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WORKING PAPER
DEPARTMENT
OF ECONOMICS
Information and Competition in
U.S. Forest Service Timber Auctions
Susan Atliey
Jonathan Levin
No. 99-12
June 1999
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
MASSACHU-;,QLOeY_
NOV 2
5 1999
LIBRARSES
Information and Competition in
U.S. Forest Service Timber Auctions*
Susan Athey
Jonathan Levin
MIT and NBER
MIT
First Draft:
November 1997
This Draft: June, 1999
Abstract
This paper studies the bidding behavior of firms in U.S. Forest Service timber auctions
When
in 1976-1990.
conducting timber auctions, the Forest Service pubhcly announces
estimates of the tract characteristics before the auction,
opportimity to inspect the tract and form
its
own
its
and each bidder additionally has an
We
private estimates.
build a model that
incorporates both differential information and the fact that bids placed in timber auctions are
multidimensional.
on
The theory
their private information,
includes both the
Our
a practice known as "skewed bidding."
Using a dataset that
pubUc ex ante Forest Service estimates and the ex post reahzations of the
tract characteristics,
information.
predicts that bidders will strategically distort their bids based
we
test
om- model and provide evidence that bidders do possess private
results suggest that private information affects Forest Service revenue
creates aUocational inefficiency. Finally,
we estabhsh
and
that risk aversion plays an important role
in bidding behavior.
JEL
*We
Numbers:
Classification
are grateful to
Glenn EUisonj' Sara
MuUin, Whitney Newey, Mike Niccolouci,
MIT, Toronto,
Patrick
Tufts, the
Wang and
DU,
NSF
D81, D82, L73.
Ellison, Phil Haile, Jerry
Hausman, Paul Joskow, Bob Marshall, Wally
Tom Stoker and seminar participants at Alabama,
Conference on Auctions and Infrastructure, and the
Leigh Linden provided exceptional research assistance.
We
Berkeley,
NBER
would also
IDEI Toulouse,
for helpful
like to
comments.
thank the numerous
employees of the U.S. Forest Service and the timber industry who took time to answer our questions. Athey acknowledges financial support from
MIT,
NSF
as well as the hospitality
Grant SBR-9631760 and the Provost's Fund
and support of the Cowles Foundation
for
for
Humanities and Social Sciences at
Economic Research at Yale University.
Introduction
1
This paper studies the extent to which bidders in auctions obtain and strategically exploit private
information. Since the private signals of bidders are not typically observable by an econometrician,
it is
in general difficult to test theories
identify a setting in
about how bidders make use of these
signals. In this
which ex post information about attributes of the object
that affect the value of the object to
all
bidders. Using this information,
we
is
common
attributes,
on behavior and the outcome of the auction. In
as well as the effect of such private information
we
available, attributes
are able to empirically
examine the extent to which bidders possess private information about these
particular,
paper we
are able to test predictions in the spirit of
Milgrom and Weber's (1982) model of the
"mineral rights" auction.
The
setting for our study
is
the U.S. Forest Service timber auction program in the Pacific
Northwest from 1976 to 1990.^ Our analysis
sale"
— used by the Forest Service.
Each
relies
on a particular auction format
—
called a "scale
tract of timber contains multiple species. Before the sale,
the Forest Service conducts a "cruise" and estimates the volume of each species (as well as other
factors such as timber quality
a month before the
sale, at
and logging
prices
is
These estimates are publicly announced
at least
which time bidders have the opportunity to hire their own cruisers and
form independent estimates. In a scale
The winner
costs).
sale, firms
bid a price per unit of volume for each species.
the firm that places the highest estimated total bid,
computed
as the
sum
of the
on each species weighted by the Forest Service quantity estimates. The contract requires
the winner to remove
Service measures the
all
designated timber from the tract. As the timber
volume of each
species,
rates specified in the bid. Thus, there
and the winner pays
may be a
for the
is
removed, the Forest
timber removed at the
gap between the average
significant
bid, weighted
by the Forest Service estimates, and the average payment, weighted by the actual proportion
each species. Such a gap
is
typical:
on
tracts with
estimate of the proportion of timber that
actual proportion that
While
in
scale sales
many government
"indefinite quantity"
ernment
specifies its
is
is
two main species of timber, the Forest Service's
the primary species turns out to be within
removed on only half of the
may appear
auctions.
5%
of the
sales.
particular to the Forest Service, similar rules are in fact used
Examples include bidding
procurement auctions for^goods and
estimated
of
demand
for
highway construction contracts and
services. In
for different goods,
such procurements, the gov-
which are used as weights
in
computing
the total bids. But, the government can also choose to order more than those estimated quantities.
Despite the prevalent use of scale sales, the equilibrium behavior of firms in such auctions has
not to our knowledge received a general theoretical or empirical analysis in the economics literature.
'Other recent analyses of Forest Service timber auctions include Hansen (1986), Leffler and Rucker (1988),
mins (1995), Baldwin, Marshall, and Richard (1997) and Haile (1998).
Cum-
Our formal model
of equilibrium bidding behavior in scale sales can be described as follows. There
are two species, 1 and 2, and each (risk-averse) bidder receives a signal about the true proportion
of species
and places bids
1,
the proportion of species
computed
timber
1
and the
xi,
is
If
62-
the Forest Service has estimated that
total quantity
Qest, then the
is
total bid
B = Qest x (^ia;i + 62(1 — ^i))- The winner, who submitted the highest
pays P = Qact x (^iPi + ^2(1 — Pi))i where Qact is the realized quantity on
the realized proportion of species
is
and
6i
as
ultimately
and pi
each species,
for
on the
1
tract.
B
is
total bid,
the tract,
In such an auction, bidders have an
incentive to "skew" their bids onto the species they believe the Forest Service has over-estimated.
The
intuition
that a bidder can achieve the
is
winning the auction) while paying
less if
same
total bid (and thus the
the total bid
is
same probability
of
allocated mainly to species that are over-
estimated, since each dollar bid on an over-estimated species increases the total bid by more than
it
increases the expected payment.
To
how
see
this
would work
in practice, consider a simple example.
Suppose that the Forest
Service announces a tract with 1000 units of volume, and estimates that half of the tract
Fir,
and half
Western Hemlock.
is
A
Douglas
bidder can achieve a total bid of $100,000 by bidding $100 on
each species. Another way to achieve the same total bid
Fir,
is
is
by bidding $150 per unit on Douglas
and $50 on Spruce. Suppose that a bidder's cruise indicates that the expected proportion of
Douglas Fir
in fact 0.25, not 0.5 as the Forest Service estimated.
is
($100,$100) yields an expected payment of $100,000.
yields
Then the bidding
strategy
However, the bidding strategy ($150,$50)
an expected payment of $75,000.
We
analyze the equilibria of sealed bid auctions and a stylized oral ascending bid auction.
The
theory exploits a basic insight, that bid decisions can be decomposed into two components:
choice of
what
species, that
total bid
is
what
B
and a decision about how to
to submit,
level of bid skew,
A6 =
61
—
62,
to select.
allocate that bid across the
We
Ixi
— Pil)
will
skew
their bids
more
aggressively,
and
will
between submitted bid and payment. As a consequence, they
oral ascending bid auction
works
in
two
show that bidders who have
higher signals about the extent to which the Forest Service has over-estimated one species
about
a
(i.e.
be more optimistic about the gap
will also
submit higher
bids.
The
a similar fashion, although some subtleties arise from the fact
that bids at any point in time have two dimensions, a total bid and a skew.
The theory we develop
has the attractive feature that "skewing behavior at each stage of the auction corresponds to the
optimal skew of a bidder
who
is
just indifferent about dropping out of the auction. Thus,
we can
give a precise interpretation to losing bids in our oral auction data.
We
test
1976-1990.
our theoretical predictions using data from sales in Oregon and Washington during
We
well as the bids
match bidding data, which includes information from the Forest Service
and bidder
identities,
cruise as
with ex post cutting data from the tract. Since we observe
both the estimated quantities of each species and the quantities that aie removed, we are able to
use the difference between the estimated and actual proportions of species
the private information potentially available to the firms.
Our
xi
— p^,
refer to this difference,
we estimate
"mis-estimate" of the Forest Service. Using this data,
about Forest Service mis-estimates
We
1,
as a proxy for
xi— pi,
as the
the extent to which information
bidding behavior and Forest Service revenue.
affects
empirical analysis broadly confirms the predictions of our theoretical model.
We find strong
evidence that bidders respond to mis-estimates by skewing their bids, and also that higher-ranked
bids tend to be better
matched to mis-estimates than lower-ranked
conditional on the level of a bidder's total bid, the
have paid, had they won)
affected
is
by mis-estimates.
decreasing in
We
l^i
—
amount paid
Likewise,
bids.
(or the
further find that in a given sale, there
is
see that
amount the bidder would
Forest Service revenue
Pi\-
we
is
also systematically
often heterogeneity in the
extent and direction of the skew, consistent with private information on the part of the bidders.
Higher bidders at an auction tend to also skew their bids more, and we find some evidence that
bidders incorporate information in the course of an oral auction. Finally,
we
frequently observe firms
using moderate skews, which can only be optimal for risk averse bidders (in or out of equilibrium)
The
results in this
paper contribute to the existing literature in several ways.
Milgrom and Weber's (1982) theory
We have extended
of mineral rights auctions to the case of "scale sales,"
as
described above. Previous papers have looked at auctions with multidimensional bids and scoring
rules,
but these studies consider settings with independent private values, and there
to the skewing behavior identified here.^
Baldwin (1995) and
Wood
is
no analog
(1989) have formally analyzed
skewing: Baldwin considers the bid allocation problem, taking the bid level as exogenous.
restricts attention to pre-selected linear bidding rules
considers both the
among
the bidders.
full
and looks
at second-price auctions. Neither
decision problem and equilibrium behavior
Baldwin
Wood
when information
is
dispersed
also empirically estimates the degree of risk aversion of bidders at
Forest Service timber auctions. Her dataset, however, does not include the ex post realizations of
the volumes of each species on the tract, and thus her conclusions are based on revealed preference
rather than direct measures of the returns to skewing.
The
effects of private
information in "mineral rights" auctions have also been empirically ana-
lyzed by Hendricks and Porter (1988) in the context of leases for off-shore oil drilling. Hendricks
and Porter
identify the effects of informational advantages using ex post information about the
tracts (in this case, the
amount 'of oil
in the tract),
have better information. In contrast to
that
it is
lease auctions, scale sales have the attractive feature
possible to distinguish the effect of information on each bidder's decision-theoretic choice
about skewing (which does not directly
^Osband and Reichelstein
in
oil
and testing to see whether "neighboring" firms
(1985),
Che
affect the probability of
(1993) and Bushnell
winning the auction) from her
and Oren (1994) study auctions where bidders can bid
two dimensions (cost and quality in the case of Che, and fixed and variable cost in the case of Bushnell and Oren).
The winner
is
determined by a scoring
rule.
strategic decision
This paper
is
about the total
organized as follows. Section 2 provides background on the Forest Service Timber
Program and describes some
of the relevant institutional details of the timber auctions.
model that captures many
3 presents a formal
sales.
bid.
of the
Section
main elements of bidding behavior
Section 4 describes the data, while Sections 5 and 6 provide our
main empirical
at scale
analysis.
Section 7 concludes.
Background: The Forest Service Timber Program
2
In the northern and western regions of the U.S., the national forests have been the primary som-ce of
timber for mills, logging companies, and forest products companies. During 1976-1990, the Forest
Service conducted well over a thousand auctions per year in these areas, generating annual revenue
Our
of around $1 billion.
empirical work will focus on Forest Service Regions 5 (California) and 6
(Oregon and Washington); in the 1980s, these regions accounted
timber sold and
80%
of
all
for two-thirds of all Forest Service
Forest Service timber receipts.
USES Timber
Auctions, 1976(1)-1990(2), Statistics
Region 5
Region 6
6009
16,857
3752.6
3994.6
184,897
329,280
542,047
682,227
110.34
151.69
Avg. Number of Bidders
3.67
5.07
%
47.5
87.1
Sales
Avg. Volume (mbf)^
Avg. Reserve Price
Avg. Winning Bid
($)
($)
Avg. Bid per unit vol ($/mbf)
Oral Auctions
Since details of the Forest Service timber program have been discussed elsewhere (see, for example,
Baldwin, Marshall, and Richard (1997)), we touch only on a few key aspects of the industry
organization,
and focus our discussion on the process through which bidders acquire information
and prepare a scale
sale bid.
/
-
Bidding in Forest Service sales
is
undertaken by a diverse collection of timber conglomerates,
smaller mills, and independent logging operations. Timber from a sale
mills
proximate to the national
types, each
demanding
forest.
is
frequently processed in
In Region 6, there are several hundred mills of different
different species
and
qualities of timber.
Weyerhauser, Boise-Cascade, and Georgia-Pacific)
own many
be able to process the whole range of species and timber
Conglomerate firms (such as
mills all over the country,
qualities
on a given
and may
tract. Smaller mills
and independent "gyppo" logging operations do not have
win Forest Service auctions, they tend to
resell
some
or
this abiUty.
When
of the timber. In the 1980s, this resale
all
through prices posted by
either through private bilateral trades or, in smaller quantities,
was done
these smaller bidders
the mills (see Haile (1998) for an analysis of resale in timber auctions).
The
Forest Service begins preparation for a sale several months prior to the sale date.
manager organizes a
forest
sale date.
The manager
cruise
and publicly announces the
also decides, based
Once the
submitting a deposit of
10%
sale is
findings at least 30 days before the
on tract characteristics and expected competition,
whether to conduct the sale by oral or sealed bidding
sales are oral auctions.
—
Region
in
6,
the great majority of the
announced, each firm must "qualify"
10%
of the bid in a sealed bid auction, or
the sale in an oral auction."* This deposit
is
The
held until the contract
is
for
an auction by
of the appraised value of
awarded.
In addition to the scale sale format already mentioned, the Forest Service occasionally uses an
alternative
"lump-sum" format. In a lump-sum
sale,
each firm makes a fixed bid, and the firm
with the highest bid wins the auction and pays that bid, irrespective of the realized volumes of
each species. As described above, bidders in a scale sale submit a bid rate for each species and
the winner pays the bid rate for the realized volume of each species.
differs slightly
between the two formats: in a lump-sum
price for the whole tract; in a scale sale, there
is
a
is
minimum
reserve price
mechanism
the Forest Service sets a fixed reserve
acceptable bid for each species.
and Western regions of the country. The main
Scale sales are used mainly in the Northern
motivation for using scale sales
sale,
The
on the part
to reduce risk
of the bidders.^
If
the bidders face
substantial risk about the value of the sale, their bidding will be less aggressive (not only do they
require a risk premium, but they
bidder
sum
who wins
is
may be more concerned about
the winner's curse, whereby the
generally the one whose cruiser gave the most optimistic estimate). In lump-
bidding, bidders bear
all
of the risk about the species composition of a tract. In contrast, a
scale auction automatically mitigates uncertainty
about the
total
volume of timber on a
tract,
and
bidders can in principle insure against uncertainty concerning the proportion of each species on the
tract
by bidding the same
profit
margin on each
species.
Before submitting their sealed bids or qualifying bids, the bidders have the opportimity to cruise
the tract and form their
own
estimates of tract characteristics. Cruising a tract has traditionally
been considered something of an
art
by industry
insiders,
and many
cruisers
have an undergraduate
degree in forestry (such a degree as well as two years of experience are requirements for admission
In oral sales, a bidder can choose to raise the bid above the reserve price in this initial qualifying
in
round of bidding,
which case the deposit must be equal to 10% of the bid.
Forest Service personnel also bear political costs
when
their ex ante estimates are incorrect.
The
scale sale system
has the potential to reduce the sensitivity of bidder outcomes to volume mis-estimates, thus making
desirable.
Firms
in the
industry have historically excercised great influence over Forest Service policy.
it
politically
made
to the industry association). According to industry sources, beginning cruisers in the 1990s
about $30,000 to $40,000 per year, while more experienced cruisers made $60,000 or more. Large
forest
product companies have in-house cruising
These
cruisers from consulting companies.
staffs,
while smaller companies
may
use for-hire
for-hire cruisers typically price their services either
by
the acre or by the hour. While the costs vary substantially from tract to tract, one firm estimated
a "typical" cost of $10/acre.
The average
tract size in our
sample
is
380 acres, putting this cost at
about .6% of the tract value. While bidders reportedly spend more resources on cruising
sum" rather than
for
"scale" sales, industry sources in the Pacific
Northwest report that
"lump
for
unusual
it is
a bidder to place a bid on a tract without cruising. Moreover, firms that have incurred the costs
of surveying a tract generally submit bids
— thus, one can think of the decision to survey a tract
as roughly equivalent to an entry decision.
To
job
is
this
cruise a tract, the cruisers go out into the forest for several days.
detecting potential defects in trees. For example, insects can
may be
difficult to verify
from external characteristics; and
caused problems in different areas of a
forest.
A
damage
difficult
aspect of the
old growth trees, but
different kinds of insects
may
have
Cruisers must guess at the defects in trees and
determine the "merchantable" timber, which will be sold at the prices bid at the auction, and the
"non-merchantable" timber, which
The Forest
(see, for
is
essentially scrap.
Service has long been aware of the problems of "skewed bidding" in scale sale auctions
example,
GAO
Report RCED-83-37, which documented revenue
losses
from skewing). In
order to limit such behavior, rules have been adopted over time that limit the scope for skewing.
For example, during the 1980s, in some forests bidders could only place bids above the reserve price
on "major
species,"
Forest Service
which account
moved
for
more than 25% of the volume. In 1995, some parts
to "proportional bidding,"
of the
whereby bidders are required to distribute
their
bids across species according to fixed proportions.
3
The Model
In this section, we build a stylized model of equilibrium bidding behavior in scale auctions.
abstract from a
number
of considerations,
most notably heterogeneity
We
in the values of firms for each
species of timber, as well as privately observed'cost or inventory shocks. Thus, our results can be
viewed as providing insight into one component of bidding behavior, while stopping short of a
analysis of the real-world auction setting. In particular, with private values as well as a
full
common
component, the tract would be allocated in part on the basis of individual preferences and in part
on the
basis of signals about the
had the same information
common
as the bidders
value. It
is
important to note, however, that
if
the
FS
about the value and extraction costs of each species of
timber (as in our formal model), there would be no need for an auction at
The model can be
commonly known
outlined as follows.
values
>
v-y
which are the same
vq,,
xi (and let
each species,
x-z
r\
=
1
and
—
We
Each bidder has an
be
strictly increasing
assume that
vi
>
for all bidders.
>
and V2
r\
—
pj
>
Q corresponds to an over-estimate of the proportion of species
We will
x^
is
species
for
which we assume to
be the difference between the estimated and actual proportion, so
that
public signal,
volume that
indicate the true, unobserved, proportion of
and
Xi
auctioneer provides
r^-
i,
8i
The
of the
species
let 6i
These species have
2.
identical utility function defined over wealth u(-),
and (weakly) concave. Let
=
and
These estimates are announced, together with the reserve prices
xi).
r2.
species, 1
Qest, and the proportion
estimates of the total volume on the tract,
1,
There are two
all.^
p^
{1)2}, which
i.
All bidders observe a
informative about the direction of the auctioneer's mis-estimate.
is
assume that x conveys enough information that
all
bidders will choose to skew in the same
This substantially simplifies the analysis, without disturbing the underlying ideas, by
direction.
allowing us to ignore the extent to which bidders learn about the direction of the mis-estimate
when they
Each bidder j
discover that they have won.
also observes a private signal, d^,
the auctioneer's mis-estimate (under the assumption below,
assume that the total volume, Qact,
necessarily equal to
independently of
(Al)
(i)
(i)
6-^.
We
on
x-)^,
=
[x^
we can
allow
Qact
and
l,a;J
E[6y\x,d^,...,d'^]
{d^, ...,d^,6-)(), {Xi^x) ^^^ strictly affiliated
implies that the public signal
>
and
random
signal).
The
affiliation
is
possible (that
is,
it
varies
for all {d^,...,d^); (ii) condi-
{d^, ....,rf^) are exchangeable.
in their assessment of the expected direction of the mis-estimate; however,
public signal, any mis-estimate
We
variables:
sufficiently informative that the bidders
is
in d^).
to be stochastic so long as
following assumptions about the
—
be increasing
and commonly known to the bidders, though not
fixed
Alternatively,
make the
support(6j^|x, d^)
tional
Part
Qest-
is
£^[6,^1^^] will
about
always agree
even after seeing the
the support does not change with the public
and exchangeability assumptions
(ii) will
allow us to establish the critical
comparative statics results in the analysis.
We now
consider the sealed bid auction and the oral ascending auctions in turn.
We
then
discuss the empirical implications of the model. Proofs are deferred to the Appendix.
3.1
The Sealed Bid Auction
Consider a
first-price, sealed
bid auction.
After observing the auctioneer's announcements, the
public signal x^ and a private signal d^, each bidder j submits a sealed bid for each species, 6]
For this reason,
set
it is
trivial to describe the
a rate for each species equal to
its
optimal mechanism in this context: the Forest Service should simply
value and
sell
to
any bidder.
the question of optimal auction design would become interesting.
If
one were to add a private value component,
and
62) to
The
the auctioneer.
total bid
computed
is
After the sale, the true total volume
wins.
Ordering the bidders j
=
I,..,
J
Qact and
= QESxY^ibl^i
B^
as
and the highest bid
proportion of species
in descending order of their total bids, the
are realized.
1, Pi,
winning bidder pays
P'=QACT^ibjPi.
It is
allocating that bid over the
two
species,
choosing
i.e.
each bidder j faces a two-part optimization:
Ab^
,
by,
— b-,y.J Given opponent
,
any total bid
allocation Ab^^{B^ d^)? Second, given that
=
any total bid B^ what
first, for
B
two parts: selecting a total bid
useful to think about each bidder's decision in
is
strategies,
the optimal bid
be allocated optimally, what
will
and
is
the
winning the auction;
his
optimal total bid B^{dy)1
Only a bidder's decision about
bid allocation
is
relevant only
if
his total bid affects his probability of
he wins. Thus, bidder j's bid allocation problem, for which we
assume a unique solution, can be written as follows
maxEi
u (qact
((A6;,
- A^^)
subject to:
bi
6y
>
(letting Av-^
+ ^^Q^) ) 4' X;
I
>
ri, b2
V=
^s5t(v
V/c 7^ j, 5'= (4)
< B^
x)):
•
,
(1)
for the winner's curse)
as well as his
,
private information.
The bid
allocation problem
as choosing his "investment,"
due to
6-)^
is
formally equivalent to a portfolio problem; think of the bidder
Ab^ —
by bidding a constant
this as "full-insurance,"
and we
Av-^^, in
profit
refer to
the risky asset,
neutral, he will invest as
much
any departure from
of his total bid
subject to the reserve price constraints.
does not depend on our assumption that
that there
is,
6-^.
is,
The
x
is
B
6-^ is
as possible
latter finding
A6^ — Av^ >
= r-,-^.
Abt - Av^ > 0.^
bid sets b^y
0,
We
refer to
if
a bidder
maximizing Aby
is
risk-
— Avy
quite robust (and in particular,
it
publicly observed). So long as the bidder believes
but in general the solution
(ii)
If the bidder is risk-averse,
= B/QBST + ^bxO-~^x)
and b^^ (and
vice- versa).
< V — (t)2 — '"2),
that Afe^ < Av^-
will
be
some investment
in the risky asset,
interior.
the bid allocation problem, (i) If a bidder is risk-neutral, then the optimal
^Note that b^
imply
x,
Av-^).
a "skew." Assumption
Thus,
positive.
on species
is
=
on average, a mis-estimate, risk-neutrality implies that skewing should be maximal.
Proposition 1 Consider
If B-'
bidder can eliminate the risk
this strategy as
In contrast, for a risk-averse bidder, the solution will involve
that
A
margin on each species (setting Ah^^
(Al) ensures that the expected value of the risky asset
6^
and
r2,
where j conditions on winning with total bid B^ (accounting
own
= v-^ — v^y.,
^"d
6^;^.
and B^
= B/Qbst — Afc^^x^i
> V—
{v2
^o choosing
a risk-averse bidder will skew maximally, setting b^^
—
'"'-^xt
—
7*2),
it is
B and Ab^
optimal to set
uniquely determines
^"^ ^^^ reserve price constraints
Proposition
risk,
1 tells
us that the solution to the bid allocation problem entails taking on some
despite the fact that a full-insurance policy
is
available. Consistent
with portfolio theory, the
next Proposition establishes that bidders with higher signals about Forest Service mis-estimates
skew
will
their bids
more
aggressively.
Proposition 2 Suppose that B^{d^
CARA
preferences satisfy
nondecreasing in d^ for
is
or lARA, any optimal AtP^{B^,dx)
Proposition 2 also establishes that the optimal skew
for this is straightforward
higher total bid
is
will he
k
^
>
and that B^
j,
nondecreasing in
B^ and
One
increasing in the total bid.
V.
If
d^.
reason
— when opponents use nondecreasing bidding functions, winning with a
means that the
also a further effect,
is
all
which
is
distribution of (defeated) opponent types
the reason
we
is
more
favorable.
require constant or increasing absolute risk aversion
(CARA or lARA). When
the total bid
(essentially) a position of
decreased wealth. Under decreasing absolute risk aversion
is
There
higher, bidders evaluate the species composition risk from
(DARA)
,
this
decrease in wealth would lead to an increase in risk-aversion and a reduced propensity to skew,
introducing a competing effect that, though presumably small, would greatly complicate the model.
Thus, while
DARA would seem to
be a reasonable assumption (and indeed, one that
by Baldwin's (1995) empirical work), we rule
We now
it
is
supported
out.
estabUsh the existence of a pure strategy Nash equilibrium in which bidders with more
optimistic signals
skew more aggressively and submit higher
that, for each bidder j,
if all
are nondecreasing in {B^^d'^
nondecreasing.
Under
opponents k
and d^
this condition,
^
total bids.
Our approach
to
is
show
j use identical strategies (A6^(B'^,d^),S'^(d^)) that
respectively, then bidder j's best response
must be
similarly
Athey (1997) has shown that a pure strategy Nash equilibrium
exists.
Proposition 3
(i)
For each bidder
j,
suppose that
the strategy 0{d^), which is nondecreasing in
lARA, any
all
d^ vnth
opponents k
/?(•)
>
V.
best response function S-'(d^) will be nondecreasing
^
j choose
B^
according to
If preferences satisfy
and have B^{-)
pure strategy Nash equilibrium exists in nondecreasing strategies where
all bids
> V.
CARA
(ii)
or
Thus, a
submitted are greater
than or equal to V.
Recall from Proposition 1 that bidders choose to
first-stage portfolio
skew
their bids as the optimal solution to their
problem. Proposition 3 implies that bidders must skew to win.
chooses a full-insurance policy that does not result in a sure loss can bid at most
thus be outbid by an opponent
who
who
who
bidder
B = V.
He
will
believes that the Forest Service estimates are incorrect, since
by skewing, such an opponent can generate a higher total
the full-insurance policy.
A
The monotonicity
bid, while still expecting to
pay
less
than
of the equilibrium bid functions implies that the bidder
gets the highest signal about the mis-estimate will
skew
his bid the
most and win the auction.
Our model assumes that bidders have
and
oil
and that
this quality
is
exogenous
Hendricks and Porter (1988) have demonstrated the importance of superior informa-
fixed.
tion in
identical signal quality
lease auctions; bidders
who have
already drilled a neighboring tract are better informed
than outsiders. In the case of timber auctions, this extreme form of information asymmetry seems
unlikely,
although
it
seems plausible that firms with greater resources could invest more in
mation gathering. Indeed, there
"cruising" the tract.
cision
is
significant evidence that bidders
Persico (1998) has
can have important consequences
statics results
we
will seek to test
would
revenue predictions. Essentially
persist in a
acquisition. Heterogeneity of information quality
it
can invest varying amounts in
shown that endogenizing the information
for
is
infor-
all
acquisition de-
of the comparative
symmetric model with endogenous information
a more
difficult
problem, and we will not pursue
here.
Equilibria in Oral Ascending Auctions
3.2
We now
describe equilibria in a stylized version of an oral ascending auction.
conducted by the Forest Service are "free-form": after each bid
is
The
oral auctions
placed (where each bid must
specify bids for each species, not just the total bid), any bidder can choose to raise the bid.
may
bidder
place a bid, stay silent for a time,
and then become
to raise her
own
more than one
active again.^ If
bidder indicates a desire to bid, the auctioneer selects one. Importantly, a bidder
A
is
not permitted
bid at the end of the auction. If an auction ends, the high bidder
is
stuck with
the skew they have chosen.
One concern with
beliefs,
this
and
oral auction data
in particular, that
is
that announced bids
may be
non-winning bids
problem might be acute since bidders have
may
not be accurate indicators of
difficult to interpret.
significant leeway to
In the scale context,
engage in strategic bidding
behavior (such as skewing dramatically or onto the wrong species) early on to try to signal to
or confuse the other bidders.
Anecdotal evidence suggests these practices are not widespread,
but nevertheless, we want to consider a more formal analysis of bidder behavior in oral auctions.
In particular,
we
are interested in interpreting the skews of non-winning bidders, since these are
observed in our data.
in the course of
We
also wish to analyze the extent to
an oral auction, as many comparative
which information
statics results (for
is
accumulated
example, the prediction
that the expected winning bid alnd winning skew will be larger in an oral auction than a first-price
sealed bid auction) follow from this information accumulation.
Our
oral auction
model
is
similar to
Milgrom and Weber's
auction, which requires total bids to rise smoothly
The
his bid
Forest Service does have a rule stating that a bidder
on any given
species.
(1982, henceforth
MW)
"EngHsh"
and does not permit bidders to drop out of the
who
hcis
placed a bid earlier in the auction cannot lower
Past bids by other bidders do not place any restriction on a given bidder's behavior,
except for the standard requirement that total bids must go up.
10
bidding and re-enter.
(English scale auction) All bidders are active
in small increments.
increment.
At any bid increment,
They then announce
The auctioneer
at zero.
bidders call out a skew (A6) for the next
all
their activity for the next increment.
dropped out can become active again.
If
raises the total bid
No
bidder
who has
only one bidder announces activity, that bidder
is
declared the winner at her last announced bid.
Observe that relative to
MW, the scale auction has an additional ambiguity.
auction, a bidder's only decision
only that his signal
a bid allocation.
auction.
A
A
is
is
whether to remain active
— and by remaining
bidder could conceivably reveal
of his information in the
all
active,
he reveals
first
round of the
simple argument, however, suggests that in equilibrium, the release of information
total bid
was
a distinct bid allocation depending on their signal.
Aby^{B,d^) that maximizes their expected
next increment.
Then each
that point on,
all
zero, conditional
on the
each bidder's strategy was to announce
i?,
For example, they might choose the value
utility conditional
on their signal and on winning at the
bidder's information could be immediately inferred
All bidders
would bid
signals of all of the bidders.
until the
expected
all
Given a history and a bid
level
the marginal signal.
conditional
from winning
is
Doing so would reduce
We
conclude
B, we define the "marginal" private signal dy to be the
utility conditional
all
same
of signal types in equilibrium.
that leaves a bidder indifferent between wirming and losing at S,
each point in time,
opponents.
opponents, and give the deviator positive expected profits.
must be some "pooling"
maximize expected
utility
all
This cannot be an equilibrium, since any
individual bidder has an incentive to deviate by announcing a lower signal.
the drop-out level of
by
bidders would have the same information and thus the
expected value for the object.
that there
MW English
above some lower bound. In the scale auction, each bidder must also announce
must be gradual. Suppose that when the
Prom
In the
active bidders
on winning
at
B.
We
mimic the optimal bid
when the skew
is
signal
chosen to
propose an equilibrium whereby, at
allocation
announcement of a firm with
Further, bidders drop out at exactly the point where their signal becomes
marginal, so that bidders with higher signals stay in the auction longer. Thus, our equilibrium has
the same qualitative features as the
and
fully reveal their informatiibn
MW equilibrium:
when they drop
are just indifferent between dropping out
bidders with lower signals drop out
out; (2) bidders
and winning the auction
information that can be inferred about active bidders
some marginal
(1)
is
=
in the auction imtil they
at the current bid; (3) the only
that their signals aro at least as great as
signal.
Formally, order the bidders in descending signal order.
Pk
remain
first,
If
k bidders have dropped out,
{piT--,Pk} be the vector of total bids at which the drop-outs occurred.
11
let
Then, define the
B*{D; k,Pk)
function
as the solution to the following problem:
= max Es."^ u[Qact [{^b -
l\v)6
Ab.
subject to
Given
earlier
{B/Qest)
:
+ A6;^(l
V-B
+ ^^—-]
|4 =
]
...
4-'=
=
Afc^xi
>
=
Qest
- xi) > ri, (B/Qest) -
D,Pfc,x
(2)
7-2
drop-out prices P^, the function B*{D;k,Pk) corresponds to the total bid level at
= D can achieve a payoff of at most zero given that all other remaining
which a bidder with signal d^
bidders also have signal D.
Proposition 4 B*{D]k,Pk)
is strictly
increasing in D, and
we have now
Defining D*{B; k, Pk) to be the inverse of B*,
drop-out points Pk, and bid level B.
D*{B]k,Pk).
solution
We
Proposition 5 Assume
>
B*{D;k,Pk)
V.
identified the "marginal" type given
Let Ab^{B;k,Pk) be the bid difference that delivers the
can describe behavior
in the auction as follows:
that utility functions are
CARA
or
lARA.
(i)
There
Bayesian equilibrium of the English scale sale auction vnth the following bidding
that k bidders have dropped out
tion until
B—
and
until another
B*{d^^;k,Pk). For each
B<
opponent drops
= max E5 ^
J5* (d^ /c,Pfc), active bidders
;
subject to
Once
total bid
({Ab-Av)6+^^^-£])
u [qact
A6^
:
I
(B/Qest)
+ ^b^^^x -
^X'
is
greater than the
maximum
is
willing to
own
more
B)
that wins the auction solves
4 = 4,4,...,4,x
announcing a deviant bid allocation
would
is
payoff-relevant only
opponents the marginal signal D*{B,Pk,k).
optimistic, they will stay in the auction,
increased optimism
lose
money
if
is
has the
It also
variety of deviations from the equilibrium strategies, the
The main
about to drop out, must beconie slightly optimistic about that bidder's
more
— any winning
aggressive, until only the bidder with the
following a larger than expected skew by a given bidder, marginal opponents,
if all
r-,-^
must skew to win
equilibrium-path beliefs required to support the equilibrium are weak.
only
(3)
the contract.
Though bidders can make a wide
is,
Given
announce Ab^{B;k,Pk).
full-ii^urance bid with positive profits.
feature that as the bid level rises, skewing becomes
highest signal
strategies.
{^IQest) - Ab^X;^ >
again, the equilibrium has the property that bidders
a Perfect
each bidder stays in the auc-
out,
(a) The player with the highest signal wins the auction, and the (A6,
exists
if
If
restriction
i.e.
signal.
those
The
is
that
who
idea
is
off-
are
that
a bidder wins at that exact bid, that
a deviation makes marginal bidders
and no gain can be
realized.
The assumption
of
reasonable since a bidder with a signal less than or equal to D*{B,Pk-,k)
the auction ended following a deviation.
12
As mentioned above, the real-world Forest Service auction
differs
from our theoretical model in
Bidders can drop out and re-enter the bidding and the bidding
several ways.
However, even in a more
rise in
jumps.
— where bidders do not necessarily have to indicate
arbitrary jumps are possible — the behavior described
model
"free- form"
their activity at every point
may
and where
above can be supported in a Perfect Bayesian Equilibrium given strong enough assumptions about
off-equilibrium-path beliefs.
Consider an auction similar to the one above, except that after bidders indicate activity at a
given round (for example, by raising their hands) one bidder
,
total bid
and
call
is
recognized and allowed to raise the
out an arbitrary skew. Bidders then indicate their activity for the next round,
and the process repeats. Moreover, bidders may drop out (not announce activity at a given bid)
and then
until
re-enter later. In this auction, there will
dropping out once and for
all,
be an equilibrium
in
which bidders remain active
minimal increments, and skew
raise the bid level only in the
according to Ab^{-) as defined above. Essentially, two assumptions are sufficient to support this
behavior as an equilibrium: following a "jump bid"
or
larger than
(i.e.
some minimum increment),
an unexpected skew, opponents assign probability one to the deviant bidder having the highest
possible signal. Moreover,
if
a bidder drops out and then re-enters, armouncing activity at a later
time, bidders similarly assign probability one to their having the highest possible signal.
off-path beliefs support the above strategies as
to
make
an equilibrium, since there
no way
for a deviator
positive expected profits.
Although bidding behavior
is
unlikely to proceed in such a structured
qualitative feature of the equilibrium
is
somewhat more
robust: bidders
consistent with having observed the lowest possible signal that
This feature
them
is
These
to bid
is
natural because inducing
more
Forest Service.
aggressively.
The FS simply
more optimistic
way
in practice, the
announce skews that are
makes them profitable
beliefs
main
if
they
win.-^"
by the opponents simply leads
It is particularly useful for interpreting
the data recorded by the
reports the highest bid placed by a particular bidder. Since that
bidder might have been willing to bid at higher prices than the last recorded price, this bid can
serve only as a
bound on the
bidder's total value. However,
if
the bidder
is
playing according to the
equilibrium described above, the bid does have an exact interpretation.
The
can be interpreted as the optimal choice by a bidder whose signal
marginal for the level of
the total bid. Although
we
will not
pursue
it liere,
this fact could
is
losing bid allocation
be used to recover parametric
estimates of the underlying risk aversion and signal distribution in a structural model.
Equilibria with this feature also could be constructed with different rules about activity announcement, as in
Harstad and Rothkopf 's (1997) "alternating recognition" model, where only two bidders at a time announce
13
activity.
Predictions of the Theory
3.3
We now
summarize a number of theoretical predictions that we
data. Since the private signals of the bidders are not observable,
of the relationship between the observed bids
will
attempt to check against the
we frame our
predictions in terms
and the Forest Service mis-estimates (the
difference
between the estimated and actual species proportions) From the perspective of the econometrician,
.
=
both the mis-estimate, 6
The
xy—pi^ and the observed
model predicts a systematic
theoretical
Proposition 6 Assume
auction, bidder j
that utility functions are
conditional on any fixed total bid
We now
some
briefly
4>
summarized
is
CARA
it is
or
are
random variables.
in Proposition 6.
lARA. In
affiliated with 6
and
1.
is
when the bid and the skew
of volume,
2.
On
statics implied
over-estimated, so that S^
—
AV(j)
>
by the model. To state
= x^ — p^>0
is
will
for
is
i.e.
=
and Ab^
AV
•
b^
is
is
— 6-,^.
positive
direction.
expected to be greater than the amount paid per unit
•
(5]
>
0.
Pr{A6^ — At)^
in the "right" direction:
>
0}
>
5;
and
further,
increasing in 6^.
be nondecreasing
in the mis-estimate,
i.e.
—
is
a fixed total bid B^, £J[A&'
ElAbrL
—
Av^f,],
ElAbrL
be
Av\B^,6]
larger,
— Av^\B^
E[AV —Av\6]
increasing in
is
increasing
6.
both unconditionally and
for
any
are decreasing in j.
Conditional on the per-unit bid, the amount paid per unit of volume, E[P^ /Qact\B^ /Qest-,
is
6.
volume
same
of the higher ranked bidders should
given bid level B,
5.
skew
0\S^}
and similarly
The skews
are in the
E[B^JQest - P^ /Qact] = E[AV
The expected skew
in 6;
4.
unit of
average, bidders
'Px{Ahi',
3.
i.e.
holds
useful to order the species so that the Forest Service mis-estimate
be the species that
The amount bid per
sealed hid
same conclusion
the
Observe that Proposition 6 holds no matter how we choose to order the species; 6
exactly
and
the oral
BK
summarize a number of comparative
of the predictions,
positive. Let
B and A6 = hi —62,
relationship,
equilibrium choice of Ah^
's
bids,
S(j,],
decreasing in the over-estimate 6^.
Under CARA, the second-highest bid should be more skewed
auction,
i.e.
in
an oral than
E[Ab\ — Av^\S^,ORAL] — E[Ab\ — Av^\6^, SEALED]
14
is
in
a sealed bid
increasing in 6^.
Data on Forest Service Auctions
4
The Data
4.1
Our data
and cutting data from Forest Service timber
set includes a subset of the bidding
described in Section
To our knowledge, we
2.
are the
first
to
combine bidding and cutting data
We
to systematically analyze bidding behavior in timber auctions. ^^
where two
species" sales,
on the
We
tract.
restrict attention to
no more than two, each comprise at
species, but
least
25%
have found the most transparent way to think about skewing
is
we
restrict attention to oral auctions in
required only a small
amount
"two
volume
of the
in the context
of two-species sales because bidders can skew their bids only along a single dimension.-'^ In
of our analysis,
sales
much
Region 6 (Oregon and Washington) that
of road construction.-'^ In Section 5.4,
we
consider sealed bid sales
Region 5 (California).
in
In the bidding data,
cruise:
its
we observe
information that the Forest Service makes public after
all
estimated volumes of timber, the reserve price, and also
and projected processing
selling values
The bidding data
costs.
bidders and their bids. In the oral auctions, the bid
annoimced
bid.-'^
The
we observe
same records used
In our empirical analysis of skewing and revenue,
X. These include the volume
not the sale was a small-business set-aside
include the
number
higher density
may
material, which
is
of
months
we
we
that bidder's last
is
sale,
to bill the winning bidder).
include a
number
of control variables,
and the average reserve
and controls
Cummins
price,
whether or
These
for other sale characteristics.
more variable volumes) the volume of per-acre
;
and the estimated
A variety of additional sale characteristics are
required.
also available
report are robust to our choice of controls. In our analyses of skewing
(1995),
life
who
number
of bidders. •'^ Last,
uses the cutting data to analyze
we include dummy
how the timing
variables for
of timber harvesting varies
of the timber contract.
analyzing skews, we simply ignore any bids,and/or mis-estimates on the remaining species. In contrast,
when we study
'
for bidder j
in the contract per unit of volume; the density of timber (where a
control for the
with market prices through the
When
also includes the identities of all
essentially scrap; the estimated logging costs for the tract;
in the data; the results
'^Although see
we
of the tract
indicate "old-growth" and thus
amoimt of road construction
(but not revenue),
estimates of end-product
cutting data allows us to observe the total volume and species proportions
actually removed from the tract (these are the
referred to as
its
its
revenue,
we consider
bids on
all
species.
In Forest Service auctions, bidders are "reimbursed" for road-building using a system of credits that can be
redeemed
for timber.
Restricting attention to sales where road construction
is
minimal allows us to more
directly
interpret the prices paid by the bidders.
'
is
As noted above, a
bidder's last observed bid might bear only a distant relationship to the highest bid the bidder
willing to place, but our theoretical
to place
'^Our
much
model
gives
higher bids, the observed skew
results
is
some guidance
in interpreting the data:
even
if
the bidder
is
willing
optimal for some "marginal" signal.
about the number of bidders should be treated with some caution, since unobserved features of the
15
the time period, for several large forests, and for several
Appendix B provides further
I
contains
species combinations.
about our data sources and sample selection
criteria.
Table
statistics.
Preliminary Observations
4.2
On
summary
details
common
average, the revenue the Forest Service collects from a given sale
To
sample, $40,692) less than the winning bid.
arises,
we can decompose
see
how
term
first
the -^'olume cut
is
is
•
Qact,
6)
than the volume estimated), and the second term
less
amount than he bid
term accounts
for
terms averages out to $10,582. Thus
for
Chart
I
for the
average thousand board feet of timber removed from
less
than
it
bid (the average winning bid
The
about one of six sales (17%). Chart
(the difference
$143).
Service estimates and in the
Forest Service estimates of the species proportions differs
5%
from the proportions removed by more than
in
is
shows the distribution of Forest Service mis-estimates, demonstrating
the potential return to information.
10%
due to the bidder
a "representative tree" from the tract. In
A second feature of the data is the significant variance in the Forest
of skewing.
is
$30,110 of the $40,692 bid-revenue gap, while the second
a tract, the auction winner pays about $4
amount
our main
the portion of the bid-revenue gap due to the volume shortfall (on average,
5.8%
first
(in
as follows:
it
systematically paying a different
our sample, the
9%
around
the gap between winning bid and revenue
B-P= {B/Qest){Qest - Qact) + (A6
where the
is
II
in nearly half the sales
(45%) and by more than
shows how the winner distributed
between the total bid and the reserve
price)
among the two
his or her overbid
species.
As
established in
the theoretical analysis, a risk-neutral bidder with information would place his or her entire overbid
on one species or the other, but the chart shows
winning overbid
Finally,
significant dispersion
and many
sales
where the
distributed across both species.
is
we note one aspect
of the data that
is,
strictly speaking, inconsistent
with our model.
While our model assumes the existence of enough public information so that in equilibrium
same
bidders skew in the
scatter- plot of the
tract
may
affect
direction, this
is
clearly not the case in practice.
skew of the wirming and second-place bids
(III. A),
both participation and skewing behavior. However, our feeUng
endogeneity problem
is
less significant
than
in
is
Chart
III
all
shows a
and of the winning and
that for skewing regressions, this
standard analyses of auctions, where the main dependent variable
is
the magnitude of the bid. Here, the unexplained portion of a bidder's skew should correspond to private information
obtainea during the cruise.
If
one thinks of the cruise as tantamount to an entry decision,
it
is
plausible that
unobserved factors leading to entry
will
any event, our
both the omission of the number of bidders as an explanatory variable, as well as
results are robust to
to instrumenting for the
as the forest
and
number
be unrelated to information obtained after the decision to cruise
of bidders using
district of the tract
some of the standard approaches from the existing
and related measures, as
16
in
Hansen (1985) and Haile
(1998)).
is
made. In
literature (such
third-place bids (III.B).
variation. In
some
1,
dependence but also significant
scatter- plots exhibit a clear positive
auctions, bidders
overbid on species
skew
the other on species
in entirely opposite directions:
one bidder has
all
of the
2.
Bid Skewing: Some Examples
4.3
To
Both
some sense
give
of the pattern of bid allocation in oral auctions,
we
briefly discuss a
few "case
studies" of bidding in oral auctions. In our theoretical model, skews increase monotonically with
oral auction bids.
Moreover, in
While
many
many
sales in the data, the
on one
an important
— at
by no means the
species, so that the reserve prices are binding.
In fact, there are
by
theory.
The
following
is
is
role
an intermediate skew, and the skews
one such case:
Example: Monotonic Bidding Behavior.
Ab - Ar
%
6
=
.019
overbid on sp. 1
(A6 - Ar)
Bidder
B/Q^.^^
1
347.8
339.39
.94
6.45
2
346.5
334.39
.94
6.35
3
285.6
50.39
.56
.96
4
277.5
33.39
.52
.63
5
250.72
.39
.48
.007
On
rule.
one of the bidders
few examples of auctions where every observed bid
are ordered exactly as predicted
it is
least
auctions, the reserve price plays
places his entire overbid
relatively
this is the case in
Sp. 1 skew:
6i
where the
occasion, skevnng can have very large consequences. Consider the following sale
bidders skewed heavily in one direction at the start, then switched later in the auction.
Example: Switching Skews.
A6 — Ar
%
—
.227
(A6 - Ar)
Bidder
B/Q^^j.
1
234.3
523.96
1
118.9
2
233.9
522.96
1
118.7
3
207.6
4
193.8
Sp. 1 skew:
;
overbid on sp. 1
-396.62
-90
-353.66
-80
In this example, earlier bids are skewed into species
turned out, the average payment was
less
2,
Forest Service would have received
which was
than half of the winning
had been awarded the contract, they would have paid
The
6
17
^i
in fact under-estimated.
As
it
bid; if the third or fourth bidder
significantly
more than
more revenue from awarding the
bidders.
•
their average bid.
tract to
one of the lower
There are a small number of cases where bidders seemed to place a "surprise" skew at the end
of
an auction.
Boise Cascade did this in two instances.
In each instance, several lower-ranked
bidders had placed large skews in one direction, and Boise Cascade
won
the auction with a very-
small increase in the average bid, but with a switch in the skew of several hundred dollars.
one case, the skew turned out to be in the right direction; in the other,
it
was wrong.
It
In
appears
that information affects allocation of the tract, since the winner pays a very different price than
the next-lower bidder would have for the
winner anticipates
same
object,
and the winner's behavior reveals that the
this.
Evidence on Skewing
5
We now
turn to testing whether the predictions of the theory are consistent with the Forest Ser-
Before presenting our empirical analysis,
vice data.
we
discuss two choices
we made
that merit
discussion.
First, the theoretical
allocation; that
model suggests thinking about the skewing
the variable of interest
is,
We
to the "values" in our model.
in the values of the species,
this
amounts to a
i.e.
A6 — Av.
However, the data contain no exact analogue
use the difference in reserve prices as a proxy for the difference
we assume
on the
restriction
is
relative to a "full-insurance"
that
coefficient of
Av =
Ar. In a linear regression of
Ab on Ar,
Ar. Such a restriction seems reasonable
—
in
a subsample of 97 auctions where mis-estimates are small and the reserve prices do not bind, a
regression of
A6 on Ar
leads to a coefficient of 1.055 on
Ar
(with a standard deviation of .171 and
ani?2of0.28).
A second point concerns the way in which we order the species.
unconditional relationship between
(A6i,<5i)
= — (A62,62)).
Ab and
Yet we would
6,
If
we were
interested only in the
the order of the species would be irrelevant (because
like to control for
the possibility that
some observable
features of the tract might explain the fact that bidders skewed in the right direction. This suggests
studying the relationship between the absolute magnitude of the mis-estimate {6^
skew
in the "right" direction
predictions. In Section 6,
is
correct, infer
and then
5.1
We
x
we
Ab^ — Ar^.
We
>
0)
and the
follow this approach in order to test the model's
consider an alternative approach: namely,
(the species that the winning'tidder believes
is
most
we assume that the model
likely to
be over-estimated),
directly estimate bidding functions A6j^(-),.B(-).
Winning Bids Reflect ex ante Information
begin with the primary question of whether winning bids in Forest Service auctions reflect ex
ante information about Forest Service mis-estimates.
18
The
following cross-tabulation suggests that
winning bids tend to be skewed in the right direction, especially when the Forest Service makes a
significant mistake in estimating the species proportions.
Mis-estimates and Skews in Region 6 Oral Auctions
Incorrect
(Ab^
Small Skew
Skew
- Ar^ < -20)
Correct Skew
< Ab^ - Ar^ <
(-20
(A6^
20)
- Ar^ >
20)
Small mis-estimate
{6^
<
mis-estimate
Larger
^
{64,
83
44
90
139
104
239
.025)
>
.025)
Overall, the winning bid
is
skewed correctly {Ab^—Ar^
>
0) in
58%
of the sales. Table 2 reports
a probit regression, confirming the theoretical prediction that, conditional on the control variables
(X) larger mis-estimates correspond to a higher probability that the winning bid
,
right direction.
An
increase in the magnitude of the mis-estimate of .01
a 1-2% higher chance of the wirming bid being skewed correctly.
relationship
is
Column
most interesting
result
is
that the
number
(2) establishes
Of the
test of
is
that the
a 2-4% increase
control variables, the
of bidders increases the probability of skewing in the
right direction. This finding supports the hypothesis that information
To obtain a more rigorous
skewed in the
associated with roughly
concave: the initial effect of an increase of .01 in the mis-estimate
in the probability of a correct skew, but the slope diminishes in S^.
value beyond
is
is
whether bidders in
what can be publicly observed, we would
is
revealed in the auction. ^^
fact
have ex ante signals about the ex post
like to
account for the competing hypothesis
that Forest Service estimates are systematically biased, and bidders have realized this and just skew
systematically in response.
To do
this,
we ask whether
unanticipated deviations in 6 help to explain
the winning bid allocation, controlling for exogenous sale characteristics (X) that might "predict"
the mis-estimate. However, theoretically
we have no reason
to expect a linear relationship between
X and the skew; X might affect the bidder's risk aversion, or her beliefs, in a variety of ways. Since
empirically, X
correlated with 6^,
important to isolate the effect of the mis-estimate from
a potential nonlinear relationship between X and the distribution of 6^. To accomplish this, we
is
it is
consider the following model:
;
Ab^ - Ar^
where
X
As dicussed
as bidders
h{X)
+ 6^j + e,
contains sale characteristics as described above.
information about S^,
'
=
we should
in footnote 15,
it
find
7
=
If
(4)
the bidders do not have ex ante
0.
seems plausible that the number of bidders
must cruise the tract to learn about mis-estimates. Our
19
is
exogenous
in the
skewing regression,
results are robust to excluding this control.
We
(as listed in
h{7i)
+ W^,
Table
2) into
where h
is
Z and dummy
continuous regressors
X
Dividing the control variables
consider a semi-parametric approach to estimation.
variables
W,
we
=
allow /i(X)
We
an arbitrary continuous function of two indices (Ziai,Z2Q!2)-
begin
by estimating
A6^ - Ar^ =
and hs take a double-index
letting /lA, f^w
model
is still
W = hwCZ) + ew and 8^ = hs{Z) +
+ ea,
/ia(Z)
somewhat
form.-'^
While
it
es,
(5)
should be noted that our double-index
our results are robust across a wide range of specifications of
restrictive,
the two indices Zi and Z2. In the specifications we report,
we
Zi be the average reserve
let
price,
while Z2 contains the remaining continuous control variables. Denoting the residuals from these
semi-parametric regressions as
eA,Gw and
GA
Our estimate
of the coefHcient
in the mis-estimate of .01
We
can
7
ss,
we then use ordinary
= ew/3+e6-f + e.
for the
estimate of
7
7
=
at the
to encourage
more
=
1%
level.
an increase
X/3.
These findings can be compared
As shown
of 383.8 with a standard error of 140.1.
magnitude of the
is,
associated with a $4 increase in the skew (in the right direction).
is
reject the null hypothesis that
OLS
(6)
406.8 with a standard error of 114.8. That
is
with a simple linear specification, assuming /i(X)
an
least squares^^ to estimate
in
Table
Table
III (2), this yields
III (3) includes
a control
bid; this captures the idea that part of the effect of the mis-estimate
aggressive bidding.
we
Indeed,
is
find a positive, significant effect of the value
bid on the skew, alongside a somewhat smaller point estimate of 7 (319.6 with a standard error
of 121.9). However,
we
are cautious about interpreting these coefficients as structural parameters
we have ignored the
of the bidding function, due to the fact that
per-species reserve prices bind in about
able to skew as
much
as they
42%
would have
issue of the reserve prices.
of our observations; this implies that bidders were not
liked,
without further raising their overbid. Further, the
magnitude of the overbid, an endogenous variable that depends on
the reserve prices bind. Thus, we interpret the coefficients in Table
deferring a
more formal
While establishing a
'^To estimate,
e.g.,
III as
—
Ar<^)
and 6^ to be
linear relationship js sufficient for testing our hypotheses
about bidder
Ab^ — Ar^ =
/i'a(Z)
-t-
ea, we used the average derivative method of Powell, Stock and Stoker
(1989) to obtain consistent (up to scale) estimates of ai,Q:2, where /ia(Z)
=
/iA(Ziai, Z20C2)-
Ab^ — Ar^
used a routine described in Stoker (1989) to choose the bandwidth (the values were
1.1 for the second),
whether or not
reduced-form correlations,
between (A6<^
kernel estimator (with a normal density) to estimate the function /ia, estimating
We
6^, determines
analysis of the reserve prices to Section 6.
Finally, all of these specifications restrict the relationship
linear.
The
and we used a trimming
rule of
5%.
Tom
=
We
then then used a
/ia(Ziq:i, Zi20C2)+£a-
.41 for the first
index and
Stoker generously provided portions of code for the
estimation.
'*As Robinson (1988) points out,
the
OLS
OLS
estimation of ca on
ew
and
es yields consistent estimates of
/3
and 7 and
standard errors are correct. See also Stoker (1989) for a clear discussion of the model we consider.
20
information,
it
is
To do
mis-estimates.
Z = (Zj,Z2) and
of Powell, Stock
more generally
interesting to inquire
W
into the relationship
we estimated the model A6^ —
this,
Ar^^
=
between skews and
/i(Za, 8^)
+
W/3-|-e (where
are defined as above), using the density- weighted average derivative
and Stoker (1989)
of specifications of the index Z,
statistically significant at
a
5%
is
as described above.
The main
finding, robust across a variety
that the average derivative with respect to 6^
level of confidence.
To
method
is
positive
and
see the shape of the functional relationship.
Chart IV plots the resulting estimate of the relationship between the skew and the mis-estimate
(/i(Za,5^), evaluated at the
mean
of Z).
The
chart illustrates the positive slope and a generally
convex shape over the region of 6^ with most of the density weight.
Bidders have Different (Private) Information
5.2
A
key aspect of our skewing model, and indeed of any "common-value" auction,
is
driven by private information rather than preferences.-'^
optimistic about the gap between bids
bids.
Thus, we would
We
in their skews.
like to
know
the right direction: because information
The top bidder should
are
more
data whether higher-ranked bidders were more aggressive
is
likely to
be skewed in
revealed during the course of an oral auction, the
more information than the
+
[j
l)«*.20
pjnally, our
j*'^
model
two bidders should exhibit very similar skewing
of information revelation implies that the top
behavior.
who
In our model, bidders
whether higher-ranked bids are more
highest bid should incorporate strictly
that allocation
and payments skew more aggressively and submit higher
in the
also seek to test
is
stay in the auction for only one bid increment past the second-
highest bider, while lower-ranked bidders will in expectation drop out at significantly lower bid
levels
with significantly
To
less aggressive skews.
more
investigate whether higher-ranked bidders allocate their bids
accurately,
we look
at
the following model:
4
Dummy(Skew
We
in right direction)jt
=
2_\ (^j
'
Dummy(Rank
+ Xt/3 + ut + Sjt-
(7)
estimated this as a fixed-effects linear probability model. Using a fixed effect for each auction
eliminates the auction-specific sale characteristics
effect of the rank.
The
results dre reported in
and the disturbance
Table FV. As expected,
Industry sources reported complaints by bidders that mis-allocation results
information
(i.e.
suggests that
ut,
we
allowing us to isolate the
find that the winning
when
their
and
opponents have "bad"
were "over-optimistic" about the skew).
As we noted in Section
it
j)
all
4.2,
our theoretical model
is
not consistent with the data on this point, since in equilibrium,
bidders will skew in the same direction (which
that bid j in an oral auction will be based on
is
not the case in practice). However, the result
more information than bid j
of models.
21
-I-
1
should be robust across a wider range
second-highest bid are significantly
and fourth-highest
We
to be
skewed
in the right direction
than the third
bids.
The second question
we
more hkely
is
whether the skews of higher-ranked bidders are more aggressive. Here,
restricted attention to sales
where none of the top four bids
is
censored by the reserve
price.-^^
estimated
4
-
{Ab^
=
Ar^).^
^
Dummy (Rank j) + Xt/3 + Ut + Cjt,
Qj
again using a fixed effects specification.
As
shown
is
in Table
IV
(8)
the winning and second-
(2),
highest bids were similarly skewed, while the third and fourth-highest bids were significantly less
skewed (by roughly $20-30).^^ The third column of Table IV reports a
the dependent variable
an alternative measure of aggressiveness: the absolute magnitude of the
is
accurately and concentrate solely on the size of the skew.
are very close, while the third
and fourth bids are
less
We
again find that the top two bids
skewed. In this case, there
between the magnitude of the third and fourth-highest
skewed than the top
Our
where
This specification separates out the fact that higher-ranked bidders skew more
bidder's skew.
difference
final specification
bids;
is
a substantial
both are significantly
less
bids.
findings are consistent with the hypothesis of differential information
among the
bidders.
Suppose, alternatively, that bidders have identical beliefs about the true volumes on the tract and
choose optimal decision-theoretic skews. In this case, observed differences in bids and skews might
be driven by differences in costs, private values
due to
different bid levels
and values
for the species, or risk aversion (either intrinsic or
for the tract)
.
For instance,
then bidders with a higher overall value
risk aversion,
aggressively.'^^
if
bidders have decreasing absolute
for the tract
would be willing to skew more
Such a model has higher-ranked bids being more skewed, though
the same direction. Idiosyncratic variation in
Av
all
bids skewed in
also could generate variation in skewing,
simply
through differences between the "full-insurance" allocation.
These private value explanations have number of shortcomings.
First, there
is
no reason to
think higher-ranked bids should be more accurate. Second, there are numerous cases in the data
where we observe a given bidder bidding on some combination of two species
fir)
in multiple sales
and skewing
in different directions in different sales.
^'Although this restriction leads to a non- representative sample,
ferences in behavior across ranks within
pronounced skewing behavior than the
One concern
reserve prices.
is
We
that
we have
an auction.
full
We
it is still
We
hemlock and
Third, reconciling the
possible to test predictions about dif-
expect that the sample of uncensored auctions displays
which none of the top four bidders were constrained by the
obtained similar results
all sales
where the top three
— no significant distinction between the top two bidders;
the third-highest bid tends to be closer to full-insurance.
We
less
sample.
relatively few sales in
performed a robustness check by enlarging the sample to include
bidders were unconstrained.
(e.g.
are grateful to Phil Haile for pointing out this potential explanation for monotonicity of the skews.
22
observed data with differences in private values or risk parameters seems to require implausible
The
differences across bidders.
some compression
can
fact that bidders
timber and subcontract logging suggests
resell
among
in values for the different species. Yet,
the subset of 72 auctions in which
the top four bids are not close to the reserve price, the average difference between the largest and
smallest skew
was substantial
— $46 (with a standard deviation
of $86
and maximum of $393) To
.
place this in perspective, the average winning bid was $167, while the average magnitude of the
winning skew was $87. Finally, while our information-based model of bid allocation
with the similar skews of the top two bidders relative to the others,
be explained in a model without some aspect of
We
informed."
is
This might
sale.
reflect either
Second,
hard to see how this could
differential information.
draw three conclusions regarding the variation
skews can be large within a given
it is
consistent
is
it
skews across bidders. First, dispersion in
in
appears that bids later in the auction are "more
more accurate
prior estimates or the fact that information
acquired during the auction. Third, given that winning bidders are effectively paying a lower
percentage of their bids than lower-ranked bidders, the logical consequence of our findings
is
that
information should have an effect on allocation.
Revenue
5.3
Effects
One simple consequence
of the scale sale format
is
that losing bidders might have actually generated
more revenue than the winners, given the actual volumes
have generated more revenue in
million,
17%
cut. In our sample, a losing bidder
of the sales; the revenue "loss" from this misallocation
would
is
$5.39
about 1.7% of the total revenue. While such discrepancies have attracted the attention of
policy-makers (see
GAO report RCED-83-37), such magnitudes should be interpreted with caution:
the losing bids as well as the wirming bids were placed taking into account the benefits of skewing,
so clearly, any change in the allocation rule would also change the equilibrium bidding strategies.
Given our model,
Service error, that
is
it is
interesting to ask
when 6^
Service per unit of volimie
is
With two
increases.
equal to
how revenue changes
^^
,
so the change in revenue
A60
increase in 6^ affects both the magnitude of the bid
•
6^. Consider first the latter effect.
We
when Ar^ >
0,
the
mean
>
of iSb^
=
'
^<t>
- ^h-
have already established that
as well. But,
$99;
is
if
Ar<^
when Ar^ <
0,
(9)
and the gap between bid and payment,
favor of a reduction in revenue. However, the sign of A6<^
then on average we expect A6<^
by the Forest
O
Q
^^Revenue = -^t-^/Qest - -w^^h
An
an arbitrary Forest
species, the revenue collected
B/Qest — ^b^
Q
in response to
<
ambiguous.
is
0,
the
we cannot
mean
>
g|-A6</,
If
Ar^
is
a force in
greater than 0,
sign A6<^. In our sample,
of A6<^
=
-$46. To capture the
net effect of the mis-estimate on the gap between the bid and the payment, Table
23
0,
V
(1) reports
the results from the following specification:^^
B/QeST - P/QaCT =
Overall, the gap between the bid
Ar<^, but derivative
is
larger
+ <5<;il{Ar^<0}72 + X/3 + £.
^.^l{Ar^>0}7l
and the payment
when Ar^ >
is
(10)
increasing in 6^ irrespective of the sign of
<
than when Ar^
($110 with a standard error of 34,
versus $82 with a standard error of 24)
Now consider -^B/Qest- Our theoretical analysis established that the level of the bid increases
when 6y^ increases. However, it is not necessarily true that (^ = x- As analyzed above, Pr(^ = x\^<l>)
is
nondecreasing in 5^; but, for those cases where
(f)
^
x^ then
is
6(j>
affiliated
with
—d-)^.
leads such bidders to be less optimistic about the magnitude of the mis-estimate,
Overall,
if Pr((/i
=
x\^<^) is
see that
when
Ar<^
<
B/Qest on
0,
may be
higher mis-estimates lead to higher bids, but not
report the relationship between revenue
An
specifications as above.
with a
{P/Qact) and the
The
is
when Ar^ >
on revenue,
on Ar<^ >
effect is relatively
less.
V
(10).
(2)
We
0.
in Table
0, is
V
(3)
same
associated
small in magnitude (a
associated with roughly a $12,500 reduction in revenue). Conditional on the
is
associated with an (even smaller
effect
on the revenue received by the
lower-valued species being over-estimated, a larger mis-estimate
in
from
mis-estimate, following the
increase in the mis-estimate, conditional
statistically significant reduction in revenue.
mis-estimate of .05
ambiguous.^^ Table
sale characteristics, following the specification
Finally, to estimate the net effect of a Forest Service mis-estimate
we
and to bid
6^
high enough, the expected value of the winning bid should be increasing
in the actual mis-estimate 5^; but, in general the relationship
reports a regression of
A higher
magnitude) increase in revenue.
Summarizing, mis-estimates appear to have a negative
we mention
Forest Service. However,
include only linear functions of
when we
X
several caveats.
and
6^.
The
While the
first is
result
functional form; the specifications
about revenue
loss
is
only reinforced
allow for richer functional forms for X,'^^ the functional relationship between
and 6^ appears to be non-monotonic, and our estimates are somewhat
B/Qest
sensitive to specification.
Second, as discussed above, we have not accounted for the fact that the reserve prices constrain
skewing behavior in
^* As
42%
of the sales.
return to this issue in Section
^^Even
if
-^B/Qest >
0,
results'
report the gap between the bid
the magnitude of
aversion of bidders and the variance of
also estimated the
6.
B/Qest — P/Qact ^ Ab^
the sales in our sample have more than two species,
approximation. The empirical
We
We
•
6^, although this
and payment including
-^-B/Qest depends on
is
a close
all species.
a number of factors, including the risk
6.
models of Table
V
(2)
and
(3)
following the two-step procedure described in equations (5)
using the semi-parametric methods from Section 5.2,
and
(6).
In the bidding equation, our estimates of 7i and
72 are 110 with a standard error of 43, and -16 with a standard error of 39, respectively. In the revenue equation, our
estimates of 7j and 72 are 49 with a standard error of 42, and -121 with a standard error of 37, respectively. Thus,
relative to the linear model, these results suggest greater revenue loss.
24
Sealed Bid Auctions
5.4
We now
some evidence on
report
We
sealed bid auctions.
auctions because the interpretation of skewing behavior
relevant in the event that the firm wins,
and so
it is
is
are especially interested in sealed bid
unambiguous - each
firm's bid
is
only
a dominant strategy for each bidder to choose
the skew that maximizes expected utility conditional on winning and the bidder's private signal.
Using a small (63
sales)
sample of "two-species" sales from Region
We
conducted on the oral auction sample.
we
5,
reprise the
main
tests
chose Region 5 because sealed-bid auctions were more
prevalent there.
The
first
column of Table VT reports sealed bid estimates of the probit model from Section
where the dependent variable
5.1,
is
the probability that the winner skewed correctly. Overall, the
65%
winner skewed in the right direction in
when the
the right direction
The
mis-estimate was large.
increasing in the mis-estimate 5^ (Table
VI
and was more
of the sealed bid sales,
(2)).
size of the winner's
skew
in
skew (Ab^ — Ar^)
is
likely to
An increase in the mis-estimate of .01
is
associated
with an additional $3.80 skew in the right direction (our point estimate in the oral auction sample
was
$3.83).
Thus
it
appears that in sealed bid auctions, as well as in oral sales, the winner's bid
does incorporate information about the species proportions beyond what
is
known from
the Forest
Service estimates.
The
last three
columns of Table VI report sealed bid estimates of the
cations, analogous to Section 5.2.
We
restricted the
sample
three bids (colimin (3)) and then (columns (4)- (5)) to a
bids are unconstrained
by the reserve
prices (that
is,
first
still
to sales where there were at least
smaller sample where the top three
where the overbid was large enough
the skew that the reserve prices do not bind). Across ranks,
we did not
in the accuracy of the skews, nor in the
magnitude of the skews in the
We
skew
did, however, find that the winner's
skews of the lower-ranked bidders. In
fact,
is
fixed-effects panel specifi-
find significant diff'erences
right direction
significantly larger in absolute
the magnitude of the skew
is
Since there
is
no learning
in a sealed bid auction,
more accurate information than other
optimistic,
it is
likely to
skew
One
larger in
theoretical prediction that
an oral
sale.
we were unable
Our samples
though not perfectly
not clear that the winner should have
larger absolute
in the right direction, this should
skews in the right direction (which we did not observe).
to test
of Region 6 oral sales
is
— Ar^).
magnitude than the
bidders. However, according to theory, the winner
and hence skews more aggressively,^leading to
Because bidders are
(A6^
ordered by rank. These
findings are fairly consistent with the logic of information-based skewing,
so.
relative to
is
more
skews (which we found).
presumably also lead to larger
^^
that the skews of the second-highest bids should be
and Region
5 sealed bid sales are sufficiently different
(Region 5 sales generally having younger, less-valuable, second-growth trees, and fewer bidders) that we did not
confident
and
is
making
direct comparisons.
Moreover, the choice of auction format
is
conditioned on sale characteristics
endogenous. In exploratory work using a mixed sample of sales from Region
25
feel
1
(Idaho and Montana), we did
Moral Hazard
5.5
An alternative story,
is
in Cutting:
An
Alternative Hypothesis
many of the empirical regularities described
that can potentially explain
that bidders do not have private information ex ante, but instead are able to destroy
timber on the tract rather than pay for
it.
some
of the
This would lead to a moral hazard problem, where sale
winners would have incentives to destroy low-quality timber rather than harvest
accounts from the industry suggest that loggers
post measurement process.
above,
may have
Anecdotal
a limited ability to manipulate the ex
This would help to explain the
and observed volumes. ^^ Moral hazard could
it.^^
6%
difference
between the estimated
also explain observed skewing, since higher bids
on a
given species would imply a larger motivation to destroy low-quality logs of that species^^ — thus
generating a correlation between skews and differences in estimated and cut species proportions.
Of
course, the presence of moral hazard
and might
is
not inconsistent with our story of ex ante information
reinforce incentives to acquire information
see whether
it
will
be easy or
difficult to
— bidders may want to examine a tract to
destroy low-quality timber of a given species.
We
can
place this in the framework of our model by thinking of easy-to-destroy low-quality timber as being
simply not present. In general, the more skewed the bidding, the higher the incentives for ex post
destruction of one of the species;^^ and, conversely, the easier
particular species ex post the
more
it is
to destroy low-quality trees of a
incentive to skew.
Several pieces of evidence weight against an explanation of the data based solely
With pure moral hazard, the
ard.
total
on moral haz-
volume of timber destroyed should be strongly correlated
with the "observed mis-estimate." But in the data, the correlation between the proportions
mated and cut and the
total
volume estimated and cut
a t-statistic of 0.32). This finding
is
very low and insignificant: 0.053 (with
confirmed when we control for observed sale characteristics.^"^
Moreover, the distribution of skews within sales
We found in
is
is
hard to rationalize with a moral hazard
some
A
difference
between skewing behavior
in oral
and sealed
sales,
skew
in the right
but perhaps due to the very small sample the
were sensitive to specification.
results
^*
story.
Section 5.2 that the second-highest bidder was just as likely to skew in the right direc-
tion as the winner, while the lower-ranked bidders were significantly less likely to
find
esti-
similar story has loggers colluding with the monitors responsible for measuring the timber as
^^Leffler
and Rucker (1988) analyze the
eflFects
of various policies on the harvest rates
showing that a higher percentage of the estimated timber
''"Only logs that
at a different rate.
low price
for
meet a certain threshold are counted
is
at
extracted
all;
when
it
leaves the tract.
on Forest Service contracts,
contracts are shorter in duration.
broken limbs and other damaged material
is
charged
Thus, a bidder could avoid "stealing" timber simply by breaking a log into pieces and paying a
it.
Interestingly,
one Forest
Servi::e official told us that cutting
was monitored intensely when bids were especially
skewed.
We
also looked at
how the gap between estimated and
Because timber prices were quite
were a primary concern.
On
volatile,
average,
it
cut total volume and species proportions varied over time.
one would expect the gap
in total
remained relatively stable over time.
26
volume to vary a
lot if
moral hazard
direction. This
is
sensible in the context of
an ex ante information
story,
but
if
the primary force, the winning bidder's cutting behavior should respond to her
moral hazard were
own skew,
while the
second-highest bidder's skew should be less closely related to the actual difference in proportions.
Bidder Heterogeneity
5.6
model assumes that bidders are homogeneous. However,
our background discus-
Our
theoretical
sion
on the timber program, we raised the issue that bidders can be a very heterogeneous group,
in
with giant conglomerates bidding against small independent logging operations. Because
seems
it
plausible that there could be systematic differences between different type of bidders in terms
of their skewing behavior,
we
our attempts to uncover observed differences between
briefly note
bidders that might be major (un-modeled) determinants of skewing behavior.
We classified the bidders into three groups
(small, mid-size
and
large)
on the basis of the number
of employees. Small bidders have less than 100 employees, mid-size bidders
300,
and large bidders have over 300.
We then looked in a variety of ways for systematic differences
in the skewing behavior of these groups of bidders.
give a brief
summary
right direction,
However,
it
somewhat more
We
also
of our findings.
No
likely to win. Overall,
specific empirical results,
particular class of bidders was
their bids
we
more
likely to
skew
likely to
in the
large, smaller bidders
were
did not find large differences between bid groups.
None
examined the individual skewing behavior of a few frequent participants.
more
we
somewhat more dramatically.
appeared that in sales where the mis-estimates were
these were significantly
skew
of
in the right direction, although a few bidders (such as
Boise Cascade) skewed more aggressively (that
We
Rather than report
though larger bidders did appear to skew
also
have between 100 and
might attribute this behavior to a lower
is,
the average magnitude of the skew was larger).
level of risk aversion, as
Boise Cascade
is
a large
conglomerate.
6
Direct Estimates of the Skewing
In the previous section,
out in the data.
correct
we
Model
tested whether predicted relationships between variables were borne
We now take
a somewhat different cut at the data, by assuming that the model
is
and attempting to directly derive and parametrize bidding functions. There are a number
of reasons to take this approach:
/of bidders
first, if
the model
skews to information; and second,
it
is
correct,
it
allows us to quantify the sensitivity
allows us to investigate in
which the reserve prices constrain skewing behavior.
27
some
detail the
way
in
Structural Bid Functions
6.1
We
by assuming that bidders have (approximately)
start
We
be ignored.
CARA
utihty, so that wealth effects
can
impose a number of functional form assumptions, beginning with an assumption
about the species values v.
(Bl)
=
tifc
rfc
+X
^fc+7?, for
fc
=
1,
2 (so
Avy = Ar^),
77
independent of
r,
X,a;^,
Sy,.
In an oral auction, the winning bid allocation solves equation (3) from Section 3.2.
ing the information that this bid
is
conditioned on as
d^^,
we can
Writ-
express the winning skew as
Ab^{B,dy,'K,r,e,x-)^), where
4^^-^)/^^^^ iind^,X,e)>^iB-R)/QEST
A6,(-)-Ar^ = (
When
otherwise
/(dj^-,X)
[
the reserve prices do not bind, our assumption that the utility function
is
that the optimal skew /(•)
is
independent of the total bid. Moreover, Proposition
any equilibrium bid A6^
>
Avy^
at
=
CARA
1
implies
implies that
Ar-^ so the reserve constraints can only prevent bidders from
skewing as much as they want.
The
function of interest
the optimal unconstrained skew /(•), which
is
we observe only when
the reserve price constraint does not bind. Because whether or not this constraint binds depends on
we need
the total bid level B,
(3) in
to
model
this selection.
Section 3.2, and can be written as
B{-)-R
^
level again solves equation
5(d^,X, r,e, 77,0;^):
(
p^(d-,x,X,r,e,r7)
1^
5"(d^,x,X,
Qest
The winning bid
r,£,
if
/(d-,x,X,£)
> ±-(B - R)/Qest
otherwise
7y)
To complete the model, we make assmnptions about functional forms. Under the assumptions
of our theoretical model, the bidders observe x, which provides information about the direction
As
of the optimal skew.
that Aby,
— Ar^ >
0.
this
can be inferred directly from the winner's skew, we order species so
Thus, both the overbid and the skew
will
always be positive in our model.
This motivates a specification of the bidding functions in logarithms (recalling that our continuous
control variables are denoted Zj
(B2)
=
~6^
(B3) dj
IniS^
+ .3); Xi^
= ln(dp =
6y^
and the
ln(Z.)
+ e,e
dummy
+ W^
independent of
/()
=
~6^ai
+ Xa2 + saia^.
(B5-) ln5(-)
=
'6^P,
+ X/32 + eP-^a, + r^P'^u^.
(B4)
In
variables are denoted
r,
X,x^, 6-^.
28
Wj):
Consider the interpretation of these equations. First, (B2) introduces logarithmic transformations of the continuous control variables; since our
low as —.27, we add
estimate
is
error term.
.3
sample includes
sales
with mis-estimates of as
(B3) states that the bidder's signal about the mis-
to the mis-estimate.
the
sum
We
interpret e as the error in the bidder's estimate of
of the (transformed) actual mis-estimate, and an independently distributed
more optimistic about the extent
of the over-estimate
on species
<5t^;
a bidder with a higher e
x- In (B4)
is
and (B5), we assume
that the log of the overbid and the log of the skew are both linear in the log of the signal.
Of course, the
and we do not attempt to
functional forms in (B2)-(B5) are ad hoc,
relate
we
the primitive utility function; but they offer several distinct advantages. First, below
an assumption on the distribution of the
thus,
we
will
assume below that e and
t]
errors,
and the normal distribution
will
them
to
will require
be convenient;
Second, although we did not
are normally distributed.
formally model the choice of transformation, several back-of-the-envelope calculations^^ indicated
that a logarithmic transformation of the skew and overbid provides a better
elasticity estimates are fairly robust to the
way
in
fit
than
linearity.
Our
which variables are scaled. Third, and most
importantly from a computational perspective, our functional form assumptions imply that the
probability that the reserve price binds
Pr(res does not bind)
Notice that
x-^,
is
a linear function of observables and unobservables:
Pr(/(d;^, x,
=
Pr(/i(d-,x,X,r,X;,)<
is
.25.
45%
full
and our focus on
sales
B
to results
The
[.25, .44]; call this
(524 sales). In group A, the reserve prices bind
when the
a crude approach
30%
group
A
and the
of the time, as opposed
to understanding the effects of the
on skews and bids
sales are divided according to
in each
group and compare
whether the reserve prices were binding
in
following table summarizes the results about the effects of mis-estimates:^^
we performed Box-Cox estimation on uncensored subsamples
ignoring censoring; however, that approach
These
due to Forest Service bidding
with two primary species, our data set includes only sales where
One-fourth of the observations (170 sales) have x^ €
In particular,
^''
u).
distribution of u. However, in practice,
of the time in group B.; This suggests
practice.
(11)
Thus, in principle, variation in Xy can
reserve prices: estimate the effects of mis-estiiriates
them
X, r, e, 77))
required to achieve the desired level of skew.
complement group
to
5"(d;^, x,
observe that for a small Xy, only a small
restrictions
>
<
its effect,
be used to identify the
Xy
X, e)
the fraction of the volume estimated on the over-estimated species Xi affects
whether the reserve price binds. To understand
overbid
—
=
elasticities
were estimated using
is
not rigorous unless censoring
OLS
is
of the data, as well as the
accounted
full
sample
for.
with robust standard errors, and using the specification of control
variables as in Table VII.
29
Contrast: low
Elasticities of:
Skew
A
Group
w.r.t.
0.065(0.025)
Overbid
Group
0.045 (0.021)
mis-est. 6^
The
A
high
v.
Group
mis-est..^^
w.r.t.
(A)
Xy,
x-^^
Contrast: Reserve prices binding
(B)
B
Not bind
Bind
0.018(0.010)
0.020(0.011)
0.032(0.011)
Group B
Not bind
Bind
0.013 (0.008)
0.010 (0.009)
0.035 (0.011)
elasticities in this table are all
some ambiguity about interpreting the
evaluated at the (overall) sample means. Note that there
elasticities
is
with respect to mis-estimates, as we are consid-
ering the effect of a larger mis-estimate in the "right" direction as perceived by the bidder. In sales
where the reserve prices bind (286
in sales
sales), bids
and skews are more
where the reserve prices did not bind. But when censoring
ding and skewing are less sensitive to mis-estimates.
when
latter
is
exogenously more
likely,
than
bid-
conforms to our theoretical model:
bidders are unconstrained in their skews, they are able to optimally adjust their skews, in
turn allowing
sales,
The
sensitive to mis-estimates
we
them
to place higher bids without expecting to pay more. Notice that in uncensored
and
find only a small
insignificant relationship
between the overbid and the mis-estimate.
This indicates that in sales where firms chose to skew "moderately," they also chose not to increase
the total bid in response to the mis-estimate.
On
the other hand,
when they skewed
aggressively,
they also increased the total bid sharply in response to higher mis-estimates.
The
fact that the
the censoring
is
way
in
which the
sales are divided affects our
endogenous. Of course, our theoretical model predicts
tion equation, u,
is
comparisons indicates that
this:
the error in the selec-
correlated (though imperfectly so) with the errors in the skewing
and overbid
equations. Formally:
-
£^[ln(A6^(-)
E[ln ( ^]^~
V Qest
]
(
most
fits
efficient
Ires
does not bind]
=
-
~6^P^
l-^^ai
+ Xa2 + aiasE[e\u >
h{-)]
+ X(3l + EleP'la^ 4- vPsf^vW >
into the
)
Ires binds]
= ~6^(3l + X/32 + E[el3l<7e + 7?/3^a^|zv <
h{-)].
(14)
approax;h to estimation entails estimating the system of equations simultaneously;
x,^
further implies that
we could allow
functional form for the distribution of the errors. However, for computational simplicity
is
(13)
K')]
framework of sample selection problems (Heckman, 1976). Of course, the
with sufficient data, the exogenous variation in
our dataset
(12)
J
^Q^~^
E[ln
The model
Ar^)|res does not bind]
limited,
selection routines (for
computed using
full
we
for
a
flexible
and because
instead estimate each equation indi\adually using standard single-equation
each equation, we used Heckman's two-step procedure, with standard errors
information
maximum
likelihood). Thus,
30
we assume that
r]
and e are standard
normal, so that -E[e|^
>
h{-)]
=
P£^j^{h{-)).^^ Also for simplicity as well as robustness, we did
not impose the cross-equation restrictions between (11) and the other equations.
Our
results
from
this estimation are reported in Table
VII
(1)
and
We use a somewhat more
(2).
parsimonious specification than in earlier tables. First consider the probit estimates. As expected,
higher values of ln(a;^) lead to a lower probability that the reserve prices do not bind.
the mis-estimate
is
close to zero, although
it is
!=a
/3").
consider the skewing regression, incorporating the selection correction Table VII (2).
parameter estimates can be interpreted as
mation of the mis-estimate
The average value
of the
elasticities,
(this requires multiplying the estimates
skew
is
by one standard deviation
by E[6y]/{E[6y^]
$103.50, while the average mis-estimate
(.075),
on average the skew
rises
between the error in the selection and skewing equations
the definitions of
Q!i is
i/
and e above,
if
e and
Our
although we must correct for the transfor-
is
by about
large
if
+ .3) =
We
.014.
is
increase in the mis-estimate leads to a .022% increase in the skew. Thus,
that
effect of
somewhat imprecisely estimated. This implies that
mis-estimates affect the overbid and the skew at approximately the same rate {ai
Now
The
.0461).
find that a
1%
the mis-estimate rises
$12. Finally, the correlation
and negative,
at —.80. Recalling
are independent or positively correlated, this implies
r]
large relative to /3". In words, the
skew
is
more
sensitive
than the overbid to the private
signal about the mis-estimate.
The selection model can be
least squares regression
The
usefully contrasted with several alternatives.
first is
on the observations where the reserve price does not bind; the second
a standard censored normal regression model (CNR), which requires that censoring
The
CNR model
E[A6^ -
^ry.]
=
_
_
6y-ai
results are reported in
hinds}0^i(^eE[e\6y.ai
+ Xa2 -
R
R
_
_
+ Xa2 + l{res.
Table VII
(3)
and
from the selection model are greater than the
Intuitively, considering
OLS
term than the
The
CNR model,
To evaluate the robustness
when polynomial
-e].
(15)
^x
estimates and smaller than the
may be
censored.
On
CNR estimates.^^
the other hand, ignoring the
^^
is
positively correlated with
leads us to exaggerate the inferred effects of both variables
censored observations.
<
-=
Notice that the coefficients on the mis-estimate
(4).
endogeneity of censoring yields an upward bias: the fact that
6-^
exogenous.
only the uncensored sample leads to a downward bias, as sales where skews
are particularly sensitive to mis-estimates
both e and
is
is
estimates:
West
The
an ordinary
on the desired skew
for
model places a smaller weight on the censoring correction
selection
more conservative estimates about the
leading to
of our results to the normality assumption,
functions of the Mills ratio
(j>/{l
—
we
verified that
effects of s
and
6y^.
our results do not change
$) are included as explanatory variables.
Such an approach
serves as a simple first pass at semi-parametric estimation.
We
also estimated (but did not report)
on the mis-estimate variables were about
an ordinary
2%
least squares regression
on the
full
smaller than the censored normal regression.
31
sample; the coefficients
Now
estimate
is
Accounting
sales;
elasticity of the overbid
approximately .01 and significantly different from
for selection (as
On the other
Sy,.
The
consider the results for overbids.
opposed to (unreported)
OLS
only at the
CNR)
or
is
approximately
.03.
effectively
and
efficiently,
and thus
their bids will
to reconcile our results with the intuition
model does not
confidence
level.
lowers the estimated effects of
This finding runs counter to the following intuition: we
would expect that when the bidders are constrained by the reserve
way
15%
hand, we find larger and significant effects of mis-estimates on overbid for censored
the elasticity
skew as
with respect to the mis-
fully
is
be
price, they will not
be able to
less sensitive to mis-estimates.
One
to consider the possibility that the econometric
account for the heterogeneity across
For example, the sales where
sales.
censoring occurs and bids are high, might be sales where the bidders are more confident about the
direction of the mis-estimate. Another hypothesis
is
that our functional form does not fully capture
nonlinear effects of the mis-estimate.
Recalling our analysis from Section 5.3,
we now use our estimates
to calculate the effect of mis-
estimates on revenue. Equation (9) indicates that an increase in 6^ affects the magnitude of the bid
as well as the gap between bid
effect of
Ab^
is
and payment, Ab^-S^- In contrast to Section
5.3,
we now examine
the
an increase in the mis-estimate in the direction of x- In our sample, the average predicted
quite large (107.8), so that an increase in the mis-estimate leads to a large increase in the gap
between the bid and the payment.
respect to mis-estimates
is
We
find that for uncensored sales, the elasticity of revenue with
-.0066; thus,
an increase in the mis-estimate of one standard deviation
(.075) leads to a reduction in revenue of $4.74 per Mbf., or
about $14,560 on the average
sale.
In sales where the reserve prices bind, the bid increases at a rate proportional to the skew. For
these (less frequent) sales,
the elasticity
is
we
find that revenue actually increases
the mis-estimate grows;
.0032.
Overall, the magnitudes of our estimates in this section should
due to the
when
fairly
still
be interpreted with caution,
strong functional form assumptions. Nonetheless, our results
insight into the direction of,
and
relative
importance
of,
may
provide some
the effects of the reserve prices; as well,
they indicate that skewing leads to small but statistically significant reductions in revenue.
7
Conclusion
In this paper,
we have considered the
effects of
information on bidding behavior in Forest Service
timber auctions. The rules of the "scale sales" create incentives for bidd,ers to distort their bids.
By
using ex post information about the value of the tract,
we
are able to provide evidence that
bidders are risk-averse, have private information about the underlying characteristics of the tract,
and
exploit this information in their bidding behavior.
in allocating the tract
This information appears to play a role
between bidders, since bidders who take larger gambles are more
32
likely to
win the auction. Further, larger realizations of the mis-estimate typically lead to lower revenue
for the Forest Service, consistent
compensate
Our
for the large
with the hypothesis that the bidders require a risk premium to
gambles they must take to remain competitive
in the auction.
use of ex post information has an additional advantage: our results provide evidence about
the validity of several of the assumptions
commonly invoked
of pure private values, which requires that
concerns the bidder's
bidding behavior
is
own
private costs
also inconsistent
and
all
The hypothesis
in auction studies.
private information on the part of a given bidder
benefits,
not supported by our analysis. Observed
is
with risk-neutrality. Finally, bidders clearly expend resources
to obtain strategically useful information, suggesting that attention should
be given to the process
of information acquisition.
In future work,
it
may be
possible to use the ex post value information in a
structural model. For example,
values for the tract
and private
we might be
able to assess the relative salience of heterogeneous
signals about mis-estimates. Alternatively,
estimates of bidder risk-aversion.
We
more complete
we could
explore direct
might also quantify the extent of the winner's curse
in this
market, and the value of additional information. However, these questions would require a number
of
demanding extensions to the theoretical and econometric models.
8
Appendix A: Proofs
The
following results are used in the formal analysis.
(1982) for basic results about affiliation
and further
treatment of these lemmas
following definitions. ^
Milgrom and Sharmon, 1994)
if for all
:
5ft^
—>
5? is bivariate
x^ > x^ and y^ > y^ g{x^ ,y^) —
>
^
increasing.
If/
is
positive, /is (strictly) log-supermodular if log(/)
and z are random variables with positive
and z are
(strictly) affiliated if
(Lebesgue measure).
A
Lemma
random
is
(strictly)
is
(strictly)
log-supermodular in
If
y
Lebesgue measure, y
(y, z)
log-supermodular in {zi,Zj) almost everywhere, for
nondecreasing in
supermodular.
almost everywhere
variables z with a positive joint density /(z)
7 (Milgrom and Shannon, 1994) If 9{x,9)
axgrnsx-x g{x,0)
is
joint density f(y, z) with respect to
and only if/
vector of
affiliated if /(z) is (strictly)
If
we need the
for a unified
(>)0 implies g{x^ ,y^) — g{x^,y^) > (>)0. A function /is swpermodular if, for all
yL^ j^yH^^^ _ ^y
j^
L ^^ jg nondecrcasing in z; f is strictly supermodular if the difference is
g{x^,y^)
yH y
and Athey (1998)
references. For our purposes,
single crossing in {x;y) (see
See the appendix of Milgrom and Weber
is (strictly)
all i j^ j.
satisfies bivariate single crossing,
x*{d)
=
OP
there are multiple optima or no optima for
some parameter
the strong set order (Milgrom and Shannon, 1994).
33
values, the set of optimizers
is
nondecreasing in
Lemma
8 Let {y,z)
be affiliated
random
variables with conditional density f{y\z). If g{x,y,z)
and super-modular in {x,z), then f g{x,y,z)f{y\z)dy
bivariate single crossing in {x;y)
is
is
bivariate
single crossing in (x\ z).
Lemitna 9 Let u{w;
ofu{w;6) (uw{w\0)
ficient of absolute risk aversion
is
and increasing in w, and suppose that 6 decreases
6) be differentiable
is
the coef-
Then J u{x y;6)f{y)dy
log-supermodular).
bivariate single crossing in (x; 9)
Lemma
10 If(y,z) are
Lemma
11 Suppose that (y,z) are
[ai,6i],z
=
G
12
modular in
[ai,bi],i
and
=
hi
for
all
is
increasing, then E[g{y)\z\ is increasing in z
i.
If g{y) is increasing, then E[g{y)\
(i)
is
2.
The
supermodular, then
.6^
"
^i
>
+ vx- -^-\ ) 4, x;
VA;
I
^2
''i,
>
is
as a function of
Moreover, from (At),
6y^.
U{A.h-^\B^ ,d]()
is
fix
d^.
An
CARA
an increase
in
it
or
i.e.
B^ has two
effects.
g^-
super-
# j,
First,
it
is
Then
be nondecreasing in B^
bidder j's
by
Lemma
8, for
of
fixed
Milgrom
nondecreasing in d^ for a fixed
acts as a
for
in
and increases A^.
Lemma
11 this effect favors
a fixed d^.
Q.E.D.
= ^v-^ (i.e. bi = Vi),
bidder j can ensure a payoff of zero. If /?(•) > V, then any total bid B < V also results in a
payoff of zero, so bidder j can restrict attention to bids that are greater than or equal to V. We
now establish monotonicity. Define m^-' = max/^j{d' }. Since the signals were assumed to be
Proof of Proposition
exchangeable, m^-'
is
3.
(i) First,
affiliated
A6j^, the bidder chooses
B
to
with
notice that
d^^
by bidding
B—V
downward movement
risk-averse
increases the set of opponent types defeated; by
will
BK
The monotonicity theorem
lARA, makes the bidder no more
A6^. So A6j^(5^,d^)
< B^
B\d^^)
crosses zero once from below
are affiliated. Thus,
then implies that A6^(B-',d^)
7)
increase in
wealth, which, under
Second,
(6;^-,<i^,d^'')
bivariate single crossing in (A6;^,d^).
and Shannon (1994) (Lemma
(Lemma 9).
is
'^2,
bivariate single crossing in (Atj^-,^;^),
payoff function, u(-)
Now
{x]bi).
J g{x,y)f{y\z)dy
Let C/(A6;^;S-', dj^) represent the objective in this optimization program. Fix
.
and
bid allocation problem can be written as:
subject to:
B^
G
(x, z)
max Es^ u (qact U^b^ - Av^)
j5^,
Zi
If g{x,y) is bivariate single crossing in
(ii)
l,..,n] is bivariate single crossing in (x;ai)
and g{x,y)
If (y,z) are affiliated,
Proof of Proposition
A6x
and g{y)
strictly affiliated,
l,..,n] is increasing in ai
{x;y), then E[g{x,y)\ Zi
Lemma
strictly affiliated,
and
Ahj^.
(Milgrom and Weber, 1982). Given an optimal choice of
maximize expected
be convenient to break the interaction between
B
34
utility.
In order to analyze the problem,
and d^ into two components, the
it
will
direct effect of
d^ on payoffs for a fixed A&^, and the indirect
of
A6^ depends on
dy,.
U{B,di^^y)
To separate the two
=
effects,
UiAV^iB,y),B,d^). f
J
and show that
To do
result.
To
we write
we show
that -^U{B,d]^,y)
begin, consider the effect of y on
fact that the optimal choice
payoffs as
^\p^^-.^^s}i^y')dFim-^\x,diJ,
[0,1]
U{B,d-'^,d--^) is bivariate single crossing in
so,
due to the
effect arising
(5;d^), which by
d\ and y
single crossing in
is
-^U{B^ dl^,y).
Lemma
a.t
y
=
7 will imply our
djj..
Letting subscripts denote partial derivatives
(and applying the envelope theorem in computing g^U{B,dl.,y)), we have
^^
-U{B,d{,y)
=
U,iAb{(B,y),B,d{)^Abi,iB,y)f{p-HB)\x,d{)
+UniAbi^{B,y),B,d^)^Ab{{B,y)
!{,(.- )<b}K')^^("^x^Ix, 4)-
•
|^^^^_^
Applying the envelope theorem again, we know that C/i(Ab^(S,y),B,<i^)
our arguments in Proposition 2 imply that [/i2(A6^(i?,y),i?,d^)
CARA/IARA and Lemma
g^U{B,d{,y)>Oaty = d{.
tion of
Now
10.
-^U {B
consider the effect of d^ on
^A6^(S,y) >
Since
,
d-'-^,y)
when y
is
>
at
y
= at y = d^.
= d^, using the
by Proposition
fixed.
We
2, it
Further,
assump-
follows that
introduce some additional
notation:
U{B,Ab{,d{,m-^)= f
u{7r{Ab{,B,6^))dF{6^\d{,m-^,x),
7[o,i]
so that
f7(B,d^,y)
By
=
J
U{B,Abi^iB,y),d{,m-^)
^_^
the envelope theorem,
-^U{B,d^,y). To apply
we can
Lemma
{B,Tn^-'). This will in turn
and bivariate
^u
7r
(
(it
single crossing
(Ab{, B, 6y^^
A6^,S,5;^.j
is
The argument
(ii)
rem
we need
8,
(5;m^'').
= -^^u'{tt).
nondecreasing in
Then, since 6^ and m^-' are
5j..
ignore the effect oi
6-^.
B
in
U through A6^ when computing
U is supermodular in {B,dy^ and
on
show that
to
imply that the integrand
in'
1{^(^-.)<b}(V)^^(VI^'4)-
U{B,dy,y)
supermodular
To'establish the desired properties of
Recall from Proposition 1 that
Since
affiliated.
is
u
is
concave,
Lemma
it
U
(B;d^)
we compute
Ab{{B,d{) > Av^, and thus
follows that
12 implies that
J7,
in
is
-^u
is
nondecreasing in
supermodular in (5;m^-').
for (5; d^) is analogous.
The above argument
for the case of
verifies
the single crossing condition in Athey's (1997) existence theo-
two bidders. The
result
can be applied to the case of J sjonmetric bidders by
35
looking for a fixed point in the best response correspondence for a single bidder,
when
all
opponents
play the same nondecreasing strategy.
Proof of Proposition
exist
some
Q.E.D.
R <
Suppose to the contrary that
4:
61,62 such that
=
'^)
Qest{^
B*{D;k,Pk) and
<
B*{D;k,Pk)
n <
<
bi
Vi.
Then
V.
there
This "safe" portfolio
has strictly positive expected profits for any resolution of uncertainty, contradicting the fact that
the optimal skew gives zero expected
>
optimum, Ab^{D;k,Pk)
((A6
— Av)6 + {V — B)/Qest) and
We
price constraints.
=
Since A6*
D;
it is
is
decreasing in
Then,
> D*{B;k,Pk)
B
B since
5.
for
only
S =
consider
announcement
are the
same
as
if all
Finally,
if d^,
<
/
^
A^;,^
if
if
j'
But
A,^.
The
is
increasing
result follows.
(J.
£'./?.
j follow the strategies described in the
all
that can be inferred
B
=^
will consider player
knows that
i3*(d^;O,0), bidder j
= d^-
0, 0)
that
is
For him to earn non-
he wins, he must announce A6-^(B;O,0). So bidder j has no incentive
B
Moreover, as
rises past
B*(d^;O,0), j can not possibly
make
he wins. So he will drop out just beyond 5*(<i^;O,0).
that
in fact,
d^-
if
j
if
had
when
B <
Consider
5*(<i^;0, 0).
first
an
that following such a deviation, opponent's beliefs
announced A6^(S;O,0), that
we know
=
>
Suppose no one has dropped out; we
bid allocation announcement Ab'
(this follows
Ab and dM. So
/.
and
opponent's signals are equal to D*(jB;
B*(dj^;O,0).
j''s
if
this event,
A6
B
active at a bid B,
still
A6 < A6^(5;O,0). Suppose
to any lower A6.
lower
11 implies the right-hand side of (16)
Suppose bidders
(i)
deviation will be payoff-relevant only
Given
Lemma
decreasing in
tt is
each opponent
positive expected profits
Now
S-^^.
opponents are
if all
negative expected profits
to deviate at
•
denote the Lagrange multiphers on the reserve
deviate from the equilibrium strategy. At
j's incentives to
he will win at
A-,;^
= Qact
Let K{Ab,B,6)
r^.
at the
+ A6;^(l - xi) - n] + A^;^ [{B/Qest) - ^b^xi - ra.]
increasing in
Proof of Proposition
d}^
and
>
v^
1,
u
Ee^
> Av^, n
Proposition.
let A;^
>
b'^
Now, by Proposition
V.
have
+A;^ {(B/Qest)
in
which implies that
Avy^,
>
So B*{-)
utility.
is,
opponents do not update at
bidder j actually wins at B,
signal
D*{B;
> D*{B; 0, 0),-so j
0, 0),
i.e.
if d^^
=
all.
Then
D*{B; 0,0)
for
he would prefer a skew of Ab^{B;
certainly prefers a
from the fact that the objective defining A6-^
is
skew of Ab^{B;
0, 0) to
the
all.
0, 0)
any
bivariate single crossing in
this deviation is not profitable.
suppose
B <
.B*(d^;O,0) and consider a deviation to
Z)*(S;O,0), then j's expected profits from bidding
opponents already believe that d^
>
theip beliefs they will revise upward.
Z)*(B;0,
To
0), it
this end,
36
Ab > Ab^(B;O,0). Note
Ab > A6-^(S;O,0)
are negative. Since
makes sense to assume that
assume that any opponent
that
if
/
they do update
who
is
marginal,
i.e.
who
plans to drop out immediately, revises his behefs about j to be that d^
where D''"(5;O,0)
D*(S;0,
0),
make no
revision in their beliefs about
>
by bidder j to A6
large
is
A6'^(S;0,
0)
opponents have more optimistic
same arguments can be applied
enough to deter
With
dl^.
after
from dropping out.
All other bidders
these off-path beliefs, a bid allocation deviation
cannot increase and
beliefs
I
> i?+(5;O,0) >
may
about his signal and
decrease j's expected payoffs (since
will stay in
the auction longer).
The
any number of bidders have dropped out, which ensures that
the proposed strategies form an equilibrium. Note that a variety of other off-equilibrium-path belief
restrictions
j,
would
also suffice.
opponents are at
will
So long as in the event of a bid allocation deviation by some bidder
least as optimistic
about his signal as in the equilibrium we have described, he
have no incentive to deviate.
(ii)
The proposed equilibrium
fully reveals
the signals of
all
players except the player with the
Q.E.D.
highest signal.
Proof of Proposition
6:
The
result follows because nondecreasing functions of affiliated
variables are affiliated. In the sealed bid auction equilibrium of Proposition 3,
and B^{d-^)
creasing in both arguments,
Since
in dy.
d-^
and
is
nondecreasing, so that Aby,{B^ (d^,)
A6^ and
are affiliated, so are
6-)^
6-)^
on dropping out. Suppose k bidders have dropped
point
B
and skew Aby^
Since each of these signals
then 6
=
6y and
increase in
—
djj.
A6 =
9
A6
is
Ab-^^
increases
and Ab. Thus 6 and
be higher, the higher
will
affiliated
so 6
A6 and
with
6-)^,
and A6 are
d. Since,
is
,
d-^) is
will
Each bidder
5.
be affiUated with
x=
Now
fully reveals their
Then the next observed drop-out
out.
affiliated conditional
conditional on
nonde-
nondecreasing
the signal of each of the lowest k
A6^
is
(and similarly for a fixed bid B).
consider the oral auction equilibrium described in Proposition
signal
A6^(5, d-^)
random
2,
^
on x
Finally,
6-^.
=
+
If
1-
and —d^ are
X =
1 signals.
if
2,
x
=
!>
then, an
affiliated, so are 6
Q.E.D.
are affiliated.
Appendix B: The Data
In this section,
we
briefly describe our
data sources and
criteria for selecting
our sample. The data
can be divided into two categories: "bidding data" and "cutting data." The bidding data contains
sale appraisal information as well as the highest bids placed
data
is
by each bidder
at the auction.
publicly available from the Forest Service immediately after the auction.
The
the ex post information about the timber actually removed from the tract. This data
available from the Forest Service.
different species of
difficult to
sources.
Because the Forett Service uses
different
This
cutting data
is
is
also publicly
coding systems for the
timber in the bidding and cutting data, the Forest Service data
is
somewhat
work with, and so we purchased "matched" data from one of the leading industry data
Timber Data Company.
37
We selected a number of forests,
two major
5,
The
species.
Forests 3,
5, 6,
forests
10, 11,
and
focusing on larger forests that had a large fraction of sales with
we obtained include Region
14-18.
Among
where the matching process met certain
number
6,
Forests 1-6, 9-12, and 15-18; Region
the sales from these forests,
reliability criteria.
We
we
consider only sales
then narrowed the sample along a
of criteria.
In Region
price, as
6,
we
ruled out sales where the average bid per Mbf. was within $1 of the reservation
such sales leave
Mbf. was
less
than
5%
little
scope for skewing; similarly, we ruled out sales where the overbid per
of the appraised diflFerence in values, Ar.
We
further ruled out sales with
extreme mis-estimates (greater than 27%) or gaps between aggregate volume sold and cut (greater
30%
than
magnitude on the
in
sale, or
circumstances (for example, cutting
outliers along a
number
25% on
may
either species), as these sales
have been aborted for some reason).
of dimensions, including only sales with
Mbf. and 25000 Mbf., and density
less
may
We
have special
then dropped
volume estimated between 100
than 115 Mbf./acre. Finally, we dropped sales where road
construction was greater than 2.5% of the value of the sale. Since the government reimburses road
construction using a complicated system of credits, bids in sales with low road construction can
be interpreted more directly
in
terms of expected payments. For Region
5,
we used
essentially the
with one notable exception: we allowed sales with road construction valued at up to
same
criteria,
100%
of the appraised value of the tract. This allows a larger
sample
size,
a problem for Region 5
sealed-bid auctions.
Finally,
we note a subtlety that
arises in interpreting the bidding data. Forest Service regulations
require that no matter
what the appraised values and
minimum value, known
as the "base rate."
costs, all per-unit bids
On a small number of sales,
must
the base rate
clear a certain
is
greater than
the reserve price. Thus, the Forest Service might accept a bid that violates the base rate rule. In
such a case, the Forest Service uses a pre-announced mechanism to lower the bid on some species
and
raise
it
on others. In
reserve price constraints.
for us to
this way, firms
can sometimes achieve skews that violate the posted
Fortunately, the Forest Service data includes the information required
compute the amount a bidder anticipates paying when the bid
as the "statistical bid,"
and
all
is
placed. This
is
known
of our results use the statistical bids. In any case, only a few sales
are affected by this rule.
References
[1]
Athey, Susan, 1997, "Single Crossing Properties and the Existence of Pure Strategy Equilibria
in
Games
of Incomplete Information,"
MIT Working
38
Paper 97-11.
[2]
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Log-Supermodularity,"
[3]
MIT Working
Paper 96-22R.
Baldwin, Laura, 1995, "Risk Aversion in Forest Service Timber Auctions," mimeo,
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[4]
Baldwin, Laura, Robert Marshall, and Jean-Francois Richard, 1997, "Bidder Collusion in U.S.
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[5]
Timber
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[7]
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Bushnell, James, and Shmuel S. Oren, 1994, "Bidder Cost Revelation in Electric Power Auctions," Journal of Regulatory
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Cummins, Jason,
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[8]
Haile, Phihp, 1998,
"Auctions with Resale:
An
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[9]
Hansen, Robert
C,
1986, "Sealed-Bid versus
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quiry, 24, 125-142.
[10]
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M. and Michael H. Rothkopf,
1997,
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'Alternating Recognition'
Model
of
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[11]
Heckman, James, 1976, "The common structure of
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Hendricks, Keimeth and Robert Porter, 1988,
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[13]
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Milgrom, Paul and Christina Sharmon, 1994, "Monotone Comparative Statics," Econometrica,
62, 157-180.
[14]
Milgrom, Paul and Robert Weber, 1982, "A Theory of Auctions and Competitive Bidding,"
Econometrica, 50, 1089-1122.
[15]
Osband, Kent, and Stefan Reichelstein, 1985, "Information-eliciting Compensation Schemes,"
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(1),
June, 107-15.
39
[16]
Persico, Nicola, 1998, "Information Acquisition in Auctions,"
Mimeo, University
of Pennsyl-
vania.
[17]
Powell,
James
L.,
James H. Stock, and Thomas M.
Stoker, 1989, "Semiparametric Estimation
of Index Coefficients," Econometrica, 57, 1403-1430.
[18]
Robinson, P. M., 1988, "Root-A'^-Consistent Semiparametric Regression," Econometrica, 56,
931-954.
[19]
Rucker, Randal R. and Keith Leffler, Keith, 1988, "To Harvest or Not to Harvest?
of Cutting Behavior
70
[20]
(2)
An Analysis
on Federal Timber Sales Contracts," Review of Economics and
Statistics
May, 207-13.
"Skewed Bidding Presents Costly Problems
for the Forest Service
Timber Program,"
GAO
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[21]
Stoker,
Thomas M.,
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on Semiparametric Econometrics, Louvain-la-Neuve: Core
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[22]
Wood, David
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A
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Component Auctions, Ph.D.
Thesis, University of California, Berkeley.
40
in Second-Price Multi-
c
o
o
.05
(0
.2
Misestimate
Chart
I:
Histogram of mis-estimates, species 1 is the high-valued species.
Sample of 699 Region 6 oral auctions.
jfei
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1
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.1
Magnitude
Chart
EI:
of
.3
Over-estimate
Percentage of overbid placed on the over-estimated species against the overestimate.
Sample of 699 Region 6
41
oral auctions.
1000
500
5
0)
CO
CM
c
<o
-500 -
-1000
r~
"1
1000
500
-500
Rank
Chart IIIA: Plot of Rank 2
Skew Against Rank
Sample of Region 6
1000
1000
Skew
1
Skew
1
(onto over-estimated species);
oral auctions with 3 or
more bidders
-
500
5
0)
CO
o
CO
o
o
O- On
c
CO
tr.
-500
1000
n
I
-1000
I
-500
500
Rank
Chart niB: Plot of Rank 3
Skew Against Rank
Sample of Region 6
1
1
Skew
(onto over-estimated species);
oral auctions with 3 or
42
1000
Skew
more bidders
ULUnLL ULUULOJIUIV
KERNEL DENSITY
do
10
C\\c^A
20
30
IV:
40
50
60
70
80
s
90
Kernel (TG^essio^.
Table
1:
Summaty
Statistics for Oral
and Sealed Auctions with Two Primary Species
Region
Regior 6 Oral
1
N
699
Mean
5
Sealed
63
Std. Dev.
Mean
std. Dev.
Bidding and Volume Variables
= b1
166.06
138.99
118.38
108.88
s2 bid rate = b2
135.34
116.42
106.00
111.42
56.37
si bid rate
=
r1
76.43
73.65
53.40
s2 reserve rate =
r2
76.54
73.65
35.86
35.83
si market value
439.90
98.16
344.18
108.27
s2 market value
438.42
88.87
309.30
97.22
3.25
3.71
1.70
3.37
3.02
3.35
1.85
3.80
74.95
55.69
43.70
33.23
0.520
0.125
0.543
0.126
0.517
0.138
0.529
0.156
467380.10
620525.50
247820.40
682193.00
426688.40
566565.30
259778.20
741056.60
142.98
83.44
104.85
56.73
139.07
81.23
99.91
52.53
si reserve rate
Volume est. (mmbf)
Volume cut (mmbf)
Average reserve price (per mbf)
Estimated % of Volume on high-valued species
Actual % of Volume on high-valued species
Tot. Bid (B)
Tot. Paid (P)
S/Q est
(Tot.
Bid)/(Volume Est.) =
(Tot.
Paid)/(Volume Cut) = P/Ocut
Skewing Variables
% of (Tot.
Bid
-
Res. Price) on over-est. species
Skew onto over-est species: Ab-A r
Revenue Shortfall from Skew
Revenue Shortfall from Undercut
Misestimate Variables
0.57
0.40
0.49
0.30
30.79
150.88
-5.35
139.15
10581.69
57819.42
9780.05
28956.45
30109.96
110740.90
-21737.85
88802.78
-0.066
0.157
0.036
0.125
0.003
0.076
0.015
0.072
0.057
0.051
0.049
0.054
# of bidders
6.06
3.12
5.32
2.60
SBA dummy
0.21
0.41
0.17
0.38
(Volume Cut-Volume Est)/(Volume
Misest. on s1=high-valued sp: Est.
Est.)
% si
-
Act.
% si
Absolute magnitude of mis-estimate
Bidder Participation Variables
Sale Characteristics
Contract Length/Volume
Density of Timber (mbf/acre)
0.01
0.02
0.02
0.02
27.98
24.31
12.32
14.74
mmbf
80.87
168.29
0.00
0.00
(thousands of $)
Road construction (thousands of $)
401.05
566.08
159.34
213.17
0.61
3.02
11.21
31.12
Volume per-acre
Logging costs
material per
est.
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DEC
2'^00
Date Due
Lib-26-67
MIT LIBRARIES
3 9080 01917 5857