web appendices to selling to overcon–dent consumers
TRANSCRIPT
Web Appendices toSelling to Overcon�dent Consumers
Michael D. Grubb
MIT Sloan School of Management
Cambridge, MA 02142
www.mit.edu/~mgrubb
May 2, 2008
B Option Pricing Intuition
This appendix provides additional intuition based on option pricing for the result in Proposition 2.
Consider the case of monopoly. At time one, the monopolist is selling a series of call options,
or equivalently units bundled with put options, rather than units themselves. The marginal price
charged for a unit q at time two is simply the strike price of the option sold on unit q at time one.
The series of call options being sold are interrelated; a call option for unit q can�t be exercised
unless the call option for unit q � 1 has already been exercised. However, it is useful to considerthe market for each option independently. According to Proposition 2 (equation 8), when there is
no pooling and the satiation constraint is not binding, the optimal marginal price for a unit q is:
P̂q (q) = Cq (q) + Vq�(q; �̂ (q))F �(�̂ (q))� F (�̂ (q))
f(�̂ (q))(21)
It turns out that this is exactly the strike price that maximizes the net value of a call or put option
on unit q given the di¤erence in priors between the two parties.37
To show this explicitly, write the net value NV of a call option on minute q as the di¤erence
between the consumers�value of the option CV and the �rm�s cost of providing that option, FV .
The option will be exercised whenever the consumer values unit q more than the strike price p,
that is whenever Vq (q; �) � p. Let � (p) denote the minimum type who exercises the call option,
characterized by the equality Vq (q; � (p)) = p.
37This parallels Mussa and Rosen�s (1978) �nding in their static screening model, that the optimal marginal pricefor unit q is identical to the optimal monopoly price for unit q if the market for unit q were treated independently ofall other units.
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The consumers� value for the option is their expected value received upon exercise, less the
expected strike price paid, where expectations are based on the consumers�prior F � (�):
CV (p) =
Z ��
�(p)Vq (q; �) f
� (�) d� � [1� F � (� (p))] p
The �rm�s cost of providing the option is the probability of exercise based on the �rm�s prior F (�)
times the di¤erence between the cost of unit q and the strike price received:
FV (p) = [1� F (� (p))] (c� p)
Putting these two pieces together, the net value of the call option is equal to the consumers�
expected value of consumption less the �rms expected cost of production plus an additional term
due to the gap in perceptions:
NV (p) =
Z ��
�(p)Vq (q; �) f
� (�) d� � [1� F (� (p))] c+ [F � (� (p))� F (� (p))] p
The additional term [F � (� (p))� F (� (p))] p represents the di¤erence between the exercise paymentthe �rm expects to receive and the consumer expects to pay. The term [F � (� (p))� F (� (p))]represents the disagreement between the parties about the probability of exercise.
As a monopolist selling call options on unit q earns the net value NV (p) of the call option
by charging the consumer CV (p) upfront, a monopolist should set the strike price p to maximize
NV (p). By the implicit function theorem, ddp� (p) =
1Vq�(q;�(p))
, so the �rst order condition which
characterizes the optimal strike price is:
f (� (p))
Vq� (q; � (p))[p� Cq (q)] = [F � (� (p))� F (� (p))] (22)
As claimed earlier, this is identical to the characterization of the optimal marginal price P̂q (q)
for the complete nonlinear pricing problem when monotonicity and satiation constraints are not
binding (equation 21).
Showing that the optimal marginal price for unit q is given by the optimal strike price for a
call option on unit q is useful, because the �rst order condition q (q; �) = 0 can be interpreted
in the option pricing framework. Consider the choice of exercise price p for an option on unit q.
A small change in the exercise price has two e¤ects. First, if a consumer is on the margin, it will
change the consumers�exercise decision. Second, it changes the payment made upon exercise by all
infra-marginal consumers. In a common-prior model, the infra-marginal e¤ect would net to zero as
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the payment is a transfer between the two parties. This is not the case here, however, as the two
parties disagree on the likelihood of exercise by [F � (� (p))� F (� (p))].Consider the �rst order condition as given above in equation (22). On the left hand side, the
term f(�(p))Vq�(q;�(p))
represents the probability that the consumer is on the margin and that a marginal
increase in the strike price p would stop the consumer exercising. The term [p� Cq (q)] is the costto the �rm if the consumer is on the margin and no longer exercises. There is no change in the
consumer�s value of the option by a change in exercise behavior at the margin, as the margin is
precisely where the consumer is indi¤erent to exercise (Vq (q; � (p)) = p).
On the right hand side, the term [F � (� (p))� F (� (p))] is the �rm�s gain on infra-marginalconsumers from charging a slightly higher exercise price. This is because consumers believe they
will pay [1� F � (� (p))] more in exercise fees, and therefore are willing to pay [1� F � (� (p))] lessupfront for the option. However the �rm believes they will actually pay [1� F (� (p))] more inexercise fees, and the di¤erence [F � (� (p))� F (� (p))] is the �rm�s perceived gain.
The �rst order condition requires that at the optimal strike price p, the cost of losing mar-
ginal consumers f(�(p))Vq�(q;�(p))
[p� Cq (q)] is exactly o¤set by the "perception arbitrage" gain on infra-marginal consumers [F � (� (p))� F (� (p))].
Setting the strike price above or below marginal cost is always costly because it reduces e¢ ciency.
In the discussion above, referring to [F � (� (p))� F (� (p))] as a "gain" to the �rm for a marginal
increase in strike price implies that the term [F � (� (p))� F (� (p))] is positive. This is the casefor � (p) > ��, when consumers underestimate their probability of exercise. In this case, from the
�rm�s perspective, raising the strike price above marginal cost increases pro�ts on infra-marginal
consumers, thereby e¤ectively exploiting the perception gap. On the other hand, for � (p) < ��, the
term [F � (� (p))� F (� (p))] is negative and consumers overestimate their probability of exercise.In this case reducing the strike price below marginal cost exploits the perception gap between
consumers and the �rm.
Fixing � and the �rm�s prior F (�), the absolute value of the perception gap is largest when the
consumer�s prior is at either of two extremes, F � (�) = 1 or F � (�) = 0. When F � (�) = 1, the
optimal marginal price reduces to the monopoly price for unit q where the market for minute q is
independent of all other units. This is because consumers believe there is zero probability that they
will want to exercise a call option for unit q. The �rm cannot charge anything for an option at time
one; essentially the �rm must wait to charge the monopoly price until time two when consumers
realize their true value.
Similarly, when F � (�) = 0, the optimal marginal price reduces to the monopsony price for unit
q. Now rather than thinking of a call option, think of the monopolist as selling a bundled unit and
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put option at time one. In this case consumers believe they will consume the unit for sure and
exercise the put option with zero probability. This means that the �rm cannot charge anything for
the put option upfront, and must wait until time two when consumers learn their true values and
buy units back from them at the monopsony price. The �rm�s ability to do so is of course limited
by free disposal which means the �rm could not buy back units for a negative price.
Marginal price can therefore be compared to three benchmarks. For all quantities q, the marginal
price will lie somewhere between the monopoly price pml (q) and the maximum of the monopsony
price pms (q) and zero, hitting either extreme when F � (�) = 1 or F � (�) = 0, respectively. When
F � (�) = F (�), marginal price is equal to marginal cost. To illustrate this point, the equilibrium
marginal price for the running example with positive marginal cost c = $0:035 and low overcon�-
dence� = 0:25 previously shown in Figure 4, plot C is replotted with the monopoly and monopsony
prices for comparison in Figure 10.
300 400 500 600 7000.1
0
0.1
0.2
0.3
quantity
$
Optimal MPMonopolyMonopsonyMC
Figure 10: Equilibrium pricing for Example 1 given c = $0:035 and � = 0:25: Marginal price isplotted along with benchmarks: (1) marginal cost, (2) ex post monopoly price - the upper bound,and (3) ex post monopsony price - the lower bound.
C Pooling
As it was omitted from the main text, a characterization of pooling quantities when the monotonic-
ity constraint binds is provided below in Lemma 4. This is useful because it facilitates the calcu-
lation of pooling quantities in numerical examples.
Lemma 4 On any interval [�1; �2] such that monotonicity (but not non-negativity) is binding in-
side, but not just outside the interval, the equilibrium allocation is constant at some level q̂ > 0 for
all � 2 [�1; �2]. Further, the pooling quantity q̂ and bounds of the pooling interval [�1; �2] must main-
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tain continuity of q̂ (�) and satisfy the �rst order condition on average:R �2�1q (q̂; �) f (�) d� = 0.
Non-negativity binds inside, but not just above, the interval [�; �2] only ifR �2� q (0; �) f (�) d� � 0
and qR (�2) = 0.
Proof. Given the result in Lemma 2, the proof is omitted as it closely follows ironing results
for the standard screening model. The result follows from the application of standard results in
optimal control theory (Seierstad and Sydsæter 1987). See for example the analogous proof given
in Fudenberg and Tirole (1991), appendix to chapter 7.
Note that because the virtual surplus function is strictly quasi-concave, but not necessarily
strictly concave, optimal control results yield necessary rather than su¢ cient conditions for the
optimal allocation. For the special case Vqq� = 0, virtual surplus is strictly-concave and applying
the ironing algorithm suggested by Fudenberg and Tirole (1991) using the conditions in Lemmas 2
and 4 is su¢ cient to identify the uniquely optimal allocation.
Proposition 2 characterizes marginal pricing at quantities for which there is no pooling, and
states that marginal price will jump discretely upwards at quantities where there is pooling. It
is therefore interesting to know when qR (�) will be locally decreasing, so that the equilibrium
allocation q̂ (�) involves pooling. First, a preliminary result is helpful. Proposition 5 compares the
relaxed allocation to the �rst best allocation, showing that the relaxed allocation is above �rst best
whenever F (�) > F � (�) and is below �rst best whenever F (�) < F � (�):
Proposition 5 Given maintained assumptions:
qR (�)
8>><>>:�� qFB (�)= qFB (�)
< qFB (�)
F (�) > F � (�)
F (�) = F � (�)
F (�) < F � (�)
(�) strict i¤ Cq�qFB (�)
�> 0
Proof. See Section C.1 at the end of this appendix.
Given that F � (�) crosses F (�) once from below, the relationship between the relaxed allocation
and the (strictly increasing) �rst best allocation given in Proposition 5 leads to the conclusion that
qR (�) is strictly increasing near the bottom � and near the top ��. More can be said about pooling
when consumers either have nearly correct beliefs, or are extremely overcon�dent. When consumers�
prior is close to that of the �rm, the relaxed solution is close to �rst best, and like �rst best is
strictly increasing. In this case the equilibrium and relaxed allocations are identical.
When consumers are extremely overcon�dent such that their prior is close to the belief that
� = �� with probability one, the relaxed solution is strictly decreasing at or just above ��. In this
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case ironing will be required and an interval of types around �� pool at the same quantity q̂ (��).
The intuition is that when the consumers�prior is exactly the belief that � = �� with probability
one, [F (�)� F � (�)] falls discontinuously below zero at ��. Thus, by Proposition 5, the relaxed
solution must drop discontinuously from weakly above �rst best just below �� to strictly below �rst
best at ��. These results and the notion of "closeness" are made precise in Proposition 6.
Proposition 6 (1) There exists " > 0 such that if jf� (�)� f (�)j < " for all � then qR (�) is
strictly increasing for all �. (2) There exists a �nite constant , such that if f� (��) > and f� (�)
is continuous at �� then qR (�) is strictly decreasing just above ��.
Proof. If neither non-negativity nor satiation constraints bind at �, Lemma 2 and the implicit
function theorem imply that dd�q
R (�) = �q�(qR(�);�)qq(qR(�);�)
. In this case the sign of the cross partial
derivative q� determines whether qR (�) is increasing or decreasing. For any � at which F � is
su¢ ciently close to F in both level and slope, q� > 0, but when f� (�) is su¢ ciently large q� < 0.
See the proof at the end of this appendix for details.
C.1 Pooling Appendix Proofs
C.1.1 Small Lemma 5
Lemma 5 Satiation qS (�) and �rst best qFB (�) quantities are continuously di¤erentiable, strictly
positive, and strictly increasing. Satiation quantity is higher than �rst best quantity, and strictly so
when marginal costs are strictly positive at qFB.
Proof. Satiation and �rst best quantities are strictly positive because by assumption: Vq (0; �) >
Cq (0) � 0. Therefore given maintained assumptions, qS (�) and qFB (�) exist, and are continu-
ous functions characterized by the �rst order conditions Vq�qS (�) ; �
�= 0 and Vq
�qFB (�) ; �
�=
Cq�qFB (�)
�respectively. Zero marginal cost at qFB (�) implies qS (�) = qFB (�). When Cq
�qFB (�)
�>
0, Vqq (q; �) < 0 implies that qS (�) > qFB (�). The implicit function theorem implies dd�q
S = �Vq�Vqq
>
0 and dd�q
FB = � Vq�Vqq�Cqq > 0.
C.1.2 Proof of Proposition 5
Proof. The relaxed allocation maximizes virtual surplus(q; �) within the constraint set�0; qS (�)
�and (q; �) is strictly quasi-concave in q (See proof of Lemma 2). Moreover, qFB 2 (0; qS (�)] andq�qFB (�) ; �
�= Vq�
�qFB (�) ; �
� F (�)�F �(�)f(�) since the �rst best allocation satis�es Vq
�qFB (�) ; �
�=
Cq�qFB (�)
�(Lemma 5). Therefore there are three cases to consider:
1. F (�) = F � (�): In this case virtual surplus and true surplus are equal so qR (�) = qFB (�).
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2. F (�) > F � (�): In this case q�qFB (�) ; �
�> 0 and therefore qR (�) � qFB (�). When
Cq�qFB (�)
�= 0, Lemma 5 shows that qS (�) = qFB (�) and therefore the satiation constraint
binds: qR (�) = qFB (�) = qS (�). When Cq�qFB (�)
�> 0, Lemma 5 shows that satiation is
not binding at �rst best, and therefore the comparison is strict: qR (�) > qFB (�).
3. F (�) < F � (�): In this case q�qFB (�) ; �
�< 0 and therefore qR (�) < qFB (�) since, by
Lemma 5, non-negativity is not binding at �rst best (qFB (�) > 0).
C.1.3 Proof of Proposition 6
Proof. If, over the interval (�1; �2), neither non-negativity nor satiation constraints bind and
q (q; �) is continuously di¤erentiable, then in the same interval q�qR; �
�= 0 (Lemma 2) and
the implicit function theorem implies dd�q
R (�) = �q�(qR(�);�)qq(qR(�);�)
. (Given maintained assumptions,
q (q; �) is continuously di¤erentiable at � where f� (�) is continuous.) Therefore, in the same inter-
val qR (�) will be strictly increasing if q��qR (�) ; �
�> 0 and strictly decreasing if q�
�qR (�) ; �
�<
0.
Part (1): As the non-negativity constraint is not binding at � (qR (�) = qFB (�) > 0), the
upper bound qS (�) is strictly increasing, and qR (�) is continuous, qR (�) will be strictly increasing
for all � if the cross partial derivative q� (q; �) is strictly positive for all (q; �) 2�0; qS
��������; ���.
De�ne ' (q; �) and ": (Note that " is well de�ned since�0; qS
��������; ���is compact and F 2 C2
and V (q; �) 2 C3 imply ' (q; �) is continuous.)
' (q; �) � 1
f (�)+(�� � �)f (�)
jVq�� (q; �)jVq� (q; �)
+
�� dd�f (�)
��f (�)
!> 0
" � min(q;�)2[0;qS(��)]�[�;��]
1
' (q; �)
By di¤erentiation:
q� (q; �) = Vq�� (q; �)F (�)� F � (�)
f (�)+ Vq� (q; �)
1 +
f (�)� f� (�)f (�)
�dd�f (�)
f2 (�)[F (�)� F � (�)]
!
As f (�) > 0 and Vq� (q; �) > 0:
q� (q; �) � � jVq�� (q; �)jjF (�)� F � (�)j
f (�)+Vq� (q; �)
1� jf (�)� f
� (�)jf (�)
��� dd�f (�)
��f2 (�)
jF (�)� F � (�)j!
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The assumption jf (�)� f� (�)j < " implies that jF (�)� F � (�)j < "(�� � �) and therefore that:
q� (q; �) > Vq� (q; �) (1� "' (q; �))
By the de�nition of ", this implies q� (q; �) > 0 for all (q; �) 2�0; qS
��������; ���.
Part (2): The �rst step is to show that q� (q; ��) < 0. De�ne :
� maxq2[0;qS(��)]
jVq�� (q; ��)jVq� (q; �
�)+ 2f (��) +
�� dd�f (�
�)��
f (��)
!
As f (�) > 0 and Vq� (q; �) > 0:
q� (q; �) � jVq�� (q; �)jjF (�)� F � (�)j
f (�)+ Vq� (q; �)
2� f
� (�)
f (�)+
�� dd�f (�)
��f2 (�)
jF (�)� F � (�)j!
since jF (�)� F � (�)j � 1:
q� (q; �) �Vq� (q; �)
f (�)
jVq�� (q; �)jVq� (q; �)
+ 2f (�) +
�� dd�f (�)
��f (�)
� f� (�)!
By de�nition of and f� (��) > , it follows that for all q 2�0; qS
�����: q� (q; ��) < 0. Given
continuity of f� (�) at ��, q� (q; �) is also continuous at ��. Therefore for some �1 > 0, q� (q; �) < 0
just above �� in the interval � 2 [��; �� + �1).The second step is to show that for some �2 > 0 neither satiation nor non-negativity constraints
are binding in the interval (��; �� + �2). First, satiation is not binding just above �� as qR is below
�rst best here (Proposition 5), which is always below the satiation bound (Lemma 5). Second,
non-negativity is not binding just to the right of �� because qR (��) = qFB (��) > 0 (Proposition 5
and Lemma 5) and qR (�) is continuous (Lemma 2).
Steps one and two imply that qR (�) is strictly decreasing in the interval (��; �� +min f�1; �2g)just above ��. Therefore, qR (�) is either strictly decreasing at �� or has a kink at �� and is
strictly decreasing just above ��. In either case, monotonicity is violated at �� and the equilibrium
allocation will involve pooling at ��.
D Monopoly Multi-Tari¤Menu Extension
In this appendix, I extend the single tari¤ model explored in detail in the main paper to a multi-
tari¤ monopoly model. The model is described in Section 5 of the paper. (Note that I replace
equation (1) by the stricter assumption Vqq� = 0.) The model and solution methods are closely
8
related to Courty and Li (2000). To work with the new problem, �rst de�ne:
De�nition 2 (1) Let qc(s; �; �0) � minfq(s; �0); qS (�)g be the consumption quantity of a consumerwho chooses tari¤ s, is of type �, and reports type �0. Consumption quantity of a consumer who
honestly reports true type � is qc (s; �) � qc (s; �; �). (2) u(s; �; �0) � V (qc(s; �; �0); �) � P (s; �0)is the utility of a consumer who chooses tari¤ s, is of type �, and reports type �0. (3) u (s; �)
� u (s; �; �) is the utility of a consumer who chooses a tari¤ s, and honestly reports true type �.
(4) U (s; s0) �R ��� u (s
0; �) f (�js) d� is the true expected utility of a consumer who receives signal s,chooses tari¤ s0, and later reports � honestly. U� (s; s0) �
R ��� u (s
0; �) f� (�js) d� is the analogousperceived expected utility. (5) U (s) � U (s; s) is the expected utility of a consumer who honestly
chooses the intended tari¤ s given signal s, and later reports � honestly. U� (s) � U� (s; s) is theanalogous perceived expected utility.
Invoking the revelation principle, the monopolist�s problem may then be written as:
maxq(s;�)�0P (s;�)
E
"Z ��
�(P (s; �)� C (q (s; �))) f (�js) d�
#such that
1. Global IC-2 u (s; �; �) � u(s; �; �0) 8s 2 S; 8�; �0 2 �2. Global IC-1 U� (s; s) � U� (s; s0) 8s; s0 2 S3. Participation U� (s) � 0 8s 2 S
As the signal s does not enter the consumers value function directly, the second period incentive
compatibility constraints may be handled just as they are in the single tari¤ model (See part 1 of
the proof of Proposition 1). In particular, second period local incentive compatibility ( @@�u (s; �) =
V� (qc (s; �) ; �)) and monotonicity ( @@�q
c (s; �) � 0) are necessary and su¢ cient for second period
global incentive compatibility. Moreover, Lemma 1 naturally extends to the multi-tari¤ setting.
Applying the same satiation re�nement q (s; �) � qS (�), the distinction between qc (s; �) and q (s; �)may be dropped.
The next step is to express payments in terms of consumer utility, and substitute in the second
period local incentive constraint, just as was done in the single-tari¤ model (See parts 2 and 3 of
the proof of Proposition 1). This yields analogs of equations (3, 15, 6):
U��s; s0
�= u
�s0; �
�+ E
�V��q�s0; �
�; �� 1� F � (�js)
f (�js)
���� s� (23)
U� (s)� U (s) = E�V� (q (s; �) ; �)
F (�js)� F � (�js)f (�js)
���� s� (24)
9
P (s; �) = V (q (s; �) ; �)�Z �
�V� (q (s; z) ; z) dz � u (s; �) (25)
In the single tari¤ model, the utility of the lowest type � was determined by the binding partic-
ipation constraint. To pin down the utilities u (s; �) in the multi-tari¤model requires consideration
of both participation and �rst-period incentive constraints given an assumed ordering of the signal
space S. I will consider signal spaces S which are ordered either by FOSD, or a more general reverse
second order stochastic dominance (RSOSD).38 Note that I assume the ordering, either FOSD or
RSOSD, applies to consumer beliefs F � (�js).In the standard single period screening model the participation constraint will bind for the
lowest type, and this guarantees that it is satis�ed for all higher types. The same is true here given
the assumed ordering of S, as stated in Lemma 6.
Lemma 6 If S is ordered by FOSD then the participation constraint binds at the bottom (U� (s) =
0). This coupled with �rst period incentive constraints are su¢ cient for participation to hold for all
higher types s >s. The same is true if S is ordered by RSOSD and V�� � 0.
Proof. It is su¢ cient to show that U� (s; s0) is non-decreasing in s. If this is true, then IC-1
and U� (s) � 0 imply participation is satis�ed: U� (s; s) � U� (s; s) � 0. Hence if U� (s) � 0
were not binding, pro�ts could be increased by raising �xed fees of all tari¤s. Now by de�nition,
U� (s; s0) � E [u (s0; �) js]. By local IC-2 and V� � 0 (which follows from V (0; �) = 0 and Vq� > 0),
it is clear that conditional on tari¤ choice s0, consumers�utility is non-decreasing in realized �:
u��s0; �
�= V�
�q�s0; �
�; ��� 0
Given a FOSD ordering, this is su¢ cient for U� (s; s0) to be non-decreasing in s (Hadar and Russell
1969, Hanoch and Levy 1969).
Taking a second derivative of consumer second period utility shows that conditional on tari¤
choice s0, consumers� utility is convex in � if V�� � 0. This follows from increasing di¤erences
Vq� > 0, monotonicity q� (s0; �) � 0, and local IC-2:
u���s0; �
�= Vq�
�q�s0; �
�; ��| {z }
(+)
q��s0; �
�| {z }(+) by IC-2
+ V���q�s0; �
�; ��| {z }
�0 (assumption)
� 0
38Courty and Li (2000) restrict attention primarily to orderings by �rst order stochastic dominance or meanpreserving spread. However, in their two type model, they do mention that orderings can be constructed from thecombination of the two which essentially allows for the more general reverse second order stochastic dominanceorderings I consider here.
10
Under RSOSD ordering, this implies that U� (s) is increasing in s. (This result is analogous to the
standard result that ifX second order stochastically dominates Y then E [h (X)] � E [h (Y )] for anyconcave utility h (Rothschild and Stiglitz 1970, Hadar and Russell 1969, Hanoch and Levy 1969).
The proof is similar and hence omitted.)
Given Lemma 6 and the preceding discussion, the monopolist�s problem can be simpli�ed as
described in Lemma 7.
Lemma 7 Given a FOSD ordering of S, or a RSOSD ordering of S and V�� � 0, the monopolist�sproblem reduces to the following constrained maximization over allocations q (s; �) and utilities
u (s; �) for U� (s) = U� (s; s) and U� (s; s0) given by equation (23):
maxq(s;�)2[0;qS(�)]
u(s;�)
E [ (s; q (s; �) ; �)]� E [U� (s)]
such that
1. Monotonicity q (s; �) non-decreasing in �
2. Global IC-1 U� (s; s) � U� (s; s0) 8s; s0 2 S3. Participation U� (s) = 0
(s; q; �) � V (q; �)� C (q) + V� (q; �)F (�js)� F � (�js)
f (�js)
Payments P (s; �) are given as a function of the allocation q (s; �) and utilities u (s; �) by equation
(25). Second period local incentive compatibility ( dd�u (q; �) = V� (q (s; �) ; �)) always holds for the
described payments.
Proof. Firm pro�ts can be re-expressed as shown in (equation 26).
E [� (s; �)] = E [S (s; �)] + E [U� (s)� U (s)]� E [U� (s)] (26)
Substituting equation (24) for �ctional surplus gives the objective function. The participation
constraint follows from Lemma 6. For handling of the satiation constraint and second period
incentive compatibility, refer to discussion in the text and proofs of Lemma 1 and Proposition 1.
To characterize the solution to the monopolist�s problem described by Lemma 7, I now proceed
to analyze a two type (s 2 fL;Hg) model and a continuum of types (s 2 [s; �s]) model separately.
11
D.1 Two-Tari¤Menu
Assume that there are two possible �rst-period signals s 2 fL;Hg, and that the probability ofsignal H is �. There are two �rst-period incentive constraints. Given Lemma 6, the downward
incentive constraint U� (H;H) � U� (H;L) (IC-H) must be binding. Otherwise the monopolist
could raise the �xed fee P (H; �), and increase pro�ts. Together with equation (23) and the binding
participation constraint U� (L) = 0 (IR-L), this pins down U� (H) as a function of the allocation
q (L; �). Substituting the binding IR-L and IC-H constraints for u (L; �) and u (H; �) into both
the monopolist�s objective function described in Lemma 7 and the remaining upward incentive
constraint U� (L;L) � U� (L;H) (IC-L) delivers a �nal simpli�cation of the monopolist�s problemin Proposition 7.
Proposition 7 De�ne a new virtual surplus � (qL; qH ; �) and function � (qL; qH ; �) by equations
(27) and (28) respectively.
� (qL; qH ; �) ��(H; qH ; �) fH (�) + (1� �) (L; qL; �) fL (�)
��V� (qL; �) (F �L (�)� F �H (�))(27)
� (qL; qH ; �) � [F �L (�)� F �H (�)] [V� (qL; �)� V� (qH ; �)] (28)
Given either a FOSD ordering of S, or a RSOSD ordering and V�� � 0:1. A monopolist�s optimal two-tari¤ menu solves the reduced problem:
maxqL(�);qH (�)2[0;qS(�)]
Z ��
�� (qL (�) ; qH (�) ; �) d�
such that
IC-2 qL (�) and qH (�) non-decreasing in �
IC-L �R ��� � (qL (�) ; qH (�) ; �) d� � 0
2. Payments are given by equations (25), and (29-30):
u (L; �) = �E�V� (qL (�) ; �)
1� F �L (�)fL (�)
����L� (29)
u (H; �) = E
�[V� (qL (�) ; �)� V� (qH (�) ; �)]
1� F �H (�)fH (�)
����H�+ u (L; �) (30)
Proof. Beginning with the result in Lemma 7, the simpli�cation is as follows: Equation (23) and
U� (L) = 0 imply equation (29). Equation (23) and U� (H;H) = U� (H;L) imply equation (30). By
equation (23) and equations (29-30) U� (H) isR ��� V� (qL (�) ; �) [F
�L (�)� F �H (�)] d�. As U� (L) = 0,
this means E [U� (s)] = �R ��� V� (qL (�) ; �) [F
�L (�)� F �H (�)] d�, which leads to the new term in the
12
revised virtual surplus function �. Further, as U� (L) = 0, the upward incentive constraint (IC-L)
is �U� (L;H) � 0. By equation (23) and equations (29-30) U� (L;H) isR ��� � (qL (�) ; qH (�) ; �) d�
which gives the new expression for the upward incentive constraint.
Given the problem described by Proposition 7, a standard approach would be to solve a relaxed
problem which ignores the upward incentive constraint, and then check that the constraint is
satis�ed. Given a FOSD ordering, a su¢ cient condition for the resulting relaxed allocation to solve
the full problem is for q (s; �) to be non-decreasing in s. (Su¢ ciency follows from IC-H binding and
Vq� > 0.) This is not the most productive approach in this case, because the su¢ cient condition is
likely to fail with high levels of overcon�dence. An alternative approach is to directly incorporate
the upward incentive constraint into the maximization problem using optimal control techniques.
Proposition 8 Given either a FOSD ordering or a RSOSD ordering and V�� � 0: The equilibriumallocations q̂L (�) and q̂H (�) are continuous and piecewise smooth. For �xed � 0, de�ne relaxedallocations (which ignore monotonicity constraints):
qRL (�) = arg maxqL 2[0;qS(�)]
�(L; qL; �)�
+ �
1� �V� (qL; �)F �L (�)� F �H (�)
fL (�)
�(31)
qRH (�) = arg maxqH2[0;qS(�)]
�(H; qH ; �) +
�V� (qH ; �)
F �L (�)� F �H (�)fH (�)
�(32)
The relaxed allocations qRL (�) and qRH (�) are continuous and piecewise smooth functions character-
ized by their respective �rst order conditions except where satiation or non-negativity constraints
bind. Moreover, there exists a non-negative constant � 0 such that:
1. On any interval over which a monotonicity constraint is not binding, the corresponding equi-
librium allocation is equal to the relaxed allocation: q̂s (�) = qRs (�).
2. On any interval [�1; �2] such that the monotonicity constraint is binding inside, but not just
outside the interval for tari¤ s, the equilibrium allocation is constant at some level q̂s (�) = q0
for all � 2 [�1; �2]. Further, the pooling quantity q0 and bounds of the pooling interval [�1; �2]must satisfy qRs (�1) = q
Rs (�2) = q
0 and meet the �rst order condition from the relaxed problem
in expectation over the interval. For instance, for qL (�) this second condition is:Z �2
�1
�q (L; q̂; �)�
+ �
1� �Vq� (q̂; �)F �L (�)� F �H (�)
fL (�)
�fL (�) d� = 0
3. Complementary slackness: = 0 or IC-L binds with equality (R ��� � (q̂L (�) ; q̂H (�) ; �) d� = 0).
13
Proof. I apply Seierstad and Sydsæter (1987) Chapter 6 Theorem 13, which gives su¢ cient
conditions for a solution. To apply the theorem, I �rst restate the problem described by Proposition
7 in the optimal control framework. This includes translating the upward incentive constraint into a
state variable k (�) =R �� �� (qL (z) ; qH (z) ; z) dz with endpoint constraints k (�) = 0 (automatically
satis�ed) and k(��) � 0. Remaining state variables are qL (�) and qH (�), which have free endpoints.Control variables are cL (�), cH (�), and ck (�), and the control set is R3. Costate variables are
�L (�), �H (�), and �k (�).
maxqL;qH
Z ��
�� (qL (�) ; qH (�) ; �) d�
State Control Costate Control - State Relation
qL (�) cL (�) �L (�) _qL (�) = cL (�)
qH (�) cH (�) �H (�) _qH (�) = cH (�)
k (�) ck (�) �k (�) _k (�) = �� (qL; qH ; �)
End Point Constraints Lagrangian Multipliers
(by de�nition) k (�) = 0
IC-L k(��) � 0 �
Control Constraints Lagrangian Multipliers
Monotonicity cL (�) � 0 �L (�)
cH (�) � 0 �H (�)
State Constraints Lagrangian Multipliers
Non-negativity qL (�) � 0 �L (�), �L, ��L
qH (�) � 0 �H (�), �H , ��H
Satiation�qS (�)� qL (�)
�� 0 �L (�), �L, ��L�
qS (�)� qH (�)�� 0 �H (�), �H , ��H
The Hamiltonian and Lagrangian for this problem are:
H = �(qL; qH ; �) + �L (�) cL (�) + �H (�) cH (�)� �k (�) � (qL; qH ; �)
L = H + �L (�) cL (�) + �H (�) cH (�) + �L (�) qL (�) + �H (�) qH (�)
+�L (�)�qS (�)� qL (�)
�+ �H (�)
�qS (�)� qH (�)
�The following 7 conditions are su¢ cient conditions for a solution:
1. Control and state constraints are quasi-concave in states qL, qH , and k, as well as controls cL,
14
cH , and ck for each �. Endpoint constraints are concave in state variables. This is satis�ed
because all constraints are linear.
2. The Hamiltonian is concave in states qL, qH , and k, as well as controls cL, cH , and ck for each
�. This is satis�ed because H is constant in k and ck, linear in cL and cH , strictly concave
in qL and qH (equations 33-34), and all other cross partials are zero. This relies on Vqq� = 0
and Sqq (q; �) < 0.
@2H
@q2L= (1� �)Sqq (qL; �) < 0 (33)
@2H
@q2H= �Sqq (qH ; �) < 0 (34)
3. State and costates are continuous and piecewise di¤erentiable in �. Lagrangian multiplier
functions as well as controls are piecewise continuous in �. All constraints are satis�ed.
4.@L̂
@cL=@L̂
@cH=@L̂
@ck= 0
5.
dd��L = �
@L̂@qL; d
d��H = �@L̂@qH
; dd��k = �
@L̂@k
6. Complementary Slackness conditions
(a) For all �:
�L; �H ; �L; �H ; �L; �H � 0
�LcL = �HcH = �LqL = �HqH = �L�qS � qL
�= �H
�qS � qH
�= 0
(b) For ��
��L; ��H ; ��L; ��H � 0
��LqL(��) = ��HqH(
��) = ��L�qS(��)� qL(��)
�= ��H
�qS(��)� qH(��)
�= 0
(c) For �
�L; �H; �L; �H � 0
�LqL (�) = �HqH (�) = �L
�qS (�)� qL (�)
�= �H
�qS (�)� qH (�)
�= 0
(d) For IC-L
� � 0
15
k (�) = � k(��) = 0
7. Transversality Conditions
�L(��) = ��L � ��L; �H(��) = ��H � ��H ; �k(��) = �
�L (�) = ��L + �L; �H (�) = ��H + �H �k (�) = �
Fix � 0. Let �k (�) = � = � = . As �k (�) is constant, it is continuous and di¤erentiable,and d
d��k = 0. Neither the state k (�) nor the control ck (�) enter the Lagrangian, so@L̂@ck
= �@L̂@k = 0.
For any allocation, k (�) = 0 by de�nition, so k (�) = 0. Putting aside for now the constraint
k(��) � 0 and the complementary slackness condition � k(��) = 0, all other conditions concerning
state k (�) are satis�ed. Moreover, the Hamiltonian and Lagrangian are both additively separable in
qL and qH . Hence the remaining conditions are identical to those for two independent maximization
problems. Namely:
maxqL2[0;qS(�)]
dd�qL�0
�(L; qL; �)�
+ �
1� �V� (qL; �)F �L (�)� F �H (�)
fL (�)
�
maxqH2[0;qS(�)]
dd�qH�0
�(H; qH ; �) +
�V� (qH ; �)
F �L (�)� F �H (�)fH (�)
�
For �xed � 0, the solution to these problems exists, satis�es Proposition 8 parts 1-2, and
meets all the relevant conditions in 3-7 above. The proof for this statement is omitted, because the
solution for each subproblem given any �xed � 0, is similar to the single-tari¤ case, and closelyparallels standard screening results. The idea is that for regions (�1; �2) where monotonicity is
not binding for allocation q̂s (�), the q̂s (�) subproblem conditions are the Kuhn-Tucker conditions
for the relaxed solution qRs (�). For regions in which monotonicity is binding, the characterization
closely parallels Fudenberg and Tirole�s (1991) treatment of ironing in the standard screening model.
For the nice properties of the relaxed solution stated in the proposition, refer to the analogous proof
of Proposition 2 part 1.
All that remains to show is that there exists a � 0, such that k(��) � 0 and k(��) = 0 givenqL (�) and qH (�) that solve the respective subproblems for that . If k(��) � 0 for = 0 there
is no problem. If k(��) < 0 for = 0, the result follows from the intermediate value theorem and
two observations: (1) k(��) varies continuously with , and (2) for su¢ ciently large k(��) > 0.
The latter point follows because each subproblem is a maximization of E [� (qL; qH ; �)] + k(��).
As increases, the weight placed on k(��) in the objective increases, and k(��) moves towards its
16
maximum. Given FOSD, k(��) is maximized (given non-negativity, satiation, and monotonicity) at
fqL (�) ; qH (�)g =�0; qS (�)
, for which qH > qL and therefore, by binding IC-H and Vq� > 0, IC-L
must be strictly satis�ed. The argument is more involved, but the constrained maximum of k(��) is
also strictly positive given RSOSD.
D.2 A Continuum of Tari¤s
For a continuum of �rst-period signals, Courty and Li (2000) characterize a necessary local condition
for �rst period incentive compatibility. Its analog here is based on consumers perceived utility U� (s)
(Lemma 8).
Lemma 8 Tari¤ fq (s; �) ; P (s; �)g satis�es period 1 incentive constraints only if
dU� (s)
ds= �
Z ��
�V� (q (s; �) ; �)
@F � (�js)@s
d�
Proof. By the envelope theorem dU�(s)ds = @U�(s)
@s . Thus dU�(s)ds =
R ��� u (s; �)
@f�(�js)@s d�. Integrating
by parts and substituting the second period local incentive condition @u(s;�)@� = V� (q (s; �) ; �) then
yields the desired result.
Now, applying the fundamental theorem of calculus, substituting the results of Lemmas 6 and 8,
taking expectations, and integrating by parts yields the following expression for expected perceived
utility E [U� (s)]:
E [U� (s)] =
Z �s
s
Z ��
�V� (q (s; �) ; �) (1�G (s))
@
@s(1� F � (�js)) d�ds (35)
Finally, combining equation (35) with Lemma 7, the monopolist�s problem may now be re-
written as given in Proposition 9:
Proposition 9 The monopolist�s problem may be re-expressed as the constrained maximization of
expected virtual surplus � (q; s; �):
maxq(s;�)2[0;qS(�)]
Z �s
s
Z ��
�� (s; q (s; �) ; �) f (�js) g (s) d�ds
such that
1. monotonicity @@�q (s; �) � 0
2. Global IC-1 U� (s; s) � U� (s; s0) 8s; s0 2 S
� (s; q; �) � (s; q; �)� V� (q; �)1�G (s)g (s)
@@s (1� F
� (�js))f (�js)
17
Proof. Outlined in text.
One can see that virtual surplus now includes the information rent term 1�G(s)g(s)
@@s[1�F �(�js)]f(�js)
which arises in Courty and Li�s (2000) model, as well as the �ctional surplus term F �(�js)�F (�js)f(�js)
which arose in the single tari¤model with overcon�dence. Ignoring the three remaining constraints,
the relaxed solution is characterized by the �rst order condition �q = 0 because virtual surplus is
concave in q (�qq < 0) under the maintained assumption Vqq� = 0.
Given a FOSD ordering, as in Courty and Li�s (2000) model, allocation q (s; �) non-decreasing in
signal s is su¢ cient but not necessary for �rst period global incentive compatibility. Now, however,
general examples for which Courty and Li (2000) were able to show the relaxed solution was non-
decreasing in signal smay violate this su¢ cient condition given high enough levels of overcon�dence.
This is because for high levels of overcon�dence, the relaxed solution involves marginal prices near
the monopoly price for each particular minute as discussed in Web Appendix B. As monopoly price
increases with demand, this implies types with higher signals should face higher marginal prices at
a given quantity, and therefore consume less for a given �. This is unfortunate, because in these
cases it is not known what the optimal tari¤ will look like.39
For speci�c cases in which the allocation of the relaxed solution is non-decreasing in signal s,
marginal price is given by equation (36) if it is well de�ned, where � = � (s; q) is the inverse of
q̂ (s; �) if the latter is invertible.
Pq(s; q) =Max
8<:0; Cq (q) + Vq� (q; �)8<:
1�G(s)g(s)
@@s[1�F �(�js)]f(�js)
+F �(�js)�F (�js)f(�js)
9=;9=; (36)
Marginal pricing for the top tari¤ intended for consumers with signal �s is the same as in the
single tari¤ model. Marginal prices for all lower tari¤s on the menu are distorted upwards by the
information rent term. Whether these tari¤s continue to o¤er initial quantities at zero marginal
price depends on the relative size of the information rent and �ctional surplus. Initial quantities
are more likely to be o¤ered at zero marginal price for all tari¤s on the menu when overcon�dence
is high, and ex ante heterogeneity is low.
E Price Discrimination with Common Priors
The model description and analysis follow directly on from where appendix D.2 left o¤above, simply
by assuming that consumers�priors are correct: F � (�js) = F (�js), and leaving o¤ all superscript
39Global incentive compatibility would need to be checked directly, and if it failed an ironing procedure could notbe used to �nd the optimal tari¤ because monotonicity is not necessary.
18
�. In this case the virtual surplus function characterized in Proposition 9 reduces to:
� (s; q; �) � V (q; �)� C (q)� V� (q; �)1�G (s)g (s)
@@s (1� F (�js))
f (�js)
It is now straight forward to characterize a solution to the relaxed problem in which the ad-
ditional global incentive constraints are ignored. Virtual surplus is concave in quantity (Equation
38) so the relaxed solution qR (s; �) is characterized by the �rst order condition (Equation 37).
�q (s; q; �) = Vq (q; �)� Cq (q)� Vq� (q; �)(1�G (s))g (s)
@@s (1� F (�js))
f (�js) = 0 (37)
�qq (s; q; �) = Vqq (q; �)| {z }(�)
� Cqq (q)| {z }(+)
� Vqq� (q; �)| {z }=0
(1�G (s))g (s)
@@s (1� F (�js))
f (�js) � 0 (38)
Unfortunately, if the relaxed solution does not satisfy global incentive compatibility, no char-
acterization of the true solution is given. This is because there is no known necessary and su¢ -
cient condition for �rst period incentive compatibility analogous to the monotonicity requirement
q� (s; �) � 0 which can be imposed in an ironing procedure.Nevertheless, there are conditions su¢ cient to verify that the relaxed solution is globally incen-
tive compatible. In particular, given the �rst order local incentive condition dU(s)ds = �
R ��� V� (q (s; �) ; �)
@F (�js)@s d�,
a positive cross partial derivative @2U(s;s0)@s@s0 � 0 is a su¢ cient second order condition to guarantee
global period one incentive compatibility. This gives the following lemma.
Lemma 9 If the relaxed solution qR (s; �) satis�es (1) monotonicity in �: @@�q
R (s; �) � 0 8s; �and (2) @
@sqR (s; �) � @@s [1� F (�js)] � 0 8s; � then q
R (s; �) and the corresponding implicit payment
function solve the original problem.
Proof. Di¤erentiating equation (23), the cross partial is:
@2U (s; s0)
@s@s0=
Z ��
�Vq��q�s0; �
�; �� @@sq�s0; �
� @@s[1� F (�js)] d�
By assumption Vq� is positive, so @@sq (s
0; �) @@s [1� F (�js)] � 0 guarantees that@2U(s;�s)@s@�s � 0. Given
the �rst period local incentive constraint, this serves as a su¢ cient second order condition for
�rst period global incentive compatibility. Finally, @@�q
R (s; �) � 0 guarantees second period globalincentive compatibility.
The conditions given in Lemma 9 are given in terms of the relaxed solution. Using standard
monotone comparative statics results, su¢ cient conditions based on the primitives of the model can
19
be derived from the cross partials �q� and �qs. Following de�nition 3, these are given in Lemma
10.
De�nition 3
� (s; �) �@@s (1� F (�js))
f (�js)
(Baron and Besanko (1984) refer to this as an "informativeness measure")
Lemma 10 Condition (1) @@�q
R (s; �) � 0 8s; � holds if and only if � (s; �) does not increase in �too quickly:
(1�G (s))g (s)
� @@�� (s; �) � 1 8s; �
If G (s) satis�es the monotone hazard condition ( @@s
�1�G(s)g(s)
�� 0) then condition (2) @@sq
R (s; �) �@@s [1� F (�js)] � 0 8s; � holds if � (s; �) and
@@s� (s; �) have opposite signs:
� (s; �) � @@s� (s; �) � 0
Proof. Follows from inspection of �q� and �qs, Vq� > 0, and standard monotone comparative
statics results.
Given the analysis thus far, it is now possible to consider whether or not menus of three-part
tari¤s of the type o¤ered by cell phone service providers would be predicted by the relaxed solution
of the model.
As absent pooling, the consumer will consume until his marginal valuation is equal to marginal
price (Pq (s; q) = Vq (q; �)), marginal prices under the relaxed solution fall out of the �rst order
condition in equation (37). In particular:
Pq (s; q) = Cq (q) + Vq� (q; �)(1�G (s))g (s)
@@s (1� F (�js))
f (�js)
By inspection, marginal price is above marginal cost whenever @@s (1� F (�js)) is positive, and
marginal price is below marginal cost whenever @@s (1� F (�js)) is negative.
First consider the special case of RSOSD where S is ordered by �rst order stochastic dominance
(FOSD) so that F (�jsH) � F (�jsL) for all � and for all sH � sL. In this case @@s (1� F (�js)) � 0
and therefore marginal prices are everywhere weakly above marginal cost.
The general su¢ cient conditions for global incentive compatibility given in Lemma 10 remain
somewhat opaque. That being said, for a simple parametrization of � it is easy to verify global
incentive compatibility.
20
Lemma 11 Assume that � is a sum of the signal s and a random variable " with smooth cumulative
density H ("):
� = s+ "
" � H (")
Then F (�js) are ordered by FOSD and if G (s) satis�es the monotone hazard rate condition@@s
�1�G(s)g(s)
�� 0, then the relaxed solution qR (s; �) satis�es global incentive compatibility.
Proof. First, the FOSD ordering follows because F (�js) = H (� � s) is decreasing in s. Second,when � is a function of s and a random variable " � H ("), and � is strictly increasing in ", it canbe shown that @
@�� (s; �) =�s"�"and @
@s� (s; �) = �ss ��s"�"�s. In this parameterization �ss = �s" = 0
which implies @@�� (s; �) =
@@�� (s; �) = 0 and therefore the conditions of Lemma 10 are met.
No absolute conclusions can be drawn without a characterization of the solution when it di¤ers
from the relaxed solution. That being said, the preceding discussion suggests that if S is ordered
by FOSD, the common-prior model cannot explain observed cell phone service pricing.
It is possible, however, to work backwards to see what type distributions might explain observed
pricing. Following the approach taken in this paper, now assume that consumers may freely dispose
of minutes beyond their �nite satiation points. As seen earlier in the paper, this essentially imposes
the constraint that marginal price be non-negative. Further, when marginal costs are assumed to
be zero, this constraint binds whenever @@s (1� F (�js)) is weakly negative.
Observed menus of tari¤s, with increasing levels of included minutes, might then correspond to
the relaxed solution if the RSOSD ordered conditional distributions satisfy equation (39) for some
cuto¤ �� (s) increasing in s. The relaxed solution would then involve zero marginal price up to
qR (s; �� (s)) on each tari¤, and positive marginal prices at higher quantities.
@
@s(1� F (�js))
8<: � 0 � � �� (s)> 0 � > �� (s)
(39)
As the su¢ cient conditions for the relaxed solution to be optimal given in Lemma 10 are
somewhat opaque, it is again useful to turn to a particular parameterization where global incentive
compatibility can be veri�ed.
Lemma 12 Assume that � is a function of the signal s and a mean zero (E ["] = 0) random
variable " with smooth cumulative distribution H (") as given below:
� = z + �s+ s1+ "
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�; � 0
Then F (�js) are ordered by RSOSD and @@s (1� F (�js))
8<: � 0 � � �� (s)> 0 � > �� (s)
for some �� (s) which
is constant for � = 0 and strictly increasing for � > 0. Moreover, if G (s) satis�es the monotone
hazard rate condition @@s
�1�G(s)g(s)
�� 0, and s is not too small (s � 1+
g(s) ), then the relaxed solution
qR (s; �) satis�es global incentive compatibility.
Proof. First, RSOSD follows as increasing s involves a mean preserving spread and a FOSD shift.
Second �� (s) is found by setting � (�; s) = �s equal to zero and solving for �. This yields
�� (s) = z + ��
1+
�s and @
@s�� (�) = �
� 1+
�which is strictly positive if � > 0.40
Third @@�� (�; s) =
�s"�e= 1+
s . Thus condition (1) reduces to(1�G(s))g(s) (1 + ) � s. Under the
monotone hazard rate assumption, this inequality is hardest to satisfy for s. Thus the assumption
s � 1+ g(s) is su¢ cient and period two global incentive compatibility is satis�ed.
Fourth, it can be shown that:
@
@s� (s; �) = �1
s� � (s; �)�
s�
Therefore � (s; �) � 0 implies that @@s� (s; �) � 0. So condition (2) of Lemma 10 is satis�ed whenever
� (s; �) � 0. (This implies that qR (s; �) is increasing in s whenever � (s; �) � 0 and corresponds tothe observed fall in overage rates on higher tari¤s.)
Of course this condition will not always be met when � (s; �) < 0. This does not matter, however,
because for � (s; �) < 0 zero marginal costs imply that the satiation (free disposal) constraint is
binding and qR (s; �) = qS (�). This in turn means that @@�q
R (s; �) = 0 and condition (2) of Lemma
9 is satis�ed, which is what counts. Thus �rst period global incentive compatibility is guaranteed.41
As an example that meets the conditions of Lemma 12, take � = 5s + s2" for s � U [2; 3], and" � N (0; 1). In this example �� (s) = 5
2s.
40The critical points �� (s) will be within the support of � given s as long as the support of " is large enough:"� � �
1+ s� .
41Note that for = 0, �rst period incentive compatibility holds without relying on the satiation constraint. In
particular, for = 0, �ss = 0 and so � (s; �) � @@s� (s; �) = �s
��ss � �s"
�"�s�� 0 since �s"; �" > 0. This means that
condition (2) of lemma 10 always holds. The fact that this works out so nicely is related to the fact that �� (s) isconstant in s and therefore all the distributions F (�js) cross once at this common point. (Essentially the case Courtyand Li (2000) focus on). However, this would correspond to a menu of tari¤s which all have the same number ofincluded minutes. To explain observed tari¤s > 0 is required so that �� (s) is increasing.
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