J.K. Horowitz, K.E. McConnell J. of Economic Behavior Org. 41 2000 221–237 229
b Economic plausibility. We take replacement cost to be less than double our cost of US 6.25 per flashlight. Observations 36, 38, 39, and 40 are eliminated this way. This
criterion leaves 33 of the 42 original observations. c
′
Positive marginal values. Subjects can reveal non-positive marginal values if CD
1
≥ CD
2
or if CD
2
≥ CD
3
. Four respondents reveal non-positive marginal values nos. 1, 12, 22, 37. This criterion leaves 29 observations.
c
′′
Bounded marginal value. Two remaining respondents reveal marginal values for the third flashlight that exceed the replacement cost bound of US 12.50 nos. 19 and 32.
d Quasi-concavity of preferences. Of the 27 remaining observations, the strict quasi- concavity criterion leaves two observations nos. 6 and 7. The weak quasi-concavity cri-
terion leaves 11 observations, the previous two plus seven subjects whose valuations are linear nos. 4, 8, 13, 14, 23, 31, 35 and two whose valuations are linear then convex nos.
3, 27.
The flashlight experiment differed in asking for compensation for just one item. This change might be expected to encourage linear responses. However, only seven observations
in the 37-observation set are completely linear.
4.3. Mugs The 41 responses for the mug experiment are shown in Table 3. As in the flashlight
experiment, subjects were given an explicit option for declining to give back any mugs. a Intuitive plausibility. The compensation demanded ranged from US 2 to 20 for
two mugs and from US 2.50 to 28.50 for three mugs. The extremes are not obviously implausible. Further, no subject stated heshe would refuse to sell a mug at any price. No
observations are excluded for this reason. b Economic plausibility. Mugs like the ones used for this experiment are widely available
for around US 5–7. We put the replacement cost at just below US 8 per mug. Therefore, we view the values revealed by Observations 37 through 41 as economically implausible.
This leaves 36 observations of the original 41 observations. c
′
Positive marginal values. All of the 36 remaining subjects asked more for three mugs than two. No observations are removed for non-positive marginal values.
c
′′
Bounded marginal value. One subject revealed a marginal value for the third mug that exceeded the replacement cost bound no. 14.
d Quasi-concavity of preferences. Of the 35 remaining observations, six have values that are strictly convex in the number of mugs relinquished nos. 4, 5, 15, 16, 17, and 20,
and 17 have values that are linear.
5. Analysis
5.1. Effects of criteria on the distribution of values The common goal of hypothetical and real valuation experiments is the estimation of a
measure of central tendency of willingness-to-pay or compensation demanded. It is clear
230 J.K. Horowitz, K.E. McConnell J. of Economic Behavior Org. 41 2000 221–237
Table 3 Compensation demanded for mugs
Observation no. CD for two mugs US
CD for three mugs US 1
2 2.50
2 2
3 3
2 3
4 3
5 5
3 5
6 3.50
5 7
4 6
8 5
7.50 9
5 7.50
10 6
9 11
6 9
12 6
9 13
6 9
14 6
15 15
6.50 10
16 7.50
12 17
7.98 12.97
18 8
10 19
8 10
20 8
15.75 21
8.50 10
22 9
13.50 23
9.95 14
24 10
14.25 25
10 15
26 10
15 27
10 15
28 10
15 29
10 15
30 10
15 31
12 18
32 12.50
18.50 33
15 18
34 15
20 35
15 21.50
36 15
22 37
15 25
38 16
22 39
17.99 25
40 20
20 41
20 28.50
that deleting observations because they are too high will reduce the mean and median of the remaining observations. Less clear, however, is the impact on other sample characteristics,
such as dispersion and skewness. Further, it is not obvious how restricting the sample to observations with positive marginal values or convex responses will influence the central
tendency. In Tables 4–6 we explore the impact of increasingly restrictive samples. In these
J.K. Horowitz, K.E. McConnell J. of Economic Behavior Org. 41 2000 221–237 231
Table 4 Summary statistics for binoculars CD N = 48
Criterion Mean US
Coefficient of Median US
Skewness Percent of
variation original
CD for two binoculars a
50.32 1.26
29.00 2.62
94 b
27.36 0.60
21.38 1.04
79 c
′
28.27 0.59
22.50 0.95
73 d
26.68 0.53
20.25 0.52
65 CD for three binoculars
a 72.63
1.27 45.00
2.88 94
b 41.40
0.58 37.25
0.60 79
c
′
43.66 0.54
40.00 0.52
73 d
42.11 0.50
39.50 0.35
65 Table 5
Summary statistics for flashlights CD N = 42 Criterion
Mean US Coefficient of
Median US Skewness
Percent of variation
original CD for one flashlight
a 6.19
0.65 5.00
1.54 95
b 5.38
0.52 5.00
0.79 88
c
′
5.38 0.50
5.00 0.78
79 c
′′
5.29 0.49
5.00 0.82
74 d
4.61 0.55
4.00 1.20
26 CD for two flashlights
a 10.82
0.67 9.25
1.94 95
b 9.24
0.49 8.50
0.70 88
c
′
9.74 0.45
9.00 0.75
79 c
′′
9.50 0.45
8.50 0.85
74 d
9.68 0.51
8.50 1.05
26 CD for three flashlights
a 16.79
0.67 14.00
1.03 88
b 13.89
0.56 12.00
0.68 79
c
′
14.82 0.50
12.75 0.82
69 c
′′
13.69 0.46
12.00 0.86
64 d
15.84 0.45
15.00 0.73
26 Table 6
Summary statistics for mugs CD N = 41 Criterion
Mean US Coefficient of
Median US Skewness
Percent of Criterion
variation original
CD for two mugs a
9.18 0.53
8.50 0.54
100 b
8.17 0.48
8.00 0.25
90 c
′′
8.23 0.48
8.00 0.20
88 d
7.25 0.46
7.00 0.26
59 CD for three mugs
a 13.33
0.49 14.00
0.29 100
b 12.32
0.45 13.24
− 0.02
93 c
′
12.11 0.45
12.97 0.02
90 c
′′
12.03 0.46
12.49 0.06
88 d
11.13 0.45
11.00 0.10
59
232 J.K. Horowitz, K.E. McConnell J. of Economic Behavior Org. 41 2000 221–237
Table 7 Summary statistics for all observations except outliers
Item Binoculars per pair
Flashlights per flashlight Mugs per mug
Mean US 24.69
5.78 4.52
Coefficient of variation 1.26
0.65 0.51
Medianmean US 0.61
0.87 1.00
Maximummean US 6.08
3.45 2.21
Skewness 2.70
1.50 0.44
tables the sample statistics are calculated with different samples, labeled a, b, c
′
, c
′′
, and d. In Sample a, the intuitively implausible observations are deleted; in b, the
economically and intuitively implausible are further deleted; in c
′
, non-positive marginal value observations; in c
′′
, the observations with too-high marginal values; and in d, valuations that are not consistent with weakly quasi-concave preferences.
2
We look at the weakest of possible restrictions for Sample d.
The number of observations that are economically plausible but fail to have positive marginal values and weakly quasi-concave preferences ranges from 14 to 69 percent of the
full sample, but the effect of the last two criteria on the central tendency and dispersion of the distributions is mixed. The means change little and the measures of dispersion appear
to decline slightly. The main effect of exclusions c and d is the loss of precision due to the reduction in observations.
5.2. Distribution of values – only outliers removed Real and hypothetical experimenters commonly analyze responses without appealing to
Criteria b, c, or d, either due to the criteria cannot be applied e.g. only one set of items is valued, which eliminates the marginal value and convexity criteria, or the item is not a
private good, which nearly eliminates the economic plausibility criterion or because the experimenters accept the reported values without applying any judgment. In both of these
cases, however, analysts are likely to remove obvious outliers. Table 7 presents statistics from all three experiments with only the intuitively implausible responses deleted.
Lognormal distributions. The lower the mean CD, the smaller the upper tail of the dis- tribution of responses. We find that a lower mean response is accompanied by: a a lower
standard deviation, relative to the mean, in other words, a lower coefficient of variation, b a median closer to the mean, c a lower maximum value, relative to the mean, and d
a lower skewness. These findings are consistent with value distributions that are roughly lognormal, conforming approximately to the shape of a lognormal although not to its range.
Number of outliers. The lower the cost of the item, the lower is the number of outliers that are removed.
2
For a and b, we remove CD
i
responses only if the response is implausible for i items. In the previous section we counted a subject only if his responses met the criterion for all i. For c and d in the flashlight case, we
remove the entire response if it violates a marginal or quasi-concavity condition anywhere.
J.K. Horowitz, K.E. McConnell J. of Economic Behavior Org. 41 2000 221–237 233
Market value and mean CD. When only outliers are removed, mean CDs are remarkably close to the item cost of US 25 binoculars, US 6.25 flashlight, and US 4 mug. A brief
look at willingness-to-pay WTP experiments shows an analogous pattern. Empirically, mean WTP is roughly half the market price, say 40–60 percent. For example, Loomis et
al. 1996 elicited willingness-to-pay for an art print using a first-price auction. Their third treatment, which is the closest to our experiments, found a mean WTP of US 14.48, which
is 41 percent of the item cost of US 35. Johannesson et al. 1997 used a second-price WTP auction for chocolates with 10 participants. The cost was 150 Swedish crowns and
the mean WTP was 87.40, which is 58 percent of the price. Neill et al. 1994 conducted a second-price WTP auction for a map. Mean WTP was either 50 or 60 percent of the US
20 cost, depending on the treatment of outliers. Horowitz and McConnell 1998 reviewed studies that collected both CD and WTP and found that for ordinary private goods, CD was
roughly 2.3 times larger than WTP. If CD is close to market value, then WTP will be 43 percent of market value.
3
These results—that mean CD values are close to market value—are striking, but they further indicate how much of the aggregate reported value, roughly half, exceeds the re-
placement cost upper bound on CD imposed by economic theory.
6. Interpreting the results
