Firms' Strategic Choices under Demand Uncertainty
(Preliminary and Incomplete)
Walter Beckert¤
University of Florida
December 2002
CORRESPONDENCE should be directed to:
Walter Beckert, Department of Economics, 224 Matherly Hall, PO Box 117140, University of
Florida, Gainesville FL 32611 - 7140, USA; Tel: 352 - 392 0113, Fax: 352 - 392 7860, e-mail:
beckert@u°.edu
¤ This
work was done while I enjoyed the hospitality of the Department of Economics at UCL. All errors are
mine.
1
Abstract
This paper investigates how ¯rms choose prices and production capacity when facing stochastic demand. With demand uncertainty, ¯rms face demand curves for their goods which
implicitly subsume the selling mechanism adopted in the markets for these goods. The selling
mechanisms considered in this analysis are bidding markets or auctions, and take-it-or-leave-it
sales. They have di®erent implications for the respective expected demand curves the ¯rms
face, and hence for the ¯rms' capacity choices. These implications entail consequences for
welfare. It is shown that auctions yield higher expected comsumer welfare than take-it-orleave-it sales when production costs are high, while take-it-or-leave-it sales enhance welfare
when production costs are low.
KEYWORDS: Stochastic Demand, Auctions, Take-It-Or-Leave-It sales, Monopoly.
2
1
Introduction
This paper investigates the behavior of ¯rms under intrinsic market uncertainty. The source of
market uncertainty considered in this paper is stochastic demand, arising from buyers' valuations
for goods being private information. The paper analyzes how ¯rms choose capacity when demand
is uncertain. The analysis of ¯rms' capacity choices builds on the insights provided by Kreps and
Scheinkman (1983) and (...) that ¯rms' capacity choices are best thought of as a two-stage process.
In their decision process, ¯rms face demand functions which implicitly subsume the mechanism
adopted to sell goods on a second stage, once ¯rms have pre-committed themselves to production
capacity on the ¯rst stage. This paper shows that, under demand uncertainty, the second-stage
selling mechanism has important implications for the ¯rms' ¯rst stage capacity decision. Bidding
markets, such as an auction market as the second stage selling mechanism, force the ¯rm to make
capacity decisions under price uncertainty, as bids are ex ante uncertain. Take-it-or-leave-it sales,
on the other hand, in which the seller quotes a price and potential buyers decide whether or not to
purchase at this price, introduce quantity uncertainty into the ¯rms' decision process, as potential
buyers willingness to pay is ex ante unknown. The paper shows that, next to production costs,
the type of uncertainty induced by the second stage selling mechanism determines ¯rms' capacity
choices. And it identi¯es important welfare implications of either mechanism when demand is
uncertain.
The results of this analysis have theoretical and policy implications. The departure from
deterministic demand, introducing stochastic demand, entails some surprising consequences. As
already alluded to, demand uncertainty can emerge in two representations: price uncertainty
in bidding markets such as auctions, and quantity uncertainty in take-it-or-leave-it sales. For
monopolistic sellers, dominance of auctions in terms of expected revenue is an easy corollary,
provided in this paper, to the well-known result that expected revenue in single item auctions
exceeds expected revenue in take-it-or-leave-it sales (see, for example, Beckert (2002), and related
Bulow and Klemperer (...)). One might intuitively suspect that bidding markets are better suited
to aim at the highest-valuation buyers, while once-and-for-all price quotes in take-it-or-leave-it sales
target a broader group of potential buyers with a wider range of valuations. Hence, this suggests
that when production costs are su±ciently high, a monopolist institutionally forced to sell in a
take-it-or-leave-it sale may ¯nd these costs prohibitive, choosing not to produce at all. If allowed
to sell in auction, on the other hand, the monopolistic ¯rm may expect to catch high valuation
buyers and, therefore, may establish production capacity, at least on a small scale. Hence, as
the paper demonstrates in general, with demand uncertainty, bidding markets as the second stage
selling mechanism and ensuing price uncertainty have obvious welfare bene¯ts in high production
cost environments, as they induce production, while take-it-or-leave-it sales may not sustain its
expected pro¯tablility.
As a converse, consider as an extreme case the absence of any production costs. Then, an
3
auction seller will keep capacity restricted because, beyond a certain point, the marginal expected
revenue from an additional successful bidder is negative. On the other hand, it turns out that
additional buyers always generate positive marginal expected revenue for the take-it-or-leave-it
seller. Therefore, the take-it-or-leave-it monopolistic seller will produce more than the auction
seller, and charge a lower price than what the auction seller may expect to receive from bids.
Hence, as the paper also establishes in some generality, with demand uncertainty, take-it-or-leaveit sales have consumer welfare bene¯ts in low cost environments, since production and expected
consumer surplus are higher. To summarize, the ¯rst part of the paper shows that monopolists'
capacity choices are not invariant to the selling mechanism when demand in stochastic, and it
highlights that this invariance has signi¯cant consumer welfare consequences.
As a dual analysis, the paper investigates procurement situations as well. The important
distinction between procurement decisions and production capacity decisions is that, while the
producer's problem is a two-stage problem with sunk capacity costs on the ¯rst stage, procurement
costs and bene¯ts typically arise jointly. Hence, typical procurement decisions are not two-stage
decision processes. Distinguishing procurement auctions and take-it-or-leave-it contract awards,
welfare results reminiscent to the ones arising from previously considered production decisions can
be identi¯ed. Ii is shown that if, for reasons of political credibility or commitment, a project
has to be undertaken once it is tendered, procurement auctions will only be held for su±ciently
high bene¯ts of the tendered project; take-it-or-leave-it awards allow the °exibility to quote a
procurement cost ceiling that always results in an expected pro¯t from tendering and hence always
leads to a tender being put out.
The welfare implications carry some signi¯cance in light of the dual, often simultaneous prevalence of these selling mechanisms in many markets, and the frequent legal mandate to adopt
bidding procedures, especially for the procurement of public goods. Real estate markets, markets
for building contracts and for heavy building equipment, and markets for airport landing slots are
among prime examples of markets on which these competing mechanisms co-exist. In the context
of building construction, the results of this paper are particularly relevant because public construction contracts are often legally bound to be awarded by holding procurement auctions. Bidding
in highway procurement has been studied econometrically by Jofre-Bonet and Pesendorfer (2000),
Feinstein et al. (1985), Porter and Zona (1993), and Ba jari (1997). Bajari et al. (2001) examine
the factors that determine whether non-residential building contracts in the private sector were
awarded by auction or negotiation, as a more re¯ned version of a take-it-or-leave-it sale. On the
basis of data from Northern California covering the period 1995-2000, they observe that auctions
tend to be counter-cyclical, while negotiations appear to be pro-cyclical. Their theoretical interpretation is that auctions are favored when buyers are more cost-sensitive. Procurement auctions,
with a monopsonistic buyer and the winning bidder submitting the lowest procurement cost as his
or her bid, are the mirror image of the type of auctions considered in this paper. The empirical
exploration by Bajari et al. suggests that auctions are favored when expected pro¯t margins are
4
critical, and this seems to coincide with economic downturns. The analysis in this paper suggests
that, if these projects were relatively low valuation projects, then awarding them by negotiations
would have created higher procurement and would have acted as counter-cyclical public policy
measure.
The results of this paper are also relevant for the recent explorations of policy options to manage
landing slots at congested airports, when considered in the joint context of capacity expansion.
Congestion management essentially always involves two components: decisions on capacity and a
mechanism to allocate it. Traditional pricing schemes for allocating landing and take-o® capacity
are akin to take-it-or-leave-it sales 1 . Recent initiatives by the U.S. Department of Justice, however,
consider landing slot auctions for congested airports, such as New York's La Guardia International
Airpot. The results of this paper identi¯e the long-term capacity repercussions that such allocation
mechanism can be expected to entail.
The paper proceeds as follows. Section two starts by laying out some auxiliary results from
mathematical statistics which permit convenient normal approximations to binomial distributions
and to the distributions of order statistics, both of which play important roles in the development
of the theory of optimal strategies under the two mechanisms and independent private valuations.
It proceeds by presenting general results for the monopoly case, illustrating all the relevant implications of the two selling mechanism on strategic monopolistic capacity choice in the context of a
simple example with independent, uniform valuations. Section three carries the analysis further
to analyze procurement situations. Section four expands the analysis to dynamic setting. And
section ¯ve concludes. Proofs are collected in the appendix of the paper.
2
Monopolistic Price and Capacity Choices
2.1
Preliminaries
This section is devoted to the analysis of a monopolist's capacity choice under demand uncertainty.
The monoplist's decision problem is recognized as a two-stage process. On the ¯rst stage, the
monoplistic producer commits to a capacity level. On the second stage, units of the good up to
the capacity level are sold, according to some selling mechanism. Potential buyers' valuations of
units the good are assumed to be private information. Two mechanisms are considered: Auctions,
in which potential buyers submit bids for untis of the good, and take-it-or-leave-it sales, in which
1 In
practice, these are more intricate than simple price quotes, involving often international agreements and
so-called \grandfather rights", as well as secondary trading. It is an open policy question how the recent debate of
landing slot auctions might address such intricacies. For a comprehensive description, see documents by the British
Civil Aviation Authority. The two most congested London airports, Heathrow and Gatwick, are also currently
subject to a debate on expanding their capacity.
5
the monopolistic seller quotes a unit price and potential buyers purchase units of the good if their
private valuations of these units exceeds the price.
To formalize the analysis, the following assumptions are maintained throughout and commented
on as they become relevant:
A1 There are n potential buyers with unit demands; their valuations Xi, i = 1; : : : ; n, for a
unit of the good are independently and identically distributed with cumulative distribution
function F (x), x 2 X , inffx : x 2 X g ¸ 0, with continuous density f (x) > 0 for all x 2 X ;
and E[X] < 1.
A2 Marginal revenue 1 ¡ F (x) ¡ xf (x) is strictly downward sloping 2 ;
A3 The production of the good under consideration involves constant marginal costs c and no
¯xed costs; production costs are sunk once cacapity is chosen.
A4 The item has zero value for the seller if it is not sold; the seller maximizes expected pro¯t,
and the buyers maximize expected surplus.
A5 The number of potential buyers, n, is large.
The auction seller faces price uncertainty at the second stage. Since from the seller's point
of view on the second stage all that matters is expected auction revenue, in light of the Revenue
Equivalence Theorem (Vickrey (1961), Riley and Samuelson (1981), Myerson (1981)) one can
remain agnostic about the format of the auction. For expositional purposes, the format is thought
of as analogous to a second-price sealed-bid auction in the single item case. If k items are sold,
then the k + 1st highest bid is paid by each of the k successful bidders. Let X(i) , i = 1; : : : ; n,
denote the ith order statistic; i.e.
X(1) = minfXi ; i = 1; : : : ; ng · : : : · X(n) = maxfXi ; i = 1; : : : ; ng:
Then, the expected revenue of the auction seller of k items is kE [X(n¡ k)], where the expectation
is taken with respect to the distribution of the (n ¡ k)th order statistic. This distribution of the
kth order statistic has a distribuion with density
f X (k) (x) =
n!
F (x)k¡1 (1 ¡ F (x))n¡ k f (x); x 2 X ;
(k ¡ 1)!(n ¡ k)!
see, e.g., Bickel and Doksum (1977). Hence, the auction seller's ¯rst-stage capacity choice problem
is
max k(E [X(n¡k ] ¡ c):
k
Notice that this embeds the solution to the second stage selling mechanism.
2 In
auction theory, this is sometimes referred to as a regularity condition; see Klemperer (2000).
6
Assumption A5 premits some convenient approximations which are heavily made use of in the
further development of the theory of capacity choice under demand uncertainty. For large n, the
scaled order statistics are asymptotically normally distributed (compare, e.g., Bickel and Docksum
(1977)), i.e. for k = [nt], t 2 [0; 1],
p
¡
¢
d
(n)(X(k) ¡ F ¡1 (t)) ! N 0; t(1 ¡ t)=(f (F ¡1 (t))) 2
so that the approximate distribution of X(k) is given by
Ã
!
µ ¶
k
k
(1
¡
)
k
1
A
n
n
X(k) » N F ¡ 1
;
; k = 1; : : : ; n;
n
n (f (F ¡1 ( nk )))2
here, F ¡ 1 (¢) denotes the inverse cumulative distribution function. 3 Hence, the auction seller's
¯rst-stage decision problem can be re-cast as
µ
µ
¶
¶
n¡k
max k F ¡1
¡c :
k
n
Considering, next, the take-it-or-leave-it seller. On the second-stage a once-and-for-all unit
price p is quoted and buyers subsequently decide whether or not to purchase a unit of the good.
Buyers are thought of as submitting purchase orders; if more purchase orders are submitted than
permitted by ¯rst-stage capacity, demand is rationed accordingly. Rationing is assumed to take
the form of each of the purchased orders being ful¯lled with equal probability. In this mechanism,
the seller faces quantity uncertainty on the second stage. The number of purchase orders at price p
has a binomial distribution bin(n; 1 ¡ F (p)), because the probability of a potential buyer actually
buying at price p equals the probability of this buyer's valuation exceeding p, Pr(X > p) = 1 ¡F (p).
Assumption A5 permits another useful approximation in the case of take-it-or-leave-it sales.
For large n, the binomial distribution can be closely approximated by the normal distribution
(see, e.g., Bickel and Doksum (1977)). Hence, for large n, the number of purchase orders N (n; p)
submitted is approximately distributed as
A
N (n; p) » N (n(1 ¡ F (p)); nF (p)(1 ¡ F (p))) :
Let ¹(n; p) = n(1 ¡F (p)) and ¾(n; p) =
p
nF (p)(1 ¡ F (p), and denote the cumulative distribution
function and the density of the standard normal distribution by © and Á, respectively. Also, let
N (n; k; p) denote the number of items sold at the second stage. Using this approximation, the
seller's expected sales, given price p at the second stage and capacity k chosen on the ¯rst stage,
are approximated by
E [N (n; k; p)]
=
=
3 Given
µ
¶
µ
µ
¶¶
1
k ¡ ¹(n; p)
k ¡ ¹(n; p)
u
Á
du + k 1 ¡ ©
¾(n; p)
¾(n; p)
¡ 1 ¾(n; p)
µ
¶
µ
µ
¶¶
µ
¶
k ¡ ¹(n; p)
k ¡ ¹(n; p)
k ¡ ¹(n; p)
¹(n; p)©
+k 1¡©
¡ ¾(n; p)Á
:
¾(n; p)
¾(n; p)
¾ (n; p)
Z
k
that A1 requires f(x) > 0 on X , F(x) has not °at regions and, therefore, the inverse CDF is well de¯ned.
7
Note that in the case of both normal approximations, the approximation error is of the order
of
1
n.
2.2
Bickel and Doksum (1977) provide further details.
Optimal Capacity Choices
This section develops some general results regarding optimal capacity choices under the two mechanisms. These are illustrated later in the context of a simple example.
As a consequence of A1, pE [N (n; k; p)] is continuous in p and ¯nite, since E [X] < 1 implies
that there is no probability mass at in¯nity if X in unbounded. Therefore, it has a maximum
over p on the interior of X . Let v(n; k) = maxp pE [N (n; k; p)], the value function associated
with the second stage optimization problem in the take-it-or-leave-it mechanism. Recall for this
interpretation that production costs ck are sunk, once the monopolistic producer has committed
capacity k on the ¯rst stage. Although the problem of capacity choice is inherently discrete under
the assumption of unit demands of n potential buyers, treating k as a continuous variable greatly
facilitates the analysis. Therefore, k is henceforth treated as a continuous variable. A number of
useful properties of this value function are summarized in the following auxiliary result.
Lemma 1: Under assumptions A1, A2, A4 and A5, (i) p k = arg maxp pE [N (n; k; p)] exists
and is unique, and (ii) the value function v(n; k) is monotonically increasing and concave in k.
The proofs of this and some subsequent results are collected in the appendix to the paper.
The second part of this result can be paraphrased as diminishing, but uniformly positive marginal
expected revenue of additional units from the perspective of the take-it-or-leave-it seller. It has
an immediate consequence for the optimal capacity choice k ? (c) = arg maxk v(n; k) ¡ ck of the
take-it-or-leave-it seller.
Theorem 1: Under assumptions A1-A5, k ? (c) < n if, and only if, c > 0.
The proof follows directly from Lemma 1 and the simplifying treatment of k as a continuous
choice variable.
Marginal expected revenue is not uniformly positive for the auction seller. Marginal expected
revenue for the auction seller is given by
£
¤
d
d
kE X(n¡k) =
kF ¡ 1
dk
dk
µ
n¡k
n
¶
=F
¡1
µ
n¡k
n
¶
· µ
¶¸¡1
k
n¡k
¡
f
;
n
n
therefore, f (x) > 0 for all x ¸ 0 = inffx : x 2 X g (A1) implies, at k = n, F ¡1 (0) ¡ [f (0)] ¡1 < 0.
£
¤
^
Letting k(c)
= arg maxk k(E X(n¡ k) ] ¡ c), this proves a counterpart to Theorem 1:
^
Theorem 2: Under assumptions A1-A5, k(c)
< n for any c.
8
The two results, taken together, might lead to the erroneous conjecture that take-it-or-leave-it
sellers always commit to higher capacity at the ¯rst stage than the auction seller. The reason why
this is not true is that the marginal expected revenue of the auction seller for small capacities k
is higher than the marginal expected revenue of the take-it-or-leave-it seller. Hence for high unit
costs, provided they are not prohibitively high, the auction seller derives an expected pro¯t from
establishing production capacity, while the take-it-or-leave-it seller does not. This is a consequence
of the next result.
Theorem 3: Under assumptions A1, A2, A4 and A5,
£
¤
^
= kE X( n¡k) ¸ v(n; k), for any k · k(0),
and
h
i
(ii) ^k(0)E X(n¡^k(0) ) ¸ p n E [N (n; n; pn )] = maxp pE[N (n; n; p)].
(i) kF ¡ 1
¡ n¡ k ¢
n
The result is proven for the single unit case k = 1 in Beckert (2002) and follows for k > 1 by
induction on k; the proof is provided in the appendix.4 As already alluded to, it has an important
consequence. Consider the case k = 1. The result states that the expected revenue from selling a
single unit in an auction is higher than the expected revenue from selling it in take-it-or-leave-it
fashion. Hence, there exists a range of high unit costs of production for which the auction seller
expects establishing production capacity to be pro¯table, while the take-it-or-leave-it seller expects
to incur a loss from producing any positive amount.
On the other hand, there clearly also exist situations in which unit costs are prohibitively high
for an auction seller to establish production capacity. These arise for marginal cost c such that
£
¤
E X(n¡1) < c. Theorem 3 implies that a take-it-or-leave-it seller, for such c, would not expect
production to be pro¯table either.
The three theorems, taken together, by continuity of the respective objective functions in c,
have then the following, immediate corollary which summarizes the ¯rst-stage optimal capacity
choices, given the second-stage selling mechanisms.
Corollary 1: Under assumptions A1-A5, there exists
(i) ¹c > 0: c ¸ ¹c ) ^k(c) = k ? (c) = 0;
(ii) c > 0; c < ¹c: c 2 (c; ¹c) ) k? (c) = 0; ^k(c) > 0 and k? (c) = 1 and ^k(c) ¸ 1;
(iii) ^c > 0; ^c < c: c · ^c ) ^k(c) · k? (c).
The signi¯cance of the corollary lies in identifying cost and mechanism constellations which,
under the maintained assumptions, permit unambigous welfare rankings. For high unit costs,
4 The
auction format in Beckert (2002) is the one of ¯rst-price sealed-bid auctions; the result applies by virtue of
the Revenue Equivalence Theorem.
9
the auction mechanism dominates the take-it-or-leave-it sale mechanism, while for low unit costs
the reverse is true. And there exists a range of costs where further welfare analysis requires an
assessment of optimal price quotes in the take-it-or-leave-it sales, and expected per-unit bids in
the case of auctions, with the aim to deduce expected consumer surplus. Notice that Theorem 3
already succinctly ranks expected revenues.
The last result of this section characterizes optimal price quotes and expected bids for interior
capacity choices.
Theorem 4: Under assumptions A1-A5, for c and c^ as in Corollary 1,
h
i
(i) E X(n¡^k(c)) · p k? (c) = p 1,
h
i
(ii) E X(n¡^k(c)) ¸ pk ? (c) for c · ^c.
The theorem, proven in the appendix, implies, in light of Corollary 1, that expected consumer
surplus at the threshold cost margin c is higher under the auction mechanism: Production capacity
is higher, and expected unit payments are lower. By Theorem 3, expected auction revenue is also
higher than expected sales revenue. Since capacity is higher under the auction mechanism, overall
expected welfare is higher. For low unit cost c · ^c, expected consumer surplus is higher under
the take-it-or-leave-it sales mechanism, since both the production capacity under this mechanims
is higher and the sales price falls below the expected winning bid. Expected revenue is lower,
however. As a consequence of capacity being higher under the sales mechanism, expected social
welfare is higher.
2.3
A Uniform Example
This section illustrates the main ideas of the analysis of monoploistic capacity choice under demand uncertainty in the context of a simple, tractable example. It uses the previously introduced
approximations and specializes the distributional assumptions. Speci¯cally, it is assumed that
the cumulative distribution of potential buyers valuations F is the uniform distribution, so that
X = [0; 1] and F (x) = x, x 2 [0; 1]. All other assumptions are retained.
Consider, ¯rst, as a benchmark the case c = 0. The take-it-or-leave-it seller chooses k ? = n,
by virtue of Lemma 1. Therefore, at price p 2 [0; 1], ¹(n; p) = n(1 ¡ p), ¾ 2 (n; p) = np(1 ¡ p), and
expected pro¯ts are
pE [N (n; n; p)]
=
"
p n(1 ¡ p)©
Ã
p
¡ np(1 ¡ p)Á
p
Ã
np
np(1 ¡ p)
!
np
p
np(1 ¡ p)
10
Ã
+ n 1¡©
!#
Ã
p
np
np(1 ¡ p)
!!
=
·
np 1 ¡ p©
µr
¶¸
np
1¡p
¡p
p
np(1 ¡ p)Á
µr
np
1¡p
¶
:
It is easy to see that this expression is maximized at p? = p n = 12 . To verify, the normal probap
p
p
bility reduces to ©( n) ! 1 and the normal density reduces to 12 nÁ( n) ! 0 as n gets large.
Therefore, the entire expression reduces to np(1 ¡ p) which is indeed maximized at the claimed
value. Note that, for c = 0, the maximizing price
price is
1
2
is independent of n. Expected demand at this
n
.
2
Now consider the auction seller. The auction seller maximizes k n¡k
, which is maximizes at
n
^k =
n
2
< n = k? , as predicted by Corollary 1. The expected winning bid is
n¡k^
n
=
1
2
= p? ,
consistent with Theorem 4. Notice that also the winning bid for c = 0 is independent of n. Since
the expected winning bid equals the optimal price quote, and expected sales equal the number of
units for auction, expected consumer surplus is equal under the two mechanisms. This can also be
formally shown, as follows. Expected surplus under the take-it-or-leave-it sale mechanism is
Z 1
3
?
?
n (E [X; X ¸ p ] ¡ p ) = n
xdx = n:
8
1=2
Expected surplus under the auction mechanism, given any k · n, is
k
k
£
¤
1X
1X
E X(n) + ¢ ¢ ¢ + X( n¡k) =
(n ¡ i) = k ¡
i:
n i=0
n i= 0
Substituting ^k =
n
2
yields
h
i
E X(n) + ¢ ¢ ¢ + X( n2 )
n=2
=
¼
=
n
1X
¡
i
2
n i=0
Z
n
1 n=2
¡
idi
2
n 0
3
n:
8
Leaving the benchmark case, consider the case of unit cost c =
1
5
and n = 50. The auction model
can still be solved analytically, but the take-it-or-leave-it sale model has to be solved numerically. 5
Following the same steps as above, ^k( 15 ) = 20, and the expected winning bid is E[X30 ] = 35 .
Therefore, expected pro¯ts are 20( 35 ¡ 15 ) = 8. Numerically solving the take-it-or-leave-it model
yields k ? ( 15 ) = 22, pk ? ( 15 ) = 0:59, and expected pro¯ts of 7:24, illustrating Corollary 1 and Theorems
h
i
3 and 4. Since now p ? 1 < E X
and k? ( 1 ) > ^k( 1 ), expected consumer surplus is higher
^ 1
k ( 5)
(n¡k( 5 )
5
5
under the take-it-or-leave-it mechanism.
For completeness, from the exact distribution6 of the maximum of n independently uniformly
£
¤
n
distributed random variables, E X(n) = n+1
= c¹; for n = 50, this yields c¹ = 0:98. Similarly, the
5 The
6 In
computations are straightforward, and a code is available upon request.
the uniform example, Xk » ¯(k; n ¡ k + 1), for k = 1; : : : ; n.
11
maximum unit price for the take-it-or-leave-it seller is maxp p(1 ¡ p n ) =
n
(n
n+1
+ 1)¡1=n = c; for
n = 50, this amounts to c = 0:91.
3
Procurement
The previous section considered auction sales and take-it-or-leave-it sales under demand uncertainty. The results presented in that section have natural counterparts in procurement. Procurement auctions have been studied by [...] in the case of [...]. Here, pro jetcs are tendered
by a procurement agency. The procurement agency takes the role of the seller, and the project
providers take the role of the buyers. The pro ject's unit value is known to the procurement agency.
The cost of procurement incurred by the potential provider of the pro ject is private information.
Assumptions A1 through A5 have the following analogues:
B1 There are n potential providers with unit supplies; their production costs Ci , i = 1; : : : ; n, are
independently and identically distributed with cumulative distribution function F (c), c 2 C,
inffc : c 2 Cg ¸ 0, with continuous density f (c) > 0 for all c 2 C; and E [C ] < 1.
B2 1 ¡ F (c) ¡ cf () is downward sloping;
B3 the provision of the pro ject under consideration produces unit value v;
B4 the project provision entails zero cost to potemtial providers they are not chosen in the
tender; the procurement agency maximizes expected surplus, and the providers maximize
expected pro¯ts.
B5 The number of potential providers, n, is large.
In standard auctions for k items, the winning bidders pay the k + 1st highest bid. In the procurement auction for k units, the winning bidders receive the k + 1st lowest bid. The procurement
agency maximizes expected surplus k(v ¡ E [X(k+1) ]).
12
4
Dynamic Choices
5
Appendix
5.1
Proof of Lemma 1
(i) Algebra yields
d
pE[N (n; k; p)]
dp
=
¶
µ
¶
k ¡ ¹(n; p)
k ¡ ¹(n; p)
+ k(1 ¡ F (p))©
¾(n; p)
¾(n; p)
µ
¶
µ
¶
1
1
k ¡ ¹(n; p)
k ¡ ¹(n; p)
¡ npf(p)(1 ¡ 2F (p))
Á
¡ ¾(n; p)Á
2
¾(n; p)
¾(n; p)
¾(n; p)
µ
µ
¶¶
k ¡ ¹(n; p)
+k 1 ¡ ©
:
(5-1)
¾(n; p)
n [(1 ¡ F (p)) ¡ pf (p)] ©
µ
Dividing by n,
1 d
pE[N (n; k; p)] = [1 ¡ F (p) ¡ pf (p)] ©
n dp
µ
k ¡ ¹(n; p)
¾(n; p)
¶
p
+ o(n) + o( n);
p
where terms of order o(n) arise from the second summand in 5 ¡ 1, and terms of order o( n) from
the last two terms. For p = inf fx : x 2 X g, this expression is positive, while for p = supfx : x 2 X g,
±
it is negative. Hence, assumption A2 implies that there exists a unique, interior p k 2X such that
d
pE [N (n; k; p)]jp=pk
dp
= 0.
(ii) Let p k = arg maxp pE [N (n; k; p)]. Since v(n; k) = maxp pE [N (n; k; p)], it follows that
@
ER(n; k; p k ) =
@k
=
@
p k E [N (n; k; pk )]
@k µ
¶
k ¡ ¹(n; pk )
1¡©
> 0;
¾(n; p k )
where the ¯rst equality follows as a consequence of the Envelope Theorem. This proves monotonicity.
Concavity follows from the concavity of E R(n; k; p)
³ = pE [N´(n; k; p)] in k and p. It follows from
2
@
@
the preceding paragraph that @k
E R(n; k; p) = 1 ¡ © k¡¹(n;p)
> 0. Therefore, @k
2 ER(n; k; p) =
¾(n;p)
³
´
n;p)
1
¡ ¾ (n;p)
Á k¡¹(
< 0. Since E R(n; k; p) has a maximum over p, given k, as argued above,
¾(n;p)
@
@p E R(n; k; p)
@2
@p2 E R(n; k; p)
d
implies that dk
pk
· (>)0 whenever p ¸ (<)p k , and as p k maximizes ER(n; k; p),
Finally, standard algebra shows
2
@
ER(n; k; p)
@ k@ p
< 0, which, incidentally,
< 0.
< 0.
Hence, the Hessian of E R(n; k; p) with respect to k and p is negative semi-de¯nite, and therefore
the concentrated value function v(n; k) is concave in k.
13
2.
5.2
Proof of Theorem 3
The proof proceeds by induction. Part (i) for k = 1 is proven in Beckert (2002). The induction step
¡
¢
^
is implied by (ii) and the monotonicity of v(n; k), provided kF ¡ 1 n¡k
is monotone for k · k(0)
n
as well. To see this, observe that
µ
¶
µ
¶
µ
¶
d2
n¡k
2 ¡1 n ¡ k
k ¡2 n ¡ k
¡1
kF
=
¡
f
¡
f
< 0;
dk2
n
n
n
n
n
¡
¢
so that kF ¡1 n¡k
is seen to be concave. Since ^k(0), by de¯nition, is its maximizer, monotonicity
n
for k · ^k(0) follows. To prove now (ii), observe that it follows from the induction hypothesis for
k = 1, the de¯nitions of pn and ^k(0) as well as the respective ¯rst-order conditions that
h
i
h
i2 ³ h
i´
^k(0)E X
=
nE
X
f
E
X
]
^
^
^
(n¡k(0))
(n¡k(0))
(n¡ k(0))
£
¤2
¸ nE X(n¡1) f (E [X(n¡1)])
¸
nv(n; 1)
¸
nv(1; 1)
=
n (1 ¡ F (pn ))2 =f (pn )
=
npn (1 ¡ F (p n ))
=
pn E [N (n; n; p n )] ;
which completes the proof.
5.3
2
Proof of Theorem 4
Let Ãn (x) = Pr(X(n) > x) ¡ xp X (n) (x), where p X(n) (x) is the normal density with mean E [X(n) ].
^ = k(c)
^
For ease of notation, let k
and p k ? = pk ? (c) = p 1 . By de¯nition of p k ? , Ã n (p k ? ) = 0. Hence,
to prove (i), it su±ces to prove that à n (E [X(n¡k)
^ ]) > 0. To see this, observe that
³
´
³
´
³
´
à n E [X(n¡k)
= Pr Xn > E[X(n¡^k) ] ¡ E[X(n¡^k) ]p X (n) E [X(n¡ ^k) ]
^ ]
¡
¢
¡
¢
¸ Pr Xn > E [X(n) ] ¡ E[X(n) ]p X (n) E [X(n) ]
=
1=2 + o(1);
^
where the inequality follows from E[X(n)] > E [X( n¡k)
^ ] for k ¸ 1 and the fact that the normal
density is maximized at its mean.
To prove (ii), using the abbreviated notation ^k = ^k(c) and k? = k ? (c) for c · ^c, observe
that E [X( n¡ ^k) ] ¸ E[X(n¡k ?+1) ]. For n ¡ k? + 1 = [nt], using the normal approximation to the
distribution of the order statistics,
à [nt ](F
¡1
s
1 p ¡1
f 2 (F ¡ 1(t))
(t)) = ¡ nF (t)
· 0 for large n:
2
t(1 ¡ t)2¼
14
Since E[X(n¡hatk) ] > F ¡1 (t) = F ¡1
theorem.
³
n¡k ?+ 1
n
´
, the result follows. This completes the proof of the
2
15
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