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How not to extend models of inequality aversion
Journal of Economic Behavior & Organization 81 (2012) 599–605
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How not to extend models of inequality aversion Dirk Engelmann a,b,c,d,∗ a b c d
Department of Economics, University of Mannheim, L7, 3-5, D-68131 Mannheim, Germany Centre for Experimental Economics, University of Copenhagen, Øster Farimagsgade 5, 1353 Copenhagen, Denmark Economics Institute of the Academy of Sciences of the Czech Republic, Politick´ ych vˇeznu˙ 7, 111 21 Prague 1, Czech Republic CESifo, Poschingerstr. 5, 81679 Munich, Germany
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 12 August 2011 Accepted 15 August 2011 Available online 22 August 2011 JEL classification: C72 C91 C92
a b s t r a c t Several authors have made attempts to improve the explanatory power of models of inequality aversion, in particular the one by Fehr and Schmidt (1999), by adding concerns for total surplus or efficiency. In this note, I point out that these attempts are misguided because they are equivalent to a much simpler change, not requiring an additional parameter, unless we simultaneously consider games with different numbers of players. In the latter case, however, such an approach yields implausible predictions. © 2011 Elsevier B.V. All rights reserved.
Keywords: Inequality aversion Efficiency Social preferences
1. Introduction Models of inequality aversion (Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999) have had substantial success in rationalizing experimental data. However, in a number of experiments they do poorly, in particular in dictator-type games where dictators appear to care to a substantial degree for maximizing the total surplus (see e.g., Kritikos and Bolle, 2001; Charness and Rabin, 2002; Engelmann and Strobel, 2004). This observation has resulted in recent papers that attempt to improve the performance of the inequality aversion models (typically using Fehr–Schmidt as a basis) by adding a term for efficiency concerns (i.e., total surplus maximization or altruism with identical weights on all subjects). These papers come about either as theoretical papers claiming to provide a useful generalization of the Fehr–Schmidt model (see e.g., Kohler, accepted) or as experimental papers, trying to get a better fit of their data by adding an extra term (see e.g., Ellison et al., 2010).1 The most simple version of this approach just adds an
∗ Correspondence address: Department of Economics, University of Mannheim, L7, 3-5, D-68131 Mannheim, Germany. Tel.: +49 621 181 1894; fax: +49 621 181 1893. 1 Typically, I came across these (anonymized) papers as a referee. Similar suggestions have been made as comments on Blanco et al. (2011). I actually explained the arguments of this note to both Kohler and Ellison et al. in referee reports. As these are the only published papers I am aware of that use such an approach and the other authors remained anonymous, I refrain from further citations here. 0167-2681/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2011.08.007
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efficiency concerns parameter (borrowed in essence from the Charness–Rabin model of social-welfare preferences) to the standard Fehr–Schmidt model, which then becomes Ui (x) = xi −
˛i ˇi max{xj − xi , 0} − max{xi − xj , 0} + i xj n−1 n−1 j= / i
j= / i
j= / i
with xi the payoff of subject i and the usual conditions of the Fehr–Schmidt model ˛i ≥ ˇi ≥ 0 and ˇi < 1, but in addition 0 ≤ i ≤ 1 (in principle, one could allow for i > 1 and thus include people who care more about others’ payoffs than their own, but that is probably empirically irrelevant).2 This note will point out that this approach is misguided, as adding the extra parameter does nothing other than allowing for ˛i < 0 and ˛i < ˇi as long as we only consider games with a constant number of players and has questionable implications if we try to fit such a utility function to a set of games with different numbers of players. Section 2 will illustrate this in the two-player setting, Section 3 will deal with the general n-player case and Section 4 with the case of simultaneously considering games with different numbers of players. While the latter sections are getting somewhat more technical, the basic point is straightforward and is best illustrated by the two-player case. It should also be stressed that a careful reading of Charness and Rabin (2002) already shows the main point for the two-player case. So this note does not have a claim of fundamental novelty. Nevertheless, the repeated occurrence of such models and suggestions indicates that an explicit statement of these results might be useful. Section 5 will briefly comment on problems with a similar approach based on the Bolton–Ockenfels model and Section 6 summarizes the main lessons. 2. The two-player case The basic logic of the argument is most easily illustrated in the two-player case. If there is only one other player, this player can have a higher or lower payoff than oneself (or the same, which can be included in either of the other cases). The suggested model then becomes
Ui (xi , xj ) =
xi − ˛i (xj − xi ) + i xj ,
if xi ≤ xj
xi − ˇi (xi − xj ) + i xj ,
if xi > xj
(1)
or Ui (xi , xj ) = xi − ˛i max {xj − xi , 0} − ˇi max {xi − xj , 0} + i xj . To understand this model, we need to answer the following question. If the other player has a higher payoff and I am envious (˛i > 0), what does it mean if I also like to maximize the (other players’) total payoff ( i > 0)? It means that while I wish to reduce j’s payoff, I also have an interest in increasing it. Depending on the relative strength of the concerns, this simply corresponds to ˛i in the standard model becoming smaller, or possibly negative, as the below reformulation of the model will show.3 If the other player has a lower payoff and I am averse to advantageous inequality (ˇi > 0) what does it mean if I also like to maximize the (other players’) total payoff ( i > 0)? It means that I like giving to j a bit more, which corresponds to ˇi in the standard model becoming larger. Let us reformulate the model to clarify this. Consider a player maximizing a utility function based on inequality aversion and total surplus maximization Ui as in (1). Simple manipulation yields
Ui (xi , xj ) =
xi − ˛i (xj − xi ) + i xj = (1 + i )xi − (˛i − i )(xj − xi ),
if xi ≤ xj
xi − ˇi (xi − xj ) + i xj = (1 + i )xi − (ˇi + i )(xi − xj ),
if xi > xj
(2)
or Ui (xi , xj ) = (1 + i )xi − (˛i − i ) max{xj − xi , 0} − (ˇi + i ) max{xi − xj , 0} Thus any preferences determined by utility function Ui can also be described by a two-parameter utility function
i (xi , xj ) = xi − ˛i max{xj − xi , 0} − ˇi max{xi − xj , 0} U i = (˛i − i )/(1 + i ) and ˇi = (ˇi + i )/(1 + i ). with ˛ 2 Alternatively, one can include xi in the last term, so that this term corresponds to the total payoff over all players instead of over all other players. This i = (˛i − (n − 1)i )/(1 + ni ) and does not change the argument in any way. The only difference is that then in the reformulated two-parameter model ˛
i = (ˇi + (n − 1)i )/(1 + ni ). See Section 3 for the interpretation of ˛i and ˇi . The particular model used in this note is the n-player generalization of the ˇ two-player model in Kohler (accepted). The approach to fit their data presented by Ellison et al. (2010) is slightly different. They combine the Fehr–Schmidt model with efficiency concerns in a model that contains only two parameters by first imposing the restriction ˛i = ˇi and then adding altruism. They claim this is both more restrictive and more general than the Fehr–Schmidt model. It is, however, at best a tedious generalization because it just corresponds to enforcing ˛i < ˇi and allowing for ˛i < 0 (more precisely, replacing ˛i ≥ 0 with ˛i ≥ − ˇi ). Moreover, they extend the model by allowing subjects to differ in their degree of altruism towards different groups of other players. 3 The three-parameter model is also psychologically and philosophically implausible. It amounts to saying: “You’re rich, so I want to hurt you. But I also like people to be wealthy, so I want to help you.” - “So do you want to hurt or help me?” - “Both, at least a bit. Maybe one more than the other, but really, really, both. So even if I want to help you a lot, I still want to hurt you a bit at the same time.” Obviously, either the helping or hurting has to dominate and thus this is what will be done.
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The three-parameter model is not only unnecessarily complicated, it is also not useful for empirical applications, because
i , ˇi ) that already the parameters cannot be identified, since we can choose infinitely many (˛i , ˇi , i ) vectors for any given (˛ completely determine the preferences.4 To understand the effects of adding efficiency concerns a bit better, let us calculate the marginal rates of substitution of own for other payoff. This also confirms that the two utility functions describe the same preferences, because if the marginal rates of substitution are identical for the two utility functions, they describe the same preferences. Consider first the case xj > xi . Then ∂Ui /∂xj −˛i + i = 1 + ˛i ∂Ui /∂xi ∂Ui /∂xj ∂Ui /∂xi
=
i −˛
i 1+˛
=
∂Ui /∂xj −˛i + i −(˛i − i )/(1 + i ) = = 1 + i + ˛i − i 1 + (˛i − i )/(1 + i ) ∂Ui /∂xi
Next consider the case xi > xj . Then
∂Ui /∂xj ˇ + i = i 1 − ˇi ∂Ui /∂xi ∂Ui /∂xj ∂Ui /∂xi
=
i ˇ
i 1−ˇ
=
∂Ui /∂xj (ˇi + i )/(1 + i ) ˇi + i = = 1 − (ˇi + i )/(1 + i ) 1 + i − ˇi − i ∂Ui /∂xi
i indeed represent the same preferences, they also provide The marginal rates of substitution not only confirm that Ui and U maybe a clearer way of understanding what having concerns for total surplus maximization means for the Fehr–Schmidt parameters. Concerning the case of disadvantageous inequality (xj > xi ), we see, in line with intuition, that efficiency concerns have the effect of reducing the marginal willingness to pay to reduce the other subject’s payoff and may even turn it into willingness to pay to increase it (if i > ˛i ). Thus, introducing efficiency concerns has the same effect as abolishing the restrictions ˛i ≥ ˇi and ˛i ≥ 0 in the Fehr–Schmidt model (or more precisely, replacing the latter with ˛i ≥ − 1/2). Concerning the case of advantageous inequality aversion, if i converges to 1 and ˇi converges to 0, the marginal rate of substitution approaches 1, which corresponds to ˇi = 1/2 in the standard Fehr–Schmidt model.5 Since ˇi = 1/2 is perfectly within the restrictions of the Fehr–Schmidt model, adding efficiency concerns provides no extension of the model for the case of advantageous inequality. i cannot One difference between the three-parameter model and a two-parameter model that allows for ˛i < 0 is that ˛ be smaller than −1/2 and hence this model puts a cap on the degree of altruism towards richer people. While we could also just add this restriction to the two-parameter model, it seems hardly a restriction in practise anyway as this only excludes choices where a person would spend z in order to increase the payoff of an already better-off person by even something slightly less than z, behavior which will most likely almost never be observed.6 So a simple two-parameter model which does not put any restrictions on ˛i < 0 is somewhat more general, but only by allowing for empirically irrelevant behavior. 3. The n-player case The previous section has shown for the two-player case that the inequality-aversion-plus-efficiency-concerns model represents the same preferences as the Fehr–Schmidt model with weaker restrictions on the parameters. The n-player case in no way differs, as long as we consider the number of players n as given, but is just a bit more tedious. So let us turn to this now for the sake of completeness. As above, let us manipulate the three-parameter model. Ui (x)
= xi −
˛i ˇi max{xj − xi , 0} − max{xi − xj , 0} + i xj n−1 n−1 j= / i
= (1 + (n − 1)i )xi −
˛ i n−1
− i
j= / i
j= / i
max{xj − xi , 0} −
j= / i
ˇi + i n−1
max{xi − xj , 0}
j= / i
4 This is nicely illustrated in Kohler (accepted) who comments on the “calibration” of his model based on the ultimatum game: “The depicted altruism and inequality aversion parameter distribution is a sample from a continuum of distributions that allow for a positive degree of altruism in the UG.” 5 Note that ˇi = 1/2 is the cut-off level for dictators to split equally in the standard dictator game. Thus those with ˇi > 1/2 are those who will split equally whenever the efficiency losses from transfers are not too large, whereas those with ˇi < 1/2 are those who will split equally only if there are some efficiency gains in transfers. In line with intuition, with increasing efficiency concerns dominating aversion towards advantageous inequality, the tolerated efficiency losses or required efficiency gains converge to zero as ˇi converges to 1/2. 6 ˛ i < −1/2 implies that i would strictly prefer to transfer a monetary unit to the better-off player j because this increases xj − xi by two monetary units. Thus by continuity i would also be happy to give away z if j received even slightly less than z.
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Note that as long as xj = / xi for all j = / i, after applying the maximum operator each xj appears exactly once in either of the two sums and xi appears a total of n − 1 times in both sums. If xj = xi for a j = / i, then the equation is still correct, because we could just include such a case in either of the two sums. Thus we see that, as for the two-player case, the preferences i determined by Ui are identical to those captured by a two-parameter utility function U
i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} n−1 n−1 j= / i
j= / i
i = (˛i − (n − 1)i )/(1 + (n − 1)i ) and ˇi = (ˇi + (n − 1)i )/(1 + (n − 1)i ). with ˛ Again, we see that introducing efficiency concerns has the consequence of reducing aversion towards disadvantageous inequality and increasing aversion towards advantageous inequality and, if sufficiently strong, yields positive willingness to pay to increase another subject’s payoff even if this subject already has a higher payoff (we can see this from the formula for i , but could also confirm this with the aid of calculating the marginal rates of substitution as above). Thus again, the effect ˛ is the same as simply abandoning the ˛i ≥ 0 and the ˛i ≥ ˇi condition. In contrast to the two-player case, already a relatively small i is sufficient in order for the efficiency concerns to dominate the aversion towards disadvantageous inequality. This is a consequence of the fact that the parameters for inequality aversion are normalized, whereas the parameter for efficiency concerns is not, following the parent models of the different components. In effect, the normalization of the inequality aversion parameters means that when I am in a game with more players, my disutility from them all being richer is not larger than in a game with fewer players, whereas since the total surplus component is not normalized, my utility derived from their total surplus is larger when more players all get the same increase. Hence the more players there are, the easier the efficiency concerns will dominate. This matter is crucial for the results in the next section and will be discussed there. 4. Considering games with different numbers of players As demonstrated in Section 3, if we want to use an inequality-aversion-plus-efficiency model to fit data for classes of games with a fixed number of players, we can achieve the same, and without running into econometric problems caused by collinearity of the parameters, by simply relaxing the restrictions on the parameters in the Fehr–Schmidt model.
i and ˇi depend, however, on n and thus if we consider trying to capture behavior of a subject (or a group The formulas for ˛ of subjects) across a range of games with different numbers of players with the basic inequality-aversion-plus-efficiency model, we cannot simply transform the three-parameter model into a two-parameter model as above. This results from the inequality-aversion parameters being normalized by dividing by n − 1, whereas the efficiency parameter is not, so that in
i and ˇi the weight on i increases in n. I chose this formulation of the inequality-aversion-plus-efficiency the formulas for ˛ model because this would amount to the most simple approach of simply putting together the Fehr–Schmidt model with the efficiency component of the Charness–Rabin model. In this case adding the third parameter actually does something other than just relaxing the conditions of the Fehr–Schmidt model. Note, however, that precisely for this reason the model has implausible implications, and this is best shown by considering the transformation of a given triple of parameters into two parameters as above. Inspecting the n-player case reveals that the i > 0 for n = 2, ˛i = 0 larger n, the more easily efficiency concerns dominate inequality aversion, e.g., for ˛i = 1 and i = 1/2, ˛ i < 0 for n > 3. Thus the model has completely different implications for different numbers of players (note for n = 3 and ˛ that this is not a consequence of the transformation of the parameters, the transformation just illustrates this property). A subject with the above parameters would be inequality averse towards one person with higher payoff in a two-player game, so would be willing to pay to reduce this person’s payoff, but would be indifferent towards the payoff of any player with a higher payoff in a three-player game, and would be willing to pay to increase the payoff of any player with a higher payoff in a game with more than three players. This rather implausible prediction (for which I am unaware of any experimental evidence) is not a consequence of the i > 0 and for all parameters chosen here for illustration. For any ˛i > i > 0 there is an n such that for all n < n we obtain ˛ i < 0. Thus the model predicts that if there are only sufficiently many rich people, I do not mind them being n > n we obtain ˛ rich anymore and want them to be even richer. Therefore, in the only case where the suggested inequality-aversion-plusefficiency-concerns model does not just capture the same preferences as the Fehr–Schmidt model with relaxed restrictions on the parameters, it yields implausible predictions. These implausible predictions result from the normalization (or lack thereof) in the parent models, which are arguably not very plausible by themselves if we take them literally (which they are certainly not meant to be) and more importantly, are not plausible in combination. Let us take the normalization in the Fehr–Schmidt model seriously. Consider a standard dictator game and a player who is just indifferent between splitting the money equally and keeping all for herself, i.e., ˇi = 1/2. Now put the same person in a dictator game with 1000 recipients and for each unit of money she gives up, all the recipients will receive one unit of money. Also in this game, she would be indifferent between giving up half her money and keeping it all, because in the first case, with pie size normalized to 1, her utility is simply 1/2 because she gives away half her money and 1000 there is no inequality. If she keeps all her money, her utility is Ui = 1 − (1/(2 · 1000)) j=1 1 = 1/2. It seems hardly plausible that a person who is sufficiently concerned with the welfare of another person that she would share with that person her
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own wealth, would not be willing to do more to make 1000 people better off. Thus, taking the normalization literally rather than as a simplified short-cut to take the number of players into account, we obtain rather implausible predictions already in the basic Fehr–Schmidt model.7 Consider in turn the lack of normalization of the efficiency component in the Charness–Rabin model. This yields similarly implausible predictions. Start with a dictator game with some efficiency gains, say for each unit of money the dictator gives up, the recipient receives two units, and a dictator who would just be indifferent between transferring any money to the recipient or not, so with (according to the notation above) i = 1/2. Now consider again a game with 1000 recipients and these recipients would in total receive two units of money for each unit the dictator gives up. The same dictator should now be indifferent between keeping one unit or giving it away so that each recipient receives 2/1000 (in both games the loss of one unit is compensated by an increase of the total payoff by 1). This again seems an implausible prediction. Thus both the extreme normalization in the Fehr–Schmidt model and the absence of normalization in the Charness–Rabin model have implausible predictions. The first ignores that generous people should be willing to pay more to help many people than they are to help just a single person. The second ignores that if the benefits of one’s actions get spread much more widely and the individual benefits thus decrease, one would most likely be less willing to give. While I am unaware of experimental evidence regarding these predictions, they both seem unintuitive and at least one of them must be wrong. The combination then of the extreme normalization of the inequality aversion parameters with an absence of normalization of the efficiency parameter yields the above result that with an increasing number of players, efficiency concerns will dominate inequality aversion, so that I dislike few rich people, but like many rich people. Given that the combination of the different approaches to take differences in the numbers of players into account was responsible for the implausible predictions of the inequality-aversion-plus-efficiency model, let us consider as an alternative that all parameters are normalized in the same way, namely we divide the efficiency component by n − 1 as well. Then we obtain: Ui (x)
= xi −
˛i ˇi i max{xj − xi , 0} − max{xi − xj , 0} + xj n−1 n−1 n−1 j= / i
˛ − i i
= (1 + i )xi −
n−1
j= / i
max{xj − xi , 0} −
j= / i
ˇi + i n−1
j= / i
max{xi − xj , 0}
j= / i
i Again the preferences determined by Ui are identical to those captured by a two-parameter utility function U i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} n−1 n−1 j= / i
j= / i
i = (˛i − i )/(1 + i ) and ˇi = (ˇi + i )/(1 + i ). In this case, the two-parameter model captures the same preferences with ˛ as the three-parameter model for all n simultaneously. While this result might look like a fatal blow to the three-parameter approach, note that it depends crucially on the normalization by dividing by n − 1. As argued above, this normalization yields implausible predictions if we simultaneously consider games with very different numbers of players. A more reasonable approach to take into account that subjects’ preferences are expected to depend also on the number of players would probably be one where the normalization would fall in between the extreme cases, such as dividing by the square root of n. For any such approach the three-parameter model does not directly collapse to a two-parameter model for all n. To see this, consider a general model where all three parameters are normalized by some function of the number of players, g(n). Then we obtain: Ui (x)
= xi −
=
˛i ˇi i max{xj − xi , 0} − max{xi − xj , 0} + xj g(n) g(n) g(n) j= / i
1+
n−1 g(n)
i xi −
˛ − i i g(n)
j= / i
max{xj − xi , 0} −
j= / i
j= / i
ˇi + i g(n)
max{xi − xj , 0}
j= / i
We can express these preferences also by a utility function
i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} g(n) g(n) j= / i
j= / i
7 Fehr and Schmidt (1999), explicitly discuss the consistency of behavior across various games with different numbers of players. Specifically, in Section 5 they compare the data from public good games with four players and market games with six players to the predictions of their model with parameters fitted based on two-player ultimatum game data. The qualitative predictions they make in other sections, however, would also survive with a different normalization.
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i = (˛i − i )/(1 + ((n − 1)/g(n))i ) and ˇi = (ˇi + i )/(1 + ((n − 1)/g(n))i ) are not independent of n and thus we cannot but ˛ simply express the three-parameter model by the same two-parameter model for different n. Therefore, if for a given number of players n, we have two decision makers with a different set of parameters (˛i , ˇi , i ) i = ˛j and ˇi = ˇj , such that they will behave in the same way, for and (˛j , ˇj , j ) that, however, yield the same parameters ˛ i =/ ˛j and ˇi =/ ˇj and thus different behavior may result. For example, let g(n) = a different number of players n , ˛
(n − 1)
i = ˛j = 1/2 and ˇi = ˇj = 1/2 and thus and let (˛i , ˇi , i ) = (2, 0, 1) and (˛j , ˇj , j ) = (1/2, 1/2, 0). Then for n = 2 we obtain ˛ both players have the same preferences (this also demonstrates again that we cannot identify the three-parameter model from observations of games with a fixed number of players). If we consider now the same decision makers i, j but games
i = (2 − 1)/(1 + (4/2)) = 1/3 =/ 1/2 = ˛j and ˇi = 1/(1 + (4/2)) = 1/3 =/ 1/2 = ˇj . with five players, then we obtain ˛ Therefore, if we simultaneously consider games with different numbers of players and a normalization other than division by n − 1, a three-parameter model could in principle have more explanatory power than a two-parameter model. Close
i and ˇi reveals, however, that they do not vary that much with n, so that if we consider inspection of the formulas for ˛ games with different, but similar, numbers of players we would get pretty close with a two-parameter model. Note that in the above example, even though we made quite a jump from 2-player to 5-player games and intentionally chose parameter combinations to generate large effects of n, the resulting parameters do not differ dramatically. Moreover, in order to fit such a model, one would probably have to estimate the normalization function g(n) as well (unless one wants to make ad hoc assumptions) and then we have actually arrived at a four-parameter model. Certainly, more elegant and logically consistent models can be found, because even in this case the implausible assumption that I simultaneously want to reduce and increase a richer person’s payoff remains the foundation of the model. The main effect of adding the efficiency parameter is again that it reduces aversion towards disadvantageous inequality and increases aversion towards advantageous inequality. That it does not disappear in the case of considering games with different numbers of players only results because it impacts on the normalization. That should be much better captured by a model that allows for heterogeneity in the way subjects adjust for the number of players, i.e., heterogeneity in the normalization parameter. 5. Combining the Bolton–Ockenfels model with efficiency concerns An alternative model of inequality aversion has been proposed by Bolton and Ockenfels (2000). The conceptual inconsistency of combining inequality aversion with efficiency concerns is even more obvious when we use this model of inequality aversion. In the Bolton–Ockenfels model, utility depends on one’s absolute payoff as well as on one’s share of the total payoff, where for given absolute payoff, utility is maximized when one’s share of the total payoff equals the “fair share”, i.e., one receives 1/n of the total payoff (with n the number of players). In this approach, holding one’s share fixed, increasing the (other players’) total payoff is equivalent to increasing one’s own absolute payoff, whereas holding one’s own absolute payoff fixed, increasing the (other players’) total payoff amounts to decreasing one’s share of the total payoff. Thus taking a given Bolton–Ockenfels utility function and adding efficiency concerns only amounts to changing the curvature and shifting the optimal own share to below the fair share. The latter results because Bolton and Ockenfels assume a smooth utility function that reaches its maximum where one’s own share equals the fair share, so if we add the ever so slightest positive concern for total payoff, this means that at the previously optimal own share, I will want to increase total payoff, so the optimal own share will be somewhat below the fair share. So again, adding efficiency concerns just amounts to relaxing one of the assumptions of the model. It also seems to be against the spirit of this model in the sense that the fundamental difference of the Bolton–Ockenfels model from the Fehr–Schmidt model is the assumption that fairness concerns only enter through the share that one receives of the total payoff. 6. Conclusions
This note has demonstrated that if we add an efficiency component to an inequality aversion model, then as long as only games with a given number of players are considered, as is done in most experimental papers, the parameters of the model cannot be identified and this approach does not add anything in terms of explanatory power that we could not also obtain by simply relaxing some of the restrictions on the inequality aversion parameters. Furthermore, it has been shown that if data from games with different numbers of players shall be fitted simultaneously, then there could potentially be a better fit by an inequality-aversion-plus-efficiency model than simply by an inequality aversion model with relaxed restrictions on the parameters. A number of observations are important here. First, the examples in Section 4 demonstrate that while in principle such a three-parameter model could add explanatory power if we consider games with different numbers of players, the simple version of adding an efficiency component to the Fehr–Schmidt model has implausible predictions (if the efficiency component is not normalized) or again collapses to a two-parameter model (if the efficiency component is normalized in the same way as the inequality aversion components). Second, the gain that could possibly be obtained by adding the efficiency component combined with some other form of normalization than simply dividing by n − 1 could be obtained in much more elegant ways by considering that individual subjects might differ in how their utility from payoff comparisons changes with
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the number of players, that is, we could consider heterogeneity in the normalization of the parameters. This should lead to a model that does not suffer from the philosophical contradictions as well as implausible predictions of a model that mixes inequality aversion with efficiency concerns. Whether such a model would be useful at all, however, is first of all an empirical question, and I am not aware of any evidence that subjects indeed differ in how they adjust for the numbers of players. From a general perspective, if we want a model that is based on the idea that subjects are generally altruistic, but less so towards people who have higher wealth with the possibility that they could become spiteful, a much better way than extending the Fehr–Schmidt model is already presented by Cox et al. (2008), with the added benefit of accounting for reciprocity. Acknowledgement The author thanks Hans-Theo Normann, Jörg Oechssler, two anonymous referees and the editor William S. Neilson for helpful comments. Financial support from the institutional research grant AV0Z70850503 of the Economics Institute of the Academy of Sciences of the Czech Republic, v.v.i. is gratefully acknowledged. References Blanco, M., Engelmann, D., Normann, H.-T., 2011. A within-subject analysis of other-regarding preferences. Games and Economics Behavior 72, 321–338. Bolton, G.E., Ockenfels, A., 2000. ERC: a theory of equity, reciprocity and competition. American Economic Review 90, 166–193. Charness, G., Rabin, M., 2002. Understanding social preferences with simple tests. Quarterly Journal of Economics 117, 817–869. Cox, J.C., Friedman, D., Sadiraj, V., 2008. Revealed altruism. Econometrica 76, 31–69. Ellison, B., Lusk, J.L., Briggeman, B., 2010. Other-regarding behavior and taxpayer preferences for farm policy. The B.E. Journal of Economic Analysis & Policy 10 (Topics), Article 96. Engelmann, D., Strobel, M., 2004. Inequality aversion, efficiency and maximin preferences in simple distribution experiments. American Economic Review 94, 857–869. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition and cooperation. Quarterly Journal of Economics 114, 817–868. Kohler, S. Altruism and fairness in experimental decisions. Journal of Economic Behavior and Organization, doi:10.1016/j.jebo.2011.02.014, in press. Kritikos, A., Bolle, F., 2001. Distributional concerns: equity- or efficiency-oriented? Economics Letters 73, 333–338.
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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo
How not to extend models of inequality aversion Dirk Engelmann a,b,c,d,∗ a b c d
Department of Economics, University of Mannheim, L7, 3-5, D-68131 Mannheim, Germany Centre for Experimental Economics, University of Copenhagen, Øster Farimagsgade 5, 1353 Copenhagen, Denmark Economics Institute of the Academy of Sciences of the Czech Republic, Politick´ ych vˇeznu˙ 7, 111 21 Prague 1, Czech Republic CESifo, Poschingerstr. 5, 81679 Munich, Germany
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 12 August 2011 Accepted 15 August 2011 Available online 22 August 2011 JEL classification: C72 C91 C92
a b s t r a c t Several authors have made attempts to improve the explanatory power of models of inequality aversion, in particular the one by Fehr and Schmidt (1999), by adding concerns for total surplus or efficiency. In this note, I point out that these attempts are misguided because they are equivalent to a much simpler change, not requiring an additional parameter, unless we simultaneously consider games with different numbers of players. In the latter case, however, such an approach yields implausible predictions. © 2011 Elsevier B.V. All rights reserved.
Keywords: Inequality aversion Efficiency Social preferences
1. Introduction Models of inequality aversion (Bolton and Ockenfels, 2000; Fehr and Schmidt, 1999) have had substantial success in rationalizing experimental data. However, in a number of experiments they do poorly, in particular in dictator-type games where dictators appear to care to a substantial degree for maximizing the total surplus (see e.g., Kritikos and Bolle, 2001; Charness and Rabin, 2002; Engelmann and Strobel, 2004). This observation has resulted in recent papers that attempt to improve the performance of the inequality aversion models (typically using Fehr–Schmidt as a basis) by adding a term for efficiency concerns (i.e., total surplus maximization or altruism with identical weights on all subjects). These papers come about either as theoretical papers claiming to provide a useful generalization of the Fehr–Schmidt model (see e.g., Kohler, accepted) or as experimental papers, trying to get a better fit of their data by adding an extra term (see e.g., Ellison et al., 2010).1 The most simple version of this approach just adds an
∗ Correspondence address: Department of Economics, University of Mannheim, L7, 3-5, D-68131 Mannheim, Germany. Tel.: +49 621 181 1894; fax: +49 621 181 1893. 1 Typically, I came across these (anonymized) papers as a referee. Similar suggestions have been made as comments on Blanco et al. (2011). I actually explained the arguments of this note to both Kohler and Ellison et al. in referee reports. As these are the only published papers I am aware of that use such an approach and the other authors remained anonymous, I refrain from further citations here. 0167-2681/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2011.08.007
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efficiency concerns parameter (borrowed in essence from the Charness–Rabin model of social-welfare preferences) to the standard Fehr–Schmidt model, which then becomes Ui (x) = xi −
˛i ˇi max{xj − xi , 0} − max{xi − xj , 0} + i xj n−1 n−1 j= / i
j= / i
j= / i
with xi the payoff of subject i and the usual conditions of the Fehr–Schmidt model ˛i ≥ ˇi ≥ 0 and ˇi < 1, but in addition 0 ≤ i ≤ 1 (in principle, one could allow for i > 1 and thus include people who care more about others’ payoffs than their own, but that is probably empirically irrelevant).2 This note will point out that this approach is misguided, as adding the extra parameter does nothing other than allowing for ˛i < 0 and ˛i < ˇi as long as we only consider games with a constant number of players and has questionable implications if we try to fit such a utility function to a set of games with different numbers of players. Section 2 will illustrate this in the two-player setting, Section 3 will deal with the general n-player case and Section 4 with the case of simultaneously considering games with different numbers of players. While the latter sections are getting somewhat more technical, the basic point is straightforward and is best illustrated by the two-player case. It should also be stressed that a careful reading of Charness and Rabin (2002) already shows the main point for the two-player case. So this note does not have a claim of fundamental novelty. Nevertheless, the repeated occurrence of such models and suggestions indicates that an explicit statement of these results might be useful. Section 5 will briefly comment on problems with a similar approach based on the Bolton–Ockenfels model and Section 6 summarizes the main lessons. 2. The two-player case The basic logic of the argument is most easily illustrated in the two-player case. If there is only one other player, this player can have a higher or lower payoff than oneself (or the same, which can be included in either of the other cases). The suggested model then becomes
Ui (xi , xj ) =
xi − ˛i (xj − xi ) + i xj ,
if xi ≤ xj
xi − ˇi (xi − xj ) + i xj ,
if xi > xj
(1)
or Ui (xi , xj ) = xi − ˛i max {xj − xi , 0} − ˇi max {xi − xj , 0} + i xj . To understand this model, we need to answer the following question. If the other player has a higher payoff and I am envious (˛i > 0), what does it mean if I also like to maximize the (other players’) total payoff ( i > 0)? It means that while I wish to reduce j’s payoff, I also have an interest in increasing it. Depending on the relative strength of the concerns, this simply corresponds to ˛i in the standard model becoming smaller, or possibly negative, as the below reformulation of the model will show.3 If the other player has a lower payoff and I am averse to advantageous inequality (ˇi > 0) what does it mean if I also like to maximize the (other players’) total payoff ( i > 0)? It means that I like giving to j a bit more, which corresponds to ˇi in the standard model becoming larger. Let us reformulate the model to clarify this. Consider a player maximizing a utility function based on inequality aversion and total surplus maximization Ui as in (1). Simple manipulation yields
Ui (xi , xj ) =
xi − ˛i (xj − xi ) + i xj = (1 + i )xi − (˛i − i )(xj − xi ),
if xi ≤ xj
xi − ˇi (xi − xj ) + i xj = (1 + i )xi − (ˇi + i )(xi − xj ),
if xi > xj
(2)
or Ui (xi , xj ) = (1 + i )xi − (˛i − i ) max{xj − xi , 0} − (ˇi + i ) max{xi − xj , 0} Thus any preferences determined by utility function Ui can also be described by a two-parameter utility function
i (xi , xj ) = xi − ˛i max{xj − xi , 0} − ˇi max{xi − xj , 0} U i = (˛i − i )/(1 + i ) and ˇi = (ˇi + i )/(1 + i ). with ˛ 2 Alternatively, one can include xi in the last term, so that this term corresponds to the total payoff over all players instead of over all other players. This i = (˛i − (n − 1)i )/(1 + ni ) and does not change the argument in any way. The only difference is that then in the reformulated two-parameter model ˛
i = (ˇi + (n − 1)i )/(1 + ni ). See Section 3 for the interpretation of ˛i and ˇi . The particular model used in this note is the n-player generalization of the ˇ two-player model in Kohler (accepted). The approach to fit their data presented by Ellison et al. (2010) is slightly different. They combine the Fehr–Schmidt model with efficiency concerns in a model that contains only two parameters by first imposing the restriction ˛i = ˇi and then adding altruism. They claim this is both more restrictive and more general than the Fehr–Schmidt model. It is, however, at best a tedious generalization because it just corresponds to enforcing ˛i < ˇi and allowing for ˛i < 0 (more precisely, replacing ˛i ≥ 0 with ˛i ≥ − ˇi ). Moreover, they extend the model by allowing subjects to differ in their degree of altruism towards different groups of other players. 3 The three-parameter model is also psychologically and philosophically implausible. It amounts to saying: “You’re rich, so I want to hurt you. But I also like people to be wealthy, so I want to help you.” - “So do you want to hurt or help me?” - “Both, at least a bit. Maybe one more than the other, but really, really, both. So even if I want to help you a lot, I still want to hurt you a bit at the same time.” Obviously, either the helping or hurting has to dominate and thus this is what will be done.
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The three-parameter model is not only unnecessarily complicated, it is also not useful for empirical applications, because
i , ˇi ) that already the parameters cannot be identified, since we can choose infinitely many (˛i , ˇi , i ) vectors for any given (˛ completely determine the preferences.4 To understand the effects of adding efficiency concerns a bit better, let us calculate the marginal rates of substitution of own for other payoff. This also confirms that the two utility functions describe the same preferences, because if the marginal rates of substitution are identical for the two utility functions, they describe the same preferences. Consider first the case xj > xi . Then ∂Ui /∂xj −˛i + i = 1 + ˛i ∂Ui /∂xi ∂Ui /∂xj ∂Ui /∂xi
=
i −˛
i 1+˛
=
∂Ui /∂xj −˛i + i −(˛i − i )/(1 + i ) = = 1 + i + ˛i − i 1 + (˛i − i )/(1 + i ) ∂Ui /∂xi
Next consider the case xi > xj . Then
∂Ui /∂xj ˇ + i = i 1 − ˇi ∂Ui /∂xi ∂Ui /∂xj ∂Ui /∂xi
=
i ˇ
i 1−ˇ
=
∂Ui /∂xj (ˇi + i )/(1 + i ) ˇi + i = = 1 − (ˇi + i )/(1 + i ) 1 + i − ˇi − i ∂Ui /∂xi
i indeed represent the same preferences, they also provide The marginal rates of substitution not only confirm that Ui and U maybe a clearer way of understanding what having concerns for total surplus maximization means for the Fehr–Schmidt parameters. Concerning the case of disadvantageous inequality (xj > xi ), we see, in line with intuition, that efficiency concerns have the effect of reducing the marginal willingness to pay to reduce the other subject’s payoff and may even turn it into willingness to pay to increase it (if i > ˛i ). Thus, introducing efficiency concerns has the same effect as abolishing the restrictions ˛i ≥ ˇi and ˛i ≥ 0 in the Fehr–Schmidt model (or more precisely, replacing the latter with ˛i ≥ − 1/2). Concerning the case of advantageous inequality aversion, if i converges to 1 and ˇi converges to 0, the marginal rate of substitution approaches 1, which corresponds to ˇi = 1/2 in the standard Fehr–Schmidt model.5 Since ˇi = 1/2 is perfectly within the restrictions of the Fehr–Schmidt model, adding efficiency concerns provides no extension of the model for the case of advantageous inequality. i cannot One difference between the three-parameter model and a two-parameter model that allows for ˛i < 0 is that ˛ be smaller than −1/2 and hence this model puts a cap on the degree of altruism towards richer people. While we could also just add this restriction to the two-parameter model, it seems hardly a restriction in practise anyway as this only excludes choices where a person would spend z in order to increase the payoff of an already better-off person by even something slightly less than z, behavior which will most likely almost never be observed.6 So a simple two-parameter model which does not put any restrictions on ˛i < 0 is somewhat more general, but only by allowing for empirically irrelevant behavior. 3. The n-player case The previous section has shown for the two-player case that the inequality-aversion-plus-efficiency-concerns model represents the same preferences as the Fehr–Schmidt model with weaker restrictions on the parameters. The n-player case in no way differs, as long as we consider the number of players n as given, but is just a bit more tedious. So let us turn to this now for the sake of completeness. As above, let us manipulate the three-parameter model. Ui (x)
= xi −
˛i ˇi max{xj − xi , 0} − max{xi − xj , 0} + i xj n−1 n−1 j= / i
= (1 + (n − 1)i )xi −
˛ i n−1
− i
j= / i
j= / i
max{xj − xi , 0} −
j= / i
ˇi + i n−1
max{xi − xj , 0}
j= / i
4 This is nicely illustrated in Kohler (accepted) who comments on the “calibration” of his model based on the ultimatum game: “The depicted altruism and inequality aversion parameter distribution is a sample from a continuum of distributions that allow for a positive degree of altruism in the UG.” 5 Note that ˇi = 1/2 is the cut-off level for dictators to split equally in the standard dictator game. Thus those with ˇi > 1/2 are those who will split equally whenever the efficiency losses from transfers are not too large, whereas those with ˇi < 1/2 are those who will split equally only if there are some efficiency gains in transfers. In line with intuition, with increasing efficiency concerns dominating aversion towards advantageous inequality, the tolerated efficiency losses or required efficiency gains converge to zero as ˇi converges to 1/2. 6 ˛ i < −1/2 implies that i would strictly prefer to transfer a monetary unit to the better-off player j because this increases xj − xi by two monetary units. Thus by continuity i would also be happy to give away z if j received even slightly less than z.
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Note that as long as xj = / xi for all j = / i, after applying the maximum operator each xj appears exactly once in either of the two sums and xi appears a total of n − 1 times in both sums. If xj = xi for a j = / i, then the equation is still correct, because we could just include such a case in either of the two sums. Thus we see that, as for the two-player case, the preferences i determined by Ui are identical to those captured by a two-parameter utility function U
i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} n−1 n−1 j= / i
j= / i
i = (˛i − (n − 1)i )/(1 + (n − 1)i ) and ˇi = (ˇi + (n − 1)i )/(1 + (n − 1)i ). with ˛ Again, we see that introducing efficiency concerns has the consequence of reducing aversion towards disadvantageous inequality and increasing aversion towards advantageous inequality and, if sufficiently strong, yields positive willingness to pay to increase another subject’s payoff even if this subject already has a higher payoff (we can see this from the formula for i , but could also confirm this with the aid of calculating the marginal rates of substitution as above). Thus again, the effect ˛ is the same as simply abandoning the ˛i ≥ 0 and the ˛i ≥ ˇi condition. In contrast to the two-player case, already a relatively small i is sufficient in order for the efficiency concerns to dominate the aversion towards disadvantageous inequality. This is a consequence of the fact that the parameters for inequality aversion are normalized, whereas the parameter for efficiency concerns is not, following the parent models of the different components. In effect, the normalization of the inequality aversion parameters means that when I am in a game with more players, my disutility from them all being richer is not larger than in a game with fewer players, whereas since the total surplus component is not normalized, my utility derived from their total surplus is larger when more players all get the same increase. Hence the more players there are, the easier the efficiency concerns will dominate. This matter is crucial for the results in the next section and will be discussed there. 4. Considering games with different numbers of players As demonstrated in Section 3, if we want to use an inequality-aversion-plus-efficiency model to fit data for classes of games with a fixed number of players, we can achieve the same, and without running into econometric problems caused by collinearity of the parameters, by simply relaxing the restrictions on the parameters in the Fehr–Schmidt model.
i and ˇi depend, however, on n and thus if we consider trying to capture behavior of a subject (or a group The formulas for ˛ of subjects) across a range of games with different numbers of players with the basic inequality-aversion-plus-efficiency model, we cannot simply transform the three-parameter model into a two-parameter model as above. This results from the inequality-aversion parameters being normalized by dividing by n − 1, whereas the efficiency parameter is not, so that in
i and ˇi the weight on i increases in n. I chose this formulation of the inequality-aversion-plus-efficiency the formulas for ˛ model because this would amount to the most simple approach of simply putting together the Fehr–Schmidt model with the efficiency component of the Charness–Rabin model. In this case adding the third parameter actually does something other than just relaxing the conditions of the Fehr–Schmidt model. Note, however, that precisely for this reason the model has implausible implications, and this is best shown by considering the transformation of a given triple of parameters into two parameters as above. Inspecting the n-player case reveals that the i > 0 for n = 2, ˛i = 0 larger n, the more easily efficiency concerns dominate inequality aversion, e.g., for ˛i = 1 and i = 1/2, ˛ i < 0 for n > 3. Thus the model has completely different implications for different numbers of players (note for n = 3 and ˛ that this is not a consequence of the transformation of the parameters, the transformation just illustrates this property). A subject with the above parameters would be inequality averse towards one person with higher payoff in a two-player game, so would be willing to pay to reduce this person’s payoff, but would be indifferent towards the payoff of any player with a higher payoff in a three-player game, and would be willing to pay to increase the payoff of any player with a higher payoff in a game with more than three players. This rather implausible prediction (for which I am unaware of any experimental evidence) is not a consequence of the i > 0 and for all parameters chosen here for illustration. For any ˛i > i > 0 there is an n such that for all n < n we obtain ˛ i < 0. Thus the model predicts that if there are only sufficiently many rich people, I do not mind them being n > n we obtain ˛ rich anymore and want them to be even richer. Therefore, in the only case where the suggested inequality-aversion-plusefficiency-concerns model does not just capture the same preferences as the Fehr–Schmidt model with relaxed restrictions on the parameters, it yields implausible predictions. These implausible predictions result from the normalization (or lack thereof) in the parent models, which are arguably not very plausible by themselves if we take them literally (which they are certainly not meant to be) and more importantly, are not plausible in combination. Let us take the normalization in the Fehr–Schmidt model seriously. Consider a standard dictator game and a player who is just indifferent between splitting the money equally and keeping all for herself, i.e., ˇi = 1/2. Now put the same person in a dictator game with 1000 recipients and for each unit of money she gives up, all the recipients will receive one unit of money. Also in this game, she would be indifferent between giving up half her money and keeping it all, because in the first case, with pie size normalized to 1, her utility is simply 1/2 because she gives away half her money and 1000 there is no inequality. If she keeps all her money, her utility is Ui = 1 − (1/(2 · 1000)) j=1 1 = 1/2. It seems hardly plausible that a person who is sufficiently concerned with the welfare of another person that she would share with that person her
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own wealth, would not be willing to do more to make 1000 people better off. Thus, taking the normalization literally rather than as a simplified short-cut to take the number of players into account, we obtain rather implausible predictions already in the basic Fehr–Schmidt model.7 Consider in turn the lack of normalization of the efficiency component in the Charness–Rabin model. This yields similarly implausible predictions. Start with a dictator game with some efficiency gains, say for each unit of money the dictator gives up, the recipient receives two units, and a dictator who would just be indifferent between transferring any money to the recipient or not, so with (according to the notation above) i = 1/2. Now consider again a game with 1000 recipients and these recipients would in total receive two units of money for each unit the dictator gives up. The same dictator should now be indifferent between keeping one unit or giving it away so that each recipient receives 2/1000 (in both games the loss of one unit is compensated by an increase of the total payoff by 1). This again seems an implausible prediction. Thus both the extreme normalization in the Fehr–Schmidt model and the absence of normalization in the Charness–Rabin model have implausible predictions. The first ignores that generous people should be willing to pay more to help many people than they are to help just a single person. The second ignores that if the benefits of one’s actions get spread much more widely and the individual benefits thus decrease, one would most likely be less willing to give. While I am unaware of experimental evidence regarding these predictions, they both seem unintuitive and at least one of them must be wrong. The combination then of the extreme normalization of the inequality aversion parameters with an absence of normalization of the efficiency parameter yields the above result that with an increasing number of players, efficiency concerns will dominate inequality aversion, so that I dislike few rich people, but like many rich people. Given that the combination of the different approaches to take differences in the numbers of players into account was responsible for the implausible predictions of the inequality-aversion-plus-efficiency model, let us consider as an alternative that all parameters are normalized in the same way, namely we divide the efficiency component by n − 1 as well. Then we obtain: Ui (x)
= xi −
˛i ˇi i max{xj − xi , 0} − max{xi − xj , 0} + xj n−1 n−1 n−1 j= / i
˛ − i i
= (1 + i )xi −
n−1
j= / i
max{xj − xi , 0} −
j= / i
ˇi + i n−1
j= / i
max{xi − xj , 0}
j= / i
i Again the preferences determined by Ui are identical to those captured by a two-parameter utility function U i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} n−1 n−1 j= / i
j= / i
i = (˛i − i )/(1 + i ) and ˇi = (ˇi + i )/(1 + i ). In this case, the two-parameter model captures the same preferences with ˛ as the three-parameter model for all n simultaneously. While this result might look like a fatal blow to the three-parameter approach, note that it depends crucially on the normalization by dividing by n − 1. As argued above, this normalization yields implausible predictions if we simultaneously consider games with very different numbers of players. A more reasonable approach to take into account that subjects’ preferences are expected to depend also on the number of players would probably be one where the normalization would fall in between the extreme cases, such as dividing by the square root of n. For any such approach the three-parameter model does not directly collapse to a two-parameter model for all n. To see this, consider a general model where all three parameters are normalized by some function of the number of players, g(n). Then we obtain: Ui (x)
= xi −
=
˛i ˇi i max{xj − xi , 0} − max{xi − xj , 0} + xj g(n) g(n) g(n) j= / i
1+
n−1 g(n)
i xi −
˛ − i i g(n)
j= / i
max{xj − xi , 0} −
j= / i
j= / i
ˇi + i g(n)
max{xi − xj , 0}
j= / i
We can express these preferences also by a utility function
i (x) = xi − U
i i ˛ ˇ max{xj − xi , 0} − max{xi − xj , 0} g(n) g(n) j= / i
j= / i
7 Fehr and Schmidt (1999), explicitly discuss the consistency of behavior across various games with different numbers of players. Specifically, in Section 5 they compare the data from public good games with four players and market games with six players to the predictions of their model with parameters fitted based on two-player ultimatum game data. The qualitative predictions they make in other sections, however, would also survive with a different normalization.
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i = (˛i − i )/(1 + ((n − 1)/g(n))i ) and ˇi = (ˇi + i )/(1 + ((n − 1)/g(n))i ) are not independent of n and thus we cannot but ˛ simply express the three-parameter model by the same two-parameter model for different n. Therefore, if for a given number of players n, we have two decision makers with a different set of parameters (˛i , ˇi , i ) i = ˛j and ˇi = ˇj , such that they will behave in the same way, for and (˛j , ˇj , j ) that, however, yield the same parameters ˛ i =/ ˛j and ˇi =/ ˇj and thus different behavior may result. For example, let g(n) = a different number of players n , ˛
(n − 1)
i = ˛j = 1/2 and ˇi = ˇj = 1/2 and thus and let (˛i , ˇi , i ) = (2, 0, 1) and (˛j , ˇj , j ) = (1/2, 1/2, 0). Then for n = 2 we obtain ˛ both players have the same preferences (this also demonstrates again that we cannot identify the three-parameter model from observations of games with a fixed number of players). If we consider now the same decision makers i, j but games
i = (2 − 1)/(1 + (4/2)) = 1/3 =/ 1/2 = ˛j and ˇi = 1/(1 + (4/2)) = 1/3 =/ 1/2 = ˇj . with five players, then we obtain ˛ Therefore, if we simultaneously consider games with different numbers of players and a normalization other than division by n − 1, a three-parameter model could in principle have more explanatory power than a two-parameter model. Close
i and ˇi reveals, however, that they do not vary that much with n, so that if we consider inspection of the formulas for ˛ games with different, but similar, numbers of players we would get pretty close with a two-parameter model. Note that in the above example, even though we made quite a jump from 2-player to 5-player games and intentionally chose parameter combinations to generate large effects of n, the resulting parameters do not differ dramatically. Moreover, in order to fit such a model, one would probably have to estimate the normalization function g(n) as well (unless one wants to make ad hoc assumptions) and then we have actually arrived at a four-parameter model. Certainly, more elegant and logically consistent models can be found, because even in this case the implausible assumption that I simultaneously want to reduce and increase a richer person’s payoff remains the foundation of the model. The main effect of adding the efficiency parameter is again that it reduces aversion towards disadvantageous inequality and increases aversion towards advantageous inequality. That it does not disappear in the case of considering games with different numbers of players only results because it impacts on the normalization. That should be much better captured by a model that allows for heterogeneity in the way subjects adjust for the number of players, i.e., heterogeneity in the normalization parameter. 5. Combining the Bolton–Ockenfels model with efficiency concerns An alternative model of inequality aversion has been proposed by Bolton and Ockenfels (2000). The conceptual inconsistency of combining inequality aversion with efficiency concerns is even more obvious when we use this model of inequality aversion. In the Bolton–Ockenfels model, utility depends on one’s absolute payoff as well as on one’s share of the total payoff, where for given absolute payoff, utility is maximized when one’s share of the total payoff equals the “fair share”, i.e., one receives 1/n of the total payoff (with n the number of players). In this approach, holding one’s share fixed, increasing the (other players’) total payoff is equivalent to increasing one’s own absolute payoff, whereas holding one’s own absolute payoff fixed, increasing the (other players’) total payoff amounts to decreasing one’s share of the total payoff. Thus taking a given Bolton–Ockenfels utility function and adding efficiency concerns only amounts to changing the curvature and shifting the optimal own share to below the fair share. The latter results because Bolton and Ockenfels assume a smooth utility function that reaches its maximum where one’s own share equals the fair share, so if we add the ever so slightest positive concern for total payoff, this means that at the previously optimal own share, I will want to increase total payoff, so the optimal own share will be somewhat below the fair share. So again, adding efficiency concerns just amounts to relaxing one of the assumptions of the model. It also seems to be against the spirit of this model in the sense that the fundamental difference of the Bolton–Ockenfels model from the Fehr–Schmidt model is the assumption that fairness concerns only enter through the share that one receives of the total payoff. 6. Conclusions
This note has demonstrated that if we add an efficiency component to an inequality aversion model, then as long as only games with a given number of players are considered, as is done in most experimental papers, the parameters of the model cannot be identified and this approach does not add anything in terms of explanatory power that we could not also obtain by simply relaxing some of the restrictions on the inequality aversion parameters. Furthermore, it has been shown that if data from games with different numbers of players shall be fitted simultaneously, then there could potentially be a better fit by an inequality-aversion-plus-efficiency model than simply by an inequality aversion model with relaxed restrictions on the parameters. A number of observations are important here. First, the examples in Section 4 demonstrate that while in principle such a three-parameter model could add explanatory power if we consider games with different numbers of players, the simple version of adding an efficiency component to the Fehr–Schmidt model has implausible predictions (if the efficiency component is not normalized) or again collapses to a two-parameter model (if the efficiency component is normalized in the same way as the inequality aversion components). Second, the gain that could possibly be obtained by adding the efficiency component combined with some other form of normalization than simply dividing by n − 1 could be obtained in much more elegant ways by considering that individual subjects might differ in how their utility from payoff comparisons changes with
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the number of players, that is, we could consider heterogeneity in the normalization of the parameters. This should lead to a model that does not suffer from the philosophical contradictions as well as implausible predictions of a model that mixes inequality aversion with efficiency concerns. Whether such a model would be useful at all, however, is first of all an empirical question, and I am not aware of any evidence that subjects indeed differ in how they adjust for the numbers of players. From a general perspective, if we want a model that is based on the idea that subjects are generally altruistic, but less so towards people who have higher wealth with the possibility that they could become spiteful, a much better way than extending the Fehr–Schmidt model is already presented by Cox et al. (2008), with the added benefit of accounting for reciprocity. Acknowledgement The author thanks Hans-Theo Normann, Jörg Oechssler, two anonymous referees and the editor William S. Neilson for helpful comments. Financial support from the institutional research grant AV0Z70850503 of the Economics Institute of the Academy of Sciences of the Czech Republic, v.v.i. is gratefully acknowledged. References Blanco, M., Engelmann, D., Normann, H.-T., 2011. A within-subject analysis of other-regarding preferences. Games and Economics Behavior 72, 321–338. Bolton, G.E., Ockenfels, A., 2000. ERC: a theory of equity, reciprocity and competition. American Economic Review 90, 166–193. Charness, G., Rabin, M., 2002. Understanding social preferences with simple tests. Quarterly Journal of Economics 117, 817–869. Cox, J.C., Friedman, D., Sadiraj, V., 2008. Revealed altruism. Econometrica 76, 31–69. Ellison, B., Lusk, J.L., Briggeman, B., 2010. Other-regarding behavior and taxpayer preferences for farm policy. The B.E. Journal of Economic Analysis & Policy 10 (Topics), Article 96. Engelmann, D., Strobel, M., 2004. Inequality aversion, efficiency and maximin preferences in simple distribution experiments. American Economic Review 94, 857–869. Fehr, E., Schmidt, K.M., 1999. A theory of fairness, competition and cooperation. Quarterly Journal of Economics 114, 817–868. Kohler, S. Altruism and fairness in experimental decisions. Journal of Economic Behavior and Organization, doi:10.1016/j.jebo.2011.02.014, in press. Kritikos, A., Bolle, F., 2001. Distributional concerns: equity- or efficiency-oriented? Economics Letters 73, 333–338.