and R(G) ∈ [0, 1], then R(H) ∈ [0, 1]. Therefore,
H ∈ F C . Hence, F C is convex.
Since the functions B and G(·|K) are continuous, Γ and ΓH are continuous in F (p). Thus, the
operator (T F )(p) is continuous. Since the operator T maps F C → F C , by Schauder’s fixed point
theorem T has a fixed point in F C (Istratescu (1981), Theorem 5.1.3). Therefore, T has a fixed
point in F C . Hence, a symmetric mixed strategy Nash equilibrium {F ∗ , k∗ } exists.
40
PROOF OF PROPOSITION 3: Taking the derivative of
h1i
1
n−1
h n ln n − n + 1 i
n(n − 1)2
n
h i
1
n
1
n−1
with respect to n yields
,
which is positive for all n ≥ 2. Therefore, 1 − F (p̂) is strictly increasing in n. Taking the limit
lim
n→∞
h1i
1
n−1
n
→ 1.
To prove that E[K] is increasing in n, we can express
E[Kn ] = Kmin +
n X
n
(A4)
(A5)
m=0
and
E[Kn+1 ] = Kmin +
n+1
X
m=0
We can re-write E[Kn+1 ] as
h
e
E[Kn+1 ] = Kmin +K(n+1)
h 1 1 im h
1 1 in−m
n−1
n−1
e
K(m)
1−
m
n
n
h 1 1 im h
1 1 in+1−m
n+1 e
n
n
K(m)
1−
.
m
n+1
n+1
1 n1 in+1 X
+
n+1
n
m=0
h 1 1 im h 1 1 in+1−m
n+1 e
n
n
K(m)
1−
.
m
n+1
n+1
(A6)
Since the last term in (A6) is greater than the last term in (A4), it follows that E[K] is increasing
in n. PROOF OF PROPOSITION 4: The expected profit for firm j as
n h
o
1 − Γi
Π(pj , kj |σ−j ) = pj Ψ Γ(1 − F (pj ))n−1 +
+ (1 − Ψ)[1 − F (pj )]n−1 .
n
(A7)
Differentiating (A7) with respect to kj and symmetry (kj = k) yields
nh ∂Γ ∂Ψ
ih
∂Π
1 io
=p Ψ
−
(1 − Γ) [1 − F (p)]n−1 −
.
∂k
∂k
∂k
n
Since Ψ ∂Γ
∂k −
∂Ψ
∂k (1
(A8)
∗ (p) = k ∗ (p).
− Γ) < 0 , kA
The expression in (6) follows from algebraic manipulation in the same way as in the proof of
Proposition 2. A solution to (6) follows from Schauder’s fixed point theorem: using the compactness
and convexity of the functional space established in Lemma A.1 and the continuity of the operator
41
in (6), a fixed point exists. The lower bound in (7) follows in the same fashion as in the proof of
Proposition 2. PROOF OF PROPOSITION 5: The profit to the advice channel is expressed as
∗
πA
= [w∗ − A(K)][1 − G(w∗ |K)].
(A9)
Taking the derivative of (A9) with respect to kj yields
h ∂w∗ ∂A(K) ih
i h
i ∂G(w∗ |K)
∗
∂πA
=
−
1 − G(w∗ |K) − w∗ − A(K)
.
∂kj
∂kj
∂kj
∂kj
The derivative
∗
∂πA
∂kj
is guaranteed to be negative as long as
∂A(K)
∂w∗
>
− [w∗ − A(K)]λG .
∂kj
∂kj
Suppose that (A10) holds for all k ∈ [k, k]. Then,
for firm j as
∂Ω
∂kj
(A10)
< 0. We can write the expected profit
h
1 − Γi Ω
Π(pj , kj |σ−j ) = pj Γ(1 − F (pj ))n−1 +
− .
n
n
Differentiating (A11) with respect to kj and symmetry (kj = k) yields
∂Π
∂Γ
1 ∂Ω
1
=p
[1 − F (p)]n−1 −
−
.
∂k
∂k
n
n ∂k
∗ (p) in (10). Since
This leads to the binary strategy for kAC
Now consider that A(K) = A is fixed. In this case,
∂Ω
∂kj
∂Ω
∂kj
∂π∗A
∂kj
(A11)
(A12)
< 0, p̂AC > p̂.
∗
|K)
= − ∂G(w
[w∗ − A] > 0. Therefore,
∂kj
∗ (p) in (10) such that p̂
> 0. Solving (A12) leads to the binary strategy for kAC
AC < p̂.
The expression in (9) follows from algebraic manipulation in the same way as in the proof of
Proposition 2. A solution to (9) follows from Schauder’s fixed point theorem: using the compactness
and convexity of the functional space established in Lemma A.1 and the continuity of the operator
in (9), a fixed point exists. The lower bound in (11) follows in the same fashion as in the proof of
Proposition 2. 42