the optimality of uniform pricing in ipos: an optimal...
TRANSCRIPT
The Optimality of Uniform Pricing in IPOs: AnOptimal Auction Approach
October 4, 2005
Abstract
This paper provides theoretical support to the use of the uniform pricing rule in the
Initial Public Offerings of shares. By using an optimal auction approach, we are able to
show that the optimality of a uniform price in IPOs crucially depends whether the seller
has full descretion on the use of quantity discrimination. This in turn depends on two
main dimensions of the IPO game: the institutional investors’ preferences and existence
or not of budget constraint on retail investors. Specifically, we show that in the standard
case, with risk neutral institutional investors and cash constrained retail investors, the
optimal IPO is implemented by a uniform price.
Keywords: Initial Public Offering, Price Discrimination, Rationing, Optimal Auc-
tion.
JEL Classification: D8, G2.
1
1 Introduction
The literature on Initial Public Offerings (IPOs) has been growing remarkably in the last
decade. Most of this literature, both theoretical (e.g. Rock, (1986), Allen and Faulhaber,
(1989), Benveniste and Spindt, (1989)) and empirical (e.g Beatty and Ritter, (1986), Ritter
(1987) Cornelli and Goldreich, (2001), Derrien and Womack (2003)) has particularly focused
on the explanation of some apparent market anomalies surrounding IPOs, such as under-
pricing, long-term underperformances and hot issues markets.1 Much less attention has been
devoted to understand whether the current IPO format which imposes uniform pricing is
efficient or not. Virtually all papers simply assumes uniform pricing to be consistent with
the current practice of IPOs. As a matter of fact, the only paper addressing this issue (Ben-
veniste and Wilhelm, (1990)) challenges the efficiency of the current regulatory constraints
and suggests that firms could improve on the IPO performance (raise the IPO proceeds) if
allowed to price discriminate across investors.2
This paper investigates more closely this issue, i.e. whether uniform price restrictions
lead to inefficient outcomes. The first result we obtain is to show that in the standard IPO
set up with risk neutral institutional investors and cash constrained retail investors uniform
pricing is the optimal pricing rule. However. the main contribution of the paper is to show
that the optimality of the uniform pricing rule crucially depends on two main dimensions
of the problem: the institutional investors’ preferences and the existence or not of budget
constraint on retail investors. Specifically, we find that as long as retail investors are not cash
constraint - i.e. they can buy all the shares - the optimal IPO can always be implemented
by a uniform price. Conversely, when retail investors are cash constraint, the institutional
investors’ preferences become determinant: price discrimination is needed in addition to
quantity discrimination if institutional investors are risk averse. The reason for this is that, if
the seller enjoys full discretion on how to allocate shares, then there is no need to also employ1Ritter and Welch (2002) provide a handy review of the IPO literature, whereas for a fully comprehensive
review of the theory and evidence on IPO activity we refer to Jenkinson and Ljungquist (2001).2Brisley (2003) in a note shows that Benveniste and Wilhelm’s result holds only when regular investors
are, on average, less informed than retail investors. Otherwise, the uniform price restriction does not affectIPO proceeds.
2
price discrimination to achieve optimality. The absence of cash constraint on retail investors
ensures the seller enough leeway on the allocation of share independently of the institutional
investors preferences. When instead institutional investors are cash constrained, the seller
may still have enough leeway on the allocation rule if institutional investors are risk neutral,
but not if they are risk averse. In this latter case quantity discrimination alone is not enough
to achieve optimality and the seller needs to employ also price discrimination.
The methodology we employ, which exploit the tools of optimal auction and mechanism
design theory for information gathering (Myerson, 1981; Maskin and Riley, 1989), represents
by itself one of the contributions of this paper.
There are few others papers in the IPO literature using the same approach (Biais, Bossaert
and Rochet (2002), Biais and Faugeron-Crouzet, (2002), Maksimovic and Pichler, (2002)).
Our paper differ from each of them in several respects, the most important of which is that,
as anticipated, all of them assume uniform pricing whereas we do not put any restriction ex
ante on the pricing rule. We endogenously derive the optimal pricing rule and show that it is
uniform. In addition to this, Bias, Bossaerts and Rochet (2002) focus on IPOs in which the
intermediary acts in the best interest of the institutional investors instead of the firm. They
derive the optimal selling mechanism that enables the seller to extract all the information
from the coalition intermediary-institutional investors and find that this mechanism specifies
a price schedule decreasing in the quantity allocated to retail investors. This price schedule
on one hand eliminates the winner’s curse for the uninformed investors and on the other hand
exhibits underpricing (in the form of informational rents) to institutional investors.
Biais and Faugeron-Crouzet instead compare the performances of alternative IPO proce-
dures and find that in general auction-like mechanisms such as Mise en Vente and Bookbuild-
ing are more informationally efficient and can lead to price discovery. They also show that
pure auction mechanisms might lead to tacit collusion among the bidders and, consequently,
to inefficient outcomes.
Maksimovic and Pichler (2002) look at the optimal amount of information gathering
by comparing private and public offering with a particular attention to the determinants of
3
underpricing. They find that in absence of any allocation constraint and under risk neutrality,
the seller is able to design a selling mechanism with zero underpricing (no rents to the informed
investors). We prove a similar result in a particular subcase of our model. Taking this result
as a benchmark, the authors then link the existence and magnitude of underpricing to various
constraint eventually imposed on the bookbuilding procedure.
In our IPO a firm wants to place a fixed number of shares. Two groups of investors take
part in the IPO: n institutional investors and, a group of atomistic investors (retail investors).
The latter typically participate only occasionally in the offering. Institutional investors have
superior information about the market value of the asset because each of them receives a
private signal, whereas retail investors have no information about the value of the asset on
sale .
The seller designs the IPO mechanism in order to elicit the information from institutional
investors and maximize the IPO proceeds. Deriving the optimal IPO means deriving the
optimal allocation rule and the optimal pricing rule. We start by looking for the optimal IPO
in the standard IPO set up where institutional investors are assumed to be risk neutral and
retail investors cash constrained.
In this context we prove that uniform pricing is the optimal pricing rule and furthermore
there exists a multiplicity of uniform prices implementing the optimal IPO. We next move
away from the standard IPO set up by challenging the assumption of risk neutral institutional
investors. We instead analyze how the optimal IPO changes if institutional investors are risk
averse, where the risk aversion we have in mind is an aversion to the inventory risk. The
results we obtain suggest the optimality of the uniform pricing rule crucially depends on
two dimensions of the problem, the existence of budget constraint to retail investors and
the institutional investors’ preferences. In particular, as long as retail investors are not
cash constrained the uniform pricing rule always supports the optimal IPO allocation no
matter what the preferences of institutional investors are. When instead retail investors
are cash constrained then institutional investors’ preferences play a role and, specifically, we
can restore the optimality of the uniform pricing rule only if they are risk neutral (but not
4
necessarily when they exhibit risk aversion). In other words, for the uniform price to be
optimal, it must be that the seller can freely choose the allocation rule, i.e. he is not subject
to any restriction on how to allocate the shares, a condition which is clearly not satisfied when
retail investors are cash constrained. Therefore, in this case, he needs to gain some leeway
elsewhere and this is achieved when informed buyers’ preferences offer sufficient flexibility
(i.e. the case of risk neutrality).
The paper is organized as follows. In the next section we set up the model and the IPO
design problem for the seller in the standard case of risk neutral institutional investors and
cash constrained retail investors. The results are presented in Section 3. The case of risk
averse institutional investors is analyzed in Section 4. Some concluding remarks are discussed
in the last section. All the proofs are presented in the Appendix.
2 The Model
A firm wants to sell Q shares in an IPO, with Q fixed and, without loss of generality, nor-
malized to 1. An intermediary is in charge of marketing the issues. He is assumed to act in
the firm’s best interest, so, hereafter, we will simply refer to the seller to denote the coalition
firm-intermediary. This is a standard way of modelling the role of intermediary in the IPO
literature. The seller wants to maximize the proceeds from the sale. He faces n(> 2) large,
institutional investors with private information about the firm’s market value, and a fringe
of retail and uninformed investors.
Institutional investors’ preferences are described by the following utility function:
ui(pi, qi, v) = z(qi, v)− qipi for all i ∈ 1, 2, ....n
where qi is the quantity assigned to investor i and pi is the price per share investor i has to
pay. We denote by Ti = piqi the total payment from investor i to the seller. v is the value of
new shares. Notice that, the above utility function is linear in the transfer Ti.3 We assume
that the valuation function z for investors satisfy some standard assumptions (subscripts are3This kind of utility functions exhibiting linearity in the total payment, are very common in the auction
literature. (See for instance Crèmer and McLean, 1988).
5
for derivatives with respect to variables): 1) z1 > 0 and z2 > 0; 2) z11 ≤ 0; 3) z(0, v) = 0 for
all v; 4) z12 > 0 (single-crossing condition); 5) z122 ≤ 0 and z112 ≥ 0; and 6) z12(0, v) ≥ 1
(see Fudenberg and Tirole, 1991).
Institutional investors have private information in that each of them receives a signal si
about the valuation of the firm by the market value. The value of the shares v is given by
the average of all the signals, that is
v(s) =1
n
Xi
si
Our choice of the informational structure is very common in auction theory (e.g. Bulow
and Klemperer, 1996, 2002). Additionally and more importantly, our informational structure
is a straightforward generalization of the simple binomial informational structure adopted
in other papers on IPOs (e.g. Benveniste and Wilhelm, 1996; Biais and Faugeron-Crouzet,
2002).4
Signals are i.i.d. according to a uniform distribution defined on Ωi = [s; s], so the cu-
mulative distribution function is Fi(si) =si − s∆s
, and the density function fi(si) =1
∆s. Let
us also denote by f(s) the joint density function so that f(s) = f(s1, ..., sn) =Qifi(si),
with s = (s1, ..., sn) ∈ Ω =NiΩi = [s, s]
n.
We assume a continuum of competitive, risk neutral retail bidders. The total mass of
these retail bidders is normalized to one. We assume that they face a cash constraint so
they cannot absorb the whole issue. In the following sections we will analyze the effect of
the existence of this cash constraint on the implementation of optimal IPO procedure with
a uniform price. We denote by qR the quantity they receive in equilibrium; by pR the price
per unit of share they are asked and by TR = qRpR the total transfer to the seller. Budget
constraint for uninformed agents stipulates that
qRpR ≤ K.4These papers assume that the signal investors receive can be either good or bad and the stock market
value is monotonic in the number of good signals. With continuous signals, as we assume, our specification of vpreserves this monotonicity property, i.e. higher signals lead to higher market value of the stock. Furthermore,our results are robust to a generalization of this function to a generic v = Ψ(s) with Ψ increasing in s . Wechose however not to use this specification because it makes notation and computation much heavier withoutadding anything in terms of insight.
6
Retail investors as the seller do not have any private information about the market value
of the asset and only observe the density f(s) of signals.
In order to extract the information from the institutional investors, the seller designs a
contract specifying the quantity and the price per share to pay to each institutional investor
and to retail investors. By using the revelation principle, we can focus on direct mechanisms
in which the firm asks each investor to announce his signal and then fixes the quantity and
the price as a function of their announcements in such a way to induce them to truthfully
reveal their information (see Fudenberg and Tirole, 1991).
We assume that all the shares issued must be allocated between institutional and retail
investors. Consequently, by choosing the vector qii the firm implicitly determines the
number of shares to allocate to retail investors, which is given by:
qR = 1−Xi
qi
The contract the seller proposes can thus be written in the following way:
Γ = pi(bsi, bs−i), qi(bsi, bs−i); pR(bsi, bs−i), qR(bsi, bs−i) for all i and all (bsi, bs−i) ∈ Ωwith bsi being institutional investor i0s announcement and bs−i the vector of the announcementsof all the other investors. The optimal IPO mechanism for the firm is the solution to the
following optimization program:
maxqi
UF = Es
"Xi
Ti(bsi, bs−i) + TR(bsi, bs−i)#
subject to the standard constraints (see Fudenberg and Tirole, 1991):
• Retail Investors’ Participation Constraint (RPC):
ES [qR(si, s−i)(v − pR(si, s−i)] ≥ 0
which requires their expected payoff to be larger than their reservation utility which,
without loss of generality, may be set equal to zero.
7
• Institutional Investors’ Participation Constraint (IPC):
Ui(bsi, bsi) = Es−i z(qi(bsi, s−i), v(bsi, s−i))− qi(bsi, s−i)pi(bsi, s−i) ≥ 0 for all si and i
which has the same meaning as the RPC. Note however that the expected profits of
participation for each informed investor is conditional on his signal.
• Institutional Investors’ Incentive Compatibility Constraint (IIC):
Ui(bsi, si) ≤ Ui(si, si) for all si, bsi and i,with
Ui(bsi, si) = Es−i z(qi(bsi, s−i), v(si, s−i))− qi(bsi, s−i))pi(bsi, s−i)which ensures that institutional investors do not have incentive to misrepresent their
type - the signal they receive - to the firm.
• Last, the Full Allocation Constraint (FAC):
Xi
qi(si, s−i) + qR(si, s−i) = 1,
the feasibility constraint:
qi(si, s−i) ≥ 0 for all i, si and s−i
and the budget constraint for uninformed investors:
qR(si, s−i)pR(si, s−i) ≤ K for all s
Also notice that the monotonic hazard rate condition is trivially satisfied by the uniform
distribution function, i.e.∂
∂si
∙fi(si)
1− Fi(si)
¸=
1
(s− si)≥ 0.
The next Lemma shows how to simplify the above maximization program.
8
Lemma 1 The seller’s optimal IPO design problem can be rewritten in the following way:
maxqin1
ZΩ
(v +
Xi
∙z(qi(s), v)−
1
n(s− si)z2(qi(s), v)− vqi(s)
¸)f(s)ds
s.t :Ui(s, s) = 0 for all i
1nEs−i
∙z12(qi(s), v)
∂qi(s)
∂si
¸≥ 0 for all i and all si
qi(s) ≥ 0 for all i and all sPiqi(s) ≤ 1 for all s.µ1−
Piqi(s)
¶pR(si, s−i) ≤ K for all s
where the second constraint is the monotonicity constraint which ensures that the SOC are
met and the mechanism is implementable.
Proof: See the Appendix.
The seller’s program in this set-up is quite different from a standard auction design prob-
lem where an uninformed seller faces usually only informed bidders. The participation to the
auction of a class of uninformed bidders, makes the problem quite different and interesting
because it mitigates the adverse selection problem and, thus, lowers the cost for the seller
to extract the informed investors’ private information.5 Furthermore, notice that prices are
present in the seller’s program only in the budget constraint of uninformed investors.
3 Risk neutral institutional investors and cash constrainedretail investors.
We start by analyzing a case similar to Benveniste and Wilhelm (1990) where institutional
investors are supposed to be risk neutral and retail investors cannot afford to buy the whole
quantity of shares, i.e. are budget constrained. Notice that this is also the most common set
up in the IPO literature.
Assume that the valuation function of informed investors is z(q, v) = vq, so that the5See Bennouri and Falconieri (2005) for a careful analysis of this issue.
9
utility function of investor i is
ui(v, qi) = v(s)qi(s)− pi(s)qi(s)
and additionally, that retail investors are subject to the following cash constraint
pR(s)qR(s) ≤ K for all s.
We assume also that K ≤ s so that the budget constraint matters. The seller’s program
becomes then
maxqin1
ZΩ
(v −
Xi
∙1
n(s− si)qi(s)
¸)f(s)ds
s.t :Ui(s, s) = 0 for all i
Es−i
∙∂qi(s)
∂si
¸≥ 0 for all i and all si
qi(s) ≥ 0 for all i and all sPiqi(s) ≤ 1 for all s.µ1−
Piqi(s)
¶pR(si, s−i) ≤ K for all s
Notice that the seller’s objective function is decreasing in qi(.) for each i. This implies
that at the optimum, the seller will allocate as much as possible to retail investors, i.e. up
to their budget constraint. The proposition below formalizes the result about the optimal
allocation rule.
Proposition 1 With risk neutral institutional investors and cash constrained retail investors,
the optimal IPO is such that retail investors are satisfied first up to their budget constraint
whereas institutional investors get the remaining share.
So in the optimal mechanism, retail investors get priority in the allocation (see also
Biais and Faugeron-Crouzet, 2002). This result highlights the important role played by the
uninformed investors in the IPO process. Their presence mitigates the adverse selection
problem vis-a-vis the institutional investors and, consequently, allows the seller to lower the
10
cost to extract their private information. The intuition for this is that the seller can always
threaten the institutional investors to allocate the shares to retail investors although in this
specific situation, the effectiveness of this threat might be limited by the fact that retail
investors are subject to cash constraint.
For the pricing rule, we know that retail investors’ budget constraint should be binding
in equilibrium, i.e.
pR(s)qR(s) = K for all s
So we now focus on the optimal pricing rule which, for each institutional investor must
satisfy the following equation:ZΩ−i
pi(s)qi(s)f−i(s−i)ds−i =
ZΩ−i
½v(s)qi(s)−
1
n
∙Z si
sqi(esi, s−i)desi¸¾ f−i(s−i)ds−i
for all si and all i (1)
and for uninformed investors the following
ZΩpR(s)qR(s)f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds = k
We can show that the next result holds:
Proposition 2 With risk neutral institutional investors and cash constrained retail investors,
the optimal IPO can be implemented with a uniform pricing rule.
Proof: see the Appendix.
The intuition behind this result is very simple. Since asymmetric information in the model
in one-dimensional, the seller needs only one tool to discriminate the informed investors. In
other word, quantity discrimination (and rationing) alone is sufficient to allow the seller to
gather information and thus achieve an optimal performance. For this to be true, however, it
must be that the seller enjoys enough discretion on how to allocate shares among investors.
In this specific case, the seller is partly limited on the use of the allocation rule because retail
investors are cash constrained, but this is offset by the flexibility gained from the institutional
investors’ risk neutrality. This is strengthened by the following Corollary:
11
Corollary 1 With risk neutral institutional investors and cash constrained retail investors,
the uniform price implementing the optimal IPO is not unique.
The fact that there are multiple uniform prices that can implement the optimal IPO
confirms the large leeway the seller has on how to design the sale.
In conclusion, the above results together suggest that in the presence of cash constraint
on retail investors, institutional investors’ preferences become crucial in order to ensure the
seller enough leeway with respect to the use of the allocation rule and, thus, make unnecessary
the use of price discrimination. We shall see more in the Section 4 how cash constraints on
retail investors and institutional investors’ preferences interact to determine the optimality
of a uniform pricing rule in IPOs.
As we know, most of the IPO literature is devoted to understand whether and why
underpricing occurs. Although this is not the focus of this paper, our model has implications
for the existence of underpricing as shown in the next proposition.
Proposition 3 The optimal IPO mechanism is characterized by underpricing, i.e. for all s,
shares are priced below their expected value v.
Proof: see the Appendix.
Our definition of underpricing is the same as in Benveniste and Spindt (1989) and Rock
(1986). The rationale for the existence of underpricing in our model is the same as in Ben-
veniste and Spindt (1989), i.e. the seller underprices the shares to lower the cost of eliciting
information from the institutional investors.
No cash constraint on retail investors
The case of risk neutral institutional investors and no cash constraint is particularly
interesting. Because of what we said before, in this case the seller has full discretion with
respect to the choice of the allocation rule, the consequence of this is stated in the next
corollary.
Corollary 2 When informed investors are risk neutral and retail investors are not cash
constrained, i.e. they could buy all the shares on sale, then the optimal offering is such that
12
all the shares are sold to retail investors irrespective of the signal reported by the informed
investors, at a marginal price equal to pR = v .
The above results implies that risk neutral institutional investors would get zero rent
in equilibrium because the seller would optimally choose to exclude them from the offering.
In other words, the institutional investors are always excluded from the offering and this is
driven by the fact that informed investors compete against uninformed investors (Bennouri
and Falconieri, 2005).
4 Risk aversion and no cash constraint
Although the benchmark IPO in the literature has always considered institutional investors to
be risk neutral, there is no reason to exclude the possibility that even institutional investors
may exhibit some kind of risk aversion. Specifically, we think it is not unreasonable that
institutional investors exhibit and aversion to the, so called, inventory risk, that is the risk
associated to the composition of their portfolio. As a matter of fact, this is well recognized
by the market microstructure literature or the foreign exchange market literature (see for
instance O’Hara (1996), Lyons (2003)). Therefore in this section we analyze how the optimal
IPO and the optimal pricing rule is affected by this assumption. In Section 3 we have
already anticipated that the optimality of the uniform pricing mostly depends on the degree
of freedom the seller has with respect to the allocation of the shares, thus, intuitively, we
might expect that the institutional investors’ risk aversion will reduce the seller’s freedom.
We shall show that this is not necessarily the case.
For the time being we will assume instead that retail investors do not bear any cash
constraint. The institutional investors’ risk aversion is interpreted as we said as an aversion
to the so called inventory risk, and is modeled, similarly to Benveniste and Wilhem (1990),
by assuming that their preferences are concave in the quantity, i.e. z(qi, v) = qi(αv− δ2qi). Our
function z is a variant of a standard Mean-Variance utility function in which the expected
value v is weighed more than the transfer Ti. The parameter α > 1 measures the regular
investor’s aggressiveness: the higher α is, the more aggressive the investor is in the IPO
13
procedure. The intuition is that, despite their risk aversion, institutional investors are very
much interested in the sale and, therefore, they will bid aggressively in the IPO. If α = 1
then the problem would be equivalent to having risk neutral institutional investors and, in
this case, we know that the optimal IPO is such that all the shares will be allocated to retail
investors no matter the information reported by the institutional investors.6
The next proposition derives the optimal allocation rule. For the details of the proof the
reader can refer to Bennouri and Falconieri (2004)
Proposition 4 The optimal IPO is characterized by the following allocation rule: there exists
a threshold value of the signals si (v−i), with v−i =Pj 6=is−j , such that,
• If all the informed investors report signals below this threshold, all the shares are awarded
to the uninformed investors; otherwise
• all the institutional investors reporting a signal below si (v−i) get zero, whereas the
others get a positive quantity which is equal to
— eqi = (α− 1)nv − α (s− si)nδ
, if the issue is not oversubscribed, with retail investors
receiving the remaining shares;
— bqi < eqi if the issue is oversubscribed, with bqi(s) = eqi(s) − β(s)/δ where β(s)/δ is
the amount of shares by which they are rationed. Retail investors get nothing.
The institutional allocation rule is such that the optimal quantity eqi(s) is increasingin the signal si reported by the institutional investor. In other words, the seller rewards
better information with a larger quantity. This is in line with the existing literature on
bookbuilding (Benveniste and Spindt (1989) and Cornelli and Goldreich (2001)), which shows
that institutions receive more shares in ”hot” issues, i.e. IPOs with strong pre-market demand
and also with the traditional Rock’s argument according to which institutions always try to
avoid ”lemons”. In a recent paper, Aggarwal et al. (2002) also provide some empirical6 In addition, the assumption reflects the practice of IPO underwriters to surround themselves of a core of
regular investors who repeatedly take part into IPOs (Benveniste and Spindt, 1989).
14
support to these predictions, confirming that institutional allocations are indeed larger in
better performing IPOs.
We can also show that the optimal mechanism obtained above satisfies the SOC as rep-
resented by the monotonicity condition. We omit the proof for the sake of simplicity.7
Previous results in the auction literature ((Maskin and Riley (1984), Eso (2005)) have
highlighted that introducing risk aversion for informed bidders generally creates a trade-off
because, on the one hand, the informed buyers must be compensated for the risk they bear in
the auction8 and the cost of screening which might instead be reduced by exposing them to
some risk. For this reason, it is interesting to see what happens to the optimal price schedule
when informed investors are risk averse.
We can however show, that despite the risk averse institutional investors, the optimal
IPO can be implemented by an uniform pricing schedule as in the standard case. The result
is stated in the next proposition.
Proposition 5 The optimal IPO mechanism can be implemented by a uniform price schedule
to all investors.
Proof: (See the Appendix).
To give an idea of the reasoning behind the proof of this Proposition, first recall that
the optimal price for institutional and retail investors must satisfy, respectively, the following
integral equationsZΩ−i
pi(s)qi(s)f−i(s−i)ds−i =
ZΩ−i
½z(qi(s), v)−
1
n
∙Z si
sz2(qi(esi, s−i), v(esi, s−i))desi¸¾ f−i(s−i)ds−i;
(2)
and
ZΩpR(s)qR(s)f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds
7See Bennouri and Falconieri (2005) for a detailed proof in a more general auction framework.8The risk comes from the fact that the payment of each bidder depends on the announcements of the other
bidders (Eso, 2005).
15
The proof is then articulated in two steps. The first step consists in proving a preliminary
result, i.e. we show the existence and uniqueness of a uniform price schedule satisfying
Equation (2) for all institutional investors. Next, we show that the same price schedule also
solve the participation constraint of retail investors.
The rationale for the above result is that, although the institutional investors’ risk aversion
may impose restrictions on the seller in how to allocate shares, the fact that retail investors are
instead not cash constrained warrants him with enough leeway. More generally, the result
implies that as long as retail investors are not cash constrained, the seller enjoys enough
discretion on how to allocate the shares no matter what institutional investors’ preferences
are. As a consequence, there will not be any need to use price discrimination in combination
with quantity discrimination to support the optimal mechanism.
To fully understand this, in the next Section we consider the case in which institutional
investors are risk averse and retail investors are cash constrained. This is the only case in
which the optimality of the uniform pricing rule breaks down.
Risk aversion and cash constraint
Let us now assume that retail investors are now budget constrained. Notice that for the
sake of tractability, in this case, we write this constraint as a quantity constraint, i.e. we
assume that retail investors can buy a maximum quantity of shares K < 1, i.e.9:
qR(s) ≤ K < 1, for all s ∈ Ω
which implies that retail investors cannot buy all the shares on sale. The quantity con-
straint on retail investors modifies the seller’s optimization problem as follows:9For the sake of tractability, we assume quantity constraint instead of a more general budget constraint.
Notice that if anything this constraint is more restrictive than the general cash constraint given that we loseone degree of freedom (the price dimension).
16
maxqin1
ZΩ
(v +
Xi
∙qi(s)
µ(α− 1) v − δ
2qi(s)−
α
n(s− si)
¶¸)f(s)ds
s.t :
U i(s, s) = 0∂qi(s)
∂si≥ 0 for all i and all s
qi(s) ≥ 0 for all i and all sPiqi(s) ≥ 1−K for all s
Piqi(s) ≤ 1 for all s.
The solution to the second problem is given in the following theorem.
Proposition 6 The optimal IPO is characterized by the following allocation rule: there exists
a threshold value of the signals bsi (v−i), with v−i = Pj 6=is−j , such that,
1. Informed investors reporting the lowest signal s get no shares;
2. All institutional investors with sufficienly high signals get a positive quantity eqi(s) =(α− 1)nv − α (s− si)
nδprovided that 1 − K ≤
Pieqi(s) ≤ 1 with the remaining shares
allotted to retail investors. If insteadPieqi(s) < 1−K, then K shares are allocated to
retail investors and all insitutional investors with a signal above the lowest one, s, get
a positive quantity eqKi (s) 6= eqi(s);3. In the case of oversubscription, all institutional investors with sufficiently high signals
are rationed and retail investors get nothing.
Proof: (See the Appendix).10
The above result is very intuitive. Whenever retail investors’ quantity constraint is hit,
the seller is forced to allocate informed investors more shares than what would be otherwise
optimal, in particular he is forced to allocate some shares also to institutional investors with
relatively bad signals (below the threshold).10We can also prove that this mechanism is implementable that is it satisfied the SOC (monotonicity
constraint).
17
Given the optimal allocation schedule we can derive the optimal pricing rule and by using
the same arguments as in Proposition 5 we can prove the following result:
Proposition 7 If retail investors are subject to quantity constraint and institutional in-
vestors are risk averse, the optimal selling mechanism as derived in Proposition 6 can be
implemented only by a discriminatory pricing schedule for all investors, such that each class
of investors pays the same price.
The proof is very much like the proof of Proposition 5. We start by showing that an
optimal uniform price exists for informed investors and that this cannot be applied to retail
investors as well because it would violate their participation constraint. The result is not
surprising: whenever the issuer faces a situation in which the quantity constraint on retail
investors is hit he will be forced to allocate to informed investors more shares than what
would be otherwise optimal for him to do and this, in turn, requires to adjust downward the
share price. However, it is important to notice that institutional investors all pay the same
price, so, in some sense, there exists a uniform price for each class of investors.
Despite the fact that the introduction of budget constraint on retail investors weakens
our result on the optimal of uniform pricing, it must be noticed that assuming that retail
investors are not budget constrained is not that restrictive for two main reasons. First,
they might be cash constrained individually but not in the aggregate (Ellul and Pagano,
2003). Secondly and most importantly, it is important to understand that it is a simplifying
assumption because in practise the seller can regain flexibility in other ways, by for instance
withholding shares whenever it is not optimal to allocate them to informed investors and
retail investors do not have the capacity to absorb them; or alternatively by using a firm-
commitment contract of underwriting, which is a common practise in the US and through
which the underwriter commits to buy all the shares not sold during the offering. It is easy to
show that incorporating this situations in our IPO framework would lead to the same result
as with no cash constraints.
18
5 Conclusions and Remarks
In this paper, we adopt an optimal auction approach to analyze the functioning of initial
offerings with the focus on the choice by the seller of the optimal instrument to allocate
efficiently the issues, this choice being price versus quantity discrimination (rationing).
Our results highlight that the seller does not need price discrimination to achieve an
optimal performance. Instead, the optimal IPO can always be implemented through a uniform
price schedule. This result holds very generally. In particular, we show that it holds in the
standard IPO framework with risk neutral institutional investors and cash constrained retail
investors.
A deeper investigation shows that as a general rule, as long as the seller can fully rely on
the allocation rule, i.e. no constraint is imposed to him on how and to whom to allocate the
shares, then uniform pricing is enough to achieve an optimal performance. The discretion
in the allocation rule in turn depends on two main dimensions: the existence or not of cash
constraints on to retail investors and the institutional investors’ preferences. In particular,
retail investors appear to play a very important role in the offering since, as long as they
are able to buy the whole quantity then rationing alone is enough to support the optimal
IPO mechanism. When this is not the case, then institutional investors’ preferences start to
matter.11 This suggests that an important empirical question is to investigate whether and to
what extent retail investors are actually cash constrained especially in the light of the recent
debate and empirical evidence which indicate distortions in the allocation of new issues due
to the intermediaries’ will to favor institutional investors at the expenses of retail investors.
Similarly, the issue of whether institutional investors may exhibit some kind of risk aversion
and the extent of it definitely deserves more empirical investigations. From a theoretical
point of view, the central role of uninformed investors in IPOs has been recently studied, in
a very similar framework, by Malakhov (2004). The author shows that the seller’s revenues11This also explains the difference between our result and the Benveniste and Wilhelm one. In their paper,
the underwriter is not able to freely ration regular investors. He can only apply a discrete rationing scheme,i.e. either he fills the full order or he gives nothing. Therefore, price discrimination is needed in addition torationing to allow the underwriter to elicit the information from the regular investors.
19
are increasing in the the number of uninformed investors participating into the offering, the
reason being that a larger number of uninformed investors lowers the outside option of the
informed ones.
6 Appendix
Proof of Lemma 1:
By the RPC and the maximand we know that the seller’s profit is increasing in the retail
investors’ payments, therefore at the optimum he will make the RPC binding. We can then
rewrite the RPC in the following way:ZΩpR(s)qR(s)f(s)ds =
ZΩvqR(s)f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds (3)
where we have put s = (si, s−i) and replaced to qR(s) from the FAC.
Now, let us consider the IPC which can be rewritten as follows:ZΩ−i
pi(s)qi(s)f−i(s−i)ds−i =
ZΩ−i
z(qi(s), v)f−i(s−i)ds−i − Ui(si, si) ≥ 0
for all si and i. (4)
by integrating over Ωi ( that is taking the expectations over si) we find:ZΩpi(s)qi(s)f(s)ds =
ZΩz(qi(s), v)f(s)ds−
ZΩi
Ui(si, si)fi(si)dsi. (5)
Let us now consider the IIC. By applying the envelope theorem to the IIC, we have:
U 0i(si, si) =1
n
ZΩ−i
z2(qi(s), v)f−i(s−i)ds−i (6)
thus,
Ui(si, si) = Ui(s, s) +
Z si
s
(1
n
ZΩ−i
z2(qi(esi, s−i), v(s))f−i(s−i)ds−i)desi. (7)
For the IPC to be satisfied we can simply set Ui(s, s) = 0 which means that the seller
leaves zero rents to the informed traders with the lowest evaluations.
20
Applying Fubini’s theorem to equation (7) yields:
Ui(si, si) =1
n
ZΩ−i
½Z si
sz2(qi(esi, s−i), v(s))desi¾ f−i(s−i)ds−i.
Integrate over Ωi, and apply again Fubini’s theorem to get the following
ZΩi
Ui(si, si)fi(si)dsi =1
n
ZΩ−i
½ZΩi
∙Z si
sz2(qi(esi, s−i), v(s))desi¸ fi(si)dsi¾ f−i(s−i)ds−i.
Last, integration by parts of the integralRΩi
hR sis z2(qi(esi, s−i), v(s))desii fi(si)dsi, yields
the following equation:
ZΩi
∙Z si
sz2(qi(esi, s−i), v(s))desi¸ fi(si)dsi =
∙½Z si
sz2(qi(esi, s−i), v(s))desi¾ (Fi(si)− 1)¸s
s
−ZΩi
z2(qi(s), v)(Fi(si)− 1)dsi.
which can be rewritten as follows:ZΩi
Ui(si, si)fi(si)dsi =1
n
ZΩ(s− si)z2(qi(s), v)f(s)ds. (8)
where (s− si) is the inverse of the hazard rate.
By replacing equation (8) into equation (5) and using equation (3) we get the seller’s
objective function. Last, in order to show the prevalence of the monotonicity condition,
notice that the (IIC )s may be written as follows
si ∈ argminbsi [Ui(bsi, bsi)− Ui(si, bsi)] for all si and i.
Writing the SOC for this problem and using equation (6) gives the result.
Proof of Proposition 2
In order to make the reasoning in the proof clear, we split out the proof in three steps. In
the first step, we derive conditions for the existence of a uniform price for informed investors.
In step 2 we derive conditions for the existence of a uniform price for all investors. Finally,
21
we show the existence of a mechanism (allocation and pricing function) satisfying these
conditions.
Step 1: The linear transfer for the institutional investors must satisfy equation (1). Now,
consider the following price function
p0i (s) = v −1n
hR sis qi(esi, s−i)desii
qi(s)(9)
for each si and each qi(s) such that qi(s) 6= 0.12
Any price satisfying equation (1) can also be written as pi(s) = p0i (s) + Φi(s) with Φi is
such that ZΩ−i
Φi(s)qi(si, s−i)f−i(s−i)ds−i = 0, for all i and si.
A uniform price function exists if and only if it satisfies the following partial differential
equation∂pi(s)
∂si=
∂pi(s)
∂sjfor all i and j
which is equivalent to∂p0i (s)
∂si+
∂Φi(s)
∂si=
∂p0i (s)
∂sj+
∂Φi(s)
∂sj. (10)
with,∂p0i (s)
∂si=
∂qi(s)
∂si
£−p0i (s) + v
¤qi(s)
(11)
and∂p0i (s)
∂sjqi(s) =
∂qi(s)
∂sj
£−p0i (s) + v
¤+1
n
∙qi(s)−
Z si
s
∂qi(esi, s−i)∂sj
desi¸ . (12)
Multiplying both sides of equation (10) by qi(s) and from equations (11) and (12) yields
1
n
∂qi(s)∂si
×R sis qi(esi, s−i)desi[qi(s)]2
− 1n+
∂Φi(s)
∂si=
∂Φi(s)
∂sj+1
n
∂qi(s)∂sj
×R sis qi(esi, s−i)desi[qi(s)]2
− 1n
Z si
s
∂qi(esi, s−i)∂sj
desiqi(s)
(13)
yet
1
n
∂qi(s)∂si
×R sis qi(esi, s−i)desi[qi(s)]2
− 1n= − 1
n
∂
∂si
"R sis qi(esi, s−i)desi
qi(s)
#12Otherwise p0i (s) = 0.
22
and
1
n
∂qi(s)∂sj
×R sis qi(esi, s−i)desi[qi(s)]2
− 1n
Z si
s
∂qi(esi, s−i)∂sj
desiqi(s)
= − 1n
∂
∂sj
"R sis qi(esi, s−i)desi
qi(s)
#
Substitution in equation (13) gives
∂
∂si
"Φi(s)−
1
n
R sis qi(esi, s−i)desi
qi(s)
#=
∂
∂sj
"Φi(s)−
1
n
R sis qi(esi, s−i)desi
qi(s)
#
This is a partial differential equation in Φi(s) −α
n
Z si
sqi(esi, s−i)desiqi(s)
. A generic solution for
this PDE is given by
Φi(s) = ϕ(s1 + ...+ sn) +1
n
R sis qi(esi, s−i)desi
qi(s), forall i
where ϕ(·) is a twice differentiable function defined on the set [ns, ns] = nΩ.
Summing up, so far we have shown that the optimal mechanism may be implemented by
a uniform price schedule for informed investors if and only if
pi(s) = p0i (s) + ϕ(s1 + ...+ sn) +1
n
Z si
sqi(esi, s−i)desiqi(s)
(14)
= v(s) + ϕ(s1 + ...+ sn) = v(s) + ϕ(v)
where ϕ is a twice differentiable function satisfying the following integral equation
ZΩ−i
⎡⎢⎢⎣ϕ(s1 + ...+ sn) + 1
n
Z si
sqi(esi, s−i)desiqi(s)
⎤⎥⎥⎦ qi(si, s−i)f−i(s−i)ds−i = 0.or alternatively Z
Ω−i
ϕ(s1 + ...+ sn)qi(si, s−i)f−i(s−i)ds−i =
− 1n
ZΩ−i
∙Z si
sqi(esi, s−i)desi¸ f−i(s−i)ds−i = g(si). (15)
The existence of a uniform price function across all informed investors is then equivalent to
the existence of the function ϕ as a solution to the integral equation defined in equation (15).
23
Step 2: Let now turn to derive the conditions for a uniform price function for all in-
vestors . For uninformed agents, the uniform price function should satisfy their participation
constraint as well as their budget constraint. This gives the following conditionsZΩv
Ã1−
Xi
qi(s)
!f(s)ds = K (16)
and
(v(s) + ϕ(v))
Ã1−
Xi
qi(s)
!= K for all s (17)
This latter equation allows to write ϕ as follows:
ϕ(v) =K
(1−Pi qi(s))
− v for all s (18)
Therefore, proving the existence of a uniform price function is equivalent to prove the existence
of a set of functions (qi(s))i=1,...,n that is the solution to equation (16) andZΩ−i
½∙K
(1−Pi qi(s))
− v¸qi(si, s−i) +
1
n
∙Z si
sqi(esi, s−i)desi¸¾ f−i(s−i)ds−i = 0
for all si and all i(19)
Notice that we can re-write equation (19) as a partial differential equation. Equation (16)
simply defines a final condition for this partial differential equation.
Additionally, from equation (17) it must be that (1−Pi qi(s)) is also a function of v as
ϕ, so in what follows we denote by
ξ(v) = 1−Xi
qi(s) = qR(s) (20)
Step 3: The remaining of the proof consists in showing the existence of functions qi and
ξ satisfying the sufficient conditions for the existence of a solution to equations (16), (19) and
(20).
A trivial solution to equation (19), after replacing 1 −Pi qi(s) by ξ(v) is obtained by
setting the value of the integrand equal to zero in each point, i.e. 13∙K
ξ(v)− v
¸qi(si, s−i) +
1
n
∙Z si
sqi(esi, s−i)desi¸ = 0 for all s and all i
13This condition reduces substantially the set of possible solutions to our equations. In a more general set
24
rearranging terms yields the following differential equation inZ si
sqi(esi, s−i)desi
qi(si, s−i)Z si
sqi(esi, s−i)desi =
1
nhv − K
ξ(v)
iand the solution is
ln
µZ si
sqi(esi, s−i)desi¶ = H(v) + C(v−i) (21)
where, H(v) is such that∂H(v)
∂si=
1
nhv − K
ξ(v)
isimple computations lead to
qi(si, s−i) =Li(v−i)
nhv − K
ξ(v)
i exp(H(v)) (22)
where L(v−i) = exp(C(v−i) is an arbitrary continuously differentiable function. Now sum-
ming up over i and using the fact that 1−Pi qi(s) = ξ(v) yields
1− ξ(v) =1
nhv − K
ξ(v)
iR(v) exp(H(v)) (23)
where14 R(v) =Pi Li(v−i). Let us take the derivatives with respect to v and use equation
(23) to substitute [1/n(v − Kξ(v))]R(v) exp(H(v)) by 1− ξ(v) and we finally get the following
differential equation
− ξ0(v)
1− ξ(v)+
Kξ0(v)
ξ(v)[vξ(v)−K] =R0(v)
R(v)(24)
We now need to determine the solution of the last differential equation in ξ in order to
construct the optimal allocations to informed investors. Note that equation (24) depends on
the “arbitrarily” chosen function R(.) and it is possible to choose such function so that the
solution of equation (24) satisfies equation (16). Then from the chosen function R we can
up we can consider the following differential equation∙K
ξ(v)− v
¸qi(si, s−i) +
1
n
∙Z si
s
qi(esi, s−i)desi¸ = zi(si, s−i) for all s and all i
where zi(.) is such thatZΩ−i
zi(si, s−i)f−i(s−i)ds−i = 0. Since we need just to prove the existence of a solution,
and in order to simplify the presentation, we restrain our analysis to the homogenous differential equation.14From equation (23) and the fact that ξ is a function of v, then R should also be a function of v.
25
construct individual Li(v−i) for each i and finally the allocations for each informed investor
(qi(s)) by using equation (22). These quantities satisfy by construction equation (19) and
equation (16) [q.d.e.]
Proof Proposition 3:
From equations (14), (18) and (20), the uniform pricing function is equal to kξ(v) . In order
to check for underpricing we just need to compare v and kξ(v) . From equation (22), we know
that (v− kξ(v)) and Li(v−i) have the same sign. However, form the definition of Li(v−i) derived
implicitly in equation (22) we have that Li(v−i) > 0 and so (v − kξ(v)).
Proof Proposition 4:
We consider the relaxed problem, i.e. we drop the monotonicity constraint in Lemma ??
and check it ex post. In this way, the objective function becomes an ordinary maximand with
the constraints defined in each point and can be maximized pointwise on Ω. The amount of
shares, qi(s), the firm must assign to investor i in order to elicit his information must solve
the following maximization problem, for each s ∈ Ω:
maxqii=1,..n
Pi
∙qi(s)
µ(α− 1) v − δ
2qi(s)−
α
n(s− si)
¶¸s.t :Ui(s, s) = 0 for all i
qi(s) ≥ 0 andPiqi(s) ≤ 1 for all i and s.
The Kuhn-Tucker conditions of this maximization program are the following:
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
(α− 1)v − δqi −α
n(s− si) + λi(s)− β(s) = 0, for all i
λi(s)qi(s) = 0
β(s)[1−Piqi(s)] = 0.
(25)
with λi and β are the Kuhn-Tucker multipliers associated respectively to the feasibility con-
straint and to the FAC.
Now let us denote by H(q, v) our objective function, that is:
26
H(q, v) =Xi
∙qi(s)
µ(α− 1) v − δ
2qi(s)−
α
n(s− si)
¶¸
with qi ∈ [0, 1] and s ∈ Ω = [s, s]n. This function is concave in qi for all i, since∂2H
∂q2i≤ 0 by
assumptions (2) and (5). Now let consider si(v−i) =αs−(α−1)v−i
(2α−1) for all i and v−i and define
the following sets for all s ∈ Ω
N−(s) = i ∈ N | si ≤ si(v−i)
N+(s) = i ∈ N | si > si(v−i)
We can easily show that for each i ∈ N−(s) it must be that qi(s) = 0. Then, for each
i ∈ N+(s), define the quantity eqi as the one solving the equation below,(α− 1)v − δeqi(s)− α
n(s− si) = 0
By the concavity of function H (with respect to qi), eqi(s) is strictly positive (for each i ∈N+(s)). Now, if
•P
i∈N+(s)
eqi(s) ≤ 1the quantity eqi(s) is the solution of our mechanism.15
•P
i∈N+(s)
eqi(s) > 1,this case corresponds to a situation of oversubscription of the new shares. The quantity
eqi(s) cannot thus be the optimal solution because it violates the FAC, so the optimalmechanism is given by the solution of the following system of equations that we denote
by bqi(s): ⎧⎪⎪⎪⎨⎪⎪⎪⎩bqi(s)[(α− 1)v − δbqi(s)− α
n(s− si)− β(s)] = 0X
i∈N+(s)
bqi(s) = 1; β(s) > 0,
which implies that bqi(s) is either zero or is positive and solves the following equation(α− 1)v − δbqi(s)− α
n(s− si)− β(s) = 0
15 In this case, the equation defining eqi(s) is the FOC of our objective function H, since the Kuhn-Tuckermultipliers, λi(s) and β(s) are both zero.
27
Clearly, in this case, all the shares are allotted to institutional investors. Retail investors
get nothing in equilibrium.
Proof of Proposition 5:
This proof is constructed in the same way as Proposition 2. We divide the proof in three
steps. In the first step, we derive conditions for the existence of a uniform pricing rule among
institutional investors. Then in step 2, we show existence of a unique pricing rule satisfying
these conditions. Finally, in step three, we show that the optimal uniform pricing rule may
be applied to retail investors.
Step 1: The linear transfer for the institutional investors must satisfy the following
equation
ZΩ−i
pi(s)qi(s)f−i(s−i)ds−i =
ZΩ−i
½z(qi(s), v)−
1
n
∙Z si
sz2(qi(esi, s−i), v(esi, s−i))desi¸¾ f−i(s−i)ds−i;
(26)
Then, consider the following price function
p0i (s) =z(qi(s), v)− 1
n
hR sis z2(qi(esi, s−i), v(esi, s−i))desii
qi(s)(27)
for each si and each qi(s) such that qi(s) 6= 0.16 Any price satisfying equation (26) can be
also written as pi(s) = p0i (s) + Φi(s) with Φi satisfies the following equationZΩ−i
Φi(s)qi(si, s−i)f−i(s−i)ds−i = 0, for all i and si.
After characterizing uniform pricing as in Proposition 2 (see equation (10)), we can show
that the optimal mechanism may be implemented with a uniform price if
¡−p0i (s) + αv − δqi(s)
¢µ∂qi(s)∂si
− ∂qi(s)
∂sj
¶− α
n
∙qi(s)−
Z si
s
∂qi(esi, s−i)∂sj
desi¸+∂Φi(s)
∂siqi(s) =
∂Φi(s)
∂sjqi(s). (28)
16Otherwise p0i (s) = 0.
28
We can easily show from Proposition 4 that the optimal quantities allocated to institutional
investors is such that∂qi(s)
∂si− ∂qi(s)
∂sj=
α
nδ
for each s and each i. This allows us to re-write equation (28) as follows
α
n
⎧⎪⎪⎪⎨⎪⎪⎪⎩£−p0i (s) + αv − δqi(s)
¤δqi(s)
− 1 +
Z si
s
∂qi(esi, s−i)∂sj
desiqi(s)
⎫⎪⎪⎪⎬⎪⎪⎪⎭+∂Φi(s)
∂si=
∂Φi(s)
∂sj.
Replacing p0i (s) by its value as defined by equation (26) yields
α
n
⎧⎪⎪⎪⎨⎪⎪⎪⎩−3
2+
α
nδ
Z si
sqi(esi, s−i)desi[qi(s)]2
+
Z si
s
∂qi(esi, s−i)∂sj
desiqi(s)
⎫⎪⎪⎪⎬⎪⎪⎪⎭+∂Φi(s)
∂si=
∂Φi(s)
∂sj.
By replacingα
nδby
∂qi(s)
∂si− ∂qi(s)
∂sjand after some simple computations, we get
∂
∂si
⎡⎢⎢⎣Φi(s)− α
n
Z si
sqi(esi, s−i)desiqi(s)
⎤⎥⎥⎦ = ∂
∂sj
⎡⎢⎢⎣Φi(s)− α
n
Z si
sqi(esi, s−i)desiqi(s)
⎤⎥⎥⎦+ α
2n, (29)
for each i and each j. This is a partial differential equation in Φi(s) −α
n
Z si
sqi(esi, s−i)desiqi(s)
whose generic solution is given by
Φi(s) = ϕ(s1 + ...+ sn) +α
n
R sis qi(esi, s−i)desi
qi(s)+
α
2nsi, for all i
where ϕ(·) is a twice differentiable function defined on the set [ns, ns] = nΩ.
Summing up, so far we have shown that the optimal mechanism may be implemented by
a uniform price schedule if and only if
pi(s) = p0i (s) + ϕ(s1 + ...+ sn) +
α
n
Z si
sqi(esi, s−i)desiqi(s)
+α
2nsi
where ϕ is a twice differentiable function satisfying the following integral equation
ZΩ−i
⎡⎢⎢⎣ϕ(s1 + ...+ sn) + α
n
Z si
sqi(esi, s−i)desiqi(s)
+α
2nsi
⎤⎥⎥⎦ qi(si, s−i)f−i(s−i)ds−i = 0.29
or alternatively ZΩ−i
ϕ(s1 + ...+ sn)qi(si, s−i)f−i(s−i)ds−i =
−ZΩ−i
⎡⎢⎢⎣αnZ si
sqi(esi, s−i)desiqi(s)
+α
2nsi
⎤⎥⎥⎦ qi(si, s−i)f−i(s−i)ds−i = g(si).(30)
Henceforth, proving the existence of a uniform price schedule is equivalent to prove the
existence of such a function ϕ.
STEP 2: We first show that equation (30) can be written as a Volterra integral equation
of the first kind.17 Then, by simply applying the general properties of this kind of integral
equations we can prove the existence and the uniqueness of a function ϕ.
To do this, we first need to transform equation (30) into a simple integral equation, i.e.
with the support defined on R.
Notice that, since qi(si, s−i) = 0 when si ≤ s0i (v−i), the support of the integral equation
(30) is equal to Ω0−i = (s1, s2, .., si−1, si+1, .., sn) |Pj 6=i sj = v−i > v0−i(si) where v0−i(si) is
defined as the inverse of s0i (v−i), i.e.
v0−i(si) =αs− (2α− 1)si
α− 1
Also, by using the definition of the optimal quantity, the LHS term of equation (30) is
equivalent toRΩ0−i
ϕ(si + v−i)qi(si, v−i)f−i(s−i)ds−i.
Now, by applying the Generalized Change Variable Theorem (GVCT) we can set v−i =
γi(s−i) for each i, which finally implies that γi(Ω0−i) = [v
0−i(si); (n− 1)s] ∈ R.18
A main implication of the GVCT is that, there exists a measure λ defined over [v0−i(si); (n−
1)s] such thatZΩ0−i
ϕ(si + v−i)qi(si, v−i)f−i(s−i)ds−i =
Z (n−1)s
v0−i(si)ϕ(si + v−i)qi(si, v−i)λ(dv−i).
17A Volterra integral equation of the the first kind is defined in the following way:Z τ(x)
y0
f(x, y)h(x, y)dy = g(x)
in other words, one of the integral extreme must depend on the variable x.18See Dunford and Schwartz (1988, 3rd Ed.), chapter 3, lemma 8, page 182.
30
Last, by applying the Radon-Nikodým theorem19, we are also able to prove the existence
of a density function ρ associated to the measure λ such thatZΩ0−i
ϕ(si + v−i)qi(si, v−i)f−i(s−i)ds−i =
Z (n−1)s
v0−i(si)ϕ(si + v−i)qi(si, v−i)ρ(v−i)dv−i.
By using the result above, the integral equation (30) reduces toZ (n−1)s
v0−i(si)ϕ(si + v−i)qi(si, v−i)ρ(v−i)dv−i = g(si)
This is a Volterra integral equation of the first kind which ensures that, as far as the
function g is well behaved, a solution in ϕ always exists. This shows the existence of a unique
uniform pricing rule for institutional investors. We denote in the following by pI(s) such
pricing rule.
Step 3:
To complete the proof we are left to show that the uniform price for institutional investors
is also applicable to retail investors. This is equivalent to show that pI(s) satisfies the retail
investors’ participation constraint. Notice that the problem is relevant only in those cases
in which both retail and institutional investors receive a positive amount of shares at the
optimum.
We start by defining the following sets:
Ω− = s | qi = 0 for all i, i.e. all the quantity is distributed to retail investors;
Ω\Ω− = s | qi(s) 6= 0 for at least one i
Recall that, retail investors’ participation constraint writes as followsZΩpR(s)qR(s)f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds
Then, proving that the pricing rule is uniform to all investors is equivalent to show the
existence of a price function pR(s) solving the above integral equation and such that:
pR(s) =
(p−(s) for all s ∈ Ω−
pI(s) for all s ∈ Ω\Ω−19See, for example, Dunford and Schwartz (1988), chapter 3, theorem 2, page 176 in the third edition.
31
which requires that retail investors are charged different prices depending on whether they
get the whole quantity or not. The problem then boils down to prove the existence of a price
p−(s) such thatZΩ−p−(s)f(s)ds+
ZΩ\Ω−
pI(s)
Ã1−
Xi
qi(s)
!f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds
or equivalentlyZΩ−p−(s)f(s)ds =
ZΩv
Ã1−
Xi
qi(s)
!f(s)ds−
ZΩ\Ω−
pI(s)
Ã1−
Xi
qi(s)
!f(s)ds (31)
Then we prove the following:
Lemma 2 A price function p−(s) as defined above exists if and only if it satisfies the fol-
lowing equationZΩ−p−(s)f(s)ds =
ZΩv +H(q, v)f(s)ds−
ZΩ\Ω−
pI(s)f(s)ds. (32)
where the function H(q, v) defines the seller’s payoff at the optimum (see the proof of Propo-
sition 4).
Proof of Lemma 2:
=⇒ Suppose there exists a price p−(s) which verifies the participation constraint of the
retail investors. This implies that the seller’s expected payoff at the optimum are given byZΩ−p−(s)f(s)ds+
ZΩ\Ω−
pI(s)f(s)ds =
ZΩv +H(q, v)f(s)ds
because of the uniform pricing rule and the fact that all the shares are always sold.
⇐= Consider the condition satisfied by the price to the institutional investors. Summing
over si and over i we getZΩ
Xi
pi(s)qi(s)f(s)ds =
ZΩ
Xi
½z(qi(s), v)−
1
n
∙Z si
sz2(qi(esi, s−i), v(esi, s−i))desi¸¾ f(s)ds
using the fact that institutional investors all pay the same price and adding −RΩ pI(s)f(s)ds
andRΩ v (1−
Pi qi(s)) f(s)ds to both sides yields the followingR
Ω p(s) (1−Pi qi(s)) f(s)ds−
RΩ v (1−
Pi qi(s)) f(s)ds =R
Ω p(s)f(s)ds−RΩv +H(q, v)f(s)ds.
32
If the uniform price exists then the RHS of the above equation is equal to zero which imme-
diately implies that the retail investors’ participation constraint, on the LHS of the equation,
is met. This ends the proof of Lemma 2
Notice that the right hand side of equation (32)does not depend the agents’ signals and
so the integral equation defined over p−(s) has many solutions. This proves that the seller
could find a pricing function where a uniform pricing rule for all investors may be applied at
the optimal mechanism. Note finally that the seller may choose among different pricing rules
that may be applied for retail investors when s belongs to Ω−.
Proof of Proposition 6
Let us define the following three sets:
N−(s) = i ∈ N | si ≤ si(v−i)
N+(s) = i ∈ N | si > si(v−i)
Ω+(s) = i ∈ N | si 6= s
and $+ = Card(Ω+(s)).
The solution of the relaxed problem is very similar to that of Proposition 4. Pointwise
maximization leads to the following Kuhn-Tucker conditions:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
(α− 1)v − δqi −α
n(s− si) + λi(s)− β(s) + γ(s) = 0, for all i
λi(s)qi(s) = 0
β(s)[1−Piqi(s)] = 0
γ(s)[1−Piqi(s)−K] = 0
(33)
where λi,β and γ are the Kuhn-Tucker multipliers associated to the three remaining con-
straints. Note that β and γ cannot be different from zero at the same time. If we consider
that γ = 0, then we get the same problem as in the case without budget constraints and
proves point (1), (2) and (4) of the Proposition.
33
For the third point, suppose thatP
i∈N+(s)
eqi(s) ≤ 1−K. So in this case β(s) = 0 and λi(s)for all i should be equal to zero otherwise some shares will remain unsold. In this case, retail
investors receive K shares and the optimal allocation rule for informed investors is the one
satisfying the following FOC:
(α− 1)v − δqi −α
n(s− si) + γ(s) = 0
for all i ∈ Ω+(s) with γ(s) solving the equation below:
γ(s) =δ(1−K)$+
+α
ns− α
n$+
Xj∈Ω+(s)
sj − (α− 1)v
By replacing γ(s) into the previous FOC, we finally obtain that the optimal quantity to
informed investors is equal to
eqKi (s) = 1
δ$+
⎧⎨⎩δ(1−K) + α
n($+ − 1)si −
α
n
Xj∈Ω+(s), j 6=i
sj
⎫⎬⎭
34
References
[1] Aggarwal R., Prabhala N. and M. Puri, 2002, “Institutional Allocations in Initial Public
Offerings: Empirical Evidence”, Journal of Finance, 57: 1421-42.
[2] Allen F. and G. Faulhaber, 1989, “Signaling by Underpricing in the IPO Market”, Jour-
nal of Financial Economics, 23 (2): 303-323.
[3] Beatty R. and J. Ritter, 1986, “Investment Banking, Reputation, and the Underpricing
of Initial Public Offerings”, Journal of Financial Economics, 15: 213-232.
[4] Bennouri M. and S. Falconieri, 2005, “Optimal Auctions with Asymmetrically Informed
Bidders”, forthcoming Economic Theory.
[5] Bennouri M. and S. Falconieri, 2004, “ Optimal Rationing in IPOs with Risk Averse
Institutional Investors”, mimeo, Tilburg University.
[6] Benveniste L.M. and P.A. Spindt, 1989, “How Investment Bankers Determine the Offer
Price and the Allocation of New Issues”, Journal of Financial Economics, 24: 213-323.
[7] Benveniste L.M. and W.J. Wilhelm, 1997, “Initial Public Offerings: Going by the Book”,
Journal of Applied Corporate Finance, 10 (1): 98-108.
[8] Benveniste, L.M. and W.J. Wilhelm, 1990, “A Comparative Analysis of IPO Proceeds
under Alternative Regulatory Environments”, Journal of Financial Economics, 28: 173-
207.
[9] Biais B. and A.M. Faugeron-Crouzet, 2002, “IPO auctions: English, Dutch, ... French
and Internet”, Journal of Financial Intermediation, 11: 9-36.
[10] Biais B., P. Bossaerts and J-C. Rochet, 2002, “An Optimal IPO Mechanism”, Review of
Economic Studies, 69: 117-146.
[11] Brisley, Neil, 2003, “A Comparative Analysis of IPO Proceeds under Alternative Regu-
latory Environments: A Comment”, Economics Bulletin, 7: 1-7
35
[12] Bulow J. and P. Klemperer, 1996, “Auctions vs. Negotiations”, American Economic
Review, 86: 180-94.
[13] Bulow J. and P. Klemperer, 2002, “Prices and the Winner’s Curse”, Rand Journal of
Economics, 33 (1): 1-21.
[14] Cornelli F. and D. Goldreich, 2001 “Bookbuilding and Strategic Allocation”, Journal of
Finance, 56: 2337-69.
[15] Cremer J. and R.P. McLean, 1988, “Full Extraction of the Surplus in Bayesian and
Dominant Strategy Auctions”, Econometrica, 56: 1247-57.
[16] Derrien, F and K.L. Womack, 2003 “Auctions vs Bookbuilding and the Control of Un-
derpricing in Hot IPO Markets”, Review of Financial Studies, 16 (1): 31-62.
[17] Dunford and J.T. Schwartz: Linear Operators: Part I: General Theory ; John Wiley and
Sons, New York, Third Edition
[18] Eso, P., 2005, “An Optimal Auction with Correlated Values and Risk Aversion”, forth-
coming, Journal of Economic Theory
[19] Ellul A. and M. Pagano, 2003, ”IPO Underpricing and After-Market Liquidity”,Working
Paper CSEF.
[20] Fudenberg D. and J. Tirole, 1991, “Game Theory”, MIT Press.
[21] Jenkinson T. and A. Ljungqvist, 2001, Going Public: The Theory and Evidence of How
Companies Raise Equity Finance, Oxford University Press.
[22] Kremer, I. and K. Nyborg, 2004, ”Divisible-Good Auctions: The Role of Allocation
Rules”, Rand Journal of Economics, 35 (1):147-159.
[23] Lyons R., 2003, “The Microstructure Approach to Exchange Rates”, MIT Press
[24] Maksimovic V. and P. Pichler, 2002 “Structuring the Initial Offering: Who to Sell To
and How to Do It”, mimeo Boston College.
36
[25] Malakhov, A., 2004, ”The Role of Uninformed Buyers in Optimal IPO Mechanisms”,
mimeo Kellog School of Management, Northwestern University.
[26] Maskin, E. and J.Riley, 1984, “Optimal Auctions with Risk Averse Buyers”, Economet-
rica, 52, 1473 - 1518
[27] Maskin E. and J. Riley, 1989, “Optimal Multi-Unit Auctions”, in The Economics of
Missing Markets, Information and Games, ed. by F.Hahn, Oxford: Oxford University
Press, 312-35.
[28] Myerson R., 1981, “Optimal Auction Design”, Mathematics of Operations Research, 6:
619-32.
[29] O’Hara M., 1995, “Market Microstructure”, Blackwell Publishers
[30] Ritter J. , 1987, “The Costs of Going Public’; Journal of Financial Economics, 19:
269-281.
[31] Ritter J. and I. Welch, 2002, “A Review of IPO Activity, Pricing, and Allocations”,
Journal of Finance, 57 (4): 1795-1828.
[32] Rock K, 1986, “Why New Issues Are Underpriced?”, Journal of Financial Economics,
15: 197-212.
37