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We expand upon the previous models of inequity aversion of Fehr and Schmidt [

The standard assumption that subjects only care about their own material payoff is frequently used to solve economic models. However, the overwhelming experimental evidence against this assumption (especially in the dictator experiment) indicates that subjects are willing to sacrifice their own material payoff so as to achieve fair allocations (see Camerer [

In the field of experimental economics, the dictator game with production (e.g., Cappelen

Although the importance of the production stage has been crucial to achieve these results, almost no studies of social preferences incorporate the source of the surplus into the theoretical analysis. The models of social preferences that have been put forward to explain dictators’ deviations from narrow self-interest usually focus on the way in which dictators divide the surplus, while leaving aside the way in which the surplus was generated (see Fehr and Schmidt [

In this paper, we propose a simple extension of Fehr and Schmidt [

To the best of our knowledge, there are only two theoretical models that consider the possibility of dictators choosing the accountability principle (Cappelen

The rest of the paper is organized as follows.

Consider the dictator game in which subjects can be labeled iϵ{d,r}, where d represents the dictator and r represents the recipient. The dictator has to divide a certain surplus _{i}≥ 0 for iϵ{d,r}, in particular,
_{i}≥ 0 represents subject i’s performance in a (previous) production stage, and p_{i} > 0 is the weight assigned to this input, for i ϵ{d,r}. In what follows, we shall think that q_{i}≥ 0 is under the subject i's control (e.g., exerted effort, time of work, money to be invested in a project, _{i }> 0 is assumed to be independent of q_{i} ≥ 0 and determines the way in which agent i’s input is transformed into money. We shall think that p_{i }> 0 is outside the subject i's control (e.g., reward level, rate of return, luck,

The dictator has to choose a division of the surplus _{d}, x_{r}) that satisfies x_{d} + x_{r }= _{i}≥ 0 denotes the monetary payoff that subject i will receive, for i ϵ{d,r}. The model of inequality aversion of Fehr and Schmidt [

This function accounts for social preferences because the dictator does not only care about her own monetary payoff. The dictator's utility also depends on the recipient’s payoffs and the relationship between both subjects’ payoffs. In particular, the model of Fehr and Schmidt [

As x_{r }= _{d} , equation (2) can be rewritten as follows:

The prediction of Fehr and Schmidt [

The utility function (4) expands upon the previous one so as to include what the authors call “just deserts”. Frohlich

The model of Frohlich _{r}

Our specification assumes that the dictator cares about her own monetary payoff, but juggles the tradeoff between subjects’ inputs and monetary contributions. This implies that the dictator suffers a cost

The utility function (5) allows dictators to take into account the way in which inputs are transformed into money so as to "compensate" for those factors outside the subjects’ control. Our prediction is that dictators might (i) behave selfishly

We derive these results in the supplementary material. Our model predicts that the dictator will behave selfishly, if 2β + ψ + ω < 1. If it is not the case, the dictator's decision depends on the subjects’ inputs (q_{i}≥ 0) as well as on the weight that is assigned to these inputs (p_{i} > 0). Consider that the dictator is at a relative advantage with regard to the accumulation of money (_{i} ≥ 0 and p_{i }> 0 determine in this framework whether these contributions (

Predictions of our model.

Note. In all the cases above, it is assumed that

Overall, our model in _{d }> p_{r}), because in this case our model predicts a larger set of transfers from the part of the dictator than previous models of inequity aversion. In this section, we present a couple of examples so as to illustrate this feature. We also mention at the end of this section some experimental papers that produce behavior that is consistent with our model, and then discuss the relevance of our approach.

To start with, let us consider a numerical example. Imagine that subjects solve a questionnaire during the production stage. In particular, assume that q_{d }= 10 and q_{r }= 15 are the number of correct answers, which are assumed to be controlled by the subjects (_{d }= 1.5 and p_{r }= 1, respectively, where the reward level is exogenously determined and is independent of performance._{d }= y_{r }=_{d }= y_{r }= _{r} in the interval [

To further illustrate that our model extends upon the previous ones, we consider _{r}/_{r}/_{r }= y_{r}. In _{r}/_{r }=_{r}=a_{r}. Therefore, allocations on this curve indicate that recipients are being transferred exactly the proportion of the surplus that is due to their effort._{d} and p_{r} establishes the concavity of the dotted curve x_{r }= a_{r} and determines those allocations that cannot be predicted by Fehr and Schmidt [_{r }= _{r }≤ max{_{r}}). However, our model takes into account the accountability principle so that our prediction includes the striped area (_{r }≤ max{_{r,}a_{r}}).

Graphical representation of our predictions if p_{d} > p_{r}.

All the allocations on the striped area give some weight to the accountability principle, such that these allocations cannot be explained with the models of inequity aversion of Fehr and Schmidt [

We find that our model generalizes the previous ones and can be used to explain dictators’ behavior, especially when the production stage involves factors within and beyond the subject's control. In that context, the accountability principle is likely to lead dictators' behavior, especially when dictators act as a third party in the distributional problem (e.g., Cappelen _{d }> p_{r}), we will find that 15-percent of the dictators behave according to the accountability principle. In _{d }> p_{r}, roughly 17-percent of the data in Rodriguez-Lara and Moreno-Garrido [_{r }≥ max{_{r}}.

One important feature of our model is that dictators are allowed to weigh three different fairness ideals instead of only one. Graphically, this implies that any allocation on the shadowed area can be derived after considering that dictators weigh the lines x_{r }= 0, x_{r }= _{r }= a_{r} and x_{r }= y_{r}. This feature of our model generalizes the idea that each dictator is motivated by a single fairness view (

We have presented a theoretical model of social preferences that expands upon Fehr and Schmidt [

Our contribution to the literature is to provide a model that predicts the accountability principle in Konow [

One novelty of our approach with respect to Cappelen

We would like to thank two anonymous referees for their comments and suggestion, which helped to shape the exposition of the paper. We are also grateful to Elisabet Rutstrom, Glenn W. Harrison, Giovanni Ponti, Paloma Ubeda, Juan D. Moreno-Ternero, Hubert Janos Kiss, Alfonso Rosa-Garcia, David Gill, Vicente Calabuig, Gonzalo Olcina and Jaromir Kovarik for valuable comments. Financial support from the Instituto Valenciano de Investigaciones Economicas (IVIE) and the Spanish Ministry of Education and Science under the projects SEJ2007-62656 and ECO2011-29230 is gratefully acknowledged.

Hereafter we focus our exposition on the dictator game with production, which has been an important device for studying social preferences and distributional justice. We note that the existence of the production stage is important to disentangle the effect of fairness concerns and property rights in the final outcome. As noted by Cherry

Frohlich

The approach in Cappelen

We acknowledge that it might be hard to disentangle which variables are under the subjects’ control and which variables are outside their control in some situations. We find, however, that the classification of factors within and beyond individuals’ control is beyond the scope of this paper. For further discussion on this topic, the interested reader can see Fleurbaey and Maniquet [

For the special case in which β = 0.5, the dictator is indifferent between any share x_{d}ϵ_{d}, x_{r}) that satisfy x_{d}ϵ_{r} = _{d}≥ 0. The models of Bolton and Ockenfelds [

We note that we have rewritten the original equation in Frohlich _{d} + y_{r }= x_{d }+ x_{r}, the latter term in equation (4) can also be thought of as the cost of not giving to the recipient her production.

Nonlinear versions of our model would predict interior results that lead to compromises between these fairness ideals. The interested reader can find a brief discussion about the linearity assumption in the supplementary material. We note that Bolton and Ockenfelds [

The experimental design in Rodriguez-Lara and Moreno-Garrido [

Recall that we focus on the case in which the dictator is rewarded at a higher rate; therefore the recipient's monetary contribution to the surplus will be below her contribution in terms of inputs (a_{r }> y_{r}). Graphically, this implies that the dotted curve (the accountability principle) is above the 45-degree line (the libertarian principle). Both principles coincide when (y_{r }/ _{r }/ _{r }= 0). The contrary is true if (y_{r }/ _{r }/ _{d }= 1.5 and p_{r }= 1. Thus, the egalitarian principle and the libertarian one coincide when (y_{r }/ _{r }= _{r }/ _{r }= _{d }= q_{r}), which implies that y_{r }/

We note that our model cannot be tested using the reported data in other experiments such as Bolton and Ockenfelds [