DOES PART-WHOLE BIAS EXIST? - AN EXPERIMENTAL INVESTIGATION by

Ian Bateman, Alistair Munro, Bruce Rhodes, Chris Starmer

and Robert Sugden

CSERGE Working Paper GEC 96-03

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DOES PART-WHOLE BIAS EXIST? - AN EXPERIMENTAL INVESTIGATION by

Ian Bateman

Centre for Social and Economic Research on the Global Environment

University of East Anglia and University College London

and

School of Environmental Sciences University of East Anglia

Alistair Munro (corresponding author)

Bruce Rhodes, Chris Starmer and Robert Sugden

School of Economic and Social Studies,

University of East Anglia

Acknowledgements The Centre for Social and Economic Research on the Global Environment (CSERGE) is a designated reserch centre of the UK Economic and Social Research Council (ESRC). ISSN 0967-8875

Abstract In the context of contingent valuation, part-whole bias exists when the sum of the valuations placed on the parts of a commodity exceed that for the whole. The existence of part-whole bias has been disputed hotly, as has its cause, but most authors agree that it is associated with non-marketed goods valued in hypothetical markets. This paper reports on an experiment into part-whole bias, designed to test an alternative hypothesis: that the bias is also present with private goods traded in real markets. Employing vouchers for parts of a restaurant meal and using an incentive compatible procedure, values of the parts and the whole were elicited. The value of the parts consistently exceeded that of the whole, providing support for the existence of part-whole bias. Given that this occurred in a real trades market using private goods, such results indicate that part-whole bias may not be confined to the realms of hypothetical markets and may be a more widespread economic phenomena. Certainly, one implication of this result is that it cannot now be assumed that further refinement of the contingent valuation method will lead to the eradication of part-whole bias.

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1. Introduction Part-whole1 bias (PHB) occurs in the context of valuation when it appears that the sum of the valuations placed by an individual on the parts of a good is larger than the valuation placed on the good as a whole. In the wake of the Exxon Valdez oil spill, PHB has recently emerged as the focus of debate on the validity of the contingent valuation (CV) method - particularly when the method attempts to measure non-use values. The argument runs through Kahneman & Knetch (1992), Smith (1992), Harrison (1992), Boyle et al (1994) as well as through the interchanges in Hausmann (1993) and between Hanemann (1994) and Diamond and Hausman (1994), for instance. While views clash on whether the existence of PHB has been convincingly demonstrated (Carson and Mitchell (1993), and on whether the bias is eradicable, there is seeming agreement that the immediate source of PHB is what Mitchell and Carson (1989) term, amenity misspecification - a conflict between the experimenter's notion of the commodity and that of the subject. The disparity between experimenter and subject occurs either because there is confusion over the scope of the good or because the commodity is seen by the respondent to be a symbol of good causes and wider concerns. One view, (e.g. Kahnemann and Knetch (1992), Diamond and Hausman (1994)) sees the confusion over the nature of the good as an inevitable consequence of applying the CV method to public goods lacking use values. On this basis, no refinement of the method will eliminate the `warm-glow' effects which are one possible source of PHB. However, while The experimental investigation of PHB reported here had two objectives. Its main goal was to provide a more rigorous test of the existence of PHB than those produced to date, using an incentive compatible procedure in which valuations of the whole, the part and its complement were estimated. At the same time, we were concerned to explore a third possible perspective on the bias. PHB, like willingness to pay/ willingness to accept disparity (see Knetch and Sinden (1984) for example) might be an indication of fundamental problems of coherent subjective valuation and, as such, present with the private goods employed in the experiment (vouchers for parts of a meal at a local pizza restaurant), as well as the public goods more familiar in contingent valuation studies. 1 The terms `part-whole' and `embedding' are employed in the cognitive psychology literature dealing with the perception of visual parts and wholes, where evidence suggests that one hemisphere of the brain is responsible for perception of wholes, while another deals with the parts of an object (Robertson & Lamb (1991), Tversky and Hemenway (1984)).

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2. Background 2.1 Theory Suppose an individual has an endowment of a group of goods, x (so that x is a vector) and spends y on all other goods. Let x¢, x¢¢ represent two other levels of endowment, with x < x¢ < x¢¢, so that (x¢¢-x¢) and (x¢-x) are the `first' and `second' parts respectively. Willingness to pay for the first part is defined by: U(x,y) = U(x¢,y-wtp1), where U is a utility function representing the individual's preferences. Willingness to pay for the second part, given that the consumer has the first part is: U(x,y) = U(x¢¢,y-wtp1-wtp2) Finally, willingness to pay for the whole is, U(x,y) = U(x¢¢,y-wtp3), Note that in field trials it is often not possible to adjust for income effects, so that wtp2 would be approximated by,2 U(x¢,y) = U(x¢¢,y-wtp2¢). PHB proper exists if the elicited values for wtpi, I = 1, 2, 3, are such that,

The method employed here to test the hypothesis finds wtp for one part and wtp for the whole, from which a wtp

for the second part, given the first part, can be derived. If this implicit value for the wtp of the second part is less than the actual wtp for the second part, then PHB exists (in the sense of (1). Formally, the bias exists if,

In field trials, however, it is not normally possible to obtain directly a figure for

wtp2. Researchers (e.g. Kahneman and Knetch (1992), Boyle et al (1994)), have therefore had to fall back on,

as a test for the existence of PHB and this creates the scope for dispute over whether

results truly indicate the presence of the bias, not least because the null hypothesis then becomes `part-whole bias', rather than `no bias'.

2 If demand for x is normal, then this leads to an overestimate of wtp2 which may yield the conclusion that part-whole bias exists when it does not.

wtp > wtp + wtp 321 0

wtp < wtp - wtp wtp Implicit 2132 ≡ 0

0. wtp - wtp but wtp wtp 1331 ≈≤ 0

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There is seeming agreement over the immediate causes of the bias, if it actually exists. First, the researcher may fail to specify the good in terms clear enough for the subject not to confuse it with the parts embedded within it or wholes of which it may be a part. For instance, the subject may not be informed that flood prevention forms part of a disaster prevention programme, or might not realise that disaster prevention includes items other than flood prevention, without some clarification from the researcher. More worryingly, CV is often applied to goods which are (i) public and (ii) have significant non-use values, both of which may produce what Mitchell and Carson (1989) term symbolic misspecification. However clear the researcher is on the scope of the good, the subject may perceive both the parts and the whole as symbols of `charity' or `deserving cases' or `sources of the warm glow of giving' and the invitation to state a value as a symbolic opportunity to purchase `moral satisfaction'. It is this latter source of bias which, according to some writers, is not potentially amenable to refinements of the CV methodology. As we stated in the introduction, it is also conceivable that part/whole bias may also arise as a result of problems of valuation even with private goods. For instance, the bias is compatible with (though not a necessary implication of) Tversky and Kahneman's (1991) theory of Loss Aversion. Figure 1 illustrates the wtp case, showing the pattern to which reference point dependent indifference curves must conform if part/whole bias is to be produced. The labels x, x¢ and x¢¢ refer to the consumption levels described above, except that, for simplicity, we suppose that the parts can be measured along the same dimension. The curve joining A, B and C is the indifference curve for someone at A. The theory of Loss Aversion states that, for an agent at B, and for the section heading back towards A, the indifference curve through B should be steeper than ABC. It has no specific prediction for the part of the indifference curve beyond B, in the direction of C. Part/whole bias requires that this section must lie below ABC as it does in the diagram, so that the sum of wtp1 and wtp2 exceeds wtp3. Alternative reasons for anticipating some form of part/whole bias, are the phenomena of attribute (Weber et al (1988)) and event splitting found in studies of decision making under uncertainty (Starmer and Sugden (1993) for example). In the latter, the splitting of an event into two identical events with a combined probability equal to the probability of the combined event, leads to a rise in the value placed by subjects on lotteries containing the event. In the former, which is perhaps more relevant here, splitting an attribute of a good (e.g. the performance of a car) in two (e.g. acceleration and top speed) leads to a higher valuation being placed on the good. Given these results, there is a reason to think that PHB could occur with private and non-hypothetical goods.

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2.2 Evidence Kahneman and Knetch (1992) report a 1986 telephone poll in which Toronto residents were asked to state a wtp to preserve fish stocks either in the small proportion of Ontario lakes around Muskoka, or for all the lakes in the province. The median wtp for the good `all lakes' was only slightly higher than that for the small proportion of lakes. Their test for bias was therefore equation (3). As a result, they concluded that PHB existed, and suggested that it arose because subjects saw both parts and whole questions as interchangeable vehicles for charitable giving. The methods employed in their study have been widely criticised (see Smith (1992) and Harrison (1992) in particular). In particular, the telephone poll left open the possibility that the commodities had not been described in sufficient detail to overcome amenity misspecification bias. Boyle et al (1994) investigated whether CV estimates were sensitive to marginal changes in an environmental commodity: the prevention of migratory waterfowl deaths in waste oil holding ponds along the Central Flyway of the United States. Subjects were located in Atlanta, Georgia, so as with the Kahneman and Knetch study, non-use values were elicited for a public good; in this case, for the prevention of 2,000, 20,000 or 200,000 bird deaths. The last of these is approximately 2% of the waterfowl population in the Central Flyway. Investigation was carried out at two malls in Atlanta, Georgia, with higher petroleum prices as the payment vehicle and an open-ended question, employed to avoid starting point bias. Subjects in the sample of 855 were shown maps of Central Flyway and given verbal descriptions of waste oil pools and the problem itself. Mean willingness to pay for the three groups were $80 (for the prevention of 2,000 bird deaths), $78 (20,000) and $88 (200,000). The test used for the existence of PHB was therefore the same one employed by Kahnemann and Knetch. Even after controlling for differences in the composition of the sub-samples there was no significant difference in the answers provided by the three sub-groups. The authors provide a number of alternative reasons for the results, including a possibility that the respondents focused on other features of the commodity (e.g. the number of waste oil ponds to be covered which was the same in each case) and that the marginal utility of bird-death prevention was zero above 2,000. They conclude that the marginal valuation of non-use value is, at best, very difficult. Diamond and Hausman (1993) also conclude in favour of the existence of PHB, after examining willingness to pay for the preservation of three wilderness areas, findings supported by Kemp and Maxwell (1993), (willingness to pay for oil spill clean-ups embedded in a longer list of government programmes), and Schulze et al

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(1993), (partial clean-up for contaminated land in Montana embedded in complete clean-up). Against this, Carson and Mitchell (1993) find a significant difference between Phila-delphia resident's willingness to pay for improvements in the Monongahela River quality and wtp for the same rise in the quality of river water nationally. They also report evidence from other studies (on mining impact in Australia and low-level risk in drinking water) which do find significant differences between parts and whole. In addition, they argue that, irrespective of the defects of the methods employed, many previous researchers did discover significant or almost significant differences between parts and whole. They draw attention to an investigation conducted by Schkade and Payne (1993) in parallel with Boyle et al's (1994) examination of wtp for the prevention of migratory fowl deaths. Using the same questionnaire and the same location, Schkade and Payne asked subjects to state their thoughts as they responded to questions. In this context, Schkade and Payne report that the mean wtp for the prevention of 200,000 deaths was actually twice the value for 2,000 deaths. Similarly, Carson and Mitchell (1993) also dispute Kahnemann and Knetch's claim of no significant difference between median wtp for the parts and the whole of Ontario lake system. The unresolved state of the debate suggests two main problems with existing experimental designs. First, the currently employed test for the existence of PHB (equation (3)) is not exact. It allows different researchers to interpret the same data in divergent ways, so that it is always possible and indeed correct, to argue that the existence of PHB has not been formally demonstrated. Secondly, in instances where PHB has apparently been found, it has not been possible to distinguish between the potential explanations. Was the closeness of the reported values of the part and the whole due to `warm glow' answers or due to confusion in the minds of the subjects over the scope of the good or was it the result of deeper problems of valuation which would also appear with private goods?3 The absence of clear answers thus far points to the potential value of an experimental approach. 3. Experimental Design 3.1 Outline In outline, the experiment for the wtp version of PHB was as follows: one group of subjects was given an endowment of money, then each individual was asked for his or her maximum wtp for one part and then the whole, using the Becker deGroot Marschak (BDM) (1962) mechanism. The production of wtp for the first part and the whole produced an implicit wtp for the second part, given ownership

3 Some clues are provided in Schkade and Payne's (1994), study. However as already noted, this particular piece of research has been claimed by both camps.

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of the first part. A second group of subjects were endowed with the first part of the good and the wtp for the remaining part elicited. The figures obtained for implicit wtp were then compared with the actual wtps for the second part, in the manner described in equation (2). By asking the questions of different sub-samples in this way, two main problems are avoided. First, asking all three questions of one sample would invite subjects to calculate the third answer on the basis of their answers to the first two questions. Asked the two parts initially, the subjects could respond with an heuristic which totted up their wtp figures to give wtp for the whole. This is not a compelling drawback if the heuristic reflected actual preferences. However, the experiment would not be able to distinguish between agents who genuinely did have consistent preferences and those who did not and simply used a heuristic which they might not employ in non-experimental conditions. More fundamentally, asking all three questions of one sample destroys the incentive compatible nature of the BDM mechanism. If one person faced all three questions and the second question was selected in the random process at the end of the experiment, the subject would receive the part from the first question and lose their reported wtp for that part. They would then face the revelation mechanism for question 2. This, though, would give them an incentive to under-report wtp in the first question, making it imperative that the question for wtp2 is posed to a sub-group different to the one which faces questions wtp1 and wtp3. 3.2 The Experimental Procedure The computer-based experiment took place in the summer and autumn of 1993 using the experimental economics laboratory in the Economics Research Centre, University of East Anglia (UEA), Norwich. Altogether 153 subjects took part, plus 43 in the pilot. Most of the subjects were undergraduates or postgraduates at UEA, members of staff or their families, recruited through advertisements posted across the campus. Each run of the experiment had up to eight subjects, all sat at their own terminals which were screened from the other workstations in the room. Subjects were instructed not to talk during the course of the experiment, except to ask questions, which were encouraged. They were guided through the experiment by an experimenter armed with a script and by the information which appeared on the screens in front of them. The commodities were vouchers for either a free main course (Red voucher) at a local restaurant, Pizza One, Pancakes Too!,4 or a free dessert with cream plus coffee (Blue voucher). The whole was therefore vouchers

4 As the name suggests, the restaurant serves mainly pizzas and pancakes as well as some of the more popular pasta dishes.

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for a complete meal, while the parts were the vouchers for individual courses5, thereby reducing the scope for confusion about the nature of the relationship between parts and whole.6 As the goods were private, important consequence of this design was that any PHB found could not be explained in terms of `warm-glow' or other, similar effects. At the start of the experiment, the vouchers were explained to the participants in some detail so that they were all aware of the terms and conditions. The participating restaurant, which is in the centre of the City of Norwich, was known to most subjects, but to ensure familiarity, a full menu for the restaurant was placed in front of each participant. They were invited to study the menu and time for this was allowed at the beginning of the experiment. A menu was left beside each person so it could be consulted at any time during the experiment. All prices were on the menu and subjects were given pencils and paper in case they wished to make calculations. 3.3 Initial Endowments and Question Order At the beginning of each run of the experiment, individuals were allocated at random by the computer into four sub-groups, defined by the possible endowments of vouchers: none, Red, Blue and both. Participants were then given their endowments of cash and vouchers (where applicable). Each possible endowment creates three possible trades providing the basis for the questions asked. Table 1 summarises the cash and voucher endowments of each group and the questions they faced subsequently. The pilot experiment had suggested that willingness to pay for the Red and Blue vouchers (given only a cash endowment) were approximately £2.00 and £1.50 respectively. In the main experiment, the random number distributions which generated the cash endowments were adjusted in line with these amounts, for subjects receiving one or more vouchers (see Table

5 Note that individual items within the Main Course and Dessert sub-sections had differing prices. Main courses were priced between £3 and £5, while Dessert plus Cream and a coffee could cost between £2 and £3.50. There was therefore no reason why willingness to accept (or willingness to pay) should converge across subjects. Vouchers could not be exchanged for cash, but no further purchase at the restaurant was necessary. The conditions of use were printed on the vouchers and they were explained to the group as a whole. 6 This could be viewed as a weakness of the experiment, since in most CV studies the whole is a simple extension of the part (e.g. many lakes versus one). However, if part-whole bias was found in a setting such as our experiment where confusion was so difficult, it is also likely to be found in cases more commonly examined in CV.

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Table 1: Summary of Sub-Group Endowments and Questions

Voucher Endowment

Cash Endowment

Summary of Questions Faced

None £Random, drawn from a rectangular

distribution, limits: £6 and £10.

wtp for a Blue voucher (wtpBlue) wtp for a Red voucher (wtpRed) wtp for both vouchers (wtpBoth)

Red £Random, drawn from a rectangular

distribution, limits: £4 and £8.

wtp for a Blue voucher (wtpBlue/Red) wta for the Red voucher (wtaRed/Red) worst terms to swap Red voucher for

Blue (swap/Red)

Blue £Random, drawn from a rectangular distribution, limits: £4.50 and £8.50.

wtp for a Red voucher (wtpRed/Blue) wta for the Blue voucher

(wtaBlue/Blue) worst terms to swap Blue voucher for

Red (swap/Blue)

Both = Red + Blue

£Random, drawn from a rectangular distribution, limits: £3.50 and £6.50.

wta for the Red voucher (wtaRed) wta for the Blue voucher (wtaBlue) wta for both vouchers (wtaBoth).

As Table 1 indicates, everyone faced a series of three questions. To avoid wealth effects, the subjects were informed at the start of the experiment that only one of these questions would be faced for real. At the end of the experiment one question was selected at random by the computer, their answer to it was retrieved from the computer's memory, then that question was asked for real. To examine the possibility of order effects (which Kahneman and Knetch (1992), and Samples and Hollyer (1990) among others report), the subgroups were further subdivided by randomising the order in which subjects received their questions. Not all of the six possible orders for asking three questions were utilised. We were particularly interested in checking whether the order of the part and whole might influence responses and whether the order of the parts might influence responses. The combinations omitted were therefore the two where the `whole' question was sandwiched between the two `part' questions.

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3.4 The questions Three types of questions were faced by the participants: opportunities to buy a voucher of a type they did not have; opportunities to sell one they already owned and opportunities to swap one voucher for another. For buying opportunities a typical question read: The next screen will offer you an opportunity to BUY the BLUE voucher. You should note that:

The BLUE voucher entitles you to a choice of any DESSERT from the menu of Pizza One Pancakes Too (including cream or ice cream) plus COFFEE.

You will be asked to state the MAXIMUM amount of money that you are willing

to pay in order to BUY the BLUE voucher. Followed by, You are being offered a chance to BUY the BLUE voucher. You must now tell us the MAXIMUM amount of money that you are

willing to pay. You should do this by using the UP and DOWN arrow keys on the keyboard.

By pressing the DOWN key, you can increase the amount you are willing to pay. By pressing the UP key, you can reduce the amount you are willing to pay.

The MAXIMUM amount that you are willing to pay is: £___ Press <ENTER> when you have set the maximum amount that you are willing to

pay. With selling opportunities, the relevant wording was changed to, You are being offered a chance to SELL the RED voucher. You must now tell us the MINIMUM amount of money that you are willing to

accept. With SWAP opportunities, the subjects were first asked to express a preference between their currently-held voucher and the other voucher. If they expressed a preference for the other voucher, then the next screen asked them their maximum willingness to pay to make the trade. If they preferred their currently-held voucher, then the next screen asked for their minimum willingness to accept compensation

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for making the trade.7 3.5 Practice questions and the BDM mechanism. Under the BDM procedure, the computer randomly generates a price. For wtp questions, if the subject's response exceeds the price, then the individual receives the good and pays the price generated by the computer. If the price generated is less than or equal to the reported wtp then no trade occurs. For minimum willingness to accept compensation (wta) questions, exchange takes place at the computer generated price, if the price set by the computer is higher than the subject's reported wta. Like the Vickrey auction, used for instance in Shogren et al (1994), the BDM mechanism is therefore incentive compatible. In order to acquaint the subjects with its incentive compatible nature, and to familiarise them with the questions the subjects faced (at least) two practice questions, for each of the three questions. The practice questions asked were identical in all respects to the real questions which followed, except the subjects were informed that the questions were just for practice and that, unlike the question asked for real, their answers were not potentially binding. After answering each practice question the subjects faced a screen, rather like a roulette wheel, around which were located sums of money, starting at £0 and rising in increments of £0.25 to £5.00.8 Sums were colour-coded in order to make it easier for subjects to understand the implications of their responses. Sums coloured green represented outcomes where a trade would be made, if the question were played out for real, given the terms set by the subject. Sums were marked red in regions where trade would not occur. For instance if a subject had stated that he or she was willing to pay £2.00 to acquire a Red voucher, all sums less than or equal to £2.00 would be marked in green and all higher sums in red. A `ball' then circled around the wheel and alighted at one sum at random. The subjects were then informed of what the consequences would have been if that question had been played for real and the ball had landed at that particular sum. Particular trouble was taken to make clear the nature of the mechanism, so that they were aware that if (for instance) their stated wtp was higher than the sum chosen from the wheel, then they would only have to pay the sum chosen on the wheel to acquire the voucher. After two practices at a particular question, the subjects were invited to practise

7 The full program for this experiment and the script used by the experimenters are available on request from the authors. 8 If subjects rounded up their answers to match the figures depicted on the wheel, this could lead to an apparent part-whole bias. For this reason, the lowest value on the wheel was randomised between £0.00, £0.05, £0.10, £0.15, £0.20 and £0.25.

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further if they wished to do so (few actually took up this opportunity). Once they had decided that they had practised sufficiently, they were warned that the question would be asked for real, the question was posed, their answers were stored without them facing the wheel and they moved on to the next question. After the three questions had been asked for real, the answer to one was retrieved at random and played out on the wheel. At that stage, any further trades of cash and vouchers necessitated by the outcome of the question played for real were executed between subjects and the experimenters and the experiment ended.

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4. Results We wished to investigate whether the sum of the values for the parts is greater than that for the wholes. If the level of the cash endowment received and the order of the questions had no effect on the reported valuations, then tests for PHB could have been done by comparing mean values from the different questions. Table 2 presents regression equations which tested for order and endowment effects. Note first that the data was recorded in the form of net payments from the researchers to the subjects, so that wtp is negative, wta positive etc. As can be seen, in most cases the equation had little or no explanatory power. In only one case was the hypothesis of no explanatory power rejected and even then, only just. Against this, in nearly every case, the endowment variable had the correct sign (assuming both vouchers to be normal goods) and, in three of the 12 regressions it was significant. Furthermore, order effects appeared to be significant in some cases. Given the overall nature of the regression results, we felt that comparison of the means was an adequate test of the part-whole hypothesis. Nevertheless, after comparing means on the assumption that the equations had no explanatory power, we then went on to make the weaker assumption that order and endowment effects were present and then compared predictions for parts and wholes on this basis. Mean valuations are given in figures 2 and 3. In figure 2, for instance, the diamond represents the routes by which individuals could pass from holding no vouchers to both, from none to one and from one to both. Alongside each vertex of each diamond, or in the interior, in the case of `Both' and `Swap' questions, mean valuations, sub-sample size and sample variances are reported. For example, the `Swap' answers show that, on average, subjects required £3.12 (36 in sub-sample, sample variance of 1.27), compensation for replacing a Red voucher with a Blue voucher (fig. 3), but were willing to pay £0.37 (sub-sample of 33, sample variance of 1.39), for the reverse trade (figure 2).

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Table 2: Regression Results for the Part/Whole Bias Experiment.

Voucher Endowment

Dependent Variable

Cash Endowment

Part/Whole Order

Red/Blue Order

R2 F

None (37)

wtpRed -0.09 (-0.46)

0.40 (0.95)

-0.36 (-0.75)

0.06 0.70

wtpBlue -0.20 (-1.09)

0.50 (1.24)

-0.41 (-0.92)

0.09 1.09

wtpboth -0.09 (-0.37)

-0.08 (-0.14)

-0.65 (-0.99)

0.03 0.34

Red (36)

wtaRed/Red 0.05 (0.42)

0.61 (1.98)

-0.35 (-1.08)

0.14 1.74

wtpBlue/Red

0.09 (0.78)

0.21 (0.69)

0.23 (0.72)

0.05 0.56

swap/Red 0.24 (1.76)

-0.09 (-0.26)

0.73 (1.98)

0.18 2.34

Blue (33)

wtpRed/Blue

-0.21 (-0.99)

0.07 (0.15)

0.54 (1.22)

0.09 0.96

wtaBlue/Blue

0.39 (1.99)

0.27 (0.64)

0.79 (1.97)

0.22 2.73*

swap/Blue 0.29 (1.44)

0.34 (0.75)

0.32 (0.78)

0.12 1.32

Both (37)

wtaRed 0.25 (1.10)

-0.53 (-1.03)

-0.30 (-0.65)

0.06

0.70

wtaBlue 0.28 (1.21)

-0.42 (-0.80)

-0.09 (-0.19)

0.05 0.58

wtaboth 0.60 (1.68)

-0.55 (-0.70)

0.98 (1.36)

0.12 1.50

Notes: The dummy variable for Part/Whole order took the value 1 if questions about the part were asked first and 0 if the question on the whole (or the swap question) was asked initially. Similarly, the Red/Blue dummy variable took a value of 1 if the question about the Red voucher was asked before that for the Blue. Figures in brackets below the coefficients are t statistics. The final column shows the F test for the hypothesis that the equation has no explanatory power (i.e. the coefficients in the equation are jointly zero). A * indicates rejection of the hypothesis at the 95% level. Dropping the least significant variables and re-estimating had little impact on the significance of the values in the final column.

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The path independent nature of consumer surplus implies that, in the absence of significant endowment and order effects, any route through the diamonds (including the SWAPs) should be consistent. We therefore tested for four possible PHBes using the method of equation (2): (i) for each subject endowed only with money, an implicit wtp for a Blue voucher given a Red voucher was calculated by taking the wtp for a Red voucher given none, from the wtp for both vouchers given none. The mean of this difference was then compared to the actual mean wtp obtained from the sample endowed with Red vouchers. (ii) An implicit wtp for a Red voucher given a Blue voucher was calculated in a similar manner and its sample mean compared to the sample mean wtp for Red vouchers given Blue voucher. (iii) For the wta questions, an implicit wta for a Red voucher given a Red voucher was calculated for subjects endowed with both vouchers, by subtracting wta for a Blue voucher given both from their wta for both vouchers given both. The mean of this variable was then compared to the sample mean of actual wta for the Red voucher for subjects endowed with a Red voucher. (iv) A similar comparison was carried out for the wta for a Blue voucher for subjects endowed with a Blue voucher.9 10 Table 3 summarises the results. In each case the implicit value of the voucher (i.e. that found by deducting one part from the whole) was less than the actual value. Thus, in each case the sum of the values of the parts exceeded that of the whole and the difference was significant at the 99% level in three cases and significant at the 95% level for the other one, for a one tailed test. In other words, there is strong evidence for the existence of PHB. We then tested the same hypothesis, but allowed for the possibility that the independent variables had some effect on declared valuations. First we estimated separate regression equations for implicit and actual valuations of parts. We then pooled the data, re-estimated and tested the hypothesis that the restrictions were

9 Note that the sample variances reported in figures 2 and 3 cannot be simply combined (in any manner) to obtain the standard error for the test statistic. A test for the bias involves a calculation of the form (z-y) - x, where x and y represent valuations of parts and z is the whole. A test statistic for the hypothesis that (z-y-x) = 0, calculated on the assumption that x, y and z are independent is inefficient, because the values of z and y were drawn from the same subjects, and so possibly correlated. The sample variance of (z-y) was therefore calculated directly. 10 The basis for our comparison of means was the claim that the regression equations reported in Table 2 had little or no explanatory power. However, it could rightly be argued that a regression for implicit valuations of parts might yield different results. Anticipating what is to follow, Table 4 reports regression results for the four implicit valuations, from which can be seen that these equations have little or no explanatory power as well.

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not binding using a Chow test (i.e. that the coefficients in the equations for actual and implicit valuation were the same). Table 3: Testing for Part/Whole Effects Using Sample Means

Actual mean

Implicit mean

Degrees of freedom.

t

WTP Blue given Red -1.32 -0.83 71 10.86**

WTP Red given Blue -1.82 -1.67 68 2.25*

WTA Red given Red 4.30 3.45 71 12.7**

WTA Blue given Blue

2.64 2.05 68 7.34**

*indicates rejection at the 95% level of the hypothesis that the means are drawn

from the same population (one tailed test); ** indicates rejection at the 99% level (one tailed test). Table 4 gives the results. It can be seen that all four restrictions were rejected at the 95% level and, in particular, for one willingness to accept equation, the difference was significant at the 99% level. The conclusion is that the implicit and explicit valuation are produced by different data generation processes. Taken with the evidence provided by Table 3, the hypothesis that there is no difference is once more rejected by the data: the sum of the parts continues to exceed the whole. However, there is a potential problem with estimation of the implicit valuation, because of the endowment variable employed. When estimating the implicit equations, the endowment variable was calculated as follows. For wtp values, the wtp for the other part was deducted from the original endowment; for wta values the wta for the other part was added to the original endowment, in line with standard consumer theory. This meant, though, that there would not necessarily be a random distribution of tastes at each endowment level. Suppose that, for example, that wtp is not correlated with endowment and that two subjects are both endowed with £10 and no vouchers. One places a low value on the vouchers and reports a wtp for a Red voucher of £1, while the other reports £4. In the implicit equation for wtp, one subject has an endowment of £9 while the other has £6. If values placed on the vouchers are positively correlated (which they were), we would obtain a spurious negative correlation between endowment and wtp in the estimated implicit equation. Thus, in general, this estimation procedure would bias the estimates for the implicit equations, since the error term would be correlated with one of the independent variables, calling for the employment of an

16

instrumental variable in the estimation procedure.

17

Table 4: Comparison of Implicit and Actual Valuations of Parts. 1. WTP for Blue voucher given Red voucher. Implicit Actual Combined Constant -1.39 -2.09 -1.83 Endowment 0.15 (1.86) 0.09 (0.75) 0.11 (1.37) Part/Whole Order -0.58 (2.32) 0.21 (0.68) -0.12 (0.60) Red/Blue Order -0.11 (0.41) 0.23 (0.72) 0.26 (1.24) R2 0.20 0.05 0.05 No. Observations 37 36 73 ESS 16.27 26.83 50.17 F 2.74* 0.52 1.25

Chow test, 2k)-M+/(NESS

)/kESS-ESS( = FUR

URC2k-M+N k, 0, where the UR = unrestricted = the sum of the

ESS for the implicit and actual equations; C = combined. Here, F4, 65 = 2.67. By interpolation, 95% critical value is 2.52; 99% critical value is 3.64 (therefore significant at the 95% level). 2. WTP for Red voucher given Blue voucher. Implicit Actual Combined Constant -2.01 -0.77 -2.10 Endowment 0.11 (0.85) -0.21 (1.00) 0.04 (0.36) Part/Whole Order -0.64 (1.64) 0.07 (0.15) -0.29 (0.97) Red/Blue Order -0.20 (0.47) 0.54 (1.23) 0.25 (0.86) R2 0.09 0.09 0.03 No. Observations 37 33 70 ESS 39.71 33.51 89.96 F 1.04 0.97 0.63 F4, 62 = 3.54. By interpolation, 95% critical value is 2.53, (therefore significant at the 95% level).

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Table 4 (continued). 3. WTA for Red voucher given Red voucher. Implicit Actual Combined Constant 1.51 3.81 3.06 Endowment 0.21 (1.75) 0.05 (0.42) 0.09 (1.00) Part/Whole Order 0.00 (0.00) 0.61 (1.97) 0.27 (0.87) Red/Blue Order 1.08 (2.30) -0.35 (1.09) 0.28 (0.90) R2 0.19 0.14 0.03 No. Observations 37 36 73 ESS 60.09 27.40 115.29 F 2.57 1.76 0.79 Chow test, F4, 65 = 5.16. By interpolation, 95% critical value is 2.52; 99% critical value is 3.64 (therefore significant at 99% level). 4. WTA for Blue voucher given Blue voucher. Implicit Actual Combined Constant -0.27 -0.25 1.28 Endowment 0.20 (1.54) 0.39 (2.05) 0.08 (0.80) Part/Whole Order 0.18 (0.36) 0.27 (0.63) 0.23 (0.64) Red/Blue Order 1.32 (2.64) 0.80 (2.00) 0.93 (2.74) R2 0.20 0.22 0.11 No. Observations 37 33 70 ESS 69.39 36.83 128.35 F 2.71* 2.73* 2.60 Chow test, F4, 62 = 3.23. By interpolation, 95% critical value is 2.53; 99% critical value is 3.64 (therefore significant at the 95% level). Note: * indicates rejection at the 95% level of the hypothesis that the equation has no explanatory power; bracketed figures are t-values. Now, as has already been seen, the correlation between endowment and valuation

19

was very weak in almost all cases, suggesting that the degree of bias would be limited. Nevertheless, using (original endowment minus estimated wtp)11 as an instrument to replace (original endowment minus actual wtp), we re-estimated the implicit and combined wtp equations, then repeated the exercise for the wta equations. The results were little changed, with the F values for the Chow test (in the same order as they appear in Table 4), becoming 2.49, 3,46, 4.55 and 2.94. The first of these, for the wtp for a Blue voucher given a Red voucher, is no longer significant (the 95% level critical value is 2.52), but the significance of the others remain unchanged, again supporting the notion of the existence of PHB. Practice Question Results. Some previous tests of CV biases suggest that biases disappear if the agents are allowed to learn (see, for instance, Shogren et al (1994)). Subjects had to undergo two practice versions of each question in our experiment, but then they could practise as many additional times as they wished. The mean number of practices per person per question was 2.2, which suggests that roughly one in five subjects opted to exercise their right to practice further. In fact, the vast majority of subjects did not exercise that right, but a few did so, often going through the practice question five or six times. It is not clear what the answers to practice questions mean in these circumstances. Subjects may be learning about their preferences, the incentive mechanism or both. They may even simply be `playing'. Visual inspection of the data suggested a large proportion whose answers differed little between practices and between practice and real questions, while a small proportion produced much more widely varying answers. Regression (unreported here) showed no systematic evidence for feedback from the outcomes of the practice questions (e.g. whether the subject would have successfully sold or bought the voucher under consideration, had the question been for real). Given the non-binding nature of the practice questions, tests for the convergence of valuations are meaningless, except that it would undermine our claim to have found a PHB, if the measured bias declined from a higher level in the initial practice rounds to something far lower once the questions were asked for real.12 There is no evidence that this was the case. Table 5 compares the estimates of part/whole bias from practice rounds of the experiment and the answers produced when questions were asked for real. Note that the computer only recorded the final three practices of subjects. This yielded over ninety-five percent of all practice answers, but means that, for those subjects 11 With the estimated wtp given by the equations estimated in Table 2. 12 There is no strong reason why the bias should be higher at the beginning of the learning process. If, for instance, learning followed the route apparent in Shogren et al (1994), where wtp rose and wta fell after repetition, we might expect the sum of the parts to be less than the whole initially for the wtp questions and the sum to exceed the parts for the wta sections of the experiment.

20

practising four or more times, their earlier answers were over-written. `Practice 1' and `Practice 2' in the table are therefore the values for practices one and two for agents who practised only twice. For subjects who practised three or more times, the figures are for their third to last and penultimate practices. As might be expected, there is considerable variation in the part/whole biases, but two things stand out: first, on only one occasion did the valuation of the parts not exceed that of the whole; second, there is no evidence for declining part/whole bias.

21

Table 5: Comparing Part/Whole Bias in Practice and Real Questions

Sample Means

Practice 1 Practice 2 Real

Implicit Wtp Blue/Red -0.59 -0.62 -0.83

Actual Wtp Blue/Red -1.26 -1.55 -1.32

Bias 0.66 0.92 0.49

Implicit Wtp Red/Blue -1.48 -1.36 -1.67

Actual Wtp Red/Blue -2.18 -2.32 -1.93

Bias 0.70 0.96 0.27

Implicit Wta Red/Red 4.89 2.84 3.45

Actual Wta Red/Red 4.28 4.59 4.30

Bias -0.61 1.74 0.85

Implicit Wta Blue/Blue 3.17 1.53 2.05

Actual Wta Blue/Blue 3.18 2.88 2.80

Bias 0.01 1.36 0.75 Note: Bias column indicates the degree to which the sum of the parts exceeds the mean valuation of the whole.

22

Order Effects. It is not uncommon to find order effects in CV exercises (see, for instance, Diamond (1994) and Kahneman and Knetch (1992)). One explanation is that the first answer anchors subsequent responses. As can be seen in table 1, we discovered little or no evidence of a systematic order effect in our experiment. The coefficients are largely insignificant and, where significant, no obvious pattern is discernible. Divergence between WTP and WTA. It is obvious from the results produced so far that we obtained a significant difference between the wtp and wta valuations placed upon the vouchers. Even if we took at face value, the income effects reported in Table 2, these would not be large enough to explain the discrepancies in figures 2 and 3. For instance, if the mean wta for both vouchers (£6.84) is added to the mean endowment of those agents (£3.38), we obtain an endowment of £10.22. The wtp for both vouchers given this endowment should equal £6.84, but if the coefficient on the endowment variable in the wtp for both vouchers given none is employed to obtain an estimate of what wtp should be given this endowment the result is £2.72 - less than half of the wta figure. These results therefore support the view that the wtp/wta divergence cannot be explained by income and substitution effects (Hanemann (1990)), but is compatible with Tversky and Kahneman's theory of loss aversion (1991). A fuller analysis of the issue is presented in Bateman et al (1995), where results from a companion experiment on the divergence between wtp and wta are analyzed. Swaps. Theoretically, different routes through the diamonds of figures 1 and 2 should yield consistent answers. Thus the acquisition of a Red voucher can be seen as a `whole' constructed from the `parts' of the gain of a Blue voucher plus a swap of the Blue voucher for a Red. Four `wholes' can be constructed in this way: to the Blue voucher beginning with either none or both vouchers and to the Red voucher starting at the same positions. Likewise, zero vouchers and both vouchers can be approached from each of the single voucher endowments, yielding another four potential comparisons of `parts' and `wholes'. The `parts' of these eight exchanges would not normally be labelled such. Rather they represent the components of indirect routes to a final endowment of vouchers which could be reached directly from the initial allocation of vouchers. Perhaps therefore, if the sum of the component valuations is different to the valuation via a direct route, a more accurate label would be indirect/direct bias. Is there any evidence of this more general phenomenon?

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Some care has to be taken over this issue. The previous section has described the divergence between valuations obtained from wtp and wta modes. Whereas in the part/whole figures reported so far in this paper all parts and the whole were in the same mode (i.e. all wta or all wtp), many of the eight indirect routes involve a mixture of wtp and wta questions and this raises the possibility of conflating biases in the results. While all the subjects holding Red vouchers, when faced with a swap question, indicated that they would demand compensation for acquiring a Blue voucher instead of a Red, twenty-two, out of the thirty-three subjects endowed with Blue vouchers, indicated that they would be willing to pay to acquire the Red instead, while eleven required compensation. Thus, although the mean value reported in figure 2 shows that, on average, these subjects were willing to pay £0.37 to replace their Blue voucher with Red, the pool of subjects contains a mixture. Out of the eight possible indirect routes, only two of them (Red to no vouchers, both vouchers to Blue voucher) yielded only wta answers and no route produced only wtp answers. Given the wta/wtp divergence reported above, it is our view that figures obtained where the indirect route involved a mixture of modes are unreliable evidence on the issue of PHB. However, for completeness, figures for all eight routes are reported below. Table 6 summarises the results. In it the eight routes have been divided into three categories: (i) exchanges where all answers represented wta; (ii) those where subjects differed as to whether compensation was required for executing one leg of the exchange; and (iii) two routes where one of the legs of the indirect route was in wtp mode with the other in wta mode. As noted above, it is our view that only the first category provides evidence on the part/whole bias issue. The final column indicates the degree to which the valuation obtained via an indirect route exceeds that of the direct estimate. For the two cases where all legs of the exchange involved wta mode, evidence of the value of the indirect exchange exceeding that of the direct exchange is clear cut. The other six results are mixed, most likely because of a conflation of part/whole and wta/wtp biases which act against one another in these circumstances.

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Table 6: Indirect versus Direct Valuation

No. Initial voucher

endowment.

Final voucher

endowment.

Indirect Valuatio

n (£).

Direct Valuation

(£).

Indirect/Direct Bias (£).

All subjects answered in WTA mode.

1 Both Blue 6.51 4.79 1.72**

2 Red None 5.76 4.30 1.46**

Indirect Routes Involving the Swap from a Blue voucher to a Red.

3 None Red -1.67 -2.18 -0.51**

4 Blue Both -1.69 -1.82 0.13*

5 Blue None 3.93 2.64 1.29**

6 Both Red 4.42 3.39 1.03**

Indirect Routes Mixing WTP and WTA Modes.

7 None Blue 0.94 -1.34 -2.28**

8 Red Both 1.30 -1.32 -2.62** Notes: 1. The final column indicates the degree to which the valuation imputed via the indirect route exceeds that obtained directly. 2. * and ** indicate significantly different from zero at the 95 and 99% (two tailed) level respectively.

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5. Conclusions By eliciting valuations for both parts and the whole, we have found clear evidence of part/whole bias for private goods, bought and sold using incentive compatible mechanisms. One view holds that part/whole bias found in the CV of public goods lacking a use value is the result of a deficiency of rigour in the methods of the researcher and can therefore be eradicated with the proper design and execution of surveys. Our results do not support such a view. Instead, they suggest that part/whole bias is more fundamental. Indeed, they do not support the view that problems of coherent valuation are confined to public goods: our results are more in line with the work of Knetch and Sinden (1984) in being at odds with standard models of consumer theory. It must be remembered that the results presented here are from only one experiment, and require confirmation. Nevertheless, if there are serious problems with the method, it is not clear why they should produce results supportive of a systematic part/whole bias, rather than some outcomes where the value of the parts exceeded that of the whole and some where the sum of the parts was less than the whole. What is lacking here is a theory of part/whole bias and we do not provide one. The phrase `part-whole' and the term `embedding' come from a result in cognitive psychology that does not seem to us to be relevant. A priori, it seems highly unlikely that our subjects valued the meal represented by the two vouchers with one hemisphere of their brains and the separate vouchers in the half hemisphere. Whatever the explanation, it cannot now be supposed that further refinement of the contingent valuation method will necessarily lead to the eradication of the part/whole bias which has plagued it so far.

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