Institute for Economic Studies, Keio University
Keio-IES Discussion Paper Series

Biased Beliefs of Consumers and Two-Part Tariff Competition
Koji Ishibashi
11 April, 2024
DP2024-009
https://ies.keio.ac.jp/en/publications/23708/

Institute for Economic Studies, Keio University
2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan
ies-office@adst.keio.ac.jp
11 April, 2024

Biased Beliefs of Consumers and Two-Part Tariff Competition
Koji Ishibashi
Keio-IES DP2024-009
11 April, 2024
JEL Classification: D42, D43, D91, L12, L13
Keywords: biased belief, overoptimistic consumers, overpessimistic consumers, twopart tariff

Abstract
This paper explores how firms respond in designing two-part tariffs to consumers'
biased beliefs about their preferences. Biased consumers could be either
overpessimistic when they underestimate their true demand or overoptimistic when
they overestimate. Assuming that unbiased consumers consist of two types with highand low valuations, I show that the effect of the presence of biased consumers on
unbiased consumers depends on market structure. The monopolist wants to educate
overpessimistic consumers while may not want to educate overoptimistic consumers.
Alternatively, in competition, firms do not have the incentive to educate any biased
consumers. A debiasing policy for either overpessimistic or overoptimistic consumers
unambiguously improves social welfare in competition but could harm social welfare inmonopoly.

Koji Ishibashi
Department of Economics, Keio University
2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japankoji@econ.keio.ac.jp

Biased Beliefs of Consumers
and Two-Part Tariﬀ Competition
Koji Ishibashi∗
April 11, 2024

Abstract
This paper explores how firms respond in designing two-part tariﬀs to consumers’ biased beliefs about
their preferences. Biased consumers could be either overpessimistic when they underestimate their true
demand or overoptimistic when they overestimate. Assuming that unbiased consumers consist of two
types with high and low valuations, I show that the eﬀect of the presence of biased consumers on unbiased
consumers depends on market structure. The monopolist wants to educate overpessimistic consumers
while may not want to educate overoptimistic consumers. Alternatively, in competition, firms do nothave the incentive to educate any biased consumers. A debiasing policy for either overpessimistic oroveroptimistic consumers unambiguously improves social welfare in competition but could harm social
welfare in monopoly.

JEL classification: D42; D43; D91; L12; L13
Keywords: biased belief; overoptimistic consumers; overpessimistic consumers; two-part tariﬀ∗ Department of Economics,
koji@econ.keio.ac.jp).

Keio

University,

2-15-45

Mita,

Minato-ku,

Tokyo

108-8345,Japan(Email:1

Introduction

The recent behavioral economics literature has explored many sources of consumer bias and their implications on consumer behavior.1 One principal source is the biased beliefs of consumers about their preferenceor willingness-to-pay; consumers may overestimate or underestimate their true demands, particularly when
they have to choose among tariﬀ plans before they actually consume goods or services. Many works on
consumer biases have investigated questions such as whether ineﬃciency arises or not under the presenceof biased consumers, is ineﬃciency due to biased consumers eliminated by competition, and does learningby consumers or educating consumers mitigate ineﬃciency.2
Most studies, however, suppose that the type of consumer is binary, biased or unbiased (put diﬀerently,
naive or sophisticated), and more importantly, unbiased consumers are homogeneous. When unbiased consumers are heterogeneous in their preferences, it is not apparent whether the existence of biased consumers
makes each type of unbiased consumer better oﬀ or worse oﬀ. The purpose of the paper is to explore themechanism of monopolistic and competitive screening via two-part tariﬀs with a high and low valuationof consumers, some of whom have biased beliefs about their own preferences, and then to investigate how
the presence of biased consumers aﬀects diﬀerent types of unbiased consumers.
There are two scenarios. One is the situation where biased consumers are overpessimistic; they mistakenly believe themselves to have a low valuation when, in fact, they have a high valuation. A naive
consumer would underestimate her demand for leisure goods with current benefits and future costs.3 For
example, credit card borrowing delivers current consumption at the expense of future consumption. Naive
consumers might underpredict their desire to use credit cards for purchases they cannot aﬀord.4 A similarbias toward overoptimistic behavior could be seen in the consumption of addicting goods or gambling andcellular phone usage.
The second scenario is where biased consumers are overoptimistic; they mistakenly believe themselvesto have a high valuation when, in fact, they have a low valuation. Consumers often overestimate theirdemand for investment goods with current costs and future benefits. For example, as to health club
attendance which incurs current eﬀort costs and provides future health benefits, they might believe thatthey will go to the gym more often than they actually do (DellaVigna and Malmendier 2006). Parents
and children may make underinvestment in the children’s education due to the present opportunity costs
1 For surveys on behavioral industrial organization, see Ellison (2006), Huck and Zhou (2011), Spiegler (2011), and Heidhues

and Kőszegi (2018).
2 The existing literature explores various sources of consumer bias such as hyperbolic discounting or time-inconsistentpreferences (DellaVigna and Malmendier 2004, Eliaz and Spiegler 2006, Heidhues and Kőszegi 2010), overconfidence (Grubb2009), unawareness or limited attention (Gabaix and Laibson 2006). Although there are many sources of misperception, acommon feature is that some consumers behave as if they mispredict the quantities they consume.
3 I follow DellaVigna and Malmendier (2004) for the definition of leisure goods and investment goods mentioned below.
4 Ausubel (1991) suggests that high interest rates on credit card debt can be explained by the presence of borrowers who
underestimate borrowing and the importance of interest rates. See also DellaVigna and Malmendier (2004).

1

of schooling which could yield significant benefits in the future. In Internet service, Lambrecht and Skiera(2006) documented evidence of a “flat-rate bias,” a tendency of consumers to prefer a flat rate when theywould save money with a pay-per-use tariﬀ, as a source of systematic overestimation of demand.
My model allows us to investigate how the presence of biased consumers aﬀects unbiased consumers inthese two scenarios in a unified manner. In the first scenario with overpessimistic consumers, consumers
with low valuation (low type) and overpessimistic consumers cannot be screened because they have thesame belief, and hence choose the same tariﬀ. Nevertheless, they choose diﬀerent quantities ex-post;
overpessimistic consumers consume more than low-type because they are, in fact, high-type consumers.
Therefore, the optimal two-part tariﬀ reflects the presence of overpessimistic consumers. For consumers
with high valuation (high type), firms set the marginal price equal to the marginal cost but a large fixed
fee. For low-type and overpessimistic consumers, in contrast, the marginal price is set higher than themarginal cost to collect variable profit from overpessimistic consumers with high ex-post valuation. Thisdistortion against biased consumers is consistent with the result of DellaVigna and Malmendier (2004)
in the context of the self-control problem. It exists regardless of market structure, that is, monopoly orcompetition.
In monopoly, another source of distortion exists due to incomplete information about consumer types.
Because a monopolist wants to reduce information rent to high-type consumers, it sets the marginal pricefor low-type and overpessimistic consumers higher than the marginal cost. A combination of this distortion
from a well-known trade-oﬀ between rent and eﬃciency and the motive mentioned above to collect variable
profit from overpessimistic consumers characterizes allocative ineﬃciency in monopoly under incomplete
information. In competition, however, incentive constraints do not bind in the design of two-part tariﬀs.
The full-information equilibrium remains an equilibrium under competition, and then the only distortionis from the incentive to collect variable profit. Thus, the allocative ineﬃciency originating from insuﬃcientquantities for low-type and overpessimistic consumers cannot be removed through competition.
In the second scenario, overoptimistic consumers choose the same tariﬀ as high-type consumers. However, the quantity overoptimistic consumers purchase ex-post becomes lower than that of high-type consumers because they are actually low-type. In the optimal two-part tariﬀ with overoptimistic consumers,
the marginal price for high-type and overoptimistic consumers is less than the marginal cost. This isbecause firms want to enlarge fixed fees by exploiting overoptimistic consumers with misperceptions, despite that this involves a loss in variable profit from high-type and overoptimistic consumers. Again, this
distortion against biased consumers is consistent with DellaVigna and Malmendier (2004), and it exists
under monopoly and competition.
The marginal price for low-type consumers, on the other hand, is set equal to the marginal cost under2

competition. Because incentive constraints are satisfied when firms compete, ineﬃciency does not arise
concerning the quantity of low-type consumers. In monopoly, however, the marginal price for low-type
consumers is distorted upward to ensure the incentive constraint for high-type consumers. Therefore, inthe presence of overoptimistic consumers, there exists a distortion in both marginal prices under monopoly
with incomplete information.
Given the above ineﬃciency due to biased consumers, I then analyze how the existence of biased consumers aﬀects unbiased consumers. By aggregating externalities between biased and unbiased consumers,
I can also infer the impacts of debiasing on consumer surplus. I further examine the incentives of firms
to educate overpessimistic or overoptimistic consumers. If firms have no incentive to educate, there is a
scope for debiasing policy by a third party or government agency. Combining eﬀects on consumer surplus
and profit allows investigating how a debiasing policy or activity educating biased consumers aﬀects social
welfare.
First, consider the scenario with overpessimistic consumers. In monopoly, the marginal price forlow-type and overpessimistic consumers is higher than the marginal cost, and it becomes lower if more
overpessimistic consumers are present. This is because the monopolist becomes more concerned with theeﬃciency of the quantity of overpessimistic consumers when their weight in the total population becomeslarge. Thus, the presence of overpessimistic consumers enlarges the information rent to high-type, yielding
a positive externality to high-type consumers. Moreover, an increase in the number of overpessimistic
consumers raises their net surplus because in equilibrium, overpessimistic consumers obtain the same levelof surplus as high-type. In other words, a debiasing policy educating overpessimistic consumers harms
high-type consumers. The net surplus of low-type consumers, in contrast, is zero because the participation
constraint is binding, and therefore, the presence of overpessimistic consumers has no externality to lowtype consumers. In sum, under monopoly, debiasing overpessimistic consumers reduces consumer surplus
but raises profit because it yields higher fixed fees from debiased (high type) consumers. The impact
on total surplus is ambiguous in general, but I show that a debiasing policy reducing the number ofoverpessimistic consumers harms social welfare under the linear demand and a wide range of parametervalues of the iso-elastic demand.
When firms compete in two-part tariﬀs, as described above, the marginal price for low-type and overpessimistic consumers is higher, while that for high-type is equal to the marginal cost. Although this distortionis similar to monopoly, the marginal price for low-type and overpessimistic consumers now becomes higher
if more overpessimistic consumers are present because collecting variable profit becomes more important.
An increase in the number of overpessimistic consumers favors low-type consumers because a lower fixed
fee is paired with a higher marginal price. This implies that, compared to monopoly, externality by the3existence of overpessimistic consumers works in the opposite way between types; low-type consumers are
better oﬀ, whereas high-type consumers are kept unchanged. Therefore, whether the presence of overpessimistic consumers makes unbiased consumers better or worse oﬀ crucially depends on market structure.
Another contrasting result in competition is that an increase in the number of overpessimistic consumers
now reduces their net surplus. To put it diﬀerently, debiasing overpessimistic consumers keeps high-typeunchanged and makes low-type worse oﬀ and overpessimistic consumers better oﬀ. It increases the consumer surplus because a negative impact on low-type is dominated by a positive impact on overpessimistic
consumers. The profit of each firm does not change; a reduction in variable profit from low-type andoverpessimistic consumers is exactly canceled out by an increase in the fixed fee for them. Therefore,
under competition, a debiasing policy educating overpessimistic consumers unambiguously raises social
welfare.
Second, consider the case of overoptimistic consumers. First, in monopoly, the aforementioned ineﬃciency is characterized by a downward distortion in the marginal price for high-type and overoptimistic
consumers and an upward distortion in the marginal price for low-type. When more overoptimistic consumers exist, the marginal price for high-type and overoptimistic consumers becomes lower because themonopolist has a stronger incentive to exploit overoptimistic consumers. At the same time, the marginal
price for low-type becomes higher because reducing the information rent to high-type, who choose the same
tariﬀ as overoptimistic consumers, is more important. Then, the presence of overoptimistic consumers
keeps low-type unchanged because their participation constraint is always binding, but makes high-typeworse oﬀ due to smaller information rent. An increase in the number of overoptimistic consumers makes
them worse oﬀ due to a larger fixed fee. In other words, high-type and overoptimistic consumers benefit
from a debiasing policy for overoptimism while low-type are kept unchanged, raising consumer surplus.
An eﬀect on profit is ambiguous; because a debiasing policy implies that some overoptimistic consumers
turn to low-type, it reduces the revenue from the fixed fee, but at the same time it enlarges the variable
profit for low-type. Hence, the impact on total surplus is also ambiguous in general.
Next, under competition, the marginal price for high-type and overoptimistic consumers is less thanthe marginal cost while that for low-type is equal to the marginal cost. Now, the overoptimistic consumers
yield a positive externality to high-type because high-type consumers can benefit from lower marginal price,and no externality to low-type. This means that the presence of overoptimistic consumers makes high-type
better oﬀ in competition, whereas it makes them worse oﬀ in monopoly. In contrast, low-type consumers
remain unchanged, independent of market structure. A debiasing policy for overoptimistic consumers
protects them from exploitation, and raises consumer surplus because a negative impact on high-type isdominated by a positive impact on debiased overoptimistic consumers. It does not change profit under4

competition; a reduction of losses in variable profit from high-type and overoptimistic consumers is exactly
canceled out by a decrease in the fixed fee. Therefore, under competition, debiasing policy unambiguously
improves social welfare.
The rest of the paper is organized as follows: After the literature review in the next section, Section 3
presents the model. The optimal two-part tariﬀ in monopoly is derived in Section 4, and two-part tariﬀ
competition is studied in Section 5. Section 6 concludes. Some proofs are in the Appendix.

2

Related Literature

In modelling consumers’ types and biased beliefs, this paper is mostly related to Sandroni and Squintani(2007, 2013). Focusing on insurance markets, they study a combination of screening with respect to consumers’ beliefs and with respect to consumers’ preferences. They investigate competitive and monopolistic
insurance markets with low-risk and high-risk consumers, some of whom are overconfident: they believe
themselves to be low-risk when, in fact, they are high-risk. Low-risk and overconfident consumers cannot
be screened because they share the same beliefs, and they buy the same insurance contract. Sandroni andSquintani (2013) show that whether or not overconfidence changes qualitative predictions in markets underasymmetric information depends on market structure. The introduction of overconfident agents overturns
fundamental relationships between observable variables in competitive insurance markets. In monopolistic
insurance markets, in contrast, overconfidence may be equivalent to changes in the risk composition in theeconomy. Although my model adopts a similar formulation of consumers’ types and beliefs to Sandroni
and Squintani (2007, 2013), there are diﬀerences in several dimensions. First, they focus on the relationship between biased beliefs and observables in insurance markets with asymmetric information, whereas
I focus on how biased beliefs of consumers aﬀect the optimal menu of two-part tariﬀs. Second, althoughoveroptimism is the only bias in their model, I allow for two-way biases where biased consumers are either overoptimistic or overpessimistic. Third, unlike insurance markets, the full-information equilibrium
remains an equilibrium under competition in this model.
Concerning market interactions between biased (naive) consumers and unbiased (sophisticated) consumers, cross-subsidization from biased consumers to unbiased consumers is first emphasized by Ellison(2005) and Gabaix and Laibson (2006) in products with shrouded attributes and by Armstrong and Vickers (2012) in the context of overdraft fee in the banking industry. More extensively, Armstrong (2015)
explored how naive and sophisticated consumers aﬀect each other in market interactions and possible
policy interventions using two models: a product with price dispersion and a product with add-on pricing.
It is assumed that there are two kinds of consumers, naive and sophisticated, and there are no systematic
diﬀerences in their preferences. In this paper, because sophisticated consumers consist of two types, the5directions of externalities of the naive consumers on two types of sophisticated consumers can be diﬀerent
from each other.
Among the literature on mechanism design and optimal contracts, this paper is related to the articles on the optimal contracting problem with non-common priors.5 Eliaz and Spiegler (2006) investigate
monopolistic screening according to prior beliefs when dynamically inconsistent consumers diﬀer in theirability to forecast their own demand. Eliaz and Spiegler (2008) study the monopolist’s design of a menu
of non-linear tariﬀs when consumers have optimistic beliefs regarding their future preferences. In theirmodel, consumers are optimistic if their prior belief assigns too much weight to states characterized bylarge gains from trade. Grubb (2009) studies monopolistic and competitive screening for overconfident
consumers who underestimate the variance of their demand forecasts in mobile phone usage, and showsthat the optimal contract can be suﬃciently approximated by three-part tariﬀs. Uthemann (2005) showsthat competing firms oﬀering menus of non-linear price schedules can screen consumers on the basis of
their mistaken priors.6 The present study is diﬀerent from these papers in two respects. First, these papers
formulate biased beliefs such that consumers and firms do not share the same distribution over the typeof consumers. Firms try to discriminate between consumers who have the same ex-post preferences but
diﬀerent ex-ante beliefs. This paper, in contrast, supposes that some consumers mistakenly believe thatthey are diﬀerent types in a deterministic sense. Because a rational consumer and a biased consumer
have the same ex-ante beliefs, they choose the same option in a menu of contracts. Although they choose
diﬀerent quantities ex-post, ex-ante screening between them is impossible in this setting; the motive fordiscrimination here is screening between a group of rational and biased consumers with the same belief andanother rational consumers with a diﬀerent belief. Second, these papers characterize the optimal menu of
non-linear tariﬀs to screen consumers’ degrees of optimism without restrictions on the space of contracts.
Although this paper restricts contracts to two-part tariﬀs, this simple structure allows me to study thetwo-way biases mentioned above and interactions between heterogeneous consumers.
There is an extensive literature on competitive non-linear pricing for rational consumers: Spulber(1989), Stole (1995), Armstrong and Vickers (2001, 2010), Rochet and Stole (2002), and Yang and Ye(2008).7 By analyzing general non-linear pricing, these papers suggest that firms oﬀer a single two-parttariﬀ with marginal price equal to marginal cost when the market is fully covered, and firms are symmetric.
5 The literature on sequential screening (Armstrong 1996, Baron and Besanko 1984, Courty and Li 2000, Miravete 1996,

2002) studies optimal non-linear pricing when consumers know only the distribution of their valuations at the time ofcontracting but subsequently learn their actual valuations. It is assumed, however, that consumers and the monopolist havecommon priors in this literature.
6 The second-degree price discrimination based on beliefs of naive consumers is also studied by Heidhues and Kőszegi(2010) and Heidhues, Kőszegi and Murooka (2017) in the model of shrouded product attributes, and by Murooka and
Schwarz (2018) and Johnen (2019) in the model of automatic renewal of contracts. I consider neither shrouded attributes
nor contract renewal in this paper.
7 For surveys on competitive non-linear pricing, see Armstrong (2016) and Stole (2007).

6

Focusing on two-part tariﬀs, Yin (2004) shows that equilibrium prices equal to marginal cost only if themarginal consumer’s demand equals to the average demand in a generalized Hotelling model with variable
demands. This paper, in the standard Hotelling model with inelastic demands, finds another source ofdistorting marginal prices in two-part tariﬀ competition with biased consumers. Other papers on two-parttariﬀ competition when consumers are rational include Hoernig and Valletti (2007) in the Hotelling model
when consumers can buy from more than one firm, Herweg (2012) in a model of vertically diﬀerentiatedduopoly, Reisinger (2014) in a model of two-sided platforms, and Griva and Vettas (2015) in a model ofhomogeneous product duopoly when consumers diﬀer with respect to their usage levels.

3

The Model

I suppose there are two kinds of consumers with a higher valuation and a lower valuation of a good, andrefer to high type (H) and low type (L). Type H and type L consumers (with fractions λH and λL ,respectively) have a correct belief about their types. In addition to these two types, there exists the third
type of consumers who have a biased belief about their own preferences. I consider two cases separately.
First, the third type is overpessimistic (type P ) consumers, with fraction λP , who mistakenly believe thatthey are type L when they are, in fact, type H. I denote the fraction of type H as λH = 1 − λP − λL
where λP , λL ∈ (0, 1). Second, the third type is overoptimistic (type O) consumers, with fraction λO , who
mistakenly believe that they are type H when they are, in fact, type L. I denote the fraction of type L as
λL = 1 − λH − λO where λH , λO ∈ (0, 1).
The utility function of type j (j = H, L) is denoted by uj (q), where uH ′ (q) > uL ′ (q) > 0 and uj ′′ (q) < 0
for j = H, L. It is also assumed that uH (0) = uL (0) = 0, which implies that uH (q) > uL (q) ∀q. The
marginal cost is constant and is denoted by c > 0.
Firms cannot observe consumers’ types and beliefs, but they know the fractions of each type of consumer. A menu of two-part tariﬀs, Tj (q) = Aj + pj q (j = H, L), consists of, in general, two diﬀerent
tariﬀs. Because type P (resp., type O) believes that they are type L (resp., type H) in choosing a tariﬀ,
firms cannot screen between type P and type L (resp., type O and type H) at the time of contracting.
The timing of the game is as follows:Stage 0: Nature chooses the type of each consumer. A realized type is private information of a consumer.
Type H and type L consumers correctly infer their true type, while type P and type O consumers have awrong belief about their own types.
Stage 1: Firms oﬀer a menu of two-part tariﬀs. Each consumer commits to choose a tariﬀ.
Stage 2: Type P or type O consumers learn their true type. Then, each consumer determines a quantity
under the menu she committed in Stage 1.
7

In Stage 1, type H chooses the tariﬀ with (AH , pH ) and type L chooses (AL , pL ). The overpessimistic(type P ) consumer chooses (AL , pL ) when she is, in fact, type H, and in contrast, the overoptimistic (typeO) consumer chooses (AH , pH ) when she is, in fact, type L.
In Stage 2, the optimal quantity is determined by uj ′ (qj ) = pj , so that it is given by qj = (uj ′ )−1 (pj ) ≡Dj (pj ). It follows that Dj ′ (pj ) < 0, because uj ′′ (q) < 0. Type H consumes DH (pH ) and type L consumes
DL (pL ), respectively. The overpessimistic type P consumes DH (pL ), and the overoptimistic type O
consumes DL (pH ).

4

Two-Part Tariﬀ in Monopoly

In this section, I study the optimal two-part tariﬀ for a monopolist in the presence of overpessimistic oroveroptimistic consumers.

4.1

Overpessimistic Consumers

I first consider the case where biased consumers are overpessimistic. Let Sj (pj ) = uj (Dj (pj )) − pj Dj (pj )denotes the consumer surplus gross of fixed fee. It follows that Sj ′ (pj ) = −Dj (pj ). In Stage 1, type H
and type L correctly infer their net consumer surplus, SH (pH ) − AH and SL (pL ) − AL , respectively. TypeP , who is overpessimistic, incorrectly perceives her net benefit is SL (pL ) − AL in Stage 1, although her
ex-post consumer surplus turns out to be SH (pL ) − AL .
The monopolist solves the following:max(AH ,pH ,AL ,pL )π = (1 − λP − λL )AH + (λP + λL )AL+ (pH − c)(1 − λP − λL )DH (pH ) + (pL − c)[λP DH (pL ) + λL DL (pL )]subject toSH (pH ) − AH ≥ SH (pL ) − AL ,(1)

SL (pL ) − AL ≥ SL (pH ) − AH ,(2)

SH (pH ) − AH ≥ 0,(3)

SL (pL ) − AL ≥ 0,

(4)

where (1) and (2) are the incentive constraints, and (3) and (4) are the participation constraints, for type
H and type L, respectively. For type P who believes they are type L in choosing a tariﬀ, the incentive
and the participation constraints are the same as type L.

8

4.1.1

Complete Information

I begin with describing the optimal two-part tariﬀ under complete information when the monopolist canobserve types of consumers. If the constraints (1) and (2) are missing, obviously the constraints (3) and(4) are binding; AH = SH (pH ) and AL = SL (pL ). The profit becomes
π = (1 − λP − λL )SH (pH ) + (λP + λL )SL (pL )+ (pH − c)(1 − λP − λL )DH (pH ) + (pL − c)[λP DH (pL ) + λL DL (pL )].
I denote the optimal marginal prices under complete information as (p∗H , p∗L ). The first-order condition
with respect to pH is∂π= (1 − λP − λL )[SH ′ (pH ) + DH (pH ) + (pH − c)DH ′ (pH )] = 0.
∂pH

(5)

From Sj ′ (pj ) = −Dj (pj ) and Dj ′ < 0, the optimal marginal price for type H is given by p∗H = c. Therefore,p∗H is eﬃcient. The first-order condition with respect to pL is∂π= (λP + λL )SL ′ (pL ) + λP DH (pL ) + λL DL (pL ) + (pL − c)[λP DH ′ (pL ) + λL DL ′ (pL )]∂pL= λP [DH (pL ) − DL (pL )] + (pL − c)[λP DH ′ (pL ) + λL DL ′ (pL )] = 0,

(6)

by using Sj ′ (pj ) = −Dj (pj ) again. Thus, the equilibrium marginal price for type L and type P satisfiesp∗L = c −λP [DH (p∗L ) − DL (p∗L )]> c.
λP DH ′ (p∗L ) + λL DL ′ (p∗L )(7)

I assume that the profit is strictly concave in (pH , pL ), so that the second-order conditions are satisfied.8
The equilibrium marginal price, p∗L , depends on the ratio λL /λP . Note that p∗L > c as long as λP > 0(p∗L = c if λP = 0). When overpessimistic consumers are present, the monopolist raises the marginal priceto collect variable profits from type P even though it reduces the revenue from the fixed fee for type L
and type P .
Let wj be the net consumer surplus of type j. It follows that wH = wL = 0, because the participation
constraints are binding. Type P ’s ex-post (actual) consumer surplus is wP = SH (p∗L ) − AL = SH (p∗L ) −SL (p∗L ) > 0, that is, type P consumers benefit from choosing the wrong tariﬀ. This optimal two-part tariﬀ
under complete information does not satisfy the incentive constraint for type H, (1), because SH (p∗H ) −A∗H = 0 and SH (p∗L ) − A∗L > 0. Contrary, the incentive constraint for type L, (2), is satisfied with strict
inequality. The optimal tariﬀs are described in Figure 1. Because both participation constraints are∗binding, the indiﬀerence curves of type H and type L cross the origin. The quantity for type H, qH, is∗eﬃcient, but both qL
and qP∗ are distorted downward.
8 Because we have ∂ 2 π/∂p

H ∂pL = 0, the conditions, ∂2 π/∂p2 < 0 and ∂ 2 π/∂p2 < 0, are suﬃcient for strict concavity.
HL

9

[Figure 1]
Proposition 1 Suppose that types of consumers are observable. In the optimal two-part tariﬀ for amonopolist under the presence of overpessimistic type P consumers, the marginal price for type H is equal
to the marginal cost (p∗H = c), and that for type L and type P is larger than the marginal cost (p∗L > c).
The fixed fee for type H is A∗H = SH (c), and that for type L and type P is A∗L = SL (p∗L ) < A∗H .
4.1.2

Incomplete Information

Now, let us turn to the case of incomplete information. By the standard procedure with ignoring the
incentive constraint for type L, (2), for a moment, we can show that only (1) and (4) are binding.
Therefore, we have AL = SL (pL ) and AH = SH (pH ) − [SH (pL ) − SL (pL )]. Using this, the monopolist’s
problem is reduced tomax π = (1 − λP − λL )[SH (pH ) − SH (pL ) + SL (pL )] + (λP + λL )SL (pL )(pH ,pL )+ (pH − c)(1 − λP − λL )DH (pH ) + (pL − c)[λP DH (pL ) + λL DL (pL )].

(8)

I denote the optimal marginal prices under incomplete information as (bpH , pbL ). The first-order condition
with respect to pH is the same as (5), and we obtain pbH = c. The first-order condition with respect to pL
is∂π= (1 − λP − λL )[SL ′ (pL ) − SH ′ (pL )] + (λP + λL )SL ′ (pL )∂pL+ λP DH (pL ) + λL DL (pL ) + (pL − c)[λP DH ′ (pL ) + λL DL ′ (pL )]= (1 − λL )[DH (pL ) − DL (pL )] + (pL − c)[λP DH ′ (pL ) + λL DL ′ (pL )] = 0.

(9)Thus, the equilibrium marginal price for type L and type P satisfies
pbL = c −(1 − λL )[DH (bpL ) − DL (b
pL )]
> c.
λP DH ′ (bpL ) + λL DL ′ (b
pL )

(10)

The net surplus of type L is wL = 0, because the participation constraint is binding. In contrast, type
H receives the information rent, R, that is, wH = R(bpL ) ≡ SH (bpL ) − SL (bpL ) > 0. Type P ’s ex-postsurplus is wP = SH (bpL ) − AL = SH (bpL ) − SL (bpL ) > 0, which is equal to the information rent to type H;
because the incentive constraint (1) is binding, the overpessimistic, type P , consumers obtain the samenet surplus as type H consumers. By comparing (10) with (7), we see that pbL > p∗L ; under incomplete
information, the monopolist raises the marginal price for type L more than complete information to reducethe information rent.
The ignored incentive constraint for type L, (2), is rewritten as∫ pbL[DL (p) − DH (p)]dp.
0 ≥ SL (bpH ) − SH (bpH ) + SH (bpL ) − SL (bpL ) =
pbH

10

Because pbH = c < pbL , this is satisfied with strict inequality. This result is due to the well-known renteﬃciency tradeoﬀ; the monopolist distorts the marginal price for type L upward to reduce the information
rent to type H (and type P ). The marginal price for type H is eﬃcient because it yields the maximum fixedfee, and the marginal price for type H does not aﬀect the information rent. The optimal two-part tariﬀs
are depicted in Figure 2. Type H consumers are indiﬀerent between the two tariﬀs; the indiﬀerence curvebH , pbH ) is also tangent to the tariﬀ (AbL , pbL ). Type L consumers receiveof type H tangent to the tariﬀ (AbL , pbL ) crosses the origin.9 The
zero net surplus; the indiﬀerence curve of type L tangent to the tariﬀ (Aquantities are qbH = DH (c), qbP = DH (bpL ), qbL = DL (bpL ), and it follows that qbL < qbP < qbH . Compared to∗∗complete information, we have qbL < qL
, qbP < qP∗ and qbH = qH.

[Figure 2]
Proposition 2 Suppose that types of consumers are unobservable. In the optimal two-part tariﬀ for amonopolist under the presence of overpessimistic type P consumers, the marginal price for type H isthe same as complete information (bpH = p∗H = c). The marginal price for type L and type P is larger
bH = SH (c) − SH (bthan complete information, pbL > p∗L > c. The fixed fee for type H is A
pL ) + SL (bpL ),bL = SL (band that for type L and type P is A
pL ). Type H and type P obtain the same net surplus,
wH = wP = SH (bpL ) − SL (bpL ) > 0, while the net surplus of type L is wL = 0.
Now, I investigate the eﬀects of a change in the fraction of overpessimistic consumers. In particular,
we are interested in the impacts of debiasing policy toward overpessimistic consumers, namely, a decreasein λP given the level of λL , which leads to an increase in λH = 1 − λP − λL . First, the marginal pricefor type H does not depend on λP . The marginal price for type L and type P , pbL , is decreasing in λP ;diﬀerentiating (9) with respect to λP , we have(bpL − c)DH ′ (b
pL )
∂ pbL
=−
< 0,
∂2π
∂λP
∂p2
L

because we are assuming that ∂ 2 π/∂p2L < 0. That is, the debiasing policy enlarges quantity distortion
for type L and type P . When more overpessimistic consumers are debiased, the monopolist becomes lessconcerned with the loss in variable profit from type P by raising the marginal price, and hence it raises
the marginal price to reduce the information rent.
The equilibrium profit becomesπ(bpH = c, pbL , λP , λL ) = (1 − λP − λL )[SH (c) − SH (bpL )] + SL (bpL ) + (bpL − c)[λP DH (bpL ) + λL DL (bpL )].
9 Because I restrict possible contracts to two-part tariﬀs, the outcomes in Figure 2 are diﬀerent from the optimal contract
described in the standard textbook argument. At the optimal contract, type H is indiﬀerent between the pairs of quantityand payment, (qH , TH ) and (qL , TL ); the indiﬀerence curve of type H across (qH , TH ) also crosses (qL , TL ). However, thisoptimal contract cannot be implemented by the two-part tariﬀs.

11

By diﬀerentiating with respect to λP , and applying the envelope theorem, we have∂π ∂ pbL∂π= SH (bpL ) − SH (c) + (bpL − c)DH (bpL ) +=−∂λP∂pL ∂λP∫ pbL[DH (p) − DH (bpL )]dp < 0,cbecause DH is strictly decreasing. A decrease in the overpessimistic consumers enlarges the revenue fromthe fixed fee while reducing the variable profit from type P . However, a gain in the fixed fee turns out tobe larger than a loss in the variable profit. Therefore, the monopolist benefits from the debiasing policyof overpessimistic consumers.
A change in λP does not aﬀect the net surplus of type L, but does aﬀect the net surplus of type H
and P . Because
∂R(b
pL )
∂ pbL
= −[DH (bpL ) − DL (b
pL )]
> 0,
∂λP
∂λP
a decrease in the number of overpessimistic consumers reduces the information rent. In this respect, thepresence of overpessimistic consumers yields a positive externality to type H. Because the monopolist
reacts to the debiasing policy by raising the marginal price for type L to reduce the information rent, it
makes type H and type P worse oﬀ, and keeps type L unchanged. In other words, the debiasing policy has
an adverse eﬀect not only on the debiased type P consumers (who turn into type H) but on the remaining
biased (type P ) consumers.
The total consumer surplus, w = (1−λP −λL )wH +λP wP +λL wL , is now w(bpL , λL ) = (1−λL )[SH (b
pL )−
SL (b
pL )], and the impact on the total consumer surplus is given by∂ pbL∂w= −(1 − λL )[DH (bpL ) − DL (b
pL )]
> 0.
∂λP
∂λP
That is, the debiasing policy reduces the total consumer surplus because it reduces the total information
rent to type H and type P . However, the impact of a decrease in the overpessimistic consumers on the
total surplus, W = w + π, is ambiguous.
Proposition 3 In monopoly with overpessimistic consumers, the debiasing policy reducing the overpessimistic consumers raises the monopolist’s profit. It makes type H and type P worse oﬀ, and keeps typeL unchanged. It reduces the total consumer surplus, but the eﬀect on social welfare is ambiguous.
To investigate further the eﬀect on social welfare, I consider two examples. The first is the quadratic
utility, uj (q) = θj q − 12 q 2 where θH > θL . This yields the linear demand, Dj (p) = θj − p, and Sj =
1
1
1
ε
2 10
The second example is uj (q) = ε−1
θj ε q 1− ε
2 (θj − p) .

(ε > 1) which yields the iso-elastic demand,jDj (p) = θj p−ε with Sj (p) = ε−1
p1−ε . In the Appendix, I show that ∂W/∂λP > 0 holds under the linearθ

demand. Therefore, the positive eﬀect of debiasing policy on profit is dominated by the negative eﬀect on
10 Another expression of the quadratic utility is given by u (q) = θ (q − 1 q 2 ). However, we can obtain the same qualitativejj2
results with this alternative formulation.

12the consumer surplus, so that a decrease in λP reduces the total surplus. Under the iso-elastic demand,
we have at least limε→1 ∂W/∂λP > 0, and a numerical analysis reveals that ∂W/∂λP > 0 holds for any
ε > 1 in a wide range of parameter values of the iso-elastic demand.
Remark 1 In monopoly with overpessimistic consumers, the debiasing policy reducing the overpessimistic
consumers deteriorates social welfare under the linear demand, and the same is true under the iso-elastic
demand when the elasticity is suﬃciently small.

4.2

Overoptimistic Consumers

Next, I consider the case where biased consumers are overoptimistic. In Stage 1, type H and L correctly
infer their consumer surplus, but type O mistakenly perceives her net benefit is SH (pH ) − AH , although
her ex-post surplus is SL (pH ) − AH .
4.2.1

Complete Information

Let us begin with complete information. Because the participation constraints are binding, the monopolist’s problem is:
max π = (λH + λO )SH (pH ) + (1 − λH − λO )SL (pL )(pH ,pL )+ (pH − c)[λH DH (pH ) + λO DL (pH )] + (pL − c)(1 − λH − λO )DL (pL ).

(11)

The first-order condition with respect to pH is∂π= (λH + λO )SH ′ (pH ) + λH DH (pH ) + λO DL (pH ) + (pH − c)[λH DH ′ (pH ) + λO DL ′ (pH )]∂pH= λO [DL (pH ) − DH (pH )] + (pH − c)[λH DH ′ (pH ) + λO DL ′ (pH )] = 0,(12)which yieldsp∗H = c +λO [DH (p∗H ) − DL (p∗H )]< c.
λH DH ′ (p∗H ) + λO DL ′ (p∗H )(13)

The equilibrium marginal price, p∗H , depends on the ratio λH /λO and is lower than the marginal cost.
Note that p∗H = c if λO = 0. When overoptimistic consumers are present, the monopolist reduces themarginal price for type H and type O to enlarge the fixed fee for them even though it makes a loss invariable profit. The first-order condition with respect to pL is∂π= (1 − λH − λO )[SL ′ (pL ) + DL (pL ) + (pL − c)DL ′ (pL )] = 0,
∂pL

(14)

which yields p∗L = c. Again, I assume that the profit is strictly concave in (pH , pL ) to satisfy the secondorder conditions.

13

The net surplus of type H and type L is wH = wL = 0, and type O’s ex-post (actual) consumer
surplus is wO = SL (p∗H ) − AH < 0. As before, this optimal two-part tariﬀ under complete information
does not satisfy the incentive constraint for type H, (1), because SH (p∗H ) − A∗H = 0 and SH (p∗L ) − A∗L =SH (c) − SL (c) > 0. In contrast, the incentive constraint for type L, (2), is satisfied with strict inequality.
Figure 3 describes the optimal two-part tariﬀs with overpessimistic consumers. Again, the indiﬀerence
curves of type L and type H cross the origin because the participation constraints are binding. The∗∗∗
quantity for type L, qL, is eﬃcient, but both qH
and qOare distorted upward.

[Figure 3]
Proposition 4 Suppose that types of consumers are observable. In the optimal two-part tariﬀ for amonopolist under the presence of overoptimistic type O consumers, the marginal price for type H and typeO is lower than the marginal cost (p∗H < c), and that for type L is equal to the marginal cost (p∗L = c).
The fixed fee for type H and type O is A∗H = SH (p∗H ), and that for type L is A∗L = SL (c).
4.2.2

Incomplete Information

Next, I turn to incomplete information. Among the constraints (1)−(4), only (1) and (4) are binding as
before. The monopolist’s problem is:
max π = (λH + λO ) [SH (pH ) − SH (pL ) + SL (pL )] + (1 − λH − λO )SL (pL )(pH ,pL )+ (pH − c)[λH DH (pH ) + λO DL (pH )] + (pL − c)(1 − λH − λO )DL (pL )(15)

The first-order condition with respect to pH is the same as (12), and therefore we have pbH = p∗H . As
before, the marginal price for type H is the same as complete information because it does not aﬀect the
information rent. The first-order condition with respect to pL is∂π= (λH + λO )[SL ′ (pL ) − SH ′ (pL )] + (1 − λH − λO )[SL ′ (pL ) + DL (pL ) + (pL − c)DL ′ (pL )]∂pL= (λH + λO )[DH (pL ) − DL (pL )] + (1 − λH − λO )(pL − c)DL ′ (pL ) = 0,(16)which yields
pbL = c −(λH + λO )[DH (bpL ) − DL (b
pL )]
> c.
′
(1 − λH − λO )DL (b
pL )

(17)

The equilibrium marginal price, pbL , depends on λL = 1 − λH − λO . It is larger than p∗L = c because themonopolist wishes to reduce the information rent.
As we have shown, the marginal price for type H and type O is smaller, and that for L is larger
than the marginal cost. Thus, the ignored incentive constraint for type L, (2), is satisfied with strict

14

inequality. The net surplus of type L is zero, wL = 0, whereas type H receives the information rent,
wH = R(bpL ) = SH (bpL ) − SL (bpL ) > 0. Type O’s ex-post (actual) consumer surplus is
wO = SL (bpH ) − AH = SL (bpH ) − SH (bpH ) + SH (bpL ) − SL (b
pL )
∫ pbL
∫ pbL
=
[SH ′ (p) − SL ′ (p)]dp =[DL (p) − DH (p)]dp < 0,
p
bH

p
bH

which means that overoptimistic consumers suﬀer a loss by choosing the wrong tariﬀ. Figure 4 depicts the
optimal tariﬀs under incomplete information. As before, type H consumers are indiﬀerent between the two
tariﬀs, and type L consumers receive zero net surplus. In contrast to the case of overpessimistic consumers,
the monopolist sets the marginal price for type H and type O below the marginal cost to enlarge a gainin the fixed fee, although it suﬀers from a loss in the variable profit. Under incomplete information, themarginal price for type L becomes larger than the marginal cost to reduce the information rent. The
quantities are qbH = DH (bpH ), qbO = DL (bpH ), qbL = DL (bpL ), and we have qbL < qbO < qbH . Compared to∗∗∗complete information, it follows that qbL < qL
, qbO = qO
and qbH = qH.

[Figure 4]
Proposition 5 Suppose that types of consumers are unobservable. In the optimal two-part tariﬀ for amonopolist under the presence of overoptimistic type O consumers, the marginal price for type H and typeO is the same as complete information (bpH = p∗H < c). The marginal price for type L is larger than themarginal cost (bpL > c). The fixed fee for type H and type O is A∗H = SH (bpH ) − SH (bpL ) + SL (bpL ), andpL ). The net surpluses are wH > 0, wL = 0, wO < 0.
that for type L is A∗L = SL (bNow, let us consider the eﬀect of a change in the fraction of overoptimistic consumers under incomplete
information. In particular, we are interested in the impact of debiasing overoptimistic consumers, adecrease in λO given the level of λH , which leads to an increase in λL .
First, consider the marginal price for type H and type O in (13). By diﬀerentiating (12) with respect
to λO , we have

∂ pbH
λH DH ′ (bpH )[DH (bpH ) − DL (b
pH )]
< 0,
= ∂2π
′
′
∂λO
· [λH DH (bpH ) + λO DL (b
pH )]
∂p2
H

because we are assuming that ∂2π/∂p2H < 0. Thus, the marginal price for type H and type O is decreasingin λO . Similarly, as to the marginal price for type L in (17), by diﬀerentiating (16) with respect to λO ,we have∂ pbL
DH (bpL ) − DL (b
pL )
> 0,
= − ∂2π∂λO· (1 − λH − λO )∂p2
Lbecause ∂ 2 π/∂p2L < 0. That is, the marginal price for type L is increasing in λO . When more overoptimistic
consumers are debiased, the marginal price for type H and type O rises while the marginal price for type15

L falls. Therefore, the debiasing policy leads to a more eﬃcient quantity for each type. The more
overoptimistic consumers are debiased, the less incentive the monopolist has to reduce the marginal pricefor type H and type O in order to enlarge revenue from the fixed fee. Moreover, because debiasing leads
to more type L consumers, the monopolist becomes more concerned with the eﬃcient quantity for type L
rather than reducing information rent for type H.
By diﬀerentiating (15) with respect to λO , and applying the envelope theorem, we have∂π∂π ∂ pbH∂π ∂ pbL= SH (bpH ) − SH (bpL ) + (bpH − c)DL (bpH ) − (bpL − c)DL (bpL ) ++∂λO∂pH ∂λO∂pL ∂λO∫ pbL=
DH (p)dp + (bpH − c)DL (bpH ) − (bpL − c)DL (b
pL )
p
bH
[∫
] [∫
]
p
bL

=

c

DH (p)dp − (bpL − c)DL (bpL ) +cpbH

DH (p)dp − (c − pbH )DL (bpH ) .

(18)

In general, the sign of (18) is ambiguous. As to the first term of (18), it follows that
∫ pbL
∫ pbL
∫ pbL
DH (p)dp − (bpL − c)DL (bpL ) >DH (p)dp − (bpL − c)DH (bpL ) =[DH (p) − DH (bpL )]dp > 0.
ccc

The first inequality follows from DH (p) > DL (p) and pbL > c > pbH , and the second follows because DH isstrictly decreasing. About the second term of (18), if DL (bpH ) < DH (c) holds, then we have
∫ c
∫ c
∫ c
DH (p)dp − (c − pbH )DL (bpH ) >DH (p)dp − (c − pbH )DH (c) =[DH (p) − DH (c)]dp > 0,
p
bH

p
bH

p
bH

and thus (18) is positive. Because a decrease in the overoptimistic consumers implies that some of themturn to type L, the debiasing policy reduces the revenue from the fixed fee, and at the same time it enlarges
the variable profit from overoptimistic consumers who changed to type L. The sign of the total eﬀect isambiguous in general. However, if the optimal marginal price satisfies DL (bpH ) < DH (c), a loss in the fixed
fee is larger than a gain in the variable profit. Therefore, a decrease in overoptimistic consumers reduces
the monopolist’s profit.
Next, consider the eﬀect on consumer surplus. A change in λO does not aﬀect the net surplus of typeL, but does aﬀect the net surplus of type H and type O. Because
∂ pbL
∂R(b
pL )
= −[DH (bpL ) − DL (b
pL )]
< 0,
∂λO
∂λO
A decrease in the number of overoptimistic consumers enlarges the information rent to type H. Put
diﬀerently, the presence of overoptimistic consumers yields a negative externality to type H. The monopolist reacts to the debiasing policy reducing λO (raising λL ) by lowering the marginal price fortype L, and hence the information rent to type H becomes large. The total consumer surplus, w =λH wH + λO wO + (1 − λH − λO )wL , is expressed asw(bpH , pbL , λH , λO ) = (λH + λO )[SH (bpL ) − SL (bpL )] + λO [SL (bpH ) − SH (bpH )].
16

It follows that∂w
= SH (bpL ) − SL (bpL ) + SL (bpH ) − SH (b
pH )
∂λO
∂ pbH
∂ pbL
+ λO [DH (bpH ) − DL (bpH )]− (λH + λO )[DH (bpL ) − DL (b
pL )]
< 0,
∂λO
∂λO
That is, a decrease in the number of overoptimistic consumers increases the total consumer surplus. When
more overoptimistic consumers are debiased, not only a debiased consumer does not suﬀer from a loss, but
the information rent to type H increases. Moreover, the debiasing also mitigates the loss of (remaining)type O’s surplus by changing both marginal prices, pbH and pbL , in the favorable direction. Therefore, thedebiasing policy makes type H and type O better oﬀ, and keeps type L unchanged. In contrast to thecase of overpessimistic consumers, it is beneficial to both the debiased type O consumers (who turn into
type L) and the remaining biased (type O) consumers.
The impact on the total surplus, W = w + π, is given by∂W= SL (bpH ) − SL (bpL ) − (c − pbH )DL (bpH ) − (bpL − c)DL (b
pL )
∂λO
∂ pbH
∂ pbL
+ λO [DH (bpH ) − DL (bpH )]− (λH + λO )[DH (bpL ) − DL (b
pL )]
.
∂λO
∂λO

(19)

Again, the sign of this expression is ambiguous.
Proposition 6 In monopoly with overoptimistic consumers, the eﬀect of the debiasing policy reducing theoveroptimistic consumers on the monopolist’s profit is ambiguous, but it reduces the profit if DL (bpH ) <DH (c) holds. It makes type H and type O better oﬀ, and keeps type L unchanged. It raises the total
consumer surplus, but the eﬀect on social welfare is ambiguous.
As before, I consider two examples to see the impact on the total surplus. In the Appendix, I show that∂W/∂λO < 0 in the linear demand example. Therefore, the negative impact of the debiasing policy on
profit is dominated by the positive eﬀect on the total consumer surplus, and hence a decrease in λO raises
the total surplus. Under the iso-elastic demand, we have at least limε→1 ∂W/∂λO < 0, and a numericalanalysis reveals that ∂W/∂λO < 0 holds for any ε > 1 in a wide range of parameter values of the iso-elastic
demand.
Remark 2 In monopoly with overoptimistic consumers, the debiasing policy reducing the overoptimistic
consumers improves social welfare under the linear demand, and the same is true under the iso-elastic
demand when the elasticity is suﬃciently small.

17

5

Two-Part Tariﬀ Competition

Following Armstrong and Vickers (2001), I consider a variation of the Hotelling duopoly with allowing
variable consumption quantity for consumers. Consumers are uniformly distributed on [0, 1] interval, withthe total mass normalized to one. Firm 1 is located at 0, and firm 2 at 1. At each location, there exist
three types of consumers, type H, type L, and the type with a biased belief (type P or type O). It is
assumed that consumers make all their purchases from a single firm (“one-stop-shopping” assumption).
iFirm i (i = 1, 2) oﬀers a menu of two-part tariﬀs, TH= AiH + piH qH for type H, and TLi = AiL + piL qL

for type L. Each firm has the same marginal cost, c. The utility function is uj (q) (j = H, L), andI maintain the same assumptions about uj (q). In Stage 2, the quantity consumed by consumers whopurchased from firm i is determined by u′j (qj ) = pij , and hence it is given by qj = (uj ′ )−1 (pij ) ≡ Dj (pij ).
Type H consumes DH (piH ) and type L consumes DL (piL ), respectively. Overpessimistic type P consumes
DH (piL ), and overoptimistic type O consumes DL (piH ).
Let Sj (pij ) = uj (Dj (pij )) − pij Dj (pij ) be the consumer surplus of type j gross of fixed fee. We have
Sj ′ (pij ) = −Dj (pij ). Denote the net consumer surplus of type j buying from firm i as wji = Sj (pij ) − Aij ;iiwHand wLare true net surplus for type H and type L, respectively. The perceived net surplus for typei
i
. I assume that each consumer receives a, and that for type O is wH
P at the time of contracting is wL

benefit r from buying a good independent of quantity and r is suﬃciently large that all consumers would
buy a good. The consumer located at x incurs transportation cost tx if she buys from firm 1 and t(1 − x)
if she buys from firm 2. The location of the marginal type j consumer is determined byr − texj + wj1 = r − t(1 − xej ) + wj2

5.1

⇐⇒

x
ej =

1
(t + wj1 − wj2 ).
2tOverpessimistic Consumers

Overpessimistic type P mistakenly believe they are type L when they are type H. First, I consider two-parttariﬀ competition under complete information where firms can observe types of consumers.
The profit of firm 1 is given by
π1 = (1 − λP − λL )exH [A1H + (p1H − c)DH (p1H )]+ λP xeL [A1L + (p1L − c)DH (p1L )] + λL x
eL [A1L + (p1L − c)DL (p1L )]1
1
2
1(t + wH− wH)[SH (p1H ) − wH
+ (p1H − c)DH (p1H )]
2t
1
1
1
2
1
1
2
+ (λP + λL ) (t + wL
− wL)[SL (p1L ) − wL
] + (t + wL
− wL)(p1L − c)[λP DH (p1L ) + λL DL (p1L )].
2t
2t= (1 − λP − λL )1
1
Firm 1 chooses (p1H , wH
, p1L , wL
) to maximize π1 . The first-order condition for p1H is

∂π1
1
1
2
= (1 − λP − λL ) (t + wH
− wH)[SH ′ (p1H ) + DH (p1H ) + (p1H − c)DH ′ (p1H )] = 0.
1
∂pH
2t
18

Since Sj ′ (pij ) = −Dj (pij ), this is reduced to (p1H − c)DH ′ (p1H ) = 0. From Dj ′ < 0, we obtain p1H = c. The
profit of firm 2 is expressed similarly, and we also obtain p2H = c. Therefore, the same as monopoly, theequilibrium marginal price satisfies p1H = p2H ≡ p∗H = c. The first-order condition for p1L is
1
∂π1
1
2
= (λP + λL ) (t + wL
− wL
)SL ′ (p1L )
1
∂pL
2t
1
2
1
− wL
)[λP DH (p1L ) + λL DL (p1L ) + (p1L − c){λP DH ′ (p1L ) + λL DL ′ (p1L )}]+ (t + wL
2t
[
]
1
1
2
= (t + wL− wL) λP DH (p1L ) − λP DL (p1L ) + (p1L − c){λP DH ′ (p1L ) + λL DL ′ (p1L )} = 0,
2t

(20)

by using Sj ′ (pij ) = −Dj (pij ) again. Thus, p1L satisfies
p1L = c −λP [DH (p1L ) − DL (p1L )]> c.
λP DH ′ (p1L ) + λL DL ′ (p1L )(21)

Therefore, the marginal price for type L and type P is larger than the marginal cost. Again, the profit offirm 2 is expressed similarly, and we can derive the same expression about p2L . The equilibrium marginal
price is given by p1L = p2L ≡ p∗L > c, which is equal to the optimal marginal price in monopoly undercomplete information, (7). As before, the marginal price for type L is also eﬃcient if type P is absent.
This is because duopolists maintain the eﬃcient marginal price to enlarge the fixed fee for each type.
When type P is present, however, duopolists want to raise the marginal price for type L and type P tocollect variable profits from type P .
1isAs to the optimal fixed fee, the first-order condition for wH

∂π1
1
1
1
1
1
1
2
1 = (1 − λP − λL ) 2t [SH (pH ) − wH + (pH − c)DH (pH ) − (t + wH − wH )] = 0.
∂wH
2
1
2
1
=
= wH
, we obtain wH= wH
From p1H = c, evaluating at the symmetric equilibrium in which wH
1is
SH (c) − t, which implies A1H = A2H = t. Finally, the condition for wL

∂π1
1 1
1
1
1
1
2
1
1
1 = (λP + λL ) 2t [SL (pL ) − wL − (t + wL − wL )] + 2t (pL − c)[λP DH (pL ) + λL DL (pL )] = 0.
∂wL
1
2
Again, evaluating at the symmetric equilibrium in which wL= wL, we obtain
1
2
wL
= wL
= SL (p1L ) − t +1(p1 − c)[λP DH (p1L ) + λL DL (p1L )].
λP + λL L

The optimal fixed fee for type L and type P is
A1L = A2L = t −1(p1 − c)[λP DH (p1L ) + λL DL (p1L )] < t,
λP + λL L(22)which implies AiL < AiH .
In choosing the fixed fee for type H, duopolists face the trade-oﬀ between enlarging revenue from fixed
fees and shrinking market share. Competition leads to the fixed fee equal to t; the firms can enjoy some19

profits because their products are diﬀerentiated. As to the fixed fee for type L and type P , in addition tothis trade-oﬀ, duopolists also take into account the gain in variable profit by expanding market share; thisleads to the fixed fee for type L and type P below t. Note that A1L = A2L = t if λP = 0. Therefore, theoptimal menu of two-part tariﬀs becomes a single two-part tariﬀ when type P is not present. This also
implies that both incentive constraints are satisfied with equality under incomplete information.
Plugging the above solutions into the profit of firm 1, we can see that the equilibrium profit is equal
to t/2, independent of λP and λL . This is because the gain in variable profit from type L and type P isexactly canceled out by the loss in a reduction of the fixed fee. Thus, the profit amounts to t/2 for any
consumer type distributions.
Proposition 7 When firms compete in two-part tariﬀs under complete information with overpessimistic
type P consumers,(i) the marginal price for type H is equal to the marginal cost (p∗H = c), and that for type L and type P islarger than the marginal cost (p∗L > c). These marginal prices are the same as those in monopoly undercomplete information. The fixed fee for type H is A∗H = t, and that for type L and type P is A∗L < t.
(ii) The net surplus of type H and type L are wH = SH (c) − t, wL = SL (p∗L ) − A∗L . The actual surplus oftype P is wP = SH (p∗L ) − A∗L > wL .
(iii) The equilibrium profit of each firm is t/2, independent of λP and λL .
(iv) When the overpessimistic consumer is not present, the optimal menu of two-part tariﬀs degenerates
to a single two-part tariﬀ in which (p∗H , A∗H ) = (p∗L , A∗L ) = (c, t).
[Figure 5]
The equilibrium menu of contracts is depicted in Figure 5. The benchmark is the single two-part tariﬀ
when type P is not present; T = t + cq. Both firms oﬀer the same tariﬀ, and type H choose DH (c)(point H) while type L chooses DL (c). When type P is present, the menu for type H remains the same,and type H continues to choose point H. As to the menu for type L and type P , TL = A∗L + p∗L q, themarginal price is larger, and the fixed fee is smaller. Type L chooses point L, and type P chooses point P ,respectively. Then, type L consumers are better oﬀ with the presence of the overpessimistic consumers.11
This contrasts to monopoly in which type L receives no externality by the overpessimistic consumers. In
addition, as we will see formally in the next proposition, the actual net surplus of type P is smaller thanthat of type H, that is, wH > wP ; type P consumers suﬀer from being biased under competition. This
result is also opposite to monopoly where type P consumers do not suﬀer from being biased.
11 Figure 5 depicts the case where the lines of the two-part tariﬀs cross at D (c). Although this happens under theLquadratic utility, it is not true in general. However, as we see in the next proposition, type L consumers strictly prefer∗∗∗∗choosing (pL , AL ) to (pH , AH ) = (c, t), so that they are better oﬀ with the presence of the overpessimistic consumers.

20

Now, let us turn to incomplete information in which firms cannot observe types of consumers. Under
incomplete information, the menu oﬀered by firm i, (piH , AiH , piL , AiL ), should satisfy the following incentive
constraints:
SH (piH ) − AiH ≥ SH (piL ) − AiL ,(23)

SL (piL ) − AiL ≥ SL (piH ) − AiH ,(24)

for i = 1, 2. The participation constraints are not necessary because it is always satisfied under duopoly
competition. We have the following:
Proposition 8 When firms compete in two-part tariﬀs under incomplete information with overpessimistic
type P consumers, the optimal two-part tariﬀ under complete information in Proposition 7 always satisfies
both (23) and (24) with strict inequality.
Proof : See Appendix.
Now, let us turn to the welfare eﬀects of the debiasing policy. The marginal price for type H does notdepend on λP . As to the marginal price for type L and type P , by diﬀerentiating (20) we have′(p∗L )λL DL∂p∗L
1 DH (p∗L ) − DL (p∗L )
=−
2π
′
∗
′ (p∗ ) > 0,
∂
1
∂λP
2
λP DH (pL ) + λL DL
1 2
L∂(pL )

because we are assuming that ∂ 2 π1 /∂(p1L )2 < 0. In contrast to the monopoly under incomplete information,
the debiasing policy leads to lower marginal price for type L and type P . The duopolists are less concernedwith collecting variable profits when there are fewer overpessimistic consumers.
As we have seen, each firm’s profit is t/2, and the producer surplus is t. In Figure 5, the debiasing
policy changes the tariﬀ for type L and type P towards the tariﬀ for type H, and hence type L is worse
oﬀ while type P is better oﬀ. The total consumer surplus, w = (1 − λP − λL )wH + λP wP + λL wL , isexressed as
w = −t + (1 − λP − λL )SH (c) + λP SH (p∗L ) + λL SL (p∗L ) + (p∗L − c)[λP DH (p∗L ) + λL DL (p∗L )].
The total surplus, W = w + π1 + π2 , amounts to
W = (1 − λP − λL )SH (c) + λP SH (p∗L ) + λL SL (p∗L ) + (p∗L − c)[λP DH (p∗L ) + λL DL (p∗L )].
By diﬀerentiating with respect to λP , we have
∂p∗
∂W
∂w
′
′
=
= SH (p∗L ) − SH (c) + (p∗L − c)DH (p∗L ) + (p∗L − c)[λP DH(p∗L ) + λL DL(p∗L )] L
∂λP
∂λP
∂λP
∫ p∗L
∗
∂p
=
[DH (p∗L ) − DH (p)]dp − λP [DH (p∗L ) − DL (p∗L )] L < 0,
∂λP
c
21

where we have used (21) and ∂p∗L /∂λP > 0. Therefore, a negative impact of the debiasing policy against
type L is dominated by a positive impact against type P .
Proposition 9 Under competition with overpessimistic type P consumers, the debiasing policy reducing
the overpessimistic consumers does not change the profit of each firm. It keeps type H unchanged, andmakes type L worse oﬀ while type P better oﬀ. The total consumer surplus is raised by the debiasing
policy. Therefore, the debiasing policy improves social welfare.
It turns out that the eﬀect of debiasing policy against overpessimistic consumers depends on market
structure. As we have seen in Proposition 3, under monopoly it is bad for both the debiased type P
consumers who turn into type L and the remaining type P consumers. Under competition, in contrast, it
is good for both the debiased and the remaining biased consumers. Moreover, it makes type H worse oﬀ andtype L unchanged in monopoly, while type H unchanged and type L worse oﬀ in competition. Debiasing
overpessimism is harmful in monopoly but beneficial in competition, not only for overall consumers but
for social welfare.

5.2

Overoptimistic Consumers

Overoptimistic type O mistakenly believe that they are type H when they are type L. First, I consider
two-part tariﬀ competition under complete information. The profit of firm 1 is given by
π1 = λ H xeH [A1H + (p1H − c)DH (p1H )] + λO xeH [A1H + (p1H − c)DL (p1H )]+ (1 − λH − λO )exL [A1L + (p1L − c)DL (p1L )]
1
1
1
2
1
1
2
(t + wH− wH)[SH (p1H ) − wH
] + (t + wH
− wH)(p1H − c)[λH DH (p1H ) + λO DL (p1H )]
2t
2t
1
1
2
1
+ (1 − λH − λO ) (t + wL
− wL)[SL (p1L ) − wL
+ (p1L − c)DL (p1L )].
2t= (λH + λO )1
1
Firm 1 chooses (p1H , wH
, p1L , wL
) to maximize π1 . The first-order condition for p1H is

∂π1
1
1
2
= (λH + λO ) (t + wH
− wH
)SH ′ (p1H )
∂p1H
2t
[
]
1
1
2
+ (t + wH− wH) λH DH (p1H ) + λO DL (p1H ) + (p1H − c){λH DH ′ (p1H ) + λO DL ′ (p1H )}
2t
[
]
1
1
2
= (t + wH− wH) λO (DL (p1H ) − DH (p1H )) + (p1H − c){λH DH ′ (p1H ) + λO DL ′ (p1H )} = 0,
2t

(25)

by using Sj ′ (pij ) = −Dj (pij ) again. Thus, p1H satisfies
p1H = c +λO [DH (p1H ) − DL (p1H )]< c.
λH DH ′ (p1H ) + λO DL ′ (p1H )(26)

The profit of firm 2 is defined similarly, and the equilibrium marginal price for type H and type O,
p1H = p2H = p∗H , is smaller than the marginal cost. This marginal price is the same as that in monopoly,22

(13). In particular, the markup is zero if λO = 0. When type O is present, as in monopoly under complete
information, duopolists want to reduce the marginal price for type H and type O to enlarge revenue fromthe fixed fee, sacrificing a loss in the variable profit.
The condition for p1L is
∂π1
1
1
2
= (1 − λH − λO ) (t + wL
− wL)[SL ′ (p1L ) + DL (p1L ) + (p1L − c)DL ′ (p1L )] = 0.
1
∂pL
2t
Since Sj ′ (pij ) = −Dj (pij ), this is reduced to (p1L − c)DL ′ (p1L ) = 0. From Dj ′ < 0, we obtain p1L = c. Again,
the profit of firm 2 is defined similarly, and we can derive that p2L = c. Therefore, similar to monopoly,each firm chooses the eﬃcient price for type L.
1
As to the optimal fixed fee, the first-order condition for wH
is

∂π1
1
1
1
1
2
= (λH + λO ) [SH (p1H ) − wH
− (t + wH− wH)] + (p1H − c)[λH DH (p1H ) + λO DL (p1H )] = 0.
1
∂wH
2t
2t
1
2
Evaluating at the symmetric equilibrium in which wH= wH, we obtain
1
2
wH
= wH
= SH (p1H ) − t +1(p1 − c)[λH DH (p1H ) + λO DL (p1H )],λH + λO H

which implies that the optimal fixed fee is given by
A1H = A2H = t −1(p1 − c)[λH DH (p1H ) + λO DL (p1H )] > t.
λH + λO H

(27)

1
is
Finally, the condition for wL

∂π1
1
1
1
1
1
1
2
1 = (1 − λH − λO ) 2t [SL (pL ) − wL + (pL − c)DL (pL ) − (t + wL − wL )] = 0.
∂wL
1
2
1
2
From p1L = c, evaluating at the symmetric equilibrium in which wL= wL, we obtain wL= wL= SL (c) − t,which implies A1L = A2L = t. Thus, the fixed fee for type H and O is larger than that for type L. Again, we
can confirm that A1H = A2H = t if λO = 0; the optimal menu of two-part tariﬀs becomes a single two-parttariﬀ when type O is not present.
Proposition 10 When firms compete in two-part tariﬀs under complete information with overoptimistic
type O consumers,(i) the marginal price for type H and type O is less than the marginal cost (p∗H < c), and that for type L
is equal to the marginal cost (p∗L = c). These marginal prices are the same as those in monopoly undercomplete information. The fixed fee for type H and type O is A∗H > t, and for type L is A∗L = t.
(ii) The net surplus of type H and type L are wH = SH (p∗H ) − A∗H , wL = SL (c) − t. The actual surplusof type O is wO = SL (p∗H ) − A∗H < wH .
(iii) The equilibrium profit of each firm is t/2, independent of λH and λO .
23

[Figure 6]
The equilibrium menu of contracts is depicted in Figure 6. Again, the benchmark is the single two-parttariﬀ when type O is absent; T = t + cq. Both firms oﬀer the same tariﬀ, and type H choose DH (c) while
type L chooses DL (c) (point L). When type O is present, the menu for type L remains the same, andtype L continues to choose point L. As to the menu for type H and type O, TH = A∗H + p∗H q, in contrast,
the marginal price is smaller and the fixed fee is larger. Type H chooses point H, and type O chooses
point O, respectively.12 Then, type H obtains a larger net surplus with the presence of overoptimistic
consumers. This result is in contrast to the case of overpessimistic consumers in which type L benefits
from positive externality by the presence of biased consumers. In addition, as we will see formally in thenext proposition, the net surplus of type O is smaller than that of type L, namely wO < wL ; similar tomonopoly, type O suﬀers from being biased under competition.
Now, let us turn to the case of incomplete information in which firms cannot observe types of consumers.
Similar to the case of overpessimistic consumers, we have the following:
Proposition 11 When firms compete in two-part tariﬀs under incomplete information with overoptimistic
type O consumers, the optimal two-part tariﬀ under complete information in Proposition 10 always satisfies
both (23) and (24) with strict inequality.
Proof : See Appendix.
Now, I study the welfare eﬀects of debiasing policy. The marginal price for type L does not depend on
λO . As to the marginal price for type H and O, by diﬀerentiating (25) we have′(p∗H )∂p∗H
λH DH
1 DH (p∗H ) − DL (p∗H )=′ (p∗ ) + λ D ′ (p∗ ) < 0,
∂ 2 π1
∂λO
2
λH DH
O L H
1 2
H
∂(pH )

since we are assuming that ∂ 2 π1 /∂(p1H )2 < 0. Like monopoly, the debiasing policy leads to higher marginal
price for type H and type O. The duopolists are less concerned with sacrificing variable profits to boost
fixed fees when there are less overoptimistic consumers.
As we have seen, each firm’s profit is t/2, so the producer surplus is t. In Figure 6, the debiasing policychanges the tariﬀ for type H and type O towards the tariﬀ for type L, and hence type H is worse oﬀ while
type O is better oﬀ. The total consumer surplus, w = λH wH + λO wO + (1 − λH − λO )wL , is expressed as
w = −t + λH SH (p∗H ) + λO SL (p∗H ) + (1 − λH − λO )SL (c) + (p∗H − c)[λH DH (p∗H ) + λO DL (p∗H )].
12 Figure 6 describes the case where the two-part tariﬀs cross at D (c). Although this happens under quadratic utility, itH
is not true in general. However, as we see in the next proposition, type H consumers strictly prefer choosing (p∗H , A∗H ) to∗∗(pL , AL ) = (c, t), so that they are better oﬀ with the presence of the overoptimistic consumers.

24

The total surplus, W = w + π1 + π2 , amounts to
W = λH SH (p∗H ) + λO SL (p∗H ) + (1 − λH − λO )SL (c) + (p∗H − c)[λH DH (p∗H ) + λO DL (p∗H )].
By diﬀerentiating with respect to λO , we have
∂W
∂w
∂p∗
′
′
=
= SL (p∗H ) − SL (c) + (p∗H − c)DL (p∗H ) + (p∗H − c)[λH DH(p∗H ) + λO DL(p∗H )] H
∂λO
∂λO
∂λO
∫ c
∗
∂p
=
[DL (p) − DL (p∗H )]dp + λO [DH (p∗H ) − DL (p∗H )] H < 0,∂λO
p∗H
where the we have used (26) and ∂p∗H /∂λO < 0. Therefore, a negative impact of the debiasing policy
against type H is dominated by a positive impact against type O.
Proposition 12 Under competition with overoptimistic type O consumers, the debiasing policy reducing
the overoptimistic consumers does not change the profit of each firm. It keeps type L unchanged, andmakes type H worse oﬀ while type O better oﬀ. The total consumer surplus is raised by the debiasing
policy. Therefore, the debiasing policy improves social welfare.
From Propositions 6 and 12, we can see that the debiasing policy against overoptimism has similareﬀect on consumers under any market structure. In monopoly as well as in competition, it is good not
only for the debiased type O consumers who turn into type H but for still biased type O consumers. The
only diﬀerence is that it makes type H better oﬀ in monopoly while makes them worse oﬀ in competition;
it keeps type L unchanged in monopoly and competition. Debiasing overoptimism is beneficial not only
for overall consumers but for social welfare, regardless of market structure.

6

Conclusion

In this article, I examined how a monopolist and competitive firms respond in designing two-part tariﬀs
when some consumers either underestimate or overestimate their true demand, and how the existence ofbiased consumers aﬀects unbiased consumers. Incomplete information about consumer types matters inmonopoly but not in competition, because full-information equilibrium remains an equilibrium in competition. When unbiased consumers are heterogeneous in their usage levels, consumers with biased beliefs
could have diﬀerent impacts on diﬀerent types of consumers, depending on market structure. I have shownthat the presence of overpessimistic consumers makes high-type consumers better oﬀ and keeps low typeunchanged in monopoly, while keeps high type unchanged and makes low type better oﬀ in competition.
Alternatively, the presence of overoptimistic consumers makes high type worse oﬀ in monopoly and betteroﬀ in competition, while always keeps low type unchanged.

25

Overall, the existence of overpessimistic consumers raises the total consumer surplus in monopoly,
but reduces in competition. In contrast, the existence of overoptimistic consumers always reduces thetotal consumer surplus. From the sellers’ viewpoint, the monopolist wants to educate overpessimistic
consumers while may not want to educate overoptimistic consumers. The competitive firms, however, do
not have the incentive to educate any biased consumers. I find that the monopolist’s desire to debias
overpessimism is harmful to overall consumers and may deteriorate social welfare. On the other hand,
educating overoptimism by the monopolist is beneficial for overall consumers and may improve social
welfare. In competition, however, my result suggests that firms do not provide eﬀort which leads tosocially desirable debiasing for either overpessimistic or overoptimistic consumers, and therefore someintervention that helps biased consumers to correct their expectations about their own demands would benecessary.
I focused on second-degree price discrimination by two-part tariﬀs without repeated interaction between
consumers and firms. In an intertemporal setting, however, firms can learn about private information or
naiveté of consumers by observing their past usage so that they can engage in third-degree (naiveté-based)discrimination. A possible extension would be to incorporate the aspect of third-degree price discrimination
by additional information about consumers (Heidhues and Kőszegi 2017) or by usage information on past
purchases in dynamic contracting (Johnen 2020).
I do not consider the aspect of learning by consumers (Goettler and Clay 2011, Grubb and Osborne2015). In this sense, the model of this paper is more prevalent to markets with an inflow of new andunsophisticated consumers or markets where switching costs could make consumers maintain the same tariﬀ
plan (Miravete 2003). Incorporating consumer learning with endogenizing fractions of biased consumers
would yield further implications about dynamic tariﬀ and usage choices by consumers and evolutions of
tariﬀ menus as a screening mechanism among heterogeneous consumers. I leave these issues for future
research.

26

Appendix
Proof of Remark 1 : First, under the linear demand, it follows thatpbL = c +1 − λL(θH − θL ),λP + λLand the quantity for each type of consumers are:qbH = θH − c, qbP = θH − c −1 − λL
1 − λL(θH − θL ), qbL = θL − c −(θH − θL ).
λP + λL
λP + λL

The impact on the total surplus is derived as[]2∂W
1 1 − λL=
(θH − θL ) > 0.
∂λP
2 λP + λL
Next, in the iso-elastic demand, we obtainpbL =ε(λP θH + λL θL )c,
A

where A ≡ ε(λP θH + λL θL ) − (1 − λL )(θH − θL ). From the second-order condition, ∂ 2 π/∂p2L < 0, it follows
that A > 0. We also have
∂ pbL
ε(1 − λL )θH (θH − θL )c
=−
< 0.
∂λP
A2
The eﬀect on the total surplus can be derived as∂W∂ pbL= −(1 − λL )[DH (bpL ) − DL (bpL )]+ SH (bpL ) − SH (c) + (bpL − c)DH (b
pL )
∂λP
∂λP
θH [ε(λP θH + λL θL )]−ε c1−ε
[K + (1 − λL )(θH − θL ){ε(λP θH + λL θL ) + (ε − 1)(1 − λL )(θH − θL )}],
=
A2−ε
(
{
}ε−1 )ε(λP θH + λL θL )εK=
(λP θH + λL θL )A 1 −
.
ε−1
A

where

Applying l’Hôpital’s rule yields([{}ε−1 )]dε(λP θH + λL θL )lim K = lim
ε(λP θH + λL θL )A 1 −ε→1ε→1 dεA(}ε−1 ){ε(λP θH +λL θL ){2ε(λP θH + λL θL ) − (1 − λL )(θH − θL )} A
 (λP θH + λL θL ) 1 −= lim 
)}

{
(
ε→1
ε(λP θH +λL θL )ε−1−{ε(λP θH + λL θL )}ε A2−ε 1−ε
+
+
ln
A
ε
A
(
)
λP θH + λL θL= −(λP θH + λL θL )A|ε=1 · ln,
A|ε=1
where A|ε=1 = λP θH + λL θL − (1 − λL )(θH − θL ) > 0. Therefore, we have
[
(
)]
θH
λP θH + λL θL∂W=(1 − λL )(θH − θL ) − A|ε=1 · ln.
ε→1 ∂λP
A|ε=1
A|ε=1
lim

27

Because the logarithmic function is strictly concave, we have ln(1 + x) < x. Thus, we obtain(())λ P θH + λ L θL
λP θH + λL θL
A|ε=1 · ln
< A|ε=1− 1 = (1 − λL )(θH − θL )
A|ε=1
A|ε=1
which implies that limε→1 ∂W/∂λP > 0. Because ∂W/∂λP is continuous in ε, ∂W/∂λP > 0 holds if ε isclose to unity. ■Proof of Remark 2 : In the linear demand example, the marginal prices are
pbH = c −λO
(θH − θL ),λH + λO

pbL = c +λH + λO(θH − θL ),1 − λH − λOand the quantity for each type of consumers are:qbH = θH − c +λOλOλH + λO(θH − θL ), qbO = θL − c +(θH − θL ), qbL = θL − c −(θH − θL ).
λH + λO
λH + λO
1 − λH − λO

The quantity of type H is larger than the eﬃcient quantity, θH − c. The condition DL (bpH ) < DH (c) issatisfied; the debiasing policy reduces the profit. The impact on the total surplus is now given by[]∂W(θH − θL )2(λH + λO )3 (2 − λH − λO ) + λO (1 − λH − λO )2 (2λH + λO ) < 0.
=−
2
2
∂λO
2(λH + λO ) (1 − λH − λO )Next, under the iso-elastic demand, the marginal prices arepbH =ε(λH θH + λO θL )
c,
B1

pbL =

ε(1 − λH − λO )θL
c
B2

where B1 ≡ ε(λH θH + λO θL ) + λO (θH − θL ) and B2 ≡ ε(1 − λH − λO )θL − (λH + λO )(θH − θL ). The
second-order condition, ∂ 2 π/∂p2H < 0, amounts to B1 > 0, and ∂ 2 π/∂p2L < 0 to B2 > 0. Because the sign
of
[
]
DL (b
pH ) − DH (c) = c−ε B1 ε θL {ε(λH θH + λO θL )}−ε − θH {ε(λH θH + λO θL ) + λO (θH − θL )}−ε
is ambiguous, the condition DL (bpH ) < DH (c) may not hold in general, but note that it does hold whenε = 1. Also, we have
∂ pbH
ελH θH (θH − θL )c
< 0,
=−
∂λO
B1 2

∂ pbL
εθL (θH − θL )c
> 0.
=
∂λO
B2 2

The impact on the total surplus in (19) now amounts to
]
[
∂W
θL
ελH λO θH (θH − θL )2 c= (bpH )−εpbH − θL (c − pbH ) −
∂λO
ε−1
B1 2
[
]
θL
ε(λH + λO )θL (θH − θL )2 c
− (bpL )−εpbL + θL (bpL − c) +ε−1B2 2{ε(λH θH + λO θL )}−ε {ε(1 − λH − λO )θL }−ε c1−ε θL εε=
L1
B1 1−ε B2 1−ε (ε − 1){ε(λH θH + λO θL )}−ε {ε(1 − λH − λO )θL }−ε c1−ε (θH − θL )2 ε1+εL2 ,−B1 2−ε B2 2−ε
28

(A.1)

whereL1 = B2 1−ε {(1 − λH − λO )θL }ε [B1 − ελO (θH − θL )] − B1 1−ε (λH θH + λO θL )ε [B2 + ε(λH + λO )(θH − θL )],L2 = B2 2−ε {(1 − λH − λO )θL }ε λH λO θH + B1 2−ε (λH θH + λO θL )ε (λH + λO )θL .
By evaluating L1 at ε = 1, we have
L1 |ε=1 = (1 − λH − λO )θL [B1 |ε=1 − λO (θH − θL )] − (λH θH + λO θL )[B2 |ε=1 + (λH + λO )(θH − θL )] = 0,
and therefore, from (A.1), we obtain∂W(θH − θL )2 [B2 |ε=1 (1 − λH − λO )λH λO θH + B1 |ε=1 (λH θH + λO θL )(λH + λO )]=−< 0.
ε→1 ∂λO(λH θH + λO θL )(λH + λO )B1 |ε=1 B2 |ε=1
lim

Because ∂W/∂λO is continuous in ε, ∂W/∂λO < 0 holds if ε is close to unity. ■Proof of Proposition 8 : First, we consider the incentive constraint for type H, (23). Plugging p∗H , A∗H , A∗L ,we obtain
SH (p∗H ) − A∗H − [SH (p∗L ) − A∗L ]1(p∗ − c)[λP DH (p∗L ) + λL DL (p∗L )]λP + λL L
]
[
∫ p∗L
λL
λP
DH (p∗L ) +DL (p∗L )=DH (p)dp − (p∗L − c)λP + λL
λP + λL
c
∫ p∗L
∫ p∗L
>
DH (p)dp − (p∗L − c)DH (p∗L ) =[DH (p) − DH (p∗L )]dp > 0,
= SH (c) − SH (p∗L ) −ccso that (23) is satisfied with strict inequality. As to the incentive constraint for type L and type P , (24),
we have
SL (p∗L ) − A∗L − [SL (p∗H ) − A∗H ] = SL (p∗L ) − SL (c) + (p∗L − c)Defining

[

]
λL
λP
DH (p∗L ) +DL (p∗L ) .
λP + λL
λP + λL]
λL
λP
DH (pL ) +DL (pL )F (pL ) = SL (pL ) − SL (c) + (pL − c)λP + λL
λP + λL[

as a function of pL , this incentive constraint amounts to F (p∗L ) ≥ 0.
From (20) we obtain
[
]
λP
λP
λL
′
′
F (pL ) =[DH (pL ) − DL (pL )] + (pL − c)
D (pL ) +D (pL )
λP + λL
λP + λL H
λP + λL L
1
2t
∂πi
.
=
i − w j ∂pi
λP + λL t + wL
L
L′i
Because the optimal marginal price satisfies ∂πi /∂pL
= 0, we have F ′ (p∗L ) = 0. Moreover, we have

F (c) = 0 and F ′ (c) > 0. Because F ′′ (pL ) < 0 follows from ∂ 2 πi /∂(piL )2 < 0, we must have F (p∗L ) > 0,
and therefore (24) is satisfied with strict inequality. ■29

Proof of Proposition 11 : First, I consider the incentive constraint for type L, (24). Plugging p∗L , A∗H , A∗L ,we obtain
SL (p∗L ) − A∗L − [SL (p∗H ) − A∗H ]1= SL (c) − SL (p∗H ) −(p∗ − c)[λH DH (p∗H ) + λO DL (p∗H )]λH + λO H
[
]
∫ c
λH
λO
∗
∗
∗
=−
DL (p)dp − (pH − c)
DH (pH ) +DL (pH )λH + λO
λH + λO
p∗
H
∫ c
∫ c
>−
DL (p)dp − (p∗H − c)DL (p∗H ) =[DL (p∗H ) − DL (p)]dp > 0,
p∗
H

p∗
H

and hence (24) is satisfied with strict inequality. As to the incentive constraint for type H and type O,(23), we haveSH (p∗H ) − A∗H − [SH (p∗L ) − A∗L ] = SH (p∗H ) − SH (c) − (c − p∗H )Defining

[

]
λH
λO
DH (p∗H ) +DL (p∗H ) .
λH + λO
λH + λO[]
λH
λO
G(pH ) = SH (pH ) − SH (c) − (c − pH )DH (pH ) +DL (pH )λH + λO
λH + λO

as a function of pH , this incentive constraint amounts to G(p∗H ) ≥ 0.
From (25) we obtain
[
]
λO
λH
λO
′
′
[DL (pH ) − DH (pH )] − (c − pH )DH(pH ) +DL(pH )λH + λO
λH + λO
λH + λO
1
2t
∂πi
=
.
i − w j ∂pi
λH + λO t + wH
H
H

G′ (pH ) =Because the optimal marginal price satisfies ∂πi /∂piH = 0, we have G′ (p∗H ) = 0. Moreover, we haveG(c) = 0 and G′ (c) < 0. Because G′′ (pH ) < 0 follows from ∂ 2 πi /∂(piH )2 < 0, we must have G(p∗H ) > 0,
and therefore (24) is satisfied with strict inequality. ■30

TA∗H + cq
Type HA∗L + p∗L q
Type P

Type L

∗
qL

qP∗

∗
qH

q

Figure 1: Monopoly with Overpessimistic Consumers; Complete InformationTbL + pbL q
A

bH + cqA

Type H and Type P

Type L

qbL

qbP

qbH

q

Figure 2: Monopoly with Overpessimistic Consumers; Incomplete Information

31

TA∗H + p∗H q
Type O
Type HA∗L + cq

Type L

∗
∗
qL
qO

∗
qH

q

Figure 3: Monopoly with Overoptimistic Consumers; Complete InformationTbL + pbL q
A

bH + pbH q
A

Type O
Type H

Type L

qbL

qbO

qbH

q

Figure 4: Monopoly with Overoptimistic Consumers; Incomplete Information

32

TA∗L + p∗L q

P

t + cqH

L∗
qL
DL (c) qP∗∗
qH= DH (c)qFigure 5: Competition with Overpessimistic ConsumersTt + cqA∗H + p∗H q
O

H

L

∗
∗
qL
= DL (c) qO

DH (c)∗qH

q

Figure 6: Competition with Overoptimistic Consumers

33

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