A Study of Sale Rates and Prices in

Impressionist and Contemporary Art Auctions

Orley Ashenfelter, Princeton University

Kathryn Graddy, University of Oxford and CEPR

Margaret Stevens, University of Oxford

December 2001¤

Abstract

This paper presents a compreshensive study of sale rates and prices in

impressionist and contemporary art auctions. Our analysis is based on a

detailed dataset of impressionist and modern art auctions and contemporary

art auctions. We use this data to estimate and test a ”search” model of

seller behavior and reserve prices, similar to that developed in the labor

economics literature. In our model, we …nd that that the reserve price

should be a constant proportion of the estimated price. Furthermore, the

reserve price and therefore sales rate, should depend only upon the variance

in log prices and the seller’s discount rate. We estimate that the secret

reserve price is set on average between 70-80% of the low estimate.

¤The authors would like to thank Jiangping Mei and Mike Moses for very useful comments.

1. Introduction

Among frequent auction-goers, it is a well-known fact that not all items that are

put up for sale are sold. Sellers of individual items will set a secret reserve price,

and if the bidding does not reach this level, the items will go unsold. An item

that has not been sold may be put up for sale at a later auction, sold elsewhere,

or taken o¤ the market. Sale rates vary tremendously across time. For example,

between 1982 and 1994, the sale rate for contemporary art varied between 52%

and 91% in di¤erent auctions. Between 1980 and 1990, in di¤erent auctions of

58 selected impressionist and modern artists at Christie’s and Sotheby’s in New

York and London, sale rates varied between 31% and 100%. Sale rates also vary

systematically over di¤erent types of auctions. During 1995 and 1996, 96% of

items put up for sale in auctions of arms and armour were sold, 89% of wine at

auction was sold, and 71% of impressionist and modern art items were sold.

This paper presents a comprehensive study of sale rates and prices in im-

pressionist and contemporary art auctions. Our analysis is based on a detailed

dataset of over 150 impressionist and modern art auctions with over 16,000 items

for sale, and another detailed dataset of over 35 contemporary art auctions with

approximately 4500 items for sale. We also have data on average sale rates and

prices for 36 di¤erent types of auctions. We begin our study by looking in detail

and sale rates and prices both across time and across di¤erent auctions. We can

conclude that while prices have trended upwards with a peak in 1989-1990, sale

rates have shown no discernible trend.

The starting point for our theoretical analysis is a model of the seller’s reserve

price. We use an approach similar to that developed in the labor economics liter-

2

ature. The seller of a painting faces a problem similar to that of an unemployed

worker; if he participates in an auction the highest bid for the painting can be

regarded as a random draw from some price distribution. When a seller sets a

reserve price, he must decide at what price he would be indi¤erent between selling

now and waiting for the next auction. We …nd that the optimal policy is to set

a reserve price that is a constant proportion of the current expected price. In

actuality, participants in auction markets often speak of the reserve as being a

percentage of the auctioneer’s estimated price. The constant proportion depends

only on the variance of log prices and the seller’s discount rate, and not on overall

price levels. We can then empirically model sale rates as being explained by

price shocks and a constant, or ”natural sale rate.” This natural sale rate again

depends only on the variance of log price and the seller’s discount rate. Using

the empirical model, we then estimate that the reserve price is generally set to be

between about 70 and 80% of the auctioneer’s low estimate.

Modeling the reserve price in this manner presents several testable implications

regarding sale rates and price variances, in addition to speci…cation tests on the

model. These tests broadly, though not completely, provide support for our

model.

In addition to providing an interesting empirical study of sales rates and prices

in art auctions, we think this paper contributes to the existing literature in various

ways. While there has been much work on strategic reasons for setting a reserve

price in an auction (see, for example, Riley and Vincent (1981), Horstmann and

LaCasse (1997), McAfee, Quan, and Vincent (2000)), we model reserve prices in

an auction as being the result of search behavior on the part of sellers. Ashenfelter

(1989) …rst mentioned this possibility of reserve prices being the result of search

3

behavior. From this model, we are able to estimate how much below the low

estimate, the reserve price is actually set. In addition, the majority of empirical

models of search behavior have been applied in a labor market context. This

paper applies a model of search in a di¤erent context.

This paper proceeds as follows. In section 2 we describe the data and look

in more detail at how prices and sale rates have behaved across time and across

di¤erent items. In sections 3 and 4 we develop and estimate an empirical model of

model of search behavior in auction markets and determine a relationship between

sales rates and unexpected price changes. In section 5 we perform various tests

on the model, and in section 6 we conclude the analysis.

2. Background: Prices and Sale Rates in Art Auctions

2.1. Art Auctions

The auction market for …ne art is dominated by two auction houses, Christie’s and

Sotheby’s. The most important …ne art auctions for these houses are generally

held in London and New York. The art auctions are separated into di¤erent

periods. For example, during the time period that the data covers, Old Masters

were grouped together, impressionist and modern art were grouped together, and

contemporary art was grouped together. Each type of auction generally occurs

about four times a year in each location.

Prior to an auction, a pre-sale catalogue is published with information on the

individual items coming up for sale. Included in the pre-sale catalogue is infor-

mation on title of painting, artist, size of painting, and medium. Furthermore,

the auction house publishes a low- and a high-price estimate for the work. The

4

auction house does not publish, and indeed is very secretive about, the seller’s

reserve price for the work of art. The auction houses observe an unwritten rule

of setting the secret reserve price at or below the low estimate.

2.2. The Data

Our data consist of objects sold in contemporary art auctions held at Christie’s

auction house located on King Street in London and objects sold in impressionist

and modern art auctions held at Christie’s and Sotheby’s auction houses in London

and New York. We also have aggregate data on sale rates and average values of

items sold in di¤erent departments at Christie’s.

The dataset on impressionist and modern art auctions was constructed by

Orley Ashenfelter and Andrew Richardson and includes sales of 58 selected im-

pressionist and modern artists that took place at Christie’s and Sotheby’s auction

houses between 1980 and 1990. The artists in this sample were selected because

their art is well represented at auction. The auction prices were collected from

public price lists, and the estimated prices and various observable characteristics

were collected from pre-sale catalogues.

The dataset on contemporary art was constructed by Kathryn Graddy and

includes all sales of contemporary art at Christie’s auction house on King Street

in London between 1982 and 1994. The data were gathered from the archives of

Christie’s auction house, and for each item, the observable characteristics were

hand-copied from the pre-sale catalogues. The information on whether or not a

lot is sold and the …nal bid from 1988 onwards was taken primarily from Christie’s

internal property system. Before 1988, many of the lots were missing from the

internal system. An assistant in the archives department said that after a certain

5

period of time, some of the lots are deleted from the system, for no predictable

reason. From December 1982 through December 1987, we had access to the auc-

tioneer’s books and were able to track the missing items in that manner.

The impressionist and modern art dataset includes over 150 auctions and

16,000 items for sale, and the contemporary art dataset includes 35 auctions and

approximately 4500 items for sale. In addition to sale prices and the low and high

price estimates for these paintings, the observable characteristics we have on each

item are title of painting, artist, size of painting, medium, and painting date. For

the impressionist and modern dataset, we also know whether or not the painting

was signed, monogrammed, or stamped. For the contemporary art dataset, we

know whether or not the item was subject to VAT and we know the highest bids

for items that were not sold. Having information on the highest bids of items

that were not sold is a primary bene…t in the contemporary art dataset. For

a further description of the contemporary art and impressionist and modern art

dataset, please see Beggs and Graddy (1997).

2.3. Average Prices and Sale Rates

Table 1 presents summary statistics on prices and sale rates for contemporary

and impressionist auctions. A simple comparison of the datasets is interesting.

The average price for impressionist and modern art is approximately nine times

the average price for contemporary art, and the standard deviation in the average

price for impressionist art is approximately …fteen times the standard deviation

in the contemporary art sample. However, these di¤erences vary tremendously

throughout the sample, with contemporary art appreciating relative to impres-

sionist art over the time of the sample. There does appear to be an upward

6

trend in prices, though prices appear to have peaked in 1989 or 1990, with a price

decline after that. It should be noted that the last years of each sample, 1991 for

impressionist art and 1994 for contemporary art, are not fully representative of

prices and sale rates in that year, as we do not have data for the full year. Dif-

ferent auctions throughout the year can systematically di¤er as to the ”quality”

of art that is sold; for example, some auctions are considered major sales where

the most expensive items are auctioned.

Sale rates clearly vary over the time of the sample, but they do not appear to

vary systematically. The average sale rate for impressionist and modern art is

71% over the period of the sample, and the average sale rate for contemporary art

is 77%.

2.4. Price and Sale Rate Indices

One concern about using average prices and sale rates to represent variation over

the time period is that price rises may be exacerbated during booms as ”better”

paintings may come up for sale, or sale rates might be in‡uenced if ”better”

paintings are easier to sell. 1In order to create a standardized measure of prices

and sale rates over time, we construct an index. For detailed information on how

the indices were constructed, and for the hedonic regressions, please see Appendix

A.

A plot of the price indices, along with simple averages of prices, is presented

for impressionist art in Figure 1 and for contemporary art in Figure 2. The1Wynne Kramarsky, whose family formerly owned ”Portrait of Dr. Gachet,” said of the

London market prior to the sale of May 15, 1990: ”’I did not think that London was poor in

terms of performance; I thought that the pictures were not up to it.”’ (Watson, 1992, p.10)

7

price indices are normalized to one in 1980. (These are real indices based on

1990 dollars). As has been documented by Pesando using a repeat sales index for

modern prints, there is a spike in 1990, followed by a drop-o¤ in prices. There are

several points to note about the relationship of the indices. First, the hedonic

index underestimates the price rise from 1985 to 1990 in relation to the repeat

sales index, and a simple graph of average prices exceeds both during the highest-

priced years of 1989-1990. This may indicate that ”better” paintings do come up

for sale during booms. For contemporary art, we also see a peak in price during

1989-1990, and the average price index exceeds the hedonic price index, by a large

margin. We do not construct a repeat sales index for contemporary art due to

lack of data. The primary di¤erence between impressionist and contemporary art

prices during the period is that the hedonic index for contemporary art rises to

about 8 times the starting index, whereas for impressionist art, it only reaches

about 3 times the starting index. Contemporary art has increased in value relative

to impressionist art.

We also construct hedonic sale rate indices for impressionist and contempo-

rary art by performing a Probit regression of whether or not a painting is sold on

the hedonic characteristics, and these are presented in Figures 3 and 4.2 It is in-

teresting to note that the hedonic sale rate index is almost identical to the average

sale rates for Impressionist art. The hedonic sale rate index for contemporary art

is smooth relative to the average sale rate index.2Constructing a sale rate index for repeat sales is not particularly informative. Paintings

that show up in the repeat sales are less likely to be sold early on in the time period (which is

the reason that many painting show up more than once), which would bias any sale rate index

downwards in the early years of the index.

8

What can we infer from looking at the above data? First, one can con-

clude that art prices have trended upwards over time, with a peak in 1989-1990.

Secondly, sale rates, while moving around, appear to have no noticeable trend.

Finally, from the above data, it is di¢cult to conclude anything about the corre-

lation of sale rates to prices. The correlation of the yearly average of sales rates

to yearly average of prices is about .26 for contemporary art, and about -.28 for

impressionist and modern art, and the correlation of the price indices to the sale

rate indrc375 0 TD 0 Tc (o) Tj 6 0 TD 0.09 0c (u) Tj 6.75 0 TD -0.168 Tc (t) Tj 9 0 TD -0.246 Tc (-) Tj 3.75 0 TD -0.336 8endeor conted e y ( s ) T j 5 . 2 5 0 T D - 0 . 1 7 5 0 T D 0 . 0 9 T c ( d ) T j 6 0 T D - 0 . 0 7 8 T c ( e ) T j 5 . 2 5 0 T 0 T c ( o ) T j 5 . 2 5 0 T D 0 . 0 9 T c ( n ) T j 6 . 7 5 0 T 0 T D 0 T c ( a ) T T j 5 . 2 5 0 T D - 0 . 1 7 5 0 T D 0 . 0 9 0 T D - 0 . 3 3 6 T c ( , ) T j 7 . 5 0 T D 0 T c ( a ) T j 5 . 2 5 0 T D 0 . 0 9 T c ( n ) T j 6 . 7 5 0 T D ( d ) T j 1 8 T D - 0 . 0 7 8 T c ( c ) T j 5 T j 6 0 T D 0 . 0 9 T c ( b ) T j 6 . 7 5 0 T D 0 T c ( o ) T j 1 6 0 T D 0 . 0 9 5 c ( u ) T j 6 . 7 5 0 T D - 0 . 1 6 8 T c ( t ) T j 9 0 T D - 0 . 2 4 6 T c ( - ) T j 3 . 7 5 0 T D - 0 . 3 3 6 8 e 28 for

impressiodereSTD 0.09 Tc (n) Tj 6 0 TD -0.168 Tc (t(r) Tj 4.5 0 T-411.078 Tc (e) Tj Tc (i) Tj 3.75 0 TD 0 Tc (o) Tj 5.25 0 TD 0.09 Tc (n) Tj 10.5 0 TD 0 Tc (o) Tj 6 Tc (a) Tj 6 0j 4.5 0 TD 0 Tc (a) Tj 6 0 TD -0.336 Tc (l) Tj 4.5 0 TD 0 Tj 6.75 0 TD -0.168 Tc (t) Tj 9 0 (t) Tj 4.5 0 TD -0.078 Tc (e) Tj46 Tc (-) Tj 3.75j 9.75 0 TD 01225 0 TD -0.228 Tc (s) Tj 4.) Tj 9 0 (t)j 9.75 0 TD 0.09 Tc (f) Tj 3.75 0 TD 0 0 TD (d) Tj 10.5 0 TD 78 Tc (e) Tj .5 -21 TD -0.336 Tc (i) Tj 3 0 TD -0.246 Tc (m) Tj 9.75 0 TD 0.09 Tc (p) Tj 6.75 0 Tj 7.5 0 TD 0 Tc (a) Tj 5.25 0 TD sseodeo

rate aleio a

indi¤erent between selling now and waiting for the next auction. However, for the

application to art auctions it would not be appropriate to assume, as is usual for

reservation wages, that the price distribution remains stationary: as described in

section 2, there were huge changes in price indices over a period of twelve years,

which may in‡uence the behavior of sellers. Instead, we will assume that the

mean of log-prices may evolve over time, according to a random walk with drift,

while the variance of log-prices (the coe¢cient of variation) remains constant. We

therefore adapt the reservation wage approach, as follows.

Suppose that the owner of painting i wishes to sell it. There is a sequence

of auctions, one in each time period, and the best price o¤er for the painting at

auction t is pt where:

lnpt = µt + ηt + εt (1)

( In this section all variables and parameters may be speci…c to painting i, but we

suppress indexation by i in order to reduce the notation.) Here, µt is the expec-

tation of lnpt, given the information available immediately before the auction:

Et [lnpt] = µt (2)

ηt and εt are mutually independent mean-zero price shocks, both independently

and identically normally distributed over time, with variances σ2η and σ2

ε respec-

tively. εt is a temporary shock, but ηt permanently a¤ects the expected price,

which follows a random walk with drift:

µt+1 = µt + g + ηt (3)

10

Thus ηt + εt » IN (0, σ2) where σ2 = σ2η + σ2

ε, and the price o¤er at auction

t has a log-normal distribution, with distribution function:

Ft(p) = © ((lnp ¡ µt) /σ) (4)

where © is the standard normal distribution function. Note that the expected

price for the painting is therefore:

Et [pt] = exp(µt + σ2/2) (5)

Before any auction, the seller sets a reserve price. If the realized price pt is

lower than the reserve price the painting will not be sold, and the seller must then

wait until the next auction. Let rt be his optimal reserve price for auction t, and

let δ0 be his discount rate. We assume that, having decided to sell the painting,

his payo¤ is simply the realized price if and when it is sold – he obtains no utility

while it remains in his possession3 .

The optimal reserve price rt for auction t is the price at which the seller is

indi¤erent between selling now, and waiting until auction t + 1 then using an

optimal policy. Hence the sequence of optimal reserve prices satis…es:

rt = e¡δ0Et

24rt+1Ft+1(rt+1) +

Z

rt+1

ydFt+1(y)

35 (6)

where the expression in square brackets is the payo¤ from using an optimal reserve3We think that this is a reasonable assumption: due to transportation costs a painting that is

not sold will often remain at the auction house for three months until the next auction. However

our theoretical results carry through if sellers do obtain some utility from an unsold painting,

provided that this utility is proportional to the current expected price.

11

price rt+1 at auction t+1. It is clear that it would not be optimal for the seller to

use a constant reserve price when the mean of the price distribution is changing

over time (that is, we cannot solve (6) by setting rt = rt+1). It can be easily shown,

however (see Appendix B) that the solution of (6) has a simple and intuitive form.

The optimal policy is to use a reserve price that is a constant proportion of the

current expected price:

rt = θEt [pt] (7)

where θ(δ, σ) is the solution of:

θ = e¡δ

0@θ +

Z

θ

¡1 ¡ ©

¡1σ ln y + σ

2

¢¢dy

1A where δ = (δ0 ¡ σ2

η/2¡ g) (8)

Thus the reserve price at auction t is set equal to the expected price o¤er,

adjusted by a factor θ which we will call the discount factor, ( although in theory it

may be more or less than one). The discount factor θ depends only on the variance

of log prices, σ, (and not on overall price levels!) and on δ, which represents the

”real” discount rate – after adjusting for the expected growth in prices. Provided

that δ > 0, equation (8) has a unique solution; otherwise, if expected price growth

were greater than the discount rate, it would be optimal to defer the sale of the

painting inde…nitely.

It should be noted that, even in the case when the distribution of prices is sta-

tionary, our result (7) appears to di¤er from the standard labor economics result

(see, for example, Devine and Kiefer, 1991) that the derivative of the reservation

wage with respect to the mean of the wage o¤er distribution lies strictly between

0 and 1. However, that result was obtained (…rst by Kiefer and Neumann, 1979)

12

under the assumption that the variance of wages is held constant. Here, we are

holding σ, the coe¢cient of variation, constant - in which case the derivative may

be more or less than one4. The latter assumption is clearly the appropriate one

for our context - it would not be sensible to assume the same price-variance for

two paintings whose expected prices di¤ered by a factor of perhaps two or more

(or for a single painting whose expected price doubled over time). Note also that

our result that the reservation price is exactly proportional to the expected price

follows from our additional assumption that either the seller obtains no current

utility from the painting, or his utility is proportional to the expected price. The

equivalent assumption for an unemployed worker would be that bene…ts are pro-

portional to expected wages (a constant replacement ratio). This is not normally

assumed, because the focus is on the e¤ect of short-run changes in expected wages

which are not re‡ected in bene…ts.

The most important implication of (7) for our analysis is that, if a painting is

o¤ered for sale at auction t, then whether or not it is sold depends on the price

shock ηt + εt, and also on the seller’s discount rate and the variability of prices,

but not on any other characteristics of the painting which a¤ect its expected price.

To see this, note that the painting is sold if and only if:

lnpt > ln rt

and using (1), (5) and (7) the condition can be written:

η t + εt > ln θ + σ2/2 (9)4This distinction has caused some confusion. Kiefer and Neumann themselves applied their

result as if it were true with the variance of log-wages held constant.

13

The role of price shocks in determining sale rates is central to the empirical model

developed in the next section. Before moving on, we brie‡y summarize the com-

parative statics for the theoretical model.

3.1. Comparative Statics

The seller’s discount factor θ depends on his personal discount rate δ0, and the

variance of both permanent and temporary prices shocks σ2η and σ2

ε. From equa-

tion (8) we can verify that θ is, as we would expect, decreasing in the discount

rate - a patient seller sets a higher reserve. It can be shown (see Appendix B) that

θ is increasing in the price-shock variances σ2η and σ2

ε . This is similarly intuitive:

if prices are more variable, the seller bene…ts from waiting, since a high price o¤er

may be realised in future.

Furthermore, from (9) we can see that a painting is more likely to be sold

when the seller’s discount rate is high, since he then sets a lower reserve, so the

threshold on the right-hand-side of (9) is low. Likewise it seems intuitive that the

probability of sale is higher when the variability of prices is low, since this also

implies a low threshold. However, there is an opposing e¤ect: when variability is

low, high values of η t+ εt are less likely to occur. It is shown in Appendix B that

the …rst e¤ect dominates, so that a painting is more likely to be sold when prices

are less variable5.5This result is similar to that of Balvers (1990), who also found that the probability of

sale decreases with the variability of the o¤er distribution. However, his result is for a mean

preserving-spread, so does not apply here since an increase in the variance of log-prices is not

mean-preserving.

14

4. An Empirical Application: Sale Rates and Unexpected

Price Changes

4.1. Random E¤ects Probit

We assume that the price shock for item i, ηit + εit » IN(0, σ2), is distributed

identically (but not independently across items), for all items in auction t. As

described above, before an art auction, the auction house publishes estimates of

the value of each item for sale. A low estimate and a high estimate is given, and

it is accepted practice that the secret reserve price should not be above the low

estimate. We will use these estimates to obtain an ”estimated price” for the item,

cpit, and interpret this as a measure of the expected price. Comparing it with the

realized price, we can infer the price shock that determines whether the item is

sold.

Let yit = 1 if item i is sold, and yit = 0 otherwise. As an item is sold if and

only if the price is great than the reserve price, our model predicts:

yit = 1 if and only if pit > θitE(pit) (10)

where θit is the discount factor of the seller of this item6. If we interpret the

estimated price as the expected price:

cpit = Et(pit)

de…ne the price shock for this item as:

psit ´ lnµ

pit

cpit

¶(11)

6Note that if estimated prices do represent expected prices, and the secret reserve is not

above the estimate, θ must be less than or equal to one.

15

and write the seller’s discount factor as:

ln θit = ln θ + ut + ωit

where θ is an ”average” discount factor, and the auction-speci…c e¤ect ut »N(0, σ2

u) and the seller-speci…c e¤ect ωit » N(0, σ2ω) are independent, then (10)

can be written:

yit = 1 if and only if psit > ln θ + ut +ω it (12)

Provided that we know the highest price o¤er both for sold and unsold items (as

we do for the contemporary art dataset) we can estimate (12) as a Random E¤ects

Probit model, using actual and estimated prices to calculate the price shock for

each item, and hence estimating the discount factor θ used to set the secret reserve

prices. In the special case when there are no auction-speci…c e¤ects (ut = 0) we

have the standard Probit model for which:

Pr[yit = 1] =©µ

psit ¡ ln θσω

¶(13)

4.2. Ordinary Least Squares

An alternative approach is to consider the relationship between the sales rate at

auction t:

St =1nt

X

i

yit

(where nt is the number of items for sale) and the average price shock for auction

t:

pst =1nt

X

i

psit

16

Decomposing the item-speci…c price shock as psit = pst ¡ ν it, where E[ν it] = 0,

we can write (12) as:

yit = 1 if and only if pst > ln θ + ut + ωit + ν it (14)

and hence:

E[St jpst, ut] = Pr[yit = 1 jpst, ut] =©µ

pst ¡ ln θ ¡ ut

σω+ν

¶(15)

This suggests estimating θ in an OLS regression:

©¡1 (St) =1

σω+ν

¡pst ¡ ln θ

¢+ πt (16)

Here, the error term πt picks up both auction-speci…c seller behavior ut, and the

error introduced by replacing the expectation of St by its actual value. If the

variation of the price shock between items within an auction is relatively small

compared with the variation in the average price shock between auctions, the loss

of e¢ciency from estimating (16) rather than the Probit model will also be small.

Moreover, this model can be used for the impressionist dataset, for which we have

prices only for sold items, if we replace pst by pssoldt , the average price shock for

sold items only. However, since pssoldt is an upwardly-biased measure of pst, we

would expect this to introduce an upward bias into the estimate of θ.

4.3. Estimation

Before we estimate the model, we …rst graph the relationship of price shocks

to the sale rate. We do this by plotting the buy-in rate (one minus the sale

rate) in relation to the price shocks. Figures 5 and 6 graph the relationship of

17

the percentage of paintings that go unsold in a particular auction to unexpected

changes in price. Figure 5 relates to impressionist and modern art, where the price

shock is calculated only across sold items, and …gure 6 relates to contemporary

art in which all items are used. As is indicated above, the price shock for an

individual item is calculated as psit ´ ln³

pitcpit

´7. These individual price shocks

are then averaged over an auction. In both …gures, there is a strong inverse

relationship between price shocks and the percentage of paintings that go unsold.

(The correlation of price shocks to the buy-in rate for contemporary art is -.86.

For impressionist art, the correlation is -.50.)

By estimating the above equations, we can derive estimates for σ and ¹θ. Es-

timating ¹θ is of special interest. ¹θ tells us the amount below the estimate that

on average the ”secret” reserve is set. Knowing ¹θ, or even a range for ¹θ, could

potentially be of use for bidders. We take two approaches to estimation.

We …rst estimate the random-e¤ects probit model, for contemporary art. We

can estimate a probit model for contemporary art as we know the highest bid

on paintings that were not sold, and we can use the highest bid for the price. In

column 1 and 3 of Table 2, psit ´ ln³

pitcpit

´where cpit is calculated as the midpoint

of the low and the high estimate as in the graphs above. In column 2 and 4,

cpit is the low estimate. The probit model in columns 1 and 2 is a special case

of the random-e¤ects probit model, where ut is zero and the contribution to the

likelihood is given by:

Pr[yit = 1] =©µ

psit ¡ ln θσω

¶

7 In these graphs, cpit is calculated as the midpoint of the low and the high estimate for an

item.

18

(13). A random-e¤ects probit model is then estimated in columns 3 and 4. The

coe¢cients are highly signi…cant in both models, and the results for both the

standard probit and the random-e¤ects probit indicate that the reserve price is

on average 71% of the expected price. The estimates of the seller-speci…c standard

deviations, σω, are also very similar in the two models, and are equal to .35 for

the models using the low-estimates. In the random e¤ects probit model, the

variance of the auction speci…c shocks, σu is estimated to be about .08. The

estimate of ρ, the correlation between reserve prices of sellers in a particular

auction, is .042. While statistically signi…cant, this indicates that the auction-

speci…c variance actually contributes very little to the total variance in the model.

We then estimate the ordinary least squares model

©¡1 (St) =1

σω+ν

¡pst ¡ ln θ

¢+ πt

for contemporary art and for impressionist art. For contemporary art, we can

estimate the equation using both the entire sample, and a sample of only sold

items. For impressionist and modern art, we use only a sample of sold items, and

thus our results will be biased upwards. Our estimation consists of an ordinary

least squares regression of the inverse normal of the sale rate on the price shocks.

In columns 5 and 6, for pst we use the average (over an auction) of the individual

price shocks for all paintings (again, for items that were not sold, we use the

high bid). In columns 7-10, for pst, we use the average (over an auction) of the

individual price shocks for paintings that were sold.8

8For contemporary art, auction sessions (i.e. morning and afternoon sessions) that were held

on the same day were grouped as one auction. Generally, auctions for contemporary art would

be held three to six months apart. For impressionist and modern art, there are often auction

19

From the results of all regressions, one can see that the coe¢cient on the

unexpected change in price and the constant is highly signi…cant in both the

impressionist and modern art and the contemporary art datasets. Furthermore,

the model appears to explain variations in the sales rates quite well. Using only

sold items, for impressionist art the R-squared is around .25. For contemporary

art, the R-squareds from the other regressions range from .62 to .78. When all

items are used in the contemporary art dataset, the R-squared increases from .62

to around .77. Clearly, there is information contained in the prices of items that

were not sold.

As would be expected, the OLS estimate of θ when all items are used in the

contemporary art data set is similar to the probit regressions (.60 vs..59 for the

average estimate, and .71 vs. .70 for the low estimates). Also as would be ex-

pected, the variance is estimated to be much lower in the Probit estimates than

in the OLS estimates. The standard deviation in the Probit estimates is σω, rep-

resenting the variation in discount rates between sellers. The standard deviation

given by the the OLS estimates is σω+υ, representing the combined variation of

sellers’ discount rates and item-speci…c price shocks as described above.

For impressionist art, we have estimated that on average the secret reserve

price is set 83% below the low estimate. For contemporary art, using only sold

items, we estimate the discount below the low estimate to be 80%. As noted above,

the estimates using sold items only are likely to be biased upward, and using allsessions are consecutive days. We grouped sessions that were held on consecutive days into one

auction. We felt this was reasonable as buyers often go long distances to attend an auction, and

as pre-sale catalogues for all auctions on consecutive days would be available before the start of

the …rst auction day.

20

items for contemporary art we estimate the discount to be 70% below the low

estimate. The relative estimates of θ for impressionist art and contemporary art

are consistent with the higher sale rate of contemporary art, than impressionist

art. The variation in prices appears to be lower for contemporary art. Our

estimates indicate that the higher sale rate for contemporary art is driven both

by a lower reserve price for contemporary art and a lower variance in the prices

and reserves.

How reasonable are our estimates of θ? Reserve prices for art are clearly set

below the low estimate. In contemporary art, out of a sample 3295 sold items,

1263 items sold at or below the low estimate. Of the items that sold at or below

the low estimate, these items sold an average of 87% below the low estimate. In

impressionist and modern art, out of 11544 sold items, 1898 items sold at or below

the low estimate. In this sample the mean percentage below the low estimate was

88%. The only evidence we could …nd on any actual reserve prices is contained in

a book by Peter Watson that documents the selling of Portrait of Dr. Gatchet.

For this picture, the secret reserve was $35,000,000, 87.5% below the low estimate

of $40,000,000.9

4.4. Implications for the E¤ective Discount Rate

Using the estimated θ, we can solve equation (8) to obtain the implied e¤ective

discount rate, δ. In order to do this, we also need to know σ, the standard9 In another context, McAfee, Quan, and Vincent (2000) construct a theoretical model and

…nd that for real estate, the optimal reserve for buildings should be at least 75% of the appraised

value, despite the Resolution Trust Corporation (RTC) and The Federal Deposit Insurance

Corporation (FDIC) using reserve prices of between 50-70%.

21

deviation of price shocks. Estimating this directly as the standard deviation of

psit gives bσ = 0.505. (The estimate is the same for both measures of cpit, and the

results for δ are not at all sensitive to bσ). A value of θ = 0.7 implies an e¤ective

discount rate δ = 0.41; the highest and lowest estimates of θ in Table 5 imply

δ = 0.29 and δ = 0.54, respectively. While at …rst glance these estimates appear

high, due to time between auctions and uncertainty, they may not be too far out

of range.

5. Testing the Model

5.1. Comparative Statics

According to the theoretical model, the probability of sale should be lower with a

higher price variance, but should not be e¤ected by the level of prices. We explore

this prediction in three di¤erent ways.

Firstly, one test of the theory would be to relate the hedonic price estimates

with the hedonic sale rate estimates in Appendix Tables 1 and 2. One explanation

for the fact that di¤erent artists have di¤erent sale rates is that the price variations

for some artists are higher than others, translating into lower sale rates. The

average price an artist receives for his paintings should not e¤ect the sale rate.

One way of exploring this idea is to regress the artist coe¢cients in the sale

rate probits on the artist coe¢cients and the standard deviations on the artist

coe¢cients from the hedonic price regressions. We do this for both Contemporary

Art and Impressionist and Modern Art; the results are presented in Table 3.

Because of the large number of di¤erent artists in the contemporary art sample,

for contemporary art, we only report the regressions in which we used artists that

22

appeared at least 40 times in the sample. The analysis is not sensitive to the

number of artists that are included. As would be expected in both datasets

the artist coe¢cients are insigni…cant – price level does not signi…cantly a¤ect

sale rate. Unfortunately, the coe¢cients on the standard deviations of the artist

coe¢cients are also insigni…cant, though the sign of the coe¢cients is negative, as

predicted, in both datasets.

Not only do sales rates di¤er between impressionist art and contemporary art;

we know empirically that sale rates di¤er across di¤erent types of auctions. While

we do not have detailed data on prices and expected prices for di¤erent types

of auctions, we do have average sales rates for all auctions that took place at

Christie’s King Street between January 1995 and September 1996. Table 4 lists

average sales rates for each department, ordered by 1996 average lot value; sales

rates range from 96% (Arms and Armour) down to 61% (Photographica). As

above, if the variance of price shocks is high for a particular type of item (such as

Photographica) then sellers will set high reserve prices, and the sales rate will be

correspondingly low.

The data in Table 4 allow us to infer a measure of the variance of price shocks.

In addition to the sales rate St for each auction in each department, we know

the percentage sold by value at each auction, Vt. This is the total value of all

the items sold in the auction, as measured by their sale price (that is, the total

auction revenue) divided by the total value of all items in the auction (measured

by the sale price for sold items and the last bid for unsold items). Since an item

is sold if and only if yit = 1, St and Vt can be written:

St =1nt

X

i

yit and Vt =P

yitpitPpit

(17)

23

Intuitively, we can see that when the variance of price shocks is high, some of

the sold items will have high prices, so the di¤erence between the percentage sold

by value and the sales rate will be high. Thus we can infer a measure of the

variability of prices by comparing Vt and St.

Speci…cally, we measure the variability of prices within a department by cal-

culating:

bφ =1n

nX

t=1

¡©¡1 (Vt) ¡ ©¡1 (St)

¢(18)

where n is the number of auctions that took place in the department. A justi…ca-

tion for the use of this measure is provided in Appendix B, where it is shown that

it can be used to estimate φ = σ2νσω+ν

which is positively related to the standard

deviation of price shocks σν, holding the variation between sellers, σω, constant

across departments. .

This estimate of φ for each department is shown in the …nal column of Table 4.

While we should be cautious in the interpretation of these values, it is reassuring

to note that the result for the contemporary art auctions is consistent with the

analysis in Table 2, where we estimated σω+ν and σω . Those estimates imply a

value of 0.35 for φ, and the value obtained using (18) is 0.36. Furthermore, the

pattern of variation of bφ across departments generally supports the hypothesis

that that higher item-speci…c price variability σν, and hence higher values of φ,

correspond to lower sales rates: there is a correlation of -0.25 between bφ and the

average sales rate for each department10.

Our third way of exploring the relationship between sale rates and price vari-

ability is to look at the spread between the high and the low price extimates for10 In the calculation of the correlation coe¢cient, observations were weighted according to the

number of auctions in the department.

24

paintings. If the auctioneer’s estimates were purely exogenous, one might suppose

that the spread between the high estimate and the low estimate is related to the

auctioneer’s estimate of the variance for each individual painting. One would

suspect that a dealer would provide a wider range of high-low estimates when

he is uncertain about what price he may fetch at auction. Suppose that the

high estimate is interpreted as the mean plus a multiple of the standard deviation

(H = µ+rσ), and the low estimate as the mean minus a multiple of the standard

deviation (L = µ ¡ rσ). Then, the high estimate minus the low estimate divided

by 2 is proportional to the standard deviation (H¡L2 = rσ) and the average of the

high estimate and the low estimate would just be the mean (H+L2 = µ). A large

di¤erence in the high estimate and the low estimate would therefore signal a high

price variance and a low probability of sale, and the mean should not e¤ect the

probability of sale. This reasoning, however, relies on the spread being a good

estimate of the auctioneer’s opinion of variance in price. In actuality, the spread

is set in conjunction with the seller’s idea of his reserve price (which in theory

should depend on his estimation of the variance along with his e¤ective discount

rate). Because of the convention that the reserve price is always set at or below

the low estimate, if the seller has a low discount rate and therefore a high reserve

price, one of two things could happen. Either the auctioneer convinces the seller

to lower his reserve price, or the auctioneer increases the lower bound on his esti-

mates. If the latter were to happen frequently enough, then a small di¤erence in

the high estimate and the low estimate would translate into a low probability of

sale and a higher mean would translate into a lower probability of sale. Therefore

correlating the di¤erence between the high estimate and the low estimate does

not present a clear test of our theory. Nonetheless, we regress the sale rates

25

on the mean of the high and low estimates and the di¤erence between the high

and the low estimates (divided by 2), where the high and the low estimates are

expressed in logs. The results are presented in Table 5, and strongly support the

interpretation of the low estimate being a re‡ection of the seller’s reserve price,

rather than an exogenous re‡ection of the auctioneer’s estimate of the variance.

5.2. Speci…cation Tests

Two natural questions arise in the context of the speci…cation of the above model.

First, the model restricts the coe¢cient on the price and expected price variables

to be identical. If the coe¢cients were not identical, that could indicate, for

example, that certain types of art (grouped by value) had di¤erent sale rates.

In the context of our model, it could also indicate that the reserve price is not

proportional to the expected price. Table 6 presents regressions that test this

restriction.

In all cases other than in columns 1 and 3, we cannot reject this restriction.

While the probit estimates that use the average of the high and low estimates

as the expected price do reject this restriction, it is not rejected when the low

estimate is used as the expected price. It is interesting that the reserve price is

often talked about as being proportional to the low estimate.

Secondly, we test for persistence in the buy-in rate that cannot be explained

by price shocks. In order to test whether there is persistence in the buy-in rate

that cannot be explained by price shocks (or the price level), we use the resid-

uals from the regressions in columns 5-10 of Table 2 and regress these residuals

on lagged residuals from these regressions. These results are presented in Table

7. For impressionist and modern art, we do …nd unexplained persistence. Al-

26

though the coe¢cients on the lagged residuals are positive, in the contemporary

art regressions, these residuals are not signi…cant. These results do suggest that

there may be other variables in‡uencing the sale rate that are not captured by

the model.

Finally, one might be concerned about correlation between the error terms and

the price shocks. However, there is little reason to believe that the error terms in

the above regression are correlated with the price shock. In almost all auctions, the

reserve prices and the estimates are set at the same time. During extraordinary

downturns, the auction houses may telephone the sellers after the pre-sale cata-

logues have been published but prior to the sale, in order to convince the sellers

to revise their secret reserve prices. This is a rare event but may have occurred

during the sharp downturn in 1990. This may induce slight correlation that could

bias downward the estimate of the coe¢cient on the price shock. However, as this

is likely to have occurred on only one or two occasions, it is unlikely to have had

much of an e¤ect on our estimates.

6. Conclusion

In this paper we have developed an empirical model of auction sale rates by using

an approach similar to that in the labor economics literature. We …nd that

auction sale-rates can be explained by price shocks and a constant, or a ”natural

sale rate.” This natural sale rate depends upon discount rates of individuals and

the variance of price-o¤ers. Using detailed data to estimate the model, we can

recover the average amount below the low estimate that the secret reserve price is

likely to be set in di¤erent types of auctions. We have estimated that the average

27

amount below the low estimate that the secret reserve price is set is between 70%

and 80% below the low estimate.

Abowd, J. and Ashenfelter, O. ”Art Auctions: Price Indices and Sale

Rates for Impressionist and Contemporary Pictures.” Mimeo, Department of

Economics, Princeton University, 1989.

Anderson, R.C. (1974) ”Paintings as an Investment. ” Economic Inquiry 12:

13-25.

Ashenfelter, O. (1989) ”How Auctions Work for Wine and Art.” Journal of

Economic Perspectives 3(3): 23-36.

Bailey, M., Muth, J.F. and Nourse, H.O. (1963) ”A Regression Method for Real

Estate Price Index Construction.” Journal of the American Statistical Association

58: 933-942.

Balvers, R. J. (1990) ”Variability and the Duration of Search.” International

Economic Review 31(3): 747-751.

Barre, M., Docclo S., and Ginsburgh, V. (1994) ”Returns of Impressionist,

Modern and Contemporary European Paintings 1962-1991.” Annales d’economie

et de Statistique 35: 143-181.

Baumol, W.J. (1986) ”Unnatural Value: or Art Investment as a Floating Crap

Game.” American Economic Review, Papers and Proceedings 76: 10-14.

Beggs, A. and Graddy, K. (1997) ”Declining Values and the Afternoon E¤ect:

Evidence from Art Auctions.” Rand Journal of Economics 28(3): 544-565.

Chanel, O., Gerard-Varet, L., and Ginsburgh, V. (1996) ”The Relevance of

Hedonic Price Indices.” Journal of Cultural Economics 20: 1-24.

Czujack, C. (1997) ”An Economic Analysis of Price Behavior in the Market

for Paintings and Prints.” Ph.D. dissertation, Universite Libre de Bruxelles.

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Devine, T. J. and Kiefer, N. M. (1991) Empirical Labor Economics, Oxford

University Press.

Frey, B.S. and Eichenberger, R. (1995) ”On the Rate of Return in the Art

Market: Survey and Evaluation.” European Economic Review 39: 528-537.

Goetzmann, W.N. (1993), ”Accounting for Taste: Art and Financial Markets

over Three Centuries.” American Economic Review 83: 1370-1376.

Horstmann, I. and LaCasse, C. (1997) ”Secret Reserve Prices in a Bidding

Model with a Resale Option.” American Economic Review 87: 553-684.

Kiefer, N. M. and Neumann, G. R. (1979) ”Estimation of Wage O¤er Distrib-

utions and Reservation Wages.” in J. J. McCall and S. A. Lippman (eds) Studies

in the Economics of Search, North Holland: New York.

Lippman, S. and McCall, J. J. (1976) ”Job Search in a Dynamic Economy.”

Journal of Economic Theory 12(3): 365-90.

Pesando, J.E. (1993) ”Art as an Investment. The Market for Modern Prints.”

American Economic Review 83: 1075-1089.

McAfee, R., Quan, D. and Vincent, D. (2000) ”How to Set Minimum Accept-

able Bids, with an Application to Real Estate Auctions.” unpublished manu-

script.

Mei, J., and Moses, M. ”Art as an Investment and the Origin of the ’Master-

piece E¤ect’ Evidence from 1875-2000.” unpublished manuscript.

Riley, J.G., and Samuelson, W.F. (1981) ”Optimal Auctions.” The American

Economic Review 71(3): 381-392.

Vincent, D. (1995) ”Bidding o¤ the Wall: Why Reserve Prices May Be Kept

Secret.” Journal of Economic Theory 65(2): 575-584.

Watson, Peter (1992) From Monet to Manhattan: The Rise of the Modern Art

29

Market, Random House: New York.

A. Appendix A

There have been many recent studies that have focused on the returns to holding

art and therefore have been concerned with price movements over time.11 We

employ two di¤erent methods to derive a price index, a repeat sales method, and

a hedonic method.12 A bene…t of using repeat sales to obtain an index is that like

is being compared with like. However, as Chanel et.al. (1996) point out, a repeat

sales index ignores all of the information on single transactions. They suggest

construction of a hedonic price index. We can construct both types of indices for

the impressionist art dataset and compare the two.

To construct a repeat-sales price index, we identi…ed 230 paintings that sold

at least twice during the period 1980-1990. To make a positive identi…cation, we

required that the paintings have an identical title, medium, artist and painting

date. As many paintings have identical titles, title and artist are not su¢cient

identi…ers. We perform an OLS regression of the log of the sale price of the

painting on a dummy variable for each painting, and the time period in which the

painting was sold. We include a dummy variable for each year. Using the antilogs11See, for example, Baumol (1986), Abowd and Ashenfelter (1989), Barre, Docclo and Gins-

burgh (1994), Goetzman (1993), Pesando (1993), Czujack (1997), Frey and Eichenberger (1995),

and Mei and Moses (2001).12Bailey, Muth, and Norse (1963) pioneered the use of repeat sales in estimating real estate

price indices. Anderson (1974), Goetzmann (1993), Pesando (1993), and Mei and Moses(2001)

have used repeat sales to construct indices for art. Chanel, Gerard-Varet, and Ginsburgh (1996)

have suggested that hedonic price indices should be used for art.

30

of the coe¢cients on the time dummies, we construct our repeat sales price index

for impressionist art. We do not have enough items to construct a repeat-sales

price index for contemporary art.

For the hedonic price index, the log of the sale price is regressed on the hedonic

painting characteristics in addition to time dummies for each period. The hedonic

characteristics used for impressionist and modern art are painting date, length,

width, signed, monogrammed, stamped, media in which it was painted, and artist.

We also include dummy variables as to whether the painting was sold at Sotheby’s

rather than Christie’s and New York rather than London. For Contemporary

art, the hedonic characteristics that are used are painting date, length, width,

medium, and artist. We also control for whether or not a painting is subject to

VAT. As many artists in the contemporary art sample have come up for sale just

one or a few times, we use a dummy variable, ”one,” for whether paintings have

been included for sale four or fewer times rather than using artist dummies for

artists with four or fewer sales. It is very interesting to look at the coe¢cients

on variables in the hedonic regressions. Looking at Appendix Table 1, which

presents the hedonic regressions for impressionist art, one sees that prices are

7% higher in New York than in London, and 5.7% higher at Sotheby’s than at

Christie’s. As might be expected, size is an important determinant in the price

of a painting, and signed paintings are more expensive than monogrammed or

stamped paintings. Oil is the most expensive medium, and van Gogh is the most

expensive artist. While we do …nd that hedonic characteristics are able to predict

whether a painting sells or not, the characteristics that in‡uence the price do not

necessarily in‡uence the sale rate in the same way. For example, while it is true

that paintings not only fetch higher prices in New York than in London and at

31

Sotheby’s rather than Christie’s, but are also more likely to sell there, size appears

to have no statistical e¤ect on whether a painting sells. Signed monogrammed,

and stamped paintings are also more likely to sell, but the medium appears to

have very little e¤ect on sale rates. Finally, it appears that Rousseau is the most

likely to sell at auction, with the prices fetched by various artists having little

e¤ect (if any) on the propensity to be sold. Note that there are di¤erent numbers

of observations for the price indices than the sale-rate indices as we only have

prices for sold items in the impressionist art sample. Also, note that we do not

have an entire set of hedonic characteristics for all items, and some that some

observations are dropped in both regressions due to collinearity.

The coe¢cients for contemporary art are reported in Appendix Table 2. For

artists who have appeared in the sample 40 times or less, we do not present the

coe¢cients on the artist dummies. For contemporary art, we …nd that more

recent paintings have lower prices, and again, that size has a strong positive e¤ect

on prices, but no e¤ect on a painting’s probability of sale. We …nd that sculp-

tures are more expensive than paintings, with pigment being the most expensive

medium, but not statistically signi…cantly di¤erent than oil. Paintings that are

subject to VAT trade for 11% less than other paintings, but the probability of sale

is not a¤ected. Finally, we …nd that Dubu¤et is the most expensive artist. While

we do have data on high bids for unsold contemporary art, which we include as

prices in that dataset, we still …nd the dataset used for the OLS regressions to

be slightly di¤erent than that for the probit regressions. The di¤erence occurs

because some artists predict success or failure (the fact that a painting has been

sold or not sold) perfectly. Again, please note that we do not have an entire set of

hedonic characteristics for all items, and some that some observations are dropped

32

in both regressions due to collinearity. Secondly, there is almost no di¤erence

between the hedonic sale rate index and the average sale rates for impressionist

art, though the two indices do di¤er for contemporary art.

B. Appendix B

B.1. Derivation of equation (8):

Integrating by parts, (6) can be written:

rt = e¡δ0Et

24rt+1 +

Z

rt+1

(1¡ Ft+1(y))dy

35

=) rt = e¡δ0Et

264rt+1 +exp(µt+1 + σ2/2)

Z

exp(¡µt+1+σ2/2)rt+1

µ1¡ ©

µln yσ

+σ2

¶¶dy

375

Now write rt ´ exp(µt + σ2/2)θt:

θt = e¡δ0Et

264exp(µt+1 ¡ µt)

0B@θt+1 +

Z

θt+1

µ1¡ ©

µln yσ

+σ2

¶¶dy

1CA

375

=) θt = e¡(δ0¡g)Et

264exp(ηt)

0B@θt+1 +

Z

θt+1

µ1¡ ©

µln yσ

+σ2

¶¶dy

1CA

375

Then, noting that Et[exp ηt] = exp(σ2η), this equation has a solution θt = θ for all

t, where θ is the solution of equation (8), and it can be veri…ed that this is the

only non-explosive solution. Hence rt = exp(µt + σ2/2)θ = E[pt]θ solves (6).

33

B.2. Comparative statics: θ is decreasing in δ and increasing in σ2ε and

σ2η

Integrating by parts, (8) can be written:

θ (exp(δ) ¡ ©(k)) =1σ

Z

θφ

µln yσ

+σ2

¶dy (19)

where:

k ´ ln θσ

+σ2

(20)

We can write θ = θ (σ(σε, ση), δ(ση)). Di¤erentiating with respect to σ, holding

δ constant:

∂θ∂σ

(exp(δ) ¡© (k)) =1σ2

Z

θ(ln y ¡ σ2/2)φ

µln yσ

+σ2

¶dy

=Z

kφ (x) (x ¡ σ)exp(σx ¡ σ2/2)dx

=Z

k(x ¡ σ)φ(x ¡ σ)dx > 0

and hence, since exp δ > ©(k), ∂θ∂σ > 0.

Since σ is increasing in σε, ∂θ∂σε

> 0 follows immediately. Di¤erentiating (19)

with respect to δ :∂θ∂δ

(exp(δ) ¡ ©(k)) + θ exp(δ) = 0

and hence ∂θ∂δ < 0.

Finally: ∂θ∂ση

= ∂θ∂σ

∂σ∂ση

+ ∂θ∂δ

∂δ∂ση

> 0, since ∂σ∂ση

> 0 and ∂δ∂ση

< 0.

34

B.3. Comparative statics: the probability of sale is increasing in δ and

decreasing in σ2ε and σ2

η

From (9), the probability of sale is P (σ, δ) ´ 1 ¡©(k(σ, δ)) where k is de…ned in

(20) above. Equation (19) can be written:

exp(δ) ¡ ©(k) =Z

kφ(x) exp(σ(x ¡ k))dx

Di¤erentiating with respect to σ (holding δ constant) gives:

0 =Z

k(x ¡ k)φ (x)exp(σ(x ¡ k))dx ¡σ

∂k∂σ

Z

kφ (x) exp(σ(x ¡ k))dx

Since both integrals are positive, ∂k∂σ > 0.

Similarly, di¤erentiating with respect to δ:

exp(δ) = ¡σ∂k∂δ

Z

kφ(x) exp(σ(x ¡ k))dx

so ∂k∂δ < 0.

Hence ∂P∂δ > 0; ∂P

∂σε= ¡∂k

∂σ∂σ∂σε

< 0 and ∂P∂ση

= ¡ ∂k∂σ

∂σ∂ση

¡ ∂k∂δ

∂δ∂ση

< 0.

B.4. Derivation of the estimator (18):

To derive a measure of price variability within a department, based on data for the

sales rate St and the percentage sold by value Vt, suppose that, as before, sellers

set reservation prices using an average discount factor θ. Ignoring the auction-

speci…c random e¤ects we know from equation (15) the expected sales rate at

auction t:

E[St jpst] = Pr[yit = 1 jpst] =©µ

pst ¡ ln θσω+ν

¶(21)

35

where ω is the seller-speci…c error term, ν is the item-speci…c price shock, and pst

is the auction-speci…c price-shock. Recall (from (17)) that the percentage sold by

value at auction t is:

Vt =P

yitpitPpit

(22)

Here the numerator is the total revenue for all the sellers in the auction. The

expected revenue for the seller of item i, conditional on the average price shock

pst, is the expectation of yitpit with respect to ω and ν. Using (14), this is given

by:

E[yitpit jpst] =ZZ

ω+ν<pst¡lnθ

pitd©µ

ωσω

¶d©

µνσν

¶

which, after a little manipulation, can be written:

E [yitpit jpst] = E [pit jpst] ©µ

pst ¡ ln θσω+ν

+ σ2ν

σω+ν

¶(23)

Comparing (22) and (23) suggests that Vt can be used to estimate©³

pst¡lnθσω+ν

+ φ´,

where φ = σ2νσω+ν

is a measure of the variance of item-speci…c shocks in the depart-

ment. It is a consistent estimator, in the sense that it would be asymptotically

unbiased if there were many items in the auction for each expected price. Since

also, from (21), St is an unbiased estimator of ©³

pst¡lnθσω+ν

´, we can estimate φ

consistently using the data for the n auctions in the department:

bφ =1n

nX

t=1

¡©¡1 (Vt) ¡ ©¡1 (St)

¢(18)

36