As argued above, under standard Hicksian reference-free preferences, selecting the dominated alternative is clearly irrational. This implies, therefore, that we would expect the share of
“mistakes”, i.e. choosing the dominated alternative, to be largest for small differences of c and t. Moreover, we would not expect the share of mistakes to differ across the four types of choices. Under reference-dependence, we would similarly expect more mistakes for small differences of c and t, but we would expect to find differences across quadrants. Observations
21 In principle these effects could be incorporated by interacting the loss aversion parameters with background variables. We have opted for the simpler model since our aim is to show that loss aversion explains the differences between quadrants.
of such mistakes are rarely analysed; in our case, they do provide a useful outside check on the theory of reference-dependence.
The dominated choice situations are labelled as shown in Figure 2. Under the standard preference model, all subjects would be expected to choose the fast and cheap alternative (to the South-West in the figure). The only way the dominated alternative can be chosen is by mistake.
time cost
EG
EL WTP
WTA
Figure 2: Labelling of dominated choice situations
As it turned out, 11.5% of subjects chose the dominated alternative. Table 5 summarises the data for the dominated choice situations. There are indeed large differences by quadrant. Independence is rejected in this table with overwhelming significance. Under reference-dependence, we would expect most mistakes in the EG-quadrant as the dominated alternative is equal to the reference in both the cost and the time dimension. Similarly, we would expect least mistakes in the EL-quadrant, since the dominating alternative is then equal to the reference. Both relationships are clearly evident from the table. For the WTP and WTA-quadrants we expect the number of “wrong” choices to be in between, as both alternatives in these choice situations match the reference on one dimension. If the loss aversion parameter for time is greater than that for cost (ηt>ηc), as found above (at least in the linear models where this inference can be made), we would expect to find more mistakes in the
WTP-quadrant than in the WTA-WTP-quadrant. These expectations are also matched by the data. So at a first glance, the predictions of reference-dependence are closely supported by the data, also for the dominated choice situations.
Table 5. Dominated choice situations
No. choosing alternative EL EG WTA WTP Total
Dominant 444 365 547 469 1825
Dominated 20 75 51 91 237
Share of mistakes 4.3% 17.0% 8.5% 16.3% 11.5%
Reference-dependence -ηt-ηc +ηt+ηc -ηt+ηc +ηt-ηc
As a further check, we estimate a series of binary logit models, letting the dependent variable be 1 if the dominated alternative is chosen and 0 otherwise. The estimation results are summarised in Table 6. The first model, denoted as D0, is specified just with a constant, such that the share of mistakes is predicted to be constant over quadrants.
Model D1 specifies constants by quadrant to allow the share of mistakes to differ by quadrant;
we find that the differences between quadrants are indeed strongly significant. Model D1R imposes the same restriction on the constants as in section 3.2. The loss aversion terms are positive as expected and time loss aversion is larger than cost loss aversion, as was also found for the non-dominated choice situations. The decrease in log-likelihood from model D1 to model D1R corresponds to a level of significance of 3.2 %.
Models D2 and D2R are similar to models D1 and D1R, but now the differences in cost and time between alternatives are used as extra controls. These variables are jointly significant and negative, indicating that the share of mistakes decreases as the cost and time differences become larger. The restriction from model D2 to D2R is significant at the 4%
level. The loss aversion terms are unaffected. In conclusion, we find that the pattern of mistakes across the four quadrants largely matches the predictions from the reference-dependence model.
Table 6. Model summary - dominated choices (t-stats in parentheses)
Model D0 D1 D1R D2 D2R
Log likelihood -735.5 -706.2 -708.5 -702.5 -704.6 Constant -2.041
(-29.6)
-2.132 (-28.4)
-1.976 (-18.9)
Constant EG -2.373
(-16.2)
-2.239 (-13.8)
Constant EL -1.640
(-14.3)
-1.475 (-10.8)
Constant WTA -1.582
(-12.5)
-1.416 (-9.5)
Constant WTP -3.100
(-13.6)
-2.951 (-12.3)
Loss aversion cost, ηc 0.129
(1.8)
0.127 (1.8) Loss aversion time, ηt 0.528
(7.0)
0.540 (7.1)
Cost difference -0.014
(-1.6)
-0.014 (-1.6)
Time difference -0.004
(-0.4)
-0.005 (-0.4)
Dof 1 4 3 6 5
LR-test 0.000 0.032 0.025 0.040
vs. D0 D1 D1 D2
4 Summary and concluding remarks
In this paper, we have specified a model of reference-dependent preferences to explain individuals´ valuation of travel time. Using data from a large-scale choice experiment, where each choice concerned a simple trade-off between travel time and travel cost, we estimate four valuation measures: willingness to pay, willingness to accept, equivalent gain and equivalent loss. We confirm the large gap between willingness to pay and willingness to accept, observed in the literature in other contexts. The implications of the theory of reference-dependence are consistently accepted against more general alternatives in tests of considerable statistical power. The results suggest that loss aversion plays an important role in explaining responses to binary choice options. Finally, we analyse data on choice situations involving “mistakes”, i.e., cases where subjects select an alternative that is dominated on both
the cost and the time dimension. These choices also show a clear pattern that to a large extent can be explained by loss aversion.
We further show that under our model it is possible to recover the underlying reference-free value of time. This is an important finding. Indeed, the large gap between the WTP and the WTA has generated a debate on which value to use for policy evaluation, and on the usefulness of contingent valuation methods as such (e.g., Diamond & Hausman 1994;Horowitz & McConnell 2003). Our model implies that the trade-off of references-free preferences is the (geometric) average of the WTP and the WTA. This conclusion hinges crucially on the assumed specification of the value functions, whereby losses are overweighted relative to the reference-free marginal utility by the same factor as gains are underweighted. We are currently investigating how this assumption may be justified. Without it, all we can say is that the reference-free marginal rate of substitution lies somewhere between the WTP and the WTA.
Our study differs from some other studies in the respect that the reference was clearly defined: Our experiment was based on a specific recent trip, identified to subjects as the specific trip hypothetical choices would concern. We are not able to say what will happen in situations where the reference is less clear. Maybe the degree of loss aversion diminishes, which would cause the four valuation measures to converge. It is a question if and how subjects form a reference and how reference-dependence can then be defined.
If one were to estimate models ignoring reference-dependence in situations where reference-dependence was present, then there would be bias. In our setup, the direction of the bias would depend on the distribution of choice situations over the four quadrants around the reference. Many WTP type choice situations would give a downward bias and so on. There would be bias even when choice situations were equally distributed over quadrants, since the resulting estimate would be some sort of average of WTP, EG, EL and WTA that would not in general be the same as the geometric average of WTP and WTA or (in our case equivalently) the geometric average of EG and EL.
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