Econ TheoryDOI 10.1007/s00199-014-0825-z

RESEARCH ARTICLE

An optimal slow Dutch auction

Artyom Shneyerov

Received: 29 October 2012 / Accepted: 5 June 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract We study slow Dutch auctions, where the clock does not fall instantaneously,but instead falls over time. Buyers are assumed less patient than the seller. In a sym-metric setting, we investigate the properties of the optimal revenue-maximizing clock.We find that the clock is genuinely dynamic and the auction involves delays.

Keywords Dynamic auctions · Optimal control

JEL Classification D44 · D82

1 Introduction

With the arrival of the Internet, we have witnessed a proliferation of new auctiondesigns in real-world marketplaces. In this paper, we study one such new design, aslow Dutch auction (SDA), where the clock does not fall instantaneously, but insteadfalls over time. For instance, Peterson CAT is a machinery company that supportsbusinesses in industries such as construction, agricultural and trucking by selling usedequipment through an SDA.1 The auction begins with an asking price that is reduceddaily until a purchase is made or the reservation price is reached. Another example

1 http://auction.petersoncat.com/.

I would like to thank Nicholas Yannelis (editor), Johannes Hörner (co-editor), as well as two anonymousreferees for their comments.

A. Shneyerov (B)CIREQ and CIRANO, Concordia University, Montreal, Canadae-mail: artyom239@gmail.com

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A. Shneyerov

is Northcore Intellectual Property Technologies that holds several SDA patents andoffers customized web-based hosting models that implement these auctions.2

But why would a seller choose to run an SDA? In a transferable utility settingwith risk-neutral buyers, Myerson (1981) shows that the seller can do no better, butinstantaneously reduce the price down to the optimal reserve. In our paper, the settingis a symmetric one essentially as in Myerson (1981), but the utility is not transferableover time: Buyers are more impatient than the seller. Specifically, a risk-neutral sellerfaces N risk-neutral buyers. The seller has one indivisible unit for sale. The sellerdiscounts time at the rate ρ, and buyers discount time at the rate r > ρ. We study anSDA where the seller commits to a price schedule, or price clock p(·), beginning witha high price, and then reducing it until the item is sold. Motivated by applications,we assume that the time of sale is exogenously constrained by some T > 0. Theprice schedule is the object of the seller’s design, with the purpose of maximizing thediscounted expected revenue.

Buyers draw their valuations from the same distribution, and we restrict attentionto symmetric equilibria. Buyers are forward looking. At any point in time, a buyer caneither purchase at the standing price p(t), or wait until the price falls further. We putessentially no restrictions on the price clock. For example, we allow the price to dropinstantaneously at any point in time. We also allow the price to remain constant overan interval, so that the good is effectively removed from the market over that period.

Our main result can be summarized as follows: The seller can do better than aninstantaneous auction by committing to a dynamic price clock. Thus, an optimal solu-tion involves delays. We use optimal control to investigate properties of the solution.As we show, the seller can equivalently implement the price clock via a time of salefunction t (·), prescribing which buyer types should buy at different points in time. Anypoint of strict monotonicity of this function corresponds to a continuous reduction inthe price, while any interval on which it is constant corresponds to an instantaneousreduction.

To gain insights into the structure of the solution, we first consider a relaxed prob-lem, with the monotonicity constraint on t (·) removed. We show that the optimalsolution will involve instantaneous auctions at the initial time t = 0 and the finaltime t = T . Otherwise, sales will effectively occur at posted prices. The posted priceschedule is shown to cross the static optimal reserve. At the final time, the price isinstantaneously reduced. Moreover, the optimal solution is continuous, implying thattemporarily removing the good from the market is not optimal. We next character-ize the solution under the monotonicity constraint on t (·). We show that almost allproperties of the relaxed-optimal solution carry over.

The intuition for our result is closely related to Stokey (1979) paper on intertemporalprice discrimination. When the seller and the buyers are equally patient, the seller doesnot benefit by selling to lower-valued buyers at later dates, i.e., from inducing delayedpurchase. The reason is that lowering the price at later dates creates an option value ofwaiting for buyers who have higher values, so the prices that can be charged to inducehigher-valued buyers to purchase at early dates must be reduced. On the other hand, if

2 http://www.northcore.com/downloads/overviewarticles/overview-ip-dutch-01.pdf.

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An optimal slow Dutch auction

the buyers are more impatient, then the option value of waiting for high-value buyersis less from the perspective of their intended early purchase date. So the rent theyexpect from waiting and mimicking a low-value buyer is more heavily discounted.This explains why intertemporal price discrimination, in the form of a declining pricepath, becomes optimal.

Subsequent to Stokey (1979), Landsberger and Meilijson (1985) have studied adurable good monopolist that sells to a population of consumers under commitment.As in our model, the monopolist is assumed to be more patient. This setting effectivelycorresponds to N = 1 in our model. Landsberger and Meilijson (1985) have also shownthe optimality of delays, assuming that the price path is continuous and differentiable.3

The solution in Landsberger and Meilijson (1985) has several features that are similarto our solution.

But the problem with N > 1, atomic buyers are quite different from the continuummodel studied in Landsberger and Meilijson (1985). The main difference is the neces-sity to allocate the good to the buyers whose assigned purchase time is the same. In oursolution, this happens at t = 0 and t = T . As one would expect, this allocation wouldneed to be done by an (instantaneous) auction. But the presence of auctions makesthe problem more difficult since, unlike in Landsberger and Meilijson (1985), herewe have a game of incomplete information among the buyers, and we use mechanismdesign tools throughout the paper.

In order to investigate the optimal time of sale schedule, we use the tools of optimalcontrol. The key technical difficulty is the need to impose a monotonicity constrainton the solution, but without imposing any additional restrictions. In particular, thesolution may have discontinuities, which in theory would correspond to a temporalremoval of the good from sale, and it may also be non-smooth as it may involveinstantaneous auctions. (Recall that an instantaneous auction would involve bunchingof types over an interval, which may result in a non-smooth solution.)

For piecewise differentiable schedules with finitely many jumps, the problem canbe analyzed using results available in the literature. In particular, Vind (1967) proposesa re-parameterization technique for such problems. The monotonicity constraint canthen be imposed as in, e.g., Guesnerie and Laffont (1984), i.e., as the sign constrainton the derivative of the schedule. But this approach is clearly restrictive as the fullyoptimal schedule may fail to be piecewise differentiable. Recently, Hellwig (2008)has proposed an extension of the Maximum Principle to arbitrary monotone controls.His approach, however, results in non-standard conditions.4

In this paper, we develop a new technique for optimal control problems with amonotonicity constraint on the control, which in the end amounts to a standard Max-imum Principle. Instead of using the time of sale schedule as a single function, were-parameterize it so that both the time of sale and the buyer type become functions ofa new “abstract” buyer type. So instead of one monotone function, we now have twomonotone functions. The reason why this helps is that according to the Lebesgue rec-

3 Wang (2001) studies pricing by a durable good monopolist and fully characterizes the optimal priceschedule, assuming infinite horizon.4 In particular, his approach entails a complementary slackness condition (f) in Theorem 3.1. It is unclearhow to apply it in the setting of our paper.

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tification theorem,5 these two functions can be chosen a.e. differentiable. This bringsthe problem back into the realm of the standard optimal control.

Moreover, this method can be used to incorporate price clocks that are not allowedto fall instantaneously.6 This is important because, in reality, when a Dutch clock fallsinstantaneously, a bidder will not be able to stop the clock at an intermediate price.So clearly, such an instantaneous Dutch auctions is a mathematical abstraction. Thisabstract clock should be viewed it as a limit of “physical” clocks that can only fall at acertain maximal speed. Using our approach, we show that the optimal unconstrainedsolution can be approximated arbitrary well by such realistic price clocks.

Further related literature. SDAs have been used on the Internet and Lucking-Reiley (1999) reported that in controlled Internet experiments, they produce greaterrevenue than sealed-bid auctions. This conforms with our results. Carare and Rothkopf(2005) study SDA with bidding costs framework and show that they may outperformsealed-bid auctions. However, they do not consider impatient buyers, nor do theyconsider an optimal design. Other than this paper, SDAs have been mostly ignored bythe theoretical literature.

Hörner and Samuelson (2011) consider a setting where the seller of a perishablegood faces N strategic buyers, as in our model, but under noncommitment and withequal discounting. They show that the deadline at t = T endows the seller withconsiderable commitment power.

In addition, there is a substantial literature on revenue management when buyersarrive over time. In operations research, it is often assumed that buyers are myopic. 7

Recent contributions in economics include Gershkov and Moldovanu (2009a,b). Withforward-looking buyers as in our paper, but without commitment, Conlisk et al. (1984)show that optimal pricing is nonstationary even in a stationary environment. Board(2008) fully characterizes the price sequence.8

Board and Skrzypacz (2010) consider a setting with a perishable good and commit-ment as in our model, but with buyers entering at random times. The optimal solutionis shown to involve posted prices for all periods except the last one. Similar to oneof our results, Board and Skrzypacz (2010) show that auctions may occur at the lastinstance, otherwise buyers buy exclusively at posted prices. Another related paper isFuchs and Skrzypacz (2010), who find that delays are optimal in a market where theseller bargains with sequentially arriving buyers. As in our model, they find that theseller “slowly screens out buyers with higher valuations.”

We should also mention two recent interesting papers on bunching without the useof optimal control: Noldeke and Samuelson (2007) and Toikka (2011).

5 See e.g. Theorem 2.2.2 in Aleksandrov and Reshetniak (1989) and Theorem 4.3 in Stein and Shakarchi(2005).6 See the discussion immediately after Proposition 3.7 This literature is extensively reviewed in Board and Skrzypacz (2010).8 In addition, Koh (2006) considers a model where infinitely-lived households face an intertemporal budgetconstraint, and finds that the price of the durable good may increase or decrease over time. Rustichini andVillamil (1996) consider a model of intertemporal pricing in a market with stochastic demand and find thatprice cycles are possible.

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An optimal slow Dutch auction

The structure of the paper is as follows. Section 2 introduces the model and also con-tains equilibrium characterization results. Section 3 characterizes the solution of therelaxed revenue maximization problem and discusses the form of the solution. Section4 shows that essentially the same characterization carries over to the monotonicallyconstrained problem. Section 5 concludes.

2 The model

A risk-neutral seller, who owns a single unit of a good, faces N ≥ 1 risk-neutralbuyers. We assume independent private values (IPV). All buyers draw their valuationsfor the good simultaneously and independently from the same distribution F(·), withsupport [0, 1] and density f (·) bounded away from 0 on the support. The seller’s costis normalized to be 0. The seller discounts future at the rate ρ ≥ 0, while the buyersdiscount at the rate r > ρ.

The seller commits to a selling mechanism, a Dutch clock P . Because we want toallow a general form of a price clock, including instantaneous price reductions, ourbasic mathematical object to model such a clock will be a planar curve rather than afunction. That is, in a parametric form,

P ≡ {(τ (w), p(w)) : w ∈ [0, 1]} (1)

where w is a parameter that describes movement of a point along the curve, andthe coordinate functions τ : [0, 1] → R+ (time) and p : [0, 1] → R+ (price) arecontinuous nondecreasing and nonincreasing, respectively.

Figure 1 exhibits an example of such a clock. There is a continuous initial segment,followed by a downward jump at t = t1. From t1 to t2, the price remains constant, andthen, over the interval [t2, T ], continuously declines again. As we shall see, the intervalsof continuous price decline may only involve sales at posted prices. A downward jumpwill always involve selling at an auction. Whenever the price is constant over an intervalof time, the good is effectively removed from the market over that interval, as no buyerwho is impatient will prefer to buy later at the same price.

Any buyer can stop the clock at any price p0 ∈ [p(0), p(1)], pay this price andobtain the good. As usual, ties are broken randomly; however, in equilibrium, therewill be no ties. The time of sale is defined as the earliest instance of the clock reaching

Fig. 1 An admissible priceschedule

t

p

T

1

p (0 )

t1 t 2

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A. Shneyerov

price; that is, using parametrization (1), we find the smallest w0 such that p0 = p(w0),and then define the time of sale t0 accordingly, t0 = τ(w0).

We restrict attention to a symmetric Bayesian Nash equilibrium, wherein each buyeremploys a nondecreasing bidding strategy b : [0, 1] → R+ that specifies the priceat which the buyer would stop the clock. Such a strategy defines a set of buyer typesA ⊆ [0, 1] that are ever able to purchase the good given clock P ,

A ≡ {v : b(v) ∈ [p(0), p(1)]}. (2)

A bidding strategy b(v) uniquely determines the time of sale function t : A → R+,where t (v) = τ(w0) is the time of sale for type v buyer, i.e., w0 is uniquely definedas the smallest w such that b(v) = p(w).

An equilibrium is further characterized in Proposition 1 below. The propositionshows, among other things, that the equilibrium time of sale function is nonincreasingand right continuous. Going in the opposite direction, the proposition also shows thatany nonincreasing, right-continuous function is implementable as an equilibrium timeof sale function with an appropriately chosen price clock. A particular parametrizationof the Dutch clock helpful in this respect is a revealed choice parametrization: Thetime coordinate of the clock is parameterized as t (v). The envelope theorem then canbe used to recover the bidding strategy; the proposition shows that it is given by

b(v) = v − ert (v)

F(v)N−1

v∫

v

e−r t (v)F(v)N−1d v, ∀v ∈ A, (3)

where v is the lowest buyer type able to obtain the object with a positive probability.(See “Appendix” for the proof of this result.) The price clock that implements t (v) isshown to be

{(t (v), b(v)) : v ∈ [v, 1]}. (4)

A difficulty with such a parametrization, however, is that the resulting curve is notconnected whenever t (v) is not continuous. It turns out that this is not a big problembecause we can connect the curve segments without changing buyers’ incentives,by following the following completion procedure. Refer to Fig. 2. Suppose v is adiscontinuity point of t (v), so that no buyer types will purchase over the time interval[t (v), t (v−)).9 We need to connect the points (t (v), b(v)) and (t (v−), b(v−)). First,we extend the price b(v) over the interval [t (v), t (v−)]. Then, at time t (v−), the priceis instantaneously reduced from b(v) to b(v−). That is, the curve (4) is augmentedwith

[t (v), t (v−)) × {b(v)} ∪ {t (v)} × [b(v−), b(v)] (5)

for each v where t (v) is discontinuous.

9 Here and below we use the notation g(x0−) to denote the limit of function g(·) as x ↑ x0.

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An optimal slow Dutch auction

Fig. 2 The completionprocedure

b(v )

t (v ) t

p

Tt(v )

b(v )

Given equilibrium play of the other buyers, the best-response problem of a buyerof type v is to choose a bid

b ∈ arg maxb≥0

(vX (b) − P(b)

),

where X (b) is the discounted probability of winning with bid b, and P(b) is theassociated discounted payment to the seller.

Proposition 1 (Properties of Equilibrium)

1. An equilibrium has the following properties:(a) The set of active buyer types is an interval, A = [v, 1], where v = p(1).(b) The bidding strategy b(v) is increasing on A, and satisfies (3), while the time

of sale t (v) is nonincreasing and right-continuous on A.2. Conversely, consider a cutoff v ∈ (0, 1) and a nonincreasing, right-continuous

function t : [v, 1] → R+. Then, the Dutch clock constructed by applying the com-pletion procedure to (4) implements t (v) as an equilibrium time of sale function.

Proof See the “Appendix”. �The intuition for this result is as follows. First, a standard single-crossing property

implies that the set of active buyer types is an interval. Second, the Envelope Theoremapplied to the buyer’s expected discounted utility implies that the expected price b(·)is given by (3). Third, incentive compatibility implies the monotonicity of the time ofsale function t (·).

We are interested in designing a price schedule p(t) that maximizes the seller’sexpected revenue. In view of Proposition 1(2), this problem can be reduced to findingan optimal time of sale function t (·). We will solve this problem using optimal controlmethods. In order to formulate it as an optimal control problem, we treat t (·) as acontrol, and the utility

U (v) = (v − b(v))e−r t (v)F(v)N−1 (6)

as the state variable. Equation (3) implies that a.e. on [v, 1]

U ′(v) = e−r t (v)F(v)N−1. (7)

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From (6),

b(v) = v − ert (v)

F(v)N−1 U (v).

Substituting this expression for b(v), the seller’s expected revenue obtained from anybuyer is

Π =1∫

v

e−ρt (v)b(v)F(v)N−1dv =1∫

v

π(v, U (v), t (v)) f (v)dv, (8)

where we denoted

π(v, U, t) ≡ e−ρt F(v)N−1v − e(r−ρ)tU.

We thus have the following optimization problem for the seller.

Problem 1 Find a cutoff type v ∈ [0, 1] and a nonincreasing function t : [v, 1] →[0, T ] that deliver a maximum to (8), where U (·) satisfies the first-order differentialequation (7) with the initial condition U (v) = 0.

The presence of the monotonicity constraint on t (·) considerably complicates theanalysis of this problem. To get useful insights, in the next section, we consider arelaxed problem, removing the monotonicity constraint. This relaxed problem can betackled by standard optimal control methods. The general problem is considered inSect. 4, where it is shown that almost all the properties of the optimal solution ofthe relaxed problem continue to hold. The only new element is that auctions may beoptimal at intermediate times.

3 Optimal solution: relaxed problem

In this section, we consider a relaxed version of Problem 1. That is, we do not imposethe monotonicity constraint on the time of sale function t (·). If the optimal solutionto the relaxed problem satisfies this monotonicity condition, it is clear that it solvesProblem 1.

Unfortunately, no closed-form solution will be available even in simple cases. Ourapproach instead is to characterize a solution implicitly, using the Maximum Principle.

The Hamiltonian that corresponds to Problem 1 is

H(v, t, U, λ) ≡(

e−ρtvF(v)N−1 − e(r−ρ)tU)

f (v) + λe−r t F(v)N−1, (9)

where λ is the costate variable associated with the state variable U whose law ofmotion is given by (7), with the initial condition U (v) = 0. The function t is viewedas a control variable. Let t∗(·) be the optimal control, and let the corresponding utilityfunction be U∗(·).

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An optimal slow Dutch auction

The Maximum Principle in the present setting implies the following.10 For anyv ∈ [v, 1],1. There exist a piecewise continuously differentiable function λ such that, except at

the points where t∗(·) is discontinuous,

λ′(v) = −∂ H

∂U= e(r−ρ)t f (v), (10)

with the boundary conditionλ(1) = 0. (11)

2. The function t∗(·) maximizes the Hamiltonian,

t∗(v) ∈ arg maxt∈[0,T ] H(v, U∗, λ, t).

3. At v = v, a transversality condition holds:

H(v, t∗(v), U∗(v), λ(v)) ≥ 0, (12)

with strict equality if v > 0.

The derivative of the Hamiltonian with respect to t is, after some re-arrangement,

∂ H

∂t= −ρF(v)N−1 f (v)e−ρtv − r F(v)N−1λe−r t − (r − ρ)e(r−ρ)tU f (v). (13)

From now on, we make the standard assumption that the Myerson virtual valuefunction is increasing.

Assumption 1 (Increasing Virtual Value) The function

J (v) ≡ v − 1 − F(v)

f (v)

is strictly increasing.

Rewrite (13) as

∂ H

∂t= e−r t h(v, t, U, λ),

where we denoted

h(v, t, U, λ) ≡ F(v)N−1 f (v)

(−ρe(r−ρ)tv − rλ

f (v)

)− (r − ρ)e(2r−ρ)tU f (v).

10 See e.g. Theorems 2 and 11 in Seierstad and Sydsæter (1987)). Theorem 2 contains the MaximumPrinciple for a fixed boundary problem. In our case, because v is chosen by the seller, we are dealing witha free boundary problem, covered in Theorem 11 in the same book.

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Fig. 3 Optimal solution

v v* v v

t

T

1

t*(v), t0 (v)

t0 (v)

v0

We now consider a solution t0(v) ∈ R+ to the F.O.C. ∂ H∂t = 0,

h (v, t, U∗(v), λ(v)) = 0, (14)

Observe that for v ∈ (0, 1),

limt→−∞ h(v, t, U∗(v), λ(v)) = −rλ(v)F(v)N−1 > 0, (15)

limt→+∞ h(v, t, U∗(v), λ(v)) = −∞. (16)

By continuity of h in t , the above inequalities imply that the solution t0(v) exists andis unique if v ∈ (0, 1). The optimal control t∗(v) ∈ [0, T ] will be a truncation of t0(v).

Define as v the minimal buyer type that is not delayed,

v ≡ inf{v : t∗(v) = 0}.

The set of non-delayed types is an interval [v, 1]. In a similar fashion, define v∗ as thelargest v that is delayed until the last instance,

v∗ ≡ sup{v : t∗(v) = T }.

Our main result in this section is the following proposition. Refer to Fig. 3.

Proposition 2 (Optimal Solution) An optimal solution to the relaxed problem is acontinuous function t∗(·) of the form

t∗(v) =

⎧⎪⎨⎪⎩

T, v ∈ [v, v∗]t0(v), v ∈ [v∗, v]0, v ∈ [v, 1]

(17)

where t0(·) is the unique solution to the F.O.C. (14), and

0 < v < v∗ < v0 < v < 1. (18)

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An optimal slow Dutch auction

Discussion. Fleshing out the dynamics of the optimal price clock, we note that theitem is always sold to the buyer with the maximal valuation vmax, provided vmax ≥ v,where v is the lowest price on the clock. The clock never enters a “quiet“ intervalI ∈ [0, T ] over which the item is never sold; this follows from the continuity of t∗(·).

The sale process begins with an instantaneous (Dutch) auction where the partici-pating buyer types v ∈ [v, 1]. If vmax ∈ [v, 1], then the item is sold at time 0, at theinstantaneous auction price equal to b(vmax). Otherwise the sale continues at postedprices until the last moment T . The “posted price“ schedule starts at the (limit) priceb(v), and then the price is continuously reduced to the (limit) p(T −) = b(v∗) at timeT . Over this period, buyers with types v ∈ (v∗, v) self-select into different time-pricepairs (t (v), b(v)). If vmax ∈ (v∗, v), then the item is sold at the time t (vmax) at theposted price. The posted price schedule penetrates “deeper” than the static optimalreserve price v0, as is evident from v∗ < v0 in (18). Otherwise, if the item is not soldat a posted price, the sale mechanism enters its final stage—the instantaneous Dutchauction at time T , with the reserve price v < p(T −). If the maximum value buyerhas type vmax ≥ v, then the item is sold at the final auction price b(vmax). Otherwise,the seller commits not to sell the item.

To get an intuition why delays are optimal when the seller is more patient, we canuse integration by parts upon the substitution of b(v) given by (3) into the seller’sprofit (8), to arrive at a familiar virtual value representation à la Myerson:

Π =1∫

v

e−ρt (v) J (v; t (·))d F(v)N , (19)

where

J (v; t (·)) ≡ v − 1

f (v)

1∫

v

e−(r−ρ)(t (v)−t (v))d F(v)

generalizes the Myerson virtual valuation function.As this formula demonstrates, time delays have two effects on the seller’s expected

profit. First, there is an obvious negative direct effect of time discounting, appearingthrough the pre-multiplying discount factor e−ρt (v) in (19). Second, and more inter-esting, there is a positive indirect strategic effect on the virtual valuations. If the selleris more patient, ρ < r , then time delays reduce the information rents of the buyers,thereby increasing the virtual values J (v; t (·)). Indeed, if a type v is time-delayedrelative to a positive measure of v types above it, then

t (v) > t (v) �⇒ e−(r−ρ)(t (v)−t (v)) < 1

and therefore his information rent is reduced and the virtual valuation increased relativeto the static setup:

J (v; t (·)) > J (v).

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A. Shneyerov

The optimal time assignment t∗(v) balances these two effects.Moreover, the optimal dynamic clock will penetrate deeper than the static optimal

reserve v0. Indeed, in the static auction, the seller finds optimal to truncate the typeslower than v0 because the presence of such types would create mimicking incentivesfor higher types, thereby increasing their information rents and reducing the seller’sexpected profit. But if the seller is patient, then the seller will find it optimal to insteaddelay the types below v0. The presence of an exogenous upper bound on the timeof sale T , however, implies that some very low types will still be truncated. This isbecause among the types that are maximally delayed, the mimicking incentives cannotbe reduced by a further delay.

The optimal solution involves an instantaneous auction at time 0 and another oneat time T . These auctions pose an interpretational issue in the continuous time setup.Given that the price falls instantaneously from say p′ to p′′ < p′, a buyer withvaluation v = b−1(p) for p ∈ (p′′, p′) will not be able to actually observe the pricep and stop the clock at that price. So how should these instantaneous Dutch auctionsbe interpreted?11

We think the right interpretation is to consider the optimal solution as a limit of solu-tions of perturbed models where the clock is not allowed to fall instantaneously. Thisconstruction is developed in the next section, with the help of a general apparatus alsodeveloped there. This apparatus allows us, among other things, to introduce a family ofε-perturbations that iron out discontinuities for an arbitrary monotone function.12 Weshow that the optimal solution can be approached in the limit of such perturbed priceclocks. Thus, such continuous perturbations arbitrary well approximate the optimalsolution.13

Proof (Proof of Proposition 2)We begin with the following simple observation thatwill be used for several results in the paper.

Lemma 1 (Quasiconcavity of Hamiltonian) The function H(v, t, U, λ) is strictlyquasiconcave in t for v ∈ (0, 1].Proof This follows from the fact that for v ∈ (0, 1], h(v, t, U, λ) is an increasingfunction of t .

The inequalities (18) are proven in a sequence of lemmas below.

Lemma 2 All sufficiently low buyer types are screened out: v > 0. The transversalitycondition is satisfied as equality and can be stated as

v −∫ 1v

e(r−ρ)(t (x)−T ) f (x)dx

f (v)= 0. (20)

11 We thank a referee for bringing up this issue to our attention.12 In particular, we do not require the function to be differentiable or even absolutely continuous.13 The situation here may be akin to the use of Brownian motion to describe security prices. It is notrealistic in a frictional market, but is used nevertheless as a useful mathematical abstraction. Moreover,more realistic random walks approximate the Brownian motion arbitrarily well.

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An optimal slow Dutch auction

Proof Assume, by the way of contradiction, that v = 0. Upon a change of variables

y = F(v)N−1

the expected profit becomes

Π = 1

N

1∫

v

(e−ρt∗(y)Q(y) − e(r−ρ)t∗(y) U (y)

yN−1

N

)dy.

where Q(·) is the inverse of F N (·). In the y variable, the law of motion for U (y) isgiven by

U ′(y) = 1

N

1

f (Q(y))e−t∗(y)

and the Hamiltonian is

H = e−ρt Q(y) − e(r−ρ)t

yN−1

N

U + λe−r t (y)

N f (Q(y)).

The adjoint variable λ(y) now satisfies the differential equation

λ′(y) = y− N−1N e(r−ρ)t∗(y)

which implies

λ(0) = −1∫

0

y− N−1N e(r−ρ)t∗(y)dy ≤ −

1∫

0

y− N−1N dy = −N < 0.

Now observe that

limy↓0

U (y)

yN−1

N

= limv↓0

U (v)

F(v)N−1 = limv↓0

U ′(v)

(N − 1) f (v)F(v)N−2

= limv↓0

F(v)N−1e−r t∗(v)

(N − 1) f (v)F(v)N−2 = 0,

where the first equality follows by L‘Hopital’s rule, the second—from the substitutionof (7), and the last one—by our assumption that the density f (v) is bounded frombelow. Therefore, the transversality condition at y = 0 is violated:

H(0, t∗(0), U∗(0), λ(0)) = λ(0)e−r t (0)

N f (0)< 0,

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A. Shneyerov

proving that the optimal solution involves v > 0, and the transversality condition (12)holds with equality.

Lemma 3 The optimal solution must involve delays: t∗(v) = T .

Proof Suppose not, i.e., t ≡ t∗(v) < T . Then either t = 0 or we have an interiormaximum. Since h is decreasing in t , in any case, we have h(v, t, U∗(v), λ(v)) ≥ 0.The last statement is equivalent to

ve(r−ρ)t − r

ρ

∫ 1v

e(r−ρ)t (x) f (x)dx

f (v)≥ 0 (21)

However, since r/ρ > 1, (21) also implies

ve(r−ρ)t −∫ 1v

e(r−ρ)t (x) f (x)dx

f (v)≥ 0

but this contradicts the transversality condition (20).

Lemma 4 It is not optimal to temporarily remove the good from the market: Thefunction t∗(·) is continuous.

Proof Because the Hamiltonian H is quasiconcave in t , the optimal solution t∗(v),which is monotone nonincreasing by assumption, must be a truncation of t0(v) as in(17). By Lemma 2, we have v > 0. Then

∂h(v, t0(v), U∗(v), λ(v))

∂t= −(r − ρ)F(v)N−1 f (v)e(r−ρ)t0(v)

−(2r − ρ)(r − ρ)e(2r−ρ)t0(v)U∗(v)

< 0

for v ∈ [v, 1]. The Implicit Function Theorem ensures that t0(·) is differentiable, andtherefore continuous. It follows that t∗(·) is a continuous function.

Lemma 5 Then, the interval of buyer types who purchase at time T , [v, v∗], has apositive measure: v < v∗, and is bounded from above by the static optimal reserve:v∗ < v0.

Proof At v = v∗, the solution satisfies the F.O.C.

h(v∗, T, U∗(v∗), λ(v∗)) = 0.

Upon an algebraic manipulation, this implies

v∗ − r

ρ

∫ 1v∗ e(r−ρ)(t (x)−T ) f (x)dx

f (v∗)+ r − ρ

ρ

erT U∗(v∗)F(v∗)N−1 = 0. (22)

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An optimal slow Dutch auction

Equation (25) implies

v∗ − r

ρ

∫ 1v∗ e(r−ρ)(t (x)−T ) f (x)dx

f (v∗)≥ 0. (23)

If we had v∗ = v, then, since r > ρ, (23) would imply

v −∫ 1v

e(r−ρ)(t (x)−T ) f (x)dx

f (v)> 0.

But this violates the transversality condition (20) for v:

v −∫ 1v

e(r−ρ)(t (x)−T ) f (x)dx

f (v)= 0.

We now show that v∗ < v0. From (6), we obtain

erT U∗(v∗)F(v∗)N−1 = v∗ − b(v∗) < v∗.

Substituting this into (23) and manipulating, we get

v∗ −∫ 1v∗ e(r−ρ)(t (x)−T ) f (x)dx

f (v∗)< 0.

Since t (x) ≤ T �⇒ e(r−ρ)(t (x)−T ) ≤ 1, the above inequality implies

v∗ − 1 − F(v∗)f (v∗)

= J (v∗) ≤ 0,

which in turn implies v∗ < v0 because, once again, J (·) is an increasing function byAssumption 1.

Lemma 6 The optimal solution involves a positive measure of types buying at time0: we have v < 1. Moreover, v > v0.

Proof In view of the continuity of h(v, t, U∗, λ) in t , it is sufficient to demonstratethat the slope of the Hamiltonian is negative at t = 0. It is sufficient to show thath(1, t, U∗(1), 0) < 0 at t = 0. But, since λ∗(1) = 0, this is immediate:

h(1, 0, U∗(1), 0) = −ρe(r−ρ)t f (1) − (r − ρ)e(2r−ρ)tU∗(1) f (v) < 0. (24)

To show that v > v0, consider the F.O.C. at v = v, which can be manipulated to theform

v∗ − r

ρ

1 − F(v)

f (v)+ r − ρ

ρ

U (v)

F(v)N−1 = 0.

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A. Shneyerov

Substituting U (v)/F(v)N−1 = v − b(v), we obtain

r

ρJ (v) = r − ρ

ρb(v) > 0, (25)

and therefore v > v0.

This last lemma completes the proof of Proposition 2.The analysis in this section has assumed that the solution to the relaxed problem

is monotone decreasing. If there is only one buyer, Landsberger and Meilijson (1985)show that the optimal solution t∗(·) is nonincreasing. Landsberger and Meilijson (1985)restrict attention to differentiable price schedules. But, a revealed preference argumenteasily shows that t∗(·) must be nonincreasing for arbitrary nonincreasing price sched-ules. Also, our analysis implies that the optimal price schedule is not differentiable att = T , as the posted reserve price exhibits a downward jump there: p(T ) = b(v) = v,p(T −) = b(v∗) > p(T ).

4 Optimal solution: general case

The results in the previous section were obtained assuming that the solution to therelaxed problem is in fact monotone. In this section, we show that essentially all theresults continue to hold without assuming monotonicity.

We will continue to use optimal control to characterize a solution. A standard tech-nique for imposing a monotonicity constraint is to differentiate the control function,and impose the non-negativity constraint on the derivative.

We will use this method in a form that allows for any monotone decreasing functiont∗(·). Our approach will require some preliminaries. The Lebesgue decompositionimplies t∗(v) = ta∗ (v) + t j∗ (v) + t s∗(v), where ta(v) is absolutely continuous, t j∗ (v) isa pure jump function (piecewise constant), and t s∗(v) is singular,, i.e., has derivativeequal 0 almost everywhere. A potential presence of the singular component in theoptimal solution creates a difficulty.

In this section, we develop a re-parameterization method that allows us to workexclusively with absolutely continuous functions. The method treats any admissiblefunction t (v) as a curve in the plane. Let

G =⋃

v∈[v,1]{v} × [t (v), t (v−)].

be a planar curve that corresponds to the graph of the function {(v, t (v) : v ∈ [0, 1]},but with the gaps that correspond to the discontinuities filled in by vertical linesegments. That is, if say v1 is a point of discontinuity, then we connect the points(v1, t (v1−)) and (v1, t (v1+)) by a linear segment.

Each such curve G admits a parametrization,

G ≡ {(V (w), t(w)) : w ∈ [0, 1]},

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An optimal slow Dutch auction

where V : [0, 1] → R+ is a continuous increasing function, and t : [0, 1] → R+ isa continuous decreasing function. Note that many such parameterizations exist for agiven t (v), depending on the choice of function V (·). Our main tool in this sectionwill be the following lemma, which shows that V (·), t (·) can be chosen absolutelycontinuous, which will allow us to apply the Maximum Principle.

Lemma 7 (Lebesgue) There exists a parametrization of G such that V (·), t(·) areabsolutely continuous, and a.e. on [0, 1],

V ′(w) = u(w) ≥ 0, t ′(w) = −q(w) ≤ 0

for some measurable non-negative functions u : [0, 1] → R+ and q : [0, 1] → R+.14

Proof We will choose the length of the curve segment as a parameter. For any w ∈[0, 1], the length of the curve (w) is defined as the supremum of the lengths ofpolygonal line approximations,

(w) ≡ sup0=w0<w2<···<wM =w

M∑i=1

||z(wi ) − z(wi−1)||,

where z(w) ≡ (V (w), t(w)) and || · || denotes the Euclidean norm. For our monotonecurve, such a bound always exists because the sum under the supremum is boundedfrom above by T + 1. Now consider a parametrization

w(w) ≡ (w)

(1),

i.e., w(w) is the normalized length of the curve segment. Define z according to z(w) =z(w). Since

||z(w1) − z(w2)|| ≤ |(w1) − (w2)| = (1) · |w1 − w2|

(the length of a curve segment connecting any two points is weakly greater than thedistance between the points), the function z ≡ (V , t) is Lipschitz continuous, andtherefore absolutely continuous, i.e., V , t have derivatives a.e. on [0, 1] and are equalto the integrals of their derivatives. The result follows.

The fact that one can fill out discontinuities is an obvious part of the result. Thenon-trivial part is that we can find an absolutely continuous parameterization even fora singular function f : [0, 1] → [0, 1] that is non-decreasing, continuous, and hasf ′(x) = 0 a.e. The intuition for this result is that a monotone function, even a highlyirregular one like the Devil’s staircase, has bounded variation. Any curve of bounded

14 This result follows from a theorem attributed to Lebesgue, see Theorem 2.2.2 on p. 34 in Aleksandrovand Reshetniak (1989) and the discussion immediately after. We present a proof for completeness, byfollowing the approach in the proof of Theorem 4.3 on p. 136 in Stein and Shakarchi (2005).

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A. Shneyerov

variation has a finite length. This length, taken from the beginning of the curve, is theparameter w in the above lemma. The proof shows that the coordinates of the pointon the curve are absolutely continuous functions of this length.

Any interval [w1, w2] such that u(w) = 0 a.e., so that V (w1) = V (w2), but t(w1) <

t(w2), corresponds to a discontinuity of the function t (·) at v = V (w1) (= V (w2)). Asw moves along this interval, t changes (in a continuous fashion as a function of w), butv stays the same. Any interval [w3, w4] such that q(w) = 0 a.e., but V (w1) < V (w2),corresponds to an interval [V (w3), V (w4)] where the monotonicity constraint on t (v)

may be binding.In this parametrized setup, the seller’s optimization problem can be stated as the

following optimal control problem.

Problem 2 Find non-negative piecewise continuous control functions u, q : [0, 1] →R+ that deliver a maximum to the expected profit of the seller

Π =1∫

v

u(w)π(V (w), U (w), t(w)) f (V (w))dw,

subject to the constraints that the associated state variables U, V, t evolve accordingto

U ′(w) = u(w) · e−r t(w)F(V (w))N−1, (26)

V ′(w) = u(w) ≥ 0, (27)

t ′(w) = −q(w) ≤ 0 (28)

a.e. on [0, 1], and obey the boundary conditions V (1) = 1, V (0) ≤ T and t(1) =0, t(0) ≥ 0.

Standard results in optimal control show that this problem always has a solution inthe set of bounded measurable (and non-negative) controls u, q. For example, one canverify the conditions of the Filippov-Cesari Theorem. 15

In this new setup, we have two new state variables V (w), t(w), whose laws ofmotion are governed by (27) and (28), subject to the boundary conditions. The Hamil-tonian is given by

H(w, U, V, t, λ, μ, χ) = u · π(w, U, V, t) + λ · u · e−r t F(V )N−1 + μ · u − χ · q

= u · (H(V, U, t, λ) + μ) − χ · q, (29)

where H is the Hamiltonian defined in the previous section, and μ and χ are thecostate variables that correspond to (27) and (28). We will employ the version of the

15 See e.g. Theorem 8 on p. 132 in Seierstad and Sydsæter (1987).

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An optimal slow Dutch auction

Maximum principle given by Theorem 2 on p. 85 in Seierstad and Sydsæter (1987). 16

At the optimal solution (u∗, q∗), the costate variables must be absolutely continuousand satisfy the adjoint system

λ′(w) = −∂H∂U

= −u∗(w)∂ H

∂U, (30)

μ′(w) = −∂H∂V

= −u∗(w)∂ H

∂V, (31)

χ ′(w) = −∂H∂t

= −u∗(w)∂ H

∂t, (32)

a.e. on [0, 1], with the boundary conditions λ(1) = 0 and

μ(0) ≤ 0, μ(0)V (0) = 0, (33)

χ(0) ≥ 0, χ(0)(T − t(0)) = 0, (34)

where the equalities are the complementary slackness conditions that correspond tothe constraints V (0) ≥ 0 and t(0) ≤ T .

The Hamiltonian H must attain the maximal value H∗ at the optimal solution(u∗, q∗). As we are allowing for arbitrary non-negative values for u, q, it follows that

H∗(w) = 0,

and, furthermore, we must have a.e. on [0, 1]

H∗(w) + μ(w) ≤ 0, (35)

u∗(w) · (H∗(w) + μ(w)) = 0, (36)

and

χ(w) ≥ 0, (37)

χ(w) · q∗(w) = 0. (38)

Equations (36) and (38) are the complementary slackness conditions that correspondto the monotonicity constraints u(w), q(w) ≥ 0.

The following lemma shows that the optimal time of sale function is a continuousfunction of v. It is parallel to Lemma 4 from the previous section.

Lemma 8 The function t∗(·) is continuous on [v, 1].

16 Theorem 2 on p. 85 in Seierstad and Sydsæter (1987) assumes piecewise continuous controls. Anextension to bounded measurable controls is given in the (detailed) footnote 9 on p. 132 of the same book,where additional references are provided. In particular, we can either apply a very general Theorem 5.3.1in Clarke (2005) or a more special Theorem 2 in Clarke (2008).

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A. Shneyerov

Proof First, observe that an optimal solution t∗(·) can always be constructed withoutdiscontinuities at v = v and v = 1. Clearly, there is no discontinuity at v = 1, as it cannever be optimal to delay trade for all buyer types. Second, a discontinuous downwardjump in t∗(·) at v = v occurs on the set of measure 0 and can be eliminated withoutaffecting the expected profit, at the same time resulting in a function that continues tobe monotone.

The rest of the proof is by contradiction: Suppose there is a discontinuity at somev1 ∈ (0, 1). In view of the above observation, this implies that

1. ∃w1, w2 such that w2 > w1 and V∗(w1) = V∗(w2) = v1 ∈ (v, 1), so thatu∗(w) = 0 a.e. on [w1, w2].

2. The function V∗(·) is strictly increasing on some open left neighborhood NL(w1)

of w1 and on some open right neighborhood of w2, so that u∗(w) > 0 a.e. on bothNL(w1) and NR(w2).

3. t∗(w1) > t∗(w2).

Equation (36) and item 2 above imply

limw↑w1

H∗(w) + μ(w) = 0, limw↓w2

H∗(w) + μ(w) = 0.

But the maximized Hamiltonian H∗(w) is a continuous function, as is the adjointμ(w), so we obtain

H∗(w1) + μ(w1) = 0, H∗(w2) + μ(w2) = 0.

Next, observe that (31) implies μ(w) does not change during the jump: μ0 ≡ μ(w1) =μ(w2), therefore H(w1) + μ0 = H(w2) + μ0. But H∗(w) is a strictly quasiconcavefunction on [w1, w2]. Indeed, as u∗(w) = 0 a.e. during the jump (item 1 above), (26)and (30) imply that U∗ and λ do not change over the jump, so H∗(w) will change onlythrough the change in t∗(w). As H(V, t, U, λ) is a strictly quasiconcave function of tby Lemma 1, and t∗(·) is a non-increasing continuous function that is not constant on[w1, w2] (item 3 above), H∗(w) + μ0 must also be strictly quasiconcave on [w1, w2].Since it is equal to 0 at the boundaries w1, w2, H∗(w) + μ0 must take on a positivevalue in [w1, w2]. But this contradicts (35).

Remark 1 The continuity of t∗(·) implies that the parameterizing function V∗(·) canbe chosen as increasing, and therefore, it will have an inverse V −1∗ (·).

The proof of our main result in the previous section, Proposition 2, relies on thefirst-order condition for the Hamiltonian. The following lemma shows that, even witha monotonicity constraint, this first-order condition remains valid at any v where t∗(·)is strictly decreasing, i.e., either t∗(v − ε) > t∗(v) or t∗(v + ε) < t∗(v), or both, forall (sufficiently small) ε > 0. Subsequently, we use this result to prove Proposition 3,the analog of Proposition 2. In this lemma, we treat the Hamiltonian as a function ofv. That is, along the optimal path, all functions in H are treated as functions of v, i.e.,V −1∗ (v) is substituted for w.

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An optimal slow Dutch auction

Lemma 9 (F.O.C.) Let v ∈ [v, 1] be a point of strict monotonicity of the optimalsolution t∗(·). Then, the Hamiltonian H satisfies the first-order condition at v,

∂ H(v, U∗(v), t∗(v), λ(v))

∂t= 0.

Proof First, consider the case when t∗ is strictly monotone both from left and right,i.e., t∗(v − ε) < t∗(v) < t∗(v + ε) for all sufficiently small ε > 0. Let w be thecorresponding value of w, which is unique because V∗(·) is chosen to be increasing.Then, there exist two sequences wn ↑ w and wn ↓ w such that V∗(wn) < v < V∗(wn)

and t ′∗(wn), t ′∗(wn) > 0. (The latter follows from the absolute continuity of t∗.) Sincethe monotonicity constraint is not binding at wn, wn , the complementary slacknesscondition (34) implies χ(wn), χ(wn) = 0, and therefore (32) implies

0 = χ(wn) − χ(wn) = −V∗(wn)∫

V∗(wn)

∂ H

∂tdv,

where in the last equality, we have used a change of variables in (32), which is justifiablebecause V∗(·) is absolutely continuous. Dividing through by V∗(wn) − V∗(wn) > 0and taking the limit as n → ∞, we obtain

0 = limn→∞

1

V∗(wn) − V∗(wn)

V∗(wn)∫

V∗(wn)

∂ H

∂tdv = ∂ H(v, U∗(v), t∗(v), λ(v))

∂t,

where the last equality follows by continuity of ∂ H/∂t in v. This verifies the result inthe lemma for any point v such that t∗(·) is strictly monotone from both sides.

Since any point v where t∗(·) decreasing from one side and is constant from the othercan be approached by a sequence of points vn such that t∗(vn) is strictly monotonefrom both sides at vn , the continuity of ∂ H/∂t in v implies that ∂ H/∂t = 0 at allpoints v where t∗(·) is decreasing.

In order to show the equivalents of Lemmas 5–6 from the previous section, we firstestablish the transversality condition (20).17 Since V∗(·) is chosen as an increasingfunction, we can pick a sequence wn ↓ 0 such that u∗(wn) > 0. Passing alongthis sequence to the limit in (36) and invoking the continuity of H∗(w) gives usH∗(0) = −μ(0). But, V∗(0) = v > 0, and the complementary slackness condition(33) therefore implies μ(0) = 0. Hence H∗(0) = 0, as desired.

Lemma 5 only relies on the first-order condition for the Hamiltonian at v = v∗,which holds by Lemma (9) because v∗ is a point of strict monotonicity of t∗(·). Lemma3 (optimality of delays) also follows by a straightforward modification of the argument.Suppose, to the contrary, that t∗(0) < T . Then, the complementary slackness condition

17 Lemma 2 does not use optimal control.

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A. Shneyerov

(34) implies χ(0) = 0, therefore

0 < χ(w) =w∫

0

χ ′(w)dw = −w∫

0

∂ H

∂tu∗(w)dw,

therefore, by continuity, ∂ H/∂t < 0 in some open (right) neighborhood of v = v. But,as in the proof of Lemma 4, this contradicts the transversality condition (20) whichimplies that for t < T ,

∂ H(v, t, 0, λ(w))

∂t> 0.

Finally, the result in Lemma 6 (v < 1) is also straightforward to obtain. If, to thecontrary, we had v = 1, then this would be a point of strict monotonicity of t∗(·),therefore ∂ H

∂t = 0 by Lemma 9. But, as in the proof of Lemma 6,

∂ H

∂t

∣∣∣∣t=0,v=1

< 0,

so we have a contradiction.To summarize, we have shown the following analog of Proposition 2.

Proposition 3 A solution to Problem 2 involves a continuous function t∗(·) and hasthe following form: there exist v, v∗, v ∈ [0, 1], 0 < v < v∗ < v0 < v < 1, such thatt∗(v) = T for v ∈ [v, v∗] and t∗(v) = 0 for v ∈ [v, 1].

Realistic price clocks. The parameterization developed in this section allows oneto incorporate realistic price clocks without jumps in a straightforward manner andto show that such clocks approximate the optimal unconstrained clock arbitrary well.Namely, let us impose an additional constraint on q(w), the slope of the time function−t(w):

q(w) ≥ ε > 0 a.e. on [0, 1].

With this constraint, the price schedule is a continuous monotonic curve, without anydownward jumps. Denote the optimal seller’s expected revenue in this ε-constrainedproblem as �(ε). Observe that the objective in Problem 2 is continuous in q(·),provided we treat q(·) as an element in the space of bounded measurable functionsendowed with a supnorm. This implies that �(ε) is continuous in ε at ε = 0: �(ε) →�(0) as ε ↓ 0, where �(0) is the optimal objective for the unperturbed problem.This implies that the optimal ε-clocks approximate the original clock. In other words,intertemporal price discrimination, a mechanism familiar in practice, is essentiallyoptimal.

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An optimal slow Dutch auction

5 Concluding remarks

It would be useful to extend this research in the following directions.Optimal Mechanisms. We have restricted attention to a specific selling mechanism

(SDA). In general, the seller may benefit from the fact that buyers are more patientby acting as a lender. If the seller’s lending capacity is unrestricted, then he can makeinfinite profit from lending, even as the lowest buyer type is also getting an infiniteutility from borrowing. However, lending requires an explicit contract with a buyer,and the cost of enforcement may be prohibitively high. SDA does not require the buyerto enter into such a long-term contractual relationship with the seller. In this sense, itis a pure selling mechanism.

So a logical question is how to generalize the notion of a selling mechanism in thiscontext. One useful restriction that would rule out lending would be to require thatall monetary transfers are made from buyers to the seller. One could then specify adirect revelation mechanism that solicits type reports and assigns the price and timeof purchase on the basis of these reports. Indeed, one can show that any equilibriumof SDA corresponds to an incentive compatible and individually rational direct mech-anism, with an additional restriction that the time of purchase only depends on ownreport. Without this restriction, we have a difficult nonlinear optimization problemthat involves multivariate functions and therefore falls outside the realm of standardoptimal control. We leave this interesting problem for future research.

Noncommitment. In our analysis, we have assumed full commitment. This assump-tion may be natural in several applications as discussed in Board and Skrzypacz (2010);it is also made in Myerson (1981), which is a natural comparison benchmark for ourSDA. But clearly, there are applications where the seller cannot commit. In a recentpaper, Skreta (2010), using the methods in Skreta (2006), completely solves a two-period optimal auction design problem under noncommitment. She shows that theoptimal solution involves two auctions: one with a reserve price in the first periodand another without a reserve in the second period. Restricting attention to dynamicpricing as we do here, Hörner and Samuelson (2011) have shown that the seller retainsa degree of monopoly power even without commitment. Given this result, it is naturalto expect that the seller will have some monopoly power in our model, even undernoncommitment. This extension is also left for future research.

6 Appendix

Proof (Proof of Proposition 1)Proof of 1(a). Suppose a buyer of type v bids b, whilethe rival buyers adopt an equilibrium bidding strategy. Let P(b) to be the expecteddiscounted payment of a buyer of type v, and let X (b) be the discounted probabilityof getting the good. The equilibrium expected utility of a type v buyer is then U (v) =vX (b(v)) − P(b(v)), and the Envelope Theorem implies U ′(v) = X (b(v)) ≥ 0.Therefore, U (v) is a nondecreasing, continuous function. This implies that the set ofbuyer types who have a positive utility is an interval (v, 1] for some v. Moreover,we must have v = p(1). Otherwise, if v < p(1), buyers with types slightly above v

cannot get a positive utility. If v > p(1), then a buyer with type v − ε slightly below v

123

A. Shneyerov

will make a positive utility by entering and waiting until T . In that case, he will haveno rivals with probability at least F(v)N−1 and will then be able to buy the good atprice p(1) < v − ε.

Proof of 1(b). We first verify functional equation (3). Since

U (v) = maxv∈[v,1]

(v − b(v))e−r t (v)F(v)N−1, (39)

the Envelope Theorem implies ∀v ∈ [v, 1]

U (v) =v∫

v

e−r t (v)F(v)N−1d v, (40)

where we have used the fact that U (v) = 0, which follows from v = p(1), as shownin (a) above. Combining (39) and (40) yields (3).

The monotone selection theorem implies that b(v) ∈ arg maxb≥0(vX (b) − P(b))

is nondecreasing. This implies that the time of sale function must be nonincreasing.Indeed, the price clock is defined in terms of monotone, respectively, nonincreasingand nondecreasing functions t(·), p(·), so a weakly lower bid implies a weakly highertime of purchase. Since we have already seen that b(·) is nondecreasing, it followsthat t (·) is nonincreasing.

We are now ready to show that the bidding strategy b(·) is increasing, as opposedto merely nondecreasing. To see why, observe that for v ∈ (v, 1), v ∈ (v, v), revealedpreference implies

e−r t∗(v)F(v)N−1(v − b(v)) ≥ e−r t∗(v)F(v)N−1(v − b(v)),

b(v)≥[

1 − e−r(t (v)−t (v))

(F(v)

F(v)

)N−1]

v+e−r(t (v)−t (v))

(F(v)

F(v)

)N−1

b(v)>b(v)

where the last inequality follows from the fact that t (v) is nondecreasing, so that

e−r(t (v)−t (v))

(F(v)

F(v)

)N−1

< 1,

and, from (3), b(v) < v.To finish the proof of item 1, we verify that t (·) must be right-continuous. Suppose,

by the way of contradiction, that for some v ∈ [v, 1), we have t (v+) < t (v). Then(3) implies

t (v+) = τ(b(v+)) = τ

⎛⎝v − ert (v+)

F(v)N−1

v∫

v

e−r t (v)F(v)N−1d v

⎞⎠

where we have used the fact that τ(·) is right-continuous by construction. (Recall thatit is defined as the first instance the price curve crosses p from above.) Since τ(·) is

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An optimal slow Dutch auction

a nonincreasing function, and we have assumed t (v+) < t (v), the above equationimplies

t (v+) ≥ τ

⎛⎝v − ert (v)

F(v)N−1

v∫

v

e−r t (v)F(v)N−1d v

⎞⎠ = t (v),

a contradiction.Proof of 2.We first consider the clock {(t (v), b(v)) : v ∈ [v, 1]}, where t (·) is a given nonin-

creasing function, and if b(v) is given by (3). To show that b(·) is in fact an equilibriumbidding strategy in a bidding game defined by this clock, we only need to demonstratethat there are no profitable deviations from v to v, where v, v ∈ [v, 1]. Assume v > v;for v < v, the argument is parallel. Then, the expected gain from the deviation is

U (v, v) − U (v) =(v − b(v))e−r t (v)F(v)N−1 −v∫

v

e−r t (v)F(v)N−1d v

= (v − b(v)

)e−r t (v)F(v)N−1 −

v∫

v

e−r t (v)F(v)N−1d v

+ (v − v)e−r t (v)F(v)N−1

=v∫

v

e−r t (v))F(v)N−1d v −v∫

v

e−r t (v)F(v)N−1d v

+ (v − v)e−r t (v)F(v)N−1

=v∫

v

e−r t (v)F(v)N−1d v − (v − v)e−r t (v)F(v)N−1

=v∫

v

(e−r t (v) − e−r t (v)

)F(v)N−1d v ≤ 0,

where the second equality follows from (3) for v, and the last inequality followsbecause t (·) is assumed to be nonincreasing.

We now show that the completion procedure described in the main text does notcreate additional incentives to deviate. Suppose that v is a point of discontinuity oft (·), so that t (v−) > t (v). First, consider the price b(v) extended to the interval[t (v), t (v−)]. Since buyers are impatient, it is clear that no buyer will prefer to purchasethe good at this price over [t (v), t (v−)], as the option to buy at time t (v) dominates.Next, consider the instantaneous price reduction from b(v) to b(v−). As b(·) is strictlyincreasing, the measure of buyer types who enter the bid equal to b(v−) is 0. Therefore,

123

A. Shneyerov

the bids in the interval (b(v−), b(v)) are dominated by b(v−). This completes theproof.

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