Determinants of the degree of loss aversion

Transcription

1 Determinants of the degree of loss aversion Katrine Hjorth Department of Transport Technical University of Denmark & Department of Economics University of Copenhagen Mogens Fosgerau Department of Transport Technical University of Denmark & Centre for Transport Studies, Sweden Paper presented at the International Choice Modelling Conference Abstract We present a framework to identify factors affecting loss aversion in a stated choice experiment where respondents trade travel time for money. Other studies have used stated choice data to estimate loss aversion separately from the marginal rate of substitution between travel time and money, but have not investigated how loss aversion varies with individual characteristics and features of the experimental design. The main contribution of the paper is thus to fill this gap. Overall, we find significant loss aversion in both the time and cost dimensions. Loss aversion in the time dimension is larger than in the cost dimension, and depends on age, education, income, and gender, while we find no significant effect of occupation. We also find evidence suggesting that loss aversion depends on how well-established the reference point is, in the sense that people tend to be more loss averse when the reference is well established. The authors are grateful to The Danish Social Science Research Council for financial support. 1

2 1 Introduction This paper presents a method to identify how the degree of loss aversion in a simple trade-off experiment depends on factors such as socio-economic characteristics and experimental design. The behavioural and experimental economics literature gives ample evidence that observed choice behaviour deviates systematically from neoclassical utility theory: Preferences are seen to depend on the framing of choice situations or the size of the choice set, or to be subject to judgmental biases, such as inability to process all relevant information or to correctly perceive correlations and probabilities (Rabin, 1998; Starmer, 2000). An important deviation from classical theory is referencedependence, where commodity bundles are valued according to how much they differ from some reference bundle, rather than their absolute level. With referencedependence the same individual s preferences vary with his reference situation, which may be his current situation, the normal state of affairs or his beliefs about the immediate future (Kahneman and Tversky, 1979; Rabin, 1998; Munro and Sugden, 2001; Köszegi and Rabin, 2006). In this paper, we focus on loss aversion: a special case of reference-dependent preferences, where people are significantly more averse to losses (compared to the reference) than they are attracted to same-sized gains. Several studies have found evidence of loss aversion in choice behaviour, in the form of a disparity between the willingness-to-pay (WTP) for a good and the willingness-to-accept (WTA), or selling price, of the good (see e.g. Bateman et al., 1997; Horowitz and McConnell, 2002; Sayman and Öncüler, 2005). Loss aversion would not be a problem if we were convinced that it is a rational phenomenon that reflects true preferences; by which we mean the preferences corresponding to the well-being that people actually experience as a consequence of their choices. 1 However, this does not seem likely, as some experiments show that loss aversion and the WTP-WTA disparity tend to diminish with choice experience (List, 2005; Plott and Zeiler, 2005). Further, Rabin (1998) argues that people tend to underestimate how quickly they adapt to changes, not foreseeing that their reference points will change. Because they do not anticipate the degree of adaptation, they exaggerate the importance of expected changes in their lives. He concludes that loss aversion often seems to be a judgmental bias, where people overweight the significance of initial losses and underestimate the long-term consequences. Rejecting that loss aversion reflects true preferences leads to serious problems when we wish to elicit preferences for a good that is not traded in the market. How can we assess people s true preferences, if we can only measure their choice preferences, i.e. preferences inferred from behaviour in experiments and surveys, which is subject to reference-dependence? To proceed, we need a framework that separates the concept of true reference-free 1 Köbberling and Wakker (2005) and Köszegi and Rabin (2006) refer to this as intrinsic preferences, while De Borger and Fosgerau (2008) use the term hedonic preferences. 2

3 preferences from judgmental biases due to reference-dependence. Our starting point for such a framework is Kahneman and Tversky s (1979) prospect theory, where the choice process is broken down into two phases: the editing phase and the evaluation phase. In the editing phase, the decision maker organises, reformulates and simplifies the available information to facilitate the subsequent evaluation. This can be in the form of coding attributes as gains and losses, or combining or ignoring attributes. The editing phase is modelled using a value function that maps attribute values to their perceived values. In case of loss aversion, the value function assigns a larger numerical value to a loss than to a gain of the same size. Several studies apply the concept of a value function to model reference-dependence, to separate loss aversion and true preferences: Köbberling and Wakker (2005), Köszegi and Rabin (2006), De Borger and Fosgerau (2008), and Fosgerau and De Borger (2008). These papers define true reference-free preferences as a normative concept based on neoclassical theory, and indicate how these true preferences may be recovered from observed choices subject to loss aversion. To identify loss aversion separately from the curvature of true preferences 2, they have to make some normalisation assumptions, see e.g. the comment in Köbberling and Wakker (2005). However, these and most other applications treat loss aversion as constant across individuals. We would expect individual heterogeneity, as well as sensibility to survey design. Very few studies have investigated the determinants of individual-level loss aversion, the most important being Johnson et al. (2006) and Gächter et al. (2007), who measure loss aversion directly as the ratio between WTA and WTP. Johnson et al. (2006) measure loss aversion in the attributes (fuel consumption, comfort, safety, and information system) of a car choice experiment, and find that loss aversion depends on socioeconomic characteristics as age, income, and occupation, and on perceived importance of and familiarity with the good in question. Gächter et al. (2007) measure loss aversion in a riskless choice (the good being a toy car) as well as in a risky lottery and find that both types of loss aversion depend (in similar ways) on age, gender, income, and occupation. In the current paper, we use data from an abstract stated preference route choice experiment, where route choice is solely a trade-off between travel time and money, and estimate loss aversion in both time and money dimensions. Note that since Johnson et al. (2006) and Gächter et al. (2007) measure loss aversion by comparing WTA and WTP, they do not separate loss aversion in the good dimension from loss aversion in the money dimension. Further, they do not separate the effect from loss aversion from that of non-constant marginal rates of substitution (preference curvature). Regarding the former issue, the current paper estimates loss aversion in the time and money dimensions separately, by using equivalent-gain (trade-off between a gain in the time dimension and a gain in the money dimension) and equivalent-loss 2 If, as is often assumed, true preferences are convex, such that people are more willing to substitute good 1 for good 2 if they have more of good 1, WTP and WTA will differ even in the absence of loss aversion. 3

4 (trade-off between a loss in the time dimension and a loss in the money dimension) valuations as well as WTA and WTP valuations. For this we use the model of De Borger and Fosgerau (2008), but we allow for heterogeneity in loss aversion by letting the degrees of loss aversion be functions of socioeconomic, trip, and design characteristics. Regarding the latter issue, we are able to at least partially control for the effect of preference curvature, by i) applying a fixed effects estimator, that uses only withinperson variation, and thus controls for the effect of person-invariant covariates and unobserved heterogeneity on the marginal rate of substitution between travel time and travel cost (which is the value of travel time, VTT), and ii) allowing the VTT of a given person to vary between different consumption bundles of travel time and cost, by letting it depend on the size of the travel time difference. The rest of the paper is organised as follows: Section 2 describes the data used in our analysis. In section 3, we develop our econometric model, while section 4 presents our estimation results and section 5 concludes. 2 Data Our data are from a survey for the Danish value of travel time study (Fosgerau et al., 2007). We use two stated choice experiments, where respondents make trade-offs between travel time and travel cost by choosing either a fast and expensive trip or a slower and cheaper one. 2.1 Interview procedure Respondents were sampled from Gallup s Danish Internet and Phone panels or contacted at educational institutions. Respondents in the Internet panel were asked to complete the questionnaire on-line themselves, while the remaining respondents were interviewed face-to-face (using the same questionnaire on a laptop). In the beginning of each interview, the respondent was asked to state the types of trips he had made within the last eight days, distributed on travel mode, trip length, and trip purpose. 3 One of these trips was selected at random as base of the experiments; we label this the base trip. In our analysis we use data for car trips only. Respondents were interviewed in detail about the base trip, giving information such as the travel time, cost, number of companions, day of travel, congestion, delays, how often the trip was made, if they had to arrive at a fixed time or had some flexibility, etc. The cost of the base trip encompassed direct driving costs only, i.e. taxes and maintenance cost were not included. Neither were road tolls and parking costs. The cost was computed by multiplying the trip length by a fixed kilometer cost (0.75 DKK 4 ). Respondents were asked if this amount was acceptable, 3 International trips, business trips, and trips shorter than five minutes were not included. 4 1 Euro 7.5 DKK. 4

5 and if not they were asked to state the amount they perceived to be the correct cost, and this measure was used instead. Respondents then participated in the first stated preference experiment (SP1), where they had to choose between variations of the base trip. This is described in detail in the section below. After this experiment, respondents were asked to state which transport mode they would use instead, if a car was unavailable; the trip with this alternative transport mode is labelled the alternative base trip. Respondents stated the travel time and cost of the alternative base trip, and indicated how often they made the alternative base trip, and how often they used the alternative mode in general. They then participated in the second experiment (SP2), with choices between variations of the alternative base trip. Only respondents who had an alternative transport mode, and had used this mode for some type of trip within the last year, participated in SP2. Finally, respondents answered a series of background questions regarding their sociodemographic characteristics. 2.2 First stated preference experiment (SP1) SP1 consisted of nine binary choices between travel alternatives described by invehicle travel time and travel cost. Respondents were instructed to pretend they were to make the base trip again, but now facing different travel times and costs. It was stressed that in every other aspect (transport mode, trip type, time-of-day, walking time, parking time, etc.), the travel alternatives were exactly as the base trip. Let t 0 and c 0 denote the travel time and cost of the base trip. Choice situations were generated by varying time and cost around (t 0, c 0 ). All respondents were presented with both gains (decreases) and losses (increases) in both attributes, defining four types of choices: Willingness-to-pay choices, comparing the reference (t 0, c 0 ) to a time decrease / cost increase (t 0 t, c 0 + c ), Willingness-to-accept choices comparing the reference (t 0, c 0 ) to a time increase / cost decrease (t 0 + t, c 0 c ), Equivalent-loss (EL) choices comparing a time increase (t 0 + t, c 0 ) to a cost increase (t 0, c 0 + c ), and Equivalent-gain (EG) choices comparing a time decrease (t 0 t, c 0 ) to a cost decrease (t 0, c 0 c ). The choice types correspond to the quadrants of the time/cost plane with origin in (t 0,c 0 ), as shown in Figure 1. The experiment had eight trade-offs, two of each choice type. The travel alternatives were generated as follows: First, a time difference t was drawn randomly from a subset of {3, 5, 10, 15, 20, 30, 45, 60} minutes; the subset depending on t 0, such that respondents on longer trips were offered larger time differences. Then 5

6 cost Willingness-to-pay Equivalent loss travel time Equivalent gain Willingness-to-accept Figure 1: Choice types, corresponding to quadrants in the time/cost plane with origin in (t 0,c 0 ). four bids v were drawn at random from the interval [0.5,3.33] DKK per minute, and for each bid the cost difference of the alternatives was computed as c = v t. Finally, the four pairs ( t, c ) were assigned randomly to the four quadrants, with one pair in each quadrant. The process was repeated to generate another four choices, and the eight choices were presented to the respondent in random order. Apart from the trade-offs, the experiment contained a check question where one alternative dominated the other. For respondents with a short base trip (t 0 10 minutes), choice situations were generated by varying time and cost around (t 0 + 2, c 0 ), such that the time attribute could still be varied both up and down. 2.3 Second stated preference experiment (SP2) Experiment 2 was similar to experiment 1, except that t 0 and c 0 were now the travel time and cost of the alternative base trip, such that alternatives varied around this trip. The experiment had eight trade-offs, two of each type. 2.4 Sample selection We use data for car trips (i.e. the base trip is a car trip with the respondent as driver), where the alternative transport mode is bus or train, or the respondent does not have an alternative mode. We exclude private trips paid by the employer, individuals who chose the dominated alternative in the check question, and individuals stating unrealistic values of e.g. speed, cost, and travel time of their base trip. The final data set contains 2001 individuals and 18,814 observations. Of these 2001 individuals, 426 participated in experiment 2. 6

7 3 Econometric Model In this section we develop an econometric model to describe the choice behaviour observed in the two experiments. We want to model the choice between two bundles of travel time and travel cost, and we consider only choices that involve a tradeoff between the attributes, i.e. one alternative is faster and more expensive than the other. In a given choice, label the alternatives 1 and 2, where alternative 1 with attributes (t 1, c 1 ) is the faster and expensive alternative, and alternative 2 with attributes (t 2, c 2 ) is the slower and cheaper alternative. An underlying assumption of our model is that both travel time and cost are non-goods, such that the (reference-dependent) utility of a choice alternative is nonincreasing in both attributes. Since the experiments are very abstract and only focus on travel time and travel cost, we assume that choices depend solely on the attributes t 1, t 2, c 1, c 2 (as perceived by the respondents) and the monetary value of travel time, VTT. 5 Our model consists of two central assumptions specifying the process of choice, based on Kahneman and Tversky s prospect theory, and a series of technical assumptions needed to transform these choice process assumptions into an econometric model that can be applied to data. Section 3.1 below presents the central behavioural assumptions, while the following three sections develop the econometric model. In sections 3.5 and 3.6 we discuss estimation and the included covariates, respectively. 3.1 Choice process Our model relies on two assumptions regarding the choice process: How choice attributes are weighted due to reference-dependence (modelled using value functions as proposed by Kahneman and Tversky (1979)), and how alternatives are compared after weighting. Traditionally, in the literature, the reference point is taken to be the individual s current situation. However, Köszegi and Rabin (2006) point out that the reference is rather the individual s expectations regarding his post-choice situation. In general, this could be quite complicated to model in an econometric framework. In our experiments, however, emphasis is put on the base trip and the alternative base trip as the normal situation, and it seems fair to assume that these trips serve as a reference of preference formation. A potential problem is if the base trip does not correspond to the respondent s idea of normal, e.g. if the base trip happens to be extraordinarily delayed. Despite this potential problem, our best guess of the reference point is the base trip of the experiment. In a given experiment, we 5 The concept of VTT has great importance in transport economics, where it appears both as a subjective value that affects individual travel behaviour and as a social value measuring the welfare effects of the time spent on travel (Mackie et al., 2001; Jara-Díaz, 2000). The relevant measure here is the subjective value of travel time, which is the marginal rate of substitution between travel time and money. The microeconomic framework of the subjective value of travel time was developed mainly by Becker (1965), Johnson (1966), Oort (1969) and DeSerpa (1971). Jara-Díaz (2000) provides a review. 7

8 therefore consider the travel time and cost of the experiment s base trip, denoted t 0 and c 0, to be the respondent s reference levels of travel time and travel cost. Assumption 1 (Reference-dependent weighting). Respondents evaluate attribute levels in terms of changes compared to the reference. The perceived value of the time attribute difference t j t 0 is V t (t j t 0 ), where V t ( ) is a value function satisfying: - V t ( ) is increasing, and V t (0) = 0. - V t ( ) is asymmetric due to loss aversion: V t ( m) < V t (m) for m > 0. Similarly, the cost difference is evaluated using a value function V c ( ) also satisfying the above. The asymmetry requirement ensures that the numerical value of a gain (compared to the reference) is smaller than the value of an equally sized loss. Assumption 2 (Comparison of alternatives). Respondents choose the alternative that maximises their reference-dependent utility. As De Borger and Fosgerau (2008), we assume the utility to be linear in the perceived values of the attribute differences V t (t j t 0 ) and V c (c j c 0 ): u(t j, c j t 0, c 0 ) = V c (c j c 0 ) V T T V t (t j t 0 ) VTT is the value of travel time, conditional on t j and c j. Maximising reference-dependent utility corresponds to choosing the faster and more expensive alternative, whenever the perceived time saving is worth more than the perceived cost saving. When we formulate an econometric model of the choice process, we want to make less strict assumptions (and allow for specification errors), so we assume only that respondents do not deviate systematically from the rule above. 3.2 Parameterisation of value functions First, we consider the econometric modelling of assumption 1. For simplicity, we apply piecewise linear value functions to map actual attribute values to perceived attribute values. To allow for asymmetry in the valuation of gains and losses, the value functions have different slopes in the positive and negative domain. The slopes themselves are not identified from our data: we cannot observe how much a loss is overweighted compared to the true attribute value, or how much a gain is underweighted, but only measure the relative value of a loss compared to a gain. We therefore have to choose some normalisation of V t ( ) and V c ( ). Based on De Borger and Fosgerau (2008) and Fosgerau and De Borger (2008), we use the following formulation, where the slopes in the positive and negative domain exhibit a certain kind of symmetry: V t (t j t 0 ) = (t j t 0 ) e ηts(t j t 0 ), j = 1, 2 (1) V c (c j c 0 ) = (c j c 0 ) e ηcs(c j c 0 ), j = 1, 2 (2) 8

9 Here S(m) denotes the sign of m. V t ( ) and V c ( ) are piecewise linear functions with slopes e ηt and e ηc in the positive domain (losses) and slopes e ηt and e ηc in the negative domain (gains). With this formulation, a time loss is valued e 2ηt times the value of an equally sized time saving, and a monetary loss e 2ηc times the value of a cost saving. In case of loss aversion, the η s will be positive. As can be seen from De Borger and Fosgerau (2008), the reference-dependent marginal rate of substitution between time and money (i.e. the observed VTT when reference-dependence is not controlled for), is given by V T T e ηt+ηc in the WTA quadrant and V T T e ηt ηc in the WTP quadrant. The WTA/WTP ratio is thus e 2ηt+2ηc. In the EL and EG quadrants, the reference-dependent marginal rate of substitution is given by V T T e ηt ηc and V T T e ηt+ηc, respectively. Thus, loss aversion in one or both dimensions will cause WTA to be larger than WTP, and loss aversion in both dimensions causes EL and EG to be smaller than WTA and larger than WTP. The relative sizes of EL and EG depend on the relative sizes of η t and η c. Fosgerau and De Borger (2008) show that this formulation is rational in a certain sense: Assuming that people s true preferences are reference-free, but that they make choices based on reference-dependent choice preferences obtained by applying linear value functions to the true preferences, then any systematic deviation from the above symmetry would lead to a decrease in expected true utility. The ratio Vc(c j c 0 ) V t(t j t 0 ) is only identified up to a scale, since any scale parameter is not identified separately from the scale of VTT: We cannot distinguish a person who values a travel time saving higher because the time saving is worth more than the cost difference, from a person who values a travel time saving higher because his perceived value of the time saving differs more from the perceived value of the cost saving. As mentioned in the introduction, this identification problem is also present in other studies. With our choice of normalisation, the entire difference between the valuation of a gain and that of a loss is ascribed to loss aversion, because value functions are linear and we assume a constant unit VTT. Hence, our estimate of loss aversion will be biased, because we ignore any effect of preference curvature (the VTT varies with the level of travel time or cost) or perception curvature (nonconstant marginal perceived value). Regarding preference curvature, we do two things to remedy the bias: - First, we limit analysis to only consider within-person variation by using the so-called fixed effects logit estimator (Chamberlain, 1984). - Second, we allow VTT to depend on the size of the time difference between the two alternatives, t = t 1 t 2, and whether the choice consideres a time loss or a time gain relative to the reference travel time. The fixed effects estimator removes the bias due to VTT varying with reference travel time and reference cost, such that the remaining bias is due to within-person variation in VTT across alternatives. Since the variation in travel cost is relatively small compared to respondents wealth, the bias from treating VTT as constant 9

10 across different cost levels is likely to be small. This is not the case for the time dimension, however, as the experiments include quite large time variations (up to 1 hour), so to eliminate preference curvature bias, we need to allow VTT to vary with the difference in travel time between the two alternatives. 3.3 Discrete choice model formulation The second part of our model is the econometric modelling of the choice behaviour in assumption 2. Here it is convenient to index variables according to individual i and choice situation s. For individual i and choice s, the attributes of alternatives 1 and 2 are (t 1,is, c 1,is ) and (t 2,is, c 2,is ), respectively. (t 0,is, c 0,is ) denotes the reference point note that the reference point also depends on s, because we assume different reference points in the two experiments. Let y is be 1 when individual i chooses the faster and more expensive alternative in choice s, and 0 otherwise. The deterministic rule in assumption 2 implies that y is = 1 u(t 1,is, c 1,is t 0,is, c 0,is ) > u(t 2,is, c 2,is t 0,is, c 0,is ) V T T is V t,is > V c,is, (3) where V t,is := V t (t 1,is t 0,is ) V t (t 2,is t 0,is ) V c,is := V c (c 1,is c 0,is ) V c (c 2,is c 0,is ) (4) We apply a stochastic discrete choice model, in which people do not deviate systematically from the rule in (3): We assume that 6 y is = 1 log V T T is log V c,is V t,is > ε is, (5) where ε is is a symmetric random error term with mean zero. The error term represents computational and judgmental errors made by the decision maker, as well as measurement and specification errors. The errors are assumed to be independent and identically distributed across all choices (also choices of the same individual), such that any systematic tendency within the individual is captured by the VTT. This implies that y i1... y is are independent conditional on VTT and the value functions. More specifically, we assume that the errors are iid logistic random variables with scale parameter µ. 7 The model in (5) applies an additive error term to the difference between the logarithms of V T T is and V c,is V t,is. Clearly, this is just one of several possible formulations of (3). Another approach would be to apply the error term directly to the difference, without logs, or to apply it directly to the formulation in (3) (yielding a 6 This formulation demands that VTT is positive, which follows from the assumption that both time and cost are non-goods such that they have negative marginal utility. 7 The scale parameter is inversely proportional to the standard deviation of the ε is s. 10

11 model in preference space rather than willingness-to-pay space, cf. Train and Weeks (2005)). In a deterministic setting, all three models would be equivalent, but for fixed distributional assumptions of the ε s and VTT, the models are different. Our choice of the model in (5) is motivated by the results in Fosgerau (2007), who use similar data, and by the fact we can apply the fixed effects logit estimator and avoid making restrictive assumptions regarding the unobserved heterogeneity in VTT. This is a consequence of the specification in (5) and the chosen parameterisation of VTT, which we discuss in the following section. 3.4 Parameterisation of VTT and loss aversion The final part of our econometric model is the specification of the parametric forms of VTT and the η s. As we do not wish to restrict the sign of the η s, we assume they are linear functions of the parameters: η t,is = γ t z t,is η c,is = γ c z c,is (6) where z t,is and z c,is are vectors of covariates related to the individual and the choice situation (both contain a constant), and γ t, γ c are parameter vectors. For notational convenience, we define t is := t 1,is + t 2,is 2t 0,is, c is := c 1,is + c 2,is 2c 0,is, Due to the experimental design, either c 1,is or c 2,is will be equal to c 0,is. Using this with (4) and (6), we can write log V c,is as log V c,is = log c is + γ cz c,is S(c is ). (7) We can do the same in the time dimension, at least when choice quadrants are determined relative to the base trip (t 0,is, c 0,is ) rather than (t 0,is + 2, c 0,is ). In this case either t 1,is or t 2,is will be equal to t 0,is, and we get 8 log V t,is = log t is + γ tz t,is S(t is ). (8) Based on Fosgerau (2007), VTT is parameterised as: { exp(β x is + u SP i 1 ) for s SP 1, V T T is = exp(β x is + u SP i 2 ) for s SP 2. where β is a parameter vector, x is is a vector of covariates related to the individual and the choice situation, and u SP i 1, u SP i 2 are individual-specific random terms that 8 For people with t 0,is 10 minutes, choice types are determined relative to (t 0,is + 2, c 0,is). In this case, log V c,is is as above, but log V t,is = log 2e γ t z t,is (t is 2)e γ t z t,iss(t is 2), since the first 2 minutes of a travel time reduction are valued at rate e η t and the remaining minutes are valued at rate e η t. 11 (9)

12 represent unobserved heterogeneity. We allow for different (potentially correlated) unobserved effects in the two experiments. In general, to estimate the parameters γ t, γ c, and β we would have to make some assumption regarding the distribution of u SP i 1 and u SP i 2 ; specifically regarding their correlation with each other and with x is. However, the fixed effects estimator that we apply does not require any such assumptions in order to be consistent. Using the parameterisations specified above in (7), (8), and (9), we can rewrite (5) as: 9 y is = 1 β x is + u SP 1 i + log t is log c is + γ tz t,is S(t is ) γ cz c,is S(c is ) > ε is, (10) for a choice s in SP1. (For a choice in SP2, u SP i 1 is replaced by u SP i 2.) The advantage of the u i s appearing linearly in the model is that they drop out of the likelihood function of the fixed effects logit estimator, such that we do not have to make any assumptions regarding the distribution of the unobserved effects. This is the subject of the next section. 3.5 Estimation To estimate the parameters in our model, we apply the fixed effects logit estimator from Chamberlain (1984). Here the likelihood contribution of individual i is the probability of choosing the sequence (y i1...y is ), conditional on the covariates and on the number of times the fast alternative is chosen in each experiment. It can be shown that since the unobserved effect (u SP i 1 or u SP i 2 ) appears linearly in (10), it drops out of the likelihood function. The same goes for all factors in x is that are invariant across observations from the same individual, since x is also appears linearly in (10). This means that x is can only contain covariates that vary within the individual the approach is therefore not suitable when the purpose is to estimate the distribution of VTT, as many important determinants (income, age, trip purpose etc.) are not identified. However, the variables of interest here, z t,is and z c,is, do not appear linearly in (10), since they are multiplied by the sign variables S(t is ) and S(c is ). Because the sign variables vary within the individual, the product variables do not drop out of the model, and hence the parameters γ t and γ c are identified. Beside the identification advantage discussed in section 3.2, the great advantage from using this approach is that because the unobserved effects u SP i 1 and u SP i 2 drop out of the model, we do not have to impose any assumption on their distribution and their relation with x is. The fixed effects logit estimator is consistent and asymptotically normal, as long as y i1... y is are independent conditional on VTT and the value functions, whereas the often applied random effects estimator that integrates 9 For observations with t 0,is 10 minutes (10) must be replaced with β x is + u SP 1 i + log 2e γ t z t,is (t is 2)e γ t z t,iss(t is 2) log c is γ cz c,iss(c is) > ε is. 12

13 out the u i s relies on independence between the u i s and x is to be consistent. This is a very strong assumption to make. The advantages come with a cost, naturally. Since we condition on the number of times the faster alternative is chosen, the individuals who consistently choose the faster alternative (or the cheaper alternative) do not contribute to the likelihood function or rather, their likelihood contribution does not depend on the parameters. Hence we throw away information by not using these in the analysis, reducing the sample to 1563 individuals and 14,813 observations. Further, by conditioning on the number of times the faster alternative is chosen, we use only variation between observations from the same person, i.e. we ignore information from interpersonal variation. This implies that the fixed effects estimator will have a larger variance than a correctly specified random effects estimator. 3.6 Covariates We now discuss which covariates to include in the model. We first consider the covariates affecting loss aversion (z t and z c ), and then the covariates affecting VTT (x) Covariates in z t and z c The results in Johnson et al. (2006) and Gächter et al. (2007) suggest to include age, income, and occupation in z t and z c. For a flexible age pattern, we use age variables of first, second and third power: age-50, (age-50) 2 /100 and (age-50) 3 / is close to the average sample age, so with this formulation the age terms capture variation around the average. Similarly, we use a demeaned income variable, namely the logarithm of personal net income minus 12. To control for occupation, we use dummies for students, retired, and people out of work (base group: workers and selfemployed). We also include a constant, a gender dummy and two education dummies indicating high school and higher education as the latest finished education (base group: primary/lower secondary school or vocational training). Other than socioeconomic characteristics, we allow for the familiarity with the reference to affect the degree of loss aversion. A possible effect of not being familiar with the reference, is that some time differences that are intended as gains are really evaluated as losses, and vice versa, thus obscuring the difference between gains and losses and leading to lower estimated loss aversion. Several factors may affect the familiarity with the reference. First, we consider it relevant to distinguish between the two experiments, because i) the first experiment concerns a transport mode the respondent has shown to prefer to the mode in the second experiment, and ii) the reference of the second experiment is of a more hypothetical nature. We therefore include a dummy for the second experiment. Another important indicator of familiarity is how often the respondent makes a trip like the base trip or alternative base trip. Trip frequency is a categorised variable with levels daily (at least 4-5 times a week), weekly (once a week or a couple of 13

14 times a week), rarely (more seldom) and never 10. We include dummies for daily, weekly and never (base group: rarely), and make the dummies experiment-specific to control for interaction between experiment and frequency. Another element that may influence the familiarity with the reference, is whether the reference travel time t 0 is a good indicator of the regular travel time on the trip. To control for this, we include the share of congestion on the trip (only recorded for car trips, i.e. in experiment 1) and two delay dummies: One for delay due to regular congestion, which the respondent anticipates, and one for extraordinary delay, which cannot be predicted. Congestion is measured as the additional travel time due to other traffic, relative to total travel time. Delays and congestion indicate a higher variability of travel time, and thus a higher tendency for the base trip travel time to be different from what the respondent considers normal Covariates in x As discussed in section 3.2, VTT must be allowed to depend on the size of the time change t = t 1 t 2 to avoid bias in the estimated loss aversion. We therefore include the logarithm of t in the vector x. By construction, t 1 < t 2, so t is always negative. However, due to preference curvature we do not expect VTT to be the same when evaluating a time loss and a time increase. To model that, we assume separate coefficients for time losses and gains. Most of the explanatory variables often used to explain VTT, such as income and trip purpose, are invariant across observations from the same individual, and thus cannot be applied. For the same reason, we cannot apply a constant term. Table 2 in the Appendix lists summary statistics of the variables used in the analysis. 10 Since the reference in the first experiment is a real trip, frequency cannot be never in this experiment. 14

15 share of obs choosing faster alternative v (in DKK per hour) Figure 2: Share of observations choosing faster alternative as a function of v for time losses (solid curve) and time savings (dotted curve) share of obs choosing faster alternative v (in DKK per hour) Figure 3: Share of observations choosing faster alternative as a function of v for cost increases (solid curve) and cost savings (dotted curve). 15

16 4 Analysis and discussion 4.1 Loss aversion in the raw data Before estimating our model, we take an initial look at the data without a model. For a given value v, we consider how often the faster alternative is chosen in a choice situation with c t = c 1 c 2 t 1 t 2 = v, depending on whether the choice represents a gain or a loss in the time and cost dimensions. Figure 2 shows the share of observations choosing the faster alternative as a function of v separately for time losses and time savings, while Figure 3 shows the same for cost increases and cost savings. In both figures, the curves are moving averages of the raw data, since the latter are very noisy. Looking at Figure 2, we see people are more likely to choose the faster alternative, if they are faced with a time loss, compared to a time saving. This indicates that the unit value assigned to a time loss exceeds the unit value assigned to a time saving. Similarly, we see in Figure 3 that people are more likely to choose the cheaper alternative, if faced with a cost increase, compared to a cost saving, indicating that the unit value of a cost increase exceeds the unit value of a cost saving. Hence, from the raw data it is obvious that losses are valued at a higher unit value than gains. We cannot tell how much of the difference that is caused by loss aversion, because we do not control for preference curvature, but the difference seems rather substantial. 4.2 What affects loss aversion? We ran an initial estimation with all the covariates mentioned in section 3.6 (Model 1). 11 The results are given in Table 3 in the Appendix, and are used as a screening device to exclude very insignificant variables. The dummies for students and retired turn out to have no explanatory power, presumably because they are entirely explained by the age variables. Further, educational variables and delay variables do not affect loss aversion in the cost dimension, and there is no difference between expected and unexpected delay in the time dimension. We therefore remove student and retired dummies from z t and z c, remove educational and delay variables from z c, and combine expected and unexpected delay in a single delay dummy in z t (the joint LR test statistic for these reductions has a p-value of 0.998). Table 1 shows the estimated parameters of the resulting model (Model 2), with ***, ** and * indicating significance at the 1, 5 and 10 percent level, respectively. To illustrate the size of loss aversion, we look at a representative person in SP1 with a representative base trip: This person is 50 years old, a male worker with a log income equal to 12, whose highest education is primary/lower secondary school or vocational training. The base trip has no congestion and is not delayed, and he makes it at most a couple of times a month. In SP1, this person has η t = and η c = This means that a time loss is valued e 2ηt = 2.4 times higher than 11 All estimations are carried out in Ox (Doornik, 2001). 16

17 Table 1: Estimation results (Model 2) Fixed effects logit estimates Parameter Estimate Std.Err. z-value β s corresponding to: log t x {t>0} *** log t x {t<0} *** γ t s corresponding to: constant *** SP *** age ** (age-50) 2 / *** (age-50) 3 / education = high school * education = higher ** female * log(income) * out of work SP1 x {freq=daily} SP1 x {freq=weekly} SP2 x {freq=daily/weekly} ** SP2 x {freq=never} share of congestion *** delay γ c s corresponding to: constant *** SP age (age-50) 2 / (age-50) 3 / female log(income) out of work ** SP1 x {freq=daily} SP1 x {freq=weekly} SP2 x {freq=daily/weekly} SP2 x {freq=never} *** share of congestion ** µ (error scale) *** Log likelihood value Number of individuals 1,563 Number of observations 14,813 17

18 an equally sized time saving, while a cost increase is valued e 2ηc = 1.5 times higher than a cost saving of the same size. If we did not control for loss aversion, we would observe a WTA-WTP ratio of e 2ηt+2ηc = 3.7. Hence the representative individual exhibits loss aversion in both the time and cost dimension, but has a higher loss aversion w.r.t. time than w.r.t. costs. This is in accordance with the empirical findings in Horowitz and McConnell (2002), where the WTA-WTP gap is large for goods that are not often traded in a market, and small for market goods and money Determinants of loss aversion in the time dimension Loss aversion in the time dimension increases with age up to around 55 years, and then declines rapidly. The effect of the third order age term is that the initial increase in loss aversion is slower than the decrease after 55 years, but the term is not significant. People with high school or higher education as latest finished education have lower loss aversion than people with lower secondary school or vocational training though for high school, the effect is only significant at the 10 percent level. Still in the time dimension, we find that women are less loss averse than men and that people with a higher income are less loss averse (both effects are significant at the 10 percent level). Occupation makes no difference. When it comes to trip frequency, there is not much difference between frequent and infrequent trips in SP1, where the reference is a recent trip. A LR test for removing the two frequency dummies for the first experiment is accepted with a p-value of For SP2, where the reference is not necessarily a recent situation, people who know the reference situation better have significantly higher loss aversion. People who never experienced the base trip have the same loss aversion as people who rarely make the base trip (base group). As indicated by the experiment dummy, loss aversion is generally lower in the second experiment, perhaps reflecting that the reference is more hypothetical and thus less integrated in choice behaviour this could have the effect we mentioned in section 3.6.1, with time differences intended as gains being evaluated as losses, or vice versa. Another potential explanation is that people during the experiments become more experienced with trading time for money some empirical studies have found that such learning decreases loss aversion (List, 2005; Plott and Zeiler, 2005). However, the learning effect found in the literature is a result of people adapting their choice behaviour to the experienced consequences of their choice. Since our experiments are purely hypothetical, and do not allow people to experience the effect of their choices, any learning effect is likely to be small. Finally, people whose base trip is delayed or very congested, have lower loss aversion, though the effect of delay is not significant. This could indicate that the reference travel time of the experiment does not coincide with the respondent s (true) reference point, when the base trip is congested. It may also be, however, that people 18

19 travelling on congested roads indeed are less loss averse than others: likely they have more experience with variable travel times, and this experience may decrease loss aversion, in line with the learning effect theory Determinants of loss aversion in the cost dimension Regarding loss aversion in the cost dimension, many of the variables are insignificant. Loss aversion increases with age up to years and then becomes roughly constant, but the variables are neither individually or jointly significant (the LR test of removing all three age variables has p-value 0.41). Gender and income also have no significant effect, but the out-of-work dummy significantly increases loss aversion. This may however be an effect of preference curvature, rather than loss aversion: The out-of-work dummy identifies a low-income group and may capture that this group has a decreasing marginal utility of money. Note that the occupational situation is an indicator of current income, while the reported income in the experiment is lagged one year thus the occupational dummy may actually be a better measure of current income. The effect of trip frequency is very similar to that in the time dimension: Loss aversion does not vary significantly with frequency in the first experiment, while in the second experiment people who know the reference situation better have higher loss aversion (though not significantly). Contrary to the time dimension, people who never make the alternative base trip have significantly lower loss aversion w.r.t. cost than the base group. Loss aversion is generally lower in SP2 than SP1, as in the time dimension, but the difference (represented by the SP2 dummy) is not significant. Finally, people with congested base trips have significantly lower cost loss aversion. 4.3 The effect of t on VTT We find that the size of the travel time saving t has a significantly positive effect on the VTT, both when considering time losses and time gains. This is not consistent with Hicksian preferences, where the marginal rate of substitution is a monotonic function of the amount of a good, but it is a common finding in value of travel time studies (Fosgerau et al., 2007; Mackie et al., 2003; Hultkrantz and Mortazavi, 2001; Gunn, 2001), where it is often seen that the VTT increases with t up to some point and then becomes constant. The effect could stem from non-hicksian preference curvature, if people consistently value small time savings at a lower unit rate than larger ones. It could also stem from curvature in the value functions. De Borger and Fosgerau (2008) show that if the true value functions are non-linear and the value function for cost bends more than the value function for time (i.e. it is more concave in the gain region and more convex in the loss region), then failing to account for the non-linearity leads to an observed VTT that is increasing in t. With the current data, we are not able to identify the two types of curvature 19

20 separately, so we do not know how much of the effect of t is due to non-monotonic preferences, and how much is due to value function curvature. 5 Conclusion In this paper, we investigated how loss aversion w.r.t. travel time and travel cost in two stated preference experiments varies with individual characteristics and between the experiments. Overall, we found significant loss aversion in both the time and cost dimensions for a representative individual, time losses are valued 2.4 times higher than equally-sized time savings, while cost increases are valued 50% higher than cost savings of the same size. This corresponds to a WTA-WTP ratio of 3.7. We found that loss aversion in the travel time dimension varies with age, education, income, and gender, but found no significant effect of occupation. We found a significant negative effect of congestion on loss aversion in both time and cost dimensions: This may be a learning effect in line with the evidence from List (2005) and Plott and Zeiler (2005), because people who travel in congested circumstances are more used to variable travel times. It could also, however, be an effect of the experimental design: We define losses and gains with respect to the respondent s observed travel time on a given day, but if the journey is very congested, this observed travel time may not coincide with the respondent s reference point (his perception of the normal travel time). When comparing an experiment with a real-life reference point and an experiment with a more hypothetical reference point, loss aversion is lower when the reference is hypothetical. With the hypothetical reference, people who do not know their reference well, because they rarely make the alternative base trip, have lower loss aversion. These two findings suggest that loss aversion depends on how established the reference is in the mind of the choice maker: People tend to be more loss averse when the reference is well established. 20