Testing the Intransitivity Explanation of the Allais Paradox - Springer Link

EBBE GROES, HANS JØRGEN JACOBSEN, BIRGITTE SLOTH and TORBEN TRANÆS

TESTING THE INTRANSITIVITY EXPLANATION OF THE ALLAIS PARADOX

ABSTRACT. This paper uses a two-dimensional version of a standard common consequence experiment to test the intransitivity explanation of Allais-paradoxtype violations of expected utility theory. We compare the common consequence effect of two choice problems differing only with respect to whether alternatives are statistically correlated or independent. We framed the experiment so that intransitive preferences could explain violating behavior when alternatives are independent, but not when they are correlated. We found the same pattern of violation in the two cases. This is evidence against intransitivity as an explanation of the Allais Paradox. The question whether violations of expected utility are mainly due to intransitivity or to violation of independence is important since it is exactly on this issue the main new decision theories differ. KEY WORDS: Allais paradox, Common consequence experiment, Sure-thing principle, Intransitive preferences

1. INTRODUCTION

There is little controversy over whether a significant fraction of decision makers violate expected utility theory (EU), the most famous type of violation being the Allais Paradox after Allais (1953); this is well established through countless experiments.1 But why is it that EU is not always a good description of people’s behavior? This is a controversial question. What is the difference between an EU – violating individual in one of these experiments and an individual in Savage’s or in von Neumann and Morgenstern’s universe? Roughly speaking, two different answers are nowadays suggested. According to the intransitivity explanation people base their decisions on standard probability measures, but their pair-wise choices are not necessarily transitive. According to the other explanation people have fixed utilities of goods, but make decisions based on modified Theory and Decision 47: 229–245, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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E. GROES, H.J. JACOBSEN, B. SLOTH AND T. TRANÆS

or twisted probability measures, and thereby violate the Sure-Thing Principle (STP) of Savage (1954) or the Substitution Axiom (SA) of von Neumann and Morgenstern (1947) (vN–M), depending on whether the framework is one of uncertainty or one of risk. In this paper we present an experimental test of the intransitivity explanation of the Allais-type of EU violation. We compare the common consequence effects of two choice experiments differing only with respect to whether alternatives are independent or correlated. The common consequence effect is a violation of EU in an experiment where subjects are choosing between two alternatives before and after a common change in consequences has been made. An alternative is a prospect or gamble where the consequence or outcome (e.g., the amount of money won) is determined by the realization of a random ‘state of the world’. The random state of the world may or may not be such that each state has an objective probability connected to it. Two alternatives are (potentially) correlated if their consequences are determined by one and the same drawing of a state of the world. Two alternatives are independent if their consequences are determined by two statistically independent drawings. Traditionally, violation of EU in experiments with independent alternatives has been attributed to subjects violating the STP or the SA. This has been the inference made from many Allais experiments, for instance. However, if individuals are choosing between statistically independent alternatives, a subject with violating behavior could as well obey the STP/SA and instead violate transitivity as will be demonstrated in the next section. In fact it is now generally accepted that violation of EU in experiments with independent alternatives could be accounted for by several new alternative theories. These theories can roughly be divided into two branches: Rank Dependent Theory (RDT), which we use as a common label for the contributions of, e.g., Kahneman and Tversky (1979), Quiggin (1982), Tversky and Kahneman (1992), Schmeidler (1989), Gilboa (1987), Wakker (1989), and Sarin and Wakker (1992); and Regret Theory (RT) which we use as a common label for such contibutions as Loomes and Sugden (1982), Fishburn (1982, 1989), Vind (1991a, b), and Sugden (1993). Both branches are in accordance with either the Savage’s or the vN–M’s axiom system, except that a single ax-


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iom is relaxed or excluded. RDT is derived by relaxing STP/SA only, while RT is derived by disposing of transitivity. Hence, RT corresponds to the intransitivity explanation of EU violation and RDT to an explanation assuming violation of STP/SA. In common consequence experiments with correlated alternatives, violation of EU can still be explained by a theory maintaining all axioms but relaxing STP/SA, while it cannot be accounted for by assuming all the axioms but transitivity, see below. Therefore we should expect to find EU-violating behavior less frequently in a correlated experiment than in an independent experiment, unless intransitivity accounts for only a negligible part of the violating behavior. Testing this is the main purpose of the paper, and our conclusion is that it cannot be rejected that intransitivity does not account for any of the EU-violations in Allais type, common consequence experiments. We find this important since the modern branches of decision theory depart exactly on the issue whether to drop transitivity or relax STP/SA. Having said that, let us emphasize that our result is evidence against the intransitivity explanation of the Allais Paradox, not against intransitivity as an explanation of any EU violation. Both RT and RDT have some experimental support.2 The paper is organized as follows. Section 2 gives the theoretical background, presents the test strategy, and discusses the reward scheme. Section 3 describes the experiment, Section 4 gives the evidence, and Section 5 concludes.

2. BACKGROUND

In this section we discuss the standard common consequence problem and the extended, or two-dimensional, version that forms the basis for our experiment. We present the latter in its two extreme versions, one where the consequence of each of the two alternatives in a choice is decided by each its independent set of states, and one where it is decided by one and the same set of states; the former has thus complete independence between alternatives, while the latter has complete correlation. Furthermore, we show that the RT or intransitivity explanation of EU violation applies only to experiments with independent alternatives, while the RDT explanation assum-

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ing violation of STP/SA applies in both cases. Finally, our testing strategy based on this point is described. (a) The common consequence problem The standard common consequence problem is illustrated in Table 1(a). Subjects are asked to choose between the alternatives A and B. An alternative is a prospect or gamble where the consequences are state dependent; in the subset of states named P both A and B yield the monetary pay-off b; in the Q-states A yields b while B yields a; etc. Assume that all states have the same probability.3 With a slight abuse of notation let the numbers of the respective states be P, Q, and R as well. The probabilities of the different subsets are then p = P/(P + Q + R), q = Q/(P + Q + R), and r = R/(P + Q + R). TABLE 1

A B

P b b

Q b a (a)

R b c

C D

P a a

Q b a (b)

R b c

Let ≺, -, and ∼ be strict preference, weak preference, and indifference, respectively. Then, EU implies that there is a utility function v on consequences such that, A % B ⇔ (p + q + r)v(b) T pv(b) + qv(a) + rv(c). ≺

(1)

The state-contingent consequences of Table 1(b) are identical to those in Table 1(a) except that the consequence in the P-states is now a rather than b for both alternatives, and EU says, C % D ⇔ pv(a) + (q + r)v(b) T (p + q)v(a) + rv(c). ≺

(2)

We can reduce (1) and (2) further, essentially using that the EU representation is linear in probabilities: A% ≺ B ⇔ (q + r)v(b) − qv(a) − rv(c) T 0, and

(3)


C % D ⇔ (q + r)v(b) − qv(a) − rv(c) T 0. ≺

233 (4)

Thus, the two choice problems, A-vs-B and C-vs-D, are equivalent under EU. In fact this extends beyond EU. For instance, if we allow preferences to be intransitive but keep the rest of the EU foundation, the representation would still be linear in probabilities and hence it would still be true that A ≺ B if and only if C ≺ D. RT is an example of just that. To a significant fraction of subjects in experiments, the choice A-vs-B is not the same as the choice C-vs-D. It is easy to calibrate a, b, and c, and p, q, and r so that in an experiment many subjects prefer A to B and D to C (having controlled for indifference). This is the so-called Allais paradox in a common consequence framing. (b) The two-dimensional common consequence problem The experimental design of Table 1 has correlated alternatives. An experiment with independent alternatives is presented in Table 2. Here, consequences are contingent upon the realizations from two different and independent state spaces, the horizontal and the vertical. For example, if the vertical realizes a state in Q, while the horizontal realizes one in T , then alternative A pays out b, and B pays a. Complete independence between alternatives is achieved if the consequence of one alternative is determined exclusively by the horizontal state and the consequence of the other is determined exclusively by the vertical state. In Table 2, C only depends on the vertical state, while D only depends on the horizontal. Thus alternatives C and D are independent, and the same is true for A and B. Table 3 illustrates the experiment with complete correlation; for all the alternatives only the vertical states matters. This makes the alternatives of Table 3 the ‘same’ as those of Table 1. In what follows we assume that the number of states S, T, and U are equal to P, Q, and R, respectively, so that the column and row probabilities become the same, i.e., p = s, q = t and r = u. This means that the alternatives in Tables 2 and 3 only differ wrt. correlation, while they give pairwise rise to the same probability distribution over consequences. For instance, both in the independent and in the correlated case al-

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TABLE 2 The common consequence problem with independent alternatives

A

B

P b b b b a c

Q b b b b a c (a)

R b b b b a c

S T U S T U

C

D

P a a a a a c

Q b b b a a c (b)

R b b b a a c

S T U S T U

TABLE 3 The common consequence problem with correlated alternatives

A

B

P b b b b b b

Q b b b a a a (a)

R b b b c c c

S T U S T U

C

D

P a a a a a a

Q b b b a a a (b)

R b b b c c c

S T U S T U

ternative D gives probability p + q of winning a and probability r of winning c. It is also assumed that 0 = a < b < c. (c) The predictions of theory Our test uses the fact that RDT giving up the STP/SA can explain EU violation in both the independent and the correlated experiment, while RT giving up transitivity can explain EU violation only in the independent experiment. To see this, we derive the predictions of both theories. The principle of RT is that utility is modified. Suppose that the two alternatives E and F of a choice result in the consequences xEj and xFj , respectively, conditioned on the state of the world


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being j . Following Loomes and Sugden (1982) the modified utility of xEj is given by a function R(xEj , xFj ) which is the joy of having xEj plus the ‘regret’, if xEj < xFj , or the ‘rejoicing’, if xEj > xFj , of forsaking xFj . Now RT suggests that in the choice between E and F the decision maker seeks to maximize the expected value of R(·, ·). The net advantage of choosing E and rejecting F in the event that state j occurs is represented by the function 9(xEj , xFj ) = R(xEj , xFj ) − R(xFj , xEj ). By definition, 9(·, ·) is skew- symmetric, that is, 9(xEj , xFj ) = −9(xEj , xFj ), implying that 9(z, z) = 0. RT suggests the following choice rule, X E% F ⇔ pj 9(xEj , xFj ) T 0, (5) ≺ j

P where pj is the probability that state j occurs, pj = 1.4 Without ‘regret’ or ‘rejoice’, R(xEj , ·) = v(xEj ), and (5) becomes the standard expected utility representation. Applying (5) to our choice problem, it turns out that it is crucial whether alternatives are independent or correlated. With independent alternatives, Table 2, RT says that, A % B ⇔ t9(b, a) + u9(b, c) T 0, and ≺ C%D ⇔ ≺

(q + r)(s + t)9(b, a) +(q + r)u9(b, c) + pu9(a, c) T 0.

Since 9(a, c) is present in the second but not in the first expression, any pattern of choices can be rationalized; A ≺ B and D ≺ C, for instance, if the ‘regret’ of getting the smallest prize a, having had the opportunity of the highest one c, is sufficiently strong. With correlated alternatives, Table 3, this is not the case. Here RT implies, A % B ⇔ q9(b, a) + r9(b, c) T 0, and ≺ C % D ⇔ q9(b, a) + r9(b, c) T 0. ≺

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Hence, A B if and only if C D. Contrary to RT, RDT ‘keeps the utilities and modifies the probabilities’ before computing mathematical expectations. Consider a prospect A = (x, p) = (xj , pj )j , where the xj ’s are outcomes in xn−1 %, . . . , % x1 , and each of n states ordered such that xn % P the pj ’s are the probabilities of the states, pj = 1. The modified probabilities over the states are given by a function ϕ(Pj ), where Pj Pj = i=1 pi , ϕ(0) = 0, and ϕ(1) = 1. There is a utility function v on consequences such that the utility of a prospect is, V (x, p) =

n X

[ϕ(Pj ) − ϕ(Pj −1 )] · v(xj ),

(6)

j =1

and the preferred prospect is simply the one with the highest utility. Consider the independent experiment, Table 2, with 0 = a < b < c, and p = s, q = t and r = u. Then v(A) = v(b), v(B) = [1 − ϕ(p + q)] · v(c) + [ϕ(p + q) − ϕ(q)] · v(b), etc. This yields, A % B ⇔ v(b) T [1 − ϕ(p + q)] · v(c) ≺ + [ϕ(p + q) − ϕ(q)] · v(b), C % D ⇔ [1 − ϕ(p + q)] · v(c) T [1 − ϕ(p)] · v(b). ≺ Thus, to rationalize violations of EU, we just need to twist the function ϕ accordingly. For instance the Allais Paradox, A B and C ≺ D, requires, ϕ(q) · v(b) > [1 − ϕ(p + q)] · [v(c) − v(b)] > [ϕ(p + q) − ϕ(p)] · v(b), which is satisfied, for instance, by the following values and parameters, v(b) = 10, v(c) = 11, ϕ(q) = 0.05, ϕ(p) = 0.80, and ϕ(p + q) = 0.81. From (6) we see that alternatives are evaluated by the utilities of the sure outcomes and a function taking the (objective) probabilities of outcomes as arguments. Therefore, to this formula the independent and the correlated choice problems are identical since the outcome probabilities and the outcomes are the same; correlation does not matter. So a RDT decision maker will have Allais-paradoxical


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behavior in the correlated experiment if and only if he has so in the independent one. Thus, RDT can explain EU violaiton in both the correlated and the independent experiment, while RT can explain this behavior only in the independent experiment. (d) The test The testing strategy is as follows. In any pool of subjects we expect to find all kinds of violating and non-violating behavior, and if we divide a given pool of subjects into two groups randomly, we expect to have the same distribution of different kinds of violating and non-violating behavior in the two groups (if the pool is large). This is what we did. We then performed an independent experiment with the one group and a correlated experiment with the other. All subjects who obey all of Savage’s or vN–M’s axioms we expect to behave according to EU in both experiments; from subjects who obey all the axioms but STP/SA we could expect violation of EU in both experiments; from subjects who obey all the axioms but transitivity we could expect violation of EU only from those in the independent experiment, while we should expect non-violating behavior in the correlated experiment. Thus, some axiom-violations can be revealed in the independent but not in the correlated experiment, namely those following from intransitive preferences, while all violations detectable in the correlated experiment are also detectable in the independent experiment. Therefore, we should expect to find a higher ratio of violation in the independent experiment than in the correlated experiment, if violation of transitivity accounts for a significant fraction of the EU-violations. (e) The reward scheme Each of the two experiments (the independent and the correlated) involves two choices, A-vs-B and C-vs-D. To avoid income effects and to avoid the possibility of obtaining insurance by diversifying choices we used the following procedure: First subjects made the choice A-vs-B. Then one subject was drawn at random, he did not participate further, and he was rewarded according to his choice at the end of the experiment. The remaining subjects, now knowing that their A-vs-B choice turned out to be fictitious, then made the

238


choice C-vs-D. The second choice was also followed by a random draw of a subject who was rewarded according to his choice. This use of random draws is not innocuous. With the random draw the state space is enlarged and the prospects chosen between are really: A yields b if the draw is won and zero otherwise, whereas B yields b if the draw is won and P occurs, a if the draw is won and Q occurs, c if the draw is won and R occurs, and zero otherwise, etc. It may be that a subject chooses between, e.g., A and B presuming that he wins the draw. In that case our experiment works as if there were real payment for that subject.5 But also if a subject perceives correctly of the involved choices our experiment has the potential of discriminating between the two competing explanations, RT and RDT. It is straightforward to check that the conclusions of Section 2(c) do not change if we extend the set of states so as to include the random draw, – Under RT a subject’s choices are independent of whether he perceives correctly of the problem or presumes to win the draw, exactly because preferences under RT fulfill the STP; in any state where the agent does not win the draw, regret will be zero, 9(z, z) = 0. Thus RT can explain Allais paradoxical behavior in the independent experiment, but not in the correlated experiment as demonstrated in Section 2(c). – Under RDT the STP is not assumed and therefore the random draw may be of importance for the subject’s choices. It may well be that a subject who obeys RDT, will choose differently between, e.g., C and D, when there is a random draw than if there were payment for everybody. However, the random draw affects the prospects, that is, the (marginal) probabilities of the different amounts, in exactly the same way in the independent and the correlated case, and under RDT choices depend only on the marginal distributions. Thus under RDT one should expect to see the same incidence of Allais paradoxical behavior in the two experiments.

3. THE EXPERIMENT

A total of 230 students participated in the experiment which took about 30 minutes of a lecture they were supposed to attend anyway.


239

The students were undergraduates studying agricultural technique and did not know anything about expected utility theory. They were able to calculate expected monetary payoffs. The experiment was run as follows. First, two identical roulettes, marked red and blue respectively, each with 37 numbers (0–36) were spun and covered before coming to rest. The subjects were divided randomly into two groups, correlated group (Group C) and independent group (Group I), of equal size. Each subject in Group C was given two questions as illustrated in the left column of Table 4. This corresponds to the design of Table 3. The first question was the choice between alternatives A and B, and the other had the choice between C and D. The second question was in a sealed envelope not to be opened until later. The subjects of Group I were given the questions in the right column of Table 4 corresponding to the design of Table 2. Thus, Group I was confronted with choices where the alternatives were bets decided by two different roulettes, whereas subjects in Group C were confronted with alternatives determined by one and the same roulette. After the subjects had handed back their answers to the first question, A-vs-B, a subject was drawn at random. This subject did not complete the part with alternatives C and D, and does not enter the statistics given below. Still, without revealing the results on the two roulettes, the subjects were asked to open the sealed envelopes and answer the second question, Cvs-D. The answers were collected, and again a subject was drawn at random. Now the two roulettes were uncovered, the position of the ball observed, and the two drawn subjects had their rewards: nothing, DKK 10,000, or DKK 11,000, according to the choices they have made (DKK 10,000 correspond to approximately US$ 1,900). The entire procedure was explained to the subjects before the experiment started.6 The EU requires that preference of A over B implies preference of C over D, and vice versa. Thus, subjects who report the combinations AC and BD are acting in accordance with the EU, whereas subjects who report the combinations AD and BC are either violating the EU, indifferent in both choices, unable to order the alternatives, or make errors. In order to control for the last three possibilities we have chosen the rewards and the probabilities so as to encourage violation in a particular direction, namely in the AD direction,

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TABLE 4 The framing of questionaires Group C(orrelated)

Group I(ndependent)

Alternative DKK 10,000 if the ball A: on the blue roulette has landed on one of: 0–36

Alternative DKK 10,000 if the ball A: on the blue roulette has landed on one of: 0–36

Alternative DKK 10,000 if the ball B: on the blue roulette has landed on one of: 0–29

Alternative DKK 10,000 if the ball B: on the red roulette has landed on one of: 0–29

nothing if the ball on the blue roulette has landed on: 30

nothing if the ball on the red roulette has landed on: 30

DKK 11,000 if the ball on the blue roulette has landed on one of: 31–36

DKK 11,000 if the ball on the red roulette has landed on one of: 31–36

Alternative nothing if the ball on the C: blue roulette has landed on one of: 0–29

Alternative nothing if the ball on the C: blue roulette has landed on one of: 0–29



Alternative nothing if the ball on the D: blue roulette has landed on one of 0–30

Alternative nothing if the ball on the D: red roulette has landed on one of: 0–30


DKK 11,000 if the ball on the red roulette has landed on one of: 31–36

which is the Allais paradox. Hence, if the combinations AD and BC are reported because subjects were indifferent, made errors, or had incomplete preferrences, we would expect them to be equally frequent, whereas a significant bias towards combination AD will be taken as evidence of systematic violation. Testing for this is of course a very crucial point in our analysis.


241

TABLE 5 The choice patterns

Combination of choices AC (consistent with EU) AD (Allais Paradox) BC (other paradox) BD (consistent with EU)

Number of subjects Z-statistic U -statistic = −0.28

Group C Pct. of subjects

Group I Pct. of subjects

Group C and I pooled Pct. of subjects

46 24 14 16

45 22 12 20

45 23 13 18

100 116

100 113

100 229

1.83

1.78

2.55

4. EVIDENCE

The choice patterns are given in Table 5. Approximately one third of the subjects made paradoxical choices (AD or BC). As argued this could simply be due to subjects being indifferent or making errors. Therefore we have tested whether the paradoxical answers go systematically in the expected AD-direction using Conlisk’s Zstatistic (1989).7 This has approximately a standard normal distribution under the null hypothesis that violation of the EU is purely random. The test is one-tailed against the alternative hypothesis that the Allais-type violation, AD, is the more frequent. In both groups the null hypothesis that violation of the EU is purely random is rejected at the 0.05 level; for the pooled data the null hypothesis is rejected even at the 0.01 level. Our main result is that it does not matter for the amount of Allais paradoxical behavior whether alternatives are statistically correlated or independent: a standard U -statistic8 does not enable us to reject at any sensible level the null hypothesis that violation is equally strong in the two groups against the single sided alternative that violation is more frequent with independent alternatives: the significance probability is 0.39. Neither can the hypothesis that the two samples are drawn from the same distribution be rejected.

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The level of EU violation, in particular of the Allais-type, is lower in our experiment that what is usually found, see Hey (1991). This may well be due to the high amount of money involved here. We performed a control experiment. Here only the Group C questionnaire was given and there were no real payoffs. A total of 55 students participated. The frequencies were: AC : 28%, AD : 31%, BC : 7%, and BD : 33%. These are significantly different from those of Table 5 (Group C). Most remarkably, the frequency of subjects choosing D over C is higher, 64 (=31+33)% against 40 (=24+16)%, which also to a large extent accounts for the higher frequency of Allais paradoxical behavior, 31% against 24%. Hence, contrary to Camerer (1989), but in line with Battalio et al. (1990), we find that real money makes a qualitative difference and seemingly tends to bring down the frequency of Allais paradoxical behavior without making it insignificant.

5. CONCLUSION

It cannot be proved that the subjects of our experiment really perceived correctly of the correlation between the involved alternatives. Even though it was made clear to the subjects of Group C, that one and the same roulette was decisive for all alternatives, one can of course insist that, nevertheless, they did perceive of the alternatives as uncorrelated, and therefore the significant frequency of EU violation observed could still be explained by intransitivity. We do not know of a method making it clearer to subjects how alternatives are related without directly giving them tables showing the simultaneous distributions. This is not necessary recommendable; it would not be even close to the way people are confronted with decisions in real life. In other words, in our experiment subjects had as much chance of getting the joint distributions correctly as people have in real decision situations. The evidence found in the experiment shows siginficant violation of EU, systematically in the AD-direction, and it does not matter for this conclusion whether alternatives are statistically correlated or independent. This is evidence against the intransitivity explanation of the Allais Paradox. According to this explanation subjects should comply to EU when alternatives are correlated but might violate EU


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when alternatives are independent. This suggests that intransitive preferences do not account for more than a negligible part of the Allais-type violations of expected utility theory.

ACKNOWLEDGMENTS

We are grateful to Karl Vind for leading us on this research and for enlightening discussions, and to Peter Wakker and an anonymous referee for comments. Financial support from the Danish Social Sciences Research Council is gratefully acknowledged.

NOTES 1. See Hey (1991) and Kagel and Roth (1995). 2. See again Hey (1991) and Kagel and Roth (1995), but also Starmer (1992) who presents experiments that partially support and partially reject both branches, RDT and RT. 3. It is only to make the explanations simpler that we assume that states have given, objective probabilities. 4. This representation is axiomatized by Fishburn (1982) and Sugden (1993). 5. A similar problem occurs in any experiment where each subject makes a number of choices and then one is chosen at random to be for real in order to control for income effects or simply to save funds. See Holt (1986) for an extensive discusion, and Starmer and Sugden (1991) for an experiment involving Allais-type choices and small monetary payoffs, which indicates that the problem is of minor importance. 6. Full translations of the questionnaires and the verbal explanations are available from the authors on request. 7. Conlisk’s (1989) Z-statistic is, √ (S − 0.5) (N − 1) Z=p 0.25(1/V ) − (S − 0.5)2 where N is sample size, S is the fraction of Allais paradoxical behaviour out of the total number of violations; S = n(AD)/(n(AD) + n(BC)), where n(XX) is the numbr of subjects responded with combination XX, and V is the fraction of subjects violating the expected utility hypothesis; V = (n(AD)+n(BC))/N. Under the null hypothesis that violation of EU is purely random, this has approximately a standard normal distribution. The test is one-tailed against the alternative hypothesis that the Allais-type violation, AD is the more frequent.

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8. The standard U -statistic used is, U=√

(V1 − V2 ) V1 (1 − V1 )/(N1 − 1) + V2 (1 − V2 )/(N2 − 1)

where N1 and N2 are the sample size of the groups being compared, and V1 and V2 the fraction of subjects violating the expected utility hypothesis in each group.

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Addresses for correspondence: Torben Tranæs, Institute of Economics, University of Copenhagen, Studiestræde 6, DK-1455 Copenhagen K., Denmark Phone: +45 35323005; Fax: +45 35323000; E-mail: torben.tranaes@econ.ku.dk