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Hitotsubashi University Repository
Title
On the Asymptotic Distribution of the Transaction
Price in a Clock Model of a Multi-Unit, Oral,
Ascending-Price Auction within the Common-Value
Paradigm
Author(s) Hong, Han; Paarsch, Harry J.; Xu, Pai
Citation
Issue Date 2010-12
Type Technical Report
Text Version publisher
URL http://hdl.handle.net/10086/18880
Right
Center for Economic Institutions
Working Paper Series
Center for Economic Institutions
Working Paper Series
Institute of Economic Research
Hitotsubashi University 2-1 Naka, Kunitachi, Tokyo, 186-8603 JAPAN
http://cei.ier.hit-u.ac.jp/English/index.html Tel:+81-42-580-8405/Fax:+81-42-580-8333
No. 2010-8
“On the Asymptotic Distribution of the Transaction Price in a Clock Model of a
Multi-Unit, Oral, Ascending-Price Auction within the Common-Value Paradigm”
Han Hong, Harry J. Paarsch and Pai Xu
January 2011
On the Asymptotic Distribution of the Transaction Pricein a Clock Model of a Multi-Unit, Oral, Ascending-Price
Auction within the Common-Value Paradigm
Han HongDepartment of Economics, Stanford University
Harry J. Paarsch∗
Department of Economics, University of MelbournePai Xu
School of Economics and Finance, University of Hong Kong
20 December 2010
Abstract
Using a clock model of a multi-unit, oral, ascending-price auction, within the common-
value paradigm, we analyse the asymptotic behaviour of the transaction price as the number
of bidders gets large. We find that even though the transaction price is determined by a
(potentially small) fraction of losing drop-out bids, thatprice converges in probability to
theex anteunknown, true value. Subsequently, we derive the asymptotic distribution of the
transaction price. Finally, we apply our methods to data from an auction of taxi license plates
held in Shenzhen, China.
Key words: common value; information aggregation; multi-unit auctions; taxis.
JEL Classification: C20, D44, L1.
∗Corresponding author: Harry J Paarsch, Level 5, Economics &Commerce Building, Parkville, 3010, Victoria,Australia. Telephone:+61 3 8344 4655; Facsimile:+61 3 8344 6899. E-mail:[email protected].
1
1 Motivation and Introduction
During the past half century, economists have made considerable progress in understanding the the-
oretical structure of equilibrium strategic behaviour under market mechanisms, such as auctions;
see Krishna [2010] for a comprehensive presentation and evaluation of progress.
One analytic device, commonly used to describe bidder motivation at single-object auctions,
is a continuous random variable that represents individual-specific signals concerning the object’s
true, but unknown, value. This true, but unknown, value willbe revealed onlyafter the auction
has ended, when the winner has been determined and the transaction price paid. Regardless of the
winner, however, the value of the object is the same to all.
The conceptual experiment involves each potential bidder’s receiving a draw from a signal
distribution. Conditional on his draw, a bidder is then assumed to act purposefully, using the
information in his signal along with Bayes’ rule to maximize either the expected profit or the
expected utility of profit from winning the auction. Anotherfrequently-made assumption is that
the signal draws of bidders are independent and that the bidders areex antesymmetric—their
draws coming from the same distribution of signals. This framework is often referred to as the
symmetric common-value paradigm(symmetric CVP).
Under these assumptions, a researcher can then focus on a representative agent’s decision rule
when characterizing equilibrium behaviour. Wilson [1977]invented this framework to illustrate
that the winner’s curse could not obtain, in equilibrium, among rational bidders. He also demon-
strated that, when the number of biddersn is large (tends to infinity), the winning bid at first-price,
sealed-bid auctions converges almost surely to the true, but unknown, value of the object. In other
words, the auction format and pricing rule play an importantrole in aggregating the disparate, indi-
vidual pieces of information held by the bidders. Milgrom [1979] subsequently provided a precise
characterization of the structure the signal distributionmust possess in order for this convergence
property to hold; Pesendorfer and Swinkels [1997] have referred to this asfull information aggre-
gation.
When several, sayk, units of a good are simultaneously for sale, at least two important ques-
2
tions arise: specifically, who will be the winning bidders and what price(s) will those winners pay?
Weber [1983] has described a number of different multi-unit auction formats as well as pricing
rules under those formats. For example, Milgrom [1981] developed a natural generalization of the
Wilson [1977] model. In Milgrom’s model, each bidder submits a price and the auctioneer then
aggregates these demands, allocating the units to those bidders with the highestk submitted bids.
The winners then pay a uniform price—specifically, the highest rejected bid.
Pesendorfer and Swinkels [1997] have built on this researchby investigating a sequence of
auctionsAr in which bothnr andkr increase. They demonstrated that a necessary and sufficient
condition for full information aggregation is thatkr → ∞ and (nr − kr) → ∞, a condition they
referred to asdouble largeness. Under this condition, non-negligible supply can be a substitute for
the strong signal structure required in Wilson [1977] as well as Milgrom [1979, 1981]. Kremer
[2002] has investigated this further.
While it is heartening to know there are conditions under which transaction prices will converge
in probability to the true, but unknown, values of objects for sale, the rate at which these prices
converge is also of interest. In particular, Hong and Shum [2004] asked the question “How large
must n be to be large enough?” and then investigated the rates of information aggregation in
common-value environments. Knowing the conditions under which the transaction price provides a
potentially useful estimate of the object’s unknown value is important to understanding the process
some refer to asprice discoverybecause, in practice, neither the number of bidders nor the number
of units for sale at an auction ever really gets to infinity.
Of course, the pricing rule investigated in Wilson [1977] and Milgrom [1979, 1981] as well
as Pesendorfer and Swinkels [1997, 2000] is not the only pricing rule that could be used under a
sealed-bid format. For example, another pricing rule wouldinvolve allocating thek units to those
bidders who tendered the highestk bids, but each winner would then pay what he bid for the unit(s)
he won. In general, at multi-unit auctions, different auction formats and different pricing rules in-
duce different equilibrium behaviour and, thus, translate into different transaction prices as well
as potentially different expected revenues for sellers. Hence, as Jackson and Kremer [2004, 2006]
3
have emphasized, understanding the effects of auction formats and pricing rules has important prac-
tical relevance. Even small changes can have effects, as has been illustrated by Mezzetti and Tsetlin
[2008, 2009].
In a companion paper to Milgrom and Weber [1982], which was published nearly two decades
later, Milgrom and Weber [2000] proposed an interesting pricing rule for multi-unit, oral, ascending-
price auctions. The model considered by Milgrom and Weber isthe multi-unit variant of the clock
model introduced by Milgrom and Weber [1982] in order to investigate behaviour at single-object,
oral, ascending-price (often referred to asEnglish) auctions. In the multi-unit model, bidders are
assumed to demand at most one unit of the good for sale; Milgrom [2004] has referred to this as
singleton demand. The current price for all units on sale rises continuously according to some
device, such as a clock. As the price rises, the drop-out prices of losing participants are recorded
when they exit the auction. The transaction price is the drop-out price of the last participant to exit
the auction. Each of the remainingk participants is then allocated one unit at the transaction price.
One attractive feature of oral, ascending-price auctionsvis-a-vissealed-bid ones, regardless
of the pricing rule, is the scope for information release at oral, ascending-price auctions. This is
particularly important in informational environments with substantial common-value components.
In such environments, by observing the actions of his competitors, a bidder can augment the in-
formation contained in his signal and, thus, may be able to reduce the uncertainty concerning the
unknown value of the object for sale. Other things being equal, this reduction in uncertainty can
induce participants to bid more aggressively than under sealed-bid formats, which means the rev-
enues the seller can expect to garner can increase. The greater is the linkage between a bidder’s
information and what he perceives others will bid, the higher the bidding; Milgrom and Weber
[1982] have referred to this as thelinkage principle. In models of single-object auctions, they used
it to rank the revenues a seller can expect to garner under thedifferent auction formats and pricing
rules. Specifically, in a theoretical model with one object for sale as well as risk-neutral potential
buyers who have affiliated signals from the same marginal distribution, Milgrom and Weber [1982]
demonstrated that the English auction format yields, on average, more revenue than first-price auc-
4
tions, such as the oral, descending-price (Dutch) format orthe first-price, sealed-bid format. One
can deduce from the structure of the proof in Milgrom and Weber [1982] that the same linkage
principle applies to the generalized Milgrom–Weber auction we study below.
Our paper is in six additional sections. In the next, we use the Milgrom–Weber clock model to
develop a theoretical framework within which to investigate the stochastic behaviour of the transac-
tion price at a multi-unit, oral, ascending-price auction within the common-value paradigm, while
in section 3, we demonstrate that the transaction price converges in probability to theex anteun-
known, true value as the number of biddersnand the number of unitsk get large in the Pesendorfer–
Swinkels sense. In section 4, we characterize the asymptotic distribution of the transaction price
when both the number of bidders and the number of units get large. We also demonstrate that the
asymptotic variance of the transaction price under the Milgrom–Weber pricing rule is less than
that under the pricing rule used by Pesendorfer and Swinkels. Thus, if the transaction prices un-
der different auction formats and pricing rules are viewed as statistical estimators of the true, but
unknown, value of the units for sale, then the Milgrom–Weberrule is a more efficient estimator of
the unknown value than the first-price, sealed-bid rule because more information is released under
the Milgrom–Weber rule than under sealed-bid ones. Note, however, that in our model, when the
number of bidders is large, no difference exists between the average transaction prices underthe
two auction formats and pricing rules because they convergeto the true value. Moreover, affiliation
in signals can only reduce the variance of the transaction price under the Milgrom–Weber pricing
rule. Thus, by working within the symmetric CVP, we provide anupper bound concerning the im-
provement in variance reduction that can obtain under the Milgrom–Weber pricing rule. In section
5, we derive the likelihood function of observed drop-out prices, while in section 6, we apply our
methods to data from an auction of taxi license plates held inShenzhen, China; our empirical work
provides an estimate of the upper bound concerning the decrease in the variances of transaction
prices when using the Milgrom–Weber pricing rule. In the final section, we summarize and con-
clude. Any details too cumbersome to be included in the text of the paper have been collected in
the appendix at the end of the paper.
5
2 Theoretical Model
Consider an oral, ascending-price auction at whichk units are for sale to a total ofn bidders, each
of whom wants at most one unit. Focus on the Milgrom and Weber [2000] pricing rule described
in the introduction. Assume that each bidder draws an independently- and identically-distributed
signalX, conditional on the true, but unknown, valuev0. Denote the cumulative distribution and
probability density functions ofX, conditional onv, by FX|V(x|v) and fX|V(x|v), respectively. Denote
by fV(v) the prior distribution ofV, the unknown value.
Consider the vector of signals (X1,X2, . . . ,Xn), a random sample ofn draws fromFX|V(x|v0).
Because this environment is symmetric, without loss of generality, focus below on bidder 1. Denote
by Yi the i th ordered signal of the opponents of bidder 1, so
Y1 ≥ Y2 ≥ · · · ≥ Yn−1.
Denote byZi the i th order statistic for all of theXis, so
Z1 ≥ Z2 ≥ · · · ≥ Zn.
The auction proceeds in roundsm = n,n − 1, . . . , k + 1. In roundm, m bidders continue to
participate in the auction. The auction ends in round (k + 1) when the (k + 1)st bidder exits the
auction. With no loss of generality, suppose that bidders are ordered in the reverse order of exit
from the auction.
LetΩm denote the information that has already been revealed in roundmby all the bidders who
have already left the auction. Hence,Ωm equalszn, zn−1, . . . , zm+1, whereΩn is the empty set∅.
According to Milgrom and Weber [2000], the symmetric equilibrium bidding rule in roundm can
be written as
βm(x) = E [V|X1 = Yk = · · · = Ym−1 = x,Ωm] (1)
whereE denotes the expectation operator. Here,Yk, . . . ,Ym−1 denote thekth through (m− 1)st order
6
statistics among the bidders who remain competing with bidder 1. On the other hand, the order
statistics in the eventΩm denote the order statistics forall the bidders who have exitted the auction.
For completeness, we describe below our reasoning behind a characterization of the equilibrium;
in their paper, Milgrom and Weber [2000] presumably omittedan argument like this because they
found it obvious.
At price p, bidder 1 is concerned with the event thatYk, . . . ,Ym−1 all drop-out simultaneously at
β−1m (p), whereβ−1
m (p) is the inverse bid function. In this event, bidder 1 will be one of the winners
of the auction, together with his remaining (k− 1) competitors. Bidder 1 should stay in the auction
at price levelp if and only if
E[
V∣
∣
∣X1 = x,Yk = · · · = Ym−1 = β−1m (p),Ωm
]
> p.
In equilibrium, p = βm(x), so x = β−1m (p)—the price level at which bidder 1 should exit—should
satisfy the relation that
p = E[
V∣
∣
∣X1 = x,Yk = · · · = Ym−1 = β−1m (p),Ωm
]
. (2)
Hence, the functional form of the bid function.
The winning price corresponds to the bid submitted by the bidder with the (k+1)st order statistic
of the signals during round (k+ 1). Hence,
p = E [V|Zk = Zk+1 = zk+1,Ωk+1] . (3)
3 Limiting Information in the Transaction Price
In this section, and the next, we have two goals: first, to study the convergence rate of the trans-
action price ˆp to the unknown true common valuev0; and, second, to characterize the limiting
distribution of the transaction price ˆp. In both of these endeavours, we assume thatn gets large,
7
tends to infinity.
In this regard, we make the following assumption concerningk, the number of units for sale
relative ton, the number of bidders at the auction.
Assumption 1 [(n− k)/n] → τ, whereτ is strictly between0 and1.
In words, the proportion of demand met has a stable limit as the numbers of bidders gets large.
Were this not the case, then the transaction price would not have a stable limit, without some po-
tentially unrealistic assumptions concerningfV(v), as was pointed out by Pesendorfer and Swinkels
[1997].
With regard to our goals, we proceed in two steps. In the first,we define ˆv, the maximum-
likelihood estimator (MLE) ofv0, based on the unobserved (to the researcher, but known to the
participants) order statisticszk+1, . . . , zn, and then we investigate the rate at which ˆv converges tov0.
In the second, we investigate the rate at which ˆp converges to ˆv. In the next section, we demonstrate
formally that the rate of convergence of the price ˆp to the true common valuev0 will be driven
(dominated) by the convergence rate of ˆv to v0. In other words, ( ˆp−v0) is op
(
v− v0)
. Therefore, to
understand the rate of information aggregation, it suffices to focus on how ˆv approachesv0 as the
“sample size”n gets large.
Under our assumptions, the MLE ˆv is defined as
v = argmaxv
log
[(
nk
)
Ln(zk+1, . . . , zn|v)
]
where the joint likelihood function of all the signals revealed under the Milgrom–Weber auction is
proportional to
Ln(zk+1, . . . , zn|v) =[
1− FX|V(zk+1|v)]k fX|V(zk+1|v) fX|V(zk+2|v) · · · fX|V(zn|v). (4)
Here, the term[
1− FX|V(zk+1|v)]k captures the fact that only limited information is known concern-
ing the signal values of thek winners—specifically, their signals are greater thanzk+1. Also,(
nk
)
8
captures the fact that there are many ways in which thek highest order statistics of signals could
exceedzk+1. Equation (4) is the joint likelihood of the lowest (n− k) order statistics—those from
zk+1 to zn.
3.1 Convergence of Price to the True Value
Given equation (4), the average log-likelihood will be a function of the lowest (n−k) order statistics.
A general function of order statistics can be difficult to analyse because of the potentially complex
correlation structure among order statistics. One possibility is to appeal to the theory ofL-statistics
to investigate the convergence properties of functions of order statistics. Fortunately, this particular
average log-likelihood function is more tractable than anL-statistic because it can be rewritten as
a function of the entire sample and the sampleτth quantile. Specifically,
Qn (v) =1n
logLn(zk+1, . . . , zn|v)
=kn
log(
1− FX|V[
F−1n (τ) |v
])
+1n
n∑
i=1
log fX|V (Xi |v) 1[
Xi ≤ F−1n (τ)
]
whereFn (·) denotes the empirical distribution function
Fn (x) =1n
n∑
i=1
1 (Xi ≤ x) .
Now, under the assumptions made formal below, the sample percentileF−1n (τ) converges in prob-
ability to the true population quantileF−1X|V
(
τ|v0)
, by a uniform law of large numbers, soQn (v)
converges uniformly in the parameter space ofv to a deterministic function
Q0(v) = (1− τ) log(
1− FX|V[F−1X|V(τ|v0)|v]
)
+
∫ F−1X|V(τ|v0)
−∞fX|V(x|v0) log fX|V(x|v) dx
where the superscript 0 is used to denote the dependence onv0: Q0 (v) ≡ Q(
v0, v)
.
In order for information to aggregate fully,Q0 (v), as a function ofv, must be uniquely maxi-
9
mized atv equalv0. As in the case of full-sample likelihood function, this canbe verified using
Jensen’s inequality. Thus, for anyv not equal tov0, Q0 (v) ≤ Q0(
v0)
. This can be shown by taking
the sum of the following two inequalities. First, by Jensen’s inequality,
∫ F−1X|V(τ|v0)
−∞ fX|V(x|v0) log fX|V(x|v) dx−∫ F−1
X|V(τ|v0)
−∞ fX|V(x|v0) log fX|V(x|v0) dx
≤ τ[
log∫ F−1
X|V(τ|v0)
−∞ fX|V(x|v) dx− logτ]
.
Second, it is easy to see that
(1− τ) log(
1− FX|V[
F−1X|V
(
τ|v0)
|v])
+ τ logFX|V[
F−1X|V
(
τ|v0)
|v]
≤ (1− τ) log(1− τ) + τ logτ
because the left-hand side, considered as a function ofFX|V[
F−1X|V
(
τ|v0)
|v]
, is maximized atτ.
Assumption 2 For v , v0, either F−1X|V (τ|v) , F−1
X|V
(
τ|v0)
or, with positive probability, X≤
F−1X|V
(
τ|v0)
under v0, fX|V (x|v) , fX|V(
x|v0)
.
This assumption mirrors a standard full-sample identification condition for likelihood analysis.
While the monotone likelihood-ratio condition used by Milgrom and Weber [1982] is required to
derive the equilibrium bidding strategy, conditional on the form of the equilibrium bidding strategy,
it is not strictly necessary for full information aggregation to hold.
The first inequality will be strict under the first condition in Assumption 2. Likewise for the
second inequality under the second condition in Assumption2. Thus, we have demonstrated that
Q0 (v) is globally and uniquely maximized atv0 provided the valuev identifiesthe signal distribu-
tion fX|V (x|v) in the sense of Assumption 2, which is stronger than the usualfull-sample identifica-
tion condition wheneverτ < 1. The usual Jensen’s inequality argument for full-sample likelihood
function is just a special case of the above whenτ is one.
10
Now, examine the following first-order condition atv0:
∂Q0(v)∂v
∣
∣
∣
∣
∣
∣
v=v0
= −
(1− τ)1− FX|V[F−1
X|V(τ|v0)|v]
∂FX|V[
F−1X|V(τ|v0)|v
]
∂v
∣
∣
∣
∣
∣
∣
v=v0
+
∫ F−1X|V(τ|v0)
−∞
∂ fX|V(x|v0)
∂vdx
= −
∂FX|V[
F−1X|V(τ|v0)|v
]
∂v
∣
∣
∣
∣
∣
∣
v=v0
+
∫ F−1X|V(τ|v0)
−∞
∂ fX|V(x|v0)
∂vdx
= 0.
Therefore, subject to the regularity conditions, which areoutlined completely in the next section,
v is a consistent estimator ofv0. To wit, (v− v0)p→ 0.
Given thatQ0 (v) is a properly-defined averaged log-likelihood function that depends linearly
on the observed sample up to a given sample quantile and that the central sample quantiles are√
n-consistent as well as distributed asymptotically normal,the information equality then holds for
v, and is related to the asymptotic variance of ˆv. Given the form ofQ0 (v), the expected Hessian is
∂2
∂v2 Q0(v)
∣
∣
∣
∣
∣
∣
v=v0
where
∂2
∂v2Q0(v) = − ∂
2
∂v2FX|V
[
F−1X|V
(
τ|v0)
|v]
− (5)
11− τ
(
∂
∂vFX|V
[
F−1X|V
(
τ|v0)
|v]
)2
+
∫ τ
0
∂2
∂v2log fX|V
[
F−1X|V
(
u|v0)
|v]
du.
3.2 Information-Matrix Equality
In full-sample likelihood models, the asymptotic varianceof the maximum-likelihood estimator
is usually calculated using an information-matrix equality. Here, we show that an analogous
information-matrix equality also holds for thepartial sampleinformation model that we consider,
which we shall use to characterize the amount of limiting information contained in the price as an
estimate of the true value.
One approach to calculating the information matrix equality is to view the limiting first-order
11
condition atv0 as an identity, and then totally differentiate it with respect tov0. Specifically, because
∂
∂vQ0 (v)
∣
∣
∣
∣
∣
∣
v=v0
= 0,
then∂
∂v0
[
∂
∂vQ0 (v)
]
∣
∣
∣
∣
∣
∣
v=v0
= 0.
This can be written as
∂2
∂v2Q0(v)∣
∣
∣
∣
∣
∣
v=v0
+∂
∂u
[
∂
∂vQu (v)
]
∣
∣
∣
∣
∣
∣
u=v=v0
= 0. (6)
In the next section, the second term on the left-hand side, which is the negative of the Hessian
given in equation (5), will be shown to equal the asymptotic variance of the score function. The
following provides a direct calculation of the second term in equation (6), which independently
verifies equation (6) and facilitates the comparison with the variance of the score function in the
next section.
To compute this term, we need to calculate
∂
∂vF−1
X|V (τ|v)
as well as∂
∂vFX|V
[
F−1X|V (τ|v) |v0
]
∣
∣
∣
∣
∣
∣
v=v0
=∂
∂vF−1
X|V (τ|v) fX|V[
F−1X|V
(
τ|v0)
|v]
.
Both can be found by totally differentiating the identity
∫ F−1X|V(τ|v)
−∞fX|V (x|v) dx = τ,
12
which leads to
∂
∂vF−1
X|V (τ|v) = −∂∂vFX|V
[
F−1X|V
(
τ|v0)
|v]
fX|V[
F−1X|V
(
τ|v0) |v
] ,∂
∂vFX|V
[
F−1X|V (τ|v) |v0
]
∣
∣
∣
∣
∣
∣
v=v0
= − ∂∂v
FX|V[
F−1X|V
(
τ|v0)
|v]
∣
∣
∣
∣
∣
∣
v=v0
.
Using these relations,
∂
∂u
[
∂
∂vQu (v)
]
∣
∣
∣
∣
∣
∣
u=v=v0
=1
1−τ
(
∂∂vFX|V
(
F−1X|V
(
τ|v0)
|v])2+
∫ F−1X|V(τ|v0)
−∞
[
∂∂v log fX|V (x|v)
]2fX|V
(
x|v0)
dx∣∣
∣
∣
∣
∣
v=v0
.
The next section formally demonstrates that the log-likelihood function of the partially observed
sample in our model has a similar statistical behaviour to the usual full-sample log-likelihood func-
tion, so√
n(
v− v0)
will converge in distribution to a normal random variable whose asymptotic
variance is the inverse of either∂∂u
[
∂∂vQu (v)
]
∣
∣
∣
∣
∣
∣
u=v=v0
or equivalently ∂2
∂v2 Q0(v)
∣
∣
∣
∣
∣
∣
v=v0
. We now need to
show that√
n (p− v) is op (1) because, then, these will also represent the asymptotic variance of√
n(
p− v0)
.
For this purpose, we employ Bayesian asymptotic analysis. First, note that
p = βk+1 (zk+1) =∫
vfZ|V(Zk = Zk+1 = zk+1,Ωk+1|v) fV(v)
∫
fZ|V(Zk = Zk+1 = zk+1,Ωk+1|u) fV(u) dudv
where the likelihood of the conditioning event in the bid function is proportional to
fZ|V(Zk = Zk+1 = zk+1,Ωk+1|v) =[
1− FX|V(zk+1|v)]k−1 fX|V(zk+1|v)2 fX|V(zk+2|v) · · · fX|V(zn|v). (7)
Recall the definition in equation (4)
fV|Z(Zk = Zk+1 = zk+1,Ωk+1|v) = Ln(zk+1, . . . , zn|v)fX|V(zk+1|v)
[1 − FX|V(zk+1|v)]
= exp[nQn(v)]fX|V(zk+1|v)
[1 − FX|V(zk+1|v)],
13
which we can write, using a change of variables,
√n(p− v) =
∫
hexp[n
(
Qn(v+ h/√
n) − Qn(v))
] fV(v+ h/√
n) fX|V(zk+1|v+h/√
n)[1−FX|V(zk+1|v+h/
√n)]
∫
exp[n(
Qn(v+ u/√
n) − Qn(v))
] fV(v+ u/√
n) fX|V(zk+1|v+u/√
n)[1−FX|V(zk+1|v+u/
√n)]
dudh.
It is demonstrated in the next section that the above renormalized posterior distribution is
asymptotically normal. Intuitively,√
n(p−v)p→ 0 obtains because the mean of the above renormal-
ized posterior distribution is asymptotically zero. It is also clear that the single-unit English-auction
model investigated by Milgrom and Weber [1982] is a special case of this result—whenτ is one,
which corresponds to the conventional full-sample maximum-likelihood analysis and Bayesian
posterior distribution. At a typical English auction, where τ is one, the only difference from full-
sample maximum-likelihood analysis is that the maximum order statistic is unobserved. However,
a single order statistic is asymptotically negligible. Likewise, the conditioning event in the bid
function in equation (7) differs from the corresponding partial-sample likelihood in equation (4)
only by a single order statistic and the difference is asymptotically negligible.
3.3 Simple Example
Consider the following example, which can be solved in closed-form. Suppose that the conditional
distribution ofX is exponential, having meanv, so
fX|V (x|v) =1v
exp(−x/v) for x ≥ 0, v > 0.
The posterior distribution needed to compute the bid function in equation (1) is proportional to
(
1v
)n−m
exp
−1v
m+1∑
j=n
zj
(
1v
)m−k+1
exp(−z/v)m−k+1 exp(−z/v)k−1 fV(v) =
(
1v
)n−k+1
exp
−1v
m+1∑
j=n
zj +mz
fV(v)
14
SupposefV (v) is a diffuse prior.1 In this case, the above posterior distribution is then an inverse
gamma distribution having parameters (n− k+ 1) and(
∑m+1j=n zj +mz
)
, which has mean
E [V|X1 = Yk = · · · = Ym−1 = z,Ωm] =
∑m+1j=n zj +mz
n− k,
which is also the bid function at roundm. Therefore, the transaction price is given by the bid
function withm equal (k+ 1) andzequalzk+1:
p =
∑k+2j=n zj + (k+ 1)zk+1
n− k.
To see why ˆp converges to the truev0, note that in this example,Zk+1p→ F−1
X|V (τ) which equals
−v0 log(1− τ). Also, by invoking a law of large numbers,
∑k+2j=n zj
n
p→ v0
[
log(1− τ)1− τ + τ
]
.
Therefore,
pp→ v01
τ
[(
log(1− τ) (1− τ) + τ)] − v0 log(1− τ) 1− ττ= v0.
The maximum-likelihood estimator ˆv, which is the mode of the posterior distribution, is
v =
∑k+2j=n zj + (k+ 1)zk+1
n− k+ 2.
Hence,
v =n− k
n− k+ 2p.
It can then be verified that√
n (p− v)p→ 0.
1We could also use a Pareto prior, but this would clutter the calculations considerably.
15
4 Asymptotic Distribution of Transaction Price
In this section, we provide formal conditions to justify theclaims made in the previous section. Our
analysis is broken into two parts: in the first, we derive the asymptotic distribution of√
n(
v− v0)
,
while in the second we show that√
n (p− v) is op (1). As Newey and McFadden [1994] as well as
Chernozhukov and Hong [2003] have pointed out, both parts depend on the stochastic equiconti-
nuity properties of the sample-averaged log-likelihood functionQn (v).
To begin, we state assumptions sufficient to the task. Instead of striving for the weakest possible
set of assumptions, we are content with potentially overly-strong sufficient conditions that illustrate
the main results. Note, too, that in theoretical models of auctions, the monotone likelihood-ratio
condition is typically imposed, which restricts how weak the conditions for equicontinuity can be.
Assumption 3 The support of the prior distribution fV (v) is a compact closed interval, and the
true common value v0 is contained in the interior of that interval.
Assumption 4 The support of fX|V (x|v) is independent of v and bounded, whilelog fX|V (x|v) is
uniformly bounded, having bounded continuous third derivatives in both arguments on its support.
These two assumptions are regularity conditions required to demonstrate uniform convergence
and stochastic equicontinuity.
Theorem 1 Under Assumptions 1 to 4, if fV (v) is continuous at v0 with a finite mean, then
√n(
v− v0) d−→ N
0,Σ (τ) =∂
∂u
[
∂
∂vQu (v)
]∣
∣
∣
∣
∣
∣
−1
u=v=v0
,
and√
n (p− v)p−→ 0,
so√
n(
p− v0) d−→ N [0,Σ (τ)] .
16
Remark 1: In the Pesendorfer–Swinkels model, under the same assumption (k/n)→ (1− τ), only
the signal of a single last-losing bidder is revealed, instead of the signals of all the losing bidders.
Therefore, intuitively, the transaction price in the Pesendorfer–Swinkels model should aggregate
less information than that in the Milgrom–Weber model. In fact, this turns out to be true. While
the prices in both the Pesendorfer–Swinkels and the Milgrom–Weber models converge tov0 at
rate√
n, the asymptotic variance of the Pesendorfer–Swinkels price is greater than the Milgrom–
Weber price. We demonstrate this result formally using the influence function representation of the
asymptotic variance. We note, first, from the proof of the theorem thatΣ (τ) equals Var[
ψ1 (X, τ)]−1,
where theinfluence functionψ1 (X, τ) is given by
ψ1 (X, τ) ≡ ∂∂v log fX|V
(
X|v0)
1[
X ≤ F−1X|V
(
τ|v0)]
−
E(
∂∂v log fX|V
(
X|v0)
1[
X ≤ F−1X|V
(
τ|v0)])
+
11−τ
(
∂∂vFX|V
[
F−1X|V (τ|v) |v
]) (
1[
X ≤ F−1X|V (τ|v) |v
]
− τ)
.
Next, we characterize the average log-likelihood functionas well as the score and influence func-
tions in the Pesendorfer–Swinkels model, and show that theyimply a variance larger thanΣ (τ).
The average log-likelihood of the Pesendorfer–Swinkels model, which depends only on a single
order statisticzk+1 = F−1n (τ), is given by
Qn (v) = kn log
[
1− FX|V (zk+1|v)]
+
(
1− kn
)
logFX|V (zk+1|v)
= (1− τ) log(
1− FX|V[
F−1n (τ) |v
])
+ τ logFX|V[
F−1n (τ) |v
]
.
Its corresponding score function is
∂
∂vQn (v) = −
(
1−τ1−FX|V[ F−1
n (τ)|v] −τ
FX|V[ F−1n (τ)|v]
)
∂∂vFX|V
[
F−1n (τ) |v
]
.
If we evaluate the first-order approximation of the score function with respect toF−1n (τ) as it
approachesF−1X|V
(
τ|v0)
at v = v0, and make use of the well-established asymptotic approximation
17
of the sample quantile
√n[
F−1n (τ) − F−1
X|V
(
τ|v0)]
= − 1√
n
n∑
i=1
1[
Xi ≤ F−1X|V
(
τ|v0)]
− τ
fX|V[
F−1X|V
(
τ|v0)
] + op (1) , (8)
then we find the following influence function representationfor the Pesendorfer–Swinkels score
function:
√n∂
∂vQn
(
v0)
=1√
n
n∑
i=1
ψ2 (Xi , τ) + op (1)
where
ψ2 (X, τ) ≡ 1τ (1− τ)
∂
∂vFX|V
[
F−1X|V (τ|v) |v
] (
1[
X ≤ F−1X|V
(
τ|v0)]
− τ)
.
Letting p to denote the transaction price in the Pesendorfer–Swinkels auction model, we have
√n(
p− v0) d−→ N
(
0,Var[
ψ2 (X, τ)]−1
)
.
In order to show that Var[
ψ2 (X, τ)] ≤ Var
[
ψ1 (X, τ)]
, we compute
ψ1 (X, τ) − ψ2 (X, τ) = ∂∂v log fX|V
(
X|v0)
1[
X ≤ F−1X|V
(
τ|v0)]
−
E(
∂∂v log fX|V
(
X|v0)
1[
X ≤ F−1X|V
(
τ|v0)])
−
1τ
(
∂∂vFX|V
[
F−1X|V
(
τ|v0)]) (
1[
X ≤ F−1X|V
(
τ|v0)]
− τ)
.
We can then easily verify that
Cov[
ψ1 (X, τ) − ψ2 (X, τ) , ψ2 (X, τ)]
= 0.
18
Hence,
Var[
ψ1 (X, τ)] ≥ Var
[
ψ2 (X, τ)]
.
Remark 2: Above, we have indexed the asymptotic variance byτ, the proportion of losing bidders.
Intuitively, the larger the fraction of losing bidders, themore information revealed at the auction.
Therefore, we expectΣ (τ) to be a monotonically decreasing function ofτ, in a matrix sense. In
other words, for any 0< τ1 ≤ τ2 ≤ 1,
Σ (τ1) ≥ Σ (τ2) .
This also turns out to be true. To wit, forτ1 > τ2, Var[
ψ1 (X, τ1)] ≤ Var
[
ψ1 (X, τ2)]
. This will, in
turn, follow from
Cov[
ψ1 (X, τ1) − ψ1 (X, τ2) , ψ1 (X, τ1)]
= 0. (9)
Verifying equation (9) is tedious, but straightforward: itdepends on the following two key rela-
tions. First,
E(
∂
∂vlog fX|V
(
X|v0)
1[
X ≤ F−1X|V
(
τ|v0)]
)
=∂
∂vFX|V
[
F−1X|V (τ|v) |v0
]
and, second, that, forτ1 > τ2,
1[
X ≤ F−1X|V (τ1|v) |v0
]
1[
X ≤ F−1X|V (τ2|v) |v0
]
= 1[
X ≤ F−1X|V (τ2|v) |v0
]
.
Hence, under the assumptions made above, especially the common support Assumption 4, for
0 < τ ≤ 1, the larger isτ, the more information is aggregated in the Milgrom–Weber model, in
the sense of having a smaller variance despite that the rate of convergence stays the same. It can
also be shown that this conclusion continues to hold withoutthe support Assumption 4. When
19
the upper support is increasing inv, while the condition still holds, the rate of convergence can
improve beyond√
n whenτ equals one. On the other hand, if the lower support is also increasing
in v, then it is possible that the convergence rate is faster than√
n even whenτ is zero. In this case,
while there will be no information loss whenτ increases above zero, there may be no additional
asymptotic information either untilτ becomes one.
This desirable monotonicity property of information aggregation in the Milgrom–Weber model
is in contrast to the Pesendorfer–Swinkels model. The amount of information aggregated asymp-
totically in the price of the Pesendorfer–Swinkels model isnot monotonic inτ. For example, when
fX|V(
x|v0)
is uniform inX, the worstτ for information aggregation is one-half in the Pesendorfer–
Swinkels model, because this involves the worst balance between the winner’s curse and the loser’s
curse. In general, the optimalτ in the Pesendorfer–Swinkels model obviously depends on theshape
of this conditional density. Intuitively, in the Pesendorfer–Swinkels model, a differentτ selects a
different information set, while in the Milgrom–Weber model, a largerτ always selects a larger
information set.
5 Deriving Likelihood Function of Observed Drop-Out Prices
In section 2, we derived the bid function of a representativebidder as well as characterized the
transaction price; see equations (1) and (3). In sections 3 and 4, we then demonstrated that the
transaction price converged in probability to the true, unknown valuev0 and derived its asymptotic
distribution. To provide a framework within which to conduct our empirical analysis in section 6,
in this section, we derive the likelihood function of the biddata observed by an econometrician.
We highlight the fact that the sampling variability of the econometrician’s estimate of the true, but
unknown valuev0 will depend on nuisance parameters unknown to the econometrician.
We first introduce some additional notation. We denote by ˆpj the j th drop-out price, soj =
1,2, . . . ,n − k. For example, in our empirical application, we haven equal forty bidders andk
equal twenty units, so there are twenty drop-out prices, thelast being the transaction price, which
20
we denoted above by ˆp, but now denote as ˆpn−k. Thus, our observables are ( ˆp1, p2, . . . , pn−k−1, pn−k).
Now, from equation (2), we can recover the signal consistentwith the first bidder’s drop-out price—
viz.,
zn = β−1n (p1).
Likewise, for each ofj = 2,3, . . . ,n−k, we can recursively recover ˜zj, the signals of the (n−k−1)
losing bidders, so
zn− j+1 = β−1n− j+1(pj;Ωn− j+1).
For thek bidders who win the auction, all we know is thatZ j exceedsβ−1n−(n−k−1)(pn−k;Ωk+1).
In the general case, the bid functionβm (x) in equation (1) takes the following form:
βm(x;Ωm) =∫
vfV(v)g (X1 = Yk = . . . = Ym−1 = x,Ωm|v)
∫
fV(u)g (X1 = Yk = . . . = Ym−1 = x,Ωm|u) dudv
where
g (X1 = Yk = . . . = Ym−1 = x,Ωm|v) =[
1− FX|V (x|v)]k−1 fX|V (x|v)m−k+1
m+1∏
j=n
fX|V(
zj |v)
.
If we assume thatfV(v) is diffuse and thatX given v is normal, having meanv and varianceσ2,
then we can write
g (X1 = Yk = . . . = Ym−1 = x,Ωm|v) =[
1− Φ(x− vσ
)]k−1 1σm−k+1
φ
(x− vσ
)m−k+1
1σn−m
m+1∏
j=n
φ
(zj − v
σ
)
.
To summarize, under the assumptions of normality as well as adiffuse prior,
βm(x;Ωm) =∫
v
[
1− Φ(
x−vσ
)]k−1.φ
(
x−vσ
)m−k+1 ∏m+1j=n φ
(
zj−vσ
)
∫ [
1− Φ(
x−uσ
)]k−1φ(
x−uσ
)m−k+1 ∏m+1j=n φ
(
zj−uσ
)
dudv.
Consider (˜zn, zn−1, . . . , zk+1), the vector of (n− k) signals consistent with the observed drop-out
21
prices as well as the transaction price. The joint likelihood function of all the signals consistent
with the drop-out prices revealed under the Milgrom–Weber auction is
Ln(zk+1, . . . , zn|v, θ) =[
1− FX|V(zk+1|v, θ)]k fX|V(zk+1|v, θ)
∣
∣
∣
∣
∣
∣
∂β−1n−k(pn−k)
∂pn−k
∣
∣
∣
∣
∣
∣
fX|V(zk+2|v, θ)∣
∣
∣
∣
∣
∣
∂β−1n−k−1(pn−k−1)
∂pn−k−1
∣
∣
∣
∣
∣
∣
· · · fX|V(zn|v, θ)∣
∣
∣
∣
∣
∣
∂β−11 (p1)
∂p1
∣
∣
∣
∣
∣
∣
(10)
Here,θ denotes a vector of unknown parameters, and captures the fact the probability density and
cumulative distribution functions of signals can depend onparameters known to the bidders, but
unknown to the econometrician.
The econometrician’s MLE ˜v is defined as
v = argmaxv
log
[(
nk
)
Ln(zk+1, . . . , zn|v, θ)]
whereθ denotes the MLE ofθ0. While knowing the true nuisance parameters inθ0 is unimportant
in demonstrating that the transaction price converges in probability to the true, unknown valuev0,
because the parameters contained inθ0 are of second-order importance, the nuisance parameters are
critical when calculating an estimate of the sampling variation in v, the econometrician’s estimate
of the true, but unknown valuev0.
6 Empirical Application
We have applied the methods described above to data from an auction of taxi license plates held in
Shenzhen, China in October 2007. At this auction, the municipal transportation bureau sold 2,000
additionalred taxi license plates. Red taxis are special in Shenzhen because they can operate both
inside and outside the Special Economic Zone (SEZ), unlikeyellow taxis which can operate only
inside the SEZ, andgreen taxis which can only operate outside the SEZ.
The city of Shenzhen had not issued any new license plates forred taxis since 1993. However,
rapid growth in Shenzhen’s population meant that patrons were experiencing a shortage of taxis,
22
leading to an increase in the number of illegally-operated taxis. In 2007, the per capita number
of taxis in Shenzhen was low when compared to other parts of China: only 10,305 taxis were
licensed in a city of 7.5 million permanent residents, about 13.74 taxis for every 10,000 residents.
The Ministry of Construction in China recommended that citiesshould have 21 taxis for every
10,000 residents.
Before the auction, the authorities reviewed the qualifications of all those who had applied to
participate at the auction. Potential bidders could be individual taxi companies or groups formed
by different companies. While fifty-one ‘firms’ apparently requested to participate, only forty
potential bidders were certified to participate at the auction. Thus,n was 40.
In written documentation, potential bidders were remindedto be aware of the risks involved.
For example, consider a translation of the text from one document:
Following this auction, more taxi license plates will be issued through auction or other
ways over the next four years. The number of taxis in Shenzhenwill reach about
20,000 by 2011. The issuance of a great number of license plates might have much
impact on the taxi industry.
Despite these warnings, representatives of taxi companiesin the city showed great interest in the
auction, perhaps because operating a taxi has been one of thehighest profit margins in the trans-
portation industry. Also, historically, taxis have provided a stable return against investment.
Before the auction, 53 out of 73 taxi companies in Shenzhen owned between 50 to 200 taxis
each. To wit, the majority of the city’s taxi companies were small- and medium-sized ones. Some
incumbent taxi drivers expressed concern that entry would erode profits. One was quoted in the
local newpaper (our translation) as saying that
Actually we are not earning much nowadays. If more taxis wereon the road, we would
have a hard time making ends meet.
In contrast, local residents supported the issuance of additional license plates. One was quoted
(again our translation from the local newspaper) as saying
23
The sooner new taxis hit the road the better. It’s too hard to hail a taxi during peak
hours and holidays.
This anecdotal evidence, along with casual observation, suggests to us that the value of a red-
taxi license plate in Shenzhen has a large common-value component. Before the auction, however,
this common value was unknown to potential bidders. Using whatever means at their disposal,
potential bidders formed estimates of the unknown common value which they then used during
bidding at the auction.
The auction in Shenzhen proceeded according to the rules described in Milgrom and Weber
[2000]. In written rules announced before the auction, the authorities informed potential bidders
that the 2,000 license plates on sale would be distributed evenly amongthe final twenty highest
bidders; each winner would be required to buy 100 license plates.
The auctioneer, Tian Tao, was a registered member of China’s auction industry association.
The reserve price was set at 150,000 yuan per license plate, but the price rose to 500,000 yuan
in fourth minute of bidding. During the auction, Tian reminded bidders repeatedly to be aware
of the risks involved. In fact, Tao took a break for ten minutes to allow the bidders “to cool their
enthusiasm.” We have translated one of his comments as “thisis one of the most intensive auctions
I’ve experienced in my career as an auctioneer.” At the closeof the auction, the price of a red-taxi
license plate was 542,500 yuan, around US$80,000.
In table 1, we present the prices called out during the auction along with the number of bidders
who exited the auction at those prices, while in figure 1 we depict the empirical survivor function
of prices. The prices in this table are in 10,000 yuan.
Zhang Hongzhi, a manager of Shenzhen Xilie Taxi Company, was reported in the newspaper
to have said that he “felt very excited after we won a bid.” Before his attending the auction, his
company had decided on 550,000 yuan as the highest they would pay for a red-taxi license plate.
To implement equation (4), we assumed thatX, conditional onv0, is distributed normally,
24
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
Drop−Out Price
Est
imat
ed S
urvi
vor
Fun
ctio
n
Figure 1: Estimated Survivor Function of Drop-Out Prices
having varianceσ2, so
FX|V(x|v0) =∫ x
−∞
1
σ√
2πexp
−
(
u− v0)2
σ2
du ≡ Φ
(
x− v0)
σ
with
fX|V(x|v0) =1
σ√
2πexp
−
(
x− v0)2
σ2
≡ 1σφ
(
x− v0)
σ
.
We also assumed thatfV(v) is a diffuse prior. In table 2, we present the MLEs ofv0 andσ as well
as their standard errors; the logarithm of the likelihood function for this empirical specification is
−55.98. Here, the units of the parameters estimates are in 10,000 yuan.
In order to understand the implications of these parameter estimates, we used these parameters
to simulate the differences between the prices in a Milgrom–Weber auction and a Pesendorfer–
Swinkels auction. A subset of these results are reported in table 3. Each entry in the table provides
the difference in the expected revenue between the Milgrom–Weber auction and the Pesendorfer–
Swinkels auction, measured in units of 10,000 yuan. In calculating table 3, we need three param-
25
eters: the prior mean and variance of the common value distribution as well as the variance of
the signal distribution conditional on the common value. Weused the estimate ofv0 to specify
the prior mean, and the estimate ofσ necessary to specify both the variance of the signal dis-
tribution and the prior variance of the value distribution.As predicted by the linkage principle
of Milgrom and Weber [1982], the Milgrom–Weber auction always generates an higher expected
revenue than the Pesendorfer–Swinkels auction. However, as table 3 illustrates, the difference in
the expected revenues is rather small when compared to both the selling price and the estimated
common value.
7 Summary and Conclusions
Using a clock model of a multi-unit, oral, ascending-price,auction, within the common-value
paradigm, under the Milgrom–Weber pricing rule, we have analysed the asymptotic behaviour of
the transaction price as the number of bidders and the numberof units get large. We have demon-
strated that even though the transaction price is determined by a (potentially small) fraction of
losing drop-out bids, that price converges almost surely totheex anteunknown, true value. Subse-
quently, we have demonstrated that the asymptotic distribution of the transaction price is Gaussian.
We also demonstrated that the asymptotic variance of the transaction price under the Milgrom–
Weber pricing rule is less than that under the pricing rule used by Pesendorfer and Swinkels. Thus,
if the transaction prices under different auction formats and pricing rules are viewed as statistical
estimators of the true, but unknown, value of the units for sale, then the Milgrom–Weber pricing
rule is a more efficient estimator of the unknown value than the first-price, sealed-bid rule because
more information is released under the Milgrom–Weber rule than under sealed-bid ones. Note,
however, that in our model, when the number of bidders is large, no difference exists between the
average transaction prices under the two auction formats and pricing rules because they converge
to the true value. Moreover, affiliation in signals can only reduce the variance of the transaction
price under the Milgrom–Weber pricing rule. Thus, by working within the symmetric CVP, we
26
have provided an upper bound concerning the improvement in variance reduction that obtains un-
der the Milgrom–Weber pricing rule. Finally, we applied ourmethods to data from an auction of
taxi license plates held in Shenzhen, China, finding that our estimate of the unknown, true value
was not significantly different from the transaction price at size 0.01, but is at size 0.05.
Acknowledgements
We should like to acknowledge Eugene Jeong for excellent research assistance. We also thank
the municipal transportation bureau in the city of Shenzhen, China for providing helpful advice
and useful information as well as the Shenzhen Auction Housewhich provided us the data from
the auction of taxi license plates. We are also grateful to Srihari Govindan, Kenneth L. Judd, and
Robert B. Wilson for providing us with helpful comments and useful suggestions on an earlier
draft of the paper. Paarsch gratefully acknowledges that much of the research for this paper was
completed while he was a visiting research scholar at the Center for Economic Institutions in the
Institute of Economic Research at Hitotsubashi University in Kunitachi, Japan and that subsequent
work on the paper was also completed while he was a visiting fellow at the Collegio Carlo Alberto
in Moncalieri, Italy.
Appendix
To reduce clutter in the text of the paper, in this appendix, we collect the proofs of the results
claimed in the text.
Proof of Main Theorem
The proof involves verifying two high-level conditions in Newey and McFadden [1994] as well
as Chernozhukov and Hong [2003]. The first condition deliversconsistency, while the second
delivers asymptotic normality of ˆv and the relation that√
n (p− v) is op (1). We first state these
27
conditions within the context of our notation.
Condition 1 For anyδ > 0, there exists anǫ > 0, such that
lim infn→∞
P∗
sup|v−v0|≥δ
[
Qn (v) − Qn
(
v0)]
≤ −ǫ
= 1.
Condition 2 There exists∆n
(
v0)
and J0 such that for v in an open neighbourhood of v0,
i. n[
Qn (v) − Qn
(
v0)]
=
(
v− v0)
∆n
(
v0)
− 12
(
v− v0)2 [
nJ0]
+ Rn (v),
ii. For any sequenceδn→ 0,
sup|v−v0|≤δn
|Rn (v) |1+ n|v− v0|2 = op(1).
iii. ∆n
(
v0)
/√
nd→ N
(
0,Ω0)
, where both J0 andΩ0 are positive definite.
Condition 1 is, in turn, implied by uniform convergence ofQn (v) to Q0 (v) and becauseQ0 (v)
is uniquely maximized atv0. The unique maximum ofQ0 (v) at v0 is a direct consequence of
the identification Assumption 2. To show that supv∈V |Qn (v) − Q0 (v) | is op (1), first note that the
individual terms in the summand of the second term consist ofthe product of logfX|V (Xi |v) and
1 (Xi ≤ ξ), whereξ equalsF−1n (τ). Given Assumption 4, the first is a type II function and the
second is a type I function defined in Andrews [1994]. Both satisfy Pollard’s entropy condition,
and are stable under multiplication. Hence,
supv,ξ
∣
∣
∣
∣
∣
1n
n∑
i=1
log fX|V (Xi |v) 1 (Xi ≤ ξ) − E[
log fX|V (Xi |v) 1 (Xi ≤ ξ)]
∣
∣
∣
∣
∣
= op (1) .
Next,E [
log fX|V (Xi |v) 1 (Xi ≤ ξ)]
is a Lipschitz function inξ and the Lipschitz constant in uniform
in v. Hence, given thatF−1n (τ)
p−→ F−1
X|V
(
τ|v0)
, we also have
supv
∣
∣
∣
∣
∣
E(
log fX|V (Xi |v) 1[
Xi ≤ F−1n (τ)
])
− E(
log fX|V (Xi |v) 1[
Xi ≤ F−1X|V
(
τ|v0)])
∣
∣
∣
∣
∣
= op (1) .
28
Therefore, the second term ofQn (v) converges uniformly inv to the second term ofQ0 (v). The
first term ofQn (v) is also a Lipschitz function ofF−1n (τ) with the Lipschitz constant being uniform
in v. Therefore, by the same argument, the first term ofQn (v) also converges uniformly inv to the
first term ofQ0 (v). Hence, Condition 1 holds.
The second condition is more involved than the first. We defineξ to beF−1n (τ) whereξ0 denotes
F−1X|V
(
τ|v0)
. We rewriteQn (v) asQn
(
v, ξ)
to emphasize its direct dependence onξ. Note that, while
Qn
(
v, ξ)
is differentiable inv, it is not in ξ, so arguments relying on stochastic continuity arguments
are required. The∆n
(
v0)
andJ0 elements in Condition 2 are given by
∆n
(
v0)
= n∂
∂vQn
(
v0, ξ0)
+ n∂2
∂v∂ξQ0
(
v0, ξ0) (
ξ − ξ0)
and
J0= − ∂
2
∂v2Q0
(
v0, ξ0)
,
respectively. We decomposeRn (v) into R1n (v) + R2
n (v) with
R1n (v) = n
[
∂
∂vQn
(
v0, ξ)
− ∂
∂vQn
(
v0, ξ0)
− ∂2
∂v∂ξQ0
(
v0, ξ0) (
ξ − ξ0)
]
(
v− v0)
and
R2n (v) = n
(
v− v0)2
[
∂2
∂v2Qn
(
v∗, ξ)
− J0
]
wherev∗ is a mean value betweenv0 andv. Because∂2
∂v2 Qn
(
v∗, ξ)
− J0 p−→ 0, it follows that
sup|v−v0|≤δn
|R2n (v) |
1+ n|v− v0|2 ≤|R2
n (v) |n|v− v0|2 = op(1).
29
Consider, next,R1n (v), and define ˆm(v, ξ) = ∂
∂vQn (v, ξ). By stochastic equicontinuity, it follows
from the entropy property of ˆm(v, ξ) that
√n[
m(
v0, ξ)
− m(
v0, ξ0)
−m(
v0, ξ)
+m(
v0, ξ0)]
= op (1) .
Note, too, by a second-order mean-value expansion ofm(
v0, ξ)
in ξ, that
√n
[
m(
v0, ξ)
−m(
v0, ξ0)
− ∂2
∂v∂ξQ0
(
v0, ξ0) (
ξ − ξ0)
]
= op (1) .
Therefore, we can write
√n
[
m(
v0, ξ)
− m(
v0, ξ0)
− ∂2
∂v∂ξQ0
(
v0, ξ0) (
ξ − ξ0)
]
= op (1) .
Consequently,R1n (v) =
√n(
v− v0)
op (1). Using the relation thatx/(
1+ x2)
≤ 1/2, we conclude
that
sup|v−v0|≤δn
|R1n (v) |
1+ n|v− v0|2 ≤ op (1) sup|v−v0|≤δn
√n|v− v0|
1+ n|v− v0|2 = op (1) .
Having verified Conditions 2.i and 2.ii, it remains to verify Condition 2.iii. The Hessian termJ0
is obviously positive definite because the limiting likelihood function is multiple-times smoothly
differentiable, and becausev0 uniquely maximizesQ0(
v0, ξ0)
. We note, next, that∆n
(
v0)
/n takes
the form
− 1−τ1−FX|V[ F−1
n (τ)|v]∂∂vFX|V
[
F−1n (τ) |v
]
+1−τ
1−FX|V[
F−1X|V(τ)|v
]
∂∂vFX|V
[
F−1X|V (τ) |v
]
+
(
E − E) (
∂∂v log fX|V
(
Xi |v0)
1[
Xi ≤ F−1n (τ)
])
+ E(
∂∂v log fX|V
(
Xi |v0)
1[
Xi ≤ F−1n (τ)
])
−
E(
∂∂v log fX|V
(
Xi |v0)
1[
Xi ≤ F−1X|V
(
τ|v0)])
+ op
(
1√n
)
30
whereE denotes the empirical mean. Because we have represented the influence function of (ξ−ξ0)
as equation (8), we can compute that
∆n
(
v0)
=
n∑
i=1
ψ (Xi) + op
(√n)
where
ψ (Xi) = ∂∂v log fX|V
(
Xi |v0)
1[
Xi ≤ F−1X|V (τ|v)
]
− E(
∂∂v log fX|V
(
Xi |v0)
1[
Xi ≤ F−1X|V (τ|v)
])
−
11−τ
(
∂∂vFX|V
[
F−1X|V (τ) |v
])
fX|V[
F−1X|V (τ) |v
] [
F−1n (τ) − F−1
X|V
(
τ|v0)]
=∂∂v log fX|V (Xi |v) 1
[
Xi ≤ F−1X|V (τ|v)
]
− E(
∂∂v log fX|V (Xi |v) 1
[
Xi ≤ F−1X|V (τ|v)
])
+
11−τ
(
∂∂vFX|V
[
F−1X|V (τ|v) |v
]
1[
Xi ≤ F−1X|V (τ|v)
]
− τ)
.
Direct calculation of the asymptotic variance in the last line, while accounting for the covariance
between the two terms, yields
Var[
ψ (Xi)]
= E[
∂
∂vlog fX|V (Xi |v)
]2
1[
Xi ≤ F−1X|V (τ‖v)
]
+1
1− τ
(
∂
∂vFX|V
[
F−1X|V (τ|v) |v
]
)2
.
By inspection, we see that its inverse coincides with the asymptotic variance given inΣ (τ),
which has been verified to equalJ0 in the information matrix equality calculation and, hence,is
also positive definite. Its inverse yields the asymptotic variance of√
n(
p− v0)
and√
n(
v− v0)
.
References
Donald W.K. Andrews. Empirical process methods in econometrics. In Robert F. Engle and
Daniel L. McFadden, editors,Handbook of Econometrics, Volume 4. New York: Elsevier, 1994.
Victor Chernozhukov and Han Hong. A MCMC approach to classicalestimation. Journal of
Econometrics, 115:293–346, 2003.
31
Han Hong and Matthew Shum. Rates of information aggregation in common value auctions.
Journal of Economic Theory, 116:1–40, 2004.
Matthew O. Jackson and Ilan Kremer. The relationship between the allocation of goods and a
seller’s revenue.Journal of Mathematical Economics, 40:371–392, 2004.
Matthew O. Jackson and Ilan Kremer. The relevance of a choiceof auction format in a competitive
environment.Review of Economic Studies, 73:961–981, 2006.
Ilan Kremer. Information aggregation in common value auctions. Econometrica, 70:1675–1682,
2002.
Vijay Krishna. Auction Theory,Second Edition. San Diego: Academic Press, 2010.
Claudio Mezzetti and Ilia Tsetlin. On the lowest-winning-bid and the highest-losing-bid auctions.
Journal of Mathematical Economics, 44:1040–1048, 2008.
Claudio Mezzetti and Ilia Tsetlin. On the lowest-winning-bid and the highest-losing-bid auctions.
Games and Economic Behavior, 66:855–864, 2009.
Paul R. Milgrom. A convergence theorem for competitive bidding with differential information.
Econometrica, 47:670–688, 1979.
Paul R. Milgrom. Rational expectations, information aggregation, and competitive bidding.Econo-
metrica, 49:921–943, 1981.
Paul R. Milgrom.Putting Auction Theory to Work. New York: Cambridge University Press, 2004.
Paul R. Milgrom and Robert J. Weber. A theory of auctions and competitive bidding.Economet-
rica, 50:1089–1122, 1982.
Paul R. Milgrom and Robert J. Weber. A theory of auctions and competitive bidding, II. In Paul
Klemperer, editor,The Theory of Auctions, II. Cheltenham: Edward Elgar, 2000.
32
Whitney K. Newey and Daniel L. McFadden. Large sample estimation and hypothesis testing. In
Robert F. Engle and Daniel L. McFadden, editors,Handbook of Econometrics, Volume 4. New
York: Elsevier, 1994.
Wolfgang Pesendorfer and Jeroen M. Swinkels. The loser’s curse and information aggregation in
common value auctions.Econometrica, 65:1247–1281, 1997.
Wolfgang Pesendorfer and Jeroen M. Swinkels. Effiency and information aggregation in auctions.
American Economic Review, 90:499–525, 2000.
Robert J. Weber. Multi-object auctions. In Richard Engelbrecht-Wiggans Martin Shubik and
Robert M. Stark, editors,Auctions, Bidding and Contracting. New York: New York University
Press, 1983.
Robert B. Wilson. A bidding model of perfect competition.Review of Economic Studies, 44:
511–518, 1977.
33
Table 1: Announced Price, Number of Exits, and Total Exits
Price 20.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00Exits 0 2 0 0 0 0 0 0Total Exits 0 2 2 2 2 2 2 2Price 29.00 30.00 31.00 32.00 33.00 34.00 35.00 36.00Exits 0 0 0 0 0 0 0 1Total Exits 2 2 2 2 2 2 2 3Price 37.00 38.00 39.00 40.00 41.00 42.00 43.00 44.00Exits 0 0 0 0 1 0 2 1Total Exits 3 3 3 3 4 4 6 7Price 45.00 46.00 47.00 48.00 49.00 50.00 50.50 51.00Exits 0 3 1 0 1 1 2 2Total Exits 7 10 11 11 12 13 15 17Price 51.20 51.40 51.50 51.60 51.70 51.80 51.90 52.00Exits 0 0 0 0 0 0 0 0Total Exits 17 17 17 17 17 17 17 17Price 52.10 52.20 52.30 52.40 52.50 52.55 52.60 52.65Exits 0 0 0 0 1 0 1 0Total Exits 17 17 17 17 18 18 19 19Price 52.70 52.75 52.80 52.85 52.90 52.95 53.00 53.05Exits 0 0 0 0 0 0 0 0Total Exits 19 19 19 19 19 19 19 19Price 53.10 53.15 53.20 53.25 53.30 53.35 53.40 53.45Exits 0 0 0 0 0 0 0 0Total Exits 19 19 19 19 19 19 19 19Price 53.50 53.55 53.60 53.65 53.70 53.75 53.80 53.85Exits 0 0 0 0 0 0 0 0Total Exits 19 19 19 19 19 19 19 19Price 53.90 53.95 54.00 54.05 54.10 54.15 54.20 54.25Exits 0 0 0 0 0 0 0 1Total Exits 19 19 19 19 19 19 19 20
Table 2: Maximum-Likelihood Estimates of Normal Specification
Parameter Estimate Std.Errorv0 56.31 0.97σ 19.35 3.95
34
Table 3: Simulated Differences in Expected RevenueNumber of Objects
Number of Bidders 10 20 3020 0.13830 0.301 0.07940 0.265 0.127 0.171
35
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2005. 2005-1 Yishay P. Yafeh and Tarun Khanna, “Business Groups in Emerging Markets: Paragons or
Parasities?,” September 2005. 2005-2 Renee B. Adams and Daniel Ferreira, “Do Directors Perform for Pay?," September 2005. 2005-3 Qun Liu, Shin-ichi Fukuda and Juro Teranishi, “What are Characteristics of Financial
Systems in East Asia as a Region?.” September 2005. 2005-4 Juro Teranishi, “Is the Financial System of Postwar Japan Bank-dominated or Market
Based?,” September 2005. 2005-5 Hasung Jang, Hyung-cheol Kang and Kyung Suh Park, “Determinants of Family
Ownership: The Choice between Control and Performance,” October 2005. 2005-6 Hasung Jang, Hyung-cheol Kang and Kyung Suh Park, “The Choice of Group Structure:
Divide and Rule,” October 2005. 2005-7 Sangwoo Lee, Kwangwoo Park and Hyun-Han Shin, “The Very Dark Side of International
Capital Markets: Evidence from Diversified Business Groups in Korea,” October 2005. 2005-8 Allen N. Berger, Richard J. Rosen and Gregory F. Udell, “Does Market Size Structure
Affect Competition? The Case of Small Business Lending,” November 2005. 2005-9 Aditya Kaul and Stephen Sapp, “Trading Activity and Foreign Exchange Market Quality,”
November 2005. 2005-10 Xin Chang, Sudipto Dasgupta and Gilles Hilary, “The Effect of Auditor Choice on
Financing Decisions,” December 2005. 2005-11 Kentaro Iwatsubo, “Adjustment Speeds of Nominal Exchange Rates and Prices toward
Purchasing Power Parity,” January 2006. 2005-12 Giovanni Barone-Adesi, Robert Engle and Loriano Mancini, “GARCH Options in
Incomplete Markets”, March 2006. 2005-13 Aditya Kaul, Vikas Mehrotra and Blake Phillips, “Ownership, Foreign Listings, and Market
Valuation”, March 2006. 2005-14 Ricard Gil, “Renegotiation, Learning and Relational Contracting”, March 2006.
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2005-15 Randall Morck, “How to Eliminate Pyramidal Business Groups -The Double Taxation of Inter-corporate Dividends and other Incisive Uses of Tax Policy-”, March 2006.
2005-16 Joseph P.H. Fan, T.J. Wong and Tianyu Zhang, “The Emergence of Corporate Pyramids in
China”, March 2006. 2005-17 Yan Du, Qianqiu Liu and S. Ghon Rhee, “An Anatomy of the Magnet Effect: Evidence
from the Korea Stock Exchange High-Frequency Data”, March 2006. 2005-18 Kentaro Iwatsubo and Junko Shimizu, “Signaling Effects of Foreign Exchange Interventions
and Expectation Heterogeneity among Traders”, March 2006. 2005-19 Kentaro Iwatsubo, “Current Account Adjustment and Exchange Rate Pass-Through(in
Japanese)”, March 2006. 2005-20 Piruna Polsiri and Yupana Wiwattanakantang, “Corporate Governance of Banks in
Thailand”, March 2006. 2006-1 Hiroyuki Okamuro and Jian Xiong Zhang, “Ownership Structure and R&D Investment of
Japanese Start-up Firms,” June 2006. 2006-2 Hiroyuki Okamuro, “Determinants of R&D Activities by Start-up Firms: Evidence from
Japan,” June 2006. 2006-3 Joseph P.H. Fan, T.J. Wong and Tianyu Zhang, “The Emergence of Corporate Pyramids in
China,” August 2006. 2006-4 Pramuan Bunkanwanicha, Jyoti Gupta and Yupana Wiwattanakantang, “Pyramiding of
Family-owned Banks in Emerging Markets,” September 2006. 2006-5 Bernardo Bortolotti and Mara Faccio, “Reluctant privatization,” September 2006. 2006-6 Jörn Kleinert and Farid Toubal, “Distance costs and Multinationals’ foreign activities”,
October 2006. 2006-7 Jörn Kleinert and Farid Toubal, “Dissecting FDI”, October 2006. 2006-8 Shin-ichi Fukuda and Satoshi Koibuchi, “The Impacts of “Shock Therapy” on Large and
Small Clients: Experiences from Two Large Bank Failures in Japan”, October 2006. 2006-9 Shin-ichi Fukuda, Munehisa Kasuya and Kentaro Akashi, “The Role of Trade Credit for
Small Firms: An Implication from Japan’s Banking Crisis”, October 2006. 2006-10 Pramuan Bunkanwanicha and Yupana Wiwattanakantang, “Big Business Owners and
Politics: Investigating the Economic Incentives of Holding Top Office”, October 2006. 2006-11 Sang Whi Lee, Seung-Woog(Austin) Kwang, Donald J. Mullineaux and Kwangwoo Park,
“Agency Conflicts, Financial Distress, and Syndicate Structure: Evidence from Japanese Borrowers”, October 2006.
2006-12 Masaharu Hanazaki and Qun Liu, “Corporate Governance and Investment in East Asian
Firms -Empirical Analysis of Family-Controlled Firms”, October 2006. 2006-13 Kentaro Iwatsubo and Konomi Tonogi, “Foreign Ownership and Firm Value: Identification
through Heteroskedasticity (in Japanese)”, December 2006.
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2006-14 Kentaro Iwatsubo and Kazuyuki Inagaki, “Measuring Financial Market Contagion Using Dually-Traded Stocks of Asian Firms”, December 2006.
2006-15 Hun-Chang Lee, “When and how did Japan catch up with Korea? –A comparative study of
the pre-industrial economies of Korea and Japan”, February 2007. 2006-16 Kyoji Fukao, Keiko Ito, Shigesaburo Kabe, Deqiang Liu and Fumihide Takeuchi, “Are
Japanese Firms Failing to Catch up in Localization? An Empirical Analysis Based on Affiliate-level Data of Japanese Firms and a Case Study of the Automobile Industry in China”, February 2007.
2006-17 Kyoji Fukao, Young Gak Kim and Hyeog Ug Kwon, “Plant Turnover and TFP Dynamics in
Japanese Manufacturing”, February 2007. 2006-18 Kyoji Fukao, Keiko Ito, Hyeg Ug Kwon and Miho Takizawa, “Cross-Border Acquisitons
and Target Firms' Performance: Evidence from Japanese Firm-Level Data”, February 2007. 2006-19 Jordan Siegel and Felix Oberholzer-Gee, “Expropriators or Turnaround Artists? The Role of
Controlling Families in South Korea (1985-2003)”, March 2007. 2006-20 Francis Kramarz and David Thesmar, “Social Networks in The Boardroom”, March 2007. 2006-21 Morten Bennedsen, Francisco Pérez-González and Daniel Wolfenzon, “Do CEOs matter?”,
March 2007. 2007-1 Ichiro Iwasaki, “Endogenous board formation and its determinants in a transition economy:
evidence from Russia*”, April 2007, Revised on October 2007. 2007-2 Joji Tokui, Tomohiko Inui, and Katsuaki Ochiai, “The Impact of Vintage Capital and R&D
on Japanese Firms’ Productivity”, April 2007. 2007-3 Yasuo Nakanishi and Tomohiko Inui, “Deregulation and Productivity in Japanese
Industries”, April 2007. 2007-4 Kyoji Fukao, “The Performance of Foreign Firms and the Macroeconomic Impact of FDI”,
May 2007. 2007-5 Taku Suzuki, “The Role of the State in Economic Growth of Post-Communist Transitional
Countries”, June 2007. 2007-6 Michiel van Leuvensteijn, Jacob A. Bikker, Adrian A.R.J.M. van Rixtel and Christoffer
Kok-Sørensen*, “A new approach to measuring competition in the loan markets of the euro area”, June 2007.
2007-7 Sea Jin Chang, Jaiho Chung, and Dean Xu, “FDI and Technology Spillovers in China”, July
2007. 2007-8 Fukunari Kimura, “The mechanics of production networks in Southeast Asia: the
fragmentation theory approach”, July 2007. 2007-9 Kyoji Fukao, Tsutomu Miyagawa, Miho Takizawa, “Productivity Growth and Resource
Reallocation in Japan”, November 2007. 2007-10 YoungGak Kim, “A Survey on Intangible Capital”, December 2007.
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2007-11 Sea-Jing Chang and Jay Hyuk Rhee, “Rapid International Expansion Strategy of Emerging Market Enterprises: The Interplay between Speed and Competitive Risks on International performance”, November 2007.
2007-12 Ishtiaq Mahmood, Will Mitchell, and Chi-Nien Chung, “The Structure of Intra-Group Ties:
Innovation in Taiwanese Business”, January 2008. 2007-13 Kyoji Fukao, Tomohiko Inui, Shigesaburo Kabe and Deqiang Liu, “ An International
Comparison of the TFP Levels of Japanese, Korean and Chinese Listed Firms“, March 2008.
2007-14 Pramuan Bunkanwanicha and Yupana Wiwattanakantang, “Allocating Risk Across
Pyramidal Tiers: Evidence from Thai Business Groups”, March 2008. 2008-1 Rüdiger Fahlenbrach and René M. Stulz, "Managerial Ownership Dynamics and Firm
Value", April 2008. 2008-2 Morten Bennedsen, Kasper Meisner Nielsen, and, Thomas Vester Nielsen, “Private
Contracting and Corporate Governance: Evidence from the Provision of Tag-Along Rights in an Emerging Market”, April 2008.
2008-3 Joseph P.H. Fan, Jun Huang, Felix Oberholzer-Gee, and Mengxin Zhao, “Corporate
Diversification in China: Causes and Consequences”, April 2008. 2008-4 Daniel Ferreira, Miguel A. Ferreira, Clara C. Raposo, “Board Structure and Price
Informativeness”, April 2008. 2008-5 Nicola Gennaioli and Stefano Rossi, “Judicial Discretion in Corporate Bankruptcy”, April
2008. 2008-6 Nicola Gennaioli and Stefano Rossi, “Optimal Resolutions of Financial Distress by
Contract”, April 2008. 2008-7 Renée B. Adams and Daniel Ferreira, “Women in the Boardroom and Their Impact on
Governance and Performance”, April 2008. 2008-8 Worawat Margsiri, Antonio S. Melloy, and Martin E. Ruckesz, “A Dynamic Analysis of
Growth via Acquisition”, April 2008. 2008-9 Pantisa Pavabutra and Sukanya Prangwattananon, “Tick Size Change on the Stock
Exchange of Thailand”, April 2008. 2008-10 Maria Boutchkova, Hitesh Doshi, Art Durnev, and Alexander Molchanov, “Politics and
Volatility”, April 2008. 2008-11 Yan-Leung Cheung, P. Raghavendra Rau, and Aris Stouraitis, “The Helping Hand, the Lazy
Hand, or the Grabbing Hand? Central vs. Local Government Shareholders in Publicly Listed Firms in China”, April 2008.
2008-12 Art Durnev and Larry Fauver, “Stealing from Thieves: Firm Governance and Performance
when States are Predatory”, April 2008. 2008-13 Kenneth Lehn, Sukesh Patro, and Mengxin Zhao, “Determinants of the Size and Structure of
Corporate Boards: 1935-2000”, April 2008.
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2008-14 Ishtiaq P. Mahmood, Hong-Jin Zhu and Edward J. Zajac, “Where Can Capabilities Come From? How the Content of Network Ties Affects Capability Acquisition”, April 2008.
2008-15 Vladimir I. Ivanov and Ronald W. Masulis, “Corporate Venture Capital, Strategic Alliances,
and the Governance of Newly Public Firms”, May 2008. 2008-16 Dick Beason, Ken Gordon, Vikas Mehrotra and Akiko Watanabe, “Does Restructuring Pay
in Japan? Evidence Following the Lost Decade”, July 2008 (revision uploaded on Oct. 2009).
2009-1 Vikas Mehrotra, Dimitri van Schaik, Jaap Spronk, and Onno Steenbeek, “Creditor-Focused
Corporate Governance: Evidence from Mergers and Acquisitions in Japan,” August, 2009.
2009-2 Debin Ma, “Law and Economic Change in Traditional China: A Comparative Perspective,” September, 2009.
2009-3 Robert C. Allen, Jean-Pascal Bassino, Debin Ma, Christine Moll-Murata, and Jan Luiten van
Zanden, “Wages, Prices, and Living Standards in China, 1738-1925: in Comparison with Europe, Japan, and India,” June 2009.
2009-4 Jung-Wook Shim, “The Existence of Nepotism: Evidence from Japanese Family Firms,”
October 2009.
2009-5 Morten Bennedsen and Kasper Meisner Nielsen, “Incentive and Entrenchment Effects in European Ownership,” March 2009.
2009-6 Joseph P.H. Fan, TJ Wong, Tianyu Zhang, “Founder Succession and Accounting Properties,”
April 2009.
2009-7 Hiroyuki Okamuro, Masatoshi Kato, and Yuji Honjo, “Determinants of R&D Cooperation in Japanese High-tech Start-ups,” November 2009.
2009-8 Bill Francis, Iftekhar Hasan, Michael Koetter, and Qiang Wu, “The Effectiveness of
Corporate Boards: Evidence from Bank Loan Contracting,” November 2009.
2009-9 Allen N. Berger, Iftekhar Hasan and Mingming Zhou, “The Effects of Focus Versus Diversification on Bank Performance: Evidence from Chinese Banks,” November 2009.
2009-10 Leonardo Becchetti, Andrea Carpentieri and Iftekhar Hasan, ”The Determinants of Option
Adjusted Delta Credit Spreads: A Comparative Analysis on US, UK and the Eurozone,” November 2009.
2009-11 Luciano I. de Castro and Harry J. Paarsch, “Testing Affiliation in Private-values Models of
First-price Auctions Using Grid Distributions,” December 2009.
2009-12 Chulwoo Baek, YoungGak Kim and Heog Ug Kwon, “Market Competition and Productivity after the Asian Financial Crisis: Evidence from Korean Firm Level Data,” December 2009.
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2009-13 Jee-Hyeong Park, Stephen J. Spurr, and Sheng-Kai Chang, “A Model of Hierarchical Professionals: Cooperation and Conflict between Anesthesiologists and CRNAs,” October 2009.
2009-14 Jee-Hyeong Park, “Enforcing International Trade Agreements with Imperfect Private
Monitoring: Private Trigger Strategies and the Possible Role of the WTO,” December 2009.
2009-15 Yuji Honjo, Masatoshi Kato and Hiroyuki Okamuro, “R&D financing of start-up firms: How much does founders’ human capital matter?”, March 2010.
2010-1 Sergei V. Ryazantsev, “Migrant Workers from Central Asian Russian Federation”, June
2010.
2010-2 Tue Gørgens, Xin Meng, and Rhema Vaithianathan, “Stunting and Selection Effects of Famine: A Case Study of the Great Chinese Famine,” October 2010.
2010-3 Masatoshi Kato and Yuji Honjo, “Heterogeneous Exits: Evidence from New Firms,”
November 2010.
2010-4 Sung-Jin Cho, Harry J. Paarsch, and John Rust, “Is the ’Linkage Principle’ Valid?: Evidence from the Field,” November 2010.
2010-5 Jean-Pascal Bassino and Noriko Kato, “Rich and slim, but relatively short Explaining the
halt in the secular trend in Japan,” November 2010.
2010-6 Robert G Gregory, Dark Corners in a Bright Economy; The Lack of Jobs for Unskilled Men,” December 2010.
2010-7 Masatoshi Kato and Hiroyuki Odagiri, “Development of University Life-Science Programs
and University-Industry Joint Research in Japan,” December 2010.
2010-8 Han Hong, Harry J. Paarsch and Pai Xu, “On the Asymptotic Distribution of the Transaction Price in a Clock Model of a Multi-Unit, Oral, Ascending-Price Auction within the Common-Value Paradigm,” January 2011.