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Working P a ~ e 9402
r
AUCTIONS WITH BUDGET-CONSTRAINED BUYERS:
A NONEQUIVALENCE RESULT
by Yeon-Koo Che and Ian Gale
Yeon-Koo Che is an assistant professor of economics
at the University of Wisconsin, and Ian Gale is an
economic advisor at the Federal Reserve Bank
of Cleveland. The authors thank Ray Deneckere,
Prajit Dutta, Joe Haubrich, Don Hausch, and Larry
Samuelson for helpful comments and suggestions.
They also thank seminar participants at the University
of Wisconsin and at the 1993 Midwest Mathematical
Economics meetings in Madison, Wisconsin.
March 1994
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Abstract
Anecdotal evidence of concern about the limited financial resources of small
firms abounds in government auctions. Recent empirical work also provides evidence
of the importance of capital constraints. In this paper, we show that the first-price
sealed-bid auction yields higher expected revenue than the second-price sealed-bid
auction if buyers face wealth constraints. Differences in the extent to which wealth
constraints bind in the different auction formats generate the revenue nonequivalence.
clevelandfed.org/research/workpaper/index.cfm
Introduction
Sellers of goods and services use a wide array of sales mechanisms, including
one-on-one bargaining, oral and sealed-bid auctions, and posted-price schemes.
Auctions are frequently used to sell goods ranging from real estate and works of art to
mineral extraction rights and timber harvesting rights. For example, in the United
States, federal mineral rights have been sold exclusively through first-price sealed-bid
auctions, where the winner pays his bid, whereas timber rights have traditionally been
sold through oral auctions. (The latter are, for our purposes, equivalent to second-price
sealed-bid auctions, where the winner pays the highest losing bid.) Given the economic
significance of these auctions, it is important to understand the relative performance of
various auction formats.
Auctions with very different rules may yield similar outcomes. Consider the
independent private-values setting with symmetric buyers, where valuations are
independently and identically distributed. A large class of auctions generates the same
expected revenue for the seller, despite the differences in rules. This "revenue
equivalence" result relies on the insight that the rule for determining the winner, and
the expected surplus that accrues to a buyer with the lowest possible valuation,
completely determine the expected surplus to a given buyer. Total surplus is the same
if the winner is the same. Since each buyer's expected surplus is also the same, the
seller's expected revenue must be equal in the different auctions.
A consequence of revenue equivalence is that a seller should be indifferent
among all auction formats within the relevant class. Yet sellers employ certain formats
more frequently than others. In this paper, we show that the first-price sealed-bid
auction yields higher expected revenue than the second-price sealed-bid auction if
buyers face wealth constraints. Differencesin the extent to which wealth constraints
bind in the different formats generate the nonequivalence.
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Many buyers face some form of wealth constraint when bidding. In the case of a
consumption good, imperfect capital markets may constrain a buyer's ability to borrow
against lifetime income (which may itself be below his subjective valuation of the object
for sale). Similarly, the buyer could be a bureaucrat who internalizes the benefits from
the acquisition, but not the costs, and who is therefore subject to tight budgetary
control.
Anecdotal evidence of concern about the limited financial resources of small
firms abounds in government auctions. For example, despite the presence of
informational economies of scale, the U.S. government has limited the length and size
of mineral leases.1 In timber rights auctions, "set-aside sales" have been made available
exclusively to small firms if such firms have not attained a specified market share in the
prior 12 months (Bergsten et al. [1987]).
More recently, a proposal was made to require a substantial nonrefundable
deposit to participate in the Federal Communication Commission's Personal
Communications Service auction? Requiring a deposit is an attempt to "pool" bidders'
budget constraints by extracting revenue from all bidders, rather than from just the
winner. Royalty payments, which are popular in mineral rights auctions, provide a
method of spreading bidders' budget constraints across periods. Between 1953 and
1982, the revenue raised from royalty payments in Outer Continental Shelf (OCS)
auctions amounted to $17.3 billion, or 41.9 percent of the revenue raised from up-front
bids.3
1 The Mineral Leasing Act and the Outer Continental Shelf Land Act explicitly limit the
size of leases, but allow consolidation of leases after bidding is complete. Leases are
limited to five and ten years for producing and nonproducing tracts, respectively. See
Bergsten et al. (1987).
2 See Edmund L. Andrews, "U.S. Lays Out Rules for a Big Auction of Radio Airwaves,"
New York Times,September 24,1993.
3 Royalty payments do not solve the problem of budget constraints completely because
an increased royalty rate lowers the incentive to develop and recover minerals.
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The empirical work of Hendricks and Porter (1992) provides additional evidence
of the importance of capital constraints. Since 1975, OCS regulations have permitted
joint bidding by all but the eight largest firms. The authors study bidding behavior on
OCS leases for the period 1954-1979. They examine the impact of joint bidding on bids
and ex post profit rates. Their findings concerning the low profitability of joint ventures
involving a large firm and small fringe firms are of particular interest. Formation of
these joint ventures apparently leads to more competitive bidding. The authors suggest
that joint ventures are "motivated primarily by capital constraints" (ibid, p. 510).
McDonald (1979, pp. 106-07) reaches a similar conclusion.
We examine buyers who face an exposure limit that fixes their maximum
feasible bid. This limit, referred to as the buyer's "wealth," is considered in two settings.
The first corresponds to situations where heterogeneity of wealth is large compared to
heterogeneity of valuations. In particular, we suppose that the value of the object, in
the absence of wealth constraints, is v for all buyers. Wealth differs across buyers and is
private inf~rmation.~
First- and second-price auctions each yield revenue of v in the
absence of wealth constraints. If the wealth constraint binds, however, expected
revenue differs.
The basic argument for nonequivalence can be developed along the following
lines. Suppose that a buyer wins the object with probability X, that the expected
payment is T, conditional upon winning, and that the maximum realized payment is
m ( ~ ) . 5In the standard first-price auction, m(T) = T, since the winner pays his bid. In
the standard second-price auction, m(T) is again the bid, but here it exceeds T.
p
p
p
p
p
-
An alternate interpretation is that this is a pure common-values case where buyers
have identical information concerning the common value. Because no transmission of
information concerning the common value takes place here, the "linkage" of bids
described by Milgrom and Weber (1982) is not present.
5 There is a one-to-one correspondence between the maximum payment (the bid) and
the expected payment in both auctions, so m(T) is well defined.
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In equilibrium, a buyer with wealth w will select the feasible (X,T) pair that
maximizes his expected surplus, (v-T)X, subject to m(T) I w. The corresponding
Lagrangean is
L(X,T,h;w) = (v-T)X + h[w-m(T)].
Let (X*(w),T*(w),h*(w))denote the optimal values, and let U*(w) = (v-T*(w))X*(w)be the
maximized expected surplus. The Envelope Theorem implies
U*YW)= aL/aw = L*(w),
and integrating yields
We immediately see that the expected surplus depends on the surplus in the
benchmark case, where w = 3 and on how tightly the constraint binds. Therefore, the
property that it depends only on the allocation rule and the expected surplus in the
benchmark case does not hold. In other words, two auctions that always give the object
to the buyer with the highest wealth, and that give zero expected surplus to a buyer
with the lowest possible wealth, need not generate the same expected revenue.
The budget constraint binds differentially across auctions, which yields different
expected surplus to the bidders as well as different expected revenues. For example, if
v is very large, all buyers bid their wealth in both auctions. The expected surplus for a
given buyer is lower in the first-price auction, since the winning bidder pays his bid. In
the second-price auction, the winner's price is determined by the second-highest bid,
and it is lower with probability one. Total surplus is the same in the two auctions,
presuming that the reserve price (minimum bid) is the same, so expected revenue is
higher in the first-price auction. In cases where buyers may or may not be constrained,
we show that low-wealth buyers receive higher expected surplus in the second-price
auction, all else equal, for the reasons just given. The same revenue ranking holds.
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The second setting that we study corresponds to the opposite situation, where
heterogeneity of valuations is large compared to heterogeneity of wealth. In particular,
we suppose that wealth is equal for all buyers. Buyers have different valuations,
however, and this is private information. Since this is an independent private-values
model, revenue equivalence holds if wealth exceeds the highest possible valuation.
Buyers with independent private values shade their bids below their valuations in a
first-price auction, in the absence of budget constraints. A consequence is that, roughly
speaking, budget constraints bind less frequently in a first-price auction. (The complete
analysis accounts for possible changes in the equilibrium bidding strategies as well.)
This again makes the seller's expected revenue lower in the second-price auction.
Although revenue nonequivalence has been noted in other contexts, few papers
have examined the impact of budget
constraint^.^
One exception is Pitchik and Schotter
(1988), who consider the case of two buyers bidding for two goods in a completeinformation sequential auction. In a second-price sequential auction, there is an
incentive to bid relatively more aggressively in the initial auction. Since the losing bid
determines the price paid in each auction, bidding aggressively in the first auction can
enable a buyer to deplete her opponent's wealth, thereby making him a weaker
competitor in the second auction. This feature leads to nonequivalence, but the revenue
ranking is opposite to that found here.
Section 1characterizes the equilibria of second-price auctions with
heterogeneous wealth, followed by first-price auctions. We then give revenue
comparisons, which are made by ranking buyers' expected surplus for each possible
wealth. The first-price auction generates higher expected revenue either if.no reserve
prices are employed or if optimal reserve prices are employed. Section 2 repeats the
6 For a comprehensive review of the literature, see McAfee and McMillan (1987). Other
sources of revenue nonequivalence include buyer risk aversion and affiliation of
valuations.
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analysis for heterogeneous valuations, with the same qualitative results. The
comparisons are made here by looking at the expected price paid to the seller for each
possible highest valuation. Section 3 considers buyers with heterogeneous valuations
and wealth.
1.
Equilibria and Revenue Comparisons with Heterogeneous Wealth
There are N ex ante identical buyers who each value one unit of the good at v, so
if a buyer's wealth exceeds v, his reservation price is v. Buyer i has wealth wi E [w,%],
which is private information. Wealth is independently and identically distributed, with
cumulative distribution function F(e) and strictly positive density f(e). Buyers are riskneutral. The seller has one indivisible unit of the good to sell, which she values at zero.
We look for Nash equilibria throughout.
One case does not require analysis. If v I w, then all buyers are unconstrained.
Standard Bertrand competition ensures that at least two buyers will bid v in either
auction format, so the seller's revenue is v. Therefore, only the case of v > w requires
analysis. A reserve price below w has no effect, while a reserve price strictly above v or
w generates no revenue, so we need only consider reserve prices r E [w, min{v,FH.
We note first that neither the first-price nor the second-price auction maximizes
expected revenue if v > y.Suppose that a buyer with wealth w has the option of
receiving the object with ex ante probability X. He will not pay more than min{Xv,w]
for this gamble. Summing over bidders, the seller's expected revenue cannot exceed
min{v,Zwi], whatever mechanism she uses. We now show that this level of revenue
can be attained, which means that we have found an optimal sales mechanism.
Consider a sales mechanism in which a buyer with wealth w has a probability
w / Z Wof~ receiving the good, and must pay a transfer equal to V W / ~ ~ X { VI,w.~ W
If ~ ]
total wealth is below v, then all wealth is extracted. If total wealth exceeds v, then the
seller's revenue is v. Overall, the mechanism generates revenue equal to min{v,Zwi]. It
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can be implemented by a lottery in which v tickets are offered for sale at $1 apiece, and
each ticket gives a 1/v chance of winning. There is random rationing of tickets in case
of oversubscription. If each buyer j demands dj tickets, then the expected surplus to
buyer i is
vdi/[min(v,Xdj}l- di[v/min(v,Xdj}l = 0,
so it is weakly optimal to demand w tickets.
The first- and second-price auctions do not extract wealth from more than one
buyer, so they cannot be optimal mechanisms. For legal reasons, however, lotteries are
not a practical alternative for private sellers. Most states prohibit gambling, except for
racetrack betting, state-run lotteries, and charity fund-raisers. It is partly for this reason
that we focus on the more common auction formats. (While governments have used
lotteries to allocate assets, there has been a movement away from them, even though
concerns have been expressed that some bidders are budget ~onstrained.~
A.
Second-Price Auctions
In a second-price auction, buyers submit bids simultaneously. The high bidder
wins (if the bid is at least equal to the reserve price) and pays the larger of the secondhighest bid and the reserve price. Ties are broken randomly, here and elsewhere.
It is a dominant strategy for buyer i to bid min(v,wi}. If v > w i then the
constraint binds, and it is dominant to bid one's wealth. (We can avoid the possibility
that a buyer bids more than his wealth, and wins as a consequence, by imposing a small
penalty on anyone who reneges on a bid.) If v I wi, then the budget constraint does not
bind, so it is a dominant strategy to bid v. As noted above, we need only analyze the
case of v >-w where the constraint may or may not bind.
7 See Edmund L. Andrews, "Airwaves Auction Bill Advances," New York Times, May
12,1993.
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Suppose that there is a reserve price r E [w-min{v,i)].
We first calculate the
expected surplus that accrues to buyers. Consider a buyer with wealth w, r I w I v.
Since all other buyers bid the smaller of v and their wealth, he will win if all other
buyers have lower wealth. The expected price paid, conditional on winning, is
r
~[rnax{w,,,,r)lw,,, = w] = [ r ~ ( r ) ~+- '
U ( N - I ) F ( U ) ~ -f' (u)du]/ [F(w)]~-',
where w(l) and w(2) denote the first and second order statistics of wealth (i.e., the
highest and second-highest wealths), respectively. Conditional on the highest wealth
being equal to w, there is probability [F(r)/F(w)lNd that all other buyers have wealth
below r, in which case the high bidder pays r. The first term on the right-hand side
gives the expected revenue generated by this event. Since w(2) is the first order statistic
of N-1 random variables that are all below w, the second term gives the component of
expected revenue generated by the event w(2) 1 r. Integrating by parts, the expected
price is
~[max{w,,,
,r)lw,,, = w] = w - [
r
F ( u ) ~ -du]
' / [F(w)lN-',
given r l w l v.
A buyer with wealth w < v wins with probability F ( w ) ~ -so
~ his
, expected
surplus is
(2)
U'(w)
-
(V - w)[F(w)lN-' +
r
F ( u ) ~ -du.
'
Recall from (1)that the multiplier on wealth is equal to Uw(w). Thus, the value of $1of
additional wealth to a buyer with w < v is
UW(w)= (v-w)[(N-l)~(w)~-~f(w)].
The price paid does not change in those cases where the buyer would have won
anyway. The gain comes from the surplus (v-w) that accrues in those cases where the
buyer would not have won previously. (The price paid in these latter cases is
approximately w.)
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A buyer with wealth w 1 v wins if all other buyers have wealth strictly below v,
and he may also win if other buyers have wealth above v, since buyers with wealth
above v bid v. Therefore, he receives nonzero surplus if and only if the second order
statistic of wealth is below v. It follows that a buyer with wealth w I v has expected
surplus
U*(w) = U*(v) =
I'
r
F (u)"' du .
If w 2 v, there is clearly no gain from additional wealth.
With a reserve price equal to r, the object sells with probability 1-F(r)N.
Expected revenue is the difference between total surplus and total (ex ante) expected
surplus for the buyers:
NI= * (w)f(w)dw.
SERS= [l - ~ ( r ) ~ -] v
r
U
If the reserve is r = v, then no surplus accrues to the buyers, and SERS= [1-F(v)~]v.
B.
First-Price Auctions
In a first-price sealed-bid auction, buyers submit bids simultaneously, and the
high bidder wins and pays his bid. Once again, we need only analyze the case of v > w.
There is not a dominant strategy in this auction, so we must characterize the
equilibrium payoffs.
Suppose that there is a reserve price r E hmin{v,iV)]. The expected surplus
from submitting a bid b I r, if all other buyers bid their wealth, is
H(b) a ( v - b ) ~ ( b ) ~ - l .
Now define
U*(w) = max bE lrYW]
Hb).
U*(w) gives the highest expected surplus that a buyer with wealth w could receive if all
other buyers bid their wealth. More important, it equals the equilibrium expected
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surplus that accrues to a buyer with wealth w 2 r. (We leave the dependence of U*(.)
on r implicit.)
Lemma 1. Suppose that v > w. If there is a reserve price r E Iw,min{v,Y}],then
a buyer with wealth w 2 r receives expected surplus of U*(w)in equilibrium.
Proof: Let U(.) denote the expected surplus in a candidate equilibrium. A bid b
wins with probability F(blN-l or more, since bids cannot exceed wealth. Therefore, the
bid gives an expected surplus of at least H(b) = (v-b)F(blN-I. It follows that U(w) 2
U*(w)for all w. Now suppose that there exists a wealth w' such that U(w') > U*(w').
We show that this provides a contradiction.
Let z 5 w' denote the smallest wealth for which the equilibrium expected surplus
equals U(w'). If z = r, then b(z) = r = z. If z > r but b(z) < z, then buyers with wealth w
E
[b(z),z)would be strictly better off bidding b(z), since U(w) < U(z) for w < z.
Therefore, b(z) = z. It likewise follows that b(w) 2 z for all w > z.
The analysis above provides the following inequalities:
U(z) = U(w') > U*(w') 2 H(z).
The first holds by definition, the second by assumption, and the third by definition.
Since U(z) > H(z) = ( v - z ) ~ ( z ) ~and
- l b(z) = z a buyer with wealth z must win with
probability greater than F(zlN-l. This requires that the first order statistic of the other
N-1 bids have a mass point at z. But then an individual buyer could get a discrete
increase in expected surplus by increasing his bid infinitesimally above z. We conclude
that U(w) = U*(w)for all w. Q.E.D.
We can now provide explicit equilibrium bids. To simpllfy the exposition, we
first impose a regularity condition on the distribution function:
w
+
F(w)
(N - l)f (w)
is strictly increasing.
(This condition is clearly weaker than the standard regularity condition in mechanismdesign problems.) Condition (Rl) ensures that there exists a critical wealth w* such that
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buyers with wealth below w* will bid their wealth in equilibrium, while those with
wealth above w* will be indifferent among a range of bids. We can implicitly define w*:
v =
W*
+
F(w*>
( N - l)f (w*)
'
(If there is no solution, set w* = Y.)Clearly, w < w* < v, since F(v) > FbvJ = 0.
By (Rl) and the definition of w*, H(*) is strictly increasing for w < w* and is
strictly decreasing for w > w*. Thus, U*(w) = H(w) for w I w*, while U*(w) = U*(w*)=
H(w*) for w > w*. An immediate consequence is that it is not optimal for all buyers
with w > w* to bid their wealth, since they would be better off individually if they bid
w* instead.
It is equilibrium behavior for all buyers to use the increasing bid function
(3)
b * ( ~=) v - u * ( w ) / F ( w ) ~ - ~ .
For w I w*, U*(w) = ( v - w ) ~ ( w ) ~so
-l,
b*(w) = w.
For w > w*, U*(w) = U*(w*), so
b*(w) = v - ( v - w * ) [ ~ ( w * ) / ~ ( w<) ]w.
~-~
To see that these are equilibrium bids, note first that a buyer who bids b*(w) wins with
probability F ( w ) ~if- all
~ other buyers use this bidding function. Expected surplus is
therefore
[v-b*(w)l~(w)~-l
= U*(w).
Since U*(w) is strictly increasing for w < w* and is constant thereafter, buyers with
wealth w I w* must bid their wealth, whereas buyers with w > w* are indifferent
among all bids between w* and min{w,b*(F)). Higher bids give a strictly lower
expected surplus, so b*(*) gives equilibrium bids. Moreover, we have found the unique
symmetric equilibrium in which bids are an increasing function of wealth.
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There exist other equilibria in which bids are not both symmetric and strictly
increasing in wealth for w > w*. These equilibria yield the same expected surplus,
however, and the same expected revenue.
We conclude the analysis by characterizing the seller's expected revenue. The
impact of reserve prices is somewhat unusual here. Setting a reserve price r < w*
excludes buyers with wealth below r, with no countervailing benefit, since buyers with
wealth below w* bid their wealth. Therefore, the seller will not select a binding reserve
price below w*. If v is sufficiently large that w* = iV, for example, then all buyers bid
their wealth, and the seller will not employ a binding reserve price. In the absence of a
binding reserve price, the object sells with probability one, and the expected revenue
can be written as
It may be optimal to set a reserve price above w* if v is small. If r E (w*,min{v,w)], a
buyer with wealth w 2 r has an expected surplus of (v-r)F(rlN-l, so expected revenue is
-
SERf r [I - F ( ~ ) ~-] N[1v F(r)](v - r ) ~ ( r ) ~ " .
In particular, if r = v, then S E R ~ [I-F(V)~]V.
Figure 1 graphs the bids for the case
without a binding reserve (i.e., r = y).
We can now provide some additional comparisons that lead to the revenue
ranking. Consider a buyer with wealth w satisfying r < w < w*. The buyer wins with
probability F(wlN-l, and he receives expected surplus
(4)
U*(w)= ( v - w ) ~ ( w ) ~ - l ,
which is below the corresponding value in the second-price auction. The value of $1 of
additional wealth is
Ue(w) = (v-w)[(N-~)F(w)~-~~(w)]
- F(wlN-l.
The first term reflects the increased probability of receiving the net surplus (v-w), while
the second reflects the fact that the bid has increased for those cases in which the buyer
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would have won anyway (i.e., without the additional wealth). This second term is not
present in the second-price auction. If w > w*, the value of additional wealth is zero in
the first-price auction. In terms of (I),therefore, the multiplier on the wealth constraint
is lower in the first-price auction than in the second-price for all wealth levels.
If (R1) is not satisfied, there may be multiple bidding regimes, with buyers
bidding their wealth on disconnected intervals. The basic intuition does not change,
however. In particular, the highest level of wealth that induces a buyer to bid his
wealth is still below v, and expected surplus does not increase with wealth thereafter.
C.
Revenue Comparison
A buyer's expected surplus is at least as high in the second-price auction, for
each possible wealth, given the same reserve price. Suppose that v exceeds w. If
wealth is below w*, then the buyer bids his wealth in both auctions. Since the
probability of winning is the same, but the expected price is lower in the second-price
auction, the expected surplus is higher in the second-price auction. For wealth beyond
w*, expected surplus does not increase in the first-price auction. Since total surplus is
the same in the two auctions, given the same reserve price, the revenue ranking follows.
Proposition 1. The first-price auction has a higher optimal expected revenue
than the second-price. Expected revenue is strictly higher in the first-price auction if the
optimal reserve in the second-price is not equal to v.
Proof: If w 1 v, then all buyers are unconstrained, and revenue is equal to v in
both auctions. Now suppose that w < v, and that the reserve price r E [w,min{v,V)]is
used in both auctions. There are three cases to examine.
First, take r < w*. In both auctions, a buyer with wealth r receives an expected
mrplus of ( v - r ) ~ ( r ) ~ -Since
l . w* < v, direct comparison of (2) and (4) indicates that the
expected surplus is strictly higher in the second-price auction for all w E (r,w*].
Moreover, expected surplus is constant in the first-price auction for all w 2 w*. Since
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expected surplus is weakly increasing in wealth in the second-price auction, expected
surplus in that auction is strictly higher for all w > r.
Second, let r satisfy w* 5 r < v. Once again, in both auctions, a buyer with
wealth r has an expected surplus of (v-r)F(rlN-l. For w > r, expected surplus is
constant in the first-price auction, but it is weakly increasing in wealth in the secondprice.
Third, take r = v. Expected surplus is zero for all buyers in both auctions.
A reserve price r generates a total surplus of v [ l - ~ ( r )in~both
]
auction formats.
The expected surplus ranking implies that the seller's expected revenue is weakly
higher in the first-price auction. Now suppose, in particular, that the optimal reserve
price in the second-price auction is not equal to v. If the same reserve price is employed
in the first-price auction, then the analysis shows that the first-price auction yields a
strictly higher seller's expected revenue.
The comparisons above assume that the same reserve price was used in the two
formats. Clearly, the optimal reserve in the first-price auction may differ from the
optimal reserve in the second-price, which only strengthens the result. Q.E.D.
We have shown that the first-price auction dominates the second-price in a
setting where budget constraints are important and there is heterogeneity of wealth.
The result holds with optimal reserve prices or no reserve prices, and the difference in
revenue can be relatively large. Suppose, for example, that there are N = 2 bidders,
with wealth uniformly distributed on [0,1
I, and v 2 2. Buyers bid their wealth in both
auction formats so that the expected revenue in the first-price auction is S E R ~= 2/3,
whereas SERS= 1/3. As v drops to 1, S E Rfalls,
~ while SERs is unchanged initially. As
v drops further, both terms fall until they each equal zero when v equals zero.
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2.
Equilibria and Revenue Comparisons with Heterogeneous Valuations
Each buyer has wealth equal to w. Buyer itsvaluation of one unit of the good, in
the absence of wealth constraints, is vi E L V ] . Valuations are private information and
are independently and identically distributed, with the cumulative distribution function
G(*) and strictly positive density g(*). Buyers are risk-neutral. The seller has one
indivisible unit of the good to sell, which she values at zero.
All buyers are unconstrained if w 2 7, in which case the model collapses to a
standard independent private-values model. Conversely, if w I 5 all buyers are
constrained and find it optimal to bid their wealth. We therefore need only consider w
E
b V ) . Moreover, we need only consider reserve prices satisfying r E L w ] .
Neither auction maximizes expected revenue, in general. For instance, consider
w I y/N. It is optimal for the seller to allocate the object to each buyer with probability
1/N, for all realizations of {vi},and to extract w from each buyer. Since the auctions
cannot extract revenue from more than one buyer, they cannot be optimal sales
mechanisms. As we noted earlier, lotteries are not a practical alternative for most
private sellers.
A.
Second-Price Auctions
Once again, there is a dominant strategy for buyers in the second-price auction.
Buyer i will bid min(vi,w). As noted above, if w I 5 it is optimal for all buyers to bid w,
while if w 2 V, wealth does not bind. The rest of this section focuses on w E b V ) .
Suppose that there is a reserve price r E L w ] . Let v(l) and v(2) denote the
highest and second-highest valuations, respectively. If v(l) c r, then revenue is zero.
Now suppose that v(l) = v, where r I v I w. Bids are equal to valuations in this range,
so the high bidder pays the second-highest valuation if it exceeds the reserve price.
Otherwise, he pays the reserve. The expected price paid to the seller is therefore
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Now suppose that r 5 w < v. If v(2) > r, then the high bidder pays min(v(2),w).
If not, then he pays r. The expected price paid to the seller is now
(6)
Irw
E[max(min(v,, w),r)lv,,,= vl = [ ~ ( r ) ~+. '
U(N - l)G(ulN-'g(u)du]1[G(v)lN-'
The inequality in (6) holds because the seller receives w < v(2) if the second order
statistic exceeds w. Note also that the left-hand side of (6) is not the expected price paid
by a buyer with valuation v, conditional on winning. If v(2) 2 w, the price paid is w.
Since ties are broken randomly, however, the high-valuation buyer does not necessarily
win. As will be seen, this method of calculating expected revenue facilitates
comparison with the first-price auction.
Given a reserve price r E L w ] , (5) and (6) imply that expected revenue can be
written as
(7)
SERS= Irv ~[max(rnin(v,,,w),r)~v(~,
= vIdG,,, (v),
where G(l)(v) = G ( v ) is
~ the distribution of the first order statistic ~ ( 1 ) In
. particular, if
r = w, then SERS= [I-G(W)~]W.
B.
First-Price Auctions
The wealth constraint does not bind if w 2 V. Conversely, if w 5
all buyers
find it optimal to bid their wealth. We now consider the intermediate case where w E
b y ) ,and buyers may or may not be constrained. Once again, there is not a dominant
strategy in the first-price auction, so we must characterize the equilibrium payoffs.
A buyer with valuation v has equilibrium expected surplus of the form
U*(V)E max b (v-b)p(b),
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where p(b) denotes the probability of winning the auction with a bid of b. The
Envelope Theorem implies that
U*'(v>= p(B*(v)),
where B*(*)denotes the equilibrium bid function. Integrating implies
We now show that buyers employ a cutoff rule in determining their bids.
Lemma 2. Suppose that w E (57)and that the reserve price is r E h w ) . In
equilibrium, there is a valuation, v* > w, such that buyers with vi E Lv*) bid strictly
below w, while those with vi E (v*,7] bid w.
Proof: Feasibility of bids requires that each buyer's bid not exceed his wealth.
Let v* I 7 denote the infimum of the valuations for which the equilibrium bid is w. If
all buyers with valuations v < V bid strictly below w, then v* = 7 ,and the proof is
complete. Now suppose that v* < V. Standard incentive-compatibility arguments show
that the probability of winning is weakly increasing in an individual buyer's valuation.
All buyers with v > v* must therefore bid w.
It is not optimal to bid more than one's valuation, so v* 2 w. If v* = w, then a
buyer with valuation v* would receive zero expected surplus by bidding w. However,
since w > 5 that buyer could get a strictly positive expected surplus by bidding below
w, because he has a strictly positive probability of winning. Buyers with valuations
infinitesimally above v* would also have an incentive to bid strictly below w, since U*(*)
is continuous, contradicting the definition of v*. It follows that v* > w. Q.E.D.
We determine the bids for buyers with valuations v I v* through their expected
surplus. (If v* < V, a buyer with valuation v* is indifferent between bidding w and
bidding strictly below. We assume that such a buyer bids below w.) If v I v*, the
probability of winning is the probability that all other buyers have lower valuations.
(As noted above, the probability of winning is weakly increasing in v. If bids are
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constant, but below w, over an interval of valuations, then individual buyers would
have an incentive to raise their bids infinitesimally.) Given a reserve price r E [xw), (8)
indicates that
(9)
U*(v) =
I'
r
G ( U ) ~ du.
-'
It follows that a buyer with valuation v 5 v* bids
A buyer with valuation v* is indifferent between bidding B(v*) and w. Suppose
that all other buyers bid w if and only if their valuation exceeds v*. A bid of w wins
with probability 1/(n+l) if there are n other buyers with valuations above v*. It also
wins if all other bids are below w. Straightforward calculations show that a buyer with
valuation v*, who bids w, has an expected surplus of
Equating this expected surplus to U*(v*)from (9) implicitly defines v*.
The seller's expected revenue is
If r = w, then S E R =
~ [l-G(w)~]w.
C.
Revenue Comparison
The first-price auction dominates if no reserve prices are employed or if optimal
reserves are employed. The proof compares the (expected) price paid for all possible
realizations of the first order statistic of valuations, given a common reserve price. For
each such valuation, the winning bid in the first-price auction is weakly higher than the
expected price in the second-price auction. We graph the equilibrium bids in figure 2
for the case without a binding reserve price.
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Proposition 2. The first-price auction has a higher optimal expected revenue. It
has a strictly higher expected revenue if w E b V ) and the optimal reserve price in the
second-price auction is not equal to w.
Proof: The wealth constraint is not binding if w 2 V. The results of Myerson
(1981) and Riley and Samuelson (1981) imply that the optimal reserve price is the same
in the two auctions, as is the seller's expected revenue. The constraint binds for all
buyers if w I s so revenue is equal to w in both auctions.
Now suppose that w E b V ) and that the reserve price is r E L w ] in both
auctions. First consider r < w. The proof consists of looking at three possible ranges for
the highest valuation. If v(l) I w, (5)and (10) show that the expected price in the
second-price auction is equal to the price in the first-price auction. If w < v(l) I v*, (6)
and (10) show that the expected price in the second-price auction is strictly below the
actual price in the first-price auction. Finally, if v* < v(l), (6) shows that the expected
price is strictly below w in the second-price auction, whereas the price is equal to w in
the first-price auction. Now consider r = w. Expected revenue is equal to [I-G(w)~]w
in both auction formats.
We conclude that the seller's expected revenue is weakly higher in the first-price
auction. If w E b V ) and the optimal reserve price in the second-price auction is not
equal to w, then the first-price auction generates a strictly higher expected revenue.
Q.E.D.
3.
Heterogeneous Wealth and Valuations
In the general case, buyers can differ in both wealth and valuation.
Unfortunately, the equilibrium of the first-price auction cannot be characterized
completely enough to make general revenue comparisons. There are regions over
which comparisons can be made, however, and the first-price auction again dominates
in those regions. We sketch the arguments below.
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Suppose that valuations and wealth are distributed independently, with
distribution functions G(*)and F(*), respectively. In the second-price auction, it is a
dominant strategy to bid min{vi,wi). In the first-price auction, for a given wealth w,
there is a critical valuation v*(w) such that the equilibrium bid is below w for v > v*(w),
and is equal to w for v > v*(w). The various regions are graphed in figure 3, where 1 =
w and 7 = 7 . (We can set density equal to zero in the appropriate regions if the limits
are not equal.) In the second-price auction, buyers bid their wealth if they are above the
45" line. In the first-price auction, they bid their wealth only if they are above a wealth-
valuation locus that is itself above the 45" line.
Clearly, if all buyers are below the 45" line (with ex ante probability one), then
revenue equivalence holds, since all buyers are unconstrained. If all buyers are above
the wealth-valuation locus, then they all bid their wealth in both auctions. The firstprice auction dominates, as was shown in section 1. Now suppose that all buyers are
between the wealth-valuation locus and the 45" line, with probability one. In the firstprice auction, the winning bid is the expectation of the second-highest valuation, given
the highest valuation. In the second-price auction, the actual price paid is the secondhighest wealth. Since valuations exceed wealth for all buyers, the first-price auction
again dominates.
The probability of winning differs across auctions for a buyer with a given
valuation-wealth pair, even with the same reserve price. Therefore, rankings for
general distributions are not possible using the techniques of sections 1 and 2.
Calculation of the wealth-constraint locus requires solving differential equations. Even
the simplest case (N = 2 and uniform distributions) requires numerical solutions.
4.
Concluding Remarks
In this paper, we have demonstrated that, in two cases, first-price sealed-bid
auctions yield higher expected revenue than second-price sealed-bid auctions when
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buyers face wealth constraints: 1) heterogeneous wealth, which is the limiting case for
settings where variation in wealth is greater than variation in valuations, and 2)
heterogeneous valuations, which is the limiting case for settings where variation in
valuations is greater than variation in wealth. We should therefore see first-price
sealed-bid auctions, rather than second-price sealed-bid or oral ascending auctions, in
settings where wealth constraints are present, all else equal. This finding is consistent
with the government's predominant use of first-price sealed-bid auction^.^
A natural question concerns the robustness of the results to the availability of
credit. Suppose that buyers have future income against which they can borrow. The
case that we have considered corresponds to a situation where buyers can borrow only
at a very high interest rate. At lower rates of interest, buyers at certain wealth levels
will borrow. As long as buyers at some wealth levels find it optimal not to borrow,
however, the first-price auction will still dominate the second-price. Once the
borrowing rate is sufficiently low that buyers at all wealth levels borrow, revenue
equivalence reappears.
8 An interesting exception to this rule occurs with the sale of timber rights. The Federal
Bureau of Land Management, which operates the auctions, experimented with firstprice sealed-bid auctions and found that average winning bids were higher. It reverted
to using oral ascending auctions, however, because of a strong preference on the part of
the industry (Mead et al. [1983.]).
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References
Bergsten, C. Fred, Kimberly Ann Elliott, Jeffrey J. Schott, and Wendy E. Takacs (1987),
Auction Quotas and United States Trade Policy. Washington, D.C.: Institute for
International Economics.
Hendricks, Kenneth and Robert Porter (1992), "Joint Bidding in Federal OCS Auctions,"
American Economic Review Papers and Proceedings, 82,506-511.
McAfee, R. Preston and John McMillan (1987), "Auctions and Bidding," Journal of
Economic Literature, 25,699-738.
McDonald, Stephen (1979), The Leasing of Federal Lands for Fossil Fuel Production.
Baltimore: Johns Hopkins University Press.
Mead, Walter, Mark Schniepp, and Richard Watson (1981), "The Effectiveness of
Competition and Appraisals in the Auction Markets for National Forest Timber in the
Pacific Northwest," prepared for the U.S. Forest Service, September 30.
Milgrom, Paul and Robert Weber (1982), ''A Theory of Auctions and Competitive
Bidding," Econometrics, 50,1089-1122.
Myerson, Roger (1981), "Optimal Auction Design," Mathematics of Operations Research, 6,
58-73.
Pitchik, Carolyn and Andrew Schotter (1988), "Perfect Equilibria in Budget-Constrained
Sequential Auctions: An Experimental Study," Rand Journal of Economics, 19,363-388.
clevelandfed.org/research/workpaper/index.cfm
Riley, John and William Samuelson (19811, "Optimal Auctions," American Economic
Review, 71,381-392.
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Figure 1.
Bids
Second-Price Auction
First-Price Auction
Wealth
Source: Authors' calculations.
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Figure 2.
Bids
Values
Source: Authors' calculations.
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Figure 3.
Source: Authors' calculations.
@FedFRASER